1815 lines
53 KiB
Python
1815 lines
53 KiB
Python
#
|
|
# Author: Travis Oliphant 2002-2011 with contributions from
|
|
# SciPy Developers 2004-2011
|
|
#
|
|
from functools import partial
|
|
import warnings
|
|
|
|
from scipy import special
|
|
from scipy.special import entr, logsumexp, betaln, gammaln as gamln, zeta
|
|
from scipy._lib._util import _lazywhere, rng_integers
|
|
from scipy.interpolate import interp1d
|
|
|
|
from numpy import floor, ceil, log, exp, sqrt, log1p, expm1, tanh, cosh, sinh
|
|
|
|
import numpy as np
|
|
|
|
from ._distn_infrastructure import (rv_discrete, get_distribution_names,
|
|
_check_shape, _ShapeInfo)
|
|
import scipy.stats._boost as _boost
|
|
from ._biasedurn import (_PyFishersNCHypergeometric,
|
|
_PyWalleniusNCHypergeometric,
|
|
_PyStochasticLib3)
|
|
|
|
|
|
def _isintegral(x):
|
|
return x == np.round(x)
|
|
|
|
|
|
class binom_gen(rv_discrete):
|
|
r"""A binomial discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `binom` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = \binom{n}{k} p^k (1-p)^{n-k}
|
|
|
|
for :math:`k \in \{0, 1, \dots, n\}`, :math:`0 \leq p \leq 1`
|
|
|
|
`binom` takes :math:`n` and :math:`p` as shape parameters,
|
|
where :math:`p` is the probability of a single success
|
|
and :math:`1-p` is the probability of a single failure.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
See Also
|
|
--------
|
|
hypergeom, nbinom, nhypergeom
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("n", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("p", False, (0, 1), (True, True))]
|
|
|
|
def _rvs(self, n, p, size=None, random_state=None):
|
|
return random_state.binomial(n, p, size)
|
|
|
|
def _argcheck(self, n, p):
|
|
return (n >= 0) & _isintegral(n) & (p >= 0) & (p <= 1)
|
|
|
|
def _get_support(self, n, p):
|
|
return self.a, n
|
|
|
|
def _logpmf(self, x, n, p):
|
|
k = floor(x)
|
|
combiln = (gamln(n+1) - (gamln(k+1) + gamln(n-k+1)))
|
|
return combiln + special.xlogy(k, p) + special.xlog1py(n-k, -p)
|
|
|
|
def _pmf(self, x, n, p):
|
|
# binom.pmf(k) = choose(n, k) * p**k * (1-p)**(n-k)
|
|
return _boost._binom_pdf(x, n, p)
|
|
|
|
def _cdf(self, x, n, p):
|
|
k = floor(x)
|
|
return _boost._binom_cdf(k, n, p)
|
|
|
|
def _sf(self, x, n, p):
|
|
k = floor(x)
|
|
return _boost._binom_sf(k, n, p)
|
|
|
|
def _isf(self, x, n, p):
|
|
return _boost._binom_isf(x, n, p)
|
|
|
|
def _ppf(self, q, n, p):
|
|
return _boost._binom_ppf(q, n, p)
|
|
|
|
def _stats(self, n, p, moments='mv'):
|
|
mu = _boost._binom_mean(n, p)
|
|
var = _boost._binom_variance(n, p)
|
|
g1, g2 = None, None
|
|
if 's' in moments:
|
|
g1 = _boost._binom_skewness(n, p)
|
|
if 'k' in moments:
|
|
g2 = _boost._binom_kurtosis_excess(n, p)
|
|
return mu, var, g1, g2
|
|
|
|
def _entropy(self, n, p):
|
|
k = np.r_[0:n + 1]
|
|
vals = self._pmf(k, n, p)
|
|
return np.sum(entr(vals), axis=0)
|
|
|
|
|
|
binom = binom_gen(name='binom')
|
|
|
|
|
|
class bernoulli_gen(binom_gen):
|
|
r"""A Bernoulli discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `bernoulli` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = \begin{cases}1-p &\text{if } k = 0\\
|
|
p &\text{if } k = 1\end{cases}
|
|
|
|
for :math:`k` in :math:`\{0, 1\}`, :math:`0 \leq p \leq 1`
|
|
|
|
`bernoulli` takes :math:`p` as shape parameter,
|
|
where :math:`p` is the probability of a single success
|
|
and :math:`1-p` is the probability of a single failure.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("p", False, (0, 1), (True, True))]
|
|
|
|
def _rvs(self, p, size=None, random_state=None):
|
|
return binom_gen._rvs(self, 1, p, size=size, random_state=random_state)
|
|
|
|
def _argcheck(self, p):
|
|
return (p >= 0) & (p <= 1)
|
|
|
|
def _get_support(self, p):
|
|
# Overrides binom_gen._get_support!x
|
|
return self.a, self.b
|
|
|
|
def _logpmf(self, x, p):
|
|
return binom._logpmf(x, 1, p)
|
|
|
|
def _pmf(self, x, p):
|
|
# bernoulli.pmf(k) = 1-p if k = 0
|
|
# = p if k = 1
|
|
return binom._pmf(x, 1, p)
|
|
|
|
def _cdf(self, x, p):
|
|
return binom._cdf(x, 1, p)
|
|
|
|
def _sf(self, x, p):
|
|
return binom._sf(x, 1, p)
|
|
|
|
def _isf(self, x, p):
|
|
return binom._isf(x, 1, p)
|
|
|
|
def _ppf(self, q, p):
|
|
return binom._ppf(q, 1, p)
|
|
|
|
def _stats(self, p):
|
|
return binom._stats(1, p)
|
|
|
|
def _entropy(self, p):
|
|
return entr(p) + entr(1-p)
|
|
|
|
|
|
bernoulli = bernoulli_gen(b=1, name='bernoulli')
|
|
|
|
|
|
class betabinom_gen(rv_discrete):
|
|
r"""A beta-binomial discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The beta-binomial distribution is a binomial distribution with a
|
|
probability of success `p` that follows a beta distribution.
|
|
|
|
The probability mass function for `betabinom` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = \binom{n}{k} \frac{B(k + a, n - k + b)}{B(a, b)}
|
|
|
|
for :math:`k \in \{0, 1, \dots, n\}`, :math:`n \geq 0`, :math:`a > 0`,
|
|
:math:`b > 0`, where :math:`B(a, b)` is the beta function.
|
|
|
|
`betabinom` takes :math:`n`, :math:`a`, and :math:`b` as shape parameters.
|
|
|
|
References
|
|
----------
|
|
.. [1] https://en.wikipedia.org/wiki/Beta-binomial_distribution
|
|
|
|
%(after_notes)s
|
|
|
|
.. versionadded:: 1.4.0
|
|
|
|
See Also
|
|
--------
|
|
beta, binom
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("n", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("a", False, (0, np.inf), (False, False)),
|
|
_ShapeInfo("b", False, (0, np.inf), (False, False))]
|
|
|
|
def _rvs(self, n, a, b, size=None, random_state=None):
|
|
p = random_state.beta(a, b, size)
|
|
return random_state.binomial(n, p, size)
|
|
|
|
def _get_support(self, n, a, b):
|
|
return 0, n
|
|
|
|
def _argcheck(self, n, a, b):
|
|
return (n >= 0) & _isintegral(n) & (a > 0) & (b > 0)
|
|
|
|
def _logpmf(self, x, n, a, b):
|
|
k = floor(x)
|
|
combiln = -log(n + 1) - betaln(n - k + 1, k + 1)
|
|
return combiln + betaln(k + a, n - k + b) - betaln(a, b)
|
|
|
|
def _pmf(self, x, n, a, b):
|
|
return exp(self._logpmf(x, n, a, b))
|
|
|
|
def _stats(self, n, a, b, moments='mv'):
|
|
e_p = a / (a + b)
|
|
e_q = 1 - e_p
|
|
mu = n * e_p
|
|
var = n * (a + b + n) * e_p * e_q / (a + b + 1)
|
|
g1, g2 = None, None
|
|
if 's' in moments:
|
|
g1 = 1.0 / sqrt(var)
|
|
g1 *= (a + b + 2 * n) * (b - a)
|
|
g1 /= (a + b + 2) * (a + b)
|
|
if 'k' in moments:
|
|
g2 = a + b
|
|
g2 *= (a + b - 1 + 6 * n)
|
|
g2 += 3 * a * b * (n - 2)
|
|
g2 += 6 * n ** 2
|
|
g2 -= 3 * e_p * b * n * (6 - n)
|
|
g2 -= 18 * e_p * e_q * n ** 2
|
|
g2 *= (a + b) ** 2 * (1 + a + b)
|
|
g2 /= (n * a * b * (a + b + 2) * (a + b + 3) * (a + b + n))
|
|
g2 -= 3
|
|
return mu, var, g1, g2
|
|
|
|
|
|
betabinom = betabinom_gen(name='betabinom')
|
|
|
|
|
|
class nbinom_gen(rv_discrete):
|
|
r"""A negative binomial discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
Negative binomial distribution describes a sequence of i.i.d. Bernoulli
|
|
trials, repeated until a predefined, non-random number of successes occurs.
