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# Functions to implement several important functions for
# various Continous and Discrete Probability Distributions
#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
import math
import warnings
from copy import copy
from scipy.misc import comb, derivative
from scipy import special
from scipy import optimize
from scipy import integrate
from scipy.special import gammaln as gamln
import inspect
from numpy import all, where, arange, putmask, \
ravel, take, ones, sum, shape, product, repeat, reshape, \
zeros, floor, logical_and, log, sqrt, exp, arctanh, tan, sin, arcsin, \
arctan, tanh, ndarray, cos, cosh, sinh, newaxis, array, log1p, expm1
from numpy import atleast_1d, polyval, ceil, place, extract, \
any, argsort, argmax, vectorize, r_, asarray, nan, inf, pi, isinf, \
power, NINF, empty
import numpy
import numpy as np
import numpy.random as mtrand
from numpy import flatnonzero as nonzero
import vonmises_cython
from _tukeylambda_stats import tukeylambda_variance as _tlvar, \
tukeylambda_kurtosis as _tlkurt
__all__ = [
'rv_continuous',
'ksone', 'kstwobign', 'norm', 'alpha', 'anglit', 'arcsine',
'beta', 'betaprime', 'bradford', 'burr', 'fisk', 'cauchy',
'chi', 'chi2', 'cosine', 'dgamma', 'dweibull', 'erlang',
'expon', 'exponweib', 'exponpow', 'fatiguelife', 'foldcauchy',
'f', 'foldnorm', 'frechet_r', 'weibull_min', 'frechet_l',
'weibull_max', 'genlogistic', 'genpareto', 'genexpon', 'genextreme',
'gamma', 'gengamma', 'genhalflogistic', 'gompertz', 'gumbel_r',
'gumbel_l', 'halfcauchy', 'halflogistic', 'halfnorm', 'hypsecant',
'gausshyper', 'invgamma', 'invgauss', 'invweibull',
'johnsonsb', 'johnsonsu', 'laplace', 'levy', 'levy_l',
'levy_stable', 'logistic', 'loggamma', 'loglaplace', 'lognorm',
'gilbrat', 'maxwell', 'mielke', 'nakagami', 'ncx2', 'ncf', 't',
'nct', 'pareto', 'lomax', 'powerlaw', 'powerlognorm', 'powernorm',
'rdist', 'rayleigh', 'reciprocal', 'rice', 'recipinvgauss',
'semicircular', 'triang', 'truncexpon', 'truncnorm',
'tukeylambda', 'uniform', 'vonmises', 'wald', 'wrapcauchy',
'entropy', 'rv_discrete', 'binom', 'bernoulli', 'nbinom', 'geom',
'hypergeom', 'logser', 'poisson', 'planck', 'boltzmann', 'randint',
'zipf', 'dlaplace', 'skellam'
]
floatinfo = numpy.finfo(float)
gam = special.gamma
random = mtrand.random_sample
import types
from scipy.misc import doccer
sgf = vectorize
try:
from new import instancemethod
except ImportError:
# Python 3
def instancemethod(func, obj, cls):
return types.MethodType(func, obj)
# These are the docstring parts used for substitution in specific
# distribution docstrings.
docheaders = {'methods':"""\nMethods\n-------\n""",
'parameters':"""\nParameters\n---------\n""",
'notes':"""\nNotes\n-----\n""",
'examples':"""\nExamples\n--------\n"""}
_doc_rvs = \
"""rvs(%(shapes)s, loc=0, scale=1, size=1)
Random variates.
"""
_doc_pdf = \
"""pdf(x, %(shapes)s, loc=0, scale=1)
Probability density function.
"""
_doc_logpdf = \
"""logpdf(x, %(shapes)s, loc=0, scale=1)
Log of the probability density function.
"""
_doc_pmf = \
"""pmf(x, %(shapes)s, loc=0, scale=1)
Probability mass function.
"""
_doc_logpmf = \
"""logpmf(x, %(shapes)s, loc=0, scale=1)
Log of the probability mass function.
"""
_doc_cdf = \
"""cdf(x, %(shapes)s, loc=0, scale=1)
Cumulative density function.
"""
_doc_logcdf = \
"""logcdf(x, %(shapes)s, loc=0, scale=1)
Log of the cumulative density function.
"""
_doc_sf = \
"""sf(x, %(shapes)s, loc=0, scale=1)
Survival function (1-cdf --- sometimes more accurate).
"""
_doc_logsf = \
"""logsf(x, %(shapes)s, loc=0, scale=1)
Log of the survival function.
"""
_doc_ppf = \
"""ppf(q, %(shapes)s, loc=0, scale=1)
Percent point function (inverse of cdf --- percentiles).
"""
_doc_isf = \
"""isf(q, %(shapes)s, loc=0, scale=1)
Inverse survival function (inverse of sf).
"""
_doc_moment = \
"""moment(n, %(shapes)s, loc=0, scale=1)
Non-central moment of order n
"""
_doc_stats = \
"""stats(%(shapes)s, loc=0, scale=1, moments='mv')
Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
"""
_doc_entropy = \
"""entropy(%(shapes)s, loc=0, scale=1)
(Differential) entropy of the RV.
"""
_doc_fit = \
"""fit(data, %(shapes)s, loc=0, scale=1)
Parameter estimates for generic data.
"""
_doc_expect = \
"""expect(func, %(shapes)s, loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)
Expected value of a function (of one argument) with respect to the distribution.
"""
_doc_expect_discrete = \
"""expect(func, %(shapes)s, loc=0, lb=None, ub=None, conditional=False)
Expected value of a function (of one argument) with respect to the distribution.
"""
_doc_median = \
"""median(%(shapes)s, loc=0, scale=1)
Median of the distribution.
"""
_doc_mean = \
"""mean(%(shapes)s, loc=0, scale=1)
Mean of the distribution.
"""
_doc_var = \
"""var(%(shapes)s, loc=0, scale=1)
Variance of the distribution.
"""
_doc_std = \
"""std(%(shapes)s, loc=0, scale=1)
Standard deviation of the distribution.
"""
_doc_interval = \
"""interval(alpha, %(shapes)s, loc=0, scale=1)
Endpoints of the range that contains alpha percent of the distribution
"""
_doc_allmethods = ''.join([docheaders['methods'], _doc_rvs, _doc_pdf,
_doc_logpdf, _doc_cdf, _doc_logcdf, _doc_sf,
_doc_logsf, _doc_ppf, _doc_isf, _doc_moment,
_doc_stats, _doc_entropy, _doc_fit,
_doc_expect, _doc_median,
_doc_mean, _doc_var, _doc_std, _doc_interval])
# Note that the two lines for %(shapes) are searched for and replaced in
# rv_continuous and rv_discrete - update there if the exact string changes
_doc_default_callparams = \
"""
Parameters
----------
x : array_like
quantiles
q : array_like
lower or upper tail probability
%(shapes)s : array_like
shape parameters
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
size : int or tuple of ints, optional
shape of random variates (default computed from input arguments )
moments : str, optional
composed of letters ['mvsk'] specifying which moments to compute where
'm' = mean, 'v' = variance, 's' = (Fisher's) skew and
'k' = (Fisher's) kurtosis. (default='mv')
"""
_doc_default_longsummary = \
"""Continuous random variables are defined from a standard form and may
require some shape parameters to complete its specification. Any
optional keyword parameters can be passed to the methods of the RV
object as given below:
"""
_doc_default_frozen_note = \
"""
Alternatively, the object may be called (as a function) to fix the shape,
location, and scale parameters returning a "frozen" continuous RV object:
rv = %(name)s(%(shapes)s, loc=0, scale=1)
- Frozen RV object with the same methods but holding the given shape,
location, and scale fixed.
"""
_doc_default_example = \
"""Examples
--------
>>> from scipy.stats import %(name)s
>>> numargs = %(name)s.numargs
>>> [ %(shapes)s ] = [0.9,] * numargs
>>> rv = %(name)s(%(shapes)s)
Display frozen pdf
>>> x = np.linspace(0, np.minimum(rv.dist.b, 3))
>>> h = plt.plot(x, rv.pdf(x))
Here, ``rv.dist.b`` is the right endpoint of the support of ``rv.dist``.
Check accuracy of cdf and ppf
>>> prb = %(name)s.cdf(x, %(shapes)s)
>>> h = plt.semilogy(np.abs(x - %(name)s.ppf(prb, %(shapes)s)) + 1e-20)
Random number generation
>>> R = %(name)s.rvs(%(shapes)s, size=100)
"""
_doc_default = ''.join([_doc_default_longsummary,
_doc_allmethods,
_doc_default_callparams,
_doc_default_frozen_note,
_doc_default_example])
_doc_default_before_notes = ''.join([_doc_default_longsummary,
_doc_allmethods,
_doc_default_callparams,
_doc_default_frozen_note])
docdict = {'rvs':_doc_rvs,
'pdf':_doc_pdf,
'logpdf':_doc_logpdf,
'cdf':_doc_cdf,
'logcdf':_doc_logcdf,
'sf':_doc_sf,
'logsf':_doc_logsf,
'ppf':_doc_ppf,
'isf':_doc_isf,
'stats':_doc_stats,
'entropy':_doc_entropy,
'fit':_doc_fit,
'moment':_doc_moment,
'expect':_doc_expect,
'interval':_doc_interval,
'mean':_doc_mean,
'std':_doc_std,
'var':_doc_var,
'median':_doc_median,
'allmethods':_doc_allmethods,
'callparams':_doc_default_callparams,
'longsummary':_doc_default_longsummary,
'frozennote':_doc_default_frozen_note,
'example':_doc_default_example,
'default':_doc_default,
'before_notes':_doc_default_before_notes}
# Reuse common content between continous and discrete docs, change some
# minor bits.
docdict_discrete = docdict.copy()
docdict_discrete['pmf'] = _doc_pmf
docdict_discrete['logpmf'] = _doc_logpmf
docdict_discrete['expect'] = _doc_expect_discrete
_doc_disc_methods = ['rvs', 'pmf', 'logpmf', 'cdf', 'logcdf', 'sf', 'logsf',
'ppf', 'isf', 'stats', 'entropy', 'expect', 'median',
'mean', 'var', 'std', 'interval']
for obj in _doc_disc_methods:
docdict_discrete[obj] = docdict_discrete[obj].replace(', scale=1', '')
docdict_discrete.pop('pdf')
docdict_discrete.pop('logpdf')
_doc_allmethods = ''.join([docdict_discrete[obj] for obj in
_doc_disc_methods])
docdict_discrete['allmethods'] = docheaders['methods'] + _doc_allmethods
docdict_discrete['longsummary'] = _doc_default_longsummary.replace(\
'Continuous', 'Discrete')
_doc_default_frozen_note = \
"""
Alternatively, the object may be called (as a function) to fix the shape and
location parameters returning a "frozen" discrete RV object:
rv = %(name)s(%(shapes)s, loc=0)
- Frozen RV object with the same methods but holding the given shape and
location fixed.
