3409 lines
116 KiB
Python
3409 lines
116 KiB
Python
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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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from __future__ import division, print_function, absolute_import
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from scipy._lib.six import string_types, exec_, PY3
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from scipy._lib._util import getargspec_no_self as _getargspec
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import sys
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import keyword
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import re
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import types
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import warnings
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from scipy.misc import doccer
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from ._distr_params import distcont, distdiscrete
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from scipy._lib._util import check_random_state, _lazywhere, _lazyselect
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from scipy._lib._util import _valarray as valarray
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from scipy.special import (comb, chndtr, entr, rel_entr, kl_div, xlogy, ive)
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# for root finding for discrete distribution ppf, and max likelihood estimation
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from scipy import optimize
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# for functions of continuous distributions (e.g. moments, entropy, cdf)
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from scipy import integrate
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# to approximate the pdf of a continuous distribution given its cdf
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from scipy.misc import derivative
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from numpy import (arange, putmask, ravel, take, ones, shape, ndarray,
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product, reshape, zeros, floor, logical_and, log, sqrt, exp)
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from numpy import (place, argsort, argmax, vectorize,
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asarray, nan, inf, isinf, NINF, empty)
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import numpy as np
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from ._constants import _XMAX
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if PY3:
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def instancemethod(func, obj, cls):
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return types.MethodType(func, obj)
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else:
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instancemethod = types.MethodType
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# These are the docstring parts used for substitution in specific
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# distribution docstrings
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docheaders = {'methods': """\nMethods\n-------\n""",
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'notes': """\nNotes\n-----\n""",
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'examples': """\nExamples\n--------\n"""}
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_doc_rvs = """\
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rvs(%(shapes)s, loc=0, scale=1, size=1, random_state=None)
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Random variates.
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"""
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_doc_pdf = """\
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pdf(x, %(shapes)s, loc=0, scale=1)
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Probability density function.
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"""
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_doc_logpdf = """\
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logpdf(x, %(shapes)s, loc=0, scale=1)
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Log of the probability density function.
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"""
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_doc_pmf = """\
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pmf(k, %(shapes)s, loc=0, scale=1)
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Probability mass function.
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"""
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_doc_logpmf = """\
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logpmf(k, %(shapes)s, loc=0, scale=1)
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Log of the probability mass function.
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"""
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_doc_cdf = """\
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cdf(x, %(shapes)s, loc=0, scale=1)
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Cumulative distribution function.
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"""
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_doc_logcdf = """\
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logcdf(x, %(shapes)s, loc=0, scale=1)
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Log of the cumulative distribution function.
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"""
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_doc_sf = """\
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sf(x, %(shapes)s, loc=0, scale=1)
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Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
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"""
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_doc_logsf = """\
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logsf(x, %(shapes)s, loc=0, scale=1)
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Log of the survival function.
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"""
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_doc_ppf = """\
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ppf(q, %(shapes)s, loc=0, scale=1)
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Percent point function (inverse of ``cdf`` --- percentiles).
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"""
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_doc_isf = """\
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isf(q, %(shapes)s, loc=0, scale=1)
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Inverse survival function (inverse of ``sf``).
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"""
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_doc_moment = """\
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moment(n, %(shapes)s, loc=0, scale=1)
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Non-central moment of order n
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"""
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_doc_stats = """\
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stats(%(shapes)s, loc=0, scale=1, moments='mv')
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Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
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"""
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_doc_entropy = """\
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entropy(%(shapes)s, loc=0, scale=1)
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(Differential) entropy of the RV.
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"""
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_doc_fit = """\
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fit(data, %(shapes)s, loc=0, scale=1)
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Parameter estimates for generic data.
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"""
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_doc_expect = """\
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expect(func, args=(%(shapes_)s), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)
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Expected value of a function (of one argument) with respect to the distribution.
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"""
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_doc_expect_discrete = """\
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expect(func, args=(%(shapes_)s), loc=0, lb=None, ub=None, conditional=False)
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Expected value of a function (of one argument) with respect to the distribution.
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"""
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_doc_median = """\
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median(%(shapes)s, loc=0, scale=1)
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Median of the distribution.
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"""
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_doc_mean = """\
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mean(%(shapes)s, loc=0, scale=1)
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Mean of the distribution.
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"""
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_doc_var = """\
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var(%(shapes)s, loc=0, scale=1)
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Variance of the distribution.
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"""
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_doc_std = """\
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std(%(shapes)s, loc=0, scale=1)
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Standard deviation of the distribution.
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"""
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_doc_interval = """\
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interval(alpha, %(shapes)s, loc=0, scale=1)
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Endpoints of the range that contains alpha percent of the distribution
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"""
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_doc_allmethods = ''.join([docheaders['methods'], _doc_rvs, _doc_pdf,
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_doc_logpdf, _doc_cdf, _doc_logcdf, _doc_sf,
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_doc_logsf, _doc_ppf, _doc_isf, _doc_moment,
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_doc_stats, _doc_entropy, _doc_fit,
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_doc_expect, _doc_median,
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_doc_mean, _doc_var, _doc_std, _doc_interval])
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_doc_default_longsummary = """\
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As an instance of the `rv_continuous` class, `%(name)s` object inherits from it
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a collection of generic methods (see below for the full list),
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and completes them with details specific for this particular distribution.
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"""
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_doc_default_frozen_note = """
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Alternatively, the object may be called (as a function) to fix the shape,
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location, and scale parameters returning a "frozen" continuous RV object:
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rv = %(name)s(%(shapes)s, loc=0, scale=1)
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- Frozen RV object with the same methods but holding the given shape,
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location, and scale fixed.
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"""
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_doc_default_example = """\
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Examples
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--------
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>>> from scipy.stats import %(name)s
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>>> import matplotlib.pyplot as plt
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>>> fig, ax = plt.subplots(1, 1)
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Calculate a few first moments:
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%(set_vals_stmt)s
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>>> mean, var, skew, kurt = %(name)s.stats(%(shapes)s, moments='mvsk')
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Display the probability density function (``pdf``):
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>>> x = np.linspace(%(name)s.ppf(0.01, %(shapes)s),
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... %(name)s.ppf(0.99, %(shapes)s), 100)
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>>> ax.plot(x, %(name)s.pdf(x, %(shapes)s),
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... 'r-', lw=5, alpha=0.6, label='%(name)s pdf')
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Alternatively, the distribution object can be called (as a function)
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to fix the shape, location and scale parameters. This returns a "frozen"
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RV object holding the given parameters fixed.
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Freeze the distribution and display the frozen ``pdf``:
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>>> rv = %(name)s(%(shapes)s)
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>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
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Check accuracy of ``cdf`` and ``ppf``:
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>>> vals = %(name)s.ppf([0.001, 0.5, 0.999], %(shapes)s)
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>>> np.allclose([0.001, 0.5, 0.999], %(name)s.cdf(vals, %(shapes)s))
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True
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Generate random numbers:
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>>> r = %(name)s.rvs(%(shapes)s, size=1000)
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And compare the histogram:
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>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
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>>> ax.legend(loc='best', frameon=False)
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>>> plt.show()
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"""
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_doc_default_locscale = """\
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The probability density above is defined in the "standardized" form. To shift
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and/or scale the distribution use the ``loc`` and ``scale`` parameters.
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Specifically, ``%(name)s.pdf(x, %(shapes)s, loc, scale)`` is identically
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equivalent to ``%(name)s.pdf(y, %(shapes)s) / scale`` with
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``y = (x - loc) / scale``.
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"""
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_doc_default = ''.join([_doc_default_longsummary,
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_doc_allmethods,
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'\n',
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_doc_default_example])
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_doc_default_before_notes = ''.join([_doc_default_longsummary,
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_doc_allmethods])
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docdict = {
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'rvs': _doc_rvs,
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'pdf': _doc_pdf,
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'logpdf': _doc_logpdf,
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'cdf': _doc_cdf,
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'logcdf': _doc_logcdf,
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'sf': _doc_sf,
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'logsf': _doc_logsf,
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'ppf': _doc_ppf,
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'isf': _doc_isf,
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'stats': _doc_stats,
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'entropy': _doc_entropy,
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'fit': _doc_fit,
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'moment': _doc_moment,
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'expect': _doc_expect,
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'interval': _doc_interval,
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'mean': _doc_mean,
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'std': _doc_std,
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'var': _doc_var,
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'median': _doc_median,
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'allmethods': _doc_allmethods,
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'longsummary': _doc_default_longsummary,
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'frozennote': _doc_default_frozen_note,
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'example': _doc_default_example,
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'default': _doc_default,
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'before_notes': _doc_default_before_notes,
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'after_notes': _doc_default_locscale
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}
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# Reuse common content between continuous and discrete docs, change some
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# minor bits.
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docdict_discrete = docdict.copy()
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docdict_discrete['pmf'] = _doc_pmf
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docdict_discrete['logpmf'] = _doc_logpmf
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docdict_discrete['expect'] = _doc_expect_discrete
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_doc_disc_methods = ['rvs', 'pmf', 'logpmf', 'cdf', 'logcdf', 'sf', 'logsf',
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'ppf', 'isf', 'stats', 'entropy', 'expect', 'median',
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'mean', 'var', 'std', 'interval']
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for obj in _doc_disc_methods:
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docdict_discrete[obj] = docdict_discrete[obj].replace(', scale=1', '')
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_doc_disc_methods_err_varname = ['cdf', 'logcdf', 'sf', 'logsf']
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for obj in _doc_disc_methods_err_varname:
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docdict_discrete[obj] = docdict_discrete[obj].replace('(x, ', '(k, ')
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docdict_discrete.pop('pdf')
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docdict_discrete.pop('logpdf')
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_doc_allmethods = ''.join([docdict_discrete[obj] for obj in _doc_disc_methods])
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docdict_discrete['allmethods'] = docheaders['methods'] + _doc_allmethods
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docdict_discrete['longsummary'] = _doc_default_longsummary.replace(
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'rv_continuous', 'rv_discrete')
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_doc_default_frozen_note = """
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Alternatively, the object may be called (as a function) to fix the shape and
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location parameters returning a "frozen" discrete RV object:
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rv = %(name)s(%(shapes)s, loc=0)
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- Frozen RV object with the same methods but holding the given shape and
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location fixed.
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"""
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docdict_discrete['frozennote'] = _doc_default_frozen_note
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_doc_default_discrete_example = """\
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Examples
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--------
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>>> from scipy.stats import %(name)s
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>>> import matplotlib.pyplot as plt
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>>> fig, ax = plt.subplots(1, 1)
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Calculate a few first moments:
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%(set_vals_stmt)s
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>>> mean, var, skew, kurt = %(name)s.stats(%(shapes)s, moments='mvsk')
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Display the probability mass function (``pmf``):
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>>> x = np.arange(%(name)s.ppf(0.01, %(shapes)s),
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... %(name)s.ppf(0.99, %(shapes)s))
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>>> ax.plot(x, %(name)s.pmf(x, %(shapes)s), 'bo', ms=8, label='%(name)s pmf')
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>>> ax.vlines(x, 0, %(name)s.pmf(x, %(shapes)s), colors='b', lw=5, alpha=0.5)
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Alternatively, the distribution object can be called (as a function)
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to fix the shape and location. This returns a "frozen" RV object holding
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the given parameters fixed.
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Freeze the distribution and display the frozen ``pmf``:
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>>> rv = %(name)s(%(shapes)s)
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>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
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... label='frozen pmf')
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>>> ax.legend(loc='best', frameon=False)
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>>> plt.show()
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Check accuracy of ``cdf`` and ``ppf``:
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>>> prob = %(name)s.cdf(x, %(shapes)s)
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>>> np.allclose(x, %(name)s.ppf(prob, %(shapes)s))
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True
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Generate random numbers:
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>>> r = %(name)s.rvs(%(shapes)s, size=1000)
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"""
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_doc_default_discrete_locscale = """\
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The probability mass function above is defined in the "standardized" form.
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To shift distribution use the ``loc`` parameter.
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Specifically, ``%(name)s.pmf(k, %(shapes)s, loc)`` is identically
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equivalent to ``%(name)s.pmf(k - loc, %(shapes)s)``.
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"""
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docdict_discrete['example'] = _doc_default_discrete_example
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docdict_discrete['after_notes'] = _doc_default_discrete_locscale
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_doc_default_before_notes = ''.join([docdict_discrete['longsummary'],
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||
|
|
docdict_discrete['allmethods']])
|
||
|
|
docdict_discrete['before_notes'] = _doc_default_before_notes
|
||
|
|
|
||
|
|
_doc_default_disc = ''.join([docdict_discrete['longsummary'],
|
||
|
|
docdict_discrete['allmethods'],
|
||
|
|
docdict_discrete['frozennote'],
|
||
|
|
docdict_discrete['example']])
|
||
|
|
docdict_discrete['default'] = _doc_default_disc
|
||
|
|
|
||
|
|
# clean up all the separate docstring elements, we do not need them anymore
|
||
|
|
for obj in [s for s in dir() if s.startswith('_doc_')]:
|
||
|
|
exec('del ' + obj)
|
||
|
|
del obj
|
||
|
|
try:
|
||
|
|
del s
|
||
|
|
except NameError:
|
||
|
|
# in Python 3, loop variables are not visible after the loop
|
||
|
|
pass
|
||
|
|
|
||
|
|
|
||
|
|
def _moment(data, n, mu=None):
|
||
|
|
if mu is None:
|
||
|
|
mu = data.mean()
|
||
|
|
return ((data - mu)**n).mean()
|
||
|
|
|
||
|
|
|
||
|
|
def _moment_from_stats(n, mu, mu2, g1, g2, moment_func, args):
|
||
|
|
if (n == 0):
|
||
|
|
return 1.0
|
||
|
|
elif (n == 1):
|
||
|
|
if mu is None:
|
||
|
|
val = moment_func(1, *args)
|
||
|
|
else:
|
||
|
|
val = mu
|
||
|
|
elif (n == 2):
|
||
|
|
if mu2 is None or mu is None:
|
||
|
|
val = moment_func(2, *args)
|
||
|
|
else:
|
||
|
|
val = mu2 + mu*mu
|
||
|
|
elif (n == 3):
|
||
|
|
if g1 is None or mu2 is None or mu is None:
|
||
|
|
val = moment_func(3, *args)
|
||
|
|
else:
|
||
|
|
mu3 = g1 * np.power(mu2, 1.5) # 3rd central moment
|
||
|
|
val = mu3+3*mu*mu2+mu*mu*mu # 3rd non-central moment
|
||
|
|
elif (n == 4):
|
||
|
|
if g1 is None or g2 is None or mu2 is None or mu is None:
|
||
|
|
val = moment_func(4, *args)
|
||
|
|
else:
|
||
|
|
mu4 = (g2+3.0)*(mu2**2.0) # 4th central moment
|
||
|
|
mu3 = g1*np.power(mu2, 1.5) # 3rd central moment
|
||
|
|
val = mu4+4*mu*mu3+6*mu*mu*mu2+mu*mu*mu*mu
|
||
|
|
else:
|
||
|
|
val = moment_func(n, *args)
|
||
|
|
|
||
|
|
return val
|
||
|
|
|
||
|
|
|
||
|
|
def _skew(data):
|
||
|
|
"""
|
||
|
|
skew is third central moment / variance**(1.5)
|
||
|
|
"""
|
||
|
|
data = np.ravel(data)
|
||
|
|
mu = data.mean()
|
||
|
|
m2 = ((data - mu)**2).mean()
|
||
|
|
m3 = ((data - mu)**3).mean()
|
||
|
|
return m3 / np.power(m2, 1.5)
|
||
|
|
|
||
|
|
|
||
|
|
def _kurtosis(data):
|
||
|
|
"""
|
||
|
|
kurtosis is fourth central moment / variance**2 - 3
|
||
|
|
"""
|
||
|
|
data = np.ravel(data)
|
||
|
|
mu = data.mean()
|
||
|
|
m2 = ((data - mu)**2).mean()
|
||
|
|
m4 = ((data - mu)**4).mean()
|
||
|
|
return m4 / m2**2 - 3
|
||
|
|
|
||
|
|
|
||
|
|
# Frozen RV class
|
||
|
|
class rv_frozen(object):
|
||
|
|
|
||
|
|
def __init__(self, dist, *args, **kwds):
|
||
|
|
self.args = args
|
||
|
|
self.kwds = kwds
|
||
|
|
|
||
|
|
# create a new instance
|
||
|
|
self.dist = dist.__class__(**dist._updated_ctor_param())
