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8862 lines
268 KiB
Python
8862 lines
268 KiB
Python
# -*- coding: utf-8 -*-
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#
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# Author: Travis Oliphant 2002-2011 with contributions from
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# SciPy Developers 2004-2011
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#
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import warnings
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from collections.abc import Iterable
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import ctypes
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import numpy as np
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from scipy._lib.doccer import (extend_notes_in_docstring,
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replace_notes_in_docstring)
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from scipy._lib._ccallback import LowLevelCallable
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from scipy import optimize
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from scipy import integrate
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from scipy import interpolate
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import scipy.special as sc
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import scipy.special._ufuncs as scu
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from scipy._lib._util import _lazyselect, _lazywhere
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from . import _stats
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from ._rvs_sampling import rvs_ratio_uniforms
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from ._tukeylambda_stats import (tukeylambda_variance as _tlvar,
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tukeylambda_kurtosis as _tlkurt)
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from ._distn_infrastructure import (
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get_distribution_names, _kurtosis, _ncx2_cdf, _ncx2_log_pdf, _ncx2_pdf,
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rv_continuous, _skew, _get_fixed_fit_value, _check_shape,
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_fit_determine_optimizer)
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from ._ksstats import kolmogn, kolmognp, kolmogni
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from ._constants import (_XMIN, _EULER, _ZETA3,
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_SQRT_2_OVER_PI, _LOG_SQRT_2_OVER_PI)
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# In numpy 1.12 and above, np.power refuses to raise integers to negative
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# powers, and `np.float_power` is a new replacement.
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try:
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float_power = np.float_power
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except AttributeError:
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float_power = np.power
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def _remove_optimizer_parameters(kwds):
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"""
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Remove the optimizer-related keyword arguments 'loc', 'scale' and
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'optimizer' from `kwds`. Then check that `kwds` is empty, and
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raise `TypeError("Unknown arguments: %s." % kwds)` if it is not.
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This function is used in the fit method of distributions that override
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the default method and do not use the default optimization code.
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`kwds` is modified in-place.
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"""
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kwds.pop('loc', None)
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kwds.pop('scale', None)
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kwds.pop('optimizer', None)
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if kwds:
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raise TypeError("Unknown arguments: %s." % kwds)
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## Kolmogorov-Smirnov one-sided and two-sided test statistics
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class ksone_gen(rv_continuous):
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r"""Kolmogorov-Smirnov one-sided test statistic distribution.
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This is the distribution of the one-sided Kolmogorov-Smirnov (KS)
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statistics :math:`D_n^+` and :math:`D_n^-`
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for a finite sample size ``n`` (the shape parameter).
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%(before_notes)s
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See Also
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--------
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kstwobign, kstwo, kstest
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Notes
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-----
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:math:`D_n^+` and :math:`D_n^-` are given by
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.. math::
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D_n^+ &= \text{sup}_x (F_n(x) - F(x)),\\
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D_n^- &= \text{sup}_x (F(x) - F_n(x)),\\
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where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
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`ksone` describes the distribution under the null hypothesis of the KS test
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that the empirical CDF corresponds to :math:`n` i.i.d. random variates
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with CDF :math:`F`.
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%(after_notes)s
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References
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----------
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.. [1] Birnbaum, Z. W. and Tingey, F.H. "One-sided confidence contours
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for probability distribution functions", The Annals of Mathematical
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Statistics, 22(4), pp 592-596 (1951).
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%(example)s
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"""
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def _pdf(self, x, n):
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return -scu._smirnovp(n, x)
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def _cdf(self, x, n):
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return scu._smirnovc(n, x)
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def _sf(self, x, n):
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return sc.smirnov(n, x)
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def _ppf(self, q, n):
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return scu._smirnovci(n, q)
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def _isf(self, q, n):
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return sc.smirnovi(n, q)
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ksone = ksone_gen(a=0.0, b=1.0, name='ksone')
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class kstwo_gen(rv_continuous):
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r"""Kolmogorov-Smirnov two-sided test statistic distribution.
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This is the distribution of the two-sided Kolmogorov-Smirnov (KS)
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statistic :math:`D_n` for a finite sample size ``n``
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(the shape parameter).
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%(before_notes)s
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See Also
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--------
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kstwobign, ksone, kstest
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Notes
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-----
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:math:`D_n` is given by
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.. math::
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D_n &= \text{sup}_x |F_n(x) - F(x)|
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where :math:`F` is a (continuous) CDF and :math:`F_n` is an empirical CDF.
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`kstwo` describes the distribution under the null hypothesis of the KS test
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that the empirical CDF corresponds to :math:`n` i.i.d. random variates
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with CDF :math:`F`.
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%(after_notes)s
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References
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----------
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.. [1] Simard, R., L'Ecuyer, P. "Computing the Two-Sided
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Kolmogorov-Smirnov Distribution", Journal of Statistical Software,
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Vol 39, 11, 1-18 (2011).
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%(example)s
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"""
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def _get_support(self, n):
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return (0.5/(n if not isinstance(n, Iterable) else np.asanyarray(n)),
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1.0)
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def _pdf(self, x, n):
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return kolmognp(n, x)
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def _cdf(self, x, n):
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return kolmogn(n, x)
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def _sf(self, x, n):
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return kolmogn(n, x, cdf=False)
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def _ppf(self, q, n):
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return kolmogni(n, q, cdf=True)
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def _isf(self, q, n):
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return kolmogni(n, q, cdf=False)
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# Use the pdf, (not the ppf) to compute moments
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kstwo = kstwo_gen(momtype=0, a=0.0, b=1.0, name='kstwo')
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class kstwobign_gen(rv_continuous):
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r"""Limiting distribution of scaled Kolmogorov-Smirnov two-sided test statistic.
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This is the asymptotic distribution of the two-sided Kolmogorov-Smirnov
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statistic :math:`\sqrt{n} D_n` that measures the maximum absolute
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distance of the theoretical (continuous) CDF from the empirical CDF.
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(see `kstest`).
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%(before_notes)s
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See Also
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--------
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ksone, kstwo, kstest
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Notes
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-----
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:math:`\sqrt{n} D_n` is given by
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.. math::
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D_n = \text{sup}_x |F_n(x) - F(x)|
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where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
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`kstwobign` describes the asymptotic distribution (i.e. the limit of
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:math:`\sqrt{n} D_n`) under the null hypothesis of the KS test that the
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empirical CDF corresponds to i.i.d. random variates with CDF :math:`F`.
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%(after_notes)s
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References
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----------
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.. [1] Feller, W. "On the Kolmogorov-Smirnov Limit Theorems for Empirical
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Distributions", Ann. Math. Statist. Vol 19, 177-189 (1948).
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%(example)s
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"""
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def _pdf(self, x):
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return -scu._kolmogp(x)
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def _cdf(self, x):
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return scu._kolmogc(x)
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def _sf(self, x):
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return sc.kolmogorov(x)
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def _ppf(self, q):
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return scu._kolmogci(q)
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def _isf(self, q):
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return sc.kolmogi(q)
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kstwobign = kstwobign_gen(a=0.0, name='kstwobign')
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## Normal distribution
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# loc = mu, scale = std
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# Keep these implementations out of the class definition so they can be reused
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# by other distributions.
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_norm_pdf_C = np.sqrt(2*np.pi)
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_norm_pdf_logC = np.log(_norm_pdf_C)
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def _norm_pdf(x):
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return np.exp(-x**2/2.0) / _norm_pdf_C
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def _norm_logpdf(x):
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return -x**2 / 2.0 - _norm_pdf_logC
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def _norm_cdf(x):
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return sc.ndtr(x)
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def _norm_logcdf(x):
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return sc.log_ndtr(x)
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def _norm_ppf(q):
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return sc.ndtri(q)
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def _norm_sf(x):
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return _norm_cdf(-x)
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def _norm_logsf(x):
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return _norm_logcdf(-x)
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def _norm_isf(q):
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return -_norm_ppf(q)
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class norm_gen(rv_continuous):
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r"""A normal continuous random variable.
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The location (``loc``) keyword specifies the mean.
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The scale (``scale``) keyword specifies the standard deviation.
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%(before_notes)s
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Notes
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-----
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The probability density function for `norm` is:
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.. math::
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f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}}
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for a real number :math:`x`.
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%(after_notes)s
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%(example)s
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"""
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def _rvs(self, size=None, random_state=None):
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return random_state.standard_normal(size)
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def _pdf(self, x):
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# norm.pdf(x) = exp(-x**2/2)/sqrt(2*pi)
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return _norm_pdf(x)
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def _logpdf(self, x):
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return _norm_logpdf(x)
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def _cdf(self, x):
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return _norm_cdf(x)
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def _logcdf(self, x):
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return _norm_logcdf(x)
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def _sf(self, x):
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return _norm_sf(x)
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def _logsf(self, x):
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return _norm_logsf(x)
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def _ppf(self, q):
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return _norm_ppf(q)
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def _isf(self, q):
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return _norm_isf(q)
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def _stats(self):
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return 0.0, 1.0, 0.0, 0.0
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def _entropy(self):
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return 0.5*(np.log(2*np.pi)+1)
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@replace_notes_in_docstring(rv_continuous, notes="""\
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This function uses explicit formulas for the maximum likelihood
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estimation of the normal distribution parameters, so the
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`optimizer` argument is ignored.\n\n""")
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def fit(self, data, **kwds):
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floc = kwds.pop('floc', None)
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fscale = kwds.pop('fscale', None)
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_remove_optimizer_parameters(kwds)
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if floc is not None and fscale is not None:
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# This check is for consistency with `rv_continuous.fit`.
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# Without this check, this function would just return the
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# parameters that were given.
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raise ValueError("All parameters fixed. There is nothing to "
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"optimize.")
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data = np.asarray(data)
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if not np.isfinite(data).all():
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raise RuntimeError("The data contains non-finite values.")
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if floc is None:
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loc = data.mean()
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else:
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loc = floc
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if fscale is None:
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scale = np.sqrt(((data - loc)**2).mean())
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else:
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scale = fscale
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return loc, scale
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def _munp(self, n):
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"""
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@returns Moments of standard normal distribution for integer n >= 0
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See eq. 16 of https://arxiv.org/abs/1209.4340v2
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"""
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if n % 2 == 0:
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return sc.factorial2(n - 1)
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else:
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return 0.
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norm = norm_gen(name='norm')
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class alpha_gen(rv_continuous):
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r"""An alpha continuous random variable.
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%(before_notes)s
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Notes
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-----
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The probability density function for `alpha` ([1]_, [2]_) is:
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.. math::
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f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} *
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\exp(-\frac{1}{2} (a-1/x)^2)
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where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`.
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`alpha` takes ``a`` as a shape parameter.
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%(after_notes)s
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References
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----------
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.. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
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Distributions, Volume 1", Second Edition, John Wiley and Sons,
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p. 173 (1994).
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.. [2] Anthony A. Salvia, "Reliability applications of the Alpha
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Distribution", IEEE Transactions on Reliability, Vol. R-34,
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No. 3, pp. 251-252 (1985).
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%(example)s
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"""
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_support_mask = rv_continuous._open_support_mask
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def _pdf(self, x, a):
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# alpha.pdf(x, a) = 1/(x**2*Phi(a)*sqrt(2*pi)) * exp(-1/2 * (a-1/x)**2)
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return 1.0/(x**2)/_norm_cdf(a)*_norm_pdf(a-1.0/x)
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def _logpdf(self, x, a):
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return -2*np.log(x) + _norm_logpdf(a-1.0/x) - np.log(_norm_cdf(a))
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def _cdf(self, x, a):
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return _norm_cdf(a-1.0/x) / _norm_cdf(a)
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def _ppf(self, q, a):
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return 1.0/np.asarray(a-sc.ndtri(q*_norm_cdf(a)))
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def _stats(self, a):
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return [np.inf]*2 + [np.nan]*2
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alpha = alpha_gen(a=0.0, name='alpha')
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class anglit_gen(rv_continuous):
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r"""An anglit continuous random variable.
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%(before_notes)s
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Notes
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-----
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The probability density function for `anglit` is:
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.. math::
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f(x) = \sin(2x + \pi/2) = \cos(2x)
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for :math:`-\pi/4 \le x \le \pi/4`.
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%(after_notes)s
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%(example)s
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"""
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def _pdf(self, x):
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# anglit.pdf(x) = sin(2*x + \pi/2) = cos(2*x)
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return np.cos(2*x)
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def _cdf(self, x):
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return np.sin(x+np.pi/4)**2.0
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def _ppf(self, q):
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return np.arcsin(np.sqrt(q))-np.pi/4
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def _stats(self):
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return 0.0, np.pi*np.pi/16-0.5, 0.0, -2*(np.pi**4 - 96)/(np.pi*np.pi-8)**2
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def _entropy(self):
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return 1-np.log(2)
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anglit = anglit_gen(a=-np.pi/4, b=np.pi/4, name='anglit')
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class arcsine_gen(rv_continuous):
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r"""An arcsine continuous random variable.
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%(before_notes)s
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Notes
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-----
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The probability density function for `arcsine` is:
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.. math::
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f(x) = \frac{1}{\pi \sqrt{x (1-x)}}
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for :math:`0 < x < 1`.
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%(after_notes)s
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%(example)s
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"""
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def _pdf(self, x):
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# arcsine.pdf(x) = 1/(pi*sqrt(x*(1-x)))
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return 1.0/np.pi/np.sqrt(x*(1-x))
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def _cdf(self, x):
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return 2.0/np.pi*np.arcsin(np.sqrt(x))
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def _ppf(self, q):
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return np.sin(np.pi/2.0*q)**2.0
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def _stats(self):
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mu = 0.5
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mu2 = 1.0/8
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g1 = 0
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g2 = -3.0/2.0
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return mu, mu2, g1, g2
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def _entropy(self):
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return -0.24156447527049044468
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arcsine = arcsine_gen(a=0.0, b=1.0, name='arcsine')
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class FitDataError(ValueError):
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# This exception is raised by, for example, beta_gen.fit when both floc
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# and fscale are fixed and there are values in the data not in the open
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# interval (floc, floc+fscale).
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def __init__(self, distr, lower, upper):
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self.args = (
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"Invalid values in `data`. Maximum likelihood "
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"estimation with {distr!r} requires that {lower!r} < "
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"(x - loc)/scale < {upper!r} for each x in `data`.".format(
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distr=distr, lower=lower, upper=upper),
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)
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class FitSolverError(RuntimeError):
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# This exception is raised by, for example, beta_gen.fit when
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# optimize.fsolve returns with ier != 1.
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def __init__(self, mesg):
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emsg = "Solver for the MLE equations failed to converge: "
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emsg += mesg.replace('\n', '')
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self.args = (emsg,)
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def _beta_mle_a(a, b, n, s1):
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# The zeros of this function give the MLE for `a`, with
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# `b`, `n` and `s1` given. `s1` is the sum of the logs of
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|
# the data. `n` is the number of data points.
|
|
psiab = sc.psi(a + b)
|
|
func = s1 - n * (-psiab + sc.psi(a))
|
|
return func
|
|
|
|
|
|
def _beta_mle_ab(theta, n, s1, s2):
|
|
# Zeros of this function are critical points of
|
|
# the maximum likelihood function. Solving this system
|
|
# for theta (which contains a and b) gives the MLE for a and b
|
|
# given `n`, `s1` and `s2`. `s1` is the sum of the logs of the data,
|
|
# and `s2` is the sum of the logs of 1 - data. `n` is the number
|
|
# of data points.
|
|
a, b = theta
|
|
psiab = sc.psi(a + b)
|
|
func = [s1 - n * (-psiab + sc.psi(a)),
|
|
s2 - n * (-psiab + sc.psi(b))]
|
|
return func
|
|
|
|
|
|
class beta_gen(rv_continuous):
|
|
r"""A beta continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `beta` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}}
|
|
{\Gamma(a) \Gamma(b)}
|
|
|
|
for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where
|
|
:math:`\Gamma` is the gamma function (`scipy.special.gamma`).
|
|
|
|
`beta` takes :math:`a` and :math:`b` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, a, b, size=None, random_state=None):
|
|
return random_state.beta(a, b, size)
|
|
|
|
def _pdf(self, x, a, b):
|
|
# gamma(a+b) * x**(a-1) * (1-x)**(b-1)
|
|
# beta.pdf(x, a, b) = ------------------------------------
|
|
# gamma(a)*gamma(b)
|
|
return np.exp(self._logpdf(x, a, b))
|
|
|
|
def _logpdf(self, x, a, b):
|
|
lPx = sc.xlog1py(b - 1.0, -x) + sc.xlogy(a - 1.0, x)
|
|
lPx -= sc.betaln(a, b)
|
|
return lPx
|
|
|
|
def _cdf(self, x, a, b):
|
|
return sc.btdtr(a, b, x)
|
|
|
|
def _ppf(self, q, a, b):
|
|
return sc.btdtri(a, b, q)
|
|
|
|
def _stats(self, a, b):
|
|
mn = a*1.0 / (a + b)
|
|
var = (a*b*1.0)/(a+b+1.0)/(a+b)**2.0
|
|
g1 = 2.0*(b-a)*np.sqrt((1.0+a+b)/(a*b)) / (2+a+b)
|
|
g2 = 6.0*(a**3 + a**2*(1-2*b) + b**2*(1+b) - 2*a*b*(2+b))
|
|
g2 /= a*b*(a+b+2)*(a+b+3)
|
|
return mn, var, g1, g2
|
|
|
|
def _fitstart(self, data):
|
|
g1 = _skew(data)
|
|
g2 = _kurtosis(data)
|
|
|
|
def func(x):
|
|
a, b = x
|
|
sk = 2*(b-a)*np.sqrt(a + b + 1) / (a + b + 2) / np.sqrt(a*b)
|
|
ku = a**3 - a**2*(2*b-1) + b**2*(b+1) - 2*a*b*(b+2)
|
|
ku /= a*b*(a+b+2)*(a+b+3)
|
|
ku *= 6
|
|
return [sk-g1, ku-g2]
|
|
a, b = optimize.fsolve(func, (1.0, 1.0))
|
|
return super(beta_gen, self)._fitstart(data, args=(a, b))
|
|
|
|
@extend_notes_in_docstring(rv_continuous, notes="""\
|
|
In the special case where both `floc` and `fscale` are given, a
|
|
`ValueError` is raised if any value `x` in `data` does not satisfy
|
|
`floc < x < floc + fscale`.\n\n""")
|
|
def fit(self, data, *args, **kwds):
|
|
# Override rv_continuous.fit, so we can more efficiently handle the
|
|
# case where floc and fscale are given.
|
|
|
|
floc = kwds.get('floc', None)
|
|
fscale = kwds.get('fscale', None)
|
|
|
|
if floc is None or fscale is None:
|
|
# do general fit
|
|
return super(beta_gen, self).fit(data, *args, **kwds)
|
|
|
|
# We already got these from kwds, so just pop them.
|
|
kwds.pop('floc', None)
|
|
kwds.pop('fscale', None)
|
|
|
|
f0 = _get_fixed_fit_value(kwds, ['f0', 'fa', 'fix_a'])
|
|
f1 = _get_fixed_fit_value(kwds, ['f1', 'fb', 'fix_b'])
|
|
|
|
_remove_optimizer_parameters(kwds)
|
|
|
|
if f0 is not None and f1 is not None:
|
|
# This check is for consistency with `rv_continuous.fit`.
|
|
raise ValueError("All parameters fixed. There is nothing to "
|
|
"optimize.")
|
|
|
|
# Special case: loc and scale are constrained, so we are fitting
|
|
# just the shape parameters. This can be done much more efficiently
|
|
# than the method used in `rv_continuous.fit`. (See the subsection
|
|
# "Two unknown parameters" in the section "Maximum likelihood" of
|
|
# the Wikipedia article on the Beta distribution for the formulas.)
|
|
|
|
if not np.isfinite(data).all():
|
|
raise RuntimeError("The data contains non-finite values.")
|
|
|
|
# Normalize the data to the interval [0, 1].
|
|
data = (np.ravel(data) - floc) / fscale
|
|
if np.any(data <= 0) or np.any(data >= 1):
|
|
raise FitDataError("beta", lower=floc, upper=floc + fscale)
|
|
|
|
xbar = data.mean()
|
|
|
|
if f0 is not None or f1 is not None:
|
|
# One of the shape parameters is fixed.
|
|
|
|
if f0 is not None:
|
|
# The shape parameter a is fixed, so swap the parameters
|
|
# and flip the data. We always solve for `a`. The result
|
|
# will be swapped back before returning.
|
|
b = f0
|
|
data = 1 - data
|
|
xbar = 1 - xbar
|
|
else:
|
|
b = f1
|
|
|
|
# Initial guess for a. Use the formula for the mean of the beta
|
|
# distribution, E[x] = a / (a + b), to generate a reasonable
|
|
# starting point based on the mean of the data and the given
|
|
# value of b.
|
|
a = b * xbar / (1 - xbar)
|
|
|
|
# Compute the MLE for `a` by solving _beta_mle_a.
|
|
theta, info, ier, mesg = optimize.fsolve(
|
|
_beta_mle_a, a,
|
|
args=(b, len(data), np.log(data).sum()),
|
|
full_output=True
|
|
)
|
|
if ier != 1:
|
|
raise FitSolverError(mesg=mesg)
|
|
a = theta[0]
|
|
|
|
if f0 is not None:
|
|
# The shape parameter a was fixed, so swap back the
|
|
# parameters.
|
|
a, b = b, a
|
|
|
|
else:
|
|
# Neither of the shape parameters is fixed.
|
|
|
|
# s1 and s2 are used in the extra arguments passed to _beta_mle_ab
|
|
# by optimize.fsolve.
|
|
s1 = np.log(data).sum()
|
|
s2 = sc.log1p(-data).sum()
|
|
|
|
# Use the "method of moments" to estimate the initial
|
|
# guess for a and b.
|
|
fac = xbar * (1 - xbar) / data.var(ddof=0) - 1
|
|
a = xbar * fac
|
|
b = (1 - xbar) * fac
|
|
|
|
# Compute the MLE for a and b by solving _beta_mle_ab.
|
|
theta, info, ier, mesg = optimize.fsolve(
|
|
_beta_mle_ab, [a, b],
|
|
args=(len(data), s1, s2),
|
|
full_output=True
|
|
)
|
|
if ier != 1:
|
|
raise FitSolverError(mesg=mesg)
|
|
a, b = theta
|
|
|
|
return a, b, floc, fscale
|
|
|
|
|
|
beta = beta_gen(a=0.0, b=1.0, name='beta')
|
|
|
|
|
|
class betaprime_gen(rv_continuous):
|
|
r"""A beta prime continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `betaprime` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)}
|
|
|
|
for :math:`x >= 0`, :math:`a > 0`, :math:`b > 0`, where
|
|
:math:`\beta(a, b)` is the beta function (see `scipy.special.beta`).
|
|
|
|
`betaprime` takes ``a`` and ``b`` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _rvs(self, a, b, size=None, random_state=None):
|
|
u1 = gamma.rvs(a, size=size, random_state=random_state)
|
|
u2 = gamma.rvs(b, size=size, random_state=random_state)
|
|
return u1 / u2
|
|
|
|
def _pdf(self, x, a, b):
|
|
# betaprime.pdf(x, a, b) = x**(a-1) * (1+x)**(-a-b) / beta(a, b)
|
|
return np.exp(self._logpdf(x, a, b))
|
|
|
|
def _logpdf(self, x, a, b):
|
|
return sc.xlogy(a - 1.0, x) - sc.xlog1py(a + b, x) - sc.betaln(a, b)
|
|
|
|
def _cdf(self, x, a, b):
|
|
return sc.betainc(a, b, x/(1.+x))
|
|
|
|
def _munp(self, n, a, b):
|
|
if n == 1.0:
|
|
return np.where(b > 1,
|
|
a/(b-1.0),
|
|
np.inf)
|
|
elif n == 2.0:
|
|
return np.where(b > 2,
|
|
a*(a+1.0)/((b-2.0)*(b-1.0)),
|
|
np.inf)
|
|
elif n == 3.0:
|
|
return np.where(b > 3,
|
|
a*(a+1.0)*(a+2.0)/((b-3.0)*(b-2.0)*(b-1.0)),
|
|
np.inf)
|
|
elif n == 4.0:
|
|
return np.where(b > 4,
|
|
(a*(a + 1.0)*(a + 2.0)*(a + 3.0) /
|
|
((b - 4.0)*(b - 3.0)*(b - 2.0)*(b - 1.0))),
|
|
np.inf)
|
|
else:
|
|
raise NotImplementedError
|
|
|
|
|
|
betaprime = betaprime_gen(a=0.0, name='betaprime')
|
|
|
|
|
|
class bradford_gen(rv_continuous):
|
|
r"""A Bradford continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `bradford` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \frac{c}{\log(1+c) (1+cx)}
|
|
|
|
for :math:`0 <= x <= 1` and :math:`c > 0`.
|
|
|
|
`bradford` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, c):
|
|
# bradford.pdf(x, c) = c / (k * (1+c*x))
|
|
return c / (c*x + 1.0) / sc.log1p(c)
|
|
|
|
def _cdf(self, x, c):
|
|
return sc.log1p(c*x) / sc.log1p(c)
|
|
|
|
def _ppf(self, q, c):
|
|
return sc.expm1(q * sc.log1p(c)) / c
|
|
|
|
def _stats(self, c, moments='mv'):
|
|
k = np.log(1.0+c)
|
|
mu = (c-k)/(c*k)
|
|
mu2 = ((c+2.0)*k-2.0*c)/(2*c*k*k)
|
|
g1 = None
|
|
g2 = None
|
|
if 's' in moments:
|
|
g1 = np.sqrt(2)*(12*c*c-9*c*k*(c+2)+2*k*k*(c*(c+3)+3))
|
|
g1 /= np.sqrt(c*(c*(k-2)+2*k))*(3*c*(k-2)+6*k)
|
|
if 'k' in moments:
|
|
g2 = (c**3*(k-3)*(k*(3*k-16)+24)+12*k*c*c*(k-4)*(k-3) +
|
|
6*c*k*k*(3*k-14) + 12*k**3)
|
|
g2 /= 3*c*(c*(k-2)+2*k)**2
|
|
return mu, mu2, g1, g2
|
|
|
|
def _entropy(self, c):
|
|
k = np.log(1+c)
|
|
return k/2.0 - np.log(c/k)
|
|
|
|
|
|
bradford = bradford_gen(a=0.0, b=1.0, name='bradford')
|
|
|
|
|
|
class burr_gen(rv_continuous):
|
|
r"""A Burr (Type III) continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
fisk : a special case of either `burr` or `burr12` with ``d=1``
|
|
burr12 : Burr Type XII distribution
|
|
mielke : Mielke Beta-Kappa / Dagum distribution
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `burr` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c, d) = c d x^{-c - 1} / (1 + x^{-c})^{d + 1}
|
|
|
|
for :math:`x >= 0` and :math:`c, d > 0`.
|
|
|
|
`burr` takes :math:`c` and :math:`d` as shape parameters.
|
|
|
|
This is the PDF corresponding to the third CDF given in Burr's list;
|
|
specifically, it is equation (11) in Burr's paper [1]_. The distribution
|
|
is also commonly referred to as the Dagum distribution [2]_. If the
|
|
parameter :math:`c < 1` then the mean of the distribution does not
|
|
exist and if :math:`c < 2` the variance does not exist [2]_.
|
|
The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 1`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] Burr, I. W. "Cumulative frequency functions", Annals of
|
|
Mathematical Statistics, 13(2), pp 215-232 (1942).
|
|
.. [2] https://en.wikipedia.org/wiki/Dagum_distribution
|
|
.. [3] Kleiber, Christian. "A guide to the Dagum distributions."
|
|
Modeling Income Distributions and Lorenz Curves pp 97-117 (2008).
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
# Do not set _support_mask to rv_continuous._open_support_mask
|
|
# Whether the left-hand endpoint is suitable for pdf evaluation is dependent
|
|
# on the values of c and d: if c*d >= 1, the pdf is finite, otherwise infinite.
|
|
|
|
def _pdf(self, x, c, d):
|
|
# burr.pdf(x, c, d) = c * d * x**(-c-1) * (1+x**(-c))**(-d-1)
|
|
output = _lazywhere(x == 0, [x, c, d],
|
|
lambda x_, c_, d_: c_ * d_ * (x_**(c_*d_-1)) / (1 + x_**c_),
|
|
f2 = lambda x_, c_, d_: (c_ * d_ * (x_ ** (-c_ - 1.0)) /
|
|
((1 + x_ ** (-c_)) ** (d_ + 1.0))))
|
|
if output.ndim == 0:
|
|
return output[()]
|
|
return output
|
|
|
|
def _logpdf(self, x, c, d):
|
|
output = _lazywhere(
|
|
x == 0, [x, c, d],
|
|
lambda x_, c_, d_: (np.log(c_) + np.log(d_) + sc.xlogy(c_*d_ - 1, x_)
|
|
- (d_+1) * sc.log1p(x_**(c_))),
|
|
f2 = lambda x_, c_, d_: (np.log(c_) + np.log(d_)
|
|
+ sc.xlogy(-c_ - 1, x_)
|
|
- sc.xlog1py(d_+1, x_**(-c_))))
|
|
if output.ndim == 0:
|
|
return output[()]
|
|
return output
|
|
|
|
def _cdf(self, x, c, d):
|
|
return (1 + x**(-c))**(-d)
|
|
|
|
def _logcdf(self, x, c, d):
|
|
return sc.log1p(x**(-c)) * (-d)
|
|
|
|
def _sf(self, x, c, d):
|
|
return np.exp(self._logsf(x, c, d))
|
|
|
|
def _logsf(self, x, c, d):
|
|
return np.log1p(- (1 + x**(-c))**(-d))
|
|
|
|
def _ppf(self, q, c, d):
|
|
return (q**(-1.0/d) - 1)**(-1.0/c)
|
|
|
|
def _stats(self, c, d):
|
|
nc = np.arange(1, 5).reshape(4,1) / c
|
|
#ek is the kth raw moment, e1 is the mean e2-e1**2 variance etc.
|
|
e1, e2, e3, e4 = sc.beta(d + nc, 1. - nc) * d
|
|
mu = np.where(c > 1.0, e1, np.nan)
|
|
mu2_if_c = e2 - mu**2
|
|
mu2 = np.where(c > 2.0, mu2_if_c, np.nan)
|
|
g1 = _lazywhere(
|
|
c > 3.0,
|
|
(c, e1, e2, e3, mu2_if_c),
|
|
lambda c, e1, e2, e3, mu2_if_c: (e3 - 3*e2*e1 + 2*e1**3) / np.sqrt((mu2_if_c)**3),
|
|
fillvalue=np.nan)
|
|
g2 = _lazywhere(
|
|
c > 4.0,
|
|
(c, e1, e2, e3, e4, mu2_if_c),
|
|
lambda c, e1, e2, e3, e4, mu2_if_c: (
|
|
((e4 - 4*e3*e1 + 6*e2*e1**2 - 3*e1**4) / mu2_if_c**2) - 3),
|
|
fillvalue=np.nan)
|
|
return mu, mu2, g1, g2
|
|
|
|
def _munp(self, n, c, d):
|
|
def __munp(n, c, d):
|
|
nc = 1. * n / c
|
|
return d * sc.beta(1.0 - nc, d + nc)
|
|
n, c, d = np.asarray(n), np.asarray(c), np.asarray(d)
|
|
return _lazywhere((c > n) & (n == n) & (d == d), (c, d, n),
|
|
lambda c, d, n: __munp(n, c, d),
|
|
np.nan)
|
|
|
|
|
|
burr = burr_gen(a=0.0, name='burr')
|
|
|
|
|
|
class burr12_gen(rv_continuous):
|
|
r"""A Burr (Type XII) continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
fisk : a special case of either `burr` or `burr12` with ``d=1``
|
|
burr : Burr Type III distribution
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `burr` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c, d) = c d x^{c-1} / (1 + x^c)^{d + 1}
|
|
|
|
for :math:`x >= 0` and :math:`c, d > 0`.
|
|
|
|
`burr12` takes ``c`` and ``d`` as shape parameters for :math:`c`
|
|
and :math:`d`.
|
|
|
|
This is the PDF corresponding to the twelfth CDF given in Burr's list;
|
|
specifically, it is equation (20) in Burr's paper [1]_.
|
|
|
|
%(after_notes)s
|
|
|
|
The Burr type 12 distribution is also sometimes referred to as
|
|
the Singh-Maddala distribution from NIST [2]_.
|
|
|
|
References
|
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----------
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.. [1] Burr, I. W. "Cumulative frequency functions", Annals of
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Mathematical Statistics, 13(2), pp 215-232 (1942).
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.. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm
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.. [3] "Burr distribution",
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https://en.wikipedia.org/wiki/Burr_distribution
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%(example)s
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"""
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def _pdf(self, x, c, d):
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# burr12.pdf(x, c, d) = c * d * x**(c-1) * (1+x**(c))**(-d-1)
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return np.exp(self._logpdf(x, c, d))
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def _logpdf(self, x, c, d):
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return np.log(c) + np.log(d) + sc.xlogy(c - 1, x) + sc.xlog1py(-d-1, x**c)
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def _cdf(self, x, c, d):
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return -sc.expm1(self._logsf(x, c, d))
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def _logcdf(self, x, c, d):
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return sc.log1p(-(1 + x**c)**(-d))
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def _sf(self, x, c, d):
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return np.exp(self._logsf(x, c, d))
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def _logsf(self, x, c, d):
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return sc.xlog1py(-d, x**c)
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def _ppf(self, q, c, d):
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# The following is an implementation of
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# ((1 - q)**(-1.0/d) - 1)**(1.0/c)
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# that does a better job handling small values of q.
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return sc.expm1(-1/d * sc.log1p(-q))**(1/c)
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def _munp(self, n, c, d):
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nc = 1. * n / c
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return d * sc.beta(1.0 + nc, d - nc)
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burr12 = burr12_gen(a=0.0, name='burr12')
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class fisk_gen(burr_gen):
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r"""A Fisk continuous random variable.
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The Fisk distribution is also known as the log-logistic distribution.
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%(before_notes)s
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See Also
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--------
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burr
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Notes
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-----
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The probability density function for `fisk` is:
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.. math::
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f(x, c) = c x^{-c-1} (1 + x^{-c})^{-2}
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for :math:`x >= 0` and :math:`c > 0`.
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`fisk` takes ``c`` as a shape parameter for :math:`c`.
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`fisk` is a special case of `burr` or `burr12` with ``d=1``.
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%(after_notes)s
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%(example)s
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"""
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def _pdf(self, x, c):
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# fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2)
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return burr._pdf(x, c, 1.0)
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def _cdf(self, x, c):
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return burr._cdf(x, c, 1.0)
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def _sf(self, x, c):
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return burr._sf(x, c, 1.0)
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def _logpdf(self, x, c):
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# fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2)
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return burr._logpdf(x, c, 1.0)
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def _logcdf(self, x, c):
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return burr._logcdf(x, c, 1.0)
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def _logsf(self, x, c):
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return burr._logsf(x, c, 1.0)
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def _ppf(self, x, c):
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return burr._ppf(x, c, 1.0)
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def _munp(self, n, c):
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return burr._munp(n, c, 1.0)
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def _stats(self, c):
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return burr._stats(c, 1.0)
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def _entropy(self, c):
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return 2 - np.log(c)
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fisk = fisk_gen(a=0.0, name='fisk')
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# median = loc
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class cauchy_gen(rv_continuous):
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r"""A Cauchy continuous random variable.
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%(before_notes)s
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Notes
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-----
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The probability density function for `cauchy` is
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.. math::
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f(x) = \frac{1}{\pi (1 + x^2)}
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for a real number :math:`x`.
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%(after_notes)s
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%(example)s
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"""
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def _pdf(self, x):
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# cauchy.pdf(x) = 1 / (pi * (1 + x**2))
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return 1.0/np.pi/(1.0+x*x)
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def _cdf(self, x):
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return 0.5 + 1.0/np.pi*np.arctan(x)
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def _ppf(self, q):
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return np.tan(np.pi*q-np.pi/2.0)
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def _sf(self, x):
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return 0.5 - 1.0/np.pi*np.arctan(x)
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def _isf(self, q):
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return np.tan(np.pi/2.0-np.pi*q)
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def _stats(self):
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return np.nan, np.nan, np.nan, np.nan
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def _entropy(self):
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return np.log(4*np.pi)
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def _fitstart(self, data, args=None):
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# Initialize ML guesses using quartiles instead of moments.
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p25, p50, p75 = np.percentile(data, [25, 50, 75])
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return p50, (p75 - p25)/2
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cauchy = cauchy_gen(name='cauchy')
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class chi_gen(rv_continuous):
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r"""A chi continuous random variable.
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%(before_notes)s
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Notes
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-----
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The probability density function for `chi` is:
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.. math::
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f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)}
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x^{k-1} \exp \left( -x^2/2 \right)
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for :math:`x >= 0` and :math:`k > 0` (degrees of freedom, denoted ``df``
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in the implementation). :math:`\Gamma` is the gamma function
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(`scipy.special.gamma`).
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Special cases of `chi` are:
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- ``chi(1, loc, scale)`` is equivalent to `halfnorm`
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- ``chi(2, 0, scale)`` is equivalent to `rayleigh`
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- ``chi(3, 0, scale)`` is equivalent to `maxwell`
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`chi` takes ``df`` as a shape parameter.
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%(after_notes)s
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%(example)s
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"""
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def _rvs(self, df, size=None, random_state=None):
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return np.sqrt(chi2.rvs(df, size=size, random_state=random_state))
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def _pdf(self, x, df):
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# x**(df-1) * exp(-x**2/2)
|
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# chi.pdf(x, df) = -------------------------
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# 2**(df/2-1) * gamma(df/2)
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return np.exp(self._logpdf(x, df))
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def _logpdf(self, x, df):
|
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l = np.log(2) - .5*np.log(2)*df - sc.gammaln(.5*df)
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return l + sc.xlogy(df - 1., x) - .5*x**2
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def _cdf(self, x, df):
|
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return sc.gammainc(.5*df, .5*x**2)
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def _ppf(self, q, df):
|
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return np.sqrt(2*sc.gammaincinv(.5*df, q))
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def _stats(self, df):
|
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mu = np.sqrt(2)*sc.gamma(df/2.0+0.5)/sc.gamma(df/2.0)
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mu2 = df - mu*mu
|
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g1 = (2*mu**3.0 + mu*(1-2*df))/np.asarray(np.power(mu2, 1.5))
|
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g2 = 2*df*(1.0-df)-6*mu**4 + 4*mu**2 * (2*df-1)
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g2 /= np.asarray(mu2**2.0)
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return mu, mu2, g1, g2
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chi = chi_gen(a=0.0, name='chi')
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|
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## Chi-squared (gamma-distributed with loc=0 and scale=2 and shape=df/2)
|
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class chi2_gen(rv_continuous):
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r"""A chi-squared continuous random variable.
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For the noncentral chi-square distribution, see `ncx2`.
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%(before_notes)s
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See Also
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--------
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ncx2
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Notes
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-----
|
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The probability density function for `chi2` is:
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.. math::
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f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)}
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x^{k/2-1} \exp \left( -x/2 \right)
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for :math:`x > 0` and :math:`k > 0` (degrees of freedom, denoted ``df``
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in the implementation).
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`chi2` takes ``df`` as a shape parameter.
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%(after_notes)s
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%(example)s
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"""
|
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def _rvs(self, df, size=None, random_state=None):
|
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return random_state.chisquare(df, size)
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def _pdf(self, x, df):
|
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# chi2.pdf(x, df) = 1 / (2*gamma(df/2)) * (x/2)**(df/2-1) * exp(-x/2)
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return np.exp(self._logpdf(x, df))
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def _logpdf(self, x, df):
|
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return sc.xlogy(df/2.-1, x) - x/2. - sc.gammaln(df/2.) - (np.log(2)*df)/2.