"""
docdict_discrete['frozennote'] = _doc_default_frozen_note
_doc_default_discrete_example = \
"""Examples
--------
>>> from scipy.stats import %(name)s
>>> [ %(shapes)s ] = [<Replace with reasonable values>]
>>> rv = %(name)s(%(shapes)s)
Display frozen pmf
>>> x = np.arange(0, np.minimum(rv.dist.b, 3))
>>> h = plt.vlines(x, 0, rv.pmf(x), lw=2)
Here, ``rv.dist.b`` is the right endpoint of the support of ``rv.dist``.
Check accuracy of cdf and ppf
>>> prb = %(name)s.cdf(x, %(shapes)s)
>>> h = plt.semilogy(np.abs(x - %(name)s.ppf(prb, %(shapes)s)) + 1e-20)
Random number generation
>>> R = %(name)s.rvs(%(shapes)s, size=100)
"""
docdict_discrete['example'] = _doc_default_discrete_example
_doc_default_before_notes = ''.join([docdict_discrete['longsummary'],
docdict_discrete['allmethods'],
docdict_discrete['callparams'],
docdict_discrete['frozennote']])
docdict_discrete['before_notes'] = _doc_default_before_notes
_doc_default_disc = ''.join([docdict_discrete['longsummary'],
docdict_discrete['allmethods'],
docdict_discrete['frozennote'],
docdict_discrete['example']])
docdict_discrete['default'] = _doc_default_disc
# clean up all the separate docstring elements, we do not need them anymore
for obj in [s for s in dir() if s.startswith('_doc_')]:
exec('del ' + obj)
del obj
try:
del s
except NameError:
# in Python 3, loop variables are not visible after the loop
pass
def _moment(data, n, mu=None):
if mu is None:
mu = data.mean()
return ((data - mu)**n).mean()
def _moment_from_stats(n, mu, mu2, g1, g2, moment_func, args):
if (n==0):
return 1.0
elif (n==1):
if mu is None:
val = moment_func(1,*args)
else:
val = mu
elif (n==2):
if mu2 is None or mu is None:
val = moment_func(2,*args)
else:
val = mu2 + mu*mu
elif (n==3):
if g1 is None or mu2 is None or mu is None:
val = moment_func(3,*args)
else:
mu3 = g1*(mu2**1.5) # 3rd central moment
val = mu3+3*mu*mu2+mu**3 # 3rd non-central moment
elif (n==4):
if g1 is None or g2 is None or mu2 is None or mu is None:
val = moment_func(4,*args)
else:
mu4 = (g2+3.0)*(mu2**2.0) # 4th central moment
mu3 = g1*(mu2**1.5) # 3rd central moment
val = mu4+4*mu*mu3+6*mu*mu*mu2+mu**4
else:
val = moment_func(n, *args)
return val
def _skew(data):
"""
skew is third central moment / variance**(1.5)
"""
data = np.ravel(data)
mu = data.mean()
m2 = ((data - mu)**2).mean()
m3 = ((data - mu)**3).mean()
return m3 / m2**1.5
def _kurtosis(data):
"""
kurtosis is fourth central moment / variance**2 - 3
"""
data = np.ravel(data)
mu = data.mean()
m2 = ((data - mu)**2).mean()
m4 = ((data - mu)**4).mean()
return m4 / m2**2 - 3
# Frozen RV class
class rv_frozen(object):
def __init__(self, dist, *args, **kwds):
self.args = args
self.kwds = kwds
self.dist = dist
def pdf(self, x): #raises AttributeError in frozen discrete distribution
return self.dist.pdf(x, *self.args, **self.kwds)
def logpdf(self, x):
return self.dist.logpdf(x, *self.args, **self.kwds)
def cdf(self, x):
return self.dist.cdf(x, *self.args, **self.kwds)
def logcdf(self, x):
return self.dist.logcdf(x, *self.args, **self.kwds)
def ppf(self, q):
return self.dist.ppf(q, *self.args, **self.kwds)
def isf(self, q):
return self.dist.isf(q, *self.args, **self.kwds)
def rvs(self, size=None):
kwds = self.kwds.copy()
kwds.update({'size':size})
return self.dist.rvs(*self.args, **kwds)
def sf(self, x):
return self.dist.sf(x, *self.args, **self.kwds)
def logsf(self, x):
return self.dist.logsf(x, *self.args, **self.kwds)
def stats(self, moments='mv'):
kwds = self.kwds.copy()
kwds.update({'moments':moments})
return self.dist.stats(*self.args, **kwds)
def median(self):
return self.dist.median(*self.args, **self.kwds)
def mean(self):
return self.dist.mean(*self.args, **self.kwds)
def var(self):
return self.dist.var(*self.args, **self.kwds)
def std(self):
return self.dist.std(*self.args, **self.kwds)
def moment(self, n):
return self.dist.moment(n, *self.args, **self.kwds)
def entropy(self):
return self.dist.entropy(*self.args, **self.kwds)
def pmf(self,k):
return self.dist.pmf(k, *self.args, **self.kwds)
def logpmf(self,k):
return self.dist.logpmf(k, *self.args, **self.kwds)
def interval(self, alpha):
return self.dist.interval(alpha, *self.args, **self.kwds)
def valarray(shape,value=nan,typecode=None):
"""Return an array of all value.
"""
out = reshape(repeat([value],product(shape,axis=0),axis=0),shape)
if typecode is not None:
out = out.astype(typecode)
if not isinstance(out, ndarray):
out = asarray(out)
return out
# This should be rewritten
def argsreduce(cond, *args):
"""Return the sequence of ravel(args[i]) where ravel(condition) is
True in 1D.
Examples
--------
>>> import numpy as np
>>> rand = np.random.random_sample
>>> A = rand((4,5))
>>> B = 2
>>> C = rand((1,5))
>>> cond = np.ones(A.shape)
>>> [A1,B1,C1] = argsreduce(cond,A,B,C)
>>> B1.shape
(20,)
>>> cond[2,:] = 0
>>> [A2,B2,C2] = argsreduce(cond,A,B,C)
>>> B2.shape
(15,)
"""
newargs = atleast_1d(*args)
if not isinstance(newargs, list):
newargs = [newargs,]
expand_arr = (cond==cond)
return [extract(cond, arr1 * expand_arr) for arr1 in newargs]
class rv_generic(object):
"""Class which encapsulates common functionality between rv_discrete
and rv_continuous.
"""
def _fix_loc_scale(self, args, loc, scale=1):
N = len(args)
if N > self.numargs:
if N == self.numargs + 1 and loc is None:
# loc is given without keyword
loc = args[-1]
if N == self.numargs + 2 and scale is None:
# loc and scale given without keyword
loc, scale = args[-2:]
args = args[:self.numargs]
if scale is None:
scale = 1.0
if loc is None:
loc = 0.0
return args, loc, scale
def _fix_loc(self, args, loc):
args, loc, scale = self._fix_loc_scale(args, loc)
return args, loc
# These are actually called, and should not be overwritten if you
# want to keep error checking.
def rvs(self,*args,**kwds):
"""
Random variates of given type.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
size : int or tuple of ints, optional
defining number of random variates (default=1)
Returns
-------
rvs : array_like
random variates of given `size`
"""
kwd_names = ['loc', 'scale', 'size', 'discrete']
loc, scale, size, discrete = map(kwds.get, kwd_names,
[None]*len(kwd_names))
args, loc, scale = self._fix_loc_scale(args, loc, scale)
cond = logical_and(self._argcheck(*args),(scale >= 0))
if not all(cond):
raise ValueError("Domain error in arguments.")
# self._size is total size of all output values
self._size = product(size, axis=0)
if self._size is not None and self._size > 1:
size = numpy.array(size, ndmin=1)
if np.all(scale == 0):
return loc*ones(size, 'd')
vals = self._rvs(*args)
if self._size is not None:
vals = reshape(vals, size)
vals = vals * scale + loc
# Cast to int if discrete
if discrete:
if numpy.isscalar(vals):
vals = int(vals)
else:
vals = vals.astype(int)
return vals
def median(self, *args, **kwds):
"""
Median of the distribution.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
median : float
the median of the distribution.
