# Copyright 2002 Gary Strangman. All rights reserved # Copyright 2002-2016 The SciPy Developers # # The original code from Gary Strangman was heavily adapted for # use in SciPy by Travis Oliphant. The original code came with the # following disclaimer: # # This software is provided "as-is". There are no expressed or implied # warranties of any kind, including, but not limited to, the warranties # of merchantability and fitness for a given application. In no event # shall Gary Strangman be liable for any direct, indirect, incidental, # special, exemplary or consequential damages (including, but not limited # to, loss of use, data or profits, or business interruption) however # caused and on any theory of liability, whether in contract, strict # liability or tort (including negligence or otherwise) arising in any way # out of the use of this software, even if advised of the possibility of # such damage. """ A collection of basic statistical functions for Python. References ---------- .. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. """ import warnings import math from math import gcd from collections import namedtuple import numpy as np from numpy import array, asarray, ma from scipy import sparse from scipy.spatial import distance_matrix from scipy.optimize import milp, LinearConstraint from scipy._lib._util import (check_random_state, _get_nan, _rename_parameter, _contains_nan, AxisError) import scipy.special as special # Import unused here but needs to stay until end of deprecation periode # See https://github.com/scipy/scipy/issues/15765#issuecomment-1875564522 from scipy import linalg # noqa: F401 from . import distributions from . import _mstats_basic as mstats_basic from ._stats_mstats_common import _find_repeats, theilslopes, siegelslopes from ._stats import _kendall_dis, _toint64, _weightedrankedtau from dataclasses import dataclass, field from ._hypotests import _all_partitions from ._stats_pythran import _compute_outer_prob_inside_method from ._resampling import (MonteCarloMethod, PermutationMethod, BootstrapMethod, monte_carlo_test, permutation_test, bootstrap, _batch_generator) from ._axis_nan_policy import (_axis_nan_policy_factory, _broadcast_concatenate, _broadcast_shapes, SmallSampleWarning) from ._binomtest import _binary_search_for_binom_tst as _binary_search from scipy._lib._bunch import _make_tuple_bunch from scipy import stats from scipy.optimize import root_scalar from scipy._lib._util import normalize_axis_index from scipy._lib._array_api import (array_namespace, is_numpy, atleast_nd, xp_clip, xp_moveaxis_to_end, xp_sign, xp_minimum) from scipy._lib.array_api_compat import size as xp_size # Functions/classes in other files should be added in `__init__.py`, not here __all__ = ['find_repeats', 'gmean', 'hmean', 'pmean', 'mode', 'tmean', 'tvar', 'tmin', 'tmax', 'tstd', 'tsem', 'moment', 'skew', 'kurtosis', 'describe', 'skewtest', 'kurtosistest', 'normaltest', 'jarque_bera', 'scoreatpercentile', 'percentileofscore', 'cumfreq', 'relfreq', 'obrientransform', 'sem', 'zmap', 'zscore', 'gzscore', 'iqr', 'gstd', 'median_abs_deviation', 'sigmaclip', 'trimboth', 'trim1', 'trim_mean', 'f_oneway', 'pearsonr', 'fisher_exact', 'spearmanr', 'pointbiserialr', 'kendalltau', 'weightedtau', 'linregress', 'siegelslopes', 'theilslopes', 'ttest_1samp', 'ttest_ind', 'ttest_ind_from_stats', 'ttest_rel', 'kstest', 'ks_1samp', 'ks_2samp', 'chisquare', 'power_divergence', 'tiecorrect', 'ranksums', 'kruskal', 'friedmanchisquare', 'rankdata', 'combine_pvalues', 'quantile_test', 'wasserstein_distance', 'wasserstein_distance_nd', 'energy_distance', 'brunnermunzel', 'alexandergovern', 'expectile'] def _chk_asarray(a, axis, *, xp=None): if xp is None: xp = array_namespace(a) if axis is None: a = xp.reshape(a, (-1,)) outaxis = 0 else: a = xp.asarray(a) outaxis = axis if a.ndim == 0: a = xp.reshape(a, (-1,)) return a, outaxis def _chk2_asarray(a, b, axis): if axis is None: a = np.ravel(a) b = np.ravel(b) outaxis = 0 else: a = np.asarray(a) b = np.asarray(b) outaxis = axis if a.ndim == 0: a = np.atleast_1d(a) if b.ndim == 0: b = np.atleast_1d(b) return a, b, outaxis SignificanceResult = _make_tuple_bunch('SignificanceResult', ['statistic', 'pvalue'], []) # note that `weights` are paired with `x` @_axis_nan_policy_factory( lambda x: x, n_samples=1, n_outputs=1, too_small=0, paired=True, result_to_tuple=lambda x: (x,), kwd_samples=['weights']) def gmean(a, axis=0, dtype=None, weights=None): r"""Compute the weighted geometric mean along the specified axis. The weighted geometric mean of the array :math:`a_i` associated to weights :math:`w_i` is: .. math:: \exp \left( \frac{ \sum_{i=1}^n w_i \ln a_i }{ \sum_{i=1}^n w_i } \right) \, , and, with equal weights, it gives: .. math:: \sqrt[n]{ \prod_{i=1}^n a_i } \, . Parameters ---------- a : array_like Input array or object that can be converted to an array. axis : int or None, optional Axis along which the geometric mean is computed. Default is 0. If None, compute over the whole array `a`. dtype : dtype, optional Type to which the input arrays are cast before the calculation is performed. weights : array_like, optional The `weights` array must be broadcastable to the same shape as `a`. Default is None, which gives each value a weight of 1.0. Returns ------- gmean : ndarray See `dtype` parameter above. See Also -------- numpy.mean : Arithmetic average numpy.average : Weighted average hmean : Harmonic mean References ---------- .. [1] "Weighted Geometric Mean", *Wikipedia*, https://en.wikipedia.org/wiki/Weighted_geometric_mean. .. [2] Grossman, J., Grossman, M., Katz, R., "Averages: A New Approach", Archimedes Foundation, 1983 Examples -------- >>> from scipy.stats import gmean >>> gmean([1, 4]) 2.0 >>> gmean([1, 2, 3, 4, 5, 6, 7]) 3.3800151591412964 >>> gmean([1, 4, 7], weights=[3, 1, 3]) 2.80668351922014 """ a = np.asarray(a, dtype=dtype) if weights is not None: weights = np.asarray(weights, dtype=dtype) with np.errstate(divide='ignore'): log_a = np.log(a) return np.exp(np.average(log_a, axis=axis, weights=weights)) @_axis_nan_policy_factory( lambda x: x, n_samples=1, n_outputs=1, too_small=0, paired=True, result_to_tuple=lambda x: (x,), kwd_samples=['weights']) def hmean(a, axis=0, dtype=None, *, weights=None): r"""Calculate the weighted harmonic mean along the specified axis. The weighted harmonic mean of the array :math:`a_i` associated to weights :math:`w_i` is: .. math:: \frac{ \sum_{i=1}^n w_i }{ \sum_{i=1}^n \frac{w_i}{a_i} } \, , and, with equal weights, it gives: .. math:: \frac{ n }{ \sum_{i=1}^n \frac{1}{a_i} } \, . Parameters ---------- a : array_like Input array, masked array or object that can be converted to an array. axis : int or None, optional Axis along which the harmonic mean is computed. Default is 0. If None, compute over the whole array `a`. dtype : dtype, optional Type of the returned array and of the accumulator in which the elements are summed. If `dtype` is not specified, it defaults to the dtype of `a`, unless `a` has an integer `dtype` with a precision less than that of the default platform integer. In that case, the default platform integer is used. weights : array_like, optional The weights array can either be 1-D (in which case its length must be the size of `a` along the given `axis`) or of the same shape as `a`. Default is None, which gives each value a weight of 1.0. .. versionadded:: 1.9 Returns ------- hmean : ndarray See `dtype` parameter above. See Also -------- numpy.mean : Arithmetic average numpy.average : Weighted average gmean : Geometric mean Notes ----- The harmonic mean is computed over a single dimension of the input array, axis=0 by default, or all values in the array if axis=None. float64 intermediate and return values are used for integer inputs. References ---------- .. [1] "Weighted Harmonic Mean", *Wikipedia*, https://en.wikipedia.org/wiki/Harmonic_mean#Weighted_harmonic_mean .. [2] Ferger, F., "The nature and use of the harmonic mean", Journal of the American Statistical Association, vol. 26, pp. 36-40, 1931 Examples -------- >>> from scipy.stats import hmean >>> hmean([1, 4]) 1.6000000000000001 >>> hmean([1, 2, 3, 4, 5, 6, 7]) 2.6997245179063363 >>> hmean([1, 4, 7], weights=[3, 1, 3]) 1.9029126213592233 """ if not isinstance(a, np.ndarray): a = np.array(a, dtype=dtype) elif dtype: # Must change the default dtype allowing array type if isinstance(a, np.ma.MaskedArray): a = np.ma.asarray(a, dtype=dtype) else: a = np.asarray(a, dtype=dtype) if np.all(a >= 0): # Harmonic mean only defined if greater than or equal to zero. if weights is not None: weights = np.asanyarray(weights, dtype=dtype) with np.errstate(divide='ignore'): return 1.0 / np.average(1.0 / a, axis=axis, weights=weights) else: raise ValueError("Harmonic mean only defined if all elements greater " "than or equal to zero") @_axis_nan_policy_factory( lambda x: x, n_samples=1, n_outputs=1, too_small=0, paired=True, result_to_tuple=lambda x: (x,), kwd_samples=['weights']) def pmean(a, p, *, axis=0, dtype=None, weights=None): r"""Calculate the weighted power mean along the specified axis. The weighted power mean of the array :math:`a_i` associated to weights :math:`w_i` is: .. math:: \left( \frac{ \sum_{i=1}^n w_i a_i^p }{ \sum_{i=1}^n w_i } \right)^{ 1 / p } \, , and, with equal weights, it gives: .. math:: \left( \frac{ 1 }{ n } \sum_{i=1}^n a_i^p \right)^{ 1 / p } \, . When ``p=0``, it returns the geometric mean. This mean is also called generalized mean or Hölder mean, and must not be confused with the Kolmogorov generalized mean, also called quasi-arithmetic mean or generalized f-mean [3]_. Parameters ---------- a : array_like Input array, masked array or object that can be converted to an array. p : int or float Exponent. axis : int or None, optional Axis along which the power mean is computed. Default is 0. If None, compute over the whole array `a`. dtype : dtype, optional Type of the returned array and of the accumulator in which the elements are summed. If `dtype` is not specified, it defaults to the dtype of `a`, unless `a` has an integer `dtype` with a precision less than that of the default platform integer. In that case, the default platform integer is used. weights : array_like, optional The weights array can either be 1-D (in which case its length must be the size of `a` along the given `axis`) or of the same shape as `a`. Default is None, which gives each value a weight of 1.0. Returns ------- pmean : ndarray, see `dtype` parameter above. Output array containing the power mean values. See Also -------- numpy.average : Weighted average gmean : Geometric mean hmean : Harmonic mean Notes ----- The power mean is computed over a single dimension of the input array, ``axis=0`` by default, or all values in the array if ``axis=None``. float64 intermediate and return values are used for integer inputs. .. versionadded:: 1.9 References ---------- .. [1] "Generalized Mean", *Wikipedia*, https://en.wikipedia.org/wiki/Generalized_mean .. [2] Norris, N., "Convexity properties of generalized mean value functions", The Annals of Mathematical Statistics, vol. 8, pp. 118-120, 1937 .. [3] Bullen, P.S., Handbook of Means and Their Inequalities, 2003 Examples -------- >>> from scipy.stats import pmean, hmean, gmean >>> pmean([1, 4], 1.3) 2.639372938300652 >>> pmean([1, 2, 3, 4, 5, 6, 7], 1.3) 4.157111214492084 >>> pmean([1, 4, 7], -2, weights=[3, 1, 3]) 1.4969684896631954 For p=-1, power mean is equal to harmonic mean: >>> pmean([1, 4, 7], -1, weights=[3, 1, 3]) 1.9029126213592233 >>> hmean([1, 4, 7], weights=[3, 1, 3]) 1.9029126213592233 For p=0, power mean is defined as the geometric mean: >>> pmean([1, 4, 7], 0, weights=[3, 1, 3]) 2.80668351922014 >>> gmean([1, 4, 7], weights=[3, 1, 3]) 2.80668351922014 """ if not isinstance(p, (int, float)): raise ValueError("Power mean only defined for exponent of type int or " "float.") if p == 0: return gmean(a, axis=axis, dtype=dtype, weights=weights) if not isinstance(a, np.ndarray): a = np.array(a, dtype=dtype) elif dtype: # Must change the default dtype allowing array type if isinstance(a, np.ma.MaskedArray): a = np.ma.asarray(a, dtype=dtype) else: a = np.asarray(a, dtype=dtype) if np.all(a >= 0): # Power mean only defined if greater than or equal to zero if weights is not None: weights = np.asanyarray(weights, dtype=dtype) with np.errstate(divide='ignore'): return np.float_power( np.average(np.float_power(a, p), axis=axis, weights=weights), 1/p) else: raise ValueError("Power mean only defined if all elements greater " "than or equal to zero") ModeResult = namedtuple('ModeResult', ('mode', 'count')) def _mode_result(mode, count): # When a slice is empty, `_axis_nan_policy` automatically produces # NaN for `mode` and `count`. This is a reasonable convention for `mode`, # but `count` should not be NaN; it should be zero. i = np.isnan(count) if i.shape == (): count = np.asarray(0, dtype=count.dtype)[()] if i else count else: count[i] = 0 return ModeResult(mode, count) @_axis_nan_policy_factory(_mode_result, override={'vectorization': True, 'nan_propagation': False}) def mode(a, axis=0, nan_policy='propagate', keepdims=False): r"""Return an array of the modal (most common) value in the passed array. If there is more than one such value, only one is returned. The bin-count for the modal bins is also returned. Parameters ---------- a : array_like Numeric, n-dimensional array of which to find mode(s). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': treats nan as it would treat any other value * 'raise': throws an error * 'omit': performs the calculations ignoring nan values keepdims : bool, optional If set to ``False``, the `axis` over which the statistic is taken is consumed (eliminated from the output array). If set to ``True``, the `axis` is retained with size one, and the result will broadcast correctly against the input array. Returns ------- mode : ndarray Array of modal values. count : ndarray Array of counts for each mode. Notes ----- The mode is calculated using `numpy.unique`. In NumPy versions 1.21 and after, all NaNs - even those with different binary representations - are treated as equivalent and counted as separate instances of the same value. By convention, the mode of an empty array is NaN, and the associated count is zero. Examples -------- >>> import numpy as np >>> a = np.array([[3, 0, 3, 7], ... [3, 2, 6, 2], ... [1, 7, 2, 8], ... [3, 0, 6, 1], ... [3, 2, 5, 5]]) >>> from scipy import stats >>> stats.mode(a, keepdims=True) ModeResult(mode=array([[3, 0, 6, 1]]), count=array([[4, 2, 2, 1]])) To get mode of whole array, specify ``axis=None``: >>> stats.mode(a, axis=None, keepdims=True) ModeResult(mode=[[3]], count=[[5]]) >>> stats.mode(a, axis=None, keepdims=False) ModeResult(mode=3, count=5) """ # `axis`, `nan_policy`, and `keepdims` are handled by `_axis_nan_policy` if not np.issubdtype(a.dtype, np.number): message = ("Argument `a` is not recognized as numeric. " "Support for input that cannot be coerced to a numeric " "array was deprecated in SciPy 1.9.0 and removed in SciPy " "1.11.0. Please consider `np.unique`.") raise TypeError(message) if a.size == 0: NaN = _get_nan(a) return ModeResult(*np.array([NaN, 0], dtype=NaN.dtype)) vals, cnts = np.unique(a, return_counts=True) modes, counts = vals[cnts.argmax()], cnts.max() return ModeResult(modes[()], counts[()]) def _put_nan_to_limits(a, limits, inclusive): """Put NaNs in an array for values outside of given limits. This is primarily a utility function. Parameters ---------- a : array limits : (float or None, float or None) A tuple consisting of the (lower limit, upper limit). Values in the input array less than the lower limit or greater than the upper limit will be replaced with `np.nan`. None implies no limit. inclusive : (bool, bool) A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to lower or upper are allowed. """ if limits is None: return a mask = np.full_like(a, False, dtype=np.bool_) lower_limit, upper_limit = limits lower_include, upper_include = inclusive if lower_limit is not None: mask |= (a < lower_limit) if lower_include else a <= lower_limit if upper_limit is not None: mask |= (a > upper_limit) if upper_include else a >= upper_limit if np.all(mask): raise ValueError("No array values within given limits") if np.any(mask): a = a.copy() if np.issubdtype(a.dtype, np.inexact) else a.astype(np.float64) a[mask] = np.nan return a @_axis_nan_policy_factory( lambda x: x, n_outputs=1, default_axis=None, result_to_tuple=lambda x: (x,) ) def tmean(a, limits=None, inclusive=(True, True), axis=None): """Compute the trimmed mean. This function finds the arithmetic mean of given values, ignoring values outside the given `limits`. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None (default), then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to compute test. Default is None. Returns ------- tmean : ndarray Trimmed mean. See Also -------- trim_mean : Returns mean after trimming a proportion from both tails. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmean(x) 9.5 >>> stats.tmean(x, (3,17)) 10.0 """ a = _put_nan_to_limits(a, limits, inclusive) return np.nanmean(a, axis=axis) @_axis_nan_policy_factory( lambda x: x, n_outputs=1, result_to_tuple=lambda x: (x,) ) def tvar(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """Compute the trimmed variance. This function computes the sample variance of an array of values, while ignoring values which are outside of given `limits`. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tvar : float Trimmed variance. Notes ----- `tvar` computes the unbiased sample variance, i.e. it uses a correction factor ``n / (n - 1)``. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = np.arange(20) >>> stats.tvar(x) 35.0 >>> stats.tvar(x, (3,17)) 20.0 """ a = _put_nan_to_limits(a, limits, inclusive) return np.nanvar(a, ddof=ddof, axis=axis) @_axis_nan_policy_factory( lambda x: x, n_outputs=1, result_to_tuple=lambda x: (x,) ) def tmin(a, lowerlimit=None, axis=0, inclusive=True, nan_policy='propagate'): """Compute the trimmed minimum. This function finds the minimum value of an array `a` along the specified axis, but only considering values greater than a specified lower limit. Parameters ---------- a : array_like Array of values. lowerlimit : None or float, optional Values in the input array less than the given limit will be ignored. When lowerlimit is None, then all values are used. The default value is None. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. inclusive : {True, False}, optional This flag determines whether values exactly equal to the lower limit are included. The default value is True. Returns ------- tmin : float, int or ndarray Trimmed minimum. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmin(x) 0 >>> stats.tmin(x, 13) 13 >>> stats.tmin(x, 13, inclusive=False) 14 """ dtype = a.dtype a = _put_nan_to_limits(a, (lowerlimit, None), (inclusive, None)) res = np.nanmin(a, axis=axis) if not np.any(np.isnan(res)): # needed if input is of integer dtype return res.astype(dtype, copy=False) return res @_axis_nan_policy_factory( lambda x: x, n_outputs=1, result_to_tuple=lambda x: (x,) ) def tmax(a, upperlimit=None, axis=0, inclusive=True, nan_policy='propagate'): """Compute the trimmed maximum. This function computes the maximum value of an array along a given axis, while ignoring values larger than a specified upper limit. Parameters ---------- a : array_like Array of values. upperlimit : None or float, optional Values in the input array greater than the given limit will be ignored. When upperlimit is None, then all values are used. The default value is None. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. inclusive : {True, False}, optional This flag determines whether values exactly equal to the upper limit are included. The default value is True. Returns ------- tmax : float, int or ndarray Trimmed maximum. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmax(x) 19 >>> stats.tmax(x, 13) 13 >>> stats.tmax(x, 13, inclusive=False) 12 """ dtype = a.dtype a = _put_nan_to_limits(a, (None, upperlimit), (None, inclusive)) res = np.nanmax(a, axis=axis) if not np.any(np.isnan(res)): # needed if input is of integer dtype return res.astype(dtype, copy=False) return res @_axis_nan_policy_factory( lambda x: x, n_outputs=1, result_to_tuple=lambda x: (x,) ) def tstd(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """Compute the trimmed sample standard deviation. This function finds the sample standard deviation of given values, ignoring values outside the given `limits`. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tstd : float Trimmed sample standard deviation. Notes ----- `tstd` computes the unbiased sample standard deviation, i.e. it uses a correction factor ``n / (n - 1)``. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = np.arange(20) >>> stats.tstd(x) 5.9160797830996161 >>> stats.tstd(x, (3,17)) 4.4721359549995796 """ return np.sqrt(tvar(a, limits, inclusive, axis, ddof, _no_deco=True)) @_axis_nan_policy_factory( lambda x: x, n_outputs=1, result_to_tuple=lambda x: (x,) ) def tsem(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """Compute the trimmed standard error of the mean. This function finds the standard error of the mean for given values, ignoring values outside the given `limits`. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tsem : float Trimmed standard error of the mean. Notes ----- `tsem` uses unbiased sample standard deviation, i.e. it uses a correction factor ``n / (n - 1)``. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = np.arange(20) >>> stats.tsem(x) 1.3228756555322954 >>> stats.tsem(x, (3,17)) 1.1547005383792515 """ a = _put_nan_to_limits(a, limits, inclusive) sd = np.sqrt(np.nanvar(a, ddof=ddof, axis=axis)) n_obs = (~np.isnan(a)).sum(axis=axis) return sd / np.sqrt(n_obs, dtype=sd.dtype) ##################################### # MOMENTS # ##################################### def _moment_outputs(kwds): order = np.atleast_1d(kwds.get('order', 1)) if order.size == 0: raise ValueError("'order' must be a scalar or a non-empty 1D " "list/array.") return len(order) def _moment_result_object(*args): if len(args) == 1: return args[0] return np.asarray(args) # `moment` fits into the `_axis_nan_policy` pattern, but it is a bit unusual # because the number of outputs is variable. Specifically, # `result_to_tuple=lambda x: (x,)` may be surprising for a function that # can produce more than one output, but it is intended here. # When `moment is called to produce the output: # - `result_to_tuple` packs the returned array into a single-element tuple, # - `_moment_result_object` extracts and returns that single element. # However, when the input array is empty, `moment` is never called. Instead, # - `_check_empty_inputs` is used to produce an empty array with the # appropriate dimensions. # - A list comprehension creates the appropriate number of copies of this # array, depending on `n_outputs`. # - This list - which may have multiple elements - is passed into # `_moment_result_object`. # - If there is a single output, `_moment_result_object` extracts and returns # the single output from the list. # - If there are multiple outputs, and therefore multiple elements in the list, # `_moment_result_object` converts the list of arrays to a single array and # returns it. # Currently this leads to a slight inconsistency: when the input array is # empty, there is no distinction between the `moment` function being called # with parameter `order=1` and `order=[1]`; the latter *should* produce # the same as the former but with a singleton zeroth dimension. @_rename_parameter('moment', 'order') @_axis_nan_policy_factory( # noqa: E302 _moment_result_object, n_samples=1, result_to_tuple=lambda x: (x,), n_outputs=_moment_outputs ) def moment(a, order=1, axis=0, nan_policy='propagate', *, center=None): r"""Calculate the nth moment about the mean for a sample. A moment is a specific quantitative measure of the shape of a set of points. It is often used to calculate coefficients of skewness and kurtosis due to its close relationship with them. Parameters ---------- a : array_like Input array. order : int or 1-D array_like of ints, optional Order of central moment that is returned. Default is 1. axis : int or None, optional Axis along which the central moment is computed. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values center : float or None, optional The point about which moments are taken. This can be the sample mean, the origin, or any other be point. If `None` (default) compute the center as the sample mean. Returns ------- n-th moment about the `center` : ndarray or float The appropriate moment along the given axis or over all values if axis is None. The denominator for the moment calculation is the number of observations, no degrees of freedom correction is done. See Also -------- kurtosis, skew, describe Notes ----- The k-th moment of a data sample is: .. math:: m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - c)^k Where `n` is the number of samples, and `c` is the center around which the moment is calculated. This function uses exponentiation by squares [1]_ for efficiency. Note that, if `a` is an empty array (``a.size == 0``), array `moment` with one element (`moment.size == 1`) is treated the same as scalar `moment` (``np.isscalar(moment)``). This might produce arrays of unexpected shape. References ---------- .. [1] https://eli.thegreenplace.net/2009/03/21/efficient-integer-exponentiation-algorithms Examples -------- >>> from scipy.stats import moment >>> moment([1, 2, 3, 4, 5], order=1) 0.0 >>> moment([1, 2, 3, 4, 5], order=2) 2.0 """ xp = array_namespace(a) a, axis = _chk_asarray(a, axis, xp=xp) if xp.isdtype(a.dtype, 'integral'): a = xp.asarray(a, dtype=xp.float64) else: a = xp.asarray(a) order = xp.asarray(order, dtype=a.dtype) if xp_size(order) == 0: # This is tested by `_moment_outputs`, which is run by the `_axis_nan_policy` # decorator. Currently, the `_axis_nan_policy` decorator is skipped when `a` # is a non-NumPy array, so we need to check again. When the decorator is # updated for array API compatibility, we can remove this second check. raise ValueError("'order' must be a scalar or a non-empty 1D list/array.") if xp.any(order != xp.round(order)): raise ValueError("All elements of `order` must be integral.") order = order[()] if order.ndim == 0 else order # for array_like order input, return a value for each. if order.ndim > 0: # Calculated the mean once at most, and only if it will be used calculate_mean = center is None and xp.any(order > 1) mean = xp.mean(a, axis=axis, keepdims=True) if calculate_mean else None mmnt = [] for i in range(order.shape[0]): order_i = order[i] if center is None and order_i > 1: mmnt.append(_moment(a, order_i, axis, mean=mean)[np.newaxis, ...]) else: mmnt.append(_moment(a, order_i, axis, mean=center)[np.newaxis, ...]) return xp.concat(mmnt, axis=0) else: return _moment(a, order, axis, mean=center) def _moment(a, order, axis, *, mean=None, xp=None): """Vectorized calculation of raw moment about specified center When `mean` is None, the mean is computed and used as the center; otherwise, the provided value is used as the center. """ xp = array_namespace(a) if xp is None else xp if xp.isdtype(a.dtype, 'integral'): a = xp.asarray(a, dtype=xp.float64) dtype = a.dtype # moment of empty array is the same regardless of order if xp_size(a) == 0: return xp.mean(a, axis=axis) if order == 0 or (order == 1 and mean is None): # By definition the zeroth moment is always 1, and the first *central* # moment is 0. shape = list(a.shape) del shape[axis] temp = (xp.ones(shape, dtype=dtype) if order == 0 else xp.zeros(shape, dtype=dtype)) return temp[()] if temp.ndim == 0 else temp # Exponentiation by squares: form exponent sequence n_list = [order] current_n = order while current_n > 2: if current_n % 2: current_n = (current_n - 1) / 2 else: current_n /= 2 n_list.append(current_n) # Starting point for exponentiation by squares mean = (xp.mean(a, axis=axis, keepdims=True) if mean is None else xp.asarray(mean, dtype=dtype)) mean = mean[()] if mean.ndim == 0 else mean a_zero_mean = a - mean eps = xp.finfo(dtype).eps * 10 with np.errstate(divide='ignore', invalid='ignore'): rel_diff = xp.max(xp.abs(a_zero_mean), axis=axis, keepdims=True) / xp.abs(mean) with np.errstate(invalid='ignore'): precision_loss = xp.any(rel_diff < eps) n = a.shape[axis] if axis is not None else a.size if precision_loss and n > 1: message = ("Precision loss occurred in moment calculation due to " "catastrophic cancellation. This occurs when the data " "are nearly identical. Results may be unreliable.") warnings.warn(message, RuntimeWarning, stacklevel=4) if n_list[-1] == 1: s = xp.asarray(a_zero_mean, copy=True) else: s = a_zero_mean**2 # Perform multiplications for n in n_list[-2::-1]: s = s**2 if n % 2: s *= a_zero_mean return xp.mean(s, axis=axis) def _var(x, axis=0, ddof=0, mean=None, xp=None): # Calculate variance of sample, warning if precision is lost xp = array_namespace(x) if xp is None else xp var = _moment(x, 2, axis, mean=mean, xp=xp) if ddof != 0: n = x.shape[axis] if axis is not None else x.size var *= np.divide(n, n-ddof) # to avoid error on division by zero return var @_axis_nan_policy_factory( lambda x: x, result_to_tuple=lambda x: (x,), n_outputs=1 ) # nan_policy handled by `_axis_nan_policy`, but needs to be left # in signature to preserve use as a positional argument def skew(a, axis=0, bias=True, nan_policy='propagate'): r"""Compute the sample skewness of a data set. For normally distributed data, the skewness should be about zero. For unimodal continuous distributions, a skewness value greater than zero means that there is more weight in the right tail of the distribution. The function `skewtest` can be used to determine if the skewness value is close enough to zero, statistically speaking. Parameters ---------- a : ndarray Input array. axis : int or None, optional Axis along which skewness is calculated. Default is 0. If None, compute over the whole array `a`. bias : bool, optional If False, then the calculations are corrected for statistical bias. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- skewness : ndarray The skewness of values along an axis, returning NaN where all values are equal. Notes ----- The sample skewness is computed as the Fisher-Pearson coefficient of skewness, i.e. .. math:: g_1=\frac{m_3}{m_2^{3/2}} where .. math:: m_i=\frac{1}{N}\sum_{n=1}^N(x[n]-\bar{x})^i is the biased sample :math:`i\texttt{th}` central moment, and :math:`\bar{x}` is the sample mean. If ``bias`` is False, the calculations are corrected for bias and the value computed is the adjusted Fisher-Pearson standardized moment coefficient, i.e. .. math:: G_1=\frac{k_3}{k_2^{3/2}}= \frac{\sqrt{N(N-1)}}{N-2}\frac{m_3}{m_2^{3/2}}. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 2.2.24.1 Examples -------- >>> from scipy.stats import skew >>> skew([1, 2, 3, 4, 5]) 0.0 >>> skew([2, 8, 0, 4, 1, 9, 9, 0]) 0.2650554122698573 """ xp = array_namespace(a) a, axis = _chk_asarray(a, axis, xp=xp) n = a.shape[axis] mean = xp.mean(a, axis=axis, keepdims=True) mean_reduced = xp.squeeze(mean, axis=axis) # needed later m2 = _moment(a, 2, axis, mean=mean, xp=xp) m3 = _moment(a, 3, axis, mean=mean, xp=xp) with np.errstate(all='ignore'): eps = xp.finfo(m2.dtype).eps zero = m2 <= (eps * mean_reduced)**2 vals = xp.where(zero, xp.asarray(xp.nan), m3 / m2**1.5) if not bias: can_correct = ~zero & (n > 2) if xp.any(can_correct): m2 = m2[can_correct] m3 = m3[can_correct] nval = ((n - 1.0) * n)**0.5 / (n - 2.0) * m3 / m2**1.5 vals[can_correct] = nval return vals[()] if vals.ndim == 0 else vals @_axis_nan_policy_factory( lambda x: x, result_to_tuple=lambda x: (x,), n_outputs=1 ) # nan_policy handled by `_axis_nan_policy`, but needs to be left # in signature to preserve use as a positional argument def kurtosis(a, axis=0, fisher=True, bias=True, nan_policy='propagate'): """Compute the kurtosis (Fisher or Pearson) of a dataset. Kurtosis is the fourth central moment divided by the square of the variance. If Fisher's definition is used, then 3.0 is subtracted from the result to give 0.0 for a normal distribution. If bias is False then the kurtosis is calculated using k statistics to eliminate bias coming from biased moment estimators Use `kurtosistest` to see if result is close enough to normal. Parameters ---------- a : array Data for which the kurtosis is calculated. axis : int or None, optional Axis along which the kurtosis is calculated. Default is 0. If None, compute over the whole array `a`. fisher : bool, optional If True, Fisher's definition is used (normal ==> 0.0). If False, Pearson's definition is used (normal ==> 3.0). bias : bool, optional If False, then the calculations are corrected for statistical bias. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. 'propagate' returns nan, 'raise' throws an error, 'omit' performs the calculations ignoring nan values. Default is 'propagate'. Returns ------- kurtosis : array The kurtosis of values along an axis, returning NaN where all values are equal. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Examples -------- In Fisher's definition, the kurtosis of the normal distribution is zero. In the following example, the kurtosis is close to zero, because it was calculated from the dataset, not from the continuous distribution. >>> import numpy as np >>> from scipy.stats import norm, kurtosis >>> data = norm.rvs(size=1000, random_state=3) >>> kurtosis(data) -0.06928694200380558 The distribution with a higher kurtosis has a heavier tail. The zero valued kurtosis of the normal distribution in Fisher's definition can serve as a reference point. >>> import matplotlib.pyplot as plt >>> import scipy.stats as stats >>> from scipy.stats import kurtosis >>> x = np.linspace(-5, 5, 100) >>> ax = plt.subplot() >>> distnames = ['laplace', 'norm', 'uniform'] >>> for distname in distnames: ... if distname == 'uniform': ... dist = getattr(stats, distname)(loc=-2, scale=4) ... else: ... dist = getattr(stats, distname) ... data = dist.rvs(size=1000) ... kur = kurtosis(data, fisher=True) ... y = dist.pdf(x) ... ax.plot(x, y, label="{}, {}".format(distname, round(kur, 3))) ... ax.legend() The Laplace distribution has a heavier tail than the normal distribution. The uniform distribution (which has negative kurtosis) has the thinnest tail. """ xp = array_namespace(a) a, axis = _chk_asarray(a, axis, xp=xp) n = a.shape[axis] mean = xp.mean(a, axis=axis, keepdims=True) mean_reduced = xp.squeeze(mean, axis=axis) # needed later m2 = _moment(a, 2, axis, mean=mean, xp=xp) m4 = _moment(a, 4, axis, mean=mean, xp=xp) with np.errstate(all='ignore'): zero = m2 <= (xp.finfo(m2.dtype).eps * mean_reduced)**2 NaN = _get_nan(m4, xp=xp) vals = xp.where(zero, NaN, m4 / m2**2.0) if not bias: can_correct = ~zero & (n > 3) if xp.any(can_correct): m2 = m2[can_correct] m4 = m4[can_correct] nval = 1.0/(n-2)/(n-3) * ((n**2-1.0)*m4/m2**2.0 - 3*(n-1)**2.0) vals[can_correct] = nval + 3.0 vals = vals - 3 if fisher else vals return vals[()] if vals.ndim == 0 else vals DescribeResult = namedtuple('DescribeResult', ('nobs', 'minmax', 'mean', 'variance', 'skewness', 'kurtosis')) def describe(a, axis=0, ddof=1, bias=True, nan_policy='propagate'): """Compute several descriptive statistics of the passed array. Parameters ---------- a : array_like Input data. axis : int or None, optional Axis along which statistics are calculated. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees of freedom (only for variance). Default is 1. bias : bool, optional If False, then the skewness and kurtosis calculations are corrected for statistical bias. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- nobs : int or ndarray of ints Number of observations (length of data along `axis`). When 'omit' is chosen as nan_policy, the length along each axis slice is counted separately. minmax: tuple of ndarrays or floats Minimum and maximum value of `a` along the given axis. mean : ndarray or float Arithmetic mean of `a` along the given axis. variance : ndarray or float Unbiased variance of `a` along the given axis; denominator is number of observations minus one. skewness : ndarray or float Skewness of `a` along the given axis, based on moment calculations with denominator equal to the number of observations, i.e. no degrees of freedom correction. kurtosis : ndarray or float Kurtosis (Fisher) of `a` along the given axis. The kurtosis is normalized so that it is zero for the normal distribution. No degrees of freedom are used. See Also -------- skew, kurtosis Examples -------- >>> import numpy as np >>> from scipy import stats >>> a = np.arange(10) >>> stats.describe(a) DescribeResult(nobs=10, minmax=(0, 9), mean=4.5, variance=9.166666666666666, skewness=0.0, kurtosis=-1.2242424242424244) >>> b = [[1, 2], [3, 4]] >>> stats.describe(b) DescribeResult(nobs=2, minmax=(array([1, 2]), array([3, 4])), mean=array([2., 3.]), variance=array([2., 2.]), skewness=array([0., 0.]), kurtosis=array([-2., -2.])) """ xp = array_namespace(a) a, axis = _chk_asarray(a, axis, xp=xp) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': # only NumPy gets here; `_contains_nan` raises error for the rest a = ma.masked_invalid(a) return mstats_basic.describe(a, axis, ddof, bias) if xp_size(a) == 0: raise ValueError("The input must not be empty.") n = a.shape[axis] mm = (xp.min(a, axis=axis), xp.max(a, axis=axis)) m = xp.mean(a, axis=axis) v = _var(a, axis=axis, ddof=ddof, xp=xp) sk = skew(a, axis, bias=bias) kurt = kurtosis(a, axis, bias=bias) return DescribeResult(n, mm, m, v, sk, kurt) ##################################### # NORMALITY TESTS # ##################################### def _get_pvalue(statistic, distribution, alternative, symmetric=True, xp=None): """Get p-value given the statistic, (continuous) distribution, and alternative""" xp = array_namespace(statistic) if xp is None else xp if alternative == 'less': pvalue = distribution.cdf(statistic) elif alternative == 'greater': pvalue = distribution.sf(statistic) elif alternative == 'two-sided': pvalue = 2 * (distribution.sf(xp.abs(statistic)) if symmetric else xp_minimum(distribution.cdf(statistic), distribution.sf(statistic), xp=xp)) else: message = "`alternative` must be 'less', 'greater', or 'two-sided'." raise ValueError(message) return pvalue SkewtestResult = namedtuple('SkewtestResult', ('statistic', 'pvalue')) @_axis_nan_policy_factory(SkewtestResult, n_samples=1, too_small=7) # nan_policy handled by `_axis_nan_policy`, but needs to be left # in signature to preserve use as a positional argument def skewtest(a, axis=0, nan_policy='propagate', alternative='two-sided'): r"""Test whether the skew is different from the normal distribution. This function tests the null hypothesis that the skewness of the population that the sample was drawn from is the same as that of a corresponding normal distribution. Parameters ---------- a : array The data to be tested. Must contain at least eight observations. axis : int or None, optional Axis along which statistics are calculated. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. Default is 'two-sided'. The following options are available: * 'two-sided': the skewness of the distribution underlying the sample is different from that of the normal distribution (i.e. 0) * 'less': the skewness of the distribution underlying the sample is less than that of the normal distribution * 'greater': the skewness of the distribution underlying the sample is greater than that of the normal distribution .. versionadded:: 1.7.0 Returns ------- statistic : float The computed z-score for this test. pvalue : float The p-value for the hypothesis test. Notes ----- The sample size must be at least 8. References ---------- .. [1] R. B. D'Agostino, A. J. Belanger and R. B. D'Agostino Jr., "A suggestion for using powerful and informative tests of normality", American Statistician 44, pp. 316-321, 1990. .. [2] Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611. .. [3] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn." Statistical Applications in Genetics and Molecular Biology 9.1 (2010). Examples -------- Suppose we wish to infer from measurements whether the weights of adult human males in a medical study are not normally distributed [2]_. The weights (lbs) are recorded in the array ``x`` below. >>> import numpy as np >>> x = np.array([148, 154, 158, 160, 161, 162, 166, 170, 182, 195, 236]) The skewness test from [1]_ begins by computing a statistic based on the sample skewness. >>> from scipy import stats >>> res = stats.skewtest(x) >>> res.statistic 2.7788579769903414 Because normal distributions have zero skewness, the magnitude of this statistic tends to be low for samples drawn from a normal distribution. The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that the weights were drawn from a normal distribution. For this test, the null distribution of the statistic for very large samples is the standard normal distribution. >>> import matplotlib.pyplot as plt >>> dist = stats.norm() >>> st_val = np.linspace(-5, 5, 100) >>> pdf = dist.pdf(st_val) >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> def st_plot(ax): # we'll reuse this ... ax.plot(st_val, pdf) ... ax.set_title("Skew Test Null Distribution") ... ax.set_xlabel("statistic") ... ax.set_ylabel("probability density") >>> st_plot(ax) >>> plt.show() The comparison is quantified by the p-value: the proportion of values in the null distribution as extreme or more extreme than the observed value of the statistic. In a two-sided test, elements of the null distribution greater than the observed statistic and elements of the null distribution less than the negative of the observed statistic are both considered "more extreme". >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> st_plot(ax) >>> pvalue = dist.cdf(-res.statistic) + dist.sf(res.statistic) >>> annotation = (f'p-value={pvalue:.3f}\n(shaded area)') >>> props = dict(facecolor='black', width=1, headwidth=5, headlength=8) >>> _ = ax.annotate(annotation, (3, 0.005), (3.25, 0.02), arrowprops=props) >>> i = st_val >= res.statistic >>> ax.fill_between(st_val[i], y1=0, y2=pdf[i], color='C0') >>> i = st_val <= -res.statistic >>> ax.fill_between(st_val[i], y1=0, y2=pdf[i], color='C0') >>> ax.set_xlim(-5, 5) >>> ax.set_ylim(0, 0.1) >>> plt.show() >>> res.pvalue 0.005455036974740185 If the p-value is "small" - that is, if there is a low probability of sampling data from a normally distributed population that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the weights were not drawn from a normal distribution. Note that: - The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis. - The threshold for values that will be considered "small" is a choice that should be made before the data is analyzed [3]_ with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis). Note that the standard normal distribution provides an asymptotic approximation of the null distribution; it is only accurate for samples with many observations. For small samples like ours, `scipy.stats.monte_carlo_test` may provide a more accurate, albeit stochastic, approximation of the exact p-value. >>> def statistic(x, axis): ... # get just the skewtest statistic; ignore the p-value ... return stats.skewtest(x, axis=axis).statistic >>> res = stats.monte_carlo_test(x, stats.norm.rvs, statistic) >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> st_plot(ax) >>> ax.hist(res.null_distribution, np.linspace(-5, 5, 50), ... density=True) >>> ax.legend(['aymptotic approximation\n(many observations)', ... 