|
||
|
|
|
||
|
|
# a, b may be set in _argcheck, depending on *args, **kwds. Ouch.
|
||
|
|
shapes, _, _ = self.dist._parse_args(*args, **kwds)
|
||
|
|
self.dist._argcheck(*shapes)
|
||
|
|
self.a, self.b = self.dist.a, self.dist.b
|
||
|
|
|
||
|
|
@property
|
||
|
|
def random_state(self):
|
||
|
|
return self.dist._random_state
|
||
|
|
|
||
|
|
@random_state.setter
|
||
|
|
def random_state(self, seed):
|
||
|
|
self.dist._random_state = check_random_state(seed)
|
||
|
|
|
||
|
|
def pdf(self, x): # raises AttributeError in frozen discrete distribution
|
||
|
|
return self.dist.pdf(x, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def logpdf(self, x):
|
||
|
|
return self.dist.logpdf(x, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def cdf(self, x):
|
||
|
|
return self.dist.cdf(x, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def logcdf(self, x):
|
||
|
|
return self.dist.logcdf(x, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def ppf(self, q):
|
||
|
|
return self.dist.ppf(q, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def isf(self, q):
|
||
|
|
return self.dist.isf(q, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def rvs(self, size=None, random_state=None):
|
||
|
|
kwds = self.kwds.copy()
|
||
|
|
kwds.update({'size': size, 'random_state': random_state})
|
||
|
|
return self.dist.rvs(*self.args, **kwds)
|
||
|
|
|
||
|
|
def sf(self, x):
|
||
|
|
return self.dist.sf(x, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def logsf(self, x):
|
||
|
|
return self.dist.logsf(x, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def stats(self, moments='mv'):
|
||
|
|
kwds = self.kwds.copy()
|
||
|
|
kwds.update({'moments': moments})
|
||
|
|
return self.dist.stats(*self.args, **kwds)
|
||
|
|
|
||
|
|
def median(self):
|
||
|
|
return self.dist.median(*self.args, **self.kwds)
|
||
|
|
|
||
|
|
def mean(self):
|
||
|
|
return self.dist.mean(*self.args, **self.kwds)
|
||
|
|
|
||
|
|
def var(self):
|
||
|
|
return self.dist.var(*self.args, **self.kwds)
|
||
|
|
|
||
|
|
def std(self):
|
||
|
|
return self.dist.std(*self.args, **self.kwds)
|
||
|
|
|
||
|
|
def moment(self, n):
|
||
|
|
return self.dist.moment(n, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def entropy(self):
|
||
|
|
return self.dist.entropy(*self.args, **self.kwds)
|
||
|
|
|
||
|
|
def pmf(self, k):
|
||
|
|
return self.dist.pmf(k, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def logpmf(self, k):
|
||
|
|
return self.dist.logpmf(k, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def interval(self, alpha):
|
||
|
|
return self.dist.interval(alpha, *self.args, **self.kwds)
|
||
|
|
|
||
|
|
def expect(self, func=None, lb=None, ub=None, conditional=False, **kwds):
|
||
|
|
# expect method only accepts shape parameters as positional args
|
||
|
|
# hence convert self.args, self.kwds, also loc/scale
|
||
|
|
# See the .expect method docstrings for the meaning of
|
||
|
|
# other parameters.
|
||
|
|
a, loc, scale = self.dist._parse_args(*self.args, **self.kwds)
|
||
|
|
if isinstance(self.dist, rv_discrete):
|
||
|
|
return self.dist.expect(func, a, loc, lb, ub, conditional, **kwds)
|
||
|
|
else:
|
||
|
|
return self.dist.expect(func, a, loc, scale, lb, ub,
|
||
|
|
conditional, **kwds)
|
||
|
|
|
||
|
|
|
||
|
|
# This should be rewritten
|
||
|
|
def argsreduce(cond, *args):
|
||
|
|
"""Return the sequence of ravel(args[i]) where ravel(condition) is
|
||
|
|
True in 1D.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> rand = np.random.random_sample
|
||
|
|
>>> A = rand((4, 5))
|
||
|
|
>>> B = 2
|
||
|
|
>>> C = rand((1, 5))
|
||
|
|
>>> cond = np.ones(A.shape)
|
||
|
|
>>> [A1, B1, C1] = argsreduce(cond, A, B, C)
|
||
|
|
>>> B1.shape
|
||
|
|
(20,)
|
||
|
|
>>> cond[2,:] = 0
|
||
|
|
>>> [A2, B2, C2] = argsreduce(cond, A, B, C)
|
||
|
|
>>> B2.shape
|
||
|
|
(15,)
|
||
|
|
|
||
|
|
"""
|
||
|
|
newargs = np.atleast_1d(*args)
|
||
|
|
if not isinstance(newargs, list):
|
||
|
|
newargs = [newargs, ]
|
||
|
|
expand_arr = (cond == cond)
|
||
|
|
return [np.extract(cond, arr1 * expand_arr) for arr1 in newargs]
|
||
|
|
|
||
|
|
|
||
|
|
parse_arg_template = """
|
||
|
|
def _parse_args(self, %(shape_arg_str)s %(locscale_in)s):
|
||
|
|
return (%(shape_arg_str)s), %(locscale_out)s
|
||
|
|
|
||
|
|
def _parse_args_rvs(self, %(shape_arg_str)s %(locscale_in)s, size=None):
|
||
|
|
return self._argcheck_rvs(%(shape_arg_str)s %(locscale_out)s, size=size)
|
||
|
|
|
||
|
|
def _parse_args_stats(self, %(shape_arg_str)s %(locscale_in)s, moments='mv'):
|
||
|
|
return (%(shape_arg_str)s), %(locscale_out)s, moments
|
||
|
|
"""
|
||
|
|
|
||
|
|
|
||
|
|
# Both the continuous and discrete distributions depend on ncx2.
|
||
|
|
# I think the function name ncx2 is an abbreviation for noncentral chi squared.
|
||
|
|
|
||
|
|
def _ncx2_log_pdf(x, df, nc):
|
||
|
|
# We use (xs**2 + ns**2)/2 = (xs - ns)**2/2 + xs*ns, and include the factor
|
||
|
|
# of exp(-xs*ns) into the ive function to improve numerical stability
|
||
|
|
# at large values of xs. See also `rice.pdf`.
|
||
|
|
df2 = df/2.0 - 1.0
|
||
|
|
xs, ns = np.sqrt(x), np.sqrt(nc)
|
||
|
|
res = xlogy(df2/2.0, x/nc) - 0.5*(xs - ns)**2
|
||
|
|
res += np.log(ive(df2, xs*ns) / 2.0)
|
||
|
|
return res
|
||
|
|
|
||
|
|
|
||
|
|
def _ncx2_pdf(x, df, nc):
|
||
|
|
return np.exp(_ncx2_log_pdf(x, df, nc))
|
||
|
|
|
||
|
|
|
||
|
|
def _ncx2_cdf(x, df, nc):
|
||
|
|
return chndtr(x, df, nc)
|
||
|
|
|
||
|
|
|
||
|
|
class rv_generic(object):
|
||
|
|
"""Class which encapsulates common functionality between rv_discrete
|
||
|
|
and rv_continuous.
|
||
|
|
|
||
|
|
"""
|
||
|
|
def __init__(self, seed=None):
|
||
|
|
super(rv_generic, self).__init__()
|
||
|
|
|
||
|
|
# figure out if _stats signature has 'moments' keyword
|
||
|
|
sign = _getargspec(self._stats)
|
||
|
|
self._stats_has_moments = ((sign[2] is not None) or
|
||
|
|
('moments' in sign[0]))
|
||
|
|
self._random_state = check_random_state(seed)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def random_state(self):
|
||
|
|
""" Get or set the RandomState object for generating random variates.
|
||
|
|
|
||
|
|
This can be either None or an existing RandomState object.
|
||
|
|
|
||
|
|
If None (or np.random), use the RandomState singleton used by np.random.
|
||
|
|
If already a RandomState instance, use it.
|
||
|
|
If an int, use a new RandomState instance seeded with seed.
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self._random_state
|
||
|
|
|
||
|
|
@random_state.setter
|
||
|
|
def random_state(self, seed):
|
||
|
|
self._random_state = check_random_state(seed)
|
||
|
|
|
||
|
|
def __getstate__(self):
|
||
|
|
return self._updated_ctor_param(), self._random_state
|
||
|
|
|
||
|
|
def __setstate__(self, state):
|
||
|
|
ctor_param, r = state
|
||
|
|
self.__init__(**ctor_param)
|
||
|
|
self._random_state = r
|
||
|
|
return self
|
||
|
|
|
||
|
|
def _construct_argparser(
|
||
|
|
self, meths_to_inspect, locscale_in, locscale_out):
|
||
|
|
"""Construct the parser for the shape arguments.
|
||
|
|
|
||
|
|
Generates the argument-parsing functions dynamically and attaches
|
||
|
|
them to the instance.
|
||
|
|
Is supposed to be called in __init__ of a class for each distribution.
|
||
|
|
|
||
|
|
If self.shapes is a non-empty string, interprets it as a
|
||
|
|
comma-separated list of shape parameters.
|
||
|
|
|
||
|
|
Otherwise inspects the call signatures of `meths_to_inspect`
|
||
|
|
and constructs the argument-parsing functions from these.
|
||
|
|
In this case also sets `shapes` and `numargs`.
|
||
|
|
"""
|
||
|
|
|
||
|
|
if self.shapes:
|
||
|
|
# sanitize the user-supplied shapes
|
||
|
|
if not isinstance(self.shapes, string_types):
|
||
|
|
raise TypeError('shapes must be a string.')
|
||
|
|
|
||
|
|
shapes = self.shapes.replace(',', ' ').split()
|
||
|
|
|
||
|
|
for field in shapes:
|
||
|
|
if keyword.iskeyword(field):
|
||
|
|
raise SyntaxError('keywords cannot be used as shapes.')
|
||
|
|
if not re.match('^[_a-zA-Z][_a-zA-Z0-9]*$', field):
|
||
|
|
raise SyntaxError(
|
||
|
|
'shapes must be valid python identifiers')
|
||
|
|
else:
|
||
|
|
# find out the call signatures (_pdf, _cdf etc), deduce shape
|
||
|
|
# arguments. Generic methods only have 'self, x', any further args
|
||
|
|
# are shapes.
|
||
|
|
shapes_list = []
|
||
|
|
for meth in meths_to_inspect:
|
||
|
|
shapes_args = _getargspec(meth) # NB: does not contain self
|
||
|
|
args = shapes_args.args[1:] # peel off 'x', too
|
||
|
|
|
||
|
|
if args:
|
||
|
|
shapes_list.append(args)
|
||
|
|
|
||
|
|
# *args or **kwargs are not allowed w/automatic shapes
|
||
|
|
if shapes_args.varargs is not None:
|
||
|
|
raise TypeError(
|
||
|
|
'*args are not allowed w/out explicit shapes')
|
||
|
|
if shapes_args.keywords is not None:
|
||
|
|
raise TypeError(
|
||
|
|
'**kwds are not allowed w/out explicit shapes')
|
||
|
|
if shapes_args.defaults is not None:
|
||
|
|
raise TypeError('defaults are not allowed for shapes')
|
||
|
|
|
||
|
|
if shapes_list:
|
||
|
|
shapes = shapes_list[0]
|
||
|
|
|
||
|
|
# make sure the signatures are consistent
|
||
|
|
for item in shapes_list:
|
||
|
|
if item != shapes:
|
||
|
|
raise TypeError('Shape arguments are inconsistent.')
|
||
|
|
else:
|
||
|
|
shapes = []
|
||
|
|
|
||
|
|
# have the arguments, construct the method from template
|
||
|
|
shapes_str = ', '.join(shapes) + ', ' if shapes else '' # NB: not None
|
||
|
|
dct = dict(shape_arg_str=shapes_str,
|
||
|
|
locscale_in=locscale_in,
|
||
|
|
locscale_out=locscale_out,
|
||
|
|
)
|
||
|
|
ns = {}
|
||
|
|
exec_(parse_arg_template % dct, ns)
|
||
|
|
# NB: attach to the instance, not class
|
||
|
|
for name in ['_parse_args', '_parse_args_stats', '_parse_args_rvs']:
|
||
|
|
setattr(self, name,
|
||
|
|
instancemethod(ns[name], self, self.__class__)
|
||
|
|
)
|
||
|
|
|
||
|
|
self.shapes = ', '.join(shapes) if shapes else None
|
||
|
|
if not hasattr(self, 'numargs'):
|
||
|
|
# allows more general subclassing with *args
|
||
|
|
self.numargs = len(shapes)
|
||
|
|
|
||
|
|
def _construct_doc(self, docdict, shapes_vals=None):
|
||
|
|
"""Construct the instance docstring with string substitutions."""
|
||
|
|
tempdict = docdict.copy()
|
||
|
|
tempdict['name'] = self.name or 'distname'
|
||
|
|
tempdict['shapes'] = self.shapes or ''
|
||
|
|
|
||
|
|
if shapes_vals is None:
|
||
|
|
shapes_vals = ()
|
||
|
|
vals = ', '.join('%.3g' % val for val in shapes_vals)
|
||
|
|
tempdict['vals'] = vals
|
||
|
|
|
||
|
|
tempdict['shapes_'] = self.shapes or ''
|
||
|
|
if self.shapes and self.numargs == 1:
|
||
|
|
tempdict['shapes_'] += ','
|
||
|
|
|
||
|
|
if self.shapes:
|
||
|
|
tempdict['set_vals_stmt'] = '>>> %s = %s' % (self.shapes, vals)
|
||
|
|
else:
|
||
|
|
tempdict['set_vals_stmt'] = ''
|
||
|
|
|
||
|
|
if self.shapes is None:
|
||
|
|
# remove shapes from call parameters if there are none
|
||
|
|
for item in ['default', 'before_notes']:
|
||
|
|
tempdict[item] = tempdict[item].replace(
|
||
|
|
"\n%(shapes)s : array_like\n shape parameters", "")
|
||
|
|
for i in range(2):
|
||
|
|
if self.shapes is None:
|
||
|
|
# necessary because we use %(shapes)s in two forms (w w/o ", ")
|
||
|
|
self.__doc__ = self.__doc__.replace("%(shapes)s, ", "")
|
||
|
|
self.__doc__ = doccer.docformat(self.__doc__, tempdict)
|
||
|
|
|
||
|
|
# correct for empty shapes
|
||
|
|
self.__doc__ = self.__doc__.replace('(, ', '(').replace(', )', ')')
|
||
|
|
|
||
|
|
def _construct_default_doc(self, longname=None, extradoc=None,
|
||
|
|
docdict=None, discrete='continuous'):
|
||
|
|
"""Construct instance docstring from the default template."""
|
||
|
|
if longname is None:
|
||
|
|
longname = 'A'
|
||
|
|
if extradoc is None:
|
||
|
|
extradoc = ''
|
||
|
|
if extradoc.startswith('\n\n'):
|
||
|
|
extradoc = extradoc[2:]
|
||
|
|
self.__doc__ = ''.join(['%s %s random variable.' % (longname, discrete),
|
||
|
|
'\n\n%(before_notes)s\n', docheaders['notes'],
|
||
|
|
extradoc, '\n%(example)s'])
|
||
|
|
self._construct_doc(docdict)
|
||
|
|
|
||
|
|
def freeze(self, *args, **kwds):
|
||
|
|
"""Freeze the distribution for the given arguments.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution. Should include all
|
||
|
|
the non-optional arguments, may include ``loc`` and ``scale``.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
rv_frozen : rv_frozen instance
|
||
|
|
The frozen distribution.