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def _cdf(self, x, df):
|
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return sc.chdtr(df, x)
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def _sf(self, x, df):
|
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return sc.chdtrc(df, x)
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|
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def _isf(self, p, df):
|
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return sc.chdtri(df, p)
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|
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def _ppf(self, p, df):
|
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return 2*sc.gammaincinv(df/2, p)
|
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|
|
def _stats(self, df):
|
|
mu = df
|
|
mu2 = 2*df
|
|
g1 = 2*np.sqrt(2.0/df)
|
|
g2 = 12.0/df
|
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return mu, mu2, g1, g2
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|
|
|
|
chi2 = chi2_gen(a=0.0, name='chi2')
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|
|
|
|
class cosine_gen(rv_continuous):
|
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r"""A cosine continuous random variable.
|
|
|
|
%(before_notes)s
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|
|
|
Notes
|
|
-----
|
|
The cosine distribution is an approximation to the normal distribution.
|
|
The probability density function for `cosine` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{1}{2\pi} (1+\cos(x))
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|
|
for :math:`-\pi \le x \le \pi`.
|
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|
|
%(after_notes)s
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|
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|
%(example)s
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|
|
|
"""
|
|
def _pdf(self, x):
|
|
# cosine.pdf(x) = 1/(2*pi) * (1+cos(x))
|
|
return 1.0/2/np.pi*(1+np.cos(x))
|
|
|
|
def _cdf(self, x):
|
|
return 1.0/2/np.pi*(np.pi + x + np.sin(x))
|
|
|
|
def _stats(self):
|
|
return 0.0, np.pi*np.pi/3.0-2.0, 0.0, -6.0*(np.pi**4-90)/(5.0*(np.pi*np.pi-6)**2)
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|
|
|
def _entropy(self):
|
|
return np.log(4*np.pi)-1.0
|
|
|
|
|
|
cosine = cosine_gen(a=-np.pi, b=np.pi, name='cosine')
|
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|
|
|
|
class dgamma_gen(rv_continuous):
|
|
r"""A double gamma continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `dgamma` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|)
|
|
|
|
for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the
|
|
gamma function (`scipy.special.gamma`).
|
|
|
|
`dgamma` takes ``a`` as a shape parameter for :math:`a`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, a, size=None, random_state=None):
|
|
u = random_state.uniform(size=size)
|
|
gm = gamma.rvs(a, size=size, random_state=random_state)
|
|
return gm * np.where(u >= 0.5, 1, -1)
|
|
|
|
def _pdf(self, x, a):
|
|
# dgamma.pdf(x, a) = 1 / (2*gamma(a)) * abs(x)**(a-1) * exp(-abs(x))
|
|
ax = abs(x)
|
|
return 1.0/(2*sc.gamma(a))*ax**(a-1.0) * np.exp(-ax)
|
|
|
|
def _logpdf(self, x, a):
|
|
ax = abs(x)
|
|
return sc.xlogy(a - 1.0, ax) - ax - np.log(2) - sc.gammaln(a)
|
|
|
|
def _cdf(self, x, a):
|
|
fac = 0.5*sc.gammainc(a, abs(x))
|
|
return np.where(x > 0, 0.5 + fac, 0.5 - fac)
|
|
|
|
def _sf(self, x, a):
|
|
fac = 0.5*sc.gammainc(a, abs(x))
|
|
return np.where(x > 0, 0.5-fac, 0.5+fac)
|
|
|
|
def _ppf(self, q, a):
|
|
fac = sc.gammainccinv(a, 1-abs(2*q-1))
|
|
return np.where(q > 0.5, fac, -fac)
|
|
|
|
def _stats(self, a):
|
|
mu2 = a*(a+1.0)
|
|
return 0.0, mu2, 0.0, (a+2.0)*(a+3.0)/mu2-3.0
|
|
|
|
|
|
dgamma = dgamma_gen(name='dgamma')
|
|
|
|
|
|
class dweibull_gen(rv_continuous):
|
|
r"""A double Weibull continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `dweibull` is given by
|
|
|
|
.. math::
|
|
|
|
f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c)
|
|
|
|
for a real number :math:`x` and :math:`c > 0`.
|
|
|
|
`dweibull` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, c, size=None, random_state=None):
|
|
u = random_state.uniform(size=size)
|
|
w = weibull_min.rvs(c, size=size, random_state=random_state)
|
|
return w * (np.where(u >= 0.5, 1, -1))
|
|
|
|
def _pdf(self, x, c):
|
|
# dweibull.pdf(x, c) = c / 2 * abs(x)**(c-1) * exp(-abs(x)**c)
|
|
ax = abs(x)
|
|
Px = c / 2.0 * ax**(c-1.0) * np.exp(-ax**c)
|
|
return Px
|
|
|
|
def _logpdf(self, x, c):
|
|
ax = abs(x)
|
|
return np.log(c) - np.log(2.0) + sc.xlogy(c - 1.0, ax) - ax**c
|
|
|
|
def _cdf(self, x, c):
|
|
Cx1 = 0.5 * np.exp(-abs(x)**c)
|
|
return np.where(x > 0, 1 - Cx1, Cx1)
|
|
|
|
def _ppf(self, q, c):
|
|
fac = 2. * np.where(q <= 0.5, q, 1. - q)
|
|
fac = np.power(-np.log(fac), 1.0 / c)
|
|
return np.where(q > 0.5, fac, -fac)
|
|
|
|
def _munp(self, n, c):
|
|
return (1 - (n % 2)) * sc.gamma(1.0 + 1.0 * n / c)
|
|
|
|
# since we know that all odd moments are zeros, return them at once.
|
|
# returning Nones from _stats makes the public stats call _munp
|
|
# so overall we're saving one or two gamma function evaluations here.
|
|
def _stats(self, c):
|
|
return 0, None, 0, None
|
|
|
|
|
|
dweibull = dweibull_gen(name='dweibull')
|
|
|
|
|
|
## Exponential (gamma distributed with a=1.0, loc=loc and scale=scale)
|
|
class expon_gen(rv_continuous):
|
|
r"""An exponential continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `expon` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \exp(-x)
|
|
|
|
for :math:`x \ge 0`.
|
|
|
|
%(after_notes)s
|
|
|
|
A common parameterization for `expon` is in terms of the rate parameter
|
|
``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This
|
|
parameterization corresponds to using ``scale = 1 / lambda``.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, size=None, random_state=None):
|
|
return random_state.standard_exponential(size)
|
|
|
|
def _pdf(self, x):
|
|
# expon.pdf(x) = exp(-x)
|
|
return np.exp(-x)
|
|
|
|
def _logpdf(self, x):
|
|
return -x
|
|
|
|
def _cdf(self, x):
|
|
return -sc.expm1(-x)
|
|
|
|
def _ppf(self, q):
|
|
return -sc.log1p(-q)
|
|
|
|
def _sf(self, x):
|
|
return np.exp(-x)
|
|
|
|
def _logsf(self, x):
|
|
return -x
|
|
|
|
def _isf(self, q):
|
|
return -np.log(q)
|
|
|
|
def _stats(self):
|
|
return 1.0, 1.0, 2.0, 6.0
|
|
|
|
def _entropy(self):
|
|
return 1.0
|
|
|
|
@replace_notes_in_docstring(rv_continuous, notes="""\
|
|
This function uses explicit formulas for the maximum likelihood
|
|
estimation of the exponential distribution parameters, so the
|
|
`optimizer`, `loc` and `scale` keyword arguments are ignored.\n\n""")
|
|
def fit(self, data, *args, **kwds):
|
|
if len(args) > 0:
|
|
raise TypeError("Too many arguments.")
|
|
|
|
floc = kwds.pop('floc', None)
|
|
fscale = kwds.pop('fscale', None)
|
|
|
|
_remove_optimizer_parameters(kwds)
|
|
|
|
if floc is not None and fscale is not None:
|
|
# This check is for consistency with `rv_continuous.fit`.
|
|
raise ValueError("All parameters fixed. There is nothing to "
|
|
"optimize.")
|
|
|
|
data = np.asarray(data)
|
|
|
|
if not np.isfinite(data).all():
|
|
raise RuntimeError("The data contains non-finite values.")
|
|
|
|
data_min = data.min()
|
|
|
|
if floc is None:
|
|
# ML estimate of the location is the minimum of the data.
|
|
loc = data_min
|
|
else:
|
|
loc = floc
|
|
if data_min < loc:
|
|
# There are values that are less than the specified loc.
|
|
raise FitDataError("expon", lower=floc, upper=np.inf)
|
|
|
|
if fscale is None:
|
|
# ML estimate of the scale is the shifted mean.
|
|
scale = data.mean() - loc
|
|
else:
|
|
scale = fscale
|
|
|
|
# We expect the return values to be floating point, so ensure it
|
|
# by explicitly converting to float.
|
|
return float(loc), float(scale)
|
|
|
|
|
|
expon = expon_gen(a=0.0, name='expon')
|
|
|
|
|
|
# Exponentially Modified Normal (exponential distribution
|
|
# convolved with a Normal).
|
|
# This is called an exponentially modified gaussian on wikipedia.
|
|
class exponnorm_gen(rv_continuous):
|
|
r"""An exponentially modified Normal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `exponnorm` is:
|
|
|
|
.. math::
|
|
|
|
f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right)
|
|
\text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)
|
|
|
|
where :math:`x` is a real number and :math:`K > 0`.
|
|
|
|
It can be thought of as the sum of a standard normal random variable
|
|
and an independent exponentially distributed random variable with rate
|
|
``1/K``.
|
|
|
|
%(after_notes)s
|
|
|
|
An alternative parameterization of this distribution (for example, in
|
|
`Wikipedia <https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution>`_)
|
|
involves three parameters, :math:`\mu`, :math:`\lambda` and
|
|
:math:`\sigma`.
|
|
In the present parameterization this corresponds to having ``loc`` and
|
|
``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and
|
|
shape parameter :math:`K = 1/(\sigma\lambda)`.
|
|
|
|
.. versionadded:: 0.16.0
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, K, size=None, random_state=None):
|
|
expval = random_state.standard_exponential(size) * K
|
|
gval = random_state.standard_normal(size)
|
|
return expval + gval
|
|
|
|
def _pdf(self, x, K):
|
|
return np.exp(self._logpdf(x, K))
|
|
|
|
def _logpdf(self, x, K):
|
|
invK = 1.0 / K
|
|
exparg = invK * (0.5 * invK - x)
|
|
return exparg + _norm_logcdf(x - invK) - np.log(K)
|
|
|
|
def _cdf(self, x, K):
|
|
invK = 1.0 / K
|
|
expval = invK * (0.5 * invK - x)
|
|
return _norm_cdf(x) - np.exp(expval) * _norm_cdf(x - invK)
|
|
|
|
def _sf(self, x, K):
|
|
invK = 1.0 / K
|
|
expval = invK * (0.5 * invK - x)
|
|
return _norm_cdf(-x) + np.exp(expval) * _norm_cdf(x - invK)
|
|
|
|
def _stats(self, K):
|
|
K2 = K * K
|
|
opK2 = 1.0 + K2
|
|
skw = 2 * K**3 * opK2**(-1.5)
|
|
krt = 6.0 * K2 * K2 * opK2**(-2)
|
|
return K, opK2, skw, krt
|
|
|
|
|
|
exponnorm = exponnorm_gen(name='exponnorm')
|
|
|
|
|
|
class exponweib_gen(rv_continuous):
|
|
r"""An exponentiated Weibull continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
weibull_min, numpy.random.RandomState.weibull
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `exponweib` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, c) = a c [1-\exp(-x^c)]^{a-1} \exp(-x^c) x^{c-1}
|
|
|
|
and its cumulative distribution function is:
|
|
|
|
.. math::
|
|
|
|
F(x, a, c) = [1-\exp(-x^c)]^a
|
|
|
|
for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`.
|
|
|
|
`exponweib` takes :math:`a` and :math:`c` as shape parameters:
|
|
|
|
* :math:`a` is the exponentiation parameter,
|
|
with the special case :math:`a=1` corresponding to the
|
|
(non-exponentiated) Weibull distribution `weibull_min`.
|
|
* :math:`c` is the shape parameter of the non-exponentiated Weibull law.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
https://en.wikipedia.org/wiki/Exponentiated_Weibull_distribution
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, a, c):
|
|
# exponweib.pdf(x, a, c) =
|
|
# a * c * (1-exp(-x**c))**(a-1) * exp(-x**c)*x**(c-1)
|
|
return np.exp(self._logpdf(x, a, c))
|
|
|
|
def _logpdf(self, x, a, c):
|
|
negxc = -x**c
|
|
exm1c = -sc.expm1(negxc)
|
|
logp = (np.log(a) + np.log(c) + sc.xlogy(a - 1.0, exm1c) +
|
|
negxc + sc.xlogy(c - 1.0, x))
|
|
return logp
|
|
|
|
def _cdf(self, x, a, c):
|
|
exm1c = -sc.expm1(-x**c)
|
|
return exm1c**a
|
|
|
|
def _ppf(self, q, a, c):
|
|
return (-sc.log1p(-q**(1.0/a)))**np.asarray(1.0/c)
|
|
|
|
|
|
exponweib = exponweib_gen(a=0.0, name='exponweib')
|
|
|
|
|
|
class exponpow_gen(rv_continuous):
|
|
r"""An exponential power continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `exponpow` is:
|
|
|
|
.. math::
|
|
|
|
f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b))
|
|
|
|
for :math:`x \ge 0`, :math:`b > 0`. Note that this is a different
|
|
distribution from the exponential power distribution that is also known
|
|
under the names "generalized normal" or "generalized Gaussian".
|
|
|
|
`exponpow` takes ``b`` as a shape parameter for :math:`b`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, b):
|
|
# exponpow.pdf(x, b) = b * x**(b-1) * exp(1 + x**b - exp(x**b))
|
|
return np.exp(self._logpdf(x, b))
|
|
|
|
def _logpdf(self, x, b):
|
|
xb = x**b
|
|
f = 1 + np.log(b) + sc.xlogy(b - 1.0, x) + xb - np.exp(xb)
|
|
return f
|
|
|
|
def _cdf(self, x, b):
|
|
return -sc.expm1(-sc.expm1(x**b))
|
|
|
|
def _sf(self, x, b):
|
|
return np.exp(-sc.expm1(x**b))
|
|
|
|
def _isf(self, x, b):
|
|
return (sc.log1p(-np.log(x)))**(1./b)
|
|
|
|
def _ppf(self, q, b):
|
|
return pow(sc.log1p(-sc.log1p(-q)), 1.0/b)
|
|
|
|
|
|
exponpow = exponpow_gen(a=0.0, name='exponpow')
|
|
|
|
|
|
class fatiguelife_gen(rv_continuous):
|
|
r"""A fatigue-life (Birnbaum-Saunders) continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `fatiguelife` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \frac{x+1}{2c\sqrt{2\pi x^3}} \exp(-\frac{(x-1)^2}{2x c^2})
|
|
|
|
for :math:`x >= 0` and :math:`c > 0`.
|
|
|
|
`fatiguelife` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] "Birnbaum-Saunders distribution",
|
|
https://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _rvs(self, c, size=None, random_state=None):
|
|
z = random_state.standard_normal(size)
|
|
x = 0.5*c*z
|
|
x2 = x*x
|
|
t = 1.0 + 2*x2 + 2*x*np.sqrt(1 + x2)
|
|
return t
|
|
|
|
def _pdf(self, x, c):
|
|
# fatiguelife.pdf(x, c) =
|
|
# (x+1) / (2*c*sqrt(2*pi*x**3)) * exp(-(x-1)**2/(2*x*c**2))
|
|
return np.exp(self._logpdf(x, c))
|
|
|
|
def _logpdf(self, x, c):
|
|
return (np.log(x+1) - (x-1)**2 / (2.0*x*c**2) - np.log(2*c) -
|
|
0.5*(np.log(2*np.pi) + 3*np.log(x)))
|
|
|
|
def _cdf(self, x, c):
|
|
return _norm_cdf(1.0 / c * (np.sqrt(x) - 1.0/np.sqrt(x)))
|
|
|
|
def _ppf(self, q, c):
|
|
tmp = c*sc.ndtri(q)
|
|
return 0.25 * (tmp + np.sqrt(tmp**2 + 4))**2
|
|
|
|
def _stats(self, c):
|
|
# NB: the formula for kurtosis in wikipedia seems to have an error:
|
|
# it's 40, not 41. At least it disagrees with the one from Wolfram
|
|
# Alpha. And the latter one, below, passes the tests, while the wiki
|
|
# one doesn't So far I didn't have the guts to actually check the
|
|
# coefficients from the expressions for the raw moments.
|
|
c2 = c*c
|
|
mu = c2 / 2.0 + 1.0
|
|
den = 5.0 * c2 + 4.0
|
|
mu2 = c2*den / 4.0
|
|
g1 = 4 * c * (11*c2 + 6.0) / np.power(den, 1.5)
|
|
g2 = 6 * c2 * (93*c2 + 40.0) / den**2.0
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
fatiguelife = fatiguelife_gen(a=0.0, name='fatiguelife')
|
|
|
|
|
|
class foldcauchy_gen(rv_continuous):
|
|
r"""A folded Cauchy continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `foldcauchy` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)}
|
|
|
|
for :math:`x \ge 0`.
|
|
|
|
`foldcauchy` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, c, size=None, random_state=None):
|
|
return abs(cauchy.rvs(loc=c, size=size,
|
|
random_state=random_state))
|
|
|
|
def _pdf(self, x, c):
|
|
# foldcauchy.pdf(x, c) = 1/(pi*(1+(x-c)**2)) + 1/(pi*(1+(x+c)**2))
|
|
return 1.0/np.pi*(1.0/(1+(x-c)**2) + 1.0/(1+(x+c)**2))
|
|
|
|
def _cdf(self, x, c):
|
|
return 1.0/np.pi*(np.arctan(x-c) + np.arctan(x+c))
|
|
|
|
def _stats(self, c):
|
|
return np.inf, np.inf, np.nan, np.nan
|
|
|
|
|
|
foldcauchy = foldcauchy_gen(a=0.0, name='foldcauchy')
|
|
|
|
|
|
class f_gen(rv_continuous):
|
|
r"""An F continuous random variable.
|
|
|
|
For the noncentral F distribution, see `ncf`.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
ncf
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `f` is:
|
|
|
|
.. math::
|
|
|
|
f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}}
|
|
{(df_2+df_1 x)^{(df_1+df_2)/2}
|
|
B(df_1/2, df_2/2)}
|
|
|
|
for :math:`x > 0`.
|
|
|
|
`f` takes ``dfn`` and ``dfd`` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, dfn, dfd, size=None, random_state=None):
|
|
return random_state.f(dfn, dfd, size)
|
|
|
|
def _pdf(self, x, dfn, dfd):
|
|
# df2**(df2/2) * df1**(df1/2) * x**(df1/2-1)
|
|
# F.pdf(x, df1, df2) = --------------------------------------------
|
|
# (df2+df1*x)**((df1+df2)/2) * B(df1/2, df2/2)
|
|
return np.exp(self._logpdf(x, dfn, dfd))
|
|
|
|
def _logpdf(self, x, dfn, dfd):
|
|
n = 1.0 * dfn
|
|
m = 1.0 * dfd
|
|
lPx = m/2 * np.log(m) + n/2 * np.log(n) + sc.xlogy(n/2 - 1, x)
|
|
lPx -= ((n+m)/2) * np.log(m + n*x) + sc.betaln(n/2, m/2)
|
|
return lPx
|
|
|
|
def _cdf(self, x, dfn, dfd):
|
|
return sc.fdtr(dfn, dfd, x)
|
|
|
|
def _sf(self, x, dfn, dfd):
|
|
return sc.fdtrc(dfn, dfd, x)
|
|
|
|
def _ppf(self, q, dfn, dfd):
|
|
return sc.fdtri(dfn, dfd, q)
|
|
|
|
def _stats(self, dfn, dfd):
|
|
v1, v2 = 1. * dfn, 1. * dfd
|
|
v2_2, v2_4, v2_6, v2_8 = v2 - 2., v2 - 4., v2 - 6., v2 - 8.
|
|
|
|
mu = _lazywhere(
|
|
v2 > 2, (v2, v2_2),
|
|
lambda v2, v2_2: v2 / v2_2,
|
|
np.inf)
|
|
|
|
mu2 = _lazywhere(
|
|
v2 > 4, (v1, v2, v2_2, v2_4),
|
|
lambda v1, v2, v2_2, v2_4:
|
|
2 * v2 * v2 * (v1 + v2_2) / (v1 * v2_2**2 * v2_4),
|
|
np.inf)
|
|
|
|
g1 = _lazywhere(
|
|
v2 > 6, (v1, v2_2, v2_4, v2_6),
|
|
lambda v1, v2_2, v2_4, v2_6:
|
|
(2 * v1 + v2_2) / v2_6 * np.sqrt(v2_4 / (v1 * (v1 + v2_2))),
|
|
np.nan)
|
|
g1 *= np.sqrt(8.)
|
|
|
|
g2 = _lazywhere(
|
|
v2 > 8, (g1, v2_6, v2_8),
|
|
lambda g1, v2_6, v2_8: (8 + g1 * g1 * v2_6) / v2_8,
|
|
np.nan)
|
|
g2 *= 3. / 2.
|
|
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
f = f_gen(a=0.0, name='f')
|
|
|
|
|
|
## Folded Normal
|
|
## abs(Z) where (Z is normal with mu=L and std=S so that c=abs(L)/S)
|
|
##
|
|
## note: regress docs have scale parameter correct, but first parameter
|
|
## he gives is a shape parameter A = c * scale
|
|
|
|
## Half-normal is folded normal with shape-parameter c=0.
|
|
|
|
class foldnorm_gen(rv_continuous):
|
|
r"""A folded normal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `foldnorm` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2})
|
|
|
|
for :math:`c \ge 0`.
|
|
|
|
`foldnorm` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, c):
|
|
return c >= 0
|
|
|
|
def _rvs(self, c, size=None, random_state=None):
|
|
return abs(random_state.standard_normal(size) + c)
|
|
|
|
def _pdf(self, x, c):
|
|
# foldnormal.pdf(x, c) = sqrt(2/pi) * cosh(c*x) * exp(-(x**2+c**2)/2)
|
|
return _norm_pdf(x + c) + _norm_pdf(x-c)
|
|
|
|
def _cdf(self, x, c):
|
|
return _norm_cdf(x-c) + _norm_cdf(x+c) - 1.0
|
|
|
|
def _stats(self, c):
|
|
# Regina C. Elandt, Technometrics 3, 551 (1961)
|
|
# https://www.jstor.org/stable/1266561
|
|
#
|
|
c2 = c*c
|
|
expfac = np.exp(-0.5*c2) / np.sqrt(2.*np.pi)
|
|
|
|
mu = 2.*expfac + c * sc.erf(c/np.sqrt(2))
|
|
mu2 = c2 + 1 - mu*mu
|
|
|
|
g1 = 2. * (mu*mu*mu - c2*mu - expfac)
|
|
g1 /= np.power(mu2, 1.5)
|
|
|
|
g2 = c2 * (c2 + 6.) + 3 + 8.*expfac*mu
|
|
g2 += (2. * (c2 - 3.) - 3. * mu**2) * mu**2
|
|
g2 = g2 / mu2**2.0 - 3.
|
|
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
foldnorm = foldnorm_gen(a=0.0, name='foldnorm')
|
|
|
|
|
|
class weibull_min_gen(rv_continuous):
|
|
r"""Weibull minimum continuous random variable.
|
|
|
|
The Weibull Minimum Extreme Value distribution, from extreme value theory
|
|
(Fisher-Gnedenko theorem), is also often simply called the Weibull
|
|
distribution. It arises as the limiting distribution of the rescaled
|
|
minimum of iid random variables.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
weibull_max, numpy.random.RandomState.weibull, exponweib
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `weibull_min` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = c x^{c-1} \exp(-x^c)
|
|
|
|
for :math:`x > 0`, :math:`c > 0`.
|
|
|
|
`weibull_min` takes ``c`` as a shape parameter for :math:`c`.
|
|
(named :math:`k` in Wikipedia article and :math:`a` in
|
|
``numpy.random.weibull``). Special shape values are :math:`c=1` and
|
|
:math:`c=2` where Weibull distribution reduces to the `expon` and
|
|
`rayleigh` distributions respectively.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
https://en.wikipedia.org/wiki/Weibull_distribution
|
|
|
|
https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _pdf(self, x, c):
|
|
# weibull_min.pdf(x, c) = c * x**(c-1) * exp(-x**c)
|
|
return c*pow(x, c-1)*np.exp(-pow(x, c))
|
|
|
|
def _logpdf(self, x, c):
|
|
return np.log(c) + sc.xlogy(c - 1, x) - pow(x, c)
|
|
|
|
def _cdf(self, x, c):
|
|
return -sc.expm1(-pow(x, c))
|
|
|
|
def _sf(self, x, c):
|
|
return np.exp(-pow(x, c))
|
|
|
|
def _logsf(self, x, c):
|
|
return -pow(x, c)
|
|
|
|
def _ppf(self, q, c):
|
|
return pow(-sc.log1p(-q), 1.0/c)
|
|
|
|
def _munp(self, n, c):
|
|
return sc.gamma(1.0+n*1.0/c)
|
|
|
|
def _entropy(self, c):
|
|
return -_EULER / c - np.log(c) + _EULER + 1
|
|
|
|
|
|
weibull_min = weibull_min_gen(a=0.0, name='weibull_min')
|
|
|
|
|
|
class weibull_max_gen(rv_continuous):
|
|
r"""Weibull maximum continuous random variable.
|
|
|
|
The Weibull Maximum Extreme Value distribution, from extreme value theory
|
|
(Fisher-Gnedenko theorem), is the limiting distribution of rescaled
|
|
maximum of iid random variables. This is the distribution of -X
|
|
if X is from the `weibull_min` function.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
weibull_min
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `weibull_max` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = c (-x)^{c-1} \exp(-(-x)^c)
|
|
|
|
for :math:`x < 0`, :math:`c > 0`.
|
|
|
|
`weibull_max` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
https://en.wikipedia.org/wiki/Weibull_distribution
|
|
|
|
https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, c):
|
|
# weibull_max.pdf(x, c) = c * (-x)**(c-1) * exp(-(-x)**c)
|
|
return c*pow(-x, c-1)*np.exp(-pow(-x, c))
|
|
|
|
def _logpdf(self, x, c):
|
|
return np.log(c) + sc.xlogy(c-1, -x) - pow(-x, c)
|
|
|
|
def _cdf(self, x, c):
|
|
return np.exp(-pow(-x, c))
|
|
|
|
def _logcdf(self, x, c):
|
|
return -pow(-x, c)
|
|
|
|
def _sf(self, x, c):
|
|
return -sc.expm1(-pow(-x, c))
|
|
|
|
def _ppf(self, q, c):
|
|
return -pow(-np.log(q), 1.0/c)
|
|
|
|
def _munp(self, n, c):
|
|
val = sc.gamma(1.0+n*1.0/c)
|
|
if int(n) % 2:
|
|
sgn = -1
|
|
else:
|
|
sgn = 1
|
|
return sgn * val
|
|
|
|
def _entropy(self, c):
|
|
return -_EULER / c - np.log(c) + _EULER + 1
|
|
|
|
|
|
weibull_max = weibull_max_gen(b=0.0, name='weibull_max')
|
|
|
|
|
|
class genlogistic_gen(rv_continuous):
|
|
r"""A generalized logistic continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `genlogistic` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = c \frac{\exp(-x)}
|
|
{(1 + \exp(-x))^{c+1}}
|
|
|
|
for :math:`x >= 0`, :math:`c > 0`.
|
|
|
|
`genlogistic` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, c):
|
|
# genlogistic.pdf(x, c) = c * exp(-x) / (1 + exp(-x))**(c+1)
|
|
return np.exp(self._logpdf(x, c))
|
|
|
|
def _logpdf(self, x, c):
|
|
# Two mathematically equivalent expressions for log(pdf(x, c)):
|
|
# log(pdf(x, c)) = log(c) - x - (c + 1)*log(1 + exp(-x))
|
|
# = log(c) + c*x - (c + 1)*log(1 + exp(x))
|
|
mult = -(c - 1) * (x < 0) - 1
|
|
absx = np.abs(x)
|
|
return np.log(c) + mult*absx - (c+1) * sc.log1p(np.exp(-absx))
|
|
|
|
def _cdf(self, x, c):
|
|
Cx = (1+np.exp(-x))**(-c)
|
|
return Cx
|
|
|
|
def _ppf(self, q, c):
|
|
vals = -np.log(pow(q, -1.0/c)-1)
|
|
return vals
|
|
|
|
def _stats(self, c):
|
|
mu = _EULER + sc.psi(c)
|
|
mu2 = np.pi*np.pi/6.0 + sc.zeta(2, c)
|
|
g1 = -2*sc.zeta(3, c) + 2*_ZETA3
|
|
g1 /= np.power(mu2, 1.5)
|
|
g2 = np.pi**4/15.0 + 6*sc.zeta(4, c)
|
|
g2 /= mu2**2.0
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
genlogistic = genlogistic_gen(name='genlogistic')
|
|
|
|
|
|
class genpareto_gen(rv_continuous):
|
|
r"""A generalized Pareto continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `genpareto` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = (1 + c x)^{-1 - 1/c}
|
|
|
|
defined for :math:`x \ge 0` if :math:`c \ge 0`, and for
|
|
:math:`0 \le x \le -1/c` if :math:`c < 0`.
|
|
|
|
`genpareto` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
For :math:`c=0`, `genpareto` reduces to the exponential
|
|
distribution, `expon`:
|
|
|
|
.. math::
|
|
|
|
f(x, 0) = \exp(-x)
|
|
|
|
For :math:`c=-1`, `genpareto` is uniform on ``[0, 1]``:
|
|
|
|
.. math::
|
|
|
|
f(x, -1) = 1
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, c):
|
|
return np.isfinite(c)
|
|
|
|
def _get_support(self, c):
|
|
c = np.asarray(c)
|
|
b = _lazywhere(c < 0, (c,),
|
|
lambda c: -1. / c,
|
|
np.inf)
|
|
a = np.where(c >= 0, self.a, self.a)
|
|
return a, b
|
|
|
|
def _pdf(self, x, c):
|
|
# genpareto.pdf(x, c) = (1 + c * x)**(-1 - 1/c)
|
|
return np.exp(self._logpdf(x, c))
|
|
|
|
def _logpdf(self, x, c):
|
|
return _lazywhere((x == x) & (c != 0), (x, c),
|
|
lambda x, c: -sc.xlog1py(c + 1., c*x) / c,
|
|
-x)
|
|
|
|
def _cdf(self, x, c):
|
|
return -sc.inv_boxcox1p(-x, -c)
|
|
|
|
def _sf(self, x, c):
|
|
return sc.inv_boxcox(-x, -c)
|
|
|
|
def _logsf(self, x, c):
|
|
return _lazywhere((x == x) & (c != 0), (x, c),
|
|
lambda x, c: -sc.log1p(c*x) / c,
|
|
-x)
|
|
|
|
def _ppf(self, q, c):
|
|
return -sc.boxcox1p(-q, -c)
|
|
|
|
def _isf(self, q, c):
|
|
return -sc.boxcox(q, -c)
|
|
|
|
def _stats(self, c, moments='mv'):
|
|
if 'm' not in moments:
|
|
m = None
|
|
else:
|
|
m = _lazywhere(c < 1, (c,),
|
|
lambda xi: 1/(1 - xi),
|
|
np.inf)
|
|
if 'v' not in moments:
|
|
v = None
|
|
else:
|
|
v = _lazywhere(c < 1/2, (c,),
|
|
lambda xi: 1 / (1 - xi)**2 / (1 - 2*xi),
|
|
np.nan)
|
|
if 's' not in moments:
|
|
s = None
|
|
else:
|
|
s = _lazywhere(c < 1/3, (c,),
|
|
lambda xi: 2 * (1 + xi) * np.sqrt(1 - 2*xi) /
|
|
(1 - 3*xi),
|
|
np.nan)
|
|
if 'k' not in moments:
|
|
k = None
|
|
else:
|
|
k = _lazywhere(c < 1/4, (c,),
|
|
lambda xi: 3 * (1 - 2*xi) * (2*xi**2 + xi + 3) /
|
|
(1 - 3*xi) / (1 - 4*xi) - 3,
|
|
np.nan)
|
|
return m, v, s, k
|
|
|
|
def _munp(self, n, c):
|
|
def __munp(n, c):
|
|
val = 0.0
|
|
k = np.arange(0, n + 1)
|
|
for ki, cnk in zip(k, sc.comb(n, k)):
|
|
val = val + cnk * (-1) ** ki / (1.0 - c * ki)
|
|
return np.where(c * n < 1, val * (-1.0 / c) ** n, np.inf)
|
|
return _lazywhere(c != 0, (c,),
|
|
lambda c: __munp(n, c),
|
|
sc.gamma(n + 1))
|
|
|
|
def _entropy(self, c):
|
|
return 1. + c
|
|
|
|
|
|
genpareto = genpareto_gen(a=0.0, name='genpareto')
|
|
|
|
|
|
class genexpon_gen(rv_continuous):
|
|
r"""A generalized exponential continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `genexpon` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, b, c) = (a + b (1 - \exp(-c x)))
|
|
\exp(-a x - b x + \frac{b}{c} (1-\exp(-c x)))
|
|
|
|
for :math:`x \ge 0`, :math:`a, b, c > 0`.
|
|
|
|
`genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential
|
|
Distribution", Journal of the American Statistical Association, 1993.
|
|
|
|
N. Balakrishnan, "The Exponential Distribution: Theory, Methods and
|
|
Applications", Asit P. Basu.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, a, b, c):
|
|
# genexpon.pdf(x, a, b, c) = (a + b * (1 - exp(-c*x))) * \
|
|
# exp(-a*x - b*x + b/c * (1-exp(-c*x)))
|
|
return (a + b*(-sc.expm1(-c*x)))*np.exp((-a-b)*x +
|
|
b*(-sc.expm1(-c*x))/c)
|
|
|
|
def _cdf(self, x, a, b, c):
|
|
return -sc.expm1((-a-b)*x + b*(-sc.expm1(-c*x))/c)
|
|
|
|
def _logpdf(self, x, a, b, c):
|
|
return np.log(a+b*(-sc.expm1(-c*x))) + (-a-b)*x+b*(-sc.expm1(-c*x))/c
|
|
|
|
|
|
genexpon = genexpon_gen(a=0.0, name='genexpon')
|
|
|
|
|
|
class genextreme_gen(rv_continuous):
|
|
r"""A generalized extreme value continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
gumbel_r
|
|
|
|
Notes
|
|
-----
|
|
For :math:`c=0`, `genextreme` is equal to `gumbel_r`.
|
|
The probability density function for `genextreme` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \begin{cases}
|
|
\exp(-\exp(-x)) \exp(-x) &\text{for } c = 0\\
|
|
\exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1} &\text{for }
|
|
x \le 1/c, c > 0
|
|
\end{cases}
|
|
|
|
|
|
Note that several sources and software packages use the opposite
|
|
convention for the sign of the shape parameter :math:`c`.
|
|
|
|
`genextreme` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, c):
|
|
return np.where(abs(c) == np.inf, 0, 1)
|
|
|
|
def _get_support(self, c):
|
|
_b = np.where(c > 0, 1.0 / np.maximum(c, _XMIN), np.inf)
|
|
_a = np.where(c < 0, 1.0 / np.minimum(c, -_XMIN), -np.inf)
|
|
return _a, _b
|
|
|
|
def _loglogcdf(self, x, c):
|
|
return _lazywhere((x == x) & (c != 0), (x, c),
|
|
lambda x, c: sc.log1p(-c*x)/c, -x)
|
|
|
|
def _pdf(self, x, c):
|
|
# genextreme.pdf(x, c) =
|
|
# exp(-exp(-x))*exp(-x), for c==0
|
|
# exp(-(1-c*x)**(1/c))*(1-c*x)**(1/c-1), for x \le 1/c, c > 0
|
|
return np.exp(self._logpdf(x, c))
|
|
|
|
def _logpdf(self, x, c):
|
|
cx = _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: c*x, 0.0)
|
|
logex2 = sc.log1p(-cx)
|
|
logpex2 = self._loglogcdf(x, c)
|
|
pex2 = np.exp(logpex2)
|
|
# Handle special cases
|
|
np.putmask(logpex2, (c == 0) & (x == -np.inf), 0.0)
|
|
logpdf = np.where((cx == 1) | (cx == -np.inf),
|
|
-np.inf,
|
|
-pex2+logpex2-logex2)
|
|
np.putmask(logpdf, (c == 1) & (x == 1), 0.0)
|
|
return logpdf
|
|
|
|
def _logcdf(self, x, c):
|
|
return -np.exp(self._loglogcdf(x, c))
|
|
|
|
def _cdf(self, x, c):
|
|
return np.exp(self._logcdf(x, c))
|
|
|
|
def _sf(self, x, c):
|
|
return -sc.expm1(self._logcdf(x, c))
|
|
|
|
def _ppf(self, q, c):
|
|
x = -np.log(-np.log(q))
|
|
return _lazywhere((x == x) & (c != 0), (x, c),
|
|
lambda x, c: -sc.expm1(-c * x) / c, x)
|
|
|
|
def _isf(self, q, c):
|
|
x = -np.log(-sc.log1p(-q))
|
|
return _lazywhere((x == x) & (c != 0), (x, c),
|
|
lambda x, c: -sc.expm1(-c * x) / c, x)
|
|
|
|
def _stats(self, c):
|
|
g = lambda n: sc.gamma(n*c + 1)
|
|
g1 = g(1)
|
|
g2 = g(2)
|
|
g3 = g(3)
|
|
g4 = g(4)
|
|
g2mg12 = np.where(abs(c) < 1e-7, (c*np.pi)**2.0/6.0, g2-g1**2.0)
|
|
gam2k = np.where(abs(c) < 1e-7, np.pi**2.0/6.0,
|
|
sc.expm1(sc.gammaln(2.0*c+1.0)-2*sc.gammaln(c + 1.0))/c**2.0)
|
|
eps = 1e-14
|
|
gamk = np.where(abs(c) < eps, -_EULER, sc.expm1(sc.gammaln(c + 1))/c)
|
|
|
|
m = np.where(c < -1.0, np.nan, -gamk)
|
|
v = np.where(c < -0.5, np.nan, g1**2.0*gam2k)
|
|
|
|
# skewness
|
|
sk1 = _lazywhere(c >= -1./3,
|
|
(c, g1, g2, g3, g2mg12),
|
|
lambda c, g1, g2, g3, g2gm12:
|
|
np.sign(c)*(-g3 + (g2 + 2*g2mg12)*g1)/g2mg12**1.5,
|
|
fillvalue=np.nan)
|
|
sk = np.where(abs(c) <= eps**0.29, 12*np.sqrt(6)*_ZETA3/np.pi**3, sk1)
|
|
|
|
# kurtosis
|
|
ku1 = _lazywhere(c >= -1./4,
|
|
(g1, g2, g3, g4, g2mg12),
|
|
lambda g1, g2, g3, g4, g2mg12:
|
|
(g4 + (-4*g3 + 3*(g2 + g2mg12)*g1)*g1)/g2mg12**2,
|
|
fillvalue=np.nan)
|
|
ku = np.where(abs(c) <= (eps)**0.23, 12.0/5.0, ku1-3.0)
|
|
return m, v, sk, ku
|
|
|
|
def _fitstart(self, data):
|
|
# This is better than the default shape of (1,).
|
|
g = _skew(data)
|
|
if g < 0:
|
|
a = 0.5
|
|
else:
|
|
a = -0.5
|
|
return super(genextreme_gen, self)._fitstart(data, args=(a,))
|
|
|
|
def _munp(self, n, c):
|
|
k = np.arange(0, n+1)
|
|
vals = 1.0/c**n * np.sum(
|
|
sc.comb(n, k) * (-1)**k * sc.gamma(c*k + 1),
|
|
axis=0)
|
|
return np.where(c*n > -1, vals, np.inf)
|
|
|
|
def _entropy(self, c):
|
|
return _EULER*(1 - c) + 1
|
|
|
|
|
|
genextreme = genextreme_gen(name='genextreme')
|
|
|
|
|
|
def _digammainv(y):
|
|
# Inverse of the digamma function (real positive arguments only).
|
|
# This function is used in the `fit` method of `gamma_gen`.
|
|
# The function uses either optimize.fsolve or optimize.newton
|
|
# to solve `sc.digamma(x) - y = 0`. There is probably room for
|
|
# improvement, but currently it works over a wide range of y:
|
|
# >>> y = 64*np.random.randn(1000000)
|
|
# >>> y.min(), y.max()
|
|
# (-311.43592651416662, 351.77388222276869)
|
|
# x = [_digammainv(t) for t in y]
|
|
# np.abs(sc.digamma(x) - y).max()
|
|
# 1.1368683772161603e-13
|
|
#
|
|
_em = 0.5772156649015328606065120
|
|
func = lambda x: sc.digamma(x) - y
|
|
if y > -0.125:
|
|
x0 = np.exp(y) + 0.5
|
|
if y < 10:
|
|
# Some experimentation shows that newton reliably converges
|
|
# must faster than fsolve in this y range. For larger y,
|
|
# newton sometimes fails to converge.
|
|
value = optimize.newton(func, x0, tol=1e-10)
|
|
return value
|
|
elif y > -3:
|
|
x0 = np.exp(y/2.332) + 0.08661
|
|
else:
|
|
x0 = 1.0 / (-y - _em)
|
|
|
|
value, info, ier, mesg = optimize.fsolve(func, x0, xtol=1e-11,
|
|
full_output=True)
|
|
if ier != 1:
|
|
raise RuntimeError("_digammainv: fsolve failed, y = %r" % y)
|
|
|
|
return value[0]
|
|
|
|
|
|
## Gamma (Use MATLAB and MATHEMATICA (b=theta=scale, a=alpha=shape) definition)
|
|
|
|
## gamma(a, loc, scale) with a an integer is the Erlang distribution
|
|
## gamma(1, loc, scale) is the Exponential distribution
|
|
## gamma(df/2, 0, 2) is the chi2 distribution with df degrees of freedom.
|
|
|
|
class gamma_gen(rv_continuous):
|
|
r"""A gamma continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
erlang, expon
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `gamma` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)}
|
|
|
|
for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the
|
|
gamma function.