'Monte Carlo approximation\n(11 observations)']) >>> plt.show() >>> res.pvalue 0.0062 # may vary In this case, the asymptotic approximation and Monte Carlo approximation agree fairly closely, even for our small sample. """ xp = array_namespace(a) a, axis = _chk_asarray(a, axis, xp=xp) b2 = skew(a, axis, _no_deco=True) n = a.shape[axis] if n < 8: message = ("`skewtest` requires at least 8 observations; " f"only {n=} observations were given.") raise ValueError(message) y = b2 * math.sqrt(((n + 1) * (n + 3)) / (6.0 * (n - 2))) beta2 = (3.0 * (n**2 + 27*n - 70) * (n+1) * (n+3) / ((n-2.0) * (n+5) * (n+7) * (n+9))) W2 = -1 + math.sqrt(2 * (beta2 - 1)) delta = 1 / math.sqrt(0.5 * math.log(W2)) alpha = math.sqrt(2.0 / (W2 - 1)) y = xp.where(y == 0, xp.asarray(1, dtype=y.dtype), y) Z = delta * xp.log(y / alpha + xp.sqrt((y / alpha)**2 + 1)) pvalue = _get_pvalue(Z, _SimpleNormal(), alternative, xp=xp) Z = Z[()] if Z.ndim == 0 else Z pvalue = pvalue[()] if pvalue.ndim == 0 else pvalue return SkewtestResult(Z, pvalue) KurtosistestResult = namedtuple('KurtosistestResult', ('statistic', 'pvalue')) @_axis_nan_policy_factory(KurtosistestResult, n_samples=1, too_small=4) def kurtosistest(a, axis=0, nan_policy='propagate', alternative='two-sided'): r"""Test whether a dataset has normal kurtosis. This function tests the null hypothesis that the kurtosis of the population from which the sample was drawn is that of the normal distribution. Parameters ---------- a : array Array of the sample data. Must contain at least five observations. axis : int or None, optional Axis along which to compute test. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided': the kurtosis of the distribution underlying the sample is different from that of the normal distribution * 'less': the kurtosis of the distribution underlying the sample is less than that of the normal distribution * 'greater': the kurtosis of the distribution underlying the sample is greater than that of the normal distribution .. versionadded:: 1.7.0 Returns ------- statistic : float The computed z-score for this test. pvalue : float The p-value for the hypothesis test. Notes ----- Valid only for n>20. This function uses the method described in [1]_. References ---------- .. [1] see e.g. F. J. Anscombe, W. J. Glynn, "Distribution of the kurtosis statistic b2 for normal samples", Biometrika, vol. 70, pp. 227-234, 1983. .. [2] Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611. .. [3] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn." Statistical Applications in Genetics and Molecular Biology 9.1 (2010). .. [4] Panagiotakos, D. B. (2008). The value of p-value in biomedical research. The open cardiovascular medicine journal, 2, 97. Examples -------- Suppose we wish to infer from measurements whether the weights of adult human males in a medical study are not normally distributed [2]_. The weights (lbs) are recorded in the array ``x`` below. >>> import numpy as np >>> x = np.array([148, 154, 158, 160, 161, 162, 166, 170, 182, 195, 236]) The kurtosis test from [1]_ begins by computing a statistic based on the sample (excess/Fisher) kurtosis. >>> from scipy import stats >>> res = stats.kurtosistest(x) >>> res.statistic 2.3048235214240873 (The test warns that our sample has too few observations to perform the test. We'll return to this at the end of the example.) Because normal distributions have zero excess kurtosis (by definition), the magnitude of this statistic tends to be low for samples drawn from a normal distribution. The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that the weights were drawn from a normal distribution. For this test, the null distribution of the statistic for very large samples is the standard normal distribution. >>> import matplotlib.pyplot as plt >>> dist = stats.norm() >>> kt_val = np.linspace(-5, 5, 100) >>> pdf = dist.pdf(kt_val) >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> def kt_plot(ax): # we'll reuse this ... ax.plot(kt_val, pdf) ... ax.set_title("Kurtosis Test Null Distribution") ... ax.set_xlabel("statistic") ... ax.set_ylabel("probability density") >>> kt_plot(ax) >>> plt.show() The comparison is quantified by the p-value: the proportion of values in the null distribution as extreme or more extreme than the observed value of the statistic. In a two-sided test in which the statistic is positive, elements of the null distribution greater than the observed statistic and elements of the null distribution less than the negative of the observed statistic are both considered "more extreme". >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> kt_plot(ax) >>> pvalue = dist.cdf(-res.statistic) + dist.sf(res.statistic) >>> annotation = (f'p-value={pvalue:.3f}\n(shaded area)') >>> props = dict(facecolor='black', width=1, headwidth=5, headlength=8) >>> _ = ax.annotate(annotation, (3, 0.005), (3.25, 0.02), arrowprops=props) >>> i = kt_val >= res.statistic >>> ax.fill_between(kt_val[i], y1=0, y2=pdf[i], color='C0') >>> i = kt_val <= -res.statistic >>> ax.fill_between(kt_val[i], y1=0, y2=pdf[i], color='C0') >>> ax.set_xlim(-5, 5) >>> ax.set_ylim(0, 0.1) >>> plt.show() >>> res.pvalue 0.0211764592113868 If the p-value is "small" - that is, if there is a low probability of sampling data from a normally distributed population that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the weights were not drawn from a normal distribution. Note that: - The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis. - The threshold for values that will be considered "small" is a choice that should be made before the data is analyzed [3]_ with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis). Note that the standard normal distribution provides an asymptotic approximation of the null distribution; it is only accurate for samples with many observations. This is the reason we received a warning at the beginning of the example; our sample is quite small. In this case, `scipy.stats.monte_carlo_test` may provide a more accurate, albeit stochastic, approximation of the exact p-value. >>> def statistic(x, axis): ... # get just the skewtest statistic; ignore the p-value ... return stats.kurtosistest(x, axis=axis).statistic >>> res = stats.monte_carlo_test(x, stats.norm.rvs, statistic) >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> kt_plot(ax) >>> ax.hist(res.null_distribution, np.linspace(-5, 5, 50), ... density=True) >>> ax.legend(['aymptotic approximation\n(many observations)', ... 'Monte Carlo approximation\n(11 observations)']) >>> plt.show() >>> res.pvalue 0.0272 # may vary Furthermore, despite their stochastic nature, p-values computed in this way can be used to exactly control the rate of false rejections of the null hypothesis [4]_. """ xp = array_namespace(a) a, axis = _chk_asarray(a, axis, xp=xp) n = a.shape[axis] if n < 5: message = ("`kurtosistest` requires at least 5 observations; " f"only {n=} observations were given.") raise ValueError(message) if n < 20: message = ("`kurtosistest` p-value may be inaccurate with fewer than 20 " f"observations; only {n=} observations were given.") warnings.warn(message, stacklevel=2) b2 = kurtosis(a, axis, fisher=False, _no_deco=True) E = 3.0*(n-1) / (n+1) varb2 = 24.0*n*(n-2)*(n-3) / ((n+1)*(n+1.)*(n+3)*(n+5)) # [1]_ Eq. 1 x = (b2-E) / varb2**0.5 # [1]_ Eq. 4 # [1]_ Eq. 2: sqrtbeta1 = 6.0*(n*n-5*n+2)/((n+7)*(n+9)) * ((6.0*(n+3)*(n+5)) / (n*(n-2)*(n-3)))**0.5 # [1]_ Eq. 3: A = 6.0 + 8.0/sqrtbeta1 * (2.0/sqrtbeta1 + (1+4.0/(sqrtbeta1**2))**0.5) term1 = 1 - 2/(9.0*A) denom = 1 + x * (2/(A-4.0))**0.5 NaN = _get_nan(x, xp=xp) term2 = xp_sign(denom) * xp.where(denom == 0.0, NaN, ((1-2.0/A)/xp.abs(denom))**(1/3)) if xp.any(denom == 0): msg = ("Test statistic not defined in some cases due to division by " "zero. Return nan in that case...") warnings.warn(msg, RuntimeWarning, stacklevel=2) Z = (term1 - term2) / (2/(9.0*A))**0.5 # [1]_ Eq. 5 pvalue = _get_pvalue(Z, _SimpleNormal(), alternative, xp=xp) Z = Z[()] if Z.ndim == 0 else Z pvalue = pvalue[()] if pvalue.ndim == 0 else pvalue return KurtosistestResult(Z, pvalue) NormaltestResult = namedtuple('NormaltestResult', ('statistic', 'pvalue')) @_axis_nan_policy_factory(NormaltestResult, n_samples=1, too_small=7) def normaltest(a, axis=0, nan_policy='propagate'): r"""Test whether a sample differs from a normal distribution. This function tests the null hypothesis that a sample comes from a normal distribution. It is based on D'Agostino and Pearson's [1]_, [2]_ test that combines skew and kurtosis to produce an omnibus test of normality. Parameters ---------- a : array_like The array containing the sample to be tested. Must contain at least eight observations. axis : int or None, optional Axis along which to compute test. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float or array ``s^2 + k^2``, where ``s`` is the z-score returned by `skewtest` and ``k`` is the z-score returned by `kurtosistest`. pvalue : float or array A 2-sided chi squared probability for the hypothesis test. References ---------- .. [1] D'Agostino, R. B. (1971), "An omnibus test of normality for moderate and large sample size", Biometrika, 58, 341-348 .. [2] D'Agostino, R. and Pearson, E. S. (1973), "Tests for departure from normality", Biometrika, 60, 613-622 .. [3] Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611. .. [4] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn." Statistical Applications in Genetics and Molecular Biology 9.1 (2010). .. [5] Panagiotakos, D. B. (2008). The value of p-value in biomedical research. The open cardiovascular medicine journal, 2, 97. Examples -------- Suppose we wish to infer from measurements whether the weights of adult human males in a medical study are not normally distributed [3]_. The weights (lbs) are recorded in the array ``x`` below. >>> import numpy as np >>> x = np.array([148, 154, 158, 160, 161, 162, 166, 170, 182, 195, 236]) The normality test of [1]_ and [2]_ begins by computing a statistic based on the sample skewness and kurtosis. >>> from scipy import stats >>> res = stats.normaltest(x) >>> res.statistic 13.034263121192582 (The test warns that our sample has too few observations to perform the test. We'll return to this at the end of the example.) Because the normal distribution has zero skewness and zero ("excess" or "Fisher") kurtosis, the value of this statistic tends to be low for samples drawn from a normal distribution. The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that the weights were drawn from a normal distribution. For this normality test, the null distribution for very large samples is the chi-squared distribution with two degrees of freedom. >>> import matplotlib.pyplot as plt >>> dist = stats.chi2(df=2) >>> stat_vals = np.linspace(0, 16, 100) >>> pdf = dist.pdf(stat_vals) >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> def plot(ax): # we'll reuse this ... ax.plot(stat_vals, pdf) ... ax.set_title("Normality Test Null Distribution") ... ax.set_xlabel("statistic") ... ax.set_ylabel("probability density") >>> plot(ax) >>> plt.show() The comparison is quantified by the p-value: the proportion of values in the null distribution greater than or equal to the observed value of the statistic. >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> plot(ax) >>> pvalue = dist.sf(res.statistic) >>> annotation = (f'p-value={pvalue:.6f}\n(shaded area)') >>> props = dict(facecolor='black', width=1, headwidth=5, headlength=8) >>> _ = ax.annotate(annotation, (13.5, 5e-4), (14, 5e-3), arrowprops=props) >>> i = stat_vals >= res.statistic # index more extreme statistic values >>> ax.fill_between(stat_vals[i], y1=0, y2=pdf[i]) >>> ax.set_xlim(8, 16) >>> ax.set_ylim(0, 0.01) >>> plt.show() >>> res.pvalue 0.0014779023013100172 If the p-value is "small" - that is, if there is a low probability of sampling data from a normally distributed population that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the weights were not drawn from a normal distribution. Note that: - The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis. - The threshold for values that will be considered "small" is a choice that should be made before the data is analyzed [4]_ with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis). Note that the chi-squared distribution provides an asymptotic approximation of the null distribution; it is only accurate for samples with many observations. This is the reason we received a warning at the beginning of the example; our sample is quite small. In this case, `scipy.stats.monte_carlo_test` may provide a more accurate, albeit stochastic, approximation of the exact p-value. >>> def statistic(x, axis): ... # Get only the `normaltest` statistic; ignore approximate p-value ... return stats.normaltest(x, axis=axis).statistic >>> res = stats.monte_carlo_test(x, stats.norm.rvs, statistic, ... alternative='greater') >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> plot(ax) >>> ax.hist(res.null_distribution, np.linspace(0, 25, 50), ... density=True) >>> ax.legend(['aymptotic approximation (many observations)', ... 'Monte Carlo approximation (11 observations)']) >>> ax.set_xlim(0, 14) >>> plt.show() >>> res.pvalue 0.0082 # may vary Furthermore, despite their stochastic nature, p-values computed in this way can be used to exactly control the rate of false rejections of the null hypothesis [5]_. """ xp = array_namespace(a) s, _ = skewtest(a, axis, _no_deco=True) k, _ = kurtosistest(a, axis, _no_deco=True) statistic = s*s + k*k chi2 = _SimpleChi2(xp.asarray(2.)) pvalue = _get_pvalue(statistic, chi2, alternative='greater', symmetric=False, xp=xp) statistic = statistic[()] if statistic.ndim == 0 else statistic pvalue = pvalue[()] if pvalue.ndim == 0 else pvalue return NormaltestResult(statistic, pvalue) @_axis_nan_policy_factory(SignificanceResult, default_axis=None) def jarque_bera(x, *, axis=None): r"""Perform the Jarque-Bera goodness of fit test on sample data. The Jarque-Bera test tests whether the sample data has the skewness and kurtosis matching a normal distribution. Note that this test only works for a large enough number of data samples (>2000) as the test statistic asymptotically has a Chi-squared distribution with 2 degrees of freedom. Parameters ---------- x : array_like Observations of a random variable. axis : int or None, default: 0 If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If ``None``, the input will be raveled before computing the statistic. Returns ------- result : SignificanceResult An object with the following attributes: statistic : float The test statistic. pvalue : float The p-value for the hypothesis test. References ---------- .. [1] Jarque, C. and Bera, A. (1980) "Efficient tests for normality, homoscedasticity and serial independence of regression residuals", 6 Econometric Letters 255-259. .. [2] Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611. .. [3] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn." Statistical Applications in Genetics and Molecular Biology 9.1 (2010). .. [4] Panagiotakos, D. B. (2008). The value of p-value in biomedical research. The open cardiovascular medicine journal, 2, 97. Examples -------- Suppose we wish to infer from measurements whether the weights of adult human males in a medical study are not normally distributed [2]_. The weights (lbs) are recorded in the array ``x`` below. >>> import numpy as np >>> x = np.array([148, 154, 158, 160, 161, 162, 166, 170, 182, 195, 236]) The Jarque-Bera test begins by computing a statistic based on the sample skewness and kurtosis. >>> from scipy import stats >>> res = stats.jarque_bera(x) >>> res.statistic 6.982848237344646 Because the normal distribution has zero skewness and zero ("excess" or "Fisher") kurtosis, the value of this statistic tends to be low for samples drawn from a normal distribution. The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that the weights were drawn from a normal distribution. For the Jarque-Bera test, the null distribution for very large samples is the chi-squared distribution with two degrees of freedom. >>> import matplotlib.pyplot as plt >>> dist = stats.chi2(df=2) >>> jb_val = np.linspace(0, 11, 100) >>> pdf = dist.pdf(jb_val) >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> def jb_plot(ax): # we'll reuse this ... ax.plot(jb_val, pdf) ... ax.set_title("Jarque-Bera Null Distribution") ... ax.set_xlabel("statistic") ... ax.set_ylabel("probability density") >>> jb_plot(ax) >>> plt.show() The comparison is quantified by the p-value: the proportion of values in the null distribution greater than or equal to the observed value of the statistic. >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> jb_plot(ax) >>> pvalue = dist.sf(res.statistic) >>> annotation = (f'p-value={pvalue:.6f}\n(shaded area)') >>> props = dict(facecolor='black', width=1, headwidth=5, headlength=8) >>> _ = ax.annotate(annotation, (7.5, 0.01), (8, 0.05), arrowprops=props) >>> i = jb_val >= res.statistic # indices of more extreme statistic values >>> ax.fill_between(jb_val[i], y1=0, y2=pdf[i]) >>> ax.set_xlim(0, 11) >>> ax.set_ylim(0, 0.3) >>> plt.show() >>> res.pvalue 0.03045746622458189 If the p-value is "small" - that is, if there is a low probability of sampling data from a normally distributed population that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the weights were not drawn from a normal distribution. Note that: - The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis. - The threshold for values that will be considered "small" is a choice that should be made before the data is analyzed [3]_ with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis). Note that the chi-squared distribution provides an asymptotic approximation of the null distribution; it is only accurate for samples with many observations. For small samples like ours, `scipy.stats.monte_carlo_test` may provide a more accurate, albeit stochastic, approximation of the exact p-value. >>> def statistic(x, axis): ... # underlying calculation of the Jarque Bera statistic ... s = stats.skew(x, axis=axis) ... k = stats.kurtosis(x, axis=axis) ... return x.shape[axis]/6 * (s**2 + k**2/4) >>> res = stats.monte_carlo_test(x, stats.norm.rvs, statistic, ... alternative='greater') >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> jb_plot(ax) >>> ax.hist(res.null_distribution, np.linspace(0, 10, 50), ... density=True) >>> ax.legend(['aymptotic approximation (many observations)', ... 'Monte Carlo approximation (11 observations)']) >>> plt.show() >>> res.pvalue 0.0097 # may vary Furthermore, despite their stochastic nature, p-values computed in this way can be used to exactly control the rate of false rejections of the null hypothesis [4]_. """ xp = array_namespace(x) x = xp.asarray(x) if axis is None: x = xp.reshape(x, (-1,)) axis = 0 n = x.shape[axis] if n == 0: raise ValueError('At least one observation is required.') mu = xp.mean(x, axis=axis, keepdims=True) diffx = x - mu s = skew(diffx, axis=axis, _no_deco=True) k = kurtosis(diffx, axis=axis, _no_deco=True) statistic = n / 6 * (s**2 + k**2 / 4) chi2 = _SimpleChi2(xp.asarray(2.)) pvalue = _get_pvalue(statistic, chi2, alternative='greater', symmetric=False, xp=xp) statistic = statistic[()] if statistic.ndim == 0 else statistic pvalue = pvalue[()] if pvalue.ndim == 0 else pvalue return SignificanceResult(statistic, pvalue) ##################################### # FREQUENCY FUNCTIONS # ##################################### def scoreatpercentile(a, per, limit=(), interpolation_method='fraction', axis=None): """Calculate the score at a given percentile of the input sequence. For example, the score at `per=50` is the median. If the desired quantile lies between two data points, we interpolate between them, according to the value of `interpolation`. If the parameter `limit` is provided, it should be a tuple (lower, upper) of two values. Parameters ---------- a : array_like A 1-D array of values from which to extract score. per : array_like Percentile(s) at which to extract score. Values should be in range [0,100]. limit : tuple, optional Tuple of two scalars, the lower and upper limits within which to compute the percentile. Values of `a` outside this (closed) interval will be ignored. interpolation_method : {'fraction', 'lower', 'higher'}, optional Specifies the interpolation method to use, when the desired quantile lies between two data points `i` and `j` The following options are available (default is 'fraction'): * 'fraction': ``i + (j - i) * fraction`` where ``fraction`` is the fractional part of the index surrounded by ``i`` and ``j`` * 'lower': ``i`` * 'higher': ``j`` axis : int, optional Axis along which the percentiles are computed. Default is None. If None, compute over the whole array `a`. Returns ------- score : float or ndarray Score at percentile(s). See Also -------- percentileofscore, numpy.percentile Notes ----- This function will become obsolete in the future. For NumPy 1.9 and higher, `numpy.percentile` provides all the functionality that `scoreatpercentile` provides. And it's significantly faster. Therefore it's recommended to use `numpy.percentile` for users that have numpy >= 1.9. Examples -------- >>> import numpy as np >>> from scipy import stats >>> a = np.arange(100) >>> stats.scoreatpercentile(a, 50) 49.5 """ # adapted from NumPy's percentile function. When we require numpy >= 1.8, # the implementation of this function can be replaced by np.percentile. a = np.asarray(a) if a.size == 0: # empty array, return nan(s) with shape matching `per` if np.isscalar(per): return np.nan else: return np.full(np.asarray(per).shape, np.nan, dtype=np.float64) if limit: a = a[(limit[0] <= a) & (a <= limit[1])] sorted_ = np.sort(a, axis=axis) if axis is None: axis = 0 return _compute_qth_percentile(sorted_, per, interpolation_method, axis) # handle sequence of per's without calling sort multiple times def _compute_qth_percentile(sorted_, per, interpolation_method, axis): if not np.isscalar(per): score = [_compute_qth_percentile(sorted_, i, interpolation_method, axis) for i in per] return np.array(score) if not (0 <= per <= 100): raise ValueError("percentile must be in the range [0, 100]") indexer = [slice(None)] * sorted_.ndim idx = per / 100. * (sorted_.shape[axis] - 1) if int(idx) != idx: # round fractional indices according to interpolation method if interpolation_method == 'lower': idx = int(np.floor(idx)) elif interpolation_method == 'higher': idx = int(np.ceil(idx)) elif interpolation_method == 'fraction': pass # keep idx as fraction and interpolate else: raise ValueError("interpolation_method can only be 'fraction', " "'lower' or 'higher'") i = int(idx) if i == idx: indexer[axis] = slice(i, i + 1) weights = array(1) sumval = 1.0 else: indexer[axis] = slice(i, i + 2) j = i + 1 weights = array([(j - idx), (idx - i)], float) wshape = [1] * sorted_.ndim wshape[axis] = 2 weights.shape = wshape sumval = weights.sum() # Use np.add.reduce (== np.sum but a little faster) to coerce data type return np.add.reduce(sorted_[tuple(indexer)] * weights, axis=axis) / sumval def percentileofscore(a, score, kind='rank', nan_policy='propagate'): """Compute the percentile rank of a score relative to a list of scores. A `percentileofscore` of, for example, 80% means that 80% of the scores in `a` are below the given score. In the case of gaps or ties, the exact definition depends on the optional keyword, `kind`. Parameters ---------- a : array_like A 1-D array to which `score` is compared. score : array_like Scores to compute percentiles for. kind : {'rank', 'weak', 'strict', 'mean'}, optional Specifies the interpretation of the resulting score. The following options are available (default is 'rank'): * 'rank': Average percentage ranking of score. In case of multiple matches, average the percentage rankings of all matching scores. * 'weak': This kind corresponds to the definition of a cumulative distribution function. A percentileofscore of 80% means that 80% of values are less than or equal to the provided score. * 'strict': Similar to "weak", except that only values that are strictly less than the given score are counted. * 'mean': The average of the "weak" and "strict" scores, often used in testing. See https://en.wikipedia.org/wiki/Percentile_rank nan_policy : {'propagate', 'raise', 'omit'}, optional Specifies how to treat `nan` values in `a`. The following options are available (default is 'propagate'): * 'propagate': returns nan (for each value in `score`). * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- pcos : float Percentile-position of score (0-100) relative to `a`. See Also -------- numpy.percentile scipy.stats.scoreatpercentile, scipy.stats.rankdata Examples -------- Three-quarters of the given values lie below a given score: >>> import numpy as np >>> from scipy import stats >>> stats.percentileofscore([1, 2, 3, 4], 3) 75.0 With multiple matches, note how the scores of the two matches, 0.6 and 0.8 respectively, are averaged: >>> stats.percentileofscore([1, 2, 3, 3, 4], 3) 70.0 Only 2/5 values are strictly less than 3: >>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='strict') 40.0 But 4/5 values are less than or equal to 3: >>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='weak') 80.0 The average between the weak and the strict scores is: >>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='mean') 60.0 Score arrays (of any dimensionality) are supported: >>> stats.percentileofscore([1, 2, 3, 3, 4], [2, 3]) array([40., 70.]) The inputs can be infinite: >>> stats.percentileofscore([-np.inf, 0, 1, np.inf], [1, 2, np.inf]) array([75., 75., 100.]) If `a` is empty, then the resulting percentiles are all `nan`: >>> stats.percentileofscore([], [1, 2]) array([nan, nan]) """ a = np.asarray(a) n = len(a) score = np.asarray(score) # Nan treatment cna, npa = _contains_nan(a, nan_policy) cns, nps = _contains_nan(score, nan_policy) if (cna or cns) and nan_policy == 'raise': raise ValueError("The input contains nan values") if cns: # If a score is nan, then the output should be nan # (also if nan_policy is "omit", because it only applies to `a`) score = ma.masked_where(np.isnan(score), score) if cna: if nan_policy == "omit": # Don't count nans a = ma.masked_where(np.isnan(a), a) n = a.count() if nan_policy == "propagate": # All outputs should be nans n = 0 # Cannot compare to empty list ==> nan if n == 0: perct = np.full_like(score, np.nan, dtype=np.float64) else: # Prepare broadcasting score = score[..., None] def count(x): return np.count_nonzero(x, -1) # Main computations/logic if kind == 'rank': left = count(a < score) right = count(a <= score) plus1 = left < right perct = (left + right + plus1) * (50.0 / n) elif kind == 'strict': perct = count(a < score) * (100.0 / n) elif kind == 'weak': perct = count(a <= score) * (100.0 / n) elif kind == 'mean': left = count(a < score) right = count(a <= score) perct = (left + right) * (50.0 / n) else: raise ValueError( "kind can only be 'rank', 'strict', 'weak' or 'mean'") # Re-insert nan values perct = ma.filled(perct, np.nan) if perct.ndim == 0: return perct[()] return perct HistogramResult = namedtuple('HistogramResult', ('count', 'lowerlimit', 'binsize', 'extrapoints')) def _histogram(a, numbins=10, defaultlimits=None, weights=None, printextras=False): """Create a histogram. Separate the range into several bins and return the number of instances in each bin. Parameters ---------- a : array_like Array of scores which will be put into bins. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultlimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically ``(a.min() - s, a.max() + s)``, where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. weights : array_like, optional The weights for each value in `a`. Default is None, which gives each value a weight of 1.0 printextras : bool, optional If True, if there are extra points (i.e. the points that fall outside the bin limits) a warning is raised saying how many of those points there are. Default is False. Returns ------- count : ndarray Number of points (or sum of weights) in each bin. lowerlimit : float Lowest value of histogram, the lower limit of the first bin. binsize : float The size of the bins (all bins have the same size). extrapoints : int The number of points outside the range of the histogram. See Also -------- numpy.histogram Notes ----- This histogram is based on numpy's histogram but has a larger range by default if default limits is not set. """ a = np.ravel(a) if defaultlimits is None: if a.size == 0: # handle empty arrays. Undetermined range, so use 0-1. defaultlimits = (0, 1) else: # no range given, so use values in `a` data_min = a.min() data_max = a.max() # Have bins extend past min and max values slightly s = (data_max - data_min) / (2. * (numbins - 1.)) defaultlimits = (data_min - s, data_max + s) # use numpy's histogram method to compute bins hist, bin_edges = np.histogram(a, bins=numbins, range=defaultlimits, weights=weights) # hist are not always floats, convert to keep with old output hist = np.array(hist, dtype=float) # fixed width for bins is assumed, as numpy's histogram gives # fixed width bins for int values for 'bins' binsize = bin_edges[1] - bin_edges[0] # calculate number of extra points extrapoints = len([v for v in a if defaultlimits[0] > v or v > defaultlimits[1]]) if extrapoints > 0 and printextras: warnings.warn(f"Points outside given histogram range = {extrapoints}", stacklevel=3,) return HistogramResult(hist, defaultlimits[0], binsize, extrapoints) CumfreqResult = namedtuple('CumfreqResult', ('cumcount', 'lowerlimit', 'binsize', 'extrapoints')) def cumfreq(a, numbins=10, defaultreallimits=None, weights=None): """Return a cumulative frequency histogram, using the histogram function. A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. Parameters ---------- a : array_like Input array. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultreallimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in `a` is used. Specifically ``(a.min() - s, a.max() + s)``, where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. weights : array_like, optional The weights for each value in `a`. Default is None, which gives each value a weight of 1.0 Returns ------- cumcount : ndarray Binned values of cumulative frequency. lowerlimit : float Lower real limit binsize : float Width of each bin. extrapoints : int Extra points. Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> rng = np.random.default_rng() >>> x = [1, 4, 2, 1, 3, 1] >>> res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5)) >>> res.cumcount array([ 1., 2., 3., 3.]) >>> res.extrapoints 3 Create a normal distribution with 1000 random values >>> samples = stats.norm.rvs(size=1000, random_state=rng) Calculate cumulative frequencies >>> res = stats.cumfreq(samples, numbins=25) Calculate space of values for x >>> x = res.lowerlimit + np.linspace(0, res.binsize*res.cumcount.size, ... res.cumcount.size) Plot histogram and cumulative histogram >>> fig = plt.figure(figsize=(10, 4)) >>> ax1 = fig.add_subplot(1, 2, 1) >>> ax2 = fig.add_subplot(1, 2, 2) >>> ax1.hist(samples, bins=25) >>> ax1.set_title('Histogram') >>> ax2.bar(x, res.cumcount, width=res.binsize) >>> ax2.set_title('Cumulative histogram') >>> ax2.set_xlim([x.min(), x.max()]) >>> plt.show() """ h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights) cumhist = np.cumsum(h * 1, axis=0) return CumfreqResult(cumhist, l, b, e) RelfreqResult = namedtuple('RelfreqResult', ('frequency', 'lowerlimit', 'binsize', 'extrapoints')) def relfreq(a, numbins=10, defaultreallimits=None, weights=None): """Return a relative frequency histogram, using the histogram function. A relative frequency histogram is a mapping of the number of observations in each of the bins relative to the total of observations. Parameters ---------- a : array_like Input array. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultreallimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically ``(a.min() - s, a.max() + s)``, where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. weights : array_like, optional The weights for each value in `a`. Default is None, which gives each value a weight of 1.0 Returns ------- frequency : ndarray Binned values of relative frequency. lowerlimit : float Lower real limit. binsize : float Width of each bin. extrapoints : int Extra points. Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> rng = np.random.default_rng() >>> a = np.array([2, 4, 1, 2, 3, 2]) >>> res = stats.relfreq(a, numbins=4) >>> res.frequency array([ 0.16666667, 0.5 , 0.16666667, 0.16666667]) >>> np.sum(res.frequency) # relative frequencies should add up to 1 1.0 Create a normal distribution with 1000 random values >>> samples = stats.norm.rvs(size=1000, random_state=rng) Calculate relative frequencies >>> res = stats.relfreq(samples, numbins=25) Calculate space of values for x >>> x = res.lowerlimit + np.linspace(0, res.binsize*res.frequency.size, ... res.frequency.size) Plot relative frequency histogram >>> fig = plt.figure(figsize=(5, 4)) >>> ax = fig.add_subplot(1, 1, 1) >>> ax.bar(x, res.frequency, width=res.binsize) >>> ax.set_title('Relative frequency histogram') >>> ax.set_xlim([x.min(), x.max()]) >>> plt.show() """ a = np.asanyarray(a) h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights) h = h / a.shape[0] return RelfreqResult(h, l, b, e) ##################################### # VARIABILITY FUNCTIONS # ##################################### def obrientransform(*samples): """Compute the O'Brien transform on input data (any number of arrays). Used to test for homogeneity of variance prior to running one-way stats. Each array in ``*samples`` is one level of a factor. If `f_oneway` is run on the transformed data and found significant, the variances are unequal. From Maxwell and Delaney [1]_, p.112. Parameters ---------- sample1, sample2, ... : array_like Any number of arrays. Returns ------- obrientransform : ndarray Transformed data for use in an ANOVA. The first dimension of the result corresponds to the sequence of transformed arrays. If the arrays given are all 1-D of the same length, the return value is a 2-D array; otherwise it is a 1-D array of type object, with each element being an ndarray. References ---------- .. [1] S. E. Maxwell and H. D. Delaney, "Designing Experiments and Analyzing Data: A Model Comparison Perspective", Wadsworth, 1990. Examples -------- We'll test the following data sets for differences in their variance. >>> x = [10, 11, 13, 9, 7, 12, 12, 9, 10] >>> y = [13, 21, 5, 10, 8, 14, 10, 12, 7, 15] Apply the O'Brien transform to the data. >>> from scipy.stats import obrientransform >>> tx, ty = obrientransform(x, y) Use `scipy.stats.f_oneway` to apply a one-way ANOVA test to the transformed data. >>> from scipy.stats import f_oneway >>> F, p = f_oneway(tx, ty) >>> p 0.1314139477040335 If we require that ``p < 0.05`` for significance, we cannot conclude that the variances are different. """ TINY = np.sqrt(np.finfo(float).eps) # `arrays` will hold the transformed arguments. arrays = [] sLast = None for sample in samples: a = np.asarray(sample) n = len(a) mu = np.mean(a) sq = (a - mu)**2 sumsq = sq.sum() # The O'Brien transform. t = ((n - 1.5) * n * sq - 0.5 * sumsq) / ((n - 1) * (n - 2)) # Check that the mean of the transformed data is equal to the # original variance. var = sumsq / (n - 1) if abs(var - np.mean(t)) > TINY: raise ValueError('Lack of convergence in obrientransform.') arrays.append(t) sLast = a.shape if sLast: for arr in arrays[:-1]: if sLast != arr.shape: return np.array(arrays, dtype=object) return np.array(arrays) @_axis_nan_policy_factory( lambda x: x, result_to_tuple=lambda x: (x,), n_outputs=1, too_small=1 ) def sem(a, axis=0, ddof=1, nan_policy='propagate'): """Compute standard error of the mean. Calculate the standard error of the mean (or standard error of measurement) of the values in the input array. Parameters ---------- a : array_like An array containing the values for which the standard error is returned. Must contain at least two observations. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees-of-freedom. How many degrees of freedom to adjust for bias in limited samples relative to the population estimate of variance. Defaults to 1. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- s : ndarray or float The standard error of the mean in the sample(s), along the input axis. Notes ----- The default value for `ddof` is different to the default (0) used by other ddof containing routines, such as np.std and np.nanstd. Examples -------- Find standard error along the first axis: >>> import numpy as np >>> from scipy import stats >>> a = np.arange(20).reshape(5,4) >>> stats.sem(a) array([ 2.8284, 2.8284, 2.8284, 2.8284]) Find standard error across the whole array, using n degrees of freedom: >>> stats.sem(a, axis=None, ddof=0) 1.2893796958227628 """ xp = array_namespace(a) if axis is None: a = xp.reshape(a, (-1,)) axis = 0 a = atleast_nd(a, ndim=1, xp=xp) n = a.shape[axis] s = xp.std(a, axis=axis, correction=ddof) / n**0.5 return s def _isconst(x): """ Check if all values in x are the same. nans are ignored. x must be a 1d array. The return value is a 1d array with length 1, so it can be used in np.apply_along_axis. """ y = x[~np.isnan(x)] if y.size == 0: return np.array([True]) else: return (y[0] == y).all(keepdims=True) def _quiet_nanmean(x): """ Compute nanmean for the 1d array x, but quietly return nan if x is all nan. The return value is a 1d array with length 1, so it can be used in np.apply_along_axis. """ y = x[~np.isnan(x)] if y.size == 0: return np.array([np.nan]) else: return np.mean(y, keepdims=True) def _quiet_nanstd(x, ddof=0): """ Compute nanstd for the 1d array x, but quietly return nan if x is all nan. The return value is a 1d array with length 1, so it can be used in np.apply_along_axis. """ y = x[~np.isnan(x)] if y.size == 0: return np.array([np.nan]) else: return np.std(y, keepdims=True, ddof=ddof) def zscore(a, axis=0, ddof=0, nan_policy='propagate'): """ Compute the z score. Compute the z score of each value in the sample, relative to the sample mean and standard deviation. Parameters ---------- a : array_like An array like object containing the sample data. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Degrees of freedom correction in the calculation of the standard deviation. Default is 0. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. 'propagate' returns nan, 'raise' throws an error, 'omit' performs the calculations ignoring nan values. Default is 'propagate'. Note that when the value is 'omit', nans in the input also propagate to the output, but they do not affect the z-scores computed for the non-nan values. Returns ------- zscore : array_like The z-scores, standardized by mean and standard deviation of input array `a`. See Also -------- numpy.mean : Arithmetic average numpy.std : Arithmetic standard deviation scipy.stats.gzscore : Geometric standard score Notes ----- This function preserves ndarray subclasses, and works also with matrices and masked arrays (it uses `asanyarray` instead of `asarray` for parameters). References ---------- .. [1] "Standard score", *Wikipedia*, https://en.wikipedia.org/wiki/Standard_score. .. [2] Huck, S. W., Cross, T. L., Clark, S. B, "Overcoming misconceptions about Z-scores", Teaching Statistics, vol. 8, pp. 38-40, 1986 Examples -------- >>> import numpy as np >>> a = np.array([ 0.7972, 0.0767, 0.4383, 0.7866, 0.8091, ... 0.1954, 0.6307, 0.6599, 0.1065, 0.0508]) >>> from scipy import stats >>> stats.zscore(a) array([ 1.1273, -1.247 , -0.0552, 1.0923, 1.1664, -0.8559, 0.5786, 0.6748, -1.1488, -1.3324]) Computing along a specified axis, using n-1 degrees of freedom (``ddof=1``) to calculate the standard deviation: >>> b = np.array([[ 0.3148, 0.0478, 0.6243, 0.4608], ... [ 0.7149, 0.0775, 0.6072, 0.9656], ... [ 0.6341, 0.1403, 0.9759, 0.4064], ... [ 0.5918, 0.6948, 0.904 , 0.3721], ... [ 0.0921, 0.2481, 0.1188, 0.1366]]) >>> stats.zscore(b, axis=1, ddof=1) array([[-0.19264823, -1.28415119, 1.07259584, 0.40420358], [ 0.33048416, -1.37380874, 0.04251374, 1.00081084], [ 0.26796377, -1.12598418, 1.23283094, -0.37481053], [-0.22095197, 0.24468594, 1.19042819, -1.21416216], [-0.82780366, 1.4457416 , -0.43867764, -0.1792603 ]]) An example with `nan_policy='omit'`: >>> x = np.array([[25.11, 30.10, np.nan, 32.02, 43.15], ... [14.95, 16.06, 121.25, 94.35, 29.81]]) >>> stats.zscore(x, axis=1, nan_policy='omit') array([[-1.13490897, -0.37830299, nan, -0.08718406, 1.60039602], [-0.91611681, -0.89090508, 1.4983032 , 0.88731639, -0.5785977 ]]) """ return zmap(a, a, axis=axis, ddof=ddof, nan_policy=nan_policy) def gzscore(a, *, axis=0, ddof=0, nan_policy='propagate'): """ Compute the geometric standard score. Compute the geometric z score of each strictly positive value in the sample, relative to the geometric mean and standard deviation. Mathematically the geometric z score can be evaluated as:: gzscore = log(a/gmu) / log(gsigma) where ``gmu`` (resp. ``gsigma``) is the geometric mean (resp. standard deviation). Parameters ---------- a : array_like Sample data. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Degrees of freedom correction in the calculation of the standard deviation. Default is 0. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. 