#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
import warnings
from collections.abc import Iterable
from functools import wraps, cached_property
import ctypes
import numpy as np
from numpy.polynomial import Polynomial
from scipy._lib.doccer import (extend_notes_in_docstring,
replace_notes_in_docstring,
inherit_docstring_from)
from scipy._lib._ccallback import LowLevelCallable
from scipy import optimize
from scipy import integrate
import scipy.special as sc
import scipy.special._ufuncs as scu
from scipy._lib._util import _lazyselect, _lazywhere
from . import _stats
from ._tukeylambda_stats import (tukeylambda_variance as _tlvar,
tukeylambda_kurtosis as _tlkurt)
from ._distn_infrastructure import (
get_distribution_names, _kurtosis,
rv_continuous, _skew, _get_fixed_fit_value, _check_shape, _ShapeInfo)
from ._ksstats import kolmogn, kolmognp, kolmogni
from ._constants import (_XMIN, _LOGXMIN, _EULER, _ZETA3, _SQRT_PI,
_SQRT_2_OVER_PI, _LOG_SQRT_2_OVER_PI)
from ._censored_data import CensoredData
import scipy.stats._boost as _boost
from scipy.optimize import root_scalar
from scipy.stats._warnings_errors import FitError
import scipy.stats as stats
def _remove_optimizer_parameters(kwds):
"""
Remove the optimizer-related keyword arguments 'loc', 'scale' and
'optimizer' from `kwds`. Then check that `kwds` is empty, and
raise `TypeError("Unknown arguments: %s." % kwds)` if it is not.
This function is used in the fit method of distributions that override
the default method and do not use the default optimization code.
`kwds` is modified in-place.
"""
kwds.pop('loc', None)
kwds.pop('scale', None)
kwds.pop('optimizer', None)
kwds.pop('method', None)
if kwds:
raise TypeError("Unknown arguments: %s." % kwds)
def _call_super_mom(fun):
# If fit method is overridden only for MLE and doesn't specify what to do
# if method == 'mm' or with censored data, this decorator calls the generic
# implementation.
@wraps(fun)
def wrapper(self, data, *args, **kwds):
method = kwds.get('method', 'mle').lower()
censored = isinstance(data, CensoredData)
if method == 'mm' or (censored and data.num_censored() > 0):
return super(type(self), self).fit(data, *args, **kwds)
else:
if censored:
# data is an instance of CensoredData, but actually holds
# no censored values, so replace it with the array of
# uncensored values.
data = data._uncensored
return fun(self, data, *args, **kwds)
return wrapper
def _get_left_bracket(fun, rbrack, lbrack=None):
# find left bracket for `root_scalar`. A guess for lbrack may be provided.
lbrack = lbrack or rbrack - 1
diff = rbrack - lbrack
# if there is no sign change in `fun` between the brackets, expand
# rbrack - lbrack until a sign change occurs
def interval_contains_root(lbrack, rbrack):
# return true if the signs disagree.
return np.sign(fun(lbrack)) != np.sign(fun(rbrack))
while not interval_contains_root(lbrack, rbrack):
diff *= 2
lbrack = rbrack - diff
msg = ("The solver could not find a bracket containing a "
"root to an MLE first order condition.")
if np.isinf(lbrack):
raise FitSolverError(msg)
return lbrack
class ksone_gen(rv_continuous):
r"""Kolmogorov-Smirnov one-sided test statistic distribution.
This is the distribution of the one-sided Kolmogorov-Smirnov (KS)
statistics :math:`D_n^+` and :math:`D_n^-`
for a finite sample size ``n >= 1`` (the shape parameter).
%(before_notes)s
See Also
--------
kstwobign, kstwo, kstest
Notes
-----
:math:`D_n^+` and :math:`D_n^-` are given by
.. math::
D_n^+ &= \text{sup}_x (F_n(x) - F(x)),\\
D_n^- &= \text{sup}_x (F(x) - F_n(x)),\\
where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
`ksone` describes the distribution under the null hypothesis of the KS test
that the empirical CDF corresponds to :math:`n` i.i.d. random variates
with CDF :math:`F`.
%(after_notes)s
References
----------
.. [1] Birnbaum, Z. W. and Tingey, F.H. "One-sided confidence contours
for probability distribution functions", The Annals of Mathematical
Statistics, 22(4), pp 592-596 (1951).
%(example)s
"""
def _argcheck(self, n):
return (n >= 1) & (n == np.round(n))
def _shape_info(self):
return [_ShapeInfo("n", True, (1, np.inf), (True, False))]
def _pdf(self, x, n):
return -scu._smirnovp(n, x)
def _cdf(self, x, n):
return scu._smirnovc(n, x)
def _sf(self, x, n):
return sc.smirnov(n, x)
def _ppf(self, q, n):
return scu._smirnovci(n, q)
def _isf(self, q, n):
return sc.smirnovi(n, q)
ksone = ksone_gen(a=0.0, b=1.0, name='ksone')
class kstwo_gen(rv_continuous):
r"""Kolmogorov-Smirnov two-sided test statistic distribution.
This is the distribution of the two-sided Kolmogorov-Smirnov (KS)
statistic :math:`D_n` for a finite sample size ``n >= 1``
(the shape parameter).
%(before_notes)s
See Also
--------
kstwobign, ksone, kstest
Notes
-----
:math:`D_n` is given by
.. math::
D_n = \text{sup}_x |F_n(x) - F(x)|
where :math:`F` is a (continuous) CDF and :math:`F_n` is an empirical CDF.
`kstwo` describes the distribution under the null hypothesis of the KS test
that the empirical CDF corresponds to :math:`n` i.i.d. random variates
with CDF :math:`F`.
%(after_notes)s
References
----------
.. [1] Simard, R., L'Ecuyer, P. "Computing the Two-Sided
Kolmogorov-Smirnov Distribution", Journal of Statistical Software,
Vol 39, 11, 1-18 (2011).
%(example)s
"""
def _argcheck(self, n):
return (n >= 1) & (n == np.round(n))
def _shape_info(self):
return [_ShapeInfo("n", True, (1, np.inf), (True, False))]
def _get_support(self, n):
return (0.5/(n if not isinstance(n, Iterable) else np.asanyarray(n)),
1.0)
def _pdf(self, x, n):
return kolmognp(n, x)
def _cdf(self, x, n):
return kolmogn(n, x)
def _sf(self, x, n):
return kolmogn(n, x, cdf=False)
def _ppf(self, q, n):
return kolmogni(n, q, cdf=True)
def _isf(self, q, n):
return kolmogni(n, q, cdf=False)
# Use the pdf, (not the ppf) to compute moments
kstwo = kstwo_gen(momtype=0, a=0.0, b=1.0, name='kstwo')
class kstwobign_gen(rv_continuous):
r"""Limiting distribution of scaled Kolmogorov-Smirnov two-sided test statistic.
This is the asymptotic distribution of the two-sided Kolmogorov-Smirnov
statistic :math:`\sqrt{n} D_n` that measures the maximum absolute
distance of the theoretical (continuous) CDF from the empirical CDF.
(see `kstest`).
%(before_notes)s
See Also
--------
ksone, kstwo, kstest
Notes
-----
:math:`\sqrt{n} D_n` is given by
.. math::
D_n = \text{sup}_x |F_n(x) - F(x)|
where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
`kstwobign` describes the asymptotic distribution (i.e. the limit of
:math:`\sqrt{n} D_n`) under the null hypothesis of the KS test that the
empirical CDF corresponds to i.i.d. random variates with CDF :math:`F`.
%(after_notes)s
References
----------
.. [1] Feller, W. "On the Kolmogorov-Smirnov Limit Theorems for Empirical
Distributions", Ann. Math. Statist. Vol 19, 177-189 (1948).
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
return -scu._kolmogp(x)
def _cdf(self, x):
return scu._kolmogc(x)
def _sf(self, x):
return sc.kolmogorov(x)
def _ppf(self, q):
return scu._kolmogci(q)
def _isf(self, q):
return sc.kolmogi(q)
kstwobign = kstwobign_gen(a=0.0, name='kstwobign')
## Normal distribution
# loc = mu, scale = std
# Keep these implementations out of the class definition so they can be reused
# by other distributions.
_norm_pdf_C = np.sqrt(2*np.pi)
_norm_pdf_logC = np.log(_norm_pdf_C)
def _norm_pdf(x):
return np.exp(-x**2/2.0) / _norm_pdf_C
def _norm_logpdf(x):
return -x**2 / 2.0 - _norm_pdf_logC
def _norm_cdf(x):
return sc.ndtr(x)
def _norm_logcdf(x):
return sc.log_ndtr(x)
def _norm_ppf(q):
return sc.ndtri(q)
def _norm_sf(x):
return _norm_cdf(-x)
def _norm_logsf(x):
return _norm_logcdf(-x)
def _norm_isf(q):
return -_norm_ppf(q)
class norm_gen(rv_continuous):
r"""A normal continuous random variable.
The location (``loc``) keyword specifies the mean.
The scale (``scale``) keyword specifies the standard deviation.
%(before_notes)s
Notes
-----
The probability density function for `norm` is:
.. math::
f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}}
for a real number :math:`x`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return random_state.standard_normal(size)
def _pdf(self, x):
# norm.pdf(x) = exp(-x**2/2)/sqrt(2*pi)
return _norm_pdf(x)
def _logpdf(self, x):
return _norm_logpdf(x)
def _cdf(self, x):
return _norm_cdf(x)
def _logcdf(self, x):
return _norm_logcdf(x)
def _sf(self, x):
return _norm_sf(x)
def _logsf(self, x):
return _norm_logsf(x)
def _ppf(self, q):
return _norm_ppf(q)
def _isf(self, q):
return _norm_isf(q)
def _stats(self):
return 0.0, 1.0, 0.0, 0.0
def _entropy(self):
return 0.5*(np.log(2*np.pi)+1)
@_call_super_mom
@replace_notes_in_docstring(rv_continuous, notes="""\
For the normal distribution, method of moments and maximum likelihood
estimation give identical fits, and explicit formulas for the estimates
are available.
This function uses these explicit formulas for the maximum likelihood
estimation of the normal distribution parameters, so the
`optimizer` and `method` arguments are ignored.\n\n""")
def fit(self, data, **kwds):
floc = kwds.pop('floc', None)
fscale = kwds.pop('fscale', None)
_remove_optimizer_parameters(kwds)
if floc is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
# Without this check, this function would just return the
# parameters that were given.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
data = np.asarray(data)
if not np.isfinite(data).all():
raise ValueError("The data contains non-finite values.")
if floc is None:
loc = data.mean()
else:
loc = floc
if fscale is None:
scale = np.sqrt(((data - loc)**2).mean())
else:
scale = fscale
return loc, scale
def _munp(self, n):
"""
@returns Moments of standard normal distribution for integer n >= 0
See eq. 16 of https://arxiv.org/abs/1209.4340v2
"""
if n % 2 == 0:
return sc.factorial2(n - 1)
else:
return 0.
norm = norm_gen(name='norm')
class alpha_gen(rv_continuous):
r"""An alpha continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `alpha` ([1]_, [2]_) is:
.. math::
f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} *
\exp(-\frac{1}{2} (a-1/x)^2)
where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`.
`alpha` takes ``a`` as a shape parameter.
%(after_notes)s
References
----------
.. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
Distributions, Volume 1", Second Edition, John Wiley and Sons,
p. 173 (1994).
.. [2] Anthony A. Salvia, "Reliability applications of the Alpha
Distribution", IEEE Transactions on Reliability, Vol. R-34,
No. 3, pp. 251-252 (1985).
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return [_ShapeInfo("a", False, (0, np.inf), (False, False))]
def _pdf(self, x, a):
# alpha.pdf(x, a) = 1/(x**2*Phi(a)*sqrt(2*pi)) * exp(-1/2 * (a-1/x)**2)
return 1.0/(x**2)/_norm_cdf(a)*_norm_pdf(a-1.0/x)
def _logpdf(self, x, a):
return -2*np.log(x) + _norm_logpdf(a-1.0/x) - np.log(_norm_cdf(a))
def _cdf(self, x, a):
return _norm_cdf(a-1.0/x) / _norm_cdf(a)
def _ppf(self, q, a):
return 1.0/np.asarray(a - _norm_ppf(q*_norm_cdf(a)))
def _stats(self, a):
return [np.inf]*2 + [np.nan]*2
alpha = alpha_gen(a=0.0, name='alpha')
class anglit_gen(rv_continuous):
r"""An anglit continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `anglit` is:
.. math::
f(x) = \sin(2x + \pi/2) = \cos(2x)
for :math:`-\pi/4 \le x \le \pi/4`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# anglit.pdf(x) = sin(2*x + \pi/2) = cos(2*x)
return np.cos(2*x)
def _cdf(self, x):
return np.sin(x+np.pi/4)**2.0
def _sf(self, x):
return np.cos(x + np.pi / 4) ** 2.0
def _ppf(self, q):
return np.arcsin(np.sqrt(q))-np.pi/4
def _stats(self):
return 0.0, np.pi*np.pi/16-0.5, 0.0, -2*(np.pi**4 - 96)/(np.pi*np.pi-8)**2
def _entropy(self):
return 1-np.log(2)
anglit = anglit_gen(a=-np.pi/4, b=np.pi/4, name='anglit')
class arcsine_gen(rv_continuous):
r"""An arcsine continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `arcsine` is:
.. math::
f(x) = \frac{1}{\pi \sqrt{x (1-x)}}
for :math:`0 < x < 1`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# arcsine.pdf(x) = 1/(pi*sqrt(x*(1-x)))
with np.errstate(divide='ignore'):
return 1.0/np.pi/np.sqrt(x*(1-x))
def _cdf(self, x):
return 2.0/np.pi*np.arcsin(np.sqrt(x))
def _ppf(self, q):
return np.sin(np.pi/2.0*q)**2.0
def _stats(self):
mu = 0.5
mu2 = 1.0/8
g1 = 0
g2 = -3.0/2.0
return mu, mu2, g1, g2
def _entropy(self):
return -0.24156447527049044468
arcsine = arcsine_gen(a=0.0, b=1.0, name='arcsine')
class FitDataError(ValueError):
"""Raised when input data is inconsistent with fixed parameters."""
# This exception is raised by, for example, beta_gen.fit when both floc
# and fscale are fixed and there are values in the data not in the open
# interval (floc, floc+fscale).
def __init__(self, distr, lower, upper):
self.args = (
"Invalid values in `data`. Maximum likelihood "
f"estimation with {distr!r} requires that {lower!r} < "
f"(x - loc)/scale < {upper!r} for each x in `data`.",
)
class FitSolverError(FitError):
"""
Raised when a solver fails to converge while fitting a distribution.
"""
# This exception is raised by, for example, beta_gen.fit when
# optimize.fsolve returns with ier != 1.
def __init__(self, mesg):
emsg = "Solver for the MLE equations failed to converge: "
emsg += mesg.replace('\n', '')
self.args = (emsg,)
def _beta_mle_a(a, b, n, s1):
# The zeros of this function give the MLE for `a`, with
# `b`, `n` and `s1` given. `s1` is the sum of the logs of
# the data. `n` is the number of data points.
psiab = sc.psi(a + b)
func = s1 - n * (-psiab + sc.psi(a))
return func
def _beta_mle_ab(theta, n, s1, s2):
# Zeros of this function are critical points of
# the maximum likelihood function. Solving this system
# for theta (which contains a and b) gives the MLE for a and b
# given `n`, `s1` and `s2`. `s1` is the sum of the logs of the data,
# and `s2` is the sum of the logs of 1 - data. `n` is the number
# of data points.
a, b = theta
psiab = sc.psi(a + b)
func = [s1 - n * (-psiab + sc.psi(a)),
s2 - n * (-psiab + sc.psi(b))]
return func
class beta_gen(rv_continuous):
r"""A beta continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `beta` is:
.. math::
f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}}
{\Gamma(a) \Gamma(b)}
for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where
:math:`\Gamma` is the gamma function (`scipy.special.gamma`).
`beta` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
return [ia, ib]
def _rvs(self, a, b, size=None, random_state=None):
return random_state.beta(a, b, size)
def _pdf(self, x, a, b):
# gamma(a+b) * x**(a-1) * (1-x)**(b-1)
# beta.pdf(x, a, b) = ------------------------------------
# gamma(a)*gamma(b)
with np.errstate(over='ignore'):
return _boost._beta_pdf(x, a, b)
def _logpdf(self, x, a, b):
lPx = sc.xlog1py(b - 1.0, -x) + sc.xlogy(a - 1.0, x)
lPx -= sc.betaln(a, b)
return lPx
def _cdf(self, x, a, b):
return _boost._beta_cdf(x, a, b)
def _sf(self, x, a, b):
return _boost._beta_sf(x, a, b)
def _isf(self, x, a, b):
with np.errstate(over='ignore'): # see gh-17432
return _boost._beta_isf(x, a, b)
def _ppf(self, q, a, b):
with np.errstate(over='ignore'): # see gh-17432
return _boost._beta_ppf(q, a, b)
def _stats(self, a, b):
return (
_boost._beta_mean(a, b),
_boost._beta_variance(a, b),
_boost._beta_skewness(a, b),
_boost._beta_kurtosis_excess(a, b))
def _fitstart(self, data):
if isinstance(data, CensoredData):
data = data._uncensor()
g1 = _skew(data)
g2 = _kurtosis(data)
def func(x):
a, b = x
sk = 2*(b-a)*np.sqrt(a + b + 1) / (a + b + 2) / np.sqrt(a*b)
ku = a**3 - a**2*(2*b-1) + b**2*(b+1) - 2*a*b*(b+2)
ku /= a*b*(a+b+2)*(a+b+3)
ku *= 6
return [sk-g1, ku-g2]
a, b = optimize.fsolve(func, (1.0, 1.0))
return super()._fitstart(data, args=(a, b))
@_call_super_mom
@extend_notes_in_docstring(rv_continuous, notes="""\
In the special case where `method="MLE"` and
both `floc` and `fscale` are given, a
`ValueError` is raised if any value `x` in `data` does not satisfy
`floc < x < floc + fscale`.\n\n""")
def fit(self, data, *args, **kwds):
# Override rv_continuous.fit, so we can more efficiently handle the
# case where floc and fscale are given.
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
if floc is None or fscale is None:
# do general fit
return super().fit(data, *args, **kwds)
# We already got these from kwds, so just pop them.
kwds.pop('floc', None)
kwds.pop('fscale', None)
f0 = _get_fixed_fit_value(kwds, ['f0', 'fa', 'fix_a'])
f1 = _get_fixed_fit_value(kwds, ['f1', 'fb', 'fix_b'])
_remove_optimizer_parameters(kwds)
if f0 is not None and f1 is not None:
# This check is for consistency with `rv_continuous.fit`.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
# Special case: loc and scale are constrained, so we are fitting
# just the shape parameters. This can be done much more efficiently
# than the method used in `rv_continuous.fit`. (See the subsection
# "Two unknown parameters" in the section "Maximum likelihood" of
# the Wikipedia article on the Beta distribution for the formulas.)
if not np.isfinite(data).all():
raise ValueError("The data contains non-finite values.")
# Normalize the data to the interval [0, 1].
data = (np.ravel(data) - floc) / fscale
if np.any(data <= 0) or np.any(data >= 1):
raise FitDataError("beta", lower=floc, upper=floc + fscale)
xbar = data.mean()
if f0 is not None or f1 is not None:
# One of the shape parameters is fixed.
if f0 is not None:
# The shape parameter a is fixed, so swap the parameters
# and flip the data. We always solve for `a`. The result
# will be swapped back before returning.
b = f0
data = 1 - data
xbar = 1 - xbar
else:
b = f1
# Initial guess for a. Use the formula for the mean of the beta
# distribution, E[x] = a / (a + b), to generate a reasonable
# starting point based on the mean of the data and the given
# value of b.
a = b * xbar / (1 - xbar)
# Compute the MLE for `a` by solving _beta_mle_a.
theta, info, ier, mesg = optimize.fsolve(
_beta_mle_a, a,
args=(b, len(data), np.log(data).sum()),
full_output=True
)
if ier != 1:
raise FitSolverError(mesg=mesg)
a = theta[0]
if f0 is not None:
# The shape parameter a was fixed, so swap back the
# parameters.
a, b = b, a
else:
# Neither of the shape parameters is fixed.
# s1 and s2 are used in the extra arguments passed to _beta_mle_ab
# by optimize.fsolve.
s1 = np.log(data).sum()
s2 = sc.log1p(-data).sum()
# Use the "method of moments" to estimate the initial
# guess for a and b.
fac = xbar * (1 - xbar) / data.var(ddof=0) - 1
a = xbar * fac
b = (1 - xbar) * fac
# Compute the MLE for a and b by solving _beta_mle_ab.
theta, info, ier, mesg = optimize.fsolve(
_beta_mle_ab, [a, b],
args=(len(data), s1, s2),
full_output=True
)
if ier != 1:
raise FitSolverError(mesg=mesg)
a, b = theta
return a, b, floc, fscale
def _entropy(self, a, b):
def regular(a, b):
return (sc.betaln(a, b) - (a - 1) * sc.psi(a) -
(b - 1) * sc.psi(b) + (a + b - 2) * sc.psi(a + b))
def asymptotic_ab_large(a, b):
sum_ab = a + b
log_term = 0.5 * (
np.log(2*np.pi) + np.log(a) + np.log(b) - 3*np.log(sum_ab) + 1
)
t1 = 110/sum_ab + 20*sum_ab**-2.0 + sum_ab**-3.0 - 2*sum_ab**-4.0
t2 = -50/a - 10*a**-2.0 - a**-3.0 + a**-4.0
t3 = -50/b - 10*b**-2.0 - b**-3.0 + b**-4.0
return log_term + (t1 + t2 + t3) / 120
def asymptotic_b_large(a, b):
sum_ab = a + b
t1 = sc.gammaln(a) - (a - 1) * sc.psi(a)
t2 = (
- 1/(2*b) + 1/(12*b) - b**-2.0/12 - b**-3.0/120 + b**-4.0/120
+ b**-5.0/252 - b**-6.0/252 + 1/sum_ab - 1/(12*sum_ab)
+ sum_ab**-2.0/6 + sum_ab**-3.0/120 - sum_ab**-4.0/60
- sum_ab**-5.0/252 + sum_ab**-6.0/126
)
log_term = sum_ab*np.log1p(a/b) + np.log(b) - 2*np.log(sum_ab)
return t1 + t2 + log_term
def threshold_large(v):
if v == 1.0:
return 1000
j = np.log10(v)
digits = int(j)
d = int(v / 10 ** digits) + 2
return d*10**(7 + j)
if a >= 4.96e6 and b >= 4.96e6:
return asymptotic_ab_large(a, b)
elif a <= 4.9e6 and b - a >= 1e6 and b >= threshold_large(a):
return asymptotic_b_large(a, b)
elif b <= 4.9e6 and a - b >= 1e6 and a >= threshold_large(b):
return asymptotic_b_large(b, a)
else:
return regular(a, b)
beta = beta_gen(a=0.0, b=1.0, name='beta')
class betaprime_gen(rv_continuous):
r"""A beta prime continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `betaprime` is:
.. math::
f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)}
for :math:`x >= 0`, :math:`a > 0`, :math:`b > 0`, where
:math:`\beta(a, b)` is the beta function (see `scipy.special.beta`).
`betaprime` takes ``a`` and ``b`` as shape parameters.
The distribution is related to the `beta` distribution as follows:
If :math:`X` follows a beta distribution with parameters :math:`a, b`,
then :math:`Y = X/(1-X)` has a beta prime distribution with
parameters :math:`a, b` ([1]_).
The beta prime distribution is a reparametrized version of the
F distribution. The beta prime distribution with shape parameters
``a`` and ``b`` and ``scale = s`` is equivalent to the F distribution
with parameters ``d1 = 2*a``, ``d2 = 2*b`` and ``scale = (a/b)*s``.
For example,
>>> from scipy.stats import betaprime, f
>>> x = [1, 2, 5, 10]
>>> a = 12
>>> b = 5
>>> betaprime.pdf(x, a, b, scale=2)
array([0.00541179, 0.08331299, 0.14669185, 0.03150079])
>>> f.pdf(x, 2*a, 2*b, scale=(a/b)*2)
array([0.00541179, 0.08331299, 0.14669185, 0.03150079])
%(after_notes)s
References
----------
.. [1] Beta prime distribution, Wikipedia,
https://en.wikipedia.org/wiki/Beta_prime_distribution
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
return [ia, ib]
def _rvs(self, a, b, size=None, random_state=None):
u1 = gamma.rvs(a, size=size, random_state=random_state)
u2 = gamma.rvs(b, size=size, random_state=random_state)
return u1 / u2
def _pdf(self, x, a, b):
# betaprime.pdf(x, a, b) = x**(a-1) * (1+x)**(-a-b) / beta(a, b)
return np.exp(self._logpdf(x, a, b))
def _logpdf(self, x, a, b):
return sc.xlogy(a - 1.0, x) - sc.xlog1py(a + b, x) - sc.betaln(a, b)
def _cdf(self, x, a, b):
# note: f2 is the direct way to compute the cdf if the relationship
# to the beta distribution is used.
# however, for very large x, x/(1+x) == 1. since the distribution
# has very fat tails if b is small, this can cause inaccurate results
# use the following relationship of the incomplete beta function:
# betainc(x, a, b) = 1 - betainc(1-x, b, a)
# see gh-17631
return _lazywhere(
x > 1, [x, a, b],
lambda x_, a_, b_: beta._sf(1/(1+x_), b_, a_),
f2=lambda x_, a_, b_: beta._cdf(x_/(1+x_), a_, b_))
def _sf(self, x, a, b):
return _lazywhere(
x > 1, [x, a, b],
lambda x_, a_, b_: beta._cdf(1/(1+x_), b_, a_),
f2=lambda x_, a_, b_: beta._sf(x_/(1+x_), a_, b_)
)
def _ppf(self, p, a, b):
p, a, b = np.broadcast_arrays(p, a, b)
# by default, compute compute the ppf by solving the following:
# p = beta._cdf(x/(1+x), a, b). This implies x = r/(1-r) with
# r = beta._ppf(p, a, b). This can cause numerical issues if r is
# very close to 1. in that case, invert the alternative expression of
# the cdf: p = beta._sf(1/(1+x), b, a).
r = stats.beta._ppf(p, a, b)
with np.errstate(divide='ignore'):
out = r / (1 - r)
i = (r > 0.9999)
out[i] = 1/stats.beta._isf(p[i], b[i], a[i]) - 1
return out
def _munp(self, n, a, b):
return _lazywhere(
b > n, (a, b),
lambda a, b: np.prod([(a+i-1)/(b-i) for i in range(1, n+1)], axis=0),
fillvalue=np.inf)
betaprime = betaprime_gen(a=0.0, name='betaprime')
class bradford_gen(rv_continuous):
r"""A Bradford continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `bradford` is:
.. math::
f(x, c) = \frac{c}{\log(1+c) (1+cx)}
for :math:`0 <= x <= 1` and :math:`c > 0`.
`bradford` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# bradford.pdf(x, c) = c / (k * (1+c*x))
return c / (c*x + 1.0) / sc.log1p(c)
def _cdf(self, x, c):
return sc.log1p(c*x) / sc.log1p(c)
def _ppf(self, q, c):
return sc.expm1(q * sc.log1p(c)) / c
def _stats(self, c, moments='mv'):
k = np.log(1.0+c)
mu = (c-k)/(c*k)
mu2 = ((c+2.0)*k-2.0*c)/(2*c*k*k)
g1 = None
g2 = None
if 's' in moments:
g1 = np.sqrt(2)*(12*c*c-9*c*k*(c+2)+2*k*k*(c*(c+3)+3))
g1 /= np.sqrt(c*(c*(k-2)+2*k))*(3*c*(k-2)+6*k)
if 'k' in moments:
g2 = (c**3*(k-3)*(k*(3*k-16)+24)+12*k*c*c*(k-4)*(k-3) +
6*c*k*k*(3*k-14) + 12*k**3)
g2 /= 3*c*(c*(k-2)+2*k)**2
return mu, mu2, g1, g2
def _entropy(self, c):
k = np.log(1+c)
return k/2.0 - np.log(c/k)
bradford = bradford_gen(a=0.0, b=1.0, name='bradford')
class burr_gen(rv_continuous):
r"""A Burr (Type III) continuous random variable.
%(before_notes)s
See Also
--------
fisk : a special case of either `burr` or `burr12` with ``d=1``
burr12 : Burr Type XII distribution
mielke : Mielke Beta-Kappa / Dagum distribution
Notes
-----
The probability density function for `burr` is:
.. math::
f(x; c, d) = c d \frac{x^{-c - 1}}
{{(1 + x^{-c})}^{d + 1}}
for :math:`x >= 0` and :math:`c, d > 0`.
`burr` takes ``c`` and ``d`` as shape parameters for :math:`c` and
:math:`d`.
This is the PDF corresponding to the third CDF given in Burr's list;
specifically, it is equation (11) in Burr's paper [1]_. The distribution
is also commonly referred to as the Dagum distribution [2]_. If the
parameter :math:`c < 1` then the mean of the distribution does not
exist and if :math:`c < 2` the variance does not exist [2]_.
The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 1`.
%(after_notes)s
References
----------
.. [1] Burr, I. W. "Cumulative frequency functions", Annals of
Mathematical Statistics, 13(2), pp 215-232 (1942).
.. [2] https://en.wikipedia.org/wiki/Dagum_distribution
.. [3] Kleiber, Christian. "A guide to the Dagum distributions."
Modeling Income Distributions and Lorenz Curves pp 97-117 (2008).
%(example)s
"""
# Do not set _support_mask to rv_continuous._open_support_mask
# Whether the left-hand endpoint is suitable for pdf evaluation is dependent
# on the values of c and d: if c*d >= 1, the pdf is finite, otherwise infinite.
def _shape_info(self):
ic = _ShapeInfo("c", False, (0, np.inf), (False, False))
id = _ShapeInfo("d", False, (0, np.inf), (False, False))
return [ic, id]
def _pdf(self, x, c, d):
# burr.pdf(x, c, d) = c * d * x**(-c-1) * (1+x**(-c))**(-d-1)
output = _lazywhere(
x == 0, [x, c, d],
lambda x_, c_, d_: c_ * d_ * (x_**(c_*d_-1)) / (1 + x_**c_),
f2=lambda x_, c_, d_: (c_ * d_ * (x_ ** (-c_ - 1.0)) /
((1 + x_ ** (-c_)) ** (d_ + 1.0))))
if output.ndim == 0:
return output[()]
return output
def _logpdf(self, x, c, d):
output = _lazywhere(
x == 0, [x, c, d],
lambda x_, c_, d_: (np.log(c_) + np.log(d_) + sc.xlogy(c_*d_ - 1, x_)
- (d_+1) * sc.log1p(x_**(c_))),
f2=lambda x_, c_, d_: (np.log(c_) + np.log(d_)
+ sc.xlogy(-c_ - 1, x_)
- sc.xlog1py(d_+1, x_**(-c_))))
if output.ndim == 0:
return output[()]
return output
def _cdf(self, x, c, d):
return (1 + x**(-c))**(-d)
def _logcdf(self, x, c, d):
return sc.log1p(x**(-c)) * (-d)
def _sf(self, x, c, d):
return np.exp(self._logsf(x, c, d))
def _logsf(self, x, c, d):
return np.log1p(- (1 + x**(-c))**(-d))
def _ppf(self, q, c, d):
return (q**(-1.0/d) - 1)**(-1.0/c)
def _isf(self, q, c, d):
_q = sc.xlog1py(-1.0 / d, -q)
return sc.expm1(_q) ** (-1.0 / c)
def _stats(self, c, d):
nc = np.arange(1, 5).reshape(4,1) / c
# ek is the kth raw moment, e1 is the mean e2-e1**2 variance etc.
e1, e2, e3, e4 = sc.beta(d + nc, 1. - nc) * d
mu = np.where(c > 1.0, e1, np.nan)
mu2_if_c = e2 - mu**2
mu2 = np.where(c > 2.0, mu2_if_c, np.nan)
g1 = _lazywhere(
c > 3.0,
(c, e1, e2, e3, mu2_if_c),
lambda c, e1, e2, e3, mu2_if_c: ((e3 - 3*e2*e1 + 2*e1**3)
/ np.sqrt((mu2_if_c)**3)),
fillvalue=np.nan)
g2 = _lazywhere(
c > 4.0,
(c, e1, e2, e3, e4, mu2_if_c),
lambda c, e1, e2, e3, e4, mu2_if_c: (
((e4 - 4*e3*e1 + 6*e2*e1**2 - 3*e1**4) / mu2_if_c**2) - 3),
fillvalue=np.nan)
if np.ndim(c) == 0:
return mu.item(), mu2.item(), g1.item(), g2.item()
return mu, mu2, g1, g2
def _munp(self, n, c, d):
def __munp(n, c, d):
nc = 1. * n / c
return d * sc.beta(1.0 - nc, d + nc)
n, c, d = np.asarray(n), np.asarray(c), np.asarray(d)
return _lazywhere((c > n) & (n == n) & (d == d), (c, d, n),
lambda c, d, n: __munp(n, c, d),
np.nan)
burr = burr_gen(a=0.0, name='burr')
class burr12_gen(rv_continuous):
r"""A Burr (Type XII) continuous random variable.
%(before_notes)s
See Also
--------
fisk : a special case of either `burr` or `burr12` with ``d=1``
burr : Burr Type III distribution
Notes
-----
The probability density function for `burr12` is:
.. math::
f(x; c, d) = c d \frac{x^{c-1}}
{(1 + x^c)^{d + 1}}
for :math:`x >= 0` and :math:`c, d > 0`.
`burr12` takes ``c`` and ``d`` as shape parameters for :math:`c`
and :math:`d`.
This is the PDF corresponding to the twelfth CDF given in Burr's list;
specifically, it is equation (20) in Burr's paper [1]_.
%(after_notes)s
The Burr type 12 distribution is also sometimes referred to as
the Singh-Maddala distribution from NIST [2]_.
References
----------
.. [1] Burr, I. W. "Cumulative frequency functions", Annals of
Mathematical Statistics, 13(2), pp 215-232 (1942).
.. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm
.. [3] "Burr distribution",
https://en.wikipedia.org/wiki/Burr_distribution
%(example)s
"""
def _shape_info(self):
ic = _ShapeInfo("c", False, (0, np.inf), (False, False))
id = _ShapeInfo("d", False, (0, np.inf), (False, False))
return [ic, id]
def _pdf(self, x, c, d):
# burr12.pdf(x, c, d) = c * d * x**(c-1) * (1+x**(c))**(-d-1)
return np.exp(self._logpdf(x, c, d))
def _logpdf(self, x, c, d):
return np.log(c) + np.log(d) + sc.xlogy(c - 1, x) + sc.xlog1py(-d-1, x**c)
def _cdf(self, x, c, d):
return -sc.expm1(self._logsf(x, c, d))
def _logcdf(self, x, c, d):
return sc.log1p(-(1 + x**c)**(-d))
def _sf(self, x, c, d):
return np.exp(self._logsf(x, c, d))
def _logsf(self, x, c, d):
return sc.xlog1py(-d, x**c)
def _ppf(self, q, c, d):
# The following is an implementation of
# ((1 - q)**(-1.0/d) - 1)**(1.0/c)
# that does a better job handling small values of q.
return sc.expm1(-1/d * sc.log1p(-q))**(1/c)
def _munp(self, n, c, d):
def moment_if_exists(n, c, d):
nc = 1. * n / c
return d * sc.beta(1.0 + nc, d - nc)
return _lazywhere(c * d > n, (n, c, d), moment_if_exists,
fillvalue=np.nan)
burr12 = burr12_gen(a=0.0, name='burr12')
class fisk_gen(burr_gen):
r"""A Fisk continuous random variable.
The Fisk distribution is also known as the log-logistic distribution.
%(before_notes)s
See Also
--------
burr
Notes
-----
The probability density function for `fisk` is:
.. math::
f(x, c) = \frac{c x^{c-1}}
{(1 + x^c)^2}
for :math:`x >= 0` and :math:`c > 0`.
Please note that the above expression can be transformed into the following
one, which is also commonly used:
.. math::
f(x, c) = \frac{c x^{-c-1}}
{(1 + x^{-c})^2}
`fisk` takes ``c`` as a shape parameter for :math:`c`.
`fisk` is a special case of `burr` or `burr12` with ``d=1``.
Suppose ``X`` is a logistic random variable with location ``l``
and scale ``s``. Then ``Y = exp(X)`` is a Fisk (log-logistic)
random variable with ``scale = exp(l)`` and shape ``c = 1/s``.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2)
return burr._pdf(x, c, 1.0)
def _cdf(self, x, c):
return burr._cdf(x, c, 1.0)
def _sf(self, x, c):
return burr._sf(x, c, 1.0)
def _logpdf(self, x, c):
# fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2)
return burr._logpdf(x, c, 1.0)
def _logcdf(self, x, c):
return burr._logcdf(x, c, 1.0)
def _logsf(self, x, c):
return burr._logsf(x, c, 1.0)
def _ppf(self, x, c):
return burr._ppf(x, c, 1.0)
def _isf(self, q, c):
return burr._isf(q, c, 1.0)
def _munp(self, n, c):
return burr._munp(n, c, 1.0)
def _stats(self, c):
return burr._stats(c, 1.0)
def _entropy(self, c):
return 2 - np.log(c)
fisk = fisk_gen(a=0.0, name='fisk')
class cauchy_gen(rv_continuous):
r"""A Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `cauchy` is
.. math::
f(x) = \frac{1}{\pi (1 + x^2)}
for a real number :math:`x`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# cauchy.pdf(x) = 1 / (pi * (1 + x**2))
return 1.0/np.pi/(1.0+x*x)
def _cdf(self, x):
return 0.5 + 1.0/np.pi*np.arctan(x)
def _ppf(self, q):
return np.tan(np.pi*q-np.pi/2.0)
def _sf(self, x):
return 0.5 - 1.0/np.pi*np.arctan(x)
def _isf(self, q):
return np.tan(np.pi/2.0-np.pi*q)
def _stats(self):
return np.nan, np.nan, np.nan, np.nan
def _entropy(self):
return np.log(4*np.pi)
def _fitstart(self, data, args=None):
# Initialize ML guesses using quartiles instead of moments.
if isinstance(data, CensoredData):
data = data._uncensor()
p25, p50, p75 = np.percentile(data, [25, 50, 75])
return p50, (p75 - p25)/2
cauchy = cauchy_gen(name='cauchy')
class chi_gen(rv_continuous):
r"""A chi continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `chi` is:
.. math::
f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)}
x^{k-1} \exp \left( -x^2/2 \right)
for :math:`x >= 0` and :math:`k > 0` (degrees of freedom, denoted ``df``
in the implementation). :math:`\Gamma` is the gamma function
(`scipy.special.gamma`).
Special cases of `chi` are:
- ``chi(1, loc, scale)`` is equivalent to `halfnorm`
- ``chi(2, 0, scale)`` is equivalent to `rayleigh`
- ``chi(3, 0, scale)`` is equivalent to `maxwell`
`chi` takes ``df`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("df", False, (0, np.inf), (False, False))]
def _rvs(self, df, size=None, random_state=None):
return np.sqrt(chi2.rvs(df, size=size, random_state=random_state))
def _pdf(self, x, df):
# x**(df-1) * exp(-x**2/2)
# chi.pdf(x, df) = -------------------------
# 2**(df/2-1) * gamma(df/2)
return np.exp(self._logpdf(x, df))
def _logpdf(self, x, df):
l = np.log(2) - .5*np.log(2)*df - sc.gammaln(.5*df)
return l + sc.xlogy(df - 1., x) - .5*x**2
def _cdf(self, x, df):
return sc.gammainc(.5*df, .5*x**2)
def _sf(self, x, df):
return sc.gammaincc(.5*df, .5*x**2)
def _ppf(self, q, df):
return np.sqrt(2*sc.gammaincinv(.5*df, q))
def _isf(self, q, df):
return np.sqrt(2*sc.gammainccinv(.5*df, q))
def _stats(self, df):
# poch(df/2, 1/2) = gamma(df/2 + 1/2) / gamma(df/2)
mu = np.sqrt(2) * sc.poch(0.5 * df, 0.5)
mu2 = df - mu*mu
g1 = (2*mu**3.0 + mu*(1-2*df))/np.asarray(np.power(mu2, 1.5))
g2 = 2*df*(1.0-df)-6*mu**4 + 4*mu**2 * (2*df-1)
g2 /= np.asarray(mu2**2.0)
return mu, mu2, g1, g2
def _entropy(self, df):
def regular_formula(df):
return (sc.gammaln(.5 * df)
+ 0.5 * (df - np.log(2) - (df - 1) * sc.digamma(0.5 * df)))
def asymptotic_formula(df):
return (0.5 + np.log(np.pi)/2 - (df**-1)/6 - (df**-2)/6
- 4/45*(df**-3) + (df**-4)/15)
return _lazywhere(df < 3e2, (df, ), regular_formula,
f2=asymptotic_formula)
chi = chi_gen(a=0.0, name='chi')
class chi2_gen(rv_continuous):
r"""A chi-squared continuous random variable.
For the noncentral chi-square distribution, see `ncx2`.
%(before_notes)s
See Also
--------
ncx2
Notes
-----
The probability density function for `chi2` is:
.. math::
f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)}
x^{k/2-1} \exp \left( -x/2 \right)
for :math:`x > 0` and :math:`k > 0` (degrees of freedom, denoted ``df``
in the implementation).
`chi2` takes ``df`` as a shape parameter.
The chi-squared distribution is a special case of the gamma
distribution, with gamma parameters ``a = df/2``, ``loc = 0`` and
``scale = 2``.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("df", False, (0, np.inf), (False, False))]
def _rvs(self, df, size=None, random_state=None):
return random_state.chisquare(df, size)
def _pdf(self, x, df):
# chi2.pdf(x, df) = 1 / (2*gamma(df/2)) * (x/2)**(df/2-1) * exp(-x/2)
return np.exp(self._logpdf(x, df))
def _logpdf(self, x, df):
return sc.xlogy(df/2.-1, x) - x/2. - sc.gammaln(df/2.) - (np.log(2)*df)/2.
def _cdf(self, x, df):
return sc.chdtr(df, x)
def _sf(self, x, df):
return sc.chdtrc(df, x)
def _isf(self, p, df):
return sc.chdtri(df, p)
def _ppf(self, p, df):
return 2*sc.gammaincinv(df/2, p)
def _stats(self, df):
mu = df
mu2 = 2*df
g1 = 2*np.sqrt(2.0/df)
g2 = 12.0/df
return mu, mu2, g1, g2
def _entropy(self, df):
half_df = 0.5 * df
def regular_formula(half_df):
return (half_df + np.log(2) + sc.gammaln(half_df) +
(1 - half_df) * sc.psi(half_df))
def asymptotic_formula(half_df):
# plug in the above formula the following asymptotic
# expansions:
# ln(gamma(a)) ~ (a - 0.5) * ln(a) - a + 0.5 * ln(2 * pi) +
# 1/(12 * a) - 1/(360 * a**3)
# psi(a) ~ ln(a) - 1/(2 * a) - 1/(3 * a**2) + 1/120 * a**4)
c = np.log(2) + 0.5*(1 + np.log(2*np.pi))
h = 0.5/half_df
return (h*(-2/3 + h*(-1/3 + h*(-4/45 + h/7.5))) +
0.5*np.log(half_df) + c)
return _lazywhere(half_df < 125, (half_df, ),
regular_formula,
f2=asymptotic_formula)
chi2 = chi2_gen(a=0.0, name='chi2')
class cosine_gen(rv_continuous):
r"""A cosine continuous random variable.
%(before_notes)s
Notes
-----
The cosine distribution is an approximation to the normal distribution.
