1355 lines
58 KiB
Python
1355 lines
58 KiB
Python
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import warnings
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from collections import namedtuple
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import numpy as np
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from scipy import optimize, stats
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from scipy._lib._util import check_random_state
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def _combine_bounds(name, user_bounds, shape_domain, integral):
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"""Intersection of user-defined bounds and distribution PDF/PMF domain"""
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user_bounds = np.atleast_1d(user_bounds)
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if user_bounds[0] > user_bounds[1]:
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message = (f"There are no values for `{name}` on the interval "
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f"{list(user_bounds)}.")
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raise ValueError(message)
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bounds = (max(user_bounds[0], shape_domain[0]),
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min(user_bounds[1], shape_domain[1]))
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if integral and (np.ceil(bounds[0]) > np.floor(bounds[1])):
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message = (f"There are no integer values for `{name}` on the interval "
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f"defined by the user-provided bounds and the domain "
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"of the distribution.")
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raise ValueError(message)
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elif not integral and (bounds[0] > bounds[1]):
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message = (f"There are no values for `{name}` on the interval "
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f"defined by the user-provided bounds and the domain "
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"of the distribution.")
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raise ValueError(message)
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if not np.all(np.isfinite(bounds)):
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message = (f"The intersection of user-provided bounds for `{name}` "
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f"and the domain of the distribution is not finite. Please "
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f"provide finite bounds for shape `{name}` in `bounds`.")
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raise ValueError(message)
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return bounds
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class FitResult:
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r"""Result of fitting a discrete or continuous distribution to data
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Attributes
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----------
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params : namedtuple
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A namedtuple containing the maximum likelihood estimates of the
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shape parameters, location, and (if applicable) scale of the
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distribution.
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success : bool or None
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Whether the optimizer considered the optimization to terminate
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successfully or not.
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message : str or None
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Any status message provided by the optimizer.
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"""
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def __init__(self, dist, data, discrete, res):
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self._dist = dist
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self._data = data
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self.discrete = discrete
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self.pxf = getattr(dist, "pmf", None) or getattr(dist, "pdf", None)
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shape_names = [] if dist.shapes is None else dist.shapes.split(", ")
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if not discrete:
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FitParams = namedtuple('FitParams', shape_names + ['loc', 'scale'])
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else:
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FitParams = namedtuple('FitParams', shape_names + ['loc'])
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self.params = FitParams(*res.x)
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# Optimizer can report success even when nllf is infinite
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if res.success and not np.isfinite(self.nllf()):
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res.success = False
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res.message = ("Optimization converged to parameter values that "
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"are inconsistent with the data.")
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self.success = getattr(res, "success", None)
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self.message = getattr(res, "message", None)
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def __repr__(self):
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keys = ["params", "success", "message"]
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m = max(map(len, keys)) + 1
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return '\n'.join([key.rjust(m) + ': ' + repr(getattr(self, key))
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for key in keys if getattr(self, key) is not None])
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def nllf(self, params=None, data=None):
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"""Negative log-likelihood function
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Evaluates the negative of the log-likelihood function of the provided
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data at the provided parameters.
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Parameters
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----------
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params : tuple, optional
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The shape parameters, location, and (if applicable) scale of the
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distribution as a single tuple. Default is the maximum likelihood
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estimates (``self.params``).
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data : array_like, optional
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The data for which the log-likelihood function is to be evaluated.
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Default is the data to which the distribution was fit.
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Returns
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-------
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nllf : float
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The negative of the log-likelihood function.
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"""
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params = params if params is not None else self.params
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data = data if data is not None else self._data
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return self._dist.nnlf(theta=params, x=data)
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def plot(self, ax=None, *, plot_type="hist"):
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"""Visually compare the data against the fitted distribution.
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Available only if `matplotlib` is installed.
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Parameters
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----------
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ax : `matplotlib.axes.Axes`
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Axes object to draw the plot onto, otherwise uses the current Axes.
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plot_type : {"hist", "qq", "pp", "cdf"}
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Type of plot to draw. Options include:
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- "hist": Superposes the PDF/PMF of the fitted distribution
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over a normalized histogram of the data.
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- "qq": Scatter plot of theoretical quantiles against the
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empirical quantiles. Specifically, the x-coordinates are the
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values of the fitted distribution PPF evaluated at the
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percentiles ``(np.arange(1, n) - 0.5)/n``, where ``n`` is the
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number of data points, and the y-coordinates are the sorted
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data points.
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- "pp": Scatter plot of theoretical percentiles against the
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observed percentiles. Specifically, the x-coordinates are the
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percentiles ``(np.arange(1, n) - 0.5)/n``, where ``n`` is
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the number of data points, and the y-coordinates are the values
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of the fitted distribution CDF evaluated at the sorted
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data points.
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- "cdf": Superposes the CDF of the fitted distribution over the
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empirical CDF. Specifically, the x-coordinates of the empirical
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CDF are the sorted data points, and the y-coordinates are the
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percentiles ``(np.arange(1, n) - 0.5)/n``, where ``n`` is
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the number of data points.
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Returns
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-------
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ax : `matplotlib.axes.Axes`
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The matplotlib Axes object on which the plot was drawn.
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Examples
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--------
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>>> import numpy as np
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>>> from scipy import stats
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>>> import matplotlib.pyplot as plt # matplotlib must be installed
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>>> rng = np.random.default_rng()
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>>> data = stats.nbinom(5, 0.5).rvs(size=1000, random_state=rng)
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>>> bounds = [(0, 30), (0, 1)]
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>>> res = stats.fit(stats.nbinom, data, bounds)
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>>> ax = res.plot() # save matplotlib Axes object
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The `matplotlib.axes.Axes` object can be used to customize the plot.
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See `matplotlib.axes.Axes` documentation for details.
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>>> ax.set_xlabel('number of trials') # customize axis label
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>>> ax.get_children()[0].set_linewidth(5) # customize line widths
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>>> ax.legend()
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>>> plt.show()
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"""
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try:
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import matplotlib # noqa: F401
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except ModuleNotFoundError as exc:
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message = "matplotlib must be installed to use method `plot`."
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raise ModuleNotFoundError(message) from exc
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plots = {'histogram': self._hist_plot, 'qq': self._qq_plot,
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'pp': self._pp_plot, 'cdf': self._cdf_plot,
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'hist': self._hist_plot}
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if plot_type.lower() not in plots:
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message = f"`plot_type` must be one of {set(plots.keys())}"
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raise ValueError(message)
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plot = plots[plot_type.lower()]
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if ax is None:
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import matplotlib.pyplot as plt
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ax = plt.gca()
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fit_params = np.atleast_1d(self.params)
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return plot(ax=ax, fit_params=fit_params)
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def _hist_plot(self, ax, fit_params):
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from matplotlib.ticker import MaxNLocator
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support = self._dist.support(*fit_params)
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lb = support[0] if np.isfinite(support[0]) else min(self._data)
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ub = support[1] if np.isfinite(support[1]) else max(self._data)
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pxf = "PMF" if self.discrete else "PDF"
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if self.discrete:
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x = np.arange(lb, ub + 2)
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y = self.pxf(x, *fit_params)
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ax.vlines(x[:-1], 0, y[:-1], label='Fitted Distribution PMF',
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color='C0')
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options = dict(density=True, bins=x, align='left', color='C1')
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ax.xaxis.set_major_locator(MaxNLocator(integer=True))
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ax.set_xlabel('k')
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ax.set_ylabel('PMF')
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else:
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x = np.linspace(lb, ub, 200)
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y = self.pxf(x, *fit_params)
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ax.plot(x, y, '--', label='Fitted Distribution PDF', color='C0')
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options = dict(density=True, bins=50, align='mid', color='C1')
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ax.set_xlabel('x')
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ax.set_ylabel('PDF')
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if len(self._data) > 50 or self.discrete:
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ax.hist(self._data, label="Histogram of Data", **options)
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else:
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ax.plot(self._data, np.zeros_like(self._data), "*",
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label='Data', color='C1')
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ax.set_title(rf"Fitted $\tt {self._dist.name}$ {pxf} and Histogram")
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ax.legend(*ax.get_legend_handles_labels())
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return ax
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def _qp_plot(self, ax, fit_params, qq):
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data = np.sort(self._data)
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ps = self._plotting_positions(len(self._data))
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if qq:
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qp = "Quantiles"
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plot_type = 'Q-Q'
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x = self._dist.ppf(ps, *fit_params)
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y = data
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else:
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qp = "Percentiles"
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plot_type = 'P-P'
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x = ps
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y = self._dist.cdf(data, *fit_params)
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ax.plot(x, y, '.', label=f'Fitted Distribution {plot_type}',
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color='C0', zorder=1)
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xlim = ax.get_xlim()
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ylim = ax.get_ylim()
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lim = [min(xlim[0], ylim[0]), max(xlim[1], ylim[1])]
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if not qq:
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lim = max(lim[0], 0), min(lim[1], 1)
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if self.discrete and qq:
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q_min, q_max = int(lim[0]), int(lim[1]+1)
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q_ideal = np.arange(q_min, q_max)
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# q_ideal = np.unique(self._dist.ppf(ps, *fit_params))
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ax.plot(q_ideal, q_ideal, 'o', label='Reference', color='k',
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alpha=0.25, markerfacecolor='none', clip_on=True)
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elif self.discrete and not qq:
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# The intent of this is to match the plot that would be produced
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# if x were continuous on [0, 1] and y were cdf(ppf(x)).
