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pywafo/wafo/stats/_continuous_distns.py

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#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
from __future__ import division, print_function, absolute_import
import warnings
from scipy.special import comb
from scipy.misc.doccer import inherit_docstring_from
from scipy import special
from scipy import optimize
from scipy import integrate
from scipy.special import (gammaln as gamln, gamma as gam, boxcox, boxcox1p,
inv_boxcox, inv_boxcox1p, erfc, chndtr, chndtrix,
log1p, expm1,
i0, i1, ndtr as _norm_cdf, log_ndtr as _norm_logcdf)
from numpy import (where, arange, putmask, ravel, shape,
log, sqrt, exp, arctanh, tan, sin, arcsin, arctan,
tanh, cos, cosh, sinh)
from numpy import polyval, place, extract, asarray, nan, inf, pi
import numpy as np
from scipy.stats.mstats_basic import mode
try:
from scipy.stats import vonmises_cython
except:
vonmises_cython = None
from scipy.stats._tukeylambda_stats import (tukeylambda_variance as _tlvar,
tukeylambda_kurtosis as _tlkurt)
from ._distn_infrastructure import (
rv_continuous, valarray, _skew, _kurtosis, _lazywhere,
_ncx2_log_pdf, _ncx2_pdf, _ncx2_cdf, get_distribution_names,
)
from ._constants import _XMIN, _EULER, _ZETA3, _XMAX, _LOGXMAX, _EPS
## Kolmogorov-Smirnov one-sided and two-sided test statistics
class ksone_gen(rv_continuous):
"""General Kolmogorov-Smirnov one-sided test.
%(default)s
"""
def _cdf(self, x, n):
return 1.0 - special.smirnov(n, x)
def _ppf(self, q, n):
return special.smirnovi(n, 1.0 - q)
ksone = ksone_gen(a=0.0, name='ksone')
class kstwobign_gen(rv_continuous):
"""Kolmogorov-Smirnov two-sided test for large N.
%(default)s
"""
def _cdf(self, x):
return 1.0 - special.kolmogorov(x)
def _sf(self, x):
return special.kolmogorov(x)
def _ppf(self, q):
return special.kolmogi(1.0-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*pi)
_norm_pdf_logC = np.log(_norm_pdf_C)
def _norm_pdf(x):
return exp(-x**2/2.0) / _norm_pdf_C
def _norm_logpdf(x):
return -x**2 / 2.0 - _norm_pdf_logC
def _norm_ppf(q):
return special.ndtri(q)
def _norm_sf(x):
return _norm_cdf(-x)
def _norm_logsf(x):
return _norm_logcdf(-x)
def _norm_isf(q):
return -special.ndtri(q)
class norm_gen(rv_continuous):
"""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::
norm.pdf(x) = exp(-x**2/2)/sqrt(2*pi)
The survival function, ``norm.sf``, is also referred to as the
Q-function in some contexts (see, e.g.,
`Wikipedia's <https://en.wikipedia.org/wiki/Q-function>`_ definition).
%(after_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.standard_normal(self._size)
def _pdf(self, x):
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*(log(2*pi)+1)
@inherit_docstring_from(rv_continuous)
def fit(self, data, **kwds):
"""%(super)s
This function (norm_gen.fit) uses explicit formulas for the maximum
likelihood estimation of the parameters, so the `optimizer` argument
is ignored.
"""
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
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 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
norm = norm_gen(name='norm')
class alpha_gen(rv_continuous):
"""An alpha continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `alpha` is::
alpha.pdf(x, a) = 1/(x**2*Phi(a)*sqrt(2*pi)) * exp(-1/2 * (a-1/x)**2),
where ``Phi(alpha)`` is the normal CDF, ``x > 0``, and ``a > 0``.
`alpha` takes ``a`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, a):
return 1.0/(x**2)/_norm_cdf(a)*_norm_pdf(a-1.0/x)
def _logpdf(self, x, a):
return -2*log(x) + _norm_logpdf(a-1.0/x) - 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/asarray(a-special.ndtri(q*special.ndtr(a)))
def _stats(self, a):
return [inf]*2 + [nan]*2
alpha = alpha_gen(a=0.0, name='alpha')
class anglit_gen(rv_continuous):
"""An anglit continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `anglit` is::
anglit.pdf(x) = sin(2*x + pi/2) = cos(2*x),
for ``-pi/4 <= x <= pi/4``.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return cos(2*x)
def _cdf(self, x):
return sin(x+pi/4)**2.0
def _ppf(self, q):
return (arcsin(sqrt(q))-pi/4)
def _stats(self):
return 0.0, pi*pi/16-0.5, 0.0, -2*(pi**4 - 96)/(pi*pi-8)**2
def _entropy(self):
return 1-log(2)
anglit = anglit_gen(a=-pi/4, b=pi/4, name='anglit')
class arcsine_gen(rv_continuous):
"""An arcsine continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `arcsine` is::
arcsine.pdf(x) = 1/(pi*sqrt(x*(1-x)))
for ``0 < x < 1``.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return 1.0/pi/sqrt(x*(1-x))
def _cdf(self, x):
return 2.0/pi*arcsin(sqrt(x))
def _ppf(self, q):
return sin(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):
# 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 "
"estimation with {distr!r} requires that {lower!r} < x "
"< {upper!r} for each x in `data`.".format(
distr=distr, lower=lower, upper=upper),
)
class FitSolverError(RuntimeError):
# 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 = special.psi(a + b)
func = s1 - n * (-psiab + special.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 = special.psi(a + b)
func = [s1 - n * (-psiab + special.psi(a)),
s2 - n * (-psiab + special.psi(b))]
return func
class beta_gen(rv_continuous):
"""A beta continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `beta` is::
gamma(a+b) * x**(a-1) * (1-x)**(b-1)
beta.pdf(x, a, b) = ------------------------------------
gamma(a)*gamma(b)
for ``0 < x < 1``, ``a > 0``, ``b > 0``, where ``gamma(z)`` is the gamma
function (`scipy.special.gamma`).
`beta` takes ``a`` and ``b`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, a, b):
return self._random_state.beta(a, b, self._size)
def _pdf(self, x, a, b):
return np.exp(self._logpdf(x, a, b))
def _logpdf(self, x, a, b):
lPx = special.xlog1py(b-1.0, -x) + special.xlogy(a-1.0, x)
lPx -= special.betaln(a, b)
return lPx
def _cdf(self, x, a, b):
return special.btdtr(a, b, x)
def _ppf(self, q, a, b):
return special.btdtri(a, b, q)
def _stats(self, a, b):
mn = a*1.0 / (a + b)
var = (a*b*1.0)/(a+b+1.0)/(a+b)**2.0
g1 = 2.0*(b-a)*sqrt((1.0+a+b)/(a*b)) / (2+a+b)
g2 = 6.0*(a**3 + a**2*(1-2*b) + b**2*(1+b) - 2*a*b*(2+b))
g2 /= a*b*(a+b+2)*(a+b+3)
return mn, var, g1, g2
def _fitstart(self, data):
g1 = _skew(data)
g2 = _kurtosis(data)
def func(x):
a, b = x
sk = 2*(b-a)*sqrt(a + b + 1) / (a + b + 2) / sqrt(a*b)
ku = a**3 - a**2*(2*b-1) + b**2*(b+1) - 2*a*b*(b+2)
ku /= a*b*(a+b+2)*(a+b+3)
ku *= 6
return [sk-g1, ku-g2]
a, b = optimize.fsolve(func, (1.0, 1.0))
return super(beta_gen, self)._fitstart(data, args=(a, b))
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
"""%(super)s
In the special case where both `floc` and `fscale` are given, a
`ValueError` is raised if any value `x` in `data` does not satisfy
`floc < x < floc + fscale`.
"""
# Override rv_continuous.fit, so we can more efficiently handle the
# case where floc and fscale are given.
f0 = (kwds.get('f0', None) or kwds.get('fa', None) or
kwds.get('fix_a', None))
f1 = (kwds.get('f1', None) or kwds.get('fb', None) or
kwds.get('fix_b', None))
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
if floc is None or fscale is None:
# do general fit
return super(beta_gen, self).fit(data, *args, **kwds)
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.)
# Normalize the data to the interval [0, 1].
data = (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 = np.log(1 - data).sum()
# Use the "method of moments" to estimate the initial
# guess for a and b.
fac = xbar * (1 - xbar) / data.var(ddof=0) - 1
a = xbar * fac
b = (1 - xbar) * fac
# Compute the MLE for a and b by solving _beta_mle_ab.
theta, info, ier, mesg = optimize.fsolve(
_beta_mle_ab, [a, b],
args=(len(data), s1, s2),
full_output=True
)
if ier != 1:
raise FitSolverError(mesg=mesg)
a, b = theta
return a, b, floc, fscale
beta = beta_gen(a=0.0, b=1.0, name='beta')
class betaprime_gen(rv_continuous):
"""A beta prime continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `betaprime` is::
betaprime.pdf(x, a, b) = x**(a-1) * (1+x)**(-a-b) / beta(a, b)
for ``x > 0``, ``a > 0``, ``b > 0``, where ``beta(a, b)`` is the beta
function (see `scipy.special.beta`).
`betaprime` takes ``a`` and ``b`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, a, b):
sz, rndm = self._size, self._random_state
u1 = gamma.rvs(a, size=sz, random_state=rndm)
u2 = gamma.rvs(b, size=sz, random_state=rndm)
return (u1 / u2)
def _pdf(self, x, a, b):
return np.exp(self._logpdf(x, a, b))
def _logpdf(self, x, a, b):
return (special.xlogy(a-1.0, x) - special.xlog1py(a+b, x) -
special.betaln(a, b))
def _cdf(self, x, a, b):
return special.betainc(a, b, x/(1.+x))
def _fitstart(self, data):
g1 = np.mean(data)
g2 = mode(data)[0]
def func(x):
a, b = x
me = a / (b - 1) if 1 < b else 1e100
mo = (a - 1) / (b + 1) if 1 <= a else 0
return [me - g1, mo - g2]
a, b = optimize.fsolve(func, (1.0, 1.5))
return super(betaprime_gen, self)._fitstart(data, args=(a, b))
def _munp(self, n, a, b):
if (n == 1.0):
return where(b > 1, a/(b-1.0), inf)
elif (n == 2.0):
return where(b > 2, a*(a+1.0)/((b-2.0)*(b-1.0)), inf)
elif (n == 3.0):
return where(b > 3, a*(a+1.0)*(a+2.0)/((b-3.0)*(b-2.0)*(b-1.0)),
inf)
elif (n == 4.0):
return where(b > 4,
a*(a+1.0)*(a+2.0)*(a+3.0)/((b-4.0)*(b-3.0)
* (b-2.0)*(b-1.0)), inf)
else:
raise NotImplementedError
betaprime = betaprime_gen(a=0.0, name='betaprime')
class bradford_gen(rv_continuous):
"""A Bradford continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `bradford` is::
bradford.pdf(x, c) = c / (k * (1+c*x)),
for ``0 < x < 1``, ``c > 0`` and ``k = log(1+c)``.
`bradford` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
return c / (c * x + 1.0) / log1p(c)
def _cdf(self, x, c):
return log1p(c * x) / log1p(c)
def _ppf(self, q, c):
return ((1.0+c)**q-1)/c
def _stats(self, c, moments='mv'):
k = log1p(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 = sqrt(2)*(12*c*c-9*c*k*(c+2)+2*k*k*(c*(c+3)+3))
g1 /= 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 = log1p(c)
return k/2.0 - log(c/k)
def _fitstart(self, data):
loc = data.min() - 1e-4
scale = (data - loc).max()
m = np.mean((data - loc) / scale)
fun = lambda c: (c - log1p(c)) / (c * log1p(c)) - m
res = optimize.root(fun, 0.3)
c = res.x
return c, loc, scale
bradford = bradford_gen(a=0.0, b=1.0, name='bradford')
class burr_gen(rv_continuous):
"""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
Notes
-----
The probability density function for `burr` is::
burr.pdf(x, c, d) = c * d * x**(-c-1) * (1+x**(-c))**(-d-1)
for ``x > 0``.
`burr` takes ``c`` and ``d`` as shape parameters.
This is the PDF corresponding to the third CDF given in Burr's list;
specifically, it is equation (11) in Burr's paper [1]_.
%(after_notes)s
References
----------
.. [1] Burr, I. W. "Cumulative frequency functions", Annals of
Mathematical Statistics, 13(2), pp 215-232 (1942).
%(example)s
"""
def _pdf(self, x, c, d):
return c * d * (x**(-c - 1.0)) * ((1 + x**(-c))**(-d - 1.0))
def _cdf(self, x, c, d):
return (1 + x**(-c))**(-d)
def _ppf(self, q, c, d):
return (q**(-1.0/d) - 1)**(-1.0/c)
def _munp(self, n, c, d):
nc = 1. * n / c
return d * special.beta(1.0 - nc, d + nc)
burr = burr_gen(a=0.0, name='burr')
class burr12_gen(rv_continuous):
"""A Burr (Type XII) continuous random variable.
%(before_notes)s
See Also
--------
fisk : a special case of either `burr` or ``burr12`` with ``d = 1``
burr : Burr Type III distribution
Notes
-----
The probability density function for `burr` is::
burr12.pdf(x, c, d) = c * d * x**(c-1) * (1+x**(c))**(-d-1)
for ``x > 0``.
`burr12` takes ``c`` and ``d`` as shape parameters.
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] http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm
%(example)s
"""
def _pdf(self, x, c, d):
return np.exp(self._logpdf(x, c, d))
def _logpdf(self, x, c, d):
return log(c) + log(d) + special.xlogy(c-1, x) + special.xlog1py(-d-1, x**c)
def _cdf(self, x, c, d):
return 1 - self._sf(x, c, d)
def _logcdf(self, x, c, d):
return special.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 special.xlog1py(-d, x**c)
def _ppf(self, q, c, d):
return ((1 - q)**(-1.0/d) - 1)**(1.0/c)
def _munp(self, n, c, d):
nc = 1. * n / c
return d * special.beta(1.0 + nc, d - nc)
burr12 = burr12_gen(a=0.0, name='burr12')
class fisk_gen(burr_gen):
"""A Fisk continuous random variable.
The Fisk distribution is also known as the log-logistic distribution, and
equals the Burr distribution with ``d == 1``.
`fisk` takes ``c`` as a shape parameter.
%(before_notes)s
Notes
-----
The probability density function for `fisk` is::
fisk.pdf(x, c) = c * x**(-c-1) * (1 + x**(-c))**(-2)
for ``x > 0``.
`fisk` takes ``c`` as a shape parameters.