The probability density function for `cosine` is:
.. math::
f(x) = \frac{1}{2\pi} (1+\cos(x))
for :math:`-\pi \le x \le \pi`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# cosine.pdf(x) = 1/(2*pi) * (1+cos(x))
return 1.0/2/np.pi*(1+np.cos(x))
def _logpdf(self, x):
c = np.cos(x)
return _lazywhere(c != -1, (c,),
lambda c: np.log1p(c) - np.log(2*np.pi),
fillvalue=-np.inf)
def _cdf(self, x):
return scu._cosine_cdf(x)
def _sf(self, x):
return scu._cosine_cdf(-x)
def _ppf(self, p):
return scu._cosine_invcdf(p)
def _isf(self, p):
return -scu._cosine_invcdf(p)
def _stats(self):
v = (np.pi * np.pi / 3.0) - 2.0
k = -6.0 * (np.pi**4 - 90) / (5.0 * (np.pi * np.pi - 6)**2)
return 0.0, v, 0.0, k
def _entropy(self):
return np.log(4*np.pi)-1.0
cosine = cosine_gen(a=-np.pi, b=np.pi, name='cosine')
class dgamma_gen(rv_continuous):
r"""A double gamma continuous random variable.
The double gamma distribution is also known as the reflected gamma
distribution [1]_.
%(before_notes)s
Notes
-----
The probability density function for `dgamma` is:
.. math::
f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|)
for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the
gamma function (`scipy.special.gamma`).
`dgamma` takes ``a`` as a shape parameter for :math:`a`.
%(after_notes)s
References
----------
.. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
Distributions, Volume 1", Second Edition, John Wiley and Sons
(1994).
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("a", False, (0, np.inf), (False, False))]
def _rvs(self, a, size=None, random_state=None):
u = random_state.uniform(size=size)
gm = gamma.rvs(a, size=size, random_state=random_state)
return gm * np.where(u >= 0.5, 1, -1)
def _pdf(self, x, a):
# dgamma.pdf(x, a) = 1 / (2*gamma(a)) * abs(x)**(a-1) * exp(-abs(x))
ax = abs(x)
return 1.0/(2*sc.gamma(a))*ax**(a-1.0) * np.exp(-ax)
def _logpdf(self, x, a):
ax = abs(x)
return sc.xlogy(a - 1.0, ax) - ax - np.log(2) - sc.gammaln(a)
def _cdf(self, x, a):
return np.where(x > 0,
0.5 + 0.5*sc.gammainc(a, x),
0.5*sc.gammaincc(a, -x))
def _sf(self, x, a):
return np.where(x > 0,
0.5*sc.gammaincc(a, x),
0.5 + 0.5*sc.gammainc(a, -x))
def _entropy(self, a):
return stats.gamma._entropy(a) - np.log(0.5)
def _ppf(self, q, a):
return np.where(q > 0.5,
sc.gammaincinv(a, 2*q - 1),
-sc.gammainccinv(a, 2*q))
def _isf(self, q, a):
return np.where(q > 0.5,
-sc.gammaincinv(a, 2*q - 1),
sc.gammainccinv(a, 2*q))
def _stats(self, a):
mu2 = a*(a+1.0)
return 0.0, mu2, 0.0, (a+2.0)*(a+3.0)/mu2-3.0
dgamma = dgamma_gen(name='dgamma')
class dweibull_gen(rv_continuous):
r"""A double Weibull continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `dweibull` is given by
.. math::
f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c)
for a real number :math:`x` and :math:`c > 0`.
`dweibull` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _rvs(self, c, size=None, random_state=None):
u = random_state.uniform(size=size)
w = weibull_min.rvs(c, size=size, random_state=random_state)
return w * (np.where(u >= 0.5, 1, -1))
def _pdf(self, x, c):
# dweibull.pdf(x, c) = c / 2 * abs(x)**(c-1) * exp(-abs(x)**c)
ax = abs(x)
Px = c / 2.0 * ax**(c-1.0) * np.exp(-ax**c)
return Px
def _logpdf(self, x, c):
ax = abs(x)
return np.log(c) - np.log(2.0) + sc.xlogy(c - 1.0, ax) - ax**c
def _cdf(self, x, c):
Cx1 = 0.5 * np.exp(-abs(x)**c)
return np.where(x > 0, 1 - Cx1, Cx1)
def _ppf(self, q, c):
fac = 2. * np.where(q <= 0.5, q, 1. - q)
fac = np.power(-np.log(fac), 1.0 / c)
return np.where(q > 0.5, fac, -fac)
def _sf(self, x, c):
half_weibull_min_sf = 0.5 * stats.weibull_min._sf(np.abs(x), c)
return np.where(x > 0, half_weibull_min_sf, 1 - half_weibull_min_sf)
def _isf(self, q, c):
double_q = 2. * np.where(q <= 0.5, q, 1. - q)
weibull_min_isf = stats.weibull_min._isf(double_q, c)
return np.where(q > 0.5, -weibull_min_isf, weibull_min_isf)
def _munp(self, n, c):
return (1 - (n % 2)) * sc.gamma(1.0 + 1.0 * n / c)
# since we know that all odd moments are zeros, return them at once.
# returning Nones from _stats makes the public stats call _munp
# so overall we're saving one or two gamma function evaluations here.
def _stats(self, c):
return 0, None, 0, None
def _entropy(self, c):
h = stats.weibull_min._entropy(c) - np.log(0.5)
return h
dweibull = dweibull_gen(name='dweibull')
class expon_gen(rv_continuous):
r"""An exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `expon` is:
.. math::
f(x) = \exp(-x)
for :math:`x \ge 0`.
%(after_notes)s
A common parameterization for `expon` is in terms of the rate parameter
``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This
parameterization corresponds to using ``scale = 1 / lambda``.
The exponential distribution is a special case of the gamma
distributions, with gamma shape parameter ``a = 1``.
%(example)s
"""
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return random_state.standard_exponential(size)
def _pdf(self, x):
# expon.pdf(x) = exp(-x)
return np.exp(-x)
def _logpdf(self, x):
return -x
def _cdf(self, x):
return -sc.expm1(-x)
def _ppf(self, q):
return -sc.log1p(-q)
def _sf(self, x):
return np.exp(-x)
def _logsf(self, x):
return -x
def _isf(self, q):
return -np.log(q)
def _stats(self):
return 1.0, 1.0, 2.0, 6.0
def _entropy(self):
return 1.0
@_call_super_mom
@replace_notes_in_docstring(rv_continuous, notes="""\
When `method='MLE'`,
this function uses explicit formulas for the maximum likelihood
estimation of the exponential distribution parameters, so the
`optimizer`, `loc` and `scale` keyword arguments are
ignored.\n\n""")
def fit(self, data, *args, **kwds):
if len(args) > 0:
raise TypeError("Too many arguments.")
floc = kwds.pop('floc', None)
fscale = kwds.pop('fscale', None)
_remove_optimizer_parameters(kwds)
if floc is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
data = np.asarray(data)
if not np.isfinite(data).all():
raise ValueError("The data contains non-finite values.")
data_min = data.min()
if floc is None:
# ML estimate of the location is the minimum of the data.
loc = data_min
else:
loc = floc
if data_min < loc:
# There are values that are less than the specified loc.
raise FitDataError("expon", lower=floc, upper=np.inf)
if fscale is None:
# ML estimate of the scale is the shifted mean.
scale = data.mean() - loc
else:
scale = fscale
# We expect the return values to be floating point, so ensure it
# by explicitly converting to float.
return float(loc), float(scale)
expon = expon_gen(a=0.0, name='expon')
class exponnorm_gen(rv_continuous):
r"""An exponentially modified Normal continuous random variable.
Also known as the exponentially modified Gaussian distribution [1]_.
%(before_notes)s
Notes
-----
The probability density function for `exponnorm` is:
.. math::
f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right)
\text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)
where :math:`x` is a real number and :math:`K > 0`.
It can be thought of as the sum of a standard normal random variable
and an independent exponentially distributed random variable with rate
``1/K``.
%(after_notes)s
An alternative parameterization of this distribution (for example, in
the Wikipedia article [1]_) involves three parameters, :math:`\mu`,
:math:`\lambda` and :math:`\sigma`.
In the present parameterization this corresponds to having ``loc`` and
``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and
shape parameter :math:`K = 1/(\sigma\lambda)`.
.. versionadded:: 0.16.0
References
----------
.. [1] Exponentially modified Gaussian distribution, Wikipedia,
https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("K", False, (0, np.inf), (False, False))]
def _rvs(self, K, size=None, random_state=None):
expval = random_state.standard_exponential(size) * K
gval = random_state.standard_normal(size)
return expval + gval
def _pdf(self, x, K):
return np.exp(self._logpdf(x, K))
def _logpdf(self, x, K):
invK = 1.0 / K
exparg = invK * (0.5 * invK - x)
return exparg + _norm_logcdf(x - invK) - np.log(K)
def _cdf(self, x, K):
invK = 1.0 / K
expval = invK * (0.5 * invK - x)
logprod = expval + _norm_logcdf(x - invK)
return _norm_cdf(x) - np.exp(logprod)
def _sf(self, x, K):
invK = 1.0 / K
expval = invK * (0.5 * invK - x)
logprod = expval + _norm_logcdf(x - invK)
return _norm_cdf(-x) + np.exp(logprod)
def _stats(self, K):
K2 = K * K
opK2 = 1.0 + K2
skw = 2 * K**3 * opK2**(-1.5)
krt = 6.0 * K2 * K2 * opK2**(-2)
return K, opK2, skw, krt
exponnorm = exponnorm_gen(name='exponnorm')
def _pow1pm1(x, y):
"""
Compute (1 + x)**y - 1.
Uses expm1 and xlog1py to avoid loss of precision when
(1 + x)**y is close to 1.
Note that the inverse of this function with respect to x is
``_pow1pm1(x, 1/y)``. That is, if
t = _pow1pm1(x, y)
then
x = _pow1pm1(t, 1/y)
"""
return np.expm1(sc.xlog1py(y, x))
class exponweib_gen(rv_continuous):
r"""An exponentiated Weibull continuous random variable.
%(before_notes)s
See Also
--------
weibull_min, numpy.random.Generator.weibull
Notes
-----
The probability density function for `exponweib` is:
.. math::
f(x, a, c) = a c [1-\exp(-x^c)]^{a-1} \exp(-x^c) x^{c-1}
and its cumulative distribution function is:
.. math::
F(x, a, c) = [1-\exp(-x^c)]^a
for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`.
`exponweib` takes :math:`a` and :math:`c` as shape parameters:
* :math:`a` is the exponentiation parameter,
with the special case :math:`a=1` corresponding to the
(non-exponentiated) Weibull distribution `weibull_min`.
* :math:`c` is the shape parameter of the non-exponentiated Weibull law.
%(after_notes)s
References
----------
https://en.wikipedia.org/wiki/Exponentiated_Weibull_distribution
%(example)s
"""
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ic = _ShapeInfo("c", False, (0, np.inf), (False, False))
return [ia, ic]
def _pdf(self, x, a, c):
# exponweib.pdf(x, a, c) =
# a * c * (1-exp(-x**c))**(a-1) * exp(-x**c)*x**(c-1)
return np.exp(self._logpdf(x, a, c))
def _logpdf(self, x, a, c):
negxc = -x**c
exm1c = -sc.expm1(negxc)
logp = (np.log(a) + np.log(c) + sc.xlogy(a - 1.0, exm1c) +
negxc + sc.xlogy(c - 1.0, x))
return logp
def _cdf(self, x, a, c):
exm1c = -sc.expm1(-x**c)
return exm1c**a
def _ppf(self, q, a, c):
return (-sc.log1p(-q**(1.0/a)))**np.asarray(1.0/c)
def _sf(self, x, a, c):
return -_pow1pm1(-np.exp(-x**c), a)
def _isf(self, p, a, c):
return (-np.log(-_pow1pm1(-p, 1/a)))**(1/c)
exponweib = exponweib_gen(a=0.0, name='exponweib')
class exponpow_gen(rv_continuous):
r"""An exponential power continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `exponpow` is:
.. math::
f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b))
for :math:`x \ge 0`, :math:`b > 0`. Note that this is a different
distribution from the exponential power distribution that is also known
under the names "generalized normal" or "generalized Gaussian".
`exponpow` takes ``b`` as a shape parameter for :math:`b`.
%(after_notes)s
References
----------
http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("b", False, (0, np.inf), (False, False))]
def _pdf(self, x, b):
# exponpow.pdf(x, b) = b * x**(b-1) * exp(1 + x**b - exp(x**b))
return np.exp(self._logpdf(x, b))
def _logpdf(self, x, b):
xb = x**b
f = 1 + np.log(b) + sc.xlogy(b - 1.0, x) + xb - np.exp(xb)
return f
def _cdf(self, x, b):
return -sc.expm1(-sc.expm1(x**b))
def _sf(self, x, b):
return np.exp(-sc.expm1(x**b))
def _isf(self, x, b):
return (sc.log1p(-np.log(x)))**(1./b)
def _ppf(self, q, b):
return pow(sc.log1p(-sc.log1p(-q)), 1.0/b)
exponpow = exponpow_gen(a=0.0, name='exponpow')
class fatiguelife_gen(rv_continuous):
r"""A fatigue-life (Birnbaum-Saunders) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `fatiguelife` is:
.. math::
f(x, c) = \frac{x+1}{2c\sqrt{2\pi x^3}} \exp(-\frac{(x-1)^2}{2x c^2})
for :math:`x >= 0` and :math:`c > 0`.
`fatiguelife` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
References
----------
.. [1] "Birnbaum-Saunders distribution",
https://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _rvs(self, c, size=None, random_state=None):
z = random_state.standard_normal(size)
x = 0.5*c*z
x2 = x*x
t = 1.0 + 2*x2 + 2*x*np.sqrt(1 + x2)
return t
def _pdf(self, x, c):
# fatiguelife.pdf(x, c) =
# (x+1) / (2*c*sqrt(2*pi*x**3)) * exp(-(x-1)**2/(2*x*c**2))
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return (np.log(x+1) - (x-1)**2 / (2.0*x*c**2) - np.log(2*c) -
0.5*(np.log(2*np.pi) + 3*np.log(x)))
def _cdf(self, x, c):
return _norm_cdf(1.0 / c * (np.sqrt(x) - 1.0/np.sqrt(x)))
def _ppf(self, q, c):
tmp = c * _norm_ppf(q)
return 0.25 * (tmp + np.sqrt(tmp**2 + 4))**2
def _sf(self, x, c):
return _norm_sf(1.0 / c * (np.sqrt(x) - 1.0/np.sqrt(x)))
def _isf(self, q, c):
tmp = -c * _norm_ppf(q)
return 0.25 * (tmp + np.sqrt(tmp**2 + 4))**2
def _stats(self, c):
# NB: the formula for kurtosis in wikipedia seems to have an error:
# it's 40, not 41. At least it disagrees with the one from Wolfram
# Alpha. And the latter one, below, passes the tests, while the wiki
# one doesn't So far I didn't have the guts to actually check the
# coefficients from the expressions for the raw moments.
c2 = c*c
mu = c2 / 2.0 + 1.0
den = 5.0 * c2 + 4.0
mu2 = c2*den / 4.0
g1 = 4 * c * (11*c2 + 6.0) / np.power(den, 1.5)
g2 = 6 * c2 * (93*c2 + 40.0) / den**2.0
return mu, mu2, g1, g2
fatiguelife = fatiguelife_gen(a=0.0, name='fatiguelife')
class foldcauchy_gen(rv_continuous):
r"""A folded Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `foldcauchy` is:
.. math::
f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)}
for :math:`x \ge 0` and :math:`c \ge 0`.
`foldcauchy` takes ``c`` as a shape parameter for :math:`c`.
%(example)s
"""
def _argcheck(self, c):
return c >= 0
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (True, False))]
def _rvs(self, c, size=None, random_state=None):
return abs(cauchy.rvs(loc=c, size=size,
random_state=random_state))
def _pdf(self, x, c):
# foldcauchy.pdf(x, c) = 1/(pi*(1+(x-c)**2)) + 1/(pi*(1+(x+c)**2))
return 1.0/np.pi*(1.0/(1+(x-c)**2) + 1.0/(1+(x+c)**2))
def _cdf(self, x, c):
return 1.0/np.pi*(np.arctan(x-c) + np.arctan(x+c))
def _sf(self, x, c):
# 1 - CDF(x, c) = 1 - (atan(x - c) + atan(x + c))/pi
# = ((pi/2 - atan(x - c)) + (pi/2 - atan(x + c)))/pi
# = (acot(x - c) + acot(x + c))/pi
# = (atan2(1, x - c) + atan2(1, x + c))/pi
return (np.arctan2(1, x - c) + np.arctan2(1, x + c))/np.pi
def _stats(self, c):
return np.inf, np.inf, np.nan, np.nan
foldcauchy = foldcauchy_gen(a=0.0, name='foldcauchy')
class f_gen(rv_continuous):
r"""An F continuous random variable.
For the noncentral F distribution, see `ncf`.
%(before_notes)s
See Also
--------
ncf
Notes
-----
The F distribution with :math:`df_1 > 0` and :math:`df_2 > 0` degrees of freedom is
the distribution of the ratio of two independent chi-squared distributions with
:math:`df_1` and :math:`df_2` degrees of freedom, after rescaling by
:math:`df_2 / df_1`.
The probability density function for `f` is:
.. math::
f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}}
{(df_2+df_1 x)^{(df_1+df_2)/2}
B(df_1/2, df_2/2)}
for :math:`x > 0`.
`f` accepts shape parameters ``dfn`` and ``dfd`` for :math:`df_1`, the degrees of
freedom of the chi-squared distribution in the numerator, and :math:`df_2`, the
degrees of freedom of the chi-squared distribution in the denominator, respectively.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
idfn = _ShapeInfo("dfn", False, (0, np.inf), (False, False))
idfd = _ShapeInfo("dfd", False, (0, np.inf), (False, False))
return [idfn, idfd]
def _rvs(self, dfn, dfd, size=None, random_state=None):
return random_state.f(dfn, dfd, size)
def _pdf(self, x, dfn, dfd):
# df2**(df2/2) * df1**(df1/2) * x**(df1/2-1)
# F.pdf(x, df1, df2) = --------------------------------------------
# (df2+df1*x)**((df1+df2)/2) * B(df1/2, df2/2)
return np.exp(self._logpdf(x, dfn, dfd))
def _logpdf(self, x, dfn, dfd):
n = 1.0 * dfn
m = 1.0 * dfd
lPx = (m/2 * np.log(m) + n/2 * np.log(n) + sc.xlogy(n/2 - 1, x)
- (((n+m)/2) * np.log(m + n*x) + sc.betaln(n/2, m/2)))
return lPx
def _cdf(self, x, dfn, dfd):
return sc.fdtr(dfn, dfd, x)
def _sf(self, x, dfn, dfd):
return sc.fdtrc(dfn, dfd, x)
def _ppf(self, q, dfn, dfd):
return sc.fdtri(dfn, dfd, q)
def _stats(self, dfn, dfd):
v1, v2 = 1. * dfn, 1. * dfd
v2_2, v2_4, v2_6, v2_8 = v2 - 2., v2 - 4., v2 - 6., v2 - 8.
mu = _lazywhere(
v2 > 2, (v2, v2_2),
lambda v2, v2_2: v2 / v2_2,
np.inf)
mu2 = _lazywhere(
v2 > 4, (v1, v2, v2_2, v2_4),
lambda v1, v2, v2_2, v2_4:
2 * v2 * v2 * (v1 + v2_2) / (v1 * v2_2**2 * v2_4),
np.inf)
g1 = _lazywhere(
v2 > 6, (v1, v2_2, v2_4, v2_6),
lambda v1, v2_2, v2_4, v2_6:
(2 * v1 + v2_2) / v2_6 * np.sqrt(v2_4 / (v1 * (v1 + v2_2))),
np.nan)
g1 *= np.sqrt(8.)
g2 = _lazywhere(
v2 > 8, (g1, v2_6, v2_8),
lambda g1, v2_6, v2_8: (8 + g1 * g1 * v2_6) / v2_8,
np.nan)
g2 *= 3. / 2.
return mu, mu2, g1, g2
def _entropy(self, dfn, dfd):
# the formula found in literature is incorrect. This one yields the
# same result as numerical integration using the generic entropy
# definition. This is also tested in tests/test_conntinous_basic
half_dfn = 0.5 * dfn
half_dfd = 0.5 * dfd
half_sum = 0.5 * (dfn + dfd)
return (np.log(dfd) - np.log(dfn) + sc.betaln(half_dfn, half_dfd) +
(1 - half_dfn) * sc.psi(half_dfn) - (1 + half_dfd) *
sc.psi(half_dfd) + half_sum * sc.psi(half_sum))
f = f_gen(a=0.0, name='f')
## Folded Normal
## abs(Z) where (Z is normal with mu=L and std=S so that c=abs(L)/S)
##
## note: regress docs have scale parameter correct, but first parameter
## he gives is a shape parameter A = c * scale
## Half-normal is folded normal with shape-parameter c=0.
class foldnorm_gen(rv_continuous):
r"""A folded normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `foldnorm` is:
.. math::
f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2})
for :math:`x \ge 0` and :math:`c \ge 0`.
`foldnorm` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
return c >= 0
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (True, False))]
def _rvs(self, c, size=None, random_state=None):
return abs(random_state.standard_normal(size) + c)
def _pdf(self, x, c):
# foldnormal.pdf(x, c) = sqrt(2/pi) * cosh(c*x) * exp(-(x**2+c**2)/2)
return _norm_pdf(x + c) + _norm_pdf(x-c)
def _cdf(self, x, c):
sqrt_two = np.sqrt(2)
return 0.5 * (sc.erf((x - c)/sqrt_two) + sc.erf((x + c)/sqrt_two))
def _sf(self, x, c):
return _norm_sf(x - c) + _norm_sf(x + c)
def _stats(self, c):
# Regina C. Elandt, Technometrics 3, 551 (1961)
# https://www.jstor.org/stable/1266561
#
c2 = c*c
expfac = np.exp(-0.5*c2) / np.sqrt(2.*np.pi)
mu = 2.*expfac + c * sc.erf(c/np.sqrt(2))
mu2 = c2 + 1 - mu*mu
g1 = 2. * (mu*mu*mu - c2*mu - expfac)
g1 /= np.power(mu2, 1.5)
g2 = c2 * (c2 + 6.) + 3 + 8.*expfac*mu
g2 += (2. * (c2 - 3.) - 3. * mu**2) * mu**2
g2 = g2 / mu2**2.0 - 3.
return mu, mu2, g1, g2
foldnorm = foldnorm_gen(a=0.0, name='foldnorm')
class weibull_min_gen(rv_continuous):
r"""Weibull minimum continuous random variable.
The Weibull Minimum Extreme Value distribution, from extreme value theory
(Fisher-Gnedenko theorem), is also often simply called the Weibull
distribution. It arises as the limiting distribution of the rescaled
minimum of iid random variables.
%(before_notes)s
See Also
--------
weibull_max, numpy.random.Generator.weibull, exponweib
Notes
-----
The probability density function for `weibull_min` is:
.. math::
f(x, c) = c x^{c-1} \exp(-x^c)
for :math:`x > 0`, :math:`c > 0`.
`weibull_min` takes ``c`` as a shape parameter for :math:`c`.
(named :math:`k` in Wikipedia article and :math:`a` in
``numpy.random.weibull``). Special shape values are :math:`c=1` and
:math:`c=2` where Weibull distribution reduces to the `expon` and
`rayleigh` distributions respectively.
Suppose ``X`` is an exponentially distributed random variable with
scale ``s``. Then ``Y = X**k`` is `weibull_min` distributed with shape
``c = 1/k`` and scale ``s**k``.
%(after_notes)s
References
----------
https://en.wikipedia.org/wiki/Weibull_distribution
https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# weibull_min.pdf(x, c) = c * x**(c-1) * exp(-x**c)
return c*pow(x, c-1)*np.exp(-pow(x, c))
def _logpdf(self, x, c):
return np.log(c) + sc.xlogy(c - 1, x) - pow(x, c)
def _cdf(self, x, c):
return -sc.expm1(-pow(x, c))
def _ppf(self, q, c):
return pow(-sc.log1p(-q), 1.0/c)
def _sf(self, x, c):
return np.exp(self._logsf(x, c))
def _logsf(self, x, c):
return -pow(x, c)
def _isf(self, q, c):
return (-np.log(q))**(1/c)
def _munp(self, n, c):
return sc.gamma(1.0+n*1.0/c)
def _entropy(self, c):
return -_EULER / c - np.log(c) + _EULER + 1
@extend_notes_in_docstring(rv_continuous, notes="""\
If ``method='mm'``, parameters fixed by the user are respected, and the
remaining parameters are used to match distribution and sample moments
where possible. For example, if the user fixes the location with
``floc``, the parameters will only match the distribution skewness and
variance to the sample skewness and variance; no attempt will be made
to match the means or minimize a norm of the errors.
\n\n""")
def fit(self, data, *args, **kwds):
if isinstance(data, CensoredData):
if data.num_censored() == 0:
data = data._uncensor()
else:
return super().fit(data, *args, **kwds)
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
# this extracts fixed shape, location, and scale however they
# are specified, and also leaves them in `kwds`
data, fc, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
method = kwds.get("method", "mle").lower()
# See https://en.wikipedia.org/wiki/Weibull_distribution#Moments for
# moment formulas.
def skew(c):
gamma1 = sc.gamma(1+1/c)
gamma2 = sc.gamma(1+2/c)
gamma3 = sc.gamma(1+3/c)
num = 2 * gamma1**3 - 3*gamma1*gamma2 + gamma3
den = (gamma2 - gamma1**2)**(3/2)
return num/den
# For c in [1e2, 3e4], population skewness appears to approach
# asymptote near -1.139, but past c > 3e4, skewness begins to vary
# wildly, and MoM won't provide a good guess. Get out early.
s = stats.skew(data)
max_c = 1e4
s_min = skew(max_c)
if s < s_min and method != "mm" and fc is None and not args:
return super().fit(data, *args, **kwds)
# If method is method of moments, we don't need the user's guesses.
# Otherwise, extract the guesses from args and kwds.
if method == "mm":
c, loc, scale = None, None, None
else:
c = args[0] if len(args) else None
loc = kwds.pop('loc', None)
scale = kwds.pop('scale', None)
if fc is None and c is None: # not fixed and no guess: use MoM
# Solve for c that matches sample distribution skewness to sample
# skewness.
# we start having numerical issues with `weibull_min` with
# parameters outside this range - and not just in this method.
# We could probably improve the situation by doing everything
# in the log space, but that is for another time.
c = root_scalar(lambda c: skew(c) - s, bracket=[0.02, max_c],
method='bisect').root
elif fc is not None: # fixed: use it
c = fc
if fscale is None and scale is None:
v = np.var(data)
scale = np.sqrt(v / (sc.gamma(1+2/c) - sc.gamma(1+1/c)**2))
elif fscale is not None:
scale = fscale
if floc is None and loc is None:
m = np.mean(data)
loc = m - scale*sc.gamma(1 + 1/c)
elif floc is not None:
loc = floc
if method == 'mm':
return c, loc, scale
else:
# At this point, parameter "guesses" may equal the fixed parameters
# in kwds. No harm in passing them as guesses, too.
return super().fit(data, c, loc=loc, scale=scale, **kwds)
weibull_min = weibull_min_gen(a=0.0, name='weibull_min')
class truncweibull_min_gen(rv_continuous):
r"""A doubly truncated Weibull minimum continuous random variable.
%(before_notes)s
See Also
--------
weibull_min, truncexpon
Notes
-----
The probability density function for `truncweibull_min` is:
.. math::
f(x, a, b, c) = \frac{c x^{c-1} \exp(-x^c)}{\exp(-a^c) - \exp(-b^c)}
for :math:`a < x <= b`, :math:`0 \le a < b` and :math:`c > 0`.
`truncweibull_min` takes :math:`a`, :math:`b`, and :math:`c` as shape
parameters.
Notice that the truncation values, :math:`a` and :math:`b`, are defined in
standardized form:
.. math::
a = (u_l - loc)/scale
b = (u_r - loc)/scale
where :math:`u_l` and :math:`u_r` are the specific left and right
truncation values, respectively. In other words, the support of the
distribution becomes :math:`(a*scale + loc) < x <= (b*scale + loc)` when
:math:`loc` and/or :math:`scale` are provided.
%(after_notes)s
References
----------
.. [1] Rinne, H. "The Weibull Distribution: A Handbook". CRC Press (2009).
%(example)s
"""
def _argcheck(self, c, a, b):
return (a >= 0.) & (b > a) & (c > 0.)
def _shape_info(self):
ic = _ShapeInfo("c", False, (0, np.inf), (False, False))
ia = _ShapeInfo("a", False, (0, np.inf), (True, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
return [ic, ia, ib]
def _fitstart(self, data):
# Arbitrary, but default a=b=c=1 is not valid
return super()._fitstart(data, args=(1, 0, 1))
def _get_support(self, c, a, b):
return a, b
def _pdf(self, x, c, a, b):
denum = (np.exp(-pow(a, c)) - np.exp(-pow(b, c)))
return (c * pow(x, c-1) * np.exp(-pow(x, c))) / denum
def _logpdf(self, x, c, a, b):
logdenum = np.log(np.exp(-pow(a, c)) - np.exp(-pow(b, c)))
return np.log(c) + sc.xlogy(c - 1, x) - pow(x, c) - logdenum
def _cdf(self, x, c, a, b):
num = (np.exp(-pow(a, c)) - np.exp(-pow(x, c)))
denum = (np.exp(-pow(a, c)) - np.exp(-pow(b, c)))
return num / denum
def _logcdf(self, x, c, a, b):
lognum = np.log(np.exp(-pow(a, c)) - np.exp(-pow(x, c)))
logdenum = np.log(np.exp(-pow(a, c)) - np.exp(-pow(b, c)))
return lognum - logdenum
def _sf(self, x, c, a, b):
num = (np.exp(-pow(x, c)) - np.exp(-pow(b, c)))
denum = (np.exp(-pow(a, c)) - np.exp(-pow(b, c)))
return num / denum
def _logsf(self, x, c, a, b):
lognum = np.log(np.exp(-pow(x, c)) - np.exp(-pow(b, c)))
logdenum = np.log(np.exp(-pow(a, c)) - np.exp(-pow(b, c)))
return lognum - logdenum
def _isf(self, q, c, a, b):
return pow(
-np.log((1 - q) * np.exp(-pow(b, c)) + q * np.exp(-pow(a, c))), 1/c
)
def _ppf(self, q, c, a, b):
return pow(
-np.log((1 - q) * np.exp(-pow(a, c)) + q * np.exp(-pow(b, c))), 1/c
)
def _munp(self, n, c, a, b):
gamma_fun = sc.gamma(n/c + 1.) * (
sc.gammainc(n/c + 1., pow(b, c)) - sc.gammainc(n/c + 1., pow(a, c))
)
denum = (np.exp(-pow(a, c)) - np.exp(-pow(b, c)))
return gamma_fun / denum
truncweibull_min = truncweibull_min_gen(name='truncweibull_min')
class weibull_max_gen(rv_continuous):
r"""Weibull maximum continuous random variable.
The Weibull Maximum Extreme Value distribution, from extreme value theory
(Fisher-Gnedenko theorem), is the limiting distribution of rescaled
maximum of iid random variables. This is the distribution of -X
if X is from the `weibull_min` function.
%(before_notes)s
See Also
--------
weibull_min
Notes
-----
The probability density function for `weibull_max` is:
.. math::
f(x, c) = c (-x)^{c-1} \exp(-(-x)^c)
for :math:`x < 0`, :math:`c > 0`.
`weibull_max` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
References
----------
https://en.wikipedia.org/wiki/Weibull_distribution
https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# weibull_max.pdf(x, c) = c * (-x)**(c-1) * exp(-(-x)**c)
return c*pow(-x, c-1)*np.exp(-pow(-x, c))
def _logpdf(self, x, c):
return np.log(c) + sc.xlogy(c-1, -x) - pow(-x, c)
def _cdf(self, x, c):
return np.exp(-pow(-x, c))
def _logcdf(self, x, c):
return -pow(-x, c)
def _sf(self, x, c):
return -sc.expm1(-pow(-x, c))
def _ppf(self, q, c):
return -pow(-np.log(q), 1.0/c)
def _munp(self, n, c):
val = sc.gamma(1.0+n*1.0/c)
if int(n) % 2:
sgn = -1
else:
sgn = 1
return sgn * val
def _entropy(self, c):
return -_EULER / c - np.log(c) + _EULER + 1
weibull_max = weibull_max_gen(b=0.0, name='weibull_max')
class genlogistic_gen(rv_continuous):
r"""A generalized logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genlogistic` is:
.. math::
f(x, c) = c \frac{\exp(-x)}
{(1 + \exp(-x))^{c+1}}
for real :math:`x` and :math:`c > 0`. In literature, different
generalizations of the logistic distribution can be found. This is the type 1
generalized logistic distribution according to [1]_. It is also referred to
as the skew-logistic distribution [2]_.
`genlogistic` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
References
----------
.. [1] Johnson et al. "Continuous Univariate Distributions", Volume 2,
Wiley. 1995.
.. [2] "Generalized Logistic Distribution", Wikipedia,
https://en.wikipedia.org/wiki/Generalized_logistic_distribution
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# genlogistic.pdf(x, c) = c * exp(-x) / (1 + exp(-x))**(c+1)
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
# Two mathematically equivalent expressions for log(pdf(x, c)):
# log(pdf(x, c)) = log(c) - x - (c + 1)*log(1 + exp(-x))
# = log(c) + c*x - (c + 1)*log(1 + exp(x))
mult = -(c - 1) * (x < 0) - 1
absx = np.abs(x)
return np.log(c) + mult*absx - (c+1) * sc.log1p(np.exp(-absx))
def _cdf(self, x, c):
Cx = (1+np.exp(-x))**(-c)
return Cx
def _logcdf(self, x, c):
return -c * np.log1p(np.exp(-x))
def _ppf(self, q, c):
return -np.log(sc.powm1(q, -1.0/c))
def _sf(self, x, c):
return -sc.expm1(self._logcdf(x, c))
def _isf(self, q, c):
return self._ppf(1 - q, c)
def _stats(self, c):
mu = _EULER + sc.psi(c)
mu2 = np.pi*np.pi/6.0 + sc.zeta(2, c)
g1 = -2*sc.zeta(3, c) + 2*_ZETA3
g1 /= np.power(mu2, 1.5)
g2 = np.pi**4/15.0 + 6*sc.zeta(4, c)
g2 /= mu2**2.0
return mu, mu2, g1, g2
def _entropy(self, c):
return _lazywhere(c < 8e6, (c, ),
lambda c: -np.log(c) + sc.psi(c + 1) + _EULER + 1,
# asymptotic expansion: psi(c) ~ log(c) - 1/(2 * c)
# a = -log(c) + psi(c + 1)
# = -log(c) + psi(c) + 1/c
# ~ -log(c) + log(c) - 1/(2 * c) + 1/c
# = 1/(2 * c)
f2=lambda c: 1/(2 * c) + _EULER + 1)
genlogistic = genlogistic_gen(name='genlogistic')
class genpareto_gen(rv_continuous):
r"""A generalized Pareto continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genpareto` is:
.. math::
f(x, c) = (1 + c x)^{-1 - 1/c}
defined for :math:`x \ge 0` if :math:`c \ge 0`, and for
:math:`0 \le x \le -1/c` if :math:`c < 0`.
`genpareto` takes ``c`` as a shape parameter for :math:`c`.
For :math:`c=0`, `genpareto` reduces to the exponential
distribution, `expon`:
.. math::
f(x, 0) = \exp(-x)
For :math:`c=-1`, `genpareto` is uniform on ``[0, 1]``:
.. math::
f(x, -1) = 1
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
return np.isfinite(c)
def _shape_info(self):
return [_ShapeInfo("c", False, (-np.inf, np.inf), (False, False))]
def _get_support(self, c):
c = np.asarray(c)
b = _lazywhere(c < 0, (c,),
lambda c: -1. / c,
np.inf)
a = np.where(c >= 0, self.a, self.a)
return a, b
def _pdf(self, x, c):
# genpareto.pdf(x, c) = (1 + c * x)**(-1 - 1/c)
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -sc.xlog1py(c + 1., c*x) / c,
-x)
def _cdf(self, x, c):
return -sc.inv_boxcox1p(-x, -c)
def _sf(self, x, c):
return sc.inv_boxcox(-x, -c)
def _logsf(self, x, c):
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -sc.log1p(c*x) / c,
-x)
def _ppf(self, q, c):
return -sc.boxcox1p(-q, -c)
def _isf(self, q, c):
return -sc.boxcox(q, -c)
def _stats(self, c, moments='mv'):
if 'm' not in moments:
m = None
else:
m = _lazywhere(c < 1, (c,),
lambda xi: 1/(1 - xi),
np.inf)
if 'v' not in moments:
v = None
else:
v = _lazywhere(c < 1/2, (c,),
lambda xi: 1 / (1 - xi)**2 / (1 - 2*xi),
np.nan)
if 's' not in moments:
s = None
else:
s = _lazywhere(c < 1/3, (c,),
lambda xi: (2 * (1 + xi) * np.sqrt(1 - 2*xi) /
(1 - 3*xi)),
np.nan)
if 'k' not in moments:
k = None
else:
k = _lazywhere(c < 1/4, (c,),
lambda xi: (3 * (1 - 2*xi) * (2*xi**2 + xi + 3) /
(1 - 3*xi) / (1 - 4*xi) - 3),
np.nan)
return m, v, s, k
def _munp(self, n, c):
def __munp(n, c):
val = 0.0
k = np.arange(0, n + 1)
for ki, cnk in zip(k, sc.comb(n, k)):
val = val + cnk * (-1) ** ki / (1.0 - c * ki)
return np.where(c * n < 1, val * (-1.0 / c) ** n, np.inf)
return _lazywhere(c != 0, (c,),
lambda c: __munp(n, c),
sc.gamma(n + 1))
def _entropy(self, c):
return 1. + c
genpareto = genpareto_gen(a=0.0, name='genpareto')
class genexpon_gen(rv_continuous):
r"""A generalized exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genexpon` is:
.. math::
f(x, a, b, c) = (a + b (1 - \exp(-c x)))
\exp(-a x - b x + \frac{b}{c} (1-\exp(-c x)))
for :math:`x \ge 0`, :math:`a, b, c > 0`.
`genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters.
%(after_notes)s
References
----------
H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential
Distribution", Journal of the American Statistical Association, 1993.
N. Balakrishnan, Asit P. Basu (editors), *The Exponential Distribution:
Theory, Methods and Applications*, Gordon and Breach, 1995.
ISBN 10: 2884491929
%(example)s
"""
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
ic = _ShapeInfo("c", False, (0, np.inf), (False, False))
return [ia, ib, ic]
def _pdf(self, x, a, b, c):
# genexpon.pdf(x, a, b, c) = (a + b * (1 - exp(-c*x))) * \
# exp(-a*x - b*x + b/c * (1-exp(-c*x)))
return (a + b*(-sc.expm1(-c*x)))*np.exp((-a-b)*x +
b*(-sc.expm1(-c*x))/c)
def _logpdf(self, x, a, b, c):
return np.log(a+b*(-sc.expm1(-c*x))) + (-a-b)*x+b*(-sc.expm1(-c*x))/c
def _cdf(self, x, a, b, c):
return -sc.expm1((-a-b)*x + b*(-sc.expm1(-c*x))/c)
def _ppf(self, p, a, b, c):
s = a + b
t = (b - c*np.log1p(-p))/s
return (t + sc.lambertw(-b/s * np.exp(-t)).real)/c
def _sf(self, x, a, b, c):
return np.exp((-a-b)*x + b*(-sc.expm1(-c*x))/c)
def _isf(self, p, a, b, c):
s = a + b
t = (b - c*np.log(p))/s
return (t + sc.lambertw(-b/s * np.exp(-t)).real)/c
genexpon = genexpon_gen(a=0.0, name='genexpon')
class genextreme_gen(rv_continuous):
r"""A generalized extreme value continuous random variable.
%(before_notes)s
See Also
--------
gumbel_r
Notes
-----
For :math:`c=0`, `genextreme` is equal to `gumbel_r` with
probability density function
.. math::
f(x) = \exp(-\exp(-x)) \exp(-x),
where :math:`-\infty < x < \infty`.
For :math:`c \ne 0`, the probability density function for `genextreme` is:
.. math::
f(x, c) = \exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1},
where :math:`-\infty < x \le 1/c` if :math:`c > 0` and
:math:`1/c \le x < \infty` if :math:`c < 0`.
Note that several sources and software packages use the opposite
convention for the sign of the shape parameter :math:`c`.
`genextreme` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
return np.isfinite(c)
def _shape_info(self):
return [_ShapeInfo("c", False, (-np.inf, np.inf), (False, False))]
def _get_support(self, c):
_b = np.where(c > 0, 1.0 / np.maximum(c, _XMIN), np.inf)
_a = np.where(c < 0, 1.0 / np.minimum(c, -_XMIN), -np.inf)
return _a, _b
def _loglogcdf(self, x, c):
# Returns log(-log(cdf(x, c)))
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: sc.log1p(-c*x)/c, -x)
def _pdf(self, x, c):
# genextreme.pdf(x, c) =
# exp(-exp(-x))*exp(-x), for c==0
# exp(-(1-c*x)**(1/c))*(1-c*x)**(1/c-1), for x \le 1/c, c > 0
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
cx = _lazywhere((x == x) & (c != 0), (x, c), lambda x, c: c*x, 0.0)
logex2 = sc.log1p(-cx)
logpex2 = self._loglogcdf(x, c)
pex2 = np.exp(logpex2)
# Handle special cases
np.putmask(logpex2, (c == 0) & (x == -np.inf), 0.0)
logpdf = _lazywhere(~((cx == 1) | (cx == -np.inf)),
(pex2, logpex2, logex2),
lambda pex2, lpex2, lex2: -pex2 + lpex2 - lex2,
fillvalue=-np.inf)
np.putmask(logpdf, (c == 1) & (x == 1), 0.0)
return logpdf
def _logcdf(self, x, c):
return -np.exp(self._loglogcdf(x, c))
def _cdf(self, x, c):
return np.exp(self._logcdf(x, c))
def _sf(self, x, c):
return -sc.expm1(self._logcdf(x, c))
def _ppf(self, q, c):
x = -np.log(-np.log(q))
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -sc.expm1(-c * x) / c, x)
def _isf(self, q, c):
x = -np.log(-sc.log1p(-q))
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -sc.expm1(-c * x) / c, x)
def _stats(self, c):
def g(n):
return sc.gamma(n * c + 1)
g1 = g(1)
g2 = g(2)
g3 = g(3)
g4 = g(4)
g2mg12 = np.where(abs(c) < 1e-7, (c*np.pi)**2.0/6.0, g2-g1**2.0)
gam2k = np.where(abs(c) < 1e-7, np.pi**2.0/6.0,
sc.expm1(sc.gammaln(2.0*c+1.0)-2*sc.gammaln(c + 1.0))/c**2.0)
eps = 1e-14
gamk = np.where(abs(c) < eps, -_EULER, sc.expm1(sc.gammaln(c + 1))/c)
m = np.where(c < -1.0, np.nan, -gamk)
v = np.where(c < -0.5, np.nan, g1**2.0*gam2k)
# skewness
sk1 = _lazywhere(c >= -1./3,
(c, g1, g2, g3, g2mg12),
lambda c, g1, g2, g3, g2mg12:
np.sign(c)*(-g3 + (g2 + 2*g2mg12)*g1)/g2mg12**1.5,
fillvalue=np.nan)
sk = np.where(abs(c) <= eps**0.29, 12*np.sqrt(6)*_ZETA3/np.pi**3, sk1)
# kurtosis
ku1 = _lazywhere(c >= -1./4,
(g1, g2, g3, g4, g2mg12),
lambda g1, g2, g3, g4, g2mg12:
(g4 + (-4*g3 + 3*(g2 + g2mg12)*g1)*g1)/g2mg12**2,
fillvalue=np.nan)
ku = np.where(abs(c) <= (eps)**0.23, 12.0/5.0, ku1-3.0)
return m, v, sk, ku
def _fitstart(self, data):
if isinstance(data, CensoredData):
data = data._uncensor()
# This is better than the default shape of (1,).
g = _skew(data)
if g < 0:
a = 0.5
else:
a = -0.5
return super()._fitstart(data, args=(a,))
def _munp(self, n, c):
k = np.arange(0, n+1)
vals = 1.0/c**n * np.sum(
sc.comb(n, k) * (-1)**k * sc.gamma(c*k + 1),
axis=0)
return np.where(c*n > -1, vals, np.inf)
def _entropy(self, c):
return _EULER*(1 - c) + 1
genextreme = genextreme_gen(name='genextreme')
def _digammainv(y):
"""Inverse of the digamma function (real positive arguments only).