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# It can be approximated by letting x = np.linspace(0, 1, 1000),
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# but this might not look great when zooming in. The vertical
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# portions are included to indicate where the transition occurs
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# where the data completely obscures the horizontal portions.
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p_min, p_max = lim
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a, b = self._dist.support(*fit_params)
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p_min = max(p_min, 0 if np.isfinite(a) else 1e-3)
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p_max = min(p_max, 1 if np.isfinite(b) else 1-1e-3)
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q_min, q_max = self._dist.ppf([p_min, p_max], *fit_params)
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qs = np.arange(q_min-1, q_max+1)
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ps = self._dist.cdf(qs, *fit_params)
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ax.step(ps, ps, '-', label='Reference', color='k', alpha=0.25,
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clip_on=True)
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else:
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ax.plot(lim, lim, '-', label='Reference', color='k', alpha=0.25,
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clip_on=True)
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ax.set_xlim(lim)
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ax.set_ylim(lim)
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ax.set_xlabel(rf"Fitted $\tt {self._dist.name}$ Theoretical {qp}")
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ax.set_ylabel(f"Data {qp}")
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ax.set_title(rf"Fitted $\tt {self._dist.name}$ {plot_type} Plot")
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ax.legend(*ax.get_legend_handles_labels())
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ax.set_aspect('equal')
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return ax
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def _qq_plot(self, **kwargs):
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return self._qp_plot(qq=True, **kwargs)
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def _pp_plot(self, **kwargs):
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return self._qp_plot(qq=False, **kwargs)
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def _plotting_positions(self, n, a=.5):
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# See https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot#Plotting_positions
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k = np.arange(1, n+1)
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return (k-a) / (n + 1 - 2*a)
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def _cdf_plot(self, ax, fit_params):
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data = np.sort(self._data)
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ecdf = self._plotting_positions(len(self._data))
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ls = '--' if len(np.unique(data)) < 30 else '.'
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xlabel = 'k' if self.discrete else 'x'
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ax.step(data, ecdf, ls, label='Empirical CDF', color='C1', zorder=0)
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xlim = ax.get_xlim()
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q = np.linspace(*xlim, 300)
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tcdf = self._dist.cdf(q, *fit_params)
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ax.plot(q, tcdf, label='Fitted Distribution CDF', color='C0', zorder=1)
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ax.set_xlim(xlim)
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ax.set_ylim(0, 1)
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ax.set_xlabel(xlabel)
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ax.set_ylabel("CDF")
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ax.set_title(rf"Fitted $\tt {self._dist.name}$ and Empirical CDF")
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handles, labels = ax.get_legend_handles_labels()
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ax.legend(handles[::-1], labels[::-1])
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return ax
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def fit(dist, data, bounds=None, *, guess=None, method='mle',
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optimizer=optimize.differential_evolution):
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r"""Fit a discrete or continuous distribution to data
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Given a distribution, data, and bounds on the parameters of the
|
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distribution, return maximum likelihood estimates of the parameters.
|
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|
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Parameters
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||
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----------
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dist : `scipy.stats.rv_continuous` or `scipy.stats.rv_discrete`
|
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The object representing the distribution to be fit to the data.
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data : 1D array_like
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The data to which the distribution is to be fit. If the data contain
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any of ``np.nan``, ``np.inf``, or -``np.inf``, the fit method will
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raise a ``ValueError``.
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bounds : dict or sequence of tuples, optional
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If a dictionary, each key is the name of a parameter of the
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distribution, and the corresponding value is a tuple containing the
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lower and upper bound on that parameter. If the distribution is
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defined only for a finite range of values of that parameter, no entry
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for that parameter is required; e.g., some distributions have
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parameters which must be on the interval [0, 1]. Bounds for parameters
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location (``loc``) and scale (``scale``) are optional; by default,
|
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they are fixed to 0 and 1, respectively.
|
||
|
|
|
||
|
|
If a sequence, element *i* is a tuple containing the lower and upper
|
||
|
|
bound on the *i*\ th parameter of the distribution. In this case,
|
||
|
|
bounds for *all* distribution shape parameters must be provided.
|
||
|
|
Optionally, bounds for location and scale may follow the
|
||
|
|
distribution shape parameters.
|
||
|
|
|
||
|
|
If a shape is to be held fixed (e.g. if it is known), the
|
||
|
|
lower and upper bounds may be equal. If a user-provided lower or upper
|
||
|
|
bound is beyond a bound of the domain for which the distribution is
|
||
|
|
defined, the bound of the distribution's domain will replace the
|
||
|
|
user-provided value. Similarly, parameters which must be integral
|
||
|
|
will be constrained to integral values within the user-provided bounds.
|
||
|
|
guess : dict or array_like, optional
|
||
|
|
If a dictionary, each key is the name of a parameter of the
|
||
|
|
distribution, and the corresponding value is a guess for the value
|
||
|
|
of the parameter.
|
||
|
|
|
||
|
|
If a sequence, element *i* is a guess for the *i*\ th parameter of the
|
||
|
|
distribution. In this case, guesses for *all* distribution shape
|
||
|
|
parameters must be provided.
|
||
|
|
|
||
|
|
If `guess` is not provided, guesses for the decision variables will
|
||
|
|
not be passed to the optimizer. If `guess` is provided, guesses for
|
||
|
|
any missing parameters will be set at the mean of the lower and
|
||
|
|
upper bounds. Guesses for parameters which must be integral will be
|
||
|
|
rounded to integral values, and guesses that lie outside the
|
||
|
|
intersection of the user-provided bounds and the domain of the
|
||
|
|
distribution will be clipped.
|
||
|
|
method : {'mle', 'mse'}
|
||
|
|
With ``method="mle"`` (default), the fit is computed by minimizing
|
||
|
|
the negative log-likelihood function. A large, finite penalty
|
||
|
|
(rather than infinite negative log-likelihood) is applied for
|
||
|
|
observations beyond the support of the distribution.
|
||
|
|
With ``method="mse"``, the fit is computed by minimizing
|
||
|
|
the negative log-product spacing function. The same penalty is applied
|
||
|
|
for observations beyond the support. We follow the approach of [1]_,
|
||
|
|
which is generalized for samples with repeated observations.
|
||
|
|
optimizer : callable, optional
|
||
|
|
`optimizer` is a callable that accepts the following positional
|
||
|
|
argument.
|
||
|
|
|
||
|
|
fun : callable
|
||
|
|
The objective function to be optimized. `fun` accepts one argument
|
||
|
|
``x``, candidate shape parameters of the distribution, and returns
|
||
|
|
the objective function value given ``x``, `dist`, and the provided
|
||
|
|
`data`.
|
||
|
|
The job of `optimizer` is to find values of the decision variables
|
||
|
|
that minimizes `fun`.
|
||
|
|
|
||
|
|
`optimizer` must also accept the following keyword argument.
|
||
|
|
|
||
|
|
bounds : sequence of tuples
|
||
|
|
The bounds on values of the decision variables; each element will
|
||
|
|
be a tuple containing the lower and upper bound on a decision
|
||
|
|
variable.
|
||
|
|
|
||
|
|
If `guess` is provided, `optimizer` must also accept the following
|
||
|
|
keyword argument.
|
||
|
|
|
||
|
|
x0 : array_like
|
||
|
|
The guesses for each decision variable.
|
||
|
|
|
||
|
|
If the distribution has any shape parameters that must be integral or
|
||
|
|
if the distribution is discrete and the location parameter is not
|
||
|
|
fixed, `optimizer` must also accept the following keyword argument.
|
||
|
|
|
||
|
|
integrality : array_like of bools
|
||
|
|
For each decision variable, True if the decision variable
|
||
|
|
must be constrained to integer values and False if the decision
|
||
|
|
variable is continuous.