See Also
--------
self.ppf --- inverse of the CDF
"""
return self.ppf(0.5, *args, **kwds)
def mean(self, *args, **kwds):
"""
Mean of the distribution
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
mean : float
the mean of the distribution
"""
kwds['moments'] = 'm'
res = self.stats(*args, **kwds)
if isinstance(res, ndarray) and res.ndim == 0:
return res[()]
return res
def var(self, *args, **kwds):
"""
Variance of the distribution
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
var : float
the variance of the distribution
"""
kwds['moments'] = 'v'
res = self.stats(*args, **kwds)
if isinstance(res, ndarray) and res.ndim == 0:
return res[()]
return res
def std(self, *args, **kwds):
"""
Standard deviation of the distribution.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
std : float
standard deviation of the distribution
"""
kwds['moments'] = 'v'
res = sqrt(self.stats(*args, **kwds))
return res
def interval(self, alpha, *args, **kwds):
"""Confidence interval with equal areas around the median
Parameters
----------
alpha : array_like float in [0,1]
Probability that an rv will be drawn from the returned range
arg1, arg2, ... : array_like
The shape parameter(s) for the distribution (see docstring of the instance
object for more information)
loc : array_like, optional
location parameter (default = 0)
scale : array_like, optional
scale paramter (default = 1)
Returns
-------
a, b : array_like (float)
end-points of range that contain alpha % of the rvs
"""
alpha = asarray(alpha)
if any((alpha > 1) | (alpha < 0)):
raise ValueError("alpha must be between 0 and 1 inclusive")
q1 = (1.0-alpha)/2
q2 = (1.0+alpha)/2
a = self.ppf(q1, *args, **kwds)
b = self.ppf(q2, *args, **kwds)
return a, b
## continuous random variables: implement maybe later
##
## hf --- Hazard Function (PDF / SF)
## chf --- Cumulative hazard function (-log(SF))
## psf --- Probability sparsity function (reciprocal of the pdf) in
## units of percent-point-function (as a function of q).
## Also, the derivative of the percent-point function.
class rv_continuous(rv_generic):
"""
A generic continuous random variable class meant for subclassing.
`rv_continuous` is a base class to construct specific distribution classes
and instances from for continuous random variables. It cannot be used
directly as a distribution.
Parameters
----------
momtype : int, optional
The type of generic moment calculation to use: 0 for pdf, 1 (default) for ppf.
a : float, optional
Lower bound of the support of the distribution, default is minus
infinity.
b : float, optional
Upper bound of the support of the distribution, default is plus
infinity.
xa : float, optional
DEPRECATED
xb : float, optional
DEPRECATED
xtol : float, optional
The tolerance for fixed point calculation for generic ppf.
badvalue : object, optional
The value in a result arrays that indicates a value that for which
some argument restriction is violated, default is np.nan.
name : str, optional
The name of the instance. This string is used to construct the default
example for distributions.
longname : str, optional
This string is used as part of the first line of the docstring returned
when a subclass has no docstring of its own. Note: `longname` exists
for backwards compatibility, do not use for new subclasses.
shapes : str, optional
The shape of the distribution. For example ``"m, n"`` for a
distribution that takes two integers as the two shape arguments for all
its methods.
extradoc : str, optional, deprecated
This string is used as the last part of the docstring returned when a
subclass has no docstring of its own. Note: `extradoc` exists for
backwards compatibility, do not use for new subclasses.
Methods
-------
rvs(<shape(s)>, loc=0, scale=1, size=1)
random variates
pdf(x, <shape(s)>, loc=0, scale=1)
probability density function
logpdf(x, <shape(s)>, loc=0, scale=1)
log of the probability density function
cdf(x, <shape(s)>, loc=0, scale=1)
cumulative density function
logcdf(x, <shape(s)>, loc=0, scale=1)
log of the cumulative density function
sf(x, <shape(s)>, loc=0, scale=1)
survival function (1-cdf --- sometimes more accurate)
logsf(x, <shape(s)>, loc=0, scale=1)
log of the survival function
ppf(q, <shape(s)>, loc=0, scale=1)
percent point function (inverse of cdf --- quantiles)
isf(q, <shape(s)>, loc=0, scale=1)
inverse survival function (inverse of sf)
moment(n, <shape(s)>, loc=0, scale=1)
non-central n-th moment of the distribution. May not work for array arguments.
stats(<shape(s)>, loc=0, scale=1, moments='mv')
mean('m'), variance('v'), skew('s'), and/or kurtosis('k')
entropy(<shape(s)>, loc=0, scale=1)
(differential) entropy of the RV.
fit(data, <shape(s)>, loc=0, scale=1)
Parameter estimates for generic data
expect(func=None, args=(), loc=0, scale=1, lb=None, ub=None,
conditional=False, **kwds)
Expected value of a function with respect to the distribution.
Additional kwd arguments passed to integrate.quad
median(<shape(s)>, loc=0, scale=1)
Median of the distribution.
mean(<shape(s)>, loc=0, scale=1)
Mean of the distribution.
std(<shape(s)>, loc=0, scale=1)
Standard deviation of the distribution.
var(<shape(s)>, loc=0, scale=1)
Variance of the distribution.
interval(alpha, <shape(s)>, loc=0, scale=1)
Interval that with `alpha` percent probability contains a random
realization of this distribution.
__call__(<shape(s)>, loc=0, scale=1)
Calling a distribution instance creates a frozen RV object with the
same methods but holding the given shape, location, and scale fixed.
See Notes section.
**Parameters for Methods**
x : array_like
quantiles
q : array_like
lower or upper tail probability
<shape(s)> : array_like
shape parameters
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
size : int or tuple of ints, optional
shape of random variates (default computed from input arguments )
moments : string, optional
composed of letters ['mvsk'] specifying which moments to compute where
'm' = mean, 'v' = variance, 's' = (Fisher's) skew and
'k' = (Fisher's) kurtosis. (default='mv')
n : int
order of moment to calculate in method moments
Notes
-----
**Methods that can be overwritten by subclasses**
::
_rvs
_pdf
_cdf
_sf
_ppf
_isf
_stats
_munp
_entropy
_argcheck
There are additional (internal and private) generic methods that can
be useful for cross-checking and for debugging, but might work in all
cases when directly called.
**Frozen Distribution**
Alternatively, the object may be called (as a function) to fix the shape,
location, and scale parameters returning a "frozen" continuous RV object:
rv = generic(<shape(s)>, loc=0, scale=1)
frozen RV object with the same methods but holding the given shape,
location, and scale fixed
**Subclassing**
New random variables can be defined by subclassing rv_continuous class
and re-defining at least the ``_pdf`` or the ``_cdf`` method (normalized
to location 0 and scale 1) which will be given clean arguments (in between
a and b) and passing the argument check method.
If positive argument checking is not correct for your RV
then you will also need to re-define the ``_argcheck`` method.
Correct, but potentially slow defaults exist for the remaining
methods but for speed and/or accuracy you can over-ride::
_logpdf, _cdf, _logcdf, _ppf, _rvs, _isf, _sf, _logsf
Rarely would you override ``_isf``, ``_sf`` or ``_logsf``, but you could.
Statistics are computed using numerical integration by default.
For speed you can redefine this using ``_stats``:
- take shape parameters and return mu, mu2, g1, g2
- If you can't compute one of these, return it as None
- Can also be defined with a keyword argument ``moments=<str>``,
where <str> is a string composed of 'm', 'v', 's',
and/or 'k'. Only the components appearing in string
should be computed and returned in the order 'm', 'v',
's', or 'k' with missing values returned as None.
Alternatively, you can override ``_munp``, which takes n and shape
parameters and returns the nth non-central moment of the distribution.
Examples
--------
To create a new Gaussian distribution, we would do the following::
class gaussian_gen(rv_continuous):
"Gaussian distribution"
def _pdf:
...
...
"""
def __init__(self, momtype=1, a=None, b=None, xa=None, xb=None,
xtol=1e-14, badvalue=None, name=None, longname=None,
shapes=None, extradoc=None):
rv_generic.__init__(self)
if badvalue is None:
badvalue = nan
if name is None:
name = 'Distribution'
self.badvalue = badvalue
self.name = name
self.a = a
self.b = b
if a is None:
self.a = -inf
if b is None:
self.b = inf
if xa is not None:
warnings.warn("The `xa` parameter is deprecated and will be "
"removed in scipy 0.12", DeprecationWarning)
if xb is not None:
warnings.warn("The `xb` parameter is deprecated and will be "
"removed in scipy 0.12", DeprecationWarning)
self.xa = xa
self.xb = xb
self.xtol = xtol
self._size = 1
self.m = 0.0
self.moment_type = momtype
self.expandarr = 1
if not hasattr(self,'numargs'):
#allows more general subclassing with *args
cdf_signature = inspect.getargspec(self._cdf.im_func)
numargs1 = len(cdf_signature[0]) - 2
pdf_signature = inspect.getargspec(self._pdf.im_func)
numargs2 = len(pdf_signature[0]) - 2
self.numargs = max(numargs1, numargs2)
#nin correction
self.vecfunc = sgf(self._ppf_single_call,otypes='d')
self.vecfunc.nin = self.numargs + 1
self.vecentropy = sgf(self._entropy,otypes='d')
self.vecentropy.nin = self.numargs + 1
self.veccdf = sgf(self._cdf_single_call,otypes='d')
self.veccdf.nin = self.numargs + 1
self.shapes = shapes
self.extradoc = extradoc
if momtype == 0:
self.generic_moment = sgf(self._mom0_sc,otypes='d')
else:
self.generic_moment = sgf(self._mom1_sc,otypes='d')
self.generic_moment.nin = self.numargs+1 # Because of the *args argument
# of _mom0_sc, vectorize cannot count the number of arguments correctly.
if longname is None:
if name[0] in ['aeiouAEIOU']:
hstr = "An "
else:
hstr = "A "
longname = hstr + name
# generate docstring for subclass instances
if self.__doc__ is None:
self._construct_default_doc(longname=longname, extradoc=extradoc)
else:
self._construct_doc()
## This only works for old-style classes...
# self.__class__.__doc__ = self.__doc__
def _construct_default_doc(self, longname=None, extradoc=None):
"""Construct instance docstring from the default template."""