This function is used in the `fit` method of `gamma_gen`.
The function uses either optimize.fsolve or optimize.newton
to solve `sc.digamma(x) - y = 0`. There is probably room for
improvement, but currently it works over a wide range of y:
>>> import numpy as np
>>> rng = np.random.default_rng()
>>> y = 64*rng.standard_normal(1000000)
>>> y.min(), y.max()
(-311.43592651416662, 351.77388222276869)
>>> x = [_digammainv(t) for t in y]
>>> np.abs(sc.digamma(x) - y).max()
1.1368683772161603e-13
"""
_em = 0.5772156649015328606065120
def func(x):
return sc.digamma(x) - y
if y > -0.125:
x0 = np.exp(y) + 0.5
if y < 10:
# Some experimentation shows that newton reliably converges
# must faster than fsolve in this y range. For larger y,
# newton sometimes fails to converge.
value = optimize.newton(func, x0, tol=1e-10)
return value
elif y > -3:
x0 = np.exp(y/2.332) + 0.08661
else:
x0 = 1.0 / (-y - _em)
value, info, ier, mesg = optimize.fsolve(func, x0, xtol=1e-11,
full_output=True)
if ier != 1:
raise RuntimeError("_digammainv: fsolve failed, y = %r" % y)
return value[0]
## Gamma (Use MATLAB and MATHEMATICA (b=theta=scale, a=alpha=shape) definition)
## gamma(a, loc, scale) with a an integer is the Erlang distribution
## gamma(1, loc, scale) is the Exponential distribution
## gamma(df/2, 0, 2) is the chi2 distribution with df degrees of freedom.
class gamma_gen(rv_continuous):
r"""A gamma continuous random variable.
%(before_notes)s
See Also
--------
erlang, expon
Notes
-----
The probability density function for `gamma` is:
.. math::
f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)}
for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the
gamma function.
`gamma` takes ``a`` as a shape parameter for :math:`a`.
When :math:`a` is an integer, `gamma` reduces to the Erlang
distribution, and when :math:`a=1` to the exponential distribution.
Gamma distributions are sometimes parameterized with two variables,
with a probability density function of:
.. math::
f(x, \alpha, \beta) =
\frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x }}{\Gamma(\alpha)}
Note that this parameterization is equivalent to the above, with
``scale = 1 / beta``.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("a", False, (0, np.inf), (False, False))]
def _rvs(self, a, size=None, random_state=None):
return random_state.standard_gamma(a, size)
def _pdf(self, x, a):
# gamma.pdf(x, a) = x**(a-1) * exp(-x) / gamma(a)
return np.exp(self._logpdf(x, a))
def _logpdf(self, x, a):
return sc.xlogy(a-1.0, x) - x - sc.gammaln(a)
def _cdf(self, x, a):
return sc.gammainc(a, x)
def _sf(self, x, a):
return sc.gammaincc(a, x)
def _ppf(self, q, a):
return sc.gammaincinv(a, q)
def _isf(self, q, a):
return sc.gammainccinv(a, q)
def _stats(self, a):
return a, a, 2.0/np.sqrt(a), 6.0/a
def _entropy(self, a):
def regular_formula(a):
return sc.psi(a) * (1-a) + a + sc.gammaln(a)
def asymptotic_formula(a):
# plug in above formula the expansions:
# psi(a) ~ ln(a) - 1/2a - 1/12a^2 + 1/120a^4
# gammaln(a) ~ a * ln(a) - a - 1/2 * ln(a) + 1/2 ln(2 * pi) +
# 1/12a - 1/360a^3
return (0.5 * (1. + np.log(2*np.pi) + np.log(a)) - 1/(3 * a)
- (a**-2.)/12 - (a**-3.)/90 + (a**-4.)/120)
return _lazywhere(a < 250, (a, ), regular_formula,
f2=asymptotic_formula)
def _fitstart(self, data):
# The skewness of the gamma distribution is `2 / np.sqrt(a)`.
# We invert that to estimate the shape `a` using the skewness
# of the data. The formula is regularized with 1e-8 in the
# denominator to allow for degenerate data where the skewness
# is close to 0.
if isinstance(data, CensoredData):
data = data._uncensor()
sk = _skew(data)
a = 4 / (1e-8 + sk**2)
return super()._fitstart(data, args=(a,))
@extend_notes_in_docstring(rv_continuous, notes="""\
When the location is fixed by using the argument `floc`
and `method='MLE'`, this
function uses explicit formulas or solves a simpler numerical
problem than the full ML optimization problem. So in that case,
the `optimizer`, `loc` and `scale` arguments are ignored.
\n\n""")
def fit(self, data, *args, **kwds):
floc = kwds.get('floc', None)
method = kwds.get('method', 'mle')
if (isinstance(data, CensoredData) or
floc is None and method.lower() != 'mm'):
# loc is not fixed or we're not doing standard MLE.
# Use the default fit method.
return super().fit(data, *args, **kwds)
# We already have this value, so just pop it from kwds.
kwds.pop('floc', None)
f0 = _get_fixed_fit_value(kwds, ['f0', 'fa', 'fix_a'])
fscale = kwds.pop('fscale', None)
_remove_optimizer_parameters(kwds)
if f0 is not None and floc is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
# Without this check, this function would just return the
# parameters that were given.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
# Fixed location is handled by shifting the data.
data = np.asarray(data)
if not np.isfinite(data).all():
raise ValueError("The data contains non-finite values.")
# Use explicit formulas for mm (gh-19884)
if method.lower() == 'mm':
m1 = np.mean(data)
m2 = np.var(data)
m3 = np.mean((data - m1) ** 3)
a, loc, scale = f0, floc, fscale
# Three unknowns
if a is None and loc is None and scale is None:
scale = m3 / (2 * m2)
# Two unknowns
if loc is None and scale is None:
scale = np.sqrt(m2 / a)
if a is None and scale is None:
scale = m2 / (m1 - loc)
if a is None and loc is None:
a = m2 / (scale ** 2)
# One unknown
if a is None:
a = (m1 - loc) / scale
if loc is None:
loc = m1 - a * scale
if scale is None:
scale = (m1 - loc) / a
return a, loc, scale
# Special case: loc is fixed.
# NB: data == loc is ok if a >= 1; the below check is more strict.
if np.any(data <= floc):
raise FitDataError("gamma", lower=floc, upper=np.inf)
if floc != 0:
# Don't do the subtraction in-place, because `data` might be a
# view of the input array.
data = data - floc
xbar = data.mean()
# Three cases to handle:
# * shape and scale both free
# * shape fixed, scale free
# * shape free, scale fixed
if fscale is None:
# scale is free
if f0 is not None:
# shape is fixed
a = f0
else:
# shape and scale are both free.
# The MLE for the shape parameter `a` is the solution to:
# np.log(a) - sc.digamma(a) - np.log(xbar) +
# np.log(data).mean() = 0
s = np.log(xbar) - np.log(data).mean()
aest = (3-s + np.sqrt((s-3)**2 + 24*s)) / (12*s)
xa = aest*(1-0.4)
xb = aest*(1+0.4)
a = optimize.brentq(lambda a: np.log(a) - sc.digamma(a) - s,
xa, xb, disp=0)
# The MLE for the scale parameter is just the data mean
# divided by the shape parameter.
scale = xbar / a
else:
# scale is fixed, shape is free
# The MLE for the shape parameter `a` is the solution to:
# sc.digamma(a) - np.log(data).mean() + np.log(fscale) = 0
c = np.log(data).mean() - np.log(fscale)
a = _digammainv(c)
scale = fscale
return a, floc, scale
gamma = gamma_gen(a=0.0, name='gamma')
class erlang_gen(gamma_gen):
"""An Erlang continuous random variable.
%(before_notes)s
See Also
--------
gamma
Notes
-----
The Erlang distribution is a special case of the Gamma distribution, with
the shape parameter `a` an integer. Note that this restriction is not
enforced by `erlang`. It will, however, generate a warning the first time
a non-integer value is used for the shape parameter.
Refer to `gamma` for examples.
"""
def _argcheck(self, a):
allint = np.all(np.floor(a) == a)
if not allint:
# An Erlang distribution shouldn't really have a non-integer
# shape parameter, so warn the user.
message = ('The shape parameter of the erlang distribution '
f'has been given a non-integer value {a!r}.')
warnings.warn(message, RuntimeWarning, stacklevel=3)
return a > 0
def _shape_info(self):
return [_ShapeInfo("a", True, (1, np.inf), (True, False))]
def _fitstart(self, data):
# Override gamma_gen_fitstart so that an integer initial value is
# used. (Also regularize the division, to avoid issues when
# _skew(data) is 0 or close to 0.)
if isinstance(data, CensoredData):
data = data._uncensor()
a = int(4.0 / (1e-8 + _skew(data)**2))
return super(gamma_gen, self)._fitstart(data, args=(a,))
# Trivial override of the fit method, so we can monkey-patch its
# docstring.
@extend_notes_in_docstring(rv_continuous, notes="""\
The Erlang distribution is generally defined to have integer values
for the shape parameter. This is not enforced by the `erlang` class.
When fitting the distribution, it will generally return a non-integer
value for the shape parameter. By using the keyword argument
`f0=<integer>`, the fit method can be constrained to fit the data to
a specific integer shape parameter.""")
def fit(self, data, *args, **kwds):
return super().fit(data, *args, **kwds)
erlang = erlang_gen(a=0.0, name='erlang')
class gengamma_gen(rv_continuous):
r"""A generalized gamma continuous random variable.
%(before_notes)s
See Also
--------
gamma, invgamma, weibull_min
Notes
-----
The probability density function for `gengamma` is ([1]_):
.. math::
f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)}
for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`.
:math:`\Gamma` is the gamma function (`scipy.special.gamma`).
`gengamma` takes :math:`a` and :math:`c` as shape parameters.
%(after_notes)s
References
----------
.. [1] E.W. Stacy, "A Generalization of the Gamma Distribution",
Annals of Mathematical Statistics, Vol 33(3), pp. 1187--1192.
%(example)s
"""
def _argcheck(self, a, c):
return (a > 0) & (c != 0)
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ic = _ShapeInfo("c", False, (-np.inf, np.inf), (False, False))
return [ia, ic]
def _pdf(self, x, a, c):
return np.exp(self._logpdf(x, a, c))
def _logpdf(self, x, a, c):
return _lazywhere((x != 0) | (c > 0), (x, c),
lambda x, c: (np.log(abs(c)) + sc.xlogy(c*a - 1, x)
- x**c - sc.gammaln(a)),
fillvalue=-np.inf)
def _cdf(self, x, a, c):
xc = x**c
val1 = sc.gammainc(a, xc)
val2 = sc.gammaincc(a, xc)
return np.where(c > 0, val1, val2)
def _rvs(self, a, c, size=None, random_state=None):
r = random_state.standard_gamma(a, size=size)
return r**(1./c)
def _sf(self, x, a, c):
xc = x**c
val1 = sc.gammainc(a, xc)
val2 = sc.gammaincc(a, xc)
return np.where(c > 0, val2, val1)
def _ppf(self, q, a, c):
val1 = sc.gammaincinv(a, q)
val2 = sc.gammainccinv(a, q)
return np.where(c > 0, val1, val2)**(1.0/c)
def _isf(self, q, a, c):
val1 = sc.gammaincinv(a, q)
val2 = sc.gammainccinv(a, q)
return np.where(c > 0, val2, val1)**(1.0/c)
def _munp(self, n, a, c):
# Pochhammer symbol: sc.pocha,n) = gamma(a+n)/gamma(a)
return sc.poch(a, n*1.0/c)
def _entropy(self, a, c):
def regular(a, c):
val = sc.psi(a)
A = a * (1 - val) + val / c
B = sc.gammaln(a) - np.log(abs(c))
h = A + B
return h
def asymptotic(a, c):
# using asymptotic expansions for gammaln and psi (see gh-18093)
return (norm._entropy() - np.log(a)/2
- np.log(np.abs(c)) + (a**-1.)/6 - (a**-3.)/90
+ (np.log(a) - (a**-1.)/2 - (a**-2.)/12 + (a**-4.)/120)/c)
h = _lazywhere(a >= 2e2, (a, c), f=asymptotic, f2=regular)
return h
gengamma = gengamma_gen(a=0.0, name='gengamma')
class genhalflogistic_gen(rv_continuous):
r"""A generalized half-logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genhalflogistic` is:
.. math::
f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2}
for :math:`0 \le x \le 1/c`, and :math:`c > 0`.
`genhalflogistic` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _get_support(self, c):
return self.a, 1.0/c
def _pdf(self, x, c):
# genhalflogistic.pdf(x, c) =
# 2 * (1-c*x)**(1/c-1) / (1+(1-c*x)**(1/c))**2
limit = 1.0/c
tmp = np.asarray(1-c*x)
tmp0 = tmp**(limit-1)
tmp2 = tmp0*tmp
return 2*tmp0 / (1+tmp2)**2
def _cdf(self, x, c):
limit = 1.0/c
tmp = np.asarray(1-c*x)
tmp2 = tmp**(limit)
return (1.0-tmp2) / (1+tmp2)
def _ppf(self, q, c):
return 1.0/c*(1-((1.0-q)/(1.0+q))**c)
def _entropy(self, c):
return 2 - (2*c+1)*np.log(2)
genhalflogistic = genhalflogistic_gen(a=0.0, name='genhalflogistic')
class genhyperbolic_gen(rv_continuous):
r"""A generalized hyperbolic continuous random variable.
%(before_notes)s
See Also
--------
t, norminvgauss, geninvgauss, laplace, cauchy
Notes
-----
The probability density function for `genhyperbolic` is:
.. math::
f(x, p, a, b) =
\frac{(a^2 - b^2)^{p/2}}
{\sqrt{2\pi}a^{p-1/2}
K_p\Big(\sqrt{a^2 - b^2}\Big)}
e^{bx} \times \frac{K_{p - 1/2}
(a \sqrt{1 + x^2})}
{(\sqrt{1 + x^2})^{1/2 - p}}
for :math:`x, p \in ( - \infty; \infty)`,
:math:`|b| < a` if :math:`p \ge 0`,
:math:`|b| \le a` if :math:`p < 0`.
:math:`K_{p}(.)` denotes the modified Bessel function of the second
kind and order :math:`p` (`scipy.special.kv`)
`genhyperbolic` takes ``p`` as a tail parameter,
``a`` as a shape parameter,
``b`` as a skewness parameter.
%(after_notes)s
The original parameterization of the Generalized Hyperbolic Distribution
is found in [1]_ as follows
.. math::
f(x, \lambda, \alpha, \beta, \delta, \mu) =
\frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)}
e^{\beta (x - \mu)} \times \frac{K_{\lambda - 1/2}
(\alpha \sqrt{\delta^2 + (x - \mu)^2})}
{(\sqrt{\delta^2 + (x - \mu)^2} / \alpha)^{1/2 - \lambda}}
for :math:`x \in ( - \infty; \infty)`,
:math:`\gamma := \sqrt{\alpha^2 - \beta^2}`,
:math:`\lambda, \mu \in ( - \infty; \infty)`,
:math:`\delta \ge 0, |\beta| < \alpha` if :math:`\lambda \ge 0`,
:math:`\delta > 0, |\beta| \le \alpha` if :math:`\lambda < 0`.
The location-scale-based parameterization implemented in
SciPy is based on [2]_, where :math:`a = \alpha\delta`,
:math:`b = \beta\delta`, :math:`p = \lambda`,
:math:`scale=\delta` and :math:`loc=\mu`
Moments are implemented based on [3]_ and [4]_.
For the distributions that are a special case such as Student's t,
it is not recommended to rely on the implementation of genhyperbolic.
To avoid potential numerical problems and for performance reasons,
the methods of the specific distributions should be used.
References
----------
.. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions
on Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
pp. 151-157, 1978. https://www.jstor.org/stable/4615705
.. [2] Eberlein E., Prause K. (2002) The Generalized Hyperbolic Model:
Financial Derivatives and Risk Measures. In: Geman H., Madan D.,
Pliska S.R., Vorst T. (eds) Mathematical Finance - Bachelier
Congress 2000. Springer Finance. Springer, Berlin, Heidelberg.
:doi:`10.1007/978-3-662-12429-1_12`
.. [3] Scott, David J, Würtz, Diethelm, Dong, Christine and Tran,
Thanh Tam, (2009), Moments of the generalized hyperbolic
distribution, MPRA Paper, University Library of Munich, Germany,
https://EconPapers.repec.org/RePEc:pra:mprapa:19081.
.. [4] E. Eberlein and E. A. von Hammerstein. Generalized hyperbolic
and inverse Gaussian distributions: Limiting cases and approximation
of processes. FDM Preprint 80, April 2003. University of Freiburg.
https://freidok.uni-freiburg.de/fedora/objects/freidok:7974/datastreams/FILE1/content
%(example)s
"""
def _argcheck(self, p, a, b):
return (np.logical_and(np.abs(b) < a, p >= 0)
| np.logical_and(np.abs(b) <= a, p < 0))
def _shape_info(self):
ip = _ShapeInfo("p", False, (-np.inf, np.inf), (False, False))
ia = _ShapeInfo("a", False, (0, np.inf), (True, False))
ib = _ShapeInfo("b", False, (-np.inf, np.inf), (False, False))
return [ip, ia, ib]
def _fitstart(self, data):
# Arbitrary, but the default p = a = b = 1 is not valid; the
# distribution requires |b| < a if p >= 0.
return super()._fitstart(data, args=(1, 1, 0.5))
def _logpdf(self, x, p, a, b):
# kve instead of kv works better for large values of p
# and smaller values of sqrt(a^2 - b^2)
@np.vectorize
def _logpdf_single(x, p, a, b):
return _stats.genhyperbolic_logpdf(x, p, a, b)
return _logpdf_single(x, p, a, b)
def _pdf(self, x, p, a, b):
# kve instead of kv works better for large values of p
# and smaller values of sqrt(a^2 - b^2)
@np.vectorize
def _pdf_single(x, p, a, b):
return _stats.genhyperbolic_pdf(x, p, a, b)
return _pdf_single(x, p, a, b)
# np.vectorize isn't currently designed to be used as a decorator,
# so use a lambda instead. This allows us to decorate the function
# with `np.vectorize` and still provide the `otypes` parameter.
# The first argument to `vectorize` is `func.__get__(object)` for
# compatibility with Python 3.9. In Python 3.10, this can be
# simplified to just `func`.
@lambda func: np.vectorize(func.__get__(object), otypes=[np.float64])
@staticmethod
def _integrate_pdf(x0, x1, p, a, b):
"""
Integrate the pdf of the genhyberbolic distribution from x0 to x1.
This is a private function used by _cdf() and _sf() only; either x0
will be -inf or x1 will be inf.
"""
user_data = np.array([p, a, b], float).ctypes.data_as(ctypes.c_void_p)
llc = LowLevelCallable.from_cython(_stats, '_genhyperbolic_pdf',
user_data)
d = np.sqrt((a + b)*(a - b))
mean = b/d * sc.kv(p + 1, d) / sc.kv(p, d)
epsrel = 1e-10
epsabs = 0
if x0 < mean < x1:
# If the interval includes the mean, integrate over the two
# intervals [x0, mean] and [mean, x1] and add. If we try to do
# the integral in one call of quad and the non-infinite endpoint
# is far in the tail, quad might return an incorrect result
# because it does not "see" the peak of the PDF.
intgrl = (integrate.quad(llc, x0, mean,
epsrel=epsrel, epsabs=epsabs)[0]
+ integrate.quad(llc, mean, x1,
epsrel=epsrel, epsabs=epsabs)[0])
else:
intgrl = integrate.quad(llc, x0, x1,
epsrel=epsrel, epsabs=epsabs)[0]
if np.isnan(intgrl):
msg = ("Infinite values encountered in scipy.special.kve. "
"Values replaced by NaN to avoid incorrect results.")
warnings.warn(msg, RuntimeWarning, stacklevel=3)
return max(0.0, min(1.0, intgrl))
def _cdf(self, x, p, a, b):
return self._integrate_pdf(-np.inf, x, p, a, b)
def _sf(self, x, p, a, b):
return self._integrate_pdf(x, np.inf, p, a, b)
def _rvs(self, p, a, b, size=None, random_state=None):
# note: X = b * V + sqrt(V) * X has a
# generalized hyperbolic distribution
# if X is standard normal and V is
# geninvgauss(p = p, b = t2, loc = loc, scale = t3)
t1 = np.float_power(a, 2) - np.float_power(b, 2)
# b in the GIG
t2 = np.float_power(t1, 0.5)
# scale in the GIG
t3 = np.float_power(t1, - 0.5)
gig = geninvgauss.rvs(
p=p,
b=t2,
scale=t3,
size=size,
random_state=random_state
)
normst = norm.rvs(size=size, random_state=random_state)
return b * gig + np.sqrt(gig) * normst
def _stats(self, p, a, b):
# https://mpra.ub.uni-muenchen.de/19081/1/MPRA_paper_19081.pdf
# https://freidok.uni-freiburg.de/fedora/objects/freidok:7974/datastreams/FILE1/content
# standardized moments
p, a, b = np.broadcast_arrays(p, a, b)
t1 = np.float_power(a, 2) - np.float_power(b, 2)
t1 = np.float_power(t1, 0.5)
t2 = np.float_power(1, 2) * np.float_power(t1, - 1)
integers = np.linspace(0, 4, 5)
# make integers perpendicular to existing dimensions
integers = integers.reshape(integers.shape + (1,) * p.ndim)
b0, b1, b2, b3, b4 = sc.kv(p + integers, t1)
r1, r2, r3, r4 = (b / b0 for b in (b1, b2, b3, b4))
m = b * t2 * r1
v = (
t2 * r1 + np.float_power(b, 2) * np.float_power(t2, 2) *
(r2 - np.float_power(r1, 2))
)
m3e = (
np.float_power(b, 3) * np.float_power(t2, 3) *
(r3 - 3 * b2 * b1 * np.float_power(b0, -2) +
2 * np.float_power(r1, 3)) +
3 * b * np.float_power(t2, 2) *
(r2 - np.float_power(r1, 2))
)
s = m3e * np.float_power(v, - 3 / 2)
m4e = (
np.float_power(b, 4) * np.float_power(t2, 4) *
(r4 - 4 * b3 * b1 * np.float_power(b0, - 2) +
6 * b2 * np.float_power(b1, 2) * np.float_power(b0, - 3) -
3 * np.float_power(r1, 4)) +
np.float_power(b, 2) * np.float_power(t2, 3) *
(6 * r3 - 12 * b2 * b1 * np.float_power(b0, - 2) +
6 * np.float_power(r1, 3)) +
3 * np.float_power(t2, 2) * r2
)
k = m4e * np.float_power(v, -2) - 3
return m, v, s, k
genhyperbolic = genhyperbolic_gen(name='genhyperbolic')
class gompertz_gen(rv_continuous):
r"""A Gompertz (or truncated Gumbel) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gompertz` is:
.. math::
f(x, c) = c \exp(x) \exp(-c (e^x-1))
for :math:`x \ge 0`, :math:`c > 0`.
`gompertz` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# gompertz.pdf(x, c) = c * exp(x) * exp(-c*(exp(x)-1))
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return np.log(c) + x - c * sc.expm1(x)
def _cdf(self, x, c):
return -sc.expm1(-c * sc.expm1(x))
def _ppf(self, q, c):
return sc.log1p(-1.0 / c * sc.log1p(-q))
def _sf(self, x, c):
return np.exp(-c * sc.expm1(x))
def _isf(self, p, c):
return sc.log1p(-np.log(p)/c)
def _entropy(self, c):
return 1.0 - np.log(c) - sc._ufuncs._scaled_exp1(c)/c
gompertz = gompertz_gen(a=0.0, name='gompertz')
def _average_with_log_weights(x, logweights):
x = np.asarray(x)
logweights = np.asarray(logweights)
maxlogw = logweights.max()
weights = np.exp(logweights - maxlogw)
return np.average(x, weights=weights)
class gumbel_r_gen(rv_continuous):
r"""A right-skewed Gumbel continuous random variable.
%(before_notes)s
See Also
--------
gumbel_l, gompertz, genextreme
Notes
-----
The probability density function for `gumbel_r` is:
.. math::
f(x) = \exp(-(x + e^{-x}))
The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
distribution. It is also related to the extreme value distribution,
log-Weibull and Gompertz distributions.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# gumbel_r.pdf(x) = exp(-(x + exp(-x)))
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return -x - np.exp(-x)
def _cdf(self, x):
return np.exp(-np.exp(-x))
def _logcdf(self, x):
return -np.exp(-x)
def _ppf(self, q):
return -np.log(-np.log(q))
def _sf(self, x):
return -sc.expm1(-np.exp(-x))
def _isf(self, p):
return -np.log(-np.log1p(-p))
def _stats(self):
return _EULER, np.pi*np.pi/6.0, 12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5
def _entropy(self):
# https://en.wikipedia.org/wiki/Gumbel_distribution
return _EULER + 1.
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
data, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
# By the method of maximum likelihood, the estimators of the
# location and scale are the roots of the equations defined in
# `func` and the value of the expression for `loc` that follows.
# The first `func` is a first order derivative of the log-likelihood
# equation and the second is from Source: Statistical Distributions,
# 3rd Edition. Evans, Hastings, and Peacock (2000), Page 101.
def get_loc_from_scale(scale):
return -scale * (sc.logsumexp(-data / scale) - np.log(len(data)))
if fscale is not None:
# if the scale is fixed, the location can be analytically
# determined.
scale = fscale
loc = get_loc_from_scale(scale)
else:
# A different function is solved depending on whether the location
# is fixed.
if floc is not None:
loc = floc
# equation to use if the location is fixed.
# note that one cannot use the equation in Evans, Hastings,
# and Peacock (2000) (since it assumes that the derivative
# w.r.t. the log-likelihood is zero). however, it is easy to
# derive the MLE condition directly if loc is fixed
def func(scale):
term1 = (loc - data) * np.exp((loc - data) / scale) + data
term2 = len(data) * (loc + scale)
return term1.sum() - term2
else:
# equation to use if both location and scale are free
def func(scale):
sdata = -data / scale
wavg = _average_with_log_weights(data, logweights=sdata)
return data.mean() - wavg - scale
# set brackets for `root_scalar` to use when optimizing over the
# scale such that a root is likely between them. Use user supplied
# guess or default 1.
brack_start = kwds.get('scale', 1)
lbrack, rbrack = brack_start / 2, brack_start * 2
# if a root is not between the brackets, iteratively expand them
# until they include a sign change, checking after each bracket is
# modified.
def interval_contains_root(lbrack, rbrack):
# return true if the signs disagree.
return (np.sign(func(lbrack)) !=
np.sign(func(rbrack)))
while (not interval_contains_root(lbrack, rbrack)
and (lbrack > 0 or rbrack < np.inf)):
lbrack /= 2
rbrack *= 2
res = optimize.root_scalar(func, bracket=(lbrack, rbrack),
rtol=1e-14, xtol=1e-14)
scale = res.root
loc = floc if floc is not None else get_loc_from_scale(scale)
return loc, scale
gumbel_r = gumbel_r_gen(name='gumbel_r')
class gumbel_l_gen(rv_continuous):
r"""A left-skewed Gumbel continuous random variable.
%(before_notes)s
See Also
--------
gumbel_r, gompertz, genextreme
Notes
-----
The probability density function for `gumbel_l` is:
.. math::
f(x) = \exp(x - e^x)
The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
distribution. It is also related to the extreme value distribution,
log-Weibull and Gompertz distributions.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# gumbel_l.pdf(x) = exp(x - exp(x))
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return x - np.exp(x)
def _cdf(self, x):
return -sc.expm1(-np.exp(x))
def _ppf(self, q):
return np.log(-sc.log1p(-q))
def _logsf(self, x):
return -np.exp(x)
def _sf(self, x):
return np.exp(-np.exp(x))
def _isf(self, x):
return np.log(-np.log(x))
def _stats(self):
return -_EULER, np.pi*np.pi/6.0, \
-12*np.sqrt(6)/np.pi**3 * _ZETA3, 12.0/5
def _entropy(self):
return _EULER + 1.
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
# The fit method of `gumbel_r` can be used for this distribution with
# small modifications. The process to do this is
# 1. pass the sign negated data into `gumbel_r.fit`
# - if the location is fixed, it should also be negated.
# 2. negate the sign of the resulting location, leaving the scale
# unmodified.
# `gumbel_r.fit` holds necessary input checks.
if kwds.get('floc') is not None:
kwds['floc'] = -kwds['floc']
loc_r, scale_r, = gumbel_r.fit(-np.asarray(data), *args, **kwds)
return -loc_r, scale_r
gumbel_l = gumbel_l_gen(name='gumbel_l')
class halfcauchy_gen(rv_continuous):
r"""A Half-Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halfcauchy` is:
.. math::
f(x) = \frac{2}{\pi (1 + x^2)}
for :math:`x \ge 0`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# halfcauchy.pdf(x) = 2 / (pi * (1 + x**2))
return 2.0/np.pi/(1.0+x*x)
def _logpdf(self, x):
return np.log(2.0/np.pi) - sc.log1p(x*x)
def _cdf(self, x):
return 2.0/np.pi*np.arctan(x)
def _ppf(self, q):
return np.tan(np.pi/2*q)
def _sf(self, x):
return 2.0/np.pi * np.arctan2(1, x)
def _isf(self, p):
return 1.0/np.tan(np.pi*p/2)
def _stats(self):
return np.inf, np.inf, np.nan, np.nan
def _entropy(self):
return np.log(2*np.pi)
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
data, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
# location is independent from the scale
data_min = np.min(data)
if floc is not None:
if data_min < floc:
# There are values that are less than the specified loc.
raise FitDataError("halfcauchy", lower=floc, upper=np.inf)
loc = floc
else:
# if not provided, location MLE is the minimal data point
loc = data_min
# find scale
def find_scale(loc, data):
shifted_data = data - loc
n = data.size
shifted_data_squared = np.square(shifted_data)
def fun_to_solve(scale):
denominator = scale**2 + shifted_data_squared
return 2 * np.sum(shifted_data_squared/denominator) - n
small = np.finfo(1.0).tiny**0.5 # avoid underflow
res = root_scalar(fun_to_solve, bracket=(small, np.max(shifted_data)))
return res.root
if fscale is not None:
scale = fscale
else:
scale = find_scale(loc, data)
return loc, scale
halfcauchy = halfcauchy_gen(a=0.0, name='halfcauchy')
class halflogistic_gen(rv_continuous):
r"""A half-logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halflogistic` is:
.. math::
f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 }
= \frac{1}{2} \text{sech}(x/2)^2
for :math:`x \ge 0`.
%(after_notes)s
References
----------
.. [1] Asgharzadeh et al (2011). "Comparisons of Methods of Estimation for the
Half-Logistic Distribution". Selcuk J. Appl. Math. 93-108.
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# halflogistic.pdf(x) = 2 * exp(-x) / (1+exp(-x))**2
# = 1/2 * sech(x/2)**2
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return np.log(2) - x - 2. * sc.log1p(np.exp(-x))
def _cdf(self, x):
return np.tanh(x/2.0)
def _ppf(self, q):
return 2*np.arctanh(q)
def _sf(self, x):
return 2 * sc.expit(-x)
def _isf(self, q):
return _lazywhere(q < 0.5, (q, ),
lambda q: -sc.logit(0.5 * q),
f2=lambda q: 2*np.arctanh(1 - q))
def _munp(self, n):
if n == 1:
return 2*np.log(2)
if n == 2:
return np.pi*np.pi/3.0
if n == 3:
return 9*_ZETA3
if n == 4:
return 7*np.pi**4 / 15.0
return 2*(1-pow(2.0, 1-n))*sc.gamma(n+1)*sc.zeta(n, 1)
def _entropy(self):
return 2-np.log(2)
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
data, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
def find_scale(data, loc):
# scale is solution to a fix point problem ([1] 2.6)
# use approximate MLE as starting point ([1] 3.1)
n_observations = data.shape[0]
sorted_data = np.sort(data, axis=0)
p = np.arange(1, n_observations + 1)/(n_observations + 1)
q = 1 - p
pp1 = 1 + p
alpha = p - 0.5 * q * pp1 * np.log(pp1 / q)
beta = 0.5 * q * pp1
sorted_data = sorted_data - loc
B = 2 * np.sum(alpha[1:] * sorted_data[1:])
C = 2 * np.sum(beta[1:] * sorted_data[1:]**2)
# starting guess
scale = ((B + np.sqrt(B**2 + 8 * n_observations * C))
/(4 * n_observations))
# relative tolerance of fix point iterator
rtol = 1e-8
relative_residual = 1
shifted_mean = sorted_data.mean() # y_mean - y_min
# find fix point by repeated application of eq. (2.6)
# simplify as
# exp(-x) / (1 + exp(-x)) = 1 / (1 + exp(x))
# = expit(-x))
while relative_residual > rtol:
sum_term = sorted_data * sc.expit(-sorted_data/scale)
scale_new = shifted_mean - 2/n_observations * sum_term.sum()
relative_residual = abs((scale - scale_new)/scale)
scale = scale_new
return scale
# location is independent from the scale
data_min = np.min(data)
if floc is not None:
if data_min < floc:
# There are values that are less than the specified loc.
raise FitDataError("halflogistic", lower=floc, upper=np.inf)
loc = floc
else:
# if not provided, location MLE is the minimal data point
loc = data_min
# scale depends on location
scale = fscale if fscale is not None else find_scale(data, loc)
return loc, scale
halflogistic = halflogistic_gen(a=0.0, name='halflogistic')
class halfnorm_gen(rv_continuous):
r"""A half-normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halfnorm` is:
.. math::
f(x) = \sqrt{2/\pi} \exp(-x^2 / 2)
for :math:`x >= 0`.
`halfnorm` is a special case of `chi` with ``df=1``.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return abs(random_state.standard_normal(size=size))
def _pdf(self, x):
# halfnorm.pdf(x) = sqrt(2/pi) * exp(-x**2/2)
return np.sqrt(2.0/np.pi)*np.exp(-x*x/2.0)
def _logpdf(self, x):
return 0.5 * np.log(2.0/np.pi) - x*x/2.0
def _cdf(self, x):
return sc.erf(x / np.sqrt(2))
def _ppf(self, q):
return _norm_ppf((1+q)/2.0)
def _sf(self, x):
return 2 * _norm_sf(x)
def _isf(self, p):
return _norm_isf(p/2)
def _stats(self):
return (np.sqrt(2.0/np.pi),
1-2.0/np.pi,
np.sqrt(2)*(4-np.pi)/(np.pi-2)**1.5,
8*(np.pi-3)/(np.pi-2)**2)
def _entropy(self):
return 0.5*np.log(np.pi/2.0)+0.5
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
data, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
data_min = np.min(data)
if floc is not None:
if data_min < floc:
# There are values that are less than the specified loc.
raise FitDataError("halfnorm", lower=floc, upper=np.inf)
loc = floc
else:
loc = data_min
if fscale is not None:
scale = fscale
else:
scale = stats.moment(data, order=2, center=loc)**0.5
return loc, scale
halfnorm = halfnorm_gen(a=0.0, name='halfnorm')
class hypsecant_gen(rv_continuous):
r"""A hyperbolic secant continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `hypsecant` is:
.. math::
f(x) = \frac{1}{\pi} \text{sech}(x)
for a real number :math:`x`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
# hypsecant.pdf(x) = 1/pi * sech(x)
return 1.0/(np.pi*np.cosh(x))
def _cdf(self, x):
return 2.0/np.pi*np.arctan(np.exp(x))
def _ppf(self, q):
return np.log(np.tan(np.pi*q/2.0))
def _sf(self, x):
return 2.0/np.pi*np.arctan(np.exp(-x))
def _isf(self, q):
return -np.log(np.tan(np.pi*q/2.0))
def _stats(self):
return 0, np.pi*np.pi/4, 0, 2
def _entropy(self):
return np.log(2*np.pi)
hypsecant = hypsecant_gen(name='hypsecant')
class gausshyper_gen(rv_continuous):
r"""A Gauss hypergeometric continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gausshyper` is:
.. math::
f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c}
for :math:`0 \le x \le 1`, :math:`a,b > 0`, :math:`c` a real number,
:math:`z > -1`, and :math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`.
:math:`F[2, 1]` is the Gauss hypergeometric function
`scipy.special.hyp2f1`.
`gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape
parameters.
%(after_notes)s
References
----------
.. [1] Armero, C., and M. J. Bayarri. "Prior Assessments for Prediction in
Queues." *Journal of the Royal Statistical Society*. Series D (The
Statistician) 43, no. 1 (1994): 139-53. doi:10.2307/2348939
%(example)s
"""
def _argcheck(self, a, b, c, z):
# z > -1 per gh-10134
return (a > 0) & (b > 0) & (c == c) & (z > -1)
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
ic = _ShapeInfo("c", False, (-np.inf, np.inf), (False, False))
iz = _ShapeInfo("z", False, (-1, np.inf), (False, False))
return [ia, ib, ic, iz]
def _pdf(self, x, a, b, c, z):
normalization_constant = sc.beta(a, b) * sc.hyp2f1(c, a, a + b, -z)
return (1./normalization_constant * x**(a - 1.) * (1. - x)**(b - 1.0)
/ (1.0 + z*x)**c)
def _munp(self, n, a, b, c, z):
fac = sc.beta(n+a, b) / sc.beta(a, b)
num = sc.hyp2f1(c, a+n, a+b+n, -z)
den = sc.hyp2f1(c, a, a+b, -z)
return fac*num / den
gausshyper = gausshyper_gen(a=0.0, b=1.0, name='gausshyper')
class invgamma_gen(rv_continuous):
r"""An inverted gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `invgamma` is:
.. math::
f(x, a) = \frac{x^{-a-1}}{\Gamma(a)} \exp(-\frac{1}{x})
for :math:`x >= 0`, :math:`a > 0`. :math:`\Gamma` is the gamma function
(`scipy.special.gamma`).
`invgamma` takes ``a`` as a shape parameter for :math:`a`.
`invgamma` is a special case of `gengamma` with ``c=-1``, and it is a
different parameterization of the scaled inverse chi-squared distribution.
Specifically, if the scaled inverse chi-squared distribution is
parameterized with degrees of freedom :math:`\nu` and scaling parameter
:math:`\tau^2`, then it can be modeled using `invgamma` with
``a=`` :math:`\nu/2` and ``scale=`` :math:`\nu \tau^2/2`.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, a):
# invgamma.pdf(x, a) = x**(-a-1) / gamma(a) * exp(-1/x)
return np.exp(self._logpdf(x, a))
def _logpdf(self, x, a):
return -(a+1) * np.log(x) - sc.gammaln(a) - 1.0/x
def _cdf(self, x, a):
return sc.gammaincc(a, 1.0 / x)
def _ppf(self, q, a):
return 1.0 / sc.gammainccinv(a, q)
def _sf(self, x, a):
return sc.gammainc(a, 1.0 / x)
def _isf(self, q, a):
return 1.0 / sc.gammaincinv(a, q)
def _stats(self, a, moments='mvsk'):
m1 = _lazywhere(a > 1, (a,), lambda x: 1. / (x - 1.), np.inf)
m2 = _lazywhere(a > 2, (a,), lambda x: 1. / (x - 1.)**2 / (x - 2.),
np.inf)
g1, g2 = None, None
if 's' in moments:
g1 = _lazywhere(
a > 3, (a,),
lambda x: 4. * np.sqrt(x - 2.) / (x - 3.), np.nan)
if 'k' in moments:
g2 = _lazywhere(
a > 4, (a,),
lambda x: 6. * (5. * x - 11.) / (x - 3.) / (x - 4.), np.nan)
return m1, m2, g1, g2
def _entropy(self, a):
def regular(a):
h = a - (a + 1.0) * sc.psi(a) + sc.gammaln(a)
return h
def asymptotic(a):
# gammaln(a) ~ a * ln(a) - a - 0.5 * ln(a) + 0.5 * ln(2 * pi)
# psi(a) ~ ln(a) - 1 / (2 * a)
h = ((1 - 3*np.log(a) + np.log(2) + np.log(np.pi))/2
+ 2/3*a**-1. + a**-2./12 - a**-3./90 - a**-4./120)
return h
h = _lazywhere(a >= 2e2, (a,), f=asymptotic, f2=regular)
return h
invgamma = invgamma_gen(a=0.0, name='invgamma')
class invgauss_gen(rv_continuous):
r"""An inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `invgauss` is:
.. math::
f(x; \mu) = \frac{1}{\sqrt{2 \pi x^3}}
\exp\left(-\frac{(x-\mu)^2}{2 \mu^2 x}\right)
for :math:`x \ge 0` and :math:`\mu > 0`.
`invgauss` takes ``mu`` as a shape parameter for :math:`\mu`.
%(after_notes)s
A common shape-scale parameterization of the inverse Gaussian distribution
has density
.. math::
f(x; \nu, \lambda) = \sqrt{\frac{\lambda}{2 \pi x^3}}
\exp\left( -\frac{\lambda(x-\nu)^2}{2 \nu^2 x}\right)
Using ``nu`` for :math:`\nu` and ``lam`` for :math:`\lambda`, this
parameterization is equivalent to the one above with ``mu = nu/lam``,
``loc = 0``, and ``scale = lam``.
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return [_ShapeInfo("mu", False, (0, np.inf), (False, False))]
def _rvs(self, mu, size=None, random_state=None):
return random_state.wald(mu, 1.0, size=size)
def _pdf(self, x, mu):
# invgauss.pdf(x, mu) =
# 1 / sqrt(2*pi*x**3) * exp(-(x-mu)**2/(2*x*mu**2))
return 1.0/np.sqrt(2*np.pi*x**3.0)*np.exp(-1.0/(2*x)*((x-mu)/mu)**2)
def _logpdf(self, x, mu):
return -0.5*np.log(2*np.pi) - 1.5*np.log(x) - ((x-mu)/mu)**2/(2*x)
# approach adapted from equations in
# https://journal.r-project.org/archive/2016-1/giner-smyth.pdf,
# not R code. see gh-13616
def _logcdf(self, x, mu):
fac = 1 / np.sqrt(x)
a = _norm_logcdf(fac * ((x / mu) - 1))
b = 2 / mu + _norm_logcdf(-fac * ((x / mu) + 1))
return a + np.log1p(np.exp(b - a))
def _logsf(self, x, mu):
fac = 1 / np.sqrt(x)
a = _norm_logsf(fac * ((x / mu) - 1))
b = 2 / mu + _norm_logcdf(-fac * (x + mu) / mu)
return a + np.log1p(-np.exp(b - a))
def _sf(self, x, mu):
return np.exp(self._logsf(x, mu))
def _cdf(self, x, mu):
return np.exp(self._logcdf(x, mu))
def _ppf(self, x, mu):
with np.errstate(divide='ignore', over='ignore', invalid='ignore'):
x, mu = np.broadcast_arrays(x, mu)
ppf = _boost._invgauss_ppf(x, mu, 1)
i_wt = x > 0.5 # "wrong tail" - sometimes too inaccurate
ppf[i_wt] = _boost._invgauss_isf(1-x[i_wt], mu[i_wt], 1)
i_nan = np.isnan(ppf)
ppf[i_nan] = super()._ppf(x[i_nan], mu[i_nan])
return ppf
def _isf(self, x, mu):
with np.errstate(divide='ignore', over='ignore', invalid='ignore'):
x, mu = np.broadcast_arrays(x, mu)
isf = _boost._invgauss_isf(x, mu, 1)
i_wt = x > 0.5 # "wrong tail" - sometimes too inaccurate
isf[i_wt] = _boost._invgauss_ppf(1-x[i_wt], mu[i_wt], 1)
i_nan = np.isnan(isf)
isf[i_nan] = super()._isf(x[i_nan], mu[i_nan])
return isf
def _stats(self, mu):
return mu, mu**3.0, 3*np.sqrt(mu), 15*mu
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
method = kwds.get('method', 'mle')
if (isinstance(data, CensoredData) or type(self) == wald_gen
or method.lower() == 'mm'):
return super().fit(data, *args, **kwds)
data, fshape_s, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
'''
Source: Statistical Distributions, 3rd Edition. Evans, Hastings,
and Peacock (2000), Page 121. Their shape parameter is equivalent to
SciPy's with the conversion `fshape_s = fshape / scale`.