|
||
|
|
|
||
|
|
`optimizer` must return an object, such as an instance of
|
||
|
|
`scipy.optimize.OptimizeResult`, which holds the optimal values of
|
||
|
|
the decision variables in an attribute ``x``. If attributes
|
||
|
|
``fun``, ``status``, or ``message`` are provided, they will be
|
||
|
|
included in the result object returned by `fit`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
result : `~scipy.stats._result_classes.FitResult`
|
||
|
|
An object with the following fields.
|
||
|
|
|
||
|
|
params : namedtuple
|
||
|
|
A namedtuple containing the maximum likelihood estimates of the
|
||
|
|
shape parameters, location, and (if applicable) scale of the
|
||
|
|
distribution.
|
||
|
|
success : bool or None
|
||
|
|
Whether the optimizer considered the optimization to terminate
|
||
|
|
successfully or not.
|
||
|
|
message : str or None
|
||
|
|
Any status message provided by the optimizer.
|
||
|
|
|
||
|
|
The object has the following method:
|
||
|
|
|
||
|
|
nllf(params=None, data=None)
|
||
|
|
By default, the negative log-likehood function at the fitted
|
||
|
|
`params` for the given `data`. Accepts a tuple containing
|
||
|
|
alternative shapes, location, and scale of the distribution and
|
||
|
|
an array of alternative data.
|
||
|
|
|
||
|
|
plot(ax=None)
|
||
|
|
Superposes the PDF/PMF of the fitted distribution over a normalized
|
||
|
|
histogram of the data.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
rv_continuous, rv_discrete
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Optimization is more likely to converge to the maximum likelihood estimate
|
||
|
|
when the user provides tight bounds containing the maximum likelihood
|
||
|
|
estimate. For example, when fitting a binomial distribution to data, the
|
||
|
|
number of experiments underlying each sample may be known, in which case
|
||
|
|
the corresponding shape parameter ``n`` can be fixed.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Shao, Yongzhao, and Marjorie G. Hahn. "Maximum product of spacings
|
||
|
|
method: a unified formulation with illustration of strong
|
||
|
|
consistency." Illinois Journal of Mathematics 43.3 (1999): 489-499.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
Suppose we wish to fit a distribution to the following data.
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from scipy import stats
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> dist = stats.nbinom
|
||
|
|
>>> shapes = (5, 0.5)
|
||
|
|
>>> data = dist.rvs(*shapes, size=1000, random_state=rng)
|
||
|
|
|
||
|
|
Suppose we do not know how the data were generated, but we suspect that
|
||
|
|
it follows a negative binomial distribution with parameters *n* and *p*\.
|
||
|
|
(See `scipy.stats.nbinom`.) We believe that the parameter *n* was fewer
|
||
|
|
than 30, and we know that the parameter *p* must lie on the interval
|
||
|
|
[0, 1]. We record this information in a variable `bounds` and pass
|
||
|
|
this information to `fit`.
|
||
|
|
|
||
|
|
>>> bounds = [(0, 30), (0, 1)]
|
||
|
|
>>> res = stats.fit(dist, data, bounds)
|
||
|
|
|
||
|
|
`fit` searches within the user-specified `bounds` for the
|
||
|
|
values that best match the data (in the sense of maximum likelihood
|
||
|
|
estimation). In this case, it found shape values similar to those
|
||
|
|
from which the data were actually generated.
|
||
|
|
|
||
|
|
>>> res.params
|
||
|
|
FitParams(n=5.0, p=0.5028157644634368, loc=0.0) # may vary
|
||
|
|
|
||
|
|
We can visualize the results by superposing the probability mass function
|
||
|
|
of the distribution (with the shapes fit to the data) over a normalized
|
||
|
|
histogram of the data.
|
||
|
|
|
||
|
|
>>> import matplotlib.pyplot as plt # matplotlib must be installed to plot
|
||
|
|
>>> res.plot()
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
Note that the estimate for *n* was exactly integral; this is because
|
||
|
|
the domain of the `nbinom` PMF includes only integral *n*, and the `nbinom`
|
||
|
|
object "knows" that. `nbinom` also knows that the shape *p* must be a
|
||
|
|
value between 0 and 1. In such a case - when the domain of the distribution
|
||
|
|
with respect to a parameter is finite - we are not required to specify
|
||
|
|
bounds for the parameter.
|
||
|
|
|
||
|
|
>>> bounds = {'n': (0, 30)} # omit parameter p using a `dict`
|
||
|
|
>>> res2 = stats.fit(dist, data, bounds)
|
||
|
|
>>> res2.params
|
||
|
|
FitParams(n=5.0, p=0.5016492009232932, loc=0.0) # may vary
|
||
|
|
|
||
|
|
If we wish to force the distribution to be fit with *n* fixed at 6, we can
|
||
|
|
set both the lower and upper bounds on *n* to 6. Note, however, that the
|
||
|
|
value of the objective function being optimized is typically worse (higher)
|
||
|
|
in this case.
|
||
|
|
|
||
|
|
>>> bounds = {'n': (6, 6)} # fix parameter `n`
|
||
|
|
>>> res3 = stats.fit(dist, data, bounds)
|
||
|
|
>>> res3.params
|
||
|
|
FitParams(n=6.0, p=0.5486556076755706, loc=0.0) # may vary
|
||
|
|
>>> res3.nllf() > res.nllf()
|
||
|
|
True # may vary
|
||
|
|
|
||
|
|
Note that the numerical results of the previous examples are typical, but
|
||
|
|
they may vary because the default optimizer used by `fit`,
|
||
|
|
`scipy.optimize.differential_evolution`, is stochastic. However, we can
|
||
|
|
customize the settings used by the optimizer to ensure reproducibility -
|
||
|
|
or even use a different optimizer entirely - using the `optimizer`
|
||
|
|
parameter.
|
||
|
|
|
||
|
|
>>> from scipy.optimize import differential_evolution
|
||
|
|
>>> rng = np.random.default_rng(767585560716548)
|
||
|
|
>>> def optimizer(fun, bounds, *, integrality):
|
||
|
|
... return differential_evolution(fun, bounds, strategy='best2bin',
|
||
|
|
... seed=rng, integrality=integrality)
|
||
|
|
>>> bounds = [(0, 30), (0, 1)]
|
||
|
|
>>> res4 = stats.fit(dist, data, bounds, optimizer=optimizer)
|
||
|
|
>>> res4.params
|
||
|
|
FitParams(n=5.0, p=0.5015183149259951, loc=0.0)
|
||
|
|
|
||
|
|
"""
|
||
|
|
# --- Input Validation / Standardization --- #
|
||
|
|
user_bounds = bounds
|
||
|
|
user_guess = guess
|
||
|
|
|
||
|
|
# distribution input validation and information collection
|
||
|
|
if hasattr(dist, "pdf"): # can't use isinstance for types
|
||
|
|
default_bounds = {'loc': (0, 0), 'scale': (1, 1)}
|
||
|
|
discrete = False
|
||
|
|
elif hasattr(dist, "pmf"):
|
||
|
|
default_bounds = {'loc': (0, 0)}
|
||
|
|
discrete = True
|
||
|
|
else:
|
||
|
|
message = ("`dist` must be an instance of `rv_continuous` "
|
||
|
|
"or `rv_discrete.`")
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
try:
|
||
|
|
param_info = dist._param_info()
|
||
|
|
except AttributeError as e:
|
||
|
|
message = (f"Distribution `{dist.name}` is not yet supported by "
|
||
|
|
"`scipy.stats.fit` because shape information has "
|
||
|
|
"not been defined.")
|
||
|
|
raise ValueError(message) from e
|
||
|
|
|
||
|
|
# data input validation
|
||
|
|
data = np.asarray(data)
|
||
|
|
if data.ndim != 1:
|
||
|
|
message = "`data` must be exactly one-dimensional."
|
||
|
|
raise ValueError(message)
|
||
|
|
if not (np.issubdtype(data.dtype, np.number)
|
||
|
|
and np.all(np.isfinite(data))):
|
||
|
|
message = "All elements of `data` must be finite numbers."