|
|
|
|
`gamma` takes ``a`` as a shape parameter for :math:`a`.
|
|
|
|
When :math:`a` is an integer, `gamma` reduces to the Erlang
|
|
distribution, and when :math:`a=1` to the exponential distribution.
|
|
|
|
Gamma distributions are sometimes parameterized with two variables,
|
|
with a probability density function of:
|
|
|
|
.. math::
|
|
|
|
f(x, \alpha, \beta) = \frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x }}{\Gamma(\alpha)}
|
|
|
|
Note that this parameterization is equivalent to the above, with
|
|
``scale = 1 / beta``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, a, size=None, random_state=None):
|
|
return random_state.standard_gamma(a, size)
|
|
|
|
def _pdf(self, x, a):
|
|
# gamma.pdf(x, a) = x**(a-1) * exp(-x) / gamma(a)
|
|
return np.exp(self._logpdf(x, a))
|
|
|
|
def _logpdf(self, x, a):
|
|
return sc.xlogy(a-1.0, x) - x - sc.gammaln(a)
|
|
|
|
def _cdf(self, x, a):
|
|
return sc.gammainc(a, x)
|
|
|
|
def _sf(self, x, a):
|
|
return sc.gammaincc(a, x)
|
|
|
|
def _ppf(self, q, a):
|
|
return sc.gammaincinv(a, q)
|
|
|
|
def _stats(self, a):
|
|
return a, a, 2.0/np.sqrt(a), 6.0/a
|
|
|
|
def _entropy(self, a):
|
|
return sc.psi(a)*(1-a) + a + sc.gammaln(a)
|
|
|
|
def _fitstart(self, data):
|
|
# The skewness of the gamma distribution is `4 / np.sqrt(a)`.
|
|
# We invert that to estimate the shape `a` using the skewness
|
|
# of the data. The formula is regularized with 1e-8 in the
|
|
# denominator to allow for degenerate data where the skewness
|
|
# is close to 0.
|
|
a = 4 / (1e-8 + _skew(data)**2)
|
|
return super(gamma_gen, self)._fitstart(data, args=(a,))
|
|
|
|
@extend_notes_in_docstring(rv_continuous, notes="""\
|
|
When the location is fixed by using the argument `floc`, this
|
|
function uses explicit formulas or solves a simpler numerical
|
|
problem than the full ML optimization problem. So in that case,
|
|
the `optimizer`, `loc` and `scale` arguments are ignored.\n\n""")
|
|
def fit(self, data, *args, **kwds):
|
|
floc = kwds.get('floc', None)
|
|
|
|
if floc is None:
|
|
# loc is not fixed. Use the default fit method.
|
|
return super(gamma_gen, self).fit(data, *args, **kwds)
|
|
|
|
# We already have this value, so just pop it from kwds.
|
|
kwds.pop('floc', None)
|
|
|
|
f0 = _get_fixed_fit_value(kwds, ['f0', 'fa', 'fix_a'])
|
|
fscale = kwds.pop('fscale', None)
|
|
|
|
_remove_optimizer_parameters(kwds)
|
|
|
|
# Special case: loc is fixed.
|
|
|
|
if f0 is not None and fscale is not None:
|
|
# This check is for consistency with `rv_continuous.fit`.
|
|
# Without this check, this function would just return the
|
|
# parameters that were given.
|
|
raise ValueError("All parameters fixed. There is nothing to "
|
|
"optimize.")
|
|
|
|
# Fixed location is handled by shifting the data.
|
|
data = np.asarray(data)
|
|
|
|
if not np.isfinite(data).all():
|
|
raise RuntimeError("The data contains non-finite values.")
|
|
|
|
if np.any(data <= floc):
|
|
raise FitDataError("gamma", lower=floc, upper=np.inf)
|
|
|
|
if floc != 0:
|
|
# Don't do the subtraction in-place, because `data` might be a
|
|
# view of the input array.
|
|
data = data - floc
|
|
xbar = data.mean()
|
|
|
|
# Three cases to handle:
|
|
# * shape and scale both free
|
|
# * shape fixed, scale free
|
|
# * shape free, scale fixed
|
|
|
|
if fscale is None:
|
|
# scale is free
|
|
if f0 is not None:
|
|
# shape is fixed
|
|
a = f0
|
|
else:
|
|
# shape and scale are both free.
|
|
# The MLE for the shape parameter `a` is the solution to:
|
|
# np.log(a) - sc.digamma(a) - np.log(xbar) +
|
|
# np.log(data).mean() = 0
|
|
s = np.log(xbar) - np.log(data).mean()
|
|
func = lambda a: np.log(a) - sc.digamma(a) - s
|
|
aest = (3-s + np.sqrt((s-3)**2 + 24*s)) / (12*s)
|
|
xa = aest*(1-0.4)
|
|
xb = aest*(1+0.4)
|
|
a = optimize.brentq(func, xa, xb, disp=0)
|
|
|
|
# The MLE for the scale parameter is just the data mean
|
|
# divided by the shape parameter.
|
|
scale = xbar / a
|
|
else:
|
|
# scale is fixed, shape is free
|
|
# The MLE for the shape parameter `a` is the solution to:
|
|
# sc.digamma(a) - np.log(data).mean() + np.log(fscale) = 0
|
|
c = np.log(data).mean() - np.log(fscale)
|
|
a = _digammainv(c)
|
|
scale = fscale
|
|
|
|
return a, floc, scale
|
|
|
|
|
|
gamma = gamma_gen(a=0.0, name='gamma')
|
|
|
|
|
|
class erlang_gen(gamma_gen):
|
|
"""An Erlang continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
gamma
|
|
|
|
Notes
|
|
-----
|
|
The Erlang distribution is a special case of the Gamma distribution, with
|
|
the shape parameter `a` an integer. Note that this restriction is not
|
|
enforced by `erlang`. It will, however, generate a warning the first time
|
|
a non-integer value is used for the shape parameter.
|
|
|
|
Refer to `gamma` for examples.
|
|
|
|
"""
|
|
|
|
def _argcheck(self, a):
|
|
allint = np.all(np.floor(a) == a)
|
|
if not allint:
|
|
# An Erlang distribution shouldn't really have a non-integer
|
|
# shape parameter, so warn the user.
|
|
warnings.warn(
|
|
'The shape parameter of the erlang distribution '
|
|
'has been given a non-integer value %r.' % (a,),
|
|
RuntimeWarning)
|
|
return a > 0
|
|
|
|
def _fitstart(self, data):
|
|
# Override gamma_gen_fitstart so that an integer initial value is
|
|
# used. (Also regularize the division, to avoid issues when
|
|
# _skew(data) is 0 or close to 0.)
|
|
a = int(4.0 / (1e-8 + _skew(data)**2))
|
|
return super(gamma_gen, self)._fitstart(data, args=(a,))
|
|
|
|
# Trivial override of the fit method, so we can monkey-patch its
|
|
# docstring.
|
|
def fit(self, data, *args, **kwds):
|
|
return super(erlang_gen, self).fit(data, *args, **kwds)
|
|
|
|
if fit.__doc__:
|
|
fit.__doc__ = (rv_continuous.fit.__doc__ +
|
|
"""
|
|
Notes
|
|
-----
|
|
The Erlang distribution is generally defined to have integer values
|
|
for the shape parameter. This is not enforced by the `erlang` class.
|
|
When fitting the distribution, it will generally return a non-integer
|
|
value for the shape parameter. By using the keyword argument
|
|
`f0=<integer>`, the fit method can be constrained to fit the data to
|
|
a specific integer shape parameter.
|
|
""")
|
|
|
|
|
|
erlang = erlang_gen(a=0.0, name='erlang')
|
|
|
|
|
|
class gengamma_gen(rv_continuous):
|
|
r"""A generalized gamma continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
gamma, invgamma, weibull_min
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `gengamma` is ([1]_):
|
|
|
|
.. math::
|
|
|
|
f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)}
|
|
|
|
for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`.
|
|
:math:`\Gamma` is the gamma function (`scipy.special.gamma`).
|
|
|
|
`gengamma` takes :math:`a` and :math:`c` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] E.W. Stacy, "A Generalization of the Gamma Distribution",
|
|
Annals of Mathematical Statistics, Vol 33(3), pp. 1187--1192.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, a, c):
|
|
return (a > 0) & (c != 0)
|
|
|
|
def _pdf(self, x, a, c):
|
|
return np.exp(self._logpdf(x, a, c))
|
|
|
|
def _logpdf(self, x, a, c):
|
|
return np.log(abs(c)) + sc.xlogy(c*a - 1, x) - x**c - sc.gammaln(a)
|
|
|
|
def _cdf(self, x, a, c):
|
|
xc = x**c
|
|
val1 = sc.gammainc(a, xc)
|
|
val2 = sc.gammaincc(a, xc)
|
|
return np.where(c > 0, val1, val2)
|
|
|
|
def _rvs(self, a, c, size=None, random_state=None):
|
|
r = random_state.standard_gamma(a, size=size)
|
|
return r**(1./c)
|
|
|
|
def _sf(self, x, a, c):
|
|
xc = x**c
|
|
val1 = sc.gammainc(a, xc)
|
|
val2 = sc.gammaincc(a, xc)
|
|
return np.where(c > 0, val2, val1)
|
|
|
|
def _ppf(self, q, a, c):
|
|
val1 = sc.gammaincinv(a, q)
|
|
val2 = sc.gammainccinv(a, q)
|
|
return np.where(c > 0, val1, val2)**(1.0/c)
|
|
|
|
def _isf(self, q, a, c):
|
|
val1 = sc.gammaincinv(a, q)
|
|
val2 = sc.gammainccinv(a, q)
|
|
return np.where(c > 0, val2, val1)**(1.0/c)
|
|
|
|
def _munp(self, n, a, c):
|
|
# Pochhammer symbol: sc.pocha,n) = gamma(a+n)/gamma(a)
|
|
return sc.poch(a, n*1.0/c)
|
|
|
|
def _entropy(self, a, c):
|
|
val = sc.psi(a)
|
|
return a*(1-val) + 1.0/c*val + sc.gammaln(a) - np.log(abs(c))
|
|
|
|
|
|
gengamma = gengamma_gen(a=0.0, name='gengamma')
|
|
|
|
|
|
class genhalflogistic_gen(rv_continuous):
|
|
r"""A generalized half-logistic continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `genhalflogistic` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2}
|
|
|
|
for :math:`0 \le x \le 1/c`, and :math:`c > 0`.
|
|
|
|
`genhalflogistic` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, c):
|
|
return c > 0
|
|
|
|
def _get_support(self, c):
|
|
return self.a, 1.0/c
|
|
|
|
def _pdf(self, x, c):
|
|
# genhalflogistic.pdf(x, c) =
|
|
# 2 * (1-c*x)**(1/c-1) / (1+(1-c*x)**(1/c))**2
|
|
limit = 1.0/c
|
|
tmp = np.asarray(1-c*x)
|
|
tmp0 = tmp**(limit-1)
|
|
tmp2 = tmp0*tmp
|
|
return 2*tmp0 / (1+tmp2)**2
|
|
|
|
def _cdf(self, x, c):
|
|
limit = 1.0/c
|
|
tmp = np.asarray(1-c*x)
|
|
tmp2 = tmp**(limit)
|
|
return (1.0-tmp2) / (1+tmp2)
|
|
|
|
def _ppf(self, q, c):
|
|
return 1.0/c*(1-((1.0-q)/(1.0+q))**c)
|
|
|
|
def _entropy(self, c):
|
|
return 2 - (2*c+1)*np.log(2)
|
|
|
|
|
|
genhalflogistic = genhalflogistic_gen(a=0.0, name='genhalflogistic')
|
|
|
|
|
|
class gompertz_gen(rv_continuous):
|
|
r"""A Gompertz (or truncated Gumbel) continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `gompertz` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = c \exp(x) \exp(-c (e^x-1))
|
|
|
|
for :math:`x \ge 0`, :math:`c > 0`.
|
|
|
|
`gompertz` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, c):
|
|
# gompertz.pdf(x, c) = c * exp(x) * exp(-c*(exp(x)-1))
|
|
return np.exp(self._logpdf(x, c))
|
|
|
|
def _logpdf(self, x, c):
|
|
return np.log(c) + x - c * sc.expm1(x)
|
|
|
|
def _cdf(self, x, c):
|
|
return -sc.expm1(-c * sc.expm1(x))
|
|
|
|
def _ppf(self, q, c):
|
|
return sc.log1p(-1.0 / c * sc.log1p(-q))
|
|
|
|
def _entropy(self, c):
|
|
return 1.0 - np.log(c) - np.exp(c)*sc.expn(1, c)
|
|
|
|
|
|
gompertz = gompertz_gen(a=0.0, name='gompertz')
|
|
|
|
|
|
def _average_with_log_weights(x, logweights):
|
|
x = np.asarray(x)
|
|
logweights = np.asarray(logweights)
|
|
maxlogw = logweights.max()
|
|
weights = np.exp(logweights - maxlogw)
|
|
return np.average(x, weights=weights)
|
|
|
|
|
|
class gumbel_r_gen(rv_continuous):
|
|
r"""A right-skewed Gumbel continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
gumbel_l, gompertz, genextreme
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `gumbel_r` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \exp(-(x + e^{-x}))
|
|
|
|
The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
|
|
distribution. It is also related to the extreme value distribution,
|
|
log-Weibull and Gompertz distributions.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x):
|
|
# gumbel_r.pdf(x) = exp(-(x + exp(-x)))
|
|
return np.exp(self._logpdf(x))
|
|
|
|
def _logpdf(self, x):
|
|
return -x - np.exp(-x)
|
|
|
|
def _cdf(self, x):
|
|
return np.exp(-np.exp(-x))
|
|
|
|
def _logcdf(self, x):
|
|
return -np.exp(-x)
|
|
|
|
def _ppf(self, q):
|
|
return -np.log(-np.log(q))
|
|
|
|
def _sf(self, x):
|
|
return -sc.expm1(-np.exp(-x))
|
|
|
|
def _isf(self, p):
|
|
return -np.log(-np.log1p(-p))
|
|
|
|
def _stats(self):
|
|
return _EULER, np.pi*np.pi/6.0, 12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5
|
|
|
|
def _entropy(self):
|
|
# https://en.wikipedia.org/wiki/Gumbel_distribution
|
|
return _EULER + 1.
|
|
|
|
def fit(self, data, *args, **kwds):
|
|
data, floc, fscale = _check_fit_input_parameters(self, data,
|
|
args, kwds)
|
|
|
|
# if user has provided `floc` or `fscale`, fall back on super fit
|
|
# method. This scenario is not suitable for solving a system of
|
|
# equations
|
|
if floc is not None or fscale is not None:
|
|
return super(gumbel_r_gen, self).fit(data, *args, **kwds)
|
|
|
|
# rv_continuous provided guesses
|
|
loc, scale = self._fitstart(data)
|
|
# account for user provided guesses
|
|
loc = kwds.pop('loc', loc)
|
|
scale = kwds.pop('scale', scale)
|
|
|
|
# By the method of maximum likelihood, the estimators of the
|
|
# location and scale are the roots of the equation defined in
|
|
# `func` and the value of the expression for `loc` that follows.
|
|
# Source: Statistical Distributions, 3rd Edition. Evans, Hastings,
|
|
# and Peacock (2000), Page 101
|
|
|
|
def func(scale, data):
|
|
sdata = -data / scale
|
|
wavg = _average_with_log_weights(data, logweights=sdata)
|
|
return data.mean() - wavg - scale
|
|
|
|
soln = optimize.root(func, scale, args=(data,),
|
|
options={'xtol': 1e-14})
|
|
scale = soln.x[0]
|
|
loc = -scale * (sc.logsumexp(-data/scale) - np.log(len(data)))
|
|
|
|
return loc, scale
|
|
|
|
|
|
gumbel_r = gumbel_r_gen(name='gumbel_r')
|
|
|
|
|
|
class gumbel_l_gen(rv_continuous):
|
|
r"""A left-skewed Gumbel continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
gumbel_r, gompertz, genextreme
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `gumbel_l` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \exp(x - e^x)
|
|
|
|
The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
|
|
distribution. It is also related to the extreme value distribution,
|
|
log-Weibull and Gompertz distributions.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x):
|
|
# gumbel_l.pdf(x) = exp(x - exp(x))
|
|
return np.exp(self._logpdf(x))
|
|
|
|
def _logpdf(self, x):
|
|
return x - np.exp(x)
|
|
|
|
def _cdf(self, x):
|
|
return -sc.expm1(-np.exp(x))
|
|
|
|
def _ppf(self, q):
|
|
return np.log(-sc.log1p(-q))
|
|
|
|
def _logsf(self, x):
|
|
return -np.exp(x)
|
|
|
|
def _sf(self, x):
|
|
return np.exp(-np.exp(x))
|
|
|
|
def _isf(self, x):
|
|
return np.log(-np.log(x))
|
|
|
|
def _stats(self):
|
|
return -_EULER, np.pi*np.pi/6.0, \
|
|
-12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5
|
|
|
|
def _entropy(self):
|
|
return _EULER + 1.
|
|
|
|
def fit(self, data, *args, **kwds):
|
|
# The fit method of `gumbel_r` can be used for this distribution with
|
|
# small modifications. The process to do this is
|
|
# 1. pass the sign negated data into `gumbel_r.fit`
|
|
# 2. negate the sign of the resulting location, leaving the scale
|
|
# unmodified.
|
|
# `gumbel_r.fit` holds necessary input checks.
|
|
|
|
loc_r, scale_r, = gumbel_r.fit(-np.asarray(data), *args, **kwds)
|
|
return (-loc_r, scale_r)
|
|
|
|
|
|
gumbel_l = gumbel_l_gen(name='gumbel_l')
|
|
|
|
|
|
class halfcauchy_gen(rv_continuous):
|
|
r"""A Half-Cauchy continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `halfcauchy` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{2}{\pi (1 + x^2)}
|
|
|
|
for :math:`x \ge 0`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x):
|
|
# halfcauchy.pdf(x) = 2 / (pi * (1 + x**2))
|
|
return 2.0/np.pi/(1.0+x*x)
|
|
|
|
def _logpdf(self, x):
|
|
return np.log(2.0/np.pi) - sc.log1p(x*x)
|
|
|
|
def _cdf(self, x):
|
|
return 2.0/np.pi*np.arctan(x)
|
|
|
|
def _ppf(self, q):
|
|
return np.tan(np.pi/2*q)
|
|
|
|
def _stats(self):
|
|
return np.inf, np.inf, np.nan, np.nan
|
|
|
|
def _entropy(self):
|
|
return np.log(2*np.pi)
|
|
|
|
|
|
halfcauchy = halfcauchy_gen(a=0.0, name='halfcauchy')
|
|
|
|
|
|
class halflogistic_gen(rv_continuous):
|
|
r"""A half-logistic continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `halflogistic` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 }
|
|
= \frac{1}{2} \text{sech}(x/2)^2
|
|
|
|
for :math:`x \ge 0`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x):
|
|
# halflogistic.pdf(x) = 2 * exp(-x) / (1+exp(-x))**2
|
|
# = 1/2 * sech(x/2)**2
|
|
return np.exp(self._logpdf(x))
|
|
|
|
def _logpdf(self, x):
|
|
return np.log(2) - x - 2. * sc.log1p(np.exp(-x))
|
|
|
|
def _cdf(self, x):
|
|
return np.tanh(x/2.0)
|
|
|
|
def _ppf(self, q):
|
|
return 2*np.arctanh(q)
|
|
|
|
def _munp(self, n):
|
|
if n == 1:
|
|
return 2*np.log(2)
|
|
if n == 2:
|
|
return np.pi*np.pi/3.0
|
|
if n == 3:
|
|
return 9*_ZETA3
|
|
if n == 4:
|
|
return 7*np.pi**4 / 15.0
|
|
return 2*(1-pow(2.0, 1-n))*sc.gamma(n+1)*sc.zeta(n, 1)
|
|
|
|
def _entropy(self):
|
|
return 2-np.log(2)
|
|
|
|
|
|
halflogistic = halflogistic_gen(a=0.0, name='halflogistic')
|
|
|
|
|
|
class halfnorm_gen(rv_continuous):
|
|
r"""A half-normal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `halfnorm` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \sqrt{2/\pi} \exp(-x^2 / 2)
|
|
|
|
for :math:`x >= 0`.
|
|
|
|
`halfnorm` is a special case of `chi` with ``df=1``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, size=None, random_state=None):
|
|
return abs(random_state.standard_normal(size=size))
|
|
|
|
def _pdf(self, x):
|
|
# halfnorm.pdf(x) = sqrt(2/pi) * exp(-x**2/2)
|
|
return np.sqrt(2.0/np.pi)*np.exp(-x*x/2.0)
|
|
|
|
def _logpdf(self, x):
|
|
return 0.5 * np.log(2.0/np.pi) - x*x/2.0
|
|
|
|
def _cdf(self, x):
|
|
return _norm_cdf(x)*2-1.0
|
|
|
|
def _ppf(self, q):
|
|
return sc.ndtri((1+q)/2.0)
|
|
|
|
def _stats(self):
|
|
return (np.sqrt(2.0/np.pi),
|
|
1-2.0/np.pi,
|
|
np.sqrt(2)*(4-np.pi)/(np.pi-2)**1.5,
|
|
8*(np.pi-3)/(np.pi-2)**2)
|
|
|
|
def _entropy(self):
|
|
return 0.5*np.log(np.pi/2.0)+0.5
|
|
|
|
|
|
halfnorm = halfnorm_gen(a=0.0, name='halfnorm')
|
|
|
|
|
|
class hypsecant_gen(rv_continuous):
|
|
r"""A hyperbolic secant continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `hypsecant` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{1}{\pi} \text{sech}(x)
|
|
|
|
for a real number :math:`x`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x):
|
|
# hypsecant.pdf(x) = 1/pi * sech(x)
|
|
return 1.0/(np.pi*np.cosh(x))
|
|
|
|
def _cdf(self, x):
|
|
return 2.0/np.pi*np.arctan(np.exp(x))
|
|
|
|
def _ppf(self, q):
|
|
return np.log(np.tan(np.pi*q/2.0))
|
|
|
|
def _stats(self):
|
|
return 0, np.pi*np.pi/4, 0, 2
|
|
|
|
def _entropy(self):
|
|
return np.log(2*np.pi)
|
|
|
|
|
|
hypsecant = hypsecant_gen(name='hypsecant')
|
|
|
|
|
|
class gausshyper_gen(rv_continuous):
|
|
r"""A Gauss hypergeometric continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `gausshyper` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c}
|
|
|
|
for :math:`0 \le x \le 1`, :math:`a > 0`, :math:`b > 0`, :math:`z > -1`,
|
|
and :math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`.
|
|
:math:`F[2, 1]` is the Gauss hypergeometric function
|
|
`scipy.special.hyp2f1`.
|
|
|
|
`gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape
|
|
parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] Armero, C., and M. J. Bayarri. "Prior Assessments for Prediction in
|
|
Queues." *Journal of the Royal Statistical Society*. Series D (The
|
|
Statistician) 43, no. 1 (1994): 139-53. doi:10.2307/2348939
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _argcheck(self, a, b, c, z):
|
|
# z > -1 per gh-10134
|
|
return (a > 0) & (b > 0) & (c == c) & (z > -1)
|
|
|
|
def _pdf(self, x, a, b, c, z):
|
|
# gausshyper.pdf(x, a, b, c, z) =
|
|
# C * x**(a-1) * (1-x)**(b-1) * (1+z*x)**(-c)
|
|
Cinv = sc.gamma(a)*sc.gamma(b)/sc.gamma(a+b)*sc.hyp2f1(c, a, a+b, -z)
|
|
return 1.0/Cinv * x**(a-1.0) * (1.0-x)**(b-1.0) / (1.0+z*x)**c
|
|
|
|
def _munp(self, n, a, b, c, z):
|
|
fac = sc.beta(n+a, b) / sc.beta(a, b)
|
|
num = sc.hyp2f1(c, a+n, a+b+n, -z)
|
|
den = sc.hyp2f1(c, a, a+b, -z)
|
|
return fac*num / den
|
|
|
|
|
|
gausshyper = gausshyper_gen(a=0.0, b=1.0, name='gausshyper')
|
|
|
|
|
|
class invgamma_gen(rv_continuous):
|
|
r"""An inverted gamma continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `invgamma` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a) = \frac{x^{-a-1}}{\Gamma(a)} \exp(-\frac{1}{x})
|
|
|
|
for :math:`x >= 0`, :math:`a > 0`. :math:`\Gamma` is the gamma function
|
|
(`scipy.special.gamma`).
|
|
|
|
`invgamma` takes ``a`` as a shape parameter for :math:`a`.
|
|
|
|
`invgamma` is a special case of `gengamma` with ``c=-1``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _pdf(self, x, a):
|
|
# invgamma.pdf(x, a) = x**(-a-1) / gamma(a) * exp(-1/x)
|
|
return np.exp(self._logpdf(x, a))
|
|
|
|
def _logpdf(self, x, a):
|
|
return -(a+1) * np.log(x) - sc.gammaln(a) - 1.0/x
|
|
|
|
def _cdf(self, x, a):
|
|
return sc.gammaincc(a, 1.0 / x)
|
|
|
|
def _ppf(self, q, a):
|
|
return 1.0 / sc.gammainccinv(a, q)
|
|
|
|
def _sf(self, x, a):
|
|
return sc.gammainc(a, 1.0 / x)
|
|
|
|
def _isf(self, q, a):
|
|
return 1.0 / sc.gammaincinv(a, q)
|
|
|
|
def _stats(self, a, moments='mvsk'):
|
|
m1 = _lazywhere(a > 1, (a,), lambda x: 1. / (x - 1.), np.inf)
|
|
m2 = _lazywhere(a > 2, (a,), lambda x: 1. / (x - 1.)**2 / (x - 2.),
|
|
np.inf)
|
|
|
|
g1, g2 = None, None
|
|
if 's' in moments:
|
|
g1 = _lazywhere(
|
|
a > 3, (a,),
|
|
lambda x: 4. * np.sqrt(x - 2.) / (x - 3.), np.nan)
|
|
if 'k' in moments:
|
|
g2 = _lazywhere(
|
|
a > 4, (a,),
|
|
lambda x: 6. * (5. * x - 11.) / (x - 3.) / (x - 4.), np.nan)
|
|
return m1, m2, g1, g2
|
|
|
|
def _entropy(self, a):
|
|
return a - (a+1.0) * sc.psi(a) + sc.gammaln(a)
|
|
|
|
|
|
invgamma = invgamma_gen(a=0.0, name='invgamma')
|
|
|
|
|
|
# scale is gamma from DATAPLOT and B from Regress
|
|
class invgauss_gen(rv_continuous):
|
|
r"""An inverse Gaussian continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `invgauss` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \mu) = \frac{1}{\sqrt{2 \pi x^3}}
|
|
\exp(-\frac{(x-\mu)^2}{2 x \mu^2})
|
|
|
|
for :math:`x >= 0` and :math:`\mu > 0`.
|
|
|
|
`invgauss` takes ``mu`` as a shape parameter for :math:`\mu`.
|
|
|
|
%(after_notes)s
|
|
|
|
When :math:`\mu` is too small, evaluating the cumulative distribution
|
|
function will be inaccurate due to ``cdf(mu -> 0) = inf * 0``.
|
|
NaNs are returned for :math:`\mu \le 0.0028`.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _rvs(self, mu, size=None, random_state=None):
|
|
return random_state.wald(mu, 1.0, size=size)
|
|
|
|
def _pdf(self, x, mu):
|
|
# invgauss.pdf(x, mu) =
|
|
# 1 / sqrt(2*pi*x**3) * exp(-(x-mu)**2/(2*x*mu**2))
|
|
return 1.0/np.sqrt(2*np.pi*x**3.0)*np.exp(-1.0/(2*x)*((x-mu)/mu)**2)
|
|
|
|
def _logpdf(self, x, mu):
|
|
return -0.5*np.log(2*np.pi) - 1.5*np.log(x) - ((x-mu)/mu)**2/(2*x)
|
|
|
|
def _cdf(self, x, mu):
|
|
fac = np.sqrt(1.0/x)
|
|
# Numerical accuracy for small `mu` is bad. See #869.
|
|
C1 = _norm_cdf(fac*(x-mu)/mu)
|
|
C1 += np.exp(1.0/mu) * _norm_cdf(-fac*(x+mu)/mu) * np.exp(1.0/mu)
|
|
return C1
|
|
|
|
def _stats(self, mu):
|
|
return mu, mu**3.0, 3*np.sqrt(mu), 15*mu
|
|
|
|
def fit(self, data, *args, **kwds):
|
|
|
|
if type(self) == wald_gen:
|
|
return super(invgauss_gen, self).fit(data, *args, **kwds)
|
|
|
|
data, fshape_s, floc, fscale = _check_fit_input_parameters(self, data,
|
|
args, kwds)
|
|
'''
|
|
Source: Statistical Distributions, 3rd Edition. Evans, Hastings,
|
|
and Peacock (2000), Page 121. Their shape parameter is equivilent to
|
|
SciPy's with the conversion `fshape_s = fshape / scale`.
|
|
|
|
MLE formulas are not used in 3 condtions:
|
|
- `loc` is not fixed
|
|
- `mu` is fixed
|
|
These cases fall back on the superclass fit method.
|
|
- `loc` is fixed but translation results in negative data raises
|
|
a `FitDataError`.
|
|
'''
|
|
if floc is None or fshape_s is not None:
|
|
return super(invgauss_gen, self).fit(data, *args, **kwds)
|
|
elif np.any(data - floc < 0):
|
|
raise FitDataError("invgauss", lower=0, upper=np.inf)
|
|
else:
|
|
data = data - floc
|
|
fshape_n = np.mean(data)
|
|
if fscale is None:
|
|
fscale = len(data) / (np.sum(data ** -1 - fshape_n ** -1))
|
|
fshape_s = fshape_n / fscale
|
|
return fshape_s, floc, fscale
|
|
|
|
|
|
invgauss = invgauss_gen(a=0.0, name='invgauss')
|
|
|
|
|
|
class geninvgauss_gen(rv_continuous):
|
|
r"""A Generalized Inverse Gaussian continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `geninvgauss` is:
|
|
|
|
.. math::
|
|
|
|
f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b))
|
|
|
|
where `x > 0`, and the parameters `p, b` satisfy `b > 0` ([1]_).
|
|
:math:`K_p` is the modified Bessel function of second kind of order `p`
|
|
(`scipy.special.kv`).
|
|
|
|
%(after_notes)s
|
|
|
|
The inverse Gaussian distribution `stats.invgauss(mu)` is a special case of
|
|
`geninvgauss` with `p = -1/2`, `b = 1 / mu` and `scale = mu`.
|
|
|
|
Generating random variates is challenging for this distribution. The
|
|
implementation is based on [2]_.
|
|
|
|
References
|
|
----------
|
|
.. [1] O. Barndorff-Nielsen, P. Blaesild, C. Halgreen, "First hitting time
|
|
models for the generalized inverse gaussian distribution",
|
|
Stochastic Processes and their Applications 7, pp. 49--54, 1978.
|
|
|
|
.. [2] W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian
|
|
random variates", Statistics and Computing, 24(4), p. 547--557, 2014.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, p, b):
|
|
return (p == p) & (b > 0)
|
|
|
|
def _logpdf(self, x, p, b):
|
|
# kve instead of kv works better for large values of b
|
|
# warn if kve produces infinite values and replace by nan
|
|
# otherwise c = -inf and the results are often incorrect
|
|
@np.vectorize
|
|
def logpdf_single(x, p, b):
|
|
return _stats.geninvgauss_logpdf(x, p, b)
|
|
|
|
z = logpdf_single(x, p, b)
|
|
if np.isnan(z).any():
|
|
msg = ("Infinite values encountered in scipy.special.kve(p, b). "
|
|
"Values replaced by NaN to avoid incorrect results.")
|
|
warnings.warn(msg, RuntimeWarning)
|
|
return z
|
|
|
|
def _pdf(self, x, p, b):
|
|
# relying on logpdf avoids overflow of x**(p-1) for large x and p
|
|
return np.exp(self._logpdf(x, p, b))
|
|
|
|
def _cdf(self, x, *args):
|
|
_a, _b = self._get_support(*args)
|
|
|
|
@np.vectorize
|
|
def _cdf_single(x, *args):
|
|
p, b = args
|
|
user_data = np.array([p, b], float).ctypes.data_as(ctypes.c_void_p)
|
|
llc = LowLevelCallable.from_cython(_stats, '_geninvgauss_pdf', user_data)
|
|
|
|
return integrate.quad(llc, _a, x)[0]
|
|
|
|
return _cdf_single(x, *args)
|
|
|
|
def _logquasipdf(self, x, p, b):
|
|
# log of the quasi-density (w/o normalizing constant) used in _rvs
|
|
return _lazywhere(x > 0, (x, p, b),
|
|
lambda x, p, b: (p - 1)*np.log(x) - b*(x + 1/x)/2,
|
|
-np.inf)
|
|
|
|
def _rvs(self, p, b, size=None, random_state=None):
|
|
# if p and b are scalar, use _rvs_scalar, otherwise need to create
|
|
# output by iterating over parameters
|
|
if np.isscalar(p) and np.isscalar(b):
|
|
out = self._rvs_scalar(p, b, size, random_state)
|
|
elif p.size == 1 and b.size == 1:
|
|
out = self._rvs_scalar(p.item(), b.item(), size, random_state)
|
|
else:
|
|
# When this method is called, size will be a (possibly empty)
|
|
# tuple of integers. It will not be None; if `size=None` is passed
|
|
# to `rvs()`, size will be the empty tuple ().
|
|
|
|
p, b = np.broadcast_arrays(p, b)
|
|
# p and b now have the same shape.
|
|
|
|
# `shp` is the shape of the blocks of random variates that are
|
|
# generated for each combination of parameters associated with
|
|
# broadcasting p and b.
|
|
# bc is a tuple the same lenth as size. The values
|
|
# in bc are bools. If bc[j] is True, it means that
|
|
# entire axis is filled in for a given combination of the
|
|
# broadcast arguments.
|
|
shp, bc = _check_shape(p.shape, size)
|
|
|
|
# `numsamples` is the total number of variates to be generated
|
|
# for each combination of the input arguments.
|
|
numsamples = int(np.prod(shp))
|
|
|
|
# `out` is the array to be returned. It is filled in in the
|
|
# loop below.
|
|
out = np.empty(size)
|
|
|
|
it = np.nditer([p, b],
|
|
flags=['multi_index'],
|
|
op_flags=[['readonly'], ['readonly']])
|
|
while not it.finished:
|
|
# Convert the iterator's multi_index into an index into the
|
|
# `out` array where the call to _rvs_scalar() will be stored.
|
|
# Where bc is True, we use a full slice; otherwise we use the
|
|
# index value from it.multi_index. len(it.multi_index) might
|
|
# be less than len(bc), and in that case we want to align these
|
|
# two sequences to the right, so the loop variable j runs from
|
|
# -len(size) to 0. This doesn't cause an IndexError, as
|
|
# bc[j] will be True in those cases where it.multi_index[j]
|
|
# would cause an IndexError.
|
|
idx = tuple((it.multi_index[j] if not bc[j] else slice(None))
|
|
for j in range(-len(size), 0))
|
|
out[idx] = self._rvs_scalar(it[0], it[1], numsamples, random_state).reshape(shp)
|
|
it.iternext()
|
|
|
|
if size == ():
|
|
out = out.item()
|
|
return out
|
|
|
|
def _rvs_scalar(self, p, b, numsamples, random_state):
|
|
# following [2], the quasi-pdf is used instead of the pdf for the
|
|
# generation of rvs
|
|
invert_res = False
|
|
if not(numsamples):
|
|
numsamples = 1
|
|
if p < 0:
|
|
# note: if X is geninvgauss(p, b), then 1/X is geninvgauss(-p, b)
|
|
p = -p
|
|
invert_res = True
|
|
m = self._mode(p, b)
|
|
|
|
# determine method to be used following [2]
|
|
ratio_unif = True
|
|
if p >= 1 or b > 1:
|
|
# ratio of uniforms with mode shift below
|
|
mode_shift = True
|
|
elif b >= min(0.5, 2 * np.sqrt(1 - p) / 3):
|
|
# ratio of uniforms without mode shift below
|
|
mode_shift = False
|
|
else:
|
|
# new algorithm in [2]
|
|
ratio_unif = False
|
|
|
|
# prepare sampling of rvs
|
|
size1d = tuple(np.atleast_1d(numsamples))
|
|
N = np.prod(size1d) # number of rvs needed, reshape upon return
|
|
x = np.zeros(N)
|
|
simulated = 0
|
|
|
|
if ratio_unif:
|
|
# use ratio of uniforms method
|
|
if mode_shift:
|
|
a2 = -2 * (p + 1) / b - m
|
|
a1 = 2 * m * (p - 1) / b - 1
|
|
# find roots of x**3 + a2*x**2 + a1*x + m (Cardano's formula)
|
|
p1 = a1 - a2**2 / 3
|
|
q1 = 2 * a2**3 / 27 - a2 * a1 / 3 + m
|
|
phi = np.arccos(-q1 * np.sqrt(-27 / p1**3) / 2)
|
|
s1 = -np.sqrt(-4 * p1 / 3)
|
|
root1 = s1 * np.cos(phi / 3 + np.pi / 3) - a2 / 3
|
|
root2 = -s1 * np.cos(phi / 3) - a2 / 3
|
|
# root3 = s1 * np.cos(phi / 3 - np.pi / 3) - a2 / 3
|
|
|
|
# if g is the quasipdf, rescale: g(x) / g(m) which we can write
|
|
# as exp(log(g(x)) - log(g(m))). This is important
|
|
# since for large values of p and b, g cannot be evaluated.
|
|
# denote the rescaled quasipdf by h
|
|
lm = self._logquasipdf(m, p, b)
|
|
d1 = self._logquasipdf(root1, p, b) - lm
|
|
d2 = self._logquasipdf(root2, p, b) - lm
|
|
# compute the bounding rectangle w.r.t. h. Note that
|
|
# np.exp(0.5*d1) = np.sqrt(g(root1)/g(m)) = np.sqrt(h(root1))
|
|
vmin = (root1 - m) * np.exp(0.5 * d1)
|
|
vmax = (root2 - m) * np.exp(0.5 * d2)
|
|
umax = 1 # umax = sqrt(h(m)) = 1
|
|
|
|
logqpdf = lambda x: self._logquasipdf(x, p, b) - lm
|
|
c = m
|
|
else:
|
|
# ratio of uniforms without mode shift
|
|
# compute np.sqrt(quasipdf(m))
|
|
umax = np.exp(0.5*self._logquasipdf(m, p, b))
|
|
xplus = ((1 + p) + np.sqrt((1 + p)**2 + b**2))/b
|
|
vmin = 0
|
|
# compute xplus * np.sqrt(quasipdf(xplus))
|
|
vmax = xplus * np.exp(0.5 * self._logquasipdf(xplus, p, b))
|
|
c = 0
|
|
logqpdf = lambda x: self._logquasipdf(x, p, b)
|
|
|
|
if vmin >= vmax:
|
|
raise ValueError("vmin must be smaller than vmax.")