|
||
|
|
|
||
|
|
"""
|
||
|
|
return rv_frozen(self, *args, **kwds)
|
||
|
|
|
||
|
|
def __call__(self, *args, **kwds):
|
||
|
|
return self.freeze(*args, **kwds)
|
||
|
|
__call__.__doc__ = freeze.__doc__
|
||
|
|
|
||
|
|
# The actual calculation functions (no basic checking need be done)
|
||
|
|
# If these are defined, the others won't be looked at.
|
||
|
|
# Otherwise, the other set can be defined.
|
||
|
|
def _stats(self, *args, **kwds):
|
||
|
|
return None, None, None, None
|
||
|
|
|
||
|
|
# Central moments
|
||
|
|
def _munp(self, n, *args):
|
||
|
|
# Silence floating point warnings from integration.
|
||
|
|
olderr = np.seterr(all='ignore')
|
||
|
|
vals = self.generic_moment(n, *args)
|
||
|
|
np.seterr(**olderr)
|
||
|
|
return vals
|
||
|
|
|
||
|
|
def _argcheck_rvs(self, *args, **kwargs):
|
||
|
|
# Handle broadcasting and size validation of the rvs method.
|
||
|
|
# Subclasses should not have to override this method.
|
||
|
|
# The rule is that if `size` is not None, then `size` gives the
|
||
|
|
# shape of the result (integer values of `size` are treated as
|
||
|
|
# tuples with length 1; i.e. `size=3` is the same as `size=(3,)`.)
|
||
|
|
#
|
||
|
|
# `args` is expected to contain the shape parameters (if any), the
|
||
|
|
# location and the scale in a flat tuple (e.g. if there are two
|
||
|
|
# shape parameters `a` and `b`, `args` will be `(a, b, loc, scale)`).
|
||
|
|
# The only keyword argument expected is 'size'.
|
||
|
|
size = kwargs.get('size', None)
|
||
|
|
all_bcast = np.broadcast_arrays(*args)
|
||
|
|
|
||
|
|
def squeeze_left(a):
|
||
|
|
while a.ndim > 0 and a.shape[0] == 1:
|
||
|
|
a = a[0]
|
||
|
|
return a
|
||
|
|
|
||
|
|
# Eliminate trivial leading dimensions. In the convention
|
||
|
|
# used by numpy's random variate generators, trivial leading
|
||
|
|
# dimensions are effectively ignored. In other words, when `size`
|
||
|
|
# is given, trivial leading dimensions of the broadcast parameters
|
||
|
|
# in excess of the number of dimensions in size are ignored, e.g.
|
||
|
|
# >>> np.random.normal([[1, 3, 5]], [[[[0.01]]]], size=3)
|
||
|
|
# array([ 1.00104267, 3.00422496, 4.99799278])
|
||
|
|
# If `size` is not given, the exact broadcast shape is preserved:
|
||
|
|
# >>> np.random.normal([[1, 3, 5]], [[[[0.01]]]])
|
||
|
|
# array([[[[ 1.00862899, 3.00061431, 4.99867122]]]])
|
||
|
|
#
|
||
|
|
all_bcast = [squeeze_left(a) for a in all_bcast]
|
||
|
|
bcast_shape = all_bcast[0].shape
|
||
|
|
bcast_ndim = all_bcast[0].ndim
|
||
|
|
|
||
|
|
if size is None:
|
||
|
|
size_ = bcast_shape
|
||
|
|
else:
|
||
|
|
size_ = tuple(np.atleast_1d(size))
|
||
|
|
|
||
|
|
# Check compatibility of size_ with the broadcast shape of all
|
||
|
|
# the parameters. This check is intended to be consistent with
|
||
|
|
# how the numpy random variate generators (e.g. np.random.normal,
|
||
|
|
# np.random.beta) handle their arguments. The rule is that, if size
|
||
|
|
# is given, it determines the shape of the output. Broadcasting
|
||
|
|
# can't change the output size.
|
||
|
|
|
||
|
|
# This is the standard broadcasting convention of extending the
|
||
|
|
# shape with fewer dimensions with enough dimensions of length 1
|
||
|
|
# so that the two shapes have the same number of dimensions.
|
||
|
|
ndiff = bcast_ndim - len(size_)
|
||
|
|
if ndiff < 0:
|
||
|
|
bcast_shape = (1,)*(-ndiff) + bcast_shape
|
||
|
|
elif ndiff > 0:
|
||
|
|
size_ = (1,)*ndiff + size_
|
||
|
|
|
||
|
|
# This compatibility test is not standard. In "regular" broadcasting,
|
||
|
|
# two shapes are compatible if for each dimension, the lengths are the
|
||
|
|
# same or one of the lengths is 1. Here, the length of a dimension in
|
||
|
|
# size_ must not be less than the corresponding length in bcast_shape.
|
||
|
|
ok = all([bcdim == 1 or bcdim == szdim
|
||
|
|
for (bcdim, szdim) in zip(bcast_shape, size_)])
|
||
|
|
if not ok:
|
||
|
|
raise ValueError("size does not match the broadcast shape of "
|
||
|
|
"the parameters.")
|
||
|
|
|
||
|
|
param_bcast = all_bcast[:-2]
|
||
|
|
loc_bcast = all_bcast[-2]
|
||
|
|
scale_bcast = all_bcast[-1]
|
||
|
|
|
||
|
|
return param_bcast, loc_bcast, scale_bcast, size_
|
||
|
|
|
||
|
|
## These are the methods you must define (standard form functions)
|
||
|
|
## NB: generic _pdf, _logpdf, _cdf are different for
|
||
|
|
## rv_continuous and rv_discrete hence are defined in there
|
||
|
|
def _argcheck(self, *args):
|
||
|
|
"""Default check for correct values on args and keywords.
|
||
|
|
|
||
|
|
Returns condition array of 1's where arguments are correct and
|
||
|
|
0's where they are not.
|
||
|
|
|
||
|
|
"""
|
||
|
|
cond = 1
|
||
|
|
for arg in args:
|
||
|
|
cond = logical_and(cond, (asarray(arg) > 0))
|
||
|
|
return cond
|
||
|
|
|
||
|
|
def _support_mask(self, x):
|
||
|
|
return (self.a <= x) & (x <= self.b)
|
||
|
|
|
||
|
|
def _open_support_mask(self, x):
|
||
|
|
return (self.a < x) & (x < self.b)
|
||
|
|
|
||
|
|
def _rvs(self, *args):
|
||
|
|
# This method must handle self._size being a tuple, and it must
|
||
|
|
# properly broadcast *args and self._size. self._size might be
|
||
|
|
# an empty tuple, which means a scalar random variate is to be
|
||
|
|
# generated.
|
||
|
|
|
||
|
|
## Use basic inverse cdf algorithm for RV generation as default.
|
||
|
|
U = self._random_state.random_sample(self._size)
|
||
|
|
Y = self._ppf(U, *args)
|
||
|
|
return Y
|
||
|
|
|
||
|
|
def _logcdf(self, x, *args):
|
||
|
|
return log(self._cdf(x, *args))
|
||
|
|
|
||
|
|
def _sf(self, x, *args):
|
||
|
|
return 1.0-self._cdf(x, *args)
|
||
|
|
|
||
|
|
def _logsf(self, x, *args):
|
||
|
|
return log(self._sf(x, *args))
|
||
|
|
|
||
|
|
def _ppf(self, q, *args):
|
||
|
|
return self._ppfvec(q, *args)
|
||
|
|
|
||
|
|
def _isf(self, q, *args):
|
||
|
|
return self._ppf(1.0-q, *args) # use correct _ppf for subclasses
|
||
|
|
|
||
|
|
# These are actually called, and should not be overwritten if you
|
||
|
|
# want to keep error checking.
|
||
|
|
def rvs(self, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Random variates of given type.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
scale : array_like, optional
|
||
|
|
Scale parameter (default=1).
|
||
|
|
size : int or tuple of ints, optional
|
||
|
|
Defining number of random variates (default is 1).
|
||
|
|
random_state : None or int or ``np.random.RandomState`` instance, optional
|
||
|
|
If int or RandomState, use it for drawing the random variates.
|
||
|
|
If None, rely on ``self.random_state``.
|
||
|
|
Default is None.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
rvs : ndarray or scalar
|
||
|
|
Random variates of given `size`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
discrete = kwds.pop('discrete', None)
|
||
|
|
rndm = kwds.pop('random_state', None)
|
||
|
|
args, loc, scale, size = self._parse_args_rvs(*args, **kwds)
|
||
|
|
cond = logical_and(self._argcheck(*args), (scale >= 0))
|
||
|
|
if not np.all(cond):
|
||
|
|
raise ValueError("Domain error in arguments.")
|
||
|
|
|
||
|
|
if np.all(scale == 0):
|
||
|
|
return loc*ones(size, 'd')
|
||
|
|
|
||
|
|
# extra gymnastics needed for a custom random_state
|
||
|
|
if rndm is not None:
|
||
|
|
random_state_saved = self._random_state
|
||
|
|
self._random_state = check_random_state(rndm)
|
||
|
|
|
||
|
|
# `size` should just be an argument to _rvs(), but for, um,
|
||
|
|
# historical reasons, it is made an attribute that is read
|
||
|
|
# by _rvs().
|
||
|
|
self._size = size
|
||
|
|
vals = self._rvs(*args)
|
||
|
|
|
||
|
|
vals = vals * scale + loc
|
||
|
|
|
||
|
|
# do not forget to restore the _random_state
|
||
|
|
if rndm is not None:
|
||
|
|
self._random_state = random_state_saved
|
||
|
|
|
||
|
|
# Cast to int if discrete
|
||
|
|
if discrete:
|
||
|
|
if size == ():
|
||
|
|
vals = int(vals)
|
||
|
|
else:
|
||
|
|
vals = vals.astype(int)
|
||
|
|
|
||
|
|
return vals
|
||
|
|
|
||
|
|
def stats(self, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Some statistics of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional (continuous RVs only)
|
||
|
|
scale parameter (default=1)
|
||
|
|
moments : str, optional
|
||
|
|
composed of letters ['mvsk'] defining which moments to compute:
|
||
|
|
'm' = mean,
|
||
|
|
'v' = variance,
|
||
|
|
's' = (Fisher's) skew,
|
||
|
|
'k' = (Fisher's) kurtosis.
|
||
|
|
(default is 'mv')
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
stats : sequence
|
||
|
|
of requested moments.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale, moments = self._parse_args_stats(*args, **kwds)
|
||
|
|
# scale = 1 by construction for discrete RVs
|
||
|
|
loc, scale = map(asarray, (loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
cond = self._argcheck(*args) & (scale > 0) & (loc == loc)
|
||
|
|
output = []
|
||
|
|
default = valarray(shape(cond), self.badvalue)
|
||
|
|
|
||
|
|
# Use only entries that are valid in calculation
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *(args+(scale, loc)))
|
||
|
|
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
|
||
|
|
|
||
|
|
if self._stats_has_moments:
|
||
|
|
mu, mu2, g1, g2 = self._stats(*goodargs,
|
||
|
|
**{'moments': moments})
|
||
|
|
else:
|
||
|
|
mu, mu2, g1, g2 = self._stats(*goodargs)
|
||
|
|
if g1 is None:
|
||
|
|
mu3 = None
|
||
|
|
else:
|
||
|
|
if mu2 is None:
|
||
|
|
mu2 = self._munp(2, *goodargs)
|
||
|
|
if g2 is None:
|
||
|
|
# (mu2**1.5) breaks down for nan and inf
|
||
|
|
mu3 = g1 * np.power(mu2, 1.5)
|
||
|
|
|
||
|
|
if 'm' in moments:
|
||
|
|
if mu is None:
|
||
|
|
mu = self._munp(1, *goodargs)
|
||
|
|
out0 = default.copy()
|
||
|
|
place(out0, cond, mu * scale + loc)
|
||
|
|
output.append(out0)
|
||
|
|
|
||
|
|
if 'v' in moments:
|
||
|
|
if mu2 is None:
|
||
|
|
mu2p = self._munp(2, *goodargs)
|
||
|
|
if mu is None:
|
||
|
|
mu = self._munp(1, *goodargs)
|
||
|
|
mu2 = mu2p - mu * mu
|
||
|
|
if np.isinf(mu):
|
||
|
|
# if mean is inf then var is also inf
|
||
|
|
mu2 = np.inf
|
||
|
|
out0 = default.copy()
|
||
|
|
place(out0, cond, mu2 * scale * scale)
|
||
|
|
output.append(out0)
|
||
|
|
|
||
|
|
if 's' in moments:
|
||
|
|
if g1 is None:
|
||
|
|
mu3p = self._munp(3, *goodargs)
|
||
|
|
if mu is None:
|
||
|
|
mu = self._munp(1, *goodargs)
|
||
|
|
if mu2 is None:
|
||
|
|
mu2p = self._munp(2, *goodargs)
|
||
|
|
mu2 = mu2p - mu * mu
|
||
|
|
with np.errstate(invalid='ignore'):
|
||
|
|
mu3 = mu3p - 3 * mu * mu2 - mu**3
|
||
|
|
g1 = mu3 / np.power(mu2, 1.5)
|
||
|
|
out0 = default.copy()
|
||
|
|
place(out0, cond, g1)
|
||
|
|
output.append(out0)
|
||
|
|
|
||
|
|
if 'k' in moments:
|
||
|
|
if g2 is None:
|
||
|
|
mu4p = self._munp(4, *goodargs)
|
||
|
|
if mu is None:
|
||
|
|
mu = self._munp(1, *goodargs)
|
||
|
|
if mu2 is None:
|
||
|
|
mu2p = self._munp(2, *goodargs)
|
||
|
|
mu2 = mu2p - mu * mu
|
||
|
|
if mu3 is None:
|
||
|
|
mu3p = self._munp(3, *goodargs)
|
||
|
|
with np.errstate(invalid='ignore'):
|
||
|
|
mu3 = mu3p - 3 * mu * mu2 - mu**3
|
||
|
|
with np.errstate(invalid='ignore'):
|
||
|
|
mu4 = mu4p - 4 * mu * mu3 - 6 * mu * mu * mu2 - mu**4
|
||
|
|
g2 = mu4 / mu2**2.0 - 3.0
|
||
|
|
out0 = default.copy()
|
||
|
|
place(out0, cond, g2)
|
||
|
|
output.append(out0)
|
||
|
|
else: # no valid args
|
||
|
|
output = []
|
||
|
|
for _ in moments:
|
||
|
|
out0 = default.copy()
|
||
|
|
output.append(out0)
|
||
|
|
|
||
|
|
if len(output) == 1:
|
||
|
|
return output[0]
|
||
|
|
else:
|
||
|
|
return tuple(output)
|
||
|
|
|
||
|
|
def entropy(self, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Differential entropy of the RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
scale : array_like, optional (continuous distributions only).
|
||
|
|
Scale parameter (default=1).
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Entropy is defined base `e`:
|
||
|
|
|
||
|
|
>>> drv = rv_discrete(values=((0, 1), (0.5, 0.5)))
|
||
|
|
>>> np.allclose(drv.entropy(), np.log(2.0))
|
||
|
|
True
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
# NB: for discrete distributions scale=1 by construction in _parse_args
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
|
||
|
|
output = zeros(shape(cond0), 'd')
|
||
|
|
place(output, (1-cond0), self.badvalue)
|
||
|
|
goodargs = argsreduce(cond0, *args)
|
||
|
|
place(output, cond0, self.vecentropy(*goodargs) + log(scale))
|
||
|
|
return output
|
||
|
|
|
||
|
|
def moment(self, n, *args, **kwds):
|
||
|
|
"""
|
||
|
|
n-th order non-central moment of distribution.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
n : int, n >= 1
|
||
|
|
Order of moment.
|
||
|
|
arg1, arg2, arg3,... : float
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
if not (self._argcheck(*args) and (scale > 0)):
|
||
|
|
return nan
|
||
|
|
if (floor(n) != n):
|
||
|
|
raise ValueError("Moment must be an integer.")
|
||
|
|
if (n < 0):
|
||
|
|
raise ValueError("Moment must be positive.")
|
||
|
|
mu, mu2, g1, g2 = None, None, None, None
|
||
|
|
if (n > 0) and (n < 5):
|
||
|
|
if self._stats_has_moments:
|
||
|
|
mdict = {'moments': {1: 'm', 2: 'v', 3: 'vs', 4: 'vk'}[n]}
|
||
|
|
else:
|
||
|
|
mdict = {}
|
||
|
|
mu, mu2, g1, g2 = self._stats(*args, **mdict)
|
||
|
|
val = _moment_from_stats(n, mu, mu2, g1, g2, self._munp, args)
|
||
|
|
|
||
|
|
# Convert to transformed X = L + S*Y
|
||
|
|
# E[X^n] = E[(L+S*Y)^n] = L^n sum(comb(n, k)*(S/L)^k E[Y^k], k=0...n)
|
||
|
|
if loc == 0:
|
||
|
|
return scale**n * val
|
||
|
|
else:
|
||
|
|
result = 0
|
||
|
|
fac = float(scale) / float(loc)
|
||
|
|
for k in range(n):
|
||
|
|
valk = _moment_from_stats(k, mu, mu2, g1, g2, self._munp, args)
|
||
|
|
result += comb(n, k, exact=True)*(fac**k) * valk
|
||
|
|
result += fac**n * val
|
||
|
|
return result * loc**n
|
||
|
|
|
||
|
|
def median(self, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Median of the distribution.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter, Default is 0.
|
||
|
|
scale : array_like, optional
|
||
|
|
Scale parameter, Default is 1.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
median : float
|
||
|
|
The median of the distribution.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
stats.distributions.rv_discrete.ppf
|
||
|
|
Inverse of the CDF
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self.ppf(0.5, *args, **kwds)
|
||
|
|
|
||
|
|
def mean(self, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Mean of the distribution.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
mean : float
|
||
|
|
the mean of the distribution
|
||
|
|
|
||
|
|
"""
|
||
|
|
kwds['moments'] = 'm'
|
||
|
|
res = self.stats(*args, **kwds)
|
||
|
|
if isinstance(res, ndarray) and res.ndim == 0:
|
||
|
|
return res[()]
|
||
|
|
return res
|
||
|
|
|
||
|
|
def var(self, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Variance of the distribution.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
var : float
|
||
|
|
the variance of the distribution
|
||
|
|
|
||
|
|
"""
|
||
|
|
kwds['moments'] = 'v'
|
||
|
|
res = self.stats(*args, **kwds)
|
||
|
|
if isinstance(res, ndarray) and res.ndim == 0:
|
||
|
|
return res[()]
|
||
|
|
return res
|
||
|
|
|
||
|
|
def std(self, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Standard deviation of the distribution.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
std : float
|
||
|
|
standard deviation of the distribution
|
||
|
|
|
||
|
|
"""
|
||
|
|
kwds['moments'] = 'v'
|
||
|
|
res = sqrt(self.stats(*args, **kwds))
|
||
|
|
return res
|
||
|
|
|
||
|
|
def interval(self, alpha, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Confidence interval with equal areas around the median.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
alpha : array_like of float
|
||
|
|
Probability that an rv will be drawn from the returned range.
|
||
|
|
Each value should be in the range [0, 1].
|
||
|
|
arg1, arg2, ... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter, Default is 0.