%(after_notes)s
See Also
--------
burr
%(example)s
"""
def _pdf(self, x, c):
return burr_gen._pdf(self, x, c, 1.0)
def _cdf(self, x, c):
return burr_gen._cdf(self, x, c, 1.0)
def _ppf(self, x, c):
return burr_gen._ppf(self, x, c, 1.0)
def _munp(self, n, c):
return burr_gen._munp(self, n, c, 1.0)
def _entropy(self, c):
return 2 - log(c)
fisk = fisk_gen(a=0.0, name='fisk')
# median = loc
class cauchy_gen(rv_continuous):
"""A Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `cauchy` is::
cauchy.pdf(x) = 1 / (pi * (1 + x**2))
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return 1.0/pi/(1.0+x*x)
def _cdf(self, x):
return 0.5 + 1.0/pi*arctan(x)
def _ppf(self, q):
return tan(pi*q-pi/2.0)
def _sf(self, x):
return 0.5 - 1.0/pi*arctan(x)
def _isf(self, q):
return tan(pi/2.0-pi*q)
def _stats(self):
return nan, nan, nan, nan
def _entropy(self):
return log(4*pi)
def _fitstart(self, data, args=None):
# Initialize ML guesses using quartiles instead of moments.
p25, p50, p75 = np.percentile(data, [25, 50, 75])
return p50, (p75 - p25)/2
cauchy = cauchy_gen(name='cauchy')
class chi_gen(rv_continuous):
"""A chi continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `chi` is::
chi.pdf(x, df) = x**(df-1) * exp(-x**2/2) / (2**(df/2-1) * gamma(df/2))
for ``x > 0``.
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 _rvs(self, df):
sz, rndm = self._size, self._random_state
return sqrt(chi2.rvs(df, size=sz, random_state=rndm))
def _pdf(self, x, df):
return np.exp(self._logpdf(x, df))
def _logpdf(self, x, df):
l = np.log(2) - .5*np.log(2)*df - special.gammaln(.5*df)
return l + special.xlogy(df-1.,x) - .5*x**2
def _cdf(self, x, df):
return special.gammainc(.5*df, .5*x**2)
def _ppf(self, q, df):
return sqrt(2*special.gammaincinv(.5*df, q))
def _stats(self, df):
mu = sqrt(2)*special.gamma(df/2.0+0.5)/special.gamma(df/2.0)
mu2 = df - mu*mu
g1 = (2*mu**3.0 + mu*(1-2*df))/asarray(np.power(mu2, 1.5))
g2 = 2*df*(1.0-df)-6*mu**4 + 4*mu**2 * (2*df-1)
g2 /= asarray(mu2**2.0)
return mu, mu2, g1, g2
def _fitstart(self, data):
m = data.mean()
v = data.var()
# Supply a starting guess with method of moments:
df = max(np.round(v + m ** 2), 1)
return super(chi_gen, self)._fitstart(data, args=(df,))
chi = chi_gen(a=0.0, name='chi')
## Chi-squared (gamma-distributed with loc=0 and scale=2 and shape=df/2)
class chi2_gen(rv_continuous):
"""A chi-squared continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `chi2` is::
chi2.pdf(x, df) = 1 / (2*gamma(df/2)) * (x/2)**(df/2-1) * exp(-x/2)
`chi2` takes ``df`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, df):
return self._random_state.chisquare(df, self._size)
def _pdf(self, x, df):
return exp(self._logpdf(x, df))
def _logpdf(self, x, df):
return special.xlogy(df/2.-1, x) - x/2. - gamln(df/2.) - (log(2)*df)/2.
def _cdf(self, x, df):
return special.chdtr(df, x)
def _sf(self, x, df):
return special.chdtrc(df, x)
def _isf(self, p, df):
return special.chdtri(df, p)
def _ppf(self, p, df):
return self._isf(1.0-p, df)
def _stats(self, df):
mu = df
mu2 = 2*df
g1 = 2*sqrt(2.0/df)
g2 = 12.0/df
return mu, mu2, g1, g2
def _fitstart(self, data):
m = data.mean()
v = data.var()
# Supply a starting guess with method of moments:
df = max(np.round((m + v / 2) / 2), 1)
return super(chi2_gen, self)._fitstart(data, args=(df,))
chi2 = chi2_gen(a=0.0, name='chi2')
class cosine_gen(rv_continuous):
"""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::
cosine.pdf(x) = 1/(2*pi) * (1+cos(x))
for ``-pi <= x <= pi``.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return 1.0/2/pi*(1+cos(x))
def _cdf(self, x):
return 1.0/2/pi*(pi + x + sin(x))
def _stats(self):
return 0.0, pi*pi/3.0-2.0, 0.0, -6.0*(pi**4-90)/(5.0*(pi*pi-6)**2)
def _entropy(self):
return log(4*pi)-1.0
cosine = cosine_gen(a=-pi, b=pi, name='cosine')
class dgamma_gen(rv_continuous):
"""A double gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `dgamma` is::
dgamma.pdf(x, a) = 1 / (2*gamma(a)) * abs(x)**(a-1) * exp(-abs(x))
for ``a > 0``.
`dgamma` takes ``a`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, a):
sz, rndm = self._size, self._random_state
u = rndm.random_sample(size=sz)
gm = gamma.rvs(a, size=sz, random_state=rndm)
return gm * where(u >= 0.5, 1, -1)
def _pdf(self, x, a):
ax = abs(x)
return 1.0/(2*special.gamma(a))*ax**(a-1.0) * exp(-ax)
def _logpdf(self, x, a):
ax = abs(x)
return special.xlogy(a-1.0, ax) - ax - log(2) - gamln(a)
def _cdf(self, x, a):
fac = 0.5*special.gammainc(a, abs(x))
return where(x > 0, 0.5 + fac, 0.5 - fac)
def _sf(self, x, a):
fac = 0.5*special.gammainc(a, abs(x))
return where(x > 0, 0.5-fac, 0.5+fac)
def _ppf(self, q, a):
fac = special.gammainccinv(a, 1-abs(2*q-1))
return where(q > 0.5, fac, -fac)
def _stats(self, a):
mu2 = a*(a+1.0)
return 0.0, mu2, 0.0, (a+2.0)*(a+3.0)/mu2-3.0
dgamma = dgamma_gen(name='dgamma')
class dweibull_gen(rv_continuous):
"""A double Weibull continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `dweibull` is::
dweibull.pdf(x, c) = c / 2 * abs(x)**(c-1) * exp(-abs(x)**c)
`dweibull` takes ``d`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, c):
sz, rndm = self._size, self._random_state
u = rndm.random_sample(size=sz)
w = weibull_min.rvs(c, size=sz, random_state=rndm)
return w * (where(u >= 0.5, 1, -1))
def _pdf(self, x, c):
ax = abs(x)
Px = c / 2.0 * ax**(c-1.0) * exp(-ax**c)
return Px
def _logpdf(self, x, c):
ax = abs(x)
return log(c) - log(2.0) + special.xlogy(c - 1.0, ax) - ax**c
def _cdf(self, x, c):
Cx1 = 0.5 * exp(-abs(x)**c)
return where(x > 0, 1 - Cx1, Cx1)
def _ppf(self, q, c):
fac = 2. * where(q <= 0.5, q, 1. - q)
fac = np.power(-log(fac), 1.0 / c)
return where(q > 0.5, fac, -fac)
def _munp(self, n, c):
return (1 - (n % 2)) * special.gamma(1.0 + 1.0 * n / c)
# since we know that all odd moments are zeros, return them at once.
# returning Nones from _stats makes the public stats call _munp
# so overall we're saving one or two gamma function evaluations here.
def _stats(self, c):
return 0, None, 0, None
dweibull = dweibull_gen(name='dweibull')
## Exponential (gamma distributed with a=1.0, loc=loc and scale=scale)
class expon_gen(rv_continuous):
"""An exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `expon` is::
expon.pdf(x) = exp(-x)
for ``x >= 0``.
%(after_notes)s
A common parameterization for `expon` is in terms of the rate parameter
``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This
parameterization corresponds to using ``scale = 1 / lambda``.
%(example)s
"""
def _link(self, x, logSF, phat, ix):
if ix == 1:
return - (x - phat[0]) / logSF
elif ix == 0:
return x + phat[1] * logSF
else:
raise IndexError('Index to the fixed parameter is out of bounds')
def _rvs(self):
return self._random_state.standard_exponential(self._size)
def _pdf(self, x):
return exp(-x)
def _logpdf(self, x):
return -x
def _cdf(self, x):
return -special.expm1(-x)
def _ppf(self, q):
return -special.log1p(-q)
def _sf(self, x):
return exp(-x)
def _logsf(self, x):
return -x
def _isf(self, q):
return -log(q)
def _stats(self):
return 1.0, 1.0, 2.0, 6.0
def _entropy(self):
return 1.0
expon = expon_gen(a=0.0, name='expon')
## Exponentially Modified Normal (exponential distribution
## convolved with a Normal).
## This is called an exponentially modified gaussian on wikipedia
class exponnorm_gen(rv_continuous):
"""An exponentially modified Normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `exponnorm` is::
exponnorm.pdf(x, K) = 1/(2*K) exp(1/(2 * K**2)) exp(-x / K) * erfc(-(x - 1/K) / sqrt(2))
where the shape parameter ``K > 0``.
It can be thought of as the sum of a normally distributed random
value with mean ``loc`` and sigma ``scale`` and an exponentially
distributed random number with a pdf proportional to ``exp(-lambda * x)``
where ``lambda = (K * scale)**(-1)``.
%(after_notes)s
An alternative parameterization of this distribution (for example, in
`Wikipedia <http://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution>`_)
involves three parameters, :math:`\mu`, :math:`\lambda` and :math:`\sigma`.
In the present parameterization this corresponds to having ``loc`` and
``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and
shape parameter :math:`K = 1/\sigma\lambda`.
.. versionadded:: 0.16.0
%(example)s
"""
def _rvs(self, K):
expval = self._random_state.standard_exponential(self._size) * K
gval = self._random_state.standard_normal(self._size)
return expval + gval
def _pdf(self, x, K):
invK = 1.0 / K
exparg = 0.5 * invK**2 - invK * x
# Avoid overflows; setting exp(exparg) to the max float works
# all right here
expval = _lazywhere(exparg < _LOGXMAX, (exparg,), exp, _XMAX)
return 0.5 * invK * expval * erfc(-(x - invK) / sqrt(2))
def _logpdf(self, x, K):
invK = 1.0 / K
exparg = 0.5 * invK**2 - invK * x
return exparg + log(0.5 * invK * erfc(-(x - invK) / sqrt(2)))
def _cdf(self, x, K):
invK = 1.0 / K
expval = invK * (0.5 * invK - x)
return _norm_cdf(x) - exp(expval) * _norm_cdf(x - invK)
def _sf(self, x, K):
invK = 1.0 / K
expval = invK * (0.5 * invK - x)
return _norm_cdf(-x) + exp(expval) * _norm_cdf(x - invK)
def _stats(self, K):
K2 = K * K
opK2 = 1.0 + K2
skw = 2 * K**3 * opK2**(-1.5)
krt = 6.0 * K2 * K2 * opK2**(-2)
return K, opK2, skw, krt
exponnorm = exponnorm_gen(name='exponnorm')
class exponweib_gen(rv_continuous):
"""An exponentiated Weibull continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `exponweib` is::
exponweib.pdf(x, a, c) =
a * c * (1-exp(-x**c))**(a-1) * exp(-x**c)*x**(c-1)
for ``x > 0``, ``a > 0``, ``c > 0``.
`exponweib` takes ``a`` and ``c`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, a, c):
return exp(self._logpdf(x, a, c))
def _logpdf(self, x, a, c):
negxc = -x**c
exm1c = -special.expm1(negxc)
logp = (log(a) + log(c) + special.xlogy(a - 1.0, exm1c) +
negxc + special.xlogy(c - 1.0, x))
return logp
def _cdf(self, x, a, c):
exm1c = -special.expm1(-x**c)
return exm1c**a
def _ppf(self, q, a, c):
return (-special.log1p(-q**(1.0/a)))**asarray(1.0/c)
exponweib = exponweib_gen(a=0.0, name='exponweib')
class exponpow_gen(rv_continuous):
"""An exponential power continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `exponpow` is::
exponpow.pdf(x, b) = b * x**(b-1) * exp(1 + x**b - exp(x**b))
for ``x >= 0``, ``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.
%(after_notes)s
References
----------
http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf
%(example)s
"""
def _pdf(self, x, b):
return exp(self._logpdf(x, b))
def _logpdf(self, x, b):
xb = x**b
f = 1 + log(b) + special.xlogy(b - 1.0, x) + xb - exp(xb)
return f
def _cdf(self, x, b):
return -special.expm1(-special.expm1(x**b))
def _sf(self, x, b):
return exp(-special.expm1(x**b))
def _isf(self, x, b):
return (special.log1p(-log(x)))**(1./b)
def _ppf(self, q, b):
return pow(special.log1p(-special.log1p(-q)), 1.0/b)
exponpow = exponpow_gen(a=0.0, name='exponpow')
class fatiguelife_gen(rv_continuous):
"""A fatigue-life (Birnbaum-Saunders) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `fatiguelife` is::
fatiguelife.pdf(x, c) =
(x+1) / (2*c*sqrt(2*pi*x**3)) * exp(-(x-1)**2/(2*x*c**2))
for ``x > 0``.
`fatiguelife` takes ``c`` as a shape parameter.
%(after_notes)s
References
----------
.. [1] "Birnbaum-Saunders distribution",
http://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution
%(example)s
"""
def _rvs(self, c):
z = self._random_state.standard_normal(self._size)
x = 0.5*c*z
x2 = x*x
t = 1.0 + 2*x2 + 2*x*sqrt(1 + x2)
return t
def _pdf(self, x, c):
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return (log(x+1) - (x-1)**2 / (2.0*x*c**2) - log(2*c) -
0.5*(log(2*pi) + 3*log(x)))
def _cdf(self, x, c):
return _norm_cdf(1.0 / c * (sqrt(x) - 1.0/sqrt(x)))
def _ppf(self, q, c):
tmp = c*special.ndtri(q)
return 0.25 * (tmp + 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):
"""A folded Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `foldcauchy` is::
foldcauchy.pdf(x, c) = 1/(pi*(1+(x-c)**2)) + 1/(pi*(1+(x+c)**2))
for ``x >= 0``.
`foldcauchy` takes ``c`` as a shape parameter.
%(example)s
"""
def _rvs(self, c):
return abs(cauchy.rvs(loc=c, size=self._size,
random_state=self._random_state))
def _pdf(self, x, c):
return 1.0/pi*(1.0/(1+(x-c)**2) + 1.0/(1+(x+c)**2))
def _cdf(self, x, c):
return 1.0/pi*(arctan(x-c) + arctan(x+c))
def _stats(self, c):
return inf, inf, nan, nan
foldcauchy = foldcauchy_gen(a=0.0, name='foldcauchy')
class f_gen(rv_continuous):
"""An F continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `f` is::
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)
for ``x > 0``.