if longname is None:
longname = 'A'
if extradoc is None:
extradoc = ''
if extradoc.startswith('\n\n'):
extradoc = extradoc[2:]
self.__doc__ = ''.join(['%s continuous random variable.'%longname,
'\n\n%(before_notes)s\n', docheaders['notes'],
extradoc, '\n%(example)s'])
self._construct_doc()
def _construct_doc(self):
"""Construct the instance docstring with string substitutions."""
tempdict = docdict.copy()
tempdict['name'] = self.name or 'distname'
tempdict['shapes'] = self.shapes or ''
if self.shapes is None:
# remove shapes from call parameters if there are none
for item in ['callparams', 'default', 'before_notes']:
tempdict[item] = tempdict[item].replace(\
"\n%(shapes)s : array_like\n shape parameters", "")
for i in range(2):
if self.shapes is None:
# necessary because we use %(shapes)s in two forms (w w/o ", ")
self.__doc__ = self.__doc__.replace("%(shapes)s, ", "")
self.__doc__ = doccer.docformat(self.__doc__, tempdict)
def _ppf_to_solve(self, x, q,*args):
return apply(self.cdf, (x, )+args)-q
def _ppf_single_call(self, q, *args):
left = right = None
if self.a > -np.inf:
left = self.a
if self.b < np.inf:
right = self.b
factor = 10.
if not left: # i.e. self.a = -inf
left = -1.*factor
while self._ppf_to_solve(left, q,*args) > 0.:
right = left
left *= factor
# left is now such that cdf(left) < q
if not right: # i.e. self.b = inf
right = factor
while self._ppf_to_solve(right, q,*args) < 0.:
left = right
right *= factor
# right is now such that cdf(right) > q
return optimize.brentq(self._ppf_to_solve, \
left, right, args=(q,)+args, xtol=self.xtol)
# moment from definition
def _mom_integ0(self, x,m,*args):
return x**m * self.pdf(x,*args)
def _mom0_sc(self, m,*args):
return integrate.quad(self._mom_integ0, self.a,
self.b, args=(m,)+args)[0]
# moment calculated using ppf
def _mom_integ1(self, q,m,*args):
return (self.ppf(q,*args))**m
def _mom1_sc(self, m,*args):
return integrate.quad(self._mom_integ1, 0, 1,args=(m,)+args)[0]
## These are the methods you must define (standard form functions)
def _argcheck(self, *args):
# Default check for correct values on args and keywords.
# Returns condition array of 1's where arguments are correct and
# 0's where they are not.
cond = 1
for arg in args:
cond = logical_and(cond,(asarray(arg) > 0))
return cond
def _pdf(self,x,*args):
return derivative(self._cdf,x,dx=1e-5,args=args,order=5)
## Could also define any of these
def _logpdf(self, x, *args):
return log(self._pdf(x, *args))
##(return 1-d using self._size to get number)
def _rvs(self, *args):
## Use basic inverse cdf algorithm for RV generation as default.
U = mtrand.sample(self._size)
Y = self._ppf(U,*args)
return Y
def _cdf_single_call(self, x, *args):
return integrate.quad(self._pdf, self.a, x, args=args)[0]
def _cdf(self, x, *args):
return self.veccdf(x,*args)
def _logcdf(self, x, *args):
return log(self._cdf(x, *args))
def _sf(self, x, *args):
return 1.0-self._cdf(x,*args)
def _logsf(self, x, *args):
return log(self._sf(x, *args))
def _ppf(self, q, *args):
return self.vecfunc(q,*args)
def _isf(self, q, *args):
return self._ppf(1.0-q,*args) #use correct _ppf for subclasses
# The actual cacluation functions (no basic checking need be done)
# If these are defined, the others won't be looked at.
# Otherwise, the other set can be defined.
def _stats(self,*args, **kwds):
return None, None, None, None
# Central moments
def _munp(self,n,*args):
return self.generic_moment(n,*args)
def pdf(self,x,*args,**kwds):
"""
Probability density function at x of the given RV.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
pdf : ndarray
Probability density function evaluated at x
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
x,loc,scale = map(asarray,(x,loc,scale))
args = tuple(map(asarray,args))
x = asarray((x-loc)*1.0/scale)
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = (scale > 0) & (x >= self.a) & (x <= self.b)
cond = cond0 & cond1
output = zeros(shape(cond),'d')
putmask(output,(1-cond0)+np.isnan(x),self.badvalue)
if any(cond):
goodargs = argsreduce(cond, *((x,)+args+(scale,)))
scale, goodargs = goodargs[-1], goodargs[:-1]
place(output,cond,self._pdf(*goodargs) / scale)
if output.ndim == 0:
return output[()]
return output
def logpdf(self, x, *args, **kwds):
"""
Log of the probability density function at x of the given RV.
This uses a more numerically accurate calculation if available.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
logpdf : array_like
Log of the probability density function evaluated at x
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
x,loc,scale = map(asarray,(x,loc,scale))
args = tuple(map(asarray,args))
x = asarray((x-loc)*1.0/scale)
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = (scale > 0) & (x >= self.a) & (x <= self.b)
cond = cond0 & cond1
output = empty(shape(cond),'d')
output.fill(NINF)
putmask(output,(1-cond0)+np.isnan(x),self.badvalue)
if any(cond):
goodargs = argsreduce(cond, *((x,)+args+(scale,)))
scale, goodargs = goodargs[-1], goodargs[:-1]
place(output,cond,self._logpdf(*goodargs) - log(scale))
if output.ndim == 0:
return output[()]
return output
def cdf(self,x,*args,**kwds):
"""
Cumulative distribution function at x of the given RV.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
cdf : array_like
Cumulative distribution function evaluated at x
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
x,loc,scale = map(asarray,(x,loc,scale))
args = tuple(map(asarray,args))
x = (x-loc)*1.0/scale
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = (scale > 0) & (x > self.a) & (x < self.b)
cond2 = (x >= self.b) & cond0
cond = cond0 & cond1
output = zeros(shape(cond),'d')
place(output,(1-cond0)+np.isnan(x),self.badvalue)
place(output,cond2,1.0)
if any(cond): #call only if at least 1 entry
goodargs = argsreduce(cond, *((x,)+args))
place(output,cond,self._cdf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def logcdf(self,x,*args,**kwds):
"""
Log of the cumulative distribution function at x of the given RV.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
logcdf : array_like
Log of the cumulative distribution function evaluated at x
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
x,loc,scale = map(asarray,(x,loc,scale))
args = tuple(map(asarray,args))
x = (x-loc)*1.0/scale
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = (scale > 0) & (x > self.a) & (x < self.b)
cond2 = (x >= self.b) & cond0
cond = cond0 & cond1
output = empty(shape(cond),'d')
output.fill(NINF)
place(output,(1-cond0)*(cond1==cond1)+np.isnan(x),self.badvalue)
place(output,cond2,0.0)
if any(cond): #call only if at least 1 entry
goodargs = argsreduce(cond, *((x,)+args))
place(output,cond,self._logcdf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def sf(self,x,*args,**kwds):
"""
Survival function (1-cdf) at x of the given RV.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
sf : array_like
Survival function evaluated at x
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
x,loc,scale = map(asarray,(x,loc,scale))
args = tuple(map(asarray,args))
x = (x-loc)*1.0/scale
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = (scale > 0) & (x > self.a) & (x < self.b)
cond2 = cond0 & (x <= self.a)
cond = cond0 & cond1
output = zeros(shape(cond),'d')
place(output,(1-cond0)+np.isnan(x),self.badvalue)
place(output,cond2,1.0)
if any(cond):
goodargs = argsreduce(cond, *((x,)+args))
place(output,cond,self._sf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def logsf(self,x,*args,**kwds):
"""
Log of the survival function of the given RV.
Returns the log of the "survival function," defined as (1 - `cdf`),
evaluated at `x`.
Parameters
----------
x : array_like
quantiles
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
logsf : ndarray
Log of the survival function evaluated at `x`.
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
x,loc,scale = map(asarray,(x,loc,scale))
args = tuple(map(asarray,args))
x = (x-loc)*1.0/scale
cond0 = self._argcheck(*args) & (scale > 0)
cond1 = (scale > 0) & (x > self.a) & (x < self.b)
cond2 = cond0 & (x <= self.a)
cond = cond0 & cond1
output = empty(shape(cond),'d')
output.fill(NINF)
place(output,(1-cond0)+np.isnan(x),self.badvalue)
place(output,cond2,0.0)
if any(cond):
goodargs = argsreduce(cond, *((x,)+args))
place(output,cond,self._logsf(*goodargs))
if output.ndim == 0:
return output[()]
return output
def ppf(self,q,*args,**kwds):
"""
Percent point function (inverse of cdf) at q of the given RV.
Parameters
----------
q : array_like
lower tail probability
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
x : array_like
quantile corresponding to the lower tail probability q.