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
# bounds input validation and information collection
|
||
|
|
n_params = len(param_info)
|
||
|
|
n_shapes = n_params - (1 if discrete else 2)
|
||
|
|
param_list = [param.name for param in param_info]
|
||
|
|
param_names = ", ".join(param_list)
|
||
|
|
shape_names = ", ".join(param_list[:n_shapes])
|
||
|
|
|
||
|
|
if user_bounds is None:
|
||
|
|
user_bounds = {}
|
||
|
|
|
||
|
|
if isinstance(user_bounds, dict):
|
||
|
|
default_bounds.update(user_bounds)
|
||
|
|
user_bounds = default_bounds
|
||
|
|
user_bounds_array = np.empty((n_params, 2))
|
||
|
|
for i in range(n_params):
|
||
|
|
param_name = param_info[i].name
|
||
|
|
user_bound = user_bounds.pop(param_name, None)
|
||
|
|
if user_bound is None:
|
||
|
|
user_bound = param_info[i].domain
|
||
|
|
user_bounds_array[i] = user_bound
|
||
|
|
if user_bounds:
|
||
|
|
message = ("Bounds provided for the following unrecognized "
|
||
|
|
f"parameters will be ignored: {set(user_bounds)}")
|
||
|
|
warnings.warn(message, RuntimeWarning, stacklevel=2)
|
||
|
|
|
||
|
|
else:
|
||
|
|
try:
|
||
|
|
user_bounds = np.asarray(user_bounds, dtype=float)
|
||
|
|
if user_bounds.size == 0:
|
||
|
|
user_bounds = np.empty((0, 2))
|
||
|
|
except ValueError as e:
|
||
|
|
message = ("Each element of a `bounds` sequence must be a tuple "
|
||
|
|
"containing two elements: the lower and upper bound of "
|
||
|
|
"a distribution parameter.")
|
||
|
|
raise ValueError(message) from e
|
||
|
|
if (user_bounds.ndim != 2 or user_bounds.shape[1] != 2):
|
||
|
|
message = ("Each element of `bounds` must be a tuple specifying "
|
||
|
|
"the lower and upper bounds of a shape parameter")
|
||
|
|
raise ValueError(message)
|
||
|
|
if user_bounds.shape[0] < n_shapes:
|
||
|
|
message = (f"A `bounds` sequence must contain at least {n_shapes} "
|
||
|
|
"elements: tuples specifying the lower and upper "
|
||
|
|
f"bounds of all shape parameters {shape_names}.")
|
||
|
|
raise ValueError(message)
|
||
|
|
if user_bounds.shape[0] > n_params:
|
||
|
|
message = ("A `bounds` sequence may not contain more than "
|
||
|
|
f"{n_params} elements: tuples specifying the lower and "
|
||
|
|
"upper bounds of distribution parameters "
|
||
|
|
f"{param_names}.")
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
user_bounds_array = np.empty((n_params, 2))
|
||
|
|
user_bounds_array[n_shapes:] = list(default_bounds.values())
|
||
|
|
user_bounds_array[:len(user_bounds)] = user_bounds
|
||
|
|
|
||
|
|
user_bounds = user_bounds_array
|
||
|
|
validated_bounds = []
|
||
|
|
for i in range(n_params):
|
||
|
|
name = param_info[i].name
|
||
|
|
user_bound = user_bounds_array[i]
|
||
|
|
param_domain = param_info[i].domain
|
||
|
|
integral = param_info[i].integrality
|
||
|
|
combined = _combine_bounds(name, user_bound, param_domain, integral)
|
||
|
|
validated_bounds.append(combined)
|
||
|
|
|
||
|
|
bounds = np.asarray(validated_bounds)
|
||
|
|
integrality = [param.integrality for param in param_info]
|
||
|
|
|
||
|
|
# guess input validation
|
||
|
|
|
||
|
|
if user_guess is None:
|
||
|
|
guess_array = None
|
||
|
|
elif isinstance(user_guess, dict):
|
||
|
|
default_guess = {param.name: np.mean(bound)
|
||
|
|
for param, bound in zip(param_info, bounds)}
|
||
|
|
unrecognized = set(user_guess) - set(default_guess)
|
||
|
|
if unrecognized:
|
||
|
|
message = ("Guesses provided for the following unrecognized "
|
||
|
|
f"parameters will be ignored: {unrecognized}")
|
||
|
|
warnings.warn(message, RuntimeWarning, stacklevel=2)
|
||
|
|
default_guess.update(user_guess)
|
||
|
|
|
||
|
|
message = ("Each element of `guess` must be a scalar "
|
||
|
|
"guess for a distribution parameter.")
|
||
|
|
try:
|
||
|
|
guess_array = np.asarray([default_guess[param.name]
|
||
|
|
for param in param_info], dtype=float)
|
||
|
|
except ValueError as e:
|
||
|
|
raise ValueError(message) from e
|
||
|
|
|
||
|
|
else:
|
||
|
|
message = ("Each element of `guess` must be a scalar "
|
||
|
|
"guess for a distribution parameter.")
|
||
|
|
try:
|
||
|
|
user_guess = np.asarray(user_guess, dtype=float)
|
||
|
|
except ValueError as e:
|
||
|
|
raise ValueError(message) from e
|
||
|
|
if user_guess.ndim != 1:
|
||
|
|
raise ValueError(message)
|
||
|
|
if user_guess.shape[0] < n_shapes:
|
||
|
|
message = (f"A `guess` sequence must contain at least {n_shapes} "
|
||
|
|
"elements: scalar guesses for the distribution shape "
|
||
|
|
f"parameters {shape_names}.")
|
||
|
|
raise ValueError(message)
|
||
|
|
if user_guess.shape[0] > n_params:
|
||
|
|
message = ("A `guess` sequence may not contain more than "
|
||
|
|
f"{n_params} elements: scalar guesses for the "
|
||
|
|
f"distribution parameters {param_names}.")
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
guess_array = np.mean(bounds, axis=1)
|
||
|
|
guess_array[:len(user_guess)] = user_guess
|
||
|
|
|
||
|
|
if guess_array is not None:
|
||
|
|
guess_rounded = guess_array.copy()
|
||
|
|
|
||
|
|
guess_rounded[integrality] = np.round(guess_rounded[integrality])
|
||
|
|
rounded = np.where(guess_rounded != guess_array)[0]
|
||
|
|
for i in rounded:
|
||
|
|
message = (f"Guess for parameter `{param_info[i].name}` "
|
||
|
|
f"rounded from {guess_array[i]} to {guess_rounded[i]}.")
|
||
|
|
warnings.warn(message, RuntimeWarning, stacklevel=2)
|
||
|
|
|
||
|
|
guess_clipped = np.clip(guess_rounded, bounds[:, 0], bounds[:, 1])
|
||
|
|
clipped = np.where(guess_clipped != guess_rounded)[0]
|
||
|
|
for i in clipped:
|
||
|
|
message = (f"Guess for parameter `{param_info[i].name}` "
|
||
|
|
f"clipped from {guess_rounded[i]} to "
|
||
|
|
f"{guess_clipped[i]}.")
|
||
|
|
warnings.warn(message, RuntimeWarning, stacklevel=2)
|
||
|
|
|
||
|
|
guess = guess_clipped
|
||
|
|
else:
|
||
|
|
guess = None
|
||
|
|
|
||
|
|
# --- Fitting --- #
|
||
|
|
def nllf(free_params, data=data): # bind data NOW
|
||
|
|
with np.errstate(invalid='ignore', divide='ignore'):
|
||
|
|
return dist._penalized_nnlf(free_params, data)
|
||
|
|
|
||
|
|
def nlpsf(free_params, data=data): # bind data NOW
|
||
|
|
with np.errstate(invalid='ignore', divide='ignore'):
|
||
|
|
return dist._penalized_nlpsf(free_params, data)
|
||
|
|
|
||
|
|
methods = {'mle': nllf, 'mse': nlpsf}
|
||
|
|
objective = methods[method.lower()]
|
||
|
|
|
||
|
|
with np.errstate(invalid='ignore', divide='ignore'):
|
||
|
|
kwds = {}
|
||
|
|
if bounds is not None:
|
||
|
|
kwds['bounds'] = bounds
|
||
|
|
if np.any(integrality):
|
||
|
|
kwds['integrality'] = integrality
|
||
|
|
if guess is not None:
|
||
|
|
kwds['x0'] = guess
|
||
|
|
res = optimizer(objective, **kwds)
|
||
|
|
|
||
|
|
return FitResult(dist, data, discrete, res)
|
||
|
|
|
||
|
|
|
||
|
|
GoodnessOfFitResult = namedtuple('GoodnessOfFitResult',
|
||
|
|
('fit_result', 'statistic', 'pvalue',
|
||
|
|
'null_distribution'))
|
||
|
|
|
||
|
|
|
||
|
|
def goodness_of_fit(dist, data, *, known_params=None, fit_params=None,
|
||
|
|
guessed_params=None, statistic='ad', n_mc_samples=9999,
|
||
|
|
random_state=None):
|
||
|
|
r"""
|
||
|
|
Perform a goodness of fit test comparing data to a distribution family.
|
||
|
|
|
||
|
|
Given a distribution family and data, perform a test of the null hypothesis
|
||
|
|
that the data were drawn from a distribution in that family. Any known
|
||
|
|
parameters of the distribution may be specified. Remaining parameters of
|
||
|
|
the distribution will be fit to the data, and the p-value of the test
|
||
|
|
is computed accordingly. Several statistics for comparing the distribution
|
||
|
|
to data are available.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
dist : `scipy.stats.rv_continuous`
|
||
|
|
The object representing the distribution family under the null
|
||
|
|
hypothesis.