MLE formulas are not used in 3 conditions:
- `loc` is not fixed
- `mu` is fixed
These cases fall back on the superclass fit method.
- `loc` is fixed but translation results in negative data raises
a `FitDataError`.
'''
if floc is None or fshape_s is not None:
return super().fit(data, *args, **kwds)
elif np.any(data - floc < 0):
raise FitDataError("invgauss", lower=0, upper=np.inf)
else:
data = data - floc
fshape_n = np.mean(data)
if fscale is None:
fscale = len(data) / (np.sum(data ** -1 - fshape_n ** -1))
fshape_s = fshape_n / fscale
return fshape_s, floc, fscale
def _entropy(self, mu):
"""
Ref.: https://moser-isi.ethz.ch/docs/papers/smos-2012-10.pdf (eq. 9)
"""
# a = log(2*pi*e*mu**3)
# = 1 + log(2*pi) + 3 * log(mu)
a = 1. + np.log(2 * np.pi) + 3 * np.log(mu)
# b = exp(2/mu) * exp1(2/mu)
# = _scaled_exp1(2/mu) / (2/mu)
r = 2/mu
b = sc._ufuncs._scaled_exp1(r)/r
return 0.5 * a - 1.5 * b
invgauss = invgauss_gen(a=0.0, name='invgauss')
class geninvgauss_gen(rv_continuous):
r"""A Generalized Inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `geninvgauss` is:
.. math::
f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b))
where `x > 0`, `p` is a real number and `b > 0`\([1]_).
:math:`K_p` is the modified Bessel function of second kind of order `p`
(`scipy.special.kv`).
%(after_notes)s
The inverse Gaussian distribution `stats.invgauss(mu)` is a special case of
`geninvgauss` with `p = -1/2`, `b = 1 / mu` and `scale = mu`.
Generating random variates is challenging for this distribution. The
implementation is based on [2]_.
References
----------
.. [1] O. Barndorff-Nielsen, P. Blaesild, C. Halgreen, "First hitting time
models for the generalized inverse gaussian distribution",
Stochastic Processes and their Applications 7, pp. 49--54, 1978.
.. [2] W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian
random variates", Statistics and Computing, 24(4), p. 547--557, 2014.
%(example)s
"""
def _argcheck(self, p, b):
return (p == p) & (b > 0)
def _shape_info(self):
ip = _ShapeInfo("p", False, (-np.inf, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
return [ip, ib]
def _logpdf(self, x, p, b):
# kve instead of kv works better for large values of b
# warn if kve produces infinite values and replace by nan
# otherwise c = -inf and the results are often incorrect
def logpdf_single(x, p, b):
return _stats.geninvgauss_logpdf(x, p, b)
logpdf_single = np.vectorize(logpdf_single, otypes=[np.float64])
z = logpdf_single(x, p, b)
if np.isnan(z).any():
msg = ("Infinite values encountered in scipy.special.kve(p, b). "
"Values replaced by NaN to avoid incorrect results.")
warnings.warn(msg, RuntimeWarning, stacklevel=3)
return z
def _pdf(self, x, p, b):
# relying on logpdf avoids overflow of x**(p-1) for large x and p
return np.exp(self._logpdf(x, p, b))
def _cdf(self, x, *args):
_a, _b = self._get_support(*args)
def _cdf_single(x, *args):
p, b = args
user_data = np.array([p, b], float).ctypes.data_as(ctypes.c_void_p)
llc = LowLevelCallable.from_cython(_stats, '_geninvgauss_pdf',
user_data)
return integrate.quad(llc, _a, x)[0]
_cdf_single = np.vectorize(_cdf_single, otypes=[np.float64])
return _cdf_single(x, *args)
def _logquasipdf(self, x, p, b):
# log of the quasi-density (w/o normalizing constant) used in _rvs
return _lazywhere(x > 0, (x, p, b),
lambda x, p, b: (p - 1)*np.log(x) - b*(x + 1/x)/2,
-np.inf)
def _rvs(self, p, b, size=None, random_state=None):
# if p and b are scalar, use _rvs_scalar, otherwise need to create
# output by iterating over parameters
if np.isscalar(p) and np.isscalar(b):
out = self._rvs_scalar(p, b, size, random_state)
elif p.size == 1 and b.size == 1:
out = self._rvs_scalar(p.item(), b.item(), size, random_state)
else:
# When this method is called, size will be a (possibly empty)
# tuple of integers. It will not be None; if `size=None` is passed
# to `rvs()`, size will be the empty tuple ().
p, b = np.broadcast_arrays(p, b)
# p and b now have the same shape.
# `shp` is the shape of the blocks of random variates that are
# generated for each combination of parameters associated with
# broadcasting p and b.
# bc is a tuple the same length as size. The values
# in bc are bools. If bc[j] is True, it means that
# entire axis is filled in for a given combination of the
# broadcast arguments.
shp, bc = _check_shape(p.shape, size)
# `numsamples` is the total number of variates to be generated
# for each combination of the input arguments.
numsamples = int(np.prod(shp))
# `out` is the array to be returned. It is filled in the
# loop below.
out = np.empty(size)
it = np.nditer([p, b],
flags=['multi_index'],
op_flags=[['readonly'], ['readonly']])
while not it.finished:
# Convert the iterator's multi_index into an index into the
# `out` array where the call to _rvs_scalar() will be stored.
# Where bc is True, we use a full slice; otherwise we use the
# index value from it.multi_index. len(it.multi_index) might
# be less than len(bc), and in that case we want to align these
# two sequences to the right, so the loop variable j runs from
# -len(size) to 0. This doesn't cause an IndexError, as
# bc[j] will be True in those cases where it.multi_index[j]
# would cause an IndexError.
idx = tuple((it.multi_index[j] if not bc[j] else slice(None))
for j in range(-len(size), 0))
out[idx] = self._rvs_scalar(it[0], it[1], numsamples,
random_state).reshape(shp)
it.iternext()
if size == ():
out = out.item()
return out
def _rvs_scalar(self, p, b, numsamples, random_state):
# following [2], the quasi-pdf is used instead of the pdf for the
# generation of rvs
invert_res = False
if not numsamples:
numsamples = 1
if p < 0:
# note: if X is geninvgauss(p, b), then 1/X is geninvgauss(-p, b)
p = -p
invert_res = True
m = self._mode(p, b)
# determine method to be used following [2]
ratio_unif = True
if p >= 1 or b > 1:
# ratio of uniforms with mode shift below
mode_shift = True
elif b >= min(0.5, 2 * np.sqrt(1 - p) / 3):
# ratio of uniforms without mode shift below
mode_shift = False
else:
# new algorithm in [2]
ratio_unif = False
# prepare sampling of rvs
size1d = tuple(np.atleast_1d(numsamples))
N = np.prod(size1d) # number of rvs needed, reshape upon return
x = np.zeros(N)
simulated = 0
if ratio_unif:
# use ratio of uniforms method
if mode_shift:
a2 = -2 * (p + 1) / b - m
a1 = 2 * m * (p - 1) / b - 1
# find roots of x**3 + a2*x**2 + a1*x + m (Cardano's formula)
p1 = a1 - a2**2 / 3
q1 = 2 * a2**3 / 27 - a2 * a1 / 3 + m
phi = np.arccos(-q1 * np.sqrt(-27 / p1**3) / 2)
s1 = -np.sqrt(-4 * p1 / 3)
root1 = s1 * np.cos(phi / 3 + np.pi / 3) - a2 / 3
root2 = -s1 * np.cos(phi / 3) - a2 / 3
# root3 = s1 * np.cos(phi / 3 - np.pi / 3) - a2 / 3
# if g is the quasipdf, rescale: g(x) / g(m) which we can write
# as exp(log(g(x)) - log(g(m))). This is important
# since for large values of p and b, g cannot be evaluated.
# denote the rescaled quasipdf by h
lm = self._logquasipdf(m, p, b)
d1 = self._logquasipdf(root1, p, b) - lm
d2 = self._logquasipdf(root2, p, b) - lm
# compute the bounding rectangle w.r.t. h. Note that
# np.exp(0.5*d1) = np.sqrt(g(root1)/g(m)) = np.sqrt(h(root1))
vmin = (root1 - m) * np.exp(0.5 * d1)
vmax = (root2 - m) * np.exp(0.5 * d2)
umax = 1 # umax = sqrt(h(m)) = 1
def logqpdf(x):
return self._logquasipdf(x, p, b) - lm
c = m
else:
# ratio of uniforms without mode shift
# compute np.sqrt(quasipdf(m))
umax = np.exp(0.5*self._logquasipdf(m, p, b))
xplus = ((1 + p) + np.sqrt((1 + p)**2 + b**2))/b
vmin = 0
# compute xplus * np.sqrt(quasipdf(xplus))
vmax = xplus * np.exp(0.5 * self._logquasipdf(xplus, p, b))
c = 0
def logqpdf(x):
return self._logquasipdf(x, p, b)
if vmin >= vmax:
raise ValueError("vmin must be smaller than vmax.")
if umax <= 0:
raise ValueError("umax must be positive.")
i = 1
while simulated < N:
k = N - simulated
# simulate uniform rvs on [0, umax] and [vmin, vmax]
u = umax * random_state.uniform(size=k)
v = random_state.uniform(size=k)
v = vmin + (vmax - vmin) * v
rvs = v / u + c
# rewrite acceptance condition u**2 <= pdf(rvs) by taking logs
accept = (2*np.log(u) <= logqpdf(rvs))
num_accept = np.sum(accept)
if num_accept > 0:
x[simulated:(simulated + num_accept)] = rvs[accept]
simulated += num_accept
if (simulated == 0) and (i*N >= 50000):
msg = ("Not a single random variate could be generated "
f"in {i*N} attempts. Sampling does not appear to "
"work for the provided parameters.")
raise RuntimeError(msg)
i += 1
else:
# use new algorithm in [2]
x0 = b / (1 - p)
xs = np.max((x0, 2 / b))
k1 = np.exp(self._logquasipdf(m, p, b))
A1 = k1 * x0
if x0 < 2 / b:
k2 = np.exp(-b)
if p > 0:
A2 = k2 * ((2 / b)**p - x0**p) / p
else:
A2 = k2 * np.log(2 / b**2)
else:
k2, A2 = 0, 0
k3 = xs**(p - 1)
A3 = 2 * k3 * np.exp(-xs * b / 2) / b
A = A1 + A2 + A3
# [2]: rejection constant is < 2.73; so expected runtime is finite
while simulated < N:
k = N - simulated
h, rvs = np.zeros(k), np.zeros(k)
# simulate uniform rvs on [x1, x2] and [0, y2]
u = random_state.uniform(size=k)
v = A * random_state.uniform(size=k)
cond1 = v <= A1
cond2 = np.logical_not(cond1) & (v <= A1 + A2)
cond3 = np.logical_not(cond1 | cond2)
# subdomain (0, x0)
rvs[cond1] = x0 * v[cond1] / A1
h[cond1] = k1
# subdomain (x0, 2 / b)
if p > 0:
rvs[cond2] = (x0**p + (v[cond2] - A1) * p / k2)**(1 / p)
else:
rvs[cond2] = b * np.exp((v[cond2] - A1) * np.exp(b))
h[cond2] = k2 * rvs[cond2]**(p - 1)
# subdomain (xs, infinity)
z = np.exp(-xs * b / 2) - b * (v[cond3] - A1 - A2) / (2 * k3)
rvs[cond3] = -2 / b * np.log(z)
h[cond3] = k3 * np.exp(-rvs[cond3] * b / 2)
# apply rejection method
accept = (np.log(u * h) <= self._logquasipdf(rvs, p, b))
num_accept = sum(accept)
if num_accept > 0:
x[simulated:(simulated + num_accept)] = rvs[accept]
simulated += num_accept
rvs = np.reshape(x, size1d)
if invert_res:
rvs = 1 / rvs
return rvs
def _mode(self, p, b):
# distinguish cases to avoid catastrophic cancellation (see [2])
if p < 1:
return b / (np.sqrt((p - 1)**2 + b**2) + 1 - p)
else:
return (np.sqrt((1 - p)**2 + b**2) - (1 - p)) / b
def _munp(self, n, p, b):
num = sc.kve(p + n, b)
denom = sc.kve(p, b)
inf_vals = np.isinf(num) | np.isinf(denom)
if inf_vals.any():
msg = ("Infinite values encountered in the moment calculation "
"involving scipy.special.kve. Values replaced by NaN to "
"avoid incorrect results.")
warnings.warn(msg, RuntimeWarning, stacklevel=3)
m = np.full_like(num, np.nan, dtype=np.float64)
m[~inf_vals] = num[~inf_vals] / denom[~inf_vals]
else:
m = num / denom
return m
geninvgauss = geninvgauss_gen(a=0.0, name="geninvgauss")
class norminvgauss_gen(rv_continuous):
r"""A Normal Inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `norminvgauss` is:
.. math::
f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \,
\exp(\sqrt{a^2 - b^2} + b x)
where :math:`x` is a real number, the parameter :math:`a` is the tail
heaviness and :math:`b` is the asymmetry parameter satisfying
:math:`a > 0` and :math:`|b| <= a`.
:math:`K_1` is the modified Bessel function of second kind
(`scipy.special.k1`).
%(after_notes)s
A normal inverse Gaussian random variable `Y` with parameters `a` and `b`
can be expressed as a normal mean-variance mixture:
`Y = b * V + sqrt(V) * X` where `X` is `norm(0,1)` and `V` is
`invgauss(mu=1/sqrt(a**2 - b**2))`. This representation is used
to generate random variates.
Another common parametrization of the distribution (see Equation 2.1 in
[2]_) is given by the following expression of the pdf:
.. math::
g(x, \alpha, \beta, \delta, \mu) =
\frac{\alpha\delta K_1\left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}
{\pi \sqrt{\delta^2 + (x - \mu)^2}} \,
e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)}
In SciPy, this corresponds to
`a = alpha * delta, b = beta * delta, loc = mu, scale=delta`.
References
----------
.. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on
Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
pp. 151-157, 1978.
.. [2] O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and
Stochastic Volatility Modelling", Scandinavian Journal of
Statistics, Vol. 24, pp. 1-13, 1997.
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _argcheck(self, a, b):
return (a > 0) & (np.absolute(b) < a)
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ib = _ShapeInfo("b", False, (-np.inf, np.inf), (False, False))
return [ia, ib]
def _fitstart(self, data):
# Arbitrary, but the default a = b = 1 is not valid; the distribution
# requires |b| < a.
return super()._fitstart(data, args=(1, 0.5))
def _pdf(self, x, a, b):
gamma = np.sqrt(a**2 - b**2)
fac1 = a / np.pi
sq = np.hypot(1, x) # reduce overflows
return fac1 * sc.k1e(a * sq) * np.exp(b*x - a*sq + gamma) / sq
def _sf(self, x, a, b):
if np.isscalar(x):
# If x is a scalar, then so are a and b.
return integrate.quad(self._pdf, x, np.inf, args=(a, b))[0]
else:
a = np.atleast_1d(a)
b = np.atleast_1d(b)
result = []
for (x0, a0, b0) in zip(x, a, b):
result.append(integrate.quad(self._pdf, x0, np.inf,
args=(a0, b0))[0])
return np.array(result)
def _isf(self, q, a, b):
def _isf_scalar(q, a, b):
def eq(x, a, b, q):
# Solve eq(x, a, b, q) = 0 to obtain isf(x, a, b) = q.
return self._sf(x, a, b) - q
# Find a bracketing interval for the root.
# Start at the mean, and grow the length of the interval
# by 2 each iteration until there is a sign change in eq.
xm = self.mean(a, b)
em = eq(xm, a, b, q)
if em == 0:
# Unlikely, but might as well check.
return xm
if em > 0:
delta = 1
left = xm
right = xm + delta
while eq(right, a, b, q) > 0:
delta = 2*delta
right = xm + delta
else:
# em < 0
delta = 1
right = xm
left = xm - delta
while eq(left, a, b, q) < 0:
delta = 2*delta
left = xm - delta
result = optimize.brentq(eq, left, right, args=(a, b, q),
xtol=self.xtol)
return result
if np.isscalar(q):
return _isf_scalar(q, a, b)
else:
result = []
for (q0, a0, b0) in zip(q, a, b):
result.append(_isf_scalar(q0, a0, b0))
return np.array(result)
def _rvs(self, a, b, size=None, random_state=None):
# note: X = b * V + sqrt(V) * X is norminvgaus(a,b) if X is standard
# normal and V is invgauss(mu=1/sqrt(a**2 - b**2))
gamma = np.sqrt(a**2 - b**2)
ig = invgauss.rvs(mu=1/gamma, size=size, random_state=random_state)
return b * ig + np.sqrt(ig) * norm.rvs(size=size,
random_state=random_state)
def _stats(self, a, b):
gamma = np.sqrt(a**2 - b**2)
mean = b / gamma
variance = a**2 / gamma**3
skewness = 3.0 * b / (a * np.sqrt(gamma))
kurtosis = 3.0 * (1 + 4 * b**2 / a**2) / gamma
return mean, variance, skewness, kurtosis
norminvgauss = norminvgauss_gen(name="norminvgauss")
class invweibull_gen(rv_continuous):
"""An inverted Weibull continuous random variable.
This distribution is also known as the Fréchet distribution or the
type II extreme value distribution.
%(before_notes)s
Notes
-----
The probability density function for `invweibull` is:
.. math::
f(x, c) = c x^{-c-1} \\exp(-x^{-c})
for :math:`x > 0`, :math:`c > 0`.
`invweibull` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
References
----------
F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse
Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011.
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# invweibull.pdf(x, c) = c * x**(-c-1) * exp(-x**(-c))
xc1 = np.power(x, -c - 1.0)
xc2 = np.power(x, -c)
xc2 = np.exp(-xc2)
return c * xc1 * xc2
def _cdf(self, x, c):
xc1 = np.power(x, -c)
return np.exp(-xc1)
def _sf(self, x, c):
return -np.expm1(-x**-c)
def _ppf(self, q, c):
return np.power(-np.log(q), -1.0/c)
def _isf(self, p, c):
return (-np.log1p(-p))**(-1/c)
def _munp(self, n, c):
return sc.gamma(1 - n / c)
def _entropy(self, c):
return 1+_EULER + _EULER / c - np.log(c)
def _fitstart(self, data, args=None):
# invweibull requires c > 1 for the first moment to exist, so use 2.0
args = (2.0,) if args is None else args
return super()._fitstart(data, args=args)
invweibull = invweibull_gen(a=0, name='invweibull')
class jf_skew_t_gen(rv_continuous):
r"""Jones and Faddy skew-t distribution.
%(before_notes)s
Notes
-----
The probability density function for `jf_skew_t` is:
.. math::
f(x; a, b) = C_{a,b}^{-1}
\left(1+\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{a+1/2}
\left(1-\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{b+1/2}
for real numbers :math:`a>0` and :math:`b>0`, where
:math:`C_{a,b} = 2^{a+b-1}B(a,b)(a+b)^{1/2}`, and :math:`B` denotes the
beta function (`scipy.special.beta`).
When :math:`a<b`, the distribution is negatively skewed, and when
:math:`a>b`, the distribution is positively skewed. If :math:`a=b`, then
we recover the `t` distribution with :math:`2a` degrees of freedom.
`jf_skew_t` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
References
----------
.. [1] M.C. Jones and M.J. Faddy. "A skew extension of the t distribution,
with applications" *Journal of the Royal Statistical Society*.
Series B (Statistical Methodology) 65, no. 1 (2003): 159-174.
:doi:`10.1111/1467-9868.00378`
%(example)s
"""
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
return [ia, ib]
def _pdf(self, x, a, b):
c = 2 ** (a + b - 1) * sc.beta(a, b) * np.sqrt(a + b)
d1 = (1 + x / np.sqrt(a + b + x ** 2)) ** (a + 0.5)
d2 = (1 - x / np.sqrt(a + b + x ** 2)) ** (b + 0.5)
return d1 * d2 / c
def _rvs(self, a, b, size=None, random_state=None):
d1 = random_state.beta(a, b, size)
d2 = (2 * d1 - 1) * np.sqrt(a + b)
d3 = 2 * np.sqrt(d1 * (1 - d1))
return d2 / d3
def _cdf(self, x, a, b):
y = (1 + x / np.sqrt(a + b + x ** 2)) * 0.5
return sc.betainc(a, b, y)
def _ppf(self, q, a, b):
d1 = beta.ppf(q, a, b)
d2 = (2 * d1 - 1) * np.sqrt(a + b)
d3 = 2 * np.sqrt(d1 * (1 - d1))
return d2 / d3
def _munp(self, n, a, b):
"""Returns the n-th moment(s) where all the following hold:
- n >= 0
- a > n / 2
- b > n / 2
The result is np.nan in all other cases.
"""
def nth_moment(n_k, a_k, b_k):
"""Computes E[T^(n_k)] where T is skew-t distributed with
parameters a_k and b_k.
"""
num = (a_k + b_k) ** (0.5 * n_k)
denom = 2 ** n_k * sc.beta(a_k, b_k)
indices = np.arange(n_k + 1)
sgn = np.where(indices % 2 > 0, -1, 1)
d = sc.beta(a_k + 0.5 * n_k - indices, b_k - 0.5 * n_k + indices)
sum_terms = sc.comb(n_k, indices) * sgn * d
return num / denom * sum_terms.sum()
nth_moment_valid = (a > 0.5 * n) & (b > 0.5 * n) & (n >= 0)
return _lazywhere(
nth_moment_valid,
(n, a, b),
np.vectorize(nth_moment, otypes=[np.float64]),
np.nan,
)
jf_skew_t = jf_skew_t_gen(name='jf_skew_t')
class johnsonsb_gen(rv_continuous):
r"""A Johnson SB continuous random variable.
%(before_notes)s
See Also
--------
johnsonsu
Notes
-----
The probability density function for `johnsonsb` is:
.. math::
f(x, a, b) = \frac{b}{x(1-x)} \phi(a + b \log \frac{x}{1-x} )
where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`
and :math:`x \in [0,1]`. :math:`\phi` is the pdf of the normal
distribution.
`johnsonsb` takes :math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _argcheck(self, a, b):
return (b > 0) & (a == a)
def _shape_info(self):
ia = _ShapeInfo("a", False, (-np.inf, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
return [ia, ib]
def _pdf(self, x, a, b):
# johnsonsb.pdf(x, a, b) = b / (x*(1-x)) * phi(a + b * log(x/(1-x)))
trm = _norm_pdf(a + b*sc.logit(x))
return b*1.0/(x*(1-x))*trm
def _cdf(self, x, a, b):
return _norm_cdf(a + b*sc.logit(x))
def _ppf(self, q, a, b):
return sc.expit(1.0 / b * (_norm_ppf(q) - a))
def _sf(self, x, a, b):
return _norm_sf(a + b*sc.logit(x))
def _isf(self, q, a, b):
return sc.expit(1.0 / b * (_norm_isf(q) - a))
johnsonsb = johnsonsb_gen(a=0.0, b=1.0, name='johnsonsb')
class johnsonsu_gen(rv_continuous):
r"""A Johnson SU continuous random variable.
%(before_notes)s
See Also
--------
johnsonsb
Notes
-----
The probability density function for `johnsonsu` is:
.. math::
f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}}
\phi(a + b \log(x + \sqrt{x^2 + 1}))
where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`.
:math:`\phi` is the pdf of the normal distribution.
`johnsonsu` takes :math:`a` and :math:`b` as shape parameters.
The first four central moments are calculated according to the formulas
in [1]_.
%(after_notes)s
References
----------
.. [1] Taylor Enterprises. "Johnson Family of Distributions".
https://variation.com/wp-content/distribution_analyzer_help/hs126.htm
%(example)s
"""
def _argcheck(self, a, b):
return (b > 0) & (a == a)
def _shape_info(self):
ia = _ShapeInfo("a", False, (-np.inf, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
return [ia, ib]
def _pdf(self, x, a, b):
# johnsonsu.pdf(x, a, b) = b / sqrt(x**2 + 1) *
# phi(a + b * log(x + sqrt(x**2 + 1)))
x2 = x*x
trm = _norm_pdf(a + b * np.arcsinh(x))
return b*1.0/np.sqrt(x2+1.0)*trm
def _cdf(self, x, a, b):
return _norm_cdf(a + b * np.arcsinh(x))
def _ppf(self, q, a, b):
return np.sinh((_norm_ppf(q) - a) / b)
def _sf(self, x, a, b):
return _norm_sf(a + b * np.arcsinh(x))
def _isf(self, x, a, b):
return np.sinh((_norm_isf(x) - a) / b)
def _stats(self, a, b, moments='mv'):
# Naive implementation of first and second moment to address gh-18071.
# https://variation.com/wp-content/distribution_analyzer_help/hs126.htm
# Numerical improvements left to future enhancements.
mu, mu2, g1, g2 = None, None, None, None
bn2 = b**-2.
expbn2 = np.exp(bn2)
a_b = a / b
if 'm' in moments:
mu = -expbn2**0.5 * np.sinh(a_b)
if 'v' in moments:
mu2 = 0.5*sc.expm1(bn2)*(expbn2*np.cosh(2*a_b) + 1)
if 's' in moments:
t1 = expbn2**.5 * sc.expm1(bn2)**0.5
t2 = 3*np.sinh(a_b)
t3 = expbn2 * (expbn2 + 2) * np.sinh(3*a_b)
denom = np.sqrt(2) * (1 + expbn2 * np.cosh(2*a_b))**(3/2)
g1 = -t1 * (t2 + t3) / denom
if 'k' in moments:
t1 = 3 + 6*expbn2
t2 = 4*expbn2**2 * (expbn2 + 2) * np.cosh(2*a_b)
t3 = expbn2**2 * np.cosh(4*a_b)
t4 = -3 + 3*expbn2**2 + 2*expbn2**3 + expbn2**4
denom = 2*(1 + expbn2*np.cosh(2*a_b))**2
g2 = (t1 + t2 + t3*t4) / denom - 3
return mu, mu2, g1, g2
johnsonsu = johnsonsu_gen(name='johnsonsu')
class laplace_gen(rv_continuous):
r"""A Laplace continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `laplace` is
.. math::
f(x) = \frac{1}{2} \exp(-|x|)
for a real number :math:`x`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return random_state.laplace(0, 1, size=size)
def _pdf(self, x):
# laplace.pdf(x) = 1/2 * exp(-abs(x))
return 0.5*np.exp(-abs(x))
def _cdf(self, x):
with np.errstate(over='ignore'):
return np.where(x > 0, 1.0 - 0.5*np.exp(-x), 0.5*np.exp(x))
def _sf(self, x):
# By symmetry...
return self._cdf(-x)
def _ppf(self, q):
return np.where(q > 0.5, -np.log(2*(1-q)), np.log(2*q))
def _isf(self, q):
# By symmetry...
return -self._ppf(q)
def _stats(self):
return 0, 2, 0, 3
def _entropy(self):
return np.log(2)+1
@_call_super_mom
@replace_notes_in_docstring(rv_continuous, notes="""\
This function uses explicit formulas for the maximum likelihood
estimation of the Laplace distribution parameters, so the keyword
arguments `loc`, `scale`, and `optimizer` are ignored.\n\n""")
def fit(self, data, *args, **kwds):
data, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
# Source: Statistical Distributions, 3rd Edition. Evans, Hastings,
# and Peacock (2000), Page 124
if floc is None:
floc = np.median(data)
if fscale is None:
fscale = (np.sum(np.abs(data - floc))) / len(data)
return floc, fscale
laplace = laplace_gen(name='laplace')
class laplace_asymmetric_gen(rv_continuous):
r"""An asymmetric Laplace continuous random variable.
%(before_notes)s
See Also
--------
laplace : Laplace distribution
Notes
-----
The probability density function for `laplace_asymmetric` is
.. math::
f(x, \kappa) &= \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0\\
&= \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0\\
for :math:`-\infty < x < \infty`, :math:`\kappa > 0`.
`laplace_asymmetric` takes ``kappa`` as a shape parameter for
:math:`\kappa`. For :math:`\kappa = 1`, it is identical to a
Laplace distribution.
%(after_notes)s
Note that the scale parameter of some references is the reciprocal of
SciPy's ``scale``. For example, :math:`\lambda = 1/2` in the
parameterization of [1]_ is equivalent to ``scale = 2`` with
`laplace_asymmetric`.
References
----------
.. [1] "Asymmetric Laplace distribution", Wikipedia
https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution
.. [2] Kozubowski TJ and Podgórski K. A Multivariate and
Asymmetric Generalization of Laplace Distribution,
Computational Statistics 15, 531--540 (2000).
:doi:`10.1007/PL00022717`
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("kappa", False, (0, np.inf), (False, False))]
def _pdf(self, x, kappa):
return np.exp(self._logpdf(x, kappa))
def _logpdf(self, x, kappa):
kapinv = 1/kappa
lPx = x * np.where(x >= 0, -kappa, kapinv)
lPx -= np.log(kappa+kapinv)
return lPx
def _cdf(self, x, kappa):
kapinv = 1/kappa
kappkapinv = kappa+kapinv
return np.where(x >= 0,
1 - np.exp(-x*kappa)*(kapinv/kappkapinv),
np.exp(x*kapinv)*(kappa/kappkapinv))
def _sf(self, x, kappa):
kapinv = 1/kappa
kappkapinv = kappa+kapinv
return np.where(x >= 0,
np.exp(-x*kappa)*(kapinv/kappkapinv),
1 - np.exp(x*kapinv)*(kappa/kappkapinv))
def _ppf(self, q, kappa):
kapinv = 1/kappa
kappkapinv = kappa+kapinv
return np.where(q >= kappa/kappkapinv,
-np.log((1 - q)*kappkapinv*kappa)*kapinv,
np.log(q*kappkapinv/kappa)*kappa)
def _isf(self, q, kappa):
kapinv = 1/kappa
kappkapinv = kappa+kapinv
return np.where(q <= kapinv/kappkapinv,
-np.log(q*kappkapinv*kappa)*kapinv,
np.log((1 - q)*kappkapinv/kappa)*kappa)
def _stats(self, kappa):
kapinv = 1/kappa
mn = kapinv - kappa
var = kapinv*kapinv + kappa*kappa
g1 = 2.0*(1-np.power(kappa, 6))/np.power(1+np.power(kappa, 4), 1.5)
g2 = 6.0*(1+np.power(kappa, 8))/np.power(1+np.power(kappa, 4), 2)
return mn, var, g1, g2
def _entropy(self, kappa):
return 1 + np.log(kappa+1/kappa)
laplace_asymmetric = laplace_asymmetric_gen(name='laplace_asymmetric')
def _check_fit_input_parameters(dist, data, args, kwds):
if not isinstance(data, CensoredData):
data = np.asarray(data)
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
num_shapes = len(dist.shapes.split(",")) if dist.shapes else 0
fshape_keys = []
fshapes = []
# user has many options for fixing the shape, so here we standardize it
# into 'f' + the number of the shape.
# Adapted from `_reduce_func` in `_distn_infrastructure.py`:
if dist.shapes:
shapes = dist.shapes.replace(',', ' ').split()
for j, s in enumerate(shapes):
key = 'f' + str(j)
names = [key, 'f' + s, 'fix_' + s]
val = _get_fixed_fit_value(kwds, names)
fshape_keys.append(key)
fshapes.append(val)
if val is not None:
kwds[key] = val
# determine if there are any unknown arguments in kwds
known_keys = {'loc', 'scale', 'optimizer', 'method',
'floc', 'fscale', *fshape_keys}
unknown_keys = set(kwds).difference(known_keys)
if unknown_keys:
raise TypeError(f"Unknown keyword arguments: {unknown_keys}.")
if len(args) > num_shapes:
raise TypeError("Too many positional arguments.")
if None not in {floc, fscale, *fshapes}:
# This check is for consistency with `rv_continuous.fit`.
# Without this check, this function would just return the
# parameters that were given.
raise RuntimeError("All parameters fixed. There is nothing to "
"optimize.")
uncensored = data._uncensor() if isinstance(data, CensoredData) else data
if not np.isfinite(uncensored).all():
raise ValueError("The data contains non-finite values.")
return (data, *fshapes, floc, fscale)
class levy_gen(rv_continuous):
r"""A Levy continuous random variable.
%(before_notes)s
See Also
--------
levy_stable, levy_l
Notes
-----
The probability density function for `levy` is:
.. math::
f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right)
for :math:`x > 0`.
This is the same as the Levy-stable distribution with :math:`a=1/2` and
:math:`b=1`.
%(after_notes)s
Examples
--------
>>> import numpy as np
>>> from scipy.stats import levy
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> mean, var, skew, kurt = levy.stats(moments='mvsk')
Display the probability density function (``pdf``):
>>> # `levy` is very heavy-tailed.
>>> # To show a nice plot, let's cut off the upper 40 percent.
>>> a, b = levy.ppf(0), levy.ppf(0.6)
>>> x = np.linspace(a, b, 100)
>>> ax.plot(x, levy.pdf(x),
... 'r-', lw=5, alpha=0.6, label='levy pdf')
Alternatively, the distribution object can be called (as a function)
to fix the shape, location and scale parameters. This returns a "frozen"
RV object holding the given parameters fixed.
Freeze the distribution and display the frozen ``pdf``:
>>> rv = levy()
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of ``cdf`` and ``ppf``:
>>> vals = levy.ppf([0.001, 0.5, 0.999])
>>> np.allclose([0.001, 0.5, 0.999], levy.cdf(vals))
True
Generate random numbers:
>>> r = levy.rvs(size=1000)
And compare the histogram:
>>> # manual binning to ignore the tail
>>> bins = np.concatenate((np.linspace(a, b, 20), [np.max(r)]))
>>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return []
def _pdf(self, x):
# levy.pdf(x) = 1 / (x * sqrt(2*pi*x)) * exp(-1/(2*x))
return 1 / np.sqrt(2*np.pi*x) / x * np.exp(-1/(2*x))
def _cdf(self, x):
# Equivalent to 2*norm.sf(np.sqrt(1/x))
return sc.erfc(np.sqrt(0.5 / x))
def _sf(self, x):
return sc.erf(np.sqrt(0.5 / x))
def _ppf(self, q):
# Equivalent to 1.0/(norm.isf(q/2)**2) or 0.5/(erfcinv(q)**2)
val = _norm_isf(q/2)
return 1.0 / (val * val)
def _isf(self, p):
return 1/(2*sc.erfinv(p)**2)
def _stats(self):
return np.inf, np.inf, np.nan, np.nan
levy = levy_gen(a=0.0, name="levy")
class levy_l_gen(rv_continuous):
r"""A left-skewed Levy continuous random variable.
%(before_notes)s
See Also
--------
levy, levy_stable
Notes
-----
The probability density function for `levy_l` is:
.. math::
f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp{ \left(-\frac{1}{2|x|} \right)}
for :math:`x < 0`.
This is the same as the Levy-stable distribution with :math:`a=1/2` and
:math:`b=-1`.
%(after_notes)s
Examples
--------
>>> import numpy as np
>>> from scipy.stats import levy_l
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> mean, var, skew, kurt = levy_l.stats(moments='mvsk')
Display the probability density function (``pdf``):
>>> # `levy_l` is very heavy-tailed.
>>> # To show a nice plot, let's cut off the lower 40 percent.
>>> a, b = levy_l.ppf(0.4), levy_l.ppf(1)
>>> x = np.linspace(a, b, 100)
>>> ax.plot(x, levy_l.pdf(x),
... 'r-', lw=5, alpha=0.6, label='levy_l pdf')
Alternatively, the distribution object can be called (as a function)
to fix the shape, location and scale parameters. This returns a "frozen"
RV object holding the given parameters fixed.
Freeze the distribution and display the frozen ``pdf``:
>>> rv = levy_l()
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of ``cdf`` and ``ppf``:
>>> vals = levy_l.ppf([0.001, 0.5, 0.999])
>>> np.allclose([0.001, 0.5, 0.999], levy_l.cdf(vals))
True
Generate random numbers:
>>> r = levy_l.rvs(size=1000)
And compare the histogram:
>>> # manual binning to ignore the tail
>>> bins = np.concatenate(([np.min(r)], np.linspace(a, b, 20)))
>>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return []
def _pdf(self, x):
# levy_l.pdf(x) = 1 / (abs(x) * sqrt(2*pi*abs(x))) * exp(-1/(2*abs(x)))
ax = abs(x)
return 1/np.sqrt(2*np.pi*ax)/ax*np.exp(-1/(2*ax))
def _cdf(self, x):
ax = abs(x)
return 2 * _norm_cdf(1 / np.sqrt(ax)) - 1
def _sf(self, x):
ax = abs(x)
return 2 * _norm_sf(1 / np.sqrt(ax))
def _ppf(self, q):
val = _norm_ppf((q + 1.0) / 2)
return -1.0 / (val * val)
def _isf(self, p):
return -1/_norm_isf(p/2)**2
def _stats(self):
return np.inf, np.inf, np.nan, np.nan
levy_l = levy_l_gen(b=0.0, name="levy_l")
class logistic_gen(rv_continuous):
r"""A logistic (or Sech-squared) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `logistic` is:
.. math::
f(x) = \frac{\exp(-x)}
{(1+\exp(-x))^2}
`logistic` is a special case of `genlogistic` with ``c=1``.
Remark that the survival function (``logistic.sf``) is equal to the
Fermi-Dirac distribution describing fermionic statistics.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return random_state.logistic(size=size)
def _pdf(self, x):
# logistic.pdf(x) = exp(-x) / (1+exp(-x))**2
return np.exp(self._logpdf(x))
def _logpdf(self, x):
y = -np.abs(x)
return y - 2. * sc.log1p(np.exp(y))
def _cdf(self, x):
return sc.expit(x)
def _logcdf(self, x):
return sc.log_expit(x)
def _ppf(self, q):
return sc.logit(q)
def _sf(self, x):
return sc.expit(-x)
def _logsf(self, x):
return sc.log_expit(-x)
def _isf(self, q):
return -sc.logit(q)
def _stats(self):
return 0, np.pi*np.pi/3.0, 0, 6.0/5.0
def _entropy(self):
# https://en.wikipedia.org/wiki/Logistic_distribution
return 2.0
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
data, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
n = len(data)
# rv_continuous provided guesses
loc, scale = self._fitstart(data)
# these are trumped by user-provided guesses
loc, scale = kwds.get('loc', loc), kwds.get('scale', scale)
# the maximum likelihood estimators `a` and `b` of the location and
# scale parameters are roots of the two equations described in `func`.
# Source: Statistical Distributions, 3rd Edition. Evans, Hastings, and
# Peacock (2000), Page 130
def dl_dloc(loc, scale=fscale):
c = (data - loc) / scale
return np.sum(sc.expit(c)) - n/2
def dl_dscale(scale, loc=floc):
c = (data - loc) / scale
return np.sum(c*np.tanh(c/2)) - n
def func(params):
loc, scale = params
return dl_dloc(loc, scale), dl_dscale(scale, loc)
if fscale is not None and floc is None:
res = optimize.root(dl_dloc, (loc,))
loc = res.x[0]
scale = fscale
elif floc is not None and fscale is None:
res = optimize.root(dl_dscale, (scale,))
scale = res.x[0]
loc = floc
else:
res = optimize.root(func, (loc, scale))
loc, scale = res.x
# Note: gh-18176 reported data for which the reported MLE had
# `scale < 0`. To fix the bug, we return abs(scale). This is OK because
# `dl_dscale` and `dl_dloc` are even and odd functions of `scale`,
# respectively, so if `-scale` is a solution, so is `scale`.
scale = abs(scale)
return ((loc, scale) if res.success
else super().fit(data, *args, **kwds))
logistic = logistic_gen(name='logistic')
class loggamma_gen(rv_continuous):
r"""A log gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `loggamma` is:
.. math::
f(x, c) = \frac{\exp(c x - \exp(x))}
{\Gamma(c)}
for all :math:`x, c > 0`. Here, :math:`\Gamma` is the
gamma function (`scipy.special.gamma`).
`loggamma` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _rvs(self, c, size=None, random_state=None):
# Use the property of the gamma distribution Gamma(c)
# Gamma(c) ~ Gamma(c + 1)*U**(1/c),
# where U is uniform on [0, 1]. (See, e.g.,
# G. Marsaglia and W.W. Tsang, "A simple method for generating gamma
# variables", https://doi.org/10.1145/358407.358414)
# So
# log(Gamma(c)) ~ log(Gamma(c + 1)) + log(U)/c
# Generating a sample with this formulation is a bit slower
# than the more obvious log(Gamma(c)), but it avoids loss
# of precision when c << 1.
return (np.log(random_state.gamma(c + 1, size=size))
+ np.log(random_state.uniform(size=size))/c)
def _pdf(self, x, c):
# loggamma.pdf(x, c) = exp(c*x-exp(x)) / gamma(c)
return np.exp(c*x-np.exp(x)-sc.gammaln(c))
def _logpdf(self, x, c):
return c*x - np.exp(x) - sc.gammaln(c)
def _cdf(self, x, c):
# This function is gammainc(c, exp(x)), where gammainc(c, z) is
# the regularized incomplete gamma function.
# The first term in a series expansion of gamminc(c, z) is
# z**c/Gamma(c+1); see 6.5.29 of Abramowitz & Stegun (and refer
# back to 6.5.1, 6.5.2 and 6.5.4 for the relevant notation).
# This can also be found in the wikipedia article
# https://en.wikipedia.org/wiki/Incomplete_gamma_function.
# Here we use that formula when x is sufficiently negative that
# exp(x) will result in subnormal numbers and lose precision.
# We evaluate the log of the expression first to allow the possible
# cancellation of the terms in the division, and then exponentiate.
# That is,
# exp(x)**c/Gamma(c+1) = exp(log(exp(x)**c/Gamma(c+1)))
# = exp(c*x - gammaln(c+1))
return _lazywhere(x < _LOGXMIN, (x, c),
lambda x, c: np.exp(c*x - sc.gammaln(c+1)),
f2=lambda x, c: sc.gammainc(c, np.exp(x)))
def _ppf(self, q, c):
# The expression used when g < _XMIN inverts the one term expansion
# given in the comments of _cdf().
g = sc.gammaincinv(c, q)
return _lazywhere(g < _XMIN, (g, q, c),
lambda g, q, c: (np.log(q) + sc.gammaln(c+1))/c,
f2=lambda g, q, c: np.log(g))
def _sf(self, x, c):
# See the comments for _cdf() for how x < _LOGXMIN is handled.
return _lazywhere(x < _LOGXMIN, (x, c),
lambda x, c: -np.expm1(c*x - sc.gammaln(c+1)),
f2=lambda x, c: sc.gammaincc(c, np.exp(x)))
def _isf(self, q, c):
# The expression used when g < _XMIN inverts the complement of
# the one term expansion given in the comments of _cdf().
g = sc.gammainccinv(c, q)
return _lazywhere(g < _XMIN, (g, q, c),
lambda g, q, c: (np.log1p(-q) + sc.gammaln(c+1))/c,
f2=lambda g, q, c: np.log(g))
def _stats(self, c):
# See, for example, "A Statistical Study of Log-Gamma Distribution", by
# Ping Shing Chan (thesis, McMaster University, 1993).
mean = sc.digamma(c)
var = sc.polygamma(1, c)
skewness = sc.polygamma(2, c) / np.power(var, 1.5)
excess_kurtosis = sc.polygamma(3, c) / (var*var)
return mean, var, skewness, excess_kurtosis
def _entropy(self, c):
def regular(c):
h = sc.gammaln(c) - c * sc.digamma(c) + c
return h
def asymptotic(c):
# using asymptotic expansions for gammaln and psi (see gh-18093)
term = -0.5*np.log(c) + c**-1./6 - c**-3./90 + c**-5./210
h = norm._entropy() + term
return h
h = _lazywhere(c >= 45, (c, ), f=asymptotic, f2=regular)
return h
loggamma = loggamma_gen(name='loggamma')
class loglaplace_gen(rv_continuous):
r"""A log-Laplace continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `loglaplace` is:
.. math::
f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1} &\text{for } 0 < x < 1\\
\frac{c}{2} x^{-c-1} &\text{for } x \ge 1
\end{cases}
for :math:`c > 0`.