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
q,loc,scale = map(asarray,(q,loc,scale))
args = tuple(map(asarray,args))
cond0 = self._argcheck(*args) & (scale > 0) & (loc==loc)
cond1 = (q > 0) & (q < 1)
cond2 = (q==1) & cond0
cond = cond0 & cond1
output = valarray(shape(cond),value=self.a*scale + loc)
place(output,(1-cond0)+(1-cond1)*(q!=0.0), self.badvalue)
place(output,cond2,self.b*scale + loc)
if any(cond): #call only if at least 1 entry
goodargs = argsreduce(cond, *((q,)+args+(scale,loc)))
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
place(output,cond,self._ppf(*goodargs)*scale + loc)
if output.ndim == 0:
return output[()]
return output
def isf(self,q,*args,**kwds):
"""
Inverse survival function at q of the given RV.
Parameters
----------
q : array_like
upper tail probability
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
Returns
-------
x : array_like
quantile corresponding to the upper tail probability q.
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
q,loc,scale = map(asarray,(q,loc,scale))
args = tuple(map(asarray,args))
cond0 = self._argcheck(*args) & (scale > 0) & (loc==loc)
cond1 = (q > 0) & (q < 1)
cond2 = (q==1) & cond0
cond = cond0 & cond1
output = valarray(shape(cond),value=self.b)
#place(output,(1-cond0)*(cond1==cond1), self.badvalue)
place(output,(1-cond0)*(cond1==cond1)+(1-cond1)*(q!=0.0), self.badvalue)
place(output,cond2,self.a)
if any(cond): #call only if at least 1 entry
goodargs = argsreduce(cond, *((q,)+args+(scale,loc))) #PB replace 1-q by q
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
place(output,cond,self._isf(*goodargs)*scale + loc) #PB use _isf instead of _ppf
if output.ndim == 0:
return output[()]
return output
def stats(self,*args,**kwds):
"""
Some statistics of the given RV
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information)
loc : array_like, optional
location parameter (default=0)
scale : array_like, optional
scale parameter (default=1)
moments : string, optional
composed of letters ['mvsk'] defining which moments to compute:
'm' = mean,
'v' = variance,
's' = (Fisher's) skew,
'k' = (Fisher's) kurtosis.
(default='mv')
Returns
-------
stats : sequence
of requested moments.
"""
loc,scale,moments=map(kwds.get,['loc','scale','moments'])
N = len(args)
if N > self.numargs:
if N == self.numargs + 1 and loc is None:
# loc is given without keyword
loc = args[-1]
if N == self.numargs + 2 and scale is None:
# loc and scale given without keyword
loc, scale = args[-2:]
if N == self.numargs + 3 and moments is None:
# loc, scale, and moments
loc, scale, moments = args[-3:]
args = args[:self.numargs]
if scale is None: scale = 1.0
if loc is None: loc = 0.0
if moments is None: moments = 'mv'
loc,scale = map(asarray,(loc,scale))
args = tuple(map(asarray,args))
cond = self._argcheck(*args) & (scale > 0) & (loc==loc)
signature = inspect.getargspec(self._stats.im_func)
if (signature[2] is not None) or ('moments' in signature[0]):
mu, mu2, g1, g2 = self._stats(*args,**{'moments':moments})
else:
mu, mu2, g1, g2 = self._stats(*args)
if g1 is None:
mu3 = None
else:
mu3 = g1*np.power(mu2,1.5) #(mu2**1.5) breaks down for nan and inf
default = valarray(shape(cond), self.badvalue)
output = []
# Use only entries that are valid in calculation
if any(cond):
goodargs = argsreduce(cond, *(args+(scale,loc)))
scale, loc, goodargs = goodargs[-2], goodargs[-1], goodargs[:-2]
if 'm' in moments:
if mu is None:
mu = self._munp(1.0,*goodargs)
out0 = default.copy()
place(out0,cond,mu*scale+loc)
output.append(out0)
if 'v' in moments:
if mu2 is None:
mu2p = self._munp(2.0,*goodargs)
if mu is None:
mu = self._munp(1.0,*goodargs)
mu2 = mu2p - mu*mu
if np.isinf(mu):
#if mean is inf then var is also inf
mu2 = np.inf
out0 = default.copy()
place(out0,cond,mu2*scale*scale)
output.append(out0)
if 's' in moments:
if g1 is None:
mu3p = self._munp(3.0,*goodargs)
if mu is None:
mu = self._munp(1.0,*goodargs)
if mu2 is None:
mu2p = self._munp(2.0,*goodargs)
mu2 = mu2p - mu*mu
mu3 = mu3p - 3*mu*mu2 - mu**3
g1 = mu3 / mu2**1.5
out0 = default.copy()
place(out0,cond,g1)
output.append(out0)
if 'k' in moments:
if g2 is None:
mu4p = self._munp(4.0,*goodargs)
if mu is None:
mu = self._munp(1.0,*goodargs)
if mu2 is None:
mu2p = self._munp(2.0,*goodargs)
mu2 = mu2p - mu*mu
if mu3 is None:
mu3p = self._munp(3.0,*goodargs)
mu3 = mu3p - 3*mu*mu2 - mu**3
mu4 = mu4p - 4*mu*mu3 - 6*mu*mu*mu2 - mu**4
g2 = mu4 / mu2**2.0 - 3.0
out0 = default.copy()
place(out0,cond,g2)
output.append(out0)
else: #no valid args
output = []
for _ in moments:
out0 = default.copy()
output.append(out0)
if len(output) == 1:
return output[0]
else:
return tuple(output)
def moment(self, n, *args, **kwds):
"""
n'th order non-central moment of distribution.
Parameters
----------
n : int, n>=1
Order of moment.
arg1, arg2, arg3,... : float
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
kwds : keyword arguments, optional
These can include "loc" and "scale", as well as other keyword
arguments relevant for a given distribution.
"""
loc = kwds.get('loc', 0)
scale = kwds.get('scale', 1)
if not (self._argcheck(*args) and (scale > 0)):
return nan
if (floor(n) != n):
raise ValueError("Moment must be an integer.")
if (n < 0): raise ValueError("Moment must be positive.")
mu, mu2, g1, g2 = None, None, None, None
if (n > 0) and (n < 5):
signature = inspect.getargspec(self._stats.im_func)
if (signature[2] is not None) or ('moments' in signature[0]):
mdict = {'moments':{1:'m',2:'v',3:'vs',4:'vk'}[n]}
else:
mdict = {}
mu, mu2, g1, g2 = self._stats(*args,**mdict)
val = _moment_from_stats(n, mu, mu2, g1, g2, self._munp, args)
# Convert to transformed X = L + S*Y
# so E[X^n] = E[(L+S*Y)^n] = L^n sum(comb(n,k)*(S/L)^k E[Y^k],k=0...n)
if loc == 0:
return scale**n * val
else:
result = 0
fac = float(scale) / float(loc)
for k in range(n):
valk = _moment_from_stats(k, mu, mu2, g1, g2, self._munp, args)
result += comb(n,k,exact=True)*(fac**k) * valk
result += fac**n * val
return result * loc**n
def _nnlf(self, x, *args):
return -sum(self._logpdf(x, *args),axis=0)
def nnlf(self, theta, x):
# - sum (log pdf(x, theta),axis=0)
# where theta are the parameters (including loc and scale)
#
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)):
return inf
else:
N = len(x)
return self._nnlf(x, *args) + N*log(scale)
# return starting point for fit (shape arguments + loc + scale)
def _fitstart(self, data, args=None):
if args is None:
args = (1.0,)*self.numargs
return args + self.fit_loc_scale(data, *args)
# Return the (possibly reduced) function to optimize in order to find MLE
# estimates for the .fit method
def _reduce_func(self, args, kwds):
args = list(args)
Nargs = len(args)
fixedn = []
index = range(Nargs)
names = ['f%d' % n for n in range(Nargs - 2)] + ['floc', 'fscale']
x0 = []
for n, key in zip(index, names):
if kwds.has_key(key):
fixedn.append(n)
args[n] = kwds[key]
else:
x0.append(args[n])
if len(fixedn) == 0:
func = self.nnlf
restore = None
else:
if len(fixedn) == len(index):
raise ValueError("All parameters fixed. There is nothing to optimize.")
def restore(args, theta):
# Replace with theta for all numbers not in fixedn
# This allows the non-fixed values to vary, but
# we still call self.nnlf with all parameters.
i = 0
for n in range(Nargs):
if n not in fixedn:
args[n] = theta[i]
i += 1
return args
def func(theta, x):
newtheta = restore(args[:], theta)
return self.nnlf(newtheta, x)
return x0, func, restore, args
def fit(self, data, *args, **kwds):
"""
Return MLEs for shape, location, and scale parameters from data.
MLE stands for Maximum Likelihood Estimate. Starting estimates for
the fit are given by input arguments; for any arguments not provided
with starting estimates, ``self._fitstart(data)`` is called to generate
such.
One can hold some parameters fixed to specific values by passing in
keyword arguments ``f0``, ``f1``, ..., ``fn`` (for shape parameters)
and ``floc`` and ``fscale`` (for location and scale parameters,
respectively).
Parameters
----------
data : array_like
Data to use in calculating the MLEs.
args : floats, optional
Starting value(s) for any shape-characterizing arguments (those not
provided will be determined by a call to ``_fitstart(data)``).
No default value.
kwds : floats, optional
Starting values for the location and scale parameters; no default.
Special keyword arguments are recognized as holding certain
parameters fixed:
f0...fn : hold respective shape parameters fixed.
floc : hold location parameter fixed to specified value.
fscale : hold scale parameter fixed to specified value.
optimizer : The optimizer to use. The optimizer must take func,
and starting position as the first two arguments,
plus args (for extra arguments to pass to the
function to be optimized) and disp=0 to suppress
output as keyword arguments.