|
|
|
|
The probability mass function of the number of failures for `nbinom` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k
|
|
|
|
for :math:`k \ge 0`, :math:`0 < p \leq 1`
|
|
|
|
`nbinom` takes :math:`n` and :math:`p` as shape parameters where :math:`n`
|
|
is the number of successes, :math:`p` is the probability of a single
|
|
success, and :math:`1-p` is the probability of a single failure.
|
|
|
|
Another common parameterization of the negative binomial distribution is
|
|
in terms of the mean number of failures :math:`\mu` to achieve :math:`n`
|
|
successes. The mean :math:`\mu` is related to the probability of success
|
|
as
|
|
|
|
.. math::
|
|
|
|
p = \frac{n}{n + \mu}
|
|
|
|
The number of successes :math:`n` may also be specified in terms of a
|
|
"dispersion", "heterogeneity", or "aggregation" parameter :math:`\alpha`,
|
|
which relates the mean :math:`\mu` to the variance :math:`\sigma^2`,
|
|
e.g. :math:`\sigma^2 = \mu + \alpha \mu^2`. Regardless of the convention
|
|
used for :math:`\alpha`,
|
|
|
|
.. math::
|
|
|
|
p &= \frac{\mu}{\sigma^2} \\
|
|
n &= \frac{\mu^2}{\sigma^2 - \mu}
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
See Also
|
|
--------
|
|
hypergeom, binom, nhypergeom
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("n", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("p", False, (0, 1), (True, True))]
|
|
|
|
def _rvs(self, n, p, size=None, random_state=None):
|
|
return random_state.negative_binomial(n, p, size)
|
|
|
|
def _argcheck(self, n, p):
|
|
return (n > 0) & (p > 0) & (p <= 1)
|
|
|
|
def _pmf(self, x, n, p):
|
|
# nbinom.pmf(k) = choose(k+n-1, n-1) * p**n * (1-p)**k
|
|
return _boost._nbinom_pdf(x, n, p)
|
|
|
|
def _logpmf(self, x, n, p):
|
|
coeff = gamln(n+x) - gamln(x+1) - gamln(n)
|
|
return coeff + n*log(p) + special.xlog1py(x, -p)
|
|
|
|
def _cdf(self, x, n, p):
|
|
k = floor(x)
|
|
return _boost._nbinom_cdf(k, n, p)
|
|
|
|
def _logcdf(self, x, n, p):
|
|
k = floor(x)
|
|
cdf = self._cdf(k, n, p)
|
|
cond = cdf > 0.5
|
|
|
|
def f1(k, n, p):
|
|
return np.log1p(-special.betainc(k + 1, n, 1 - p))
|
|
|
|
# do calc in place
|
|
logcdf = cdf
|
|
with np.errstate(divide='ignore'):
|
|
logcdf[cond] = f1(k[cond], n[cond], p[cond])
|
|
logcdf[~cond] = np.log(cdf[~cond])
|
|
return logcdf
|
|
|
|
def _sf(self, x, n, p):
|
|
k = floor(x)
|
|
return _boost._nbinom_sf(k, n, p)
|
|
|
|
def _isf(self, x, n, p):
|
|
with warnings.catch_warnings():
|
|
# See gh-14901
|
|
message = "overflow encountered in _nbinom_isf"
|
|
warnings.filterwarnings('ignore', message=message)
|
|
return _boost._nbinom_isf(x, n, p)
|
|
|
|
def _ppf(self, q, n, p):
|
|
with warnings.catch_warnings():
|
|
message = "overflow encountered in _nbinom_ppf"
|
|
warnings.filterwarnings('ignore', message=message)
|
|
return _boost._nbinom_ppf(q, n, p)
|
|
|
|
def _stats(self, n, p):
|
|
return (
|
|
_boost._nbinom_mean(n, p),
|
|
_boost._nbinom_variance(n, p),
|
|
_boost._nbinom_skewness(n, p),
|
|
_boost._nbinom_kurtosis_excess(n, p),
|
|
)
|
|
|
|
|
|
nbinom = nbinom_gen(name='nbinom')
|
|
|
|
|
|
class geom_gen(rv_discrete):
|
|
r"""A geometric discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `geom` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = (1-p)^{k-1} p
|
|
|
|
for :math:`k \ge 1`, :math:`0 < p \leq 1`
|
|
|
|
`geom` takes :math:`p` as shape parameter,
|
|
where :math:`p` is the probability of a single success
|
|
and :math:`1-p` is the probability of a single failure.
|
|
|
|
%(after_notes)s
|
|
|
|
See Also
|
|
--------
|
|
planck
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("p", False, (0, 1), (True, True))]
|
|
|
|
def _rvs(self, p, size=None, random_state=None):
|
|
return random_state.geometric(p, size=size)
|
|
|
|
def _argcheck(self, p):
|
|
return (p <= 1) & (p > 0)
|
|
|
|
def _pmf(self, k, p):
|
|
return np.power(1-p, k-1) * p
|
|
|
|
def _logpmf(self, k, p):
|
|
return special.xlog1py(k - 1, -p) + log(p)
|
|
|
|
def _cdf(self, x, p):
|
|
k = floor(x)
|
|
return -expm1(log1p(-p)*k)
|
|
|
|
def _sf(self, x, p):
|
|
return np.exp(self._logsf(x, p))
|
|
|
|
def _logsf(self, x, p):
|
|
k = floor(x)
|
|
return k*log1p(-p)
|
|
|
|
def _ppf(self, q, p):
|
|
vals = ceil(log1p(-q) / log1p(-p))
|
|
temp = self._cdf(vals-1, p)
|
|
return np.where((temp >= q) & (vals > 0), vals-1, vals)
|
|
|
|
def _stats(self, p):
|
|
mu = 1.0/p
|
|
qr = 1.0-p
|
|
var = qr / p / p
|
|
g1 = (2.0-p) / sqrt(qr)
|
|
g2 = np.polyval([1, -6, 6], p)/(1.0-p)
|
|
return mu, var, g1, g2
|
|
|
|
|
|
geom = geom_gen(a=1, name='geom', longname="A geometric")
|
|
|
|
|
|
class hypergeom_gen(rv_discrete):
|
|
r"""A hypergeometric discrete random variable.
|
|
|
|
The hypergeometric distribution models drawing objects from a bin.
|
|
`M` is the total number of objects, `n` is total number of Type I objects.
|
|
The random variate represents the number of Type I objects in `N` drawn
|
|
without replacement from the total population.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The symbols used to denote the shape parameters (`M`, `n`, and `N`) are not
|
|
universally accepted. See the Examples for a clarification of the
|
|
definitions used here.