`f` takes ``dfn`` and ``dfd`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, dfn, dfd):
return self._random_state.f(dfn, dfd, self._size)
def _pdf(self, x, dfn, dfd):
return exp(self._logpdf(x, dfn, dfd))
def _logpdf(self, x, dfn, dfd):
n = 1.0 * dfn
m = 1.0 * dfd
lPx = m/2 * log(m) + n/2 * log(n) + (n/2 - 1) * log(x)
lPx -= ((n+m)/2) * log(m + n*x) + special.betaln(n/2, m/2)
return lPx
def _cdf(self, x, dfn, dfd):
return special.fdtr(dfn, dfd, x)
def _sf(self, x, dfn, dfd):
return special.fdtrc(dfn, dfd, x)
def _ppf(self, q, dfn, dfd):
return special.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 * 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 _fitstart(self, data):
m = data.mean()
v = data.var()
# Supply a starting guess with method of moments:
dfd = max(np.round(2 * m / (m - 1)), 5)
dfn = max(
np.round(2 * dfd * dfd * (dfd - 2) /
(v * (dfd - 4) * (dfd - 2) ** 2 - 2 * dfd * dfd)), 1)
return super(f_gen, self)._fitstart(data, args=(dfn, dfd,))
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):
"""A folded normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `foldnorm` is::
foldnormal.pdf(x, c) = sqrt(2/pi) * cosh(c*x) * exp(-(x**2+c**2)/2)
for ``c >= 0``.
`foldnorm` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
return (c >= 0)
def _rvs(self, c):
return abs(self._random_state.standard_normal(self._size) + c)
def _pdf(self, x, c):
return _norm_pdf(x + c) + _norm_pdf(x-c)
def _cdf(self, x, c):
return _norm_cdf(x-c) + _norm_cdf(x+c) - 1.0
def _stats(self, c):
# Regina C. Elandt, Technometrics 3, 551 (1961)
# http://www.jstor.org/stable/1266561
#
c2 = c*c
expfac = np.exp(-0.5*c2) / np.sqrt(2.*pi)
mu = 2.*expfac + c * special.erf(c/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')
## Extreme Value Type II or Frechet
## (defined in Regress+ documentation as Extreme LB) as
## a limiting value distribution.
##
class frechet_r_gen(rv_continuous):
"""A Frechet right (or Weibull minimum) continuous random variable.
%(before_notes)s
See Also
--------
weibull_min : The same distribution as `frechet_r`.
frechet_l, weibull_max
Notes
-----
The probability density function for `frechet_r` is::
frechet_r.pdf(x, c) = c * x**(c-1) * exp(-x**c)
for ``x > 0``, ``c > 0``.
`frechet_r` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _link(self, x, logSF, phat, ix):
if ix == 0:
phati = log(-logSF) / log((x - phat[1]) / phat[2])
elif ix == 1:
phati = x - phat[2] * (-logSF) ** (1. / phat[0])
elif ix == 2:
phati = (x - phat[1]) / (-logSF) ** (1. / phat[0])
else:
raise IndexError('Index to the fixed parameter is out of bounds')
return phati
def _pdf(self, x, c):
return c*pow(x, c-1)*exp(-pow(x, c))
def _logpdf(self, x, c):
return log(c) + special.xlogy(c - 1, x) - pow(x, c)
def _cdf(self, x, c):
return -special.expm1(-pow(x, c))
def _ppf(self, q, c):
return pow(-special.log1p(-q), 1.0/c)
def _munp(self, n, c):
return special.gamma(1.0+n*1.0/c)
def _entropy(self, c):
return -_EULER / c - log(c) + _EULER + 1
def _fitstart(self, data):
loc = data.min() - 0.01 # *np.std(data)
chat = 1. / (6 ** (1 / 2) / pi * np.std(log(data - loc)))
scale = np.mean((data - loc) ** chat) ** (1. / chat)
return chat, loc, scale
frechet_r = frechet_r_gen(a=0.0, name='frechet_r')
weibull_min = frechet_r_gen(a=0.0, name='weibull_min')
class frechet_l_gen(rv_continuous):
"""A Frechet left (or Weibull maximum) continuous random variable.
%(before_notes)s
See Also
--------
weibull_max : The same distribution as `frechet_l`.
frechet_r, weibull_min
Notes
-----
The probability density function for `frechet_l` is::
frechet_l.pdf(x, c) = c * (-x)**(c-1) * exp(-(-x)**c)
for ``x < 0``, ``c > 0``.
`frechet_l` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
return c*pow(-x, c-1)*exp(-pow(-x, c))
def _cdf(self, x, c):
return exp(-pow(-x, c))
def _ppf(self, q, c):
return -pow(-log(q), 1.0/c)
def _munp(self, n, c):
val = special.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 - log(c) + _EULER + 1
def _fitstart(self, data):
loc = data.max() + 0.1 * np.std(data)
chat = 1. / (6 ** (1 / 2) / pi * np.std(log(loc - data)))
scale = np.mean((loc - data) ** chat) ** (1. / chat)
return chat, loc, scale
frechet_l = frechet_l_gen(b=0.0, name='frechet_l')
weibull_max = frechet_l_gen(b=0.0, name='weibull_max')
class genlogistic_gen(rv_continuous):
"""A generalized logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genlogistic` is::
genlogistic.pdf(x, c) = c * exp(-x) / (1 + exp(-x))**(c+1)
for ``x > 0``, ``c > 0``.
`genlogistic` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
return exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return log(c) - x - (c+1.0)*special.log1p(exp(-x))
def _cdf(self, x, c):
Cx = (1+exp(-x))**(-c)
return Cx
def _ppf(self, q, c):
vals = -log(pow(q, -1.0/c)-1)
return vals
def _stats(self, c):
zeta = special.zeta
mu = _EULER + special.psi(c)
mu2 = pi*pi/6.0 + zeta(2, c)
g1 = -2*zeta(3, c) + 2*_ZETA3
g1 /= np.power(mu2, 1.5)
g2 = pi**4/15.0 + 6*zeta(4, c)
g2 /= mu2**2.0
return mu, mu2, g1, g2
genlogistic = genlogistic_gen(name='genlogistic')
def log1pxdx(x):
'''Computes Log(1+x)/x
'''
xd = where((x == 0) | (x == inf), 1.0, x) # avoid 0/0 or inf/inf
y = where(x == 0, 1.0, log1p(x) / xd)
return where(x == inf, 0.0, y)
class genpareto_gen(rv_continuous):
"""A generalized Pareto continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genpareto` is::
genpareto.pdf(x, c) = (1 + c * x)**(-1 - 1/c)
defined for ``x >= 0`` if ``c >=0``, and for
``0 <= x <= -1/c`` if ``c < 0``.
`genpareto` takes ``c`` as a shape parameter.
For ``c == 0``, `genpareto` reduces to the exponential
distribution, `expon`::
genpareto.pdf(x, c=0) = exp(-x)
For ``c == -1``, `genpareto` is uniform on ``[0, 1]``::
genpareto.cdf(x, c=-1) = x
%(after_notes)s
%(example)s
"""
def _link(self, x, logSF, phat, ix):
# Reference
# Stuart Coles (2004)
# "An introduction to statistical modelling of extreme values".
# Springer series in statistics
c, loc, scale = phat
if ix == 2:
# Reorganizing w.r.t.scale, Eq. 4.13 and 4.14, pp 81 in
# Coles (2004) gives
# link = -(x-loc)*c/expm1(-c*logSF)
if c != 0.0:
phati = (x - loc) * c / expm1(-c * logSF)
else:
phati = -(x - loc) / logSF
elif ix == 1:
if c != 0:
phati = x + scale * expm1(c * logSF) / c
else:
phati = x + scale * logSF
elif ix == 0:
raise NotImplementedError(
'link(x,logSF,phat,i) where i=0 is not implemented!')
else:
raise IndexError('Index to the fixed parameter is out of bounds')
return phati
def _argcheck(self, c):
c = asarray(c)
self.b = _lazywhere(c < 0, (c,),
lambda c: -1. / c, np.inf)
return where(abs(c) == inf, False, True)
def _pdf(self, x, c):
return np.exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -special.xlog1py(c+1., c*x) / c,
-x)
def _cdf(self, x, c):
return -inv_boxcox1p(-x, -c)
def _sf(self, x, c):
return inv_boxcox(-x, -c)
def _logsf(self, x, c):
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -special.log1p(c*x) / c,
-x)
def _ppf(self, q, c):
return -boxcox1p(-q, -c)
def _isf(self, q, c):
return -boxcox(q, -c)
def _fitstart(self, data):
d = asarray(data)
loc = d.min() - 0.01 * d.std()
# moments estimator
d1 = d - loc
m = d1.mean()
s = d1.std()
shape = ((m / s) ** 2 - 1) / 2
scale = m * ((m / s) ** 2 + 1) / 2
return shape, loc, scale
def hessian_nnlf(self, theta, x, eps=None):
try:
loc = theta[-2]
scale = theta[-1]
args = tuple(theta[:-2])
except IndexError:
raise ValueError("Not enough input arguments.")
if not self._argcheck(*args) or scale <= 0:
return inf
x = asarray((x - loc) / scale)
cond0 = (x <= self.a) | (x >= self.b)
if any(cond0):
np = self.numargs + 2
return valarray((np, np), value=nan)
eps = _EPS
c = args[0]
n = len(x)
if abs(c) > eps:
cx = c * x
sumlog1pcx = sum(log1p(cx))
# LL = n*log(scale) + (1-1/k)*sumlog1mkxn
r = x / (1.0 + cx)
sumix = sum(1.0 / (1.0 + cx) ** 2.0)
sumr = sum(r)
sumr2 = sum(r ** 2.0)
H11 = -2 * sumlog1pcx / c ** 3 + 2 * \
sumr / c ** 2 + (1.0 + 1.0 / c) * sumr2
H22 = c * (c + 1) * sumix / scale ** 2.0
H33 = (n - 2 * (c + 1) * sumr +
c * (c + 1) * sumr2) / scale ** 2.0
H12 = -sum((1 - x) / ((1 + cx) ** 2.0)) / scale
H23 = -(c + 1) * sumix / scale ** 2.0
H13 = -(sumr - (c + 1) * sumr2) / scale
else: # c == 0
sumx = sum(x)
# LL = n*log(scale) + sumx;
sumx2 = sum(x ** 2.0)
H11 = -(2 / 3) * sum(x ** 3.0) + sumx2
H22 = 0.0
H12 = -(n - sum(x)) / scale
H23 = -n * 1.0 / scale ** 2.0
H33 = (n - 2 * sumx) / scale ** 2.0
H13 = -(sumx - sumx2) / scale
# Hessian matrix
H = [[H11, H12, H13], [H12, H22, H23], [H13, H23, H33]]
return asarray(H)
def __stats(self, c):
# return None,None,None,None
k = -c
m = where(k < -1.0, inf, 1.0 / (1 + k))
v = where(k < -0.5, nan, 1.0 / ((1 + k) ** 2.0 * (1 + 2 * k)))
sk = where(k < -1.0 / 3, nan, 2. * (1 - k)
* sqrt(1 + 2.0 * k) / (1.0 + 3. * k))
# E(X^r) = s^r*(-k)^-(r+1)*gamma(1+r)*gamma(-1/k-r)/gamma(1-1/k)
# = s^r*gamma(1+r)./( (1+k)*(1+2*k).*....*(1+r*k))
# E[(1-k(X-m0)/s)^r] = 1/(1+k*r)
# Ex3 = (sk.*sqrt(v)+3*m).*v+m^3
# Ex3 = 6.*s.^3/((1+k).*(1+2*k).*(1+3*k))
r = 4.0
Ex4 = gam(1. + r) / \
((1. + k) * (1. + 2. * k) * (1. + 3. * k) * (1 + 4. * k))
m1 = m
ku = where(k < -1. / 4, nan,
(Ex4 - 4. * sk * v ** (3. / 2)
* m1 - 6 * m1 ** 2. * v - m1 ** 4.) / v ** 2. - 3.0)
return m, v, sk, ku
def _munp(self, n, c):
def __munp(n, c):
val = 0.0
k = arange(0, n + 1)
for ki, cnk in zip(k, comb(n, k)):
val = val + cnk * (-1) ** ki / (1.0 - c * ki)
return where(c * n < 1, val * (-1.0 / c) ** n, inf)
return _lazywhere(c != 0, (c,),
lambda c: __munp(n, c),
gam(n + 1))
def _entropy(self, c):
return 1. + c
genpareto = genpareto_gen(a=0.0, name='genpareto')
class genexpon_gen(rv_continuous):
"""A generalized exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genexpon` is::
genexpon.pdf(x, a, b, c) = (a + b * (1 - exp(-c*x))) * \
exp(-a*x - b*x + b/c * (1-exp(-c*x)))
for ``x >= 0``, ``a, b, c > 0``.
`genexpon` takes ``a``, ``b`` and ``c`` as shape parameters.
%(after_notes)s
References
----------
H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential
Distribution", Journal of the American Statistical Association, 1993.
N. Balakrishnan, "The Exponential Distribution: Theory, Methods and
Applications", Asit P. Basu.
%(example)s
"""
def _link(self, x, logSF, phat, ix):
_a, b, c, loc, scale = phat
xn = (x - loc) / scale
fact1 = (xn + expm1(-c * xn) / c)
if ix == 0:
phati = b * fact1 + logSF
elif ix == 1:
phati = (phat[0] - logSF) / fact1
elif ix in [2, 3, 4]:
raise NotImplementedError('Only implemented for index in [0,1]!')
else:
raise IndexError('Index to the fixed parameter is out of bounds')
return phati
def _pdf(self, x, a, b, c):
return (a + b*(-special.expm1(-c*x)))*exp((-a-b)*x +
b*(-special.expm1(-c*x))/c)
def _cdf(self, x, a, b, c):
return -special.expm1((-a-b)*x + b*(-special.expm1(-c*x))/c)
def _logpdf(self, x, a, b, c):
return np.log(a+b*(-special.expm1(-c*x))) + \
(-a-b)*x+b*(-special.expm1(-c*x))/c
genexpon = genexpon_gen(a=0.0, name='genexpon')
class genextreme_gen(rv_continuous):
"""A generalized extreme value continuous random variable.
%(before_notes)s
See Also
--------
gumbel_r
Notes
-----
For ``c=0``, `genextreme` is equal to `gumbel_r`.
The probability density function for `genextreme` is::
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 <= 1/c, c > 0
Note that several sources and software packages use the opposite
convention for the sign of the shape parameter ``c``.