|
|
if umax <= 0:
|
|
raise ValueError("umax must be positive.")
|
|
|
|
i = 1
|
|
while simulated < N:
|
|
k = N - simulated
|
|
# simulate uniform rvs on [0, umax] and [vmin, vmax]
|
|
u = umax * random_state.uniform(size=k)
|
|
v = random_state.uniform(size=k)
|
|
v = vmin + (vmax - vmin) * v
|
|
rvs = v / u + c
|
|
# rewrite acceptance condition u**2 <= pdf(rvs) by taking logs
|
|
accept = (2*np.log(u) <= logqpdf(rvs))
|
|
num_accept = np.sum(accept)
|
|
if num_accept > 0:
|
|
x[simulated:(simulated + num_accept)] = rvs[accept]
|
|
simulated += num_accept
|
|
|
|
if (simulated == 0) and (i*N >= 50000):
|
|
msg = ("Not a single random variate could be generated "
|
|
"in {} attempts. Sampling does not appear to "
|
|
"work for the provided parameters.".format(i*N))
|
|
raise RuntimeError(msg)
|
|
i += 1
|
|
else:
|
|
# use new algorithm in [2]
|
|
x0 = b / (1 - p)
|
|
xs = np.max((x0, 2 / b))
|
|
k1 = np.exp(self._logquasipdf(m, p, b))
|
|
A1 = k1 * x0
|
|
if x0 < 2 / b:
|
|
k2 = np.exp(-b)
|
|
if p > 0:
|
|
A2 = k2 * ((2 / b)**p - x0**p) / p
|
|
else:
|
|
A2 = k2 * np.log(2 / b**2)
|
|
else:
|
|
k2, A2 = 0, 0
|
|
k3 = xs**(p - 1)
|
|
A3 = 2 * k3 * np.exp(-xs * b / 2) / b
|
|
A = A1 + A2 + A3
|
|
|
|
# [2]: rejection constant is < 2.73; so expected runtime is finite
|
|
while simulated < N:
|
|
k = N - simulated
|
|
h, rvs = np.zeros(k), np.zeros(k)
|
|
# simulate uniform rvs on [x1, x2] and [0, y2]
|
|
u = random_state.uniform(size=k)
|
|
v = A * random_state.uniform(size=k)
|
|
cond1 = v <= A1
|
|
cond2 = np.logical_not(cond1) & (v <= A1 + A2)
|
|
cond3 = np.logical_not(cond1 | cond2)
|
|
# subdomain (0, x0)
|
|
rvs[cond1] = x0 * v[cond1] / A1
|
|
h[cond1] = k1
|
|
# subdomain (x0, 2 / b)
|
|
if p > 0:
|
|
rvs[cond2] = (x0**p + (v[cond2] - A1) * p / k2)**(1 / p)
|
|
else:
|
|
rvs[cond2] = b * np.exp((v[cond2] - A1) * np.exp(b))
|
|
h[cond2] = k2 * rvs[cond2]**(p - 1)
|
|
# subdomain (xs, infinity)
|
|
z = np.exp(-xs * b / 2) - b * (v[cond3] - A1 - A2) / (2 * k3)
|
|
rvs[cond3] = -2 / b * np.log(z)
|
|
h[cond3] = k3 * np.exp(-rvs[cond3] * b / 2)
|
|
# apply rejection method
|
|
accept = (np.log(u * h) <= self._logquasipdf(rvs, p, b))
|
|
num_accept = sum(accept)
|
|
if num_accept > 0:
|
|
x[simulated:(simulated + num_accept)] = rvs[accept]
|
|
simulated += num_accept
|
|
|
|
rvs = np.reshape(x, size1d)
|
|
if invert_res:
|
|
rvs = 1 / rvs
|
|
return rvs
|
|
|
|
def _mode(self, p, b):
|
|
# distinguish cases to avoid catastrophic cancellation (see [2])
|
|
if p < 1:
|
|
return b / (np.sqrt((p - 1)**2 + b**2) + 1 - p)
|
|
else:
|
|
return (np.sqrt((1 - p)**2 + b**2) - (1 - p)) / b
|
|
|
|
def _munp(self, n, p, b):
|
|
num = sc.kve(p + n, b)
|
|
denom = sc.kve(p, b)
|
|
inf_vals = np.isinf(num) | np.isinf(denom)
|
|
if inf_vals.any():
|
|
msg = ("Infinite values encountered in the moment calculation "
|
|
"involving scipy.special.kve. Values replaced by NaN to "
|
|
"avoid incorrect results.")
|
|
warnings.warn(msg, RuntimeWarning)
|
|
m = np.full_like(num, np.nan, dtype=np.double)
|
|
m[~inf_vals] = num[~inf_vals] / denom[~inf_vals]
|
|
else:
|
|
m = num / denom
|
|
return m
|
|
|
|
|
|
geninvgauss = geninvgauss_gen(a=0.0, name="geninvgauss")
|
|
|
|
|
|
class norminvgauss_gen(rv_continuous):
|
|
r"""A Normal Inverse Gaussian continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `norminvgauss` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \,
|
|
\exp(\sqrt{a^2 - b^2} + b x)
|
|
|
|
where :math:`x` is a real number, the parameter :math:`a` is the tail
|
|
heaviness and :math:`b` is the asymmetry parameter satisfying
|
|
:math:`a > 0` and :math:`|b| <= a`.
|
|
:math:`K_1` is the modified Bessel function of second kind
|
|
(`scipy.special.k1`).
|
|
|
|
%(after_notes)s
|
|
|
|
A normal inverse Gaussian random variable `Y` with parameters `a` and `b`
|
|
can be expressed as a normal mean-variance mixture:
|
|
`Y = b * V + sqrt(V) * X` where `X` is `norm(0,1)` and `V` is
|
|
`invgauss(mu=1/sqrt(a**2 - b**2))`. This representation is used
|
|
to generate random variates.
|
|
|
|
References
|
|
----------
|
|
O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on
|
|
Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
|
|
pp. 151-157, 1978.
|
|
|
|
O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and Stochastic
|
|
Volatility Modelling", Scandinavian Journal of Statistics, Vol. 24,
|
|
pp. 1-13, 1997.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _argcheck(self, a, b):
|
|
return (a > 0) & (np.absolute(b) < a)
|
|
|
|
def _pdf(self, x, a, b):
|
|
gamma = np.sqrt(a**2 - b**2)
|
|
fac1 = a / np.pi * np.exp(gamma)
|
|
sq = np.hypot(1, x) # reduce overflows
|
|
return fac1 * sc.k1e(a * sq) * np.exp(b*x - a*sq) / sq
|
|
|
|
def _rvs(self, a, b, size=None, random_state=None):
|
|
# note: X = b * V + sqrt(V) * X is norminvgaus(a,b) if X is standard
|
|
# normal and V is invgauss(mu=1/sqrt(a**2 - b**2))
|
|
gamma = np.sqrt(a**2 - b**2)
|
|
ig = invgauss.rvs(mu=1/gamma, size=size, random_state=random_state)
|
|
return b * ig + np.sqrt(ig) * norm.rvs(size=size, random_state=random_state)
|
|
|
|
def _stats(self, a, b):
|
|
gamma = np.sqrt(a**2 - b**2)
|
|
mean = b / gamma
|
|
variance = a**2 / gamma**3
|
|
skewness = 3.0 * b / (a * np.sqrt(gamma))
|
|
kurtosis = 3.0 * (1 + 4 * b**2 / a**2) / gamma
|
|
return mean, variance, skewness, kurtosis
|
|
|
|
|
|
norminvgauss = norminvgauss_gen(name="norminvgauss")
|
|
|
|
|
|
class invweibull_gen(rv_continuous):
|
|
u"""An inverted Weibull continuous random variable.
|
|
|
|
This distribution is also known as the Fréchet distribution or the
|
|
type II extreme value distribution.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `invweibull` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = c x^{-c-1} \\exp(-x^{-c})
|
|
|
|
for :math:`x > 0`, :math:`c > 0`.
|
|
|
|
`invweibull` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse
|
|
Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _pdf(self, x, c):
|
|
# invweibull.pdf(x, c) = c * x**(-c-1) * exp(-x**(-c))
|
|
xc1 = np.power(x, -c - 1.0)
|
|
xc2 = np.power(x, -c)
|
|
xc2 = np.exp(-xc2)
|
|
return c * xc1 * xc2
|
|
|
|
def _cdf(self, x, c):
|
|
xc1 = np.power(x, -c)
|
|
return np.exp(-xc1)
|
|
|
|
def _ppf(self, q, c):
|
|
return np.power(-np.log(q), -1.0/c)
|
|
|
|
def _munp(self, n, c):
|
|
return sc.gamma(1 - n / c)
|
|
|
|
def _entropy(self, c):
|
|
return 1+_EULER + _EULER / c - np.log(c)
|
|
|
|
|
|
invweibull = invweibull_gen(a=0, name='invweibull')
|
|
|
|
|
|
class johnsonsb_gen(rv_continuous):
|
|
r"""A Johnson SB continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
johnsonsu
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `johnsonsb` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, b) = \frac{b}{x(1-x)} \phi(a + b \log \frac{x}{1-x} )
|
|
|
|
for :math:`0 <= x < =1` and :math:`a, b > 0`, and :math:`\phi` is the normal
|
|
pdf.
|
|
|
|
`johnsonsb` takes :math:`a` and :math:`b` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _argcheck(self, a, b):
|
|
return (b > 0) & (a == a)
|
|
|
|
def _pdf(self, x, a, b):
|
|
# johnsonsb.pdf(x, a, b) = b / (x*(1-x)) * phi(a + b * log(x/(1-x)))
|
|
trm = _norm_pdf(a + b*np.log(x/(1.0-x)))
|
|
return b*1.0/(x*(1-x))*trm
|
|
|
|
def _cdf(self, x, a, b):
|
|
return _norm_cdf(a + b*np.log(x/(1.0-x)))
|
|
|
|
def _ppf(self, q, a, b):
|
|
return 1.0 / (1 + np.exp(-1.0 / b * (_norm_ppf(q) - a)))
|
|
|
|
|
|
johnsonsb = johnsonsb_gen(a=0.0, b=1.0, name='johnsonsb')
|
|
|
|
|
|
class johnsonsu_gen(rv_continuous):
|
|
r"""A Johnson SU continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
johnsonsb
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `johnsonsu` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}}
|
|
\phi(a + b \log(x + \sqrt{x^2 + 1}))
|
|
|
|
for all :math:`x, a, b > 0`, and :math:`\phi` is the normal pdf.
|
|
|
|
`johnsonsu` takes :math:`a` and :math:`b` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, a, b):
|
|
return (b > 0) & (a == a)
|
|
|
|
def _pdf(self, x, a, b):
|
|
# johnsonsu.pdf(x, a, b) = b / sqrt(x**2 + 1) *
|
|
# phi(a + b * log(x + sqrt(x**2 + 1)))
|
|
x2 = x*x
|
|
trm = _norm_pdf(a + b * np.log(x + np.sqrt(x2+1)))
|
|
return b*1.0/np.sqrt(x2+1.0)*trm
|
|
|
|
def _cdf(self, x, a, b):
|
|
return _norm_cdf(a + b * np.log(x + np.sqrt(x*x + 1)))
|
|
|
|
def _ppf(self, q, a, b):
|
|
return np.sinh((_norm_ppf(q) - a) / b)
|
|
|
|
|
|
johnsonsu = johnsonsu_gen(name='johnsonsu')
|
|
|
|
|
|
class laplace_gen(rv_continuous):
|
|
r"""A Laplace continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `laplace` is
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{1}{2} \exp(-|x|)
|
|
|
|
for a real number :math:`x`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, size=None, random_state=None):
|
|
return random_state.laplace(0, 1, size=size)
|
|
|
|
def _pdf(self, x):
|
|
# laplace.pdf(x) = 1/2 * exp(-abs(x))
|
|
return 0.5*np.exp(-abs(x))
|
|
|
|
def _cdf(self, x):
|
|
return np.where(x > 0, 1.0-0.5*np.exp(-x), 0.5*np.exp(x))
|
|
|
|
def _ppf(self, q):
|
|
return np.where(q > 0.5, -np.log(2*(1-q)), np.log(2*q))
|
|
|
|
def _stats(self):
|
|
return 0, 2, 0, 3
|
|
|
|
def _entropy(self):
|
|
return np.log(2)+1
|
|
|
|
@replace_notes_in_docstring(rv_continuous, notes="""\
|
|
This function uses explicit formulas for the maximum likelihood
|
|
estimation of the Laplace distribution parameters, so the keyword
|
|
arguments `loc`, `scale`, and `optimizer` are ignored.\n\n""")
|
|
def fit(self, data, *args, **kwds):
|
|
data, floc, fscale = _check_fit_input_parameters(self, data,
|
|
args, kwds)
|
|
|
|
# Source: Statistical Distributions, 3rd Edition. Evans, Hastings,
|
|
# and Peacock (2000), Page 124
|
|
|
|
if floc is None:
|
|
floc = np.median(data)
|
|
|
|
if fscale is None:
|
|
fscale = (np.sum(np.abs(data - floc))) / len(data)
|
|
|
|
return floc, fscale
|
|
|
|
|
|
laplace = laplace_gen(name='laplace')
|
|
|
|
|
|
class laplace_asymmetric_gen(rv_continuous):
|
|
r"""An asymmetric Laplace continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
laplace : Laplace distribution
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `laplace_asymmetric` is
|
|
|
|
.. math::
|
|
|
|
f(x, \kappa) &= \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0\\
|
|
&= \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0\\
|
|
|
|
for :math:`-\infty < x < \infty`, :math:`\kappa > 0`.
|
|
|
|
`laplace_asymmetric` takes ``kappa`` as a shape parameter for
|
|
:math:`\kappa`. For :math:`\kappa = 1`, it is identical to a
|
|
Laplace distribution.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] "Asymmetric Laplace distribution", Wikipedia
|
|
https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution
|
|
|
|
.. [2] Kozubowski TJ and Podgórski K. A Multivariate and
|
|
Asymmetric Generalization of Laplace Distribution,
|
|
Computational Statistics 15, 531--540 (2000).
|
|
:doi:`10.1007/PL00022717`
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, kappa):
|
|
return np.exp(self._logpdf(x, kappa))
|
|
|
|
def _logpdf(self, x, kappa):
|
|
kapinv = 1/kappa
|
|
lPx = x * np.where(x >= 0, -kappa, kapinv)
|
|
lPx -= np.log(kappa+kapinv)
|
|
return lPx
|
|
|
|
def _cdf(self, x, kappa):
|
|
kapinv = 1/kappa
|
|
kappkapinv = kappa+kapinv
|
|
return np.where(x >= 0,
|
|
1 - np.exp(-x*kappa)*(kapinv/kappkapinv),
|
|
np.exp(x*kapinv)*(kappa/kappkapinv))
|
|
|
|
def _sf(self, x, kappa):
|
|
kapinv = 1/kappa
|
|
kappkapinv = kappa+kapinv
|
|
return np.where(x >= 0,
|
|
np.exp(-x*kappa)*(kapinv/kappkapinv),
|
|
1 - np.exp(x*kapinv)*(kappa/kappkapinv))
|
|
|
|
def _ppf(self, q, kappa):
|
|
kapinv = 1/kappa
|
|
kappkapinv = kappa+kapinv
|
|
return np.where(q >= kappa/kappkapinv,
|
|
-np.log((1 - q)*kappkapinv*kappa)*kapinv,
|
|
np.log(q*kappkapinv/kappa)*kappa)
|
|
|
|
def _isf(self, q, kappa):
|
|
kapinv = 1/kappa
|
|
kappkapinv = kappa+kapinv
|
|
return np.where(q <= kapinv/kappkapinv,
|
|
-np.log(q*kappkapinv*kappa)*kapinv,
|
|
np.log((1 - q)*kappkapinv/kappa)*kappa)
|
|
|
|
def _stats(self, kappa):
|
|
kapinv = 1/kappa
|
|
mn = kapinv - kappa
|
|
var = kapinv*kapinv + kappa*kappa
|
|
g1 = 2.0*(1-np.power(kappa, 6))/np.power(1+np.power(kappa, 4), 1.5)
|
|
g2 = 6.0*(1+np.power(kappa, 8))/np.power(1+np.power(kappa, 4), 2)
|
|
return mn, var, g1, g2
|
|
|
|
def _entropy(self, kappa):
|
|
return 1 + np.log(kappa+1/kappa)
|
|
|
|
|
|
laplace_asymmetric = laplace_asymmetric_gen(name='laplace_asymmetric')
|
|
|
|
|
|
def _check_fit_input_parameters(dist, data, args, kwds):
|
|
data = np.asarray(data)
|
|
floc = kwds.get('floc', None)
|
|
fscale = kwds.get('fscale', None)
|
|
|
|
num_shapes = len(dist.shapes.split(",")) if dist.shapes else 0
|
|
fshape_keys = []
|
|
fshapes = []
|
|
|
|
# user has many options for fixing the shape, so here we standardize it
|
|
# into 'f' + the number of the shape.
|
|
# Adapted from `_reduce_func` in `_distn_infrastructure.py`:
|
|
if dist.shapes:
|
|
shapes = dist.shapes.replace(',', ' ').split()
|
|
for j, s in enumerate(shapes):
|
|
key = 'f' + str(j)
|
|
names = [key, 'f' + s, 'fix_' + s]
|
|
val = _get_fixed_fit_value(kwds, names)
|
|
fshape_keys.append(key)
|
|
fshapes.append(val)
|
|
if val is not None:
|
|
kwds[key] = val
|
|
|
|
# determine if there are any unknown arguments in kwds
|
|
known_keys = {'loc', 'scale', 'optimizer', 'floc', 'fscale', *fshape_keys}
|
|
unknown_keys = set(kwds).difference(known_keys)
|
|
if unknown_keys:
|
|
raise TypeError(f"Unknown keyword arguments: {unknown_keys}.")
|
|
|
|
if len(args) > num_shapes:
|
|
raise TypeError("Too many positional arguments.")
|
|
|
|
if None not in {floc, fscale, *fshapes}:
|
|
# This check is for consistency with `rv_continuous.fit`.
|
|
# Without this check, this function would just return the
|
|
# parameters that were given.
|
|
raise RuntimeError("All parameters fixed. There is nothing to "
|
|
"optimize.")
|
|
|
|
if not np.isfinite(data).all():
|
|
raise RuntimeError("The data contains non-finite values.")
|
|
|
|
return (data, *fshapes, floc, fscale)
|
|
|
|
|
|
class levy_gen(rv_continuous):
|
|
r"""A Levy continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
levy_stable, levy_l
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `levy` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right)
|
|
|
|
for :math:`x >= 0`.
|
|
|
|
This is the same as the Levy-stable distribution with :math:`a=1/2` and
|
|
:math:`b=1`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _pdf(self, x):
|
|
# levy.pdf(x) = 1 / (x * sqrt(2*pi*x)) * exp(-1/(2*x))
|
|
return 1 / np.sqrt(2*np.pi*x) / x * np.exp(-1/(2*x))
|
|
|
|
def _cdf(self, x):
|
|
# Equivalent to 2*norm.sf(np.sqrt(1/x))
|
|
return sc.erfc(np.sqrt(0.5 / x))
|
|
|
|
def _sf(self, x):
|
|
return sc.erf(np.sqrt(0.5 / x))
|
|
|
|
def _ppf(self, q):
|
|
# Equivalent to 1.0/(norm.isf(q/2)**2) or 0.5/(erfcinv(q)**2)
|
|
val = -sc.ndtri(q/2)
|
|
return 1.0 / (val * val)
|
|
|
|
def _stats(self):
|
|
return np.inf, np.inf, np.nan, np.nan
|
|
|
|
|
|
levy = levy_gen(a=0.0, name="levy")
|
|
|
|
|
|
class levy_l_gen(rv_continuous):
|
|
r"""A left-skewed Levy continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
levy, levy_stable
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `levy_l` is:
|
|
|
|
.. math::
|
|
f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp{ \left(-\frac{1}{2|x|} \right)}
|
|
|
|
for :math:`x <= 0`.
|
|
|
|
This is the same as the Levy-stable distribution with :math:`a=1/2` and
|
|
:math:`b=-1`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _pdf(self, x):
|
|
# levy_l.pdf(x) = 1 / (abs(x) * sqrt(2*pi*abs(x))) * exp(-1/(2*abs(x)))
|
|
ax = abs(x)
|
|
return 1/np.sqrt(2*np.pi*ax)/ax*np.exp(-1/(2*ax))
|
|
|
|
def _cdf(self, x):
|
|
ax = abs(x)
|
|
return 2 * _norm_cdf(1 / np.sqrt(ax)) - 1
|
|
|
|
def _sf(self, x):
|
|
ax = abs(x)
|
|
return 2 * _norm_sf(1 / np.sqrt(ax))
|
|
|
|
def _ppf(self, q):
|
|
val = _norm_ppf((q + 1.0) / 2)
|
|
return -1.0 / (val * val)
|
|
|
|
def _isf(self, p):
|
|
return -1/_norm_isf(p/2)**2
|
|
|
|
def _stats(self):
|
|
return np.inf, np.inf, np.nan, np.nan
|
|
|
|
|
|
levy_l = levy_l_gen(b=0.0, name="levy_l")
|
|
|
|
|
|
class levy_stable_gen(rv_continuous):
|
|
r"""A Levy-stable continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
levy, levy_l
|
|
|
|
Notes
|
|
-----
|
|
The distribution for `levy_stable` has characteristic function:
|
|
|
|
.. math::
|
|
|
|
\varphi(t, \alpha, \beta, c, \mu) =
|
|
e^{it\mu -|ct|^{\alpha}(1-i\beta \operatorname{sign}(t)\Phi(\alpha, t))}
|
|
|
|
where:
|
|
|
|
.. math::
|
|
|
|
\Phi = \begin{cases}
|
|
\tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\
|
|
-{\frac {2}{\pi }}\log |t|&\alpha =1
|
|
\end{cases}
|
|
|
|
The probability density function for `levy_stable` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty \varphi(t)e^{-ixt}\,dt
|
|
|
|
where :math:`-\infty < t < \infty`. This integral does not have a known closed form.
|
|
|
|
For evaluation of pdf we use either Zolotarev :math:`S_0` parameterization with integration,
|
|
direct integration of standard parameterization of characteristic function or FFT of
|
|
characteristic function. If set to other than None and if number of points is greater than
|
|
``levy_stable.pdf_fft_min_points_threshold`` (defaults to None) we use FFT otherwise we use one
|
|
of the other methods.
|
|
|
|
The default method is 'best' which uses Zolotarev's method if alpha = 1 and integration of
|
|
characteristic function otherwise. The default method can be changed by setting
|
|
``levy_stable.pdf_default_method`` to either 'zolotarev', 'quadrature' or 'best'.
|
|
|
|
To increase accuracy of FFT calculation one can specify ``levy_stable.pdf_fft_grid_spacing``
|
|
(defaults to 0.001) and ``pdf_fft_n_points_two_power`` (defaults to a value that covers the
|
|
input range * 4). Setting ``pdf_fft_n_points_two_power`` to 16 should be sufficiently accurate
|
|
in most cases at the expense of CPU time.
|
|
|
|
For evaluation of cdf we use Zolatarev :math:`S_0` parameterization with integration or integral of
|
|
the pdf FFT interpolated spline. The settings affecting FFT calculation are the same as
|
|
for pdf calculation. Setting the threshold to ``None`` (default) will disable FFT. For cdf
|
|
calculations the Zolatarev method is superior in accuracy, so FFT is disabled by default.
|
|
|
|
Fitting estimate uses quantile estimation method in [MC]. MLE estimation of parameters in
|
|
fit method uses this quantile estimate initially. Note that MLE doesn't always converge if
|
|
using FFT for pdf calculations; so it's best that ``pdf_fft_min_points_threshold`` is left unset.
|
|
|
|
.. warning::
|
|
|
|
For pdf calculations implementation of Zolatarev is unstable for values where alpha = 1 and
|
|
beta != 0. In this case the quadrature method is recommended. FFT calculation is also
|
|
considered experimental.
|
|
|
|
For cdf calculations FFT calculation is considered experimental. Use Zolatarev's method
|
|
instead (default).
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [MC] McCulloch, J., 1986. Simple consistent estimators of stable distribution parameters.
|
|
Communications in Statistics - Simulation and Computation 15, 11091136.
|
|
.. [MS] Mittnik, S.T. Rachev, T. Doganoglu, D. Chenyao, 1999. Maximum likelihood estimation
|
|
of stable Paretian models, Mathematical and Computer Modelling, Volume 29, Issue 10,
|
|
1999, Pages 275-293.
|
|
.. [BS] Borak, S., Hardle, W., Rafal, W. 2005. Stable distributions, Economic Risk.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _rvs(self, alpha, beta, size=None, random_state=None):
|
|
|
|
def alpha1func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W):
|
|
return (2/np.pi*(np.pi/2 + bTH)*tanTH -
|
|
beta*np.log((np.pi/2*W*cosTH)/(np.pi/2 + bTH)))
|
|
|
|
def beta0func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W):
|
|
return (W/(cosTH/np.tan(aTH) + np.sin(TH)) *
|
|
((np.cos(aTH) + np.sin(aTH)*tanTH)/W)**(1.0/alpha))
|
|
|
|
def otherwise(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W):
|
|
# alpha is not 1 and beta is not 0
|
|
val0 = beta*np.tan(np.pi*alpha/2)
|
|
th0 = np.arctan(val0)/alpha
|
|
val3 = W/(cosTH/np.tan(alpha*(th0 + TH)) + np.sin(TH))
|
|
res3 = val3*((np.cos(aTH) + np.sin(aTH)*tanTH -
|
|
val0*(np.sin(aTH) - np.cos(aTH)*tanTH))/W)**(1.0/alpha)
|
|
return res3
|
|
|
|
def alphanot1func(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W):
|
|
res = _lazywhere(beta == 0,
|
|
(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W),
|
|
beta0func, f2=otherwise)
|
|
return res
|
|
|
|
alpha = np.broadcast_to(alpha, size)
|
|
beta = np.broadcast_to(beta, size)
|
|
TH = uniform.rvs(loc=-np.pi/2.0, scale=np.pi, size=size,
|
|
random_state=random_state)
|
|
W = expon.rvs(size=size, random_state=random_state)
|
|
aTH = alpha*TH
|
|
bTH = beta*TH
|
|
cosTH = np.cos(TH)
|
|
tanTH = np.tan(TH)
|
|
res = _lazywhere(alpha == 1,
|
|
(alpha, beta, TH, aTH, bTH, cosTH, tanTH, W),
|
|
alpha1func, f2=alphanot1func)
|
|
return res
|
|
|
|
def _argcheck(self, alpha, beta):
|
|
return (alpha > 0) & (alpha <= 2) & (beta <= 1) & (beta >= -1)
|
|
|
|
@staticmethod
|
|
def _cf(t, alpha, beta):
|
|
Phi = lambda alpha, t: np.tan(np.pi*alpha/2) if alpha != 1 else -2.0*np.log(np.abs(t))/np.pi
|
|
return np.exp(-(np.abs(t)**alpha)*(1-1j*beta*np.sign(t)*Phi(alpha, t)))
|
|
|
|
@staticmethod
|
|
def _pdf_from_cf_with_fft(cf, h=0.01, q=9):
|
|
"""Calculates pdf from cf using fft. Using region around 0 with N=2**q points
|
|
separated by distance h. As suggested by [MS].
|
|
"""
|
|
N = 2**q
|
|
n = np.arange(1,N+1)
|
|
density = ((-1)**(n-1-N/2))*np.fft.fft(((-1)**(n-1))*cf(2*np.pi*(n-1-N/2)/h/N))/h/N
|
|
x = (n-1-N/2)*h
|
|
return (x, density)
|
|
|
|
@staticmethod
|
|
def _pdf_single_value_best(x, alpha, beta):
|
|
if alpha != 1. or (alpha == 1. and beta == 0.):
|
|
return levy_stable_gen._pdf_single_value_zolotarev(x, alpha, beta)
|
|
else:
|
|
return levy_stable_gen._pdf_single_value_cf_integrate(x, alpha, beta)
|
|
|
|
@staticmethod
|
|
def _pdf_single_value_cf_integrate(x, alpha, beta):
|
|
cf = lambda t: levy_stable_gen._cf(t, alpha, beta)
|
|
return integrate.quad(lambda t: np.real(np.exp(-1j*t*x)*cf(t)), -np.inf, np.inf, limit=1000)[0]/np.pi/2
|
|
|
|
@staticmethod
|
|
def _pdf_single_value_zolotarev(x, alpha, beta):
|
|
"""Calculate pdf using Zolotarev's methods as detailed in [BS].
|
|
"""
|
|
zeta = -beta*np.tan(np.pi*alpha/2.)
|
|
if alpha != 1:
|
|
x0 = x + zeta # convert to S_0 parameterization
|
|
xi = np.arctan(-zeta)/alpha
|
|
|
|
def V(theta):
|
|
return np.cos(alpha*xi)**(1/(alpha-1)) * \
|
|
(np.cos(theta)/np.sin(alpha*(xi+theta)))**(alpha/(alpha-1)) * \
|
|
(np.cos(alpha*xi+(alpha-1)*theta)/np.cos(theta))
|
|
if x0 > zeta:
|
|
def g(theta):
|
|
return (V(theta) *
|
|
np.real(np.complex128(x0-zeta)**(alpha/(alpha-1))))
|
|
|
|
def f(theta):
|
|
return g(theta) * np.exp(-g(theta))
|
|
|
|
# spare calculating integral on null set
|
|
# use isclose as macos has fp differences
|
|
if np.isclose(-xi, np.pi/2, rtol=1e-014, atol=1e-014):
|
|
return 0.
|
|
|
|
with np.errstate(all="ignore"):
|
|
intg_max = optimize.minimize_scalar(lambda theta: -f(theta), bounds=[-xi, np.pi/2])
|
|
intg_kwargs = {}
|
|
# windows quadpack less forgiving with points out of bounds
|
|
if intg_max.success and not np.isnan(intg_max.fun)\
|
|
and intg_max.x > -xi and intg_max.x < np.pi/2:
|
|
intg_kwargs["points"] = [intg_max.x]
|
|
intg = integrate.quad(f, -xi, np.pi/2, **intg_kwargs)[0]
|
|
return alpha * intg / np.pi / np.abs(alpha-1) / (x0-zeta)
|
|
elif x0 == zeta:
|
|
return sc.gamma(1+1/alpha)*np.cos(xi)/np.pi/((1+zeta**2)**(1/alpha/2))
|
|
else:
|
|
return levy_stable_gen._pdf_single_value_zolotarev(-x, alpha, -beta)
|
|
else:
|
|
# since location zero, no need to reposition x for S_0 parameterization
|
|
xi = np.pi/2
|
|
if beta != 0:
|
|
warnings.warn('Density calculation unstable for alpha=1 and beta!=0.' +
|
|
' Use quadrature method instead.', RuntimeWarning)
|
|
|
|
def V(theta):
|
|
expr_1 = np.pi/2+beta*theta
|
|
return 2. * expr_1 * np.exp(expr_1*np.tan(theta)/beta) / np.cos(theta) / np.pi
|
|
|
|
def g(theta):
|
|
return np.exp(-np.pi * x / 2. / beta) * V(theta)
|
|
|
|
def f(theta):
|
|
return g(theta) * np.exp(-g(theta))
|
|
|
|
with np.errstate(all="ignore"):
|
|
intg_max = optimize.minimize_scalar(lambda theta: -f(theta), bounds=[-np.pi/2, np.pi/2])
|
|
intg = integrate.fixed_quad(f, -np.pi/2, intg_max.x)[0] + integrate.fixed_quad(f, intg_max.x, np.pi/2)[0]
|
|
return intg / np.abs(beta) / 2.
|
|
else:
|
|
return 1/(1+x**2)/np.pi
|
|
|
|
@staticmethod
|
|
def _cdf_single_value_zolotarev(x, alpha, beta):
|
|
"""Calculate cdf using Zolotarev's methods as detailed in [BS].
|
|
"""
|
|
zeta = -beta*np.tan(np.pi*alpha/2.)
|
|
if alpha != 1:
|
|
x0 = x + zeta # convert to S_0 parameterization
|
|
xi = np.arctan(-zeta)/alpha
|
|
|
|
def V(theta):
|
|
return np.cos(alpha*xi)**(1/(alpha-1)) * \
|
|
(np.cos(theta)/np.sin(alpha*(xi+theta)))**(alpha/(alpha-1)) * \
|
|
(np.cos(alpha*xi+(alpha-1)*theta)/np.cos(theta))
|
|
if x0 > zeta:
|
|
c_1 = 1 if alpha > 1 else .5 - xi/np.pi
|
|
|
|
def f(theta):
|
|
z = np.complex128(x0 - zeta)
|
|
return np.exp(-V(theta) * np.real(z**(alpha/(alpha-1))))
|
|
|
|
with np.errstate(all="ignore"):
|
|
# spare calculating integral on null set
|
|
# use isclose as macos has fp differences
|
|
if np.isclose(-xi, np.pi/2, rtol=1e-014, atol=1e-014):
|
|
intg = 0
|
|
else:
|
|
intg = integrate.quad(f, -xi, np.pi/2)[0]
|
|
return c_1 + np.sign(1-alpha) * intg / np.pi
|
|
elif x0 == zeta:
|
|
return .5 - xi/np.pi
|
|
else:
|
|
return 1 - levy_stable_gen._cdf_single_value_zolotarev(-x, alpha, -beta)
|
|
|
|
else:
|
|
# since location zero, no need to reposition x for S_0 parameterization
|
|
xi = np.pi/2
|
|
if beta > 0:
|
|
|
|
def V(theta):
|
|
expr_1 = np.pi/2+beta*theta
|
|
return 2. * expr_1 * np.exp(expr_1*np.tan(theta)/beta) / np.cos(theta) / np.pi
|
|
|
|
with np.errstate(all="ignore"):
|
|
expr_1 = np.exp(-np.pi*x/beta/2.)