Returns
-------
shape, loc, scale : tuple of floats
MLEs for any shape statistics, followed by those for location and
scale.
"""
Narg = len(args)
if Narg > self.numargs:
raise ValueError("Too many input arguments.")
start = [None]*2
if (Narg < self.numargs) or not (kwds.has_key('loc') and
kwds.has_key('scale')):
start = self._fitstart(data) # get distribution specific starting locations
args += start[Narg:-2]
loc = kwds.get('loc', start[-2])
scale = kwds.get('scale', start[-1])
args += (loc, scale)
x0, func, restore, args = self._reduce_func(args, kwds)
optimizer = kwds.get('optimizer', optimize.fmin)
# convert string to function in scipy.optimize
if not callable(optimizer) and isinstance(optimizer, (str, unicode)):
if not optimizer.startswith('fmin_'):
optimizer = "fmin_"+optimizer
if optimizer == 'fmin_':
optimizer = 'fmin'
try:
optimizer = getattr(optimize, optimizer)
except AttributeError:
raise ValueError("%s is not a valid optimizer" % optimizer)
vals = optimizer(func,x0,args=(ravel(data),),disp=0)
if restore is not None:
vals = restore(args, vals)
vals = tuple(vals)
return vals
def fit_loc_scale(self, data, *args):
"""
Estimate loc and scale parameters from data using 1st and 2nd moments.
Parameters
----------
data : array_like
Data to fit.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
Returns
-------
Lhat : float
Estimated location parameter for the data.
Shat : float
Estimated scale parameter for the data.
"""
mu, mu2 = self.stats(*args,**{'moments':'mv'})
tmp = asarray(data)
muhat = tmp.mean()
mu2hat = tmp.var()
Shat = sqrt(mu2hat / mu2)
Lhat = muhat - Shat*mu
return Lhat, Shat
@np.deprecate
def est_loc_scale(self, data, *args):
"""This function is deprecated, use self.fit_loc_scale(data) instead."""
return self.fit_loc_scale(data, *args)
def freeze(self,*args,**kwds):
"""Freeze the distribution for the given arguments.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution. Should include all
the non-optional arguments, may include ``loc`` and ``scale``.
Returns
-------
rv_frozen : rv_frozen instance
The frozen distribution.
"""
return rv_frozen(self,*args,**kwds)
def __call__(self, *args, **kwds):
return self.freeze(*args, **kwds)
def _entropy(self, *args):
def integ(x):
val = self._pdf(x, *args)
return val*log(val)
entr = -integrate.quad(integ,self.a,self.b)[0]
if not np.isnan(entr):
return entr
else: # try with different limits if integration problems
low,upp = self.ppf([0.001,0.999],*args)
if np.isinf(self.b):
upper = upp
else:
upper = self.b
if np.isinf(self.a):
lower = low
else:
lower = self.a
return -integrate.quad(integ,lower,upper)[0]
def entropy(self, *args, **kwds):
"""
Differential entropy of the RV.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
scale : array_like, optional
Scale parameter (default=1).
"""
loc,scale=map(kwds.get,['loc','scale'])
args, loc, scale = self._fix_loc_scale(args, loc, scale)
args = tuple(map(asarray,args))
cond0 = self._argcheck(*args) & (scale > 0) & (loc==loc)
output = zeros(shape(cond0),'d')
place(output,(1-cond0),self.badvalue)
goodargs = argsreduce(cond0, *args)
#I don't know when or why vecentropy got broken when numargs == 0
if self.numargs == 0:
place(output,cond0,self._entropy()+log(scale))
else:
place(output,cond0,self.vecentropy(*goodargs)+log(scale))
return output
def expect(self, func=None, args=(), loc=0, scale=1, lb=None, ub=None,
conditional=False, **kwds):
"""Calculate expected value of a function with respect to the distribution
Location and scale only tested on a few examples.
Parameters
----------
func : callable, optional
Function for which integral is calculated. Takes only one argument.
The default is the identity mapping f(x) = x.
args : tuple, optional
Argument (parameters) of the distribution.
lb, ub : scalar, optional
Lower and upper bound for integration. default is set to the support
of the distribution.
conditional : bool, optional
If True, the integral is corrected by the conditional probability
of the integration interval. The return value is the expectation
of the function, conditional on being in the given interval.
Default is False.
Additional keyword arguments are passed to the integration routine.
Returns
-------
expected value : float
Notes
-----
This function has not been checked for it's behavior when the integral is
not finite. The integration behavior is inherited from integrate.quad.
"""
lockwds = {'loc': loc,
'scale':scale}
if func is None:
def fun(x, *args):
return x*self.pdf(x, *args, **lockwds)
else:
def fun(x, *args):
return func(x)*self.pdf(x, *args, **lockwds)
if lb is None:
lb = loc + self.a * scale
if ub is None:
ub = loc + self.b * scale
if conditional:
invfac = (self.sf(lb, *args, **lockwds)
- self.sf(ub, *args, **lockwds))
else:
invfac = 1.0
kwds['args'] = args
return integrate.quad(fun, lb, ub, **kwds)[0] / invfac
_EULER = 0.577215664901532860606512090082402431042 # -special.psi(1)
_ZETA3 = 1.202056903159594285399738161511449990765 # special.zeta(3,1) Apery's constant
## 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', shapes="n")
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 = math.sqrt(2*pi)
_norm_pdf_logC = math.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_cdf(x):
return special.ndtr(x)
def _norm_logcdf(x):
return special.log_ndtr(x)
def _norm_ppf(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)
%(example)s
"""
def _rvs(self):
return mtrand.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_cdf(-x)
def _logsf(self, x):
return _norm_logcdf(-x)
def _ppf(self,q):
return _norm_ppf(q)
def _isf(self,q):
return -_norm_ppf(q)
def _stats(self):
return 0.0, 1.0, 0.0, 0.0
def _entropy(self):
return 0.5*(log(2*pi)+1)
norm = norm_gen(name='norm')
## Alpha distribution
##
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``.
%(example)s
"""
def _pdf(self, x, a):
return 1.0/(x**2)/special.ndtr(a)*_norm_pdf(a-1.0/x)
def _logpdf(self, x, a):
return -2*log(x) + _norm_logpdf(a-1.0/x) - log(special.ndtr(a))
def _cdf(self, x, a):
return special.ndtr(a-1.0/x) / special.ndtr(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', shapes='a')
## Anglit distribution
##
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``.
%(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')
## Arcsine distribution
##
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.
%(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):
#mup = 0.5, 3.0/8.0, 15.0/48.0, 35.0/128.0
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')
## Beta distribution
##
class beta_gen(rv_continuous):
"""A beta continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `beta` is::
beta.pdf(x, a, b) = gamma(a+b)/(gamma(a)*gamma(b)) * x**(a-1) *
(1-x)**(b-1),
for ``0 < x < 1``, ``a > 0``, ``b > 0``.
%(example)s
"""
def _rvs(self, a, b):
return mtrand.beta(a,b,self._size)
def _pdf(self, x, a, b):
Px = (1.0-x)**(b-1.0) * x**(a-1.0)
Px /= special.beta(a,b)
return Px
def _logpdf(self, x, a, b):
lPx = (b-1.0)*log(1.0-x) + (a-1.0)*log(x)
lPx -= log(special.beta(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))
def fit(self, data, *args, **kwds):
floc = kwds.get('floc', None)
fscale = kwds.get('fscale', None)
if floc is not None and fscale is not None:
# special case
data = (ravel(data)-floc)/fscale
xbar = data.mean()
v = data.var(ddof=0)
fac = xbar*(1-xbar)/v - 1
a = xbar * fac
b = (1-xbar) * fac
return a, b, floc, fscale
else: # do general fit
return super(beta_gen, self).fit(data, *args, **kwds)
beta = beta_gen(a=0.0, b=1.0, name='beta', shapes='a, b')
## Beta Prime
class betaprime_gen(rv_continuous):
"""A beta prima continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `betaprime` is::
betaprime.pdf(x, a, b) =
gamma(a+b) / (gamma(a)*gamma(b)) * x**(a-1) * (1-x)**(-a-b)
for ``x > 0``, ``a > 0``, ``b > 0``.
%(example)s
"""
def _rvs(self, a, b):
u1 = gamma.rvs(a,size=self._size)
u2 = gamma.rvs(b,size=self._size)
return (u1 / u2)
def _pdf(self, x, a, b):
return 1.0/special.beta(a,b)*x**(a-1.0)/(1+x)**(a+b)
def _logpdf(self, x, a, b):
return (a-1.0)*log(x) - (a+b)*log(1+x) - log(special.beta(a,b))
def _cdf_skip(self, x, a, b):
# remove for now: special.hyp2f1 is incorrect for large a
x = where(x==1.0, 1.0-1e-6,x)
return pow(x,a)*special.hyp2f1(a+b,a,1+a,-x)/a/special.beta(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, b=500.0, name='betaprime', shapes='a, b')
## Bradford
##
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)``.
%(example)s
"""
def _pdf(self, x, c):
return c / (c*x + 1.0) / log(1.0+c)
def _cdf(self, x, c):
return log(1.0+c*x) / log(c+1.0)
def _ppf(self, q, c):
return ((1.0+c)**q-1)/c
def _stats(self, c, moments='mv'):
k = log(1.0+c)
mu = (c-k)/(c*k)
mu2 = ((c+2.0)*k-2.0*c)/(2*c*k*k)
g1 = None
g2 = None
if 's' in moments:
g1 = 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 = log(1+c)
return k/2.0 - log(c/k)
bradford = bradford_gen(a=0.0, b=1.0, name='bradford', shapes='c')
## Burr
# burr with d=1 is called the fisk distribution
class burr_gen(rv_continuous):
"""A Burr continuous random variable.