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter, Default is 1.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
a, b : ndarray of float
|
||
|
|
end-points of range that contain ``100 * alpha %`` of the rv's
|
||
|
|
possible values.
|
||
|
|
|
||
|
|
"""
|
||
|
|
alpha = asarray(alpha)
|
||
|
|
if np.any((alpha > 1) | (alpha < 0)):
|
||
|
|
raise ValueError("alpha must be between 0 and 1 inclusive")
|
||
|
|
q1 = (1.0-alpha)/2
|
||
|
|
q2 = (1.0+alpha)/2
|
||
|
|
a = self.ppf(q1, *args, **kwds)
|
||
|
|
b = self.ppf(q2, *args, **kwds)
|
||
|
|
return a, b
|
||
|
|
|
||
|
|
|
||
|
|
## continuous random variables: implement maybe later
|
||
|
|
##
|
||
|
|
## hf --- Hazard Function (PDF / SF)
|
||
|
|
## chf --- Cumulative hazard function (-log(SF))
|
||
|
|
## psf --- Probability sparsity function (reciprocal of the pdf) in
|
||
|
|
## units of percent-point-function (as a function of q).
|
||
|
|
## Also, the derivative of the percent-point function.
|
||
|
|
|
||
|
|
class rv_continuous(rv_generic):
|
||
|
|
"""
|
||
|
|
A generic continuous random variable class meant for subclassing.
|
||
|
|
|
||
|
|
`rv_continuous` is a base class to construct specific distribution classes
|
||
|
|
and instances for continuous random variables. It cannot be used
|
||
|
|
directly as a distribution.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
momtype : int, optional
|
||
|
|
The type of generic moment calculation to use: 0 for pdf, 1 (default)
|
||
|
|
for ppf.
|
||
|
|
a : float, optional
|
||
|
|
Lower bound of the support of the distribution, default is minus
|
||
|
|
infinity.
|
||
|
|
b : float, optional
|
||
|
|
Upper bound of the support of the distribution, default is plus
|
||
|
|
infinity.
|
||
|
|
xtol : float, optional
|
||
|
|
The tolerance for fixed point calculation for generic ppf.
|
||
|
|
badvalue : float, optional
|
||
|
|
The value in a result arrays that indicates a value that for which
|
||
|
|
some argument restriction is violated, default is np.nan.
|
||
|
|
name : str, optional
|
||
|
|
The name of the instance. This string is used to construct the default
|
||
|
|
example for distributions.
|
||
|
|
longname : str, optional
|
||
|
|
This string is used as part of the first line of the docstring returned
|
||
|
|
when a subclass has no docstring of its own. Note: `longname` exists
|
||
|
|
for backwards compatibility, do not use for new subclasses.
|
||
|
|
shapes : str, optional
|
||
|
|
The shape of the distribution. For example ``"m, n"`` for a
|
||
|
|
distribution that takes two integers as the two shape arguments for all
|
||
|
|
its methods. If not provided, shape parameters will be inferred from
|
||
|
|
the signature of the private methods, ``_pdf`` and ``_cdf`` of the
|
||
|
|
instance.
|
||
|
|
extradoc : str, optional, deprecated
|
||
|
|
This string is used as the last part of the docstring returned when a
|
||
|
|
subclass has no docstring of its own. Note: `extradoc` exists for
|
||
|
|
backwards compatibility, do not use for new subclasses.
|
||
|
|
seed : None or int or ``numpy.random.RandomState`` instance, optional
|
||
|
|
This parameter defines the RandomState object to use for drawing
|
||
|
|
random variates.
|
||
|
|
If None (or np.random), the global np.random state is used.
|
||
|
|
If integer, it is used to seed the local RandomState instance.
|
||
|
|
Default is None.
|
||
|
|
|
||
|
|
Methods
|
||
|
|
-------
|
||
|
|
rvs
|
||
|
|
pdf
|
||
|
|
logpdf
|
||
|
|
cdf
|
||
|
|
logcdf
|
||
|
|
sf
|
||
|
|
logsf
|
||
|
|
ppf
|
||
|
|
isf
|
||
|
|
moment
|
||
|
|
stats
|
||
|
|
entropy
|
||
|
|
expect
|
||
|
|
median
|
||
|
|
mean
|
||
|
|
std
|
||
|
|
var
|
||
|
|
interval
|
||
|
|
__call__
|
||
|
|
fit
|
||
|
|
fit_loc_scale
|
||
|
|
nnlf
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Public methods of an instance of a distribution class (e.g., ``pdf``,
|
||
|
|
``cdf``) check their arguments and pass valid arguments to private,
|
||
|
|
computational methods (``_pdf``, ``_cdf``). For ``pdf(x)``, ``x`` is valid
|
||
|
|
if it is within the support of a distribution, ``self.a <= x <= self.b``.
|
||
|
|
Whether a shape parameter is valid is decided by an ``_argcheck`` method
|
||
|
|
(which defaults to checking that its arguments are strictly positive.)
|
||
|
|
|
||
|
|
**Subclassing**
|
||
|
|
|
||
|
|
New random variables can be defined by subclassing the `rv_continuous` class
|
||
|
|
and re-defining at least the ``_pdf`` or the ``_cdf`` method (normalized
|
||
|
|
to location 0 and scale 1).
|
||
|
|
|
||
|
|
If positive argument checking is not correct for your RV
|
||
|
|
then you will also need to re-define the ``_argcheck`` method.
|
||
|
|
|
||
|
|
Correct, but potentially slow defaults exist for the remaining
|
||
|
|
methods but for speed and/or accuracy you can over-ride::
|
||
|
|
|
||
|
|
_logpdf, _cdf, _logcdf, _ppf, _rvs, _isf, _sf, _logsf
|
||
|
|
|
||
|
|
Rarely would you override ``_isf``, ``_sf`` or ``_logsf``, but you could.
|
||
|
|
|
||
|
|
**Methods that can be overwritten by subclasses**
|
||
|
|
::
|
||
|
|
|
||
|
|
_rvs
|
||
|
|
_pdf
|
||
|
|
_cdf
|
||
|
|
_sf
|
||
|
|
_ppf
|
||
|
|
_isf
|
||
|
|
_stats
|
||
|
|
_munp
|
||
|
|
_entropy
|
||
|
|
_argcheck
|
||
|
|
|
||
|
|
There are additional (internal and private) generic methods that can
|
||
|
|
be useful for cross-checking and for debugging, but might work in all
|
||
|
|
cases when directly called.
|
||
|
|
|
||
|
|
A note on ``shapes``: subclasses need not specify them explicitly. In this
|
||
|
|
case, `shapes` will be automatically deduced from the signatures of the
|
||
|
|
overridden methods (`pdf`, `cdf` etc).
|
||
|
|
If, for some reason, you prefer to avoid relying on introspection, you can
|
||
|
|
specify ``shapes`` explicitly as an argument to the instance constructor.
|
||
|
|
|
||
|
|
|
||
|
|
**Frozen Distributions**
|
||
|
|
|
||
|
|
Normally, you must provide shape parameters (and, optionally, location and
|
||
|
|
scale parameters to each call of a method of a distribution.
|
||
|
|
|
||
|
|
Alternatively, the object may be called (as a function) to fix the shape,
|
||
|
|
location, and scale parameters returning a "frozen" continuous RV object:
|
||
|
|
|
||
|
|
rv = generic(<shape(s)>, loc=0, scale=1)
|
||
|
|
frozen RV object with the same methods but holding the given shape,
|
||
|
|
location, and scale fixed
|
||
|
|
|
||
|
|
**Statistics**
|
||
|
|
|
||
|
|
Statistics are computed using numerical integration by default.
|
||
|
|
For speed you can redefine this using ``_stats``:
|
||
|
|
|
||
|
|
- take shape parameters and return mu, mu2, g1, g2
|
||
|
|
- If you can't compute one of these, return it as None
|
||
|
|
- Can also be defined with a keyword argument ``moments``, which is a
|
||
|
|
string composed of "m", "v", "s", and/or "k".
|
||
|
|
Only the components appearing in string should be computed and
|
||
|
|
returned in the order "m", "v", "s", or "k" with missing values
|
||
|
|
returned as None.
|
||
|
|
|
||
|
|
Alternatively, you can override ``_munp``, which takes ``n`` and shape
|
||
|
|
parameters and returns the n-th non-central moment of the distribution.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
To create a new Gaussian distribution, we would do the following:
|
||
|
|
|
||
|
|
>>> from scipy.stats import rv_continuous
|
||
|
|
>>> class gaussian_gen(rv_continuous):
|
||
|
|
... "Gaussian distribution"
|
||
|
|
... def _pdf(self, x):
|
||
|
|
... return np.exp(-x**2 / 2.) / np.sqrt(2.0 * np.pi)
|
||
|
|
>>> gaussian = gaussian_gen(name='gaussian')
|
||
|
|
|
||
|
|
``scipy.stats`` distributions are *instances*, so here we subclass
|
||
|
|
`rv_continuous` and create an instance. With this, we now have
|
||
|
|
a fully functional distribution with all relevant methods automagically
|
||
|
|
generated by the framework.
|
||
|
|
|
||
|
|
Note that above we defined a standard normal distribution, with zero mean
|
||
|
|
and unit variance. Shifting and scaling of the distribution can be done
|
||
|
|
by using ``loc`` and ``scale`` parameters: ``gaussian.pdf(x, loc, scale)``
|
||
|
|
essentially computes ``y = (x - loc) / scale`` and
|
||
|
|
``gaussian._pdf(y) / scale``.
|
||
|
|
|
||
|
|
"""
|
||
|
|
def __init__(self, momtype=1, a=None, b=None, xtol=1e-14,
|
||
|
|
badvalue=None, name=None, longname=None,
|
||
|
|
shapes=None, extradoc=None, seed=None):
|
||
|
|
|
||
|
|
super(rv_continuous, self).__init__(seed)
|
||
|
|
|
||
|
|
# save the ctor parameters, cf generic freeze
|
||
|
|
self._ctor_param = dict(
|
||
|
|
momtype=momtype, a=a, b=b, xtol=xtol,
|
||
|
|
badvalue=badvalue, name=name, longname=longname,
|
||
|
|
shapes=shapes, extradoc=extradoc, seed=seed)
|
||
|
|
|
||
|
|
if badvalue is None:
|
||
|
|
badvalue = nan
|
||
|
|
if name is None:
|
||
|
|
name = 'Distribution'
|
||
|
|
self.badvalue = badvalue
|
||
|
|
self.name = name
|
||
|
|
self.a = a
|
||
|
|
self.b = b
|
||
|
|
if a is None:
|
||
|
|
self.a = -inf
|
||
|
|
if b is None:
|
||
|
|
self.b = inf
|
||
|
|
self.xtol = xtol
|
||
|
|
self.moment_type = momtype
|
||
|
|
self.shapes = shapes
|
||
|
|
self._construct_argparser(meths_to_inspect=[self._pdf, self._cdf],
|
||
|
|
locscale_in='loc=0, scale=1',
|
||
|
|
locscale_out='loc, scale')
|
||
|
|
|
||
|
|
# nin correction
|
||
|
|
self._ppfvec = vectorize(self._ppf_single, otypes='d')
|
||
|
|
self._ppfvec.nin = self.numargs + 1
|
||
|
|
self.vecentropy = vectorize(self._entropy, otypes='d')
|
||
|
|
self._cdfvec = vectorize(self._cdf_single, otypes='d')
|
||
|
|
self._cdfvec.nin = self.numargs + 1
|
||
|
|
|
||
|
|
self.extradoc = extradoc
|
||
|
|
if momtype == 0:
|
||
|
|
self.generic_moment = vectorize(self._mom0_sc, otypes='d')
|
||
|
|
else:
|
||
|
|
self.generic_moment = vectorize(self._mom1_sc, otypes='d')
|
||
|
|
# Because of the *args argument of _mom0_sc, vectorize cannot count the
|
||
|
|
# number of arguments correctly.
|
||
|
|
self.generic_moment.nin = self.numargs + 1
|
||
|
|
|
||
|
|
if longname is None:
|
||
|
|
if name[0] in ['aeiouAEIOU']:
|
||
|
|
hstr = "An "
|
||
|
|
else:
|
||
|
|
hstr = "A "
|
||
|
|
longname = hstr + name
|
||
|
|
|
||
|
|
if sys.flags.optimize < 2:
|
||
|
|
# Skip adding docstrings if interpreter is run with -OO
|
||
|
|
if self.__doc__ is None:
|
||
|
|
self._construct_default_doc(longname=longname,
|
||
|
|
extradoc=extradoc,
|
||
|
|
docdict=docdict,
|
||
|
|
discrete='continuous')
|
||
|
|
else:
|
||
|
|
dct = dict(distcont)
|
||
|
|
self._construct_doc(docdict, dct.get(self.name))
|
||
|
|
|
||
|
|
def _updated_ctor_param(self):
|
||
|
|
""" Return the current version of _ctor_param, possibly updated by user.
|
||
|
|
|
||
|
|
Used by freezing and pickling.
|
||
|
|
Keep this in sync with the signature of __init__.
|
||
|
|
"""
|
||
|
|
dct = self._ctor_param.copy()
|
||
|
|
dct['a'] = self.a
|
||
|
|
dct['b'] = self.b
|
||
|
|
dct['xtol'] = self.xtol
|
||
|
|
dct['badvalue'] = self.badvalue
|
||
|
|
dct['name'] = self.name
|
||
|
|
dct['shapes'] = self.shapes
|
||
|
|
dct['extradoc'] = self.extradoc
|
||
|
|
return dct
|
||
|
|
|
||
|
|
def _ppf_to_solve(self, x, q, *args):
|
||
|
|
return self.cdf(*(x, )+args)-q
|
||
|
|
|
||
|
|
def _ppf_single(self, q, *args):
|
||
|
|
left = right = None
|
||
|
|
if self.a > -np.inf:
|
||
|
|
left = self.a
|
||
|
|
if self.b < np.inf:
|
||
|
|
right = self.b
|
||
|
|
|
||
|
|
factor = 10.
|
||
|
|
if not left: # i.e. self.a = -inf
|
||
|
|
left = -1.*factor
|
||
|
|
while self._ppf_to_solve(left, q, *args) > 0.:
|
||
|
|
right = left
|
||
|
|
left *= factor
|
||
|
|
# left is now such that cdf(left) < q
|
||
|
|
if not right: # i.e. self.b = inf
|
||
|
|
right = factor
|
||
|
|
while self._ppf_to_solve(right, q, *args) < 0.:
|
||
|
|
left = right
|
||
|
|
right *= factor
|
||
|
|
# right is now such that cdf(right) > q
|
||
|
|
|
||
|
|
return optimize.brentq(self._ppf_to_solve,
|
||
|
|
left, right, args=(q,)+args, xtol=self.xtol)
|
||
|
|
|
||
|
|
# moment from definition
|
||
|
|
def _mom_integ0(self, x, m, *args):
|
||
|
|
return x**m * self.pdf(x, *args)
|
||
|
|
|
||
|
|
def _mom0_sc(self, m, *args):
|
||
|
|
return integrate.quad(self._mom_integ0, self.a, self.b,
|
||
|
|
args=(m,)+args)[0]
|
||
|
|
|
||
|
|
# moment calculated using ppf
|
||
|
|
def _mom_integ1(self, q, m, *args):
|
||
|
|
return (self.ppf(q, *args))**m
|
||
|
|
|
||
|
|
def _mom1_sc(self, m, *args):
|
||
|
|
return integrate.quad(self._mom_integ1, 0, 1, args=(m,)+args)[0]
|
||
|
|
|
||
|
|
def _pdf(self, x, *args):
|
||
|
|
return derivative(self._cdf, x, dx=1e-5, args=args, order=5)
|
||
|
|
|
||
|
|
## Could also define any of these
|
||
|
|
def _logpdf(self, x, *args):
|
||
|
|
return log(self._pdf(x, *args))
|
||
|
|
|
||
|
|
def _cdf_single(self, x, *args):
|
||
|
|
return integrate.quad(self._pdf, self.a, x, args=args)[0]
|
||
|
|
|
||
|
|
def _cdf(self, x, *args):
|
||
|
|
return self._cdfvec(x, *args)
|
||
|
|
|
||
|
|
## generic _argcheck, _logcdf, _sf, _logsf, _ppf, _isf, _rvs are defined
|
||
|
|
## in rv_generic
|
||
|
|
|
||
|
|
def pdf(self, x, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Probability density function at x of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
quantiles
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
pdf : ndarray
|
||
|
|
Probability density function evaluated at x
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
x, loc, scale = map(asarray, (x, loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
dtyp = np.find_common_type([x.dtype, np.float64], [])
|
||
|
|
x = np.asarray((x - loc)/scale, dtype=dtyp)
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0)
|
||
|
|
cond1 = self._support_mask(x) & (scale > 0)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = zeros(shape(cond), dtyp)
|
||
|
|
putmask(output, (1-cond0)+np.isnan(x), self.badvalue)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((x,)+args+(scale,)))
|
||
|
|
scale, goodargs = goodargs[-1], goodargs[:-1]
|
||
|
|
place(output, cond, self._pdf(*goodargs) / scale)
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def logpdf(self, x, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Log of the probability density function at x of the given RV.