|
|
|
|
The probability mass function is defined as,
|
|
|
|
.. math:: p(k, M, n, N) = \frac{\binom{n}{k} \binom{M - n}{N - k}}
|
|
{\binom{M}{N}}
|
|
|
|
for :math:`k \in [\max(0, N - M + n), \min(n, N)]`, where the binomial
|
|
coefficients are defined as,
|
|
|
|
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
|
|
|
|
%(after_notes)s
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import hypergeom
|
|
>>> import matplotlib.pyplot as plt
|
|
|
|
Suppose we have a collection of 20 animals, of which 7 are dogs. Then if
|
|
we want to know the probability of finding a given number of dogs if we
|
|
choose at random 12 of the 20 animals, we can initialize a frozen
|
|
distribution and plot the probability mass function:
|
|
|
|
>>> [M, n, N] = [20, 7, 12]
|
|
>>> rv = hypergeom(M, n, N)
|
|
>>> x = np.arange(0, n+1)
|
|
>>> pmf_dogs = rv.pmf(x)
|
|
|
|
>>> fig = plt.figure()
|
|
>>> ax = fig.add_subplot(111)
|
|
>>> ax.plot(x, pmf_dogs, 'bo')
|
|
>>> ax.vlines(x, 0, pmf_dogs, lw=2)
|
|
>>> ax.set_xlabel('# of dogs in our group of chosen animals')
|
|
>>> ax.set_ylabel('hypergeom PMF')
|
|
>>> plt.show()
|
|
|
|
Instead of using a frozen distribution we can also use `hypergeom`
|
|
methods directly. To for example obtain the cumulative distribution
|
|
function, use:
|
|
|
|
>>> prb = hypergeom.cdf(x, M, n, N)
|
|
|
|
And to generate random numbers:
|
|
|
|
>>> R = hypergeom.rvs(M, n, N, size=10)
|
|
|
|
See Also
|
|
--------
|
|
nhypergeom, binom, nbinom
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("M", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("n", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("N", True, (0, np.inf), (True, False))]
|
|
|
|
def _rvs(self, M, n, N, size=None, random_state=None):
|
|
return random_state.hypergeometric(n, M-n, N, size=size)
|
|
|
|
def _get_support(self, M, n, N):
|
|
return np.maximum(N-(M-n), 0), np.minimum(n, N)
|
|
|
|
def _argcheck(self, M, n, N):
|
|
cond = (M > 0) & (n >= 0) & (N >= 0)
|
|
cond &= (n <= M) & (N <= M)
|
|
cond &= _isintegral(M) & _isintegral(n) & _isintegral(N)
|
|
return cond
|
|
|
|
def _logpmf(self, k, M, n, N):
|
|
tot, good = M, n
|
|
bad = tot - good
|
|
result = (betaln(good+1, 1) + betaln(bad+1, 1) + betaln(tot-N+1, N+1) -
|
|
betaln(k+1, good-k+1) - betaln(N-k+1, bad-N+k+1) -
|
|
betaln(tot+1, 1))
|
|
return result
|
|
|
|
def _pmf(self, k, M, n, N):
|
|
return _boost._hypergeom_pdf(k, n, N, M)
|
|
|
|
def _cdf(self, k, M, n, N):
|
|
return _boost._hypergeom_cdf(k, n, N, M)
|
|
|
|
def _stats(self, M, n, N):
|
|
M, n, N = 1. * M, 1. * n, 1. * N
|
|
m = M - n
|
|
|
|
# Boost kurtosis_excess doesn't return the same as the value
|
|
# computed here.
|
|
g2 = M * (M + 1) - 6. * N * (M - N) - 6. * n * m
|
|
g2 *= (M - 1) * M * M
|
|
g2 += 6. * n * N * (M - N) * m * (5. * M - 6)
|
|
g2 /= n * N * (M - N) * m * (M - 2.) * (M - 3.)
|
|
return (
|
|
_boost._hypergeom_mean(n, N, M),
|
|
_boost._hypergeom_variance(n, N, M),
|
|
_boost._hypergeom_skewness(n, N, M),
|
|
g2,
|
|
)
|
|
|
|
def _entropy(self, M, n, N):
|
|
k = np.r_[N - (M - n):min(n, N) + 1]
|
|
vals = self.pmf(k, M, n, N)
|
|
return np.sum(entr(vals), axis=0)
|
|
|
|
def _sf(self, k, M, n, N):
|
|
return _boost._hypergeom_sf(k, n, N, M)
|
|
|
|
def _logsf(self, k, M, n, N):
|
|
res = []
|
|
for quant, tot, good, draw in zip(*np.broadcast_arrays(k, M, n, N)):
|
|
if (quant + 0.5) * (tot + 0.5) < (good - 0.5) * (draw - 0.5):
|
|
# Less terms to sum if we calculate log(1-cdf)
|
|
res.append(log1p(-exp(self.logcdf(quant, tot, good, draw))))
|
|
else:
|
|
# Integration over probability mass function using logsumexp
|
|
k2 = np.arange(quant + 1, draw + 1)
|
|
res.append(logsumexp(self._logpmf(k2, tot, good, draw)))
|
|
return np.asarray(res)
|
|
|
|
def _logcdf(self, k, M, n, N):
|
|
res = []
|
|
for quant, tot, good, draw in zip(*np.broadcast_arrays(k, M, n, N)):
|
|
if (quant + 0.5) * (tot + 0.5) > (good - 0.5) * (draw - 0.5):
|
|
# Less terms to sum if we calculate log(1-sf)
|
|
res.append(log1p(-exp(self.logsf(quant, tot, good, draw))))
|
|
else:
|
|
# Integration over probability mass function using logsumexp
|
|
k2 = np.arange(0, quant + 1)
|
|
res.append(logsumexp(self._logpmf(k2, tot, good, draw)))
|
|
return np.asarray(res)
|
|
|
|
|
|
hypergeom = hypergeom_gen(name='hypergeom')
|
|
|
|
|
|
class nhypergeom_gen(rv_discrete):
|
|
r"""A negative hypergeometric discrete random variable.
|
|
|
|
Consider a box containing :math:`M` balls:, :math:`n` red and
|
|
:math:`M-n` blue. We randomly sample balls from the box, one
|
|
at a time and *without* replacement, until we have picked :math:`r`
|
|
blue balls. `nhypergeom` is the distribution of the number of
|
|
red balls :math:`k` we have picked.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The symbols used to denote the shape parameters (`M`, `n`, and `r`) are not
|
|
universally accepted. See the Examples for a clarification of the
|
|
definitions used here.
|
|
|
|
The probability mass function is defined as,
|
|
|
|
.. math:: f(k; M, n, r) = \frac{{{k+r-1}\choose{k}}{{M-r-k}\choose{n-k}}}
|
|
{{M \choose n}}
|
|
|
|
for :math:`k \in [0, n]`, :math:`n \in [0, M]`, :math:`r \in [0, M-n]`,
|
|
and the binomial coefficient is:
|
|
|
|
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
|
|
|
|
It is equivalent to observing :math:`k` successes in :math:`k+r-1`
|
|
samples with :math:`k+r`'th sample being a failure. The former
|
|
can be modelled as a hypergeometric distribution. The probability
|
|
of the latter is simply the number of failures remaining
|
|
:math:`M-n-(r-1)` divided by the size of the remaining population
|
|
:math:`M-(k+r-1)`. This relationship can be shown as:
|
|
|
|
.. math:: NHG(k;M,n,r) = HG(k;M,n,k+r-1)\frac{(M-n-(r-1))}{(M-(k+r-1))}
|
|
|
|
where :math:`NHG` is probability mass function (PMF) of the
|
|
negative hypergeometric distribution and :math:`HG` is the
|
|
PMF of the hypergeometric distribution.
|
|
|
|
%(after_notes)s
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import nhypergeom
|
|
>>> import matplotlib.pyplot as plt
|
|
|
|
Suppose we have a collection of 20 animals, of which 7 are dogs.