%(before_notes)s
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``.
%(example)s
"""
def _pdf(self, x, c, d):
return c*d*(x**(-c-1.0))*((1+x**(-c*1.0))**(-d-1.0))
def _cdf(self, x, c, d):
return (1+x**(-c*1.0))**(-d**1.0)
def _ppf(self, q, c, d):
return (q**(-1.0/d)-1)**(-1.0/c)
def _stats(self, c, d, moments='mv'):
g2c, g2cd = gam(1-2.0/c), gam(2.0/c+d)
g1c, g1cd = gam(1-1.0/c), gam(1.0/c+d)
gd = gam(d)
k = gd*g2c*g2cd - g1c**2 * g1cd**2
mu = g1c*g1cd / gd
mu2 = k / gd**2.0
g1, g2 = None, None
g3c, g3cd = None, None
if 's' in moments:
g3c, g3cd = gam(1-3.0/c), gam(3.0/c+d)
g1 = 2*g1c**3 * g1cd**3 + gd*gd*g3c*g3cd - 3*gd*g2c*g1c*g1cd*g2cd
g1 /= sqrt(k**3)
if 'k' in moments:
if g3c is None:
g3c = gam(1-3.0/c)
if g3cd is None:
g3cd = gam(3.0/c+d)
g4c, g4cd = gam(1-4.0/c), gam(4.0/c+d)
g2 = 6*gd*g2c*g2cd * g1c**2 * g1cd**2 + gd**3 * g4c*g4cd
g2 -= 3*g1c**4 * g1cd**4 -4*gd**2*g3c*g1c*g1cd*g3cd
return mu, mu2, g1, g2
burr = burr_gen(a=0.0, name='burr', shapes="c, d")
# Fisk distribution
# burr is a generalization
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``.
%(before_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 _stats(self, c):
return burr_gen._stats(self, c, 1.0)
def _entropy(self, c):
return 2 - log(c)
fisk = fisk_gen(a=0.0, name='fisk', shapes='c')
## Cauchy
# 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))
%(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 inf, inf, nan, nan
def _entropy(self):
return log(4*pi)
def _fitstart(data, args=None):
return (0, 1)
cauchy = cauchy_gen(name='cauchy')
## Chi
## (positive square-root of chi-square)
## chi(1, loc, scale) = halfnormal
## chi(2, 0, scale) = Rayleigh
## chi(3, 0, scale) = MaxWell
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``.
%(example)s
"""
def _rvs(self, df):
return sqrt(chi2.rvs(df,size=self._size))
def _pdf(self, x, df):
return x**(df-1.)*exp(-x*x*0.5)/(2.0)**(df*0.5-1)/gam(df*0.5)
def _cdf(self, x, df):
return special.gammainc(df*0.5,0.5*x*x)
def _ppf(self, q, df):
return sqrt(2*special.gammaincinv(df*0.5,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(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
chi = chi_gen(a=0.0, name='chi', shapes='df')
## 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)
%(example)s
"""
def _rvs(self, df):
return mtrand.chisquare(df,self._size)
def _pdf(self, x, df):
return exp(self._logpdf(x, df))
def _logpdf(self, x, df):
#term1 = (df/2.-1)*log(x)
#term1[(df==2)*(x==0)] = 0
#avoid 0*log(0)==nan
return (df/2.-1)*log(x+1e-300) - x/2. - gamln(df/2.) - (log(2)*df)/2.
## Px = x**(df/2.0-1)*exp(-x/2.0)
## Px /= special.gamma(df/2.0)* 2**(df/2.0)
## return log(Px)
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
chi2 = chi2_gen(a=0.0, name='chi2', shapes='df')
## Cosine (Approximation to the Normal)
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``.
%(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')
## Double Gamma distribution
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``.
%(example)s
"""
def _rvs(self, a):
u = random(size=self._size)
return (gamma.rvs(a,size=self._size)*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 (a-1.0)*log(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>0,0.5-0.5*fac,0.5+0.5*fac)
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', shapes='a')
## Double Weibull distribution
##
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)
%(example)s
"""
def _rvs(self, c):
u = random(size=self._size)
return weibull_min.rvs(c, size=self._size)*(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) + (c-1.0)*log(ax) - ax**c
def _cdf(self, x, c):
Cx1 = 0.5*exp(-abs(x)**c)
return where(x > 0, 1-Cx1, Cx1)
def _ppf_skip(self, q, c):
fac = where(q<=0.5,2*q,2*q-1)
fac = pow(asarray(log(1.0/fac)),1.0/c)
return where(q>0.5,fac,-fac)
def _stats(self, c):
var = gam(1+2.0/c)
return 0.0, var, 0.0, gam(1+4.0/c)/var
dweibull = dweibull_gen(name='dweibull', shapes='c')
## ERLANG
##
## Special case of the Gamma distribution with shape parameter an integer.
##
class erlang_gen(rv_continuous):
"""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. Refer to
the ``gamma`` distribution for further examples.
"""
def _rvs(self, a):
return gamma.rvs(a, size=self._size)
def _arg_check(self, a):
return (a > 0) & (floor(a)==a)
def _pdf(self, x, a):
Px = (x)**(a-1.0)*exp(-x)/special.gamma(a)
return Px
def _logpdf(self, x, a):
return (a-1.0)*log(x) - x - gamln(a)
def _cdf(self, x, a):
return special.gdtr(1.0,a,x)
def _sf(self, x, a):
return special.gdtrc(1.0,a,x)
def _ppf(self, q, a):
return special.gdtrix(1.0, a, q)
def _stats(self, a):
a = a*1.0
return a, a, 2/sqrt(a), 6/a
def _entropy(self, a):
return special.psi(a)*(1-a) + 1 + gamln(a)
erlang = erlang_gen(a=0.0, name='erlang', shapes='a')
## Exponential (gamma distributed with a=1.0, loc=loc and scale=scale)
## scale == 1.0 / lambda
class expon_gen(rv_continuous):
"""An exponential continuous random variable.
%(before_notes)s
Notes
-----
The probability density function for `expon` is::
expon.pdf(x) = lambda * exp(- lambda*x)
for ``x >= 0``.
The scale parameter is equal to ``scale = 1.0 / lambda``.
`expon` does not have shape parameters.
%(example)s
"""
def _rvs(self):
return mtrand.standard_exponential(self._size)
def _pdf(self, x):
return exp(-x)
def _logpdf(self, x):
return -x
def _cdf(self, x):
return -expm1(-x)
def _ppf(self, q):
return -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')
## Exponentiated Weibull
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``.
%(example)s
"""
def _pdf(self, x, a, c):
exc = exp(-x**c)
return a*c*(1-exc)**asarray(a-1) * exc * x**(c-1)
def _logpdf(self, x, a, c):
exc = exp(-x**c)
return log(a) + log(c) + (a-1.)*log(1-exc) - x**c + (c-1.0)*log(x)
def _cdf(self, x, a, c):
exm1c = -expm1(-x**c)
return (exm1c)**a
def _ppf(self, q, a, c):
return (-log1p(-q**(1.0/a)))**asarray(1.0/c)
exponweib = exponweib_gen(a=0.0, name='exponweib', shapes="a, c")
## Exponential Power
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``.
%(example)s
"""
def _pdf(self, x, b):
xbm1 = x**(b-1.0)
xb = xbm1 * x
return exp(1)*b*xbm1 * exp(xb - exp(xb))
def _logpdf(self, x, b):
xb = x**(b-1.0)*x
return 1 + log(b) + (b-1.0)*log(x) + xb - exp(xb)
def _cdf(self, x, b):
return -expm1(-expm1(x**b))
def _sf(self, x, b):
return exp(-expm1(x**b))
def _isf(self, x, b):
return (log1p(-log(x)))**(1./b)
def _ppf(self, q, b):
return pow(log1p(-log1p(-q)), 1.0/b)
exponpow = exponpow_gen(a=0.0, name='exponpow', shapes='b')
## Fatigue-Life (Birnbaum-Sanders)
class fatiguelife_gen(rv_continuous):
"""A fatigue-life (Birnbaum-Sanders) 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``.
%(example)s
"""
def _rvs(self, c):
z = norm.rvs(size=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 (x+1)/asarray(2*c*sqrt(2*pi*x**3))*exp(-(x-1)**2/asarray((2.0*x*c**2)))
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 special.ndtr(1.0/c*(sqrt(x)-1.0/asarray(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):
c2 = c*c
mu = c2 / 2.0 + 1
den = 5*c2 + 4
mu2 = c2*den /4.0
g1 = 4*c*sqrt(11*c2+6.0)/den**1.5
g2 = 6*c2*(93*c2+41.0) / den**2.0
return mu, mu2, g1, g2
fatiguelife = fatiguelife_gen(a=0.0, name='fatiguelife', shapes='c')
## Folded Cauchy
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``.
%(example)s
"""
def _rvs(self, c):
return abs(cauchy.rvs(loc=c,size=self._size))
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', shapes='c')
## F
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``.