`genextreme` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
min = np.minimum
max = np.maximum
self.b = where(c > 0, 1.0 / max(c, _XMIN), inf)
self.a = where(c < 0, 1.0 / min(c, -_XMIN), -inf)
return where(abs(c) == inf, 0, 1)
def _pdf(self, x, c):
return exp(self._logpdf(x, c))
def _logpdf(self, x, c):
x1 = where((c == 0) & (x == inf), 0.0, x)
cx = c * x1
cond1 = (c == 0) * (x == x)
logex2 = where(cond1, 0.0, log1p(-cx))
logpex2 = -x * log1pxdx(-cx)
# logpex2 = where(cond1,-x, logex2/c)
pex2 = exp(logpex2)
# Handle special cases
logpdf = where(
(cx == 1) | (cx == -inf), -inf, -pex2 + logpex2 - logex2)
putmask(logpdf, (c == 1) & (x == 1), 0.0)
return logpdf
def _cdf(self, x, c):
return exp(self._logcdf(x, c))
def _logcdf(self, x, c):
x1 = where((c == 0) & (x == inf), 0.0, x)
cx = c * x1
loglogcdf = -x * log1pxdx(-cx)
# loglogcdf = where((c==0)*(x==x),-x,log1p(-cx)/c)
return -exp(loglogcdf)
def _sf(self, x, c):
return -expm1(self._logcdf(x, c))
def _ppf(self, q, c):
x = -log(-log(q))
return _lazywhere((x == x) & (c != 0), (x, c),
lambda x, c: -expm1(-c * x) / c, x)
def _isf(self, q, c):
x = -log(-special.log1p(-q))
result = _lazywhere((c == 0)*(x == x), (x, c),
f=lambda x, c: x,
f2=lambda x, c: -special.expm1(-c*x)/c)
return result
def _stats(self, c):
g = lambda n: gam(n*c+1)
g1 = g(1)
g2 = g(2)
g3 = g(3)
g4 = g(4)
g2mg12 = where(abs(c) < 1e-7, (c*pi)**2.0/6.0, g2-g1**2.0)
gam2k = where(abs(c) < 1e-7, pi**2.0/6.0,
special.expm1(gamln(2.0*c+1.0)-2*gamln(c+1.0))/c**2.0)
eps = 1e-14
gamk = where(abs(c) < eps, -_EULER, special.expm1(gamln(c+1))/c)
m = where(c < -1.0, nan, -gamk)
v = where(c < -0.5, nan, g1**2.0*gam2k)
# skewness
sk1 = where(c < -1./3, nan,
np.sign(c)*(-g3+(g2+2*g2mg12)*g1)/((g2mg12)**(3./2.)))
sk = where(abs(c) <= eps**0.29, 12*sqrt(6)*_ZETA3/pi**3, sk1)
# kurtosis
ku1 = where(c < -1./4, nan,
(g4+(-4*g3+3*(g2+g2mg12)*g1)*g1)/((g2mg12)**2))
ku = where(abs(c) <= (eps)**0.23, 12.0/5.0, ku1-3.0)
return m, v, sk, ku
def _fitstart(self, data):
d = asarray(data)
# Probability weighted moments
log = np.log
n = len(d)
d.sort()
koeff1 = np.r_[0:n] / (n - 1)
koeff2 = koeff1 * (np.r_[0:n] - 1) / (n - 2)
b2 = np.dot(koeff2, d) / n
b1 = np.dot(koeff1, d) / n
b0 = d.mean()
z = (2 * b1 - b0) / (3 * b2 - b0) - log(2) / log(3)
shape = 7.8590 * z + 2.9554 * z ** 2
scale = (2 * b1 - b0) * shape / \
(exp(gamln(1 + shape)) * (1 - 2 ** (-shape)))
loc = b0 + scale * (expm1(gamln(1 + shape))) / shape
return shape, loc, scale
def _munp(self, n, c):
k = arange(0, n+1)
vals = 1.0/c**n * np.sum(
comb(n, k) * (-1)**k * special.gamma(c*k + 1),
axis=0)
return where(c*n > -1, vals, 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 `digamma(x) - y = 0`. There is probably room for
# improvement, but currently it works over a wide range of y:
# >>> y = 64*np.random.randn(1000000)
# >>> y.min(), y.max()
# (-311.43592651416662, 351.77388222276869)
# x = [_digammainv(t) for t in y]
# np.abs(digamma(x) - y).max()
# 1.1368683772161603e-13
#
_em = 0.5772156649015328606065120
func = lambda x: special.digamma(x) - y
if y > -0.125:
x0 = 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 = 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):
"""A gamma continuous random variable.
%(before_notes)s
See Also
--------
erlang, expon
Notes
-----
The probability density function for `gamma` is::
gamma.pdf(x, a) = x**(a-1) * exp(-x) / gamma(a)
for ``x >= 0``, ``a > 0``. Here ``gamma(a)`` refers to the gamma function.
`gamma` has a shape parameter `a` which needs to be set explicitly.
When ``a`` is an integer, `gamma` reduces to the Erlang
distribution, and when ``a=1`` to the exponential distribution.
%(after_notes)s
%(example)s
"""
def _rvs(self, a):
return self._random_state.standard_gamma(a, self._size)
def _pdf(self, x, a):
return exp(self._logpdf(x, a))
def _logpdf(self, x, a):
return special.xlogy(a-1.0, x) - x - gamln(a)
def _cdf(self, x, a):
return special.gammainc(a, x)
def _sf(self, x, a):
return special.gammaincc(a, x)
def _ppf(self, q, a):
return special.gammaincinv(a, q)
def _stats(self, a):
return a, a, 2.0/sqrt(a), 6.0/a
def _entropy(self, a):
return special.psi(a)*(1-a) + a + gamln(a)
def _fitstart(self, data):
# The skewness of the gamma distribution is `4 / sqrt(a)`.
# We invert that to estimate the shape `a` using the skewness
# of the data. The formula is regularized with 1e-8 in the
# denominator to allow for degenerate data where the skewness
# is close to 0.
a = 4 / (1e-8 + _skew(data)**2)
return super(gamma_gen, self)._fitstart(data, args=(a,))
@inherit_docstring_from(rv_continuous)
def fit(self, data, *args, **kwds):
f0 = (kwds.get('f0', None) or kwds.get('fa', None) or
kwds.get('fix_a', None))
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
if floc is None:
# loc is not fixed. Use the default fit method.
return super(gamma_gen, self).fit(data, *args, **kwds)
# Special case: loc is fixed.
if f0 is not None and fscale is not None:
# This check is for consistency with `rv_continuous.fit`.
# Without this check, this function would just return the
# parameters that were given.
raise ValueError("All parameters fixed. There is nothing to "
"optimize.")
# Fixed location is handled by shifting the data.
data = np.asarray(data)
if 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:
# log(a) - special.digamma(a) - log(xbar) + log(data.mean) = 0
s = log(xbar) - log(data).mean()
func = lambda a: log(a) - special.digamma(a) - s
aest = (3-s + np.sqrt((s-3)**2 + 24*s)) / (12*s)
xa = aest*(1-0.4)
xb = aest*(1+0.4)
a = optimize.brentq(func, xa, xb, disp=0)
# The MLE for the scale parameter is just the data mean
# divided by the shape parameter.
scale = xbar / a
else:
# scale is fixed, shape is free
# The MLE for the shape parameter `a` is the solution to:
# special.digamma(a) - log(data).mean() + log(fscale) = 0
c = log(data).mean() - 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)
allpos = np.all(a > 0)
if not allint:
# An Erlang distribution shouldn't really have a non-integer
# shape parameter, so warn the user.
warnings.warn(
'The shape parameter of the erlang distribution '
'has been given a non-integer value %r.' % (a,),
RuntimeWarning)
return allpos
def _fitstart(self, data):
# Override gamma_gen_fitstart so that an integer initial value is
# used. (Also regularize the division, to avoid issues when
# _skew(data) is 0 or close to 0.)
a = int(4.0 / (1e-8 + _skew(data)**2))
return super(gamma_gen, self)._fitstart(data, args=(a,))
# Trivial override of the fit method, so we can monkey-patch its
# docstring.
def fit(self, data, *args, **kwds):
return super(erlang_gen, self).fit(data, *args, **kwds)
if fit.__doc__ is not None:
fit.__doc__ = (rv_continuous.fit.__doc__ +
"""
Notes
-----
The Erlang distribution is generally defined to have integer values
for the shape parameter. This is not enforced by the `erlang` class.
When fitting the distribution, it will generally return a non-integer
value for the shape parameter. By using the keyword argument
`f0=<integer>`, the fit method can be constrained to fit the data to
a specific integer shape parameter.
""")
erlang = erlang_gen(a=0.0, name='erlang')
class gengamma_gen(rv_continuous):
"""A generalized gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gengamma` is::
gengamma.pdf(x, a, c) = abs(c) * x**(c*a-1) * exp(-x**c) / gamma(a)
for ``x > 0``, ``a > 0``, and ``c != 0``.
`gengamma` takes ``a`` and ``c`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, c):
return (a > 0) & (c != 0)
def _pdf(self, x, a, c):
return np.exp(self._logpdf(x, a, c))
def _logpdf(self, x, a, c):
return np.log(abs(c)) + special.xlogy(c*a - 1, x) - x**c - special.gammaln(a)
def _cdf(self, x, a, c):
xc = x**c
val1 = special.gammainc(a, xc)
val2 = special.gammaincc(a, xc)
return np.where(c > 0, val1, val2)
def _sf(self, x, a, c):
xc = x**c
val1 = special.gammainc(a, xc)
val2 = special.gammaincc(a, xc)
return np.where(c > 0, val2, val1)
def _ppf(self, q, a, c):
val1 = special.gammaincinv(a, q)
val2 = special.gammainccinv(a, q)
return np.where(c > 0, val1, val2)**(1.0/c)
def _isf(self, q, a, c):
val1 = special.gammaincinv(a, q)
val2 = special.gammainccinv(a, q)
return np.where(c > 0, val2, val1)**(1.0/c)
def _munp(self, n, a, c):
# Pochhammer symbol: poch(a,n) = gamma(a+n)/gamma(a)
return special.poch(a, n*1.0/c)
def _entropy(self, a, c):
val = special.psi(a)
return a*(1-val) + 1.0/c*val + special.gammaln(a) - np.log(abs(c))
gengamma = gengamma_gen(a=0.0, name='gengamma')
class genhalflogistic_gen(rv_continuous):
"""A generalized half-logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `genhalflogistic` is::
genhalflogistic.pdf(x, c) = 2 * (1-c*x)**(1/c-1) / (1+(1-c*x)**(1/c))**2
for ``0 <= x <= 1/c``, and ``c > 0``.
`genhalflogistic` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
self.b = 1.0 / c
return (c > 0)
def _pdf(self, x, c):
limit = 1.0/c
tmp = 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 = 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)*log(2)
genhalflogistic = genhalflogistic_gen(a=0.0, name='genhalflogistic')
class gompertz_gen(rv_continuous):
"""A Gompertz (or truncated Gumbel) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gompertz` is::
gompertz.pdf(x, c) = c * exp(x) * exp(-c*(exp(x)-1))
for ``x >= 0``, ``c > 0``.
`gompertz` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
return exp(self._logpdf(x, c))
def _logpdf(self, x, c):
return log(c) + x - c * special.expm1(x)
def _cdf(self, x, c):
return -special.expm1(-c * special.expm1(x))
def _ppf(self, q, c):
return special.log1p(-1.0 / c * special.log1p(-q))
def _entropy(self, c):
return 1.0 - log(c) - exp(c)*special.expn(1, c)
gompertz = gompertz_gen(a=0.0, name='gompertz')
class gumbel_r_gen(rv_continuous):
"""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::
gumbel_r.pdf(x) = exp(-(x + exp(-x)))
The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
distribution. It is also related to the extreme value distribution,
log-Weibull and Gompertz distributions.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return exp(self._logpdf(x))
def _logpdf(self, x):
return -x - exp(-x)
def _cdf(self, x):
return exp(-exp(-x))
def _logcdf(self, x):
return -exp(-x)
def _ppf(self, q):
return -log(-log(q))
def _stats(self):
return _EULER, pi*pi/6.0, 12*sqrt(6)/pi**3 * _ZETA3, 12.0/5
def _entropy(self):
# http://en.wikipedia.org/wiki/Gumbel_distribution
return _EULER + 1.
gumbel_r = gumbel_r_gen(name='gumbel_r')
class gumbel_l_gen(rv_continuous):
"""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::
gumbel_l.pdf(x) = exp(x - exp(x))
The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
distribution. It is also related to the extreme value distribution,
log-Weibull and Gompertz distributions.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return exp(self._logpdf(x))
def _logpdf(self, x):
return x - exp(x)
def _cdf(self, x):
return -expm1(-exp(x))
def _ppf(self, q):
return log(-log1p(-q))
def _stats(self):
return -_EULER, pi*pi/6.0, \
-12*sqrt(6)/pi**3 * _ZETA3, 12.0/5
def _entropy(self):
return _EULER + 1.
gumbel_l = gumbel_l_gen(name='gumbel_l')
class halfcauchy_gen(rv_continuous):
"""A Half-Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halfcauchy` is::
halfcauchy.pdf(x) = 2 / (pi * (1 + x**2))
for ``x >= 0``.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return 2.0/pi/(1.0+x*x)
def _logpdf(self, x):
return np.log(2.0/pi) - special.log1p(x*x)
def _cdf(self, x):
return 2.0/pi*arctan(x)
def _ppf(self, q):
return tan(pi/2*q)
def _stats(self):
return inf, inf, nan, nan
def _entropy(self):
return log(2*pi)
halfcauchy = halfcauchy_gen(a=0.0, name='halfcauchy')
class halflogistic_gen(rv_continuous):
"""A half-logistic continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halflogistic` is::
halflogistic.pdf(x) = 2 * exp(-x) / (1+exp(-x))**2 = 1/2 * sech(x/2)**2
for ``x >= 0``.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return exp(self._logpdf(x))
def _logpdf(self, x):
return log(2) - x - 2. * special.log1p(exp(-x))
def _cdf(self, x):
return tanh(x/2.0)
def _ppf(self, q):
return 2*arctanh(q)
def _munp(self, n):
if n == 1:
return 2*log(2)
if n == 2:
return pi*pi/3.0
if n == 3:
return 9*_ZETA3
if n == 4:
return 7*pi**4 / 15.0
return 2*(1-pow(2.0, 1-n))*special.gamma(n+1)*special.zeta(n, 1)
def _entropy(self):
return 2-log(2)
halflogistic = halflogistic_gen(a=0.0, name='halflogistic')
class halfnorm_gen(rv_continuous):
"""A half-normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halfnorm` is::
halfnorm.pdf(x) = sqrt(2/pi) * exp(-x**2/2)
for ``x > 0``.
`halfnorm` is a special case of `chi` with ``df == 1``.