|
||
|
|
|
||
|
|
This uses a more numerically accurate calculation if available.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
quantiles
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
logpdf : array_like
|
||
|
|
Log of the probability density function evaluated at x
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
x, loc, scale = map(asarray, (x, loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
dtyp = np.find_common_type([x.dtype, np.float64], [])
|
||
|
|
x = np.asarray((x - loc)/scale, dtype=dtyp)
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0)
|
||
|
|
cond1 = self._support_mask(x) & (scale > 0)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = empty(shape(cond), dtyp)
|
||
|
|
output.fill(NINF)
|
||
|
|
putmask(output, (1-cond0)+np.isnan(x), self.badvalue)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((x,)+args+(scale,)))
|
||
|
|
scale, goodargs = goodargs[-1], goodargs[:-1]
|
||
|
|
place(output, cond, self._logpdf(*goodargs) - log(scale))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def cdf(self, x, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Cumulative distribution function of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
quantiles
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
cdf : ndarray
|
||
|
|
Cumulative distribution function evaluated at `x`
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
x, loc, scale = map(asarray, (x, loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
dtyp = np.find_common_type([x.dtype, np.float64], [])
|
||
|
|
x = np.asarray((x - loc)/scale, dtype=dtyp)
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0)
|
||
|
|
cond1 = self._open_support_mask(x) & (scale > 0)
|
||
|
|
cond2 = (x >= self.b) & cond0
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = zeros(shape(cond), dtyp)
|
||
|
|
place(output, (1-cond0)+np.isnan(x), self.badvalue)
|
||
|
|
place(output, cond2, 1.0)
|
||
|
|
if np.any(cond): # call only if at least 1 entry
|
||
|
|
goodargs = argsreduce(cond, *((x,)+args))
|
||
|
|
place(output, cond, self._cdf(*goodargs))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def logcdf(self, x, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Log of the cumulative distribution function at x of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
quantiles
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
logcdf : array_like
|
||
|
|
Log of the cumulative distribution function evaluated at x
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
x, loc, scale = map(asarray, (x, loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
dtyp = np.find_common_type([x.dtype, np.float64], [])
|
||
|
|
x = np.asarray((x - loc)/scale, dtype=dtyp)
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0)
|
||
|
|
cond1 = self._open_support_mask(x) & (scale > 0)
|
||
|
|
cond2 = (x >= self.b) & cond0
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = empty(shape(cond), dtyp)
|
||
|
|
output.fill(NINF)
|
||
|
|
place(output, (1-cond0)*(cond1 == cond1)+np.isnan(x), self.badvalue)
|
||
|
|
place(output, cond2, 0.0)
|
||
|
|
if np.any(cond): # call only if at least 1 entry
|
||
|
|
goodargs = argsreduce(cond, *((x,)+args))
|
||
|
|
place(output, cond, self._logcdf(*goodargs))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def sf(self, x, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Survival function (1 - `cdf`) at x of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
quantiles
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
sf : array_like
|
||
|
|
Survival function evaluated at x
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
x, loc, scale = map(asarray, (x, loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
dtyp = np.find_common_type([x.dtype, np.float64], [])
|
||
|
|
x = np.asarray((x - loc)/scale, dtype=dtyp)
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0)
|
||
|
|
cond1 = self._open_support_mask(x) & (scale > 0)
|
||
|
|
cond2 = cond0 & (x <= self.a)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = zeros(shape(cond), dtyp)
|
||
|
|
place(output, (1-cond0)+np.isnan(x), self.badvalue)
|
||
|
|
place(output, cond2, 1.0)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((x,)+args))
|
||
|
|
place(output, cond, self._sf(*goodargs))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def logsf(self, x, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Log of the survival function of the given RV.
|
||
|
|
|
||
|
|
Returns the log of the "survival function," defined as (1 - `cdf`),
|
||
|
|
evaluated at `x`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
x : array_like
|
||
|
|
quantiles
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
logsf : ndarray
|
||
|
|
Log of the survival function evaluated at `x`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
x, loc, scale = map(asarray, (x, loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
dtyp = np.find_common_type([x.dtype, np.float64], [])
|
||
|
|
x = np.asarray((x - loc)/scale, dtype=dtyp)
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0)
|
||
|
|
cond1 = self._open_support_mask(x) & (scale > 0)
|
||
|
|
cond2 = cond0 & (x <= self.a)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = empty(shape(cond), dtyp)
|
||
|
|
output.fill(NINF)
|
||
|
|
place(output, (1-cond0)+np.isnan(x), self.badvalue)
|
||
|
|
place(output, cond2, 0.0)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((x,)+args))
|
||
|
|
place(output, cond, self._logsf(*goodargs))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def ppf(self, q, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Percent point function (inverse of `cdf`) at q of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
q : array_like
|
||
|
|
lower tail probability
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
x : array_like
|
||
|
|
quantile corresponding to the lower tail probability q.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
q, loc, scale = map(asarray, (q, loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
|
||
|
|
cond1 = (0 < q) & (q < 1)
|
||
|
|
cond2 = cond0 & (q == 0)
|
||
|
|
cond3 = cond0 & (q == 1)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = valarray(shape(cond), value=self.badvalue)
|
||
|
|
|
||
|
|
lower_bound = self.a * scale + loc
|
||
|
|
upper_bound = self.b * scale + loc
|
||
|
|
place(output, cond2, argsreduce(cond2, lower_bound)[0])
|
||
|
|
place(output, cond3, argsreduce(cond3, upper_bound)[0])
|
||
|
|
|
||
|
|
if np.any(cond): # call only if at least 1 entry
|
||
|
|
goodargs = argsreduce(cond, *((q,)+args+(scale, loc)))
|
||
|
|
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
|
||
|
|
place(output, cond, self._ppf(*goodargs) * scale + loc)
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def isf(self, q, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Inverse survival function (inverse of `sf`) at q of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
q : array_like
|
||
|
|
upper tail probability
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
location parameter (default=0)
|
||
|
|
scale : array_like, optional
|
||
|
|
scale parameter (default=1)
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
x : ndarray or scalar
|
||
|
|
Quantile corresponding to the upper tail probability q.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, scale = self._parse_args(*args, **kwds)
|
||
|
|
q, loc, scale = map(asarray, (q, loc, scale))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
cond0 = self._argcheck(*args) & (scale > 0) & (loc == loc)
|
||
|
|
cond1 = (0 < q) & (q < 1)
|
||
|
|
cond2 = cond0 & (q == 1)
|
||
|
|
cond3 = cond0 & (q == 0)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = valarray(shape(cond), value=self.badvalue)
|
||
|
|
|
||
|
|
lower_bound = self.a * scale + loc
|
||
|
|
upper_bound = self.b * scale + loc
|
||
|
|
place(output, cond2, argsreduce(cond2, lower_bound)[0])
|
||
|
|
place(output, cond3, argsreduce(cond3, upper_bound)[0])
|
||
|
|
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((q,)+args+(scale, loc)))
|
||
|
|
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
|
||
|
|
place(output, cond, self._isf(*goodargs) * scale + loc)
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def _nnlf(self, x, *args):
|
||
|
|
return -np.sum(self._logpdf(x, *args), axis=0)
|
||
|
|
|
||
|
|
def _unpack_loc_scale(self, theta):
|
||
|
|
try:
|
||
|
|
loc = theta[-2]
|
||
|
|
scale = theta[-1]
|
||
|
|
args = tuple(theta[:-2])
|
||
|
|
except IndexError:
|
||
|
|
raise ValueError("Not enough input arguments.")
|
||
|
|
return loc, scale, args
|
||
|
|
|
||
|
|
def nnlf(self, theta, x):
|
||
|
|
'''Return negative loglikelihood function.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is ``-sum(log pdf(x, theta), axis=0)`` where `theta` are the
|
||
|
|
parameters (including loc and scale).
|
||
|
|
'''
|
||
|
|
loc, scale, args = self._unpack_loc_scale(theta)
|
||
|
|
if not self._argcheck(*args) or scale <= 0:
|
||
|
|
return inf
|
||
|
|
x = asarray((x-loc) / scale)
|
||
|
|
n_log_scale = len(x) * log(scale)
|
||
|
|
if np.any(~self._support_mask(x)):
|
||
|
|
return inf
|
||
|
|
return self._nnlf(x, *args) + n_log_scale
|
||
|
|
|
||
|
|
def _nnlf_and_penalty(self, x, args):
|
||
|
|
cond0 = ~self._support_mask(x)
|
||
|
|
n_bad = np.count_nonzero(cond0, axis=0)
|
||
|
|
if n_bad > 0:
|
||
|
|
x = argsreduce(~cond0, x)[0]
|
||
|
|
logpdf = self._logpdf(x, *args)
|
||
|
|
finite_logpdf = np.isfinite(logpdf)
|
||
|
|
n_bad += np.sum(~finite_logpdf, axis=0)
|
||
|
|
if n_bad > 0:
|
||
|
|
penalty = n_bad * log(_XMAX) * 100
|
||
|
|
return -np.sum(logpdf[finite_logpdf], axis=0) + penalty
|
||
|
|
return -np.sum(logpdf, axis=0)
|
||
|
|
|
||
|
|
def _penalized_nnlf(self, theta, x):
|
||
|
|
''' Return penalized negative loglikelihood function,
|
||
|
|
i.e., - sum (log pdf(x, theta), axis=0) + penalty
|
||
|
|
where theta are the parameters (including loc and scale)
|
||
|
|
'''
|
||
|
|
loc, scale, args = self._unpack_loc_scale(theta)
|
||
|
|
if not self._argcheck(*args) or scale <= 0:
|
||
|
|
return inf
|
||
|
|
x = asarray((x-loc) / scale)
|
||
|
|
n_log_scale = len(x) * log(scale)
|
||
|
|
return self._nnlf_and_penalty(x, args) + n_log_scale
|
||
|
|
|
||
|
|
# return starting point for fit (shape arguments + loc + scale)
|
||
|
|
def _fitstart(self, data, args=None):
|
||
|
|
if args is None:
|
||
|
|
args = (1.0,)*self.numargs
|
||
|
|
loc, scale = self._fit_loc_scale_support(data, *args)
|
||
|
|
return args + (loc, scale)
|
||
|
|
|
||
|
|
# Return the (possibly reduced) function to optimize in order to find MLE
|
||
|
|
# estimates for the .fit method
|
||
|
|
def _reduce_func(self, args, kwds):
|
||
|
|
# First of all, convert fshapes params to fnum: eg for stats.beta,
|
||
|
|
# shapes='a, b'. To fix `a`, can specify either `f1` or `fa`.
|
||
|
|
# Convert the latter into the former.
|
||
|
|
if self.shapes:
|
||
|
|
shapes = self.shapes.replace(',', ' ').split()
|
||
|
|
for j, s in enumerate(shapes):
|
||
|
|
val = kwds.pop('f' + s, None) or kwds.pop('fix_' + s, None)
|
||
|
|
if val is not None:
|
||
|
|
key = 'f%d' % j
|
||
|
|
if key in kwds:
|
||
|
|
raise ValueError("Duplicate entry for %s." % key)
|
||
|
|
else:
|
||
|
|
kwds[key] = val
|
||
|
|
|
||
|
|
args = list(args)
|
||
|
|
Nargs = len(args)
|
||
|
|
fixedn = []
|
||
|
|
names = ['f%d' % n for n in range(Nargs - 2)] + ['floc', 'fscale']
|
||
|
|
x0 = []
|
||
|
|
for n, key in enumerate(names):
|
||
|
|
if key in kwds:
|
||
|
|
fixedn.append(n)
|
||
|
|
args[n] = kwds.pop(key)
|
||
|
|
else:
|
||
|
|
x0.append(args[n])
|
||
|
|
|
||
|
|
if len(fixedn) == 0:
|
||
|
|
func = self._penalized_nnlf
|
||
|
|
restore = None
|
||
|
|
else:
|
||
|
|
if len(fixedn) == Nargs:
|
||
|
|
raise ValueError(
|
||
|
|
"All parameters fixed. There is nothing to optimize.")
|
||
|
|
|
||
|
|
def restore(args, theta):
|
||
|
|
# Replace with theta for all numbers not in fixedn
|
||
|
|
# This allows the non-fixed values to vary, but
|
||
|
|
# we still call self.nnlf with all parameters.
|
||
|
|
i = 0
|
||
|
|
for n in range(Nargs):
|
||
|
|
if n not in fixedn:
|
||
|
|
args[n] = theta[i]
|
||
|
|
i += 1
|
||
|
|
return args
|
||
|
|
|
||
|
|
def func(theta, x):
|
||
|
|
newtheta = restore(args[:], theta)
|
||
|
|
return self._penalized_nnlf(newtheta, x)
|
||
|
|
|
||
|
|
return x0, func, restore, args
|
||
|
|
|
||
|
|
def fit(self, data, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Return MLEs for shape (if applicable), location, and scale
|
||
|
|
parameters from data.
|
||
|
|
|
||
|
|
MLE stands for Maximum Likelihood Estimate. Starting estimates for
|
||
|
|
the fit are given by input arguments; for any arguments not provided
|
||
|
|
with starting estimates, ``self._fitstart(data)`` is called to generate
|
||
|
|
such.
|
||
|
|
|
||
|
|
One can hold some parameters fixed to specific values by passing in
|
||
|
|
keyword arguments ``f0``, ``f1``, ..., ``fn`` (for shape parameters)
|
||
|
|
and ``floc`` and ``fscale`` (for location and scale parameters,
|
||
|
|
respectively).
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : array_like
|
||
|
|
Data to use in calculating the MLEs.
|
||
|
|
args : floats, optional
|
||
|
|
Starting value(s) for any shape-characterizing arguments (those not
|
||
|
|
provided will be determined by a call to ``_fitstart(data)``).
|
||
|
|
No default value.
|
||
|
|
kwds : floats, optional
|
||
|
|
Starting values for the location and scale parameters; no default.
|
||
|
|
Special keyword arguments are recognized as holding certain
|
||
|
|
parameters fixed:
|
||
|
|
|
||
|
|
- f0...fn : hold respective shape parameters fixed.
|
||
|
|
Alternatively, shape parameters to fix can be specified by name.
|
||
|
|
For example, if ``self.shapes == "a, b"``, ``fa``and ``fix_a``
|
||
|
|
are equivalent to ``f0``, and ``fb`` and ``fix_b`` are
|
||
|
|
equivalent to ``f1``.
|
||
|
|
|
||
|
|
- floc : hold location parameter fixed to specified value.
|
||
|
|
|
||
|
|
- fscale : hold scale parameter fixed to specified value.
|
||
|
|
|
||
|
|
- optimizer : The optimizer to use. The optimizer must take ``func``,
|
||
|
|
and starting position as the first two arguments,
|
||
|
|
plus ``args`` (for extra arguments to pass to the
|
||
|
|
function to be optimized) and ``disp=0`` to suppress
|
||
|
|
output as keyword arguments.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
mle_tuple : tuple of floats
|
||
|
|
MLEs for any shape parameters (if applicable), followed by those
|
||
|
|
for location and scale. For most random variables, shape statistics
|
||
|
|
will be returned, but there are exceptions (e.g. ``norm``).