|
|
int_1 = integrate.quad(lambda theta: np.exp(-expr_1 * V(theta)), -np.pi/2, np.pi/2)[0]
|
|
return int_1 / np.pi
|
|
elif beta == 0:
|
|
return .5 + np.arctan(x)/np.pi
|
|
else:
|
|
return 1 - levy_stable_gen._cdf_single_value_zolotarev(-x, 1, -beta)
|
|
|
|
def _pdf(self, x, alpha, beta):
|
|
|
|
x = np.asarray(x).reshape(1, -1)[0,:]
|
|
|
|
x, alpha, beta = np.broadcast_arrays(x, alpha, beta)
|
|
|
|
data_in = np.dstack((x, alpha, beta))[0]
|
|
data_out = np.empty(shape=(len(data_in),1))
|
|
|
|
pdf_default_method_name = getattr(self, 'pdf_default_method', 'best')
|
|
if pdf_default_method_name == 'best':
|
|
pdf_single_value_method = levy_stable_gen._pdf_single_value_best
|
|
elif pdf_default_method_name == 'zolotarev':
|
|
pdf_single_value_method = levy_stable_gen._pdf_single_value_zolotarev
|
|
else:
|
|
pdf_single_value_method = levy_stable_gen._pdf_single_value_cf_integrate
|
|
|
|
fft_min_points_threshold = getattr(self, 'pdf_fft_min_points_threshold', None)
|
|
fft_grid_spacing = getattr(self, 'pdf_fft_grid_spacing', 0.001)
|
|
fft_n_points_two_power = getattr(self, 'pdf_fft_n_points_two_power', None)
|
|
|
|
# group data in unique arrays of alpha, beta pairs
|
|
uniq_param_pairs = np.vstack(list({tuple(row) for row in
|
|
data_in[:, 1:]}))
|
|
for pair in uniq_param_pairs:
|
|
data_mask = np.all(data_in[:,1:] == pair, axis=-1)
|
|
data_subset = data_in[data_mask]
|
|
if fft_min_points_threshold is None or len(data_subset) < fft_min_points_threshold:
|
|
data_out[data_mask] = np.array([pdf_single_value_method(_x, _alpha, _beta)
|
|
for _x, _alpha, _beta in data_subset]).reshape(len(data_subset), 1)
|
|
else:
|
|
warnings.warn('Density calculations experimental for FFT method.' +
|
|
' Use combination of zolatarev and quadrature methods instead.', RuntimeWarning)
|
|
_alpha, _beta = pair
|
|
_x = data_subset[:,(0,)]
|
|
|
|
# need enough points to "cover" _x for interpolation
|
|
h = fft_grid_spacing
|
|
q = np.ceil(np.log(2*np.max(np.abs(_x))/h)/np.log(2)) + 2 if fft_n_points_two_power is None else int(fft_n_points_two_power)
|
|
|
|
density_x, density = levy_stable_gen._pdf_from_cf_with_fft(lambda t: levy_stable_gen._cf(t, _alpha, _beta), h=h, q=q)
|
|
f = interpolate.interp1d(density_x, np.real(density))
|
|
data_out[data_mask] = f(_x)
|
|
|
|
return data_out.T[0]
|
|
|
|
def _cdf(self, x, alpha, beta):
|
|
|
|
x = np.asarray(x).reshape(1, -1)[0,:]
|
|
|
|
x, alpha, beta = np.broadcast_arrays(x, alpha, beta)
|
|
|
|
data_in = np.dstack((x, alpha, beta))[0]
|
|
data_out = np.empty(shape=(len(data_in),1))
|
|
|
|
fft_min_points_threshold = getattr(self, 'pdf_fft_min_points_threshold', None)
|
|
fft_grid_spacing = getattr(self, 'pdf_fft_grid_spacing', 0.001)
|
|
fft_n_points_two_power = getattr(self, 'pdf_fft_n_points_two_power', None)
|
|
|
|
# group data in unique arrays of alpha, beta pairs
|
|
uniq_param_pairs = np.vstack(
|
|
list({tuple(row) for row in data_in[:,1:]}))
|
|
for pair in uniq_param_pairs:
|
|
data_mask = np.all(data_in[:,1:] == pair, axis=-1)
|
|
data_subset = data_in[data_mask]
|
|
if fft_min_points_threshold is None or len(data_subset) < fft_min_points_threshold:
|
|
data_out[data_mask] = np.array([levy_stable._cdf_single_value_zolotarev(_x, _alpha, _beta)
|
|
for _x, _alpha, _beta in data_subset]).reshape(len(data_subset), 1)
|
|
else:
|
|
warnings.warn("FFT method is considered experimental for "
|
|
"cumulative distribution function "
|
|
"evaluations. Use Zolotarev's method instead.",
|
|
RuntimeWarning)
|
|
_alpha, _beta = pair
|
|
_x = data_subset[:,(0,)]
|
|
|
|
# need enough points to "cover" _x for interpolation
|
|
h = fft_grid_spacing
|
|
q = 16 if fft_n_points_two_power is None else int(fft_n_points_two_power)
|
|
|
|
density_x, density = levy_stable_gen._pdf_from_cf_with_fft(lambda t: levy_stable_gen._cf(t, _alpha, _beta), h=h, q=q)
|
|
f = interpolate.InterpolatedUnivariateSpline(density_x, np.real(density))
|
|
data_out[data_mask] = np.array([f.integral(self.a, x_1) for x_1 in _x]).reshape(data_out[data_mask].shape)
|
|
|
|
return data_out.T[0]
|
|
|
|
def _fitstart(self, data):
|
|
# We follow McCullock 1986 method - Simple Consistent Estimators
|
|
# of Stable Distribution Parameters
|
|
|
|
# Table III and IV
|
|
nu_alpha_range = [2.439, 2.5, 2.6, 2.7, 2.8, 3, 3.2, 3.5, 4, 5, 6, 8, 10, 15, 25]
|
|
nu_beta_range = [0, 0.1, 0.2, 0.3, 0.5, 0.7, 1]
|
|
|
|
# table III - alpha = psi_1(nu_alpha, nu_beta)
|
|
alpha_table = [
|
|
[2.000, 2.000, 2.000, 2.000, 2.000, 2.000, 2.000],
|
|
[1.916, 1.924, 1.924, 1.924, 1.924, 1.924, 1.924],
|
|
[1.808, 1.813, 1.829, 1.829, 1.829, 1.829, 1.829],
|
|
[1.729, 1.730, 1.737, 1.745, 1.745, 1.745, 1.745],
|
|
[1.664, 1.663, 1.663, 1.668, 1.676, 1.676, 1.676],
|
|
[1.563, 1.560, 1.553, 1.548, 1.547, 1.547, 1.547],
|
|
[1.484, 1.480, 1.471, 1.460, 1.448, 1.438, 1.438],
|
|
[1.391, 1.386, 1.378, 1.364, 1.337, 1.318, 1.318],
|
|
[1.279, 1.273, 1.266, 1.250, 1.210, 1.184, 1.150],
|
|
[1.128, 1.121, 1.114, 1.101, 1.067, 1.027, 0.973],
|
|
[1.029, 1.021, 1.014, 1.004, 0.974, 0.935, 0.874],
|
|
[0.896, 0.892, 0.884, 0.883, 0.855, 0.823, 0.769],
|
|
[0.818, 0.812, 0.806, 0.801, 0.780, 0.756, 0.691],
|
|
[0.698, 0.695, 0.692, 0.689, 0.676, 0.656, 0.597],
|
|
[0.593, 0.590, 0.588, 0.586, 0.579, 0.563, 0.513]]
|
|
|
|
# table IV - beta = psi_2(nu_alpha, nu_beta)
|
|
beta_table = [
|
|
[0, 2.160, 1.000, 1.000, 1.000, 1.000, 1.000],
|
|
[0, 1.592, 3.390, 1.000, 1.000, 1.000, 1.000],
|
|
[0, 0.759, 1.800, 1.000, 1.000, 1.000, 1.000],
|
|
[0, 0.482, 1.048, 1.694, 1.000, 1.000, 1.000],
|
|
[0, 0.360, 0.760, 1.232, 2.229, 1.000, 1.000],
|
|
[0, 0.253, 0.518, 0.823, 1.575, 1.000, 1.000],
|
|
[0, 0.203, 0.410, 0.632, 1.244, 1.906, 1.000],
|
|
[0, 0.165, 0.332, 0.499, 0.943, 1.560, 1.000],
|
|
[0, 0.136, 0.271, 0.404, 0.689, 1.230, 2.195],
|
|
[0, 0.109, 0.216, 0.323, 0.539, 0.827, 1.917],
|
|
[0, 0.096, 0.190, 0.284, 0.472, 0.693, 1.759],
|
|
[0, 0.082, 0.163, 0.243, 0.412, 0.601, 1.596],
|
|
[0, 0.074, 0.147, 0.220, 0.377, 0.546, 1.482],
|
|
[0, 0.064, 0.128, 0.191, 0.330, 0.478, 1.362],
|
|
[0, 0.056, 0.112, 0.167, 0.285, 0.428, 1.274]]
|
|
|
|
# Table V and VII
|
|
alpha_range = [2, 1.9, 1.8, 1.7, 1.6, 1.5, 1.4, 1.3, 1.2, 1.1, 1, 0.9, 0.8, 0.7, 0.6, 0.5]
|
|
beta_range = [0, 0.25, 0.5, 0.75, 1]
|
|
|
|
# Table V - nu_c = psi_3(alpha, beta)
|
|
nu_c_table = [
|
|
[1.908, 1.908, 1.908, 1.908, 1.908],
|
|
[1.914, 1.915, 1.916, 1.918, 1.921],
|
|
[1.921, 1.922, 1.927, 1.936, 1.947],
|
|
[1.927, 1.930, 1.943, 1.961, 1.987],
|
|
[1.933, 1.940, 1.962, 1.997, 2.043],
|
|
[1.939, 1.952, 1.988, 2.045, 2.116],
|
|
[1.946, 1.967, 2.022, 2.106, 2.211],
|
|
[1.955, 1.984, 2.067, 2.188, 2.333],
|
|
[1.965, 2.007, 2.125, 2.294, 2.491],
|
|
[1.980, 2.040, 2.205, 2.435, 2.696],
|
|
[2.000, 2.085, 2.311, 2.624, 2.973],
|
|
[2.040, 2.149, 2.461, 2.886, 3.356],
|
|
[2.098, 2.244, 2.676, 3.265, 3.912],
|
|
[2.189, 2.392, 3.004, 3.844, 4.775],
|
|
[2.337, 2.634, 3.542, 4.808, 6.247],
|
|
[2.588, 3.073, 4.534, 6.636, 9.144]]
|
|
|
|
# Table VII - nu_zeta = psi_5(alpha, beta)
|
|
nu_zeta_table = [
|
|
[0, 0.000, 0.000, 0.000, 0.000],
|
|
[0, -0.017, -0.032, -0.049, -0.064],
|
|
[0, -0.030, -0.061, -0.092, -0.123],
|
|
[0, -0.043, -0.088, -0.132, -0.179],
|
|
[0, -0.056, -0.111, -0.170, -0.232],
|
|
[0, -0.066, -0.134, -0.206, -0.283],
|
|
[0, -0.075, -0.154, -0.241, -0.335],
|
|
[0, -0.084, -0.173, -0.276, -0.390],
|
|
[0, -0.090, -0.192, -0.310, -0.447],
|
|
[0, -0.095, -0.208, -0.346, -0.508],
|
|
[0, -0.098, -0.223, -0.380, -0.576],
|
|
[0, -0.099, -0.237, -0.424, -0.652],
|
|
[0, -0.096, -0.250, -0.469, -0.742],
|
|
[0, -0.089, -0.262, -0.520, -0.853],
|
|
[0, -0.078, -0.272, -0.581, -0.997],
|
|
[0, -0.061, -0.279, -0.659, -1.198]]
|
|
|
|
psi_1 = interpolate.interp2d(nu_beta_range, nu_alpha_range, alpha_table, kind='linear')
|
|
psi_2 = interpolate.interp2d(nu_beta_range, nu_alpha_range, beta_table, kind='linear')
|
|
psi_2_1 = lambda nu_beta, nu_alpha: psi_2(nu_beta, nu_alpha) if nu_beta > 0 else -psi_2(-nu_beta, nu_alpha)
|
|
|
|
phi_3 = interpolate.interp2d(beta_range, alpha_range, nu_c_table, kind='linear')
|
|
phi_3_1 = lambda beta, alpha: phi_3(beta, alpha) if beta > 0 else phi_3(-beta, alpha)
|
|
phi_5 = interpolate.interp2d(beta_range, alpha_range, nu_zeta_table, kind='linear')
|
|
phi_5_1 = lambda beta, alpha: phi_5(beta, alpha) if beta > 0 else -phi_5(-beta, alpha)
|
|
|
|
# quantiles
|
|
p05 = np.percentile(data, 5)
|
|
p50 = np.percentile(data, 50)
|
|
p95 = np.percentile(data, 95)
|
|
p25 = np.percentile(data, 25)
|
|
p75 = np.percentile(data, 75)
|
|
|
|
nu_alpha = (p95 - p05)/(p75 - p25)
|
|
nu_beta = (p95 + p05 - 2*p50)/(p95 - p05)
|
|
|
|
if nu_alpha >= 2.439:
|
|
alpha = np.clip(psi_1(nu_beta, nu_alpha)[0], np.finfo(float).eps, 2.)
|
|
beta = np.clip(psi_2_1(nu_beta, nu_alpha)[0], -1., 1.)
|
|
else:
|
|
alpha = 2.0
|
|
beta = np.sign(nu_beta)
|
|
c = (p75 - p25) / phi_3_1(beta, alpha)[0]
|
|
zeta = p50 + c*phi_5_1(beta, alpha)[0]
|
|
delta = np.clip(zeta-beta*c*np.tan(np.pi*alpha/2.) if alpha == 1. else zeta, np.finfo(float).eps, np.inf)
|
|
|
|
return (alpha, beta, delta, c)
|
|
|
|
def _stats(self, alpha, beta):
|
|
mu = 0 if alpha > 1 else np.nan
|
|
mu2 = 2 if alpha == 2 else np.inf
|
|
g1 = 0. if alpha == 2. else np.NaN
|
|
g2 = 0. if alpha == 2. else np.NaN
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
levy_stable = levy_stable_gen(name='levy_stable')
|
|
|
|
|
|
class logistic_gen(rv_continuous):
|
|
r"""A logistic (or Sech-squared) continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `logistic` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{\exp(-x)}
|
|
{(1+\exp(-x))^2}
|
|
|
|
`logistic` is a special case of `genlogistic` with ``c=1``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, size=None, random_state=None):
|
|
return random_state.logistic(size=size)
|
|
|
|
def _pdf(self, x):
|
|
# logistic.pdf(x) = exp(-x) / (1+exp(-x))**2
|
|
return np.exp(self._logpdf(x))
|
|
|
|
def _logpdf(self, x):
|
|
y = -np.abs(x)
|
|
return y - 2. * sc.log1p(np.exp(y))
|
|
|
|
def _cdf(self, x):
|
|
return sc.expit(x)
|
|
|
|
def _ppf(self, q):
|
|
return sc.logit(q)
|
|
|
|
def _sf(self, x):
|
|
return sc.expit(-x)
|
|
|
|
def _isf(self, q):
|
|
return -sc.logit(q)
|
|
|
|
def _stats(self):
|
|
return 0, np.pi*np.pi/3.0, 0, 6.0/5.0
|
|
|
|
def _entropy(self):
|
|
# https://en.wikipedia.org/wiki/Logistic_distribution
|
|
return 2.0
|
|
|
|
def fit(self, data, *args, **kwds):
|
|
data, floc, fscale = _check_fit_input_parameters(self, data,
|
|
args, kwds)
|
|
|
|
# if user has provided `floc` or `fscale`, fall back on super fit
|
|
# method. This scenario is not suitable for solving a system of
|
|
# equations
|
|
if floc is not None or fscale is not None:
|
|
return super(logistic_gen, self).fit(data, *args, **kwds)
|
|
|
|
# rv_continuous provided guesses
|
|
loc, scale = self._fitstart(data)
|
|
# account for user provided guesses
|
|
loc = kwds.pop('loc', loc)
|
|
scale = kwds.pop('scale', scale)
|
|
|
|
# the maximum likelihood estimators `a` and `b` of the location and
|
|
# scale parameters are roots of the two equations described in `func`.
|
|
# Source: Statistical Distributions, 3rd Edition. Evans, Hastings, and
|
|
# Peacock (2000), Page 130
|
|
def func(params, data):
|
|
a, b = params
|
|
n = len(data)
|
|
c = (data - a) / b
|
|
x1 = np.sum(sc.expit(c)) - n/2
|
|
x2 = np.sum(c*np.tanh(c/2)) - n
|
|
return x1, x2
|
|
|
|
return tuple(optimize.root(func, (loc, scale), args=(data,)).x)
|
|
|
|
|
|
logistic = logistic_gen(name='logistic')
|
|
|
|
|
|
class loggamma_gen(rv_continuous):
|
|
r"""A log gamma continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `loggamma` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \frac{\exp(c x - \exp(x))}
|
|
{\Gamma(c)}
|
|
|
|
for all :math:`x, c > 0`. Here, :math:`\Gamma` is the
|
|
gamma function (`scipy.special.gamma`).
|
|
|
|
`loggamma` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, c, size=None, random_state=None):
|
|
return np.log(random_state.gamma(c, size=size))
|
|
|
|
def _pdf(self, x, c):
|
|
# loggamma.pdf(x, c) = exp(c*x-exp(x)) / gamma(c)
|
|
return np.exp(c*x-np.exp(x)-sc.gammaln(c))
|
|
|
|
def _cdf(self, x, c):
|
|
return sc.gammainc(c, np.exp(x))
|
|
|
|
def _ppf(self, q, c):
|
|
return np.log(sc.gammaincinv(c, q))
|
|
|
|
def _stats(self, c):
|
|
# See, for example, "A Statistical Study of Log-Gamma Distribution", by
|
|
# Ping Shing Chan (thesis, McMaster University, 1993).
|
|
mean = sc.digamma(c)
|
|
var = sc.polygamma(1, c)
|
|
skewness = sc.polygamma(2, c) / np.power(var, 1.5)
|
|
excess_kurtosis = sc.polygamma(3, c) / (var*var)
|
|
return mean, var, skewness, excess_kurtosis
|
|
|
|
|
|
loggamma = loggamma_gen(name='loggamma')
|
|
|
|
|
|
class loglaplace_gen(rv_continuous):
|
|
r"""A log-Laplace continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `loglaplace` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1} &\text{for } 0 < x < 1\\
|
|
\frac{c}{2} x^{-c-1} &\text{for } x \ge 1
|
|
\end{cases}
|
|
|
|
for :math:`c > 0`.
|
|
|
|
`loglaplace` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model",
|
|
The Mathematical Scientist, vol. 28, pp. 49-60, 2003.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, c):
|
|
# loglaplace.pdf(x, c) = c / 2 * x**(c-1), for 0 < x < 1
|
|
# = c / 2 * x**(-c-1), for x >= 1
|
|
cd2 = c/2.0
|
|
c = np.where(x < 1, c, -c)
|
|
return cd2*x**(c-1)
|
|
|
|
def _cdf(self, x, c):
|
|
return np.where(x < 1, 0.5*x**c, 1-0.5*x**(-c))
|
|
|
|
def _ppf(self, q, c):
|
|
return np.where(q < 0.5, (2.0*q)**(1.0/c), (2*(1.0-q))**(-1.0/c))
|
|
|
|
def _munp(self, n, c):
|
|
return c**2 / (c**2 - n**2)
|
|
|
|
def _entropy(self, c):
|
|
return np.log(2.0/c) + 1.0
|
|
|
|
|
|
loglaplace = loglaplace_gen(a=0.0, name='loglaplace')
|
|
|
|
|
|
def _lognorm_logpdf(x, s):
|
|
return _lazywhere(x != 0, (x, s),
|
|
lambda x, s: -np.log(x)**2 / (2*s**2) - np.log(s*x*np.sqrt(2*np.pi)),
|
|
-np.inf)
|
|
|
|
|
|
class lognorm_gen(rv_continuous):
|
|
r"""A lognormal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `lognorm` is:
|
|
|
|
.. math::
|
|
|
|
f(x, s) = \frac{1}{s x \sqrt{2\pi}}
|
|
\exp\left(-\frac{\log^2(x)}{2s^2}\right)
|
|
|
|
for :math:`x > 0`, :math:`s > 0`.
|
|
|
|
`lognorm` takes ``s`` as a shape parameter for :math:`s`.
|
|
|
|
%(after_notes)s
|
|
|
|
A common parametrization for a lognormal random variable ``Y`` is in
|
|
terms of the mean, ``mu``, and standard deviation, ``sigma``, of the
|
|
unique normally distributed random variable ``X`` such that exp(X) = Y.
|
|
This parametrization corresponds to setting ``s = sigma`` and ``scale =
|
|
exp(mu)``.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _rvs(self, s, size=None, random_state=None):
|
|
return np.exp(s * random_state.standard_normal(size))
|
|
|
|
def _pdf(self, x, s):
|
|
# lognorm.pdf(x, s) = 1 / (s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2)
|
|
return np.exp(self._logpdf(x, s))
|
|
|
|
def _logpdf(self, x, s):
|
|
return _lognorm_logpdf(x, s)
|
|
|
|
def _cdf(self, x, s):
|
|
return _norm_cdf(np.log(x) / s)
|
|
|
|
def _logcdf(self, x, s):
|
|
return _norm_logcdf(np.log(x) / s)
|
|
|
|
def _ppf(self, q, s):
|
|
return np.exp(s * _norm_ppf(q))
|
|
|
|
def _sf(self, x, s):
|
|
return _norm_sf(np.log(x) / s)
|
|
|
|
def _logsf(self, x, s):
|
|
return _norm_logsf(np.log(x) / s)
|
|
|
|
def _stats(self, s):
|
|
p = np.exp(s*s)
|
|
mu = np.sqrt(p)
|
|
mu2 = p*(p-1)
|
|
g1 = np.sqrt((p-1))*(2+p)
|
|
g2 = np.polyval([1, 2, 3, 0, -6.0], p)
|
|
return mu, mu2, g1, g2
|
|
|
|
def _entropy(self, s):
|
|
return 0.5 * (1 + np.log(2*np.pi) + 2 * np.log(s))
|
|
|
|
@extend_notes_in_docstring(rv_continuous, notes="""\
|
|
When the location parameter is fixed by using the `floc` argument,
|
|
this function uses explicit formulas for the maximum likelihood
|
|
estimation of the log-normal shape and scale parameters, so the
|
|
`optimizer`, `loc` and `scale` keyword arguments are ignored.\n\n""")
|
|
def fit(self, data, *args, **kwds):
|
|
floc = kwds.get('floc', None)
|
|
if floc is None:
|
|
# loc is not fixed. Use the default fit method.
|
|
return super(lognorm_gen, self).fit(data, *args, **kwds)
|
|
|
|
f0 = (kwds.get('f0', None) or kwds.get('fs', None) or
|
|
kwds.get('fix_s', None))
|
|
fscale = kwds.get('fscale', None)
|
|
|
|
if len(args) > 1:
|
|
raise TypeError("Too many input arguments.")
|
|
for name in ['f0', 'fs', 'fix_s', 'floc', 'fscale', 'loc', 'scale',
|
|
'optimizer']:
|
|
kwds.pop(name, None)
|
|
if kwds:
|
|
raise TypeError("Unknown arguments: %s." % kwds)
|
|
|
|
# Special case: loc is fixed. Use the maximum likelihood formulas
|
|
# instead of the numerical solver.
|
|
|
|
if f0 is not None and fscale is not None:
|
|
# This check is for consistency with `rv_continuous.fit`.
|
|
raise ValueError("All parameters fixed. There is nothing to "
|
|
"optimize.")
|
|
|
|
data = np.asarray(data)
|
|
|
|
if not np.isfinite(data).all():
|
|
raise RuntimeError("The data contains non-finite values.")
|
|
|
|
floc = float(floc)
|
|
if floc != 0:
|
|
# Shifting the data by floc. Don't do the subtraction in-place,
|
|
# because `data` might be a view of the input array.
|
|
data = data - floc
|
|
if np.any(data <= 0):
|
|
raise FitDataError("lognorm", lower=floc, upper=np.inf)
|
|
lndata = np.log(data)
|
|
|
|
# Three cases to handle:
|
|
# * shape and scale both free
|
|
# * shape fixed, scale free
|
|
# * shape free, scale fixed
|
|
|
|
if fscale is None:
|
|
# scale is free.
|
|
scale = np.exp(lndata.mean())
|
|
if f0 is None:
|
|
# shape is free.
|
|
shape = lndata.std()
|
|
else:
|
|
# shape is fixed.
|
|
shape = float(f0)
|
|
else:
|
|
# scale is fixed, shape is free
|
|
scale = float(fscale)
|
|
shape = np.sqrt(((lndata - np.log(scale))**2).mean())
|
|
|
|
return shape, floc, scale
|
|
|
|
|
|
lognorm = lognorm_gen(a=0.0, name='lognorm')
|
|
|
|
|
|
class gilbrat_gen(rv_continuous):
|
|
r"""A Gilbrat continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `gilbrat` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2)
|
|
|
|
`gilbrat` is a special case of `lognorm` with ``s=1``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _rvs(self, size=None, random_state=None):
|
|
return np.exp(random_state.standard_normal(size))
|
|
|
|
def _pdf(self, x):
|
|
# gilbrat.pdf(x) = 1/(x*sqrt(2*pi)) * exp(-1/2*(log(x))**2)
|
|
return np.exp(self._logpdf(x))
|
|
|
|
def _logpdf(self, x):
|
|
return _lognorm_logpdf(x, 1.0)
|
|
|
|
def _cdf(self, x):
|
|
return _norm_cdf(np.log(x))
|
|
|
|
def _ppf(self, q):
|
|
return np.exp(_norm_ppf(q))
|
|
|
|
def _stats(self):
|
|
p = np.e
|
|
mu = np.sqrt(p)
|
|
mu2 = p * (p - 1)
|
|
g1 = np.sqrt((p - 1)) * (2 + p)
|
|
g2 = np.polyval([1, 2, 3, 0, -6.0], p)
|
|
return mu, mu2, g1, g2
|
|
|
|
def _entropy(self):
|
|
return 0.5 * np.log(2 * np.pi) + 0.5
|
|
|
|
|
|
gilbrat = gilbrat_gen(a=0.0, name='gilbrat')
|
|
|
|
|
|
class maxwell_gen(rv_continuous):
|
|
r"""A Maxwell continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
A special case of a `chi` distribution, with ``df=3``, ``loc=0.0``,
|
|
and given ``scale = a``, where ``a`` is the parameter used in the
|
|
Mathworld description [1]_.
|
|
|
|
The probability density function for `maxwell` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2)
|
|
|
|
for :math:`x >= 0`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] http://mathworld.wolfram.com/MaxwellDistribution.html
|
|
|
|
%(example)s
|
|
"""
|
|
def _rvs(self, size=None, random_state=None):
|
|
return chi.rvs(3.0, size=size, random_state=random_state)
|
|
|
|
def _pdf(self, x):
|
|
# maxwell.pdf(x) = sqrt(2/pi)x**2 * exp(-x**2/2)
|
|
return _SQRT_2_OVER_PI*x*x*np.exp(-x*x/2.0)
|
|
|
|
def _logpdf(self, x):
|
|
return _LOG_SQRT_2_OVER_PI + 2*np.log(x) - 0.5*x*x
|
|
|
|
def _cdf(self, x):
|
|
return sc.gammainc(1.5, x*x/2.0)
|
|
|
|
def _ppf(self, q):
|
|
return np.sqrt(2*sc.gammaincinv(1.5, q))
|
|
|
|
def _stats(self):
|
|
val = 3*np.pi-8
|
|
return (2*np.sqrt(2.0/np.pi),
|
|
3-8/np.pi,
|
|
np.sqrt(2)*(32-10*np.pi)/val**1.5,
|
|
(-12*np.pi*np.pi + 160*np.pi - 384) / val**2.0)
|
|
|
|
def _entropy(self):
|
|
return _EULER + 0.5*np.log(2*np.pi)-0.5
|
|
|
|
|
|
maxwell = maxwell_gen(a=0.0, name='maxwell')
|
|
|
|
|
|
class mielke_gen(rv_continuous):
|
|
r"""A Mielke Beta-Kappa / Dagum continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `mielke` is:
|
|
|
|
.. math::
|
|
|
|
f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}
|
|
|
|
for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes
|
|
called Dagum distribution ([2]_). It was already defined in [3]_, called
|
|
a Burr Type III distribution (`burr` with parameters ``c=s`` and
|
|
``d=k/s``).
|
|
|
|
`mielke` takes ``k`` and ``s`` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] Mielke, P.W., 1973 "Another Family of Distributions for Describing
|
|
and Analyzing Precipitation Data." J. Appl. Meteor., 12, 275-280
|
|
.. [2] Dagum, C., 1977 "A new model for personal income distribution."
|
|
Economie Appliquee, 33, 327-367.
|
|
.. [3] Burr, I. W. "Cumulative frequency functions", Annals of
|
|
Mathematical Statistics, 13(2), pp 215-232 (1942).
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, k, s):
|
|
return (k > 0) & (s > 0)
|
|
|
|
def _pdf(self, x, k, s):
|
|
return k*x**(k-1.0) / (1.0+x**s)**(1.0+k*1.0/s)
|
|
|
|
def _logpdf(self, x, k, s):
|
|
return np.log(k) + np.log(x)*(k-1.0) - np.log1p(x**s)*(1.0+k*1.0/s)
|
|
|
|
def _cdf(self, x, k, s):
|
|
return x**k / (1.0+x**s)**(k*1.0/s)
|
|
|
|
def _ppf(self, q, k, s):
|
|
qsk = pow(q, s*1.0/k)
|
|
return pow(qsk/(1.0-qsk), 1.0/s)
|
|
|
|
def _munp(self, n, k, s):
|
|
def nth_moment(n, k, s):
|
|
# n-th moment is defined for -k < n < s
|
|
return sc.gamma((k+n)/s)*sc.gamma(1-n/s)/sc.gamma(k/s)
|
|
|
|
return _lazywhere(n < s, (n, k, s), nth_moment, np.inf)
|
|
|
|
|
|
mielke = mielke_gen(a=0.0, name='mielke')
|
|
|
|
|
|
class kappa4_gen(rv_continuous):
|
|
r"""Kappa 4 parameter distribution.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for kappa4 is:
|
|
|
|
.. math::
|
|
|
|
f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}
|
|
|
|
if :math:`h` and :math:`k` are not equal to 0.
|
|
|
|
If :math:`h` or :math:`k` are zero then the pdf can be simplified:
|
|
|
|
h = 0 and k != 0::
|
|
|
|
kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
|
|
exp(-(1.0 - k*x)**(1.0/k))
|
|
|
|
h != 0 and k = 0::
|
|
|
|
kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)
|
|
|
|
h = 0 and k = 0::
|
|
|
|
kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))
|
|
|
|
kappa4 takes :math:`h` and :math:`k` as shape parameters.
|
|
|
|
The kappa4 distribution returns other distributions when certain
|
|
:math:`h` and :math:`k` values are used.
|
|
|
|
+------+-------------+----------------+------------------+
|
|
| h | k=0.0 | k=1.0 | -inf<=k<=inf |
|
|
+======+=============+================+==================+
|
|
| -1.0 | Logistic | | Generalized |
|
|
| | | | Logistic(1) |
|
|
| | | | |
|
|
| | logistic(x) | | |
|
|
+------+-------------+----------------+------------------+
|
|
| 0.0 | Gumbel | Reverse | Generalized |
|
|
| | | Exponential(2) | Extreme Value |
|
|
| | | | |
|
|
| | gumbel_r(x) | | genextreme(x, k) |
|
|
+------+-------------+----------------+------------------+
|
|
| 1.0 | Exponential | Uniform | Generalized |
|
|
| | | | Pareto |
|
|
| | | | |
|
|
| | expon(x) | uniform(x) | genpareto(x, -k) |
|
|
+------+-------------+----------------+------------------+
|
|
|
|
(1) There are at least five generalized logistic distributions.
|
|
Four are described here:
|
|
https://en.wikipedia.org/wiki/Generalized_logistic_distribution
|
|
The "fifth" one is the one kappa4 should match which currently
|
|
isn't implemented in scipy:
|
|
https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution
|
|
https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html
|
|
(2) This distribution is currently not in scipy.
|
|
|
|
References
|
|
----------
|
|
J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect
|
|
to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate
|
|
Faculty of the Louisiana State University and Agricultural and Mechanical
|
|
College, (August, 2004),
|
|
https://digitalcommons.lsu.edu/gradschool_dissertations/3672
|
|
|
|
J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res.
|
|
Develop. 38 (3), 25 1-258 (1994).
|
|
|
|
B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao
|
|
Site in the Chi River Basin, Thailand", Journal of Water Resource and
|
|
Protection, vol. 4, 866-869, (2012).
|
|
:doi:`10.4236/jwarp.2012.410101`
|
|
|
|
C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A
|
|
Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March
|
|
2000).
|
|
http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, h, k):
|
|
return h == h
|
|
|
|
def _get_support(self, h, k):
|
|
condlist = [np.logical_and(h > 0, k > 0),
|
|
np.logical_and(h > 0, k == 0),
|
|
np.logical_and(h > 0, k < 0),
|
|
np.logical_and(h <= 0, k > 0),
|
|
np.logical_and(h <= 0, k == 0),
|
|
np.logical_and(h <= 0, k < 0)]
|
|
|
|
def f0(h, k):
|
|
return (1.0 - float_power(h, -k))/k
|
|
|
|
def f1(h, k):
|
|
return np.log(h)
|
|
|
|
def f3(h, k):
|
|
a = np.empty(np.shape(h))
|
|
a[:] = -np.inf
|
|
return a
|
|
|
|
def f5(h, k):
|
|
return 1.0/k
|
|
|
|
_a = _lazyselect(condlist,
|
|
[f0, f1, f0, f3, f3, f5],
|
|
[h, k],
|
|
default=np.nan)
|
|
|
|
def f0(h, k):
|
|
return 1.0/k
|
|
|
|
def f1(h, k):
|
|
a = np.empty(np.shape(h))
|
|
a[:] = np.inf
|
|
return a
|
|
|
|
_b = _lazyselect(condlist,
|
|
[f0, f1, f1, f0, f1, f1],
|
|
[h, k],
|
|
default=np.nan)
|
|
return _a, _b
|
|
|
|
def _pdf(self, x, h, k):
|
|
# kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
|
|
# (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1)
|
|
return np.exp(self._logpdf(x, h, k))
|
|
|
|
def _logpdf(self, x, h, k):
|
|
condlist = [np.logical_and(h != 0, k != 0),
|
|
np.logical_and(h == 0, k != 0),
|
|
np.logical_and(h != 0, k == 0),
|
|
np.logical_and(h == 0, k == 0)]
|
|
|
|
def f0(x, h, k):
|
|
'''pdf = (1.0 - k*x)**(1.0/k - 1.0)*(
|
|
1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0)
|
|
logpdf = ...
|
|
'''
|
|
return (sc.xlog1py(1.0/k - 1.0, -k*x) +
|
|
sc.xlog1py(1.0/h - 1.0, -h*(1.0 - k*x)**(1.0/k)))
|
|
|
|
def f1(x, h, k):
|
|
'''pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-(
|
|
1.0 - k*x)**(1.0/k))
|
|
logpdf = ...
|
|
'''
|
|
return sc.xlog1py(1.0/k - 1.0, -k*x) - (1.0 - k*x)**(1.0/k)
|
|
|
|
def f2(x, h, k):
|
|
'''pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0)
|
|
logpdf = ...
|
|
'''
|
|
return -x + sc.xlog1py(1.0/h - 1.0, -h*np.exp(-x))
|
|
|
|
def f3(x, h, k):
|
|
'''pdf = np.exp(-x-np.exp(-x))
|
|
logpdf = ...
|
|
'''
|
|
return -x - np.exp(-x)
|
|
|
|
return _lazyselect(condlist,
|
|
[f0, f1, f2, f3],
|
|
[x, h, k],
|
|
default=np.nan)
|
|
|
|
def _cdf(self, x, h, k):
|
|
return np.exp(self._logcdf(x, h, k))
|
|
|
|
def _logcdf(self, x, h, k):
|
|
condlist = [np.logical_and(h != 0, k != 0),
|
|
np.logical_and(h == 0, k != 0),
|
|
np.logical_and(h != 0, k == 0),
|
|
np.logical_and(h == 0, k == 0)]
|
|
|
|
def f0(x, h, k):
|
|
'''cdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h)
|
|
logcdf = ...
|
|
'''
|
|
return (1.0/h)*sc.log1p(-h*(1.0 - k*x)**(1.0/k))
|
|
|
|
def f1(x, h, k):
|
|
'''cdf = np.exp(-(1.0 - k*x)**(1.0/k))
|
|
logcdf = ...
|
|
'''
|
|
return -(1.0 - k*x)**(1.0/k)
|
|
|
|
def f2(x, h, k):
|
|
'''cdf = (1.0 - h*np.exp(-x))**(1.0/h)
|
|
logcdf = ...
|
|
'''
|
|
return (1.0/h)*sc.log1p(-h*np.exp(-x))
|
|
|
|
def f3(x, h, k):
|
|
'''cdf = np.exp(-np.exp(-x))
|
|
logcdf = ...
|
|
'''
|
|
return -np.exp(-x)
|
|
|
|
return _lazyselect(condlist,
|
|
[f0, f1, f2, f3],
|
|
[x, h, k],
|
|
default=np.nan)
|
|
|
|
def _ppf(self, q, h, k):
|
|
condlist = [np.logical_and(h != 0, k != 0),
|
|
np.logical_and(h == 0, k != 0),
|
|
np.logical_and(h != 0, k == 0),
|
|
np.logical_and(h == 0, k == 0)]
|
|
|
|
def f0(q, h, k):
|
|
return 1.0/k*(1.0 - ((1.0 - (q**h))/h)**k)
|
|
|
|
def f1(q, h, k):
|
|
return 1.0/k*(1.0 - (-np.log(q))**k)
|
|
|
|
def f2(q, h, k):
|
|
'''ppf = -np.log((1.0 - (q**h))/h)
|
|
'''
|
|
return -sc.log1p(-(q**h)) + np.log(h)
|
|
|
|
def f3(q, h, k):
|
|
return -np.log(-np.log(q))
|
|
|
|
return _lazyselect(condlist,
|
|
[f0, f1, f2, f3],
|
|
[q, h, k],
|
|
default=np.nan)
|
|
|
|
def _stats(self, h, k):
|
|
if h >= 0 and k >= 0:
|
|
maxr = 5
|
|
elif h < 0 and k >= 0:
|
|
maxr = int(-1.0/h*k)
|
|
elif k < 0:
|
|
maxr = int(-1.0/k)
|
|
else:
|
|
maxr = 5
|
|
|
|
outputs = [None if r < maxr else np.nan for r in range(1, 5)]
|
|
return outputs[:]
|
|
|
|
|
|
kappa4 = kappa4_gen(name='kappa4')
|
|
|
|
|
|
class kappa3_gen(rv_continuous):
|
|
r"""Kappa 3 parameter distribution.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `kappa3` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a) = a (a + x^a)^{-(a + 1)/a}
|
|
|
|
for :math:`x > 0` and :math:`a > 0`.
|
|
|
|
`kappa3` takes ``a`` as a shape parameter for :math:`a`.
|
|
|
|
References
|
|
----------
|
|
P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum
|
|
Likelihood and Likelihood Ratio Tests", Methods in Weather Research,
|
|
701-707, (September, 1973),
|
|
:doi:`10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2`
|
|
|
|
B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the
|
|
Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2,
|
|
415-419 (2012), :doi:`10.4236/ojs.2012.24050`
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, a):
|
|
return a > 0
|
|
|
|
def _pdf(self, x, a):
|
|
# kappa3.pdf(x, a) = a*(a + x**a)**(-(a + 1)/a), for x > 0
|
|
return a*(a + x**a)**(-1.0/a-1)
|
|
|
|
def _cdf(self, x, a):
|
|
return x*(a + x**a)**(-1.0/a)
|
|
|
|
def _ppf(self, q, a):
|
|
return (a/(q**-a - 1.0))**(1.0/a)
|
|
|
|
def _stats(self, a):
|
|
outputs = [None if i < a else np.nan for i in range(1, 5)]
|
|
return outputs[:]
|
|
|
|
|
|
kappa3 = kappa3_gen(a=0.0, name='kappa3')
|
|
|
|
class moyal_gen(rv_continuous):
|
|
r"""A Moyal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `moyal` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}
|
|
|
|
for a real number :math:`x`.
|
|
|
|
%(after_notes)s
|
|
|
|
This distribution has utility in high-energy physics and radiation
|
|
detection. It describes the energy loss of a charged relativistic
|
|
particle due to ionization of the medium [1]_. It also provides an
|
|
approximation for the Landau distribution. For an in depth description
|
|
see [2]_. For additional description, see [3]_.
|
|
|
|
References
|
|
----------
|
|
.. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations",
|
|
The London, Edinburgh, and Dublin Philosophical Magazine
|
|
and Journal of Science, vol 46, 263-280, (1955).
|
|
:doi:`10.1080/14786440308521076` (gated)
|
|
.. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution",
|
|
International Journal of Research and Reviews in Applied Sciences,
|
|
vol 10, 171-192, (2012).
|
|
http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf
|
|
.. [3] C. Walck, "Handbook on Statistical Distributions for
|
|
Experimentalists; International Report SUF-PFY/96-01", Chapter 26,
|
|
University of Stockholm: Stockholm, Sweden, (2007).
|
|
http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf
|
|
|
|
.. versionadded:: 1.1.0
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, size=None, random_state=None):
|
|
u1 = gamma.rvs(a = 0.5, scale = 2, size=size, random_state=random_state)
|
|
return -np.log(u1)
|
|
|
|
def _pdf(self, x):
|
|
return np.exp(-0.5 * (x + np.exp(-x))) / np.sqrt(2*np.pi)
|
|
|
|
def _cdf(self, x):
|
|
return sc.erfc(np.exp(-0.5 * x) / np.sqrt(2))
|
|
|
|
def _sf(self, x):
|
|
return sc.erf(np.exp(-0.5 * x) / np.sqrt(2))
|
|
|
|
def _ppf(self, x):
|
|
return -np.log(2 * sc.erfcinv(x)**2)
|
|
|
|
def _stats(self):
|
|
mu = np.log(2) + np.euler_gamma
|
|
mu2 = np.pi**2 / 2
|
|
g1 = 28 * np.sqrt(2) * sc.zeta(3) / np.pi**3
|
|
g2 = 4.
|
|
return mu, mu2, g1, g2
|
|
|
|
def _munp(self, n):
|
|
if n == 1.0:
|
|
return np.log(2) + np.euler_gamma
|
|
elif n == 2.0:
|
|
return np.pi**2 / 2 + (np.log(2) + np.euler_gamma)**2
|
|
elif n == 3.0:
|
|
tmp1 = 1.5 * np.pi**2 * (np.log(2)+np.euler_gamma)
|
|
tmp2 = (np.log(2)+np.euler_gamma)**3
|
|
tmp3 = 14 * sc.zeta(3)
|
|
return tmp1 + tmp2 + tmp3
|
|
elif n == 4.0:
|
|
tmp1 = 4 * 14 * sc.zeta(3) * (np.log(2) + np.euler_gamma)
|
|
tmp2 = 3 * np.pi**2 * (np.log(2) + np.euler_gamma)**2
|
|
tmp3 = (np.log(2) + np.euler_gamma)**4
|
|
tmp4 = 7 * np.pi**4 / 4
|
|
return tmp1 + tmp2 + tmp3 + tmp4
|
|
else:
|
|
# return generic for higher moments
|
|
# return rv_continuous._mom1_sc(self, n, b)
|
|
return self._mom1_sc(n)
|
|
|
|
|
|
moyal = moyal_gen(name="moyal")
|
|
|
|
|
|
class nakagami_gen(rv_continuous):
|
|
r"""A Nakagami continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `nakagami` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2)
|
|
|
|
for :math:`x >= 0`, :math:`\nu > 0`.