%(after_notes)s
%(example)s
"""
def _rvs(self):
return abs(self._random_state.standard_normal(size=self._size))
def _pdf(self, x):
return sqrt(2.0/pi)*exp(-x*x/2.0)
def _logpdf(self, x):
return 0.5 * np.log(2.0/pi) - x*x/2.0
def _cdf(self, x):
return _norm_cdf(x)*2-1.0
def _ppf(self, q):
return special.ndtri((1+q)/2.0)
def _stats(self):
return (sqrt(2.0/pi), 1-2.0/pi, sqrt(2)*(4-pi)/(pi-2)**1.5,
8*(pi-3)/(pi-2)**2)
def _entropy(self):
return 0.5*log(pi/2.0)+0.5
halfnorm = halfnorm_gen(a=0.0, name='halfnorm')
class hypsecant_gen(rv_continuous):
"""A hyperbolic secant continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `hypsecant` is::
hypsecant.pdf(x) = 1/pi * sech(x)
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return 1.0/(pi*cosh(x))
def _cdf(self, x):
return 2.0/pi*arctan(exp(x))
def _ppf(self, q):
return log(tan(pi*q/2.0))
def _stats(self):
return 0, pi*pi/4, 0, 2
def _entropy(self):
return log(2*pi)
hypsecant = hypsecant_gen(name='hypsecant')
class gausshyper_gen(rv_continuous):
"""A Gauss hypergeometric continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gausshyper` is::
gausshyper.pdf(x, a, b, c, z) =
C * x**(a-1) * (1-x)**(b-1) * (1+z*x)**(-c)
for ``0 <= x <= 1``, ``a > 0``, ``b > 0``, and
``C = 1 / (B(a, b) F[2, 1](c, a; a+b; -z))``
`gausshyper` takes ``a``, ``b``, ``c`` and ``z`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b, c, z):
return (a > 0) & (b > 0) & (c == c) & (z == z)
def _pdf(self, x, a, b, c, z):
Cinv = gam(a)*gam(b)/gam(a+b)*special.hyp2f1(c, a, a+b, -z)
return 1.0/Cinv * x**(a-1.0) * (1.0-x)**(b-1.0) / (1.0+z*x)**c
def _munp(self, n, a, b, c, z):
fac = special.beta(n+a, b) / special.beta(a, b)
num = special.hyp2f1(c, a+n, a+b+n, -z)
den = special.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):
"""An inverted gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `invgamma` is::
invgamma.pdf(x, a) = x**(-a-1) / gamma(a) * exp(-1/x)
for x > 0, a > 0.
`invgamma` takes ``a`` as a shape parameter.
`invgamma` is a special case of `gengamma` with ``c == -1``.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, a):
return exp(self._logpdf(x, a))
def _logpdf(self, x, a):
return (-(a+1) * log(x) - gamln(a) - 1.0/x)
def _cdf(self, x, a):
return 1.0 - special.gammainc(a, 1.0/x)
def _ppf(self, q, a):
return 1.0 / special.gammaincinv(a, 1.-q)
def _stats(self, a, moments='mvsk'):
m1 = _lazywhere(a > 1, (a,), lambda x: 1. / (x - 1.), np.inf)
m2 = _lazywhere(a > 2, (a,), lambda x: 1. / (x - 1.)**2 / (x - 2.),
np.inf)
g1, g2 = None, None
if 's' in moments:
g1 = _lazywhere(
a > 3, (a,),
lambda x: 4. * np.sqrt(x - 2.) / (x - 3.), np.nan)
if 'k' in moments:
g2 = _lazywhere(
a > 4, (a,),
lambda x: 6. * (5. * x - 11.) / (x - 3.) / (x - 4.), np.nan)
return m1, m2, g1, g2
def _entropy(self, a):
return a - (a+1.0) * special.psi(a) + gamln(a)
invgamma = invgamma_gen(a=0.0, name='invgamma')
# scale is gamma from DATAPLOT and B from Regress
class invgauss_gen(rv_continuous):
"""An inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `invgauss` is::
invgauss.pdf(x, mu) = 1 / sqrt(2*pi*x**3) * exp(-(x-mu)**2/(2*x*mu**2))
for ``x > 0``.
`invgauss` takes ``mu`` as a shape parameter.
%(after_notes)s
When `mu` is too small, evaluating the cumulative distribution function will be
inaccurate due to ``cdf(mu -> 0) = inf * 0``.
NaNs are returned for ``mu <= 0.0028``.
%(example)s
"""
def _rvs(self, mu):
return self._random_state.wald(mu, 1.0, size=self._size)
def _pdf(self, x, mu):
return 1.0/sqrt(2*pi*x**3.0)*exp(-1.0/(2*x)*((x-mu)/mu)**2)
def _logpdf(self, x, mu):
return -0.5*log(2*pi) - 1.5*log(x) - ((x-mu)/mu)**2/(2*x)
def _cdf(self, x, mu):
fac = sqrt(1.0/x)
# Numerical accuracy for small `mu` is bad. See #869.
C1 = _norm_cdf(fac*(x-mu)/mu)
C1 += exp(1.0/mu) * _norm_cdf(-fac*(x+mu)/mu) * exp(1.0/mu)
return C1
def _stats(self, mu):
return mu, mu**3.0, 3*sqrt(mu), 15*mu
invgauss = invgauss_gen(a=0.0, name='invgauss')
class invweibull_gen(rv_continuous):
"""An inverted Weibull continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `invweibull` is::
invweibull.pdf(x, c) = c * x**(-c-1) * exp(-x**(-c))
for ``x > 0``, ``c > 0``.
`invweibull` takes ``c`` as a shape parameter.
%(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
"""
def _pdf(self, x, c):
xc1 = np.power(x, -c - 1.0)
xc2 = np.power(x, -c)
xc2 = exp(-xc2)
return c * xc1 * xc2
def _cdf(self, x, c):
xc1 = np.power(x, -c)
return exp(-xc1)
def _ppf(self, q, c):
return np.power(-log(q), -1.0/c)
def _munp(self, n, c):
return special.gamma(1 - n / c)
def _entropy(self, c):
return 1+_EULER + _EULER / c - log(c)
invweibull = invweibull_gen(a=0, name='invweibull')
class johnsonsb_gen(rv_continuous):
"""A Johnson SB continuous random variable.
%(before_notes)s
See Also
--------
johnsonsu
Notes
-----
The probability density function for `johnsonsb` is::
johnsonsb.pdf(x, a, b) = b / (x*(1-x)) * phi(a + b * log(x/(1-x)))
for ``0 < x < 1`` and ``a, b > 0``, and ``phi`` is the normal pdf.
`johnsonsb` takes ``a`` and ``b`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b):
return (b > 0) & (a == a)
def _pdf(self, x, a, b):
trm = _norm_pdf(a + b*log(x/(1.0-x)))
return b*1.0/(x*(1-x))*trm
def _cdf(self, x, a, b):
return _norm_cdf(a + b*log(x/(1.0-x)))
def _ppf(self, q, a, b):
return 1.0 / (1 + exp(-1.0 / b * (_norm_ppf(q) - a)))
johnsonsb = johnsonsb_gen(a=0.0, b=1.0, name='johnsonsb')
class johnsonsu_gen(rv_continuous):
"""A Johnson SU continuous random variable.
%(before_notes)s
See Also
--------
johnsonsb
Notes
-----
The probability density function for `johnsonsu` is::
johnsonsu.pdf(x, a, b) = b / sqrt(x**2 + 1) *
phi(a + b * log(x + sqrt(x**2 + 1)))
for all ``x, a, b > 0``, and `phi` is the normal pdf.
`johnsonsu` takes ``a`` and ``b`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b):
return (b > 0) & (a == a)
def _pdf(self, x, a, b):
x2 = x*x
trm = _norm_pdf(a + b * log(x + sqrt(x2+1)))
return b*1.0/sqrt(x2+1.0)*trm
def _cdf(self, x, a, b):
return _norm_cdf(a + b * log(x + sqrt(x*x + 1)))
def _ppf(self, q, a, b):
return sinh((_norm_ppf(q) - a) / b)
johnsonsu = johnsonsu_gen(name='johnsonsu')
class laplace_gen(rv_continuous):
"""A Laplace continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `laplace` is::
laplace.pdf(x) = 1/2 * exp(-abs(x))
%(after_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.laplace(0, 1, size=self._size)
def _pdf(self, x):
return 0.5*exp(-abs(x))
def _cdf(self, x):
return where(x > 0, 1.0-0.5*exp(-x), 0.5*exp(x))
def _ppf(self, q):
return where(q > 0.5, -log(2) - log1p(-q), log(2 * q))
def _stats(self):
return 0, 2, 0, 3
def _entropy(self):
return log(2)+1
laplace = laplace_gen(name='laplace')
class levy_gen(rv_continuous):
"""A Levy continuous random variable.
%(before_notes)s
See Also
--------
levy_stable, levy_l
Notes
-----
The probability density function for `levy` is::
levy.pdf(x) = 1 / (x * sqrt(2*pi*x)) * exp(-1/(2*x))
for ``x > 0``.
This is the same as the Levy-stable distribution with a=1/2 and b=1.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return 1 / sqrt(2*pi*x) / x * exp(-1/(2*x))
def _cdf(self, x):
# Equivalent to 2*norm.sf(sqrt(1/x))
return special.erfc(sqrt(0.5 / x))
def _ppf(self, q):
# Equivalent to 1.0/(norm.isf(q/2)**2) or 0.5/(erfcinv(q)**2)
val = -special.ndtri(q/2)
return 1.0 / (val * val)
def _stats(self):
return inf, inf, nan, nan
levy = levy_gen(a=0.0, name="levy")
class levy_l_gen(rv_continuous):
"""A left-skewed Levy continuous random variable.
%(before_notes)s
See Also
--------
levy, levy_stable
Notes
-----
The probability density function for `levy_l` is::
levy_l.pdf(x) = 1 / (abs(x) * sqrt(2*pi*abs(x))) * exp(-1/(2*abs(x)))
for ``x < 0``.
This is the same as the Levy-stable distribution with a=1/2 and b=-1.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
ax = abs(x)
return 1/sqrt(2*pi*ax)/ax*exp(-1/(2*ax))
def _cdf(self, x):
ax = abs(x)
return 2 * _norm_cdf(1 / sqrt(ax)) - 1
def _ppf(self, q):
val = _norm_ppf((q + 1.0) / 2)
return -1.0 / (val * val)
def _stats(self):
return inf, inf, nan, nan
levy_l = levy_l_gen(b=0.0, name="levy_l")
class levy_stable_gen(rv_continuous):
"""A Levy-stable continuous random variable.
%(before_notes)s
See Also
--------
levy, levy_l
Notes
-----
Levy-stable distribution (only random variates available -- ignore other
docs)
%(after_notes)s
%(example)s
"""
def _rvs(self, alpha, beta):
sz = self._size
TH = uniform.rvs(loc=-pi/2.0, scale=pi, size=sz)
W = expon.rvs(size=sz)
if alpha == 1:
return 2/pi*(pi/2+beta*TH)*tan(TH)-beta*log((pi/2*W*cos(TH))/(pi/2+beta*TH))
ialpha = 1.0/alpha
aTH = alpha*TH
if beta == 0:
return W/(cos(TH)/tan(aTH)+sin(TH))*((cos(aTH)+sin(aTH)*tan(TH))/W)**ialpha
val0 = beta*tan(pi*alpha/2)
th0 = arctan(val0)/alpha
val3 = W/(cos(TH)/tan(alpha*(th0+TH))+sin(TH))
res3 = val3*((cos(aTH)+sin(aTH)*tan(TH)-val0*(sin(aTH)-cos(aTH)*tan(TH)))/W)**ialpha
return res3
def _argcheck(self, alpha, beta):
if beta == -1:
self.b = 0.0
elif beta == 1:
self.a = 0.0
return (alpha > 0) & (alpha <= 2) & (beta <= 1) & (beta >= -1)
def _pdf(self, x, alpha, beta):
raise NotImplementedError
levy_stable = levy_stable_gen(name='levy_stable')
class logistic_gen(rv_continuous):
"""A logistic (or Sech-squared) continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `logistic` is::
logistic.pdf(x) = exp(-x) / (1+exp(-x))**2
`logistic` is a special case of `genlogistic` with ``c == 1``.
%(after_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.logistic(size=self._size)
def _pdf(self, x):
return exp(self._logpdf(x))
def _logpdf(self, x):
return -x - 2. * special.log1p(exp(-x))
def _cdf(self, x):
return special.expit(x)
def _ppf(self, q):
return -log1p(-q) + log(q)
def _stats(self):
return 0, pi*pi/3.0, 0, 6.0/5.0
def _entropy(self):
# http://en.wikipedia.org/wiki/Logistic_distribution
return 2.0
logistic = logistic_gen(name='logistic')
class loggamma_gen(rv_continuous):
"""A log gamma continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `loggamma` is::
loggamma.pdf(x, c) = exp(c*x-exp(x)) / gamma(c)
for all ``x, c > 0``.
`loggamma` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, c):
return log(self._random_state.gamma(c, size=self._size))
def _pdf(self, x, c):
return exp(c*x-exp(x)-gamln(c))
def _cdf(self, x, c):
return special.gammainc(c, exp(x))
def _ppf(self, q, c):
return log(special.gammaincinv(c, q))
def _stats(self, c):
# See, for example, "A Statistical Study of Log-Gamma Distribution", by
# Ping Shing Chan (thesis, McMaster University, 1993).
mean = special.digamma(c)
var = special.polygamma(1, c)
skewness = special.polygamma(2, c) / np.power(var, 1.5)
excess_kurtosis = special.polygamma(3, c) / (var*var)
return mean, var, skewness, excess_kurtosis
loggamma = loggamma_gen(name='loggamma')
class loglaplace_gen(rv_continuous):
"""A log-Laplace continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `loglaplace` is::
loglaplace.pdf(x, c) = c / 2 * x**(c-1), for 0 < x < 1
= c / 2 * x**(-c-1), for x >= 1
for ``c > 0``.
`loglaplace` takes ``c`` as a shape parameter.
%(after_notes)s
References
----------
T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model",
The Mathematical Scientist, vol. 28, pp. 49-60, 2003.
%(example)s
"""
def _pdf(self, x, c):
cd2 = c/2.0
c = where(x < 1, c, -c)
return cd2*x**(c-1)
def _cdf(self, x, c):
return where(x < 1, 0.5*x**c, 1-0.5*x**(-c))
def _ppf(self, q, c):
return where(q < 0.5, (2.0*q)**(1.0/c), (2*(1.0-q))**(-1.0/c))
def _munp(self, n, c):
return c**2 / (c**2 - n**2)
def _entropy(self, c):
return log(2.0/c) + 1.0
loglaplace = loglaplace_gen(a=0.0, name='loglaplace')
def _lognorm_logpdf(x, s):
return _lazywhere(x != 0, (x, s),
lambda x, s: -log(x)**2 / (2*s**2) - log(s*x*sqrt(2*pi)),
-np.inf)
class lognorm_gen(rv_continuous):
"""A lognormal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `lognorm` is::
lognorm.pdf(x, s) = 1 / (s*x*sqrt(2*pi)) * exp(-1/2*(log(x)/s)**2)
for ``x > 0``, ``s > 0``.
`lognorm` takes ``s`` as a shape parameter.
%(after_notes)s
A common parametrization for a lognormal random variable ``Y`` is in
terms of the mean, ``mu``, and standard deviation, ``sigma``, of the
unique normally distributed random variable ``X`` such that exp(X) = Y.