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This fit is computed by maximizing a log-likelihood function, with
|
||
|
|
penalty applied for samples outside of range of the distribution. The
|
||
|
|
returned answer is not guaranteed to be the globally optimal MLE, it
|
||
|
|
may only be locally optimal, or the optimization may fail altogether.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
|
||
|
|
Generate some data to fit: draw random variates from the `beta`
|
||
|
|
distribution
|
||
|
|
|
||
|
|
>>> from scipy.stats import beta
|
||
|
|
>>> a, b = 1., 2.
|
||
|
|
>>> x = beta.rvs(a, b, size=1000)
|
||
|
|
|
||
|
|
Now we can fit all four parameters (``a``, ``b``, ``loc`` and ``scale``):
|
||
|
|
|
||
|
|
>>> a1, b1, loc1, scale1 = beta.fit(x)
|
||
|
|
|
||
|
|
We can also use some prior knowledge about the dataset: let's keep
|
||
|
|
``loc`` and ``scale`` fixed:
|
||
|
|
|
||
|
|
>>> a1, b1, loc1, scale1 = beta.fit(x, floc=0, fscale=1)
|
||
|
|
>>> loc1, scale1
|
||
|
|
(0, 1)
|
||
|
|
|
||
|
|
We can also keep shape parameters fixed by using ``f``-keywords. To
|
||
|
|
keep the zero-th shape parameter ``a`` equal 1, use ``f0=1`` or,
|
||
|
|
equivalently, ``fa=1``:
|
||
|
|
|
||
|
|
>>> a1, b1, loc1, scale1 = beta.fit(x, fa=1, floc=0, fscale=1)
|
||
|
|
>>> a1
|
||
|
|
1
|
||
|
|
|
||
|
|
Not all distributions return estimates for the shape parameters.
|
||
|
|
``norm`` for example just returns estimates for location and scale:
|
||
|
|
|
||
|
|
>>> from scipy.stats import norm
|
||
|
|
>>> x = norm.rvs(a, b, size=1000, random_state=123)
|
||
|
|
>>> loc1, scale1 = norm.fit(x)
|
||
|
|
>>> loc1, scale1
|
||
|
|
(0.92087172783841631, 2.0015750750324668)
|
||
|
|
"""
|
||
|
|
Narg = len(args)
|
||
|
|
if Narg > self.numargs:
|
||
|
|
raise TypeError("Too many input arguments.")
|
||
|
|
|
||
|
|
start = [None]*2
|
||
|
|
if (Narg < self.numargs) or not ('loc' in kwds and
|
||
|
|
'scale' in kwds):
|
||
|
|
# get distribution specific starting locations
|
||
|
|
start = self._fitstart(data)
|
||
|
|
args += start[Narg:-2]
|
||
|
|
loc = kwds.pop('loc', start[-2])
|
||
|
|
scale = kwds.pop('scale', start[-1])
|
||
|
|
args += (loc, scale)
|
||
|
|
x0, func, restore, args = self._reduce_func(args, kwds)
|
||
|
|
|
||
|
|
optimizer = kwds.pop('optimizer', optimize.fmin)
|
||
|
|
# convert string to function in scipy.optimize
|
||
|
|
if not callable(optimizer) and isinstance(optimizer, string_types):
|
||
|
|
if not optimizer.startswith('fmin_'):
|
||
|
|
optimizer = "fmin_"+optimizer
|
||
|
|
if optimizer == 'fmin_':
|
||
|
|
optimizer = 'fmin'
|
||
|
|
try:
|
||
|
|
optimizer = getattr(optimize, optimizer)
|
||
|
|
except AttributeError:
|
||
|
|
raise ValueError("%s is not a valid optimizer" % optimizer)
|
||
|
|
|
||
|
|
# by now kwds must be empty, since everybody took what they needed
|
||
|
|
if kwds:
|
||
|
|
raise TypeError("Unknown arguments: %s." % kwds)
|
||
|
|
|
||
|
|
vals = optimizer(func, x0, args=(ravel(data),), disp=0)
|
||
|
|
if restore is not None:
|
||
|
|
vals = restore(args, vals)
|
||
|
|
vals = tuple(vals)
|
||
|
|
return vals
|
||
|
|
|
||
|
|
def _fit_loc_scale_support(self, data, *args):
|
||
|
|
"""
|
||
|
|
Estimate loc and scale parameters from data accounting for support.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : array_like
|
||
|
|
Data to fit.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
Lhat : float
|
||
|
|
Estimated location parameter for the data.
|
||
|
|
Shat : float
|
||
|
|
Estimated scale parameter for the data.
|
||
|
|
|
||
|
|
"""
|
||
|
|
data = np.asarray(data)
|
||
|
|
|
||
|
|
# Estimate location and scale according to the method of moments.
|
||
|
|
loc_hat, scale_hat = self.fit_loc_scale(data, *args)
|
||
|
|
|
||
|
|
# Compute the support according to the shape parameters.
|
||
|
|
self._argcheck(*args)
|
||
|
|
a, b = self.a, self.b
|
||
|
|
support_width = b - a
|
||
|
|
|
||
|
|
# If the support is empty then return the moment-based estimates.
|
||
|
|
if support_width <= 0:
|
||
|
|
return loc_hat, scale_hat
|
||
|
|
|
||
|
|
# Compute the proposed support according to the loc and scale estimates.
|
||
|
|
a_hat = loc_hat + a * scale_hat
|
||
|
|
b_hat = loc_hat + b * scale_hat
|
||
|
|
|
||
|
|
# Use the moment-based estimates if they are compatible with the data.
|
||
|
|
data_a = np.min(data)
|
||
|
|
data_b = np.max(data)
|
||
|
|
if a_hat < data_a and data_b < b_hat:
|
||
|
|
return loc_hat, scale_hat
|
||
|
|
|
||
|
|
# Otherwise find other estimates that are compatible with the data.
|
||
|
|
data_width = data_b - data_a
|
||
|
|
rel_margin = 0.1
|
||
|
|
margin = data_width * rel_margin
|
||
|
|
|
||
|
|
# For a finite interval, both the location and scale
|
||
|
|
# should have interesting values.
|
||
|
|
if support_width < np.inf:
|
||
|
|
loc_hat = (data_a - a) - margin
|
||
|
|
scale_hat = (data_width + 2 * margin) / support_width
|
||
|
|
return loc_hat, scale_hat
|
||
|
|
|
||
|
|
# For a one-sided interval, use only an interesting location parameter.
|
||
|
|
if a > -np.inf:
|
||
|
|
return (data_a - a) - margin, 1
|
||
|
|
elif b < np.inf:
|
||
|
|
return (data_b - b) + margin, 1
|
||
|
|
else:
|
||
|
|
raise RuntimeError
|
||
|
|
|
||
|
|
def fit_loc_scale(self, data, *args):
|
||
|
|
"""
|
||
|
|
Estimate loc and scale parameters from data using 1st and 2nd moments.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : array_like
|
||
|
|
Data to fit.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
Lhat : float
|
||
|
|
Estimated location parameter for the data.
|
||
|
|
Shat : float
|
||
|
|
Estimated scale parameter for the data.
|
||
|
|
|
||
|
|
"""
|
||
|
|
mu, mu2 = self.stats(*args, **{'moments': 'mv'})
|
||
|
|
tmp = asarray(data)
|
||
|
|
muhat = tmp.mean()
|
||
|
|
mu2hat = tmp.var()
|
||
|
|
Shat = sqrt(mu2hat / mu2)
|
||
|
|
Lhat = muhat - Shat*mu
|
||
|
|
if not np.isfinite(Lhat):
|
||
|
|
Lhat = 0
|
||
|
|
if not (np.isfinite(Shat) and (0 < Shat)):
|
||
|
|
Shat = 1
|
||
|
|
return Lhat, Shat
|
||
|
|
|
||
|
|
def _entropy(self, *args):
|
||
|
|
def integ(x):
|
||
|
|
val = self._pdf(x, *args)
|
||
|
|
return entr(val)
|
||
|
|
|
||
|
|
# upper limit is often inf, so suppress warnings when integrating
|
||
|
|
olderr = np.seterr(over='ignore')
|
||
|
|
h = integrate.quad(integ, self.a, self.b)[0]
|
||
|
|
np.seterr(**olderr)
|
||
|
|
|
||
|
|
if not np.isnan(h):
|
||
|
|
return h
|
||
|
|
else:
|
||
|
|
# try with different limits if integration problems
|
||
|
|
low, upp = self.ppf([1e-10, 1. - 1e-10], *args)
|
||
|
|
if np.isinf(self.b):
|
||
|
|
upper = upp
|
||
|
|
else:
|
||
|
|
upper = self.b
|
||
|
|
if np.isinf(self.a):
|
||
|
|
lower = low
|
||
|
|
else:
|
||
|
|
lower = self.a
|
||
|
|
return integrate.quad(integ, lower, upper)[0]
|
||
|
|
|
||
|
|
def expect(self, func=None, args=(), loc=0, scale=1, lb=None, ub=None,
|
||
|
|
conditional=False, **kwds):
|
||
|
|
"""Calculate expected value of a function with respect to the
|
||
|
|
distribution.
|
||
|
|
|
||
|
|
The expected value of a function ``f(x)`` with respect to a
|
||
|
|
distribution ``dist`` is defined as::
|
||
|
|
|
||
|
|
ubound
|
||
|
|
E[x] = Integral(f(x) * dist.pdf(x))
|
||
|
|
lbound
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable, optional
|
||
|
|
Function for which integral is calculated. Takes only one argument.
|
||
|
|
The default is the identity mapping f(x) = x.
|
||
|
|
args : tuple, optional
|
||
|
|
Shape parameters of the distribution.
|
||
|
|
loc : float, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
scale : float, optional
|
||
|
|
Scale parameter (default=1).
|
||
|
|
lb, ub : scalar, optional
|
||
|
|
Lower and upper bound for integration. Default is set to the
|
||
|
|
support of the distribution.
|
||
|
|
conditional : bool, optional
|
||
|
|
If True, the integral is corrected by the conditional probability
|
||
|
|
of the integration interval. The return value is the expectation
|
||
|
|
of the function, conditional on being in the given interval.
|
||
|
|
Default is False.
|
||
|
|
|
||
|
|
Additional keyword arguments are passed to the integration routine.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
expect : float
|
||
|
|
The calculated expected value.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The integration behavior of this function is inherited from
|
||
|
|
`integrate.quad`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
lockwds = {'loc': loc,
|
||
|
|
'scale': scale}
|
||
|
|
self._argcheck(*args)
|
||
|
|
if func is None:
|
||
|
|
def fun(x, *args):
|
||
|
|
return x * self.pdf(x, *args, **lockwds)
|
||
|
|
else:
|
||
|
|
def fun(x, *args):
|
||
|
|
return func(x) * self.pdf(x, *args, **lockwds)
|
||
|
|
if lb is None:
|
||
|
|
lb = loc + self.a * scale
|
||
|
|
if ub is None:
|
||
|
|
ub = loc + self.b * scale
|
||
|
|
if conditional:
|
||
|
|
invfac = (self.sf(lb, *args, **lockwds)
|
||
|
|
- self.sf(ub, *args, **lockwds))
|
||
|
|
else:
|
||
|
|
invfac = 1.0
|
||
|
|
kwds['args'] = args
|
||
|
|
# Silence floating point warnings from integration.
|
||
|
|
olderr = np.seterr(all='ignore')
|
||
|
|
vals = integrate.quad(fun, lb, ub, **kwds)[0] / invfac
|
||
|
|
np.seterr(**olderr)
|
||
|
|
return vals
|
||
|
|
|
||
|
|
|
||
|
|
# Helpers for the discrete distributions
|
||
|
|
def _drv2_moment(self, n, *args):
|
||
|
|
"""Non-central moment of discrete distribution."""
|
||
|
|
def fun(x):
|
||
|
|
return np.power(x, n) * self._pmf(x, *args)
|
||
|
|
return _expect(fun, self.a, self.b, self.ppf(0.5, *args), self.inc)
|
||
|
|
|
||
|
|
|
||
|
|
def _drv2_ppfsingle(self, q, *args): # Use basic bisection algorithm
|
||
|
|
b = self.b
|
||
|
|
a = self.a
|
||
|
|
if isinf(b): # Be sure ending point is > q
|
||
|
|
b = int(max(100*q, 10))
|
||
|
|
while 1:
|
||
|
|
if b >= self.b:
|
||
|
|
qb = 1.0
|
||
|
|
break
|
||
|
|
qb = self._cdf(b, *args)
|
||
|
|
if (qb < q):
|
||
|
|
b += 10
|
||
|
|
else:
|
||
|
|
break
|
||
|
|
else:
|
||
|
|
qb = 1.0
|
||
|
|
if isinf(a): # be sure starting point < q
|
||
|
|
a = int(min(-100*q, -10))
|
||
|
|
while 1:
|
||
|
|
if a <= self.a:
|
||
|
|
qb = 0.0
|
||
|
|
break
|
||
|
|
qa = self._cdf(a, *args)
|
||
|
|
if (qa > q):
|
||
|
|
a -= 10
|
||
|
|
else:
|
||
|
|
break
|
||
|
|
else:
|
||
|
|
qa = self._cdf(a, *args)
|
||
|
|
|
||
|
|
while 1:
|
||
|
|
if (qa == q):
|
||
|
|
return a
|
||
|
|
if (qb == q):
|
||
|
|
return b
|
||
|
|
if b <= a+1:
|
||
|
|
# testcase: return wrong number at lower index
|
||
|
|
# python -c "from scipy.stats import zipf;print zipf.ppf(0.01, 2)" wrong
|
||
|
|
# python -c "from scipy.stats import zipf;print zipf.ppf([0.01, 0.61, 0.77, 0.83], 2)"
|
||
|
|
# python -c "from scipy.stats import logser;print logser.ppf([0.1, 0.66, 0.86, 0.93], 0.6)"
|
||
|
|
if qa > q:
|
||
|
|
return a
|
||
|
|
else:
|
||
|
|
return b
|
||
|
|
c = int((a+b)/2.0)
|
||
|
|
qc = self._cdf(c, *args)
|
||
|
|
if (qc < q):
|
||
|
|
if a != c:
|
||
|
|
a = c
|
||
|
|
else:
|
||
|
|
raise RuntimeError('updating stopped, endless loop')
|
||
|
|
qa = qc
|
||
|
|
elif (qc > q):
|
||
|
|
if b != c:
|
||
|
|
b = c
|
||
|
|
else:
|
||
|
|
raise RuntimeError('updating stopped, endless loop')
|
||
|
|
qb = qc
|
||
|
|
else:
|
||
|
|
return c
|
||
|
|
|
||
|
|
|
||
|
|
def entropy(pk, qk=None, base=None):
|
||
|
|
"""Calculate the entropy of a distribution for given probability values.
|
||
|
|
|
||
|
|
If only probabilities `pk` are given, the entropy is calculated as
|
||
|
|
``S = -sum(pk * log(pk), axis=0)``.
|
||
|
|
|
||
|
|
If `qk` is not None, then compute the Kullback-Leibler divergence
|
||
|
|
``S = sum(pk * log(pk / qk), axis=0)``.
|
||
|
|
|
||
|
|
This routine will normalize `pk` and `qk` if they don't sum to 1.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
pk : sequence
|
||
|
|
Defines the (discrete) distribution. ``pk[i]`` is the (possibly
|
||
|
|
unnormalized) probability of event ``i``.
|
||
|
|
qk : sequence, optional
|
||
|
|
Sequence against which the relative entropy is computed. Should be in
|
||
|
|
the same format as `pk`.
|
||
|
|
base : float, optional
|
||
|
|
The logarithmic base to use, defaults to ``e`` (natural logarithm).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
S : float
|
||
|
|
The calculated entropy.
|
||
|
|
|
||
|
|
"""
|
||
|
|
pk = asarray(pk)
|
||
|
|
pk = 1.0*pk / np.sum(pk, axis=0)
|
||
|
|
if qk is None:
|
||
|
|
vec = entr(pk)
|
||
|
|
else:
|
||
|
|
qk = asarray(qk)
|
||
|
|
if len(qk) != len(pk):
|
||
|
|
raise ValueError("qk and pk must have same length.")
|
||
|
|
qk = 1.0*qk / np.sum(qk, axis=0)
|
||
|
|
vec = rel_entr(pk, qk)
|
||
|
|
S = np.sum(vec, axis=0)
|
||
|
|
if base is not None:
|
||
|
|
S /= log(base)
|
||
|
|
return S
|
||
|
|
|
||
|
|
|
||
|
|
# Must over-ride one of _pmf or _cdf or pass in
|
||
|
|
# x_k, p(x_k) lists in initialization
|
||
|
|
|
||
|
|
class rv_discrete(rv_generic):
|
||
|
|
"""
|
||
|
|
A generic discrete random variable class meant for subclassing.
|
||
|
|
|
||
|
|
`rv_discrete` is a base class to construct specific distribution classes
|
||
|
|
and instances for discrete random variables. It can also be used
|
||
|
|
to construct an arbitrary distribution defined by a list of support
|
||
|
|
points and corresponding probabilities.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
a : float, optional
|
||
|
|
Lower bound of the support of the distribution, default: 0
|
||
|
|
b : float, optional
|
||
|
|
Upper bound of the support of the distribution, default: plus infinity
|
||
|
|
moment_tol : float, optional
|
||
|
|
The tolerance for the generic calculation of moments.
|
||
|
|
values : tuple of two array_like, optional
|
||
|
|
``(xk, pk)`` where ``xk`` are integers with non-zero
|
||
|
|
probabilities ``pk`` with ``sum(pk) = 1``.
|
||
|
|
inc : integer, optional
|
||
|
|
Increment for the support of the distribution.
|
||
|
|
Default is 1. (other values have not been tested)
|
||
|
|
badvalue : float, optional
|
||
|
|
The value in a result arrays that indicates a value that for which
|
||
|
|
some argument restriction is violated, default is np.nan.