|
|
Then if we want to know the probability of finding a given number
|
|
of dogs (successes) in a sample with exactly 12 animals that
|
|
aren't dogs (failures), we can initialize a frozen distribution
|
|
and plot the probability mass function:
|
|
|
|
>>> M, n, r = [20, 7, 12]
|
|
>>> rv = nhypergeom(M, n, r)
|
|
>>> x = np.arange(0, n+2)
|
|
>>> pmf_dogs = rv.pmf(x)
|
|
|
|
>>> fig = plt.figure()
|
|
>>> ax = fig.add_subplot(111)
|
|
>>> ax.plot(x, pmf_dogs, 'bo')
|
|
>>> ax.vlines(x, 0, pmf_dogs, lw=2)
|
|
>>> ax.set_xlabel('# of dogs in our group with given 12 failures')
|
|
>>> ax.set_ylabel('nhypergeom PMF')
|
|
>>> plt.show()
|
|
|
|
Instead of using a frozen distribution we can also use `nhypergeom`
|
|
methods directly. To for example obtain the probability mass
|
|
function, use:
|
|
|
|
>>> prb = nhypergeom.pmf(x, M, n, r)
|
|
|
|
And to generate random numbers:
|
|
|
|
>>> R = nhypergeom.rvs(M, n, r, size=10)
|
|
|
|
To verify the relationship between `hypergeom` and `nhypergeom`, use:
|
|
|
|
>>> from scipy.stats import hypergeom, nhypergeom
|
|
>>> M, n, r = 45, 13, 8
|
|
>>> k = 6
|
|
>>> nhypergeom.pmf(k, M, n, r)
|
|
0.06180776620271643
|
|
>>> hypergeom.pmf(k, M, n, k+r-1) * (M - n - (r-1)) / (M - (k+r-1))
|
|
0.06180776620271644
|
|
|
|
See Also
|
|
--------
|
|
hypergeom, binom, nbinom
|
|
|
|
References
|
|
----------
|
|
.. [1] Negative Hypergeometric Distribution on Wikipedia
|
|
https://en.wikipedia.org/wiki/Negative_hypergeometric_distribution
|
|
|
|
.. [2] Negative Hypergeometric Distribution from
|
|
http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Negativehypergeometric.pdf
|
|
|
|
"""
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("M", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("n", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("r", True, (0, np.inf), (True, False))]
|
|
|
|
def _get_support(self, M, n, r):
|
|
return 0, n
|
|
|
|
def _argcheck(self, M, n, r):
|
|
cond = (n >= 0) & (n <= M) & (r >= 0) & (r <= M-n)
|
|
cond &= _isintegral(M) & _isintegral(n) & _isintegral(r)
|
|
return cond
|
|
|
|
def _rvs(self, M, n, r, size=None, random_state=None):
|
|
|
|
@_vectorize_rvs_over_shapes
|
|
def _rvs1(M, n, r, size, random_state):
|
|
# invert cdf by calculating all values in support, scalar M, n, r
|
|
a, b = self.support(M, n, r)
|
|
ks = np.arange(a, b+1)
|
|
cdf = self.cdf(ks, M, n, r)
|
|
ppf = interp1d(cdf, ks, kind='next', fill_value='extrapolate')
|
|
rvs = ppf(random_state.uniform(size=size)).astype(int)
|
|
if size is None:
|
|
return rvs.item()
|
|
return rvs
|
|
|
|
return _rvs1(M, n, r, size=size, random_state=random_state)
|
|
|
|
def _logpmf(self, k, M, n, r):
|
|
cond = ((r == 0) & (k == 0))
|
|
result = _lazywhere(~cond, (k, M, n, r),
|
|
lambda k, M, n, r:
|
|
(-betaln(k+1, r) + betaln(k+r, 1) -
|
|
betaln(n-k+1, M-r-n+1) + betaln(M-r-k+1, 1) +
|
|
betaln(n+1, M-n+1) - betaln(M+1, 1)),
|
|
fillvalue=0.0)
|
|
return result
|
|
|
|
def _pmf(self, k, M, n, r):
|
|
# same as the following but numerically more precise
|
|
# return comb(k+r-1, k) * comb(M-r-k, n-k) / comb(M, n)
|
|
return exp(self._logpmf(k, M, n, r))
|
|
|
|
def _stats(self, M, n, r):
|
|
# Promote the datatype to at least float
|
|
# mu = rn / (M-n+1)
|
|
M, n, r = 1.*M, 1.*n, 1.*r
|
|
mu = r*n / (M-n+1)
|
|
|
|
var = r*(M+1)*n / ((M-n+1)*(M-n+2)) * (1 - r / (M-n+1))
|
|
|
|
# The skew and kurtosis are mathematically
|
|
# intractable so return `None`. See [2]_.
|
|
g1, g2 = None, None
|
|
return mu, var, g1, g2
|
|
|
|
|
|
nhypergeom = nhypergeom_gen(name='nhypergeom')
|
|
|
|
|
|
# FIXME: Fails _cdfvec
|
|
class logser_gen(rv_discrete):
|
|
r"""A Logarithmic (Log-Series, Series) discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `logser` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = - \frac{p^k}{k \log(1-p)}
|
|
|
|
for :math:`k \ge 1`, :math:`0 < p < 1`
|
|
|
|
`logser` takes :math:`p` as shape parameter,
|
|
where :math:`p` is the probability of a single success
|
|
and :math:`1-p` is the probability of a single failure.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("p", False, (0, 1), (True, True))]
|
|
|
|
def _rvs(self, p, size=None, random_state=None):
|
|
# looks wrong for p>0.5, too few k=1
|
|
# trying to use generic is worse, no k=1 at all
|
|
return random_state.logseries(p, size=size)
|
|
|
|
def _argcheck(self, p):
|
|
return (p > 0) & (p < 1)
|
|
|
|
def _pmf(self, k, p):
|
|
# logser.pmf(k) = - p**k / (k*log(1-p))
|
|
return -np.power(p, k) * 1.0 / k / special.log1p(-p)
|
|
|
|
def _stats(self, p):
|
|
r = special.log1p(-p)
|
|
mu = p / (p - 1.0) / r
|
|
mu2p = -p / r / (p - 1.0)**2
|
|
var = mu2p - mu*mu
|
|
mu3p = -p / r * (1.0+p) / (1.0 - p)**3
|
|
mu3 = mu3p - 3*mu*mu2p + 2*mu**3
|
|
g1 = mu3 / np.power(var, 1.5)
|
|
|
|
mu4p = -p / r * (
|
|
1.0 / (p-1)**2 - 6*p / (p - 1)**3 + 6*p*p / (p-1)**4)
|
|
mu4 = mu4p - 4*mu3p*mu + 6*mu2p*mu*mu - 3*mu**4
|
|
g2 = mu4 / var**2 - 3.0
|
|
return mu, var, g1, g2
|
|
|
|
|
|
logser = logser_gen(a=1, name='logser', longname='A logarithmic')
|
|
|
|
|
|
class poisson_gen(rv_discrete):
|
|
r"""A Poisson discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `poisson` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = \exp(-\mu) \frac{\mu^k}{k!}
|
|
|
|
for :math:`k \ge 0`.
|
|
|
|
`poisson` takes :math:`\mu \geq 0` as shape parameter.
|
|
When :math:`\mu = 0`, the ``pmf`` method
|
|
returns ``1.0`` at quantile :math:`k = 0`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("mu", False, (0, np.inf), (True, False))]
|
|
|
|
# Override rv_discrete._argcheck to allow mu=0.
|
|
def _argcheck(self, mu):
|
|
return mu >= 0
|
|
|
|
def _rvs(self, mu, size=None, random_state=None):
|
|
return random_state.poisson(mu, size)
|
|
|
|
def _logpmf(self, k, mu):
|
|
Pk = special.xlogy(k, mu) - gamln(k + 1) - mu
|
|
return Pk
|
|
|
|
def _pmf(self, k, mu):
|
|
# poisson.pmf(k) = exp(-mu) * mu**k / k!
|
|
return exp(self._logpmf(k, mu))
|
|
|
|
def _cdf(self, x, mu):
|
|
k = floor(x)
|
|
return special.pdtr(k, mu)
|
|
|
|
def _sf(self, x, mu):
|
|
k = floor(x)
|
|
return special.pdtrc(k, mu)
|
|
|
|
def _ppf(self, q, mu):
|
|
vals = ceil(special.pdtrik(q, mu))
|
|
vals1 = np.maximum(vals - 1, 0)
|
|
temp = special.pdtr(vals1, mu)
|
|
return np.where(temp >= q, vals1, vals)
|
|
|
|
def _stats(self, mu):
|
|
var = mu
|
|
tmp = np.asarray(mu)
|
|
mu_nonzero = tmp > 0
|
|
g1 = _lazywhere(mu_nonzero, (tmp,), lambda x: sqrt(1.0/x), np.inf)
|
|
g2 = _lazywhere(mu_nonzero, (tmp,), lambda x: 1.0/x, np.inf)
|
|
return mu, var, g1, g2
|
|
|
|
|
|
poisson = poisson_gen(name="poisson", longname='A Poisson')
|
|
|
|
|
|
class planck_gen(rv_discrete):
|
|
r"""A Planck discrete exponential random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `planck` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = (1-\exp(-\lambda)) \exp(-\lambda k)
|
|
|
|
for :math:`k \ge 0` and :math:`\lambda > 0`.
|
|
|
|
`planck` takes :math:`\lambda` as shape parameter. The Planck distribution
|
|
can be written as a geometric distribution (`geom`) with
|
|
:math:`p = 1 - \exp(-\lambda)` shifted by ``loc = -1``.
