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1270 lines
36 KiB
Python
1270 lines
36 KiB
Python
#
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# Author: Travis Oliphant 2002-2011 with contributions from
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# SciPy Developers 2004-2011
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#
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from functools import partial
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from scipy import special
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from scipy.special import entr, logsumexp, betaln, gammaln as gamln
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from scipy._lib._util import _lazywhere, rng_integers
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from numpy import floor, ceil, log, exp, sqrt, log1p, expm1, tanh, cosh, sinh
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import numpy as np
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from ._distn_infrastructure import (
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rv_discrete, _ncx2_pdf, _ncx2_cdf, get_distribution_names)
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class binom_gen(rv_discrete):
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r"""A binomial discrete random variable.
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%(before_notes)s
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Notes
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-----
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The probability mass function for `binom` is:
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.. math::
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f(k) = \binom{n}{k} p^k (1-p)^{n-k}
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for ``k`` in ``{0, 1,..., n}``, :math:`0 \leq p \leq 1`
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`binom` takes ``n`` and ``p`` as shape parameters,
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where :math:`p` is the probability of a single success
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and :math:`1-p` is the probability of a single failure.
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%(after_notes)s
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%(example)s
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See Also
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--------
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hypergeom, nbinom, nhypergeom
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"""
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def _rvs(self, n, p, size=None, random_state=None):
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return random_state.binomial(n, p, size)
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def _argcheck(self, n, p):
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return (n >= 0) & (p >= 0) & (p <= 1)
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def _get_support(self, n, p):
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return self.a, n
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def _logpmf(self, x, n, p):
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k = floor(x)
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combiln = (gamln(n+1) - (gamln(k+1) + gamln(n-k+1)))
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return combiln + special.xlogy(k, p) + special.xlog1py(n-k, -p)
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def _pmf(self, x, n, p):
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# binom.pmf(k) = choose(n, k) * p**k * (1-p)**(n-k)
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return exp(self._logpmf(x, n, p))
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def _cdf(self, x, n, p):
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k = floor(x)
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vals = special.bdtr(k, n, p)
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return vals
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def _sf(self, x, n, p):
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k = floor(x)
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return special.bdtrc(k, n, p)
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def _ppf(self, q, n, p):
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vals = ceil(special.bdtrik(q, n, p))
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vals1 = np.maximum(vals - 1, 0)
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temp = special.bdtr(vals1, n, p)
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return np.where(temp >= q, vals1, vals)
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def _stats(self, n, p, moments='mv'):
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q = 1.0 - p
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mu = n * p
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var = n * p * q
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g1, g2 = None, None
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if 's' in moments:
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g1 = (q - p) / sqrt(var)
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if 'k' in moments:
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g2 = (1.0 - 6*p*q) / var
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return mu, var, g1, g2
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def _entropy(self, n, p):
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k = np.r_[0:n + 1]
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vals = self._pmf(k, n, p)
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return np.sum(entr(vals), axis=0)
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binom = binom_gen(name='binom')
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class bernoulli_gen(binom_gen):
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r"""A Bernoulli discrete random variable.
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%(before_notes)s
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Notes
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-----
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The probability mass function for `bernoulli` is:
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.. math::
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f(k) = \begin{cases}1-p &\text{if } k = 0\\
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p &\text{if } k = 1\end{cases}
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for :math:`k` in :math:`\{0, 1\}`, :math:`0 \leq p \leq 1`
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`bernoulli` takes :math:`p` as shape parameter,
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where :math:`p` is the probability of a single success
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and :math:`1-p` is the probability of a single failure.
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%(after_notes)s
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%(example)s
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"""
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def _rvs(self, p, size=None, random_state=None):
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return binom_gen._rvs(self, 1, p, size=size, random_state=random_state)
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def _argcheck(self, p):
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return (p >= 0) & (p <= 1)
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def _get_support(self, p):
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# Overrides binom_gen._get_support!x
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return self.a, self.b
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def _logpmf(self, x, p):
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return binom._logpmf(x, 1, p)
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def _pmf(self, x, p):
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# bernoulli.pmf(k) = 1-p if k = 0
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# = p if k = 1
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return binom._pmf(x, 1, p)
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def _cdf(self, x, p):
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return binom._cdf(x, 1, p)
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def _sf(self, x, p):
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return binom._sf(x, 1, p)
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def _ppf(self, q, p):
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return binom._ppf(q, 1, p)
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def _stats(self, p):
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return binom._stats(1, p)
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def _entropy(self, p):
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return entr(p) + entr(1-p)
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bernoulli = bernoulli_gen(b=1, name='bernoulli')
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class betabinom_gen(rv_discrete):
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r"""A beta-binomial discrete random variable.