This parametrization corresponds to setting ``s = sigma`` and ``scale =
exp(mu)``.
%(example)s
"""
def _rvs(self, s):
return exp(s * self._random_state.standard_normal(self._size))
def _pdf(self, x, s):
return exp(self._logpdf(x, s))
def _logpdf(self, x, s):
return _lognorm_logpdf(x, s)
def _cdf(self, x, s):
return _norm_cdf(log(x) / s)
def _logcdf(self, x, s):
return _norm_logcdf(log(x) / s)
def _ppf(self, q, s):
return exp(s * _norm_ppf(q))
def _sf(self, x, s):
return _norm_sf(log(x) / s)
def _logsf(self, x, s):
return _norm_logsf(log(x) / s)
def _stats(self, s):
p = exp(s*s)
mu = sqrt(p)
mu2 = p*(p-1)
g1 = 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 + log(2*pi) + 2 * log(s))
def _fitstart(self, data):
scale = data.std()
loc = data.min() - 0.001
logd = log(data - loc)
m = logd.mean()
s = sqrt((logd ** 2).mean() - m ** 2)
return s, loc, scale
lognorm = lognorm_gen(a=0.0, name='lognorm')
class gilbrat_gen(rv_continuous):
"""A Gilbrat continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gilbrat` is::
gilbrat.pdf(x) = 1/(x*sqrt(2*pi)) * exp(-1/2*(log(x))**2)
`gilbrat` is a special case of `lognorm` with ``s = 1``.
%(after_notes)s
%(example)s
"""
def _rvs(self):
return exp(self._random_state.standard_normal(self._size))
def _pdf(self, x):
return exp(self._logpdf(x))
def _logpdf(self, x):
return _lognorm_logpdf(x, 1.0)
def _cdf(self, x):
return _norm_cdf(log(x))
def _ppf(self, q):
return exp(_norm_ppf(q))
def _stats(self):
p = np.e
mu = sqrt(p)
mu2 = p * (p - 1)
g1 = 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 * log(2 * pi) + 0.5
def _fitstart(self, data):
scale = data.std()
loc = data.min() - 0.001
return loc, scale
gilbrat = gilbrat_gen(a=0.0, name='gilbrat')
class maxwell_gen(rv_continuous):
"""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::
maxwell.pdf(x) = sqrt(2/pi)x**2 * exp(-x**2/2)
for ``x > 0``.
%(after_notes)s
References
----------
.. [1] http://mathworld.wolfram.com/MaxwellDistribution.html
%(example)s
"""
def _rvs(self):
return chi.rvs(3.0, size=self._size, random_state=self._random_state)
def _pdf(self, x):
return sqrt(2.0/pi)*x*x*exp(-x*x/2.0)
def _cdf(self, x):
return special.gammainc(1.5, x*x/2.0)
def _ppf(self, q):
return sqrt(2*special.gammaincinv(1.5, q))
def _stats(self):
val = 3*pi-8
return (2*sqrt(2.0/pi), 3-8/pi, sqrt(2)*(32-10*pi)/val**1.5,
(-12*pi*pi + 160*pi - 384) / val**2.0)
def _entropy(self):
return _EULER + 0.5*log(2*pi)-0.5
maxwell = maxwell_gen(a=0.0, name='maxwell')
class mielke_gen(rv_continuous):
"""A Mielke's Beta-Kappa continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `mielke` is::
mielke.pdf(x, k, s) = k * x**(k-1) / (1+x**s)**(1+k/s)
for ``x > 0``.
`mielke` takes ``k`` and ``s`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, k, s):
return k*x**(k-1.0) / (1.0+x**s)**(1.0+k*1.0/s)
def _cdf(self, x, k, s):
return x**k / (1.0+x**s)**(k*1.0/s)
def _ppf(self, q, k, s):
qsk = pow(q, s*1.0/k)
return pow(qsk/(1.0-qsk), 1.0/s)
mielke = mielke_gen(a=0.0, name='mielke')
class nakagami_gen(rv_continuous):
"""A Nakagami continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `nakagami` is::
nakagami.pdf(x, nu) = 2 * nu**nu / gamma(nu) *
x**(2*nu-1) * exp(-nu*x**2)
for ``x > 0``, ``nu > 0``.
`nakagami` takes ``nu`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, nu):
return 2*nu**nu/gam(nu)*(x**(2*nu-1.0))*exp(-nu*x*x)
def _cdf(self, x, nu):
return special.gammainc(nu, nu*x*x)
def _ppf(self, q, nu):
return sqrt(1.0/nu*special.gammaincinv(nu, q))
def _stats(self, nu):
mu = gam(nu+0.5)/gam(nu)/sqrt(nu)
mu2 = 1.0-mu*mu
g1 = mu * (1 - 4*nu*mu2) / 2.0 / nu / np.power(mu2, 1.5)
g2 = -6*mu**4*nu + (8*nu-2)*mu**2-2*nu + 1
g2 /= nu*mu2**2.0
return mu, mu2, g1, g2
nakagami = nakagami_gen(a=0.0, name="nakagami")
class ncx2_gen(rv_continuous):
"""A non-central chi-squared continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `ncx2` is::
ncx2.pdf(x, df, nc) = exp(-(nc+x)/2) * 1/2 * (x/nc)**((df-2)/4)
* I[(df-2)/2](sqrt(nc*x))
for ``x > 0``.
`ncx2` takes ``df`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, df, nc):
return self._random_state.noncentral_chisquare(df, nc, self._size)
def _logpdf(self, x, df, nc):
return _ncx2_log_pdf(x, df, nc)
def _pdf(self, x, df, nc):
return _ncx2_pdf(x, df, nc)
def _cdf(self, x, df, nc):
return _ncx2_cdf(x, df, nc)
def _ppf(self, q, df, nc):
return special.chndtrix(q, df, nc)
def _stats(self, df, nc):
val = df + 2.0*nc
return (df + nc, 2*val, sqrt(8)*(val+nc)/val**1.5,
12.0*(val+2*nc)/val**2.0)
def _fitstart(self, data):
m = data.mean()
v = data.var()
# Supply a starting guess with method of moments:
nc = (v / 2 - m) / 2
df = m - nc
return super(ncx2_gen, self)._fitstart(data, args=(df, nc))
ncx2 = ncx2_gen(a=0.0, name='ncx2')
class ncf_gen(rv_continuous):
"""A non-central F distribution continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `ncf` is::
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))
for ``df1, df2, nc > 0``.
`ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _rvs(self, dfn, dfd, nc):
return self._random_state.noncentral_f(dfn, dfd, nc, self._size)
def _pdf_skip(self, x, dfn, dfd, nc):
n1, n2 = dfn, dfd
term = -nc/2+nc*n1*x/(2*(n2+n1*x)) + gamln(n1/2.)+gamln(1+n2/2.)
term -= gamln((n1+n2)/2.0)
Px = exp(term)
Px *= n1**(n1/2) * n2**(n2/2) * x**(n1/2-1)
Px *= (n2+n1*x)**(-(n1+n2)/2)
Px *= special.assoc_laguerre(-nc*n1*x/(2.0*(n2+n1*x)), n2/2, n1/2-1)
Px /= special.beta(n1/2, n2/2)
# This function does not have a return. Drop it for now, the generic
# function seems to work OK.
def _cdf(self, x, dfn, dfd, nc):
return special.ncfdtr(dfn, dfd, nc, x)
def _ppf(self, q, dfn, dfd, nc):
return special.ncfdtri(dfn, dfd, nc, q)
def _munp(self, n, dfn, dfd, nc):
val = (dfn * 1.0/dfd)**n
term = gamln(n+0.5*dfn) + gamln(0.5*dfd-n) - gamln(dfd*0.5)
val *= exp(-nc / 2.0+term)
val *= special.hyp1f1(n+0.5*dfn, 0.5*dfn, 0.5*nc)
return val
def _stats(self, dfn, dfd, nc):
mu = where(dfd <= 2, inf, dfd / (dfd-2.0)*(1+nc*1.0/dfn))
mu2 = where(dfd <= 4, inf, 2*(dfd*1.0/dfn)**2.0 *
((dfn+nc/2.0)**2.0 + (dfn+nc)*(dfd-2.0)) /
((dfd-2.0)**2.0 * (dfd-4.0)))
return mu, mu2, None, None
ncf = ncf_gen(a=0.0, name='ncf')
class t_gen(rv_continuous):
"""A Student's T continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `t` is::
gamma((df+1)/2)
t.pdf(x, df) = ---------------------------------------------------
sqrt(pi*df) * gamma(df/2) * (1+x**2/df)**((df+1)/2)
for ``df > 0``.
`t` takes ``df`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, df):
return self._random_state.standard_t(df, size=self._size)
def _pdf(self, x, df):
r = asarray(df*1.0)
Px = exp(gamln((r+1)/2)-gamln(r/2))
Px /= sqrt(r*pi)*(1+(x**2)/r)**((r+1)/2)
return Px
def _logpdf(self, x, df):
r = df*1.0
lPx = gamln((r+1)/2)-gamln(r/2)
lPx -= 0.5*log(r*pi) + (r+1)/2*log(1+(x**2)/r)
return lPx
def _cdf(self, x, df):
return special.stdtr(df, x)
def _sf(self, x, df):
return special.stdtr(df, -x)
def _ppf(self, q, df):
return special.stdtrit(df, q)
def _isf(self, q, df):
return -special.stdtrit(df, q)
def _stats(self, df):
mu2 = _lazywhere(df > 2, (df,),
lambda df: df / (df-2.0),
np.inf)
g1 = where(df > 3, 0.0, np.nan)
g2 = _lazywhere(df > 4, (df,),
lambda df: 6.0 / (df-4.0),
np.nan)
return 0, mu2, g1, g2
t = t_gen(name='t')
class nct_gen(rv_continuous):
"""A non-central Student's T continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `nct` is::
df**(df/2) * gamma(df+1)
nct.pdf(x, df, nc) = ----------------------------------------------------
2**df*exp(nc**2/2) * (df+x**2)**(df/2) * gamma(df/2)
for ``df > 0``.
`nct` takes ``df`` and ``nc`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, df, nc):
return (df > 0) & (nc == nc)
def _rvs(self, df, nc):
sz, rndm = self._size, self._random_state
n = norm.rvs(loc=nc, size=sz, random_state=rndm)
c2 = chi2.rvs(df, size=sz, random_state=rndm)
return n * sqrt(df) / sqrt(c2)
def _pdf(self, x, df, nc):
n = df*1.0
nc = nc*1.0
x2 = x*x
ncx2 = nc*nc*x2
fac1 = n + x2
trm1 = n/2.*log(n) + gamln(n+1)
trm1 -= n*log(2)+nc*nc/2.+(n/2.)*log(fac1)+gamln(n/2.)
Px = exp(trm1)
valF = ncx2 / (2*fac1)
trm1 = sqrt(2)*nc*x*special.hyp1f1(n/2+1, 1.5, valF)
trm1 /= asarray(fac1*special.gamma((n+1)/2))
trm2 = special.hyp1f1((n+1)/2, 0.5, valF)
trm2 /= asarray(sqrt(fac1)*special.gamma(n/2+1))
Px *= trm1+trm2
return Px
def _cdf(self, x, df, nc):
return special.nctdtr(df, nc, x)
def _ppf(self, q, df, nc):
return special.nctdtrit(df, nc, q)
def _fitstart(self, data):
me = np.mean(data)
# g2 = mode(data)[0]
sa = np.std(data)
def func(df):
return ((df - 2) * (4 * df - 1) -
(4 * df - 1) * df / (sa ** 2 + me ** 2) +
me ** 2 / (sa ** 2 + me ** 2) * (df * (4 * df - 1) -
3 * df))
df0 = np.maximum(2 * sa / (sa - 1), 1)
df = optimize.fsolve(func, df0)
mu = me * (1 - 3 / (4 * df - 1))
return super(nct_gen, self)._fitstart(data, args=(df, mu))
def _stats(self, df, nc, moments='mv'):
#
# See D. Hogben, R.S. Pinkham, and M.B. Wilk,
# 'The moments of the non-central t-distribution'
# Biometrika 48, p. 465 (2961).
# e.g. http://www.jstor.org/stable/2332772 (gated)
#
mu, mu2, g1, g2 = None, None, None, None
gfac = gam(df/2.-0.5) / gam(df/2.)
c11 = sqrt(df/2.) * gfac
c20 = df / (df-2.)
c22 = c20 - c11*c11
mu = np.where(df > 1, nc*c11, np.inf)
mu2 = np.where(df > 2, c22*nc*nc + c20, np.inf)
if 's' in moments:
c33t = df * (7.-2.*df) / (df-2.) / (df-3.) + 2.*c11*c11
c31t = 3.*df / (df-2.) / (df-3.)
mu3 = (c33t*nc*nc + c31t) * c11*nc
g1 = np.where(df > 3, mu3 / np.power(mu2, 1.5), np.nan)
#kurtosis
if 'k' in moments:
c44 = df*df / (df-2.) / (df-4.)
c44 -= c11*c11 * 2.*df*(5.-df) / (df-2.) / (df-3.)
c44 -= 3.*c11**4
c42 = df / (df-4.) - c11*c11 * (df-1.) / (df-3.)
c42 *= 6.*df / (df-2.)
c40 = 3.*df*df / (df-2.) / (df-4.)
mu4 = c44 * nc**4 + c42*nc**2 + c40
g2 = np.where(df > 4, mu4/mu2**2 - 3., np.nan)
return mu, mu2, g1, g2
nct = nct_gen(name="nct")
class pareto_gen(rv_continuous):
"""A Pareto continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `pareto` is::
pareto.pdf(x, b) = b / x**(b+1)
for ``x >= 1``, ``b > 0``.
`pareto` takes ``b`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, b):
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 _stats(self, b, moments='mv'):
mu, mu2, g1, g2 = None, None, None, None
if 'm' in moments:
mask = b > 1
bt = extract(mask, b)
mu = valarray(shape(b), value=inf)
place(mu, mask, bt / (bt-1.0))
if 'v' in moments:
mask = b > 2
bt = extract(mask, b)
mu2 = valarray(shape(b), value=inf)
place(mu2, mask, bt / (bt-2.0) / (bt-1.0)**2)
if 's' in moments:
mask = b > 3
bt = extract(mask, b)
g1 = valarray(shape(b), value=nan)
vals = 2 * (bt + 1.0) * sqrt(bt - 2.0) / ((bt - 3.0) * sqrt(bt))
place(g1, mask, vals)
if 'k' in moments:
mask = b > 4
bt = extract(mask, b)
g2 = valarray(shape(b), value=nan)
vals = (6.0*polyval([1.0, 1.0, -6, -2], bt) /
polyval([1.0, -7.0, 12.0, 0.0], bt))
place(g2, mask, vals)
return mu, mu2, g1, g2
def _entropy(self, c):
return 1 + 1.0/c - log(c)
pareto = pareto_gen(a=1.0, name="pareto")
class lomax_gen(rv_continuous):
"""A Lomax (Pareto of the second kind) continuous random variable.