%(example)s
"""
def _rvs(self, dfn, dfd):
return mtrand.f(dfn, dfd, self._size)
def _pdf(self, x, dfn, dfd):
# n = asarray(1.0*dfn)
# m = asarray(1.0*dfd)
# Px = m**(m/2) * n**(n/2) * x**(n/2-1)
# Px /= (m+n*x)**((n+m)/2)*special.beta(n/2,m/2)
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):
v2 = asarray(dfd*1.0)
v1 = asarray(dfn*1.0)
mu = where (v2 > 2, v2 / asarray(v2 - 2), inf)
mu2 = 2*v2*v2*(v2+v1-2)/(v1*(v2-2)**2 * (v2-4))
mu2 = where(v2 > 4, mu2, inf)
g1 = 2*(v2+2*v1-2)/(v2-6)*sqrt((2*v2-4)/(v1*(v2+v1-2)))
g1 = where(v2 > 6, g1, nan)
g2 = 3/(2*v2-16)*(8+g1*g1*(v2-6))
g2 = where(v2 > 8, g2, nan)
return mu, mu2, g1, g2
f = f_gen(a=0.0, name='f', shapes="dfn, dfd")
## 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``.
%(example)s
"""
def _rvs(self, c):
return abs(norm.rvs(loc=c,size=self._size))
def _pdf(self, x, c):
return sqrt(2.0/pi)*cosh(c*x)*exp(-(x*x+c*c)/2.0)
def _cdf(self, x, c,):
return special.ndtr(x-c) + special.ndtr(x+c) - 1.0
def _stats(self, c):
fac = special.erf(c/sqrt(2))
mu = sqrt(2.0/pi)*exp(-0.5*c*c)+c*fac
mu2 = c*c + 1 - mu*mu
c2 = c*c
g1 = sqrt(2/pi)*exp(-1.5*c2)*(4-pi*exp(c2)*(2*c2+1.0))
g1 += 2*c*fac*(6*exp(-c2) + 3*sqrt(2*pi)*c*exp(-c2/2.0)*fac + \
pi*c*(fac*fac-1))
g1 /= pi*mu2**1.5
g2 = c2*c2+6*c2+3+6*(c2+1)*mu*mu - 3*mu**4
g2 -= 4*exp(-c2/2.0)*mu*(sqrt(2.0/pi)*(c2+2)+c*(c2+3)*exp(c2/2.0)*fac)
g2 /= mu2**2.0
return mu, mu2, g1, g2
foldnorm = foldnorm_gen(a=0.0, name='foldnorm', shapes='c')
## 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``.
%(example)s
"""
def _pdf(self, x, c):
return c*pow(x,c-1)*exp(-pow(x,c))
def _logpdf(self, x, c):
return log(c) + (c-1)*log(x) - pow(x,c)
def _cdf(self, x, c):
return -expm1(-pow(x,c))
def _ppf(self, q, c):
return pow(-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
frechet_r = frechet_r_gen(a=0.0, name='frechet_r', shapes='c')
weibull_min = frechet_r_gen(a=0.0, name='weibull_min', shapes='c')
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``.
%(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
frechet_l = frechet_l_gen(b=0.0, name='frechet_l', shapes='c')
weibull_max = frechet_l_gen(b=0.0, name='weibull_max', shapes='c')
## Generalized Logistic
##
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``.
%(example)s
"""
def _pdf(self, x, c):
Px = c*exp(-x)/(1+exp(-x))**(c+1.0)
return Px
def _logpdf(self, x, c):
return log(c) - x - (c+1.0)*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 /= 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', shapes='c')
## Generalized Pareto
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)
for ``c != 0``, and for ``x >= 0`` for all c,
and ``x < 1/abs(c)`` for ``c < 0``.
%(example)s
"""
def _argcheck(self, c):
c = asarray(c)
self.b = where(c < 0, 1.0/abs(c), inf)
return where(c==0, 0, 1)
def _pdf(self, x, c):
Px = pow(1+c*x,asarray(-1.0-1.0/c))
return Px
def _logpdf(self, x, c):
return (-1.0-1.0/c) * np.log1p(c*x)
def _cdf(self, x, c):
return 1.0 - pow(1+c*x,asarray(-1.0/c))
def _ppf(self, q, c):
vals = 1.0/c * (pow(1-q, -c)-1)
return vals
def _munp(self, n, c):
k = arange(0,n+1)
val = (-1.0/c)**n * sum(comb(n,k)*(-1)**k / (1.0-c*k),axis=0)
return where(c*n < 1, val, inf)
def _entropy(self, c):
if (c > 0):
return 1+c
else:
self.b = -1.0 / c
return rv_continuous._entropy(self, c)
genpareto = genpareto_gen(a=0.0, name='genpareto', shapes='c')
## Generalized Exponential
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``.
References
----------
H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential
Distribution", Journal of the American Statistical Association, 1993.
N. Balakrishnan, "The Exponential Distribution: Theory, Methods and
Applications", Asit P. Basu.
%(example)s
"""
def _pdf(self, x, a, b, c):
return (a+b*(-expm1(-c*x)))*exp((-a-b)*x+b*(-expm1(-c*x))/c)
def _cdf(self, x, a, b, c):
return -expm1((-a-b)*x + b*(-expm1(-c*x))/c)
def _logpdf(self, x, a, b, c):
return np.log(a+b*(-expm1(-c*x))) + (-a-b)*x+b*(-expm1(-c*x))/c
genexpon = genexpon_gen(a=0.0, name='genexpon', shapes='a, b, c')
## Generalized Extreme Value
## c=0 is just gumbel distribution.
## This version does now accept c==0
## Use gumbel_r for c==0
# new version by Per Brodtkorb, see ticket:767
# also works for c==0, special case is gumbel_r
# increased precision for small c
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
%(example)s
"""
def _argcheck(self, c):
min = np.minimum
max = np.maximum
sml = floatinfo.machar.xmin
#self.b = where(c > 0, 1.0 / c,inf)
#self.a = where(c < 0, 1.0 / c, -inf)
self.b = where(c > 0, 1.0 / max(c, sml),inf)
self.a = where(c < 0, 1.0 / min(c,-sml), -inf)
return where(abs(c)==inf, 0, 1) #True #(c!=0)
def _pdf(self, x, c):
## ex2 = 1-c*x
## pex2 = pow(ex2,1.0/c)
## p2 = exp(-pex2)*pex2/ex2
## return p2
cx = c*x
logex2 = where((c==0)*(x==x),0.0,log1p(-cx))
logpex2 = where((c==0)*(x==x),-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) # logpdf(c==1 & x==1) = 0; % 0^0 situation
return exp(logpdf)
def _cdf(self, x, c):
#return exp(-pow(1-c*x,1.0/c))
loglogcdf = where((c==0)*(x==x),-x,log1p(-c*x)/c)
return exp(-exp(loglogcdf))
def _ppf(self, q, c):
#return 1.0/c*(1.-(-log(q))**c)
x = -log(-log(q))
return where((c==0)*(x==x),x,-expm1(-c*x)/c)
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, expm1(gamln(2.0*c+1.0)-2*gamln(c+1.0))/c**2.0);
eps = 1e-14
gamk = where(abs(c)<eps,-_EULER,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)
#% The kurtosis is:
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 _munp(self, n, c):
k = arange(0,n+1)
vals = 1.0/c**n * sum(comb(n,k) * (-1)**k * special.gamma(c*k + 1),axis=0)
return where(c*n > -1, vals, inf)
genextreme = genextreme_gen(name='genextreme', shapes='c')
## 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) = lambda**a * x**(a-1) * exp(-lambda*x) / gamma(a)
for ``x >= 0``, ``a > 0``. Here ``gamma(a)`` refers to the gamma function.
The scale parameter is equal to ``scale = 1.0 / lambda``.
`gamma` has a shape parameter `a` which needs to be set explicitly. For instance:
>>> from scipy.stats import gamma
>>> rv = gamma(3., loc = 0., scale = 2.)
produces a frozen form of `gamma` with shape ``a = 3.``, ``loc =
0.`` and ``lambda = 1./scale = 1./2.``.
When ``a`` is an integer, `gamma` reduces to the Erlang
distribution, and when ``a=1`` to the exponential distribution.
%(example)s
"""
def _rvs(self, a):
return mtrand.standard_gamma(a, self._size)
def _pdf(self, x, a):
return exp(self._logpdf(x, a))
def _logpdf(self, x, a):
return (a-1)*log(x) - x - gamln(a)
def _cdf(self, x, a):
return special.gammainc(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) + 1 + gamln(a)
def _fitstart(self, data):
a = 4 / _skew(data)**2
return super(gamma_gen, self)._fitstart(data, args=(a,))
def fit(self, data, *args, **kwds):
floc = kwds.get('floc', None)
if floc == 0:
xbar = ravel(data).mean()
logx_bar = ravel(log(data)).mean()
s = log(xbar) - logx_bar
def func(a):
return log(a) - special.digamma(a) - s
aest = (3-s + math.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)
scale = xbar / a
return a, floc, scale
else:
return super(gamma_gen, self).fit(data, *args, **kwds)
gamma = gamma_gen(a=0.0, name='gamma', shapes='a')
# Generalized Gamma
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``.
%(example)s
"""
def _argcheck(self, a, c):
return (a > 0) & (c != 0)
def _pdf(self, x, a, c):
return abs(c)* exp((c*a-1)*log(x)-x**c- gamln(a))
def _cdf(self, x, a, c):
val = special.gammainc(a,x**c)
cond = c + 0*val
return where(cond>0,val,1-val)
def _ppf(self, q, a, c):
val1 = special.gammaincinv(a,q)
val2 = special.gammaincinv(a,1.0-q)
ic = 1.0/c
cond = c+0*val1
return where(cond > 0,val1**ic,val2**ic)
def _munp(self, n, a, c):
return special.gamma(a+n*1.0/c) / special.gamma(a)
def _entropy(self, a,c):
val = special.psi(a)
return a*(1-val) + 1.0/c*val + gamln(a)-log(abs(c))
gengamma = gengamma_gen(a=0.0, name='gengamma', shapes="a, c")
## Generalized Half-Logistic
##
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``.
%(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)