|
||
|
|
name : str, optional
|
||
|
|
The name of the instance. This string is used to construct the default
|
||
|
|
example for distributions.
|
||
|
|
longname : str, optional
|
||
|
|
This string is used as part of the first line of the docstring returned
|
||
|
|
when a subclass has no docstring of its own. Note: `longname` exists
|
||
|
|
for backwards compatibility, do not use for new subclasses.
|
||
|
|
shapes : str, optional
|
||
|
|
The shape of the distribution. For example "m, n" for a distribution
|
||
|
|
that takes two integers as the two shape arguments for all its methods
|
||
|
|
If not provided, shape parameters will be inferred from
|
||
|
|
the signatures of the private methods, ``_pmf`` and ``_cdf`` of
|
||
|
|
the instance.
|
||
|
|
extradoc : str, optional
|
||
|
|
This string is used as the last part of the docstring returned when a
|
||
|
|
subclass has no docstring of its own. Note: `extradoc` exists for
|
||
|
|
backwards compatibility, do not use for new subclasses.
|
||
|
|
seed : None or int or ``numpy.random.RandomState`` instance, optional
|
||
|
|
This parameter defines the RandomState object to use for drawing
|
||
|
|
random variates.
|
||
|
|
If None, the global np.random state is used.
|
||
|
|
If integer, it is used to seed the local RandomState instance.
|
||
|
|
Default is None.
|
||
|
|
|
||
|
|
Methods
|
||
|
|
-------
|
||
|
|
rvs
|
||
|
|
pmf
|
||
|
|
logpmf
|
||
|
|
cdf
|
||
|
|
logcdf
|
||
|
|
sf
|
||
|
|
logsf
|
||
|
|
ppf
|
||
|
|
isf
|
||
|
|
moment
|
||
|
|
stats
|
||
|
|
entropy
|
||
|
|
expect
|
||
|
|
median
|
||
|
|
mean
|
||
|
|
std
|
||
|
|
var
|
||
|
|
interval
|
||
|
|
__call__
|
||
|
|
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
|
||
|
|
This class is similar to `rv_continuous`. Whether a shape parameter is
|
||
|
|
valid is decided by an ``_argcheck`` method (which defaults to checking
|
||
|
|
that its arguments are strictly positive.)
|
||
|
|
The main differences are:
|
||
|
|
|
||
|
|
- the support of the distribution is a set of integers
|
||
|
|
- instead of the probability density function, ``pdf`` (and the
|
||
|
|
corresponding private ``_pdf``), this class defines the
|
||
|
|
*probability mass function*, `pmf` (and the corresponding
|
||
|
|
private ``_pmf``.)
|
||
|
|
- scale parameter is not defined.
|
||
|
|
|
||
|
|
To create a new discrete distribution, we would do the following:
|
||
|
|
|
||
|
|
>>> from scipy.stats import rv_discrete
|
||
|
|
>>> class poisson_gen(rv_discrete):
|
||
|
|
... "Poisson distribution"
|
||
|
|
... def _pmf(self, k, mu):
|
||
|
|
... return exp(-mu) * mu**k / factorial(k)
|
||
|
|
|
||
|
|
and create an instance::
|
||
|
|
|
||
|
|
>>> poisson = poisson_gen(name="poisson")
|
||
|
|
|
||
|
|
Note that above we defined the Poisson distribution in the standard form.
|
||
|
|
Shifting the distribution can be done by providing the ``loc`` parameter
|
||
|
|
to the methods of the instance. For example, ``poisson.pmf(x, mu, loc)``
|
||
|
|
delegates the work to ``poisson._pmf(x-loc, mu)``.
|
||
|
|
|
||
|
|
**Discrete distributions from a list of probabilities**
|
||
|
|
|
||
|
|
Alternatively, you can construct an arbitrary discrete rv defined
|
||
|
|
on a finite set of values ``xk`` with ``Prob{X=xk} = pk`` by using the
|
||
|
|
``values`` keyword argument to the `rv_discrete` constructor.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
|
||
|
|
Custom made discrete distribution:
|
||
|
|
|
||
|
|
>>> from scipy import stats
|
||
|
|
>>> xk = np.arange(7)
|
||
|
|
>>> pk = (0.1, 0.2, 0.3, 0.1, 0.1, 0.0, 0.2)
|
||
|
|
>>> custm = stats.rv_discrete(name='custm', values=(xk, pk))
|
||
|
|
>>>
|
||
|
|
>>> import matplotlib.pyplot as plt
|
||
|
|
>>> fig, ax = plt.subplots(1, 1)
|
||
|
|
>>> ax.plot(xk, custm.pmf(xk), 'ro', ms=12, mec='r')
|
||
|
|
>>> ax.vlines(xk, 0, custm.pmf(xk), colors='r', lw=4)
|
||
|
|
>>> plt.show()
|
||
|
|
|
||
|
|
Random number generation:
|
||
|
|
|
||
|
|
>>> R = custm.rvs(size=100)
|
||
|
|
|
||
|
|
"""
|
||
|
|
def __new__(cls, a=0, b=inf, name=None, badvalue=None,
|
||
|
|
moment_tol=1e-8, values=None, inc=1, longname=None,
|
||
|
|
shapes=None, extradoc=None, seed=None):
|
||
|
|
|
||
|
|
if values is not None:
|
||
|
|
# dispatch to a subclass
|
||
|
|
return super(rv_discrete, cls).__new__(rv_sample)
|
||
|
|
else:
|
||
|
|
# business as usual
|
||
|
|
return super(rv_discrete, cls).__new__(cls)
|
||
|
|
|
||
|
|
def __init__(self, a=0, b=inf, name=None, badvalue=None,
|
||
|
|
moment_tol=1e-8, values=None, inc=1, longname=None,
|
||
|
|
shapes=None, extradoc=None, seed=None):
|
||
|
|
|
||
|
|
super(rv_discrete, self).__init__(seed)
|
||
|
|
|
||
|
|
# cf generic freeze
|
||
|
|
self._ctor_param = dict(
|
||
|
|
a=a, b=b, name=name, badvalue=badvalue,
|
||
|
|
moment_tol=moment_tol, values=values, inc=inc,
|
||
|
|
longname=longname, shapes=shapes, extradoc=extradoc, seed=seed)
|
||
|
|
|
||
|
|
if badvalue is None:
|
||
|
|
badvalue = nan
|
||
|
|
self.badvalue = badvalue
|
||
|
|
self.a = a
|
||
|
|
self.b = b
|
||
|
|
self.moment_tol = moment_tol
|
||
|
|
self.inc = inc
|
||
|
|
self._cdfvec = vectorize(self._cdf_single, otypes='d')
|
||
|
|
self.vecentropy = vectorize(self._entropy)
|
||
|
|
self.shapes = shapes
|
||
|
|
|
||
|
|
if values is not None:
|
||
|
|
raise ValueError("rv_discrete.__init__(..., values != None, ...)")
|
||
|
|
|
||
|
|
self._construct_argparser(meths_to_inspect=[self._pmf, self._cdf],
|
||
|
|
locscale_in='loc=0',
|
||
|
|
# scale=1 for discrete RVs
|
||
|
|
locscale_out='loc, 1')
|
||
|
|
|
||
|
|
# nin correction needs to be after we know numargs
|
||
|
|
# correct nin for generic moment vectorization
|
||
|
|
_vec_generic_moment = vectorize(_drv2_moment, otypes='d')
|
||
|
|
_vec_generic_moment.nin = self.numargs + 2
|
||
|
|
self.generic_moment = instancemethod(_vec_generic_moment,
|
||
|
|
self, rv_discrete)
|
||
|
|
|
||
|
|
# correct nin for ppf vectorization
|
||
|
|
_vppf = vectorize(_drv2_ppfsingle, otypes='d')
|
||
|
|
_vppf.nin = self.numargs + 2
|
||
|
|
self._ppfvec = instancemethod(_vppf,
|
||
|
|
self, rv_discrete)
|
||
|
|
|
||
|
|
# now that self.numargs is defined, we can adjust nin
|
||
|
|
self._cdfvec.nin = self.numargs + 1
|
||
|
|
|
||
|
|
self._construct_docstrings(name, longname, extradoc)
|
||
|
|
|
||
|
|
def _construct_docstrings(self, name, longname, extradoc):
|
||
|
|
if name is None:
|
||
|
|
name = 'Distribution'
|
||
|
|
self.name = name
|
||
|
|
self.extradoc = extradoc
|
||
|
|
|
||
|
|
# generate docstring for subclass instances
|
||
|
|
if longname is None:
|
||
|
|
if name[0] in ['aeiouAEIOU']:
|
||
|
|
hstr = "An "
|
||
|
|
else:
|
||
|
|
hstr = "A "
|
||
|
|
longname = hstr + name
|
||
|
|
|
||
|
|
if sys.flags.optimize < 2:
|
||
|
|
# Skip adding docstrings if interpreter is run with -OO
|
||
|
|
if self.__doc__ is None:
|
||
|
|
self._construct_default_doc(longname=longname,
|
||
|
|
extradoc=extradoc,
|
||
|
|
docdict=docdict_discrete,
|
||
|
|
discrete='discrete')
|
||
|
|
else:
|
||
|
|
dct = dict(distdiscrete)
|
||
|
|
self._construct_doc(docdict_discrete, dct.get(self.name))
|
||
|
|
|
||
|
|
# discrete RV do not have the scale parameter, remove it
|
||
|
|
self.__doc__ = self.__doc__.replace(
|
||
|
|
'\n scale : array_like, '
|
||
|
|
'optional\n scale parameter (default=1)', '')
|
||
|
|
|
||
|
|
def _updated_ctor_param(self):
|
||
|
|
""" Return the current version of _ctor_param, possibly updated by user.
|
||
|
|
|
||
|
|
Used by freezing and pickling.
|
||
|
|
Keep this in sync with the signature of __init__.
|
||
|
|
"""
|
||
|
|
dct = self._ctor_param.copy()
|
||
|
|
dct['a'] = self.a
|
||
|
|
dct['b'] = self.b
|
||
|
|
dct['badvalue'] = self.badvalue
|
||
|
|
dct['moment_tol'] = self.moment_tol
|
||
|
|
dct['inc'] = self.inc
|
||
|
|
dct['name'] = self.name
|
||
|
|
dct['shapes'] = self.shapes
|
||
|
|
dct['extradoc'] = self.extradoc
|
||
|
|
return dct
|
||
|
|
|
||
|
|
def _nonzero(self, k, *args):
|
||
|
|
return floor(k) == k
|
||
|
|
|
||
|
|
def _pmf(self, k, *args):
|
||
|
|
return self._cdf(k, *args) - self._cdf(k-1, *args)
|
||
|
|
|
||
|
|
def _logpmf(self, k, *args):
|
||
|
|
return log(self._pmf(k, *args))
|
||
|
|
|
||
|
|
def _cdf_single(self, k, *args):
|
||
|
|
m = arange(int(self.a), k+1)
|
||
|
|
return np.sum(self._pmf(m, *args), axis=0)
|
||
|
|
|
||
|
|
def _cdf(self, x, *args):
|
||
|
|
k = floor(x)
|
||
|
|
return self._cdfvec(k, *args)
|
||
|
|
|
||
|
|
# generic _logcdf, _sf, _logsf, _ppf, _isf, _rvs defined in rv_generic
|
||
|
|
|
||
|
|
def rvs(self, *args, **kwargs):
|
||
|
|
"""
|
||
|
|
Random variates of given type.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
size : int or tuple of ints, optional
|
||
|
|
Defining number of random variates (Default is 1). Note that `size`
|
||
|
|
has to be given as keyword, not as positional argument.
|
||
|
|
random_state : None or int or ``np.random.RandomState`` instance, optional
|
||
|
|
If int or RandomState, use it for drawing the random variates.
|
||
|
|
If None, rely on ``self.random_state``.
|
||
|
|
Default is None.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
rvs : ndarray or scalar
|
||
|
|
Random variates of given `size`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
kwargs['discrete'] = True
|
||
|
|
return super(rv_discrete, self).rvs(*args, **kwargs)
|
||
|
|
|
||
|
|
def pmf(self, k, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Probability mass function at k of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
k : array_like
|
||
|
|
Quantiles.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information)
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
pmf : array_like
|
||
|
|
Probability mass function evaluated at k
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, _ = self._parse_args(*args, **kwds)
|
||
|
|
k, loc = map(asarray, (k, loc))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
k = asarray((k-loc))
|
||
|
|
cond0 = self._argcheck(*args)
|
||
|
|
cond1 = (k >= self.a) & (k <= self.b) & self._nonzero(k, *args)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = zeros(shape(cond), 'd')
|
||
|
|
place(output, (1-cond0) + np.isnan(k), self.badvalue)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((k,)+args))
|
||
|
|
place(output, cond, np.clip(self._pmf(*goodargs), 0, 1))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def logpmf(self, k, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Log of the probability mass function at k of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
k : array_like
|
||
|
|
Quantiles.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter. Default is 0.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
logpmf : array_like
|
||
|
|
Log of the probability mass function evaluated at k.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, _ = self._parse_args(*args, **kwds)
|
||
|
|
k, loc = map(asarray, (k, loc))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
k = asarray((k-loc))
|
||
|
|
cond0 = self._argcheck(*args)
|
||
|
|
cond1 = (k >= self.a) & (k <= self.b) & self._nonzero(k, *args)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = empty(shape(cond), 'd')
|
||
|
|
output.fill(NINF)
|
||
|
|
place(output, (1-cond0) + np.isnan(k), self.badvalue)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((k,)+args))
|
||
|
|
place(output, cond, self._logpmf(*goodargs))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def cdf(self, k, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Cumulative distribution function of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
k : array_like, int
|
||
|
|
Quantiles.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
cdf : ndarray
|
||
|
|
Cumulative distribution function evaluated at `k`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, _ = self._parse_args(*args, **kwds)
|
||
|
|
k, loc = map(asarray, (k, loc))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
k = asarray((k-loc))
|
||
|
|
cond0 = self._argcheck(*args)
|
||
|
|
cond1 = (k >= self.a) & (k < self.b)
|
||
|
|
cond2 = (k >= self.b)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = zeros(shape(cond), 'd')
|
||
|
|
place(output, (1-cond0) + np.isnan(k), self.badvalue)
|
||
|
|
place(output, cond2*(cond0 == cond0), 1.0)
|
||
|
|
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((k,)+args))
|
||
|
|
place(output, cond, np.clip(self._cdf(*goodargs), 0, 1))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def logcdf(self, k, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Log of the cumulative distribution function at k of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
k : array_like, int
|
||
|
|
Quantiles.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
logcdf : array_like
|
||
|
|
Log of the cumulative distribution function evaluated at k.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, _ = self._parse_args(*args, **kwds)
|
||
|
|
k, loc = map(asarray, (k, loc))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
k = asarray((k-loc))
|
||
|
|
cond0 = self._argcheck(*args)
|
||
|
|
cond1 = (k >= self.a) & (k < self.b)
|
||
|
|
cond2 = (k >= self.b)
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = empty(shape(cond), 'd')
|
||
|
|
output.fill(NINF)
|
||
|
|
place(output, (1-cond0) + np.isnan(k), self.badvalue)
|
||
|
|
place(output, cond2*(cond0 == cond0), 0.0)
|
||
|
|
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((k,)+args))
|
||
|
|
place(output, cond, self._logcdf(*goodargs))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def sf(self, k, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Survival function (1 - `cdf`) at k of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
k : array_like
|
||
|
|
Quantiles.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
sf : array_like
|
||
|
|
Survival function evaluated at k.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, _ = self._parse_args(*args, **kwds)
|
||
|
|
k, loc = map(asarray, (k, loc))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
k = asarray(k-loc)
|
||
|
|
cond0 = self._argcheck(*args)
|
||
|
|
cond1 = (k >= self.a) & (k < self.b)
|
||
|
|
cond2 = (k < self.a) & cond0
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = zeros(shape(cond), 'd')
|
||
|
|
place(output, (1-cond0) + np.isnan(k), self.badvalue)
|
||
|
|
place(output, cond2, 1.0)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((k,)+args))
|
||
|
|
place(output, cond, np.clip(self._sf(*goodargs), 0, 1))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def logsf(self, k, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Log of the survival function of the given RV.
|
||
|
|
|
||
|
|
Returns the log of the "survival function," defined as 1 - `cdf`,
|
||
|
|
evaluated at `k`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
k : array_like
|
||
|
|
Quantiles.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
logsf : ndarray
|
||
|
|
Log of the survival function evaluated at `k`.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, _ = self._parse_args(*args, **kwds)
|
||
|
|
k, loc = map(asarray, (k, loc))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
k = asarray(k-loc)
|
||
|
|
cond0 = self._argcheck(*args)
|
||
|
|
cond1 = (k >= self.a) & (k < self.b)
|
||
|
|
cond2 = (k < self.a) & cond0
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = empty(shape(cond), 'd')
|
||
|
|
output.fill(NINF)
|
||
|
|
place(output, (1-cond0) + np.isnan(k), self.badvalue)
|
||
|
|
place(output, cond2, 0.0)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((k,)+args))
|
||
|
|
place(output, cond, self._logsf(*goodargs))
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def ppf(self, q, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Percent point function (inverse of `cdf`) at q of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
q : array_like
|
||
|
|
Lower tail probability.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
k : array_like
|
||
|
|
Quantile corresponding to the lower tail probability, q.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, _ = self._parse_args(*args, **kwds)
|
||
|
|
q, loc = map(asarray, (q, loc))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
cond0 = self._argcheck(*args) & (loc == loc)
|
||
|
|
cond1 = (q > 0) & (q < 1)
|
||
|
|
cond2 = (q == 1) & cond0
|
||
|
|
cond = cond0 & cond1
|
||
|
|
output = valarray(shape(cond), value=self.badvalue, typecode='d')
|
||
|
|
# output type 'd' to handle nin and inf
|
||
|
|
place(output, (q == 0)*(cond == cond), self.a-1)
|
||
|
|
place(output, cond2, self.b)
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((q,)+args+(loc,)))
|
||
|
|
loc, goodargs = goodargs[-1], goodargs[:-1]
|
||
|
|
place(output, cond, self._ppf(*goodargs) + loc)
|
||
|
|
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def isf(self, q, *args, **kwds):
|
||
|
|
"""
|
||
|
|
Inverse survival function (inverse of `sf`) at q of the given RV.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
q : array_like
|
||
|
|
Upper tail probability.
|
||
|
|
arg1, arg2, arg3,... : array_like
|
||
|
|
The shape parameter(s) for the distribution (see docstring of the
|
||
|
|
instance object for more information).
|
||
|
|
loc : array_like, optional
|
||
|
|
Location parameter (default=0).