|
||
|
|
data : 1D array_like
|
||
|
|
Finite, uncensored data to be tested.
|
||
|
|
known_params : dict, optional
|
||
|
|
A dictionary containing name-value pairs of known distribution
|
||
|
|
parameters. Monte Carlo samples are randomly drawn from the
|
||
|
|
null-hypothesized distribution with these values of the parameters.
|
||
|
|
Before the statistic is evaluated for each Monte Carlo sample, only
|
||
|
|
remaining unknown parameters of the null-hypothesized distribution
|
||
|
|
family are fit to the samples; the known parameters are held fixed.
|
||
|
|
If all parameters of the distribution family are known, then the step
|
||
|
|
of fitting the distribution family to each sample is omitted.
|
||
|
|
fit_params : dict, optional
|
||
|
|
A dictionary containing name-value pairs of distribution parameters
|
||
|
|
that have already been fit to the data, e.g. using `scipy.stats.fit`
|
||
|
|
or the ``fit`` method of `dist`. Monte Carlo samples are drawn from the
|
||
|
|
null-hypothesized distribution with these specified values of the
|
||
|
|
parameter. On those Monte Carlo samples, however, these and all other
|
||
|
|
unknown parameters of the null-hypothesized distribution family are
|
||
|
|
fit before the statistic is evaluated.
|
||
|
|
guessed_params : dict, optional
|
||
|
|
A dictionary containing name-value pairs of distribution parameters
|
||
|
|
which have been guessed. These parameters are always considered as
|
||
|
|
free parameters and are fit both to the provided `data` as well as
|
||
|
|
to the Monte Carlo samples drawn from the null-hypothesized
|
||
|
|
distribution. The purpose of these `guessed_params` is to be used as
|
||
|
|
initial values for the numerical fitting procedure.
|
||
|
|
statistic : {"ad", "ks", "cvm", "filliben"} or callable, optional
|
||
|
|
The statistic used to compare data to a distribution after fitting
|
||
|
|
unknown parameters of the distribution family to the data. The
|
||
|
|
Anderson-Darling ("ad") [1]_, Kolmogorov-Smirnov ("ks") [1]_,
|
||
|
|
Cramer-von Mises ("cvm") [1]_, and Filliben ("filliben") [7]_
|
||
|
|
statistics are available. Alternatively, a callable with signature
|
||
|
|
``(dist, data, axis)`` may be supplied to compute the statistic. Here
|
||
|
|
``dist`` is a frozen distribution object (potentially with array
|
||
|
|
parameters), ``data`` is an array of Monte Carlo samples (of
|
||
|
|
compatible shape), and ``axis`` is the axis of ``data`` along which
|
||
|
|
the statistic must be computed.
|
||
|
|
n_mc_samples : int, default: 9999
|
||
|
|
The number of Monte Carlo samples drawn from the null hypothesized
|
||
|
|
distribution to form the null distribution of the statistic. The
|
||
|
|
sample size of each is the same as the given `data`.
|
||
|
|
random_state : {None, int, `numpy.random.Generator`,
|
||
|
|
`numpy.random.RandomState`}, optional
|
||
|
|
|
||
|
|
Pseudorandom number generator state used to generate the Monte Carlo
|
||
|
|
samples.
|
||
|
|
|
||
|
|
If `random_state` is ``None`` (default), the
|
||
|
|
`numpy.random.RandomState` singleton is used.
|
||
|
|
If `random_state` is an int, a new ``RandomState`` instance is used,
|
||
|
|
seeded with `random_state`.
|
||
|
|
If `random_state` is already a ``Generator`` or ``RandomState``
|
||
|
|
instance, then the provided instance is used.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : GoodnessOfFitResult
|
||
|
|
An object with the following attributes.
|
||
|
|
|
||
|
|
fit_result : `~scipy.stats._result_classes.FitResult`
|
||
|
|
An object representing the fit of the provided `dist` to `data`.
|
||
|
|
This object includes the values of distribution family parameters
|
||
|
|
that fully define the null-hypothesized distribution, that is,
|
||
|
|
the distribution from which Monte Carlo samples are drawn.
|
||
|
|
statistic : float
|
||
|
|
The value of the statistic comparing provided `data` to the
|
||
|
|
null-hypothesized distribution.
|
||
|
|
pvalue : float
|
||
|
|
The proportion of elements in the null distribution with
|
||
|
|
statistic values at least as extreme as the statistic value of the
|
||
|
|
provided `data`.
|
||
|
|
null_distribution : ndarray
|
||
|
|
The value of the statistic for each Monte Carlo sample
|
||
|
|
drawn from the null-hypothesized distribution.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is a generalized Monte Carlo goodness-of-fit procedure, special cases
|
||
|
|
of which correspond with various Anderson-Darling tests, Lilliefors' test,
|
||
|
|
etc. The test is described in [2]_, [3]_, and [4]_ as a parametric
|
||
|
|
bootstrap test. This is a Monte Carlo test in which parameters that
|
||
|
|
specify the distribution from which samples are drawn have been estimated
|
||
|
|
from the data. We describe the test using "Monte Carlo" rather than
|
||
|
|
"parametric bootstrap" throughout to avoid confusion with the more familiar
|
||
|
|
nonparametric bootstrap, and describe how the test is performed below.
|
||
|
|
|
||
|
|
*Traditional goodness of fit tests*
|
||
|
|
|
||
|
|
Traditionally, critical values corresponding with a fixed set of
|
||
|
|
significance levels are pre-calculated using Monte Carlo methods. Users
|
||
|
|
perform the test by calculating the value of the test statistic only for
|
||
|
|
their observed `data` and comparing this value to tabulated critical
|
||
|
|
values. This practice is not very flexible, as tables are not available for
|
||
|
|
all distributions and combinations of known and unknown parameter values.
|
||
|
|
Also, results can be inaccurate when critical values are interpolated from
|
||
|
|
limited tabulated data to correspond with the user's sample size and
|
||
|
|
fitted parameter values. To overcome these shortcomings, this function
|
||
|
|
allows the user to perform the Monte Carlo trials adapted to their
|
||
|
|
particular data.
|
||
|
|
|
||
|
|
*Algorithmic overview*
|
||
|
|
|
||
|
|
In brief, this routine executes the following steps:
|
||
|
|
|
||
|
|
1. Fit unknown parameters to the given `data`, thereby forming the
|
||
|
|
"null-hypothesized" distribution, and compute the statistic of
|
||
|
|
this pair of data and distribution.
|
||
|
|
2. Draw random samples from this null-hypothesized distribution.
|
||
|
|
3. Fit the unknown parameters to each random sample.
|
||
|
|
4. Calculate the statistic between each sample and the distribution that
|
||
|
|
has been fit to the sample.
|
||
|
|
5. Compare the value of the statistic corresponding with `data` from (1)
|
||
|
|
against the values of the statistic corresponding with the random
|
||
|
|
samples from (4). The p-value is the proportion of samples with a
|
||
|
|
statistic value greater than or equal to the statistic of the observed
|
||
|
|
data.
|
||
|
|
|
||
|
|
In more detail, the steps are as follows.
|
||
|
|
|
||
|
|
First, any unknown parameters of the distribution family specified by
|
||
|
|
`dist` are fit to the provided `data` using maximum likelihood estimation.
|
||
|
|
(One exception is the normal distribution with unknown location and scale:
|
||
|
|
we use the bias-corrected standard deviation ``np.std(data, ddof=1)`` for
|
||
|
|
the scale as recommended in [1]_.)
|
||
|
|
These values of the parameters specify a particular member of the
|
||
|
|
distribution family referred to as the "null-hypothesized distribution",
|
||
|
|
that is, the distribution from which the data were sampled under the null
|
||
|
|
hypothesis. The `statistic`, which compares data to a distribution, is
|
||
|
|
computed between `data` and the null-hypothesized distribution.
|
||
|
|
|
||
|
|
Next, many (specifically `n_mc_samples`) new samples, each containing the
|
||
|
|
same number of observations as `data`, are drawn from the
|
||
|
|
null-hypothesized distribution. All unknown parameters of the distribution
|
||
|
|
family `dist` are fit to *each resample*, and the `statistic` is computed
|
||
|
|
between each sample and its corresponding fitted distribution. These
|
||
|
|
values of the statistic form the Monte Carlo null distribution (not to be
|
||
|
|
confused with the "null-hypothesized distribution" above).