|
|
|
|
`nakagami` takes ``nu`` as a shape parameter for :math:`\nu`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, nu):
|
|
return np.exp(self._logpdf(x, nu))
|
|
|
|
def _logpdf(self, x, nu):
|
|
# nakagami.pdf(x, nu) = 2 * nu**nu / gamma(nu) *
|
|
# x**(2*nu-1) * exp(-nu*x**2)
|
|
return (np.log(2) + sc.xlogy(nu, nu) - sc.gammaln(nu) +
|
|
sc.xlogy(2*nu - 1, x) - nu*x**2)
|
|
|
|
def _cdf(self, x, nu):
|
|
return sc.gammainc(nu, nu*x*x)
|
|
|
|
def _ppf(self, q, nu):
|
|
return np.sqrt(1.0/nu*sc.gammaincinv(nu, q))
|
|
|
|
def _sf(self, x, nu):
|
|
return sc.gammaincc(nu, nu*x*x)
|
|
|
|
def _isf(self, p, nu):
|
|
return np.sqrt(1/nu * sc.gammainccinv(nu, p))
|
|
|
|
def _stats(self, nu):
|
|
mu = sc.gamma(nu+0.5)/sc.gamma(nu)/np.sqrt(nu)
|
|
mu2 = 1.0-mu*mu
|
|
g1 = mu * (1 - 4*nu*mu2) / 2.0 / nu / np.power(mu2, 1.5)
|
|
g2 = -6*mu**4*nu + (8*nu-2)*mu**2-2*nu + 1
|
|
g2 /= nu*mu2**2.0
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
nakagami = nakagami_gen(a=0.0, name="nakagami")
|
|
|
|
|
|
class ncx2_gen(rv_continuous):
|
|
r"""A non-central chi-squared continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `ncx2` is:
|
|
|
|
.. math::
|
|
|
|
f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2)
|
|
(x/\lambda)^{(k-2)/4} I_{(k-2)/2}(\sqrt{\lambda x})
|
|
|
|
for :math:`x >= 0` and :math:`k, \lambda > 0`. :math:`k` specifies the
|
|
degrees of freedom (denoted ``df`` in the implementation) and
|
|
:math:`\lambda` is the non-centrality parameter (denoted ``nc`` in the
|
|
implementation). :math:`I_\nu` denotes the modified Bessel function of
|
|
first order of degree :math:`\nu` (`scipy.special.iv`).
|
|
|
|
`ncx2` takes ``df`` and ``nc`` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, df, nc):
|
|
return (df > 0) & (nc >= 0)
|
|
|
|
def _rvs(self, df, nc, size=None, random_state=None):
|
|
return random_state.noncentral_chisquare(df, nc, size)
|
|
|
|
def _logpdf(self, x, df, nc):
|
|
cond = np.ones_like(x, dtype=bool) & (nc != 0)
|
|
return _lazywhere(cond, (x, df, nc), f=_ncx2_log_pdf, f2=chi2.logpdf)
|
|
|
|
def _pdf(self, x, df, nc):
|
|
# ncx2.pdf(x, df, nc) = exp(-(nc+x)/2) * 1/2 * (x/nc)**((df-2)/4)
|
|
# * I[(df-2)/2](sqrt(nc*x))
|
|
cond = np.ones_like(x, dtype=bool) & (nc != 0)
|
|
return _lazywhere(cond, (x, df, nc), f=_ncx2_pdf, f2=chi2.pdf)
|
|
|
|
def _cdf(self, x, df, nc):
|
|
cond = np.ones_like(x, dtype=bool) & (nc != 0)
|
|
return _lazywhere(cond, (x, df, nc), f=_ncx2_cdf, f2=chi2.cdf)
|
|
|
|
def _ppf(self, q, df, nc):
|
|
cond = np.ones_like(q, dtype=bool) & (nc != 0)
|
|
return _lazywhere(cond, (q, df, nc), f=sc.chndtrix, f2=chi2.ppf)
|
|
|
|
def _stats(self, df, nc):
|
|
val = df + 2.0*nc
|
|
return (df + nc,
|
|
2*val,
|
|
np.sqrt(8)*(val+nc)/val**1.5,
|
|
12.0*(val+2*nc)/val**2.0)
|
|
|
|
|
|
ncx2 = ncx2_gen(a=0.0, name='ncx2')
|
|
|
|
|
|
class ncf_gen(rv_continuous):
|
|
r"""A non-central F distribution continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
scipy.stats.f : Fisher distribution
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `ncf` is:
|
|
|
|
.. math::
|
|
|
|
f(x, n_1, n_2, \lambda) =
|
|
\exp\left(\frac{\lambda}{2} +
|
|
\lambda n_1 \frac{x}{2(n_1 x + n_2)}
|
|
\right)
|
|
n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\
|
|
(n_2 + n_1 x)^{-(n_1 + n_2)/2}
|
|
\gamma(n_1/2) \gamma(1 + n_2/2) \\
|
|
\frac{L^{\frac{n_1}{2}-1}_{n_2/2}
|
|
\left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)}
|
|
{B(n_1/2, n_2/2)
|
|
\gamma\left(\frac{n_1 + n_2}{2}\right)}
|
|
|
|
for :math:`n_1, n_2 > 0`, :math:`\lambda\geq 0`. Here :math:`n_1` is the
|
|
degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in
|
|
the denominator, :math:`\lambda` the non-centrality parameter,
|
|
:math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a
|
|
generalized Laguerre polynomial and :math:`B` is the beta function.
|
|
|
|
`ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters. If ``nc=0``,
|
|
the distribution becomes equivalent to the Fisher distribution.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, df1, df2, nc):
|
|
return (df1 > 0) & (df2 > 0) & (nc >= 0)
|
|
|
|
def _rvs(self, dfn, dfd, nc, size=None, random_state=None):
|
|
return random_state.noncentral_f(dfn, dfd, nc, size)
|
|
|
|
def _pdf_skip(self, x, dfn, dfd, nc):
|
|
# ncf.pdf(x, df1, df2, nc) = exp(nc/2 + nc*df1*x/(2*(df1*x+df2))) *
|
|
# df1**(df1/2) * df2**(df2/2) * x**(df1/2-1) *
|
|
# (df2+df1*x)**(-(df1+df2)/2) *
|
|
# gamma(df1/2)*gamma(1+df2/2) *
|
|
# L^{v1/2-1}^{v2/2}(-nc*v1*x/(2*(v1*x+v2))) /
|
|
# (B(v1/2, v2/2) * gamma((v1+v2)/2))
|
|
n1, n2 = dfn, dfd
|
|
term = -nc/2+nc*n1*x/(2*(n2+n1*x)) + sc.gammaln(n1/2.)+sc.gammaln(1+n2/2.)
|
|
term -= sc.gammaln((n1+n2)/2.0)
|
|
Px = np.exp(term)
|
|
Px *= n1**(n1/2) * n2**(n2/2) * x**(n1/2-1)
|
|
Px *= (n2+n1*x)**(-(n1+n2)/2)
|
|
Px *= sc.assoc_laguerre(-nc*n1*x/(2.0*(n2+n1*x)), n2/2, n1/2-1)
|
|
Px /= sc.beta(n1/2, n2/2)
|
|
# This function does not have a return. Drop it for now, the generic
|
|
# function seems to work OK.
|
|
|
|
def _cdf(self, x, dfn, dfd, nc):
|
|
return sc.ncfdtr(dfn, dfd, nc, x)
|
|
|
|
def _ppf(self, q, dfn, dfd, nc):
|
|
return sc.ncfdtri(dfn, dfd, nc, q)
|
|
|
|
def _munp(self, n, dfn, dfd, nc):
|
|
val = (dfn * 1.0/dfd)**n
|
|
term = sc.gammaln(n+0.5*dfn) + sc.gammaln(0.5*dfd-n) - sc.gammaln(dfd*0.5)
|
|
val *= np.exp(-nc / 2.0+term)
|
|
val *= sc.hyp1f1(n+0.5*dfn, 0.5*dfn, 0.5*nc)
|
|
return val
|
|
|
|
def _stats(self, dfn, dfd, nc):
|
|
# Note: the rv_continuous class ensures that dfn > 0 when this function
|
|
# is called, so we don't have to check for division by zero with dfn
|
|
# in the following.
|
|
mu_num = dfd * (dfn + nc)
|
|
mu_den = dfn * (dfd - 2)
|
|
mu = np.full_like(mu_num, dtype=np.float64, fill_value=np.inf)
|
|
np.true_divide(mu_num, mu_den, where=dfd > 2, out=mu)
|
|
|
|
mu2_num = 2*((dfn + nc)**2 + (dfn + 2*nc)*(dfd - 2))*(dfd/dfn)**2
|
|
mu2_den = (dfd - 2)**2 * (dfd - 4)
|
|
mu2 = np.full_like(mu2_num, dtype=np.float64, fill_value=np.inf)
|
|
np.true_divide(mu2_num, mu2_den, where=dfd > 4, out=mu2)
|
|
|
|
return mu, mu2, None, None
|
|
|
|
|
|
ncf = ncf_gen(a=0.0, name='ncf')
|
|
|
|
|
|
class t_gen(rv_continuous):
|
|
r"""A Student's t continuous random variable.
|
|
|
|
For the noncentral t distribution, see `nct`.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
nct
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `t` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \nu) = \frac{\Gamma((\nu+1)/2)}
|
|
{\sqrt{\pi \nu} \Gamma(\nu/2)}
|
|
(1+x^2/\nu)^{-(\nu+1)/2}
|
|
|
|
where :math:`x` is a real number and the degrees of freedom parameter
|
|
:math:`\nu` (denoted ``df`` in the implementation) satisfies
|
|
:math:`\nu > 0`. :math:`\Gamma` is the gamma function
|
|
(`scipy.special.gamma`).
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, df):
|
|
return df > 0
|
|
|
|
def _rvs(self, df, size=None, random_state=None):
|
|
return random_state.standard_t(df, size=size)
|
|
|
|
def _pdf(self, x, df):
|
|
# gamma((df+1)/2)
|
|
# t.pdf(x, df) = ---------------------------------------------------
|
|
# sqrt(pi*df) * gamma(df/2) * (1+x**2/df)**((df+1)/2)
|
|
r = np.asarray(df*1.0)
|
|
Px = np.exp(sc.gammaln((r+1)/2)-sc.gammaln(r/2))
|
|
Px /= np.sqrt(r*np.pi)*(1+(x**2)/r)**((r+1)/2)
|
|
return Px
|
|
|
|
def _logpdf(self, x, df):
|
|
r = df*1.0
|
|
lPx = sc.gammaln((r+1)/2)-sc.gammaln(r/2)
|
|
lPx -= 0.5*np.log(r*np.pi) + (r+1)/2*np.log(1+(x**2)/r)
|
|
return lPx
|
|
|
|
def _cdf(self, x, df):
|
|
return sc.stdtr(df, x)
|
|
|
|
def _sf(self, x, df):
|
|
return sc.stdtr(df, -x)
|
|
|
|
def _ppf(self, q, df):
|
|
return sc.stdtrit(df, q)
|
|
|
|
def _isf(self, q, df):
|
|
return -sc.stdtrit(df, q)
|
|
|
|
def _stats(self, df):
|
|
mu = np.where(df > 1, 0.0, np.inf)
|
|
mu2 = _lazywhere(df > 2, (df,),
|
|
lambda df: df / (df-2.0),
|
|
np.inf)
|
|
mu2 = np.where(df <= 1, np.nan, mu2)
|
|
g1 = np.where(df > 3, 0.0, np.nan)
|
|
g2 = _lazywhere(df > 4, (df,),
|
|
lambda df: 6.0 / (df-4.0),
|
|
np.inf)
|
|
g2 = np.where(df <= 2, np.nan, g2)
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
t = t_gen(name='t')
|
|
|
|
|
|
class nct_gen(rv_continuous):
|
|
r"""A non-central Student's t continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
If :math:`Y` is a standard normal random variable and :math:`V` is
|
|
an independent chi-square random variable (`chi2`) with :math:`k` degrees
|
|
of freedom, then
|
|
|
|
.. math::
|
|
|
|
X = \frac{Y + c}{\sqrt{V/k}}
|
|
|
|
has a non-central Student's t distribution on the real line.
|
|
The degrees of freedom parameter :math:`k` (denoted ``df`` in the
|
|
implementation) satisfies :math:`k > 0` and the noncentrality parameter
|
|
:math:`c` (denoted ``nc`` in the implementation) is a real number.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, df, nc):
|
|
return (df > 0) & (nc == nc)
|
|
|
|
def _rvs(self, df, nc, size=None, random_state=None):
|
|
n = norm.rvs(loc=nc, size=size, random_state=random_state)
|
|
c2 = chi2.rvs(df, size=size, random_state=random_state)
|
|
return n * np.sqrt(df) / np.sqrt(c2)
|
|
|
|
def _pdf(self, x, df, nc):
|
|
n = df*1.0
|
|
nc = nc*1.0
|
|
x2 = x*x
|
|
ncx2 = nc*nc*x2
|
|
fac1 = n + x2
|
|
trm1 = n/2.*np.log(n) + sc.gammaln(n+1)
|
|
trm1 -= n*np.log(2)+nc*nc/2.+(n/2.)*np.log(fac1)+sc.gammaln(n/2.)
|
|
Px = np.exp(trm1)
|
|
valF = ncx2 / (2*fac1)
|
|
trm1 = np.sqrt(2)*nc*x*sc.hyp1f1(n/2+1, 1.5, valF)
|
|
trm1 /= np.asarray(fac1*sc.gamma((n+1)/2))
|
|
trm2 = sc.hyp1f1((n+1)/2, 0.5, valF)
|
|
trm2 /= np.asarray(np.sqrt(fac1)*sc.gamma(n/2+1))
|
|
Px *= trm1+trm2
|
|
return Px
|
|
|
|
def _cdf(self, x, df, nc):
|
|
return sc.nctdtr(df, nc, x)
|
|
|
|
def _ppf(self, q, df, nc):
|
|
return sc.nctdtrit(df, nc, q)
|
|
|
|
def _stats(self, df, nc, moments='mv'):
|
|
#
|
|
# See D. Hogben, R.S. Pinkham, and M.B. Wilk,
|
|
# 'The moments of the non-central t-distribution'
|
|
# Biometrika 48, p. 465 (2961).
|
|
# e.g. https://www.jstor.org/stable/2332772 (gated)
|
|
#
|
|
mu, mu2, g1, g2 = None, None, None, None
|
|
|
|
gfac = np.exp(sc.betaln(df/2-0.5, 0.5) - sc.gammaln(0.5))
|
|
c11 = np.sqrt(df/2.) * gfac
|
|
c20 = np.where(df > 2., df / (df-2.), np.nan)
|
|
c22 = c20 - c11*c11
|
|
mu = np.where(df > 1, nc*c11, np.nan)
|
|
mu2 = np.where(df > 2, c22*nc*nc + c20, np.nan)
|
|
if 's' in moments:
|
|
c33t = df * (7.-2.*df) / (df-2.) / (df-3.) + 2.*c11*c11
|
|
c31t = 3.*df / (df-2.) / (df-3.)
|
|
mu3 = (c33t*nc*nc + c31t) * c11*nc
|
|
g1 = np.where(df > 3, mu3 / np.power(mu2, 1.5), np.nan)
|
|
# kurtosis
|
|
if 'k' in moments:
|
|
c44 = df*df / (df-2.) / (df-4.)
|
|
c44 -= c11*c11 * 2.*df*(5.-df) / (df-2.) / (df-3.)
|
|
c44 -= 3.*c11**4
|
|
c42 = df / (df-4.) - c11*c11 * (df-1.) / (df-3.)
|
|
c42 *= 6.*df / (df-2.)
|
|
c40 = 3.*df*df / (df-2.) / (df-4.)
|
|
|
|
mu4 = c44 * nc**4 + c42*nc**2 + c40
|
|
g2 = np.where(df > 4, mu4/mu2**2 - 3., np.nan)
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
nct = nct_gen(name="nct")
|
|
|
|
|
|
class pareto_gen(rv_continuous):
|
|
r"""A Pareto continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `pareto` is:
|
|
|
|
.. math::
|
|
|
|
f(x, b) = \frac{b}{x^{b+1}}
|
|
|
|
for :math:`x \ge 1`, :math:`b > 0`.
|
|
|
|
`pareto` takes ``b`` as a shape parameter for :math:`b`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, b):
|
|
# pareto.pdf(x, b) = b / x**(b+1)
|
|
return b * x**(-b-1)
|
|
|
|
def _cdf(self, x, b):
|
|
return 1 - x**(-b)
|
|
|
|
def _ppf(self, q, b):
|
|
return pow(1-q, -1.0/b)
|
|
|
|
def _sf(self, x, b):
|
|
return x**(-b)
|
|
|
|
def _stats(self, b, moments='mv'):
|
|
mu, mu2, g1, g2 = None, None, None, None
|
|
if 'm' in moments:
|
|
mask = b > 1
|
|
bt = np.extract(mask, b)
|
|
mu = np.full(np.shape(b), fill_value=np.inf)
|
|
np.place(mu, mask, bt / (bt-1.0))
|
|
if 'v' in moments:
|
|
mask = b > 2
|
|
bt = np.extract(mask, b)
|
|
mu2 = np.full(np.shape(b), fill_value=np.inf)
|
|
np.place(mu2, mask, bt / (bt-2.0) / (bt-1.0)**2)
|
|
if 's' in moments:
|
|
mask = b > 3
|
|
bt = np.extract(mask, b)
|
|
g1 = np.full(np.shape(b), fill_value=np.nan)
|
|
vals = 2 * (bt + 1.0) * np.sqrt(bt - 2.0) / ((bt - 3.0) * np.sqrt(bt))
|
|
np.place(g1, mask, vals)
|
|
if 'k' in moments:
|
|
mask = b > 4
|
|
bt = np.extract(mask, b)
|
|
g2 = np.full(np.shape(b), fill_value=np.nan)
|
|
vals = (6.0*np.polyval([1.0, 1.0, -6, -2], bt) /
|
|
np.polyval([1.0, -7.0, 12.0, 0.0], bt))
|
|
np.place(g2, mask, vals)
|
|
return mu, mu2, g1, g2
|
|
|
|
def _entropy(self, c):
|
|
return 1 + 1.0/c - np.log(c)
|
|
|
|
def fit(self, data, *args, **kwds):
|
|
parameters = _check_fit_input_parameters(self, data, args, kwds)
|
|
data, fshape, floc, fscale = parameters
|
|
if floc is None:
|
|
return super(pareto_gen, self).fit(data, **kwds)
|
|
if np.any(data - floc < (fscale if fscale else 0)):
|
|
raise FitDataError("pareto", lower=1, upper=np.inf)
|
|
data = data - floc
|
|
|
|
# Source: Evans, Hastings, and Peacock (2000), Statistical
|
|
# Distributions, 3rd. Ed., John Wiley and Sons. Page 149.
|
|
|
|
if fscale is None:
|
|
fscale = np.min(data)
|
|
if fshape is None:
|
|
fshape = 1/((1/len(data)) * np.sum(np.log(data/fscale)))
|
|
return fshape, floc, fscale
|
|
|
|
|
|
pareto = pareto_gen(a=1.0, name="pareto")
|
|
|
|
|
|
class lomax_gen(rv_continuous):
|
|
r"""A Lomax (Pareto of the second kind) continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `lomax` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \frac{c}{(1+x)^{c+1}}
|
|
|
|
for :math:`x \ge 0`, :math:`c > 0`.
|
|
|
|
`lomax` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
`lomax` is a special case of `pareto` with ``loc=-1.0``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, c):
|
|
# lomax.pdf(x, c) = c / (1+x)**(c+1)
|
|
return c*1.0/(1.0+x)**(c+1.0)
|
|
|
|
def _logpdf(self, x, c):
|
|
return np.log(c) - (c+1)*sc.log1p(x)
|
|
|
|
def _cdf(self, x, c):
|
|
return -sc.expm1(-c*sc.log1p(x))
|
|
|
|
def _sf(self, x, c):
|
|
return np.exp(-c*sc.log1p(x))
|
|
|
|
def _logsf(self, x, c):
|
|
return -c*sc.log1p(x)
|
|
|
|
def _ppf(self, q, c):
|
|
return sc.expm1(-sc.log1p(-q)/c)
|
|
|
|
def _stats(self, c):
|
|
mu, mu2, g1, g2 = pareto.stats(c, loc=-1.0, moments='mvsk')
|
|
return mu, mu2, g1, g2
|
|
|
|
def _entropy(self, c):
|
|
return 1+1.0/c-np.log(c)
|
|
|
|
|
|
lomax = lomax_gen(a=0.0, name="lomax")
|
|
|
|
|
|
class pearson3_gen(rv_continuous):
|
|
r"""A pearson type III continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `pearson3` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)}
|
|
(\beta (x - \zeta))^{\alpha - 1}
|
|
\exp(-\beta (x - \zeta))
|
|
|
|
where:
|
|
|
|
.. math::
|
|
|
|
\beta = \frac{2}{\kappa}
|
|
|
|
\alpha = \beta^2 = \frac{4}{\kappa^2}
|
|
|
|
\zeta = -\frac{\alpha}{\beta} = -\beta
|
|
|
|
:math:`\Gamma` is the gamma function (`scipy.special.gamma`).
|
|
Pass the skew :math:`\kappa` into `pearson3` as the shape parameter
|
|
``skew``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
References
|
|
----------
|
|
R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and
|
|
Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water
|
|
Resources Research, Vol.27, 3149-3158 (1991).
|
|
|
|
L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist.,
|
|
Vol.1, 191-198 (1930).
|
|
|
|
"Using Modern Computing Tools to Fit the Pearson Type III Distribution to
|
|
Aviation Loads Data", Office of Aviation Research (2003).
|
|
|
|
"""
|
|
def _preprocess(self, x, skew):
|
|
# The real 'loc' and 'scale' are handled in the calling pdf(...). The
|
|
# local variables 'loc' and 'scale' within pearson3._pdf are set to
|
|
# the defaults just to keep them as part of the equations for
|
|
# documentation.
|
|
loc = 0.0
|
|
scale = 1.0
|
|
|
|
# If skew is small, return _norm_pdf. The divide between pearson3
|
|
# and norm was found by brute force and is approximately a skew of
|
|
# 0.000016. No one, I hope, would actually use a skew value even
|
|
# close to this small.
|
|
norm2pearson_transition = 0.000016
|
|
|
|
ans, x, skew = np.broadcast_arrays([1.0], x, skew)
|
|
ans = ans.copy()
|
|
|
|
# mask is True where skew is small enough to use the normal approx.
|
|
mask = np.absolute(skew) < norm2pearson_transition
|
|
invmask = ~mask
|
|
|
|
beta = 2.0 / (skew[invmask] * scale)
|
|
alpha = (scale * beta)**2
|
|
zeta = loc - alpha / beta
|
|
|
|
transx = beta * (x[invmask] - zeta)
|
|
return ans, x, transx, mask, invmask, beta, alpha, zeta
|
|
|
|
def _argcheck(self, skew):
|
|
# The _argcheck function in rv_continuous only allows positive
|
|
# arguments. The skew argument for pearson3 can be zero (which I want
|
|
# to handle inside pearson3._pdf) or negative. So just return True
|
|
# for all skew args.
|
|
return np.ones(np.shape(skew), dtype=bool)
|
|
|
|
def _stats(self, skew):
|
|
m = 0.0
|
|
v = 1.0
|
|
s = skew
|
|
k = 1.5*skew**2
|
|
return m, v, s, k
|
|
|
|
def _pdf(self, x, skew):
|
|
# pearson3.pdf(x, skew) = abs(beta) / gamma(alpha) *
|
|
# (beta * (x - zeta))**(alpha - 1) * exp(-beta*(x - zeta))
|
|
# Do the calculation in _logpdf since helps to limit
|
|
# overflow/underflow problems
|
|
ans = np.exp(self._logpdf(x, skew))
|
|
if ans.ndim == 0:
|
|
if np.isnan(ans):
|
|
return 0.0
|
|
return ans
|
|
ans[np.isnan(ans)] = 0.0
|
|
return ans
|
|
|
|
def _logpdf(self, x, skew):
|
|
# PEARSON3 logpdf GAMMA logpdf
|
|
# np.log(abs(beta))
|
|
# + (alpha - 1)*np.log(beta*(x - zeta)) + (a - 1)*np.log(x)
|
|
# - beta*(x - zeta) - x
|
|
# - sc.gammalnalpha) - sc.gammalna)
|
|
ans, x, transx, mask, invmask, beta, alpha, _ = (
|
|
self._preprocess(x, skew))
|
|
|
|
ans[mask] = np.log(_norm_pdf(x[mask]))
|
|
# use logpdf instead of _logpdf to fix issue mentioned in gh-12640
|
|
# (_logpdf does not return correct result for alpha = 1)
|
|
ans[invmask] = np.log(abs(beta)) + gamma.logpdf(transx, alpha)
|
|
return ans
|
|
|
|
def _cdf(self, x, skew):
|
|
ans, x, transx, mask, invmask, _, alpha, _ = (
|
|
self._preprocess(x, skew))
|
|
|
|
ans[mask] = _norm_cdf(x[mask])
|
|
|
|
invmask1a = np.logical_and(invmask, skew > 0)
|
|
invmask1b = skew[invmask] > 0
|
|
# use cdf instead of _cdf to fix issue mentioned in gh-12640
|
|
# (_cdf produces NaNs for inputs outside support)
|
|
ans[invmask1a] = gamma.cdf(transx[invmask1b], alpha[invmask1b])
|
|
|
|
# The gamma._cdf approach wasn't working with negative skew.
|
|
# Note that multiplying the skew by -1 reflects about x=0.
|
|
# So instead of evaluating the CDF with negative skew at x,
|
|
# evaluate the SF with positive skew at -x.
|
|
invmask2a = np.logical_and(invmask, skew < 0)
|
|
invmask2b = skew[invmask] < 0
|
|
# gamma._sf produces NaNs when transx < 0, so use gamma.sf
|
|
ans[invmask2a] = gamma.sf(transx[invmask2b], alpha[invmask2b])
|
|
|
|
return ans
|
|
|
|
def _rvs(self, skew, size=None, random_state=None):
|
|
skew = np.broadcast_to(skew, size)
|
|
ans, _, _, mask, invmask, beta, alpha, zeta = (
|
|
self._preprocess([0], skew))
|
|
|
|
nsmall = mask.sum()
|
|
nbig = mask.size - nsmall
|
|
ans[mask] = random_state.standard_normal(nsmall)
|
|
ans[invmask] = random_state.standard_gamma(alpha, nbig)/beta + zeta
|
|
|
|
if size == ():
|
|
ans = ans[0]
|
|
return ans
|
|
|
|
def _ppf(self, q, skew):
|
|
ans, q, _, mask, invmask, beta, alpha, zeta = (
|
|
self._preprocess(q, skew))
|
|
ans[mask] = _norm_ppf(q[mask])
|
|
ans[invmask] = sc.gammaincinv(alpha, q[invmask])/beta + zeta
|
|
return ans
|
|
|
|
|
|
pearson3 = pearson3_gen(name="pearson3")
|
|
|
|
|
|
class powerlaw_gen(rv_continuous):
|
|
r"""A power-function continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `powerlaw` is:
|
|
|
|
.. math::
|
|
|
|
f(x, a) = a x^{a-1}
|
|
|
|
for :math:`0 \le x \le 1`, :math:`a > 0`.
|
|
|
|
`powerlaw` takes ``a`` as a shape parameter for :math:`a`.
|
|
|
|
%(after_notes)s
|
|
|
|
`powerlaw` is a special case of `beta` with ``b=1``.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, a):
|
|
# powerlaw.pdf(x, a) = a * x**(a-1)
|
|
return a*x**(a-1.0)
|
|
|
|
def _logpdf(self, x, a):
|
|
return np.log(a) + sc.xlogy(a - 1, x)
|
|
|
|
def _cdf(self, x, a):
|
|
return x**(a*1.0)
|
|
|
|
def _logcdf(self, x, a):
|
|
return a*np.log(x)
|
|
|
|
def _ppf(self, q, a):
|
|
return pow(q, 1.0/a)
|
|
|
|
def _stats(self, a):
|
|
return (a / (a + 1.0),
|
|
a / (a + 2.0) / (a + 1.0) ** 2,
|
|
-2.0 * ((a - 1.0) / (a + 3.0)) * np.sqrt((a + 2.0) / a),
|
|
6 * np.polyval([1, -1, -6, 2], a) / (a * (a + 3.0) * (a + 4)))
|
|
|
|
def _entropy(self, a):
|
|
return 1 - 1.0/a - np.log(a)
|
|
|
|
|
|
powerlaw = powerlaw_gen(a=0.0, b=1.0, name="powerlaw")
|
|
|
|
|
|
class powerlognorm_gen(rv_continuous):
|
|
r"""A power log-normal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `powerlognorm` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s)
|
|
(\Phi(-\log(x)/s))^{c-1}
|
|
|
|
where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
|
|
and :math:`x > 0`, :math:`s, c > 0`.
|
|
|
|
`powerlognorm` takes :math:`c` and :math:`s` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _pdf(self, x, c, s):
|
|
# powerlognorm.pdf(x, c, s) = c / (x*s) * phi(log(x)/s) *
|
|
# (Phi(-log(x)/s))**(c-1),
|
|
return (c/(x*s) * _norm_pdf(np.log(x)/s) *
|
|
pow(_norm_cdf(-np.log(x)/s), c*1.0-1.0))
|
|
|
|
def _cdf(self, x, c, s):
|
|
return 1.0 - pow(_norm_cdf(-np.log(x)/s), c*1.0)
|
|
|
|
def _ppf(self, q, c, s):
|
|
return np.exp(-s * _norm_ppf(pow(1.0 - q, 1.0 / c)))
|
|
|
|
|
|
powerlognorm = powerlognorm_gen(a=0.0, name="powerlognorm")
|
|
|
|
|
|
class powernorm_gen(rv_continuous):
|
|
r"""A power normal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `powernorm` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = c \phi(x) (\Phi(-x))^{c-1}
|
|
|
|
where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
|
|
and :math:`x >= 0`, :math:`c > 0`.
|
|
|
|
`powernorm` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x, c):
|
|
# powernorm.pdf(x, c) = c * phi(x) * (Phi(-x))**(c-1)
|
|
return c*_norm_pdf(x) * (_norm_cdf(-x)**(c-1.0))
|
|
|
|
def _logpdf(self, x, c):
|
|
return np.log(c) + _norm_logpdf(x) + (c-1)*_norm_logcdf(-x)
|
|
|
|
def _cdf(self, x, c):
|
|
return 1.0-_norm_cdf(-x)**(c*1.0)
|
|
|
|
def _ppf(self, q, c):
|
|
return -_norm_ppf(pow(1.0 - q, 1.0 / c))
|
|
|
|
|
|
powernorm = powernorm_gen(name='powernorm')
|
|
|
|
|
|
class rdist_gen(rv_continuous):
|
|
r"""An R-distributed (symmetric beta) continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `rdist` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)}
|
|
|
|
for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the
|
|
symmetric beta distribution: if B has a `beta` distribution with
|
|
parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with
|
|
parameter c.
|
|
|
|
`rdist` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
This distribution includes the following distribution kernels as
|
|
special cases::
|
|
|
|
c = 2: uniform
|
|
c = 3: `semicircular`
|
|
c = 4: Epanechnikov (parabolic)
|
|
c = 6: quartic (biweight)
|
|
c = 8: triweight
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
# use relation to the beta distribution for pdf, cdf, etc
|
|
def _pdf(self, x, c):
|
|
return 0.5*beta._pdf((x + 1)/2, c/2, c/2)
|
|
|
|
def _logpdf(self, x, c):
|
|
return -np.log(2) + beta._logpdf((x + 1)/2, c/2, c/2)
|
|
|
|
def _cdf(self, x, c):
|
|
return beta._cdf((x + 1)/2, c/2, c/2)
|
|
|
|
def _ppf(self, q, c):
|
|
return 2*beta._ppf(q, c/2, c/2) - 1
|
|
|
|
def _rvs(self, c, size=None, random_state=None):
|
|
return 2 * random_state.beta(c/2, c/2, size) - 1
|
|
|
|
def _munp(self, n, c):
|
|
numerator = (1 - (n % 2)) * sc.beta((n + 1.0) / 2, c / 2.0)
|
|
return numerator / sc.beta(1. / 2, c / 2.)
|
|
|
|
|
|
rdist = rdist_gen(a=-1.0, b=1.0, name="rdist")
|
|
|
|
|
|
def _rayleigh_fit_check_error(ier, msg):
|
|
if ier != 1:
|
|
raise RuntimeError('rayleigh.fit: fsolve failed to find the root of '
|
|
'the first-order conditions of the log-likelihood '
|
|
f'function: {msg} (ier={ier})')
|
|
|
|
|
|
class rayleigh_gen(rv_continuous):
|
|
r"""A Rayleigh continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `rayleigh` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = x \exp(-x^2/2)
|
|
|
|
for :math:`x \ge 0`.
|
|
|
|
`rayleigh` is a special case of `chi` with ``df=2``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _rvs(self, size=None, random_state=None):
|
|
return chi.rvs(2, size=size, random_state=random_state)
|
|
|
|
def _pdf(self, r):
|
|
# rayleigh.pdf(r) = r * exp(-r**2/2)
|
|
return np.exp(self._logpdf(r))
|
|
|
|
def _logpdf(self, r):
|
|
return np.log(r) - 0.5 * r * r
|
|
|
|
def _cdf(self, r):
|
|
return -sc.expm1(-0.5 * r**2)
|
|
|
|
def _ppf(self, q):
|
|
return np.sqrt(-2 * sc.log1p(-q))
|
|
|
|
def _sf(self, r):
|
|
return np.exp(self._logsf(r))
|
|
|
|
def _logsf(self, r):
|
|
return -0.5 * r * r
|
|
|
|
def _isf(self, q):
|
|
return np.sqrt(-2 * np.log(q))
|
|
|
|
def _stats(self):
|
|
val = 4 - np.pi
|
|
return (np.sqrt(np.pi/2),
|
|
val/2,
|
|
2*(np.pi-3)*np.sqrt(np.pi)/val**1.5,
|
|
6*np.pi/val-16/val**2)
|
|
|
|
def _entropy(self):
|
|
return _EULER/2.0 + 1 - 0.5*np.log(2)
|
|
|
|
@extend_notes_in_docstring(rv_continuous, notes="""\
|
|
Notes specifically for ``rayleigh.fit``: If the location is fixed with
|
|
the `floc` parameter, this method uses an analytical formula to find
|
|
the scale. Otherwise, this function uses a numerical root finder on
|
|
the first order conditions of the log-likelihood function to find the
|
|
MLE. Only the (optional) `loc` parameter is used as the initial guess
|
|
for the root finder; the `scale` parameter and any other parameters
|
|
for the optimizer are ignored.\n\n""")
|
|
def fit(self, data, *args, **kwds):
|
|
data, floc, fscale = _check_fit_input_parameters(self, data,
|
|
args, kwds)
|
|
|
|
def scale_mle(loc, data):
|
|
# Source: Statistical Distributions, 3rd Edition. Evans, Hastings,
|
|
# and Peacock (2000), Page 175
|
|
return (np.sum((data - loc) ** 2) / (2 * len(data))) ** .5
|
|
|
|
def loc_mle(loc, data):
|
|
# This implicit equation for `loc` is used when
|
|
# both `loc` and `scale` are free.
|
|
xm = data - loc
|
|
s1 = xm.sum()
|
|
s2 = (xm**2).sum()
|
|
s3 = (1/xm).sum()
|
|
return s1 - s2/(2*len(data))*s3
|
|
|
|
def loc_mle_scale_fixed(loc, scale, data):
|
|
# This implicit equation for `loc` is used when
|
|
# `scale` is fixed but `loc` is not.
|
|
xm = data - loc
|
|
return xm.sum() - scale**2 * (1/xm).sum()
|
|
|
|
if floc is not None:
|
|
# `loc` is fixed, analytically determine `scale`.
|
|
if np.any(data - floc <= 0):
|
|
raise FitDataError("rayleigh", lower=1, upper=np.inf)
|
|
else:
|
|
return floc, scale_mle(floc, data)
|
|
|
|
# Account for user provided guess of `loc`.
|
|
loc0 = kwds.get('loc')
|
|
if loc0 is None:
|
|
# Use _fitstart to estimate loc; ignore the returned scale.
|
|
loc0 = self._fitstart(data)[0]
|
|
|
|
if fscale is not None:
|
|
# `scale` is fixed
|
|
x, info, ier, msg = optimize.fsolve(loc_mle_scale_fixed, x0=loc0,
|
|
args=(fscale, data,),
|
|
xtol=1e-10, full_output=True)
|
|
_rayleigh_fit_check_error(ier, msg)
|
|
return x[0], fscale
|
|
else:
|
|
# Neither `loc` nor `scale` are fixed.
|
|
x, info, ier, msg = optimize.fsolve(loc_mle, x0=loc0, args=(data,),
|
|
xtol=1e-10, full_output=True)
|
|
_rayleigh_fit_check_error(ier, msg)
|
|
return x[0], scale_mle(x[0], data)
|
|
|
|
|
|
rayleigh = rayleigh_gen(a=0.0, name="rayleigh")
|
|
|
|
|
|
class reciprocal_gen(rv_continuous):
|
|
r"""A loguniform or reciprocal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for this class is:
|
|
|
|
.. math::
|
|
|
|
f(x, a, b) = \frac{1}{x \log(b/a)}
|
|
|
|
for :math:`a \le x \le b`, :math:`b > a > 0`. This class takes
|
|
:math:`a` and :math:`b` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
This doesn't show the equal probability of ``0.01``, ``0.1`` and
|
|
``1``. This is best when the x-axis is log-scaled:
|
|
|
|
>>> import numpy as np
|
|
>>> fig, ax = plt.subplots(1, 1)
|
|
>>> ax.hist(np.log10(r))
|
|
>>> ax.set_ylabel("Frequency")
|
|
>>> ax.set_xlabel("Value of random variable")
|
|
>>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
|
|
>>> ticks = ["$10^{{ {} }}$".format(i) for i in [-2, -1, 0]]
|
|
>>> ax.set_xticklabels(ticks) # doctest: +SKIP
|
|
>>> plt.show()
|
|
|
|
This random variable will be log-uniform regardless of the base chosen for
|
|
``a`` and ``b``. Let's specify with base ``2`` instead:
|
|
|
|
>>> rvs = %(name)s(2**-2, 2**0).rvs(size=1000)
|
|
|
|
Values of ``1/4``, ``1/2`` and ``1`` are equally likely with this random
|
|
variable. Here's the histogram:
|
|
|
|
>>> fig, ax = plt.subplots(1, 1)
|
|
>>> ax.hist(np.log2(rvs))
|
|
>>> ax.set_ylabel("Frequency")
|
|
>>> ax.set_xlabel("Value of random variable")
|
|
>>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
|
|
>>> ticks = ["$2^{{ {} }}$".format(i) for i in [-2, -1, 0]]
|
|
>>> ax.set_xticklabels(ticks) # doctest: +SKIP
|
|
>>> plt.show()
|
|
|
|
"""
|
|
def _argcheck(self, a, b):
|
|
return (a > 0) & (b > a)
|
|
|
|
def _get_support(self, a, b):
|
|
return a, b
|
|
|
|
def _pdf(self, x, a, b):
|
|
# reciprocal.pdf(x, a, b) = 1 / (x*log(b/a))
|
|
return 1.0 / (x * np.log(b * 1.0 / a))
|
|
|
|
def _logpdf(self, x, a, b):
|
|
return -np.log(x) - np.log(np.log(b * 1.0 / a))
|
|
|
|
def _cdf(self, x, a, b):
|
|
return (np.log(x)-np.log(a)) / np.log(b * 1.0 / a)
|
|
|
|
def _ppf(self, q, a, b):
|
|
return a*pow(b*1.0/a, q)
|
|
|
|
def _munp(self, n, a, b):
|
|
return 1.0/np.log(b*1.0/a) / n * (pow(b*1.0, n) - pow(a*1.0, n))
|
|
|
|
def _entropy(self, a, b):
|
|
return 0.5*np.log(a*b)+np.log(np.log(b*1.0/a))
|
|
|
|
|
|
loguniform = reciprocal_gen(name="loguniform")
|
|
reciprocal = reciprocal_gen(name="reciprocal")
|
|
|
|
|
|
class rice_gen(rv_continuous):
|
|
r"""A Rice continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `rice` is:
|
|
|
|
.. math::
|
|
|
|
f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I_0(x b)
|
|
|
|
for :math:`x >= 0`, :math:`b > 0`. :math:`I_0` is the modified Bessel
|
|
function of order zero (`scipy.special.i0`).
|
|
|
|
`rice` takes ``b`` as a shape parameter for :math:`b`.