|
|
|
|
%(after_notes)s
|
|
|
|
See Also
|
|
--------
|
|
geom
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("lambda", False, (0, np.inf), (False, False))]
|
|
|
|
def _argcheck(self, lambda_):
|
|
return lambda_ > 0
|
|
|
|
def _pmf(self, k, lambda_):
|
|
return -expm1(-lambda_)*exp(-lambda_*k)
|
|
|
|
def _cdf(self, x, lambda_):
|
|
k = floor(x)
|
|
return -expm1(-lambda_*(k+1))
|
|
|
|
def _sf(self, x, lambda_):
|
|
return exp(self._logsf(x, lambda_))
|
|
|
|
def _logsf(self, x, lambda_):
|
|
k = floor(x)
|
|
return -lambda_*(k+1)
|
|
|
|
def _ppf(self, q, lambda_):
|
|
vals = ceil(-1.0/lambda_ * log1p(-q)-1)
|
|
vals1 = (vals-1).clip(*(self._get_support(lambda_)))
|
|
temp = self._cdf(vals1, lambda_)
|
|
return np.where(temp >= q, vals1, vals)
|
|
|
|
def _rvs(self, lambda_, size=None, random_state=None):
|
|
# use relation to geometric distribution for sampling
|
|
p = -expm1(-lambda_)
|
|
return random_state.geometric(p, size=size) - 1.0
|
|
|
|
def _stats(self, lambda_):
|
|
mu = 1/expm1(lambda_)
|
|
var = exp(-lambda_)/(expm1(-lambda_))**2
|
|
g1 = 2*cosh(lambda_/2.0)
|
|
g2 = 4+2*cosh(lambda_)
|
|
return mu, var, g1, g2
|
|
|
|
def _entropy(self, lambda_):
|
|
C = -expm1(-lambda_)
|
|
return lambda_*exp(-lambda_)/C - log(C)
|
|
|
|
|
|
planck = planck_gen(a=0, name='planck', longname='A discrete exponential ')
|
|
|
|
|
|
class boltzmann_gen(rv_discrete):
|
|
r"""A Boltzmann (Truncated Discrete Exponential) random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `boltzmann` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = (1-\exp(-\lambda)) \exp(-\lambda k) / (1-\exp(-\lambda N))
|
|
|
|
for :math:`k = 0,..., N-1`.
|
|
|
|
`boltzmann` takes :math:`\lambda > 0` and :math:`N > 0` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("lambda_", False, (0, np.inf), (False, False)),
|
|
_ShapeInfo("N", True, (0, np.inf), (False, False))]
|
|
|
|
def _argcheck(self, lambda_, N):
|
|
return (lambda_ > 0) & (N > 0) & _isintegral(N)
|
|
|
|
def _get_support(self, lambda_, N):
|
|
return self.a, N - 1
|
|
|
|
def _pmf(self, k, lambda_, N):
|
|
# boltzmann.pmf(k) =
|
|
# (1-exp(-lambda_)*exp(-lambda_*k)/(1-exp(-lambda_*N))
|
|
fact = (1-exp(-lambda_))/(1-exp(-lambda_*N))
|
|
return fact*exp(-lambda_*k)
|
|
|
|
def _cdf(self, x, lambda_, N):
|
|
k = floor(x)
|
|
return (1-exp(-lambda_*(k+1)))/(1-exp(-lambda_*N))
|
|
|
|
def _ppf(self, q, lambda_, N):
|
|
qnew = q*(1-exp(-lambda_*N))
|
|
vals = ceil(-1.0/lambda_ * log(1-qnew)-1)
|
|
vals1 = (vals-1).clip(0.0, np.inf)
|
|
temp = self._cdf(vals1, lambda_, N)
|
|
return np.where(temp >= q, vals1, vals)
|
|
|
|
def _stats(self, lambda_, N):
|
|
z = exp(-lambda_)
|
|
zN = exp(-lambda_*N)
|
|
mu = z/(1.0-z)-N*zN/(1-zN)
|
|
var = z/(1.0-z)**2 - N*N*zN/(1-zN)**2
|
|
trm = (1-zN)/(1-z)
|
|
trm2 = (z*trm**2 - N*N*zN)
|
|
g1 = z*(1+z)*trm**3 - N**3*zN*(1+zN)
|
|
g1 = g1 / trm2**(1.5)
|
|
g2 = z*(1+4*z+z*z)*trm**4 - N**4 * zN*(1+4*zN+zN*zN)
|
|
g2 = g2 / trm2 / trm2
|
|
return mu, var, g1, g2
|
|
|
|
|
|
boltzmann = boltzmann_gen(name='boltzmann', a=0,
|
|
longname='A truncated discrete exponential ')
|
|
|
|
|
|
class randint_gen(rv_discrete):
|
|
r"""A uniform discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `randint` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = \frac{1}{\texttt{high} - \texttt{low}}
|
|
|
|
for :math:`k \in \{\texttt{low}, \dots, \texttt{high} - 1\}`.
|
|
|
|
`randint` takes :math:`\texttt{low}` and :math:`\texttt{high}` as shape
|
|
parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("low", True, (-np.inf, np.inf), (False, False)),
|
|
_ShapeInfo("high", True, (-np.inf, np.inf), (False, False))]
|
|
|
|
def _argcheck(self, low, high):
|
|
return (high > low) & _isintegral(low) & _isintegral(high)
|
|
|
|
def _get_support(self, low, high):
|
|
return low, high-1
|
|
|
|
def _pmf(self, k, low, high):
|
|
# randint.pmf(k) = 1./(high - low)
|
|
p = np.ones_like(k) / (high - low)
|
|
return np.where((k >= low) & (k < high), p, 0.)
|
|
|
|
def _cdf(self, x, low, high):
|
|
k = floor(x)
|
|
return (k - low + 1.) / (high - low)
|
|
|
|
def _ppf(self, q, low, high):
|
|
vals = ceil(q * (high - low) + low) - 1
|
|
vals1 = (vals - 1).clip(low, high)
|
|
temp = self._cdf(vals1, low, high)
|
|
return np.where(temp >= q, vals1, vals)
|
|
|
|
def _stats(self, low, high):
|
|
m2, m1 = np.asarray(high), np.asarray(low)
|
|
mu = (m2 + m1 - 1.0) / 2
|
|
d = m2 - m1
|
|
var = (d*d - 1) / 12.0
|
|
g1 = 0.0
|
|
g2 = -6.0/5.0 * (d*d + 1.0) / (d*d - 1.0)
|
|
return mu, var, g1, g2
|
|
|
|
def _rvs(self, low, high, size=None, random_state=None):
|
|
"""An array of *size* random integers >= ``low`` and < ``high``."""
|
|
if np.asarray(low).size == 1 and np.asarray(high).size == 1:
|
|
# no need to vectorize in that case
|
|
return rng_integers(random_state, low, high, size=size)
|
|
|
|
if size is not None:
|
|
# NumPy's RandomState.randint() doesn't broadcast its arguments.
|
|
# Use `broadcast_to()` to extend the shapes of low and high
|
|
# up to size. Then we can use the numpy.vectorize'd
|
|
# randint without needing to pass it a `size` argument.
|
|
low = np.broadcast_to(low, size)
|
|
high = np.broadcast_to(high, size)
|
|
randint = np.vectorize(partial(rng_integers, random_state),
|
|
otypes=[np.int_])
|
|
return randint(low, high)
|
|
|
|
def _entropy(self, low, high):
|
|
return log(high - low)
|
|
|
|
|
|
randint = randint_gen(name='randint', longname='A discrete uniform '
|
|
'(random integer)')
|
|
|
|
|
|
# FIXME: problems sampling.
|
|
class zipf_gen(rv_discrete):
|
|
r"""A Zipf (Zeta) discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
zipfian
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `zipf` is:
|
|
|
|
.. math::
|
|
|
|
f(k, a) = \frac{1}{\zeta(a) k^a}
|
|
|
|
for :math:`k \ge 1`, :math:`a > 1`.
|
|
|
|
`zipf` takes :math:`a > 1` as shape parameter. :math:`\zeta` is the
|
|
Riemann zeta function (`scipy.special.zeta`)
|
|
|
|
The Zipf distribution is also known as the zeta distribution, which is
|
|
a special case of the Zipfian distribution (`zipfian`).