|
||
|
|
|
||
|
|
The p-value of the test is the proportion of statistic values in the Monte
|
||
|
|
Carlo null distribution that are at least as extreme as the statistic value
|
||
|
|
of the provided `data`. More precisely, the p-value is given by
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
p = \frac{b + 1}
|
||
|
|
{m + 1}
|
||
|
|
|
||
|
|
where :math:`b` is the number of statistic values in the Monte Carlo null
|
||
|
|
distribution that are greater than or equal to the statistic value
|
||
|
|
calculated for `data`, and :math:`m` is the number of elements in the
|
||
|
|
Monte Carlo null distribution (`n_mc_samples`). The addition of :math:`1`
|
||
|
|
to the numerator and denominator can be thought of as including the
|
||
|
|
value of the statistic corresponding with `data` in the null distribution,
|
||
|
|
but a more formal explanation is given in [5]_.
|
||
|
|
|
||
|
|
*Limitations*
|
||
|
|
|
||
|
|
The test can be very slow for some distribution families because unknown
|
||
|
|
parameters of the distribution family must be fit to each of the Monte
|
||
|
|
Carlo samples, and for most distributions in SciPy, distribution fitting
|
||
|
|
performed via numerical optimization.
|
||
|
|
|
||
|
|
*Anti-Pattern*
|
||
|
|
|
||
|
|
For this reason, it may be tempting
|
||
|
|
to treat parameters of the distribution pre-fit to `data` (by the user)
|
||
|
|
as though they were `known_params`, as specification of all parameters of
|
||
|
|
the distribution precludes the need to fit the distribution to each Monte
|
||
|
|
Carlo sample. (This is essentially how the original Kilmogorov-Smirnov
|
||
|
|
test is performed.) Although such a test can provide evidence against the
|
||
|
|
null hypothesis, the test is conservative in the sense that small p-values
|
||
|
|
will tend to (greatly) *overestimate* the probability of making a type I
|
||
|
|
error (that is, rejecting the null hypothesis although it is true), and the
|
||
|
|
power of the test is low (that is, it is less likely to reject the null
|
||
|
|
hypothesis even when the null hypothesis is false).
|
||
|
|
This is because the Monte Carlo samples are less likely to agree with the
|
||
|
|
null-hypothesized distribution as well as `data`. This tends to increase
|
||
|
|
the values of the statistic recorded in the null distribution, so that a
|
||
|
|
larger number of them exceed the value of statistic for `data`, thereby
|
||
|
|
inflating the p-value.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] M. A. Stephens (1974). "EDF Statistics for Goodness of Fit and
|
||
|
|
Some Comparisons." Journal of the American Statistical Association,
|
||
|
|
Vol. 69, pp. 730-737.
|
||
|
|
.. [2] W. Stute, W. G. Manteiga, and M. P. Quindimil (1993).
|
||
|
|
"Bootstrap based goodness-of-fit-tests." Metrika 40.1: 243-256.
|
||
|
|
.. [3] C. Genest, & B Rémillard. (2008). "Validity of the parametric
|
||
|
|
bootstrap for goodness-of-fit testing in semiparametric models."
|
||
|
|
Annales de l'IHP Probabilités et statistiques. Vol. 44. No. 6.
|
||
|
|
.. [4] I. Kojadinovic and J. Yan (2012). "Goodness-of-fit testing based on
|
||
|
|
a weighted bootstrap: A fast large-sample alternative to the
|
||
|
|
parametric bootstrap." Canadian Journal of Statistics 40.3: 480-500.
|
||
|
|
.. [5] B. Phipson and G. K. Smyth (2010). "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.
|
||
|
|
.. [6] H. W. Lilliefors (1967). "On the Kolmogorov-Smirnov test for
|
||
|
|
normality with mean and variance unknown." Journal of the American
|
||
|
|
statistical Association 62.318: 399-402.
|
||
|
|
.. [7] Filliben, James J. "The probability plot correlation coefficient
|
||
|
|
test for normality." Technometrics 17.1 (1975): 111-117.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
A well-known test of the null hypothesis that data were drawn from a
|
||
|
|
given distribution is the Kolmogorov-Smirnov (KS) test, available in SciPy
|
||
|
|
as `scipy.stats.ks_1samp`. Suppose we wish to test whether the following
|
||
|
|
data:
|
||
|
|
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from scipy import stats
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> x = stats.uniform.rvs(size=75, random_state=rng)
|
||
|
|
|
||
|
|
were sampled from a normal distribution. To perform a KS test, the
|
||
|
|
empirical distribution function of the observed data will be compared
|
||
|
|
against the (theoretical) cumulative distribution function of a normal
|
||
|
|
distribution. Of course, to do this, the normal distribution under the null
|
||
|
|
hypothesis must be fully specified. This is commonly done by first fitting
|
||
|
|
the ``loc`` and ``scale`` parameters of the distribution to the observed
|
||
|
|
data, then performing the test.
|
||
|
|
|
||
|
|
>>> loc, scale = np.mean(x), np.std(x, ddof=1)
|
||
|
|
>>> cdf = stats.norm(loc, scale).cdf
|
||
|
|
>>> stats.ks_1samp(x, cdf)
|
||
|
|
KstestResult(statistic=0.1119257570456813,
|
||
|
|
pvalue=0.2827756409939257,
|
||
|
|
statistic_location=0.7751845155861765,
|
||
|
|
statistic_sign=-1)
|
||
|
|
|
||
|
|
An advantage of the KS-test is that the p-value - the probability of
|
||
|
|
obtaining a value of the test statistic under the null hypothesis as
|
||
|
|
extreme as the value obtained from the observed data - can be calculated
|
||
|
|
exactly and efficiently. `goodness_of_fit` can only approximate these
|
||
|
|
results.
|
||
|
|
|
||
|
|
>>> known_params = {'loc': loc, 'scale': scale}
|
||
|
|
>>> res = stats.goodness_of_fit(stats.norm, x, known_params=known_params,
|
||
|
|
... statistic='ks', random_state=rng)
|
||
|
|
>>> res.statistic, res.pvalue
|
||
|
|
(0.1119257570456813, 0.2788)
|
||
|
|
|
||
|
|
The statistic matches exactly, but the p-value is estimated by forming
|
||
|
|
a "Monte Carlo null distribution", that is, by explicitly drawing random
|
||
|
|
samples from `scipy.stats.norm` with the provided parameters and
|
||
|
|
calculating the stastic for each. The fraction of these statistic values
|
||
|
|
at least as extreme as ``res.statistic`` approximates the exact p-value
|
||
|
|
calculated by `scipy.stats.ks_1samp`.
|
||
|
|
|
||
|
|
However, in many cases, we would prefer to test only that the data were
|
||
|
|
sampled from one of *any* member of the normal distribution family, not
|
||
|
|
specifically from the normal distribution with the location and scale
|
||
|
|
fitted to the observed sample. In this case, Lilliefors [6]_ argued that
|
||
|
|
the KS test is far too conservative (that is, the p-value overstates
|
||
|
|
the actual probability of rejecting a true null hypothesis) and thus lacks
|
||
|
|
power - the ability to reject the null hypothesis when the null hypothesis
|
||
|
|
is actually false.
|
||
|
|
Indeed, our p-value above is approximately 0.28, which is far too large
|
||
|
|
to reject the null hypothesis at any common significance level.
|
||
|
|
|
||
|
|
Consider why this might be. Note that in the KS test above, the statistic
|
||
|
|
always compares data against the CDF of a normal distribution fitted to the
|
||
|
|
*observed data*. This tends to reduce the value of the statistic for the
|
||
|
|
observed data, but it is "unfair" when computing the statistic for other
|
||
|
|
samples, such as those we randomly draw to form the Monte Carlo null
|
||
|
|
distribution. It is easy to correct for this: whenever we compute the KS
|
||
|
|
statistic of a sample, we use the CDF of a normal distribution fitted
|
||
|
|
to *that sample*. The null distribution in this case has not been
|
||
|
|
calculated exactly and is tyically approximated using Monte Carlo methods
|
||
|
|
as described above. This is where `goodness_of_fit` excels.
|
||
|
|
|
||
|
|
>>> res = stats.goodness_of_fit(stats.norm, x, statistic='ks',
|
||
|
|
... random_state=rng)
|
||
|
|
>>> res.statistic, res.pvalue
|
||
|
|
(0.1119257570456813, 0.0196)
|
||
|
|
|
||
|
|
Indeed, this p-value is much smaller, and small enough to (correctly)
|
||
|
|
reject the null hypothesis at common significance levels, including 5% and
|
||
|
|
2.5%.