|
|
|
|
%(after_notes)s
|
|
|
|
The Rice distribution describes the length, :math:`r`, of a 2-D vector with
|
|
components :math:`(U+u, V+v)`, where :math:`U, V` are constant, :math:`u,
|
|
v` are independent Gaussian random variables with standard deviation
|
|
:math:`s`. Let :math:`R = \sqrt{U^2 + V^2}`. Then the pdf of :math:`r` is
|
|
``rice.pdf(x, R/s, scale=s)``.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, b):
|
|
return b >= 0
|
|
|
|
def _rvs(self, b, size=None, random_state=None):
|
|
# https://en.wikipedia.org/wiki/Rice_distribution
|
|
t = b/np.sqrt(2) + random_state.standard_normal(size=(2,) + size)
|
|
return np.sqrt((t*t).sum(axis=0))
|
|
|
|
def _cdf(self, x, b):
|
|
return sc.chndtr(np.square(x), 2, np.square(b))
|
|
|
|
def _ppf(self, q, b):
|
|
return np.sqrt(sc.chndtrix(q, 2, np.square(b)))
|
|
|
|
def _pdf(self, x, b):
|
|
# rice.pdf(x, b) = x * exp(-(x**2+b**2)/2) * I[0](x*b)
|
|
#
|
|
# We use (x**2 + b**2)/2 = ((x-b)**2)/2 + xb.
|
|
# The factor of np.exp(-xb) is then included in the i0e function
|
|
# in place of the modified Bessel function, i0, improving
|
|
# numerical stability for large values of xb.
|
|
return x * np.exp(-(x-b)*(x-b)/2.0) * sc.i0e(x*b)
|
|
|
|
def _munp(self, n, b):
|
|
nd2 = n/2.0
|
|
n1 = 1 + nd2
|
|
b2 = b*b/2.0
|
|
return (2.0**(nd2) * np.exp(-b2) * sc.gamma(n1) *
|
|
sc.hyp1f1(n1, 1, b2))
|
|
|
|
|
|
rice = rice_gen(a=0.0, name="rice")
|
|
|
|
|
|
# FIXME: PPF does not work.
|
|
class recipinvgauss_gen(rv_continuous):
|
|
r"""A reciprocal inverse Gaussian continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `recipinvgauss` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \mu) = \frac{1}{\sqrt{2\pi x}}
|
|
\exp\left(\frac{-(1-\mu x)^2}{2\mu^2x}\right)
|
|
|
|
for :math:`x \ge 0`.
|
|
|
|
`recipinvgauss` takes ``mu`` as a shape parameter for :math:`\mu`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _pdf(self, x, mu):
|
|
# recipinvgauss.pdf(x, mu) =
|
|
# 1/sqrt(2*pi*x) * exp(-(1-mu*x)**2/(2*x*mu**2))
|
|
return 1.0/np.sqrt(2*np.pi*x)*np.exp(-(1-mu*x)**2.0 / (2*x*mu**2.0))
|
|
|
|
def _logpdf(self, x, mu):
|
|
return -(1-mu*x)**2.0 / (2*x*mu**2.0) - 0.5*np.log(2*np.pi*x)
|
|
|
|
def _cdf(self, x, mu):
|
|
trm1 = 1.0/mu - x
|
|
trm2 = 1.0/mu + x
|
|
isqx = 1.0/np.sqrt(x)
|
|
return 1.0-_norm_cdf(isqx*trm1)-np.exp(2.0/mu)*_norm_cdf(-isqx*trm2)
|
|
|
|
def _rvs(self, mu, size=None, random_state=None):
|
|
return 1.0/random_state.wald(mu, 1.0, size=size)
|
|
|
|
|
|
recipinvgauss = recipinvgauss_gen(a=0.0, name='recipinvgauss')
|
|
|
|
|
|
class semicircular_gen(rv_continuous):
|
|
r"""A semicircular continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
rdist
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `semicircular` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{2}{\pi} \sqrt{1-x^2}
|
|
|
|
for :math:`-1 \le x \le 1`.
|
|
|
|
The distribution is a special case of `rdist` with `c = 3`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] "Wigner semicircle distribution",
|
|
https://en.wikipedia.org/wiki/Wigner_semicircle_distribution
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _pdf(self, x):
|
|
return 2.0/np.pi*np.sqrt(1-x*x)
|
|
|
|
def _logpdf(self, x):
|
|
return np.log(2/np.pi) + 0.5*np.log1p(-x*x)
|
|
|
|
def _cdf(self, x):
|
|
return 0.5+1.0/np.pi*(x*np.sqrt(1-x*x) + np.arcsin(x))
|
|
|
|
def _ppf(self, q):
|
|
return rdist._ppf(q, 3)
|
|
|
|
def _rvs(self, size=None, random_state=None):
|
|
# generate values uniformly distributed on the area under the pdf
|
|
# (semi-circle) by randomly generating the radius and angle
|
|
r = np.sqrt(random_state.uniform(size=size))
|
|
a = np.cos(np.pi * random_state.uniform(size=size))
|
|
return r * a
|
|
|
|
def _stats(self):
|
|
return 0, 0.25, 0, -1.0
|
|
|
|
def _entropy(self):
|
|
return 0.64472988584940017414
|
|
|
|
|
|
semicircular = semicircular_gen(a=-1.0, b=1.0, name="semicircular")
|
|
|
|
|
|
class skew_norm_gen(rv_continuous):
|
|
r"""A skew-normal random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The pdf is::
|
|
|
|
skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x)
|
|
|
|
`skewnorm` takes a real number :math:`a` as a skewness parameter
|
|
When ``a = 0`` the distribution is identical to a normal distribution
|
|
(`norm`). `rvs` implements the method of [1]_.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
References
|
|
----------
|
|
.. [1] A. Azzalini and A. Capitanio (1999). Statistical applications of the
|
|
multivariate skew-normal distribution. J. Roy. Statist. Soc., B 61, 579-602.
|
|
:arxiv:`0911.2093`
|
|
|
|
"""
|
|
def _argcheck(self, a):
|
|
return np.isfinite(a)
|
|
|
|
def _pdf(self, x, a):
|
|
return 2.*_norm_pdf(x)*_norm_cdf(a*x)
|
|
|
|
def _cdf_single(self, x, *args):
|
|
_a, _b = self._get_support(*args)
|
|
if x <= 0:
|
|
cdf = integrate.quad(self._pdf, _a, x, args=args)[0]
|
|
else:
|
|
t1 = integrate.quad(self._pdf, _a, 0, args=args)[0]
|
|
t2 = integrate.quad(self._pdf, 0, x, args=args)[0]
|
|
cdf = t1 + t2
|
|
if cdf > 1:
|
|
# Presumably numerical noise, e.g. 1.0000000000000002
|
|
cdf = 1.0
|
|
return cdf
|
|
|
|
def _sf(self, x, a):
|
|
return self._cdf(-x, -a)
|
|
|
|
def _rvs(self, a, size=None, random_state=None):
|
|
u0 = random_state.normal(size=size)
|
|
v = random_state.normal(size=size)
|
|
d = a/np.sqrt(1 + a**2)
|
|
u1 = d*u0 + v*np.sqrt(1 - d**2)
|
|
return np.where(u0 >= 0, u1, -u1)
|
|
|
|
def _stats(self, a, moments='mvsk'):
|
|
output = [None, None, None, None]
|
|
const = np.sqrt(2/np.pi) * a/np.sqrt(1 + a**2)
|
|
|
|
if 'm' in moments:
|
|
output[0] = const
|
|
if 'v' in moments:
|
|
output[1] = 1 - const**2
|
|
if 's' in moments:
|
|
output[2] = ((4 - np.pi)/2) * (const/np.sqrt(1 - const**2))**3
|
|
if 'k' in moments:
|
|
output[3] = (2*(np.pi - 3)) * (const**4/(1 - const**2)**2)
|
|
|
|
return output
|
|
|
|
|
|
skewnorm = skew_norm_gen(name='skewnorm')
|
|
|
|
|
|
class trapezoid_gen(rv_continuous):
|
|
r"""A trapezoidal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The trapezoidal distribution can be represented with an up-sloping line
|
|
from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)``
|
|
and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``. This
|
|
defines the trapezoid base from ``loc`` to ``(loc+scale)`` and the flat
|
|
top from ``c`` to ``d`` proportional to the position along the base
|
|
with ``0 <= c <= d <= 1``. When ``c=d``, this is equivalent to `triang`
|
|
with the same values for `loc`, `scale` and `c`.
|
|
The method of [1]_ is used for computing moments.
|
|
|
|
`trapezoid` takes :math:`c` and :math:`d` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
The standard form is in the range [0, 1] with c the mode.
|
|
The location parameter shifts the start to `loc`.
|
|
The scale parameter changes the width from 1 to `scale`.
|
|
|
|
%(example)s
|
|
|
|
References
|
|
----------
|
|
.. [1] Kacker, R.N. and Lawrence, J.F. (2007). Trapezoidal and triangular
|
|
distributions for Type B evaluation of standard uncertainty.
|
|
Metrologia 44, 117-127. :doi:`10.1088/0026-1394/44/2/003`
|
|
|
|
|
|
"""
|
|
def _argcheck(self, c, d):
|
|
return (c >= 0) & (c <= 1) & (d >= 0) & (d <= 1) & (d >= c)
|
|
|
|
def _pdf(self, x, c, d):
|
|
u = 2 / (d-c+1)
|
|
|
|
return _lazyselect([x < c,
|
|
(c <= x) & (x <= d),
|
|
x > d],
|
|
[lambda x, c, d, u: u * x / c,
|
|
lambda x, c, d, u: u,
|
|
lambda x, c, d, u: u * (1-x) / (1-d)],
|
|
(x, c, d, u))
|
|
|
|
def _cdf(self, x, c, d):
|
|
return _lazyselect([x < c,
|
|
(c <= x) & (x <= d),
|
|
x > d],
|
|
[lambda x, c, d: x**2 / c / (d-c+1),
|
|
lambda x, c, d: (c + 2 * (x-c)) / (d-c+1),
|
|
lambda x, c, d: 1-((1-x) ** 2
|
|
/ (d-c+1) / (1-d))],
|
|
(x, c, d))
|
|
|
|
def _ppf(self, q, c, d):
|
|
qc, qd = self._cdf(c, c, d), self._cdf(d, c, d)
|
|
condlist = [q < qc, q <= qd, q > qd]
|
|
choicelist = [np.sqrt(q * c * (1 + d - c)),
|
|
0.5 * q * (1 + d - c) + 0.5 * c,
|
|
1 - np.sqrt((1 - q) * (d - c + 1) * (1 - d))]
|
|
return np.select(condlist, choicelist)
|
|
|
|
def _munp(self, n, c, d):
|
|
# Using the parameterization from Kacker, 2007, with
|
|
# a=bottom left, c=top left, d=top right, b=bottom right, then
|
|
# E[X^n] = h/(n+1)/(n+2) [(b^{n+2}-d^{n+2})/(b-d)
|
|
# - ((c^{n+2} - a^{n+2})/(c-a)]
|
|
# with h = 2/((b-a) - (d-c)). The corresponding parameterization
|
|
# in scipy, has a'=loc, c'=loc+c*scale, d'=loc+d*scale, b'=loc+scale,
|
|
# which for standard form reduces to a'=0, b'=1, c'=c, d'=d.
|
|
# Substituting into E[X^n] gives the bd' term as (1 - d^{n+2})/(1 - d)
|
|
# and the ac' term as c^{n-1} for the standard form. The bd' term has
|
|
# numerical difficulties near d=1, so replace (1 - d^{n+2})/(1-d)
|
|
# with expm1((n+2)*log(d))/(d-1).
|
|
# Testing with n=18 for c=(1e-30,1-eps) shows that this is stable.
|
|
# We still require an explicit test for d=1 to prevent divide by zero,
|
|
# and now a test for d=0 to prevent log(0).
|
|
ab_term = c**(n+1)
|
|
dc_term = _lazyselect(
|
|
[d == 0.0, (0.0 < d) & (d < 1.0), d == 1.0],
|
|
[lambda d: 1.0,
|
|
lambda d: np.expm1((n+2) * np.log(d)) / (d-1.0),
|
|
lambda d: n+2],
|
|
[d])
|
|
val = 2.0 / (1.0+d-c) * (dc_term - ab_term) / ((n+1) * (n+2))
|
|
return val
|
|
|
|
def _entropy(self, c, d):
|
|
# Using the parameterization from Wikipedia (van Dorp, 2003)
|
|
# with a=bottom left, c=top left, d=top right, b=bottom right
|
|
# gives a'=loc, b'=loc+c*scale, c'=loc+d*scale, d'=loc+scale,
|
|
# which for loc=0, scale=1 is a'=0, b'=c, c'=d, d'=1.
|
|
# Substituting into the entropy formula from Wikipedia gives
|
|
# the following result.
|
|
return 0.5 * (1.0-d+c) / (1.0+d-c) + np.log(0.5 * (1.0+d-c))
|
|
|
|
|
|
trapezoid = trapezoid_gen(a=0.0, b=1.0, name="trapezoid")
|
|
# Note: alias kept for backwards compatibility. Rename was done
|
|
# because trapz is a slur in colloquial English (see gh-12924).
|
|
trapz = trapezoid_gen(a=0.0, b=1.0, name="trapz")
|
|
if trapz.__doc__:
|
|
trapz.__doc__ = "trapz is an alias for `trapezoid`"
|
|
|
|
|
|
class triang_gen(rv_continuous):
|
|
r"""A triangular continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The triangular distribution can be represented with an up-sloping line from
|
|
``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)``
|
|
to ``(loc + scale)``.
|
|
|
|
`triang` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
The standard form is in the range [0, 1] with c the mode.
|
|
The location parameter shifts the start to `loc`.
|
|
The scale parameter changes the width from 1 to `scale`.
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, c, size=None, random_state=None):
|
|
return random_state.triangular(0, c, 1, size)
|
|
|
|
def _argcheck(self, c):
|
|
return (c >= 0) & (c <= 1)
|
|
|
|
def _pdf(self, x, c):
|
|
# 0: edge case where c=0
|
|
# 1: generalised case for x < c, don't use x <= c, as it doesn't cope
|
|
# with c = 0.
|
|
# 2: generalised case for x >= c, but doesn't cope with c = 1
|
|
# 3: edge case where c=1
|
|
r = _lazyselect([c == 0,
|
|
x < c,
|
|
(x >= c) & (c != 1),
|
|
c == 1],
|
|
[lambda x, c: 2 - 2 * x,
|
|
lambda x, c: 2 * x / c,
|
|
lambda x, c: 2 * (1 - x) / (1 - c),
|
|
lambda x, c: 2 * x],
|
|
(x, c))
|
|
return r
|
|
|
|
def _cdf(self, x, c):
|
|
r = _lazyselect([c == 0,
|
|
x < c,
|
|
(x >= c) & (c != 1),
|
|
c == 1],
|
|
[lambda x, c: 2*x - x*x,
|
|
lambda x, c: x * x / c,
|
|
lambda x, c: (x*x - 2*x + c) / (c-1),
|
|
lambda x, c: x * x],
|
|
(x, c))
|
|
return r
|
|
|
|
def _ppf(self, q, c):
|
|
return np.where(q < c, np.sqrt(c * q), 1-np.sqrt((1-c) * (1-q)))
|
|
|
|
def _stats(self, c):
|
|
return ((c+1.0)/3.0,
|
|
(1.0-c+c*c)/18,
|
|
np.sqrt(2)*(2*c-1)*(c+1)*(c-2) / (5*np.power((1.0-c+c*c), 1.5)),
|
|
-3.0/5.0)
|
|
|
|
def _entropy(self, c):
|
|
return 0.5-np.log(2)
|
|
|
|
|
|
triang = triang_gen(a=0.0, b=1.0, name="triang")
|
|
|
|
|
|
class truncexpon_gen(rv_continuous):
|
|
r"""A truncated exponential continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `truncexpon` is:
|
|
|
|
.. math::
|
|
|
|
f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)}
|
|
|
|
for :math:`0 <= x <= b`.
|
|
|
|
`truncexpon` takes ``b`` as a shape parameter for :math:`b`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, b):
|
|
return b > 0
|
|
|
|
def _get_support(self, b):
|
|
return self.a, b
|
|
|
|
def _pdf(self, x, b):
|
|
# truncexpon.pdf(x, b) = exp(-x) / (1-exp(-b))
|
|
return np.exp(-x)/(-sc.expm1(-b))
|
|
|
|
def _logpdf(self, x, b):
|
|
return -x - np.log(-sc.expm1(-b))
|
|
|
|
def _cdf(self, x, b):
|
|
return sc.expm1(-x)/sc.expm1(-b)
|
|
|
|
def _ppf(self, q, b):
|
|
return -sc.log1p(q*sc.expm1(-b))
|
|
|
|
def _munp(self, n, b):
|
|
# wrong answer with formula, same as in continuous.pdf
|
|
# return sc.gamman+1)-sc.gammainc1+n, b)
|
|
if n == 1:
|
|
return (1-(b+1)*np.exp(-b))/(-sc.expm1(-b))
|
|
elif n == 2:
|
|
return 2*(1-0.5*(b*b+2*b+2)*np.exp(-b))/(-sc.expm1(-b))
|
|
else:
|
|
# return generic for higher moments
|
|
# return rv_continuous._mom1_sc(self, n, b)
|
|
return self._mom1_sc(n, b)
|
|
|
|
def _entropy(self, b):
|
|
eB = np.exp(b)
|
|
return np.log(eB-1)+(1+eB*(b-1.0))/(1.0-eB)
|
|
|
|
|
|
truncexpon = truncexpon_gen(a=0.0, name='truncexpon')
|
|
|
|
|
|
TRUNCNORM_TAIL_X = 30
|
|
TRUNCNORM_MAX_BRENT_ITERS = 40
|
|
|
|
def _truncnorm_get_delta_scalar(a, b):
|
|
if (a > TRUNCNORM_TAIL_X) or (b < -TRUNCNORM_TAIL_X):
|
|
return 0
|
|
if a > 0:
|
|
delta = _norm_sf(a) - _norm_sf(b)
|
|
else:
|
|
delta = _norm_cdf(b) - _norm_cdf(a)
|
|
delta = max(delta, 0)
|
|
return delta
|
|
|
|
def _truncnorm_get_delta(a, b):
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
return _truncnorm_get_delta_scalar(a, b)
|
|
a, b = np.atleast_1d(a), np.atleast_1d(b)
|
|
if a.size == 1 and b.size == 1:
|
|
return _truncnorm_get_delta_scalar(a.item(), b.item())
|
|
delta = np.zeros(np.shape(a))
|
|
condinner = (a <= TRUNCNORM_TAIL_X) & (b >= -TRUNCNORM_TAIL_X)
|
|
conda = (a > 0) & condinner
|
|
condb = (a <= 0) & condinner
|
|
if np.any(conda):
|
|
np.place(delta, conda, _norm_sf(a[conda]) - _norm_sf(b[conda]))
|
|
if np.any(condb):
|
|
np.place(delta, condb, _norm_cdf(b[condb]) - _norm_cdf(a[condb]))
|
|
delta[delta < 0] = 0
|
|
return delta
|
|
|
|
def _truncnorm_get_logdelta_scalar(a, b):
|
|
if (a <= TRUNCNORM_TAIL_X) and (b >= -TRUNCNORM_TAIL_X):
|
|
if a > 0:
|
|
delta = _norm_sf(a) - _norm_sf(b)
|
|
else:
|
|
delta = _norm_cdf(b) - _norm_cdf(a)
|
|
delta = max(delta, 0)
|
|
if delta > 0:
|
|
return np.log(delta)
|
|
|
|
if b < 0 or (np.abs(a) >= np.abs(b)):
|
|
nla, nlb = _norm_logcdf(a), _norm_logcdf(b)
|
|
logdelta = nlb + np.log1p(-np.exp(nla - nlb))
|
|
else:
|
|
sla, slb = _norm_logsf(a), _norm_logsf(b)
|
|
logdelta = sla + np.log1p(-np.exp(slb - sla))
|
|
return logdelta
|
|
|
|
|
|
def _truncnorm_logpdf_scalar(x, a, b):
|
|
with np.errstate(invalid='ignore'):
|
|
if np.isscalar(x):
|
|
if x < a:
|
|
return -np.inf
|
|
if x > b:
|
|
return -np.inf
|
|
shp = np.shape(x)
|
|
x = np.atleast_1d(x)
|
|
out = np.full_like(x, np.nan, dtype=np.double)
|
|
condlta, condgtb = (x < a), (x > b)
|
|
if np.any(condlta):
|
|
np.place(out, condlta, -np.inf)
|
|
if np.any(condgtb):
|
|
np.place(out, condgtb, -np.inf)
|
|
cond_inner = ~condlta & ~condgtb
|
|
if np.any(cond_inner):
|
|
_logdelta = _truncnorm_get_logdelta_scalar(a, b)
|
|
np.place(out, cond_inner, _norm_logpdf(x[cond_inner]) - _logdelta)
|
|
return (out[0] if (shp == ()) else out)
|
|
|
|
|
|
def _truncnorm_pdf_scalar(x, a, b):
|
|
with np.errstate(invalid='ignore'):
|
|
if np.isscalar(x):
|
|
if x < a:
|
|
return 0.0
|
|
if x > b:
|
|
return 0.0
|
|
shp = np.shape(x)
|
|
x = np.atleast_1d(x)
|
|
out = np.full_like(x, np.nan, dtype=np.double)
|
|
condlta, condgtb = (x < a), (x > b)
|
|
if np.any(condlta):
|
|
np.place(out, condlta, 0.0)
|
|
if np.any(condgtb):
|
|
np.place(out, condgtb, 0.0)
|
|
cond_inner = ~condlta & ~condgtb
|
|
if np.any(cond_inner):
|
|
delta = _truncnorm_get_delta_scalar(a, b)
|
|
if delta > 0:
|
|
np.place(out, cond_inner, _norm_pdf(x[cond_inner]) / delta)
|
|
else:
|
|
np.place(out, cond_inner,
|
|
np.exp(_truncnorm_logpdf_scalar(x[cond_inner], a, b)))
|
|
return (out[0] if (shp == ()) else out)
|
|
|
|
|
|
def _truncnorm_logcdf_scalar(x, a, b):
|
|
with np.errstate(invalid='ignore'):
|
|
if np.isscalar(x):
|
|
if x <= a:
|
|
return -np.inf
|
|
if x >= b:
|
|
return 0
|
|
shp = np.shape(x)
|
|
x = np.atleast_1d(x)
|
|
out = np.full_like(x, np.nan, dtype=np.double)
|
|
condlea, condgeb = (x <= a), (x >= b)
|
|
if np.any(condlea):
|
|
np.place(out, condlea, -np.inf)
|
|
if np.any(condgeb):
|
|
np.place(out, condgeb, 0.0)
|
|
cond_inner = ~condlea & ~condgeb
|
|
if np.any(cond_inner):
|
|
delta = _truncnorm_get_delta_scalar(a, b)
|
|
if delta > 0:
|
|
np.place(out, cond_inner,
|
|
np.log((_norm_cdf(x[cond_inner]) - _norm_cdf(a)) / delta))
|
|
else:
|
|
with np.errstate(divide='ignore'):
|
|
if a < 0:
|
|
nla, nlb = _norm_logcdf(a), _norm_logcdf(b)
|
|
tab = np.log1p(-np.exp(nla - nlb))
|
|
nlx = _norm_logcdf(x[cond_inner])
|
|
tax = np.log1p(-np.exp(nla - nlx))
|
|
np.place(out, cond_inner, nlx + tax - (nlb + tab))
|
|
else:
|
|
sla = _norm_logsf(a)
|
|
slb = _norm_logsf(b)
|
|
np.place(out, cond_inner,
|
|
np.log1p(-np.exp(_norm_logsf(x[cond_inner]) - sla))
|
|
- np.log1p(-np.exp(slb - sla)))
|
|
return (out[0] if (shp == ()) else out)
|
|
|
|
|
|
def _truncnorm_cdf_scalar(x, a, b):
|
|
with np.errstate(invalid='ignore'):
|
|
if np.isscalar(x):
|
|
if x <= a:
|
|
return -0
|
|
if x >= b:
|
|
return 1
|
|
shp = np.shape(x)
|
|
x = np.atleast_1d(x)
|
|
out = np.full_like(x, np.nan, dtype=np.double)
|
|
condlea, condgeb = (x <= a), (x >= b)
|
|
if np.any(condlea):
|
|
np.place(out, condlea, 0)
|
|
if np.any(condgeb):
|
|
np.place(out, condgeb, 1.0)
|
|
cond_inner = ~condlea & ~condgeb
|
|
if np.any(cond_inner):
|
|
delta = _truncnorm_get_delta_scalar(a, b)
|
|
if delta > 0:
|
|
np.place(out, cond_inner,
|
|
(_norm_cdf(x[cond_inner]) - _norm_cdf(a)) / delta)
|
|
else:
|
|
with np.errstate(divide='ignore'):
|
|
np.place(out, cond_inner,
|
|
np.exp(_truncnorm_logcdf_scalar(x[cond_inner], a, b)))
|
|
return (out[0] if (shp == ()) else out)
|
|
|
|
|
|
def _truncnorm_logsf_scalar(x, a, b):
|
|
with np.errstate(invalid='ignore'):
|
|
if np.isscalar(x):
|
|
if x <= a:
|
|
return 0.0
|
|
if x >= b:
|
|
return -np.inf
|
|
shp = np.shape(x)
|
|
x = np.atleast_1d(x)
|
|
out = np.full_like(x, np.nan, dtype=np.double)
|
|
|
|
condlea, condgeb = (x <= a), (x >= b)
|
|
if np.any(condlea):
|
|
np.place(out, condlea, 0)
|
|
if np.any(condgeb):
|
|
np.place(out, condgeb, -np.inf)
|
|
cond_inner = ~condlea & ~condgeb
|
|
if np.any(cond_inner):
|
|
delta = _truncnorm_get_delta_scalar(a, b)
|
|
if delta > 0:
|
|
np.place(out, cond_inner, np.log((_norm_sf(x[cond_inner]) - _norm_sf(b)) / delta))
|
|
else:
|
|
with np.errstate(divide='ignore'):
|
|
if b < 0:
|
|
nla, nlb = _norm_logcdf(a), _norm_logcdf(b)
|
|
np.place(out, cond_inner,
|
|
np.log1p(-np.exp(_norm_logcdf(x[cond_inner]) - nlb))
|
|
- np.log1p(-np.exp(nla - nlb)))
|
|
else:
|
|
sla, slb = _norm_logsf(a), _norm_logsf(b)
|
|
tab = np.log1p(-np.exp(slb - sla))
|
|
slx = _norm_logsf(x[cond_inner])
|
|
tax = np.log1p(-np.exp(slb - slx))
|
|
np.place(out, cond_inner, slx + tax - (sla + tab))
|
|
return (out[0] if (shp == ()) else out)
|
|
|
|
|
|
def _truncnorm_sf_scalar(x, a, b):
|
|
with np.errstate(invalid='ignore'):
|
|
if np.isscalar(x):
|
|
if x <= a:
|
|
return 1.0
|
|
if x >= b:
|
|
return 0.0
|
|
shp = np.shape(x)
|
|
x = np.atleast_1d(x)
|
|
out = np.full_like(x, np.nan, dtype=np.double)
|
|
|
|
condlea, condgeb = (x <= a), (x >= b)
|
|
if np.any(condlea):
|
|
np.place(out, condlea, 1.0)
|
|
if np.any(condgeb):
|
|
np.place(out, condgeb, 0.0)
|
|
cond_inner = ~condlea & ~condgeb
|
|
if np.any(cond_inner):
|
|
delta = _truncnorm_get_delta_scalar(a, b)
|
|
if delta > 0:
|
|
np.place(out, cond_inner, (_norm_sf(x[cond_inner]) - _norm_sf(b)) / delta)
|
|
else:
|
|
np.place(out, cond_inner, np.exp(_truncnorm_logsf_scalar(x[cond_inner], a, b)))
|
|
return (out[0] if (shp == ()) else out)
|
|
|
|
|
|
def _norm_logcdfprime(z):
|
|
# derivative of special.log_ndtr (See special/cephes/ndtr.c)
|
|
# Differentiate formula for log Phi(z)_truncnorm_ppf
|
|
# log Phi(z) = -z^2/2 - log(-z) - log(2pi)/2 + log(1 + sum (-1)^n (2n-1)!! / z^(2n))
|
|
# Convergence of series is slow for |z| < 10, but can use d(log Phi(z))/dz = dPhi(z)/dz / Phi(z)
|
|
# Just take the first 10 terms because that is sufficient for use in _norm_ilogcdf
|
|
assert np.all(z <= -10)
|
|
lhs = -z - 1/z
|
|
denom_cons = 1/z**2
|
|
numerator = 1
|
|
pwr = 1.0
|
|
denom_total, numerator_total = 0, 0
|
|
sign = -1
|
|
for i in range(1, 11):
|
|
pwr *= denom_cons
|
|
numerator *= 2 * i - 1
|
|
term = sign * numerator * pwr
|
|
denom_total += term
|
|
numerator_total += term * (2 * i) / z
|
|
sign = -sign
|
|
return lhs - numerator_total / (1 + denom_total)
|
|
|
|
def _norm_ilogcdf(y):
|
|
"""Inverse function to _norm_logcdf==sc.log_ndtr."""
|
|
# Apply approximate Newton-Raphson
|
|
# Only use for very negative values of y.
|
|
# At minimum requires y <= -(log(2pi)+2^2)/2 ~= -2.9
|
|
# Much better convergence for y <= -10
|
|
z = -np.sqrt(-2 * (y + np.log(2*np.pi)/2))
|
|
for _ in range(4):
|
|
z = z - (_norm_logcdf(z) - y) / _norm_logcdfprime(z)
|
|
return z
|
|
|
|
|
|
def _truncnorm_ppf_scalar(q, a, b):
|
|
shp = np.shape(q)
|
|
q = np.atleast_1d(q)
|
|
out = np.zeros(np.shape(q))
|
|
condle0, condge1 = (q <= 0), (q >= 1)
|
|
if np.any(condle0):
|
|
out[condle0] = a
|
|
if np.any(condge1):
|
|
out[condge1] = b
|
|
delta = _truncnorm_get_delta_scalar(a, b)
|
|
cond_inner = ~condle0 & ~condge1
|
|
if np.any(cond_inner):
|
|
qinner = q[cond_inner]
|
|
if delta > 0:
|
|
if a > 0:
|
|
sa, sb = _norm_sf(a), _norm_sf(b)
|
|
np.place(out, cond_inner,
|
|
_norm_isf(qinner * sb + sa * (1.0 - qinner)))
|
|
else:
|
|
na, nb = _norm_cdf(a), _norm_cdf(b)
|
|
np.place(out, cond_inner, _norm_ppf(qinner * nb + na * (1.0 - qinner)))
|
|
elif np.isinf(b):
|
|
np.place(out, cond_inner,
|
|
-_norm_ilogcdf(np.log1p(-qinner) + _norm_logsf(a)))
|
|
elif np.isinf(a):
|
|
np.place(out, cond_inner,
|
|
_norm_ilogcdf(np.log(q) + _norm_logcdf(b)))
|
|
else:
|
|
if b < 0:
|
|
# Solve norm_logcdf(x) = norm_logcdf(a) + log1p(q * (expm1(norm_logcdf(b) - norm_logcdf(a)))
|
|
# = nla + log1p(q * expm1(nlb - nla))
|
|
# = nlb + log(q) + log1p((1-q) * exp(nla - nlb)/q)
|
|
def _f_cdf(x, c):
|
|
return _norm_logcdf(x) - c
|
|
|
|
nla, nlb = _norm_logcdf(a), _norm_logcdf(b)
|
|
values = nlb + np.log(q[cond_inner])
|
|
C = np.exp(nla - nlb)
|
|
if C:
|
|
one_minus_q = (1 - q)[cond_inner]
|
|
values += np.log1p(one_minus_q * C / q[cond_inner])
|
|
x = [optimize.zeros.brentq(_f_cdf, a, b, args=(c,),
|
|
maxiter=TRUNCNORM_MAX_BRENT_ITERS)for c in values]
|
|
np.place(out, cond_inner, x)
|
|
else:
|
|
# Solve norm_logsf(x) = norm_logsf(b) + log1p((1-q) * (expm1(norm_logsf(a) - norm_logsf(b)))
|
|
# = slb + log1p((1-q)[cond_inner] * expm1(sla - slb))
|
|
# = sla + log(1-q) + log1p(q * np.exp(slb - sla)/(1-q))
|
|
def _f_sf(x, c):
|
|
return _norm_logsf(x) - c
|
|
|
|
sla, slb = _norm_logsf(a), _norm_logsf(b)
|
|
one_minus_q = (1-q)[cond_inner]
|
|
values = sla + np.log(one_minus_q)
|
|
C = np.exp(slb - sla)
|
|
if C:
|
|
values += np.log1p(q[cond_inner] * C / one_minus_q)
|
|
x = [optimize.zeros.brentq(_f_sf, a, b, args=(c,),
|
|
maxiter=TRUNCNORM_MAX_BRENT_ITERS) for c in values]
|
|
np.place(out, cond_inner, x)
|
|
out[out < a] = a
|
|
out[out > b] = b
|
|
return (out[0] if (shp == ()) else out)
|
|
|
|
|
|
class truncnorm_gen(rv_continuous):
|
|
r"""A truncated normal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The standard form of this distribution is a standard normal truncated to
|
|
the range [a, b] --- notice that a and b are defined over the domain of the
|
|
standard normal. To convert clip values for a specific mean and standard
|
|
deviation, use::
|
|
|
|
a, b = (myclip_a - my_mean) / my_std, (myclip_b - my_mean) / my_std
|
|
|
|
`truncnorm` takes :math:`a` and :math:`b` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, a, b):
|
|
return a < b
|
|
|
|
def _get_support(self, a, b):
|
|
return a, b
|
|
|
|
def _pdf(self, x, a, b):
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
return _truncnorm_pdf_scalar(x, a, b)
|
|
a, b = np.atleast_1d(a), np.atleast_1d(b)
|
|
if a.size == 1 and b.size == 1:
|
|
return _truncnorm_pdf_scalar(x, a.item(), b.item())
|
|
it = np.nditer([x, a, b, None], [],
|
|
[['readonly'], ['readonly'], ['readonly'], ['writeonly','allocate']])
|
|
for (_x, _a, _b, _ld) in it:
|
|
_ld[...] = _truncnorm_pdf_scalar(_x, _a, _b)
|
|
return it.operands[3]
|
|
|
|
def _logpdf(self, x, a, b):
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
return _truncnorm_logpdf_scalar(x, a, b)
|
|
a, b = np.atleast_1d(a), np.atleast_1d(b)
|
|
if a.size == 1 and b.size == 1:
|
|
return _truncnorm_logpdf_scalar(x, a.item(), b.item())
|
|
it = np.nditer([x, a, b, None], [],
|
|
[['readonly'], ['readonly'], ['readonly'], ['writeonly','allocate']])
|
|
for (_x, _a, _b, _ld) in it:
|
|
_ld[...] = _truncnorm_logpdf_scalar(_x, _a, _b)
|
|
return it.operands[3]
|
|
|
|
def _cdf(self, x, a, b):
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
return _truncnorm_cdf_scalar(x, a, b)
|
|
a, b = np.atleast_1d(a), np.atleast_1d(b)
|
|
if a.size == 1 and b.size == 1:
|
|
return _truncnorm_cdf_scalar(x, a.item(), b.item())
|
|
out = None
|
|
it = np.nditer([x, a, b, out], [],
|
|
[['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']])
|
|
for (_x, _a, _b, _p) in it:
|
|
_p[...] = _truncnorm_cdf_scalar(_x, _a, _b)
|
|
return it.operands[3]
|
|
|
|
def _logcdf(self, x, a, b):
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
return _truncnorm_logcdf_scalar(x, a, b)
|
|
a, b = np.atleast_1d(a), np.atleast_1d(b)
|
|
if a.size == 1 and b.size == 1:
|
|
return _truncnorm_logcdf_scalar(x, a.item(), b.item())
|
|
it = np.nditer([x, a, b, None], [],
|
|
[['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']])
|
|
for (_x, _a, _b, _p) in it:
|
|
_p[...] = _truncnorm_logcdf_scalar(_x, _a, _b)
|
|
return it.operands[3]
|
|
|
|
def _sf(self, x, a, b):
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
return _truncnorm_sf_scalar(x, a, b)
|
|
a, b = np.atleast_1d(a), np.atleast_1d(b)
|
|
if a.size == 1 and b.size == 1:
|
|
return _truncnorm_sf_scalar(x, a.item(), b.item())
|
|
out = None
|
|
it = np.nditer([x, a, b, out], [],
|
|
[['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']])
|
|
for (_x, _a, _b, _p) in it:
|
|
_p[...] = _truncnorm_sf_scalar(_x, _a, _b)
|
|
return it.operands[3]
|
|
|
|
def _logsf(self, x, a, b):
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
return _truncnorm_logsf_scalar(x, a, b)
|
|
a, b = np.atleast_1d(a), np.atleast_1d(b)
|
|
if a.size == 1 and b.size == 1:
|
|
return _truncnorm_logsf_scalar(x, a.item(), b.item())
|
|
out = None
|
|
it = np.nditer([x, a, b, out], [],
|
|
[['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']])
|
|
for (_x, _a, _b, _p) in it:
|
|
_p[...] = _truncnorm_logsf_scalar(_x, _a, _b)
|
|
return it.operands[3]
|
|
|
|
def _ppf(self, q, a, b):
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
return _truncnorm_ppf_scalar(q, a, b)
|
|
a, b = np.atleast_1d(a), np.atleast_1d(b)
|
|
if a.size == 1 and b.size == 1:
|
|
return _truncnorm_ppf_scalar(q, a.item(), b.item())
|
|
|
|
out = None
|
|
it = np.nditer([q, a, b, out], [],
|
|
[['readonly'], ['readonly'], ['readonly'], ['writeonly', 'allocate']])
|
|
for (_q, _a, _b, _x) in it:
|
|
_x[...] = _truncnorm_ppf_scalar(_q, _a, _b)
|
|
return it.operands[3]
|
|
|
|
def _munp(self, n, a, b):
|
|
def n_th_moment(n, a, b):
|
|
"""
|
|
Returns n-th moment. Defined only if n >= 0.