%(before_notes)s
Notes
-----
The Lomax distribution is a special case of the Pareto distribution, with
(loc=-1.0).
The probability density function for `lomax` is::
lomax.pdf(x, c) = c / (1+x)**(c+1)
for ``x >= 0``, ``c > 0``.
`lomax` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
return c*1.0/(1.0+x)**(c+1.0)
def _logpdf(self, x, c):
return log(c) - (c+1)*special.log1p(x)
def _cdf(self, x, c):
return -special.expm1(-c*special.log1p(x))
def _sf(self, x, c):
return exp(-c*special.log1p(x))
def _logsf(self, x, c):
return -c*special.log1p(x)
def _ppf(self, q, c):
return special.expm1(-special.log1p(-q)/c)
def _stats(self, c):
mu, mu2, g1, g2 = pareto.stats(c, loc=-1.0, moments='mvsk')
return mu, mu2, g1, g2
def _entropy(self, c):
return 1+1.0/c-log(c)
lomax = lomax_gen(a=0.0, name="lomax")
class pearson3_gen(rv_continuous):
"""A pearson type III continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `pearson3` is::
pearson3.pdf(x, skew) = abs(beta) / gamma(alpha) *
(beta * (x - zeta))**(alpha - 1) * exp(-beta*(x - zeta))
where::
beta = 2 / (skew * stddev)
alpha = (stddev * beta)**2
zeta = loc - alpha / beta
`pearson3` takes ``skew`` as a shape parameter.
%(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 = 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, skew, mask, invmask, beta, alpha, zeta
def _argcheck(self, skew):
# The _argcheck function in rv_continuous only allows positive
# arguments. The skew argument for pearson3 can be zero (which I want
# to handle inside pearson3._pdf) or negative. So just return True
# for all skew args.
return np.ones(np.shape(skew), dtype=bool)
def _stats(self, skew):
ans, x, transx, skew, mask, invmask, beta, alpha, zeta = (
self._preprocess([1], skew))
m = zeta + alpha / beta
v = alpha / (beta**2)
s = 2.0 / (alpha**0.5) * np.sign(beta)
k = 6.0 / alpha
return m, v, s, k
def _pdf(self, x, skew):
# Do the calculation in _logpdf since helps to limit
# overflow/underflow problems
ans = 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)*log(beta*(x - zeta)) + (a - 1)*log(x)
# - beta*(x - zeta) - x
# - gamln(alpha) - gamln(a)
ans, x, transx, skew, mask, invmask, beta, alpha, zeta = (
self._preprocess(x, skew))
ans[mask] = np.log(_norm_pdf(x[mask]))
ans[invmask] = log(abs(beta)) + gamma._logpdf(transx, alpha)
return ans
def _cdf(self, x, skew):
ans, x, transx, skew, mask, invmask, beta, alpha, zeta = (
self._preprocess(x, skew))
ans[mask] = _norm_cdf(x[mask])
ans[invmask] = gamma._cdf(transx, alpha)
return ans
def _rvs(self, skew):
ans, x, transx, skew, mask, invmask, beta, alpha, zeta = (
self._preprocess([0], skew))
if mask[0]:
return self._random_state.standard_normal(self._size)
ans = self._random_state.standard_gamma(alpha, self._size)/beta + zeta
if ans.size == 1:
return ans[0]
return ans
def _ppf(self, q, skew):
ans, q, transq, skew, mask, invmask, beta, alpha, zeta = (
self._preprocess(q, skew))
ans[mask] = _norm_ppf(q[mask])
ans[invmask] = special.gammaincinv(alpha, q[invmask])/beta + zeta
return ans
pearson3 = pearson3_gen(name="pearson3")
class powerlaw_gen(rv_continuous):
"""A power-function continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `powerlaw` is::
powerlaw.pdf(x, a) = a * x**(a-1)
for ``0 <= x <= 1``, ``a > 0``.
`powerlaw` takes ``a`` as a shape parameter.
%(after_notes)s
`powerlaw` is a special case of `beta` with ``b == 1``.
%(example)s
"""
def _pdf(self, x, a):
return a*x**(a-1.0)
def _logpdf(self, x, a):
return log(a) + special.xlogy(a - 1, x)
def _cdf(self, x, a):
return x**(a*1.0)
def _logcdf(self, x, a):
return a*log(x)
def _ppf(self, q, a):
return pow(q, 1.0/a)
def _stats(self, a):
return (a / (a + 1.0),
a / (a + 2.0) / (a + 1.0) ** 2,
-2.0 * ((a - 1.0) / (a + 3.0)) * sqrt((a + 2.0) / a),
6 * polyval([1, -1, -6, 2], a) / (a * (a + 3.0) * (a + 4)))
def _entropy(self, a):
return 1 - 1.0/a - log(a)
powerlaw = powerlaw_gen(a=0.0, b=1.0, name="powerlaw")
class powerlognorm_gen(rv_continuous):
"""A power log-normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `powerlognorm` is::
powerlognorm.pdf(x, c, s) = c / (x*s) * phi(log(x)/s) *
(Phi(-log(x)/s))**(c-1),
where ``phi`` is the normal pdf, and ``Phi`` is the normal cdf,
and ``x > 0``, ``s, c > 0``.
`powerlognorm` takes ``c`` and ``s`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c, s):
return (c/(x*s) * _norm_pdf(log(x)/s) *
pow(_norm_cdf(-log(x)/s), c*1.0-1.0))
def _cdf(self, x, c, s):
return 1.0 - pow(_norm_cdf(-log(x)/s), c*1.0)
def _ppf(self, q, c, s):
return exp(-s * _norm_ppf(pow(1.0 - q, 1.0 / c)))
powerlognorm = powerlognorm_gen(a=0.0, name="powerlognorm")
class powernorm_gen(rv_continuous):
"""A power normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `powernorm` is::
powernorm.pdf(x, c) = c * phi(x) * (Phi(-x))**(c-1)
where ``phi`` is the normal pdf, and ``Phi`` is the normal cdf,
and ``x > 0``, ``c > 0``.
`powernorm` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
return (c*_norm_pdf(x) * (_norm_cdf(-x)**(c-1.0)))
def _logpdf(self, x, c):
return log(c) + _norm_logpdf(x) + (c-1)*_norm_logcdf(-x)
def _cdf(self, x, c):
return 1.0-_norm_cdf(-x)**(c*1.0)
def _ppf(self, q, c):
return -_norm_ppf(pow(1.0 - q, 1.0 / c))
powernorm = powernorm_gen(name='powernorm')
class rdist_gen(rv_continuous):
"""An R-distributed continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rdist` is::
rdist.pdf(x, c) = (1-x**2)**(c/2-1) / B(1/2, c/2)
for ``-1 <= x <= 1``, ``c > 0``.
`rdist` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _pdf(self, x, c):
return np.power((1.0 - x**2), c / 2.0 - 1) / special.beta(0.5, c / 2.0)
def _cdf(self, x, c):
term1 = x / special.beta(0.5, c / 2.0)
res = 0.5 + term1 * special.hyp2f1(0.5, 1 - c / 2.0, 1.5, x**2)
# There's an issue with hyp2f1, it returns nans near x = +-1, c > 100.
# Use the generic implementation in that case. See gh-1285 for
# background.
if np.any(np.isnan(res)):
return rv_continuous._cdf(self, x, c)
return res
def _munp(self, n, c):
numerator = (1 - (n % 2)) * special.beta((n + 1.0) / 2, c / 2.0)
return numerator / special.beta(1. / 2, c / 2.)
rdist = rdist_gen(a=-1.0, b=1.0, name="rdist")
class rayleigh_gen(rv_continuous):
"""A Rayleigh continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rayleigh` is::
rayleigh.pdf(r) = r * exp(-r**2/2)
for ``x >= 0``.
`rayleigh` is a special case of `chi` with ``df == 2``.
%(after_notes)s
%(example)s
"""
def _link(self, x, logSF, phat, ix):
if ix == 1:
return x - phat[0] / sqrt(-2.0 * logSF)
elif ix == 0:
return x - phat[1] * sqrt(-2.0 * logSF)
else:
raise IndexError('Index to the fixed parameter is out of bounds')
def _rvs(self):
return chi.rvs(2, size=self._size, random_state=self._random_state)
def _pdf(self, r):
return exp(self._logpdf(r))
def _logpdf(self, r):
rr2 = r * r / 2.0
return where(rr2 == inf, - rr2, log(r) - rr2)
def _cdf(self, r):
return -special.expm1(-0.5 * r**2)
def _ppf(self, q):
return sqrt(-2 * special.log1p(-q))
def _sf(self, r):
return exp(-0.5 * r**2)
def _isf(self, q):
return sqrt(-2 * log(q))
def _stats(self):
val = 4 - pi
return (np.sqrt(pi/2), val/2, 2*(pi-3)*sqrt(pi)/val**1.5,
6*pi/val-16/val**2)
def _entropy(self):
return _EULER/2.0 + 1 - 0.5*log(2)
rayleigh = rayleigh_gen(a=0.0, name="rayleigh")
class truncrayleigh_gen(rv_continuous):
"""A truncated Rayleigh continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `truncrayleigh` is::
truncrayleigh.cdf(r) = 1 - exp(-((r+c)**2-c**2)/2)
for ``x >= 0, c>=0``.
%(example)s
"""
def _argcheck(self, c):
return (c >= 0)
def _link(self, x, logSF, phat, ix):
c, loc, scale = phat
if ix == 2:
return x - loc / (sqrt(c * c - 2 * logSF) - c)
elif ix == 1:
return x - scale * (sqrt(c * c - 2 * logSF) - c)
elif ix == 0:
xn = (x - loc) / scale
return - 2 * logSF / xn - xn / 2.0
else:
raise IndexError('Index to the fixed parameter is out of bounds')
def _fitstart(self, data, args=None):
if args is None:
args = (0.0,) * self.numargs
return args + self.fit_loc_scale(data, *args)
def _pdf(self, r, c):
rc = r + c
return rc * exp(-(rc * rc - c * c) / 2.0)
def _logpdf(self, r, c):
rc = r + c
return log(rc) - (rc * rc - c * c) / 2.0
def _cdf(self, r, c):
rc = r + c
return - expm1(-(rc * rc - c * c) / 2.0)
def _logsf(self, r, c):
rc = r + c
return -(rc * rc - c * c) / 2.0
def _sf(self, r, c):
return exp(self._logsf(r, c))
def _ppf(self, q, c):
return sqrt(c * c - 2 * log1p(-q)) - c
def _stats(self, c):
# TODO: correct this it is wrong!
val = 4 - pi
return (np.sqrt(pi / 2),
val / 2,
2 * (pi - 3) * sqrt(pi) / val ** 1.5,
6 * pi / val - 16 / val ** 2)
def _entropy(self, c):
# TODO: correct this it is wrong!
return _EULER / 2.0 + 1 - 0.5 * log(2)
truncrayleigh = truncrayleigh_gen(a=0.0, name="truncrayleigh", shapes='c')
# Reciprocal Distribution
class reciprocal_gen(rv_continuous):
"""A reciprocal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `reciprocal` is::
reciprocal.pdf(x, a, b) = 1 / (x*log(b/a))
for ``a <= x <= b``, ``a, b > 0``.
`reciprocal` takes ``a`` and ``b`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b):
self.a = a
self.b = b
self.d = log(b*1.0 / a)
return (a > 0) & (b > 0) & (b > a)
def _pdf(self, x, a, b):
return 1.0 / (x * self.d)
def _logpdf(self, x, a, b):
return -log(x) - log(self.d)
def _cdf(self, x, a, b):
return (log(x)-log(a)) / self.d
def _ppf(self, q, a, b):
return a*pow(b*1.0/a, q)
def _munp(self, n, a, b):
return 1.0/self.d / n * (pow(b*1.0, n) - pow(a*1.0, n))
def _fitstart(self, data):
a = np.min(data)
a -= 0.01 * np.abs(a)
b = np.max(data)
b += 0.01 * np.abs(b)
if a <= 0:
da = np.abs(a) + 0.001
a += da
b += da
return super(reciprocal_gen, self)._fitstart(data, args=(a, b))
def _entropy(self, a, b):
return 0.5*log(a*b)+log(log(b/a))
reciprocal = reciprocal_gen(name="reciprocal")
class rice_gen(rv_continuous):
"""A Rice continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `rice` is::
rice.pdf(x, b) = x * exp(-(x**2+b**2)/2) * I[0](x*b)
for ``x > 0``, ``b > 0``.
`rice` takes ``b`` as a shape parameter.
%(after_notes)s
The Rice distribution describes the length, ``r``, of a 2-D vector
with components ``(U+u, V+v)``, where ``U, V`` are constant, ``u, v``
are independent Gaussian random variables with standard deviation
``s``. Let ``R = (U**2 + V**2)**0.5``. Then the pdf of ``r`` is
``rice.pdf(x, R/s, scale=s)``.
%(example)s
"""
def _argcheck(self, b):
return b >= 0
def _rvs(self, b):
# http://en.wikipedia.org/wiki/Rice_distribution
sz = self._size if self._size else 1
t = b/np.sqrt(2) + self._random_state.standard_normal(size=(2, sz))
return np.sqrt((t*t).sum(axis=0))
def _cdf(self, x, b):
return chndtr(np.square(x), 2, np.square(b))
def _ppf(self, q, b):
return np.sqrt(chndtrix(q, 2, np.square(b)))
def _pdf(self, x, b):
# We use (x**2 + b**2)/2 = ((x-b)**2)/2 + xb.
# The factor of 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 * exp(-(x-b)*(x-b)/2.0) * special.i0e(x*b)
def _munp(self, n, b):
nd2 = n/2.0
n1 = 1 + nd2
b2 = b*b/2.0
return (2.0**(nd2) * exp(-b2) * special.gamma(n1) *
special.hyp1f1(n1, 1, b2))
rice = rice_gen(a=0.0, name="rice")
# FIXME: PPF does not work.
class recipinvgauss_gen(rv_continuous):
"""A reciprocal inverse Gaussian continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `recipinvgauss` is::
recipinvgauss.pdf(x, mu) = 1/sqrt(2*pi*x) * exp(-(1-mu*x)**2/(2*x*mu**2))
for ``x >= 0``.
`recipinvgauss` takes ``mu`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _rvs(self, mu):
return 1.0/self._random_state.wald(mu, 1.0, size=self._size)
def _pdf(self, x, mu):
return 1.0/sqrt(2*pi*x)*exp(-(1-mu*x)**2.0 / (2*x*mu**2.0))
def _logpdf(self, x, mu):
return -(1-mu*x)**2.0 / (2*x*mu**2.0) - 0.5*log(2*pi*x)
def _cdf(self, x, mu):
trm1 = 1.0/mu - x
trm2 = 1.0/mu + x
isqx = 1.0/sqrt(x)
return 1.0-_norm_cdf(isqx*trm1)-exp(2.0/mu)*_norm_cdf(-isqx*trm2)
recipinvgauss = recipinvgauss_gen(a=0.0, name='recipinvgauss')
class semicircular_gen(rv_continuous):
"""A semicircular continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `semicircular` is::
semicircular.pdf(x) = 2/pi * sqrt(1-x**2)
for ``-1 <= x <= 1``.
