227 lines
8.0 KiB
Python
227 lines
8.0 KiB
Python
|
import numpy as np
|
||
|
|
from statsmodels.robust import mad
|
||
|
|
from scipy.optimize import minimize_scalar
|
||
|
|
|
||
|
|
|
||
|
|
class BoxCox:
|
||
|
|
"""
|
||
|
|
Mixin class to allow for a Box-Cox transformation.
|
||
|
|
"""
|
||
|
|
|
||
|
|
def transform_boxcox(self, x, lmbda=None, method='guerrero', **kwargs):
|
||
|
|
"""
|
||
|
|
Performs a Box-Cox transformation on the data array x. If lmbda is None,
|
||
|
|
the indicated method is used to estimate a suitable lambda parameter.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
lmbda : float
|
||
|
|
The lambda parameter for the Box-Cox transform. If None, a value
|
||
|
|
will be estimated by means of the specified method.
|
||
|
|
method : {'guerrero', 'loglik'}
|
||
|
|
The method to estimate the lambda parameter. Will only be used if
|
||
|
|
lmbda is None, and defaults to 'guerrero', detailed in Guerrero
|
||
|
|
(1993). 'loglik' maximizes the profile likelihood.
|
||
|
|
**kwargs
|
||
|
|
Options for the specified method.
|
||
|
|
* For 'guerrero', this entails window_length, the grouping
|
||
|
|
parameter, scale, the dispersion measure, and options, to be
|
||
|
|
passed to the optimizer.
|
||
|
|
* For 'loglik': options, to be passed to the optimizer.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
y : array_like
|
||
|
|
The transformed series.
|
||
|
|
lmbda : float
|
||
|
|
The lmbda parameter used to transform the series.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
Guerrero, Victor M. 1993. "Time-series analysis supported by power
|
||
|
|
transformations". `Journal of Forecasting`. 12 (1): 37-48.
|
||
|
|
|
||
|
|
Guerrero, Victor M. and Perera, Rafael. 2004. "Variance Stabilizing
|
||
|
|
Power Transformation for Time Series," `Journal of Modern Applied
|
||
|
|
Statistical Methods`. 3 (2): 357-369.
|
||
|
|
|
||
|
|
Box, G. E. P., and D. R. Cox. 1964. "An Analysis of Transformations".
|
||
|
|
`Journal of the Royal Statistical Society`. 26 (2): 211-252.
|
||
|
|
"""
|
||
|
|
x = np.asarray(x)
|
||
|
|
|
||
|
|
if np.any(x <= 0):
|
||
|
|
raise ValueError("Non-positive x.")
|
||
|
|
|
||
|
|
if lmbda is None:
|
||
|
|
lmbda = self._est_lambda(x,
|
||
|
|
method=method,
|
||
|
|
**kwargs)
|
||
|
|
|
||
|
|
# if less than 0.01, treat lambda as zero.
|
||
|
|
if np.isclose(lmbda, 0.):
|
||
|
|
y = np.log(x)
|
||
|
|
else:
|
||
|
|
y = (np.power(x, lmbda) - 1.) / lmbda
|
||
|
|
|
||
|
|
return y, lmbda
|
||
|
|
|
||
|
|
def untransform_boxcox(self, x, lmbda, method='naive'):
|
||
|
|
"""
|
||
|
|
Back-transforms the Box-Cox transformed data array, by means of the
|
||
|
|
indicated method. The provided argument lmbda should be the lambda
|
||
|
|
parameter that was used to initially transform the data.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
The transformed series.
|
||
|
|
lmbda : float
|
||
|
|
The lambda parameter that was used to transform the series.
|
||
|
|
method : {'naive'}
|
||
|
|
Indicates the method to be used in the untransformation. Defaults
|
||
|
|
to 'naive', which reverses the transformation.
|
||
|
|
|
||
|
|
NOTE: 'naive' is implemented natively, while other methods may be
|
||
|
|
available in subclasses!
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
y : array_like
|
||
|
|
The untransformed series.
|
||
|
|
"""
|
||
|
|
method = method.lower()
|
||
|
|
x = np.asarray(x)
|
||
|
|
|
||
|
|
if method == 'naive':
|
||
|
|
if np.isclose(lmbda, 0.):
|
||
|
|
y = np.exp(x)
|
||
|
|
else:
|
||
|
|
y = np.power(lmbda * x + 1, 1. / lmbda)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Method '{method}' not understood.")
|
||
|
|
|
||
|
|
return y
|
||
|
|
|
||
|
|
def _est_lambda(self, x, bounds=(-1, 2), method='guerrero', **kwargs):
|
||
|
|
"""
|
||
|
|
Computes an estimate for the lambda parameter in the Box-Cox
|
||
|
|
transformation using method.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
The untransformed data.
|
||
|
|
bounds : tuple
|
||
|
|
Numeric 2-tuple, that indicate the solution space for the lambda
|
||
|
|
parameter. Default (-1, 2).