|
|
Function cannot broadcast due to the loop over n
|
|
"""
|
|
pA, pB = self._pdf([a, b], a, b)
|
|
probs = [pA, -pB]
|
|
moments = [0, 1]
|
|
for k in range(1, n+1):
|
|
# a or b might be infinite, and the corresponding pdf value
|
|
# is 0 in that case, but nan is returned for the
|
|
# multiplication. However, as b->infinity, pdf(b)*b**k -> 0.
|
|
# So it is safe to use _lazywhere to avoid the nan.
|
|
vals = _lazywhere(probs, [probs, [a, b]],
|
|
lambda x, y: x * y**(k-1), fillvalue=0)
|
|
mk = np.sum(vals) + (k-1) * moments[-2]
|
|
moments.append(mk)
|
|
return moments[-1]
|
|
|
|
return _lazywhere((n >= 0) & (a == a) & (b == b), (n, a, b),
|
|
np.vectorize(n_th_moment, otypes=[np.float64]),
|
|
np.nan)
|
|
|
|
def _stats(self, a, b, moments='mv'):
|
|
pA, pB = self._pdf(np.array([a, b]), a, b)
|
|
m1 = pA - pB
|
|
mu = m1
|
|
# use _lazywhere to avoid nan (See detailed comment in _munp)
|
|
probs = [pA, -pB]
|
|
vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y,
|
|
fillvalue=0)
|
|
m2 = 1 + np.sum(vals)
|
|
vals = _lazywhere(probs, [probs, [a-mu, b-mu]], lambda x, y: x*y,
|
|
fillvalue=0)
|
|
# mu2 = m2 - mu**2, but not as numerically stable as:
|
|
# mu2 = (a-mu)*pA - (b-mu)*pB + 1
|
|
mu2 = 1 + np.sum(vals)
|
|
vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y**2,
|
|
fillvalue=0)
|
|
m3 = 2*m1 + np.sum(vals)
|
|
vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y**3,
|
|
fillvalue=0)
|
|
m4 = 3*m2 + np.sum(vals)
|
|
|
|
mu3 = m3 + m1 * (-3*m2 + 2*m1**2)
|
|
g1 = mu3 / np.power(mu2, 1.5)
|
|
mu4 = m4 + m1*(-4*m3 + 3*m1*(2*m2 - m1**2))
|
|
g2 = mu4 / mu2**2 - 3
|
|
return mu, mu2, g1, g2
|
|
|
|
def _rvs(self, a, b, size=None, random_state=None):
|
|
# if a and b are scalar, use _rvs_scalar, otherwise need to create
|
|
# output by iterating over parameters
|
|
if np.isscalar(a) and np.isscalar(b):
|
|
out = self._rvs_scalar(a, b, size, random_state=random_state)
|
|
elif a.size == 1 and b.size == 1:
|
|
out = self._rvs_scalar(a.item(), b.item(), size, random_state=random_state)
|
|
else:
|
|
# When this method is called, size will be a (possibly empty)
|
|
# tuple of integers. It will not be None; if `size=None` is passed
|
|
# to `rvs()`, size will be the empty tuple ().
|
|
|
|
a, b = np.broadcast_arrays(a, b)
|
|
# a and b now have the same shape.
|
|
|
|
# `shp` is the shape of the blocks of random variates that are
|
|
# generated for each combination of parameters associated with
|
|
# broadcasting a and b.
|
|
# bc is a tuple the same length as size. The values
|
|
# in bc are bools. If bc[j] is True, it means that
|
|
# entire axis is filled in for a given combination of the
|
|
# broadcast arguments.
|
|
shp, bc = _check_shape(a.shape, size)
|
|
|
|
# `numsamples` is the total number of variates to be generated
|
|
# for each combination of the input arguments.
|
|
numsamples = int(np.prod(shp))
|
|
|
|
# `out` is the array to be returned. It is filled in in the
|
|
# loop below.
|
|
out = np.empty(size)
|
|
|
|
it = np.nditer([a, b],
|
|
flags=['multi_index'],
|
|
op_flags=[['readonly'], ['readonly']])
|
|
while not it.finished:
|
|
# Convert the iterator's multi_index into an index into the
|
|
# `out` array where the call to _rvs_scalar() will be stored.
|
|
# Where bc is True, we use a full slice; otherwise we use the
|
|
# index value from it.multi_index. len(it.multi_index) might
|
|
# be less than len(bc), and in that case we want to align these
|
|
# two sequences to the right, so the loop variable j runs from
|
|
# -len(size) to 0. This doesn't cause an IndexError, as
|
|
# bc[j] will be True in those cases where it.multi_index[j]
|
|
# would cause an IndexError.
|
|
idx = tuple((it.multi_index[j] if not bc[j] else slice(None))
|
|
for j in range(-len(size), 0))
|
|
out[idx] = self._rvs_scalar(it[0], it[1], numsamples, random_state).reshape(shp)
|
|
it.iternext()
|
|
|
|
if size == ():
|
|
out = out.item()
|
|
return out
|
|
|
|
def _rvs_scalar(self, a, b, numsamples=None, random_state=None):
|
|
if not numsamples:
|
|
numsamples = 1
|
|
|
|
# prepare sampling of rvs
|
|
size1d = tuple(np.atleast_1d(numsamples))
|
|
N = np.prod(size1d) # number of rvs needed, reshape upon return
|
|
# Calculate some rvs
|
|
U = random_state.uniform(low=0, high=1, size=N)
|
|
x = self._ppf(U, a, b)
|
|
rvs = np.reshape(x, size1d)
|
|
return rvs
|
|
|
|
|
|
truncnorm = truncnorm_gen(name='truncnorm', momtype=1)
|
|
|
|
|
|
# FIXME: RVS does not work.
|
|
class tukeylambda_gen(rv_continuous):
|
|
r"""A Tukey-Lamdba continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
A flexible distribution, able to represent and interpolate between the
|
|
following distributions:
|
|
|
|
- Cauchy (:math:`lambda = -1`)
|
|
- logistic (:math:`lambda = 0`)
|
|
- approx Normal (:math:`lambda = 0.14`)
|
|
- uniform from -1 to 1 (:math:`lambda = 1`)
|
|
|
|
`tukeylambda` takes a real number :math:`lambda` (denoted ``lam``
|
|
in the implementation) as a shape parameter.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, lam):
|
|
return np.ones(np.shape(lam), dtype=bool)
|
|
|
|
def _pdf(self, x, lam):
|
|
Fx = np.asarray(sc.tklmbda(x, lam))
|
|
Px = Fx**(lam-1.0) + (np.asarray(1-Fx))**(lam-1.0)
|
|
Px = 1.0/np.asarray(Px)
|
|
return np.where((lam <= 0) | (abs(x) < 1.0/np.asarray(lam)), Px, 0.0)
|
|
|
|
def _cdf(self, x, lam):
|
|
return sc.tklmbda(x, lam)
|
|
|
|
def _ppf(self, q, lam):
|
|
return sc.boxcox(q, lam) - sc.boxcox1p(-q, lam)
|
|
|
|
def _stats(self, lam):
|
|
return 0, _tlvar(lam), 0, _tlkurt(lam)
|
|
|
|
def _entropy(self, lam):
|
|
def integ(p):
|
|
return np.log(pow(p, lam-1)+pow(1-p, lam-1))
|
|
return integrate.quad(integ, 0, 1)[0]
|
|
|
|
|
|
tukeylambda = tukeylambda_gen(name='tukeylambda')
|
|
|
|
|
|
class FitUniformFixedScaleDataError(FitDataError):
|
|
def __init__(self, ptp, fscale):
|
|
self.args = (
|
|
"Invalid values in `data`. Maximum likelihood estimation with "
|
|
"the uniform distribution and fixed scale requires that "
|
|
"data.ptp() <= fscale, but data.ptp() = %r and fscale = %r." %
|
|
(ptp, fscale),
|
|
)
|
|
|
|
|
|
class uniform_gen(rv_continuous):
|
|
r"""A uniform continuous random variable.
|
|
|
|
In the standard form, the distribution is uniform on ``[0, 1]``. Using
|
|
the parameters ``loc`` and ``scale``, one obtains the uniform distribution
|
|
on ``[loc, loc + scale]``.
|
|
|
|
%(before_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, size=None, random_state=None):
|
|
return random_state.uniform(0.0, 1.0, size)
|
|
|
|
def _pdf(self, x):
|
|
return 1.0*(x == x)
|
|
|
|
def _cdf(self, x):
|
|
return x
|
|
|
|
def _ppf(self, q):
|
|
return q
|
|
|
|
def _stats(self):
|
|
return 0.5, 1.0/12, 0, -1.2
|
|
|
|
def _entropy(self):
|
|
return 0.0
|
|
|
|
def fit(self, data, *args, **kwds):
|
|
"""
|
|
Maximum likelihood estimate for the location and scale parameters.
|
|
|
|
`uniform.fit` uses only the following parameters. Because exact
|
|
formulas are used, the parameters related to optimization that are
|
|
available in the `fit` method of other distributions are ignored
|
|
here. The only positional argument accepted is `data`.
|
|
|
|
Parameters
|
|
----------
|
|
data : array_like
|
|
Data to use in calculating the maximum likelihood estimate.
|
|
floc : float, optional
|
|
Hold the location parameter fixed to the specified value.
|
|
fscale : float, optional
|
|
Hold the scale parameter fixed to the specified value.
|
|
|
|
Returns
|
|
-------
|
|
loc, scale : float
|
|
Maximum likelihood estimates for the location and scale.
|
|
|
|
Notes
|
|
-----
|
|
An error is raised if `floc` is given and any values in `data` are
|
|
less than `floc`, or if `fscale` is given and `fscale` is less
|
|
than ``data.max() - data.min()``. An error is also raised if both
|
|
`floc` and `fscale` are given.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.stats import uniform
|
|
|
|
We'll fit the uniform distribution to `x`:
|
|
|
|
>>> x = np.array([2, 2.5, 3.1, 9.5, 13.0])
|
|
|
|
For a uniform distribution MLE, the location is the minimum of the
|
|
data, and the scale is the maximum minus the minimum.
|
|
|
|
>>> loc, scale = uniform.fit(x)
|
|
>>> loc
|
|
2.0
|
|
>>> scale
|
|
11.0
|
|
|
|
If we know the data comes from a uniform distribution where the support
|
|
starts at 0, we can use `floc=0`:
|
|
|
|
>>> loc, scale = uniform.fit(x, floc=0)
|
|
>>> loc
|
|
0.0
|
|
>>> scale
|
|
13.0
|
|
|
|
Alternatively, if we know the length of the support is 12, we can use
|
|
`fscale=12`:
|
|
|
|
>>> loc, scale = uniform.fit(x, fscale=12)
|
|
>>> loc
|
|
1.5
|
|
>>> scale
|
|
12.0
|
|
|
|
In that last example, the support interval is [1.5, 13.5]. This
|
|
solution is not unique. For example, the distribution with ``loc=2``
|
|
and ``scale=12`` has the same likelihood as the one above. When
|
|
`fscale` is given and it is larger than ``data.max() - data.min()``,
|
|
the parameters returned by the `fit` method center the support over
|
|
the interval ``[data.min(), data.max()]``.
|
|
|
|
"""
|
|
if len(args) > 0:
|
|
raise TypeError("Too many arguments.")
|
|
|
|
floc = kwds.pop('floc', None)
|
|
fscale = kwds.pop('fscale', None)
|
|
|
|
_remove_optimizer_parameters(kwds)
|
|
|
|
if floc is not None and fscale is not None:
|
|
# This check is for consistency with `rv_continuous.fit`.
|
|
raise ValueError("All parameters fixed. There is nothing to "
|
|
"optimize.")
|
|
|
|
data = np.asarray(data)
|
|
|
|
if not np.isfinite(data).all():
|
|
raise RuntimeError("The data contains non-finite values.")
|
|
|
|
# MLE for the uniform distribution
|
|
# --------------------------------
|
|
# The PDF is
|
|
#
|
|
# f(x, loc, scale) = {1/scale for loc <= x <= loc + scale
|
|
# {0 otherwise}
|
|
#
|
|
# The likelihood function is
|
|
# L(x, loc, scale) = (1/scale)**n
|
|
# where n is len(x), assuming loc <= x <= loc + scale for all x.
|
|
# The log-likelihood is
|
|
# l(x, loc, scale) = -n*log(scale)
|
|
# The log-likelihood is maximized by making scale as small as possible,
|
|
# while keeping loc <= x <= loc + scale. So if neither loc nor scale
|
|
# are fixed, the log-likelihood is maximized by choosing
|
|
# loc = x.min()
|
|
# scale = x.ptp()
|
|
# If loc is fixed, it must be less than or equal to x.min(), and then
|
|
# the scale is
|
|
# scale = x.max() - loc
|
|
# If scale is fixed, it must not be less than x.ptp(). If scale is
|
|
# greater than x.ptp(), the solution is not unique. Note that the
|
|
# likelihood does not depend on loc, except for the requirement that
|
|
# loc <= x <= loc + scale. All choices of loc for which
|
|
# x.max() - scale <= loc <= x.min()
|
|
# have the same log-likelihood. In this case, we choose loc such that
|
|
# the support is centered over the interval [data.min(), data.max()]:
|
|
# loc = x.min() = 0.5*(scale - x.ptp())
|
|
|
|
if fscale is None:
|
|
# scale is not fixed.
|
|
if floc is None:
|
|
# loc is not fixed, scale is not fixed.
|
|
loc = data.min()
|
|
scale = data.ptp()
|
|
else:
|
|
# loc is fixed, scale is not fixed.
|
|
loc = floc
|
|
scale = data.max() - loc
|
|
if data.min() < loc:
|
|
raise FitDataError("uniform", lower=loc, upper=loc + scale)
|
|
else:
|
|
# loc is not fixed, scale is fixed.
|
|
ptp = data.ptp()
|
|
if ptp > fscale:
|
|
raise FitUniformFixedScaleDataError(ptp=ptp, fscale=fscale)
|
|
# If ptp < fscale, the ML estimate is not unique; see the comments
|
|
# above. We choose the distribution for which the support is
|
|
# centered over the interval [data.min(), data.max()].
|
|
loc = data.min() - 0.5*(fscale - ptp)
|
|
scale = fscale
|
|
|
|
# We expect the return values to be floating point, so ensure it
|
|
# by explicitly converting to float.
|
|
return float(loc), float(scale)
|
|
|
|
|
|
uniform = uniform_gen(a=0.0, b=1.0, name='uniform')
|
|
|
|
|
|
class vonmises_gen(rv_continuous):
|
|
r"""A Von Mises continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `vonmises` and `vonmises_line` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }
|
|
|
|
for :math:`-\pi \le x \le \pi`, :math:`\kappa > 0`. :math:`I_0` is the
|
|
modified Bessel function of order zero (`scipy.special.i0`).
|
|
|
|
`vonmises` is a circular distribution which does not restrict the
|
|
distribution to a fixed interval. Currently, there is no circular
|
|
distribution framework in scipy. The ``cdf`` is implemented such that
|
|
``cdf(x + 2*np.pi) == cdf(x) + 1``.
|
|
|
|
`vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]`
|
|
on the real line. This is a regular (i.e. non-circular) distribution.
|
|
|
|
`vonmises` and `vonmises_line` take ``kappa`` as a shape parameter.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _rvs(self, kappa, size=None, random_state=None):
|
|
return random_state.vonmises(0.0, kappa, size=size)
|
|
|
|
def _pdf(self, x, kappa):
|
|
# vonmises.pdf(x, kappa) = exp(kappa * cos(x)) / (2*pi*I[0](kappa))
|
|
# = exp(kappa * (cos(x) - 1)) /
|
|
# (2*pi*exp(-kappa)*I[0](kappa))
|
|
# = exp(kappa * cosm1(x)) / (2*pi*i0e(kappa))
|
|
return np.exp(kappa*sc.cosm1(x)) / (2*np.pi*sc.i0e(kappa))
|
|
|
|
def _cdf(self, x, kappa):
|
|
return _stats.von_mises_cdf(kappa, x)
|
|
|
|
def _stats_skip(self, kappa):
|
|
return 0, None, 0, None
|
|
|
|
def _entropy(self, kappa):
|
|
return (-kappa * sc.i1(kappa) / sc.i0(kappa) +
|
|
np.log(2 * np.pi * sc.i0(kappa)))
|
|
|
|
|
|
vonmises = vonmises_gen(name='vonmises')
|
|
vonmises_line = vonmises_gen(a=-np.pi, b=np.pi, name='vonmises_line')
|
|
|
|
|
|
class wald_gen(invgauss_gen):
|
|
r"""A Wald continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `wald` is:
|
|
|
|
.. math::
|
|
|
|
f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x })
|
|
|
|
for :math:`x >= 0`.
|
|
|
|
`wald` is a special case of `invgauss` with ``mu=1``.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
"""
|
|
_support_mask = rv_continuous._open_support_mask
|
|
|
|
def _rvs(self, size=None, random_state=None):
|
|
return random_state.wald(1.0, 1.0, size=size)
|
|
|
|
def _pdf(self, x):
|
|
# wald.pdf(x) = 1/sqrt(2*pi*x**3) * exp(-(x-1)**2/(2*x))
|
|
return invgauss._pdf(x, 1.0)
|
|
|
|
def _logpdf(self, x):
|
|
return invgauss._logpdf(x, 1.0)
|
|
|
|
def _cdf(self, x):
|
|
return invgauss._cdf(x, 1.0)
|
|
|
|
def _stats(self):
|
|
return 1.0, 1.0, 3.0, 15.0
|
|
|
|
|
|
wald = wald_gen(a=0.0, name="wald")
|
|
|
|
|
|
class wrapcauchy_gen(rv_continuous):
|
|
r"""A wrapped Cauchy continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `wrapcauchy` is:
|
|
|
|
.. math::
|
|
|
|
f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))}
|
|
|
|
for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`.
|
|
|
|
`wrapcauchy` takes ``c`` as a shape parameter for :math:`c`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, c):
|
|
return (c > 0) & (c < 1)
|
|
|
|
def _pdf(self, x, c):
|
|
# wrapcauchy.pdf(x, c) = (1-c**2) / (2*pi*(1+c**2-2*c*cos(x)))
|
|
return (1.0-c*c)/(2*np.pi*(1+c*c-2*c*np.cos(x)))
|
|
|
|
def _cdf(self, x, c):
|
|
output = np.zeros(x.shape, dtype=x.dtype)
|
|
val = (1.0+c)/(1.0-c)
|
|
c1 = x < np.pi
|
|
c2 = 1-c1
|
|
xp = np.extract(c1, x)
|
|
xn = np.extract(c2, x)
|
|
if np.any(xn):
|
|
valn = np.extract(c2, np.ones_like(x)*val)
|
|
xn = 2*np.pi - xn
|
|
yn = np.tan(xn/2.0)
|
|
on = 1.0-1.0/np.pi*np.arctan(valn*yn)
|
|
np.place(output, c2, on)
|
|
if np.any(xp):
|
|
valp = np.extract(c1, np.ones_like(x)*val)
|
|
yp = np.tan(xp/2.0)
|
|
op = 1.0/np.pi*np.arctan(valp*yp)
|
|
np.place(output, c1, op)
|
|
return output
|
|
|
|
def _ppf(self, q, c):
|
|
val = (1.0-c)/(1.0+c)
|
|
rcq = 2*np.arctan(val*np.tan(np.pi*q))
|
|
rcmq = 2*np.pi-2*np.arctan(val*np.tan(np.pi*(1-q)))
|
|
return np.where(q < 1.0/2, rcq, rcmq)
|
|
|
|
def _entropy(self, c):
|
|
return np.log(2*np.pi*(1-c*c))
|
|
|
|
|
|
wrapcauchy = wrapcauchy_gen(a=0.0, b=2*np.pi, name='wrapcauchy')
|
|
|
|
|
|
class gennorm_gen(rv_continuous):
|
|
r"""A generalized normal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
laplace : Laplace distribution
|
|
norm : normal distribution
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `gennorm` is [1]_:
|
|
|
|
.. math::
|
|
|
|
f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta)
|
|
|
|
:math:`\Gamma` is the gamma function (`scipy.special.gamma`).
|
|
|
|
`gennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
|
|
For :math:`\beta = 1`, it is identical to a Laplace distribution.
|
|
For :math:`\beta = 2`, it is identical to a normal distribution
|
|
(with ``scale=1/sqrt(2)``).
|
|
|
|
References
|
|
----------
|
|
|
|
.. [1] "Generalized normal distribution, Version 1",
|
|
https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _pdf(self, x, beta):
|
|
return np.exp(self._logpdf(x, beta))
|
|
|
|
def _logpdf(self, x, beta):
|
|
return np.log(0.5*beta) - sc.gammaln(1.0/beta) - abs(x)**beta
|
|
|
|
def _cdf(self, x, beta):
|
|
c = 0.5 * np.sign(x)
|
|
# evaluating (.5 + c) first prevents numerical cancellation
|
|
return (0.5 + c) - c * sc.gammaincc(1.0/beta, abs(x)**beta)
|
|
|
|
def _ppf(self, x, beta):
|
|
c = np.sign(x - 0.5)
|
|
# evaluating (1. + c) first prevents numerical cancellation
|
|
return c * sc.gammainccinv(1.0/beta, (1.0 + c) - 2.0*c*x)**(1.0/beta)
|
|
|
|
def _sf(self, x, beta):
|
|
return self._cdf(-x, beta)
|
|
|
|
def _isf(self, x, beta):
|
|
return -self._ppf(x, beta)
|
|
|
|
def _stats(self, beta):
|
|
c1, c3, c5 = sc.gammaln([1.0/beta, 3.0/beta, 5.0/beta])
|
|
return 0., np.exp(c3 - c1), 0., np.exp(c5 + c1 - 2.0*c3) - 3.
|
|
|
|
def _entropy(self, beta):
|
|
return 1. / beta - np.log(.5 * beta) + sc.gammaln(1. / beta)
|
|
|
|
|
|
gennorm = gennorm_gen(name='gennorm')
|
|
|
|
|
|
class halfgennorm_gen(rv_continuous):
|
|
r"""The upper half of a generalized normal continuous random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
gennorm : generalized normal distribution
|
|
expon : exponential distribution
|
|
halfnorm : half normal distribution
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `halfgennorm` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \beta) = \frac{\beta}{\Gamma(1/\beta)} \exp(-|x|^\beta)
|
|
|
|
for :math:`x > 0`. :math:`\Gamma` is the gamma function
|
|
(`scipy.special.gamma`).
|
|
|
|
`gennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
|
|
For :math:`\beta = 1`, it is identical to an exponential distribution.
|
|
For :math:`\beta = 2`, it is identical to a half normal distribution
|
|
(with ``scale=1/sqrt(2)``).
|
|
|
|
References
|
|
----------
|
|
|
|
.. [1] "Generalized normal distribution, Version 1",
|
|
https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _pdf(self, x, beta):
|
|
# beta
|
|
# halfgennorm.pdf(x, beta) = ------------- exp(-|x|**beta)
|
|
# gamma(1/beta)
|
|
return np.exp(self._logpdf(x, beta))
|
|
|
|
def _logpdf(self, x, beta):
|
|
return np.log(beta) - sc.gammaln(1.0/beta) - x**beta
|
|
|
|
def _cdf(self, x, beta):
|
|
return sc.gammainc(1.0/beta, x**beta)
|
|
|
|
def _ppf(self, x, beta):
|
|
return sc.gammaincinv(1.0/beta, x)**(1.0/beta)
|
|
|
|
def _sf(self, x, beta):
|
|
return sc.gammaincc(1.0/beta, x**beta)
|
|
|
|
def _isf(self, x, beta):
|
|
return sc.gammainccinv(1.0/beta, x)**(1.0/beta)
|
|
|
|
def _entropy(self, beta):
|
|
return 1.0/beta - np.log(beta) + sc.gammaln(1.0/beta)
|
|
|
|
|
|
halfgennorm = halfgennorm_gen(a=0, name='halfgennorm')
|
|
|
|
|
|
class crystalball_gen(rv_continuous):
|
|
r"""
|
|
Crystalball distribution
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `crystalball` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \beta, m) = \begin{cases}
|
|
N \exp(-x^2 / 2), &\text{for } x > -\beta\\
|
|
N A (B - x)^{-m} &\text{for } x \le -\beta
|
|
\end{cases}
|
|
|
|
where :math:`A = (m / |\beta|)^n \exp(-\beta^2 / 2)`,
|
|
:math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant.
|
|
|
|
`crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape
|
|
parameters. :math:`\beta` defines the point where the pdf changes
|
|
from a power-law to a Gaussian distribution. :math:`m` is the power
|
|
of the power-law tail.
|
|
|
|
References
|
|
----------
|
|
.. [1] "Crystal Ball Function",
|
|
https://en.wikipedia.org/wiki/Crystal_Ball_function
|
|
|
|
%(after_notes)s
|
|
|
|
.. versionadded:: 0.19.0
|
|
|
|
%(example)s
|
|
"""
|
|
|
|
def _pdf(self, x, beta, m):
|
|
"""
|
|
Return PDF of the crystalball function.
|
|
|
|
--
|
|
| exp(-x**2 / 2), for x > -beta
|
|
crystalball.pdf(x, beta, m) = N * |
|
|
| A * (B - x)**(-m), for x <= -beta
|
|
--
|
|
"""
|
|
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
|
|
_norm_pdf_C * _norm_cdf(beta))
|
|
|
|
def rhs(x, beta, m):
|
|
return np.exp(-x**2 / 2)
|
|
|
|
def lhs(x, beta, m):
|
|
return ((m/beta)**m * np.exp(-beta**2 / 2.0) *
|
|
(m/beta - beta - x)**(-m))
|
|
|
|
return N * _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs)
|
|
|
|
def _logpdf(self, x, beta, m):
|
|
"""
|
|
Return the log of the PDF of the crystalball function.
|
|
"""
|
|
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
|
|
_norm_pdf_C * _norm_cdf(beta))
|
|
|
|
def rhs(x, beta, m):
|
|
return -x**2/2
|
|
|
|
def lhs(x, beta, m):
|
|
return m*np.log(m/beta) - beta**2/2 - m*np.log(m/beta - beta - x)
|
|
|
|
return np.log(N) + _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs)
|
|
|
|
def _cdf(self, x, beta, m):
|
|
"""
|
|
Return CDF of the crystalball function
|
|
"""
|
|
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
|
|
_norm_pdf_C * _norm_cdf(beta))
|
|
|
|
def rhs(x, beta, m):
|
|
return ((m/beta) * np.exp(-beta**2 / 2.0) / (m-1) +
|
|
_norm_pdf_C * (_norm_cdf(x) - _norm_cdf(-beta)))
|
|
|
|
def lhs(x, beta, m):
|
|
return ((m/beta)**m * np.exp(-beta**2 / 2.0) *
|
|
(m/beta - beta - x)**(-m+1) / (m-1))
|
|
|
|
return N * _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs)
|
|
|
|
def _ppf(self, p, beta, m):
|
|
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
|
|
_norm_pdf_C * _norm_cdf(beta))
|
|
pbeta = N * (m/beta) * np.exp(-beta**2/2) / (m - 1)
|
|
|
|
def ppf_less(p, beta, m):
|
|
eb2 = np.exp(-beta**2/2)
|
|
C = (m/beta) * eb2 / (m-1)
|
|
N = 1/(C + _norm_pdf_C * _norm_cdf(beta))
|
|
return (m/beta - beta -
|
|
((m - 1)*(m/beta)**(-m)/eb2*p/N)**(1/(1-m)))
|
|
|
|
def ppf_greater(p, beta, m):
|
|
eb2 = np.exp(-beta**2/2)
|
|
C = (m/beta) * eb2 / (m-1)
|
|
N = 1/(C + _norm_pdf_C * _norm_cdf(beta))
|
|
return _norm_ppf(_norm_cdf(-beta) + (1/_norm_pdf_C)*(p/N - C))
|
|
|
|
return _lazywhere(p < pbeta, (p, beta, m), f=ppf_less, f2=ppf_greater)
|
|
|
|
def _munp(self, n, beta, m):
|
|
"""
|
|
Returns the n-th non-central moment of the crystalball function.
|
|
"""
|
|
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
|
|
_norm_pdf_C * _norm_cdf(beta))
|
|
|
|
def n_th_moment(n, beta, m):
|
|
"""
|
|
Returns n-th moment. Defined only if n+1 < m
|
|
Function cannot broadcast due to the loop over n
|
|
"""
|
|
A = (m/beta)**m * np.exp(-beta**2 / 2.0)
|
|
B = m/beta - beta
|
|
rhs = (2**((n-1)/2.0) * sc.gamma((n+1)/2) *
|
|
(1.0 + (-1)**n * sc.gammainc((n+1)/2, beta**2 / 2)))
|
|
lhs = np.zeros(rhs.shape)
|
|
for k in range(n + 1):
|
|
lhs += (sc.binom(n, k) * B**(n-k) * (-1)**k / (m - k - 1) *
|
|
(m/beta)**(-m + k + 1))
|
|
return A * lhs + rhs
|
|
|
|
return N * _lazywhere(n + 1 < m, (n, beta, m),
|
|
np.vectorize(n_th_moment, otypes=[np.float64]),
|
|
np.inf)
|
|
|
|
def _argcheck(self, beta, m):
|
|
"""
|
|
Shape parameter bounds are m > 1 and beta > 0.
|
|
"""
|
|
return (m > 1) & (beta > 0)
|
|
|
|
|
|
crystalball = crystalball_gen(name='crystalball', longname="A Crystalball Function")
|
|
|
|
|
|
def _argus_phi(chi):
|
|
"""
|
|
Utility function for the argus distribution
|
|
used in the CDF and norm of the Argus Funktion
|
|
"""
|
|
return _norm_cdf(chi) - chi * _norm_pdf(chi) - 0.5
|
|
|
|
|
|
class argus_gen(rv_continuous):
|
|
r"""
|
|
Argus distribution
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability density function for `argus` is:
|
|
|
|
.. math::
|
|
|
|
f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2}
|
|
\exp(-\chi^2 (1 - x^2)/2)
|
|
|
|
for :math:`0 < x < 1` and :math:`\chi > 0`, where
|
|
|
|
.. math::
|
|
|
|
\Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2
|
|
|
|
with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard
|
|
normal distribution, respectively.
|
|
|
|
`argus` takes :math:`\chi` as shape a parameter.
|
|
|
|
%(after_notes)s
|
|
|
|
.. versionadded:: 0.19.0
|
|
|
|
References
|
|
----------
|
|
.. [1] "ARGUS distribution",
|
|
https://en.wikipedia.org/wiki/ARGUS_distribution
|
|
|
|
%(example)s
|
|
"""
|
|
def _pdf(self, x, chi):
|
|
y = 1.0 - x**2
|
|
A = chi**3 / (_norm_pdf_C * _argus_phi(chi))
|
|
return A * x * np.sqrt(y) * np.exp(-chi**2 * y / 2)
|
|
|
|
def _cdf(self, x, chi):
|
|
return 1.0 - self._sf(x, chi)
|
|
|
|
def _sf(self, x, chi):
|
|
return _argus_phi(chi * np.sqrt(1 - x**2)) / _argus_phi(chi)
|
|
|
|
def _rvs(self, chi, size=None, random_state=None):
|
|
chi = np.asarray(chi)
|
|
if chi.size == 1:
|
|
out = self._rvs_scalar(chi, numsamples=size,
|
|
random_state=random_state)
|
|
else:
|
|
shp, bc = _check_shape(chi.shape, size)
|
|
numsamples = int(np.prod(shp))
|
|
out = np.empty(size)
|
|
it = np.nditer([chi],
|
|
flags=['multi_index'],
|
|
op_flags=[['readonly']])
|
|
while not it.finished:
|
|
idx = tuple((it.multi_index[j] if not bc[j] else slice(None))
|
|
for j in range(-len(size), 0))
|
|
r = self._rvs_scalar(it[0], numsamples=numsamples,
|
|
random_state=random_state)
|
|
out[idx] = r.reshape(shp)
|
|
it.iternext()
|
|
|
|
if size == ():
|
|
out = out[()]
|
|
return out
|
|
|
|
def _rvs_scalar(self, chi, numsamples=None, random_state=None):
|
|
# if chi <= 2.611:
|
|
# use rejection method, see Devroye:
|
|
# Non-Uniform Random Variate Generation, 1986, section II.3.2.
|
|
# write: self.pdf = c * g(x) * h(x), where
|
|
# h is [0,1]-valued and g is a density
|
|
# g(x) = d1 * chi**2 * x * exp(-chi**2 * (1 - x**2) / 2), 0 <= x <= 1
|
|
# h(x) = sqrt(1 - x**2), 0 <= x <= 1
|
|
# Integrating g, we get:
|
|
# G(x) = d1 * exp(-chi**2 * (1 - x**2) / 2) - d2
|
|
# d1 and d2 are determined by G(0) = 0 and G(1) = 1
|
|
# d1 = 1 / (1 - exp(-0.5 * chi**2))
|
|
# d2 = 1 / (exp(0.5 * chi**2) - 1)
|
|
# => G(x) = (exp(chi**2 * x**2 /2) - 1) / (exp(chi**2 / 2) - 1)
|
|
# expected number of iterations is c with
|
|
# c = -np.expm1(-0.5 * chi**2) * chi / (_norm_pdf_C * _argus_phi(chi))
|
|
# note that G can be inverted easily, so we can sample
|
|
# rvs from this distribution
|
|
# G_inv(y) = sqrt(2 * log(1 + (exp(chi**2 / 2) - 1) * y) / chi**2)
|
|
# to avoid an overflow of exp(chi**2 / 2), it is convenient to write
|
|
# G_inv(y) = sqrt(1 + 2 * log(exp(-chi**2 / 2) * (1-y) + y) / chi**2)
|
|
#
|
|
# if chi > 2.611:
|
|
# use ratio of uniforms method applied to a transformed variable of X
|
|
# (X is ARGUS with parameter chi):
|
|
# Y = chi * sqrt(1 - X**2) has density proportional to
|
|
# u**2 * exp(-u**2 / 2) on [0, chi] (Maxwell distribution conditioned
|
|
# on [0, chi]). Apply ratio of uniforms to this density to generate
|
|
# samples of Y and convert back to X
|
|
#
|
|
# The expected number of iterations using the rejection method
|
|
# increases with increasing chi, whereas the expected number of
|
|
# iterations using the ratio of uniforms method decreases with
|
|
# increasing chi. The crossover occurs where
|
|
# chi*(1 - exp(-0.5*chi**2)) = 8*sqrt(2)*exp(-1.5) => chi ~ 2.611
|
|
# Switching algorithms at chi=2.611 means that the expected number of
|
|
# iterations is always below 2.2.
|
|
|
|
if chi <= 2.611:
|
|
# use rejection method
|
|
size1d = tuple(np.atleast_1d(numsamples))
|
|
N = int(np.prod(size1d))
|
|
x = np.zeros(N)
|
|
echi = np.exp(-chi**2 / 2)
|
|
simulated = 0
|
|
while simulated < N:
|
|
k = N - simulated
|
|
u = random_state.uniform(size=k)
|
|
v = random_state.uniform(size=k)
|
|
# acceptance condition: u <= h(G_inv(v)). This simplifies to
|
|
z = 2 * np.log(echi * (1 - v) + v) / chi**2
|
|
accept = (u**2 + z <= 0)
|
|
num_accept = np.sum(accept)
|
|
if num_accept > 0:
|
|
# rvs follow a distribution with density g: rvs = G_inv(v)
|
|
rvs = np.sqrt(1 + z[accept])
|
|
x[simulated:(simulated + num_accept)] = rvs
|
|
simulated += num_accept
|
|
|
|
return np.reshape(x, size1d)
|
|
else:
|
|
# use ratio of uniforms method
|
|
def f(x):
|
|
return np.where((x >= 0) & (x <= chi),
|
|
np.exp(2*np.log(x) - x**2/2), 0)
|
|
|
|
umax = np.sqrt(2) / np.exp(0.5)
|
|
vmax = 4 / np.exp(1)
|
|
z = rvs_ratio_uniforms(f, umax, 0, vmax, size=numsamples,
|
|
random_state=random_state)
|
|
return np.sqrt(1 - z*z / chi**2)
|
|
|
|
def _stats(self, chi):
|
|
chi2 = chi**2
|
|
phi = _argus_phi(chi)
|
|
m = np.sqrt(np.pi/8) * chi * sc.ive(1, chi2/4) / phi
|
|
v = (1 - 3 / chi2 + chi * _norm_pdf(chi) / phi) - m**2
|
|
return m, v, None, None
|
|
|
|
|
|
argus = argus_gen(name='argus', longname="An Argus Function", a=0.0, b=1.0)
|
|
|
|
|
|
class rv_histogram(rv_continuous):
|
|
"""
|
|
Generates a distribution given by a histogram.
|
|
This is useful to generate a template distribution from a binned
|
|
datasample.
|
|
|
|
As a subclass of the `rv_continuous` class, `rv_histogram` inherits from it
|
|
a collection of generic methods (see `rv_continuous` for the full list),
|
|
and implements them based on the properties of the provided binned
|
|
datasample.
|
|
|
|
Parameters
|
|
----------
|
|
histogram : tuple of array_like
|
|
Tuple containing two array_like objects
|
|
The first containing the content of n bins
|
|
The second containing the (n+1) bin boundaries
|
|
In particular the return value np.histogram is accepted
|
|
|
|
Notes
|
|
-----
|
|
There are no additional shape parameters except for the loc and scale.
|
|
The pdf is defined as a stepwise function from the provided histogram
|
|
The cdf is a linear interpolation of the pdf.
|
|
|
|
.. versionadded:: 0.19.0
|
|
|
|
Examples
|
|
--------
|
|
|
|
Create a scipy.stats distribution from a numpy histogram
|
|
|
|
>>> import scipy.stats
|
|
>>> import numpy as np
|
|
>>> data = scipy.stats.norm.rvs(size=100000, loc=0, scale=1.5, random_state=123)
|
|
>>> hist = np.histogram(data, bins=100)
|
|
>>> hist_dist = scipy.stats.rv_histogram(hist)
|
|
|
|
Behaves like an ordinary scipy rv_continuous distribution
|
|
|
|
>>> hist_dist.pdf(1.0)
|
|
0.20538577847618705
|
|
>>> hist_dist.cdf(2.0)
|
|
0.90818568543056499
|
|
|
|
PDF is zero above (below) the highest (lowest) bin of the histogram,
|
|
defined by the max (min) of the original dataset
|
|
|
|
>>> hist_dist.pdf(np.max(data))
|
|
0.0
|
|
>>> hist_dist.cdf(np.max(data))
|
|
1.0
|
|
>>> hist_dist.pdf(np.min(data))
|
|
7.7591907244498314e-05
|
|
>>> hist_dist.cdf(np.min(data))
|
|
0.0
|
|
|
|
PDF and CDF follow the histogram
|
|
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> X = np.linspace(-5.0, 5.0, 100)
|
|
>>> plt.title("PDF from Template")
|
|
>>> plt.hist(data, density=True, bins=100)
|
|
>>> plt.plot(X, hist_dist.pdf(X), label='PDF')
|
|
>>> plt.plot(X, hist_dist.cdf(X), label='CDF')
|
|
>>> plt.show()
|
|
|
|
"""
|
|
_support_mask = rv_continuous._support_mask
|
|
|
|
def __init__(self, histogram, *args, **kwargs):
|
|
"""
|
|
Create a new distribution using the given histogram
|
|
|
|
Parameters
|
|
----------
|
|
histogram : tuple of array_like
|
|
Tuple containing two array_like objects
|
|
The first containing the content of n bins
|
|
The second containing the (n+1) bin boundaries
|
|
In particular the return value np.histogram is accepted
|
|
"""
|
|
self._histogram = histogram
|
|
if len(histogram) != 2:
|
|
raise ValueError("Expected length 2 for parameter histogram")
|
|
self._hpdf = np.asarray(histogram[0])
|
|
self._hbins = np.asarray(histogram[1])
|
|
if len(self._hpdf) + 1 != len(self._hbins):
|
|
raise ValueError("Number of elements in histogram content "
|
|
"and histogram boundaries do not match, "
|
|
"expected n and n+1.")
|
|
self._hbin_widths = self._hbins[1:] - self._hbins[:-1]
|
|
self._hpdf = self._hpdf / float(np.sum(self._hpdf * self._hbin_widths))
|
|
self._hcdf = np.cumsum(self._hpdf * self._hbin_widths)
|
|
self._hpdf = np.hstack([0.0, self._hpdf, 0.0])
|
|
self._hcdf = np.hstack([0.0, self._hcdf])
|
|
# Set support
|
|
kwargs['a'] = self.a = self._hbins[0]
|
|
kwargs['b'] = self.b = self._hbins[-1]
|
|
super(rv_histogram, self).__init__(*args, **kwargs)
|
|
|
|
def _pdf(self, x):
|
|
"""
|
|
PDF of the histogram
|
|
"""
|
|
return self._hpdf[np.searchsorted(self._hbins, x, side='right')]
|
|
|
|
def _cdf(self, x):
|
|
"""
|
|
CDF calculated from the histogram
|
|
"""
|
|
return np.interp(x, self._hbins, self._hcdf)
|
|
|
|
def _ppf(self, x):
|
|
"""
|
|
Percentile function calculated from the histogram
|
|
"""
|
|
return np.interp(x, self._hcdf, self._hbins)
|
|
|
|
def _munp(self, n):
|
|
"""Compute the n-th non-central moment."""
|
|
integrals = (self._hbins[1:]**(n+1) - self._hbins[:-1]**(n+1)) / (n+1)
|
|
return np.sum(self._hpdf[1:-1] * integrals)
|
|
|
|
def _entropy(self):
|
|
"""Compute entropy of distribution"""
|
|
res = _lazywhere(self._hpdf[1:-1] > 0.0,
|
|
(self._hpdf[1:-1],),
|
|
np.log,
|
|
0.0)
|
|
return -np.sum(self._hpdf[1:-1] * res * self._hbin_widths)
|
|
|
|
def _updated_ctor_param(self):
|
|
"""
|
|
Set the histogram as additional constructor argument
|
|
"""
|
|
dct = super(rv_histogram, self)._updated_ctor_param()
|
|
dct['histogram'] = self._histogram
|
|
return dct
|
|
|
|
|
|
# Collect names of classes and objects in this module.
|
|
pairs = list(globals().copy().items())
|
|
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_continuous)
|
|
|
|
__all__ = _distn_names + _distn_gen_names + ['rv_histogram']
|