%(after_notes)s
%(example)s
"""
def _pdf(self, x):
return 2.0/pi*sqrt(1-x*x)
def _cdf(self, x):
return 0.5+1.0/pi*(x*sqrt(1-x*x) + arcsin(x))
def _stats(self):
return 0, 0.25, 0, -1.0
def _entropy(self):
return 0.64472988584940017414
semicircular = semicircular_gen(a=-1.0, b=1.0, name="semicircular")
class skew_norm_gen(rv_continuous):
"""A skew-normal random variable.
%(before_notes)s
Notes
-----
The pdf is::
skewnorm.pdf(x, a) = 2*norm.pdf(x)*norm.cdf(ax)
`skewnorm` takes ``a`` as a skewness parameter
When a=0 the distribution is identical to a normal distribution.
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.
http://azzalini.stat.unipd.it/SN/faq-r.html
"""
def _argcheck(self, a):
return np.isfinite(a)
def _pdf(self, x, a):
return 2.*_norm_pdf(x)*_norm_cdf(a*x)
def _rvs(self, a):
u0 = self._random_state.normal(size=self._size)
v = self._random_state.normal(size=self._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/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 - pi)/2) * (const/np.sqrt(1 - const**2))**3
if 'k' in moments:
output[3] = (2*(pi - 3)) * (const**4/(1 - const**2)**2)
return output
skewnorm = skew_norm_gen(name='skewnorm')
class trapz_gen(rv_continuous):
"""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)``.
`trapz` takes ``c`` and ``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
"""
def _argcheck(self, c, d):
return (c >= 0) & (c <= 1) & (d >= 0) & (d <= 1) & (d >= c)
def _pdf(self, x, c, d):
u = 2 / (d - c + 1)
condlist = [x < c, x <= d, x > d]
choicelist = [u * x / c, u, u * (1 - x) / (1 - d)]
return np.select(condlist, choicelist)
def _cdf(self, x, c, d):
condlist = [x < c, x <= d, x > d]
choicelist = [x**2 / c / (d - c + 1),
(c + 2 * (x - c)) / (d - c + 1),
1 - ((1 - x)**2 / (d - c + 1) / (1 - d))]
return np.select(condlist, choicelist)
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 - sqrt((1 - q) * (d - c + 1) * (1 - d))]
return np.select(condlist, choicelist)
trapz = trapz_gen(a=0.0, b=1.0, name="trapz")
class triang_gen(rv_continuous):
"""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.
%(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):
return self._random_state.triangular(0, c, 1, self._size)
def _argcheck(self, c):
return (c >= 0) & (c <= 1)
def _pdf(self, x, c):
return where(x < c, 2*x/c, 2*(1-x)/(1-c))
def _cdf(self, x, c):
return where(x < c, x*x/c, (x*x-2*x+c)/(c-1))
def _ppf(self, q, c):
return where(q < c, sqrt(c*q), 1-sqrt((1-c)*(1-q)))
def _stats(self, c):
return (c+1.0)/3.0, (1.0-c+c*c)/18, 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-log(2)
triang = triang_gen(a=0.0, b=1.0, name="triang")
class truncexpon_gen(rv_continuous):
"""A truncated exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `truncexpon` is::
truncexpon.pdf(x, b) = exp(-x) / (1-exp(-b))
for ``0 < x < b``.
`truncexpon` takes ``b`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, b):
self.b = b
return (b > 0)
def _pdf(self, x, b):
return exp(-x)/(-special.expm1(-b))
def _logpdf(self, x, b):
return -x - log(-special.expm1(-b))
def _cdf(self, x, b):
return special.expm1(-x)/special.expm1(-b)
def _ppf(self, q, b):
return -special.log1p(q*special.expm1(-b))
def _munp(self, n, b):
# wrong answer with formula, same as in continuous.pdf
# return gam(n+1)-special.gammainc(1+n, b)
if n == 1:
return (1-(b+1)*exp(-b))/(-special.expm1(-b))
elif n == 2:
return 2*(1-0.5*(b*b+2*b+2)*exp(-b))/(-special.expm1(-b))
else:
# return generic for higher moments
# return rv_continuous._mom1_sc(self, n, b)
return self._mom1_sc(n, b)
def _entropy(self, b):
eB = exp(b)
return log(eB-1)+(1+eB*(b-1.0))/(1.0-eB)
truncexpon = truncexpon_gen(a=0.0, name='truncexpon')
class truncnorm_gen(rv_continuous):
"""A truncated normal continuous random variable.
%(before_notes)s
Notes
-----
The standard form of this distribution is a standard normal truncated to
the range [a, b] --- notice that a and b are defined over the domain of the
standard normal. To convert clip values for a specific mean and standard
deviation, use::
a, b = (myclip_a - my_mean) / my_std, (myclip_b - my_mean) / my_std
`truncnorm` takes ``a`` and ``b`` as shape parameters.
%(after_notes)s
%(example)s
"""
def _argcheck(self, a, b):
self.a = a
self.b = b
self._nb = _norm_cdf(b)
self._na = _norm_cdf(a)
self._sb = _norm_sf(b)
self._sa = _norm_sf(a)
if self.a > 0:
self._delta = -(self._sb - self._sa)
else:
self._delta = self._nb - self._na
self._logdelta = log(self._delta)
return (a != b)
def _pdf(self, x, a, b):
return _norm_pdf(x) / self._delta
def _logpdf(self, x, a, b):
return _norm_logpdf(x) - self._logdelta
def _cdf(self, x, a, b):
return (_norm_cdf(x) - self._na) / self._delta
def _ppf(self, q, a, b):
if self.a > 0:
return _norm_isf(q*self._sb + self._sa*(1.0-q))
else:
return _norm_ppf(q*self._nb + self._na*(1.0-q))
def _stats(self, a, b):
nA, nB = self._na, self._nb
d = nB - nA
pA, pB = _norm_pdf(a), _norm_pdf(b)
mu = (pA - pB) / d # correction sign
mu2 = 1 + (a*pA - b*pB) / d - mu*mu
return mu, mu2, None, None
truncnorm = truncnorm_gen(name='truncnorm')
# FIXME: RVS does not work.
class tukeylambda_gen(rv_continuous):
"""A Tukey-Lamdba continuous random variable.
%(before_notes)s
Notes
-----
A flexible distribution, able to represent and interpolate between the
following distributions:
- Cauchy (lam=-1)
- logistic (lam=0.0)
- approx Normal (lam=0.14)
- u-shape (lam = 0.5)
- uniform from -1 to 1 (lam = 1)
`tukeylambda` takes ``lam`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, lam):
return np.ones(np.shape(lam), dtype=bool)
def _pdf(self, x, lam):
Fx = asarray(special.tklmbda(x, lam))
Px = Fx**(lam-1.0) + (asarray(1-Fx))**(lam-1.0)
Px = 1.0/asarray(Px)
return where((lam <= 0) | (abs(x) < 1.0/asarray(lam)), Px, 0.0)
def _cdf(self, x, lam):
return special.tklmbda(x, lam)
def _ppf(self, q, lam):
return special.boxcox(q, lam) - special.boxcox1p(-q, lam)
def _stats(self, lam):
return 0, _tlvar(lam), 0, _tlkurt(lam)
def _entropy(self, lam):
def integ(p):
return log(pow(p, lam-1)+pow(1-p, lam-1))
return integrate.quad(integ, 0, 1)[0]
tukeylambda = tukeylambda_gen(name='tukeylambda')
class uniform_gen(rv_continuous):
"""A uniform continuous random variable.
This distribution is constant between `loc` and ``loc + scale``.
%(before_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.uniform(0.0, 1.0, self._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
uniform = uniform_gen(a=0.0, b=1.0, name='uniform')
class vonmises_gen(rv_continuous):
"""A Von Mises continuous random variable.
%(before_notes)s
Notes
-----
If `x` is not in range or `loc` is not in range it assumes they are angles
and converts them to [-pi, pi] equivalents.
The probability density function for `vonmises` is::
vonmises.pdf(x, kappa) = exp(kappa * cos(x)) / (2*pi*I[0](kappa))
for ``-pi <= x <= pi``, ``kappa > 0``.
`vonmises` takes ``kappa`` as a shape parameter.
%(after_notes)s
See Also
--------
vonmises_line : The same distribution, defined on a [-pi, pi] segment
of the real line.
%(example)s
"""
def _rvs(self, kappa):
return self._random_state.vonmises(0.0, kappa, size=self._size)
def _pdf(self, x, kappa):
return exp(kappa * cos(x)) / (2*pi*i0(kappa))
def _cdf(self, x, kappa):
return vonmises_cython.von_mises_cdf(kappa, x)
def _stats_skip(self, kappa):
return 0, None, 0, None
def _entropy(self, kappa):
return (-kappa * i1(kappa) / i0(kappa)
+ np.log(2 * np.pi * i0(kappa)))
vonmises = vonmises_gen(name='vonmises')
vonmises_line = vonmises_gen(a=-np.pi, b=np.pi, name='vonmises_line')
class wald_gen(invgauss_gen):
"""A Wald continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `wald` is::
wald.pdf(x) = 1/sqrt(2*pi*x**3) * exp(-(x-1)**2/(2*x))
for ``x > 0``.
`wald` is a special case of `invgauss` with ``mu == 1``.
%(after_notes)s
%(example)s
"""
def _rvs(self):
return self._random_state.wald(1.0, 1.0, size=self._size)
def _pdf(self, x):
return invgauss._pdf(x, 1.0)
def _logpdf(self, x):
return invgauss._logpdf(x, 1.0)
def _cdf(self, x):
return invgauss._cdf(x, 1.0)
def _stats(self):
return 1.0, 1.0, 3.0, 15.0
wald = wald_gen(a=0.0, name="wald")
class wrapcauchy_gen(rv_continuous):
"""A wrapped Cauchy continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `wrapcauchy` is::
wrapcauchy.pdf(x, c) = (1-c**2) / (2*pi*(1+c**2-2*c*cos(x)))
for ``0 <= x <= 2*pi``, ``0 < c < 1``.
`wrapcauchy` takes ``c`` as a shape parameter.
%(after_notes)s
%(example)s
"""
def _argcheck(self, c):
return (c > 0) & (c < 1)
def _pdf(self, x, c):
return (1.0-c*c)/(2*pi*(1+c*c-2*c*cos(x)))
def _cdf(self, x, c):
output = np.zeros(x.shape, dtype=x.dtype)
val = (1.0+c)/(1.0-c)
c1 = x < pi
c2 = 1-c1
xp = extract(c1, x)
xn = extract(c2, x)
if np.any(xn):
valn = extract(c2, np.ones_like(x)*val)
xn = 2*pi - xn
yn = tan(xn/2.0)
on = 1.0-1.0/pi*arctan(valn*yn)
place(output, c2, on)
if np.any(xp):
valp = extract(c1, np.ones_like(x)*val)
yp = tan(xp/2.0)
op = 1.0/pi*arctan(valp*yp)
place(output, c1, op)
return output
def _ppf(self, q, c):
val = (1.0-c)/(1.0+c)
rcq = 2*arctan(val*tan(pi*q))
rcmq = 2*pi-2*arctan(val*tan(pi*(1-q)))
return where(q < 1.0/2, rcq, rcmq)
def _entropy(self, c):
return log(2*pi*(1-c*c))
wrapcauchy = wrapcauchy_gen(a=0.0, b=2*pi, name='wrapcauchy')
class gennorm_gen(rv_continuous):
"""A generalized normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `gennorm` is [1]_::
beta
gennorm.pdf(x, beta) = --------------- exp(-|x|**beta)
2 gamma(1/beta)
`gennorm` takes ``beta`` as a shape parameter.
For ``beta = 1``, it is identical to a Laplace distribution.
For ``beta = 2``, it is identical to a normal distribution
(with ``scale=1/sqrt(2)``).
See Also
--------
laplace : Laplace distribution
norm : normal distribution
References
----------
.. [1] "Generalized normal distribution, Version 1",
https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
%(example)s
"""
def _pdf(self, x, beta):
return np.exp(self._logpdf(x, beta))
def _logpdf(self, x, beta):
return np.log(.5 * beta) - special.gammaln(1. / beta) - abs(x)**beta
def _cdf(self, x, beta):
c = .5 * np.sign(x)
# evaluating (.5 + c) first prevents numerical cancellation
return (.5 + c) - c * special.gammaincc(1. / beta, abs(x)**beta)
def _ppf(self, x, beta):
c = np.sign(x - .5)
# evaluating (1. + c) first prevents numerical cancellation
return c * special.gammainccinv(1. / beta, (1. + c) - 2.*c*x)**(1. / 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 = special.gammaln([1./beta, 3./beta, 5./beta])
return 0., np.exp(c3 - c1), 0., np.exp(c5 + c1 - 2. * c3) - 3.
def _entropy(self, beta):
return 1. / beta - np.log(.5 * beta) + special.gammaln(1. / beta)
gennorm = gennorm_gen(name='gennorm')
class halfgennorm_gen(rv_continuous):
"""The upper half of a generalized normal continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `halfgennorm` is::
beta
halfgennorm.pdf(x, beta) = ------------- exp(-|x|**beta)
gamma(1/beta)
`gennorm` takes ``beta`` as a shape parameter.
For ``beta = 1``, it is identical to an exponential distribution.
For ``beta = 2``, it is identical to a half normal distribution
(with ``scale=1/sqrt(2)``).
See Also
--------
gennorm : generalized normal distribution
expon : exponential distribution
halfnorm : half normal distribution
References
----------
.. [1] "Generalized normal distribution, Version 1",
https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
%(example)s
"""
def _pdf(self, x, beta):
return np.exp(self._logpdf(x, beta))
def _logpdf(self, x, beta):
return np.log(beta) - special.gammaln(1. / beta) - x**beta
def _cdf(self, x, beta):
return special.gammainc(1. / beta, x**beta)
def _ppf(self, x, beta):
return special.gammaincinv(1. / beta, x)**(1. / beta)
def _sf(self, x, beta):
return special.gammaincc(1. / beta, x**beta)
def _isf(self, x, beta):
return special.gammainccinv(1. / beta, x)**(1. / beta)
def _entropy(self, beta):
return 1. / beta - np.log(beta) + special.gammaln(1. / beta)
halfgennorm = halfgennorm_gen(a=0, name='halfgennorm')
# Collect names of classes and objects in this module.
pairs = list(globals().items())
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_continuous)
__all__ = _distn_names + _distn_gen_names
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