ALINA_KURBANOVA/AIM-PIbd-32-Kurbanova-A-A
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
d82b74e2e8
AIM-PIbd-32-Kurbanova-A-A/aimenv/Lib/site-packages/scipy/stats/_discrete_distns.py

1923 lines
56 KiB
Python
Raw Normal View History

lab 1 is done
2024-10-02 22:15:59 +04:00
#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
from functools import partial
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,
_vectorize_rvs_over_shapes,
_ShapeInfo, _isintegral)
from ._biasedurn import (_PyFishersNCHypergeometric,
_PyWalleniusNCHypergeometric,
_PyStochasticLib3)
import scipy.special._ufuncs as scu
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 scu._binom_pmf(x, n, p)
def _cdf(self, x, n, p):
k = floor(x)
return scu._binom_cdf(k, n, p)
def _sf(self, x, n, p):
k = floor(x)
return scu._binom_sf(k, n, p)
def _isf(self, x, n, p):
return scu._binom_isf(x, n, p)
def _ppf(self, q, n, p):
return scu._binom_ppf(q, n, p)
def _stats(self, n, p, moments='mv'):
mu = n * p
var = mu - n * np.square(p)
g1, g2 = None, None
if 's' in moments:
pq = p - np.square(p)
npq_sqrt = np.sqrt(n * pq)
t1 = np.reciprocal(npq_sqrt)
t2 = (2.0 * p) / npq_sqrt
g1 = t1 - t2
if 'k' in moments:
pq = p - np.square(p)
npq = n * pq
t1 = np.reciprocal(npq)
t2 = 6.0/n
g2 = t1 - t2
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).astype(e_p.dtype)
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 scu._nbinom_pmf(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 scu._nbinom_cdf(k, n, p)
def _logcdf(self, x, n, p):
k = floor(x)
k, n, p = np.broadcast_arrays(k, n, p)
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 scu._nbinom_sf(k, n, p)
def _isf(self, x, n, p):
with np.errstate(over='ignore'): # see gh-17432
return scu._nbinom_isf(x, n, p)
def _ppf(self, q, n, p):
with np.errstate(over='ignore'): # see gh-17432
return scu._nbinom_ppf(q, n, p)
def _stats(self, n, p):
return (
scu._nbinom_mean(n, p),
scu._nbinom_variance(n, p),
scu._nbinom_skewness(n, p),
scu._nbinom_kurtosis_excess(n, p),
)
nbinom = nbinom_gen(name='nbinom')
class betanbinom_gen(rv_discrete):
r"""A beta-negative-binomial discrete random variable.
%(before_notes)s
Notes
-----
The beta-negative-binomial distribution is a negative binomial
distribution with a probability of success `p` that follows a
beta distribution.
The probability mass function for `betanbinom` is:
.. math::
f(k) = \binom{n + k - 1}{k} \frac{B(a + n, b + k)}{B(a, b)}
for :math:`k \ge 0`, :math:`n \geq 0`, :math:`a > 0`,
:math:`b > 0`, where :math:`B(a, b)` is the beta function.
`betanbinom` takes :math:`n`, :math:`a`, and :math:`b` as shape parameters.
References
----------
.. [1] https://en.wikipedia.org/wiki/Beta_negative_binomial_distribution
%(after_notes)s
.. versionadded:: 1.12.0
See Also
--------
betabinom : Beta binomial distribution
%(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.negative_binomial(n, p, size)
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 = -np.log(n + k) - betaln(n, k + 1)
return combiln + betaln(a + n, b + k) - 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'):
# reference: Wolfram Alpha input
# BetaNegativeBinomialDistribution[a, b, n]
def mean(n, a, b):
return n * b / (a - 1.)
mu = _lazywhere(a > 1, (n, a, b), f=mean, fillvalue=np.inf)
def var(n, a, b):
return (n * b * (n + a - 1.) * (a + b - 1.)
/ ((a - 2.) * (a - 1.)**2.))
var = _lazywhere(a > 2, (n, a, b), f=var, fillvalue=np.inf)
g1, g2 = None, None
def skew(n, a, b):
return ((2 * n + a - 1.) * (2 * b + a - 1.)
/ (a - 3.) / sqrt(n * b * (n + a - 1.) * (b + a - 1.)
/ (a - 2.)))
if 's' in moments:
g1 = _lazywhere(a > 3, (n, a, b), f=skew, fillvalue=np.inf)
def kurtosis(n, a, b):
term = (a - 2.)
term_2 = ((a - 1.)**2. * (a**2. + a * (6 * b - 1.)
+ 6. * (b - 1.) * b)
+ 3. * n**2. * ((a + 5.) * b**2. + (a + 5.)
* (a - 1.) * b + 2. * (a - 1.)**2)
+ 3 * (a - 1.) * n
* ((a + 5.) * b**2. + (a + 5.) * (a - 1.) * b
+ 2. * (a - 1.)**2.))
denominator = ((a - 4.) * (a - 3.) * b * n
* (a + b - 1.) * (a + n - 1.))
# Wolfram Alpha uses Pearson kurtosis, so we substract 3 to get
# scipy's Fisher kurtosis
return term * term_2 / denominator - 3.
if 'k' in moments:
g2 = _lazywhere(a > 4, (n, a, b), f=kurtosis, fillvalue=np.inf)
return mu, var, g1, g2
betanbinom = betanbinom_gen(name='betanbinom')
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
def _entropy(self, p):
return -np.log(p) - np.log1p(-p) * (1.0-p) / p
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 scu._hypergeom_pmf(k, n, N, M)
def _cdf(self, k, M, n, N):
return scu._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 (
scu._hypergeom_mean(n, N, M),
scu._hypergeom_variance(n, N, M),
scu._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 scu._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
Examples
--------
>>> import numpy as np
>>> from scipy.stats import randint
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> low, high = 7, 31
>>> mean, var, skew, kurt = randint.stats(low, high, moments='mvsk')
Display the probability mass function (``pmf``):
>>> x = np.arange(low - 5, high + 5)
>>> ax.plot(x, randint.pmf(x, low, high), 'bo', ms=8, label='randint pmf')
>>> ax.vlines(x, 0, randint.pmf(x, low, high), colors='b', lw=5, alpha=0.5)
Alternatively, the distribution object can be called (as a function) to
fix the shape and location. This returns a "frozen" RV object holding the
given parameters fixed.
Freeze the distribution and display the frozen ``pmf``:
>>> rv = randint(low, high)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-',
... lw=1, label='frozen pmf')
>>> ax.legend(loc='lower center')
>>> plt.show()
Check the relationship between the cumulative distribution function
(``cdf``) and its inverse, the percent point function (``ppf``):
>>> q = np.arange(low, high)
>>> p = randint.cdf(q, low, high)
>>> np.allclose(q, randint.ppf(p, low, high))
True
Generate random numbers:
>>> r = randint.rvs(low, high, size=1000)
"""
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.dtype(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 zipf, 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):
k = k.astype(np.float64)
# 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, zipfian
>>> 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):
k = k.astype(np.float64)
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)
def f(k, a):
return 1.0 - exp(-a * k) / (exp(a) + 1)
def f2(k, a):
return 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 np.errstate(over='ignore'): # see gh-17432
px = np.where(x < 0,
scu._ncx2_pdf(2*mu2, 2*(1-x), 2*mu1)*2,
scu._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)
with np.errstate(over='ignore'): # see gh-17432
px = np.where(x < 0,
scu._ncx2_cdf(2*mu2, -2*x, 2*mu1),
1 - scu._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 + ((11 * 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)
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
Reference in New Issue Copy Permalink