|
||
|
|
|
||
|
|
However, the KS statistic is not very sensitive to all deviations from
|
||
|
|
normality. The original advantage of the KS statistic was the ability
|
||
|
|
to compute the null distribution theoretically, but a more sensitive
|
||
|
|
statistic - resulting in a higher test power - can be used now that we can
|
||
|
|
approximate the null distribution
|
||
|
|
computationally. The Anderson-Darling statistic [1]_ tends to be more
|
||
|
|
sensitive, and critical values of the this statistic have been tabulated
|
||
|
|
for various significance levels and sample sizes using Monte Carlo methods.
|
||
|
|
|
||
|
|
>>> res = stats.anderson(x, 'norm')
|
||
|
|
>>> print(res.statistic)
|
||
|
|
1.2139573337497467
|
||
|
|
>>> print(res.critical_values)
|
||
|
|
[0.549 0.625 0.75 0.875 1.041]
|
||
|
|
>>> print(res.significance_level)
|
||
|
|
[15. 10. 5. 2.5 1. ]
|
||
|
|
|
||
|
|
Here, the observed value of the statistic exceeds the critical value
|
||
|
|
corresponding with a 1% significance level. This tells us that the p-value
|
||
|
|
of the observed data is less than 1%, but what is it? We could interpolate
|
||
|
|
from these (already-interpolated) values, but `goodness_of_fit` can
|
||
|
|
estimate it directly.
|
||
|
|
|
||
|
|
>>> res = stats.goodness_of_fit(stats.norm, x, statistic='ad',
|
||
|
|
... random_state=rng)
|
||
|
|
>>> res.statistic, res.pvalue
|
||
|
|
(1.2139573337497467, 0.0034)
|
||
|
|
|
||
|
|
A further advantage is that use of `goodness_of_fit` is not limited to
|
||
|
|
a particular set of distributions or conditions on which parameters
|
||
|
|
are known versus which must be estimated from data. Instead,
|
||
|
|
`goodness_of_fit` can estimate p-values relatively quickly for any
|
||
|
|
distribution with a sufficiently fast and reliable ``fit`` method. For
|
||
|
|
instance, here we perform a goodness of fit test using the Cramer-von Mises
|
||
|
|
statistic against the Rayleigh distribution with known location and unknown
|
||
|
|
scale.
|
||
|
|
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> x = stats.chi(df=2.2, loc=0, scale=2).rvs(size=1000, random_state=rng)
|
||
|
|
>>> res = stats.goodness_of_fit(stats.rayleigh, x, statistic='cvm',
|
||
|
|
... known_params={'loc': 0}, random_state=rng)
|
||
|
|
|
||
|
|
This executes fairly quickly, but to check the reliability of the ``fit``
|
||
|
|
method, we should inspect the fit result.
|
||
|
|
|
||
|
|
>>> res.fit_result # location is as specified, and scale is reasonable
|
||
|
|
params: FitParams(loc=0.0, scale=2.1026719844231243)
|
||
|
|
success: True
|
||
|
|
message: 'The fit was performed successfully.'
|
||
|
|
>>> import matplotlib.pyplot as plt # matplotlib must be installed to plot
|
||
|
|
>>> res.fit_result.plot()
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
If the distribution is not fit to the observed data as well as possible,
|
||
|
|
the test may not control the type I error rate, that is, the chance of
|
||
|
|
rejecting the null hypothesis even when it is true.
|
||
|
|
|
||
|
|
We should also look for extreme outliers in the null distribution that
|
||
|
|
may be caused by unreliable fitting. These do not necessarily invalidate
|
||
|
|
the result, but they tend to reduce the test's power.
|
||
|
|
|
||
|
|
>>> _, ax = plt.subplots()
|
||
|
|
>>> ax.hist(np.log10(res.null_distribution))
|
||
|
|
>>> ax.set_xlabel("log10 of CVM statistic under the null hypothesis")
|
||
|
|
>>> ax.set_ylabel("Frequency")
|
||
|
|
>>> ax.set_title("Histogram of the Monte Carlo null distribution")
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
This plot seems reassuring.
|
||
|
|
|
||
|
|
If ``fit`` method is working reliably, and if the distribution of the test
|
||
|
|
statistic is not particularly sensitive to the values of the fitted
|
||
|
|
parameters, then the p-value provided by `goodness_of_fit` is expected to
|
||
|
|
be a good approximation.
|
||
|
|
|
||
|
|
>>> res.statistic, res.pvalue
|
||
|
|
(0.2231991510248692, 0.0525)
|
||
|
|
|
||
|
|
"""
|
||
|
|
args = _gof_iv(dist, data, known_params, fit_params, guessed_params,
|
||
|
|
statistic, n_mc_samples, random_state)
|
||
|
|
(dist, data, fixed_nhd_params, fixed_rfd_params, guessed_nhd_params,
|
||
|
|
guessed_rfd_params, statistic, n_mc_samples_int, random_state) = args
|
||
|
|
|
||
|
|
# Fit null hypothesis distribution to data
|
||
|
|
nhd_fit_fun = _get_fit_fun(dist, data, guessed_nhd_params,
|
||
|
|
fixed_nhd_params)
|
||
|
|
nhd_vals = nhd_fit_fun(data)
|
||
|
|
nhd_dist = dist(*nhd_vals)
|
||
|
|
|
||
|
|
def rvs(size):
|
||
|
|
return nhd_dist.rvs(size=size, random_state=random_state)
|
||
|
|
|
||
|
|
# Define statistic
|
||
|
|
fit_fun = _get_fit_fun(dist, data, guessed_rfd_params, fixed_rfd_params)
|
||
|
|
if callable(statistic):
|
||
|
|
compare_fun = statistic
|
||
|
|
else:
|
||
|
|
compare_fun = _compare_dict[statistic]
|
||
|
|
alternative = getattr(compare_fun, 'alternative', 'greater')
|
||
|
|
|
||
|
|
def statistic_fun(data, axis):
|
||
|
|
# Make things simple by always working along the last axis.
|
||
|
|
data = np.moveaxis(data, axis, -1)
|
||
|
|
rfd_vals = fit_fun(data)
|
||
|
|
rfd_dist = dist(*rfd_vals)
|
||
|
|
return compare_fun(rfd_dist, data, axis=-1)
|
||
|
|
|
||
|
|
res = stats.monte_carlo_test(data, rvs, statistic_fun, vectorized=True,
|
||
|
|
n_resamples=n_mc_samples, axis=-1,
|
||
|
|
alternative=alternative)
|
||
|
|
opt_res = optimize.OptimizeResult()
|
||
|
|
opt_res.success = True
|
||
|
|
opt_res.message = "The fit was performed successfully."
|
||
|
|
opt_res.x = nhd_vals
|
||
|
|
# Only continuous distributions for now, hence discrete=False
|
||
|
|
# There's no fundamental limitation; it's just that we're not using
|
||
|
|
# stats.fit, discrete distributions don't have `fit` method, and
|
||
|
|
# we haven't written any vectorized fit functions for a discrete
|
||
|
|
# distribution yet.
|
||
|
|
return GoodnessOfFitResult(FitResult(dist, data, False, opt_res),
|
||
|
|
res.statistic, res.pvalue,
|
||
|
|
res.null_distribution)
|
||
|
|
|
||
|
|
|
||
|
|
def _get_fit_fun(dist, data, guessed_params, fixed_params):
|
||
|
|
|
||
|
|
shape_names = [] if dist.shapes is None else dist.shapes.split(", ")
|
||
|
|
param_names = shape_names + ['loc', 'scale']
|
||
|
|
fparam_names = ['f'+name for name in param_names]
|
||
|
|
all_fixed = not set(fparam_names).difference(fixed_params)
|
||
|
|
guessed_shapes = [guessed_params.pop(x, None)
|
||
|
|
for x in shape_names if x in guessed_params]
|
||
|
|
|
||
|
|
if all_fixed:
|
||
|
|
def fit_fun(data):
|
||
|
|
return [fixed_params[name] for name in fparam_names]
|
||
|
|
# Define statistic, including fitting distribution to data
|
||
|
|
elif dist in _fit_funs:
|
||
|
|
def fit_fun(data):
|
||
|
|
params = _fit_funs[dist](data, **fixed_params)
|
||
|
|
params = np.asarray(np.broadcast_arrays(*params))
|
||
|
|
if params.ndim > 1:
|
||
|
|
params = params[..., np.newaxis]
|
||
|
|
return params
|
||
|
|
else:
|
||
|
|
def fit_fun_1d(data):
|
||
|
|
return dist.fit(data, *guessed_shapes, **guessed_params,
|
||
|
|
**fixed_params)
|
||
|
|
|
||
|
|
def fit_fun(data):
|
||
|
|
params = np.apply_along_axis(fit_fun_1d, axis=-1, arr=data)
|
||
|
|
if params.ndim > 1:
|
||
|
|
params = params.T[..., np.newaxis]