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] "Zeta Distribution", Wikipedia,
|
|
https://en.wikipedia.org/wiki/Zeta_distribution
|
|
|
|
%(example)s
|
|
|
|
Confirm that `zipf` is the large `n` limit of `zipfian`.
|
|
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import zipfian
|
|
>>> k = np.arange(11)
|
|
>>> np.allclose(zipf.pmf(k, a), zipfian.pmf(k, a, n=10000000))
|
|
True
|
|
|
|
"""
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("a", False, (1, np.inf), (False, False))]
|
|
|
|
def _rvs(self, a, size=None, random_state=None):
|
|
return random_state.zipf(a, size=size)
|
|
|
|
def _argcheck(self, a):
|
|
return a > 1
|
|
|
|
def _pmf(self, k, a):
|
|
# zipf.pmf(k, a) = 1/(zeta(a) * k**a)
|
|
Pk = 1.0 / special.zeta(a, 1) / k**a
|
|
return Pk
|
|
|
|
def _munp(self, n, a):
|
|
return _lazywhere(
|
|
a > n + 1, (a, n),
|
|
lambda a, n: special.zeta(a - n, 1) / special.zeta(a, 1),
|
|
np.inf)
|
|
|
|
|
|
zipf = zipf_gen(a=1, name='zipf', longname='A Zipf')
|
|
|
|
|
|
def _gen_harmonic_gt1(n, a):
|
|
"""Generalized harmonic number, a > 1"""
|
|
# See https://en.wikipedia.org/wiki/Harmonic_number; search for "hurwitz"
|
|
return zeta(a, 1) - zeta(a, n+1)
|
|
|
|
|
|
def _gen_harmonic_leq1(n, a):
|
|
"""Generalized harmonic number, a <= 1"""
|
|
if not np.size(n):
|
|
return n
|
|
n_max = np.max(n) # loop starts at maximum of all n
|
|
out = np.zeros_like(a, dtype=float)
|
|
# add terms of harmonic series; starting from smallest to avoid roundoff
|
|
for i in np.arange(n_max, 0, -1, dtype=float):
|
|
mask = i <= n # don't add terms after nth
|
|
out[mask] += 1/i**a[mask]
|
|
return out
|
|
|
|
|
|
def _gen_harmonic(n, a):
|
|
"""Generalized harmonic number"""
|
|
n, a = np.broadcast_arrays(n, a)
|
|
return _lazywhere(a > 1, (n, a),
|
|
f=_gen_harmonic_gt1, f2=_gen_harmonic_leq1)
|
|
|
|
|
|
class zipfian_gen(rv_discrete):
|
|
r"""A Zipfian discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
zipf
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `zipfian` is:
|
|
|
|
.. math::
|
|
|
|
f(k, a, n) = \frac{1}{H_{n,a} k^a}
|
|
|
|
for :math:`k \in \{1, 2, \dots, n-1, n\}`, :math:`a \ge 0`,
|
|
:math:`n \in \{1, 2, 3, \dots\}`.
|
|
|
|
`zipfian` takes :math:`a` and :math:`n` as shape parameters.
|
|
:math:`H_{n,a}` is the :math:`n`:sup:`th` generalized harmonic
|
|
number of order :math:`a`.
|
|
|
|
The Zipfian distribution reduces to the Zipf (zeta) distribution as
|
|
:math:`n \rightarrow \infty`.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] "Zipf's Law", Wikipedia, https://en.wikipedia.org/wiki/Zipf's_law
|
|
.. [2] Larry Leemis, "Zipf Distribution", Univariate Distribution
|
|
Relationships. http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf
|
|
|
|
%(example)s
|
|
|
|
Confirm that `zipfian` reduces to `zipf` for large `n`, `a > 1`.
|
|
|
|
>>> import numpy as np
|
|
>>> from scipy.stats import zipf
|
|
>>> k = np.arange(11)
|
|
>>> np.allclose(zipfian.pmf(k, a=3.5, n=10000000), zipf.pmf(k, a=3.5))
|
|
True
|
|
|
|
"""
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("a", False, (0, np.inf), (True, False)),
|
|
_ShapeInfo("n", True, (0, np.inf), (False, False))]
|
|
|
|
def _argcheck(self, a, n):
|
|
# we need np.asarray here because moment (maybe others) don't convert
|
|
return (a >= 0) & (n > 0) & (n == np.asarray(n, dtype=int))
|
|
|
|
def _get_support(self, a, n):
|
|
return 1, n
|
|
|
|
def _pmf(self, k, a, n):
|
|
return 1.0 / _gen_harmonic(n, a) / k**a
|
|
|
|
def _cdf(self, k, a, n):
|
|
return _gen_harmonic(k, a) / _gen_harmonic(n, a)
|
|
|
|
def _sf(self, k, a, n):
|
|
k = k + 1 # # to match SciPy convention
|
|
# see http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf
|
|
return ((k**a*(_gen_harmonic(n, a) - _gen_harmonic(k, a)) + 1)
|
|
/ (k**a*_gen_harmonic(n, a)))
|
|
|
|
def _stats(self, a, n):
|
|
# see # see http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf
|
|
Hna = _gen_harmonic(n, a)
|
|
Hna1 = _gen_harmonic(n, a-1)
|
|
Hna2 = _gen_harmonic(n, a-2)
|
|
Hna3 = _gen_harmonic(n, a-3)
|
|
Hna4 = _gen_harmonic(n, a-4)
|
|
mu1 = Hna1/Hna
|
|
mu2n = (Hna2*Hna - Hna1**2)
|
|
mu2d = Hna**2
|
|
mu2 = mu2n / mu2d
|
|
g1 = (Hna3/Hna - 3*Hna1*Hna2/Hna**2 + 2*Hna1**3/Hna**3)/mu2**(3/2)
|
|
g2 = (Hna**3*Hna4 - 4*Hna**2*Hna1*Hna3 + 6*Hna*Hna1**2*Hna2
|
|
- 3*Hna1**4) / mu2n**2
|
|
g2 -= 3
|
|
return mu1, mu2, g1, g2
|
|
|
|
|
|
zipfian = zipfian_gen(a=1, name='zipfian', longname='A Zipfian')
|
|
|
|
|
|
class dlaplace_gen(rv_discrete):
|
|
r"""A Laplacian discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `dlaplace` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = \tanh(a/2) \exp(-a |k|)
|
|
|
|
for integers :math:`k` and :math:`a > 0`.
|
|
|
|
`dlaplace` takes :math:`a` as shape parameter.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("a", False, (0, np.inf), (False, False))]
|
|
|
|
def _pmf(self, k, a):
|
|
# dlaplace.pmf(k) = tanh(a/2) * exp(-a*abs(k))
|
|
return tanh(a/2.0) * exp(-a * abs(k))
|
|
|
|
def _cdf(self, x, a):
|
|
k = floor(x)
|
|
f = lambda k, a: 1.0 - exp(-a * k) / (exp(a) + 1)
|
|
f2 = lambda k, a: exp(a * (k+1)) / (exp(a) + 1)
|
|
return _lazywhere(k >= 0, (k, a), f=f, f2=f2)
|
|
|
|
def _ppf(self, q, a):
|
|
const = 1 + exp(a)
|
|
vals = ceil(np.where(q < 1.0 / (1 + exp(-a)),
|
|
log(q*const) / a - 1,
|
|
-log((1-q) * const) / a))
|
|
vals1 = vals - 1
|
|
return np.where(self._cdf(vals1, a) >= q, vals1, vals)
|
|
|
|
def _stats(self, a):
|
|
ea = exp(a)
|
|
mu2 = 2.*ea/(ea-1.)**2
|
|
mu4 = 2.*ea*(ea**2+10.*ea+1.) / (ea-1.)**4
|
|
return 0., mu2, 0., mu4/mu2**2 - 3.
|
|
|
|
def _entropy(self, a):
|
|
return a / sinh(a) - log(tanh(a/2.0))
|
|
|
|
def _rvs(self, a, size=None, random_state=None):
|
|
# The discrete Laplace is equivalent to the two-sided geometric
|
|
# distribution with PMF:
|
|
# f(k) = (1 - alpha)/(1 + alpha) * alpha^abs(k)
|
|
# Reference:
|
|
# https://www.sciencedirect.com/science/
|
|
# article/abs/pii/S0378375804003519
|
|
# Furthermore, the two-sided geometric distribution is
|
|
# equivalent to the difference between two iid geometric
|
|
# distributions.
|
|
# Reference (page 179):
|
|
# https://pdfs.semanticscholar.org/61b3/
|
|
# b99f466815808fd0d03f5d2791eea8b541a1.pdf
|
|
# Thus, we can leverage the following:
|
|
# 1) alpha = e^-a
|
|
# 2) probability_of_success = 1 - alpha (Bernoulli trial)
|
|
probOfSuccess = -np.expm1(-np.asarray(a))
|
|
x = random_state.geometric(probOfSuccess, size=size)
|
|
y = random_state.geometric(probOfSuccess, size=size)
|
|
return x - y
|
|
|
|
|
|
dlaplace = dlaplace_gen(a=-np.inf,
|
|
name='dlaplace', longname='A discrete Laplacian')
|
|
|
|
|
|
class skellam_gen(rv_discrete):
|
|
r"""A Skellam discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
Probability distribution of the difference of two correlated or
|
|
uncorrelated Poisson random variables.