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
k : ndarray or scalar
|
||
|
|
Quantile corresponding to the upper tail probability, q.
|
||
|
|
|
||
|
|
"""
|
||
|
|
args, loc, _ = self._parse_args(*args, **kwds)
|
||
|
|
q, loc = map(asarray, (q, loc))
|
||
|
|
args = tuple(map(asarray, args))
|
||
|
|
cond0 = self._argcheck(*args) & (loc == loc)
|
||
|
|
cond1 = (q > 0) & (q < 1)
|
||
|
|
cond2 = (q == 1) & cond0
|
||
|
|
cond = cond0 & cond1
|
||
|
|
|
||
|
|
# same problem as with ppf; copied from ppf and changed
|
||
|
|
output = valarray(shape(cond), value=self.badvalue, typecode='d')
|
||
|
|
# output type 'd' to handle nin and inf
|
||
|
|
place(output, (q == 0)*(cond == cond), self.b)
|
||
|
|
place(output, cond2, self.a-1)
|
||
|
|
|
||
|
|
# call place only if at least 1 valid argument
|
||
|
|
if np.any(cond):
|
||
|
|
goodargs = argsreduce(cond, *((q,)+args+(loc,)))
|
||
|
|
loc, goodargs = goodargs[-1], goodargs[:-1]
|
||
|
|
# PB same as ticket 766
|
||
|
|
place(output, cond, self._isf(*goodargs) + loc)
|
||
|
|
|
||
|
|
if output.ndim == 0:
|
||
|
|
return output[()]
|
||
|
|
return output
|
||
|
|
|
||
|
|
def _entropy(self, *args):
|
||
|
|
if hasattr(self, 'pk'):
|
||
|
|
return entropy(self.pk)
|
||
|
|
else:
|
||
|
|
return _expect(lambda x: entr(self.pmf(x, *args)),
|
||
|
|
self.a, self.b, self.ppf(0.5, *args), self.inc)
|
||
|
|
|
||
|
|
def expect(self, func=None, args=(), loc=0, lb=None, ub=None,
|
||
|
|
conditional=False, maxcount=1000, tolerance=1e-10, chunksize=32):
|
||
|
|
"""
|
||
|
|
Calculate expected value of a function with respect to the distribution
|
||
|
|
for discrete distribution.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable, optional
|
||
|
|
Function for which the expectation value is calculated.
|
||
|
|
Takes only one argument.
|
||
|
|
The default is the identity mapping f(k) = k.
|
||
|
|
args : tuple, optional
|
||
|
|
Shape parameters of the distribution.
|
||
|
|
loc : float, optional
|
||
|
|
Location parameter.
|
||
|
|
Default is 0.
|
||
|
|
lb, ub : int, optional
|
||
|
|
Lower and upper bound for the summation, default is set to the
|
||
|
|
support of the distribution, inclusive (``ul <= k <= ub``).
|
||
|
|
conditional : bool, optional
|
||
|
|
If true then the expectation is corrected by the conditional
|
||
|
|
probability of the summation interval. The return value is the
|
||
|
|
expectation of the function, `func`, conditional on being in
|
||
|
|
the given interval (k such that ``ul <= k <= ub``).
|
||
|
|
Default is False.
|
||
|
|
maxcount : int, optional
|
||
|
|
Maximal number of terms to evaluate (to avoid an endless loop for
|
||
|
|
an infinite sum). Default is 1000.
|
||
|
|
tolerance : float, optional
|
||
|
|
Absolute tolerance for the summation. Default is 1e-10.
|
||
|
|
chunksize : int, optional
|
||
|
|
Iterate over the support of a distributions in chunks of this size.
|
||
|
|
Default is 32.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
expect : float
|
||
|
|
Expected value.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
For heavy-tailed distributions, the expected value may or may not exist,
|
||
|
|
depending on the function, `func`. If it does exist, but the sum converges
|
||
|
|
slowly, the accuracy of the result may be rather low. For instance, for
|
||
|
|
``zipf(4)``, accuracy for mean, variance in example is only 1e-5.
|
||
|
|
increasing `maxcount` and/or `chunksize` may improve the result, but may also
|
||
|
|
make zipf very slow.
|
||
|
|
|
||
|
|
The function is not vectorized.
|
||
|
|
|
||
|
|
"""
|
||
|
|
if func is None:
|
||
|
|
def fun(x):
|
||
|
|
# loc and args from outer scope
|
||
|
|
return (x+loc)*self._pmf(x, *args)
|
||
|
|
else:
|
||
|
|
def fun(x):
|
||
|
|
# loc and args from outer scope
|
||
|
|
return func(x+loc)*self._pmf(x, *args)
|
||
|
|
# used pmf because _pmf does not check support in randint and there
|
||
|
|
# might be problems(?) with correct self.a, self.b at this stage maybe
|
||
|
|
# not anymore, seems to work now with _pmf
|
||
|
|
|
||
|
|
self._argcheck(*args) # (re)generate scalar self.a and self.b
|
||
|
|
if lb is None:
|
||
|
|
lb = self.a
|
||
|
|
else:
|
||
|
|
lb = lb - loc # convert bound for standardized distribution
|
||
|
|
if ub is None:
|
||
|
|
ub = self.b
|
||
|
|
else:
|
||
|
|
ub = ub - loc # convert bound for standardized distribution
|
||
|
|
if conditional:
|
||
|
|
invfac = self.sf(lb-1, *args) - self.sf(ub, *args)
|
||
|
|
else:
|
||
|
|
invfac = 1.0
|
||
|
|
|
||
|
|
# iterate over the support, starting from the median
|
||
|
|
x0 = self.ppf(0.5, *args)
|
||
|
|
res = _expect(fun, lb, ub, x0, self.inc, maxcount, tolerance, chunksize)
|
||
|
|
return res / invfac
|
||
|
|
|
||
|
|
|
||
|
|
def _expect(fun, lb, ub, x0, inc, maxcount=1000, tolerance=1e-10,
|
||
|
|
chunksize=32):
|
||
|
|
"""Helper for computing the expectation value of `fun`."""
|
||
|
|
|
||
|
|
# short-circuit if the support size is small enough
|
||
|
|
if (ub - lb) <= chunksize:
|
||
|
|
supp = np.arange(lb, ub+1, inc)
|
||
|
|
vals = fun(supp)
|
||
|
|
return np.sum(vals)
|
||
|
|
|
||
|
|
# otherwise, iterate starting from x0
|
||
|
|
if x0 < lb:
|
||
|
|
x0 = lb
|
||
|
|
if x0 > ub:
|
||
|
|
x0 = ub
|
||
|
|
|
||
|
|
count, tot = 0, 0.
|
||
|
|
# iterate over [x0, ub] inclusive
|
||
|
|
for x in _iter_chunked(x0, ub+1, chunksize=chunksize, inc=inc):
|
||
|
|
count += x.size
|
||
|
|
delta = np.sum(fun(x))
|
||
|
|
tot += delta
|
||
|
|
if abs(delta) < tolerance * x.size:
|
||
|
|
break
|
||
|
|
if count > maxcount:
|
||
|
|
warnings.warn('expect(): sum did not converge', RuntimeWarning)
|
||
|
|
return tot
|
||
|
|
|
||
|
|
# iterate over [lb, x0)
|
||
|
|
for x in _iter_chunked(x0-1, lb-1, chunksize=chunksize, inc=-inc):
|
||
|
|
count += x.size
|
||
|
|
delta = np.sum(fun(x))
|
||
|
|
tot += delta
|
||
|
|
if abs(delta) < tolerance * x.size:
|
||
|
|
break
|
||
|
|
if count > maxcount:
|
||
|
|
warnings.warn('expect(): sum did not converge', RuntimeWarning)
|
||
|
|
break
|
||
|
|
|
||
|
|
return tot
|
||
|
|
|
||
|
|
|
||
|
|
def _iter_chunked(x0, x1, chunksize=4, inc=1):
|
||
|
|
"""Iterate from x0 to x1 in chunks of chunksize and steps inc.
|
||
|
|
|
||
|
|
x0 must be finite, x1 need not be. In the latter case, the iterator is infinite.
|
||
|
|
Handles both x0 < x1 and x0 > x1. In the latter case, iterates downwards
|
||
|
|
(make sure to set inc < 0.)
|
||
|
|
|
||
|
|
>>> [x for x in _iter_chunked(2, 5, inc=2)]
|
||
|
|
[array([2, 4])]
|
||
|
|
>>> [x for x in _iter_chunked(2, 11, inc=2)]
|
||
|
|
[array([2, 4, 6, 8]), array([10])]
|
||
|
|
>>> [x for x in _iter_chunked(2, -5, inc=-2)]
|
||
|
|
[array([ 2, 0, -2, -4])]
|
||
|
|
>>> [x for x in _iter_chunked(2, -9, inc=-2)]
|
||
|
|
[array([ 2, 0, -2, -4]), array([-6, -8])]
|
||
|
|
|
||
|
|
"""
|
||
|
|
if inc == 0:
|
||
|
|
raise ValueError('Cannot increment by zero.')
|
||
|
|
if chunksize <= 0:
|
||
|
|
raise ValueError('Chunk size must be positive; got %s.' % chunksize)
|
||
|
|
|
||
|
|
s = 1 if inc > 0 else -1
|
||
|
|
stepsize = abs(chunksize * inc)
|
||
|
|
|
||
|
|
x = x0
|
||
|
|
while (x - x1) * inc < 0:
|
||
|
|
delta = min(stepsize, abs(x - x1))
|
||
|
|
step = delta * s
|
||
|
|
supp = np.arange(x, x + step, inc)
|
||
|
|
x += step
|
||
|
|
yield supp
|
||
|
|
|
||
|
|
|
||
|
|
class rv_sample(rv_discrete):
|
||
|
|
"""A 'sample' discrete distribution defined by the support and values.
|
||
|
|
|
||
|
|
The ctor ignores most of the arguments, only needs the `values` argument.
|
||
|
|
"""
|
||
|
|
def __init__(self, a=0, b=inf, name=None, badvalue=None,
|
||
|
|
moment_tol=1e-8, values=None, inc=1, longname=None,
|
||
|
|
shapes=None, extradoc=None, seed=None):
|
||
|
|
|
||
|
|
super(rv_discrete, self).__init__(seed)
|
||
|
|
|
||
|
|
if values is None:
|
||
|
|
raise ValueError("rv_sample.__init__(..., values=None,...)")
|
||
|
|
|
||
|
|
# cf generic freeze
|
||
|
|
self._ctor_param = dict(
|
||
|
|
a=a, b=b, name=name, badvalue=badvalue,
|
||
|
|
moment_tol=moment_tol, values=values, inc=inc,
|
||
|
|
longname=longname, shapes=shapes, extradoc=extradoc, seed=seed)
|
||
|
|
|
||
|
|
if badvalue is None:
|
||
|
|
badvalue = nan
|
||
|
|
self.badvalue = badvalue
|
||
|
|
self.moment_tol = moment_tol
|
||
|
|
self.inc = inc
|
||
|
|
self.shapes = shapes
|
||
|
|
self.vecentropy = self._entropy
|
||
|
|
|
||
|
|
xk, pk = values
|
||
|
|
|
||
|
|
if len(xk) != len(pk):
|
||
|
|
raise ValueError("xk and pk need to have the same length.")
|
||
|
|
if not np.allclose(np.sum(pk), 1):
|
||
|
|
raise ValueError("The sum of provided pk is not 1.")
|
||
|
|
|
||
|
|
indx = np.argsort(np.ravel(xk))
|
||
|
|
self.xk = np.take(np.ravel(xk), indx, 0)
|
||
|
|
self.pk = np.take(np.ravel(pk), indx, 0)
|
||
|
|
self.a = self.xk[0]
|
||
|
|
self.b = self.xk[-1]
|
||
|
|
self.qvals = np.cumsum(self.pk, axis=0)
|
||
|
|
|
||
|
|
self.shapes = ' ' # bypass inspection
|
||
|
|
self._construct_argparser(meths_to_inspect=[self._pmf],
|
||
|
|
locscale_in='loc=0',
|
||
|
|
# scale=1 for discrete RVs
|
||
|
|
locscale_out='loc, 1')
|
||
|
|
|
||
|
|
self._construct_docstrings(name, longname, extradoc)
|
||
|
|
|
||
|
|
def _pmf(self, x):
|
||
|
|
return np.select([x == k for k in self.xk],
|
||
|
|
[np.broadcast_arrays(p, x)[0] for p in self.pk], 0)
|
||
|
|
|
||
|
|
def _cdf(self, x):
|
||
|
|
xx, xxk = np.broadcast_arrays(x[:, None], self.xk)
|
||
|
|
indx = np.argmax(xxk > xx, axis=-1) - 1
|
||
|
|
return self.qvals[indx]
|
||
|
|
|
||
|
|
def _ppf(self, q):
|
||
|
|
qq, sqq = np.broadcast_arrays(q[..., None], self.qvals)
|
||
|
|
indx = argmax(sqq >= qq, axis=-1)
|
||
|
|
return self.xk[indx]
|
||
|
|
|
||
|
|
def _rvs(self):
|
||
|
|
# Need to define it explicitly, otherwise .rvs() with size=None
|
||
|
|
# fails due to explicit broadcasting in _ppf
|
||
|
|
U = self._random_state.random_sample(self._size)
|
||
|
|
if self._size is None:
|
||
|
|
U = np.array(U, ndmin=1)
|
||
|
|
Y = self._ppf(U)[0]
|
||
|
|
else:
|
||
|
|
Y = self._ppf(U)
|
||
|
|
return Y
|
||
|
|
|
||
|
|
def _entropy(self):
|
||
|
|
return entropy(self.pk)
|
||
|
|
|
||
|
|
def generic_moment(self, n):
|
||
|
|
n = asarray(n)
|
||
|
|
return np.sum(self.xk**n[np.newaxis, ...] * self.pk, axis=0)
|
||
|
|
|
||
|
|
|
||
|
|
def get_distribution_names(namespace_pairs, rv_base_class):
|
||
|
|
"""
|
||
|
|
Collect names of statistical distributions and their generators.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
namespace_pairs : sequence
|
||
|
|
A snapshot of (name, value) pairs in the namespace of a module.
|
||
|
|
rv_base_class : class
|
||
|
|
The base class of random variable generator classes in a module.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
distn_names : list of strings
|
||
|
|
Names of the statistical distributions.
|
||
|
|
distn_gen_names : list of strings
|
||
|
|
Names of the generators of the statistical distributions.
|
||
|
|
Note that these are not simply the names of the statistical
|
||
|
|
distributions, with a _gen suffix added.
|
||
|
|
|
||
|
|
"""
|
||
|
|
distn_names = []
|
||
|
|
distn_gen_names = []
|
||
|
|
for name, value in namespace_pairs:
|
||
|
|
if name.startswith('_'):
|
||
|
|
continue
|
||
|
|
if name.endswith('_gen') and issubclass(value, rv_base_class):
|
||
|
|
distn_gen_names.append(name)
|
||
|
|
if isinstance(value, rv_base_class):
|
||
|
|
distn_names.append(name)
|
||
|
|
return distn_names, distn_gen_names
|