|
||
|
|
return params
|
||
|
|
|
||
|
|
return fit_fun
|
||
|
|
|
||
|
|
|
||
|
|
# Vectorized fitting functions. These are to accept ND `data` in which each
|
||
|
|
# row (slice along last axis) is a sample to fit and scalar fixed parameters.
|
||
|
|
# They return a tuple of shape parameter arrays, each of shape data.shape[:-1].
|
||
|
|
def _fit_norm(data, floc=None, fscale=None):
|
||
|
|
loc = floc
|
||
|
|
scale = fscale
|
||
|
|
if loc is None and scale is None:
|
||
|
|
loc = np.mean(data, axis=-1)
|
||
|
|
scale = np.std(data, ddof=1, axis=-1)
|
||
|
|
elif loc is None:
|
||
|
|
loc = np.mean(data, axis=-1)
|
||
|
|
elif scale is None:
|
||
|
|
scale = np.sqrt(((data - loc)**2).mean(axis=-1))
|
||
|
|
return loc, scale
|
||
|
|
|
||
|
|
|
||
|
|
_fit_funs = {stats.norm: _fit_norm} # type: ignore[attr-defined]
|
||
|
|
|
||
|
|
|
||
|
|
# Vectorized goodness of fit statistic functions. These accept a frozen
|
||
|
|
# distribution object and `data` in which each row (slice along last axis) is
|
||
|
|
# a sample.
|
||
|
|
|
||
|
|
|
||
|
|
def _anderson_darling(dist, data, axis):
|
||
|
|
x = np.sort(data, axis=-1)
|
||
|
|
n = data.shape[-1]
|
||
|
|
i = np.arange(1, n+1)
|
||
|
|
Si = (2*i - 1)/n * (dist.logcdf(x) + dist.logsf(x[..., ::-1]))
|
||
|
|
S = np.sum(Si, axis=-1)
|
||
|
|
return -n - S
|
||
|
|
|
||
|
|
|
||
|
|
def _compute_dplus(cdfvals): # adapted from _stats_py before gh-17062
|
||
|
|
n = cdfvals.shape[-1]
|
||
|
|
return (np.arange(1.0, n + 1) / n - cdfvals).max(axis=-1)
|
||
|
|
|
||
|
|
|
||
|
|
def _compute_dminus(cdfvals):
|
||
|
|
n = cdfvals.shape[-1]
|
||
|
|
return (cdfvals - np.arange(0.0, n)/n).max(axis=-1)
|
||
|
|
|
||
|
|
|
||
|
|
def _kolmogorov_smirnov(dist, data, axis):
|
||
|
|
x = np.sort(data, axis=-1)
|
||
|
|
cdfvals = dist.cdf(x)
|
||
|
|
Dplus = _compute_dplus(cdfvals) # always works along last axis
|
||
|
|
Dminus = _compute_dminus(cdfvals)
|
||
|
|
return np.maximum(Dplus, Dminus)
|
||
|
|
|
||
|
|
|
||
|
|
def _corr(X, M):
|
||
|
|
# Correlation coefficient r, simplified and vectorized as we need it.
|
||
|
|
# See [7] Equation (2). Lemma 1/2 are only for distributions symmetric
|
||
|
|
# about 0.
|
||
|
|
Xm = X.mean(axis=-1, keepdims=True)
|
||
|
|
Mm = M.mean(axis=-1, keepdims=True)
|
||
|
|
num = np.sum((X - Xm) * (M - Mm), axis=-1)
|
||
|
|
den = np.sqrt(np.sum((X - Xm)**2, axis=-1) * np.sum((M - Mm)**2, axis=-1))
|
||
|
|
return num/den
|
||
|
|
|
||
|
|
|
||
|
|
def _filliben(dist, data, axis):
|
||
|
|
# [7] Section 8 # 1
|
||
|
|
X = np.sort(data, axis=-1)
|
||
|
|
|
||
|
|
# [7] Section 8 # 2
|
||
|
|
n = data.shape[-1]
|
||
|
|
k = np.arange(1, n+1)
|
||
|
|
# Filliben used an approximation for the uniform distribution order
|
||
|
|
# statistic medians.
|
||
|
|
# m = (k - .3175)/(n + 0.365)
|
||
|
|
# m[-1] = 0.5**(1/n)
|
||
|
|
# m[0] = 1 - m[-1]
|
||
|
|
# We can just as easily use the (theoretically) exact values. See e.g.
|
||
|
|
# https://en.wikipedia.org/wiki/Order_statistic
|
||
|
|
# "Order statistics sampled from a uniform distribution"
|
||
|
|
m = stats.beta(k, n + 1 - k).median()
|
||
|
|
|
||
|
|
# [7] Section 8 # 3
|
||
|
|
M = dist.ppf(m)
|
||
|
|
|
||
|
|
# [7] Section 8 # 4
|
||
|
|
return _corr(X, M)
|
||
|
|
_filliben.alternative = 'less' # type: ignore[attr-defined]
|
||
|
|
|
||
|
|
|
||
|
|
def _cramer_von_mises(dist, data, axis):
|
||
|
|
x = np.sort(data, axis=-1)
|
||
|
|
n = data.shape[-1]
|
||
|
|
cdfvals = dist.cdf(x)
|
||
|
|
u = (2*np.arange(1, n+1) - 1)/(2*n)
|
||
|
|
w = 1 / (12*n) + np.sum((u - cdfvals)**2, axis=-1)
|
||
|
|
return w
|
||
|
|
|
||
|
|
|
||
|
|
_compare_dict = {"ad": _anderson_darling, "ks": _kolmogorov_smirnov,
|
||
|
|
"cvm": _cramer_von_mises, "filliben": _filliben}
|
||
|
|
|
||
|
|
|
||
|
|
def _gof_iv(dist, data, known_params, fit_params, guessed_params, statistic,
|
||
|
|
n_mc_samples, random_state):
|
||
|
|
|
||
|
|
if not isinstance(dist, stats.rv_continuous):
|
||
|
|
message = ("`dist` must be a (non-frozen) instance of "
|
||
|
|
"`stats.rv_continuous`.")
|
||
|
|
raise TypeError(message)
|
||
|
|
|
||
|
|
data = np.asarray(data, dtype=float)
|
||
|
|
if not data.ndim == 1:
|
||
|
|
message = "`data` must be a one-dimensional array of numbers."
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
# Leave validation of these key/value pairs to the `fit` method,
|
||
|
|
# but collect these into dictionaries that will be used
|
||
|
|
known_params = known_params or dict()
|
||
|
|
fit_params = fit_params or dict()
|
||
|
|
guessed_params = guessed_params or dict()
|
||
|
|
|
||
|
|
known_params_f = {("f"+key): val for key, val in known_params.items()}
|
||
|
|
fit_params_f = {("f"+key): val for key, val in fit_params.items()}
|
||
|
|
|
||
|
|
# These are the values of parameters of the null distribution family
|
||
|
|
# with which resamples are drawn
|
||
|
|
fixed_nhd_params = known_params_f.copy()
|
||
|
|
fixed_nhd_params.update(fit_params_f)
|
||
|
|
|
||
|
|
# These are fixed when fitting the distribution family to resamples
|
||
|
|
fixed_rfd_params = known_params_f.copy()
|
||
|
|
|
||
|
|
# These are used as guesses when fitting the distribution family to
|
||
|
|
# the original data
|
||
|
|
guessed_nhd_params = guessed_params.copy()
|
||
|
|
|
||
|
|
# These are used as guesses when fitting the distribution family to
|
||
|
|
# resamples
|
||
|
|
guessed_rfd_params = fit_params.copy()
|
||
|
|
guessed_rfd_params.update(guessed_params)
|
||
|
|
|
||
|
|
if not callable(statistic):
|
||
|
|
statistic = statistic.lower()
|
||
|
|
statistics = {'ad', 'ks', 'cvm', 'filliben'}
|
||
|
|
if statistic not in statistics:
|
||
|
|
message = f"`statistic` must be one of {statistics}."
|
||
|
|
raise ValueError(message)
|
||
|
|
|
||
|
|
n_mc_samples_int = int(n_mc_samples)
|
||
|
|
if n_mc_samples_int != n_mc_samples:
|
||
|
|
message = "`n_mc_samples` must be an integer."
|
||
|
|
raise TypeError(message)
|
||
|
|
|
||
|
|
random_state = check_random_state(random_state)
|
||
|
|
|
||
|
|
return (dist, data, fixed_nhd_params, fixed_rfd_params, guessed_nhd_params,
|
||
|
|
guessed_rfd_params, statistic, n_mc_samples_int, random_state)
|