|
|
|
|
Let :math:`k_1` and :math:`k_2` be two Poisson-distributed r.v. with
|
|
expected values :math:`\lambda_1` and :math:`\lambda_2`. Then,
|
|
:math:`k_1 - k_2` follows a Skellam distribution with parameters
|
|
:math:`\mu_1 = \lambda_1 - \rho \sqrt{\lambda_1 \lambda_2}` and
|
|
:math:`\mu_2 = \lambda_2 - \rho \sqrt{\lambda_1 \lambda_2}`, where
|
|
:math:`\rho` is the correlation coefficient between :math:`k_1` and
|
|
:math:`k_2`. If the two Poisson-distributed r.v. are independent then
|
|
:math:`\rho = 0`.
|
|
|
|
Parameters :math:`\mu_1` and :math:`\mu_2` must be strictly positive.
|
|
|
|
For details see: https://en.wikipedia.org/wiki/Skellam_distribution
|
|
|
|
`skellam` takes :math:`\mu_1` and :math:`\mu_2` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("mu1", False, (0, np.inf), (False, False)),
|
|
_ShapeInfo("mu2", False, (0, np.inf), (False, False))]
|
|
|
|
def _rvs(self, mu1, mu2, size=None, random_state=None):
|
|
n = size
|
|
return (random_state.poisson(mu1, n) -
|
|
random_state.poisson(mu2, n))
|
|
|
|
def _pmf(self, x, mu1, mu2):
|
|
with warnings.catch_warnings():
|
|
message = "overflow encountered in _ncx2_pdf"
|
|
warnings.filterwarnings("ignore", message=message)
|
|
px = np.where(x < 0,
|
|
_boost._ncx2_pdf(2*mu2, 2*(1-x), 2*mu1)*2,
|
|
_boost._ncx2_pdf(2*mu1, 2*(1+x), 2*mu2)*2)
|
|
# ncx2.pdf() returns nan's for extremely low probabilities
|
|
return px
|
|
|
|
def _cdf(self, x, mu1, mu2):
|
|
x = floor(x)
|
|
px = np.where(x < 0,
|
|
_boost._ncx2_cdf(2*mu2, -2*x, 2*mu1),
|
|
1 - _boost._ncx2_cdf(2*mu1, 2*(x+1), 2*mu2))
|
|
return px
|
|
|
|
def _stats(self, mu1, mu2):
|
|
mean = mu1 - mu2
|
|
var = mu1 + mu2
|
|
g1 = mean / sqrt((var)**3)
|
|
g2 = 1 / var
|
|
return mean, var, g1, g2
|
|
|
|
|
|
skellam = skellam_gen(a=-np.inf, name="skellam", longname='A Skellam')
|
|
|
|
|
|
class yulesimon_gen(rv_discrete):
|
|
r"""A Yule-Simon discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
|
|
The probability mass function for the `yulesimon` is:
|
|
|
|
.. math::
|
|
|
|
f(k) = \alpha B(k, \alpha+1)
|
|
|
|
for :math:`k=1,2,3,...`, where :math:`\alpha>0`.
|
|
Here :math:`B` refers to the `scipy.special.beta` function.
|
|
|
|
The sampling of random variates is based on pg 553, Section 6.3 of [1]_.
|
|
Our notation maps to the referenced logic via :math:`\alpha=a-1`.
|
|
|
|
For details see the wikipedia entry [2]_.
|
|
|
|
References
|
|
----------
|
|
.. [1] Devroye, Luc. "Non-uniform Random Variate Generation",
|
|
(1986) Springer, New York.
|
|
|
|
.. [2] https://en.wikipedia.org/wiki/Yule-Simon_distribution
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("alpha", False, (0, np.inf), (False, False))]
|
|
|
|
def _rvs(self, alpha, size=None, random_state=None):
|
|
E1 = random_state.standard_exponential(size)
|
|
E2 = random_state.standard_exponential(size)
|
|
ans = ceil(-E1 / log1p(-exp(-E2 / alpha)))
|
|
return ans
|
|
|
|
def _pmf(self, x, alpha):
|
|
return alpha * special.beta(x, alpha + 1)
|
|
|
|
def _argcheck(self, alpha):
|
|
return (alpha > 0)
|
|
|
|
def _logpmf(self, x, alpha):
|
|
return log(alpha) + special.betaln(x, alpha + 1)
|
|
|
|
def _cdf(self, x, alpha):
|
|
return 1 - x * special.beta(x, alpha + 1)
|
|
|
|
def _sf(self, x, alpha):
|
|
return x * special.beta(x, alpha + 1)
|
|
|
|
def _logsf(self, x, alpha):
|
|
return log(x) + special.betaln(x, alpha + 1)
|
|
|
|
def _stats(self, alpha):
|
|
mu = np.where(alpha <= 1, np.inf, alpha / (alpha - 1))
|
|
mu2 = np.where(alpha > 2,
|
|
alpha**2 / ((alpha - 2.0) * (alpha - 1)**2),
|
|
np.inf)
|
|
mu2 = np.where(alpha <= 1, np.nan, mu2)
|
|
g1 = np.where(alpha > 3,
|
|
sqrt(alpha - 2) * (alpha + 1)**2 / (alpha * (alpha - 3)),
|
|
np.inf)
|
|
g1 = np.where(alpha <= 2, np.nan, g1)
|
|
g2 = np.where(alpha > 4,
|
|
(alpha + 3) + (alpha**3 - 49 * alpha - 22) / (alpha *
|
|
(alpha - 4) * (alpha - 3)), np.inf)
|
|
g2 = np.where(alpha <= 2, np.nan, g2)
|
|
return mu, mu2, g1, g2
|
|
|
|
|
|
yulesimon = yulesimon_gen(name='yulesimon', a=1)
|
|
|
|
|
|
def _vectorize_rvs_over_shapes(_rvs1):
|
|
"""Decorator that vectorizes _rvs method to work on ndarray shapes"""
|
|
# _rvs1 must be a _function_ that accepts _scalar_ args as positional
|
|
# arguments, `size` and `random_state` as keyword arguments.
|
|
# _rvs1 must return a random variate array with shape `size`. If `size` is
|
|
# None, _rvs1 must return a scalar.
|
|
# When applied to _rvs1, this decorator broadcasts ndarray args
|
|
# and loops over them, calling _rvs1 for each set of scalar args.
|
|
# For usage example, see _nchypergeom_gen
|
|
def _rvs(*args, size, random_state):
|
|
_rvs1_size, _rvs1_indices = _check_shape(args[0].shape, size)
|
|
|
|
size = np.array(size)
|
|
_rvs1_size = np.array(_rvs1_size)
|
|
_rvs1_indices = np.array(_rvs1_indices)
|
|
|
|
if np.all(_rvs1_indices): # all args are scalars
|
|
return _rvs1(*args, size, random_state)
|
|
|
|
out = np.empty(size)
|
|
|
|
# out.shape can mix dimensions associated with arg_shape and _rvs1_size
|
|
# Sort them to arg_shape + _rvs1_size for easy indexing of dimensions
|
|
# corresponding with the different sets of scalar args
|
|
j0 = np.arange(out.ndim)
|
|
j1 = np.hstack((j0[~_rvs1_indices], j0[_rvs1_indices]))
|
|
out = np.moveaxis(out, j1, j0)
|
|
|
|
for i in np.ndindex(*size[~_rvs1_indices]):
|
|
# arg can be squeezed because singleton dimensions will be
|
|
# associated with _rvs1_size, not arg_shape per _check_shape
|
|
out[i] = _rvs1(*[np.squeeze(arg)[i] for arg in args],
|
|
_rvs1_size, random_state)
|
|
|
|
return np.moveaxis(out, j0, j1) # move axes back before returning
|
|
return _rvs
|
|
|
|
|
|
class _nchypergeom_gen(rv_discrete):
|
|
r"""A noncentral hypergeometric discrete random variable.
|
|
|
|
For subclassing by nchypergeom_fisher_gen and nchypergeom_wallenius_gen.