`loglaplace` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
Suppose a random variable ``X`` follows the Laplace distribution with
location ``a`` and scale ``b``. Then ``Y = exp(X)`` follows the
log-Laplace distribution with ``c = 1 / b`` and ``scale = exp(a)``.
References
----------
T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model",
The Mathematical Scientist, vol. 28, pp. 49-60, 2003.
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# loglaplace.pdf(x, c) = c / 2 * x**(c-1), for 0 < x < 1
# = c / 2 * x**(-c-1), for x >= 1
cd2 = c/2.0
c = np.where(x < 1, c, -c)
return cd2*x**(c-1)
def _cdf(self, x, c):
return np.where(x < 1, 0.5*x**c, 1-0.5*x**(-c))
def _sf(self, x, c):
return np.where(x < 1, 1 - 0.5*x**c, 0.5*x**(-c))
def _ppf(self, q, c):
return np.where(q < 0.5, (2.0*q)**(1.0/c), (2*(1.0-q))**(-1.0/c))
def _isf(self, q, c):
return np.where(q > 0.5, (2.0*(1.0 - q))**(1.0/c), (2*q)**(-1.0/c))
def _munp(self, n, c):
with np.errstate(divide='ignore'):
c2, n2 = c**2, n**2
return np.where(n2 < c2, c2 / (c2 - n2), np.inf)
def _entropy(self, c):
return np.log(2.0/c) + 1.0
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
data, fc, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
# Specialize MLE only when location is known.
if floc is None:
return super(type(self), self).fit(data, *args, **kwds)
# Raise an error if any observation has zero likelihood.
if np.any(data <= floc):
raise FitDataError("loglaplace", lower=floc, upper=np.inf)
# Remove location from data.
if floc != 0:
data = data - floc
# When location is zero, the log-Laplace distribution is related to
# the Laplace distribution in that if X ~ Laplace(loc=a, scale=b),
# then Y = exp(X) ~ LogLaplace(c=1/b, loc=0, scale=exp(a)). It can
# be shown that the MLE for Y is the same as the MLE for X = ln(Y).
# Therefore, we reuse the formulas from laplace.fit() and transform
# the result back into log-laplace's parameter space.
a, b = laplace.fit(np.log(data),
floc=np.log(fscale) if fscale is not None else None,
fscale=1/fc if fc is not None else None,
method='mle')
loc = floc
scale = np.exp(a) if fscale is None else fscale
c = 1 / b if fc is None else fc
return c, loc, scale
loglaplace = loglaplace_gen(a=0.0, name='loglaplace')
def _lognorm_logpdf(x, s):
return _lazywhere(x != 0, (x, s),
lambda x, s: (-np.log(x)**2 / (2 * s**2)
- np.log(s * x * np.sqrt(2 * np.pi))),
-np.inf)
class lognorm_gen(rv_continuous):
r"""A lognormal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `lognorm` is:
.. math::
f(x, s) = \frac{1}{s x \sqrt{2\pi}}
\exp\left(-\frac{\log^2(x)}{2s^2}\right)
for :math:`x > 0`, :math:`s > 0`.
`lognorm` takes ``s`` as a shape parameter for :math:`s`.
%(after_notes)s
Suppose a normally distributed random variable ``X`` has mean ``mu`` and
standard deviation ``sigma``. Then ``Y = exp(X)`` is lognormally
distributed with ``s = sigma`` and ``scale = exp(mu)``.
%(example)s
The logarithm of a log-normally distributed random variable is
normally distributed:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> fig, ax = plt.subplots(1, 1)
>>> mu, sigma = 2, 0.5
>>> X = stats.norm(loc=mu, scale=sigma)
>>> Y = stats.lognorm(s=sigma, scale=np.exp(mu))
>>> x = np.linspace(*X.interval(0.999))
>>> y = Y.rvs(size=10000)
>>> ax.plot(x, X.pdf(x), label='X (pdf)')
>>> ax.hist(np.log(y), density=True, bins=x, label='log(Y) (histogram)')
>>> ax.legend()
>>> plt.show()
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return [_ShapeInfo("s", False, (0, np.inf), (False, False))]
def _rvs(self, s, size=None, random_state=None):
return np.exp(s * random_state.standard_normal(size))
def _pdf(self, x, s):
# lognorm.pdf(x, s) = 1 / (s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2)
return np.exp(self._logpdf(x, s))
def _logpdf(self, x, s):
return _lognorm_logpdf(x, s)
def _cdf(self, x, s):
return _norm_cdf(np.log(x) / s)
def _logcdf(self, x, s):
return _norm_logcdf(np.log(x) / s)
def _ppf(self, q, s):
return np.exp(s * _norm_ppf(q))
def _sf(self, x, s):
return _norm_sf(np.log(x) / s)
def _logsf(self, x, s):
return _norm_logsf(np.log(x) / s)
def _isf(self, q, s):
return np.exp(s * _norm_isf(q))
def _stats(self, s):
p = np.exp(s*s)
mu = np.sqrt(p)
mu2 = p*(p-1)
g1 = np.sqrt(p-1)*(2+p)
g2 = np.polyval([1, 2, 3, 0, -6.0], p)
return mu, mu2, g1, g2
def _entropy(self, s):
return 0.5 * (1 + np.log(2*np.pi) + 2 * np.log(s))
@_call_super_mom
@extend_notes_in_docstring(rv_continuous, notes="""\
When `method='MLE'` and
the location parameter is fixed by using the `floc` argument,
this function uses explicit formulas for the maximum likelihood
estimation of the log-normal shape and scale parameters, so the
`optimizer`, `loc` and `scale` keyword arguments are ignored.
If the location is free, a likelihood maximum is found by
setting its partial derivative wrt to location to 0, and
solving by substituting the analytical expressions of shape
and scale (or provided parameters).
See, e.g., equation 3.1 in
A. Clifford Cohen & Betty Jones Whitten (1980)
Estimation in the Three-Parameter Lognormal Distribution,
Journal of the American Statistical Association, 75:370, 399-404
https://doi.org/10.2307/2287466
\n\n""")
def fit(self, data, *args, **kwds):
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
parameters = _check_fit_input_parameters(self, data, args, kwds)
data, fshape, floc, fscale = parameters
data_min = np.min(data)
def get_shape_scale(loc):
# Calculate maximum likelihood scale and shape with analytical
# formulas unless provided by the user
if fshape is None or fscale is None:
lndata = np.log(data - loc)
scale = fscale or np.exp(lndata.mean())
shape = fshape or np.sqrt(np.mean((lndata - np.log(scale))**2))
return shape, scale
def dL_dLoc(loc):
# Derivative of (positive) LL w.r.t. loc
shape, scale = get_shape_scale(loc)
shifted = data - loc
return np.sum((1 + np.log(shifted/scale)/shape**2)/shifted)
def ll(loc):
# (Positive) log-likelihood
shape, scale = get_shape_scale(loc)
return -self.nnlf((shape, loc, scale), data)
if floc is None:
# The location must be less than the minimum of the data.
# Back off a bit to avoid numerical issues.
spacing = np.spacing(data_min)
rbrack = data_min - spacing
# Find the right end of the bracket by successive doubling of the
# distance to data_min. We're interested in a maximum LL, so the
# slope dL_dLoc_rbrack should be negative at the right end.
# optimization for later: share shape, scale
dL_dLoc_rbrack = dL_dLoc(rbrack)
ll_rbrack = ll(rbrack)
delta = 2 * spacing # 2 * (data_min - rbrack)
while dL_dLoc_rbrack >= -1e-6:
rbrack = data_min - delta
dL_dLoc_rbrack = dL_dLoc(rbrack)
delta *= 2
if not np.isfinite(rbrack) or not np.isfinite(dL_dLoc_rbrack):
# If we never find a negative slope, either we missed it or the
# slope is always positive. It's usually the latter,
# which means
# loc = data_min - spacing
# But sometimes when shape and/or scale are fixed there are
# other issues, so be cautious.
return super().fit(data, *args, **kwds)
# Now find the left end of the bracket. Guess is `rbrack-1`
# unless that is too small of a difference to resolve. Double
# the size of the interval until the left end is found.
lbrack = np.minimum(np.nextafter(rbrack, -np.inf), rbrack-1)
dL_dLoc_lbrack = dL_dLoc(lbrack)
delta = 2 * (rbrack - lbrack)
while (np.isfinite(lbrack) and np.isfinite(dL_dLoc_lbrack)
and np.sign(dL_dLoc_lbrack) == np.sign(dL_dLoc_rbrack)):
lbrack = rbrack - delta
dL_dLoc_lbrack = dL_dLoc(lbrack)
delta *= 2
# I don't recall observing this, but just in case...
if not np.isfinite(lbrack) or not np.isfinite(dL_dLoc_lbrack):
return super().fit(data, *args, **kwds)
# If we have a valid bracket, find the root
res = root_scalar(dL_dLoc, bracket=(lbrack, rbrack))
if not res.converged:
return super().fit(data, *args, **kwds)
# If the slope was positive near the minimum of the data,
# the maximum LL could be there instead of at the root. Compare
# the LL of the two points to decide.
ll_root = ll(res.root)
loc = res.root if ll_root > ll_rbrack else data_min-spacing
else:
if floc >= data_min:
raise FitDataError("lognorm", lower=0., upper=np.inf)
loc = floc
shape, scale = get_shape_scale(loc)
if not (self._argcheck(shape) and scale > 0):
return super().fit(data, *args, **kwds)
return shape, loc, scale
lognorm = lognorm_gen(a=0.0, name='lognorm')
class gibrat_gen(rv_continuous):
r"""A Gibrat continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gibrat` is:
.. math::
f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2)
`gibrat` is a special case of `lognorm` with ``s=1``.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return np.exp(random_state.standard_normal(size))
def _pdf(self, x):
# gibrat.pdf(x) = 1/(x*sqrt(2*pi)) * exp(-1/2*(log(x))**2)
return np.exp(self._logpdf(x))
def _logpdf(self, x):
return _lognorm_logpdf(x, 1.0)
def _cdf(self, x):
return _norm_cdf(np.log(x))
def _ppf(self, q):
return np.exp(_norm_ppf(q))
def _sf(self, x):
return _norm_sf(np.log(x))
def _isf(self, p):
return np.exp(_norm_isf(p))
def _stats(self):
p = np.e
mu = np.sqrt(p)
mu2 = p * (p - 1)
g1 = np.sqrt(p - 1) * (2 + p)
g2 = np.polyval([1, 2, 3, 0, -6.0], p)
return mu, mu2, g1, g2
def _entropy(self):
return 0.5 * np.log(2 * np.pi) + 0.5
gibrat = gibrat_gen(a=0.0, name='gibrat')
class maxwell_gen(rv_continuous):
r"""A Maxwell continuous random variable.
%(before_notes)s
Notes
-----
A special case of a `chi` distribution, with ``df=3``, ``loc=0.0``,
and given ``scale = a``, where ``a`` is the parameter used in the
Mathworld description [1]_.
The probability density function for `maxwell` is:
.. math::
f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2)
for :math:`x >= 0`.
%(after_notes)s
References
----------
.. [1] http://mathworld.wolfram.com/MaxwellDistribution.html
%(example)s
"""
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return chi.rvs(3.0, size=size, random_state=random_state)
def _pdf(self, x):
# maxwell.pdf(x) = sqrt(2/pi)x**2 * exp(-x**2/2)
return _SQRT_2_OVER_PI*x*x*np.exp(-x*x/2.0)
def _logpdf(self, x):
# Allow x=0 without 'divide by zero' warnings
with np.errstate(divide='ignore'):
return _LOG_SQRT_2_OVER_PI + 2*np.log(x) - 0.5*x*x
def _cdf(self, x):
return sc.gammainc(1.5, x*x/2.0)
def _ppf(self, q):
return np.sqrt(2*sc.gammaincinv(1.5, q))
def _sf(self, x):
return sc.gammaincc(1.5, x*x/2.0)
def _isf(self, q):
return np.sqrt(2*sc.gammainccinv(1.5, q))
def _stats(self):
val = 3*np.pi-8
return (2*np.sqrt(2.0/np.pi),
3-8/np.pi,
np.sqrt(2)*(32-10*np.pi)/val**1.5,
(-12*np.pi*np.pi + 160*np.pi - 384) / val**2.0)
def _entropy(self):
return _EULER + 0.5*np.log(2*np.pi)-0.5
maxwell = maxwell_gen(a=0.0, name='maxwell')
class mielke_gen(rv_continuous):
r"""A Mielke Beta-Kappa / Dagum continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `mielke` is:
.. math::
f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}
for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes
called Dagum distribution ([2]_). It was already defined in [3]_, called
a Burr Type III distribution (`burr` with parameters ``c=s`` and
``d=k/s``).
`mielke` takes ``k`` and ``s`` as shape parameters.
%(after_notes)s
References
----------
.. [1] Mielke, P.W., 1973 "Another Family of Distributions for Describing
and Analyzing Precipitation Data." J. Appl. Meteor., 12, 275-280
.. [2] Dagum, C., 1977 "A new model for personal income distribution."
Economie Appliquee, 33, 327-367.
.. [3] Burr, I. W. "Cumulative frequency functions", Annals of
Mathematical Statistics, 13(2), pp 215-232 (1942).
%(example)s
"""
def _shape_info(self):
ik = _ShapeInfo("k", False, (0, np.inf), (False, False))
i_s = _ShapeInfo("s", False, (0, np.inf), (False, False))
return [ik, i_s]
def _pdf(self, x, k, s):
return k*x**(k-1.0) / (1.0+x**s)**(1.0+k*1.0/s)
def _logpdf(self, x, k, s):
# Allow x=0 without 'divide by zero' warnings.
with np.errstate(divide='ignore'):
return np.log(k) + np.log(x)*(k - 1) - np.log1p(x**s)*(1 + k/s)
def _cdf(self, x, k, s):
return x**k / (1.0+x**s)**(k*1.0/s)
def _ppf(self, q, k, s):
qsk = pow(q, s*1.0/k)
return pow(qsk/(1.0-qsk), 1.0/s)
def _munp(self, n, k, s):
def nth_moment(n, k, s):
# n-th moment is defined for -k < n < s
return sc.gamma((k+n)/s)*sc.gamma(1-n/s)/sc.gamma(k/s)
return _lazywhere(n < s, (n, k, s), nth_moment, np.inf)
mielke = mielke_gen(a=0.0, name='mielke')
class kappa4_gen(rv_continuous):
r"""Kappa 4 parameter distribution.
%(before_notes)s
Notes
-----
The probability density function for kappa4 is:
.. math::
f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}
if :math:`h` and :math:`k` are not equal to 0.
If :math:`h` or :math:`k` are zero then the pdf can be simplified:
h = 0 and k != 0::
kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
exp(-(1.0 - k*x)**(1.0/k))
h != 0 and k = 0::
kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)
h = 0 and k = 0::
kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))
kappa4 takes :math:`h` and :math:`k` as shape parameters.
The kappa4 distribution returns other distributions when certain
:math:`h` and :math:`k` values are used.
+------+-------------+----------------+------------------+
| h | k=0.0 | k=1.0 | -inf<=k<=inf |
+======+=============+================+==================+
| -1.0 | Logistic | | Generalized |
| | | | Logistic(1) |
| | | | |
| | logistic(x) | | |
+------+-------------+----------------+------------------+
| 0.0 | Gumbel | Reverse | Generalized |
| | | Exponential(2) | Extreme Value |
| | | | |
| | gumbel_r(x) | | genextreme(x, k) |
+------+-------------+----------------+------------------+
| 1.0 | Exponential | Uniform | Generalized |
| | | | Pareto |
| | | | |
| | expon(x) | uniform(x) | genpareto(x, -k) |
+------+-------------+----------------+------------------+
(1) There are at least five generalized logistic distributions.
Four are described here:
https://en.wikipedia.org/wiki/Generalized_logistic_distribution
The "fifth" one is the one kappa4 should match which currently
isn't implemented in scipy:
https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution
https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html
(2) This distribution is currently not in scipy.
References
----------
J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect
to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate
Faculty of the Louisiana State University and Agricultural and Mechanical
College, (August, 2004),
https://digitalcommons.lsu.edu/gradschool_dissertations/3672
J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res.
Develop. 38 (3), 25 1-258 (1994).
B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao
Site in the Chi River Basin, Thailand", Journal of Water Resource and
Protection, vol. 4, 866-869, (2012).
:doi:`10.4236/jwarp.2012.410101`
C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A
Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March
2000).
http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf
%(after_notes)s
%(example)s
"""
def _argcheck(self, h, k):
shape = np.broadcast_arrays(h, k)[0].shape
return np.full(shape, fill_value=True)
def _shape_info(self):
ih = _ShapeInfo("h", False, (-np.inf, np.inf), (False, False))
ik = _ShapeInfo("k", False, (-np.inf, np.inf), (False, False))
return [ih, ik]
def _get_support(self, h, k):
condlist = [np.logical_and(h > 0, k > 0),
np.logical_and(h > 0, k == 0),
np.logical_and(h > 0, k < 0),
np.logical_and(h <= 0, k > 0),
np.logical_and(h <= 0, k == 0),
np.logical_and(h <= 0, k < 0)]
def f0(h, k):
return (1.0 - np.float_power(h, -k))/k
def f1(h, k):
return np.log(h)
def f3(h, k):
a = np.empty(np.shape(h))
a[:] = -np.inf
return a
def f5(h, k):
return 1.0/k
_a = _lazyselect(condlist,
[f0, f1, f0, f3, f3, f5],
[h, k],
default=np.nan)
def f0(h, k):
return 1.0/k
def f1(h, k):
a = np.empty(np.shape(h))
a[:] = np.inf
return a
_b = _lazyselect(condlist,
[f0, f1, f1, f0, f1, f1],
[h, k],
default=np.nan)
return _a, _b
def _pdf(self, x, h, k):
# kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
# (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1)
return np.exp(self._logpdf(x, h, k))
def _logpdf(self, x, h, k):
condlist = [np.logical_and(h != 0, k != 0),
np.logical_and(h == 0, k != 0),
np.logical_and(h != 0, k == 0),
np.logical_and(h == 0, k == 0)]
def f0(x, h, k):
'''pdf = (1.0 - k*x)**(1.0/k - 1.0)*(
1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0)
logpdf = ...
'''
return (sc.xlog1py(1.0/k - 1.0, -k*x) +
sc.xlog1py(1.0/h - 1.0, -h*(1.0 - k*x)**(1.0/k)))
def f1(x, h, k):
'''pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-(
1.0 - k*x)**(1.0/k))
logpdf = ...
'''
return sc.xlog1py(1.0/k - 1.0, -k*x) - (1.0 - k*x)**(1.0/k)
def f2(x, h, k):
'''pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0)
logpdf = ...
'''
return -x + sc.xlog1py(1.0/h - 1.0, -h*np.exp(-x))
def f3(x, h, k):
'''pdf = np.exp(-x-np.exp(-x))
logpdf = ...
'''
return -x - np.exp(-x)
return _lazyselect(condlist,
[f0, f1, f2, f3],
[x, h, k],
default=np.nan)
def _cdf(self, x, h, k):
return np.exp(self._logcdf(x, h, k))
def _logcdf(self, x, h, k):
condlist = [np.logical_and(h != 0, k != 0),
np.logical_and(h == 0, k != 0),
np.logical_and(h != 0, k == 0),
np.logical_and(h == 0, k == 0)]
def f0(x, h, k):
'''cdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h)
logcdf = ...
'''
return (1.0/h)*sc.log1p(-h*(1.0 - k*x)**(1.0/k))
def f1(x, h, k):
'''cdf = np.exp(-(1.0 - k*x)**(1.0/k))
logcdf = ...
'''
return -(1.0 - k*x)**(1.0/k)
def f2(x, h, k):
'''cdf = (1.0 - h*np.exp(-x))**(1.0/h)
logcdf = ...
'''
return (1.0/h)*sc.log1p(-h*np.exp(-x))
def f3(x, h, k):
'''cdf = np.exp(-np.exp(-x))
logcdf = ...
'''
return -np.exp(-x)
return _lazyselect(condlist,
[f0, f1, f2, f3],
[x, h, k],
default=np.nan)
def _ppf(self, q, h, k):
condlist = [np.logical_and(h != 0, k != 0),
np.logical_and(h == 0, k != 0),
np.logical_and(h != 0, k == 0),
np.logical_and(h == 0, k == 0)]
def f0(q, h, k):
return 1.0/k*(1.0 - ((1.0 - (q**h))/h)**k)
def f1(q, h, k):
return 1.0/k*(1.0 - (-np.log(q))**k)
def f2(q, h, k):
'''ppf = -np.log((1.0 - (q**h))/h)
'''
return -sc.log1p(-(q**h)) + np.log(h)
def f3(q, h, k):
return -np.log(-np.log(q))
return _lazyselect(condlist,
[f0, f1, f2, f3],
[q, h, k],
default=np.nan)
def _get_stats_info(self, h, k):
condlist = [
np.logical_and(h < 0, k >= 0),
k < 0,
]
def f0(h, k):
return (-1.0/h*k).astype(int)
def f1(h, k):
return (-1.0/k).astype(int)
return _lazyselect(condlist, [f0, f1], [h, k], default=5)
def _stats(self, h, k):
maxr = self._get_stats_info(h, k)
outputs = [None if np.any(r < maxr) else np.nan for r in range(1, 5)]
return outputs[:]
def _mom1_sc(self, m, *args):
maxr = self._get_stats_info(args[0], args[1])
if m >= maxr:
return np.nan
return integrate.quad(self._mom_integ1, 0, 1, args=(m,)+args)[0]
kappa4 = kappa4_gen(name='kappa4')
class kappa3_gen(rv_continuous):
r"""Kappa 3 parameter distribution.
%(before_notes)s
Notes
-----
The probability density function for `kappa3` is:
.. math::
f(x, a) = a (a + x^a)^{-(a + 1)/a}
for :math:`x > 0` and :math:`a > 0`.
`kappa3` takes ``a`` as a shape parameter for :math:`a`.
References
----------
P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum
Likelihood and Likelihood Ratio Tests", Methods in Weather Research,
701-707, (September, 1973),
:doi:`10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2`
B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the
Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2,
415-419 (2012), :doi:`10.4236/ojs.2012.24050`
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("a", False, (0, np.inf), (False, False))]
def _pdf(self, x, a):
# kappa3.pdf(x, a) = a*(a + x**a)**(-(a + 1)/a), for x > 0
return a*(a + x**a)**(-1.0/a-1)
def _cdf(self, x, a):
return x*(a + x**a)**(-1.0/a)
def _sf(self, x, a):
x, a = np.broadcast_arrays(x, a) # some code paths pass scalars
sf = super()._sf(x, a)
# When the SF is small, another formulation is typically more accurate.
# However, it blows up for large `a`, so use it only if it also returns
# a small value of the SF.
cutoff = 0.01
i = sf < cutoff
sf2 = -sc.expm1(sc.xlog1py(-1.0 / a[i], a[i] * x[i]**-a[i]))
i2 = sf2 > cutoff
sf2[i2] = sf[i][i2] # replace bad values with original values
sf[i] = sf2
return sf
def _ppf(self, q, a):
return (a/(q**-a - 1.0))**(1.0/a)
def _isf(self, q, a):
lg = sc.xlog1py(-a, -q)
denom = sc.expm1(lg)
return (a / denom)**(1.0 / a)
def _stats(self, a):
outputs = [None if np.any(i < a) else np.nan for i in range(1, 5)]
return outputs[:]
def _mom1_sc(self, m, *args):
if np.any(m >= args[0]):
return np.nan
return integrate.quad(self._mom_integ1, 0, 1, args=(m,)+args)[0]
kappa3 = kappa3_gen(a=0.0, name='kappa3')
class moyal_gen(rv_continuous):
r"""A Moyal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `moyal` is:
.. math::
f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}
for a real number :math:`x`.
%(after_notes)s
This distribution has utility in high-energy physics and radiation
detection. It describes the energy loss of a charged relativistic
particle due to ionization of the medium [1]_. It also provides an
approximation for the Landau distribution. For an in depth description
see [2]_. For additional description, see [3]_.
References
----------
.. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations",
The London, Edinburgh, and Dublin Philosophical Magazine
and Journal of Science, vol 46, 263-280, (1955).
:doi:`10.1080/14786440308521076` (gated)
.. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution",
International Journal of Research and Reviews in Applied Sciences,
vol 10, 171-192, (2012).
http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf
.. [3] C. Walck, "Handbook on Statistical Distributions for
Experimentalists; International Report SUF-PFY/96-01", Chapter 26,
University of Stockholm: Stockholm, Sweden, (2007).
http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf
.. versionadded:: 1.1.0
%(example)s
"""
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
u1 = gamma.rvs(a=0.5, scale=2, size=size,
random_state=random_state)
return -np.log(u1)
def _pdf(self, x):
return np.exp(-0.5 * (x + np.exp(-x))) / np.sqrt(2*np.pi)
def _cdf(self, x):
return sc.erfc(np.exp(-0.5 * x) / np.sqrt(2))
def _sf(self, x):
return sc.erf(np.exp(-0.5 * x) / np.sqrt(2))
def _ppf(self, x):
return -np.log(2 * sc.erfcinv(x)**2)
def _stats(self):
mu = np.log(2) + np.euler_gamma
mu2 = np.pi**2 / 2
g1 = 28 * np.sqrt(2) * sc.zeta(3) / np.pi**3
g2 = 4.
return mu, mu2, g1, g2
def _munp(self, n):
if n == 1.0:
return np.log(2) + np.euler_gamma
elif n == 2.0:
return np.pi**2 / 2 + (np.log(2) + np.euler_gamma)**2
elif n == 3.0:
tmp1 = 1.5 * np.pi**2 * (np.log(2)+np.euler_gamma)
tmp2 = (np.log(2)+np.euler_gamma)**3
tmp3 = 14 * sc.zeta(3)
return tmp1 + tmp2 + tmp3
elif n == 4.0:
tmp1 = 4 * 14 * sc.zeta(3) * (np.log(2) + np.euler_gamma)
tmp2 = 3 * np.pi**2 * (np.log(2) + np.euler_gamma)**2
tmp3 = (np.log(2) + np.euler_gamma)**4
tmp4 = 7 * np.pi**4 / 4
return tmp1 + tmp2 + tmp3 + tmp4
else:
# return generic for higher moments
# return rv_continuous._mom1_sc(self, n, b)
return self._mom1_sc(n)
moyal = moyal_gen(name="moyal")
class nakagami_gen(rv_continuous):
r"""A Nakagami continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `nakagami` is:
.. math::
f(x, \nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2)
for :math:`x >= 0`, :math:`\nu > 0`. The distribution was introduced in
[2]_, see also [1]_ for further information.
`nakagami` takes ``nu`` as a shape parameter for :math:`\nu`.
%(after_notes)s
References
----------
.. [1] "Nakagami distribution", Wikipedia
https://en.wikipedia.org/wiki/Nakagami_distribution
.. [2] M. Nakagami, "The m-distribution - A general formula of intensity
distribution of rapid fading", Statistical methods in radio wave
propagation, Pergamon Press, 1960, 3-36.
:doi:`10.1016/B978-0-08-009306-2.50005-4`
%(example)s
"""
def _argcheck(self, nu):
return nu > 0
def _shape_info(self):
return [_ShapeInfo("nu", False, (0, np.inf), (False, False))]
def _pdf(self, x, nu):
return np.exp(self._logpdf(x, nu))
def _logpdf(self, x, nu):
# nakagami.pdf(x, nu) = 2 * nu**nu / gamma(nu) *
# x**(2*nu-1) * exp(-nu*x**2)
return (np.log(2) + sc.xlogy(nu, nu) - sc.gammaln(nu) +
sc.xlogy(2*nu - 1, x) - nu*x**2)
def _cdf(self, x, nu):
return sc.gammainc(nu, nu*x*x)
def _ppf(self, q, nu):
return np.sqrt(1.0/nu*sc.gammaincinv(nu, q))
def _sf(self, x, nu):
return sc.gammaincc(nu, nu*x*x)
def _isf(self, p, nu):
return np.sqrt(1/nu * sc.gammainccinv(nu, p))
def _stats(self, nu):
mu = sc.poch(nu, 0.5)/np.sqrt(nu)
mu2 = 1.0-mu*mu
g1 = mu * (1 - 4*nu*mu2) / 2.0 / nu / np.power(mu2, 1.5)
g2 = -6*mu**4*nu + (8*nu-2)*mu**2-2*nu + 1
g2 /= nu*mu2**2.0
return mu, mu2, g1, g2
def _entropy(self, nu):
shape = np.shape(nu)
# because somehow this isn't taken care of by the infrastructure...
nu = np.atleast_1d(nu)
A = sc.gammaln(nu)
B = nu - (nu - 0.5) * sc.digamma(nu)
C = -0.5 * np.log(nu) - np.log(2)
h = A + B + C
# This is the asymptotic sum of A and B (see gh-17868)
norm_entropy = stats.norm._entropy()
# Above, this is lost to rounding error for large nu, so use the
# asymptotic sum when the approximation becomes accurate
i = nu > 5e4 # roundoff error ~ approximation error
# -1 / (12 * nu) is the O(1/nu) term; see gh-17929
h[i] = C[i] + norm_entropy - 1/(12*nu[i])
return h.reshape(shape)[()]
def _rvs(self, nu, size=None, random_state=None):
# this relationship can be found in [1] or by a direct calculation
return np.sqrt(random_state.standard_gamma(nu, size=size) / nu)
def _fitstart(self, data, args=None):
if isinstance(data, CensoredData):
data = data._uncensor()
if args is None:
args = (1.0,) * self.numargs
# Analytical justified estimates
# see: https://docs.scipy.org/doc/scipy/reference/tutorial/stats/continuous_nakagami.html
loc = np.min(data)
scale = np.sqrt(np.sum((data - loc)**2) / len(data))
return args + (loc, scale)
nakagami = nakagami_gen(a=0.0, name="nakagami")
# 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 = sc.xlogy(df2/2.0, x/nc) - 0.5*(xs - ns)**2
corr = sc.ive(df2, xs*ns) / 2.0
# Return res + np.log(corr) avoiding np.log(0)
return _lazywhere(
corr > 0,
(res, corr),
f=lambda r, c: r + np.log(c),
fillvalue=-np.inf)
class ncx2_gen(rv_continuous):
r"""A non-central chi-squared continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `ncx2` is:
.. math::
f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2)
(x/\lambda)^{(k-2)/4} I_{(k-2)/2}(\sqrt{\lambda x})
for :math:`x >= 0`, :math:`k > 0` and :math:`\lambda \ge 0`.
:math:`k` specifies the degrees of freedom (denoted ``df`` in the
implementation) and :math:`\lambda` is the non-centrality parameter
(denoted ``nc`` in the implementation). :math:`I_\nu` denotes the
modified Bessel function of first order of degree :math:`\nu`
(`scipy.special.iv`).
`ncx2` takes ``df`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, df, nc):
return (df > 0) & np.isfinite(df) & (nc >= 0)
def _shape_info(self):
idf = _ShapeInfo("df", False, (0, np.inf), (False, False))
inc = _ShapeInfo("nc", False, (0, np.inf), (True, False))
return [idf, inc]
def _rvs(self, df, nc, size=None, random_state=None):
return random_state.noncentral_chisquare(df, nc, size)
def _logpdf(self, x, df, nc):
cond = np.ones_like(x, dtype=bool) & (nc != 0)
return _lazywhere(cond, (x, df, nc), f=_ncx2_log_pdf,
f2=lambda x, df, _: chi2._logpdf(x, df))
def _pdf(self, x, df, nc):
cond = np.ones_like(x, dtype=bool) & (nc != 0)
with np.errstate(over='ignore'): # see gh-17432
return _lazywhere(cond, (x, df, nc), f=_boost._ncx2_pdf,
f2=lambda x, df, _: chi2._pdf(x, df))
def _cdf(self, x, df, nc):
cond = np.ones_like(x, dtype=bool) & (nc != 0)
with np.errstate(over='ignore'): # see gh-17432
return _lazywhere(cond, (x, df, nc), f=_boost._ncx2_cdf,
f2=lambda x, df, _: chi2._cdf(x, df))
def _ppf(self, q, df, nc):
cond = np.ones_like(q, dtype=bool) & (nc != 0)
with np.errstate(over='ignore'): # see gh-17432
return _lazywhere(cond, (q, df, nc), f=_boost._ncx2_ppf,
f2=lambda x, df, _: chi2._ppf(x, df))
def _sf(self, x, df, nc):
cond = np.ones_like(x, dtype=bool) & (nc != 0)
with np.errstate(over='ignore'): # see gh-17432
return _lazywhere(cond, (x, df, nc), f=_boost._ncx2_sf,
f2=lambda x, df, _: chi2._sf(x, df))
def _isf(self, x, df, nc):
cond = np.ones_like(x, dtype=bool) & (nc != 0)
with np.errstate(over='ignore'): # see gh-17432
return _lazywhere(cond, (x, df, nc), f=_boost._ncx2_isf,
f2=lambda x, df, _: chi2._isf(x, df))
def _stats(self, df, nc):
return (
_boost._ncx2_mean(df, nc),
_boost._ncx2_variance(df, nc),
_boost._ncx2_skewness(df, nc),
_boost._ncx2_kurtosis_excess(df, nc),
)
ncx2 = ncx2_gen(a=0.0, name='ncx2')
class ncf_gen(rv_continuous):
r"""A non-central F distribution continuous random variable.
%(before_notes)s
See Also
--------
scipy.stats.f : Fisher distribution
Notes
-----
The probability density function for `ncf` is:
.. math::
f(x, n_1, n_2, \lambda) =
\exp\left(\frac{\lambda}{2} +
\lambda n_1 \frac{x}{2(n_1 x + n_2)}
\right)
n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\
(n_2 + n_1 x)^{-(n_1 + n_2)/2}
\gamma(n_1/2) \gamma(1 + n_2/2) \\
\frac{L^{\frac{n_1}{2}-1}_{n_2/2}
\left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)}
{B(n_1/2, n_2/2)
\gamma\left(\frac{n_1 + n_2}{2}\right)}
for :math:`n_1, n_2 > 0`, :math:`\lambda \ge 0`. Here :math:`n_1` is the
degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in
the denominator, :math:`\lambda` the non-centrality parameter,
:math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a
generalized Laguerre polynomial and :math:`B` is the beta function.
`ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters. If ``nc=0``,
the distribution becomes equivalent to the Fisher distribution.
%(after_notes)s
%(example)s
"""
def _argcheck(self, df1, df2, nc):
return (df1 > 0) & (df2 > 0) & (nc >= 0)
def _shape_info(self):
idf1 = _ShapeInfo("df1", False, (0, np.inf), (False, False))
idf2 = _ShapeInfo("df2", False, (0, np.inf), (False, False))
inc = _ShapeInfo("nc", False, (0, np.inf), (True, False))
return [idf1, idf2, inc]
def _rvs(self, dfn, dfd, nc, size=None, random_state=None):
return random_state.noncentral_f(dfn, dfd, nc, size)
def _pdf(self, x, dfn, dfd, nc):
# ncf.pdf(x, df1, df2, nc) = exp(nc/2 + nc*df1*x/(2*(df1*x+df2))) *
# df1**(df1/2) * df2**(df2/2) * x**(df1/2-1) *
# (df2+df1*x)**(-(df1+df2)/2) *
# gamma(df1/2)*gamma(1+df2/2) *
# L^{v1/2-1}^{v2/2}(-nc*v1*x/(2*(v1*x+v2))) /
# (B(v1/2, v2/2) * gamma((v1+v2)/2))
return _boost._ncf_pdf(x, dfn, dfd, nc)
def _cdf(self, x, dfn, dfd, nc):
return _boost._ncf_cdf(x, dfn, dfd, nc)
def _ppf(self, q, dfn, dfd, nc):
with np.errstate(over='ignore'): # see gh-17432
return _boost._ncf_ppf(q, dfn, dfd, nc)
def _sf(self, x, dfn, dfd, nc):
return _boost._ncf_sf(x, dfn, dfd, nc)
def _isf(self, x, dfn, dfd, nc):
with np.errstate(over='ignore'): # see gh-17432
return _boost._ncf_isf(x, dfn, dfd, nc)
def _munp(self, n, dfn, dfd, nc):
val = (dfn * 1.0/dfd)**n
term = sc.gammaln(n+0.5*dfn) + sc.gammaln(0.5*dfd-n) - sc.gammaln(dfd*0.5)
val *= np.exp(-nc / 2.0+term)
val *= sc.hyp1f1(n+0.5*dfn, 0.5*dfn, 0.5*nc)
return val
def _stats(self, dfn, dfd, nc, moments='mv'):
mu = _boost._ncf_mean(dfn, dfd, nc)
mu2 = _boost._ncf_variance(dfn, dfd, nc)
g1 = _boost._ncf_skewness(dfn, dfd, nc) if 's' in moments else None
g2 = _boost._ncf_kurtosis_excess(
dfn, dfd, nc) if 'k' in moments else None
return mu, mu2, g1, g2
ncf = ncf_gen(a=0.0, name='ncf')
class t_gen(rv_continuous):
r"""A Student's t continuous random variable.
For the noncentral t distribution, see `nct`.
%(before_notes)s
See Also
--------
nct
Notes
-----
The probability density function for `t` is:
.. math::
f(x, \nu) = \frac{\Gamma((\nu+1)/2)}
{\sqrt{\pi \nu} \Gamma(\nu/2)}
(1+x^2/\nu)^{-(\nu+1)/2}
where :math:`x` is a real number and the degrees of freedom parameter
:math:`\nu` (denoted ``df`` in the implementation) satisfies
:math:`\nu > 0`. :math:`\Gamma` is the gamma function
(`scipy.special.gamma`).
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("df", False, (0, np.inf), (False, False))]
def _rvs(self, df, size=None, random_state=None):
return random_state.standard_t(df, size=size)
def _pdf(self, x, df):
return _lazywhere(
df == np.inf, (x, df),
f=lambda x, df: norm._pdf(x),
f2=lambda x, df: (
np.exp(self._logpdf(x, df))
)
)
def _logpdf(self, x, df):
def t_logpdf(x, df):
return (np.log(sc.poch(0.5 * df, 0.5))
- 0.5 * (np.log(df) + np.log(np.pi))
- (df + 1)/2*np.log1p(x * x/df))
def norm_logpdf(x, df):
return norm._logpdf(x)
return _lazywhere(df == np.inf, (x, df, ), f=norm_logpdf, f2=t_logpdf)
def _cdf(self, x, df):
return sc.stdtr(df, x)
def _sf(self, x, df):
return sc.stdtr(df, -x)
def _ppf(self, q, df):
return sc.stdtrit(df, q)
def _isf(self, q, df):
return -sc.stdtrit(df, q)
def _stats(self, df):
# infinite df -> normal distribution (0.0, 1.0, 0.0, 0.0)
infinite_df = np.isposinf(df)
mu = np.where(df > 1, 0.0, np.inf)
condlist = ((df > 1) & (df <= 2),
(df > 2) & np.isfinite(df),
infinite_df)
choicelist = (lambda df: np.broadcast_to(np.inf, df.shape),
lambda df: df / (df-2.0),
lambda df: np.broadcast_to(1, df.shape))
mu2 = _lazyselect(condlist, choicelist, (df,), np.nan)
g1 = np.where(df > 3, 0.0, np.nan)
condlist = ((df > 2) & (df <= 4),
(df > 4) & np.isfinite(df),
infinite_df)
choicelist = (lambda df: np.broadcast_to(np.inf, df.shape),
lambda df: 6.0 / (df-4.0),
lambda df: np.broadcast_to(0, df.shape))
g2 = _lazyselect(condlist, choicelist, (df,), np.nan)
return mu, mu2, g1, g2
def _entropy(self, df):
if df == np.inf:
return norm._entropy()
def regular(df):
half = df/2
half1 = (df + 1)/2
return (half1*(sc.digamma(half1) - sc.digamma(half))
+ np.log(np.sqrt(df)*sc.beta(half, 0.5)))
def asymptotic(df):
# Formula from Wolfram Alpha:
# "asymptotic expansion (d+1)/2 * (digamma((d+1)/2) - digamma(d/2))
# + log(sqrt(d) * beta(d/2, 1/2))"
h = (norm._entropy() + 1/df + (df**-2.)/4 - (df**-3.)/6
- (df**-4.)/8 + 3/10*(df**-5.) + (df**-6.)/4)
return h
h = _lazywhere(df >= 100, (df, ), f=asymptotic, f2=regular)
return h
t = t_gen(name='t')
class nct_gen(rv_continuous):
r"""A non-central Student's t continuous random variable.
%(before_notes)s
Notes
-----
If :math:`Y` is a standard normal random variable and :math:`V` is
an independent chi-square random variable (`chi2`) with :math:`k` degrees
of freedom, then
.. math::
X = \frac{Y + c}{\sqrt{V/k}}
has a non-central Student's t distribution on the real line.
The degrees of freedom parameter :math:`k` (denoted ``df`` in the
implementation) satisfies :math:`k > 0` and the noncentrality parameter
:math:`c` (denoted ``nc`` in the implementation) is a real number.
%(after_notes)s
%(example)s
"""
def _argcheck(self, df, nc):
return (df > 0) & (nc == nc)
def _shape_info(self):
idf = _ShapeInfo("df", False, (0, np.inf), (False, False))
inc = _ShapeInfo("nc", False, (-np.inf, np.inf), (False, False))
return [idf, inc]
def _rvs(self, df, nc, size=None, random_state=None):
n = norm.rvs(loc=nc, size=size, random_state=random_state)
c2 = chi2.rvs(df, size=size, random_state=random_state)
return n * np.sqrt(df) / np.sqrt(c2)
def _pdf(self, x, df, nc):
# Boost version has accuracy issues in left tail; see gh-16591
n = df*1.0
nc = nc*1.0
x2 = x*x
ncx2 = nc*nc*x2
fac1 = n + x2
trm1 = (n/2.*np.log(n) + sc.gammaln(n+1)
- (n*np.log(2) + nc*nc/2 + (n/2)*np.log(fac1)
+ sc.gammaln(n/2)))
Px = np.exp(trm1)
valF = ncx2 / (2*fac1)
trm1 = (np.sqrt(2)*nc*x*sc.hyp1f1(n/2+1, 1.5, valF)
/ np.asarray(fac1*sc.gamma((n+1)/2)))
trm2 = (sc.hyp1f1((n+1)/2, 0.5, valF)
/ np.asarray(np.sqrt(fac1)*sc.gamma(n/2+1)))
Px *= trm1+trm2
return np.clip(Px, 0, None)
def _cdf(self, x, df, nc):
with np.errstate(over='ignore'): # see gh-17432
return np.clip(_boost._nct_cdf(x, df, nc), 0, 1)
def _ppf(self, q, df, nc):
with np.errstate(over='ignore'): # see gh-17432
return _boost._nct_ppf(q, df, nc)
def _sf(self, x, df, nc):
with np.errstate(over='ignore'): # see gh-17432
return np.clip(_boost._nct_sf(x, df, nc), 0, 1)
def _isf(self, x, df, nc):
with np.errstate(over='ignore'): # see gh-17432
return _boost._nct_isf(x, df, nc)
def _stats(self, df, nc, moments='mv'):
mu = _boost._nct_mean(df, nc)
mu2 = _boost._nct_variance(df, nc)
g1 = _boost._nct_skewness(df, nc) if 's' in moments else None
g2 = _boost._nct_kurtosis_excess(df, nc) if 'k' in moments else None
return mu, mu2, g1, g2
nct = nct_gen(name="nct")
class pareto_gen(rv_continuous):
r"""A Pareto continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `pareto` is:
.. math::
f(x, b) = \frac{b}{x^{b+1}}
for :math:`x \ge 1`, :math:`b > 0`.