|
||
|
|
method : {'guerrero', 'loglik'}
|
||
|
|
The method by which to estimate lambda. Defaults to 'guerrero', but
|
||
|
|
the profile likelihood ('loglik') is also available.
|
||
|
|
**kwargs
|
||
|
|
Options for the specified method.
|
||
|
|
* For 'guerrero': window_length (int), the seasonality/grouping
|
||
|
|
parameter. Scale ({'mad', 'sd'}), the dispersion measure. Options
|
||
|
|
(dict), to be passed to the optimizer.
|
||
|
|
* For 'loglik': Options (dict), to be passed to the optimizer.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
lmbda : float
|
||
|
|
The lambda parameter.
|
||
|
|
"""
|
||
|
|
method = method.lower()
|
||
|
|
|
||
|
|
if len(bounds) != 2:
|
||
|
|
raise ValueError("Bounds of length {} not understood."
|
||
|
|
.format(len(bounds)))
|
||
|
|
elif bounds[0] >= bounds[1]:
|
||
|
|
raise ValueError("Lower bound exceeds upper bound.")
|
||
|
|
|
||
|
|
if method == 'guerrero':
|
||
|
|
lmbda = self._guerrero_cv(x, bounds=bounds, **kwargs)
|
||
|
|
elif method == 'loglik':
|
||
|
|
lmbda = self._loglik_boxcox(x, bounds=bounds, **kwargs)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Method '{method}' not understood.")
|
||
|
|
|
||
|
|
return lmbda
|
||
|
|
|
||
|
|
def _guerrero_cv(self, x, bounds, window_length=4, scale='sd',
|
||
|
|
options={'maxiter': 25}):
|
||
|
|
"""
|
||
|
|
Computes lambda using guerrero's coefficient of variation. If no
|
||
|
|
seasonality is present in the data, window_length is set to 4 (as
|
||
|
|
per Guerrero and Perera, (2004)).
|
||
|
|
|
||
|
|
NOTE: Seasonality-specific auxiliaries *should* provide their own
|
||
|
|
seasonality parameter.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
bounds : tuple
|
||
|
|
Numeric 2-tuple, that indicate the solution space for the lambda
|
||
|
|
parameter.
|
||
|
|
window_length : int
|
||
|
|
Seasonality/grouping parameter. Default 4, as per Guerrero and
|
||
|
|
Perera (2004). NOTE: this indicates the length of the individual
|
||
|
|
groups, not the total number of groups!
|
||
|
|
scale : {'sd', 'mad'}
|
||
|
|
The dispersion measure to be used. 'sd' indicates the sample
|
||
|
|
standard deviation, but the more robust 'mad' is also available.
|
||
|
|
options : dict
|
||
|
|
The options (as a dict) to be passed to the optimizer.
|
||
|
|
"""
|
||
|
|
nobs = len(x)
|
||
|
|
groups = int(nobs / window_length)
|
||
|
|
|
||
|
|
# remove the first n < window_length observations from consideration.
|
||
|
|
grouped_data = np.reshape(x[nobs - (groups * window_length): nobs],
|
||
|
|
(groups, window_length))
|
||
|
|
mean = np.mean(grouped_data, 1)
|
||
|
|
|
||
|
|
scale = scale.lower()
|
||
|
|
if scale == 'sd':
|
||
|
|
dispersion = np.std(grouped_data, 1, ddof=1)
|
||
|
|
elif scale == 'mad':
|
||
|
|
dispersion = mad(grouped_data, axis=1)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Scale '{scale}' not understood.")
|
||
|
|
|
||
|
|
def optim(lmbda):
|
||
|
|
rat = np.divide(dispersion, np.power(mean, 1 - lmbda)) # eq 6, p 40
|
||
|
|
return np.std(rat, ddof=1) / np.mean(rat)
|
||
|
|
|
||
|
|
res = minimize_scalar(optim,
|
||
|
|
bounds=bounds,
|
||
|
|
method='bounded',
|
||
|
|
options=options)
|
||
|
|
return res.x
|
||
|
|
|
||
|
|
def _loglik_boxcox(self, x, bounds, options={'maxiter': 25}):
|
||
|
|
"""
|
||
|
|
Taken from the Stata manual on Box-Cox regressions, where this is the
|
||
|
|
special case of 'lhs only'. As an estimator for the variance, the
|
||
|
|
sample variance is used, by means of the well-known formula.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
options : dict
|
||
|
|
The options (as a dict) to be passed to the optimizer.
|
||
|
|
"""
|
||
|
|
sum_x = np.sum(np.log(x))
|
||
|
|
nobs = len(x)
|
||
|
|
|
||
|
|
def optim(lmbda):
|
||
|
|
y, lmbda = self.transform_boxcox(x, lmbda)
|
||
|
|
return (1 - lmbda) * sum_x + (nobs / 2.) * np.log(np.var(y))
|
||
|
|
|
||
|
|
res = minimize_scalar(optim,
|
||
|
|
bounds=bounds,
|
||
|
|
method='bounded',
|
||
|
|
options=options)
|
||
|
|
return res.x
|