|
|
|
|
"""
|
|
|
|
rvs_name = None
|
|
dist = None
|
|
|
|
def _shape_info(self):
|
|
return [_ShapeInfo("M", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("n", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("N", True, (0, np.inf), (True, False)),
|
|
_ShapeInfo("odds", False, (0, np.inf), (False, False))]
|
|
|
|
def _get_support(self, M, n, N, odds):
|
|
N, m1, n = M, n, N # follow Wikipedia notation
|
|
m2 = N - m1
|
|
x_min = np.maximum(0, n - m2)
|
|
x_max = np.minimum(n, m1)
|
|
return x_min, x_max
|
|
|
|
def _argcheck(self, M, n, N, odds):
|
|
M, n = np.asarray(M), np.asarray(n),
|
|
N, odds = np.asarray(N), np.asarray(odds)
|
|
cond1 = (M.astype(int) == M) & (M >= 0)
|
|
cond2 = (n.astype(int) == n) & (n >= 0)
|
|
cond3 = (N.astype(int) == N) & (N >= 0)
|
|
cond4 = odds > 0
|
|
cond5 = N <= M
|
|
cond6 = n <= M
|
|
return cond1 & cond2 & cond3 & cond4 & cond5 & cond6
|
|
|
|
def _rvs(self, M, n, N, odds, size=None, random_state=None):
|
|
|
|
@_vectorize_rvs_over_shapes
|
|
def _rvs1(M, n, N, odds, size, random_state):
|
|
length = np.prod(size)
|
|
urn = _PyStochasticLib3()
|
|
rv_gen = getattr(urn, self.rvs_name)
|
|
rvs = rv_gen(N, n, M, odds, length, random_state)
|
|
rvs = rvs.reshape(size)
|
|
return rvs
|
|
|
|
return _rvs1(M, n, N, odds, size=size, random_state=random_state)
|
|
|
|
def _pmf(self, x, M, n, N, odds):
|
|
|
|
x, M, n, N, odds = np.broadcast_arrays(x, M, n, N, odds)
|
|
if x.size == 0: # np.vectorize doesn't work with zero size input
|
|
return np.empty_like(x)
|
|
|
|
@np.vectorize
|
|
def _pmf1(x, M, n, N, odds):
|
|
urn = self.dist(N, n, M, odds, 1e-12)
|
|
return urn.probability(x)
|
|
|
|
return _pmf1(x, M, n, N, odds)
|
|
|
|
def _stats(self, M, n, N, odds, moments):
|
|
|
|
@np.vectorize
|
|
def _moments1(M, n, N, odds):
|
|
urn = self.dist(N, n, M, odds, 1e-12)
|
|
return urn.moments()
|
|
|
|
m, v = (_moments1(M, n, N, odds) if ("m" in moments or "v" in moments)
|
|
else (None, None))
|
|
s, k = None, None
|
|
return m, v, s, k
|
|
|
|
|
|
class nchypergeom_fisher_gen(_nchypergeom_gen):
|
|
r"""A Fisher's noncentral hypergeometric discrete random variable.
|
|
|
|
Fisher's noncentral hypergeometric distribution models drawing objects of
|
|
two types from a bin. `M` is the total number of objects, `n` is the
|
|
number of Type I objects, and `odds` is the odds ratio: the odds of
|
|
selecting a Type I object rather than a Type II object when there is only
|
|
one object of each type.
|
|
The random variate represents the number of Type I objects drawn if we
|
|
take a handful of objects from the bin at once and find out afterwards
|
|
that we took `N` objects.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
nchypergeom_wallenius, hypergeom, nhypergeom
|
|
|
|
Notes
|
|
-----
|
|
Let mathematical symbols :math:`N`, :math:`n`, and :math:`M` correspond
|
|
with parameters `N`, `n`, and `M` (respectively) as defined above.
|
|
|
|
The probability mass function is defined as
|
|
|
|
.. math::
|
|
|
|
p(x; M, n, N, \omega) =
|
|
\frac{\binom{n}{x}\binom{M - n}{N-x}\omega^x}{P_0},
|
|
|
|
for
|
|
:math:`x \in [x_l, x_u]`,
|
|
:math:`M \in {\mathbb N}`,
|
|
:math:`n \in [0, M]`,
|
|
:math:`N \in [0, M]`,
|
|
:math:`\omega > 0`,
|
|
where
|
|
:math:`x_l = \max(0, N - (M - n))`,
|
|
:math:`x_u = \min(N, n)`,
|
|
|
|
.. math::
|
|
|
|
P_0 = \sum_{y=x_l}^{x_u} \binom{n}{y}\binom{M - n}{N-y}\omega^y,
|
|
|
|
and the binomial coefficients are defined as
|
|
|
|
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
|
|
|
|
`nchypergeom_fisher` uses the BiasedUrn package by Agner Fog with
|
|
permission for it to be distributed under SciPy's license.
|
|
|
|
The symbols used to denote the shape parameters (`N`, `n`, and `M`) are not
|
|
universally accepted; they are chosen for consistency with `hypergeom`.
|
|
|
|
Note that Fisher's noncentral hypergeometric distribution is distinct
|
|
from Wallenius' noncentral hypergeometric distribution, which models
|
|
drawing a pre-determined `N` objects from a bin one by one.
|
|
When the odds ratio is unity, however, both distributions reduce to the
|
|
ordinary hypergeometric distribution.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] Agner Fog, "Biased Urn Theory".
|
|
https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf
|
|
|
|
.. [2] "Fisher's noncentral hypergeometric distribution", Wikipedia,
|
|
https://en.wikipedia.org/wiki/Fisher's_noncentral_hypergeometric_distribution
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
rvs_name = "rvs_fisher"
|
|
dist = _PyFishersNCHypergeometric
|
|
|
|
|
|
nchypergeom_fisher = nchypergeom_fisher_gen(
|
|
name='nchypergeom_fisher',
|
|
longname="A Fisher's noncentral hypergeometric")
|
|
|
|
|
|
class nchypergeom_wallenius_gen(_nchypergeom_gen):
|
|
r"""A Wallenius' noncentral hypergeometric discrete random variable.
|
|
|
|
Wallenius' noncentral hypergeometric distribution models drawing objects of
|
|
two types from a bin. `M` is the total number of objects, `n` is the
|
|
number of Type I objects, and `odds` is the odds ratio: the odds of
|
|
selecting a Type I object rather than a Type II object when there is only
|
|
one object of each type.
|
|
The random variate represents the number of Type I objects drawn if we
|
|
draw a pre-determined `N` objects from a bin one by one.
|
|
|
|
%(before_notes)s
|
|
|
|
See Also
|
|
--------
|
|
nchypergeom_fisher, hypergeom, nhypergeom
|
|
|
|
Notes
|
|
-----
|
|
Let mathematical symbols :math:`N`, :math:`n`, and :math:`M` correspond
|
|
with parameters `N`, `n`, and `M` (respectively) as defined above.
|
|
|
|
The probability mass function is defined as
|
|
|
|
.. math::
|
|
|
|
p(x; N, n, M) = \binom{n}{x} \binom{M - n}{N-x}
|
|
\int_0^1 \left(1-t^{\omega/D}\right)^x\left(1-t^{1/D}\right)^{N-x} dt
|
|
|
|
for
|
|
:math:`x \in [x_l, x_u]`,
|
|
:math:`M \in {\mathbb N}`,
|
|
:math:`n \in [0, M]`,
|
|
:math:`N \in [0, M]`,
|
|
:math:`\omega > 0`,
|
|
where
|
|
:math:`x_l = \max(0, N - (M - n))`,
|
|
:math:`x_u = \min(N, n)`,
|
|
|
|
.. math::
|
|
|
|
D = \omega(n - x) + ((M - n)-(N-x)),
|
|
|
|
and the binomial coefficients are defined as
|
|
|
|
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
|
|
|
|
`nchypergeom_wallenius` uses the BiasedUrn package by Agner Fog with
|
|
permission for it to be distributed under SciPy's license.
|
|
|
|
The symbols used to denote the shape parameters (`N`, `n`, and `M`) are not
|
|
universally accepted; they are chosen for consistency with `hypergeom`.
|
|
|
|
Note that Wallenius' noncentral hypergeometric distribution is distinct
|
|
from Fisher's noncentral hypergeometric distribution, which models
|
|
take a handful of objects from the bin at once, finding out afterwards
|
|
that `N` objects were taken.
|
|
When the odds ratio is unity, however, both distributions reduce to the
|
|
ordinary hypergeometric distribution.
|
|
|
|
%(after_notes)s
|
|
|
|
References
|
|
----------
|
|
.. [1] Agner Fog, "Biased Urn Theory".
|
|
https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf
|
|
|
|
.. [2] "Wallenius' noncentral hypergeometric distribution", Wikipedia,
|
|
https://en.wikipedia.org/wiki/Wallenius'_noncentral_hypergeometric_distribution
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
rvs_name = "rvs_wallenius"
|
|
dist = _PyWalleniusNCHypergeometric
|
|
|
|
|
|
nchypergeom_wallenius = nchypergeom_wallenius_gen(
|
|
name='nchypergeom_wallenius',
|
|
longname="A Wallenius' noncentral hypergeometric")
|
|
|
|
|
|
# Collect names of classes and objects in this module.
|
|
pairs = list(globals().copy().items())
|
|
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_discrete)
|
|
|
|
__all__ = _distn_names + _distn_gen_names
|