`pareto` takes ``b`` as a shape parameter for :math:`b`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("b", False, (0, np.inf), (False, False))]
def _pdf(self, x, b):
# pareto.pdf(x, b) = b / x**(b+1)
return b * x**(-b-1)
def _cdf(self, x, b):
return 1 - x**(-b)
def _ppf(self, q, b):
return pow(1-q, -1.0/b)
def _sf(self, x, b):
return x**(-b)
def _isf(self, q, b):
return np.power(q, -1.0 / b)
def _stats(self, b, moments='mv'):
mu, mu2, g1, g2 = None, None, None, None
if 'm' in moments:
mask = b > 1
bt = np.extract(mask, b)
mu = np.full(np.shape(b), fill_value=np.inf)
np.place(mu, mask, bt / (bt-1.0))
if 'v' in moments:
mask = b > 2
bt = np.extract(mask, b)
mu2 = np.full(np.shape(b), fill_value=np.inf)
np.place(mu2, mask, bt / (bt-2.0) / (bt-1.0)**2)
if 's' in moments:
mask = b > 3
bt = np.extract(mask, b)
g1 = np.full(np.shape(b), fill_value=np.nan)
vals = 2 * (bt + 1.0) * np.sqrt(bt - 2.0) / ((bt - 3.0) * np.sqrt(bt))
np.place(g1, mask, vals)
if 'k' in moments:
mask = b > 4
bt = np.extract(mask, b)
g2 = np.full(np.shape(b), fill_value=np.nan)
vals = (6.0*np.polyval([1.0, 1.0, -6, -2], bt) /
np.polyval([1.0, -7.0, 12.0, 0.0], bt))
np.place(g2, mask, vals)
return mu, mu2, g1, g2
def _entropy(self, c):
return 1 + 1.0/c - np.log(c)
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
parameters = _check_fit_input_parameters(self, data, args, kwds)
data, fshape, floc, fscale = parameters
# ensure that any fixed parameters don't violate constraints of the
# distribution before continuing.
if floc is not None and np.min(data) - floc < (fscale or 0):
raise FitDataError("pareto", lower=1, upper=np.inf)
ndata = data.shape[0]
def get_shape(scale, location):
# The first-order necessary condition on `shape` can be solved in
# closed form
return ndata / np.sum(np.log((data - location) / scale))
if floc is fscale is None:
# The support of the distribution is `(x - loc)/scale > 0`.
# The method of Lagrange multipliers turns this constraint
# into an equation that can be solved numerically.
# See gh-12545 for details.
def dL_dScale(shape, scale):
# The partial derivative of the log-likelihood function w.r.t.
# the scale.
return ndata * shape / scale
def dL_dLocation(shape, location):
# The partial derivative of the log-likelihood function w.r.t.
# the location.
return (shape + 1) * np.sum(1 / (data - location))
def fun_to_solve(scale):
# optimize the scale by setting the partial derivatives
# w.r.t. to location and scale equal and solving.
location = np.min(data) - scale
shape = fshape or get_shape(scale, location)
return dL_dLocation(shape, location) - dL_dScale(shape, scale)
def interval_contains_root(lbrack, rbrack):
# return true if the signs disagree.
return (np.sign(fun_to_solve(lbrack)) !=
np.sign(fun_to_solve(rbrack)))
# set brackets for `root_scalar` to use when optimizing over the
# scale such that a root is likely between them. Use user supplied
# guess or default 1.
brack_start = float(kwds.get('scale', 1))
lbrack, rbrack = brack_start / 2, brack_start * 2
# if a root is not between the brackets, iteratively expand them
# until they include a sign change, checking after each bracket is
# modified.
while (not interval_contains_root(lbrack, rbrack)
and (lbrack > 0 or rbrack < np.inf)):
lbrack /= 2
rbrack *= 2
res = root_scalar(fun_to_solve, bracket=[lbrack, rbrack])
if res.converged:
scale = res.root
loc = np.min(data) - scale
shape = fshape or get_shape(scale, loc)
# The Pareto distribution requires that its parameters satisfy
# the condition `fscale + floc <= min(data)`. However, to
# avoid numerical issues, we require that `fscale + floc`
# is strictly less than `min(data)`. If this condition
# is not satisfied, reduce the scale with `np.nextafter` to
# ensure that data does not fall outside of the support.
if not (scale + loc) < np.min(data):
scale = np.min(data) - loc
scale = np.nextafter(scale, 0)
return shape, loc, scale
else:
return super().fit(data, **kwds)
elif floc is None:
loc = np.min(data) - fscale
else:
loc = floc
# Source: Evans, Hastings, and Peacock (2000), Statistical
# Distributions, 3rd. Ed., John Wiley and Sons. Page 149.
scale = fscale or np.min(data) - loc
shape = fshape or get_shape(scale, loc)
return shape, loc, scale
pareto = pareto_gen(a=1.0, name="pareto")
class lomax_gen(rv_continuous):
r"""A Lomax (Pareto of the second kind) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `lomax` is:
.. math::
f(x, c) = \frac{c}{(1+x)^{c+1}}
for :math:`x \ge 0`, :math:`c > 0`.
`lomax` takes ``c`` as a shape parameter for :math:`c`.
`lomax` is a special case of `pareto` with ``loc=-1.0``.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# lomax.pdf(x, c) = c / (1+x)**(c+1)
return c*1.0/(1.0+x)**(c+1.0)
def _logpdf(self, x, c):
return np.log(c) - (c+1)*sc.log1p(x)
def _cdf(self, x, c):
return -sc.expm1(-c*sc.log1p(x))
def _sf(self, x, c):
return np.exp(-c*sc.log1p(x))
def _logsf(self, x, c):
return -c*sc.log1p(x)
def _ppf(self, q, c):
return sc.expm1(-sc.log1p(-q)/c)
def _isf(self, q, c):
return q**(-1.0 / c) - 1
def _stats(self, c):
mu, mu2, g1, g2 = pareto.stats(c, loc=-1.0, moments='mvsk')
return mu, mu2, g1, g2
def _entropy(self, c):
return 1+1.0/c-np.log(c)
lomax = lomax_gen(a=0.0, name="lomax")
class pearson3_gen(rv_continuous):
r"""A pearson type III continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `pearson3` is:
.. math::
f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)}
(\beta (x - \zeta))^{\alpha - 1}
\exp(-\beta (x - \zeta))
where:
.. math::
\beta = \frac{2}{\kappa}
\alpha = \beta^2 = \frac{4}{\kappa^2}
\zeta = -\frac{\alpha}{\beta} = -\beta
:math:`\Gamma` is the gamma function (`scipy.special.gamma`).
Pass the skew :math:`\kappa` into `pearson3` as the shape parameter
``skew``.
%(after_notes)s
%(example)s
References
----------
R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and
Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water
Resources Research, Vol.27, 3149-3158 (1991).
L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist.,
Vol.1, 191-198 (1930).
"Using Modern Computing Tools to Fit the Pearson Type III Distribution to
Aviation Loads Data", Office of Aviation Research (2003).
"""
def _preprocess(self, x, skew):
# The real 'loc' and 'scale' are handled in the calling pdf(...). The
# local variables 'loc' and 'scale' within pearson3._pdf are set to
# the defaults just to keep them as part of the equations for
# documentation.
loc = 0.0
scale = 1.0
# If skew is small, return _norm_pdf. The divide between pearson3
# and norm was found by brute force and is approximately a skew of
# 0.000016. No one, I hope, would actually use a skew value even
# close to this small.
norm2pearson_transition = 0.000016
ans, x, skew = np.broadcast_arrays(1.0, x, skew)
ans = ans.copy()
# mask is True where skew is small enough to use the normal approx.
mask = np.absolute(skew) < norm2pearson_transition
invmask = ~mask
beta = 2.0 / (skew[invmask] * scale)
alpha = (scale * beta)**2
zeta = loc - alpha / beta
transx = beta * (x[invmask] - zeta)
return ans, x, transx, mask, invmask, beta, alpha, zeta
def _argcheck(self, skew):
# The _argcheck function in rv_continuous only allows positive
# arguments. The skew argument for pearson3 can be zero (which I want
# to handle inside pearson3._pdf) or negative. So just return True
# for all skew args.
return np.isfinite(skew)
def _shape_info(self):
return [_ShapeInfo("skew", False, (-np.inf, np.inf), (False, False))]
def _stats(self, skew):
m = 0.0
v = 1.0
s = skew
k = 1.5*skew**2
return m, v, s, k
def _pdf(self, x, skew):
# pearson3.pdf(x, skew) = abs(beta) / gamma(alpha) *
# (beta * (x - zeta))**(alpha - 1) * exp(-beta*(x - zeta))
# Do the calculation in _logpdf since helps to limit
# overflow/underflow problems
ans = np.exp(self._logpdf(x, skew))
if ans.ndim == 0:
if np.isnan(ans):
return 0.0
return ans
ans[np.isnan(ans)] = 0.0
return ans
def _logpdf(self, x, skew):
# PEARSON3 logpdf GAMMA logpdf
# np.log(abs(beta))
# + (alpha - 1)*np.log(beta*(x - zeta)) + (a - 1)*np.log(x)
# - beta*(x - zeta) - x
# - sc.gammalnalpha) - sc.gammalna)
ans, x, transx, mask, invmask, beta, alpha, _ = (
self._preprocess(x, skew))
ans[mask] = np.log(_norm_pdf(x[mask]))
# use logpdf instead of _logpdf to fix issue mentioned in gh-12640
# (_logpdf does not return correct result for alpha = 1)
ans[invmask] = np.log(abs(beta)) + gamma.logpdf(transx, alpha)
return ans
def _cdf(self, x, skew):
ans, x, transx, mask, invmask, _, alpha, _ = (
self._preprocess(x, skew))
ans[mask] = _norm_cdf(x[mask])
skew = np.broadcast_to(skew, invmask.shape)
invmask1a = np.logical_and(invmask, skew > 0)
invmask1b = skew[invmask] > 0
# use cdf instead of _cdf to fix issue mentioned in gh-12640
# (_cdf produces NaNs for inputs outside support)
ans[invmask1a] = gamma.cdf(transx[invmask1b], alpha[invmask1b])
# The gamma._cdf approach wasn't working with negative skew.
# Note that multiplying the skew by -1 reflects about x=0.
# So instead of evaluating the CDF with negative skew at x,
# evaluate the SF with positive skew at -x.
invmask2a = np.logical_and(invmask, skew < 0)
invmask2b = skew[invmask] < 0
# gamma._sf produces NaNs when transx < 0, so use gamma.sf
ans[invmask2a] = gamma.sf(transx[invmask2b], alpha[invmask2b])
return ans
def _sf(self, x, skew):
ans, x, transx, mask, invmask, _, alpha, _ = (
self._preprocess(x, skew))
ans[mask] = _norm_sf(x[mask])
skew = np.broadcast_to(skew, invmask.shape)
invmask1a = np.logical_and(invmask, skew > 0)
invmask1b = skew[invmask] > 0
ans[invmask1a] = gamma.sf(transx[invmask1b], alpha[invmask1b])
invmask2a = np.logical_and(invmask, skew < 0)
invmask2b = skew[invmask] < 0
ans[invmask2a] = gamma.cdf(transx[invmask2b], alpha[invmask2b])
return ans
def _rvs(self, skew, size=None, random_state=None):
skew = np.broadcast_to(skew, size)
ans, _, _, mask, invmask, beta, alpha, zeta = (
self._preprocess([0], skew))
nsmall = mask.sum()
nbig = mask.size - nsmall
ans[mask] = random_state.standard_normal(nsmall)
ans[invmask] = random_state.standard_gamma(alpha, nbig)/beta + zeta
if size == ():
ans = ans[0]
return ans
def _ppf(self, q, skew):
ans, q, _, mask, invmask, beta, alpha, zeta = (
self._preprocess(q, skew))
ans[mask] = _norm_ppf(q[mask])
q = q[invmask]
q[beta < 0] = 1 - q[beta < 0] # for negative skew; see gh-17050
ans[invmask] = sc.gammaincinv(alpha, q)/beta + zeta
return ans
@_call_super_mom
@extend_notes_in_docstring(rv_continuous, notes="""\
Note that method of moments (`method='MM'`) is not
available for this distribution.\n\n""")
def fit(self, data, *args, **kwds):
if kwds.get("method", None) == 'MM':
raise NotImplementedError("Fit `method='MM'` is not available for "
"the Pearson3 distribution. Please try "
"the default `method='MLE'`.")
else:
return super(type(self), self).fit(data, *args, **kwds)
pearson3 = pearson3_gen(name="pearson3")
class powerlaw_gen(rv_continuous):
r"""A power-function continuous random variable.
%(before_notes)s
See Also
--------
pareto
Notes
-----
The probability density function for `powerlaw` is:
.. math::
f(x, a) = a x^{a-1}
for :math:`0 \le x \le 1`, :math:`a > 0`.
`powerlaw` takes ``a`` as a shape parameter for :math:`a`.
%(after_notes)s
For example, the support of `powerlaw` can be adjusted from the default
interval ``[0, 1]`` to the interval ``[c, c+d]`` by setting ``loc=c`` and
``scale=d``. For a power-law distribution with infinite support, see
`pareto`.
`powerlaw` is a special case of `beta` with ``b=1``.
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("a", False, (0, np.inf), (False, False))]
def _pdf(self, x, a):
# powerlaw.pdf(x, a) = a * x**(a-1)
return a*x**(a-1.0)
def _logpdf(self, x, a):
return np.log(a) + sc.xlogy(a - 1, x)
def _cdf(self, x, a):
return x**(a*1.0)
def _logcdf(self, x, a):
return a*np.log(x)
def _ppf(self, q, a):
return pow(q, 1.0/a)
def _sf(self, p, a):
return -sc.powm1(p, a)
def _munp(self, n, a):
# The following expression is correct for all real n (provided a > 0).
return a / (a + n)
def _stats(self, a):
return (a / (a + 1.0),
a / (a + 2.0) / (a + 1.0) ** 2,
-2.0 * ((a - 1.0) / (a + 3.0)) * np.sqrt((a + 2.0) / a),
6 * np.polyval([1, -1, -6, 2], a) / (a * (a + 3.0) * (a + 4)))
def _entropy(self, a):
return 1 - 1.0/a - np.log(a)
def _support_mask(self, x, a):
return (super()._support_mask(x, a)
& ((x != 0) | (a >= 1)))
@_call_super_mom
@extend_notes_in_docstring(rv_continuous, notes="""\
Notes specifically for ``powerlaw.fit``: If the location is a free
parameter and the value returned for the shape parameter is less than
one, the true maximum likelihood approaches infinity. This causes
numerical difficulties, and the resulting estimates are approximate.
\n\n""")
def fit(self, data, *args, **kwds):
# Summary of the strategy:
#
# 1) If the scale and location are fixed, return the shape according
# to a formula.
#
# 2) If the scale is fixed, there are two possibilities for the other
# parameters - one corresponding with shape less than one, and
# another with shape greater than one. Calculate both, and return
# whichever has the better log-likelihood.
#
# At this point, the scale is known to be free.
#
# 3) If the location is fixed, return the scale and shape according to
# formulas (or, if the shape is fixed, the fixed shape).
#
# At this point, the location and scale are both free. There are
# separate equations depending on whether the shape is less than one or
# greater than one.
#
# 4a) If the shape is less than one, there are formulas for shape,
# location, and scale.
# 4b) If the shape is greater than one, there are formulas for shape
# and scale, but there is a condition for location to be solved
# numerically.
#
# If the shape is fixed and less than one, we use 4a.
# If the shape is fixed and greater than one, we use 4b.
# If the shape is also free, we calculate fits using both 4a and 4b
# and choose the one that results a better log-likelihood.
#
# In many cases, the use of `np.nextafter` is used to avoid numerical
# issues.
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
if len(np.unique(data)) == 1:
return super().fit(data, *args, **kwds)
data, fshape, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
penalized_nllf_args = [data, (self._fitstart(data),)]
penalized_nllf = self._reduce_func(penalized_nllf_args, {})[1]
# ensure that any fixed parameters don't violate constraints of the
# distribution before continuing. The support of the distribution
# is `0 < (x - loc)/scale < 1`.
if floc is not None:
if not data.min() > floc:
raise FitDataError('powerlaw', 0, 1)
if fscale is not None and not data.max() <= floc + fscale:
raise FitDataError('powerlaw', 0, 1)
if fscale is not None:
if fscale <= 0:
raise ValueError("Negative or zero `fscale` is outside the "
"range allowed by the distribution.")
if fscale <= np.ptp(data):
msg = "`fscale` must be greater than the range of data."
raise ValueError(msg)
def get_shape(data, loc, scale):
# The first-order necessary condition on `shape` can be solved in
# closed form. It can be used no matter the assumption of the
# value of the shape.
N = len(data)
return - N / (np.sum(np.log(data - loc)) - N*np.log(scale))
def get_scale(data, loc):
# analytical solution for `scale` based on the location.
# It can be used no matter the assumption of the value of the
# shape.
return data.max() - loc
# 1) The location and scale are both fixed. Analytically determine the
# shape.
if fscale is not None and floc is not None:
return get_shape(data, floc, fscale), floc, fscale
# 2) The scale is fixed. There are two possibilities for the other
# parameters. Choose the option with better log-likelihood.
if fscale is not None:
# using `data.min()` as the optimal location
loc_lt1 = np.nextafter(data.min(), -np.inf)
shape_lt1 = fshape or get_shape(data, loc_lt1, fscale)
ll_lt1 = penalized_nllf((shape_lt1, loc_lt1, fscale), data)
# using `data.max() - scale` as the optimal location
loc_gt1 = np.nextafter(data.max() - fscale, np.inf)
shape_gt1 = fshape or get_shape(data, loc_gt1, fscale)
ll_gt1 = penalized_nllf((shape_gt1, loc_gt1, fscale), data)
if ll_lt1 < ll_gt1:
return shape_lt1, loc_lt1, fscale
else:
return shape_gt1, loc_gt1, fscale
# 3) The location is fixed. Return the analytical scale and the
# analytical (or fixed) shape.
if floc is not None:
scale = get_scale(data, floc)
shape = fshape or get_shape(data, floc, scale)
return shape, floc, scale
# 4) Location and scale are both free
# 4a) Use formulas that assume `shape <= 1`.
def fit_loc_scale_w_shape_lt_1():
loc = np.nextafter(data.min(), -np.inf)
if np.abs(loc) < np.finfo(loc.dtype).tiny:
loc = np.sign(loc) * np.finfo(loc.dtype).tiny
scale = np.nextafter(get_scale(data, loc), np.inf)
shape = fshape or get_shape(data, loc, scale)
return shape, loc, scale
# 4b) Fit under the assumption that `shape > 1`. The support
# of the distribution is `(x - loc)/scale <= 1`. The method of Lagrange
# multipliers turns this constraint into the condition that
# dL_dScale - dL_dLocation must be zero, which is solved numerically.
# (Alternatively, substitute the constraint into the objective
# function before deriving the likelihood equation for location.)
def dL_dScale(data, shape, scale):
# The partial derivative of the log-likelihood function w.r.t.
# the scale.
return -data.shape[0] * shape / scale
def dL_dLocation(data, shape, loc):
# The partial derivative of the log-likelihood function w.r.t.
# the location.
return (shape - 1) * np.sum(1 / (loc - data)) # -1/(data-loc)
def dL_dLocation_star(loc):
# The derivative of the log-likelihood function w.r.t.
# the location, given optimal shape and scale
scale = np.nextafter(get_scale(data, loc), -np.inf)
shape = fshape or get_shape(data, loc, scale)
return dL_dLocation(data, shape, loc)
def fun_to_solve(loc):
# optimize the location by setting the partial derivatives
# w.r.t. to location and scale equal and solving.
scale = np.nextafter(get_scale(data, loc), -np.inf)
shape = fshape or get_shape(data, loc, scale)
return (dL_dScale(data, shape, scale)
- dL_dLocation(data, shape, loc))
def fit_loc_scale_w_shape_gt_1():
# set brackets for `root_scalar` to use when optimizing over the
# location such that a root is likely between them.
rbrack = np.nextafter(data.min(), -np.inf)
# if the sign of `dL_dLocation_star` is positive at rbrack,
# we're not going to find the root we're looking for
delta = (data.min() - rbrack)
while dL_dLocation_star(rbrack) > 0:
rbrack = data.min() - delta
delta *= 2
def interval_contains_root(lbrack, rbrack):
# Check if the interval (lbrack, rbrack) contains the root.
return (np.sign(fun_to_solve(lbrack))
!= np.sign(fun_to_solve(rbrack)))
lbrack = rbrack - 1
# if the sign doesn't change between the brackets, move the left
# bracket until it does. (The right bracket remains fixed at the
# maximum permissible value.)
i = 1.0
while (not interval_contains_root(lbrack, rbrack)
and lbrack != -np.inf):
lbrack = (data.min() - i)
i *= 2
root = optimize.root_scalar(fun_to_solve, bracket=(lbrack, rbrack))
loc = np.nextafter(root.root, -np.inf)
scale = np.nextafter(get_scale(data, loc), np.inf)
shape = fshape or get_shape(data, loc, scale)
return shape, loc, scale
# Shape is fixed - choose 4a or 4b accordingly.
if fshape is not None and fshape <= 1:
return fit_loc_scale_w_shape_lt_1()
elif fshape is not None and fshape > 1:
return fit_loc_scale_w_shape_gt_1()
# Shape is free
fit_shape_lt1 = fit_loc_scale_w_shape_lt_1()
ll_lt1 = self.nnlf(fit_shape_lt1, data)
fit_shape_gt1 = fit_loc_scale_w_shape_gt_1()
ll_gt1 = self.nnlf(fit_shape_gt1, data)
if ll_lt1 <= ll_gt1 and fit_shape_lt1[0] <= 1:
return fit_shape_lt1
elif ll_lt1 > ll_gt1 and fit_shape_gt1[0] > 1:
return fit_shape_gt1
else:
return super().fit(data, *args, **kwds)
powerlaw = powerlaw_gen(a=0.0, b=1.0, name="powerlaw")
class powerlognorm_gen(rv_continuous):
r"""A power log-normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `powerlognorm` is:
.. math::
f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s)
(\Phi(-\log(x)/s))^{c-1}
where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
and :math:`x > 0`, :math:`s, c > 0`.
`powerlognorm` takes :math:`c` and :math:`s` as shape parameters.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
ic = _ShapeInfo("c", False, (0, np.inf), (False, False))
i_s = _ShapeInfo("s", False, (0, np.inf), (False, False))
return [ic, i_s]
def _pdf(self, x, c, s):
return np.exp(self._logpdf(x, c, s))
def _logpdf(self, x, c, s):
return (np.log(c) - np.log(x) - np.log(s) +
_norm_logpdf(np.log(x) / s) +
_norm_logcdf(-np.log(x) / s) * (c - 1.))
def _cdf(self, x, c, s):
return -sc.expm1(self._logsf(x, c, s))
def _ppf(self, q, c, s):
return self._isf(1 - q, c, s)
def _sf(self, x, c, s):
return np.exp(self._logsf(x, c, s))
def _logsf(self, x, c, s):
return _norm_logcdf(-np.log(x) / s) * c
def _isf(self, q, c, s):
return np.exp(-_norm_ppf(q**(1/c)) * s)
powerlognorm = powerlognorm_gen(a=0.0, name="powerlognorm")
class powernorm_gen(rv_continuous):
r"""A power normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `powernorm` is:
.. math::
f(x, c) = c \phi(x) (\Phi(-x))^{c-1}
where :math:`\phi` is the normal pdf, :math:`\Phi` is the normal cdf,
:math:`x` is any real, and :math:`c > 0` [1]_.
`powernorm` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
References
----------
.. [1] NIST Engineering Statistics Handbook, Section 1.3.6.6.13,
https://www.itl.nist.gov/div898/handbook//eda/section3/eda366d.htm
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
def _pdf(self, x, c):
# powernorm.pdf(x, c) = c * phi(x) * (Phi(-x))**(c-1)
return c*_norm_pdf(x) * (_norm_cdf(-x)**(c-1.0))
def _logpdf(self, x, c):
return np.log(c) + _norm_logpdf(x) + (c-1)*_norm_logcdf(-x)
def _cdf(self, x, c):
return -sc.expm1(self._logsf(x, c))
def _ppf(self, q, c):
return -_norm_ppf(pow(1.0 - q, 1.0 / c))
def _sf(self, x, c):
return np.exp(self._logsf(x, c))
def _logsf(self, x, c):
return c * _norm_logcdf(-x)
def _isf(self, q, c):
return -_norm_ppf(np.exp(np.log(q) / c))
powernorm = powernorm_gen(name='powernorm')
class rdist_gen(rv_continuous):
r"""An R-distributed (symmetric beta) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rdist` is:
.. math::
f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)}
for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the
symmetric beta distribution: if B has a `beta` distribution with
parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with
parameter c.
`rdist` takes ``c`` as a shape parameter for :math:`c`.
This distribution includes the following distribution kernels as
special cases::
c = 2: uniform
c = 3: `semicircular`
c = 4: Epanechnikov (parabolic)
c = 6: quartic (biweight)
c = 8: triweight
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("c", False, (0, np.inf), (False, False))]
# use relation to the beta distribution for pdf, cdf, etc
def _pdf(self, x, c):
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return -np.log(2) + beta._logpdf((x + 1)/2, c/2, c/2)
def _cdf(self, x, c):
return beta._cdf((x + 1)/2, c/2, c/2)
def _sf(self, x, c):
return beta._sf((x + 1)/2, c/2, c/2)
def _ppf(self, q, c):
return 2*beta._ppf(q, c/2, c/2) - 1
def _rvs(self, c, size=None, random_state=None):
return 2 * random_state.beta(c/2, c/2, size) - 1
def _munp(self, n, c):
numerator = (1 - (n % 2)) * sc.beta((n + 1.0) / 2, c / 2.0)
return numerator / sc.beta(1. / 2, c / 2.)
rdist = rdist_gen(a=-1.0, b=1.0, name="rdist")
class rayleigh_gen(rv_continuous):
r"""A Rayleigh continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rayleigh` is:
.. math::
f(x) = x \exp(-x^2/2)
for :math:`x \ge 0`.
`rayleigh` is a special case of `chi` with ``df=2``.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return chi.rvs(2, size=size, random_state=random_state)
def _pdf(self, r):
# rayleigh.pdf(r) = r * exp(-r**2/2)
return np.exp(self._logpdf(r))
def _logpdf(self, r):
return np.log(r) - 0.5 * r * r
def _cdf(self, r):
return -sc.expm1(-0.5 * r**2)
def _ppf(self, q):
return np.sqrt(-2 * sc.log1p(-q))
def _sf(self, r):
return np.exp(self._logsf(r))
def _logsf(self, r):
return -0.5 * r * r
def _isf(self, q):
return np.sqrt(-2 * np.log(q))
def _stats(self):
val = 4 - np.pi
return (np.sqrt(np.pi/2),
val/2,
2*(np.pi-3)*np.sqrt(np.pi)/val**1.5,
6*np.pi/val-16/val**2)
def _entropy(self):
return _EULER/2.0 + 1 - 0.5*np.log(2)
@_call_super_mom
@extend_notes_in_docstring(rv_continuous, notes="""\
Notes specifically for ``rayleigh.fit``: If the location is fixed with
the `floc` parameter, this method uses an analytical formula to find
the scale. Otherwise, this function uses a numerical root finder on
the first order conditions of the log-likelihood function to find the
MLE. Only the (optional) `loc` parameter is used as the initial guess
for the root finder; the `scale` parameter and any other parameters
for the optimizer are ignored.\n\n""")
def fit(self, data, *args, **kwds):
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
data, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
def scale_mle(loc):
# Source: Statistical Distributions, 3rd Edition. Evans, Hastings,
# and Peacock (2000), Page 175
return (np.sum((data - loc) ** 2) / (2 * len(data))) ** .5
def loc_mle(loc):
# This implicit equation for `loc` is used when
# both `loc` and `scale` are free.
xm = data - loc
s1 = xm.sum()
s2 = (xm**2).sum()
s3 = (1/xm).sum()
return s1 - s2/(2*len(data))*s3
def loc_mle_scale_fixed(loc, scale=fscale):
# This implicit equation for `loc` is used when
# `scale` is fixed but `loc` is not.
xm = data - loc
return xm.sum() - scale**2 * (1/xm).sum()
if floc is not None:
# `loc` is fixed, analytically determine `scale`.
if np.any(data - floc <= 0):
raise FitDataError("rayleigh", lower=1, upper=np.inf)
else:
return floc, scale_mle(floc)
# Account for user provided guess of `loc`.
loc0 = kwds.get('loc')
if loc0 is None:
# Use _fitstart to estimate loc; ignore the returned scale.
loc0 = self._fitstart(data)[0]
fun = loc_mle if fscale is None else loc_mle_scale_fixed
rbrack = np.nextafter(np.min(data), -np.inf)
lbrack = _get_left_bracket(fun, rbrack)
res = optimize.root_scalar(fun, bracket=(lbrack, rbrack))
if not res.converged:
raise FitSolverError(res.flag)
loc = res.root
scale = fscale or scale_mle(loc)
return loc, scale
rayleigh = rayleigh_gen(a=0.0, name="rayleigh")
class reciprocal_gen(rv_continuous):
r"""A loguniform or reciprocal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for this class is:
.. math::
f(x, a, b) = \frac{1}{x \log(b/a)}
for :math:`a \le x \le b`, :math:`b > a > 0`. This class takes
:math:`a` and :math:`b` as shape parameters.
%(after_notes)s
%(example)s
This doesn't show the equal probability of ``0.01``, ``0.1`` and
``1``. This is best when the x-axis is log-scaled:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
>>> ax.hist(np.log10(r))
>>> ax.set_ylabel("Frequency")
>>> ax.set_xlabel("Value of random variable")
>>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
>>> ticks = ["$10^{{ {} }}$".format(i) for i in [-2, -1, 0]]
>>> ax.set_xticklabels(ticks) # doctest: +SKIP
>>> plt.show()
This random variable will be log-uniform regardless of the base chosen for
``a`` and ``b``. Let's specify with base ``2`` instead:
>>> rvs = %(name)s(2**-2, 2**0).rvs(size=1000)
Values of ``1/4``, ``1/2`` and ``1`` are equally likely with this random
variable. Here's the histogram:
>>> fig, ax = plt.subplots(1, 1)
>>> ax.hist(np.log2(rvs))
>>> ax.set_ylabel("Frequency")
>>> ax.set_xlabel("Value of random variable")
>>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
>>> ticks = ["$2^{{ {} }}$".format(i) for i in [-2, -1, 0]]
>>> ax.set_xticklabels(ticks) # doctest: +SKIP
>>> plt.show()
"""
def _argcheck(self, a, b):
return (a > 0) & (b > a)
def _shape_info(self):
ia = _ShapeInfo("a", False, (0, np.inf), (False, False))
ib = _ShapeInfo("b", False, (0, np.inf), (False, False))
return [ia, ib]
def _fitstart(self, data):
if isinstance(data, CensoredData):
data = data._uncensor()
# Reasonable, since support is [a, b]
return super()._fitstart(data, args=(np.min(data), np.max(data)))
def _get_support(self, a, b):
return a, b
def _pdf(self, x, a, b):
# reciprocal.pdf(x, a, b) = 1 / (x*(log(b) - log(a)))
return np.exp(self._logpdf(x, a, b))
def _logpdf(self, x, a, b):
return -np.log(x) - np.log(np.log(b) - np.log(a))
def _cdf(self, x, a, b):
return (np.log(x)-np.log(a)) / (np.log(b) - np.log(a))
def _ppf(self, q, a, b):
return np.exp(np.log(a) + q*(np.log(b) - np.log(a)))
def _munp(self, n, a, b):
t1 = 1 / (np.log(b) - np.log(a)) / n
t2 = np.real(np.exp(_log_diff(n * np.log(b), n*np.log(a))))
return t1 * t2
def _entropy(self, a, b):
return 0.5*(np.log(a) + np.log(b)) + np.log(np.log(b) - np.log(a))
fit_note = """\
`loguniform`/`reciprocal` is over-parameterized. `fit` automatically
fixes `scale` to 1 unless `fscale` is provided by the user.\n\n"""
@extend_notes_in_docstring(rv_continuous, notes=fit_note)
def fit(self, data, *args, **kwds):
fscale = kwds.pop('fscale', 1)
return super().fit(data, *args, fscale=fscale, **kwds)
# Details related to the decision of not defining
# the survival function for this distribution can be
# found in the PR: https://github.com/scipy/scipy/pull/18614
loguniform = reciprocal_gen(name="loguniform")
reciprocal = reciprocal_gen(name="reciprocal")
class rice_gen(rv_continuous):
r"""A Rice continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rice` is:
.. math::
f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I_0(x b)
for :math:`x >= 0`, :math:`b > 0`. :math:`I_0` is the modified Bessel
function of order zero (`scipy.special.i0`).
`rice` takes ``b`` as a shape parameter for :math:`b`.
%(after_notes)s
The Rice distribution describes the length, :math:`r`, of a 2-D vector with
components :math:`(U+u, V+v)`, where :math:`U, V` are constant, :math:`u,
v` are independent Gaussian random variables with standard deviation
:math:`s`. Let :math:`R = \sqrt{U^2 + V^2}`. Then the pdf of :math:`r` is
``rice.pdf(x, R/s, scale=s)``.
%(example)s
"""
def _argcheck(self, b):
return b >= 0
def _shape_info(self):
return [_ShapeInfo("b", False, (0, np.inf), (True, False))]
def _rvs(self, b, size=None, random_state=None):
# https://en.wikipedia.org/wiki/Rice_distribution
t = b/np.sqrt(2) + random_state.standard_normal(size=(2,) + size)
return np.sqrt((t*t).sum(axis=0))
def _cdf(self, x, b):
return sc.chndtr(np.square(x), 2, np.square(b))
def _ppf(self, q, b):
return np.sqrt(sc.chndtrix(q, 2, np.square(b)))
def _pdf(self, x, b):
# rice.pdf(x, b) = x * exp(-(x**2+b**2)/2) * I[0](x*b)
#
# We use (x**2 + b**2)/2 = ((x-b)**2)/2 + xb.
# The factor of np.exp(-xb) is then included in the i0e function
# in place of the modified Bessel function, i0, improving
# numerical stability for large values of xb.
return x * np.exp(-(x-b)*(x-b)/2.0) * sc.i0e(x*b)
def _munp(self, n, b):
nd2 = n/2.0
n1 = 1 + nd2
b2 = b*b/2.0
return (2.0**(nd2) * np.exp(-b2) * sc.gamma(n1) *
sc.hyp1f1(n1, 1, b2))
rice = rice_gen(a=0.0, name="rice")
class recipinvgauss_gen(rv_continuous):
r"""A reciprocal inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `recipinvgauss` is:
.. math::
f(x, \mu) = \frac{1}{\sqrt{2\pi x}}
\exp\left(\frac{-(1-\mu x)^2}{2\mu^2x}\right)
for :math:`x \ge 0`.
`recipinvgauss` takes ``mu`` as a shape parameter for :math:`\mu`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("mu", False, (0, np.inf), (False, False))]
def _pdf(self, x, mu):
# recipinvgauss.pdf(x, mu) =
# 1/sqrt(2*pi*x) * exp(-(1-mu*x)**2/(2*x*mu**2))
return np.exp(self._logpdf(x, mu))
def _logpdf(self, x, mu):
return _lazywhere(x > 0, (x, mu),
lambda x, mu: (-(1 - mu*x)**2.0 / (2*x*mu**2.0)
- 0.5*np.log(2*np.pi*x)),
fillvalue=-np.inf)
def _cdf(self, x, mu):
trm1 = 1.0/mu - x
trm2 = 1.0/mu + x
isqx = 1.0/np.sqrt(x)
return _norm_cdf(-isqx*trm1) - np.exp(2.0/mu)*_norm_cdf(-isqx*trm2)
def _sf(self, x, mu):
trm1 = 1.0/mu - x
trm2 = 1.0/mu + x
isqx = 1.0/np.sqrt(x)
return _norm_cdf(isqx*trm1) + np.exp(2.0/mu)*_norm_cdf(-isqx*trm2)
def _rvs(self, mu, size=None, random_state=None):
return 1.0/random_state.wald(mu, 1.0, size=size)
recipinvgauss = recipinvgauss_gen(a=0.0, name='recipinvgauss')
class semicircular_gen(rv_continuous):
r"""A semicircular continuous random variable.
%(before_notes)s
See Also
--------
rdist
Notes
-----
The probability density function for `semicircular` is:
.. math::
f(x) = \frac{2}{\pi} \sqrt{1-x^2}
for :math:`-1 \le x \le 1`.
The distribution is a special case of `rdist` with `c = 3`.
%(after_notes)s
References
----------
.. [1] "Wigner semicircle distribution",
https://en.wikipedia.org/wiki/Wigner_semicircle_distribution
%(example)s
"""
def _shape_info(self):
return []
def _pdf(self, x):
return 2.0/np.pi*np.sqrt(1-x*x)
def _logpdf(self, x):
return np.log(2/np.pi) + 0.5*sc.log1p(-x*x)
def _cdf(self, x):
return 0.5+1.0/np.pi*(x*np.sqrt(1-x*x) + np.arcsin(x))
def _ppf(self, q):
return rdist._ppf(q, 3)
def _rvs(self, size=None, random_state=None):
# generate values uniformly distributed on the area under the pdf
# (semi-circle) by randomly generating the radius and angle
r = np.sqrt(random_state.uniform(size=size))
a = np.cos(np.pi * random_state.uniform(size=size))
return r * a
def _stats(self):
return 0, 0.25, 0, -1.0
def _entropy(self):
return 0.64472988584940017414
semicircular = semicircular_gen(a=-1.0, b=1.0, name="semicircular")
class skewcauchy_gen(rv_continuous):
r"""A skewed Cauchy random variable.
%(before_notes)s
See Also
--------
cauchy : Cauchy distribution
Notes
-----
The probability density function for `skewcauchy` is:
.. math::
f(x) = \frac{1}{\pi \left(\frac{x^2}{\left(a\, \text{sign}(x) + 1
\right)^2} + 1 \right)}
for a real number :math:`x` and skewness parameter :math:`-1 < a < 1`.
When :math:`a=0`, the distribution reduces to the usual Cauchy
distribution.
%(after_notes)s
References
----------
.. [1] "Skewed generalized *t* distribution", Wikipedia
https://en.wikipedia.org/wiki/Skewed_generalized_t_distribution#Skewed_Cauchy_distribution
%(example)s
"""
def _argcheck(self, a):
return np.abs(a) < 1
def _shape_info(self):
return [_ShapeInfo("a", False, (-1.0, 1.0), (False, False))]
def _pdf(self, x, a):
return 1 / (np.pi * (x**2 / (a * np.sign(x) + 1)**2 + 1))
def _cdf(self, x, a):
return np.where(x <= 0,
(1 - a) / 2 + (1 - a) / np.pi * np.arctan(x / (1 - a)),
(1 - a) / 2 + (1 + a) / np.pi * np.arctan(x / (1 + a)))
def _ppf(self, x, a):
i = x < self._cdf(0, a)
return np.where(i,
np.tan(np.pi / (1 - a) * (x - (1 - a) / 2)) * (1 - a),
np.tan(np.pi / (1 + a) * (x - (1 - a) / 2)) * (1 + a))
def _stats(self, a, moments='mvsk'):
return np.nan, np.nan, np.nan, np.nan
def _fitstart(self, data):
# Use 0 as the initial guess of the skewness shape parameter.
# For the location and scale, estimate using the median and
# quartiles.
if isinstance(data, CensoredData):
data = data._uncensor()
p25, p50, p75 = np.percentile(data, [25, 50, 75])
return 0.0, p50, (p75 - p25)/2
skewcauchy = skewcauchy_gen(name='skewcauchy')
class skewnorm_gen(rv_continuous):
r"""A skew-normal random variable.
%(before_notes)s
Notes
-----
The pdf is::
skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x)
`skewnorm` takes a real number :math:`a` as a skewness parameter
When ``a = 0`` the distribution is identical to a normal distribution
(`norm`). `rvs` implements the method of [1]_.
%(after_notes)s
%(example)s
References
----------
.. [1] A. Azzalini and A. Capitanio (1999). Statistical applications of
the multivariate skew-normal distribution. J. Roy. Statist. Soc.,
B 61, 579-602. :arxiv:`0911.2093`
"""
def _argcheck(self, a):
return np.isfinite(a)
def _shape_info(self):
return [_ShapeInfo("a", False, (-np.inf, np.inf), (False, False))]
def _pdf(self, x, a):
return _lazywhere(
a == 0, (x, a), lambda x, a: _norm_pdf(x),
f2=lambda x, a: 2.*_norm_pdf(x)*_norm_cdf(a*x)
)
def _logpdf(self, x, a):
return _lazywhere(
a == 0, (x, a), lambda x, a: _norm_logpdf(x),
f2=lambda x, a: np.log(2)+_norm_logpdf(x)+_norm_logcdf(a*x),
)
def _cdf(self, x, a):
a = np.atleast_1d(a)
cdf = _boost._skewnorm_cdf(x, 0, 1, a)
# for some reason, a isn't broadcasted if some of x are invalid
a = np.broadcast_to(a, cdf.shape)
# Boost is not accurate in left tail when a > 0
i_small_cdf = (cdf < 1e-6) & (a > 0)
cdf[i_small_cdf] = super()._cdf(x[i_small_cdf], a[i_small_cdf])
return np.clip(cdf, 0, 1)
def _ppf(self, x, a):
return _boost._skewnorm_ppf(x, 0, 1, a)
def _sf(self, x, a):
# Boost's SF is implemented this way. Use whatever customizations
# we made in the _cdf.
return self._cdf(-x, -a)
def _isf(self, x, a):
return _boost._skewnorm_isf(x, 0, 1, a)
def _rvs(self, a, size=None, random_state=None):
u0 = random_state.normal(size=size)
v = random_state.normal(size=size)
d = a/np.sqrt(1 + a**2)
u1 = d*u0 + v*np.sqrt(1 - d**2)
return np.where(u0 >= 0, u1, -u1)
def _stats(self, a, moments='mvsk'):
output = [None, None, None, None]
const = np.sqrt(2/np.pi) * a/np.sqrt(1 + a**2)
if 'm' in moments:
output[0] = const
if 'v' in moments:
output[1] = 1 - const**2
if 's' in moments:
output[2] = ((4 - np.pi)/2) * (const/np.sqrt(1 - const**2))**3
if 'k' in moments:
output[3] = (2*(np.pi - 3)) * (const**4/(1 - const**2)**2)
return output
# For odd order, the each noncentral moment of the skew-normal distribution
# with location 0 and scale 1 can be expressed as a polynomial in delta,
# where delta = a/sqrt(1 + a**2) and `a` is the skew-normal shape
# parameter. The dictionary _skewnorm_odd_moments defines those
# polynomials for orders up to 19. The dict is implemented as a cached
# property to reduce the impact of the creation of the dict on import time.
@cached_property
def _skewnorm_odd_moments(self):
skewnorm_odd_moments = {
1: Polynomial([1]),
3: Polynomial([3, -1]),
5: Polynomial([15, -10, 3]),
7: Polynomial([105, -105, 63, -15]),
9: Polynomial([945, -1260, 1134, -540, 105]),
11: Polynomial([10395, -17325, 20790, -14850, 5775, -945]),
13: Polynomial([135135, -270270, 405405, -386100, 225225, -73710,
10395]),
15: Polynomial([2027025, -4729725, 8513505, -10135125, 7882875,
-3869775, 1091475, -135135]),
17: Polynomial([34459425, -91891800, 192972780, -275675400,
268017750, -175429800, 74220300, -18378360,
2027025]),
19: Polynomial([654729075, -1964187225, 4714049340, -7856748900,
9166207050, -7499623950, 4230557100, -1571349780,
346621275, -34459425]),
}
return skewnorm_odd_moments
def _munp(self, order, a):
if order & 1:
if order > 19:
raise NotImplementedError("skewnorm noncentral moments not "
"implemented for odd orders greater "
"than 19.")