'propagate' returns nan, 'raise' throws an error, 'omit' performs the calculations ignoring nan values. Default is 'propagate'. Note that when the value is 'omit', nans in the input also propagate to the output, but they do not affect the geometric z scores computed for the non-nan values. Returns ------- gzscore : array_like The geometric z scores, standardized by geometric mean and geometric standard deviation of input array `a`. See Also -------- gmean : Geometric mean gstd : Geometric standard deviation zscore : Standard score Notes ----- This function preserves ndarray subclasses, and works also with matrices and masked arrays (it uses ``asanyarray`` instead of ``asarray`` for parameters). .. versionadded:: 1.8 References ---------- .. [1] "Geometric standard score", *Wikipedia*, https://en.wikipedia.org/wiki/Geometric_standard_deviation#Geometric_standard_score. Examples -------- Draw samples from a log-normal distribution: >>> import numpy as np >>> from scipy.stats import zscore, gzscore >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> mu, sigma = 3., 1. # mean and standard deviation >>> x = rng.lognormal(mu, sigma, size=500) Display the histogram of the samples: >>> fig, ax = plt.subplots() >>> ax.hist(x, 50) >>> plt.show() Display the histogram of the samples standardized by the classical zscore. Distribution is rescaled but its shape is unchanged. >>> fig, ax = plt.subplots() >>> ax.hist(zscore(x), 50) >>> plt.show() Demonstrate that the distribution of geometric zscores is rescaled and quasinormal: >>> fig, ax = plt.subplots() >>> ax.hist(gzscore(x), 50) >>> plt.show() """ a = np.asanyarray(a) log = ma.log if isinstance(a, ma.MaskedArray) else np.log return zscore(log(a), axis=axis, ddof=ddof, nan_policy=nan_policy) def zmap(scores, compare, axis=0, ddof=0, nan_policy='propagate'): """ Calculate the relative z-scores. Return an array of z-scores, i.e., scores that are standardized to zero mean and unit variance, where mean and variance are calculated from the comparison array. Parameters ---------- scores : array_like The input for which z-scores are calculated. compare : array_like The input from which the mean and standard deviation of the normalization are taken; assumed to have the same dimension as `scores`. axis : int or None, optional Axis over which mean and variance of `compare` are calculated. Default is 0. If None, compute over the whole array `scores`. ddof : int, optional Degrees of freedom correction in the calculation of the standard deviation. Default is 0. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle the occurrence of nans in `compare`. 'propagate' returns nan, 'raise' raises an exception, 'omit' performs the calculations ignoring nan values. Default is 'propagate'. Note that when the value is 'omit', nans in `scores` also propagate to the output, but they do not affect the z-scores computed for the non-nan values. Returns ------- zscore : array_like Z-scores, in the same shape as `scores`. Notes ----- This function preserves ndarray subclasses, and works also with matrices and masked arrays (it uses `asanyarray` instead of `asarray` for parameters). Examples -------- >>> from scipy.stats import zmap >>> a = [0.5, 2.0, 2.5, 3] >>> b = [0, 1, 2, 3, 4] >>> zmap(a, b) array([-1.06066017, 0. , 0.35355339, 0.70710678]) """ a = np.asanyarray(compare) if a.size == 0: return np.empty(a.shape) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': if axis is None: mn = _quiet_nanmean(a.ravel()) std = _quiet_nanstd(a.ravel(), ddof=ddof) isconst = _isconst(a.ravel()) else: mn = np.apply_along_axis(_quiet_nanmean, axis, a) std = np.apply_along_axis(_quiet_nanstd, axis, a, ddof=ddof) isconst = np.apply_along_axis(_isconst, axis, a) else: mn = a.mean(axis=axis, keepdims=True) std = a.std(axis=axis, ddof=ddof, keepdims=True) # The intent is to check whether all elements of `a` along `axis` are # identical. Due to finite precision arithmetic, comparing elements # against `mn` doesn't work. Previously, this compared elements to # `_first`, but that extracts the element at index 0 regardless of # whether it is masked. As a simple fix, compare against `min`. a0 = a.min(axis=axis, keepdims=True) isconst = (a == a0).all(axis=axis, keepdims=True) # Set std deviations that are 0 to 1 to avoid division by 0. std[isconst] = 1.0 z = (scores - mn) / std # Set the outputs associated with a constant input to nan. z[np.broadcast_to(isconst, z.shape)] = np.nan return z def gstd(a, axis=0, ddof=1): r""" Calculate the geometric standard deviation of an array. The geometric standard deviation describes the spread of a set of numbers where the geometric mean is preferred. It is a multiplicative factor, and so a dimensionless quantity. It is defined as the exponential of the standard deviation of the natural logarithms of the observations. Parameters ---------- a : array_like An array containing finite, strictly positive, real numbers. .. deprecated:: 1.14.0 Support for masked array input was deprecated in SciPy 1.14.0 and will be removed in version 1.16.0. axis : int, tuple or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Degree of freedom correction in the calculation of the geometric standard deviation. Default is 1. Returns ------- gstd : ndarray or float An array of the geometric standard deviation. If `axis` is None or `a` is a 1d array a float is returned. See Also -------- gmean : Geometric mean numpy.std : Standard deviation gzscore : Geometric standard score Notes ----- Mathematically, the sample geometric standard deviation :math:`s_G` can be defined in terms of the natural logarithms of the observations :math:`y_i = \log(x_i)`: .. math:: s_G = \exp(s), \quad s = \sqrt{\frac{1}{n - d} \sum_{i=1}^n (y_i - \bar y)^2} where :math:`n` is the number of observations, :math:`d` is the adjustment `ddof` to the degrees of freedom, and :math:`\bar y` denotes the mean of the natural logarithms of the observations. Note that the default ``ddof=1`` is different from the default value used by similar functions, such as `numpy.std` and `numpy.var`. When an observation is infinite, the geometric standard deviation is NaN (undefined). Non-positive observations will also produce NaNs in the output because the *natural* logarithm (as opposed to the *complex* logarithm) is defined and finite only for positive reals. The geometric standard deviation is sometimes confused with the exponential of the standard deviation, ``exp(std(a))``. Instead, the geometric standard deviation is ``exp(std(log(a)))``. References ---------- .. [1] "Geometric standard deviation", *Wikipedia*, https://en.wikipedia.org/wiki/Geometric_standard_deviation. .. [2] Kirkwood, T. B., "Geometric means and measures of dispersion", Biometrics, vol. 35, pp. 908-909, 1979 Examples -------- Find the geometric standard deviation of a log-normally distributed sample. Note that the standard deviation of the distribution is one; on a log scale this evaluates to approximately ``exp(1)``. >>> import numpy as np >>> from scipy.stats import gstd >>> rng = np.random.default_rng() >>> sample = rng.lognormal(mean=0, sigma=1, size=1000) >>> gstd(sample) 2.810010162475324 Compute the geometric standard deviation of a multidimensional array and of a given axis. >>> a = np.arange(1, 25).reshape(2, 3, 4) >>> gstd(a, axis=None) 2.2944076136018947 >>> gstd(a, axis=2) array([[1.82424757, 1.22436866, 1.13183117], [1.09348306, 1.07244798, 1.05914985]]) >>> gstd(a, axis=(1,2)) array([2.12939215, 1.22120169]) """ a = np.asanyarray(a) if isinstance(a, ma.MaskedArray): message = ("`gstd` support for masked array input was deprecated in " "SciPy 1.14.0 and will be removed in version 1.16.0.") warnings.warn(message, DeprecationWarning, stacklevel=2) log = ma.log else: log = np.log with np.errstate(invalid='ignore', divide='ignore'): res = np.exp(np.std(log(a), axis=axis, ddof=ddof)) if (a <= 0).any(): message = ("The geometric standard deviation is only defined if all elements " "are greater than or equal to zero; otherwise, the result is NaN.") warnings.warn(message, RuntimeWarning, stacklevel=2) return res # Private dictionary initialized only once at module level # See https://en.wikipedia.org/wiki/Robust_measures_of_scale _scale_conversions = {'normal': special.erfinv(0.5) * 2.0 * math.sqrt(2.0)} @_axis_nan_policy_factory( lambda x: x, result_to_tuple=lambda x: (x,), n_outputs=1, default_axis=None, override={'nan_propagation': False} ) def iqr(x, axis=None, rng=(25, 75), scale=1.0, nan_policy='propagate', interpolation='linear', keepdims=False): r""" Compute the interquartile range of the data along the specified axis. The interquartile range (IQR) is the difference between the 75th and 25th percentile of the data. It is a measure of the dispersion similar to standard deviation or variance, but is much more robust against outliers [2]_. The ``rng`` parameter allows this function to compute other percentile ranges than the actual IQR. For example, setting ``rng=(0, 100)`` is equivalent to `numpy.ptp`. The IQR of an empty array is `np.nan`. .. versionadded:: 0.18.0 Parameters ---------- x : array_like Input array or object that can be converted to an array. axis : int or sequence of int, optional Axis along which the range is computed. The default is to compute the IQR for the entire array. rng : Two-element sequence containing floats in range of [0,100] optional Percentiles over which to compute the range. Each must be between 0 and 100, inclusive. The default is the true IQR: ``(25, 75)``. The order of the elements is not important. scale : scalar or str or array_like of reals, optional The numerical value of scale will be divided out of the final result. The following string value is also recognized: * 'normal' : Scale by :math:`2 \sqrt{2} erf^{-1}(\frac{1}{2}) \approx 1.349`. The default is 1.0. Array-like `scale` of real dtype is also allowed, as long as it broadcasts correctly to the output such that ``out / scale`` is a valid operation. The output dimensions depend on the input array, `x`, the `axis` argument, and the `keepdims` flag. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values interpolation : str, optional Specifies the interpolation method to use when the percentile boundaries lie between two data points ``i`` and ``j``. The following options are available (default is 'linear'): * 'linear': ``i + (j - i)*fraction``, where ``fraction`` is the fractional part of the index surrounded by ``i`` and ``j``. * 'lower': ``i``. * 'higher': ``j``. * 'nearest': ``i`` or ``j`` whichever is nearest. * 'midpoint': ``(i + j)/2``. For NumPy >= 1.22.0, the additional options provided by the ``method`` keyword of `numpy.percentile` are also valid. keepdims : bool, optional If this is set to True, the reduced axes are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original array `x`. Returns ------- iqr : scalar or ndarray If ``axis=None``, a scalar is returned. If the input contains integers or floats of smaller precision than ``np.float64``, then the output data-type is ``np.float64``. Otherwise, the output data-type is the same as that of the input. See Also -------- numpy.std, numpy.var References ---------- .. [1] "Interquartile range" https://en.wikipedia.org/wiki/Interquartile_range .. [2] "Robust measures of scale" https://en.wikipedia.org/wiki/Robust_measures_of_scale .. [3] "Quantile" https://en.wikipedia.org/wiki/Quantile Examples -------- >>> import numpy as np >>> from scipy.stats import iqr >>> x = np.array([[10, 7, 4], [3, 2, 1]]) >>> x array([[10, 7, 4], [ 3, 2, 1]]) >>> iqr(x) 4.0 >>> iqr(x, axis=0) array([ 3.5, 2.5, 1.5]) >>> iqr(x, axis=1) array([ 3., 1.]) >>> iqr(x, axis=1, keepdims=True) array([[ 3.], [ 1.]]) """ x = asarray(x) # This check prevents percentile from raising an error later. Also, it is # consistent with `np.var` and `np.std`. if not x.size: return _get_nan(x) # An error may be raised here, so fail-fast, before doing lengthy # computations, even though `scale` is not used until later if isinstance(scale, str): scale_key = scale.lower() if scale_key not in _scale_conversions: raise ValueError(f"{scale} not a valid scale for `iqr`") scale = _scale_conversions[scale_key] # Select the percentile function to use based on nans and policy contains_nan, nan_policy = _contains_nan(x, nan_policy) if contains_nan and nan_policy == 'omit': percentile_func = np.nanpercentile else: percentile_func = np.percentile if len(rng) != 2: raise TypeError("quantile range must be two element sequence") if np.isnan(rng).any(): raise ValueError("range must not contain NaNs") rng = sorted(rng) pct = percentile_func(x, rng, axis=axis, method=interpolation, keepdims=keepdims) out = np.subtract(pct[1], pct[0]) if scale != 1.0: out /= scale return out def _mad_1d(x, center, nan_policy): # Median absolute deviation for 1-d array x. # This is a helper function for `median_abs_deviation`; it assumes its # arguments have been validated already. In particular, x must be a # 1-d numpy array, center must be callable, and if nan_policy is not # 'propagate', it is assumed to be 'omit', because 'raise' is handled # in `median_abs_deviation`. # No warning is generated if x is empty or all nan. isnan = np.isnan(x) if isnan.any(): if nan_policy == 'propagate': return np.nan x = x[~isnan] if x.size == 0: # MAD of an empty array is nan. return np.nan # Edge cases have been handled, so do the basic MAD calculation. med = center(x) mad = np.median(np.abs(x - med)) return mad def median_abs_deviation(x, axis=0, center=np.median, scale=1.0, nan_policy='propagate'): r""" Compute the median absolute deviation of the data along the given axis. The median absolute deviation (MAD, [1]_) computes the median over the absolute deviations from the median. It is a measure of dispersion similar to the standard deviation but more robust to outliers [2]_. The MAD of an empty array is ``np.nan``. .. versionadded:: 1.5.0 Parameters ---------- x : array_like Input array or object that can be converted to an array. axis : int or None, optional Axis along which the range is computed. Default is 0. If None, compute the MAD over the entire array. center : callable, optional A function that will return the central value. The default is to use np.median. Any user defined function used will need to have the function signature ``func(arr, axis)``. scale : scalar or str, optional The numerical value of scale will be divided out of the final result. The default is 1.0. The string "normal" is also accepted, and results in `scale` being the inverse of the standard normal quantile function at 0.75, which is approximately 0.67449. Array-like scale is also allowed, as long as it broadcasts correctly to the output such that ``out / scale`` is a valid operation. The output dimensions depend on the input array, `x`, and the `axis` argument. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- mad : scalar or ndarray If ``axis=None``, a scalar is returned. If the input contains integers or floats of smaller precision than ``np.float64``, then the output data-type is ``np.float64``. Otherwise, the output data-type is the same as that of the input. See Also -------- numpy.std, numpy.var, numpy.median, scipy.stats.iqr, scipy.stats.tmean, scipy.stats.tstd, scipy.stats.tvar Notes ----- The `center` argument only affects the calculation of the central value around which the MAD is calculated. That is, passing in ``center=np.mean`` will calculate the MAD around the mean - it will not calculate the *mean* absolute deviation. The input array may contain `inf`, but if `center` returns `inf`, the corresponding MAD for that data will be `nan`. References ---------- .. [1] "Median absolute deviation", https://en.wikipedia.org/wiki/Median_absolute_deviation .. [2] "Robust measures of scale", https://en.wikipedia.org/wiki/Robust_measures_of_scale Examples -------- When comparing the behavior of `median_abs_deviation` with ``np.std``, the latter is affected when we change a single value of an array to have an outlier value while the MAD hardly changes: >>> import numpy as np >>> from scipy import stats >>> x = stats.norm.rvs(size=100, scale=1, random_state=123456) >>> x.std() 0.9973906394005013 >>> stats.median_abs_deviation(x) 0.82832610097857 >>> x[0] = 345.6 >>> x.std() 34.42304872314415 >>> stats.median_abs_deviation(x) 0.8323442311590675 Axis handling example: >>> x = np.array([[10, 7, 4], [3, 2, 1]]) >>> x array([[10, 7, 4], [ 3, 2, 1]]) >>> stats.median_abs_deviation(x) array([3.5, 2.5, 1.5]) >>> stats.median_abs_deviation(x, axis=None) 2.0 Scale normal example: >>> x = stats.norm.rvs(size=1000000, scale=2, random_state=123456) >>> stats.median_abs_deviation(x) 1.3487398527041636 >>> stats.median_abs_deviation(x, scale='normal') 1.9996446978061115 """ if not callable(center): raise TypeError("The argument 'center' must be callable. The given " f"value {repr(center)} is not callable.") # An error may be raised here, so fail-fast, before doing lengthy # computations, even though `scale` is not used until later if isinstance(scale, str): if scale.lower() == 'normal': scale = 0.6744897501960817 # special.ndtri(0.75) else: raise ValueError(f"{scale} is not a valid scale value.") x = asarray(x) # Consistent with `np.var` and `np.std`. if not x.size: if axis is None: return np.nan nan_shape = tuple(item for i, item in enumerate(x.shape) if i != axis) if nan_shape == (): # Return nan, not array(nan) return np.nan return np.full(nan_shape, np.nan) contains_nan, nan_policy = _contains_nan(x, nan_policy) if contains_nan: if axis is None: mad = _mad_1d(x.ravel(), center, nan_policy) else: mad = np.apply_along_axis(_mad_1d, axis, x, center, nan_policy) else: if axis is None: med = center(x, axis=None) mad = np.median(np.abs(x - med)) else: # Wrap the call to center() in expand_dims() so it acts like # keepdims=True was used. med = np.expand_dims(center(x, axis=axis), axis) mad = np.median(np.abs(x - med), axis=axis) return mad / scale ##################################### # TRIMMING FUNCTIONS # ##################################### SigmaclipResult = namedtuple('SigmaclipResult', ('clipped', 'lower', 'upper')) def sigmaclip(a, low=4., high=4.): """Perform iterative sigma-clipping of array elements. Starting from the full sample, all elements outside the critical range are removed, i.e. all elements of the input array `c` that satisfy either of the following conditions:: c < mean(c) - std(c)*low c > mean(c) + std(c)*high The iteration continues with the updated sample until no elements are outside the (updated) range. Parameters ---------- a : array_like Data array, will be raveled if not 1-D. low : float, optional Lower bound factor of sigma clipping. Default is 4. high : float, optional Upper bound factor of sigma clipping. Default is 4. Returns ------- clipped : ndarray Input array with clipped elements removed. lower : float Lower threshold value use for clipping. upper : float Upper threshold value use for clipping. Examples -------- >>> import numpy as np >>> from scipy.stats import sigmaclip >>> a = np.concatenate((np.linspace(9.5, 10.5, 31), ... np.linspace(0, 20, 5))) >>> fact = 1.5 >>> c, low, upp = sigmaclip(a, fact, fact) >>> c array([ 9.96666667, 10. , 10.03333333, 10. ]) >>> c.var(), c.std() (0.00055555555555555165, 0.023570226039551501) >>> low, c.mean() - fact*c.std(), c.min() (9.9646446609406727, 9.9646446609406727, 9.9666666666666668) >>> upp, c.mean() + fact*c.std(), c.max() (10.035355339059327, 10.035355339059327, 10.033333333333333) >>> a = np.concatenate((np.linspace(9.5, 10.5, 11), ... np.linspace(-100, -50, 3))) >>> c, low, upp = sigmaclip(a, 1.8, 1.8) >>> (c == np.linspace(9.5, 10.5, 11)).all() True """ c = np.asarray(a).ravel() delta = 1 while delta: c_std = c.std() c_mean = c.mean() size = c.size critlower = c_mean - c_std * low critupper = c_mean + c_std * high c = c[(c >= critlower) & (c <= critupper)] delta = size - c.size return SigmaclipResult(c, critlower, critupper) def trimboth(a, proportiontocut, axis=0): """Slice off a proportion of items from both ends of an array. Slice off the passed proportion of items from both ends of the passed array (i.e., with `proportiontocut` = 0.1, slices leftmost 10% **and** rightmost 10% of scores). The trimmed values are the lowest and highest ones. Slice off less if proportion results in a non-integer slice index (i.e. conservatively slices off `proportiontocut`). Parameters ---------- a : array_like Data to trim. proportiontocut : float Proportion (in range 0-1) of total data set to trim of each end. axis : int or None, optional Axis along which to trim data. Default is 0. If None, compute over the whole array `a`. Returns ------- out : ndarray Trimmed version of array `a`. The order of the trimmed content is undefined. See Also -------- trim_mean Examples -------- Create an array of 10 values and trim 10% of those values from each end: >>> import numpy as np >>> from scipy import stats >>> a = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] >>> stats.trimboth(a, 0.1) array([1, 3, 2, 4, 5, 6, 7, 8]) Note that the elements of the input array are trimmed by value, but the output array is not necessarily sorted. The proportion to trim is rounded down to the nearest integer. For instance, trimming 25% of the values from each end of an array of 10 values will return an array of 6 values: >>> b = np.arange(10) >>> stats.trimboth(b, 1/4).shape (6,) Multidimensional arrays can be trimmed along any axis or across the entire array: >>> c = [2, 4, 6, 8, 0, 1, 3, 5, 7, 9] >>> d = np.array([a, b, c]) >>> stats.trimboth(d, 0.4, axis=0).shape (1, 10) >>> stats.trimboth(d, 0.4, axis=1).shape (3, 2) >>> stats.trimboth(d, 0.4, axis=None).shape (6,) """ a = np.asarray(a) if a.size == 0: return a if axis is None: a = a.ravel() axis = 0 nobs = a.shape[axis] lowercut = int(proportiontocut * nobs) uppercut = nobs - lowercut if (lowercut >= uppercut): raise ValueError("Proportion too big.") atmp = np.partition(a, (lowercut, uppercut - 1), axis) sl = [slice(None)] * atmp.ndim sl[axis] = slice(lowercut, uppercut) return atmp[tuple(sl)] def trim1(a, proportiontocut, tail='right', axis=0): """Slice off a proportion from ONE end of the passed array distribution. If `proportiontocut` = 0.1, slices off 'leftmost' or 'rightmost' 10% of scores. The lowest or highest values are trimmed (depending on the tail). Slice off less if proportion results in a non-integer slice index (i.e. conservatively slices off `proportiontocut` ). Parameters ---------- a : array_like Input array. proportiontocut : float Fraction to cut off of 'left' or 'right' of distribution. tail : {'left', 'right'}, optional Defaults to 'right'. axis : int or None, optional Axis along which to trim data. Default is 0. If None, compute over the whole array `a`. Returns ------- trim1 : ndarray Trimmed version of array `a`. The order of the trimmed content is undefined. Examples -------- Create an array of 10 values and trim 20% of its lowest values: >>> import numpy as np >>> from scipy import stats >>> a = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] >>> stats.trim1(a, 0.2, 'left') array([2, 4, 3, 5, 6, 7, 8, 9]) Note that the elements of the input array are trimmed by value, but the output array is not necessarily sorted. The proportion to trim is rounded down to the nearest integer. For instance, trimming 25% of the values from an array of 10 values will return an array of 8 values: >>> b = np.arange(10) >>> stats.trim1(b, 1/4).shape (8,) Multidimensional arrays can be trimmed along any axis or across the entire array: >>> c = [2, 4, 6, 8, 0, 1, 3, 5, 7, 9] >>> d = np.array([a, b, c]) >>> stats.trim1(d, 0.8, axis=0).shape (1, 10) >>> stats.trim1(d, 0.8, axis=1).shape (3, 2) >>> stats.trim1(d, 0.8, axis=None).shape (6,) """ a = np.asarray(a) if axis is None: a = a.ravel() axis = 0 nobs = a.shape[axis] # avoid possible corner case if proportiontocut >= 1: return [] if tail.lower() == 'right': lowercut = 0 uppercut = nobs - int(proportiontocut * nobs) elif tail.lower() == 'left': lowercut = int(proportiontocut * nobs) uppercut = nobs atmp = np.partition(a, (lowercut, uppercut - 1), axis) sl = [slice(None)] * atmp.ndim sl[axis] = slice(lowercut, uppercut) return atmp[tuple(sl)] def trim_mean(a, proportiontocut, axis=0): """Return mean of array after trimming a specified fraction of extreme values Removes the specified proportion of elements from *each* end of the sorted array, then computes the mean of the remaining elements. Parameters ---------- a : array_like Input array. proportiontocut : float Fraction of the most positive and most negative elements to remove. When the specified proportion does not result in an integer number of elements, the number of elements to trim is rounded down. axis : int or None, default: 0 Axis along which the trimmed means are computed. If None, compute over the raveled array. Returns ------- trim_mean : ndarray Mean of trimmed array. See Also -------- trimboth : Remove a proportion of elements from each end of an array. tmean : Compute the mean after trimming values outside specified limits. Notes ----- For 1-D array `a`, `trim_mean` is approximately equivalent to the following calculation:: import numpy as np a = np.sort(a) m = int(proportiontocut * len(a)) np.mean(a[m: len(a) - m]) Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = [1, 2, 3, 5] >>> stats.trim_mean(x, 0.25) 2.5 When the specified proportion does not result in an integer number of elements, the number of elements to trim is rounded down. >>> stats.trim_mean(x, 0.24999) == np.mean(x) True Use `axis` to specify the axis along which the calculation is performed. >>> x2 = [[1, 2, 3, 5], ... [10, 20, 30, 50]] >>> stats.trim_mean(x2, 0.25) array([ 5.5, 11. , 16.5, 27.5]) >>> stats.trim_mean(x2, 0.25, axis=1) array([ 2.5, 25. ]) """ a = np.asarray(a) if a.size == 0: return np.nan if axis is None: a = a.ravel() axis = 0 nobs = a.shape[axis] lowercut = int(proportiontocut * nobs) uppercut = nobs - lowercut if (lowercut > uppercut): raise ValueError("Proportion too big.") atmp = np.partition(a, (lowercut, uppercut - 1), axis) sl = [slice(None)] * atmp.ndim sl[axis] = slice(lowercut, uppercut) return np.mean(atmp[tuple(sl)], axis=axis) F_onewayResult = namedtuple('F_onewayResult', ('statistic', 'pvalue')) def _create_f_oneway_nan_result(shape, axis, samples): """ This is a helper function for f_oneway for creating the return values in certain degenerate conditions. It creates return values that are all nan with the appropriate shape for the given `shape` and `axis`. """ axis = normalize_axis_index(axis, len(shape)) shp = shape[:axis] + shape[axis+1:] f = np.full(shp, fill_value=_get_nan(*samples)) prob = f.copy() return F_onewayResult(f[()], prob[()]) def _first(arr, axis): """Return arr[..., 0:1, ...] where 0:1 is in the `axis` position.""" return np.take_along_axis(arr, np.array(0, ndmin=arr.ndim), axis) def _f_oneway_is_too_small(samples, kwargs={}, axis=-1): message = f"At least two samples are required; got {len(samples)}." if len(samples) < 2: raise TypeError(message) # Check this after forming alldata, so shape errors are detected # and reported before checking for 0 length inputs. if any(sample.shape[axis] == 0 for sample in samples): return True # Must have at least one group with length greater than 1. if all(sample.shape[axis] == 1 for sample in samples): msg = ('all input arrays have length 1. f_oneway requires that at ' 'least one input has length greater than 1.') warnings.warn(SmallSampleWarning(msg), stacklevel=2) return True return False @_axis_nan_policy_factory( F_onewayResult, n_samples=None, too_small=_f_oneway_is_too_small) def f_oneway(*samples, axis=0): """Perform one-way ANOVA. The one-way ANOVA tests the null hypothesis that two or more groups have the same population mean. The test is applied to samples from two or more groups, possibly with differing sizes. Parameters ---------- sample1, sample2, ... : array_like The sample measurements for each group. There must be at least two arguments. If the arrays are multidimensional, then all the dimensions of the array must be the same except for `axis`. axis : int, optional Axis of the input arrays along which the test is applied. Default is 0. Returns ------- statistic : float The computed F statistic of the test. pvalue : float The associated p-value from the F distribution. Warns ----- `~scipy.stats.ConstantInputWarning` Emitted if all values within each of the input arrays are identical. In this case the F statistic is either infinite or isn't defined, so ``np.inf`` or ``np.nan`` is returned. RuntimeWarning Emitted if the length of any input array is 0, or if all the input arrays have length 1. ``np.nan`` is returned for the F statistic and the p-value in these cases. Notes ----- The ANOVA test has important assumptions that must be satisfied in order for the associated p-value to be valid. 1. The samples are independent. 2. Each sample is from a normally distributed population. 3. The population standard deviations of the groups are all equal. This property is known as homoscedasticity. If these assumptions are not true for a given set of data, it may still be possible to use the Kruskal-Wallis H-test (`scipy.stats.kruskal`) or the Alexander-Govern test (`scipy.stats.alexandergovern`) although with some loss of power. The length of each group must be at least one, and there must be at least one group with length greater than one. If these conditions are not satisfied, a warning is generated and (``np.nan``, ``np.nan``) is returned. If all values in each group are identical, and there exist at least two groups with different values, the function generates a warning and returns (``np.inf``, 0). If all values in all groups are the same, function generates a warning and returns (``np.nan``, ``np.nan``). The algorithm is from Heiman [2]_, pp.394-7. References ---------- .. [1] R. Lowry, "Concepts and Applications of Inferential Statistics", Chapter 14, 2014, http://vassarstats.net/textbook/ .. [2] G.W. Heiman, "Understanding research methods and statistics: An integrated introduction for psychology", Houghton, Mifflin and Company, 2001. .. [3] G.H. McDonald, "Handbook of Biological Statistics", One-way ANOVA. http://www.biostathandbook.com/onewayanova.html Examples -------- >>> import numpy as np >>> from scipy.stats import f_oneway Here are some data [3]_ on a shell measurement (the length of the anterior adductor muscle scar, standardized by dividing by length) in the mussel Mytilus trossulus from five locations: Tillamook, Oregon; Newport, Oregon; Petersburg, Alaska; Magadan, Russia; and Tvarminne, Finland, taken from a much larger data set used in McDonald et al. (1991). >>> tillamook = [0.0571, 0.0813, 0.0831, 0.0976, 0.0817, 0.0859, 0.0735, ... 0.0659, 0.0923, 0.0836] >>> newport = [0.0873, 0.0662, 0.0672, 0.0819, 0.0749, 0.0649, 0.0835, ... 0.0725] >>> petersburg = [0.0974, 0.1352, 0.0817, 0.1016, 0.0968, 0.1064, 0.105] >>> magadan = [0.1033, 0.0915, 0.0781, 0.0685, 0.0677, 0.0697, 0.0764, ... 0.0689] >>> tvarminne = [0.0703, 0.1026, 0.0956, 0.0973, 0.1039, 0.1045] >>> f_oneway(tillamook, newport, petersburg, magadan, tvarminne) F_onewayResult(statistic=7.121019471642447, pvalue=0.0002812242314534544) `f_oneway` accepts multidimensional input arrays. When the inputs are multidimensional and `axis` is not given, the test is performed along the first axis of the input arrays. For the following data, the test is performed three times, once for each column. >>> a = np.array([[9.87, 9.03, 6.81], ... [7.18, 8.35, 7.00], ... [8.39, 7.58, 7.68], ... [7.45, 6.33, 9.35], ... [6.41, 7.10, 9.33], ... [8.00, 8.24, 8.44]]) >>> b = np.array([[6.35, 7.30, 7.16], ... [6.65, 6.68, 7.63], ... [5.72, 7.73, 6.72], ... [7.01, 9.19, 7.41], ... [7.75, 7.87, 8.30], ... [6.90, 7.97, 6.97]]) >>> c = np.array([[3.31, 8.77, 1.01], ... [8.25, 3.24, 3.62], ... [6.32, 8.81, 5.19], ... [7.48, 8.83, 8.91], ... [8.59, 6.01, 6.07], ... [3.07, 9.72, 7.48]]) >>> F, p = f_oneway(a, b, c) >>> F array([1.75676344, 0.03701228, 3.76439349]) >>> p array([0.20630784, 0.96375203, 0.04733157]) """ if len(samples) < 2: raise TypeError('at least two inputs are required;' f' got {len(samples)}.') # ANOVA on N groups, each in its own array num_groups = len(samples) # We haven't explicitly validated axis, but if it is bad, this call of # np.concatenate will raise np.exceptions.AxisError. The call will raise # ValueError if the dimensions of all the arrays, except the axis # dimension, are not the same. alldata = np.concatenate(samples, axis=axis) bign = alldata.shape[axis] # Check if the inputs are too small if _f_oneway_is_too_small(samples): return _create_f_oneway_nan_result(alldata.shape, axis, samples) # Check if all values within each group are identical, and if the common # value in at least one group is different from that in another group. # Based on https://github.com/scipy/scipy/issues/11669 # If axis=0, say, and the groups have shape (n0, ...), (n1, ...), ..., # then is_const is a boolean array with shape (num_groups, ...). # It is True if the values within the groups along the axis slice are # identical. In the typical case where each input array is 1-d, is_const is # a 1-d array with length num_groups. is_const = np.concatenate( [(_first(sample, axis) == sample).all(axis=axis, keepdims=True) for sample in samples], axis=axis ) # all_const is a boolean array with shape (...) (see previous comment). # It is True if the values within each group along the axis slice are # the same (e.g. [[3, 3, 3], [5, 5, 5, 5], [4, 4, 4]]). all_const = is_const.all(axis=axis) if all_const.any(): msg = ("Each of the input arrays is constant; " "the F statistic is not defined or infinite") warnings.warn(stats.ConstantInputWarning(msg), stacklevel=2) # all_same_const is True if all the values in the groups along the axis=0 # slice are the same (e.g. [[3, 3, 3], [3, 3, 3, 3], [3, 3, 3]]). all_same_const = (_first(alldata, axis) == alldata).all(axis=axis) # Determine the mean of the data, and subtract that from all inputs to a # variance (via sum_of_sq / sq_of_sum) calculation. Variance is invariant # to a shift in location, and centering all data around zero vastly # improves numerical stability. offset = alldata.mean(axis=axis, keepdims=True) alldata = alldata - offset normalized_ss = _square_of_sums(alldata, axis=axis) / bign sstot = _sum_of_squares(alldata, axis=axis) - normalized_ss ssbn = 0 for sample in samples: smo_ss = _square_of_sums(sample - offset, axis=axis) ssbn = ssbn + smo_ss / sample.shape[axis] # Naming: variables ending in bn/b are for "between treatments", wn/w are # for "within treatments" ssbn = ssbn - normalized_ss sswn = sstot - ssbn dfbn = num_groups - 1 dfwn = bign - num_groups msb = ssbn / dfbn msw = sswn / dfwn with np.errstate(divide='ignore', invalid='ignore'): f = msb / msw prob = special.fdtrc(dfbn, dfwn, f) # equivalent to stats.f.sf # Fix any f values that should be inf or nan because the corresponding # inputs were constant. if np.isscalar(f): if all_same_const: f = np.nan prob = np.nan elif all_const: f = np.inf prob = 0.0 else: f[all_const] = np.inf prob[all_const] = 0.0 f[all_same_const] = np.nan prob[all_same_const] = np.nan return F_onewayResult(f, prob) @dataclass class AlexanderGovernResult: statistic: float pvalue: float @_axis_nan_policy_factory( AlexanderGovernResult, n_samples=None, result_to_tuple=lambda x: (x.statistic, x.pvalue), too_small=1 ) def alexandergovern(*samples, nan_policy='propagate'): """Performs the Alexander Govern test. The Alexander-Govern approximation tests the equality of k independent means in the face of heterogeneity of variance. The test is applied to samples from two or more groups, possibly with differing sizes. Parameters ---------- sample1, sample2, ... : array_like The sample measurements for each group. There must be at least two samples, and each sample must contain at least two observations. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- res : AlexanderGovernResult An object with attributes: statistic : float The computed A statistic of the test. pvalue : float The associated p-value from the chi-squared distribution. Warns ----- `~scipy.stats.ConstantInputWarning` Raised if an input is a constant array. The statistic is not defined in this case, so ``np.nan`` is returned. See Also -------- f_oneway : one-way ANOVA Notes ----- The use of this test relies on several assumptions. 1. The samples are independent. 2. Each sample is from a normally distributed population. 3. Unlike `f_oneway`, this test does not assume on homoscedasticity, instead relaxing the assumption of equal variances. Input samples must be finite, one dimensional, and with size greater than one. References ---------- .. [1] Alexander, Ralph A., and Diane M. Govern. "A New and Simpler Approximation for ANOVA under Variance Heterogeneity." Journal of Educational Statistics, vol. 19, no. 2, 1994, pp. 91-101. JSTOR, www.jstor.org/stable/1165140. Accessed 12 Sept. 2020. Examples -------- >>> from scipy.stats import alexandergovern Here are some data on annual percentage rate of interest charged on new car loans at nine of the largest banks in four American cities taken from the National Institute of Standards and Technology's ANOVA dataset. We use `alexandergovern` to test the null hypothesis that all cities have the same mean APR against the alternative that the cities do not all have the same mean APR. We decide that a significance level of 5% is required to reject the null hypothesis in favor of the alternative. >>> atlanta = [13.75, 13.75, 13.5, 13.5, 13.0, 13.0, 13.0, 12.75, 12.5] >>> chicago = [14.25, 13.0, 12.75, 12.5, 12.5, 12.4, 12.3, 11.9, 11.9] >>> houston = [14.0, 14.0, 13.51, 13.5, 13.5, 13.25, 13.0, 12.5, 12.5] >>> memphis = [15.0, 14.0, 13.75, 13.59, 13.25, 12.97, 12.5, 12.25, ... 11.89] >>> alexandergovern(atlanta, chicago, houston, memphis) AlexanderGovernResult(statistic=4.65087071883494, pvalue=0.19922132490385214) The p-value is 0.1992, indicating a nearly 20% chance of observing such an extreme value of the test statistic under the null hypothesis. This exceeds 5%, so we do not reject the null hypothesis in favor of the alternative. """ samples = _alexandergovern_input_validation(samples, nan_policy) if np.any([(sample == sample[0]).all() for sample in samples]): msg = "An input array is constant; the statistic is not defined." warnings.warn(stats.ConstantInputWarning(msg), stacklevel=2) return AlexanderGovernResult(np.nan, np.nan) # The following formula numbers reference the equation described on # page 92 by Alexander, Govern. Formulas 5, 6, and 7 describe other # tests that serve as the basis for equation (8) but are not needed # to perform the test. # precalculate mean and length of each sample lengths = np.array([len(sample) for sample in samples]) means = np.array([np.mean(sample) for sample in samples]) # (1) determine standard error of the mean for each sample standard_errors = [np.std(sample, ddof=1) / np.sqrt(length) for sample, length in zip(samples, lengths)] # (2) define a weight for each sample inv_sq_se = 1 / np.square(standard_errors) weights = inv_sq_se / np.sum(inv_sq_se) # (3) determine variance-weighted estimate of the common mean var_w = np.sum(weights * means) # (4) determine one-sample t statistic for each group t_stats = (means - var_w)/standard_errors # calculate parameters to be used in transformation v = lengths - 1 a = v - .5 b = 48 * a**2 c = (a * np.log(1 + (t_stats ** 2)/v))**.5 # (8) perform a normalizing transformation on t statistic z = (c + ((c**3 + 3*c)/b) - ((4*c**7 + 33*c**5 + 240*c**3 + 855*c) / (b**2*10 + 8*b*c**4 + 1000*b))) # (9) calculate statistic A = np.sum(np.square(z)) # "[the p value is determined from] central chi-square random deviates # with k - 1 degrees of freedom". Alexander, Govern (94) df = len(samples) - 1 chi2 = _SimpleChi2(df) p = _get_pvalue(A, chi2, alternative='greater', symmetric=False, xp=np) return AlexanderGovernResult(A, p) def _alexandergovern_input_validation(samples, nan_policy): if len(samples) < 2: raise TypeError(f"2 or more inputs required, got {len(samples)}") for sample in samples: if np.size(sample) <= 1: raise ValueError("Input sample size must be greater than one.") if np.isinf(sample).any(): raise ValueError("Input samples must be finite.") return samples def _pearsonr_fisher_ci(r, n, confidence_level, alternative): """ Compute the confidence interval for Pearson's R. Fisher's transformation is used to compute the confidence interval (https://en.wikipedia.org/wiki/Fisher_transformation). """ xp = array_namespace(r) with np.errstate(divide='ignore'): zr = xp.atanh(r) ones = xp.ones_like(r) n = xp.asarray(n, dtype=r.dtype) confidence_level = xp.asarray(confidence_level, dtype=r.dtype) if n > 3: se = xp.sqrt(1 / (n - 3)) if alternative == "two-sided": h = special.ndtri(0.5 + confidence_level/2) zlo = zr - h*se zhi = zr + h*se rlo = xp.tanh(zlo) rhi = xp.tanh(zhi) elif alternative == "less": h = special.ndtri(confidence_level) zhi = zr + h*se rhi = xp.tanh(zhi) rlo = -ones else: # alternative == "greater": h = special.ndtri(confidence_level) zlo = zr - h*se rlo = xp.tanh(zlo) rhi = ones else: rlo, rhi = -ones, ones rlo = rlo[()] if rlo.ndim == 0 else rlo rhi = rhi[()] if rhi.ndim == 0 else rhi return ConfidenceInterval(low=rlo, high=rhi) def _pearsonr_bootstrap_ci(confidence_level, method, x, y, alternative, axis): """ Compute the confidence interval for Pearson's R using the bootstrap. """ def statistic(x, y, axis): statistic, _ = pearsonr(x, y, axis=axis) return statistic res = bootstrap((x, y), statistic, confidence_level=confidence_level, axis=axis, paired=True, alternative=alternative, **method._asdict()) # for one-sided confidence intervals, bootstrap gives +/- inf on one side res.confidence_interval = np.clip(res.confidence_interval, -1, 1) return ConfidenceInterval(*res.confidence_interval) ConfidenceInterval = namedtuple('ConfidenceInterval', ['low', 'high']) PearsonRResultBase = _make_tuple_bunch('PearsonRResultBase', ['statistic', 'pvalue'], []) class PearsonRResult(PearsonRResultBase): """ Result of `scipy.stats.pearsonr` Attributes ---------- statistic : float Pearson product-moment correlation coefficient. pvalue : float The p-value associated with the chosen alternative. Methods ------- confidence_interval Computes the confidence interval of the correlation coefficient `statistic` for the given confidence level. """ def __init__(self, statistic, pvalue, alternative, n, x, y, axis): super().__init__(statistic, pvalue) self._alternative = alternative self._n = n self._x = x self._y = y self._axis = axis # add alias for consistency with other correlation functions self.correlation = statistic def confidence_interval(self, confidence_level=0.95, method=None): """ The confidence interval for the correlation coefficient. Compute the confidence interval for the correlation coefficient ``statistic`` with the given confidence level. If `method` is not provided, The confidence interval is computed using the Fisher transformation F(r) = arctanh(r) [1]_. When the sample pairs are drawn from a bivariate normal distribution, F(r) approximately follows a normal distribution with standard error ``1/sqrt(n - 3)``, where ``n`` is the length of the original samples along the calculation axis. When ``n <= 3``, this approximation does not yield a finite, real standard error, so we define the confidence interval to be -1 to 1. If `method` is an instance of `BootstrapMethod`, the confidence interval is computed using `scipy.stats.bootstrap` with the provided configuration options and other appropriate settings. In some cases, confidence limits may be NaN due to a degenerate resample, and this is typical for very small samples (~6 observations). Parameters ---------- confidence_level : float The confidence level for the calculation of the correlation coefficient confidence interval. Default is 0.95. method : BootstrapMethod, optional Defines the method used to compute the confidence interval. See method description for details. .. versionadded:: 1.11.0 Returns ------- ci : namedtuple The confidence interval is returned in a ``namedtuple`` with fields `low` and `high`. References ---------- .. [1] "Pearson correlation coefficient", Wikipedia, https://en.wikipedia.org/wiki/Pearson_correlation_coefficient """ if isinstance(method, BootstrapMethod): xp = array_namespace(self._x) message = ('`method` must be `None` if `pearsonr` ' 'arguments were not NumPy arrays.') if not is_numpy(xp): raise ValueError(message) ci = _pearsonr_bootstrap_ci(confidence_level, method, self._x, self._y, self._alternative, self._axis) elif method is None: ci = _pearsonr_fisher_ci(self.statistic, self._n, confidence_level, self._alternative) else: message = ('`method` must be an instance of `BootstrapMethod` ' 'or None.') raise ValueError(message) return ci def pearsonr(x, y, *, alternative='two-sided', method=None, axis=0): r""" Pearson correlation coefficient and p-value for testing non-correlation. The Pearson correlation coefficient [1]_ measures the linear relationship between two datasets. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact linear relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases. This function also performs a test of the null hypothesis that the distributions underlying the samples are uncorrelated and normally distributed. (See Kowalski [3]_ for a discussion of the effects of non-normality of the input on the distribution of the correlation coefficient.) The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Pearson correlation at least as extreme as the one computed from these datasets. Parameters ---------- x : array_like Input array. y : array_like Input array. axis : int or None, default Axis along which to perform the calculation. Default is 0. If None, ravel both arrays before performing the calculation. .. versionadded:: 1.13.0 alternative : {'two-sided', 'greater', 'less'}, optional Defines the alternative hypothesis. Default is 'two-sided'. The following options are available: * 'two-sided': the correlation is nonzero * 'less': the correlation is negative (less than zero) * 'greater': the correlation is positive (greater than zero) .. versionadded:: 1.9.0 method : ResamplingMethod, optional Defines the method used to compute the p-value. If `method` is an instance of `PermutationMethod`/`MonteCarloMethod`, the p-value is computed using `scipy.stats.permutation_test`/`scipy.stats.monte_carlo_test` with the provided configuration options and other appropriate settings. Otherwise, the p-value is computed as documented in the notes. .. versionadded:: 1.11.0 Returns ------- result : `~scipy.stats._result_classes.PearsonRResult` An object with the following attributes: statistic : float Pearson product-moment correlation coefficient. pvalue : float The p-value associated with the chosen alternative. The object has the following method: confidence_interval(confidence_level, method) This computes the confidence interval of the correlation coefficient `statistic` for the given confidence level. The confidence interval is returned in a ``namedtuple`` with fields `low` and `high`. If `method` is not provided, the confidence interval is computed using the Fisher transformation [1]_. If `method` is an instance of `BootstrapMethod`, the confidence interval is computed using `scipy.stats.bootstrap` with the provided configuration options and other appropriate settings. In some cases, confidence limits may be NaN due to a degenerate resample, and this is typical for very small samples (~6 observations). Warns ----- `~scipy.stats.ConstantInputWarning` Raised if an input is a constant array. The correlation coefficient is not defined in this case, so ``np.nan`` is returned. `~scipy.stats.NearConstantInputWarning` Raised if an input is "nearly" constant. The array ``x`` is considered nearly constant if ``norm(x - mean(x)) < 1e-13 * abs(mean(x))``. Numerical errors in the calculation ``x - mean(x)`` in this case might result in an inaccurate calculation of r. See Also -------- spearmanr : Spearman rank-order correlation coefficient. kendalltau : Kendall's tau, a correlation measure for ordinal data. Notes ----- The correlation coefficient is calculated as follows: .. math:: r = \frac{\sum (x - m_x) (y - m_y)} {\sqrt{\sum (x - m_x)^2 \sum (y - m_y)^2}} where :math:`m_x` is the mean of the vector x and :math:`m_y` is the mean of the vector y. Under the assumption that x and y are drawn from independent normal distributions (so the population correlation coefficient is 0), the probability density function of the sample correlation coefficient r is ([1]_, [2]_): .. math:: f(r) = \frac{{(1-r^2)}^{n/2-2}}{\mathrm{B}(\frac{1}{2},\frac{n}{2}-1)} where n is the number of samples, and B is the beta function. This is sometimes referred to as the exact distribution of r. This is the distribution that is used in `pearsonr` to compute the p-value when the `method` parameter is left at its default value (None). The distribution is a beta distribution on the interval [-1, 1], with equal shape parameters a = b = n/2 - 1. In terms of SciPy's implementation of the beta distribution, the distribution of r is:: dist = scipy.stats.beta(n/2 - 1, n/2 - 1, loc=-1, scale=2) The default p-value returned by `pearsonr` is a two-sided p-value. For a given sample with correlation coefficient r, the p-value is the probability that abs(r') of a random sample x' and y' drawn from the population with zero correlation would be greater than or equal to abs(r). In terms of the object ``dist`` shown above, the p-value for a given r and length n can be computed as:: p = 2*dist.cdf(-abs(r)) When n is 2, the above continuous distribution is not well-defined. One can interpret the limit of the beta distribution as the shape parameters a and b approach a = b = 0 as a discrete distribution with equal probability masses at r = 1 and r = -1. More directly, one can observe that, given the data x = [x1, x2] and y = [y1, y2], and assuming x1 != x2 and y1 != y2, the only possible values for r are 1 and -1. Because abs(r') for any sample x' and y' with length 2 will be 1, the two-sided p-value for a sample of length 2 is always 1. For backwards compatibility, the object that is returned also behaves like a tuple of length two that holds the statistic and the p-value. References ---------- .. [1] "Pearson correlation coefficient", Wikipedia, https://en.wikipedia.org/wiki/Pearson_correlation_coefficient .. [2] Student, "Probable error of a correlation coefficient", Biometrika, Volume 6, Issue 2-3, 1 September 1908, pp. 302-310. .. [3] C. J. Kowalski, "On the Effects of Non-Normality on the Distribution of the Sample Product-Moment Correlation Coefficient" Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 21, No. 1 (1972), pp. 1-12. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x, y = [1, 2, 3, 4, 5, 6, 7], [10, 9, 2.5, 6, 4, 3, 2] >>> res = stats.pearsonr(x, y) >>> res PearsonRResult(statistic=-0.828503883588428, pvalue=0.021280260007523286) To perform an exact permutation version of the test: >>> rng = np.random.default_rng(7796654889291491997) >>> method = stats.PermutationMethod(n_resamples=np.inf, random_state=rng) >>> stats.pearsonr(x, y, method=method) PearsonRResult(statistic=-0.828503883588428, pvalue=0.028174603174603175) To perform the test under the null hypothesis that the data were drawn from *uniform* distributions: >>> method = stats.MonteCarloMethod(rvs=(rng.uniform, rng.uniform)) >>> stats.pearsonr(x, y, method=method) PearsonRResult(statistic=-0.828503883588428, pvalue=0.0188) To produce an asymptotic 90% confidence interval: >>> res.confidence_interval(confidence_level=0.9) ConfidenceInterval(low=-0.9644331982722841, high=-0.3460237473272273) And for a bootstrap confidence interval: >>> method = stats.BootstrapMethod(method='BCa', random_state=rng) >>> res.confidence_interval(confidence_level=0.9, method=method) ConfidenceInterval(low=-0.9983163756488651, high=-0.22771001702132443) # may vary If N-dimensional arrays are provided, multiple tests are performed in a single call according to the same conventions as most `scipy.stats` functions: >>> rng = np.random.default_rng(2348246935601934321) >>> x = rng.standard_normal((8, 15)) >>> y = rng.standard_normal((8, 15)) >>> stats.pearsonr(x, y, axis=0).statistic.shape # between corresponding columns (15,) >>> stats.pearsonr(x, y, axis=1).statistic.shape # between corresponding rows (8,) To perform all pairwise comparisons between slices of the arrays, use standard NumPy broadcasting techniques. For instance, to compute the correlation between all pairs of rows: >>> stats.pearsonr(x[:, np.newaxis, :], y, axis=-1).statistic.shape (8, 8) There is a linear dependence between x and y if y = a + b*x + e, where a,b are constants and e is a random error term, assumed to be independent of x. For simplicity, assume that x is standard normal, a=0, b=1 and let e follow a normal distribution with mean zero and standard deviation s>0. >>> rng = np.random.default_rng() >>> s = 0.5 >>> x = stats.norm.rvs(size=500, random_state=rng) >>> e = stats.norm.rvs(scale=s, size=500, random_state=rng) >>> y = x + e >>> stats.pearsonr(x, y).statistic 0.9001942438244763 This should be close to the exact value given by >>> 1/np.sqrt(1 + s**2) 0.8944271909999159 For s=0.5, we observe a high level of correlation. In general, a large variance of the noise reduces the correlation, while the correlation approaches one as the variance of the error goes to zero. It is important to keep in mind that no correlation does not imply independence unless (x, y) is jointly normal. Correlation can even be zero when there is a very simple dependence structure: if X follows a standard normal distribution, let y = abs(x). Note that the correlation between x and y is zero. Indeed, since the expectation of x is zero, cov(x, y) = E[x*y]. By definition, this equals E[x*abs(x)] which is zero by symmetry. The following lines of code illustrate this observation: >>> y = np.abs(x) >>> stats.pearsonr(x, y) PearsonRResult(statistic=-0.05444919272687482, pvalue=0.22422294836207743) A non-zero correlation coefficient can be misleading. For example, if X has a standard normal distribution, define y = x if x < 0 and y = 0 otherwise. A simple calculation shows that corr(x, y) = sqrt(2/Pi) = 0.797..., implying a high level of correlation: >>> y = np.where(x < 0, x, 0) >>> stats.pearsonr(x, y) PearsonRResult(statistic=0.861985781588, pvalue=4.813432002751103e-149) This is unintuitive since there is no dependence of x and y if x is larger than zero which happens in about half of the cases if we sample x and y. """ xp = array_namespace(x, y) x = xp.asarray(x) y = xp.asarray(y) if not is_numpy(xp) and method is not None: method = 'invalid' if axis is None: x = xp.reshape(x, (-1,)) y = xp.reshape(y, (-1,)) axis = -1 axis_int = int(axis) if axis_int != axis: raise ValueError('`axis` must be an integer.') axis = axis_int n = x.shape[axis] if n != y.shape[axis]: raise ValueError('`x` and `y` must have the same length along `axis`.') if n < 2: raise ValueError('`x` and `y` must have length at least 2.') try: x, y = xp.broadcast_arrays(x, y) except (ValueError, RuntimeError) as e: message = '`x` and `y` must be broadcastable.' raise ValueError(message) from e # `moveaxis` only recently added to array API, so it's not yey available in # array_api_strict. Replace with e.g. `xp.moveaxis(x, axis, -1)` when available. x = xp_moveaxis_to_end(x, axis, xp=xp) y = xp_moveaxis_to_end(y, axis, xp=xp) axis = -1 dtype = xp.result_type(x.dtype, y.dtype) if xp.isdtype(dtype, "integral"): dtype = xp.asarray(1.).dtype if xp.isdtype(dtype, "complex floating"): raise ValueError('This function does not support complex data') x = xp.astype(x, dtype, copy=False) y = xp.astype(y, dtype, copy=False) threshold = xp.finfo(dtype).eps ** 0.75 # If an input is constant, the correlation coefficient is not defined. const_x = xp.all(x == x[..., 0:1], axis=-1) const_y = xp.all(y == y[..., 0:1], axis=-1) const_xy = const_x | const_y if xp.any(const_xy): msg = ("An input array is constant; the correlation coefficient " "is not defined.") warnings.warn(stats.ConstantInputWarning(msg), stacklevel=2) if isinstance(method, PermutationMethod): def statistic(y, axis): statistic, _ = pearsonr(x, y, axis=axis, alternative=alternative) return statistic res = permutation_test((y,), statistic, permutation_type='pairings', axis=axis, alternative=alternative, **method._asdict()) return PearsonRResult(statistic=res.statistic, pvalue=res.pvalue, n=n, alternative=alternative, x=x, y=y, axis=axis) elif isinstance(method, MonteCarloMethod): def statistic(x, y, axis): statistic, _ = pearsonr(x, y, axis=axis, alternative=alternative) return statistic if method.rvs is None: rng = np.random.default_rng() method.rvs = rng.normal, rng.normal res = monte_carlo_test((x, y,), statistic=statistic, axis=axis, alternative=alternative, **method._asdict()) return PearsonRResult(statistic=res.statistic, pvalue=res.pvalue, n=n, alternative=alternative, x=x, y=y, axis=axis) elif method == 'invalid': message = '`method` must be `None` if arguments are not NumPy arrays.' raise ValueError(message) elif method is not None: message = ('`method` must be an instance of `PermutationMethod`,' '`MonteCarloMethod`, or None.') raise ValueError(message) if n == 2: r = (xp.sign(x[..., 1] - x[..., 0])*xp.sign(y[..., 1] - y[..., 0])) r = r[()] if r.ndim == 0 else r pvalue = xp.ones_like(r) pvalue = pvalue[()] if pvalue.ndim == 0 else pvalue result = PearsonRResult(statistic=r, pvalue=pvalue, n=n, alternative=alternative, x=x, y=y, axis=axis) return result xmean = xp.mean(x, axis=axis, keepdims=True) ymean = xp.mean(y, axis=axis, keepdims=True) xm = x - xmean ym = y - ymean # scipy.linalg.norm(xm) avoids premature overflow when xm is e.g. # [-5e210, 5e210, 3e200, -3e200] # but not when `axis` is provided, so scale manually. scipy.linalg.norm # also raises an error with NaN input rather than returning NaN, so # use np.linalg.norm. xmax = xp.max(xp.abs(xm), axis=axis, keepdims=True) ymax = xp.max(xp.abs(ym), axis=axis, keepdims=True) with np.errstate(invalid='ignore'): normxm = xmax * xp.linalg.vector_norm(xm/xmax, axis=axis, keepdims=True) normym = ymax * xp.linalg.vector_norm(ym/ymax, axis=axis, keepdims=True) nconst_x = xp.any(normxm < threshold*xp.abs(xmean), axis=axis) nconst_y = xp.any(normym < threshold*xp.abs(ymean), axis=axis) nconst_xy = nconst_x | nconst_y if xp.any(nconst_xy & (~const_xy)): # If all the values in x (likewise y) are very close to the mean, # the loss of precision that occurs in the subtraction xm = x - xmean # might result in large errors in r. msg = ("An input array is nearly constant; the computed " "correlation coefficient may be inaccurate.") warnings.warn(stats.NearConstantInputWarning(msg), stacklevel=2) with np.errstate(invalid='ignore', divide='ignore'): r = xp.sum(xm/normxm * ym/normym, axis=axis) # Presumably, if abs(r) > 1, then it is only some small artifact of # floating point arithmetic. one = xp.asarray(1, dtype=dtype) # `clip` only recently added to array API, so it's not yet available in # array_api_strict. Replace with e.g. `xp.clip(r, -one, one)` when available. r = xp.asarray(xp_clip(r, -one, one, xp=xp)) r[const_xy] = xp.nan # As explained in the docstring, the distribution of `r` under the null # hypothesis is the beta distribution on (-1, 1) with a = b = n/2 - 1. # This needs to be done with NumPy arrays given the existing infrastructure. ab = n/2 - 1 dist = stats.beta(ab, ab, loc=-1, scale=2) pvalue = _get_pvalue(np.asarray(r), dist, alternative, xp=np) pvalue = xp.asarray(pvalue, dtype=dtype) r = r[()] if r.ndim == 0 else r pvalue = pvalue[()] if pvalue.ndim == 0 else pvalue return PearsonRResult(statistic=r, pvalue=pvalue, n=n, alternative=alternative, x=x, y=y, axis=axis) def fisher_exact(table, alternative='two-sided'): """Perform a Fisher exact test on a 2x2 contingency table. The null hypothesis is that the true odds ratio of the populations underlying the observations is one, and the observations were sampled from these populations under a condition: the marginals of the resulting table must equal those of the observed table. The statistic returned is the unconditional maximum likelihood estimate of the odds ratio, and the p-value is the probability under the null hypothesis of obtaining a table at least as extreme as the one that was actually observed. There are other possible choices of statistic and two-sided p-value definition associated with Fisher's exact test; please see the Notes for more information. Parameters ---------- table : array_like of ints A 2x2 contingency table. Elements must be non-negative integers. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided': the odds ratio of the underlying population is not one * 'less': the odds ratio of the underlying population is less than one * 'greater': the odds ratio of the underlying population is greater than one See the Notes for more details. Returns ------- res : SignificanceResult An object containing attributes: statistic : float This is the prior odds ratio, not a posterior estimate. pvalue : float The probability under the null hypothesis of obtaining a table at least as extreme as the one that was actually observed. See Also -------- chi2_contingency : Chi-square test of independence of variables in a contingency table. This can be used as an alternative to `fisher_exact` when the numbers in the table are large. contingency.odds_ratio : Compute the odds ratio (sample or conditional MLE) for a 2x2 contingency table. barnard_exact : Barnard's exact test, which is a more powerful alternative than Fisher's exact test for 2x2 contingency tables. boschloo_exact : Boschloo's exact test, which is a more powerful alternative than Fisher's exact test for 2x2 contingency tables. Notes ----- *Null hypothesis and p-values* The null hypothesis is that the true odds ratio of the populations underlying the observations is one, and the observations were sampled at random from these populations under a condition: the marginals of the resulting table must equal those of the observed table. Equivalently, the null hypothesis is that the input table is from the hypergeometric distribution with parameters (as used in `hypergeom`) ``M = a + b + c + d``, ``n = a + b`` and ``N = a + c``, where the input table is ``[[a, b], [c, d]]``. This distribution has support ``max(0, N + n - M) <= x <= min(N, n)``, or, in terms of the values in the input table, ``min(0, a - d) <= x <= a + min(b, c)``. ``x`` can be interpreted as the upper-left element of a 2x2 table, so the tables in the distribution have form:: [ x n - x ] [N - x M - (n + N) + x] For example, if:: table = [6 2] [1 4] then the support is ``2 <= x <= 7``, and the tables in the distribution are:: [2 6] [3 5] [4 4] [5 3] [6 2] [7 1] [5 0] [4 1] [3 2] [2 3] [1 4] [0 5] The probability of each table is given by the hypergeometric distribution ``hypergeom.pmf(x, M, n, N)``. For this example, these are (rounded to three significant digits):: x 2 3 4 5 6 7 p 0.0163 0.163 0.408 0.326 0.0816 0.00466 These can be computed with:: >>> import numpy as np >>> from scipy.stats import hypergeom >>> table = np.array([[6, 2], [1, 4]]) >>> M = table.sum() >>> n = table[0].sum() >>> N = table[:, 0].sum() >>> start, end = hypergeom.support(M, n, N) >>> hypergeom.pmf(np.arange(start, end+1), M, n, N) array([0.01631702, 0.16317016, 0.40792541, 0.32634033, 0.08158508, 0.004662 ]) The two-sided p-value is the probability that, under the null hypothesis, a random table would have a probability equal to or less than the probability of the input table. For our example, the probability of the input table (where ``x = 6``) is 0.0816. The x values where the probability does not exceed this are 2, 6 and 7, so the two-sided p-value is ``0.0163 + 0.0816 + 0.00466 ~= 0.10256``:: >>> from scipy.stats import fisher_exact >>> res = fisher_exact(table, alternative='two-sided') >>> res.pvalue 0.10256410256410257 The one-sided p-value for ``alternative='greater'`` is the probability that a random table has ``x >= a``, which in our example is ``x >= 6``, or ``0.0816 + 0.00466 ~= 0.08626``:: >>> res = fisher_exact(table, alternative='greater') >>> res.pvalue 0.08624708624708627 This is equivalent to computing the survival function of the distribution at ``x = 5`` (one less than ``x`` from the input table, because we want to include the probability of ``x = 6`` in the sum):: >>> hypergeom.sf(5, M, n, N) 0.08624708624708627 For ``alternative='less'``, the one-sided p-value is the probability that a random table has ``x <= a``, (i.e. ``x <= 6`` in our example), or ``0.0163 + 0.163 + 0.408 + 0.326 + 0.0816 ~= 0.9949``:: >>> res = fisher_exact(table, alternative='less') >>> res.pvalue 0.9953379953379957 This is equivalent to computing the cumulative distribution function of the distribution at ``x = 6``: >>> hypergeom.cdf(6, M, n, N) 0.9953379953379957 *Odds ratio* The calculated odds ratio is different from the value computed by the R function ``fisher.test``. This implementation returns the "sample" or "unconditional" maximum likelihood estimate, while ``fisher.test`` in R uses the conditional maximum likelihood estimate. To compute the conditional maximum likelihood estimate of the odds ratio, use `scipy.stats.contingency.odds_ratio`. References ---------- .. [1] Fisher, Sir Ronald A, "The Design of Experiments: Mathematics of a Lady Tasting Tea." ISBN 978-0-486-41151-4, 1935. .. [2] "Fisher's exact test", https://en.wikipedia.org/wiki/Fisher's_exact_test .. [3] Emma V. Low et al. "Identifying the lowest effective dose of acetazolamide for the prophylaxis of acute mountain sickness: systematic review and meta-analysis." BMJ, 345, :doi:`10.1136/bmj.e6779`, 2012. Examples -------- In [3]_, the effective dose of acetazolamide for the prophylaxis of acute mountain sickness was investigated. The study notably concluded: Acetazolamide 250 mg, 500 mg, and 750 mg daily were all efficacious for preventing acute mountain sickness. Acetazolamide 250 mg was the lowest effective dose with available evidence for this indication. The following table summarizes the results of the experiment in which some participants took a daily dose of acetazolamide 250 mg while others took a placebo. Cases of acute mountain sickness were recorded:: Acetazolamide Control/Placebo Acute mountain sickness 7 17 No 15 5 Is there evidence that the acetazolamide 250 mg reduces the risk of acute mountain sickness? We begin by formulating a null hypothesis :math:`H_0`: The odds of experiencing acute mountain sickness are the same with the acetazolamide treatment as they are with placebo. Let's assess the plausibility of this hypothesis with Fisher's test. >>> from scipy.stats import fisher_exact >>> res = fisher_exact([[7, 17], [15, 5]], alternative='less') >>> res.statistic 0.13725490196078433 >>> res.pvalue 0.0028841933752349743 Using a significance level of 5%, we would reject the null hypothesis in favor of the alternative hypothesis: "The odds of experiencing acute mountain sickness with acetazolamide treatment are less than the odds of experiencing acute mountain sickness with placebo." .. note:: Because the null distribution of Fisher's exact test is formed under the assumption that both row and column sums are fixed, the result of the test are conservative when applied to an experiment in which the row sums are not fixed. In this case, the column sums are fixed; there are 22 subjects in each group. But the number of cases of acute mountain sickness is not (and cannot be) fixed before conducting the experiment. It is a consequence. Boschloo's test does not depend on the assumption that the row sums are fixed, and consequently, it provides a more powerful test in this situation. >>> from scipy.stats import boschloo_exact >>> res = boschloo_exact([[7, 17], [15, 5]], alternative='less') >>> res.statistic 0.0028841933752349743 >>> res.pvalue 0.0015141406667567101 We verify that the p-value is less than with `fisher_exact`. """ hypergeom = distributions.hypergeom # int32 is not enough for the algorithm c = np.asarray(table, dtype=np.int64) if not c.shape == (2, 2): raise ValueError("The input `table` must be of shape (2, 2).") if np.any(c < 0): raise ValueError("All values in `table` must be nonnegative.") if 0 in c.sum(axis=0) or 0 in c.sum(axis=1): # If both values in a row or column are zero, the p-value is 1 and # the odds ratio is NaN. return SignificanceResult(np.nan, 1.0) if c[1, 0] > 0 and c[0, 1] > 0: oddsratio = c[0, 0] * c[1, 1] / (c[1, 0] * c[0, 1]) else: oddsratio = np.inf n1 = c[0, 0] + c[0, 1] n2 = c[1, 0] + c[1, 1] n = c[0, 0] + c[1, 0] def pmf(x): return hypergeom.pmf(x, n1 + n2, n1, n) if alternative == 'less': pvalue = hypergeom.cdf(c[0, 0], n1 + n2, n1, n) elif alternative == 'greater': # Same formula as the 'less' case, but with the second column. pvalue = hypergeom.cdf(c[0, 1], n1 + n2, n1, c[0, 1] + c[1, 1]) elif alternative == 'two-sided': mode = int((n + 1) * (n1 + 1) / (n1 + n2 + 2)) pexact = hypergeom.pmf(c[0, 0], n1 + n2, n1, n) pmode = hypergeom.pmf(mode, n1 + n2, n1, n) epsilon = 1e-14 gamma = 1 + epsilon if np.abs(pexact - pmode) / np.maximum(pexact, pmode) <= epsilon: return SignificanceResult(oddsratio, 1.) elif c[0, 0] < mode: plower = hypergeom.cdf(c[0, 0], n1 + n2, n1, n) if hypergeom.pmf(n, n1 + n2, n1, n) > pexact * gamma: return SignificanceResult(oddsratio, plower) guess = _binary_search(lambda x: -pmf(x), -pexact * gamma, mode, n) pvalue = plower + hypergeom.sf(guess, n1 + n2, n1, n) else: pupper = hypergeom.sf(c[0, 0] - 1, n1 + n2, n1, n) if hypergeom.pmf(0, n1 + n2, n1, n) > pexact * gamma: return SignificanceResult(oddsratio, pupper) guess = _binary_search(pmf, pexact * gamma, 0, mode) pvalue = pupper + hypergeom.cdf(guess, n1 + n2, n1, n) else: msg = "`alternative` should be one of {'two-sided', 'less', 'greater'}" raise ValueError(msg) pvalue = min(pvalue, 1.0) return SignificanceResult(oddsratio, pvalue) def spearmanr(a, b=None, axis=0, nan_policy='propagate', alternative='two-sided'): r"""Calculate a Spearman correlation coefficient with associated p-value. The Spearman rank-order correlation coefficient is a nonparametric measure of the monotonicity of the relationship between two datasets. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact monotonic relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases. The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Spearman correlation at least as extreme as the one computed from these datasets. Although calculation of the p-value does not make strong assumptions about the distributions underlying the samples, it is only accurate for very large samples (>500 observations). For smaller sample sizes, consider a permutation test (see Examples section below). Parameters ---------- a, b : 1D or 2D array_like, b is optional One or two 1-D or 2-D arrays containing multiple variables and observations. When these are 1-D, each represents a vector of observations of a single variable. For the behavior in the 2-D case, see under ``axis``, below. Both arrays need to have the same length in the ``axis`` dimension. axis : int or None, optional If axis=0 (default), then each column represents a variable, with observations in the rows. If axis=1, the relationship is transposed: each row represents a variable, while the columns contain observations. If axis=None, then both arrays will be raveled. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. Default is 'two-sided'. The following options are available: * 'two-sided': the correlation is nonzero * 'less': the correlation is negative (less than zero) * 'greater': the correlation is positive (greater than zero) .. versionadded:: 1.7.0 Returns ------- res : SignificanceResult An object containing attributes: statistic : float or ndarray (2-D square) Spearman correlation matrix or correlation coefficient (if only 2 variables are given as parameters). Correlation matrix is square with length equal to total number of variables (columns or rows) in ``a`` and ``b`` combined. pvalue : float The p-value for a hypothesis test whose null hypothesis is that two samples have no ordinal correlation. See `alternative` above for alternative hypotheses. `pvalue` has the same shape as `statistic`. Warns ----- `~scipy.stats.ConstantInputWarning` Raised if an input is a constant array. The correlation coefficient is not defined in this case, so ``np.nan`` is returned. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 14.7 .. [2] Kendall, M. G. and Stuart, A. (1973). The Advanced Theory of Statistics, Volume 2: Inference and Relationship. Griffin. 1973. Section 31.18 .. [3] Kershenobich, D., Fierro, F. J., & Rojkind, M. (1970). The relationship between the free pool of proline and collagen content in human liver cirrhosis. The Journal of Clinical Investigation, 49(12), 2246-2249. .. [4] Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric statistical methods. John Wiley & Sons. .. [5] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn." Statistical Applications in Genetics and Molecular Biology 9.1 (2010). .. [6] Ludbrook, J., & Dudley, H. (1998). Why permutation tests are superior to t and F tests in biomedical research. The American Statistician, 52(2), 127-132. Examples -------- Consider the following data from [3]_, which studied the relationship between free proline (an amino acid) and total collagen (a protein often found in connective tissue) in unhealthy human livers. The ``x`` and ``y`` arrays below record measurements of the two compounds. The observations are paired: each free proline measurement was taken from the same liver as the total collagen measurement at the same index. >>> import numpy as np >>> # total collagen (mg/g dry weight of liver) >>> x = np.array([7.1, 7.1, 7.2, 8.3, 9.4, 10.5, 11.4]) >>> # free proline (μ mole/g dry weight of liver) >>> y = np.array([2.8, 2.9, 2.8, 2.6, 3.5, 4.6, 5.0]) These data were analyzed in [4]_ using Spearman's correlation coefficient, a statistic sensitive to monotonic correlation between the samples. >>> from scipy import stats >>> res = stats.spearmanr(x, y) >>> res.statistic 0.7000000000000001 The value of this statistic tends to be high (close to 1) for samples with a strongly positive ordinal correlation, low (close to -1) for samples with a strongly negative ordinal correlation, and small in magnitude (close to zero) for samples with weak ordinal correlation. The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that total collagen and free proline measurements are independent. For this test, the statistic can be transformed such that the null distribution for large samples is Student's t distribution with ``len(x) - 2`` degrees of freedom. >>> import matplotlib.pyplot as plt >>> dof = len(x)-2 # len(x) == len(y) >>> dist = stats.t(df=dof) >>> t_vals = np.linspace(-5, 5, 100) >>> pdf = dist.pdf(t_vals) >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> def plot(ax): # we'll reuse this ... ax.plot(t_vals, pdf) ... ax.set_title("Spearman's Rho Test Null Distribution") ... ax.set_xlabel("statistic") ... ax.set_ylabel("probability density") >>> plot(ax) >>> plt.show() The comparison is quantified by the p-value: the proportion of values in the null distribution as extreme or more extreme than the observed value of the statistic. In a two-sided test in which the statistic is positive, elements of the null distribution greater than the transformed statistic and elements of the null distribution less than the negative of the observed statistic are both considered "more extreme". >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> plot(ax) >>> rs = res.statistic # original statistic >>> transformed = rs * np.sqrt(dof / ((rs+1.0)*(1.0-rs))) >>> pvalue = dist.cdf(-transformed) + dist.sf(transformed) >>> annotation = (f'p-value={pvalue:.4f}\n(shaded area)') >>> props = dict(facecolor='black', width=1, headwidth=5, headlength=8) >>> _ = ax.annotate(annotation, (2.7, 0.025), (3, 0.03), arrowprops=props) >>> i = t_vals >= transformed >>> ax.fill_between(t_vals[i], y1=0, y2=pdf[i], color='C0') >>> i = t_vals <= -transformed >>> ax.fill_between(t_vals[i], y1=0, y2=pdf[i], color='C0') >>> ax.set_xlim(-5, 5) >>> ax.set_ylim(0, 0.1) >>> plt.show() >>> res.pvalue 0.07991669030889909 # two-sided p-value If the p-value is "small" - that is, if there is a low probability of sampling data from independent distributions that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the distribution of total collagen and free proline are *not* independent. Note that: - The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis. - The threshold for values that will be considered "small" is a choice that should be made before the data is analyzed [5]_ with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis). - Small p-values are not evidence for a *large* effect; rather, they can only provide evidence for a "significant" effect, meaning that they are unlikely to have occurred under the null hypothesis. Suppose that before performing the experiment, the authors had reason to predict a positive correlation between the total collagen and free proline measurements, and that they had chosen to assess the plausibility of the null hypothesis against a one-sided alternative: free proline has a positive ordinal correlation with total collagen. In this case, only those values in the null distribution that are as great or greater than the observed statistic are considered to be more extreme. >>> res = stats.spearmanr(x, y, alternative='greater') >>> res.statistic 0.7000000000000001 # same statistic >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> plot(ax) >>> pvalue = dist.sf(transformed) >>> annotation = (f'p-value={pvalue:.6f}\n(shaded area)') >>> props = dict(facecolor='black', width=1, headwidth=5, headlength=8) >>> _ = ax.annotate(annotation, (3, 0.018), (3.5, 0.03), arrowprops=props) >>> i = t_vals >= transformed >>> ax.fill_between(t_vals[i], y1=0, y2=pdf[i], color='C0') >>> ax.set_xlim(1, 5) >>> ax.set_ylim(0, 0.1) >>> plt.show() >>> res.pvalue 0.03995834515444954 # one-sided p-value; half of the two-sided p-value Note that the t-distribution provides an asymptotic approximation of the null distribution; it is only accurate for samples with many observations. For small samples, it may be more appropriate to perform a permutation test: Under the null hypothesis that total collagen and free proline are independent, each of the free proline measurements were equally likely to have been observed with any of the total collagen measurements. Therefore, we can form an *exact* null distribution by calculating the statistic under each possible pairing of elements between ``x`` and ``y``. >>> def statistic(x): # explore all possible pairings by permuting `x` ... rs = stats.spearmanr(x, y).statistic # ignore pvalue ... transformed = rs * np.sqrt(dof / ((rs+1.0)*(1.0-rs))) ... return transformed >>> ref = stats.permutation_test((x,), statistic, alternative='greater', ... permutation_type='pairings') >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> plot(ax) >>> ax.hist(ref.null_distribution, np.linspace(-5, 5, 26), ... density=True) >>> ax.legend(['aymptotic approximation\n(many observations)', ... f'exact \n({len(ref.null_distribution)} permutations)']) >>> plt.show() >>> ref.pvalue 0.04563492063492063 # exact one-sided p-value """ if axis is not None and axis > 1: raise ValueError("spearmanr only handles 1-D or 2-D arrays, " f"supplied axis argument {axis}, please use only " "values 0, 1 or None for axis") a, axisout = _chk_asarray(a, axis) if a.ndim > 2: raise ValueError("spearmanr only handles 1-D or 2-D arrays") if b is None: if a.ndim < 2: raise ValueError("`spearmanr` needs at least 2 " "variables to compare") else: # Concatenate a and b, so that we now only have to handle the case # of a 2-D `a`. b, _ = _chk_asarray(b, axis) if axisout == 0: a = np.column_stack((a, b)) else: a = np.vstack((a, b)) n_vars = a.shape[1 - axisout] n_obs = a.shape[axisout] if n_obs <= 1: # Handle empty arrays or single observations. res = SignificanceResult(np.nan, np.nan) res.correlation = np.nan return res warn_msg = ("An input array is constant; the correlation coefficient " "is not defined.") if axisout == 0: if (a[:, 0][0] == a[:, 0]).all() or (a[:, 1][0] == a[:, 1]).all(): # If an input is constant, the correlation coefficient # is not defined. warnings.warn(stats.ConstantInputWarning(warn_msg), stacklevel=2) res = SignificanceResult(np.nan, np.nan) res.correlation = np.nan return res else: # case when axisout == 1 b/c a is 2 dim only if (a[0, :][0] == a[0, :]).all() or (a[1, :][0] == a[1, :]).all(): # If an input is constant, the correlation coefficient # is not defined. warnings.warn(stats.ConstantInputWarning(warn_msg), stacklevel=2) res = SignificanceResult(np.nan, np.nan) res.correlation = np.nan return res a_contains_nan, nan_policy = _contains_nan(a, nan_policy) variable_has_nan = np.zeros(n_vars, dtype=bool) if a_contains_nan: if nan_policy == 'omit': return mstats_basic.spearmanr(a, axis=axis, nan_policy=nan_policy, alternative=alternative) elif nan_policy == 'propagate': if a.ndim == 1 or n_vars <= 2: res = SignificanceResult(np.nan, np.nan) res.correlation = np.nan return res else: # Keep track of variables with NaNs, set the outputs to NaN # only for those variables variable_has_nan = np.isnan(a).any(axis=axisout) a_ranked = np.apply_along_axis(rankdata, axisout, a) rs = np.corrcoef(a_ranked, rowvar=axisout) dof = n_obs - 2 # degrees of freedom # rs can have elements equal to 1, so avoid zero division warnings with np.errstate(divide='ignore'): # clip the small negative values possibly caused by rounding # errors before taking the square root t = rs * np.sqrt((dof/((rs+1.0)*(1.0-rs))).clip(0)) prob = _get_pvalue(t, distributions.t(dof), alternative, xp=np) # For backwards compatibility, return scalars when comparing 2 columns if rs.shape == (2, 2): res = SignificanceResult(rs[1, 0], prob[1, 0]) res.correlation = rs[1, 0] return res else: rs[variable_has_nan, :] = np.nan rs[:, variable_has_nan] = np.nan res = SignificanceResult(rs[()], prob[()]) res.correlation = rs return res def pointbiserialr(x, y): r"""Calculate a point biserial correlation coefficient and its p-value. The point biserial correlation is used to measure the relationship between a binary variable, x, and a continuous variable, y. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply a determinative relationship. This function may be computed using a shortcut formula but produces the same result as `pearsonr`. Parameters ---------- x : array_like of bools Input array. y : array_like Input array. Returns ------- res: SignificanceResult An object containing attributes: statistic : float The R value. pvalue : float The two-sided p-value. Notes ----- `pointbiserialr` uses a t-test with ``n-1`` degrees of freedom. It is equivalent to `pearsonr`. The value of the point-biserial correlation can be calculated from: .. math:: r_{pb} = \frac{\overline{Y_1} - \overline{Y_0}} {s_y} \sqrt{\frac{N_0 N_1} {N (N - 1)}} Where :math:`\overline{Y_{0}}` and :math:`\overline{Y_{1}}` are means of the metric observations coded 0 and 1 respectively; :math:`N_{0}` and :math:`N_{1}` are number of observations coded 0 and 1 respectively; :math:`N` is the total number of observations and :math:`s_{y}` is the standard deviation of all the metric observations. A value of :math:`r_{pb}` that is significantly different from zero is completely equivalent to a significant difference in means between the two groups. Thus, an independent groups t Test with :math:`N-2` degrees of freedom may be used to test whether :math:`r_{pb}` is nonzero. The relation between the t-statistic for comparing two independent groups and :math:`r_{pb}` is given by: .. math:: t = \sqrt{N - 2}\frac{r_{pb}}{\sqrt{1 - r^{2}_{pb}}} References ---------- .. [1] J. Lev, "The Point Biserial Coefficient of Correlation", Ann. Math. Statist., Vol. 20, no.1, pp. 125-126, 1949. .. [2] R.F. Tate, "Correlation Between a Discrete and a Continuous Variable. Point-Biserial Correlation.", Ann. Math. Statist., Vol. 25, np. 3, pp. 603-607, 1954. .. [3] D. Kornbrot "Point Biserial Correlation", In Wiley StatsRef: Statistics Reference Online (eds N. Balakrishnan, et al.), 2014. :doi:`10.1002/9781118445112.stat06227` Examples -------- >>> import numpy as np >>> from scipy import stats >>> a = np.array([0, 0, 0, 1, 1, 1, 1]) >>> b = np.arange(7) >>> stats.pointbiserialr(a, b) (0.8660254037844386, 0.011724811003954652) >>> stats.pearsonr(a, b) (0.86602540378443871, 0.011724811003954626) >>> np.corrcoef(a, b) array([[ 1. , 0.8660254], [ 0.8660254, 1. ]]) """ rpb, prob = pearsonr(x, y) # create result object with alias for backward compatibility res = SignificanceResult(rpb, prob) res.correlation = rpb return res def kendalltau(x, y, *, nan_policy='propagate', method='auto', variant='b', alternative='two-sided'): r"""Calculate Kendall's tau, a correlation measure for ordinal data. Kendall's tau is a measure of the correspondence between two rankings. Values close to 1 indicate strong agreement, and values close to -1 indicate strong disagreement. This implements two variants of Kendall's tau: tau-b (the default) and tau-c (also known as Stuart's tau-c). These differ only in how they are normalized to lie within the range -1 to 1; the hypothesis tests (their p-values) are identical. Kendall's original tau-a is not implemented separately because both tau-b and tau-c reduce to tau-a in the absence of ties. Parameters ---------- x, y : array_like Arrays of rankings, of the same shape. If arrays are not 1-D, they will be flattened to 1-D. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values method : {'auto', 'asymptotic', 'exact'}, optional Defines which method is used to calculate the p-value [5]_. The following options are available (default is 'auto'): * 'auto': selects the appropriate method based on a trade-off between speed and accuracy * 'asymptotic': uses a normal approximation valid for large samples * 'exact': computes the exact p-value, but can only be used if no ties are present. As the sample size increases, the 'exact' computation time may grow and the result may lose some precision. variant : {'b', 'c'}, optional Defines which variant of Kendall's tau is returned. Default is 'b'. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. Default is 'two-sided'. The following options are available: * 'two-sided': the rank correlation is nonzero * 'less': the rank correlation is negative (less than zero) * 'greater': the rank correlation is positive (greater than zero) Returns ------- res : SignificanceResult An object containing attributes: statistic : float The tau statistic. pvalue : float The p-value for a hypothesis test whose null hypothesis is an absence of association, tau = 0. See Also -------- spearmanr : Calculates a Spearman rank-order correlation coefficient. theilslopes : Computes the Theil-Sen estimator for a set of points (x, y). weightedtau : Computes a weighted version of Kendall's tau. Notes ----- The definition of Kendall's tau that is used is [2]_:: tau_b = (P - Q) / sqrt((P + Q + T) * (P + Q + U)) tau_c = 2 (P - Q) / (n**2 * (m - 1) / m) where P is the number of concordant pairs, Q the number of discordant pairs, T the number of ties only in `x`, and U the number of ties only in `y`. If a tie occurs for the same pair in both `x` and `y`, it is not added to either T or U. n is the total number of samples, and m is the number of unique values in either `x` or `y`, whichever is smaller. References ---------- .. [1] Maurice G. Kendall, "A New Measure of Rank Correlation", Biometrika Vol. 30, No. 1/2, pp. 81-93, 1938. .. [2] Maurice G. Kendall, "The treatment of ties in ranking problems", Biometrika Vol. 33, No. 3, pp. 239-251. 