# Use the precomputed polynomials that were derived from the
# moment generating function.
delta = a/np.sqrt(1 + a**2)
return (delta * self._skewnorm_odd_moments[order](delta**2)
* _SQRT_2_OVER_PI)
else:
# For even order, the moment is just (order-1)!!, where !! is the
# notation for the double factorial; for an odd integer m, m!! is
# m*(m-2)*...*3*1.
# We could use special.factorial2, but we know the argument is odd,
# so avoid the overhead of that function and compute the result
# directly here.
return sc.gamma((order + 1)/2) * 2**(order/2) / _SQRT_PI
@extend_notes_in_docstring(rv_continuous, notes="""\
If ``method='mm'``, parameters fixed by the user are respected, and the
remaining parameters are used to match distribution and sample moments
where possible. For example, if the user fixes the location with
``floc``, the parameters will only match the distribution skewness and
variance to the sample skewness and variance; no attempt will be made
to match the means or minimize a norm of the errors.
Note that the maximum possible skewness magnitude of a
`scipy.stats.skewnorm` distribution is approximately 0.9952717; if the
magnitude of the data's sample skewness exceeds this, the returned
shape parameter ``a`` will be infinite.
\n\n""")
def fit(self, data, *args, **kwds):
if kwds.pop("superfit", False):
return super().fit(data, *args, **kwds)
if isinstance(data, CensoredData):
if data.num_censored() == 0:
data = data._uncensor()
else:
return super().fit(data, *args, **kwds)
# this extracts fixed shape, location, and scale however they
# are specified, and also leaves them in `kwds`
data, fa, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
method = kwds.get("method", "mle").lower()
# See https://en.wikipedia.org/wiki/Skew_normal_distribution for
# moment formulas.
def skew_d(d): # skewness in terms of delta
return (4-np.pi)/2 * ((d * np.sqrt(2 / np.pi))**3
/ (1 - 2*d**2 / np.pi)**(3/2))
def d_skew(skew): # delta in terms of skewness
s_23 = np.abs(skew)**(2/3)
return np.sign(skew) * np.sqrt(
np.pi/2 * s_23 / (s_23 + ((4 - np.pi)/2)**(2/3))
)
# If method is method of moments, we don't need the user's guesses.
# Otherwise, extract the guesses from args and kwds.
if method == "mm":
a, loc, scale = None, None, None
else:
a = args[0] if len(args) else None
loc = kwds.pop('loc', None)
scale = kwds.pop('scale', None)
if fa is None and a is None: # not fixed and no guess: use MoM
# Solve for a that matches sample distribution skewness to sample
# skewness.
s = stats.skew(data)
if method == 'mle':
# For MLE initial conditions, clip skewness to a large but
# reasonable value in case the data skewness is out-of-range.
s = np.clip(s, -0.99, 0.99)
else:
s_max = skew_d(1)
s = np.clip(s, -s_max, s_max)
d = d_skew(s)
with np.errstate(divide='ignore'):
a = np.sqrt(np.divide(d**2, (1-d**2)))*np.sign(s)
else:
a = fa if fa is not None else a
d = a / np.sqrt(1 + a**2)
if fscale is None and scale is None:
v = np.var(data)
scale = np.sqrt(v / (1 - 2*d**2/np.pi))
elif fscale is not None:
scale = fscale
if floc is None and loc is None:
m = np.mean(data)
loc = m - scale*d*np.sqrt(2/np.pi)
elif floc is not None:
loc = floc
if method == 'mm':
return a, loc, scale
else:
# At this point, parameter "guesses" may equal the fixed parameters
# in kwds. No harm in passing them as guesses, too.
return super().fit(data, a, loc=loc, scale=scale, **kwds)
skewnorm = skewnorm_gen(name='skewnorm')
class trapezoid_gen(rv_continuous):
r"""A trapezoidal continuous random variable.
%(before_notes)s
Notes
-----
The trapezoidal distribution can be represented with an up-sloping line
from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)``
and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``. This
defines the trapezoid base from ``loc`` to ``(loc+scale)`` and the flat
top from ``c`` to ``d`` proportional to the position along the base
with ``0 <= c <= d <= 1``. When ``c=d``, this is equivalent to `triang`
with the same values for `loc`, `scale` and `c`.
The method of [1]_ is used for computing moments.
`trapezoid` takes :math:`c` and :math:`d` as shape parameters.
%(after_notes)s
The standard form is in the range [0, 1] with c the mode.
The location parameter shifts the start to `loc`.
The scale parameter changes the width from 1 to `scale`.
%(example)s
References
----------
.. [1] Kacker, R.N. and Lawrence, J.F. (2007). Trapezoidal and triangular
distributions for Type B evaluation of standard uncertainty.
Metrologia 44, 117-127. :doi:`10.1088/0026-1394/44/2/003`
"""
def _argcheck(self, c, d):
return (c >= 0) & (c <= 1) & (d >= 0) & (d <= 1) & (d >= c)
def _shape_info(self):
ic = _ShapeInfo("c", False, (0, 1.0), (True, True))
id = _ShapeInfo("d", False, (0, 1.0), (True, True))
return [ic, id]
def _pdf(self, x, c, d):
u = 2 / (d-c+1)
return _lazyselect([x < c,
(c <= x) & (x <= d),
x > d],
[lambda x, c, d, u: u * x / c,
lambda x, c, d, u: u,
lambda x, c, d, u: u * (1-x) / (1-d)],
(x, c, d, u))
def _cdf(self, x, c, d):
return _lazyselect([x < c,
(c <= x) & (x <= d),
x > d],
[lambda x, c, d: x**2 / c / (d-c+1),
lambda x, c, d: (c + 2 * (x-c)) / (d-c+1),
lambda x, c, d: 1-((1-x) ** 2
/ (d-c+1) / (1-d))],
(x, c, d))
def _ppf(self, q, c, d):
qc, qd = self._cdf(c, c, d), self._cdf(d, c, d)
condlist = [q < qc, q <= qd, q > qd]
choicelist = [np.sqrt(q * c * (1 + d - c)),
0.5 * q * (1 + d - c) + 0.5 * c,
1 - np.sqrt((1 - q) * (d - c + 1) * (1 - d))]
return np.select(condlist, choicelist)
def _munp(self, n, c, d):
# Using the parameterization from Kacker, 2007, with
# a=bottom left, c=top left, d=top right, b=bottom right, then
# E[X^n] = h/(n+1)/(n+2) [(b^{n+2}-d^{n+2})/(b-d)
# - ((c^{n+2} - a^{n+2})/(c-a)]
# with h = 2/((b-a) - (d-c)). The corresponding parameterization
# in scipy, has a'=loc, c'=loc+c*scale, d'=loc+d*scale, b'=loc+scale,
# which for standard form reduces to a'=0, b'=1, c'=c, d'=d.
# Substituting into E[X^n] gives the bd' term as (1 - d^{n+2})/(1 - d)
# and the ac' term as c^{n-1} for the standard form. The bd' term has
# numerical difficulties near d=1, so replace (1 - d^{n+2})/(1-d)
# with expm1((n+2)*log(d))/(d-1).
# Testing with n=18 for c=(1e-30,1-eps) shows that this is stable.
# We still require an explicit test for d=1 to prevent divide by zero,
# and now a test for d=0 to prevent log(0).
ab_term = c**(n+1)
dc_term = _lazyselect(
[d == 0.0, (0.0 < d) & (d < 1.0), d == 1.0],
[lambda d: 1.0,
lambda d: np.expm1((n+2) * np.log(d)) / (d-1.0),
lambda d: n+2],
[d])
val = 2.0 / (1.0+d-c) * (dc_term - ab_term) / ((n+1) * (n+2))
return val
def _entropy(self, c, d):
# Using the parameterization from Wikipedia (van Dorp, 2003)
# with a=bottom left, c=top left, d=top right, b=bottom right
# gives a'=loc, b'=loc+c*scale, c'=loc+d*scale, d'=loc+scale,
# which for loc=0, scale=1 is a'=0, b'=c, c'=d, d'=1.
# Substituting into the entropy formula from Wikipedia gives
# the following result.
return 0.5 * (1.0-d+c) / (1.0+d-c) + np.log(0.5 * (1.0+d-c))
trapezoid = trapezoid_gen(a=0.0, b=1.0, name="trapezoid")
# Note: alias kept for backwards compatibility. Rename was done
# because trapz is a slur in colloquial English (see gh-12924).
trapz = trapezoid_gen(a=0.0, b=1.0, name="trapz")
if trapz.__doc__:
trapz.__doc__ = "trapz is an alias for `trapezoid`"
class triang_gen(rv_continuous):
r"""A triangular continuous random variable.
%(before_notes)s
Notes
-----
The triangular distribution can be represented with an up-sloping line from
``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)``
to ``(loc + scale)``.
`triang` takes ``c`` as a shape parameter for :math:`0 \le c \le 1`.
%(after_notes)s
The standard form is in the range [0, 1] with c the mode.
The location parameter shifts the start to `loc`.
The scale parameter changes the width from 1 to `scale`.
%(example)s
"""
def _rvs(self, c, size=None, random_state=None):
return random_state.triangular(0, c, 1, size)
def _argcheck(self, c):
return (c >= 0) & (c <= 1)
def _shape_info(self):
return [_ShapeInfo("c", False, (0, 1.0), (True, True))]
def _pdf(self, x, c):
# 0: edge case where c=0
# 1: generalised case for x < c, don't use x <= c, as it doesn't cope
# with c = 0.
# 2: generalised case for x >= c, but doesn't cope with c = 1
# 3: edge case where c=1
r = _lazyselect([c == 0,
x < c,
(x >= c) & (c != 1),
c == 1],
[lambda x, c: 2 - 2 * x,
lambda x, c: 2 * x / c,
lambda x, c: 2 * (1 - x) / (1 - c),
lambda x, c: 2 * x],
(x, c))
return r
def _cdf(self, x, c):
r = _lazyselect([c == 0,
x < c,
(x >= c) & (c != 1),
c == 1],
[lambda x, c: 2*x - x*x,
lambda x, c: x * x / c,
lambda x, c: (x*x - 2*x + c) / (c-1),
lambda x, c: x * x],
(x, c))
return r
def _ppf(self, q, c):
return np.where(q < c, np.sqrt(c * q), 1-np.sqrt((1-c) * (1-q)))
def _stats(self, c):
return ((c+1.0)/3.0,
(1.0-c+c*c)/18,
np.sqrt(2)*(2*c-1)*(c+1)*(c-2) / (5*np.power((1.0-c+c*c), 1.5)),
-3.0/5.0)
def _entropy(self, c):
return 0.5-np.log(2)
triang = triang_gen(a=0.0, b=1.0, name="triang")
class truncexpon_gen(rv_continuous):
r"""A truncated exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `truncexpon` is:
.. math::
f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)}
for :math:`0 <= x <= b`.
`truncexpon` takes ``b`` as a shape parameter for :math:`b`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("b", False, (0, np.inf), (False, False))]
def _get_support(self, b):
return self.a, b
def _pdf(self, x, b):
# truncexpon.pdf(x, b) = exp(-x) / (1-exp(-b))
return np.exp(-x)/(-sc.expm1(-b))
def _logpdf(self, x, b):
return -x - np.log(-sc.expm1(-b))
def _cdf(self, x, b):
return sc.expm1(-x)/sc.expm1(-b)
def _ppf(self, q, b):
return -sc.log1p(q*sc.expm1(-b))
def _sf(self, x, b):
return (np.exp(-b) - np.exp(-x))/sc.expm1(-b)
def _isf(self, q, b):
return -np.log(np.exp(-b) - q * sc.expm1(-b))
def _munp(self, n, b):
# wrong answer with formula, same as in continuous.pdf
# return sc.gamman+1)-sc.gammainc1+n, b)
if n == 1:
return (1-(b+1)*np.exp(-b))/(-sc.expm1(-b))
elif n == 2:
return 2*(1-0.5*(b*b+2*b+2)*np.exp(-b))/(-sc.expm1(-b))
else:
# return generic for higher moments
return super()._munp(n, b)
def _entropy(self, b):
eB = np.exp(b)
return np.log(eB-1)+(1+eB*(b-1.0))/(1.0-eB)
truncexpon = truncexpon_gen(a=0.0, name='truncexpon')
# logsumexp trick for log(p + q) with only log(p) and log(q)
def _log_sum(log_p, log_q):
return sc.logsumexp([log_p, log_q], axis=0)
# same as above, but using -exp(x) = exp(x + πi)
def _log_diff(log_p, log_q):
return sc.logsumexp([log_p, log_q+np.pi*1j], axis=0)
def _log_gauss_mass(a, b):
"""Log of Gaussian probability mass within an interval"""
a, b = np.broadcast_arrays(a, b)
# Calculations in right tail are inaccurate, so we'll exploit the
# symmetry and work only in the left tail
case_left = b <= 0
case_right = a > 0
case_central = ~(case_left | case_right)
def mass_case_left(a, b):
return _log_diff(_norm_logcdf(b), _norm_logcdf(a))
def mass_case_right(a, b):
return mass_case_left(-b, -a)
def mass_case_central(a, b):
# Previously, this was implemented as:
# left_mass = mass_case_left(a, 0)
# right_mass = mass_case_right(0, b)
# return _log_sum(left_mass, right_mass)
# Catastrophic cancellation occurs as np.exp(log_mass) approaches 1.
# Correct for this with an alternative formulation.
# We're not concerned with underflow here: if only one term
# underflows, it was insignificant; if both terms underflow,
# the result can't accurately be represented in logspace anyway
# because sc.log1p(x) ~ x for small x.
return sc.log1p(-_norm_cdf(a) - _norm_cdf(-b))
# _lazyselect not working; don't care to debug it
out = np.full_like(a, fill_value=np.nan, dtype=np.complex128)
if a[case_left].size:
out[case_left] = mass_case_left(a[case_left], b[case_left])
if a[case_right].size:
out[case_right] = mass_case_right(a[case_right], b[case_right])
if a[case_central].size:
out[case_central] = mass_case_central(a[case_central], b[case_central])
return np.real(out) # discard ~0j
class truncnorm_gen(rv_continuous):
r"""A truncated normal continuous random variable.
%(before_notes)s
Notes
-----
This distribution is the normal distribution centered on ``loc`` (default
0), with standard deviation ``scale`` (default 1), and truncated at ``a``
and ``b`` *standard deviations* from ``loc``. For arbitrary ``loc`` and
``scale``, ``a`` and ``b`` are *not* the abscissae at which the shifted
and scaled distribution is truncated.
.. note::
If ``a_trunc`` and ``b_trunc`` are the abscissae at which we wish
to truncate the distribution (as opposed to the number of standard
deviations from ``loc``), then we can calculate the distribution
parameters ``a`` and ``b`` as follows::
a, b = (a_trunc - loc) / scale, (b_trunc - loc) / scale
This is a common point of confusion. For additional clarification,
please see the example below.
%(example)s
In the examples above, ``loc=0`` and ``scale=1``, so the plot is truncated
at ``a`` on the left and ``b`` on the right. However, suppose we were to
produce the same histogram with ``loc = 1`` and ``scale=0.5``.
>>> loc, scale = 1, 0.5
>>> rv = truncnorm(a, b, loc=loc, scale=scale)
>>> x = np.linspace(truncnorm.ppf(0.01, a, b),
... truncnorm.ppf(0.99, a, b), 100)
>>> r = rv.rvs(size=1000)
>>> fig, ax = plt.subplots(1, 1)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim(a, b)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
Note that the distribution is no longer appears to be truncated at
abscissae ``a`` and ``b``. That is because the *standard* normal
distribution is first truncated at ``a`` and ``b``, *then* the resulting
distribution is scaled by ``scale`` and shifted by ``loc``. If we instead
want the shifted and scaled distribution to be truncated at ``a`` and
``b``, we need to transform these values before passing them as the
distribution parameters.
>>> a_transformed, b_transformed = (a - loc) / scale, (b - loc) / scale
>>> rv = truncnorm(a_transformed, b_transformed, loc=loc, scale=scale)
>>> x = np.linspace(truncnorm.ppf(0.01, a, b),
... truncnorm.ppf(0.99, a, b), 100)
>>> r = rv.rvs(size=10000)
>>> fig, ax = plt.subplots(1, 1)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim(a-0.1, b+0.1)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
"""
def _argcheck(self, a, b):
return a < b
def _shape_info(self):
ia = _ShapeInfo("a", False, (-np.inf, np.inf), (True, False))
ib = _ShapeInfo("b", False, (-np.inf, np.inf), (False, True))
return [ia, ib]
def _fitstart(self, data):
# Reasonable, since support is [a, b]
if isinstance(data, CensoredData):
data = data._uncensor()
return super()._fitstart(data, args=(np.min(data), np.max(data)))
def _get_support(self, a, b):
return a, b
def _pdf(self, x, a, b):
return np.exp(self._logpdf(x, a, b))
def _logpdf(self, x, a, b):
return _norm_logpdf(x) - _log_gauss_mass(a, b)
def _cdf(self, x, a, b):
return np.exp(self._logcdf(x, a, b))
def _logcdf(self, x, a, b):
x, a, b = np.broadcast_arrays(x, a, b)
logcdf = np.asarray(_log_gauss_mass(a, x) - _log_gauss_mass(a, b))
i = logcdf > -0.1 # avoid catastrophic cancellation
if np.any(i):
logcdf[i] = np.log1p(-np.exp(self._logsf(x[i], a[i], b[i])))
return logcdf
def _sf(self, x, a, b):
return np.exp(self._logsf(x, a, b))
def _logsf(self, x, a, b):
x, a, b = np.broadcast_arrays(x, a, b)
logsf = np.asarray(_log_gauss_mass(x, b) - _log_gauss_mass(a, b))
i = logsf > -0.1 # avoid catastrophic cancellation
if np.any(i):
logsf[i] = np.log1p(-np.exp(self._logcdf(x[i], a[i], b[i])))
return logsf
def _entropy(self, a, b):
A = _norm_cdf(a)
B = _norm_cdf(b)
Z = B - A
C = np.log(np.sqrt(2 * np.pi * np.e) * Z)
D = (a * _norm_pdf(a) - b * _norm_pdf(b)) / (2 * Z)
h = C + D
return h
def _ppf(self, q, a, b):
q, a, b = np.broadcast_arrays(q, a, b)
case_left = a < 0
case_right = ~case_left
def ppf_left(q, a, b):
log_Phi_x = _log_sum(_norm_logcdf(a),
np.log(q) + _log_gauss_mass(a, b))
return sc.ndtri_exp(log_Phi_x)
def ppf_right(q, a, b):
log_Phi_x = _log_sum(_norm_logcdf(-b),
np.log1p(-q) + _log_gauss_mass(a, b))
return -sc.ndtri_exp(log_Phi_x)
out = np.empty_like(q)
q_left = q[case_left]
q_right = q[case_right]
if q_left.size:
out[case_left] = ppf_left(q_left, a[case_left], b[case_left])
if q_right.size:
out[case_right] = ppf_right(q_right, a[case_right], b[case_right])
return out
def _isf(self, q, a, b):
# Mostly copy-paste of _ppf, but I think this is simpler than combining
q, a, b = np.broadcast_arrays(q, a, b)
case_left = b < 0
case_right = ~case_left
def isf_left(q, a, b):
log_Phi_x = _log_diff(_norm_logcdf(b),
np.log(q) + _log_gauss_mass(a, b))
return sc.ndtri_exp(np.real(log_Phi_x))
def isf_right(q, a, b):
log_Phi_x = _log_diff(_norm_logcdf(-a),
np.log1p(-q) + _log_gauss_mass(a, b))
return -sc.ndtri_exp(np.real(log_Phi_x))
out = np.empty_like(q)
q_left = q[case_left]
q_right = q[case_right]
if q_left.size:
out[case_left] = isf_left(q_left, a[case_left], b[case_left])
if q_right.size:
out[case_right] = isf_right(q_right, a[case_right], b[case_right])
return out
def _munp(self, n, a, b):
def n_th_moment(n, a, b):
"""
Returns n-th moment. Defined only if n >= 0.
Function cannot broadcast due to the loop over n
"""
pA, pB = self._pdf(np.asarray([a, b]), a, b)
probs = [pA, -pB]
moments = [0, 1]
for k in range(1, n+1):
# a or b might be infinite, and the corresponding pdf value
# is 0 in that case, but nan is returned for the
# multiplication. However, as b->infinity, pdf(b)*b**k -> 0.
# So it is safe to use _lazywhere to avoid the nan.
vals = _lazywhere(probs, [probs, [a, b]],
lambda x, y: x * y**(k-1), fillvalue=0)
mk = np.sum(vals) + (k-1) * moments[-2]
moments.append(mk)
return moments[-1]
return _lazywhere((n >= 0) & (a == a) & (b == b), (n, a, b),
np.vectorize(n_th_moment, otypes=[np.float64]),
np.nan)
def _stats(self, a, b, moments='mv'):
pA, pB = self.pdf(np.array([a, b]), a, b)
def _truncnorm_stats_scalar(a, b, pA, pB, moments):
m1 = pA - pB
mu = m1
# use _lazywhere to avoid nan (See detailed comment in _munp)
probs = [pA, -pB]
vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y,
fillvalue=0)
m2 = 1 + np.sum(vals)
vals = _lazywhere(probs, [probs, [a-mu, b-mu]], lambda x, y: x*y,
fillvalue=0)
# mu2 = m2 - mu**2, but not as numerically stable as:
# mu2 = (a-mu)*pA - (b-mu)*pB + 1
mu2 = 1 + np.sum(vals)
vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y**2,
fillvalue=0)
m3 = 2*m1 + np.sum(vals)
vals = _lazywhere(probs, [probs, [a, b]], lambda x, y: x*y**3,
fillvalue=0)
m4 = 3*m2 + np.sum(vals)
mu3 = m3 + m1 * (-3*m2 + 2*m1**2)
g1 = mu3 / np.power(mu2, 1.5)
mu4 = m4 + m1*(-4*m3 + 3*m1*(2*m2 - m1**2))
g2 = mu4 / mu2**2 - 3
return mu, mu2, g1, g2
_truncnorm_stats = np.vectorize(_truncnorm_stats_scalar,
excluded=('moments',))
return _truncnorm_stats(a, b, pA, pB, moments)
truncnorm = truncnorm_gen(name='truncnorm', momtype=1)
class truncpareto_gen(rv_continuous):
r"""An upper truncated Pareto continuous random variable.
%(before_notes)s
See Also
--------
pareto : Pareto distribution
Notes
-----
The probability density function for `truncpareto` is:
.. math::
f(x, b, c) = \frac{b}{1 - c^{-b}} \frac{1}{x^{b+1}}
for :math:`b > 0`, :math:`c > 1` and :math:`1 \le x \le c`.
`truncpareto` takes `b` and `c` as shape parameters for :math:`b` and
:math:`c`.
Notice that the upper truncation value :math:`c` is defined in
standardized form so that random values of an unscaled, unshifted variable
are within the range ``[1, c]``.
If ``u_r`` is the upper bound to a scaled and/or shifted variable,
then ``c = (u_r - loc) / scale``. In other words, the support of the
distribution becomes ``(scale + loc) <= x <= (c*scale + loc)`` when
`scale` and/or `loc` are provided.
%(after_notes)s
References
----------
.. [1] Burroughs, S. M., and Tebbens S. F.
"Upper-truncated power laws in natural systems."
Pure and Applied Geophysics 158.4 (2001): 741-757.
%(example)s
"""
def _shape_info(self):
ib = _ShapeInfo("b", False, (0.0, np.inf), (False, False))
ic = _ShapeInfo("c", False, (1.0, np.inf), (False, False))
return [ib, ic]
def _argcheck(self, b, c):
return (b > 0.) & (c > 1.)
def _get_support(self, b, c):
return self.a, c
def _pdf(self, x, b, c):
return b * x**-(b+1) / (1 - 1/c**b)
def _logpdf(self, x, b, c):
return np.log(b) - np.log(-np.expm1(-b*np.log(c))) - (b+1)*np.log(x)
def _cdf(self, x, b, c):
return (1 - x**-b) / (1 - 1/c**b)
def _logcdf(self, x, b, c):
return np.log1p(-x**-b) - np.log1p(-1/c**b)
def _ppf(self, q, b, c):
return pow(1 - (1 - 1/c**b)*q, -1/b)
def _sf(self, x, b, c):
return (x**-b - 1/c**b) / (1 - 1/c**b)
def _logsf(self, x, b, c):
return np.log(x**-b - 1/c**b) - np.log1p(-1/c**b)
def _isf(self, q, b, c):
return pow(1/c**b + (1 - 1/c**b)*q, -1/b)
def _entropy(self, b, c):
return -(np.log(b/(1 - 1/c**b))
+ (b+1)*(np.log(c)/(c**b - 1) - 1/b))
def _munp(self, n, b, c):
if (n == b).all():
return b*np.log(c) / (1 - 1/c**b)
else:
return b / (b-n) * (c**b - c**n) / (c**b - 1)
def _fitstart(self, data):
if isinstance(data, CensoredData):
data = data._uncensor()
b, loc, scale = pareto.fit(data)
c = (max(data) - loc)/scale
return b, c, loc, scale
@_call_super_mom
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
if kwds.pop("superfit", False):
return super().fit(data, *args, **kwds)
def log_mean(x):
return np.mean(np.log(x))
def harm_mean(x):
return 1/np.mean(1/x)
def get_b(c, loc, scale):
u = (data-loc)/scale
harm_m = harm_mean(u)
log_m = log_mean(u)
quot = (harm_m-1)/log_m
return (1 - (quot-1) / (quot - (1 - 1/c)*harm_m/np.log(c)))/log_m
def get_c(loc, scale):
return (mx - loc)/scale
def get_loc(fc, fscale):
if fscale: # (fscale and fc) or (fscale and not fc)
loc = mn - fscale
return loc
if fc:
loc = (fc*mn - mx)/(fc - 1)
return loc
def get_scale(loc):
return mn - loc
# Functions used for optimisation; partial derivatives of
# the Lagrangian, set to equal 0.
def dL_dLoc(loc, b_=None):
# Partial derivative wrt location.
# Optimised upon when no parameters, or only b, are fixed.
scale = get_scale(loc)
c = get_c(loc, scale)
b = get_b(c, loc, scale) if b_ is None else b_
harm_m = harm_mean((data - loc)/scale)
return 1 - (1 + (c - 1)/(c**(b+1) - c)) * (1 - 1/(b+1)) * harm_m
def dL_dB(b, logc, logm):
# Partial derivative wrt b.
# Optimised upon whenever at least one parameter but b is fixed,
# and b is free.
return b - np.log1p(b*logc / (1 - b*logm)) / logc
def fallback(data, *args, **kwargs):
# Should any issue arise, default to the general fit method.
return super(truncpareto_gen, self).fit(data, *args, **kwargs)
parameters = _check_fit_input_parameters(self, data, args, kwds)
data, fb, fc, floc, fscale = parameters
mn, mx = data.min(), data.max()
mn_inf = np.nextafter(mn, -np.inf)
if (fb is not None
and fc is not None
and floc is not None
and fscale is not None):
raise ValueError("All parameters fixed."
"There is nothing to optimize.")
elif fc is None and floc is None and fscale is None:
if fb is None:
def cond_b(loc):
# b is positive only if this function is positive
scale = get_scale(loc)
c = get_c(loc, scale)
harm_m = harm_mean((data - loc)/scale)
return (1 + 1/(c-1)) * np.log(c) / harm_m - 1
# This gives an upper bound on loc allowing for a positive b.
# Iteratively look for a bracket for root_scalar.
mn_inf = np.nextafter(mn, -np.inf)
rbrack = mn_inf
i = 0
lbrack = rbrack - 1
while ((lbrack > -np.inf)
and (cond_b(lbrack)*cond_b(rbrack) >= 0)):
i += 1
lbrack = rbrack - np.power(2., i)
if not lbrack > -np.inf:
return fallback(data, *args, **kwds)
res = root_scalar(cond_b, bracket=(lbrack, rbrack))
if not res.converged:
return fallback(data, *args, **kwds)
# Determine the MLE for loc.
# Iteratively look for a bracket for root_scalar.
rbrack = res.root - 1e-3 # grad_loc is numerically ill-behaved
lbrack = rbrack - 1
i = 0
while ((lbrack > -np.inf)
and (dL_dLoc(lbrack)*dL_dLoc(rbrack) >= 0)):
i += 1
lbrack = rbrack - np.power(2., i)
if not lbrack > -np.inf:
return fallback(data, *args, **kwds)
res = root_scalar(dL_dLoc, bracket=(lbrack, rbrack))
if not res.converged:
return fallback(data, *args, **kwds)
loc = res.root
scale = get_scale(loc)
c = get_c(loc, scale)
b = get_b(c, loc, scale)
std_data = (data - loc)/scale
# The expression of b relies on b being bounded above.
up_bound_b = min(1/log_mean(std_data),
1/(harm_mean(std_data)-1))
if not (b < up_bound_b):
return fallback(data, *args, **kwds)
else:
# We know b is positive (or a FitError will be triggered)
# so we let loc get close to min(data).
rbrack = mn_inf
lbrack = mn_inf - 1
i = 0
# Iteratively look for a bracket for root_scalar.
while (lbrack > -np.inf
and (dL_dLoc(lbrack, fb)
* dL_dLoc(rbrack, fb) >= 0)):
i += 1
lbrack = rbrack - 2**i
if not lbrack > -np.inf:
return fallback(data, *args, **kwds)
res = root_scalar(dL_dLoc, (fb,),
bracket=(lbrack, rbrack))
if not res.converged:
return fallback(data, *args, **kwds)
loc = res.root
scale = get_scale(loc)
c = get_c(loc, scale)
b = fb
else:
# At least one of the parameters determining the support is fixed;
# the others then have analytical expressions from the constraints.
# The completely determined case (fixed c, loc and scale)
# has to be checked for not overflowing the support.
# If not fixed, b has to be determined numerically.
loc = floc if floc is not None else get_loc(fc, fscale)
scale = fscale or get_scale(loc)
c = fc or get_c(loc, scale)
# Unscaled, translated values should be positive when the location
# is fixed. If it is not the case, we end up with negative `scale`
# and `c`, which would trigger a FitError before exiting the
# method.
if floc is not None and data.min() - floc < 0:
raise FitDataError("truncpareto", lower=1, upper=c)
# Standardised values should be within the distribution support
# when all parameters controlling it are fixed. If it not the case,
# `fc` is overridden by `c` determined from `floc` and `fscale` when
# raising the exception.
if fc and (floc is not None) and fscale:
if data.max() > fc*fscale + floc:
raise FitDataError("truncpareto", lower=1,
upper=get_c(loc, scale))
# The other constraints should be automatically satisfied
# from the analytical expressions of the parameters.
# If fc or fscale are respectively less than one or less than 0,
# a FitError is triggered before exiting the method.
if fb is None:
std_data = (data - loc)/scale
logm = log_mean(std_data)
logc = np.log(c)
# Condition for a positive root to exist.
if not (2*logm < logc):
return fallback(data, *args, **kwds)
lbrack = 1/logm + 1/(logm - logc)
rbrack = np.nextafter(1/logm, 0)
try:
res = root_scalar(dL_dB, (logc, logm),
bracket=(lbrack, rbrack))
# we should then never get there
if not res.converged:
return fallback(data, *args, **kwds)
b = res.root
except ValueError:
b = rbrack
else:
b = fb
# The distribution requires that `scale+loc <= data <= c*scale+loc`.
# To avoid numerical issues, some tuning may be necessary.
# We adjust `scale` to satisfy the lower bound, and we adjust
# `c` to satisfy the upper bound.
if not (scale+loc) < mn:
if fscale:
loc = np.nextafter(loc, -np.inf)
else:
scale = get_scale(loc)
scale = np.nextafter(scale, 0)
if not (c*scale+loc) > mx:
c = get_c(loc, scale)
c = np.nextafter(c, np.inf)
if not (np.all(self._argcheck(b, c)) and (scale > 0)):
return fallback(data, *args, **kwds)
params_override = b, c, loc, scale
if floc is None and fscale is None:
# Based on testing in gh-16782, the following methods are only
# reliable if either `floc` or `fscale` are provided. They are
# fast, though, so might as well see if they are better than the
# generic method.
params_super = fallback(data, *args, **kwds)
nllf_override = self.nnlf(params_override, data)
nllf_super = self.nnlf(params_super, data)
if nllf_super < nllf_override:
return params_super
return params_override
truncpareto = truncpareto_gen(a=1.0, name='truncpareto')
class tukeylambda_gen(rv_continuous):
r"""A Tukey-Lamdba continuous random variable.
%(before_notes)s
Notes
-----
A flexible distribution, able to represent and interpolate between the
following distributions:
- Cauchy (:math:`lambda = -1`)
- logistic (:math:`lambda = 0`)
- approx Normal (:math:`lambda = 0.14`)
- uniform from -1 to 1 (:math:`lambda = 1`)
`tukeylambda` takes a real number :math:`lambda` (denoted ``lam``
in the implementation) as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, lam):
return np.isfinite(lam)
def _shape_info(self):
return [_ShapeInfo("lam", False, (-np.inf, np.inf), (False, False))]
def _pdf(self, x, lam):
Fx = np.asarray(sc.tklmbda(x, lam))
Px = Fx**(lam-1.0) + (np.asarray(1-Fx))**(lam-1.0)
Px = 1.0/np.asarray(Px)
return np.where((lam <= 0) | (abs(x) < 1.0/np.asarray(lam)), Px, 0.0)
def _cdf(self, x, lam):
return sc.tklmbda(x, lam)
def _ppf(self, q, lam):
return sc.boxcox(q, lam) - sc.boxcox1p(-q, lam)
def _stats(self, lam):
return 0, _tlvar(lam), 0, _tlkurt(lam)
def _entropy(self, lam):
def integ(p):
return np.log(pow(p, lam-1)+pow(1-p, lam-1))
return integrate.quad(integ, 0, 1)[0]
tukeylambda = tukeylambda_gen(name='tukeylambda')
class FitUniformFixedScaleDataError(FitDataError):
def __init__(self, ptp, fscale):
self.args = (
"Invalid values in `data`. Maximum likelihood estimation with "
"the uniform distribution and fixed scale requires that "
f"np.ptp(data) <= fscale, but np.ptp(data) = {ptp} and "
f"fscale = {fscale}."
)
class uniform_gen(rv_continuous):
r"""A uniform continuous random variable.
In the standard form, the distribution is uniform on ``[0, 1]``. Using
the parameters ``loc`` and ``scale``, one obtains the uniform distribution
on ``[loc, loc + scale]``.
%(before_notes)s
%(example)s
"""
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return random_state.uniform(0.0, 1.0, size)
def _pdf(self, x):
return 1.0*(x == x)
def _cdf(self, x):
return x
def _ppf(self, q):
return q
def _stats(self):
return 0.5, 1.0/12, 0, -1.2
def _entropy(self):
return 0.0
@_call_super_mom
def fit(self, data, *args, **kwds):
"""
Maximum likelihood estimate for the location and scale parameters.
`uniform.fit` uses only the following parameters. Because exact
formulas are used, the parameters related to optimization that are
available in the `fit` method of other distributions are ignored
here. The only positional argument accepted is `data`.
Parameters
----------
data : array_like
Data to use in calculating the maximum likelihood estimate.
floc : float, optional
Hold the location parameter fixed to the specified value.
fscale : float, optional
Hold the scale parameter fixed to the specified value.
Returns
-------
loc, scale : float
Maximum likelihood estimates for the location and scale.
Notes
-----
An error is raised if `floc` is given and any values in `data` are
less than `floc`, or if `fscale` is given and `fscale` is less
than ``data.max() - data.min()``. An error is also raised if both
`floc` and `fscale` are given.
Examples
--------
>>> import numpy as np
>>> from scipy.stats import uniform
We'll fit the uniform distribution to `x`:
>>> x = np.array([2, 2.5, 3.1, 9.5, 13.0])
For a uniform distribution MLE, the location is the minimum of the
data, and the scale is the maximum minus the minimum.
>>> loc, scale = uniform.fit(x)
>>> loc
2.0
>>> scale
11.0
If we know the data comes from a uniform distribution where the support
starts at 0, we can use `floc=0`:
>>> loc, scale = uniform.fit(x, floc=0)
>>> loc
0.0
>>> scale
13.0
Alternatively, if we know the length of the support is 12, we can use
`fscale=12`:
>>> loc, scale = uniform.fit(x, fscale=12)
>>> loc
1.5
>>> scale
12.0
In that last example, the support interval is [1.5, 13.5]. This
solution is not unique. For example, the distribution with ``loc=2``
and ``scale=12`` has the same likelihood as the one above. When
`fscale` is given and it is larger than ``data.max() - data.min()``,
the parameters returned by the `fit` method center the support over
the interval ``[data.min(), data.max()]``.
"""
if len(args) > 0:
raise TypeError("Too many arguments.")
floc = kwds.pop('floc', None)
fscale = kwds.pop('fscale', None)
_remove_optimizer_parameters(kwds)
if floc is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
data = np.asarray(data)
if not np.isfinite(data).all():
raise ValueError("The data contains non-finite values.")
# MLE for the uniform distribution
# --------------------------------
# The PDF is
#
# f(x, loc, scale) = {1/scale for loc <= x <= loc + scale
# {0 otherwise}
#
# The likelihood function is
# L(x, loc, scale) = (1/scale)**n
# where n is len(x), assuming loc <= x <= loc + scale for all x.
# The log-likelihood is
# l(x, loc, scale) = -n*log(scale)
# The log-likelihood is maximized by making scale as small as possible,
# while keeping loc <= x <= loc + scale. So if neither loc nor scale
# are fixed, the log-likelihood is maximized by choosing
# loc = x.min()
# scale = np.ptp(x)
# If loc is fixed, it must be less than or equal to x.min(), and then
# the scale is
# scale = x.max() - loc
# If scale is fixed, it must not be less than np.ptp(x). If scale is
# greater than np.ptp(x), the solution is not unique. Note that the
# likelihood does not depend on loc, except for the requirement that
# loc <= x <= loc + scale. All choices of loc for which
# x.max() - scale <= loc <= x.min()
# have the same log-likelihood. In this case, we choose loc such that
# the support is centered over the interval [data.min(), data.max()]:
# loc = x.min() = 0.5*(scale - np.ptp(x))
if fscale is None:
# scale is not fixed.
if floc is None:
# loc is not fixed, scale is not fixed.
loc = data.min()
scale = np.ptp(data)
else:
# loc is fixed, scale is not fixed.
loc = floc
scale = data.max() - loc
if data.min() < loc:
raise FitDataError("uniform", lower=loc, upper=loc + scale)
else:
# loc is not fixed, scale is fixed.
ptp = np.ptp(data)
if ptp > fscale:
raise FitUniformFixedScaleDataError(ptp=ptp, fscale=fscale)
# If ptp < fscale, the ML estimate is not unique; see the comments
# above. We choose the distribution for which the support is
# centered over the interval [data.min(), data.max()].
loc = data.min() - 0.5*(fscale - ptp)
scale = fscale
# We expect the return values to be floating point, so ensure it
# by explicitly converting to float.
return float(loc), float(scale)
uniform = uniform_gen(a=0.0, b=1.0, name='uniform')
class vonmises_gen(rv_continuous):
r"""A Von Mises continuous random variable.
%(before_notes)s
See Also
--------
scipy.stats.vonmises_fisher : Von-Mises Fisher distribution on a
hypersphere
Notes
-----
The probability density function for `vonmises` and `vonmises_line` is:
.. math::
f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }
for :math:`-\pi \le x \le \pi`, :math:`\kappa \ge 0`. :math:`I_0` is the
modified Bessel function of order zero (`scipy.special.i0`).
`vonmises` is a circular distribution which does not restrict the
distribution to a fixed interval. Currently, there is no circular
distribution framework in SciPy. The ``cdf`` is implemented such that
``cdf(x + 2*np.pi) == cdf(x) + 1``.
`vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]`
on the real line. This is a regular (i.e. non-circular) distribution.
Note about distribution parameters: `vonmises` and `vonmises_line` take
``kappa`` as a shape parameter (concentration) and ``loc`` as the location
(circular mean). A ``scale`` parameter is accepted but does not have any
effect.
Examples
--------
Import the necessary modules.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import vonmises
Define distribution parameters.
>>> loc = 0.5 * np.pi # circular mean
>>> kappa = 1 # concentration
Compute the probability density at ``x=0`` via the ``pdf`` method.
>>> vonmises.pdf(0, loc=loc, kappa=kappa)
0.12570826359722018
Verify that the percentile function ``ppf`` inverts the cumulative
distribution function ``cdf`` up to floating point accuracy.
>>> x = 1
>>> cdf_value = vonmises.cdf(x, loc=loc, kappa=kappa)
>>> ppf_value = vonmises.ppf(cdf_value, loc=loc, kappa=kappa)
>>> x, cdf_value, ppf_value
(1, 0.31489339900904967, 1.0000000000000004)
Draw 1000 random variates by calling the ``rvs`` method.
>>> sample_size = 1000
>>> sample = vonmises(loc=loc, kappa=kappa).rvs(sample_size)
Plot the von Mises density on a Cartesian and polar grid to emphasize
that it is a circular distribution.
>>> fig = plt.figure(figsize=(12, 6))
>>> left = plt.subplot(121)
>>> right = plt.subplot(122, projection='polar')
>>> x = np.linspace(-np.pi, np.pi, 500)
>>> vonmises_pdf = vonmises.pdf(x, loc=loc, kappa=kappa)
>>> ticks = [0, 0.15, 0.3]
The left image contains the Cartesian plot.
>>> left.plot(x, vonmises_pdf)
>>> left.set_yticks(ticks)
>>> number_of_bins = int(np.sqrt(sample_size))
>>> left.hist(sample, density=True, bins=number_of_bins)
>>> left.set_title("Cartesian plot")
>>> left.set_xlim(-np.pi, np.pi)
>>> left.grid(True)
The right image contains the polar plot.