1945. .. [3] Gottfried E. Noether, "Elements of Nonparametric Statistics", John Wiley & Sons, 1967. .. [4] Peter M. Fenwick, "A new data structure for cumulative frequency tables", Software: Practice and Experience, Vol. 24, No. 3, pp. 327-336, 1994. .. [5] Maurice G. Kendall, "Rank Correlation Methods" (4th Edition), Charles Griffin & Co., 1970. .. [6] Kershenobich, D., Fierro, F. J., & Rojkind, M. (1970). The relationship between the free pool of proline and collagen content in human liver cirrhosis. The Journal of Clinical Investigation, 49(12), 2246-2249. .. [7] Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric statistical methods. John Wiley & Sons. .. [8] B. Phipson and G. K. Smyth. "Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn." Statistical Applications in Genetics and Molecular Biology 9.1 (2010). Examples -------- Consider the following data from [6]_, which studied the relationship between free proline (an amino acid) and total collagen (a protein often found in connective tissue) in unhealthy human livers. The ``x`` and ``y`` arrays below record measurements of the two compounds. The observations are paired: each free proline measurement was taken from the same liver as the total collagen measurement at the same index. >>> import numpy as np >>> # total collagen (mg/g dry weight of liver) >>> x = np.array([7.1, 7.1, 7.2, 8.3, 9.4, 10.5, 11.4]) >>> # free proline (μ mole/g dry weight of liver) >>> y = np.array([2.8, 2.9, 2.8, 2.6, 3.5, 4.6, 5.0]) These data were analyzed in [7]_ using Spearman's correlation coefficient, a statistic similar to Kendall's tau in that it is also sensitive to ordinal correlation between the samples. Let's perform an analogous study using Kendall's tau. >>> from scipy import stats >>> res = stats.kendalltau(x, y) >>> res.statistic 0.5499999999999999 The value of this statistic tends to be high (close to 1) for samples with a strongly positive ordinal correlation, low (close to -1) for samples with a strongly negative ordinal correlation, and small in magnitude (close to zero) for samples with weak ordinal correlation. The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that total collagen and free proline measurements are independent. For this test, the null distribution for large samples without ties is approximated as the normal distribution with variance ``(2*(2*n + 5))/(9*n*(n - 1))``, where ``n = len(x)``. >>> import matplotlib.pyplot as plt >>> n = len(x) # len(x) == len(y) >>> var = (2*(2*n + 5))/(9*n*(n - 1)) >>> dist = stats.norm(scale=np.sqrt(var)) >>> z_vals = np.linspace(-1.25, 1.25, 100) >>> pdf = dist.pdf(z_vals) >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> def plot(ax): # we'll reuse this ... ax.plot(z_vals, pdf) ... ax.set_title("Kendall Tau Test Null Distribution") ... ax.set_xlabel("statistic") ... ax.set_ylabel("probability density") >>> plot(ax) >>> plt.show() The comparison is quantified by the p-value: the proportion of values in the null distribution as extreme or more extreme than the observed value of the statistic. In a two-sided test in which the statistic is positive, elements of the null distribution greater than the transformed statistic and elements of the null distribution less than the negative of the observed statistic are both considered "more extreme". >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> plot(ax) >>> pvalue = dist.cdf(-res.statistic) + dist.sf(res.statistic) >>> annotation = (f'p-value={pvalue:.4f}\n(shaded area)') >>> props = dict(facecolor='black', width=1, headwidth=5, headlength=8) >>> _ = ax.annotate(annotation, (0.65, 0.15), (0.8, 0.3), arrowprops=props) >>> i = z_vals >= res.statistic >>> ax.fill_between(z_vals[i], y1=0, y2=pdf[i], color='C0') >>> i = z_vals <= -res.statistic >>> ax.fill_between(z_vals[i], y1=0, y2=pdf[i], color='C0') >>> ax.set_xlim(-1.25, 1.25) >>> ax.set_ylim(0, 0.5) >>> plt.show() >>> res.pvalue 0.09108705741631495 # approximate p-value Note that there is slight disagreement between the shaded area of the curve and the p-value returned by `kendalltau`. This is because our data has ties, and we have neglected a tie correction to the null distribution variance that `kendalltau` performs. For samples without ties, the shaded areas of our plot and p-value returned by `kendalltau` would match exactly. If the p-value is "small" - that is, if there is a low probability of sampling data from independent distributions that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the distribution of total collagen and free proline are *not* independent. Note that: - The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis. - The threshold for values that will be considered "small" is a choice that should be made before the data is analyzed [8]_ with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis). - Small p-values are not evidence for a *large* effect; rather, they can only provide evidence for a "significant" effect, meaning that they are unlikely to have occurred under the null hypothesis. For samples without ties of moderate size, `kendalltau` can compute the p-value exactly. However, in the presence of ties, `kendalltau` resorts to an asymptotic approximation. Nonetheles, we can use a permutation test to compute the null distribution exactly: Under the null hypothesis that total collagen and free proline are independent, each of the free proline measurements were equally likely to have been observed with any of the total collagen measurements. Therefore, we can form an *exact* null distribution by calculating the statistic under each possible pairing of elements between ``x`` and ``y``. >>> def statistic(x): # explore all possible pairings by permuting `x` ... return stats.kendalltau(x, y).statistic # ignore pvalue >>> ref = stats.permutation_test((x,), statistic, ... permutation_type='pairings') >>> fig, ax = plt.subplots(figsize=(8, 5)) >>> plot(ax) >>> bins = np.linspace(-1.25, 1.25, 25) >>> ax.hist(ref.null_distribution, bins=bins, density=True) >>> ax.legend(['aymptotic approximation\n(many observations)', ... 'exact null distribution']) >>> plot(ax) >>> plt.show() >>> ref.pvalue 0.12222222222222222 # exact p-value Note that there is significant disagreement between the exact p-value calculated here and the approximation returned by `kendalltau` above. For small samples with ties, consider performing a permutation test for more accurate results. """ x = np.asarray(x).ravel() y = np.asarray(y).ravel() if x.size != y.size: raise ValueError("All inputs to `kendalltau` must be of the same " f"size, found x-size {x.size} and y-size {y.size}") elif not x.size or not y.size: # Return NaN if arrays are empty res = SignificanceResult(np.nan, np.nan) res.correlation = np.nan return res # check both x and y cnx, npx = _contains_nan(x, nan_policy) cny, npy = _contains_nan(y, nan_policy) contains_nan = cnx or cny if npx == 'omit' or npy == 'omit': nan_policy = 'omit' if contains_nan and nan_policy == 'propagate': res = SignificanceResult(np.nan, np.nan) res.correlation = np.nan return res elif contains_nan and nan_policy == 'omit': x = ma.masked_invalid(x) y = ma.masked_invalid(y) if variant == 'b': return mstats_basic.kendalltau(x, y, method=method, use_ties=True, alternative=alternative) else: message = ("nan_policy='omit' is currently compatible only with " "variant='b'.") raise ValueError(message) def count_rank_tie(ranks): cnt = np.bincount(ranks).astype('int64', copy=False) cnt = cnt[cnt > 1] # Python ints to avoid overflow down the line return (int((cnt * (cnt - 1) // 2).sum()), int((cnt * (cnt - 1.) * (cnt - 2)).sum()), int((cnt * (cnt - 1.) * (2*cnt + 5)).sum())) size = x.size perm = np.argsort(y) # sort on y and convert y to dense ranks x, y = x[perm], y[perm] y = np.r_[True, y[1:] != y[:-1]].cumsum(dtype=np.intp) # stable sort on x and convert x to dense ranks perm = np.argsort(x, kind='mergesort') x, y = x[perm], y[perm] x = np.r_[True, x[1:] != x[:-1]].cumsum(dtype=np.intp) dis = _kendall_dis(x, y) # discordant pairs obs = np.r_[True, (x[1:] != x[:-1]) | (y[1:] != y[:-1]), True] cnt = np.diff(np.nonzero(obs)[0]).astype('int64', copy=False) ntie = int((cnt * (cnt - 1) // 2).sum()) # joint ties xtie, x0, x1 = count_rank_tie(x) # ties in x, stats ytie, y0, y1 = count_rank_tie(y) # ties in y, stats tot = (size * (size - 1)) // 2 if xtie == tot or ytie == tot: res = SignificanceResult(np.nan, np.nan) res.correlation = np.nan return res # Note that tot = con + dis + (xtie - ntie) + (ytie - ntie) + ntie # = con + dis + xtie + ytie - ntie con_minus_dis = tot - xtie - ytie + ntie - 2 * dis if variant == 'b': tau = con_minus_dis / np.sqrt(tot - xtie) / np.sqrt(tot - ytie) elif variant == 'c': minclasses = min(len(set(x)), len(set(y))) tau = 2*con_minus_dis / (size**2 * (minclasses-1)/minclasses) else: raise ValueError(f"Unknown variant of the method chosen: {variant}. " "variant must be 'b' or 'c'.") # Limit range to fix computational errors tau = np.minimum(1., max(-1., tau)) # The p-value calculation is the same for all variants since the p-value # depends only on con_minus_dis. if method == 'exact' and (xtie != 0 or ytie != 0): raise ValueError("Ties found, exact method cannot be used.") if method == 'auto': if (xtie == 0 and ytie == 0) and (size <= 33 or min(dis, tot-dis) <= 1): method = 'exact' else: method = 'asymptotic' if xtie == 0 and ytie == 0 and method == 'exact': pvalue = mstats_basic._kendall_p_exact(size, tot-dis, alternative) elif method == 'asymptotic': # con_minus_dis is approx normally distributed with this variance [3]_ m = size * (size - 1.) var = ((m * (2*size + 5) - x1 - y1) / 18 + (2 * xtie * ytie) / m + x0 * y0 / (9 * m * (size - 2))) z = con_minus_dis / np.sqrt(var) pvalue = _get_pvalue(z, _SimpleNormal(), alternative, xp=np) else: raise ValueError(f"Unknown method {method} specified. Use 'auto', " "'exact' or 'asymptotic'.") # create result object with alias for backward compatibility res = SignificanceResult(tau[()], pvalue[()]) res.correlation = tau[()] return res def weightedtau(x, y, rank=True, weigher=None, additive=True): r"""Compute a weighted version of Kendall's :math:`\tau`. The weighted :math:`\tau` is a weighted version of Kendall's :math:`\tau` in which exchanges of high weight are more influential than exchanges of low weight. The default parameters compute the additive hyperbolic version of the index, :math:`\tau_\mathrm h`, which has been shown to provide the best balance between important and unimportant elements [1]_. The weighting is defined by means of a rank array, which assigns a nonnegative rank to each element (higher importance ranks being associated with smaller values, e.g., 0 is the highest possible rank), and a weigher function, which assigns a weight based on the rank to each element. The weight of an exchange is then the sum or the product of the weights of the ranks of the exchanged elements. The default parameters compute :math:`\tau_\mathrm h`: an exchange between elements with rank :math:`r` and :math:`s` (starting from zero) has weight :math:`1/(r+1) + 1/(s+1)`. Specifying a rank array is meaningful only if you have in mind an external criterion of importance. If, as it usually happens, you do not have in mind a specific rank, the weighted :math:`\tau` is defined by averaging the values obtained using the decreasing lexicographical rank by (`x`, `y`) and by (`y`, `x`). This is the behavior with default parameters. Note that the convention used here for ranking (lower values imply higher importance) is opposite to that used by other SciPy statistical functions. Parameters ---------- x, y : array_like Arrays of scores, of the same shape. If arrays are not 1-D, they will be flattened to 1-D. rank : array_like of ints or bool, optional A nonnegative rank assigned to each element. If it is None, the decreasing lexicographical rank by (`x`, `y`) will be used: elements of higher rank will be those with larger `x`-values, using `y`-values to break ties (in particular, swapping `x` and `y` will give a different result). If it is False, the element indices will be used directly as ranks. The default is True, in which case this function returns the average of the values obtained using the decreasing lexicographical rank by (`x`, `y`) and by (`y`, `x`). weigher : callable, optional The weigher function. Must map nonnegative integers (zero representing the most important element) to a nonnegative weight. The default, None, provides hyperbolic weighing, that is, rank :math:`r` is mapped to weight :math:`1/(r+1)`. additive : bool, optional If True, the weight of an exchange is computed by adding the weights of the ranks of the exchanged elements; otherwise, the weights are multiplied. The default is True. Returns ------- res: SignificanceResult An object containing attributes: statistic : float The weighted :math:`\tau` correlation index. pvalue : float Presently ``np.nan``, as the null distribution of the statistic is unknown (even in the additive hyperbolic case). See Also -------- kendalltau : Calculates Kendall's tau. spearmanr : Calculates a Spearman rank-order correlation coefficient. theilslopes : Computes the Theil-Sen estimator for a set of points (x, y). Notes ----- This function uses an :math:`O(n \log n)`, mergesort-based algorithm [1]_ that is a weighted extension of Knight's algorithm for Kendall's :math:`\tau` [2]_. It can compute Shieh's weighted :math:`\tau` [3]_ between rankings without ties (i.e., permutations) by setting `additive` and `rank` to False, as the definition given in [1]_ is a generalization of Shieh's. NaNs are considered the smallest possible score. .. versionadded:: 0.19.0 References ---------- .. [1] Sebastiano Vigna, "A weighted correlation index for rankings with ties", Proceedings of the 24th international conference on World Wide Web, pp. 1166-1176, ACM, 2015. .. [2] W.R. Knight, "A Computer Method for Calculating Kendall's Tau with Ungrouped Data", Journal of the American Statistical Association, Vol. 61, No. 314, Part 1, pp. 436-439, 1966. .. [3] Grace S. Shieh. "A weighted Kendall's tau statistic", Statistics & Probability Letters, Vol. 39, No. 1, pp. 17-24, 1998. Examples -------- >>> import numpy as np >>> from scipy import stats >>> x = [12, 2, 1, 12, 2] >>> y = [1, 4, 7, 1, 0] >>> res = stats.weightedtau(x, y) >>> res.statistic -0.56694968153682723 >>> res.pvalue nan >>> res = stats.weightedtau(x, y, additive=False) >>> res.statistic -0.62205716951801038 NaNs are considered the smallest possible score: >>> x = [12, 2, 1, 12, 2] >>> y = [1, 4, 7, 1, np.nan] >>> res = stats.weightedtau(x, y) >>> res.statistic -0.56694968153682723 This is exactly Kendall's tau: >>> x = [12, 2, 1, 12, 2] >>> y = [1, 4, 7, 1, 0] >>> res = stats.weightedtau(x, y, weigher=lambda x: 1) >>> res.statistic -0.47140452079103173 >>> x = [12, 2, 1, 12, 2] >>> y = [1, 4, 7, 1, 0] >>> stats.weightedtau(x, y, rank=None) SignificanceResult(statistic=-0.4157652301037516, pvalue=nan) >>> stats.weightedtau(y, x, rank=None) SignificanceResult(statistic=-0.7181341329699028, pvalue=nan) """ x = np.asarray(x).ravel() y = np.asarray(y).ravel() if x.size != y.size: raise ValueError("All inputs to `weightedtau` must be " "of the same size, " f"found x-size {x.size} and y-size {y.size}") if not x.size: # Return NaN if arrays are empty res = SignificanceResult(np.nan, np.nan) res.correlation = np.nan return res # If there are NaNs we apply _toint64() if np.isnan(np.sum(x)): x = _toint64(x) if np.isnan(np.sum(y)): y = _toint64(y) # Reduce to ranks unsupported types if x.dtype != y.dtype: if x.dtype != np.int64: x = _toint64(x) if y.dtype != np.int64: y = _toint64(y) else: if x.dtype not in (np.int32, np.int64, np.float32, np.float64): x = _toint64(x) y = _toint64(y) if rank is True: tau = ( _weightedrankedtau(x, y, None, weigher, additive) + _weightedrankedtau(y, x, None, weigher, additive) ) / 2 res = SignificanceResult(tau, np.nan) res.correlation = tau return res if rank is False: rank = np.arange(x.size, dtype=np.intp) elif rank is not None: rank = np.asarray(rank).ravel() if rank.size != x.size: raise ValueError( "All inputs to `weightedtau` must be of the same size, " f"found x-size {x.size} and rank-size {rank.size}" ) tau = _weightedrankedtau(x, y, rank, weigher, additive) res = SignificanceResult(tau, np.nan) res.correlation = tau return res ##################################### # INFERENTIAL STATISTICS # ##################################### TtestResultBase = _make_tuple_bunch('TtestResultBase', ['statistic', 'pvalue'], ['df']) class TtestResult(TtestResultBase): """ Result of a t-test. See the documentation of the particular t-test function for more information about the definition of the statistic and meaning of the confidence interval. Attributes ---------- statistic : float or array The t-statistic of the sample. pvalue : float or array The p-value associated with the given alternative. df : float or array The number of degrees of freedom used in calculation of the t-statistic; this is one less than the size of the sample (``a.shape[axis]-1`` if there are no masked elements or omitted NaNs). Methods ------- confidence_interval Computes a confidence interval around the population statistic for the given confidence level. The confidence interval is returned in a ``namedtuple`` with fields `low` and `high`. """ def __init__(self, statistic, pvalue, df, # public alternative, standard_error, estimate, # private statistic_np=None, xp=None): # private super().__init__(statistic, pvalue, df=df) self._alternative = alternative self._standard_error = standard_error # denominator of t-statistic self._estimate = estimate # point estimate of sample mean self._statistic_np = statistic if statistic_np is None else statistic_np self._dtype = statistic.dtype self._xp = array_namespace(statistic, pvalue) if xp is None else xp def confidence_interval(self, confidence_level=0.95): """ Parameters ---------- confidence_level : float The confidence level for the calculation of the population mean confidence interval. Default is 0.95. Returns ------- ci : namedtuple The confidence interval is returned in a ``namedtuple`` with fields `low` and `high`. """ low, high = _t_confidence_interval(self.df, self._statistic_np, confidence_level, self._alternative, self._dtype, self._xp) low = low * self._standard_error + self._estimate high = high * self._standard_error + self._estimate return ConfidenceInterval(low=low, high=high) def pack_TtestResult(statistic, pvalue, df, alternative, standard_error, estimate): # this could be any number of dimensions (including 0d), but there is # at most one unique non-NaN value alternative = np.atleast_1d(alternative) # can't index 0D object alternative = alternative[np.isfinite(alternative)] alternative = alternative[0] if alternative.size else np.nan return TtestResult(statistic, pvalue, df=df, alternative=alternative, standard_error=standard_error, estimate=estimate) def unpack_TtestResult(res): return (res.statistic, res.pvalue, res.df, res._alternative, res._standard_error, res._estimate) @_axis_nan_policy_factory(pack_TtestResult, default_axis=0, n_samples=2, result_to_tuple=unpack_TtestResult, n_outputs=6) # nan_policy handled by `_axis_nan_policy`, but needs to be left # in signature to preserve use as a positional argument def ttest_1samp(a, popmean, axis=0, nan_policy="propagate", alternative="two-sided"): """Calculate the T-test for the mean of ONE group of scores. This is a test for the null hypothesis that the expected value (mean) of a sample of independent observations `a` is equal to the given population mean, `popmean`. Parameters ---------- a : array_like Sample observations. popmean : float or array_like Expected value in null hypothesis. If array_like, then its length along `axis` must equal 1, and it must otherwise be broadcastable with `a`. axis : int or None, optional Axis along which to compute test; default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided': the mean of the underlying distribution of the sample is different than the given population mean (`popmean`) * 'less': the mean of the underlying distribution of the sample is less than the given population mean (`popmean`) * 'greater': the mean of the underlying distribution of the sample is greater than the given population mean (`popmean`) Returns ------- result : `~scipy.stats._result_classes.TtestResult` An object with the following attributes: statistic : float or array The t-statistic. pvalue : float or array The p-value associated with the given alternative. df : float or array The number of degrees of freedom used in calculation of the t-statistic; this is one less than the size of the sample (``a.shape[axis]``). .. versionadded:: 1.10.0 The object also has the following method: confidence_interval(confidence_level=0.95) Computes a confidence interval around the population mean for the given confidence level. The confidence interval is returned in a ``namedtuple`` with fields `low` and `high`. .. versionadded:: 1.10.0 Notes ----- The statistic is calculated as ``(np.mean(a) - popmean)/se``, where ``se`` is the standard error. Therefore, the statistic will be positive when the sample mean is greater than the population mean and negative when the sample mean is less than the population mean. Examples -------- Suppose we wish to test the null hypothesis that the mean of a population is equal to 0.5. We choose a confidence level of 99%; that is, we will reject the null hypothesis in favor of the alternative if the p-value is less than 0.01. When testing random variates from the standard uniform distribution, which has a mean of 0.5, we expect the data to be consistent with the null hypothesis most of the time. >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng() >>> rvs = stats.uniform.rvs(size=50, random_state=rng) >>> stats.ttest_1samp(rvs, popmean=0.5) TtestResult(statistic=2.456308468440, pvalue=0.017628209047638, df=49) As expected, the p-value of 0.017 is not below our threshold of 0.01, so we cannot reject the null hypothesis. When testing data from the standard *normal* distribution, which has a mean of 0, we would expect the null hypothesis to be rejected. >>> rvs = stats.norm.rvs(size=50, random_state=rng) >>> stats.ttest_1samp(rvs, popmean=0.5) TtestResult(statistic=-7.433605518875, pvalue=1.416760157221e-09, df=49) Indeed, the p-value is lower than our threshold of 0.01, so we reject the null hypothesis in favor of the default "two-sided" alternative: the mean of the population is *not* equal to 0.5. However, suppose we were to test the null hypothesis against the one-sided alternative that the mean of the population is *greater* than 0.5. Since the mean of the standard normal is less than 0.5, we would not expect the null hypothesis to be rejected. >>> stats.ttest_1samp(rvs, popmean=0.5, alternative='greater') TtestResult(statistic=-7.433605518875, pvalue=0.99999999929, df=49) Unsurprisingly, with a p-value greater than our threshold, we would not reject the null hypothesis. Note that when working with a confidence level of 99%, a true null hypothesis will be rejected approximately 1% of the time. >>> rvs = stats.uniform.rvs(size=(100, 50), random_state=rng) >>> res = stats.ttest_1samp(rvs, popmean=0.5, axis=1) >>> np.sum(res.pvalue < 0.01) 1 Indeed, even though all 100 samples above were drawn from the standard uniform distribution, which *does* have a population mean of 0.5, we would mistakenly reject the null hypothesis for one of them. `ttest_1samp` can also compute a confidence interval around the population mean. >>> rvs = stats.norm.rvs(size=50, random_state=rng) >>> res = stats.ttest_1samp(rvs, popmean=0) >>> ci = res.confidence_interval(confidence_level=0.95) >>> ci ConfidenceInterval(low=-0.3193887540880017, high=0.2898583388980972) The bounds of the 95% confidence interval are the minimum and maximum values of the parameter `popmean` for which the p-value of the test would be 0.05. >>> res = stats.ttest_1samp(rvs, popmean=ci.low) >>> np.testing.assert_allclose(res.pvalue, 0.05) >>> res = stats.ttest_1samp(rvs, popmean=ci.high) >>> np.testing.assert_allclose(res.pvalue, 0.05) Under certain assumptions about the population from which a sample is drawn, the confidence interval with confidence level 95% is expected to contain the true population mean in 95% of sample replications. >>> rvs = stats.norm.rvs(size=(50, 1000), loc=1, random_state=rng) >>> res = stats.ttest_1samp(rvs, popmean=0) >>> ci = res.confidence_interval() >>> contains_pop_mean = (ci.low < 1) & (ci.high > 1) >>> contains_pop_mean.sum() 953 """ xp = array_namespace(a) a, axis = _chk_asarray(a, axis, xp=xp) n = a.shape[axis] df = n - 1 if n == 0: # This is really only needed for *testing* _axis_nan_policy decorator # It won't happen when the decorator is used. NaN = _get_nan(a) return TtestResult(NaN, NaN, df=NaN, alternative=NaN, standard_error=NaN, estimate=NaN) mean = xp.mean(a, axis=axis) try: popmean = xp.asarray(popmean) popmean = xp.squeeze(popmean, axis=axis) if popmean.ndim > 0 else popmean except ValueError as e: raise ValueError("`popmean.shape[axis]` must equal 1.") from e d = mean - popmean v = _var(a, axis=axis, ddof=1) denom = xp.sqrt(v / n) with np.errstate(divide='ignore', invalid='ignore'): t = xp.divide(d, denom) t = t[()] if t.ndim == 0 else t # This will only work for CPU backends for now. That's OK. In time, # `from_dlpack` will enable the transfer from other devices, and # `_get_pvalue` will even be reworked to support the native backend. t_np = np.asarray(t) prob = _get_pvalue(t_np, distributions.t(df), alternative, xp=np) prob = xp.asarray(prob, dtype=t.dtype) prob = prob[()] if prob.ndim == 0 else prob # when nan_policy='omit', `df` can be different for different axis-slices df = xp.broadcast_to(xp.asarray(df), t.shape) df = df[()] if df.ndim == 0 else df # _axis_nan_policy decorator doesn't play well with strings alternative_num = {"less": -1, "two-sided": 0, "greater": 1}[alternative] return TtestResult(t, prob, df=df, alternative=alternative_num, standard_error=denom, estimate=mean, statistic_np=t_np, xp=xp) def _t_confidence_interval(df, t, confidence_level, alternative, dtype=None, xp=None): # Input validation on `alternative` is already done # We just need IV on confidence_level dtype = t.dtype if dtype is None else dtype xp = array_namespace(t) if xp is None else xp # stdtrit not dispatched yet; use NumPy df, t = np.asarray(df), np.asarray(t) if confidence_level < 0 or confidence_level > 1: message = "`confidence_level` must be a number between 0 and 1." raise ValueError(message) if alternative < 0: # 'less' p = confidence_level low, high = np.broadcast_arrays(-np.inf, special.stdtrit(df, p)) elif alternative > 0: # 'greater' p = 1 - confidence_level low, high = np.broadcast_arrays(special.stdtrit(df, p), np.inf) elif alternative == 0: # 'two-sided' tail_probability = (1 - confidence_level)/2 p = tail_probability, 1-tail_probability # axis of p must be the zeroth and orthogonal to all the rest p = np.reshape(p, [2] + [1]*np.asarray(df).ndim) low, high = special.stdtrit(df, p) else: # alternative is NaN when input is empty (see _axis_nan_policy) p, nans = np.broadcast_arrays(t, np.nan) low, high = nans, nans low = xp.asarray(low, dtype=dtype) low = low[()] if low.ndim == 0 else low high = xp.asarray(high, dtype=dtype) high = high[()] if high.ndim == 0 else high return low, high def _ttest_ind_from_stats(mean1, mean2, denom, df, alternative): d = mean1 - mean2 with np.errstate(divide='ignore', invalid='ignore'): t = np.divide(d, denom)[()] prob = _get_pvalue(t, distributions.t(df), alternative, xp=np) return (t, prob) def _unequal_var_ttest_denom(v1, n1, v2, n2): vn1 = v1 / n1 vn2 = v2 / n2 with np.errstate(divide='ignore', invalid='ignore'): df = (vn1 + vn2)**2 / (vn1**2 / (n1 - 1) + vn2**2 / (n2 - 1)) # If df is undefined, variances are zero (assumes n1 > 0 & n2 > 0). # Hence it doesn't matter what df is as long as it's not NaN. df = np.where(np.isnan(df), 1, df) denom = np.sqrt(vn1 + vn2) return df, denom def _equal_var_ttest_denom(v1, n1, v2, n2): # If there is a single observation in one sample, this formula for pooled # variance breaks down because the variance of that sample is undefined. # The pooled variance is still defined, though, because the (n-1) in the # numerator should cancel with the (n-1) in the denominator, leaving only # the sum of squared differences from the mean: zero. v1 = np.where(n1 == 1, 0, v1)[()] v2 = np.where(n2 == 1, 0, v2)[()] df = n1 + n2 - 2.0 svar = ((n1 - 1) * v1 + (n2 - 1) * v2) / df denom = np.sqrt(svar * (1.0 / n1 + 1.0 / n2)) return df, denom Ttest_indResult = namedtuple('Ttest_indResult', ('statistic', 'pvalue')) def ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2, equal_var=True, alternative="two-sided"): r""" T-test for means of two independent samples from descriptive statistics. This is a test for the null hypothesis that two independent samples have identical average (expected) values. Parameters ---------- mean1 : array_like The mean(s) of sample 1. std1 : array_like The corrected sample standard deviation of sample 1 (i.e. ``ddof=1``). nobs1 : array_like The number(s) of observations of sample 1. mean2 : array_like The mean(s) of sample 2. std2 : array_like The corrected sample standard deviation of sample 2 (i.e. ``ddof=1``). nobs2 : array_like The number(s) of observations of sample 2. equal_var : bool, optional If True (default), perform a standard independent 2 sample test that assumes equal population variances [1]_. If False, perform Welch's t-test, which does not assume equal population variance [2]_. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided': the means of the distributions are unequal. * 'less': the mean of the first distribution is less than the mean of the second distribution. * 'greater': the mean of the first distribution is greater than the mean of the second distribution. .. versionadded:: 1.6.0 Returns ------- statistic : float or array The calculated t-statistics. pvalue : float or array The two-tailed p-value. See Also -------- scipy.stats.ttest_ind Notes ----- The statistic is calculated as ``(mean1 - mean2)/se``, where ``se`` is the standard error. Therefore, the statistic will be positive when `mean1` is greater than `mean2` and negative when `mean1` is less than `mean2`. This method does not check whether any of the elements of `std1` or `std2` are negative. If any elements of the `std1` or `std2` parameters are negative in a call to this method, this method will return the same result as if it were passed ``numpy.abs(std1)`` and ``numpy.abs(std2)``, respectively, instead; no exceptions or warnings will be emitted. References ---------- .. [1] https://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test .. [2] https://en.wikipedia.org/wiki/Welch%27s_t-test Examples -------- Suppose we have the summary data for two samples, as follows (with the Sample Variance being the corrected sample variance):: Sample Sample Size Mean Variance Sample 1 13 15.0 87.5 Sample 2 11 12.0 39.0 Apply the t-test to this data (with the assumption that the population variances are equal): >>> import numpy as np >>> from scipy.stats import ttest_ind_from_stats >>> ttest_ind_from_stats(mean1=15.0, std1=np.sqrt(87.5), nobs1=13, ... mean2=12.0, std2=np.sqrt(39.0), nobs2=11) Ttest_indResult(statistic=0.9051358093310269, pvalue=0.3751996797581487) For comparison, here is the data from which those summary statistics were taken. With this data, we can compute the same result using `scipy.stats.ttest_ind`: >>> a = np.array([1, 3, 4, 6, 11, 13, 15, 19, 22, 24, 25, 26, 26]) >>> b = np.array([2, 4, 6, 9, 11, 13, 14, 15, 18, 19, 21]) >>> from scipy.stats import ttest_ind >>> ttest_ind(a, b) TtestResult(statistic=0.905135809331027, pvalue=0.3751996797581486, df=22.0) Suppose we instead have binary data and would like to apply a t-test to compare the proportion of 1s in two independent groups:: Number of Sample Sample Size ones Mean Variance Sample 1 150 30 0.2 0.161073 Sample 2 200 45 0.225 0.175251 The sample mean :math:`\hat{p}` is the proportion of ones in the sample and the variance for a binary observation is estimated by :math:`\hat{p}(1-\hat{p})`. >>> ttest_ind_from_stats(mean1=0.2, std1=np.sqrt(0.161073), nobs1=150, ... mean2=0.225, std2=np.sqrt(0.175251), nobs2=200) Ttest_indResult(statistic=-0.5627187905196761, pvalue=0.5739887114209541) For comparison, we could compute the t statistic and p-value using arrays of 0s and 1s and `scipy.stat.ttest_ind`, as above. >>> group1 = np.array([1]*30 + [0]*(150-30)) >>> group2 = np.array([1]*45 + [0]*(200-45)) >>> ttest_ind(group1, group2) TtestResult(statistic=-0.5627179589855622, pvalue=0.573989277115258, df=348.0) """ mean1 = np.asarray(mean1) std1 = np.asarray(std1) mean2 = np.asarray(mean2) std2 = np.asarray(std2) if equal_var: df, denom = _equal_var_ttest_denom(std1**2, nobs1, std2**2, nobs2) else: df, denom = _unequal_var_ttest_denom(std1**2, nobs1, std2**2, nobs2) res = _ttest_ind_from_stats(mean1, mean2, denom, df, alternative) return Ttest_indResult(*res) @_axis_nan_policy_factory(pack_TtestResult, default_axis=0, n_samples=2, result_to_tuple=unpack_TtestResult, n_outputs=6) def ttest_ind(a, b, axis=0, equal_var=True, nan_policy='propagate', permutations=None, random_state=None, alternative="two-sided", trim=0): """ Calculate the T-test for the means of *two independent* samples of scores. This is a test for the null hypothesis that 2 independent samples have identical average (expected) values. This test assumes that the populations have identical variances by default. Parameters ---------- a, b : array_like The arrays must have the same shape, except in the dimension corresponding to `axis` (the first, by default). axis : int or None, optional Axis along which to compute test. If None, compute over the whole arrays, `a`, and `b`. equal_var : bool, optional If True (default), perform a standard independent 2 sample test that assumes equal population variances [1]_. If False, perform Welch's t-test, which does not assume equal population variance [2]_. .. versionadded:: 0.11.0 nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values The 'omit' option is not currently available for permutation tests or one-sided asympyotic tests. permutations : non-negative int, np.inf, or None (default), optional If 0 or None (default), use the t-distribution to calculate p-values. Otherwise, `permutations` is the number of random permutations that will be used to estimate p-values using a permutation test. If `permutations` equals or exceeds the number of distinct partitions of the pooled data, an exact test is performed instead (i.e. each distinct partition is used exactly once). See Notes for details. .. versionadded:: 1.7.0 random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional If `seed` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `seed` is an int, a new ``RandomState`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Pseudorandom number generator state used to generate permutations (used only when `permutations` is not None). .. versionadded:: 1.7.0 alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided': the means of the distributions underlying the samples are unequal. * 'less': the mean of the distribution underlying the first sample is less than the mean of the distribution underlying the second sample. * 'greater': the mean of the distribution underlying the first sample is greater than the mean of the distribution underlying the second sample. .. versionadded:: 1.6.0 trim : float, optional If nonzero, performs a trimmed (Yuen's) t-test. Defines the fraction of elements to be trimmed from each end of the input samples. If 0 (default), no elements will be trimmed from either side. The number of trimmed elements from each tail is the floor of the trim times the number of elements. Valid range is [0, .5). .. versionadded:: 1.7 Returns ------- result : `~scipy.stats._result_classes.TtestResult` An object with the following attributes: statistic : float or ndarray The t-statistic. pvalue : float or ndarray The p-value associated with the given alternative. df : float or ndarray The number of degrees of freedom used in calculation of the t-statistic. This is always NaN for a permutation t-test. .. versionadded:: 1.11.0 The object also has the following method: confidence_interval(confidence_level=0.95) Computes a confidence interval around the difference in population means for the given confidence level. The confidence interval is returned in a ``namedtuple`` with fields ``low`` and ``high``. When a permutation t-test is performed, the confidence interval is not computed, and fields ``low`` and ``high`` contain NaN. .. versionadded:: 1.11.0 Notes ----- Suppose we observe two independent samples, e.g. flower petal lengths, and we are considering whether the two samples were drawn from the same population (e.g. the same species of flower or two species with similar petal characteristics) or two different populations. The t-test quantifies the difference between the arithmetic means of the two samples. The p-value quantifies the probability of observing as or more extreme values assuming the null hypothesis, that the samples are drawn from populations with the same population means, is true. A p-value larger than a chosen threshold (e.g. 5% or 1%) indicates that our observation is not so unlikely to have occurred by chance. Therefore, we do not reject the null hypothesis of equal population means. If the p-value is smaller than our threshold, then we have evidence against the null hypothesis of equal population means. By default, the p-value is determined by comparing the t-statistic of the observed data against a theoretical t-distribution. When ``1 < permutations < binom(n, k)``, where * ``k`` is the number of observations in `a`, * ``n`` is the total number of observations in `a` and `b`, and * ``binom(n, k)`` is the binomial coefficient (``n`` choose ``k``), the data are pooled (concatenated), randomly assigned to either group `a` or `b`, and the t-statistic is calculated. This process is performed repeatedly (`permutation` times), generating a distribution of the t-statistic under the null hypothesis, and the t-statistic of the observed data is compared to this distribution to determine the p-value. Specifically, the p-value reported is the "achieved significance level" (ASL) as defined in 4.4 of [3]_. Note that there are other ways of estimating p-values using randomized permutation tests; for other options, see the more general `permutation_test`. When ``permutations >= binom(n, k)``, an exact test is performed: the data are partitioned between the groups in each distinct way exactly once. The permutation test can be computationally expensive and not necessarily more accurate than the analytical test, but it does not make strong assumptions about the shape of the underlying distribution. Use of trimming is commonly referred to as the trimmed t-test. At times called Yuen's t-test, this is an extension of Welch's t-test, with the difference being the use of winsorized means in calculation of the variance and the trimmed sample size in calculation of the statistic. Trimming is recommended if the underlying distribution is long-tailed or contaminated with outliers [4]_. The statistic is calculated as ``(np.mean(a) - np.mean(b))/se``, where ``se`` is the standard error. Therefore, the statistic will be positive when the sample mean of `a` is greater than the sample mean of `b` and negative when the sample mean of `a` is less than the sample mean of `b`. References ---------- .. [1] https://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test .. [2] https://en.wikipedia.org/wiki/Welch%27s_t-test .. [3] B. Efron and T. Hastie. Computer Age Statistical Inference. (2016). .. [4] Yuen, Karen K. "The Two-Sample Trimmed t for Unequal Population Variances." Biometrika, vol. 61, no. 1, 1974, pp. 165-170. JSTOR, www.jstor.org/stable/2334299. Accessed 30 Mar. 2021. .. [5] Yuen, Karen K., and W. J. Dixon. "The Approximate Behaviour and Performance of the Two-Sample Trimmed t." Biometrika, vol. 60, no. 2, 1973, pp. 369-374. JSTOR, www.jstor.org/stable/2334550. Accessed 30 Mar. 2021. Examples -------- >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng() Test with sample with identical means: >>> rvs1 = stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng) >>> rvs2 = stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng) >>> stats.ttest_ind(rvs1, rvs2) TtestResult(statistic=-0.4390847099199348, pvalue=0.6606952038870015, df=998.0) >>> stats.ttest_ind(rvs1, rvs2, equal_var=False) TtestResult(statistic=-0.4390847099199348, pvalue=0.6606952553131064, df=997.4602304121448) `ttest_ind` underestimates p for unequal variances: >>> rvs3 = stats.norm.rvs(loc=5, scale=20, size=500, random_state=rng) >>> stats.ttest_ind(rvs1, rvs3) TtestResult(statistic=-1.6370984482905417, pvalue=0.1019251574705033, df=998.0) >>> stats.ttest_ind(rvs1, rvs3, equal_var=False) TtestResult(statistic=-1.637098448290542, pvalue=0.10202110497954867, df=765.1098655246868) When ``n1 != n2``, the equal variance t-statistic is no longer equal to the unequal variance t-statistic: >>> rvs4 = stats.norm.rvs(loc=5, scale=20, size=100, random_state=rng) >>> stats.ttest_ind(rvs1, rvs4) TtestResult(statistic=-1.9481646859513422, pvalue=0.05186270935842703, df=598.0) >>> stats.ttest_ind(rvs1, rvs4, equal_var=False) TtestResult(statistic=-1.3146566100751664, pvalue=0.1913495266513811, df=110.41349083985212) T-test with different means, variance, and n: >>> rvs5 = stats.norm.rvs(loc=8, scale=20, size=100, random_state=rng) >>> stats.ttest_ind(rvs1, rvs5) TtestResult(statistic=-2.8415950600298774, pvalue=0.0046418707568707885, df=598.0) >>> stats.ttest_ind(rvs1, rvs5, equal_var=False) TtestResult(statistic=-1.8686598649188084, pvalue=0.06434714193919686, df=109.32167496550137) When performing a permutation test, more permutations typically yields more accurate results. Use a ``np.random.Generator`` to ensure reproducibility: >>> stats.ttest_ind(rvs1, rvs5, permutations=10000, ... random_state=rng) TtestResult(statistic=-2.8415950600298774, pvalue=0.0052994700529947, df=nan) Take these two samples, one of which has an extreme tail. >>> a = (56, 128.6, 12, 123.8, 64.34, 78, 763.3) >>> b = (1.1, 2.9, 4.2) Use the `trim` keyword to perform a trimmed (Yuen) t-test. For example, using 20% trimming, ``trim=.2``, the test will reduce the impact of one (``np.floor(trim*len(a))``) element from each tail of sample `a`. It will have no effect on sample `b` because ``np.floor(trim*len(b))`` is 0. >>> stats.ttest_ind(a, b, trim=.2) TtestResult(statistic=3.4463884028073513, pvalue=0.01369338726499547, df=6.0) """ if not (0 <= trim < .5): raise ValueError("Trimming percentage should be 0 <= `trim` < .5.") NaN = _get_nan(a, b) if a.size == 0 or b.size == 0: # _axis_nan_policy decorator ensures this only happens with 1d input return TtestResult(NaN, NaN, df=NaN, alternative=NaN, standard_error=NaN, estimate=NaN) if permutations is not None and permutations != 0: if trim != 0: raise ValueError("Permutations are currently not supported " "with trimming.") if permutations < 0 or (np.isfinite(permutations) and int(permutations) != permutations): raise ValueError("Permutations must be a non-negative integer.") t, prob = _permutation_ttest(a, b, permutations=permutations, axis=axis, equal_var=equal_var, nan_policy=nan_policy, random_state=random_state, alternative=alternative) df, denom, estimate = NaN, NaN, NaN else: n1 = a.shape[axis] n2 = b.shape[axis] if trim == 0: if equal_var: old_errstate = np.geterr() np.seterr(divide='ignore', invalid='ignore') v1 = _var(a, axis, ddof=1) v2 = _var(b, axis, ddof=1) if equal_var: np.seterr(**old_errstate) m1 = np.mean(a, axis) m2 = np.mean(b, axis) else: v1, m1, n1 = _ttest_trim_var_mean_len(a, trim, axis) v2, m2, n2 = _ttest_trim_var_mean_len(b, trim, axis) if equal_var: df, denom = _equal_var_ttest_denom(v1, n1, v2, n2) else: df, denom = _unequal_var_ttest_denom(v1, n1, v2, n2) t, prob = _ttest_ind_from_stats(m1, m2, denom, df, alternative) # when nan_policy='omit', `df` can be different for different axis-slices df = np.broadcast_to(df, t.shape)[()] estimate = m1-m2 # _axis_nan_policy decorator doesn't play well with strings alternative_num = {"less": -1, "two-sided": 0, "greater": 1}[alternative] return TtestResult(t, prob, df=df, alternative=alternative_num, standard_error=denom, estimate=estimate) def _ttest_trim_var_mean_len(a, trim, axis): """Variance, mean, and length of winsorized input along specified axis""" # for use with `ttest_ind` when trimming. # further calculations in this test assume that the inputs are sorted. # From [4] Section 1 "Let x_1, ..., x_n be n ordered observations..." a = np.sort(a, axis=axis) # `g` is the number of elements to be replaced on each tail, converted # from a percentage amount of trimming n = a.shape[axis] g = int(n * trim) # Calculate the Winsorized variance of the input samples according to # specified `g` v = _calculate_winsorized_variance(a, g, axis) # the total number of elements in the trimmed samples n -= 2 * g # calculate the g-times trimmed mean, as defined in [4] (1-1) m = trim_mean(a, trim, axis=axis) return v, m, n def _calculate_winsorized_variance(a, g, axis): """Calculates g-times winsorized variance along specified axis""" # it is expected that the input `a` is sorted along the correct axis if g == 0: return _var(a, ddof=1, axis=axis) # move the intended axis to the end that way it is easier to manipulate a_win = np.moveaxis(a, axis, -1) # save where NaNs are for later use. nans_indices = np.any(np.isnan(a_win), axis=-1) # Winsorization and variance calculation are done in one step in [4] # (1-3), but here winsorization is done first; replace the left and # right sides with the repeating value. This can be see in effect in ( # 1-3) in [4], where the leftmost and rightmost tails are replaced with # `(g + 1) * x_{g + 1}` on the left and `(g + 1) * x_{n - g}` on the # right. Zero-indexing turns `g + 1` to `g`, and `n - g` to `- g - 1` in # array indexing. a_win[..., :g] = a_win[..., [g]] a_win[..., -g:] = a_win[..., [-g - 1]] # Determine the variance. In [4], the degrees of freedom is expressed as # `h - 1`, where `h = n - 2g` (unnumbered equations in Section 1, end of # page 369, beginning of page 370). This is converted to NumPy's format, # `n - ddof` for use with `np.var`. The result is converted to an # array to accommodate indexing later. var_win = np.asarray(_var(a_win, ddof=(2 * g + 1), axis=-1)) # with `nan_policy='propagate'`, NaNs may be completely trimmed out # because they were sorted into the tail of the array. In these cases, # replace computed variances with `np.nan`. var_win[nans_indices] = np.nan return var_win def _permutation_distribution_t(data, permutations, size_a, equal_var, random_state=None): """Generation permutation distribution of t statistic""" random_state = check_random_state(random_state) # prepare permutation indices size = data.shape[-1] # number of distinct combinations n_max = special.comb(size, size_a) if permutations < n_max: perm_generator = (random_state.permutation(size) for i in range(permutations)) else: permutations = n_max perm_generator = (np.concatenate(z) for z in _all_partitions(size_a, size-size_a)) t_stat = [] for indices in _batch_generator(perm_generator, batch=50): # get one batch from perm_generator at a time as a list indices = np.array(indices) # generate permutations data_perm = data[..., indices] # move axis indexing permutations to position 0 to broadcast # nicely with t_stat_observed, which doesn't have this dimension data_perm = np.moveaxis(data_perm, -2, 0) a = data_perm[..., :size_a] b = data_perm[..., size_a:] t_stat.append(_calc_t_stat(a, b, equal_var)) t_stat = np.concatenate(t_stat, axis=0) return t_stat, permutations, n_max def _calc_t_stat(a, b, equal_var, axis=-1): """Calculate the t statistic along the given dimension.""" na = a.shape[axis] nb = b.shape[axis] avg_a = np.mean(a, axis=axis) avg_b = np.mean(b, axis=axis) var_a = _var(a, axis=axis, ddof=1) var_b = _var(b, axis=axis, ddof=1) if not equal_var: denom = _unequal_var_ttest_denom(var_a, na, var_b, nb)[1] else: denom = _equal_var_ttest_denom(var_a, na, var_b, nb)[1] return (avg_a-avg_b)/denom def _permutation_ttest(a, b, permutations, axis=0, equal_var=True, nan_policy='propagate', random_state=None, alternative="two-sided"): """ Calculates the T-test for the means of TWO INDEPENDENT samples of scores using permutation methods. This test is similar to `stats.ttest_ind`, except it doesn't rely on an approximate normality assumption since it uses a permutation test. This function is only called from ttest_ind when permutations is not None. Parameters ---------- a, b : array_like The arrays must be broadcastable, except along the dimension corresponding to `axis` (the zeroth, by default). axis : int, optional The axis over which to operate on a and b. permutations : int, optional Number of permutations used to calculate p-value. If greater than or equal to the number of distinct permutations, perform an exact test. equal_var : bool, optional If False, an equal variance (Welch's) t-test is conducted. Otherwise, an ordinary t-test is conducted. random_state : {None, int, `numpy.random.Generator`}, optional If `seed` is None the `numpy.random.Generator` singleton is used. If `seed` is an int, a new ``Generator`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` instance then that instance is used. Pseudorandom number generator state used for generating random permutations. Returns ------- statistic : float or array The calculated t-statistic. pvalue : float or array The p-value. """ random_state = check_random_state(random_state) t_stat_observed = _calc_t_stat(a, b, equal_var, axis=axis) na = a.shape[axis] mat = _broadcast_concatenate((a, b), axis=axis) mat = np.moveaxis(mat, axis, -1) t_stat, permutations, n_max = _permutation_distribution_t( mat, permutations, size_a=na, equal_var=equal_var, random_state=random_state) compare = {"less": np.less_equal, "greater": np.greater_equal, "two-sided": lambda x, y: (x <= -np.abs(y)) | (x >= np.abs(y))} # Calculate the p-values cmps = compare[alternative](t_stat, t_stat_observed) # Randomized test p-value calculation should use biased estimate; see e.g. # https://www.degruyter.com/document/doi/10.2202/1544-6115.1585/ adjustment = 1 if n_max > permutations else 0 pvalues = (cmps.sum(axis=0) + adjustment) / (permutations + adjustment) # nans propagate naturally in statistic calculation, but need to be # propagated manually into pvalues if nan_policy == 'propagate' and np.isnan(t_stat_observed).any(): if np.ndim(pvalues) == 0: pvalues = np.float64(np.nan) else: pvalues[np.isnan(t_stat_observed)] = np.nan return (t_stat_observed, pvalues) def _get_len(a, axis, msg): try: n = a.shape[axis] except IndexError: raise AxisError(axis, a.ndim, msg) from None return n @_axis_nan_policy_factory(pack_TtestResult, default_axis=0, n_samples=2, result_to_tuple=unpack_TtestResult, n_outputs=6, paired=True) def ttest_rel(a, b, axis=0, nan_policy='propagate', alternative="two-sided"): """Calculate the t-test on TWO RELATED samples of scores, a and b. This is a test for the null hypothesis that two related or repeated samples have identical average (expected) values. Parameters ---------- a, b : array_like The arrays must have the same shape. axis : int or None, optional Axis along which to compute test. If None, compute over the whole arrays, `a`, and `b`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided': the means of the distributions underlying the samples are unequal. * 'less': the mean of the distribution underlying the first sample is less than the mean of the distribution underlying the second sample. * 'greater': the mean of the distribution underlying the first sample is greater than the mean of the distribution underlying the second sample. .. versionadded:: 1.6.0 Returns ------- result : `~scipy.stats._result_classes.TtestResult` An object with the following attributes: statistic : float or array The t-statistic. pvalue : float or array The p-value associated with the given alternative. df : float or array The number of degrees of freedom used in calculation of the t-statistic; this is one less than the size of the sample (``a.shape[axis]``). .. versionadded:: 1.10.0 The object also has the following method: confidence_interval(confidence_level=0.95) Computes a confidence interval around the difference in population means for the given confidence level. The confidence interval is returned in a ``namedtuple`` with fields `low` and `high`. .. versionadded:: 1.10.0 Notes ----- Examples for use are scores of the same set of student in different exams, or repeated sampling from the same units. The test measures whether the average score differs significantly across samples (e.g. exams). If we observe a large p-value, for example greater than 0.05 or 0.1 then we cannot reject the null hypothesis of identical average scores. If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%, then we reject the null hypothesis of equal averages. Small p-values are associated with large t-statistics. The t-statistic is calculated as ``np.mean(a - b)/se``, where ``se`` is the standard error. Therefore, the t-statistic will be positive when the sample mean of ``a - b`` is greater than zero and negative when the sample mean of ``a - b`` is less than zero. References ---------- https://en.wikipedia.org/wiki/T-test#Dependent_t-test_for_paired_samples Examples -------- >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng() >>> rvs1 = stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng) >>> rvs2 = (stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng) ... + stats.norm.rvs(scale=0.2, size=500, random_state=rng)) >>> stats.ttest_rel(rvs1, rvs2) TtestResult(statistic=-0.4549717054410304, pvalue=0.6493274702088672, df=499) >>> rvs3 = (stats.norm.rvs(loc=8, scale=10, size=500, random_state=rng) ... + stats.norm.rvs(scale=0.2, size=500, random_state=rng)) >>> stats.ttest_rel(rvs1, rvs3) TtestResult(statistic=-5.879467544540889, pvalue=7.540777129099917e-09, df=499) """ a, b, axis = _chk2_asarray(a, b, axis) na = _get_len(a, axis, "first argument") nb = _get_len(b, axis, "second argument") if na != nb: raise ValueError('unequal length arrays') if na == 0 or nb == 0: # _axis_nan_policy decorator ensures this only happens with 1d input NaN = _get_nan(a, b) return TtestResult(NaN, NaN, df=NaN, alternative=NaN, standard_error=NaN, estimate=NaN) n = a.shape[axis] df = n - 1 d = (a - b).astype(np.float64) v = _var(d, axis, ddof=1) dm = np.mean(d, axis) denom = np.sqrt(v / n) with np.errstate(divide='ignore', invalid='ignore'): t = np.divide(dm, denom)[()] prob = _get_pvalue(t, distributions.t(df), alternative, xp=np) # when nan_policy='omit', `df` can be different for different axis-slices df = np.broadcast_to(df, t.shape)[()] # _axis_nan_policy decorator doesn't play well with strings alternative_num = {"less": -1, "two-sided": 0, "greater": 1}[alternative] return TtestResult(t, prob, df=df, alternative=alternative_num, standard_error=denom, estimate=dm) # Map from names to lambda_ values used in power_divergence(). _power_div_lambda_names = { "pearson": 1, "log-likelihood": 0, "freeman-tukey": -0.5, "mod-log-likelihood": -1, "neyman": -2, "cressie-read": 2/3, } def _m_count(a, *, axis, xp): """Count the number of non-masked elements of an array. This function behaves like `np.ma.count`, but is much faster for ndarrays. """ if hasattr(a, 'count'): num = a.count(axis=axis) if isinstance(num, np.ndarray) and num.ndim == 0: # In some cases, the `count` method returns a scalar array (e.g. # np.array(3)), but we want a plain integer. num = int(num) else: if axis is None: num = xp_size(a) else: num = a.shape[axis] return num def _m_broadcast_to(a, shape, *, xp): if np.ma.isMaskedArray(a): return np.ma.masked_array(np.broadcast_to(a, shape), mask=np.broadcast_to(a.mask, shape)) return xp.broadcast_to(a, shape) def _m_sum(a, *, axis, preserve_mask, xp): if np.ma.isMaskedArray(a): sum = a.sum(axis) return sum if preserve_mask else np.asarray(sum) return xp.sum(a, axis=axis) def _m_mean(a, *, axis, keepdims, xp): if np.ma.isMaskedArray(a): return np.asarray(a.mean(axis=axis, keepdims=keepdims)) return xp.mean(a, axis=axis, keepdims=keepdims) Power_divergenceResult = namedtuple('Power_divergenceResult', ('statistic', 'pvalue')) def power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None): """Cressie-Read power divergence statistic and goodness of fit test. This function tests the null hypothesis that the categorical data has the given frequencies, using the Cressie-Read power divergence statistic. Parameters ---------- f_obs : array_like Observed frequencies in each category. .. deprecated:: 1.14.0 Support for masked array input was deprecated in SciPy 1.14.0 and will be removed in version 1.16.0. f_exp : array_like, optional Expected frequencies in each category. By default the categories are assumed to be equally likely. .. deprecated:: 1.14.0 Support for masked array input was deprecated in SciPy 1.14.0 and will be removed in version 1.16.0. ddof : int, optional "Delta degrees of freedom": adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with ``k - 1 - ddof`` degrees of freedom, where `k` is the number of observed frequencies. The default value of `ddof` is 0. axis : int or None, optional The axis of the broadcast result of `f_obs` and `f_exp` along which to apply the test. If axis is None, all values in `f_obs` are treated as a single data set. Default is 0. lambda_ : float or str, optional The power in the Cressie-Read power divergence statistic. The default is 1. For convenience, `lambda_` may be assigned one of the following strings, in which case the corresponding numerical value is used: * ``"pearson"`` (value 1) Pearson's chi-squared statistic. In this case, the function is equivalent to `chisquare`. * ``"log-likelihood"`` (value 0) Log-likelihood ratio. Also known as the G-test [3]_. * ``"freeman-tukey"`` (value -1/2) Freeman-Tukey statistic. * ``"mod-log-likelihood"`` (value -1) Modified log-likelihood ratio. * ``"neyman"`` (value -2) Neyman's statistic. * ``"cressie-read"`` (value 2/3) The power recommended in [5]_. Returns ------- res: Power_divergenceResult An object containing attributes: statistic : float or ndarray The Cressie-Read power divergence test statistic. The value is a float if `axis` is None or if` `f_obs` and `f_exp` are 1-D. pvalue : float or ndarray The p-value of the test. The value is a float if `ddof` and the return value `stat` are scalars. See Also -------- chisquare Notes ----- This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5. Also, the sum of the observed and expected frequencies must be the same for the test to be valid; `power_divergence` raises an error if the sums do not agree within a relative tolerance of ``eps**0.5``, where ``eps`` is the precision of the input dtype. When `lambda_` is less than zero, the formula for the statistic involves dividing by `f_obs`, so a warning or error may be generated if any value in `f_obs` is 0. Similarly, a warning or error may be generated if any value in `f_exp` is zero when `lambda_` >= 0. The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not a chisquare, in which case this test is not appropriate. References ---------- .. [1] Lowry, Richard. "Concepts and Applications of Inferential Statistics". Chapter 8. https://web.archive.org/web/20171015035606/http://faculty.vassar.edu/lowry/ch8pt1.html .. [2] "Chi-squared test", https://en.wikipedia.org/wiki/Chi-squared_test .. [3] "G-test", https://en.wikipedia.org/wiki/G-test .. [4] Sokal, R. R. and Rohlf, F. J. "Biometry: the principles and practice of statistics in biological research", New York: Freeman (1981) .. [5] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984), pp. 440-464. Examples -------- (See `chisquare` for more examples.) When just `f_obs` is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies. Here we perform a G-test (i.e. use the log-likelihood ratio statistic): >>> import numpy as np >>> from scipy.stats import power_divergence >>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood') (2.006573162632538, 0.84823476779463769) The expected frequencies can be given with the `f_exp` argument: >>> power_divergence([16, 18, 16, 14, 12, 12], ... f_exp=[16, 16, 16, 16, 16, 8], ... lambda_='log-likelihood') (3.3281031458963746, 0.6495419288047497) When `f_obs` is 2-D, by default the test is applied to each column. >>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T >>> obs.shape (6, 2) >>> power_divergence(obs, lambda_="log-likelihood") (array([ 2.00657316, 6.77634498]), array([ 0.84823477, 0.23781225])) By setting ``axis=None``, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array. >>> power_divergence(obs, axis=None) (23.31034482758621, 0.015975692534127565) >>> power_divergence(obs.ravel()) (23.31034482758621, 0.015975692534127565) `ddof` is the change to make to the default degrees of freedom. >>> power_divergence([16, 18, 16, 14, 12, 12], ddof=1) (2.0, 0.73575888234288467) The calculation of the p-values is done by broadcasting the test statistic with `ddof`. >>> power_divergence([16, 18, 16, 14, 12, 12], ddof=[0,1,2]) (2.0, array([ 0.84914504, 0.73575888, 0.5724067 ])) `f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting `f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared statistics, we must use ``axis=1``: >>> power_divergence([16, 18, 16, 14, 12, 12], ... f_exp=[[16, 16, 16, 16, 16, 8], ... [8, 20, 20, 16, 12, 12]], ... axis=1) (array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846])) """ xp = array_namespace(f_obs) default_float = xp.asarray(1.).dtype # Convert the input argument `lambda_` to a numerical value. if isinstance(lambda_, str): if lambda_ not in _power_div_lambda_names: names = repr(list(_power_div_lambda_names.keys()))[1:-1] raise ValueError(f"invalid string for lambda_: {lambda_!r}. " f"Valid strings are {names}") lambda_ = _power_div_lambda_names[lambda_] elif lambda_ is None: lambda_ = 1 def warn_masked(arg): if isinstance(arg, ma.MaskedArray): message = ( "`power_divergence` and `chisquare` support for masked array input was " "deprecated in SciPy 1.14.0 and will be removed in version 1.16.0.") warnings.warn(message, DeprecationWarning, stacklevel=2) warn_masked(f_obs) f_obs = f_obs if np.ma.isMaskedArray(f_obs) else xp.asarray(f_obs) dtype = default_float if xp.isdtype(f_obs.dtype, 'integral') else f_obs.dtype f_obs = (f_obs.astype(dtype) if np.ma.isMaskedArray(f_obs) else xp.asarray(f_obs, dtype=dtype)) f_obs_float = (f_obs.astype(np.float64) if hasattr(f_obs, 'mask') else xp.asarray(f_obs, dtype=xp.float64)) if f_exp is not None: warn_masked(f_exp) f_exp = f_exp if np.ma.isMaskedArray(f_obs) else xp.asarray(f_exp) dtype = default_float if xp.isdtype(f_exp.dtype, 'integral') else f_exp.dtype f_exp = (f_exp.astype(dtype) if np.ma.isMaskedArray(f_exp) else xp.asarray(f_exp, dtype=dtype)) bshape = _broadcast_shapes((f_obs_float.shape, f_exp.shape)) f_obs_float = _m_broadcast_to(f_obs_float, bshape, xp=xp) f_exp = _m_broadcast_to(f_exp, bshape, xp=xp) dtype_res = xp.result_type(f_obs.dtype, f_exp.dtype) rtol = xp.finfo(dtype_res).eps**0.5 # to pass existing tests with np.errstate(invalid='ignore'): f_obs_sum = _m_sum(f_obs_float, axis=axis, preserve_mask=False, xp=xp) f_exp_sum = _m_sum(f_exp, axis=axis, preserve_mask=False, xp=xp) relative_diff = (xp.abs(f_obs_sum - f_exp_sum) / xp_minimum(f_obs_sum, f_exp_sum)) diff_gt_tol = xp.any(relative_diff > rtol, axis=None) if diff_gt_tol: msg = (f"For each axis slice, the sum of the observed " f"frequencies must agree with the sum of the " f"expected frequencies to a relative tolerance " f"of {rtol}, but the percent differences are:\n" f"{relative_diff}") raise ValueError(msg) else: # Ignore 'invalid' errors so the edge case of a data set with length 0 # is handled without spurious warnings. with np.errstate(invalid='ignore'): f_exp = _m_mean(f_obs, axis=axis, keepdims=True, xp=xp) # `terms` is the array of terms that are summed along `axis` to create # the test statistic. We use some specialized code for a few special # cases of lambda_. if lambda_ == 1: # Pearson's chi-squared statistic terms = (f_obs - f_exp)**2 / f_exp elif lambda_ == 0: # Log-likelihood ratio (i.e. G-test) terms = 2.0 * special.xlogy(f_obs, f_obs / f_exp) elif lambda_ == -1: # Modified log-likelihood ratio terms = 2.0 * special.xlogy(f_exp, f_exp / f_obs) else: # General Cressie-Read power divergence. terms = f_obs * ((f_obs / f_exp)**lambda_ - 1) terms /= 0.5 * lambda_ * (lambda_ + 1) stat = _m_sum(terms, axis=axis, preserve_mask=True, xp=xp) num_obs = _m_count(terms, axis=axis, xp=xp) ddof = xp.asarray(ddof) df = xp.asarray(num_obs - 1 - ddof) chi2 = _SimpleChi2(df) pvalue = _get_pvalue(stat, chi2 , alternative='greater', symmetric=False, xp=xp) stat = stat[()] if stat.ndim == 0 else stat pvalue = pvalue[()] if pvalue.ndim == 0 else pvalue return Power_divergenceResult(stat, pvalue) def chisquare(f_obs, f_exp=None, ddof=0, axis=0): """Calculate a one-way chi-square test. The chi-square test tests the null hypothesis that the categorical data has the given frequencies. Parameters ---------- f_obs : array_like Observed frequencies in each category. f_exp : array_like, optional Expected frequencies in each category. By default the categories are assumed to be equally likely. ddof : int, optional "Delta degrees of freedom": adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with ``k - 1 - ddof`` degrees of freedom, where `k` is the number of observed frequencies. The default value of `ddof` is 0. axis : int or None, optional The axis of the broadcast result of `f_obs` and `f_exp` along which to apply the test. If axis is None, all values in `f_obs` are treated as a single data set. Default is 0. Returns ------- res: Power_divergenceResult An object containing attributes: statistic : float or ndarray The chi-squared test statistic. The value is a float if `axis` is None or `f_obs` and `f_exp` are 1-D. pvalue : float or ndarray The p-value of the test. The value is a float if `ddof` and the result attribute `statistic` are scalars. See Also -------- scipy.stats.power_divergence scipy.stats.fisher_exact : Fisher exact test on a 2x2 contingency table. scipy.stats.barnard_exact : An unconditional exact test. An alternative to chi-squared test for small sample sizes. Notes ----- This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5. According to [3]_, the total number of samples is recommended to be greater than 13, otherwise exact tests (such as Barnard's Exact test) should be used because they do not overreject. Also, the sum of the observed and expected frequencies must be the same for the test to be valid; `chisquare` raises an error if the sums do not agree within a relative tolerance of ``1e-8``. The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not chi-square, in which case this test is not appropriate. References ---------- .. [1] Lowry, Richard. "Concepts and Applications of Inferential Statistics". Chapter 8. https://web.archive.org/web/20171022032306/http://vassarstats.net:80/textbook/ch8pt1.html .. [2] "Chi-squared test", https://en.wikipedia.org/wiki/Chi-squared_test .. [3] Pearson, Karl. "On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling", Philosophical Magazine. Series 5. 50 (1900), pp. 157-175. .. [4] Mannan, R. William and E. Charles. Meslow. "Bird populations and vegetation characteristics in managed and old-growth forests, northeastern Oregon." Journal of Wildlife Management 48, 1219-1238, :doi:`10.2307/3801783`, 1984. Examples -------- In [4]_, bird foraging behavior was investigated in an old-growth forest of Oregon. In the forest, 44% of the canopy volume was Douglas fir, 24% was ponderosa pine, 29% was grand fir, and 3% was western larch. The authors observed the behavior of several species of birds, one of which was the red-breasted nuthatch. They made 189 observations of this species foraging, recording 43 ("23%") of observations in Douglas fir, 52 ("28%") in ponderosa pine, 54 ("29%") in grand fir, and 40 ("21%") in western larch. Using a chi-square test, we can test the null hypothesis that the proportions of foraging events are equal to the proportions of canopy volume. The authors of the paper considered a p-value less than 1% to be significant. Using the above proportions of canopy volume and observed events, we can infer expected frequencies. >>> import numpy as np >>> f_exp = np.array([44, 24, 29, 3]) / 100 * 189 The observed frequencies of foraging were: >>> f_obs = np.array([43, 52, 54, 40]) We can now compare the observed frequencies with the expected frequencies. >>> from scipy.stats import chisquare >>> chisquare(f_obs=f_obs, f_exp=f_exp) Power_divergenceResult(statistic=228.23515947653874, pvalue=3.3295585338846486e-49) The p-value is well below the chosen significance level. Hence, the authors considered the difference to be significant and concluded that the relative proportions of foraging events were not the same as the relative proportions of tree canopy volume. Following are other generic examples to demonstrate how the other parameters can be used. When just `f_obs` is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies. >>> chisquare([16, 18, 16, 14, 12, 12]) Power_divergenceResult(statistic=2.0, pvalue=0.84914503608460956) With `f_exp` the expected frequencies can be given. >>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8]) Power_divergenceResult(statistic=3.5, pvalue=0.62338762774958223) When `f_obs` is 2-D, by default the test is applied to each column. >>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T >>> obs.shape (6, 2) >>> chisquare(obs) Power_divergenceResult(statistic=array([2. , 6.66666667]), pvalue=array([0.84914504, 0.24663415])) By setting ``axis=None``, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array. >>> chisquare(obs, axis=None) Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565) >>> chisquare(obs.ravel()) Power_divergenceResult(statistic=23.310344827586206, pvalue=0.01597569253412758) `ddof` is the change to make to the default degrees of freedom. >>> chisquare([16, 18, 16, 14, 12, 12], ddof=1) Power_divergenceResult(statistic=2.0, pvalue=0.7357588823428847) The calculation of the p-values is done by broadcasting the chi-squared statistic with `ddof`. >>> chisquare([16, 18, 16, 14, 12, 12], ddof=[0,1,2]) Power_divergenceResult(statistic=2.0, pvalue=array([0.84914504, 0.73575888, 0.5724067 ])) `f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting `f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared statistics, we use ``axis=1``: >>> chisquare([16, 18, 16, 14, 12, 12], ... f_exp=[[16, 16, 16, 16, 16, 8], [8, 20, 20, 16, 12, 12]], ... axis=1) Power_divergenceResult(statistic=array([3.5 , 9.25]), pvalue=array([0.62338763, 0.09949846])) """ # noqa: E501 return power_divergence(f_obs, f_exp=f_exp, ddof=ddof, axis=axis, lambda_="pearson") KstestResult = _make_tuple_bunch('KstestResult', ['statistic', 'pvalue'], ['statistic_location', 'statistic_sign']) def _compute_dplus(cdfvals, x): """Computes D+ as used in the Kolmogorov-Smirnov test. Parameters ---------- cdfvals : array_like Sorted array of CDF values between 0 and 1 x: array_like Sorted array of the stochastic variable itself Returns ------- res: Pair with the following elements: - The maximum distance of the CDF values below Uniform(0, 1). - The location at which the maximum is reached. """ n = len(cdfvals) dplus = (np.arange(1.0, n + 1) / n - cdfvals) amax = dplus.argmax() loc_max = x[amax] return (dplus[amax], loc_max) def _compute_dminus(cdfvals, x): """Computes D- as used in the Kolmogorov-Smirnov test. Parameters ---------- cdfvals : array_like Sorted array of CDF values between 0 and 1 x: array_like Sorted array of the stochastic variable itself Returns ------- res: Pair with the following elements: - Maximum distance of the CDF values above Uniform(0, 1) - The location at which the maximum is reached. """ n = len(cdfvals) dminus = (cdfvals - np.arange(0.0, n)/n) amax = dminus.argmax() loc_max = x[amax] return (dminus[amax], loc_max) def _tuple_to_KstestResult(statistic, pvalue, statistic_location, statistic_sign): return KstestResult(statistic, pvalue, statistic_location=statistic_location, statistic_sign=statistic_sign) def _KstestResult_to_tuple(res): return *res, res.statistic_location, res.statistic_sign @_axis_nan_policy_factory(_tuple_to_KstestResult, n_samples=1, n_outputs=4, result_to_tuple=_KstestResult_to_tuple) @_rename_parameter("mode", "method") def ks_1samp(x, cdf, args=(), alternative='two-sided', method='auto'): """ Performs the one-sample Kolmogorov-Smirnov test for goodness of fit. This test compares the underlying distribution F(x) of a sample against a given continuous distribution G(x). See Notes for a description of the available null and alternative hypotheses. Parameters ---------- x : array_like a 1-D array of observations of iid random variables. cdf : callable callable used to calculate the cdf. args : tuple, sequence, optional Distribution parameters, used with `cdf`. alternative : {'two-sided', 'less', 'greater'}, optional Defines the null and alternative hypotheses. Default is 'two-sided'. Please see explanations in the Notes below. method : {'auto', 'exact', 'approx', 'asymp'}, optional Defines the distribution used for calculating the p-value. The following options are available (default is 'auto'): * 'auto' : selects one of the other options. * 'exact' : uses the exact distribution of test statistic. * 'approx' : approximates the two-sided probability with twice the one-sided probability * 'asymp': uses asymptotic distribution of test statistic Returns ------- res: KstestResult An object containing attributes: statistic : float KS test statistic, either D+, D-, or D (the maximum of the two) pvalue : float One-tailed or two-tailed p-value. statistic_location : float Value of `x` corresponding with the KS statistic; i.e., the distance between the empirical distribution function and the hypothesized cumulative distribution function is measured at this observation. statistic_sign : int +1 if the KS statistic is the maximum positive difference between the empirical distribution function and the hypothesized cumulative distribution function (D+); -1 if the KS statistic is the maximum negative difference (D-). See Also -------- ks_2samp, kstest Notes ----- There are three options for the null and corresponding alternative hypothesis that can be selected using the `alternative` parameter. - `two-sided`: The null hypothesis is that the two distributions are identical, F(x)=G(x) for all x; the alternative is that they are not identical. - `less`: The null hypothesis is that F(x) >= G(x) for all x; the alternative is that F(x) < G(x) for at least one x. - `greater`: The null hypothesis is that F(x) <= G(x) for all x; the alternative is that F(x) > G(x) for at least one x. Note that the alternative hypotheses describe the *CDFs* of the underlying distributions, not the observed values. For example, suppose x1 ~ F and x2 ~ G. If F(x) > G(x) for all x, the values in x1 tend to be less than those in x2. Examples -------- Suppose we wish to test the null hypothesis that a sample is distributed according to the standard normal. We choose a confidence level of 95%; that is, we will reject the null hypothesis in favor of the alternative if the p-value is less than 0.05. When testing uniformly distributed data, we would expect the null hypothesis to be rejected. >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng() >>> stats.ks_1samp(stats.uniform.rvs(size=100, random_state=rng), ... stats.norm.cdf) KstestResult(statistic=0.5001899973268688, pvalue=1.1616392184763533e-23, statistic_location=0.00047625268963724654, statistic_sign=-1) Indeed, the p-value is lower than our threshold of 0.05, so we reject the null hypothesis in favor of the default "two-sided" alternative: the data are *not* distributed according to the standard normal. When testing random variates from the standard normal distribution, we expect the data to be consistent with the null hypothesis most of the time. >>> x = stats.norm.rvs(size=100, random_state=rng) >>> stats.ks_1samp(x, stats.norm.cdf) KstestResult(statistic=0.05345882212970396, pvalue=0.9227159037744717, statistic_location=-1.2451343873745018, statistic_sign=1) As expected, the p-value of 0.92 is not below our threshold of 0.05, so we cannot reject the null hypothesis. Suppose, however, that the random variates are distributed according to a normal distribution that is shifted toward greater values. In this case, the cumulative density function (CDF) of the underlying distribution tends to be *less* than the CDF of the standard normal. Therefore, we would expect the null hypothesis to be rejected with ``alternative='less'``: >>> x = stats.norm.rvs(size=100, loc=0.5, random_state=rng) >>> stats.ks_1samp(x, stats.norm.cdf, alternative='less') KstestResult(statistic=0.17482387821055168, pvalue=0.001913921057766743, statistic_location=0.3713830565352756, statistic_sign=-1) and indeed, with p-value smaller than our threshold, we reject the null hypothesis in favor of the alternative. """ mode = method alternative = {'t': 'two-sided', 'g': 'greater', 'l': 'less'}.get( alternative.lower()[0], alternative) if alternative not in ['two-sided', 'greater', 'less']: raise ValueError(f"Unexpected value {alternative=}") N = len(x) x = np.sort(x) cdfvals = cdf(x, *args) np_one = np.int8(1) if alternative == 'greater': Dplus, d_location = _compute_dplus(cdfvals, x) return KstestResult(Dplus, distributions.ksone.sf(Dplus, N), statistic_location=d_location, statistic_sign=np_one) if alternative == 'less': Dminus, d_location = _compute_dminus(cdfvals, x) return KstestResult(Dminus, distributions.ksone.sf(Dminus, N), statistic_location=d_location, statistic_sign=-np_one) # alternative == 'two-sided': Dplus, dplus_location = _compute_dplus(cdfvals, x) Dminus, dminus_location = _compute_dminus(cdfvals, x) if Dplus > Dminus: D = Dplus d_location = dplus_location d_sign = np_one else: D = Dminus d_location = dminus_location d_sign = -np_one if mode == 'auto': # Always select exact mode = 'exact' if mode == 'exact': prob = distributions.kstwo.sf(D, N) elif mode == 'asymp': prob = distributions.kstwobign.sf(D * np.sqrt(N)) else: # mode == 'approx' prob = 2 * distributions.ksone.sf(D, N) prob = np.clip(prob, 0, 1) return KstestResult(D, prob, statistic_location=d_location, statistic_sign=d_sign) Ks_2sampResult = KstestResult def _compute_prob_outside_square(n, h): """ Compute the proportion of paths that pass outside the two diagonal lines. Parameters ---------- n : integer n > 0 h : integer 0 <= h <= n Returns ------- p : float The proportion of paths that pass outside the lines x-y = +/-h. """ # Compute Pr(D_{n,n} >= h/n) # Prob = 2 * ( binom(2n, n-h) - binom(2n, n-2a) + binom(2n, n-3a) - ... ) # / binom(2n, n) # This formulation exhibits subtractive cancellation. # Instead divide each term by binom(2n, n), then factor common terms # and use a Horner-like algorithm # P = 2 * A0 * (1 - A1*(1 - A2*(1 - A3*(1 - A4*(...))))) P = 0.0 k = int(np.floor(n / h)) while k >= 0: p1 = 1.0 # Each of the Ai terms has numerator and denominator with # h simple terms. for j in range(h): p1 = (n - k * h - j) * p1 / (n + k * h + j + 1) P = p1 * (1.0 - P) k -= 1 return 2 * P def _count_paths_outside_method(m, n, g, h): """Count the number of paths that pass outside the specified diagonal. Parameters ---------- m : integer m > 0 n : integer n > 0 g : integer g is greatest common divisor of m and n h : integer 0 <= h <= lcm(m,n) Returns ------- p : float The number of paths that go low. The calculation may overflow - check for a finite answer. Notes ----- Count the integer lattice paths from (0, 0) to (m, n), which at some point (x, y) along the path, satisfy: m*y <= n*x - h*g The paths make steps of size +1 in either positive x or positive y directions. We generally follow Hodges' treatment of Drion/Gnedenko/Korolyuk. Hodges, J.L. Jr., "The Significance Probability of the Smirnov Two-Sample Test," Arkiv fiur Matematik, 3, No. 43 (1958), 469-86. """ # Compute #paths which stay lower than x/m-y/n = h/lcm(m,n) # B(x, y) = #{paths from (0,0) to (x,y) without # previously crossing the boundary} # = binom(x, y) - #{paths which already reached the boundary} # Multiply by the number of path extensions going from (x, y) to (m, n) # Sum. # Probability is symmetrical in m, n. Computation below assumes m >= n. if m < n: m, n = n, m mg = m // g ng = n // g # Not every x needs to be considered. # xj holds the list of x values to be checked. # Wherever n*x/m + ng*h crosses an integer lxj = n + (mg-h)//mg xj = [(h + mg * j + ng-1)//ng for j in range(lxj)] # B is an array just holding a few values of B(x,y), the ones needed. # B[j] == B(x_j, j) if lxj == 0: return special.binom(m + n, n) B = np.zeros(lxj) B[0] = 1 # Compute the B(x, y) terms for j in range(1, lxj): Bj = special.binom(xj[j] + j, j) for i in range(j): bin = special.binom(xj[j] - xj[i] + j - i, j-i) Bj -= bin * B[i] B[j] = Bj # Compute the number of path extensions... num_paths = 0 for j in range(lxj): bin = special.binom((m-xj[j]) + (n - j), n-j) term = B[j] * bin num_paths += term return num_paths def _attempt_exact_2kssamp(n1, n2, g, d, alternative): """Attempts to compute the exact 2sample probability. n1, n2 are the sample sizes g is the gcd(n1, n2) d is the computed max difference in ECDFs Returns (success, d, probability) """ lcm = (n1 // g) * n2 h = int(np.round(d * lcm)) d = h * 1.0 / lcm if h == 0: return True, d, 1.0 saw_fp_error, prob = False, np.nan try: with np.errstate(invalid="raise", over="raise"): if alternative == 'two-sided': if n1 == n2: prob = _compute_prob_outside_square(n1, h) else: prob = _compute_outer_prob_inside_method(n1, n2, g, h) else: if n1 == n2: # prob = binom(2n, n-h) / binom(2n, n) # Evaluating in that form incurs roundoff errors # from special.binom. Instead calculate directly jrange = np.arange(h) prob = np.prod((n1 - jrange) / (n1 + jrange + 1.0)) else: with np.errstate(over='raise'): num_paths = _count_paths_outside_method(n1, n2, g, h) bin = special.binom(n1 + n2, n1) if num_paths > bin or np.isinf(bin): saw_fp_error = True else: prob = num_paths / bin except (FloatingPointError, OverflowError): saw_fp_error = True if saw_fp_error: return False, d, np.nan if not (0 <= prob <= 1): return False, d, prob return True, d, prob @_axis_nan_policy_factory(_tuple_to_KstestResult, n_samples=2, n_outputs=4, result_to_tuple=_KstestResult_to_tuple) @_rename_parameter("mode", "method") def ks_2samp(data1, data2, alternative='two-sided', method='auto'): """ Performs the two-sample Kolmogorov-Smirnov test for goodness of fit. This test compares the underlying continuous distributions F(x) and G(x) of two independent samples. See Notes for a description of the available null and alternative hypotheses. Parameters ---------- data1, data2 : array_like, 1-Dimensional Two arrays of sample observations assumed to be drawn from a continuous distribution, sample sizes can be different. alternative : {'two-sided', 'less', 'greater'}, optional Defines the null and alternative hypotheses. Default is 'two-sided'. Please see explanations in the Notes below. method : {'auto', 'exact', 'asymp'}, optional Defines the method used for calculating the p-value. The following options are available (default is 'auto'): * 'auto' : use 'exact' for small size arrays, 'asymp' for large * 'exact' : use exact distribution of test statistic * 'asymp' : use asymptotic distribution of test statistic Returns ------- res: KstestResult An object containing attributes: statistic : float KS test statistic. pvalue : float One-tailed or two-tailed p-value. statistic_location : float Value from `data1` or `data2` corresponding with the KS statistic; i.e., the distance between the empirical distribution functions is measured at this observation. statistic_sign : int +1 if the empirical distribution function of `data1` exceeds the empirical distribution function of `data2` at `statistic_location`, otherwise -1. See Also -------- kstest, ks_1samp, epps_singleton_2samp, anderson_ksamp Notes ----- There are three options for the null and corresponding alternative hypothesis that can be selected using the `alternative` parameter. - `less`: The null hypothesis is that F(x) >= G(x) for all x; the alternative is that F(x) < G(x) for at least one x. The statistic is the magnitude of the minimum (most negative) difference between the empirical distribution functions of the samples. - `greater`: The null hypothesis is that F(x) <= G(x) for all x; the alternative is that F(x) > G(x) for at least one x. The statistic is the maximum (most positive) difference between the empirical distribution functions of the samples. - `two-sided`: The null hypothesis is that the two distributions are identical, F(x)=G(x) for all x; the alternative is that they are not identical. The statistic is the maximum absolute difference between the empirical distribution functions of the samples. Note that the alternative hypotheses describe the *CDFs* of the underlying