>>> right.plot(x, vonmises_pdf, label="PDF")
>>> right.set_yticks(ticks)
>>> right.hist(sample, density=True, bins=number_of_bins,
... label="Histogram")
>>> right.set_title("Polar plot")
>>> right.legend(bbox_to_anchor=(0.15, 1.06))
"""
def _shape_info(self):
return [_ShapeInfo("kappa", False, (0, np.inf), (True, False))]
def _argcheck(self, kappa):
return kappa >= 0
def _rvs(self, kappa, size=None, random_state=None):
return random_state.vonmises(0.0, kappa, size=size)
@inherit_docstring_from(rv_continuous)
def rvs(self, *args, **kwds):
rvs = super().rvs(*args, **kwds)
return np.mod(rvs + np.pi, 2*np.pi) - np.pi
def _pdf(self, x, kappa):
# vonmises.pdf(x, kappa) = exp(kappa * cos(x)) / (2*pi*I[0](kappa))
# = exp(kappa * (cos(x) - 1)) /
# (2*pi*exp(-kappa)*I[0](kappa))
# = exp(kappa * cosm1(x)) / (2*pi*i0e(kappa))
return np.exp(kappa*sc.cosm1(x)) / (2*np.pi*sc.i0e(kappa))
def _logpdf(self, x, kappa):
# vonmises.pdf(x, kappa) = exp(kappa * cosm1(x)) / (2*pi*i0e(kappa))
return kappa * sc.cosm1(x) - np.log(2*np.pi) - np.log(sc.i0e(kappa))
def _cdf(self, x, kappa):
return _stats.von_mises_cdf(kappa, x)
def _stats_skip(self, kappa):
return 0, None, 0, None
def _entropy(self, kappa):
# vonmises.entropy(kappa) = -kappa * I[1](kappa) / I[0](kappa) +
# log(2 * np.pi * I[0](kappa))
# = -kappa * I[1](kappa) * exp(-kappa) /
# (I[0](kappa) * exp(-kappa)) +
# log(2 * np.pi *
# I[0](kappa) * exp(-kappa) / exp(-kappa))
# = -kappa * sc.i1e(kappa) / sc.i0e(kappa) +
# log(2 * np.pi * i0e(kappa)) + kappa
return (-kappa * sc.i1e(kappa) / sc.i0e(kappa) +
np.log(2 * np.pi * sc.i0e(kappa)) + kappa)
@extend_notes_in_docstring(rv_continuous, notes="""\
The default limits of integration are endpoints of the interval
of width ``2*pi`` centered at `loc` (e.g. ``[-pi, pi]`` when
``loc=0``).\n\n""")
def expect(self, func=None, args=(), loc=0, scale=1, lb=None, ub=None,
conditional=False, **kwds):
_a, _b = -np.pi, np.pi
if lb is None:
lb = loc + _a
if ub is None:
ub = loc + _b
return super().expect(func, args, loc,
scale, lb, ub, conditional, **kwds)
@_call_super_mom
@extend_notes_in_docstring(rv_continuous, notes="""\
Fit data is assumed to represent angles and will be wrapped onto the
unit circle. `f0` and `fscale` are ignored; the returned shape is
always the maximum likelihood estimate and the scale is always
1. Initial guesses are ignored.\n\n""")
def fit(self, data, *args, **kwds):
if kwds.pop('superfit', False):
return super().fit(data, *args, **kwds)
data, fshape, floc, fscale = _check_fit_input_parameters(self, data,
args, kwds)
if self.a == -np.pi:
# vonmises line case, here the default fit method will be used
return super().fit(data, *args, **kwds)
# wrap data to interval [0, 2*pi]
data = np.mod(data, 2 * np.pi)
def find_mu(data):
return stats.circmean(data)
def find_kappa(data, loc):
# Usually, sources list the following as the equation to solve for
# the MLE of the shape parameter:
# r = I[1](kappa)/I[0](kappa), where r = mean resultant length
# This is valid when the location is the MLE of location.
# More generally, when the location may be fixed at an arbitrary
# value, r should be defined as follows:
r = np.sum(np.cos(loc - data))/len(data)
# See gh-18128 for more information.
# The function r[0](kappa) := I[1](kappa)/I[0](kappa) is monotonic
# increasing from r[0](0) = 0 to r[0](+inf) = 1. The partial
# derivative of the log likelihood function with respect to kappa
# is monotonic decreasing in kappa.
if r == 1:
# All observations are (almost) equal to the mean. Return
# some large kappa such that r[0](kappa) = 1.0 numerically.
return 1e16
elif r > 0:
def solve_for_kappa(kappa):
return sc.i1e(kappa)/sc.i0e(kappa) - r
# The bounds of the root of r[0](kappa) = r are derived from
# selected bounds of r[0](x) given in [1, Eq. 11 & 16]. See
# gh-20102 for details.
#
# [1] Amos, D. E. (1973). Computation of Modified Bessel
# Functions and Their Ratios. Mathematics of Computation,
# 28(125): 239-251.
lower_bound = r/(1-r)/(1+r)
upper_bound = 2*lower_bound
# The bounds are violated numerically for certain values of r,
# where solve_for_kappa evaluated at the bounds have the same
# sign. This indicates numerical imprecision of i1e()/i0e().
# Return the violated bound in this case as it's more accurate.
if solve_for_kappa(lower_bound) >= 0:
return lower_bound
elif solve_for_kappa(upper_bound) <= 0:
return upper_bound
else:
root_res = root_scalar(solve_for_kappa, method="brentq",
bracket=(lower_bound, upper_bound))
return root_res.root
else:
# if the provided floc is very far from the circular mean,
# the mean resultant length r can become negative.
# In that case, the equation
# I[1](kappa)/I[0](kappa) = r does not have a solution.
# The maximum likelihood kappa is then 0 which practically
# results in the uniform distribution on the circle. As
# vonmises is defined for kappa > 0, return instead the
# smallest floating point value.
# See gh-18190 for more information
return np.finfo(float).tiny
# location likelihood equation has a solution independent of kappa
loc = floc if floc is not None else find_mu(data)
# shape likelihood equation depends on location
shape = fshape if fshape is not None else find_kappa(data, loc)
loc = np.mod(loc + np.pi, 2 * np.pi) - np.pi # ensure in [-pi, pi]
return shape, loc, 1 # scale is not handled
vonmises = vonmises_gen(name='vonmises')
vonmises_line = vonmises_gen(a=-np.pi, b=np.pi, name='vonmises_line')
class wald_gen(invgauss_gen):
r"""A Wald continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `wald` is:
.. math::
f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x })
for :math:`x >= 0`.
`wald` is a special case of `invgauss` with ``mu=1``.
%(after_notes)s
%(example)s
"""
_support_mask = rv_continuous._open_support_mask
def _shape_info(self):
return []
def _rvs(self, size=None, random_state=None):
return random_state.wald(1.0, 1.0, size=size)
def _pdf(self, x):
# wald.pdf(x) = 1/sqrt(2*pi*x**3) * exp(-(x-1)**2/(2*x))
return invgauss._pdf(x, 1.0)
def _cdf(self, x):
return invgauss._cdf(x, 1.0)
def _sf(self, x):
return invgauss._sf(x, 1.0)
def _ppf(self, x):
return invgauss._ppf(x, 1.0)
def _isf(self, x):
return invgauss._isf(x, 1.0)
def _logpdf(self, x):
return invgauss._logpdf(x, 1.0)
def _logcdf(self, x):
return invgauss._logcdf(x, 1.0)
def _logsf(self, x):
return invgauss._logsf(x, 1.0)
def _stats(self):
return 1.0, 1.0, 3.0, 15.0
def _entropy(self):
return invgauss._entropy(1.0)
wald = wald_gen(a=0.0, name="wald")
class wrapcauchy_gen(rv_continuous):
r"""A wrapped Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `wrapcauchy` is:
.. math::
f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))}
for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`.
`wrapcauchy` takes ``c`` as a shape parameter for :math:`c`.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
return (c > 0) & (c < 1)
def _shape_info(self):
return [_ShapeInfo("c", False, (0, 1), (False, False))]
def _pdf(self, x, c):
# wrapcauchy.pdf(x, c) = (1-c**2) / (2*pi*(1+c**2-2*c*cos(x)))
return (1.0-c*c)/(2*np.pi*(1+c*c-2*c*np.cos(x)))
def _cdf(self, x, c):
def f1(x, cr):
# CDF for 0 <= x < pi
return 1/np.pi * np.arctan(cr*np.tan(x/2))
def f2(x, cr):
# CDF for pi <= x <= 2*pi
return 1 - 1/np.pi * np.arctan(cr*np.tan((2*np.pi - x)/2))
cr = (1 + c)/(1 - c)
return _lazywhere(x < np.pi, (x, cr), f=f1, f2=f2)
def _ppf(self, q, c):
val = (1.0-c)/(1.0+c)
rcq = 2*np.arctan(val*np.tan(np.pi*q))
rcmq = 2*np.pi-2*np.arctan(val*np.tan(np.pi*(1-q)))
return np.where(q < 1.0/2, rcq, rcmq)
def _entropy(self, c):
return np.log(2*np.pi*(1-c*c))
def _fitstart(self, data):
# Use 0.5 as the initial guess of the shape parameter.
# For the location and scale, use the minimum and
# peak-to-peak/(2*pi), respectively.
if isinstance(data, CensoredData):
data = data._uncensor()
return 0.5, np.min(data), np.ptp(data)/(2*np.pi)
wrapcauchy = wrapcauchy_gen(a=0.0, b=2*np.pi, name='wrapcauchy')
class gennorm_gen(rv_continuous):
r"""A generalized normal continuous random variable.
%(before_notes)s
See Also
--------
laplace : Laplace distribution
norm : normal distribution
Notes
-----
The probability density function for `gennorm` is [1]_:
.. math::
f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta),
where :math:`x` is a real number, :math:`\beta > 0` and
:math:`\Gamma` is the gamma function (`scipy.special.gamma`).
`gennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
For :math:`\beta = 1`, it is identical to a Laplace distribution.
For :math:`\beta = 2`, it is identical to a normal distribution
(with ``scale=1/sqrt(2)``).
References
----------
.. [1] "Generalized normal distribution, Version 1",
https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
.. [2] Nardon, Martina, and Paolo Pianca. "Simulation techniques for
generalized Gaussian densities." Journal of Statistical
Computation and Simulation 79.11 (2009): 1317-1329
.. [3] Wicklin, Rick. "Simulate data from a generalized Gaussian
distribution" in The DO Loop blog, September 21, 2016,
https://blogs.sas.com/content/iml/2016/09/21/simulate-generalized-gaussian-sas.html
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("beta", False, (0, np.inf), (False, False))]
def _pdf(self, x, beta):
return np.exp(self._logpdf(x, beta))
def _logpdf(self, x, beta):
return np.log(0.5*beta) - sc.gammaln(1.0/beta) - abs(x)**beta
def _cdf(self, x, beta):
c = 0.5 * np.sign(x)
# evaluating (.5 + c) first prevents numerical cancellation
return (0.5 + c) - c * sc.gammaincc(1.0/beta, abs(x)**beta)
def _ppf(self, x, beta):
c = np.sign(x - 0.5)
# evaluating (1. + c) first prevents numerical cancellation
return c * sc.gammainccinv(1.0/beta, (1.0 + c) - 2.0*c*x)**(1.0/beta)
def _sf(self, x, beta):
return self._cdf(-x, beta)
def _isf(self, x, beta):
return -self._ppf(x, beta)
def _stats(self, beta):
c1, c3, c5 = sc.gammaln([1.0/beta, 3.0/beta, 5.0/beta])
return 0., np.exp(c3 - c1), 0., np.exp(c5 + c1 - 2.0*c3) - 3.
def _entropy(self, beta):
return 1. / beta - np.log(.5 * beta) + sc.gammaln(1. / beta)
def _rvs(self, beta, size=None, random_state=None):
# see [2]_ for the algorithm
# see [3]_ for reference implementation in SAS
z = random_state.gamma(1/beta, size=size)
y = z ** (1/beta)
# convert y to array to ensure masking support
y = np.asarray(y)
mask = random_state.random(size=y.shape) < 0.5
y[mask] = -y[mask]
return y
gennorm = gennorm_gen(name='gennorm')
class halfgennorm_gen(rv_continuous):
r"""The upper half of a generalized normal continuous random variable.
%(before_notes)s
See Also
--------
gennorm : generalized normal distribution
expon : exponential distribution
halfnorm : half normal distribution
Notes
-----
The probability density function for `halfgennorm` is:
.. math::
f(x, \beta) = \frac{\beta}{\Gamma(1/\beta)} \exp(-|x|^\beta)
for :math:`x, \beta > 0`. :math:`\Gamma` is the gamma function
(`scipy.special.gamma`).
`halfgennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
For :math:`\beta = 1`, it is identical to an exponential distribution.
For :math:`\beta = 2`, it is identical to a half normal distribution
(with ``scale=1/sqrt(2)``).
References
----------
.. [1] "Generalized normal distribution, Version 1",
https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("beta", False, (0, np.inf), (False, False))]
def _pdf(self, x, beta):
# beta
# halfgennorm.pdf(x, beta) = ------------- exp(-|x|**beta)
# gamma(1/beta)
return np.exp(self._logpdf(x, beta))
def _logpdf(self, x, beta):
return np.log(beta) - sc.gammaln(1.0/beta) - x**beta
def _cdf(self, x, beta):
return sc.gammainc(1.0/beta, x**beta)
def _ppf(self, x, beta):
return sc.gammaincinv(1.0/beta, x)**(1.0/beta)
def _sf(self, x, beta):
return sc.gammaincc(1.0/beta, x**beta)
def _isf(self, x, beta):
return sc.gammainccinv(1.0/beta, x)**(1.0/beta)
def _entropy(self, beta):
return 1.0/beta - np.log(beta) + sc.gammaln(1.0/beta)
halfgennorm = halfgennorm_gen(a=0, name='halfgennorm')
class crystalball_gen(rv_continuous):
r"""
Crystalball distribution
%(before_notes)s
Notes
-----
The probability density function for `crystalball` is:
.. math::
f(x, \beta, m) = \begin{cases}
N \exp(-x^2 / 2), &\text{for } x > -\beta\\
N A (B - x)^{-m} &\text{for } x \le -\beta
\end{cases}
where :math:`A = (m / |\beta|)^m \exp(-\beta^2 / 2)`,
:math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant.
`crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape
parameters. :math:`\beta` defines the point where the pdf changes
from a power-law to a Gaussian distribution. :math:`m` is the power
of the power-law tail.
%(after_notes)s
.. versionadded:: 0.19.0
References
----------
.. [1] "Crystal Ball Function",
https://en.wikipedia.org/wiki/Crystal_Ball_function
%(example)s
"""
def _argcheck(self, beta, m):
"""
Shape parameter bounds are m > 1 and beta > 0.
"""
return (m > 1) & (beta > 0)
def _shape_info(self):
ibeta = _ShapeInfo("beta", False, (0, np.inf), (False, False))
im = _ShapeInfo("m", False, (1, np.inf), (False, False))
return [ibeta, im]
def _fitstart(self, data):
# Arbitrary, but the default m=1 is not valid
return super()._fitstart(data, args=(1, 1.5))
def _pdf(self, x, beta, m):
"""
Return PDF of the crystalball function.
--
| exp(-x**2 / 2), for x > -beta
crystalball.pdf(x, beta, m) = N * |
| A * (B - x)**(-m), for x <= -beta
--
"""
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
_norm_pdf_C * _norm_cdf(beta))
def rhs(x, beta, m):
return np.exp(-x**2 / 2)
def lhs(x, beta, m):
return ((m/beta)**m * np.exp(-beta**2 / 2.0) *
(m/beta - beta - x)**(-m))
return N * _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs)
def _logpdf(self, x, beta, m):
"""
Return the log of the PDF of the crystalball function.
"""
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
_norm_pdf_C * _norm_cdf(beta))
def rhs(x, beta, m):
return -x**2/2
def lhs(x, beta, m):
return m*np.log(m/beta) - beta**2/2 - m*np.log(m/beta - beta - x)
return np.log(N) + _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs)
def _cdf(self, x, beta, m):
"""
Return CDF of the crystalball function
"""
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
_norm_pdf_C * _norm_cdf(beta))
def rhs(x, beta, m):
return ((m/beta) * np.exp(-beta**2 / 2.0) / (m-1) +
_norm_pdf_C * (_norm_cdf(x) - _norm_cdf(-beta)))
def lhs(x, beta, m):
return ((m/beta)**m * np.exp(-beta**2 / 2.0) *
(m/beta - beta - x)**(-m+1) / (m-1))
return N * _lazywhere(x > -beta, (x, beta, m), f=rhs, f2=lhs)
def _ppf(self, p, beta, m):
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
_norm_pdf_C * _norm_cdf(beta))
pbeta = N * (m/beta) * np.exp(-beta**2/2) / (m - 1)
def ppf_less(p, beta, m):
eb2 = np.exp(-beta**2/2)
C = (m/beta) * eb2 / (m-1)
N = 1/(C + _norm_pdf_C * _norm_cdf(beta))
return (m/beta - beta -
((m - 1)*(m/beta)**(-m)/eb2*p/N)**(1/(1-m)))
def ppf_greater(p, beta, m):
eb2 = np.exp(-beta**2/2)
C = (m/beta) * eb2 / (m-1)
N = 1/(C + _norm_pdf_C * _norm_cdf(beta))
return _norm_ppf(_norm_cdf(-beta) + (1/_norm_pdf_C)*(p/N - C))
return _lazywhere(p < pbeta, (p, beta, m), f=ppf_less, f2=ppf_greater)
def _munp(self, n, beta, m):
"""
Returns the n-th non-central moment of the crystalball function.
"""
N = 1.0 / (m/beta / (m-1) * np.exp(-beta**2 / 2.0) +
_norm_pdf_C * _norm_cdf(beta))
def n_th_moment(n, beta, m):
"""
Returns n-th moment. Defined only if n+1 < m
Function cannot broadcast due to the loop over n
"""
A = (m/beta)**m * np.exp(-beta**2 / 2.0)
B = m/beta - beta
rhs = (2**((n-1)/2.0) * sc.gamma((n+1)/2) *
(1.0 + (-1)**n * sc.gammainc((n+1)/2, beta**2 / 2)))
lhs = np.zeros(rhs.shape)
for k in range(n + 1):
lhs += (sc.binom(n, k) * B**(n-k) * (-1)**k / (m - k - 1) *
(m/beta)**(-m + k + 1))
return A * lhs + rhs
return N * _lazywhere(n + 1 < m, (n, beta, m),
np.vectorize(n_th_moment, otypes=[np.float64]),
np.inf)
crystalball = crystalball_gen(name='crystalball', longname="A Crystalball Function")
def _argus_phi(chi):
"""
Utility function for the argus distribution used in the pdf, sf and
moment calculation.
Note that for all x > 0:
gammainc(1.5, x**2/2) = 2 * (_norm_cdf(x) - x * _norm_pdf(x) - 0.5).
This can be verified directly by noting that the cdf of Gamma(1.5) can
be written as erf(sqrt(x)) - 2*sqrt(x)*exp(-x)/sqrt(Pi).
We use gammainc instead of the usual definition because it is more precise
for small chi.
"""
return sc.gammainc(1.5, chi**2/2) / 2
class argus_gen(rv_continuous):
r"""
Argus distribution
%(before_notes)s
Notes
-----
The probability density function for `argus` is:
.. math::
f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2}
\exp(-\chi^2 (1 - x^2)/2)
for :math:`0 < x < 1` and :math:`\chi > 0`, where
.. math::
\Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2
with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard
normal distribution, respectively.
`argus` takes :math:`\chi` as shape a parameter. Details about sampling
from the ARGUS distribution can be found in [2]_.
%(after_notes)s
References
----------
.. [1] "ARGUS distribution",
https://en.wikipedia.org/wiki/ARGUS_distribution
.. [2] Christoph Baumgarten "Random variate generation by fast numerical
inversion in the varying parameter case." Research in Statistics,
vol. 1, 2023, doi:10.1080/27684520.2023.2279060.
.. versionadded:: 0.19.0
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("chi", False, (0, np.inf), (False, False))]
def _logpdf(self, x, chi):
# for x = 0 or 1, logpdf returns -np.inf
with np.errstate(divide='ignore'):
y = 1.0 - x*x
A = 3*np.log(chi) - _norm_pdf_logC - np.log(_argus_phi(chi))
return A + np.log(x) + 0.5*np.log1p(-x*x) - chi**2 * y / 2
def _pdf(self, x, chi):
return np.exp(self._logpdf(x, chi))
def _cdf(self, x, chi):
return 1.0 - self._sf(x, chi)
def _sf(self, x, chi):
return _argus_phi(chi * np.sqrt(1 - x**2)) / _argus_phi(chi)
def _rvs(self, chi, size=None, random_state=None):
chi = np.asarray(chi)
if chi.size == 1:
out = self._rvs_scalar(chi, numsamples=size,
random_state=random_state)
else:
shp, bc = _check_shape(chi.shape, size)
numsamples = int(np.prod(shp))
out = np.empty(size)
it = np.nditer([chi],
flags=['multi_index'],
op_flags=[['readonly']])
while not it.finished:
idx = tuple((it.multi_index[j] if not bc[j] else slice(None))
for j in range(-len(size), 0))
r = self._rvs_scalar(it[0], numsamples=numsamples,
random_state=random_state)
out[idx] = r.reshape(shp)
it.iternext()
if size == ():
out = out[()]
return out
def _rvs_scalar(self, chi, numsamples=None, random_state=None):
# if chi <= 1.8:
# use rejection method, see Devroye:
# Non-Uniform Random Variate Generation, 1986, section II.3.2.
# write: PDF f(x) = c * g(x) * h(x), where
# h is [0,1]-valued and g is a density
# we use two ways to write f
#
# Case 1:
# write g(x) = 3*x*sqrt(1-x**2), h(x) = exp(-chi**2 (1-x**2) / 2)
# If X has a distribution with density g its ppf G_inv is given by:
# G_inv(u) = np.sqrt(1 - u**(2/3))
#
# Case 2:
# g(x) = chi**2 * x * exp(-chi**2 * (1-x**2)/2) / (1 - exp(-chi**2 /2))
# h(x) = sqrt(1 - x**2), 0 <= x <= 1
# one can show that
# G_inv(u) = np.sqrt(2*np.log(u*(np.exp(chi**2/2)-1)+1))/chi
# = np.sqrt(1 + 2*np.log(np.exp(-chi**2/2)*(1-u)+u)/chi**2)
# the latter expression is used for precision with small chi
#
# In both cases, the inverse cdf of g can be written analytically, and
# we can apply the rejection method:
#
# REPEAT
# Generate U uniformly distributed on [0, 1]
# Generate X with density g (e.g. via inverse transform sampling:
# X = G_inv(V) with V uniformly distributed on [0, 1])
# UNTIL X <= h(X)
# RETURN X
#
# We use case 1 for chi <= 0.5 as it maintains precision for small chi
# and case 2 for 0.5 < chi <= 1.8 due to its speed for moderate chi.
#
# if chi > 1.8:
# use relation to the Gamma distribution: if X is ARGUS with parameter
# chi), then Y = chi**2 * (1 - X**2) / 2 has density proportional to
# sqrt(u) * exp(-u) on [0, chi**2 / 2], i.e. a Gamma(3/2) distribution
# conditioned on [0, chi**2 / 2]). Therefore, to sample X from the
# ARGUS distribution, we sample Y from the gamma distribution, keeping
# only samples on [0, chi**2 / 2], and apply the inverse
# transformation X = (1 - 2*Y/chi**2)**(1/2). Since we only
# look at chi > 1.8, gamma(1.5).cdf(chi**2/2) is large enough such
# Y falls in the interval [0, chi**2 / 2] with a high probability:
# stats.gamma(1.5).cdf(1.8**2/2) = 0.644...
#
# The points to switch between the different methods are determined
# by a comparison of the runtime of the different methods. However,
# the runtime is platform-dependent. The implemented values should
# ensure a good overall performance and are supported by an analysis
# of the rejection constants of different methods.
size1d = tuple(np.atleast_1d(numsamples))
N = int(np.prod(size1d))
x = np.zeros(N)
simulated = 0
chi2 = chi * chi
if chi <= 0.5:
d = -chi2 / 2
while simulated < N:
k = N - simulated
u = random_state.uniform(size=k)
v = random_state.uniform(size=k)
z = v**(2/3)
# acceptance condition: u <= h(G_inv(v)). This simplifies to
accept = (np.log(u) <= d * z)
num_accept = np.sum(accept)
if num_accept > 0:
# we still need to transform z=v**(2/3) to X = G_inv(v)
rvs = np.sqrt(1 - z[accept])
x[simulated:(simulated + num_accept)] = rvs
simulated += num_accept
elif chi <= 1.8:
echi = np.exp(-chi2 / 2)
while simulated < N:
k = N - simulated
u = random_state.uniform(size=k)
v = random_state.uniform(size=k)
z = 2 * np.log(echi * (1 - v) + v) / chi2
# as in case one, simplify u <= h(G_inv(v)) and then transform
# z to the target distribution X = G_inv(v)
accept = (u**2 + z <= 0)
num_accept = np.sum(accept)
if num_accept > 0:
rvs = np.sqrt(1 + z[accept])
x[simulated:(simulated + num_accept)] = rvs
simulated += num_accept
else:
# conditional Gamma for chi > 1.8
while simulated < N:
k = N - simulated
g = random_state.standard_gamma(1.5, size=k)
accept = (g <= chi2 / 2)
num_accept = np.sum(accept)
if num_accept > 0:
x[simulated:(simulated + num_accept)] = g[accept]
simulated += num_accept
x = np.sqrt(1 - 2 * x / chi2)
return np.reshape(x, size1d)
def _stats(self, chi):
# need to ensure that dtype is float
# otherwise the mask below does not work for integers
chi = np.asarray(chi, dtype=float)
phi = _argus_phi(chi)
m = np.sqrt(np.pi/8) * chi * sc.ive(1, chi**2/4) / phi
# compute second moment, use Taylor expansion for small chi (<= 0.1)
mu2 = np.empty_like(chi)
mask = chi > 0.1
c = chi[mask]
mu2[mask] = 1 - 3 / c**2 + c * _norm_pdf(c) / phi[mask]
c = chi[~mask]
coef = [-358/65690625, 0, -94/1010625, 0, 2/2625, 0, 6/175, 0, 0.4]
mu2[~mask] = np.polyval(coef, c)
return m, mu2 - m**2, None, None
argus = argus_gen(name='argus', longname="An Argus Function", a=0.0, b=1.0)
class rv_histogram(rv_continuous):
"""
Generates a distribution given by a histogram.
This is useful to generate a template distribution from a binned
datasample.
As a subclass of the `rv_continuous` class, `rv_histogram` inherits from it
a collection of generic methods (see `rv_continuous` for the full list),
and implements them based on the properties of the provided binned
datasample.
Parameters
----------
histogram : tuple of array_like
Tuple containing two array_like objects.
The first containing the content of n bins,
the second containing the (n+1) bin boundaries.
In particular, the return value of `numpy.histogram` is accepted.
density : bool, optional
If False, assumes the histogram is proportional to counts per bin;
otherwise, assumes it is proportional to a density.
For constant bin widths, these are equivalent, but the distinction
is important when bin widths vary (see Notes).
If None (default), sets ``density=True`` for backwards compatibility,
but warns if the bin widths are variable. Set `density` explicitly
to silence the warning.
.. versionadded:: 1.10.0
Notes
-----
When a histogram has unequal bin widths, there is a distinction between
histograms that are proportional to counts per bin and histograms that are
proportional to probability density over a bin. If `numpy.histogram` is
called with its default ``density=False``, the resulting histogram is the
number of counts per bin, so ``density=False`` should be passed to
`rv_histogram`. If `numpy.histogram` is called with ``density=True``, the
resulting histogram is in terms of probability density, so ``density=True``
should be passed to `rv_histogram`. To avoid warnings, always pass
``density`` explicitly when the input histogram has unequal bin widths.
There are no additional shape parameters except for the loc and scale.
The pdf is defined as a stepwise function from the provided histogram.
The cdf is a linear interpolation of the pdf.
.. versionadded:: 0.19.0
Examples
--------
Create a scipy.stats distribution from a numpy histogram
>>> import scipy.stats
>>> import numpy as np
>>> data = scipy.stats.norm.rvs(size=100000, loc=0, scale=1.5,
... random_state=123)
>>> hist = np.histogram(data, bins=100)
>>> hist_dist = scipy.stats.rv_histogram(hist, density=False)
Behaves like an ordinary scipy rv_continuous distribution
>>> hist_dist.pdf(1.0)
0.20538577847618705
>>> hist_dist.cdf(2.0)
0.90818568543056499
PDF is zero above (below) the highest (lowest) bin of the histogram,
defined by the max (min) of the original dataset
>>> hist_dist.pdf(np.max(data))
0.0
>>> hist_dist.cdf(np.max(data))
1.0
>>> hist_dist.pdf(np.min(data))
7.7591907244498314e-05
>>> hist_dist.cdf(np.min(data))
0.0
PDF and CDF follow the histogram
>>> import matplotlib.pyplot as plt
>>> X = np.linspace(-5.0, 5.0, 100)
>>> fig, ax = plt.subplots()
>>> ax.set_title("PDF from Template")
>>> ax.hist(data, density=True, bins=100)
>>> ax.plot(X, hist_dist.pdf(X), label='PDF')
>>> ax.plot(X, hist_dist.cdf(X), label='CDF')
>>> ax.legend()
>>> fig.show()
"""
_support_mask = rv_continuous._support_mask
def __init__(self, histogram, *args, density=None, **kwargs):
"""
Create a new distribution using the given histogram
Parameters
----------
histogram : tuple of array_like
Tuple containing two array_like objects.
The first containing the content of n bins,
the second containing the (n+1) bin boundaries.
In particular, the return value of np.histogram is accepted.
density : bool, optional
If False, assumes the histogram is proportional to counts per bin;
otherwise, assumes it is proportional to a density.
For constant bin widths, these are equivalent.
If None (default), sets ``density=True`` for backward
compatibility, but warns if the bin widths are variable. Set
`density` explicitly to silence the warning.
"""
self._histogram = histogram
self._density = density
if len(histogram) != 2:
raise ValueError("Expected length 2 for parameter histogram")
self._hpdf = np.asarray(histogram[0])
self._hbins = np.asarray(histogram[1])
if len(self._hpdf) + 1 != len(self._hbins):
raise ValueError("Number of elements in histogram content "
"and histogram boundaries do not match, "
"expected n and n+1.")
self._hbin_widths = self._hbins[1:] - self._hbins[:-1]
bins_vary = not np.allclose(self._hbin_widths, self._hbin_widths[0])
if density is None and bins_vary:
message = ("Bin widths are not constant. Assuming `density=True`."
"Specify `density` explicitly to silence this warning.")
warnings.warn(message, RuntimeWarning, stacklevel=2)
density = True
elif not density:
self._hpdf = self._hpdf / self._hbin_widths
self._hpdf = self._hpdf / float(np.sum(self._hpdf * self._hbin_widths))
self._hcdf = np.cumsum(self._hpdf * self._hbin_widths)
self._hpdf = np.hstack([0.0, self._hpdf, 0.0])
self._hcdf = np.hstack([0.0, self._hcdf])
# Set support
kwargs['a'] = self.a = self._hbins[0]
kwargs['b'] = self.b = self._hbins[-1]
super().__init__(*args, **kwargs)
def _pdf(self, x):
"""
PDF of the histogram
"""
return self._hpdf[np.searchsorted(self._hbins, x, side='right')]
def _cdf(self, x):
"""
CDF calculated from the histogram
"""
return np.interp(x, self._hbins, self._hcdf)
def _ppf(self, x):
"""
Percentile function calculated from the histogram
"""
return np.interp(x, self._hcdf, self._hbins)
def _munp(self, n):
"""Compute the n-th non-central moment."""
integrals = (self._hbins[1:]**(n+1) - self._hbins[:-1]**(n+1)) / (n+1)
return np.sum(self._hpdf[1:-1] * integrals)
def _entropy(self):
"""Compute entropy of distribution"""
res = _lazywhere(self._hpdf[1:-1] > 0.0,
(self._hpdf[1:-1],),
np.log,
0.0)
return -np.sum(self._hpdf[1:-1] * res * self._hbin_widths)
def _updated_ctor_param(self):
"""
Set the histogram as additional constructor argument
"""
dct = super()._updated_ctor_param()
dct['histogram'] = self._histogram
dct['density'] = self._density
return dct
class studentized_range_gen(rv_continuous):
r"""A studentized range continuous random variable.
%(before_notes)s
See Also
--------
t: Student's t distribution
Notes
-----
The probability density function for `studentized_range` is:
.. math::
f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2)
2^{\nu/2-1}} \int_{0}^{\infty} \int_{-\infty}^{\infty}
s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z)
[\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds
for :math:`x ≥ 0`, :math:`k > 1`, and :math:`\nu > 0`.
`studentized_range` takes ``k`` for :math:`k` and ``df`` for :math:`\nu`
as shape parameters.
When :math:`\nu` exceeds 100,000, an asymptotic approximation (infinite
degrees of freedom) is used to compute the cumulative distribution
function [4]_ and probability distribution function.
%(after_notes)s
References
----------
.. [1] "Studentized range distribution",
https://en.wikipedia.org/wiki/Studentized_range_distribution
.. [2] Batista, Ben Dêivide, et al. "Externally Studentized Normal Midrange
Distribution." Ciência e Agrotecnologia, vol. 41, no. 4, 2017, pp.
378-389., doi:10.1590/1413-70542017414047716.
.. [3] Harter, H. Leon. "Tables of Range and Studentized Range." The Annals
of Mathematical Statistics, vol. 31, no. 4, 1960, pp. 1122-1147.
JSTOR, www.jstor.org/stable/2237810. Accessed 18 Feb. 2021.
.. [4] Lund, R. E., and J. R. Lund. "Algorithm AS 190: Probabilities and
Upper Quantiles for the Studentized Range." Journal of the Royal
Statistical Society. Series C (Applied Statistics), vol. 32, no. 2,
1983, pp. 204-210. JSTOR, www.jstor.org/stable/2347300. Accessed 18
Feb. 2021.
Examples
--------
>>> import numpy as np
>>> from scipy.stats import studentized_range
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> k, df = 3, 10
>>> mean, var, skew, kurt = studentized_range.stats(k, df, moments='mvsk')
Display the probability density function (``pdf``):
>>> x = np.linspace(studentized_range.ppf(0.01, k, df),
... studentized_range.ppf(0.99, k, df), 100)
>>> ax.plot(x, studentized_range.pdf(x, k, df),
... 'r-', lw=5, alpha=0.6, label='studentized_range pdf')
Alternatively, the distribution object can be called (as a function)
to fix the shape, location and scale parameters. This returns a "frozen"
RV object holding the given parameters fixed.
Freeze the distribution and display the frozen ``pdf``:
>>> rv = studentized_range(k, df)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of ``cdf`` and ``ppf``:
>>> vals = studentized_range.ppf([0.001, 0.5, 0.999], k, df)
>>> np.allclose([0.001, 0.5, 0.999], studentized_range.cdf(vals, k, df))
True
Rather than using (``studentized_range.rvs``) to generate random variates,
which is very slow for this distribution, we can approximate the inverse
CDF using an interpolator, and then perform inverse transform sampling
with this approximate inverse CDF.
This distribution has an infinite but thin right tail, so we focus our
attention on the leftmost 99.9 percent.
>>> a, b = studentized_range.ppf([0, .999], k, df)
>>> a, b
0, 7.41058083802274
>>> from scipy.interpolate import interp1d
>>> rng = np.random.default_rng()
>>> xs = np.linspace(a, b, 50)
>>> cdf = studentized_range.cdf(xs, k, df)
# Create an interpolant of the inverse CDF
>>> ppf = interp1d(cdf, xs, fill_value='extrapolate')
# Perform inverse transform sampling using the interpolant
>>> r = ppf(rng.uniform(size=1000))
And compare the histogram:
>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
"""
def _argcheck(self, k, df):
return (k > 1) & (df > 0)
def _shape_info(self):
ik = _ShapeInfo("k", False, (1, np.inf), (False, False))
idf = _ShapeInfo("df", False, (0, np.inf), (False, False))
return [ik, idf]
def _fitstart(self, data):
# Default is k=1, but that is not a valid value of the parameter.
return super()._fitstart(data, args=(2, 1))
def _munp(self, K, k, df):
cython_symbol = '_studentized_range_moment'
_a, _b = self._get_support()
# all three of these are used to create a numpy array so they must
# be the same shape.
def _single_moment(K, k, df):
log_const = _stats._studentized_range_pdf_logconst(k, df)
arg = [K, k, df, log_const]
usr_data = np.array(arg, float).ctypes.data_as(ctypes.c_void_p)
llc = LowLevelCallable.from_cython(_stats, cython_symbol, usr_data)
ranges = [(-np.inf, np.inf), (0, np.inf), (_a, _b)]
opts = dict(epsabs=1e-11, epsrel=1e-12)
return integrate.nquad(llc, ranges=ranges, opts=opts)[0]
ufunc = np.frompyfunc(_single_moment, 3, 1)
return np.asarray(ufunc(K, k, df), dtype=np.float64)[()]
def _pdf(self, x, k, df):
def _single_pdf(q, k, df):
# The infinite form of the PDF is derived from the infinite
# CDF.
if df < 100000:
cython_symbol = '_studentized_range_pdf'
log_const = _stats._studentized_range_pdf_logconst(k, df)
arg = [q, k, df, log_const]
usr_data = np.array(arg, float).ctypes.data_as(ctypes.c_void_p)
ranges = [(-np.inf, np.inf), (0, np.inf)]
else:
cython_symbol = '_studentized_range_pdf_asymptotic'
arg = [q, k]
usr_data = np.array(arg, float).ctypes.data_as(ctypes.c_void_p)
ranges = [(-np.inf, np.inf)]
llc = LowLevelCallable.from_cython(_stats, cython_symbol, usr_data)
opts = dict(epsabs=1e-11, epsrel=1e-12)
return integrate.nquad(llc, ranges=ranges, opts=opts)[0]
ufunc = np.frompyfunc(_single_pdf, 3, 1)
return np.asarray(ufunc(x, k, df), dtype=np.float64)[()]
def _cdf(self, x, k, df):
def _single_cdf(q, k, df):
# "When the degrees of freedom V are infinite the probability
# integral takes [on a] simpler form," and a single asymptotic
# integral is evaluated rather than the standard double integral.
# (Lund, Lund, page 205)
if df < 100000:
cython_symbol = '_studentized_range_cdf'
log_const = _stats._studentized_range_cdf_logconst(k, df)
arg = [q, k, df, log_const]
usr_data = np.array(arg, float).ctypes.data_as(ctypes.c_void_p)
ranges = [(-np.inf, np.inf), (0, np.inf)]
else:
cython_symbol = '_studentized_range_cdf_asymptotic'
arg = [q, k]
usr_data = np.array(arg, float).ctypes.data_as(ctypes.c_void_p)
ranges = [(-np.inf, np.inf)]
llc = LowLevelCallable.from_cython(_stats, cython_symbol, usr_data)
opts = dict(epsabs=1e-11, epsrel=1e-12)
return integrate.nquad(llc, ranges=ranges, opts=opts)[0]
ufunc = np.frompyfunc(_single_cdf, 3, 1)
# clip p-values to ensure they are in [0, 1].
return np.clip(np.asarray(ufunc(x, k, df), dtype=np.float64)[()], 0, 1)
studentized_range = studentized_range_gen(name='studentized_range', a=0,
b=np.inf)
class rel_breitwigner_gen(rv_continuous):
r"""A relativistic Breit-Wigner random variable.
%(before_notes)s
See Also
--------
cauchy: Cauchy distribution, also known as the Breit-Wigner distribution.
Notes
-----
The probability density function for `rel_breitwigner` is
.. math::
f(x, \rho) = \frac{k}{(x^2 - \rho^2)^2 + \rho^2}
where
.. math::
k = \frac{2\sqrt{2}\rho^2\sqrt{\rho^2 + 1}}
{\pi\sqrt{\rho^2 + \rho\sqrt{\rho^2 + 1}}}
The relativistic Breit-Wigner distribution is used in high energy physics
to model resonances [1]_. It gives the uncertainty in the invariant mass,
:math:`M` [2]_, of a resonance with characteristic mass :math:`M_0` and
decay-width :math:`\Gamma`, where :math:`M`, :math:`M_0` and :math:`\Gamma`
are expressed in natural units. In SciPy's parametrization, the shape
parameter :math:`\rho` is equal to :math:`M_0/\Gamma` and takes values in
:math:`(0, \infty)`.
Equivalently, the relativistic Breit-Wigner distribution is said to give
the uncertainty in the center-of-mass energy :math:`E_{\text{cm}}`. In
natural units, the speed of light :math:`c` is equal to 1 and the invariant
mass :math:`M` is equal to the rest energy :math:`Mc^2`. In the
center-of-mass frame, the rest energy is equal to the total energy [3]_.
%(after_notes)s
:math:`\rho = M/\Gamma` and :math:`\Gamma` is the scale parameter. For
example, if one seeks to model the :math:`Z^0` boson with :math:`M_0
\approx 91.1876 \text{ GeV}` and :math:`\Gamma \approx 2.4952\text{ GeV}`
[4]_ one can set ``rho=91.1876/2.4952`` and ``scale=2.4952``.
To ensure a physically meaningful result when using the `fit` method, one
should set ``floc=0`` to fix the location parameter to 0.
References
----------
.. [1] Relativistic Breit-Wigner distribution, Wikipedia,
https://en.wikipedia.org/wiki/Relativistic_Breit-Wigner_distribution
.. [2] Invariant mass, Wikipedia,
https://en.wikipedia.org/wiki/Invariant_mass
.. [3] Center-of-momentum frame, Wikipedia,
https://en.wikipedia.org/wiki/Center-of-momentum_frame
.. [4] M. Tanabashi et al. (Particle Data Group) Phys. Rev. D 98, 030001 -
Published 17 August 2018
%(example)s
"""
def _argcheck(self, rho):
return rho > 0
def _shape_info(self):
return [_ShapeInfo("rho", False, (0, np.inf), (False, False))]
def _pdf(self, x, rho):
# C = k / rho**2
C = np.sqrt(
2 * (1 + 1/rho**2) / (1 + np.sqrt(1 + 1/rho**2))
) * 2 / np.pi
with np.errstate(over='ignore'):
return C / (((x - rho)*(x + rho)/rho)**2 + 1)
def _cdf(self, x, rho):
# C = k / (2 * rho**2) / np.sqrt(1 + 1/rho**2)
C = np.sqrt(2/(1 + np.sqrt(1 + 1/rho**2)))/np.pi
result = (
np.sqrt(-1 + 1j/rho)
* np.arctan(x/np.sqrt(-rho*(rho + 1j)))
)
result = C * 2 * np.imag(result)
# Sometimes above formula produces values greater than 1.
return np.clip(result, None, 1)
def _munp(self, n, rho):
if n == 1:
# C = k / (2 * rho)
C = np.sqrt(
2 * (1 + 1/rho**2) / (1 + np.sqrt(1 + 1/rho**2))
) / np.pi * rho
return C * (np.pi/2 + np.arctan(rho))
if n == 2:
# C = pi * k / (4 * rho)
C = np.sqrt(
(1 + 1/rho**2) / (2 * (1 + np.sqrt(1 + 1/rho**2)))
) * rho
result = (1 - rho * 1j) / np.sqrt(-1 - 1j/rho)
return 2 * C * np.real(result)
else:
return np.inf
def _stats(self, rho):
# Returning None from stats makes public stats use _munp.
# nan values will be omitted from public stats. Skew and
# kurtosis are actually infinite.
return None, None, np.nan, np.nan
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
# Override rv_continuous.fit to better handle case where floc is set.
data, _, floc, fscale = _check_fit_input_parameters(
self, data, args, kwds
)
censored = isinstance(data, CensoredData)
if censored:
if data.num_censored() == 0:
# There are no censored values in data, so replace the
# CensoredData instance with a regular array.
data = data._uncensored
censored = False
if floc is None or censored:
return super().fit(data, *args, **kwds)
if fscale is None:
# The interquartile range approximates the scale parameter gamma.
# The median approximates rho * gamma.
p25, p50, p75 = np.quantile(data - floc, [0.25, 0.5, 0.75])
scale_0 = p75 - p25
rho_0 = p50 / scale_0
if not args:
args = [rho_0]
if "scale" not in kwds:
kwds["scale"] = scale_0
else:
M_0 = np.median(data - floc)
rho_0 = M_0 / fscale
if not args:
args = [rho_0]
return super().fit(data, *args, **kwds)
rel_breitwigner = rel_breitwigner_gen(a=0.0, name="rel_breitwigner")
# Collect names of classes and objects in this module.
pairs = list(globals().copy().items())
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_continuous)
__all__ = _distn_names + _distn_gen_names + ['rv_histogram']