distributions, not the observed values of the data. For example, suppose x1 ~ F and x2 ~ G. If F(x) > G(x) for all x, the values in x1 tend to be less than those in x2. If the KS statistic is large, then the p-value will be small, and this may be taken as evidence against the null hypothesis in favor of the alternative. If ``method='exact'``, `ks_2samp` attempts to compute an exact p-value, that is, the probability under the null hypothesis of obtaining a test statistic value as extreme as the value computed from the data. If ``method='asymp'``, the asymptotic Kolmogorov-Smirnov distribution is used to compute an approximate p-value. If ``method='auto'``, an exact p-value computation is attempted if both sample sizes are less than 10000; otherwise, the asymptotic method is used. In any case, if an exact p-value calculation is attempted and fails, a warning will be emitted, and the asymptotic p-value will be returned. The 'two-sided' 'exact' computation computes the complementary probability and then subtracts from 1. As such, the minimum probability it can return is about 1e-16. While the algorithm itself is exact, numerical errors may accumulate for large sample sizes. It is most suited to situations in which one of the sample sizes is only a few thousand. We generally follow Hodges' treatment of Drion/Gnedenko/Korolyuk [1]_. References ---------- .. [1] Hodges, J.L. Jr., "The Significance Probability of the Smirnov Two-Sample Test," Arkiv fiur Matematik, 3, No. 43 (1958), 469-486. Examples -------- Suppose we wish to test the null hypothesis that two samples were drawn from the same distribution. We choose a confidence level of 95%; that is, we will reject the null hypothesis in favor of the alternative if the p-value is less than 0.05. If the first sample were drawn from a uniform distribution and the second were drawn from the standard normal, we would expect the null hypothesis to be rejected. >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng() >>> sample1 = stats.uniform.rvs(size=100, random_state=rng) >>> sample2 = stats.norm.rvs(size=110, random_state=rng) >>> stats.ks_2samp(sample1, sample2) KstestResult(statistic=0.5454545454545454, pvalue=7.37417839555191e-15, statistic_location=-0.014071496412861274, statistic_sign=-1) Indeed, the p-value is lower than our threshold of 0.05, so we reject the null hypothesis in favor of the default "two-sided" alternative: the data were *not* drawn from the same distribution. When both samples are drawn from the same distribution, we expect the data to be consistent with the null hypothesis most of the time. >>> sample1 = stats.norm.rvs(size=105, random_state=rng) >>> sample2 = stats.norm.rvs(size=95, random_state=rng) >>> stats.ks_2samp(sample1, sample2) KstestResult(statistic=0.10927318295739348, pvalue=0.5438289009927495, statistic_location=-0.1670157701848795, statistic_sign=-1) As expected, the p-value of 0.54 is not below our threshold of 0.05, so we cannot reject the null hypothesis. Suppose, however, that the first sample were drawn from a normal distribution shifted toward greater values. In this case, the cumulative density function (CDF) of the underlying distribution tends to be *less* than the CDF underlying the second sample. Therefore, we would expect the null hypothesis to be rejected with ``alternative='less'``: >>> sample1 = stats.norm.rvs(size=105, loc=0.5, random_state=rng) >>> stats.ks_2samp(sample1, sample2, alternative='less') KstestResult(statistic=0.4055137844611529, pvalue=3.5474563068855554e-08, statistic_location=-0.13249370614972575, statistic_sign=-1) and indeed, with p-value smaller than our threshold, we reject the null hypothesis in favor of the alternative. """ mode = method if mode not in ['auto', 'exact', 'asymp']: raise ValueError(f'Invalid value for mode: {mode}') alternative = {'t': 'two-sided', 'g': 'greater', 'l': 'less'}.get( alternative.lower()[0], alternative) if alternative not in ['two-sided', 'less', 'greater']: raise ValueError(f'Invalid value for alternative: {alternative}') MAX_AUTO_N = 10000 # 'auto' will attempt to be exact if n1,n2 <= MAX_AUTO_N if np.ma.is_masked(data1): data1 = data1.compressed() if np.ma.is_masked(data2): data2 = data2.compressed() data1 = np.sort(data1) data2 = np.sort(data2) n1 = data1.shape[0] n2 = data2.shape[0] if min(n1, n2) == 0: raise ValueError('Data passed to ks_2samp must not be empty') data_all = np.concatenate([data1, data2]) # using searchsorted solves equal data problem cdf1 = np.searchsorted(data1, data_all, side='right') / n1 cdf2 = np.searchsorted(data2, data_all, side='right') / n2 cddiffs = cdf1 - cdf2 # Identify the location of the statistic argminS = np.argmin(cddiffs) argmaxS = np.argmax(cddiffs) loc_minS = data_all[argminS] loc_maxS = data_all[argmaxS] # Ensure sign of minS is not negative. minS = np.clip(-cddiffs[argminS], 0, 1) maxS = cddiffs[argmaxS] if alternative == 'less' or (alternative == 'two-sided' and minS > maxS): d = minS d_location = loc_minS d_sign = -1 else: d = maxS d_location = loc_maxS d_sign = 1 g = gcd(n1, n2) n1g = n1 // g n2g = n2 // g prob = -np.inf if mode == 'auto': mode = 'exact' if max(n1, n2) <= MAX_AUTO_N else 'asymp' elif mode == 'exact': # If lcm(n1, n2) is too big, switch from exact to asymp if n1g >= np.iinfo(np.int32).max / n2g: mode = 'asymp' warnings.warn( f"Exact ks_2samp calculation not possible with samples sizes " f"{n1} and {n2}. Switching to 'asymp'.", RuntimeWarning, stacklevel=3) if mode == 'exact': success, d, prob = _attempt_exact_2kssamp(n1, n2, g, d, alternative) if not success: mode = 'asymp' warnings.warn(f"ks_2samp: Exact calculation unsuccessful. " f"Switching to method={mode}.", RuntimeWarning, stacklevel=3) if mode == 'asymp': # The product n1*n2 is large. Use Smirnov's asymptoptic formula. # Ensure float to avoid overflow in multiplication # sorted because the one-sided formula is not symmetric in n1, n2 m, n = sorted([float(n1), float(n2)], reverse=True) en = m * n / (m + n) if alternative == 'two-sided': prob = distributions.kstwo.sf(d, np.round(en)) else: z = np.sqrt(en) * d # Use Hodges' suggested approximation Eqn 5.3 # Requires m to be the larger of (n1, n2) expt = -2 * z**2 - 2 * z * (m + 2*n)/np.sqrt(m*n*(m+n))/3.0 prob = np.exp(expt) prob = np.clip(prob, 0, 1) # Currently, `d` is a Python float. We want it to be a NumPy type, so # float64 is appropriate. An enhancement would be for `d` to respect the # dtype of the input. return KstestResult(np.float64(d), prob, statistic_location=d_location, statistic_sign=np.int8(d_sign)) def _parse_kstest_args(data1, data2, args, N): # kstest allows many different variations of arguments. # Pull out the parsing into a separate function # (xvals, yvals, ) # 2sample # (xvals, cdf function,..) # (xvals, name of distribution, ...) # (name of distribution, name of distribution, ...) # Returns xvals, yvals, cdf # where cdf is a cdf function, or None # and yvals is either an array_like of values, or None # and xvals is array_like. rvsfunc, cdf = None, None if isinstance(data1, str): rvsfunc = getattr(distributions, data1).rvs elif callable(data1): rvsfunc = data1 if isinstance(data2, str): cdf = getattr(distributions, data2).cdf data2 = None elif callable(data2): cdf = data2 data2 = None data1 = np.sort(rvsfunc(*args, size=N) if rvsfunc else data1) return data1, data2, cdf def _kstest_n_samples(kwargs): cdf = kwargs['cdf'] return 1 if (isinstance(cdf, str) or callable(cdf)) else 2 @_axis_nan_policy_factory(_tuple_to_KstestResult, n_samples=_kstest_n_samples, n_outputs=4, result_to_tuple=_KstestResult_to_tuple) @_rename_parameter("mode", "method") def kstest(rvs, cdf, args=(), N=20, alternative='two-sided', method='auto'): """ Performs the (one-sample or two-sample) Kolmogorov-Smirnov test for goodness of fit. The one-sample test compares the underlying distribution F(x) of a sample against a given distribution G(x). The two-sample test compares the underlying distributions of two independent samples. Both tests are valid only for continuous distributions. Parameters ---------- rvs : str, array_like, or callable If an array, it should be a 1-D array of observations of random variables. If a callable, it should be a function to generate random variables; it is required to have a keyword argument `size`. If a string, it should be the name of a distribution in `scipy.stats`, which will be used to generate random variables. cdf : str, array_like or callable If array_like, it should be a 1-D array of observations of random variables, and the two-sample test is performed (and rvs must be array_like). If a callable, that callable is used to calculate the cdf. If a string, it should be the name of a distribution in `scipy.stats`, which will be used as the cdf function. args : tuple, sequence, optional Distribution parameters, used if `rvs` or `cdf` are strings or callables. N : int, optional Sample size if `rvs` is string or callable. Default is 20. alternative : {'two-sided', 'less', 'greater'}, optional Defines the null and alternative hypotheses. Default is 'two-sided'. Please see explanations in the Notes below. method : {'auto', 'exact', 'approx', 'asymp'}, optional Defines the distribution used for calculating the p-value. The following options are available (default is 'auto'): * 'auto' : selects one of the other options. * 'exact' : uses the exact distribution of test statistic. * 'approx' : approximates the two-sided probability with twice the one-sided probability * 'asymp': uses asymptotic distribution of test statistic Returns ------- res: KstestResult An object containing attributes: statistic : float KS test statistic, either D+, D-, or D (the maximum of the two) pvalue : float One-tailed or two-tailed p-value. statistic_location : float In a one-sample test, this is the value of `rvs` corresponding with the KS statistic; i.e., the distance between the empirical distribution function and the hypothesized cumulative distribution function is measured at this observation. In a two-sample test, this is the value from `rvs` or `cdf` corresponding with the KS statistic; i.e., the distance between the empirical distribution functions is measured at this observation. statistic_sign : int In a one-sample test, this is +1 if the KS statistic is the maximum positive difference between the empirical distribution function and the hypothesized cumulative distribution function (D+); it is -1 if the KS statistic is the maximum negative difference (D-). In a two-sample test, this is +1 if the empirical distribution function of `rvs` exceeds the empirical distribution function of `cdf` at `statistic_location`, otherwise -1. See Also -------- ks_1samp, ks_2samp Notes ----- There are three options for the null and corresponding alternative hypothesis that can be selected using the `alternative` parameter. - `two-sided`: The null hypothesis is that the two distributions are identical, F(x)=G(x) for all x; the alternative is that they are not identical. - `less`: The null hypothesis is that F(x) >= G(x) for all x; the alternative is that F(x) < G(x) for at least one x. - `greater`: The null hypothesis is that F(x) <= G(x) for all x; the alternative is that F(x) > G(x) for at least one x. Note that the alternative hypotheses describe the *CDFs* of the underlying distributions, not the observed values. For example, suppose x1 ~ F and x2 ~ G. If F(x) > G(x) for all x, the values in x1 tend to be less than those in x2. Examples -------- Suppose we wish to test the null hypothesis that a sample is distributed according to the standard normal. We choose a confidence level of 95%; that is, we will reject the null hypothesis in favor of the alternative if the p-value is less than 0.05. When testing uniformly distributed data, we would expect the null hypothesis to be rejected. >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng() >>> stats.kstest(stats.uniform.rvs(size=100, random_state=rng), ... stats.norm.cdf) KstestResult(statistic=0.5001899973268688, pvalue=1.1616392184763533e-23, statistic_location=0.00047625268963724654, statistic_sign=-1) Indeed, the p-value is lower than our threshold of 0.05, so we reject the null hypothesis in favor of the default "two-sided" alternative: the data are *not* distributed according to the standard normal. When testing random variates from the standard normal distribution, we expect the data to be consistent with the null hypothesis most of the time. >>> x = stats.norm.rvs(size=100, random_state=rng) >>> stats.kstest(x, stats.norm.cdf) KstestResult(statistic=0.05345882212970396, pvalue=0.9227159037744717, statistic_location=-1.2451343873745018, statistic_sign=1) As expected, the p-value of 0.92 is not below our threshold of 0.05, so we cannot reject the null hypothesis. Suppose, however, that the random variates are distributed according to a normal distribution that is shifted toward greater values. In this case, the cumulative density function (CDF) of the underlying distribution tends to be *less* than the CDF of the standard normal. Therefore, we would expect the null hypothesis to be rejected with ``alternative='less'``: >>> x = stats.norm.rvs(size=100, loc=0.5, random_state=rng) >>> stats.kstest(x, stats.norm.cdf, alternative='less') KstestResult(statistic=0.17482387821055168, pvalue=0.001913921057766743, statistic_location=0.3713830565352756, statistic_sign=-1) and indeed, with p-value smaller than our threshold, we reject the null hypothesis in favor of the alternative. For convenience, the previous test can be performed using the name of the distribution as the second argument. >>> stats.kstest(x, "norm", alternative='less') KstestResult(statistic=0.17482387821055168, pvalue=0.001913921057766743, statistic_location=0.3713830565352756, statistic_sign=-1) The examples above have all been one-sample tests identical to those performed by `ks_1samp`. Note that `kstest` can also perform two-sample tests identical to those performed by `ks_2samp`. For example, when two samples are drawn from the same distribution, we expect the data to be consistent with the null hypothesis most of the time. >>> sample1 = stats.laplace.rvs(size=105, random_state=rng) >>> sample2 = stats.laplace.rvs(size=95, random_state=rng) >>> stats.kstest(sample1, sample2) KstestResult(statistic=0.11779448621553884, pvalue=0.4494256912629795, statistic_location=0.6138814275424155, statistic_sign=1) As expected, the p-value of 0.45 is not below our threshold of 0.05, so we cannot reject the null hypothesis. """ # to not break compatibility with existing code if alternative == 'two_sided': alternative = 'two-sided' if alternative not in ['two-sided', 'greater', 'less']: raise ValueError(f"Unexpected alternative: {alternative}") xvals, yvals, cdf = _parse_kstest_args(rvs, cdf, args, N) if cdf: return ks_1samp(xvals, cdf, args=args, alternative=alternative, method=method, _no_deco=True) return ks_2samp(xvals, yvals, alternative=alternative, method=method, _no_deco=True) def tiecorrect(rankvals): """Tie correction factor for Mann-Whitney U and Kruskal-Wallis H tests. Parameters ---------- rankvals : array_like A 1-D sequence of ranks. Typically this will be the array returned by `~scipy.stats.rankdata`. Returns ------- factor : float Correction factor for U or H. See Also -------- rankdata : Assign ranks to the data mannwhitneyu : Mann-Whitney rank test kruskal : Kruskal-Wallis H test References ---------- .. [1] Siegel, S. (1956) Nonparametric Statistics for the Behavioral Sciences. New York: McGraw-Hill. Examples -------- >>> from scipy.stats import tiecorrect, rankdata >>> tiecorrect([1, 2.5, 2.5, 4]) 0.9 >>> ranks = rankdata([1, 3, 2, 4, 5, 7, 2, 8, 4]) >>> ranks array([ 1. , 4. , 2.5, 5.5, 7. , 8. , 2.5, 9. , 5.5]) >>> tiecorrect(ranks) 0.9833333333333333 """ arr = np.sort(rankvals) idx = np.nonzero(np.r_[True, arr[1:] != arr[:-1], True])[0] cnt = np.diff(idx).astype(np.float64) size = np.float64(arr.size) return 1.0 if size < 2 else 1.0 - (cnt**3 - cnt).sum() / (size**3 - size) RanksumsResult = namedtuple('RanksumsResult', ('statistic', 'pvalue')) @_axis_nan_policy_factory(RanksumsResult, n_samples=2) def ranksums(x, y, alternative='two-sided'): """Compute the Wilcoxon rank-sum statistic for two samples. The Wilcoxon rank-sum test tests the null hypothesis that two sets of measurements are drawn from the same distribution. The alternative hypothesis is that values in one sample are more likely to be larger than the values in the other sample. This test should be used to compare two samples from continuous distributions. It does not handle ties between measurements in x and y. For tie-handling and an optional continuity correction see `scipy.stats.mannwhitneyu`. Parameters ---------- x,y : array_like The data from the two samples. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. Default is 'two-sided'. The following options are available: * 'two-sided': one of the distributions (underlying `x` or `y`) is stochastically greater than the other. * 'less': the distribution underlying `x` is stochastically less than the distribution underlying `y`. * 'greater': the distribution underlying `x` is stochastically greater than the distribution underlying `y`. .. versionadded:: 1.7.0 Returns ------- statistic : float The test statistic under the large-sample approximation that the rank sum statistic is normally distributed. pvalue : float The p-value of the test. References ---------- .. [1] https://en.wikipedia.org/wiki/Wilcoxon_rank-sum_test Examples -------- We can test the hypothesis that two independent unequal-sized samples are drawn from the same distribution with computing the Wilcoxon rank-sum statistic. >>> import numpy as np >>> from scipy.stats import ranksums >>> rng = np.random.default_rng() >>> sample1 = rng.uniform(-1, 1, 200) >>> sample2 = rng.uniform(-0.5, 1.5, 300) # a shifted distribution >>> ranksums(sample1, sample2) RanksumsResult(statistic=-7.887059, pvalue=3.09390448e-15) # may vary >>> ranksums(sample1, sample2, alternative='less') RanksumsResult(statistic=-7.750585297581713, pvalue=4.573497606342543e-15) # may vary >>> ranksums(sample1, sample2, alternative='greater') RanksumsResult(statistic=-7.750585297581713, pvalue=0.9999999999999954) # may vary The p-value of less than ``0.05`` indicates that this test rejects the hypothesis at the 5% significance level. """ x, y = map(np.asarray, (x, y)) n1 = len(x) n2 = len(y) alldata = np.concatenate((x, y)) ranked = rankdata(alldata) x = ranked[:n1] s = np.sum(x, axis=0) expected = n1 * (n1+n2+1) / 2.0 z = (s - expected) / np.sqrt(n1*n2*(n1+n2+1)/12.0) pvalue = _get_pvalue(z, _SimpleNormal(), alternative, xp=np) return RanksumsResult(z[()], pvalue[()]) KruskalResult = namedtuple('KruskalResult', ('statistic', 'pvalue')) @_axis_nan_policy_factory(KruskalResult, n_samples=None) def kruskal(*samples, nan_policy='propagate'): """Compute the Kruskal-Wallis H-test for independent samples. The Kruskal-Wallis H-test tests the null hypothesis that the population median of all of the groups are equal. It is a non-parametric version of ANOVA. The test works on 2 or more independent samples, which may have different sizes. Note that rejecting the null hypothesis does not indicate which of the groups differs. Post hoc comparisons between groups are required to determine which groups are different. Parameters ---------- sample1, sample2, ... : array_like Two or more arrays with the sample measurements can be given as arguments. Samples must be one-dimensional. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float The Kruskal-Wallis H statistic, corrected for ties. pvalue : float The p-value for the test using the assumption that H has a chi square distribution. The p-value returned is the survival function of the chi square distribution evaluated at H. See Also -------- f_oneway : 1-way ANOVA. mannwhitneyu : Mann-Whitney rank test on two samples. friedmanchisquare : Friedman test for repeated measurements. Notes ----- Due to the assumption that H has a chi square distribution, the number of samples in each group must not be too small. A typical rule is that each sample must have at least 5 measurements. References ---------- .. [1] W. H. Kruskal & W. W. Wallis, "Use of Ranks in One-Criterion Variance Analysis", Journal of the American Statistical Association, Vol. 47, Issue 260, pp. 583-621, 1952. .. [2] https://en.wikipedia.org/wiki/Kruskal-Wallis_one-way_analysis_of_variance Examples -------- >>> from scipy import stats >>> x = [1, 3, 5, 7, 9] >>> y = [2, 4, 6, 8, 10] >>> stats.kruskal(x, y) KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895) >>> x = [1, 1, 1] >>> y = [2, 2, 2] >>> z = [2, 2] >>> stats.kruskal(x, y, z) KruskalResult(statistic=7.0, pvalue=0.0301973834223185) """ samples = list(map(np.asarray, samples)) num_groups = len(samples) if num_groups < 2: raise ValueError("Need at least two groups in stats.kruskal()") n = np.asarray(list(map(len, samples))) alldata = np.concatenate(samples) ranked = rankdata(alldata) ties = tiecorrect(ranked) if ties == 0: raise ValueError('All numbers are identical in kruskal') # Compute sum^2/n for each group and sum j = np.insert(np.cumsum(n), 0, 0) ssbn = 0 for i in range(num_groups): ssbn += _square_of_sums(ranked[j[i]:j[i+1]]) / n[i] totaln = np.sum(n, dtype=float) h = 12.0 / (totaln * (totaln + 1)) * ssbn - 3 * (totaln + 1) df = num_groups - 1 h /= ties chi2 = _SimpleChi2(df) pvalue = _get_pvalue(h, chi2, alternative='greater', symmetric=False, xp=np) return KruskalResult(h, pvalue) FriedmanchisquareResult = namedtuple('FriedmanchisquareResult', ('statistic', 'pvalue')) @_axis_nan_policy_factory(FriedmanchisquareResult, n_samples=None, paired=True) def friedmanchisquare(*samples): """Compute the Friedman test for repeated samples. The Friedman test tests the null hypothesis that repeated samples of the same individuals have the same distribution. It is often used to test for consistency among samples obtained in different ways. For example, if two sampling techniques are used on the same set of individuals, the Friedman test can be used to determine if the two sampling techniques are consistent. Parameters ---------- sample1, sample2, sample3... : array_like Arrays of observations. All of the arrays must have the same number of elements. At least three samples must be given. Returns ------- statistic : float The test statistic, correcting for ties. pvalue : float The associated p-value assuming that the test statistic has a chi squared distribution. Notes ----- Due to the assumption that the test statistic has a chi squared distribution, the p-value is only reliable for n > 10 and more than 6 repeated samples. References ---------- .. [1] https://en.wikipedia.org/wiki/Friedman_test .. [2] P. Sprent and N.C. Smeeton, "Applied Nonparametric Statistical Methods, Third Edition". Chapter 6, Section 6.3.2. Examples -------- In [2]_, the pulse rate (per minute) of a group of seven students was measured before exercise, immediately after exercise and 5 minutes after exercise. Is there evidence to suggest that the pulse rates on these three occasions are similar? We begin by formulating a null hypothesis :math:`H_0`: The pulse rates are identical on these three occasions. Let's assess the plausibility of this hypothesis with a Friedman test. >>> from scipy.stats import friedmanchisquare >>> before = [72, 96, 88, 92, 74, 76, 82] >>> immediately_after = [120, 120, 132, 120, 101, 96, 112] >>> five_min_after = [76, 95, 104, 96, 84, 72, 76] >>> res = friedmanchisquare(before, immediately_after, five_min_after) >>> res.statistic 10.57142857142857 >>> res.pvalue 0.005063414171757498 Using a significance level of 5%, we would reject the null hypothesis in favor of the alternative hypothesis: "the pulse rates are different on these three occasions". """ k = len(samples) if k < 3: raise ValueError('At least 3 sets of samples must be given ' f'for Friedman test, got {k}.') n = len(samples[0]) for i in range(1, k): if len(samples[i]) != n: raise ValueError('Unequal N in friedmanchisquare. Aborting.') # Rank data data = np.vstack(samples).T data = data.astype(float) for i in range(len(data)): data[i] = rankdata(data[i]) # Handle ties ties = 0 for d in data: replist, repnum = find_repeats(array(d)) for t in repnum: ties += t * (t*t - 1) c = 1 - ties / (k*(k*k - 1)*n) ssbn = np.sum(data.sum(axis=0)**2) statistic = (12.0 / (k*n*(k+1)) * ssbn - 3*n*(k+1)) / c chi2 = _SimpleChi2(k - 1) pvalue = _get_pvalue(statistic, chi2, alternative='greater', symmetric=False, xp=np) return FriedmanchisquareResult(statistic, pvalue) BrunnerMunzelResult = namedtuple('BrunnerMunzelResult', ('statistic', 'pvalue')) @_axis_nan_policy_factory(BrunnerMunzelResult, n_samples=2) def brunnermunzel(x, y, alternative="two-sided", distribution="t", nan_policy='propagate'): """Compute the Brunner-Munzel test on samples x and y. The Brunner-Munzel test is a nonparametric test of the null hypothesis that when values are taken one by one from each group, the probabilities of getting large values in both groups are equal. Unlike the Wilcoxon-Mann-Whitney's U test, this does not require the assumption of equivariance of two groups. Note that this does not assume the distributions are same. This test works on two independent samples, which may have different sizes. Parameters ---------- x, y : array_like Array of samples, should be one-dimensional. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided * 'greater': one-sided distribution : {'t', 'normal'}, optional Defines how to get the p-value. The following options are available (default is 't'): * 't': get the p-value by t-distribution * 'normal': get the p-value by standard normal distribution. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float The Brunner-Munzer W statistic. pvalue : float p-value assuming an t distribution. One-sided or two-sided, depending on the choice of `alternative` and `distribution`. See Also -------- mannwhitneyu : Mann-Whitney rank test on two samples. Notes ----- Brunner and Munzel recommended to estimate the p-value by t-distribution when the size of data is 50 or less. If the size is lower than 10, it would be better to use permuted Brunner Munzel test (see [2]_). References ---------- .. [1] Brunner, E. and Munzel, U. "The nonparametric Benhrens-Fisher problem: Asymptotic theory and a small-sample approximation". Biometrical Journal. Vol. 42(2000): 17-25. .. [2] Neubert, K. and Brunner, E. "A studentized permutation test for the non-parametric Behrens-Fisher problem". Computational Statistics and Data Analysis. Vol. 51(2007): 5192-5204. Examples -------- >>> from scipy import stats >>> x1 = [1,2,1,1,1,1,1,1,1,1,2,4,1,1] >>> x2 = [3,3,4,3,1,2,3,1,1,5,4] >>> w, p_value = stats.brunnermunzel(x1, x2) >>> w 3.1374674823029505 >>> p_value 0.0057862086661515377 """ nx = len(x) ny = len(y) rankc = rankdata(np.concatenate((x, y))) rankcx = rankc[0:nx] rankcy = rankc[nx:nx+ny] rankcx_mean = np.mean(rankcx) rankcy_mean = np.mean(rankcy) rankx = rankdata(x) ranky = rankdata(y) rankx_mean = np.mean(rankx) ranky_mean = np.mean(ranky) Sx = np.sum(np.power(rankcx - rankx - rankcx_mean + rankx_mean, 2.0)) Sx /= nx - 1 Sy = np.sum(np.power(rankcy - ranky - rankcy_mean + ranky_mean, 2.0)) Sy /= ny - 1 wbfn = nx * ny * (rankcy_mean - rankcx_mean) wbfn /= (nx + ny) * np.sqrt(nx * Sx + ny * Sy) if distribution == "t": df_numer = np.power(nx * Sx + ny * Sy, 2.0) df_denom = np.power(nx * Sx, 2.0) / (nx - 1) df_denom += np.power(ny * Sy, 2.0) / (ny - 1) df = df_numer / df_denom if (df_numer == 0) and (df_denom == 0): message = ("p-value cannot be estimated with `distribution='t' " "because degrees of freedom parameter is undefined " "(0/0). Try using `distribution='normal'") warnings.warn(message, RuntimeWarning, stacklevel=2) distribution = distributions.t(df) elif distribution == "normal": distribution = _SimpleNormal() else: raise ValueError( "distribution should be 't' or 'normal'") p = _get_pvalue(-wbfn, distribution, alternative, xp=np) return BrunnerMunzelResult(wbfn, p) @_axis_nan_policy_factory(SignificanceResult, kwd_samples=['weights'], paired=True) def combine_pvalues(pvalues, method='fisher', weights=None): """ Combine p-values from independent tests that bear upon the same hypothesis. These methods are intended only for combining p-values from hypothesis tests based upon continuous distributions. Each method assumes that under the null hypothesis, the p-values are sampled independently and uniformly from the interval [0, 1]. A test statistic (different for each method) is computed and a combined p-value is calculated based upon the distribution of this test statistic under the null hypothesis. Parameters ---------- pvalues : array_like Array of p-values assumed to come from independent tests based on continuous distributions. method : {'fisher', 'pearson', 'tippett', 'stouffer', 'mudholkar_george'} Name of method to use to combine p-values. The available methods are (see Notes for details): * 'fisher': Fisher's method (Fisher's combined probability test) * 'pearson': Pearson's method * 'mudholkar_george': Mudholkar's and George's method * 'tippett': Tippett's method * 'stouffer': Stouffer's Z-score method weights : array_like, optional Optional array of weights used only for Stouffer's Z-score method. Ignored by other methods. Returns ------- res : SignificanceResult An object containing attributes: statistic : float The statistic calculated by the specified method. pvalue : float The combined p-value. Examples -------- Suppose we wish to combine p-values from four independent tests of the same null hypothesis using Fisher's method (default). >>> from scipy.stats import combine_pvalues >>> pvalues = [0.1, 0.05, 0.02, 0.3] >>> combine_pvalues(pvalues) SignificanceResult(statistic=20.828626352604235, pvalue=0.007616871850449092) When the individual p-values carry different weights, consider Stouffer's method. >>> weights = [1, 2, 3, 4] >>> res = combine_pvalues(pvalues, method='stouffer', weights=weights) >>> res.pvalue 0.009578891494533616 Notes ----- If this function is applied to tests with a discrete statistics such as any rank test or contingency-table test, it will yield systematically wrong results, e.g. Fisher's method will systematically overestimate the p-value [1]_. This problem becomes less severe for large sample sizes when the discrete distributions become approximately continuous. The differences between the methods can be best illustrated by their statistics and what aspects of a combination of p-values they emphasise when considering significance [2]_. For example, methods emphasising large p-values are more sensitive to strong false and true negatives; conversely methods focussing on small p-values are sensitive to positives. * The statistics of Fisher's method (also known as Fisher's combined probability test) [3]_ is :math:`-2\\sum_i \\log(p_i)`, which is equivalent (as a test statistics) to the product of individual p-values: :math:`\\prod_i p_i`. Under the null hypothesis, this statistics follows a :math:`\\chi^2` distribution. This method emphasises small p-values. * Pearson's method uses :math:`-2\\sum_i\\log(1-p_i)`, which is equivalent to :math:`\\prod_i \\frac{1}{1-p_i}` [2]_. It thus emphasises large p-values. * Mudholkar and George compromise between Fisher's and Pearson's method by averaging their statistics [4]_. Their method emphasises extreme p-values, both close to 1 and 0. * Stouffer's method [5]_ uses Z-scores and the statistic: :math:`\\sum_i \\Phi^{-1} (p_i)`, where :math:`\\Phi` is the CDF of the standard normal distribution. The advantage of this method is that it is straightforward to introduce weights, which can make Stouffer's method more powerful than Fisher's method when the p-values are from studies of different size [6]_ [7]_. * Tippett's method uses the smallest p-value as a statistic. (Mind that this minimum is not the combined p-value.) Fisher's method may be extended to combine p-values from dependent tests [8]_. Extensions such as Brown's method and Kost's method are not currently implemented. .. versionadded:: 0.15.0 References ---------- .. [1] Kincaid, W. M., "The Combination of Tests Based on Discrete Distributions." Journal of the American Statistical Association 57, no. 297 (1962), 10-19. .. [2] Heard, N. and Rubin-Delanchey, P. "Choosing between methods of combining p-values." Biometrika 105.1 (2018): 239-246. .. [3] https://en.wikipedia.org/wiki/Fisher%27s_method .. [4] George, E. O., and G. S. Mudholkar. "On the convolution of logistic random variables." Metrika 30.1 (1983): 1-13. .. [5] https://en.wikipedia.org/wiki/Fisher%27s_method#Relation_to_Stouffer.27s_Z-score_method .. [6] Whitlock, M. C. "Combining probability from independent tests: the weighted Z-method is superior to Fisher's approach." Journal of Evolutionary Biology 18, no. 5 (2005): 1368-1373. .. [7] Zaykin, Dmitri V. "Optimally weighted Z-test is a powerful method for combining probabilities in meta-analysis." Journal of Evolutionary Biology 24, no. 8 (2011): 1836-1841. .. [8] https://en.wikipedia.org/wiki/Extensions_of_Fisher%27s_method """ if pvalues.size == 0: NaN = _get_nan(pvalues) return SignificanceResult(NaN, NaN) if method == 'fisher': statistic = -2 * np.sum(np.log(pvalues)) chi2 = _SimpleChi2(2 * len(pvalues)) pval = _get_pvalue(statistic, chi2, alternative='greater', symmetric=False, xp=np) elif method == 'pearson': statistic = 2 * np.sum(np.log1p(-pvalues)) # _SimpleChi2 doesn't have `cdf` yet; # add it when `combine_pvalues` is converted to array API pval = distributions.chi2.cdf(-statistic, 2 * len(pvalues)) elif method == 'mudholkar_george': normalizing_factor = np.sqrt(3/len(pvalues))/np.pi statistic = -np.sum(np.log(pvalues)) + np.sum(np.log1p(-pvalues)) nu = 5 * len(pvalues) + 4 approx_factor = np.sqrt(nu / (nu - 2)) pval = distributions.t.sf(statistic * normalizing_factor * approx_factor, nu) elif method == 'tippett': statistic = np.min(pvalues) pval = distributions.beta.cdf(statistic, 1, len(pvalues)) elif method == 'stouffer': if weights is None: weights = np.ones_like(pvalues) elif len(weights) != len(pvalues): raise ValueError("pvalues and weights must be of the same size.") Zi = distributions.norm.isf(pvalues) statistic = np.dot(weights, Zi) / np.linalg.norm(weights) pval = distributions.norm.sf(statistic) else: raise ValueError( f"Invalid method {method!r}. Valid methods are 'fisher', " "'pearson', 'mudholkar_george', 'tippett', and 'stouffer'" ) return SignificanceResult(statistic, pval) @dataclass class QuantileTestResult: r""" Result of `scipy.stats.quantile_test`. Attributes ---------- statistic: float The statistic used to calculate the p-value; either ``T1``, the number of observations less than or equal to the hypothesized quantile, or ``T2``, the number of observations strictly less than the hypothesized quantile. Two test statistics are required to handle the possibility the data was generated from a discrete or mixed distribution. statistic_type : int ``1`` or ``2`` depending on which of ``T1`` or ``T2`` was used to calculate the p-value respectively. ``T1`` corresponds to the ``"greater"`` alternative hypothesis and ``T2`` to the ``"less"``. For the ``"two-sided"`` case, the statistic type that leads to smallest p-value is used. For significant tests, ``statistic_type = 1`` means there is evidence that the population quantile is significantly greater than the hypothesized value and ``statistic_type = 2`` means there is evidence that it is significantly less than the hypothesized value. pvalue : float The p-value of the hypothesis test. """ statistic: float statistic_type: int pvalue: float _alternative: list[str] = field(repr=False) _x : np.ndarray = field(repr=False) _p : float = field(repr=False) def confidence_interval(self, confidence_level=0.95): """ Compute the confidence interval of the quantile. Parameters ---------- confidence_level : float, default: 0.95 Confidence level for the computed confidence interval of the quantile. Default is 0.95. Returns ------- ci : ``ConfidenceInterval`` object The object has attributes ``low`` and ``high`` that hold the lower and upper bounds of the confidence interval. Examples -------- >>> import numpy as np >>> import scipy.stats as stats >>> p = 0.75 # quantile of interest >>> q = 0 # hypothesized value of the quantile >>> x = np.exp(np.arange(0, 1.01, 0.01)) >>> res = stats.quantile_test(x, q=q, p=p, alternative='less') >>> lb, ub = res.confidence_interval() >>> lb, ub (-inf, 2.293318740264183) >>> res = stats.quantile_test(x, q=q, p=p, alternative='two-sided') >>> lb, ub = res.confidence_interval(0.9) >>> lb, ub (1.9542373206359396, 2.293318740264183) """ alternative = self._alternative p = self._p x = np.sort(self._x) n = len(x) bd = stats.binom(n, p) if confidence_level <= 0 or confidence_level >= 1: message = "`confidence_level` must be a number between 0 and 1." raise ValueError(message) low_index = np.nan high_index = np.nan if alternative == 'less': p = 1 - confidence_level low = -np.inf high_index = int(bd.isf(p)) high = x[high_index] if high_index < n else np.nan elif alternative == 'greater': p = 1 - confidence_level low_index = int(bd.ppf(p)) - 1 low = x[low_index] if low_index >= 0 else np.nan high = np.inf elif alternative == 'two-sided': p = (1 - confidence_level) / 2 low_index = int(bd.ppf(p)) - 1 low = x[low_index] if low_index >= 0 else np.nan high_index = int(bd.isf(p)) high = x[high_index] if high_index < n else np.nan return ConfidenceInterval(low, high) def quantile_test_iv(x, q, p, alternative): x = np.atleast_1d(x) message = '`x` must be a one-dimensional array of numbers.' if x.ndim != 1 or not np.issubdtype(x.dtype, np.number): raise ValueError(message) q = np.array(q)[()] message = "`q` must be a scalar." if q.ndim != 0 or not np.issubdtype(q.dtype, np.number): raise ValueError(message) p = np.array(p)[()] message = "`p` must be a float strictly between 0 and 1." if p.ndim != 0 or p >= 1 or p <= 0: raise ValueError(message) alternatives = {'two-sided', 'less', 'greater'} message = f"`alternative` must be one of {alternatives}" if alternative not in alternatives: raise ValueError(message) return x, q, p, alternative def quantile_test(x, *, q=0, p=0.5, alternative='two-sided'): r""" Perform a quantile test and compute a confidence interval of the quantile. This function tests the null hypothesis that `q` is the value of the quantile associated with probability `p` of the population underlying sample `x`. For example, with default parameters, it tests that the median of the population underlying `x` is zero. The function returns an object including the test statistic, a p-value, and a method for computing the confidence interval around the quantile. Parameters ---------- x : array_like A one-dimensional sample. q : float, default: 0 The hypothesized value of the quantile. p : float, default: 0.5 The probability associated with the quantile; i.e. the proportion of the population less than `q` is `p`. Must be strictly between 0 and 1. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided': the quantile associated with the probability `p` is not `q`. * 'less': the quantile associated with the probability `p` is less than `q`. * 'greater': the quantile associated with the probability `p` is greater than `q`. Returns ------- result : QuantileTestResult An object with the following attributes: statistic : float One of two test statistics that may be used in the quantile test. The first test statistic, ``T1``, is the proportion of samples in `x` that are less than or equal to the hypothesized quantile `q`. The second test statistic, ``T2``, is the proportion of samples in `x` that are strictly less than the hypothesized quantile `q`. When ``alternative = 'greater'``, ``T1`` is used to calculate the p-value and ``statistic`` is set to ``T1``. When ``alternative = 'less'``, ``T2`` is used to calculate the p-value and ``statistic`` is set to ``T2``. When ``alternative = 'two-sided'``, both ``T1`` and ``T2`` are considered, and the one that leads to the smallest p-value is used. statistic_type : int Either `1` or `2` depending on which of ``T1`` or ``T2`` was used to calculate the p-value. pvalue : float The p-value associated with the given alternative. The object also has the following method: confidence_interval(confidence_level=0.95) Computes a confidence interval around the the population quantile associated with the probability `p`. The confidence interval is returned in a ``namedtuple`` with fields `low` and `high`. Values are `nan` when there are not enough observations to compute the confidence interval at the desired confidence. Notes ----- This test and its method for computing confidence intervals are non-parametric. They are valid if and only if the observations are i.i.d. The implementation of the test follows Conover [1]_. Two test statistics are considered. ``T1``: The number of observations in `x` less than or equal to `q`. ``T1 = (x <= q).sum()`` ``T2``: The number of observations in `x` strictly less than `q`. ``T2 = (x < q).sum()`` The use of two test statistics is necessary to handle the possibility that `x` was generated from a discrete or mixed distribution. The null hypothesis for the test is: H0: The :math:`p^{\mathrm{th}}` population quantile is `q`. and the null distribution for each test statistic is :math:`\mathrm{binom}\left(n, p\right)`. When ``alternative='less'``, the alternative hypothesis is: H1: The :math:`p^{\mathrm{th}}` population quantile is less than `q`. and the p-value is the probability that the binomial random variable .. math:: Y \sim \mathrm{binom}\left(n, p\right) is greater than or equal to the observed value ``T2``. When ``alternative='greater'``, the alternative hypothesis is: H1: The :math:`p^{\mathrm{th}}` population quantile is greater than `q` and the p-value is the probability that the binomial random variable Y is less than or equal to the observed value ``T1``. When ``alternative='two-sided'``, the alternative hypothesis is H1: `q` is not the :math:`p^{\mathrm{th}}` population quantile. and the p-value is twice the smaller of the p-values for the ``'less'`` and ``'greater'`` cases. Both of these p-values can exceed 0.5 for the same data, so the value is clipped into the interval :math:`[0, 1]`. The approach for confidence intervals is attributed to Thompson [2]_ and later proven to be applicable to any set of i.i.d. samples [3]_. The computation is based on the observation that the probability of a quantile :math:`q` to be larger than any observations :math:`x_m (1\leq m \leq N)` can be computed as .. math:: \mathbb{P}(x_m \leq q) = 1 - \sum_{k=0}^{m-1} \binom{N}{k} q^k(1-q)^{N-k} By default, confidence intervals are computed for a 95% confidence level. A common interpretation of a 95% confidence intervals is that if i.i.d. samples are drawn repeatedly from the same population and confidence intervals are formed each time, the confidence interval will contain the true value of the specified quantile in approximately 95% of trials. A similar function is available in the QuantileNPCI R package [4]_. The foundation is the same, but it computes the confidence interval bounds by doing interpolations between the sample values, whereas this function uses only sample values as bounds. Thus, ``quantile_test.confidence_interval`` returns more conservative intervals (i.e., larger). The same computation of confidence intervals for quantiles is included in the confintr package [5]_. Two-sided confidence intervals are not guaranteed to be optimal; i.e., there may exist a tighter interval that may contain the quantile of interest with probability larger than the confidence level. Without further assumption on the samples (e.g., the nature of the underlying distribution), the one-sided intervals are optimally tight. References ---------- .. [1] W. J. Conover. Practical Nonparametric Statistics, 3rd Ed. 1999. .. [2] W. R. Thompson, "On Confidence Ranges for the Median and Other Expectation Distributions for Populations of Unknown Distribution Form," The Annals of Mathematical Statistics, vol. 7, no. 3, pp. 122-128, 1936, Accessed: Sep. 18, 2019. [Online]. Available: https://www.jstor.org/stable/2957563. .. [3] H. A. David and H. N. Nagaraja, "Order Statistics in Nonparametric Inference" in Order Statistics, John Wiley & Sons, Ltd, 2005, pp. 159-170. Available: https://onlinelibrary.wiley.com/doi/10.1002/0471722162.ch7. .. [4] N. Hutson, A. Hutson, L. Yan, "QuantileNPCI: Nonparametric Confidence Intervals for Quantiles," R package, https://cran.r-project.org/package=QuantileNPCI .. [5] M. Mayer, "confintr: Confidence Intervals," R package, https://cran.r-project.org/package=confintr Examples -------- Suppose we wish to test the null hypothesis that the median of a population is equal to 0.5. We choose a confidence level of 99%; that is, we will reject the null hypothesis in favor of the alternative if the p-value is less than 0.01. When testing random variates from the standard uniform distribution, which has a median of 0.5, we expect the data to be consistent with the null hypothesis most of the time. >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng(6981396440634228121) >>> rvs = stats.uniform.rvs(size=100, random_state=rng) >>> stats.quantile_test(rvs, q=0.5, p=0.5) QuantileTestResult(statistic=45, statistic_type=1, pvalue=0.36820161732669576) As expected, the p-value is not below our threshold of 0.01, so we cannot reject the null hypothesis. When testing data from the standard *normal* distribution, which has a median of 0, we would expect the null hypothesis to be rejected. >>> rvs = stats.norm.rvs(size=100, random_state=rng) >>> stats.quantile_test(rvs, q=0.5, p=0.5) QuantileTestResult(statistic=67, statistic_type=2, pvalue=0.0008737198369123724) Indeed, the p-value is lower than our threshold of 0.01, so we reject the null hypothesis in favor of the default "two-sided" alternative: the median of the population is *not* equal to 0.5. However, suppose we were to test the null hypothesis against the one-sided alternative that the median of the population is *greater* than 0.5. Since the median of the standard normal is less than 0.5, we would not expect the null hypothesis to be rejected. >>> stats.quantile_test(rvs, q=0.5, p=0.5, alternative='greater') QuantileTestResult(statistic=67, statistic_type=1, pvalue=0.9997956114162866) Unsurprisingly, with a p-value greater than our threshold, we would not reject the null hypothesis in favor of the chosen alternative. The quantile test can be used for any quantile, not only the median. For example, we can test whether the third quartile of the distribution underlying the sample is greater than 0.6. >>> rvs = stats.uniform.rvs(size=100, random_state=rng) >>> stats.quantile_test(rvs, q=0.6, p=0.75, alternative='greater') QuantileTestResult(statistic=64, statistic_type=1, pvalue=0.00940696592998271) The p-value is lower than the threshold. We reject the null hypothesis in favor of the alternative: the third quartile of the distribution underlying our sample is greater than 0.6. `quantile_test` can also compute confidence intervals for any quantile. >>> rvs = stats.norm.rvs(size=100, random_state=rng) >>> res = stats.quantile_test(rvs, q=0.6, p=0.75) >>> ci = res.confidence_interval(confidence_level=0.95) >>> ci ConfidenceInterval(low=0.284491604437432, high=0.8912531024914844) When testing a one-sided alternative, the confidence interval contains all observations such that if passed as `q`, the p-value of the test would be greater than 0.05, and therefore the null hypothesis would not be rejected. For example: >>> rvs.sort() >>> q, p, alpha = 0.6, 0.75, 0.95 >>> res = stats.quantile_test(rvs, q=q, p=p, alternative='less') >>> ci = res.confidence_interval(confidence_level=alpha) >>> for x in rvs[rvs <= ci.high]: ... res = stats.quantile_test(rvs, q=x, p=p, alternative='less') ... assert res.pvalue > 1-alpha >>> for x in rvs[rvs > ci.high]: ... res = stats.quantile_test(rvs, q=x, p=p, alternative='less') ... assert res.pvalue < 1-alpha Also, if a 95% confidence interval is repeatedly generated for random samples, the confidence interval will contain the true quantile value in approximately 95% of replications. >>> dist = stats.rayleigh() # our "unknown" distribution >>> p = 0.2 >>> true_stat = dist.ppf(p) # the true value of the statistic >>> n_trials = 1000 >>> quantile_ci_contains_true_stat = 0 >>> for i in range(n_trials): ... data = dist.rvs(size=100, random_state=rng) ... res = stats.quantile_test(data, p=p) ... ci = res.confidence_interval(0.95) ... if ci[0] < true_stat < ci[1]: ... quantile_ci_contains_true_stat += 1 >>> quantile_ci_contains_true_stat >= 950 True This works with any distribution and any quantile, as long as the samples are i.i.d. """ # Implementation carefully follows [1] 3.2 # "H0: the p*th quantile of X is x*" # To facilitate comparison with [1], we'll use variable names that # best match Conover's notation X, x_star, p_star, H1 = quantile_test_iv(x, q, p, alternative) # "We will use two test statistics in this test. Let T1 equal " # "the number of observations less than or equal to x*, and " # "let T2 equal the number of observations less than x*." T1 = (X <= x_star).sum() T2 = (X < x_star).sum() # "The null distribution of the test statistics T1 and T2 is " # "the binomial distribution, with parameters n = sample size, and " # "p = p* as given in the null hypothesis.... Y has the binomial " # "distribution with parameters n and p*." n = len(X) Y = stats.binom(n=n, p=p_star) # "H1: the p* population quantile is less than x*" if H1 == 'less': # "The p-value is the probability that a binomial random variable Y " # "is greater than *or equal to* the observed value of T2...using p=p*" pvalue = Y.sf(T2-1) # Y.pmf(T2) + Y.sf(T2) statistic = T2 statistic_type = 2 # "H1: the p* population quantile is greater than x*" elif H1 == 'greater': # "The p-value is the probability that a binomial random variable Y " # "is less than or equal to the observed value of T1... using p = p*" pvalue = Y.cdf(T1) statistic = T1 statistic_type = 1 # "H1: x* is not the p*th population quantile" elif H1 == 'two-sided': # "The p-value is twice the smaller of the probabilities that a # binomial random variable Y is less than or equal to the observed # value of T1 or greater than or equal to the observed value of T2 # using p=p*." # Note: both one-sided p-values can exceed 0.5 for the same data, so # `clip` pvalues = [Y.cdf(T1), Y.sf(T2 - 1)] # [greater, less] sorted_idx = np.argsort(pvalues) pvalue = np.clip(2*pvalues[sorted_idx[0]], 0, 1) if sorted_idx[0]: statistic, statistic_type = T2, 2 else: statistic, statistic_type = T1, 1 return QuantileTestResult( statistic=statistic, statistic_type=statistic_type, pvalue=pvalue, _alternative=H1, _x=X, _p=p_star ) ##################################### # STATISTICAL DISTANCES # ##################################### def wasserstein_distance_nd(u_values, v_values, u_weights=None, v_weights=None): r""" Compute the Wasserstein-1 distance between two N-D discrete distributions. The Wasserstein distance, also called the Earth mover's distance or the optimal transport distance, is a similarity metric between two probability distributions [1]_. In the discrete case, the Wasserstein distance can be understood as the cost of an optimal transport plan to convert one distribution into the other. The cost is calculated as the product of the amount of probability mass being moved and the distance it is being moved. A brief and intuitive introduction can be found at [2]_. .. versionadded:: 1.13.0 Parameters ---------- u_values : 2d array_like A sample from a probability distribution or the support (set of all possible values) of a probability distribution. Each element along axis 0 is an observation or possible value, and axis 1 represents the dimensionality of the distribution; i.e., each row is a vector observation or possible value. v_values : 2d array_like A sample from or the support of a second distribution. u_weights, v_weights : 1d array_like, optional Weights or counts corresponding with the sample or probability masses corresponding with the support values. Sum of elements must be positive and finite. If unspecified, each value is assigned the same weight. Returns ------- distance : float The computed distance between the distributions. Notes ----- Given two probability mass functions, :math:`u` and :math:`v`, the first Wasserstein distance between the distributions using the Euclidean norm is: .. math:: l_1 (u, v) = \inf_{\pi \in \Gamma (u, v)} \int \| x-y \|_2 \mathrm{d} \pi (x, y) where :math:`\Gamma (u, v)` is the set of (probability) distributions on :math:`\mathbb{R}^n \times \mathbb{R}^n` whose marginals are :math:`u` and :math:`v` on the first and second factors respectively. For a given value :math:`x`, :math:`u(x)` gives the probabilty of :math:`u` at position :math:`x`, and the same for :math:`v(x)`. This is also called the optimal transport problem or the Monge problem. Let the finite point sets :math:`\{x_i\}` and :math:`\{y_j\}` denote the support set of probability mass function :math:`u` and :math:`v` respectively. The Monge problem can be expressed as follows, Let :math:`\Gamma` denote the transport plan, :math:`D` denote the distance matrix and, .. math:: x = \text{vec}(\Gamma) \\ c = \text{vec}(D) \\ b = \begin{bmatrix} u\\ v\\ \end{bmatrix} The :math:`\text{vec}()` function denotes the Vectorization function that transforms a matrix into a column vector by vertically stacking the columns of the matrix. The tranport plan :math:`\Gamma` is a matrix :math:`[\gamma_{ij}]` in which :math:`\gamma_{ij}` is a positive value representing the amount of probability mass transported from :math:`u(x_i)` to :math:`v(y_i)`. Summing over the rows of :math:`\Gamma` should give the source distribution :math:`u` : :math:`\sum_j \gamma_{ij} = u(x_i)` holds for all :math:`i` and summing over the columns of :math:`\Gamma` should give the target distribution :math:`v`: :math:`\sum_i \gamma_{ij} = v(y_j)` holds for all :math:`j`. The distance matrix :math:`D` is a matrix :math:`[d_{ij}]`, in which :math:`d_{ij} = d(x_i, y_j)`. Given :math:`\Gamma`, :math:`D`, :math:`b`, the Monge problem can be tranformed into a linear programming problem by taking :math:`A x = b` as constraints and :math:`z = c^T x` as minimization target (sum of costs) , where matrix :math:`A` has the form .. math:: \begin{array} {rrrr|rrrr|r|rrrr} 1 & 1 & \dots & 1 & 0 & 0 & \dots & 0 & \dots & 0 & 0 & \dots & 0 \cr 0 & 0 & \dots & 0 & 1 & 1 & \dots & 1 & \dots & 0 & 0 &\dots & 0 \cr \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr 0 & 0 & \dots & 0 & 0 & 0 & \dots & 0 & \dots & 1 & 1 & \dots & 1 \cr \hline 1 & 0 & \dots & 0 & 1 & 0 & \dots & \dots & \dots & 1 & 0 & \dots & 0 \cr 0 & 1 & \dots & 0 & 0 & 1 & \dots & \dots & \dots & 0 & 1 & \dots & 0 \cr \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr 0 & 0 & \dots & 1 & 0 & 0 & \dots & 1 & \dots & 0 & 0 & \dots & 1 \end{array} By solving the dual form of the above linear programming problem (with solution :math:`y^*`), the Wasserstein distance :math:`l_1 (u, v)` can be computed as :math:`b^T y^*`. The above solution is inspired by Vincent Herrmann's blog [3]_ . For a more thorough explanation, see [4]_ . The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values. References ---------- .. [1] "Wasserstein metric", https://en.wikipedia.org/wiki/Wasserstein_metric .. [2] Lili Weng, "What is Wasserstein distance?", Lil'log, https://lilianweng.github.io/posts/2017-08-20-gan/#what-is-wasserstein-distance. .. [3] Hermann, Vincent. "Wasserstein GAN and the Kantorovich-Rubinstein Duality". https://vincentherrmann.github.io/blog/wasserstein/. .. [4] Peyré, Gabriel, and Marco Cuturi. "Computational optimal transport." Center for Research in Economics and Statistics Working Papers 2017-86 (2017). See Also -------- wasserstein_distance: Compute the Wasserstein-1 distance between two 1D discrete distributions. Examples -------- Compute the Wasserstein distance between two three-dimensional samples, each with two observations. >>> from scipy.stats import wasserstein_distance_nd >>> wasserstein_distance_nd([[0, 2, 3], [1, 2, 5]], [[3, 2, 3], [4, 2, 5]]) 3.0 Compute the Wasserstein distance between two two-dimensional distributions with three and two weighted observations, respectively. >>> wasserstein_distance_nd([[0, 2.75], [2, 209.3], [0, 0]], ... [[0.2, 0.322], [4.5, 25.1808]], ... [0.4, 5.2, 0.114], [0.8, 1.5]) 174.15840245217169 """ m, n = len(u_values), len(v_values) u_values = asarray(u_values) v_values = asarray(v_values) if u_values.ndim > 2 or v_values.ndim > 2: raise ValueError('Invalid input values. The inputs must have either ' 'one or two dimensions.') # if dimensions are not equal throw error if u_values.ndim != v_values.ndim: raise ValueError('Invalid input values. Dimensions of inputs must be ' 'equal.') # if data is 1D then call the cdf_distance function if u_values.ndim == 1 and v_values.ndim == 1: return _cdf_distance(1, u_values, v_values, u_weights, v_weights) u_values, u_weights = _validate_distribution(u_values, u_weights) v_values, v_weights = _validate_distribution(v_values, v_weights) # if number of columns is not equal throw error if u_values.shape[1] != v_values.shape[1]: raise ValueError('Invalid input values. If two-dimensional, ' '`u_values` and `v_values` must have the same ' 'number of columns.') # if data contains np.inf then return inf or nan if np.any(np.isinf(u_values)) ^ np.any(np.isinf(v_values)): return np.inf elif np.any(np.isinf(u_values)) and np.any(np.isinf(v_values)): return np.nan # create constraints A_upper_part = sparse.block_diag((np.ones((1, n)), ) * m) A_lower_part = sparse.hstack((sparse.eye(n), ) * m) # sparse constraint matrix of size (m + n)*(m * n) A = sparse.vstack((A_upper_part, A_lower_part)) A = sparse.coo_array(A) # get cost matrix D = distance_matrix(u_values, v_values, p=2) cost = D.ravel() # create the minimization target p_u = np.full(m, 1/m) if u_weights is None else u_weights/np.sum(u_weights) p_v = np.full(n, 1/n) if v_weights is None else v_weights/np.sum(v_weights) b = np.concatenate((p_u, p_v), axis=0) # solving LP constraints = LinearConstraint(A=A.T, ub=cost) opt_res = milp(c=-b, constraints=constraints, bounds=(-np.inf, np.inf)) return -opt_res.fun def wasserstein_distance(u_values, v_values, u_weights=None, v_weights=None): r""" Compute the Wasserstein-1 distance between two 1D discrete distributions. The Wasserstein distance, also called the Earth mover's distance or the optimal transport distance, is a similarity metric between two probability distributions [1]_. In the discrete case, the Wasserstein distance can be understood as the cost of an optimal transport plan to convert one distribution into the other. The cost is calculated as the product of the amount of probability mass being moved and the distance it is being moved. A brief and intuitive introduction can be found at [2]_. .. versionadded:: 1.0.0 Parameters ---------- u_values : 1d array_like A sample from a probability distribution or the support (set of all possible values) of a probability distribution. Each element is an observation or possible value. v_values : 1d array_like A sample from or the support of a second distribution. u_weights, v_weights : 1d array_like, optional Weights or counts corresponding with the sample or probability masses corresponding with the support values. Sum of elements must be positive and finite. If unspecified, each value is assigned the same weight. Returns ------- distance : float The computed distance between the distributions. Notes ----- Given two 1D probability mass functions, :math:`u` and :math:`v`, the first Wasserstein distance between the distributions is: .. math:: l_1 (u, v) = \inf_{\pi \in \Gamma (u, v)} \int_{\mathbb{R} \times \mathbb{R}} |x-y| \mathrm{d} \pi (x, y) where :math:`\Gamma (u, v)` is the set of (probability) distributions on :math:`\mathbb{R} \times \mathbb{R}` whose marginals are :math:`u` and :math:`v` on the first and second factors respectively. For a given value :math:`x`, :math:`u(x)` gives the probabilty of :math:`u` at position :math:`x`, and the same for :math:`v(x)`. If :math:`U` and :math:`V` are the respective CDFs of :math:`u` and :math:`v`, this distance also equals to: .. math:: l_1(u, v) = \int_{-\infty}^{+\infty} |U-V| See [3]_ for a proof of the equivalence of both definitions. The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values. References ---------- .. [1] "Wasserstein metric", https://en.wikipedia.org/wiki/Wasserstein_metric .. [2] Lili Weng, "What is Wasserstein distance?", Lil'log, https://lilianweng.github.io/posts/2017-08-20-gan/#what-is-wasserstein-distance. .. [3] Ramdas, Garcia, Cuturi "On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests" (2015). :arXiv:`1509.02237`. See Also -------- wasserstein_distance_nd: Compute the Wasserstein-1 distance between two N-D discrete distributions. Examples -------- >>> from scipy.stats import wasserstein_distance >>> wasserstein_distance([0, 1, 3], [5, 6, 8]) 5.0 >>> wasserstein_distance([0, 1], [0, 1], [3, 1], [2, 2]) 0.25 >>> wasserstein_distance([3.4, 3.9, 7.5, 7.8], [4.5, 1.4], ... [1.4, 0.9, 3.1, 7.2], [3.2, 3.5]) 4.0781331438047861 """ return _cdf_distance(1, u_values, v_values, u_weights, v_weights) def energy_distance(u_values, v_values, u_weights=None, v_weights=None): r"""Compute the energy distance between two 1D distributions. .. versionadded:: 1.0.0 Parameters ---------- u_values, v_values : array_like Values observed in the (empirical) distribution. u_weights, v_weights : array_like, optional Weight for each value. If unspecified, each value is assigned the same weight. `u_weights` (resp. `v_weights`) must have the same length as `u_values` (resp. `v_values`). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1. Returns ------- distance : float The computed distance between the distributions. Notes ----- The energy distance between two distributions :math:`u` and :math:`v`, whose respective CDFs are :math:`U` and :math:`V`, equals to: .. math:: D(u, v) = \left( 2\mathbb E|X - Y| - \mathbb E|X - X'| - \mathbb E|Y - Y'| \right)^{1/2} where :math:`X` and :math:`X'` (resp. :math:`Y` and :math:`Y'`) are independent random variables whose probability distribution is :math:`u` (resp. :math:`v`). Sometimes the square of this quantity is referred to as the "energy distance" (e.g. in [2]_, [4]_), but as noted in [1]_ and [3]_, only the definition above satisfies the axioms of a distance function (metric). As shown in [2]_, for one-dimensional real-valued variables, the energy distance is linked to the non-distribution-free version of the Cramér-von Mises distance: .. math:: D(u, v) = \sqrt{2} l_2(u, v) = \left( 2 \int_{-\infty}^{+\infty} (U-V)^2 \right)^{1/2} Note that the common Cramér-von Mises criterion uses the distribution-free version of the distance. See [2]_ (section 2), for more details about both versions of the distance. The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values. References ---------- .. [1] Rizzo, Szekely "Energy distance." Wiley Interdisciplinary Reviews: Computational Statistics, 8(1):27-38 (2015). .. [2] Szekely "E-statistics: The energy of statistical samples." Bowling Green State University, Department of Mathematics and Statistics, Technical Report 02-16 (2002). .. [3] "Energy distance", https://en.wikipedia.org/wiki/Energy_distance .. [4] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer, Munos "The Cramer Distance as a Solution to Biased Wasserstein Gradients" (2017). :arXiv:`1705.10743`. Examples -------- >>> from scipy.stats import energy_distance >>> energy_distance([0], [2]) 2.0000000000000004 >>> energy_distance([0, 8], [0, 8], [3, 1], [2, 2]) 1.0000000000000002 >>> energy_distance([0.7, 7.4, 2.4, 6.8], [1.4, 8. ], ... [2.1, 4.2, 7.4, 8. ], [7.6, 8.8]) 0.88003340976158217 """ return np.sqrt(2) * _cdf_distance(2, u_values, v_values, u_weights, v_weights) def _cdf_distance(p, u_values, v_values, u_weights=None, v_weights=None): r""" Compute, between two one-dimensional distributions :math:`u` and :math:`v`, whose respective CDFs are :math:`U` and :math:`V`, the statistical distance that is defined as: .. math:: l_p(u, v) = \left( \int_{-\infty}^{+\infty} |U-V|^p \right)^{1/p} p is a positive parameter; p = 1 gives the Wasserstein distance, p = 2 gives the energy distance. Parameters ---------- u_values, v_values : array_like Values observed in the (empirical) distribution. u_weights, v_weights : array_like, optional Weight for each value. If unspecified, each value is assigned the same weight. `u_weights` (resp. `v_weights`) must have the same length as `u_values` (resp. `v_values`). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1. Returns ------- distance : float The computed distance between the distributions. Notes ----- The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values. References ---------- .. [1] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer, Munos "The Cramer Distance as a Solution to Biased Wasserstein Gradients" (2017). :arXiv:`1705.10743`. """ u_values, u_weights = _validate_distribution(u_values, u_weights) v_values, v_weights = _validate_distribution(v_values, v_weights) u_sorter = np.argsort(u_values) v_sorter = np.argsort(v_values) all_values = np.concatenate((u_values, v_values)) all_values.sort(kind='mergesort') # Compute the differences between pairs of successive values of u and v. deltas = np.diff(all_values) # Get the respective positions of the values of u and v among the values of # both distributions. u_cdf_indices = u_values[u_sorter].searchsorted(all_values[:-1], 'right') v_cdf_indices = v_values[v_sorter].searchsorted(all_values[:-1], 'right') # Calculate the CDFs of u and v using their weights, if specified. if u_weights is None: u_cdf = u_cdf_indices / u_values.size else: u_sorted_cumweights = np.concatenate(([0], np.cumsum(u_weights[u_sorter]))) u_cdf = u_sorted_cumweights[u_cdf_indices] / u_sorted_cumweights[-1] if v_weights is None: v_cdf = v_cdf_indices / v_values.size else: v_sorted_cumweights = np.concatenate(([0], np.cumsum(v_weights[v_sorter]))) v_cdf = v_sorted_cumweights[v_cdf_indices] / v_sorted_cumweights[-1] # Compute the value of the integral based on the CDFs. # If p = 1 or p = 2, we avoid using np.power, which introduces an overhead # of about 15%. if p == 1: return np.sum(np.multiply(np.abs(u_cdf - v_cdf), deltas)) if p == 2: return np.sqrt(np.sum(np.multiply(np.square(u_cdf - v_cdf), deltas))) return np.power(np.sum(np.multiply(np.power(np.abs(u_cdf - v_cdf), p), deltas)), 1/p) def _validate_distribution(values, weights): """ Validate the values and weights from a distribution input of `cdf_distance` and return them as ndarray objects. Parameters ---------- values : array_like Values observed in the (empirical) distribution. weights : array_like Weight for each value. Returns ------- values : ndarray Values as ndarray. weights : ndarray Weights as ndarray. """ # Validate the value array. values = np.asarray(values, dtype=float) if len(values) == 0: raise ValueError("Distribution can't be empty.") # Validate the weight array, if specified. if weights is not None: weights = np.asarray(weights, dtype=float) if len(weights) != len(values): raise ValueError('Value and weight array-likes for the same ' 'empirical distribution must be of the same size.') if np.any(weights < 0): raise ValueError('All weights must be non-negative.') if not 0 < np.sum(weights) < np.inf: raise ValueError('Weight array-like sum must be positive and ' 'finite. Set as None for an equal distribution of ' 'weight.') return values, weights return values, None ##################################### # SUPPORT FUNCTIONS # ##################################### RepeatedResults = namedtuple('RepeatedResults', ('values', 'counts')) def find_repeats(arr): """Find repeats and repeat counts. Parameters ---------- arr : array_like Input array. This is cast to float64. Returns ------- values : ndarray The unique values from the (flattened) input that are repeated. counts : ndarray Number of times the corresponding 'value' is repeated. Notes ----- In numpy >= 1.9 `numpy.unique` provides similar functionality. The main difference is that `find_repeats` only returns repeated values. Examples -------- >>> from scipy import stats >>> stats.find_repeats([2, 1, 2, 3, 2, 2, 5]) RepeatedResults(values=array([2.]), counts=array([4])) >>> stats.find_repeats([[10, 20, 1, 2], [5, 5, 4, 4]]) RepeatedResults(values=array([4., 5.]), counts=array([2, 2])) """ # Note: always copies. return RepeatedResults(*_find_repeats(np.array(arr, dtype=np.float64))) def _sum_of_squares(a, axis=0): """Square each element of the input array, and return the sum(s) of that. Parameters ---------- a : array_like Input array. axis : int or None, optional Axis along which to calculate. Default is 0. If None, compute over the whole array `a`. Returns ------- sum_of_squares : ndarray The sum along the given axis for (a**2). See Also -------- _square_of_sums : The square(s) of the sum(s) (the opposite of `_sum_of_squares`). """ a, axis = _chk_asarray(a, axis) return np.sum(a*a, axis) def _square_of_sums(a, axis=0): """Sum elements of the input array, and return the square(s) of that sum. Parameters ---------- a : array_like Input array. axis : int or None, optional Axis along which to calculate. Default is 0. If None, compute over the whole array `a`. Returns ------- square_of_sums : float or ndarray The square of the sum over `axis`. See Also -------- _sum_of_squares : The sum of squares (the opposite of `square_of_sums`). """ a, axis = _chk_asarray(a, axis) s = np.sum(a, axis) if not np.isscalar(s): return s.astype(float) * s else: return float(s) * s def rankdata(a, method='average', *, axis=None, nan_policy='propagate'): """Assign ranks to data, dealing with ties appropriately. By default (``axis=None``), the data array is first flattened, and a flat array of ranks is returned. Separately reshape the rank array to the shape of the data array if desired (see Examples). Ranks begin at 1. The `method` argument controls how ranks are assigned to equal values. See [1]_ for further discussion of ranking methods. Parameters ---------- a : array_like The array of values to be ranked. method : {'average', 'min', 'max', 'dense', 'ordinal'}, optional The method used to assign ranks to tied elements. The following methods are available (default is 'average'): * 'average': The average of the ranks that would have been assigned to all the tied values is assigned to each value. * 'min': The minimum of the ranks that would have been assigned to all the tied values is assigned to each value. (This is also referred to as "competition" ranking.) * 'max': The maximum of the ranks that would have been assigned to all the tied values is assigned to each value. * 'dense': Like 'min', but the rank of the next highest element is assigned the rank immediately after those assigned to the tied elements. * 'ordinal': All values are given a distinct rank, corresponding to the order that the values occur in `a`. axis : {None, int}, optional Axis along which to perform the ranking. If ``None``, the data array is first flattened. nan_policy : {'propagate', 'omit', 'raise'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': propagates nans through the rank calculation * 'omit': performs the calculations ignoring nan values * 'raise': raises an error .. note:: When `nan_policy` is 'propagate', the output is an array of *all* nans because ranks relative to nans in the input are undefined. When `nan_policy` is 'omit', nans in `a` are ignored when ranking the other values, and the corresponding locations of the output are nan. .. versionadded:: 1.10 Returns ------- ranks : ndarray An array of size equal to the size of `a`, containing rank scores. References ---------- .. [1] "Ranking", https://en.wikipedia.org/wiki/Ranking Examples -------- >>> import numpy as np >>> from scipy.stats import rankdata >>> rankdata([0, 2, 3, 2]) array([ 1. , 2.5, 4. , 2.5]) >>> rankdata([0, 2, 3, 2], method='min') array([ 1, 2, 4, 2]) >>> rankdata([0, 2, 3, 2], method='max') array([ 1, 3, 4, 3]) >>> rankdata([0, 2, 3, 2], method='dense') array([ 1, 2, 3, 2]) >>> rankdata([0, 2, 3, 2], method='ordinal') array([ 1, 2, 4, 3]) >>> rankdata([[0, 2], [3, 2]]).reshape(2,2) array([[1. , 2.5], [4. , 2.5]]) >>> rankdata([[0, 2, 2], [3, 2, 5]], axis=1) array([[1. , 2.5, 2.5], [2. , 1. , 3. ]]) >>> rankdata([0, 2, 3, np.nan, -2, np.nan], nan_policy="propagate") array([nan, nan, nan, nan, nan, nan]) >>> rankdata([0, 2, 3, np.nan, -2, np.nan], nan_policy="omit") array([ 2., 3., 4., nan, 1., nan]) """ methods = ('average', 'min', 'max', 'dense', 'ordinal') if method not in methods: raise ValueError(f'unknown method "{method}"') x = np.asarray(a) if axis is None: x = x.ravel() axis = -1 if x.size == 0: dtype = float if method == 'average' else np.dtype("long") return np.empty(x.shape, dtype=dtype) contains_nan, nan_policy = _contains_nan(x, nan_policy) x = np.swapaxes(x, axis, -1) ranks = _rankdata(x, method) if contains_nan: i_nan = (np.isnan(x) if nan_policy == 'omit' else np.isnan(x).any(axis=-1)) ranks = ranks.astype(float, copy=False) ranks[i_nan] = np.nan ranks = np.swapaxes(ranks, axis, -1) return ranks def _order_ranks(ranks, j): # Reorder ascending order `ranks` according to `j` ordered_ranks = np.empty(j.shape, dtype=ranks.dtype) np.put_along_axis(ordered_ranks, j, ranks, axis=-1) return ordered_ranks def _rankdata(x, method, return_ties=False): # Rank data `x` by desired `method`; `return_ties` if desired shape = x.shape # Get sort order kind = 'mergesort' if method == 'ordinal' else 'quicksort' j = np.argsort(x, axis=-1, kind=kind) ordinal_ranks = np.broadcast_to(np.arange(1, shape[-1]+1, dtype=int), shape) # Ordinal ranks is very easy because ties don't matter. We're done. if method == 'ordinal': return _order_ranks(ordinal_ranks, j) # never return ties # Sort array y = np.take_along_axis(x, j, axis=-1) # Logical indices of unique elements i = np.concatenate([np.ones(shape[:-1] + (1,), dtype=np.bool_), y[..., :-1] != y[..., 1:]], axis=-1) # Integer indices of unique elements indices = np.arange(y.size)[i.ravel()] # Counts of unique elements counts = np.diff(indices, append=y.size) # Compute `'min'`, `'max'`, and `'mid'` ranks of unique elements if method == 'min': ranks = ordinal_ranks[i] elif method == 'max': ranks = ordinal_ranks[i] + counts - 1 elif method == 'average': ranks = ordinal_ranks[i] + (counts - 1)/2 elif method == 'dense': ranks = np.cumsum(i, axis=-1)[i] ranks = np.repeat(ranks, counts).reshape(shape) ranks = _order_ranks(ranks, j) if return_ties: # Tie information is returned in a format that is useful to functions that # rely on this (private) function. Example: # >>> x = np.asarray([3, 2, 1, 2, 2, 2, 1]) # >>> _, t = _rankdata(x, 'average', return_ties=True) # >>> t # array([2., 0., 4., 0., 0., 0., 1.]) # two 1s, four 2s, and one 3 # Unlike ranks, tie counts are *not* reordered to correspond with the order of # the input; e.g. the number of appearances of the lowest rank element comes # first. This is a useful format because: # - The shape of the result is the shape of the input. Different slices can # have different numbers of tied elements but not result in a ragged array. # - Functions that use `t` usually don't need to which each element of the # original array is associated with each tie count; they perform a reduction # over the tie counts onnly. The tie counts are naturally computed in a # sorted order, so this does not unnecesarily reorder them. # - One exception is `wilcoxon`, which needs the number of zeros. Zeros always # have the lowest rank, so it is easy to find them at the zeroth index. t = np.zeros(shape, dtype=float) t[i] = counts return ranks, t return ranks def expectile(a, alpha=0.5, *, weights=None): r"""Compute the expectile at the specified level. Expectiles are a generalization of the expectation in the same way as quantiles are a generalization of the median. The expectile at level `alpha = 0.5` is the mean (average). See Notes for more details. Parameters ---------- a : array_like Array containing numbers whose expectile is desired. alpha : float, default: 0.5 The level of the expectile; e.g., `alpha=0.5` gives the mean. weights : array_like, optional An array of weights associated with the values in `a`. The `weights` must be broadcastable to the same shape as `a`. Default is None, which gives each value a weight of 1.0. An integer valued weight element acts like repeating the corresponding observation in `a` that many times. See Notes for more details. Returns ------- expectile : ndarray The empirical expectile at level `alpha`. See Also -------- numpy.mean : Arithmetic average numpy.quantile : Quantile Notes ----- In general, the expectile at level :math:`\alpha` of a random variable :math:`X` with cumulative distribution function (CDF) :math:`F` is given by the unique solution :math:`t` of: .. math:: \alpha E((X - t)_+) = (1 - \alpha) E((t - X)_+) \,. Here, :math:`(x)_+ = \max(0, x)` is the positive part of :math:`x`. This equation can be equivalently written as: .. math:: \alpha \int_t^\infty (x - t)\mathrm{d}F(x) = (1 - \alpha) \int_{-\infty}^t (t - x)\mathrm{d}F(x) \,. The empirical expectile at level :math:`\alpha` (`alpha`) of a sample :math:`a_i` (the array `a`) is defined by plugging in the empirical CDF of `a`. Given sample or case weights :math:`w` (the array `weights`), it reads :math:`F_a(x) = \frac{1}{\sum_i w_i} \sum_i w_i 1_{a_i \leq x}` with indicator function :math:`1_{A}`. This leads to the definition of the empirical expectile at level `alpha` as the unique solution :math:`t` of: .. math:: \alpha \sum_{i=1}^n w_i (a_i - t)_+ = (1 - \alpha) \sum_{i=1}^n w_i (t - a_i)_+ \,. For :math:`\alpha=0.5`, this simplifies to the weighted average. Furthermore, the larger :math:`\alpha`, the larger the value of the expectile. As a final remark, the expectile at level :math:`\alpha` can also be written as a minimization problem. One often used choice is .. math:: \operatorname{argmin}_t E(\lvert 1_{t\geq X} - \alpha\rvert(t - X)^2) \,. References ---------- .. [1] W. K. Newey and J. L. Powell (1987), "Asymmetric Least Squares Estimation and Testing," Econometrica, 55, 819-847. .. [2] T. Gneiting (2009). "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, 106, 746 - 762. :doi:`10.48550/arXiv.0912.0902` Examples -------- >>> import numpy as np >>> from scipy.stats import expectile >>> a = [1, 4, 2, -1] >>> expectile(a, alpha=0.5) == np.mean(a) True >>> expectile(a, alpha=0.2) 0.42857142857142855 >>> expectile(a, alpha=0.8) 2.5714285714285716 >>> weights = [1, 3, 1, 1] """ if alpha < 0 or alpha > 1: raise ValueError( "The expectile level alpha must be in the range [0, 1]." ) a = np.asarray(a) if weights is not None: weights = np.broadcast_to(weights, a.shape) # This is the empirical equivalent of Eq. (13) with identification # function from Table 9 (omitting a factor of 2) in [2] (their y is our # data a, their x is our t) def first_order(t): return np.average(np.abs((a <= t) - alpha) * (t - a), weights=weights) if alpha >= 0.5: x0 = np.average(a, weights=weights) x1 = np.amax(a) else: x1 = np.average(a, weights=weights) x0 = np.amin(a) if x0 == x1: # a has a single unique element return x0 # Note that the expectile is the unique solution, so no worries about # finding a wrong root. res = root_scalar(first_order, x0=x0, x1=x1) return res.root LinregressResult = _make_tuple_bunch('LinregressResult', ['slope', 'intercept', 'rvalue', 'pvalue', 'stderr'], extra_field_names=['intercept_stderr']) def linregress(x, y=None, alternative='two-sided'): """ Calculate a linear least-squares regression for two sets of measurements. Parameters ---------- x, y : array_like Two sets of measurements. Both arrays should have the same length N. If only `x` is given (and ``y=None``), then it must be a two-dimensional array where one dimension has length 2. The two sets of measurements are then found by splitting the array along the length-2 dimension. In the case where ``y=None`` and `x` is a 2xN array, ``linregress(x)`` is equivalent to ``linregress(x[0], x[1])``. .. deprecated:: 1.14.0 Inference of the two sets of measurements from a single argument `x` is deprecated will result in an error in SciPy 1.16.0; the sets must be specified separately as `x` and `y`. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. Default is 'two-sided'. The following options are available: * 'two-sided': the slope of the regression line is nonzero * 'less': the slope of the regression line is less than zero * 'greater': the slope of the regression line is greater than zero .. versionadded:: 1.7.0 Returns ------- result : ``LinregressResult`` instance The return value is an object with the following attributes: slope : float Slope of the regression line. intercept : float Intercept of the regression line. rvalue : float The Pearson correlation coefficient. The square of ``rvalue`` is equal to the coefficient of determination. pvalue : float The p-value for a hypothesis test whose null hypothesis is that the slope is zero, using Wald Test with t-distribution of the test statistic. See `alternative` above for alternative hypotheses. stderr : float Standard error of the estimated slope (gradient), under the assumption of residual normality. intercept_stderr : float Standard error of the estimated intercept, under the assumption of residual normality. See Also -------- scipy.optimize.curve_fit : Use non-linear least squares to fit a function to data. scipy.optimize.leastsq : Minimize the sum of squares of a set of equations. Notes ----- For compatibility with older versions of SciPy, the return value acts like a ``namedtuple`` of length 5, with fields ``slope``, ``intercept``, ``rvalue``, ``pvalue`` and ``stderr``, so one can continue to write:: slope, intercept, r, p, se = linregress(x, y) With that style, however, the standard error of the intercept is not available. To have access to all the computed values, including the standard error of the intercept, use the return value as an object with attributes, e.g.:: result = linregress(x, y) print(result.intercept, result.intercept_stderr) Examples -------- >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> rng = np.random.default_rng() Generate some data: >>> x = rng.random(10) >>> y = 1.6*x + rng.random(10) Perform the linear regression: >>> res = stats.linregress(x, y) Coefficient of determination (R-squared): >>> print(f"R-squared: {res.rvalue**2:.6f}") R-squared: 0.717533 Plot the data along with the fitted line: >>> plt.plot(x, y, 'o', label='original data') >>> plt.plot(x, res.intercept + res.slope*x, 'r', label='fitted line') >>> plt.legend() >>> plt.show() Calculate 95% confidence interval on slope and intercept: >>> # Two-sided inverse Students t-distribution >>> # p - probability, df - degrees of freedom >>> from scipy.stats import t >>> tinv = lambda p, df: abs(t.ppf(p/2, df)) >>> ts = tinv(0.05, len(x)-2) >>> print(f"slope (95%): {res.slope:.6f} +/- {ts*res.stderr:.6f}") slope (95%): 1.453392 +/- 0.743465 >>> print(f"intercept (95%): {res.intercept:.6f}" ... f" +/- {ts*res.intercept_stderr:.6f}") intercept (95%): 0.616950 +/- 0.544475 """ TINY = 1.0e-20 if y is None: # x is a (2, N) or (N, 2) shaped array_like message = ('Inference of the two sets of measurements from a single "' 'argument `x` is deprecated will result in an error in "' 'SciPy 1.16.0; the sets must be specified separately as "' '`x` and `y`.') warnings.warn(message, DeprecationWarning, stacklevel=2) x = np.asarray(x) if x.shape[0] == 2: x, y = x elif x.shape[1] == 2: x, y = x.T else: raise ValueError("If only `x` is given as input, it has to " "be of shape (2, N) or (N, 2); provided shape " f"was {x.shape}.") else: x = np.asarray(x) y = np.asarray(y) if x.size == 0 or y.size == 0: raise ValueError("Inputs must not be empty.") if np.amax(x) == np.amin(x) and len(x) > 1: raise ValueError("Cannot calculate a linear regression " "if all x values are identical") n = len(x) xmean = np.mean(x, None) ymean = np.mean(y, None) # Average sums of square differences from the mean # ssxm = mean( (x-mean(x))^2 ) # ssxym = mean( (x-mean(x)) * (y-mean(y)) ) ssxm, ssxym, _, ssym = np.cov(x, y, bias=1).flat # R-value # r = ssxym / sqrt( ssxm * ssym ) if ssxm == 0.0 or ssym == 0.0: # If the denominator was going to be 0 r = 0.0 else: r = ssxym / np.sqrt(ssxm * ssym) # Test for numerical error propagation (make sure -1 < r < 1) if r > 1.0: r = 1.0 elif r < -1.0: r = -1.0 slope = ssxym / ssxm intercept = ymean - slope*xmean if n == 2: # handle case when only two points are passed in if y[0] == y[1]: prob = 1.0 else: prob = 0.0 slope_stderr = 0.0 intercept_stderr = 0.0 else: df = n - 2 # Number of degrees of freedom # n-2 degrees of freedom because 2 has been used up # to estimate the mean and standard deviation t = r * np.sqrt(df / ((1.0 - r + TINY)*(1.0 + r + TINY))) prob = _get_pvalue(t, distributions.t(df), alternative) slope_stderr = np.sqrt((1 - r**2) * ssym / ssxm / df) # Also calculate the standard error of the intercept # The following relationship is used: # ssxm = mean( (x-mean(x))^2 ) # = ssx - sx*sx # = mean( x^2 ) - mean(x)^2 intercept_stderr = slope_stderr * np.sqrt(ssxm + xmean**2) return LinregressResult(slope=slope, intercept=intercept, rvalue=r, pvalue=prob, stderr=slope_stderr, intercept_stderr=intercept_stderr) class _SimpleNormal: # A very simple, array-API compatible normal distribution for use in # hypothesis tests. May be replaced by new infrastructure Normal # distribution in due time. def cdf(self, x): return special.ndtr(x) def sf(self, x): return special.ndtr(-x) class _SimpleChi2: # A very simple, array-API compatible chi-squared distribution for use in # hypothesis tests. May be replaced by new infrastructure chi-squared # distribution in due time. def __init__(self, df): self.df = df def sf(self, x): return special.chdtrc(self.df, x)