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%(before_notes)s
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Notes
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-----
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The beta-binomial distribution is a binomial distribution with a
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probability of success `p` that follows a beta distribution.
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The probability mass function for `betabinom` is:
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.. math::
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f(k) = \binom{n}{k} \frac{B(k + a, n - k + b)}{B(a, b)}
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for ``k`` in ``{0, 1,..., n}``, :math:`n \geq 0`, :math:`a > 0`,
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:math:`b > 0`, where :math:`B(a, b)` is the beta function.
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`betabinom` takes :math:`n`, :math:`a`, and :math:`b` as shape parameters.
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References
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----------
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.. [1] https://en.wikipedia.org/wiki/Beta-binomial_distribution
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%(after_notes)s
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.. versionadded:: 1.4.0
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See Also
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--------
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beta, binom
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%(example)s
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"""
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def _rvs(self, n, a, b, size=None, random_state=None):
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p = random_state.beta(a, b, size)
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return random_state.binomial(n, p, size)
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def _get_support(self, n, a, b):
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return 0, n
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def _argcheck(self, n, a, b):
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return (n >= 0) & (a > 0) & (b > 0)
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def _logpmf(self, x, n, a, b):
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k = floor(x)
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combiln = -log(n + 1) - betaln(n - k + 1, k + 1)
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return combiln + betaln(k + a, n - k + b) - betaln(a, b)
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def _pmf(self, x, n, a, b):
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return exp(self._logpmf(x, n, a, b))
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def _stats(self, n, a, b, moments='mv'):
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e_p = a / (a + b)
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e_q = 1 - e_p
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mu = n * e_p
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var = n * (a + b + n) * e_p * e_q / (a + b + 1)
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g1, g2 = None, None
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if 's' in moments:
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g1 = 1.0 / sqrt(var)
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g1 *= (a + b + 2 * n) * (b - a)
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g1 /= (a + b + 2) * (a + b)
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if 'k' in moments:
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g2 = a + b
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g2 *= (a + b - 1 + 6 * n)
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g2 += 3 * a * b * (n - 2)
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g2 += 6 * n ** 2
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g2 -= 3 * e_p * b * n * (6 - n)
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g2 -= 18 * e_p * e_q * n ** 2
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g2 *= (a + b) ** 2 * (1 + a + b)
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g2 /= (n * a * b * (a + b + 2) * (a + b + 3) * (a + b + n))
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g2 -= 3
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return mu, var, g1, g2
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betabinom = betabinom_gen(name='betabinom')
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class nbinom_gen(rv_discrete):
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r"""A negative binomial discrete random variable.
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%(before_notes)s
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Notes
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-----
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Negative binomial distribution describes a sequence of i.i.d. Bernoulli
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trials, repeated until a predefined, non-random number of successes occurs.
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The probability mass function of the number of failures for `nbinom` is:
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.. math::
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f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k
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for :math:`k \ge 0`, :math:`0 < p \leq 1`
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`nbinom` takes :math:`n` and :math:`p` as shape parameters where n is the
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number of successes, :math:`p` is the probability of a single success,
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and :math:`1-p` is the probability of a single failure.
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%(after_notes)s
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%(example)s
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See Also
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--------
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hypergeom, binom, nhypergeom
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"""
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def _rvs(self, n, p, size=None, random_state=None):
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return random_state.negative_binomial(n, p, size)
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def _argcheck(self, n, p):
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return (n > 0) & (p > 0) & (p <= 1)
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def _pmf(self, x, n, p):
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# nbinom.pmf(k) = choose(k+n-1, n-1) * p**n * (1-p)**k
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return exp(self._logpmf(x, n, p))
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def _logpmf(self, x, n, p):
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coeff = gamln(n+x) - gamln(x+1) - gamln(n)
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return coeff + n*log(p) + special.xlog1py(x, -p)
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def _cdf(self, x, n, p):
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k = floor(x)
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return special.betainc(n, k+1, p)
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def _sf_skip(self, x, n, p):
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# skip because special.nbdtrc doesn't work for 0<n<1
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k = floor(x)
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return special.nbdtrc(k, n, p)
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def _ppf(self, q, n, p):
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vals = ceil(special.nbdtrik(q, n, p))
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vals1 = (vals-1).clip(0.0, np.inf)
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temp = self._cdf(vals1, n, p)
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return np.where(temp >= q, vals1, vals)
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def _stats(self, n, p):
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Q = 1.0 / p
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P = Q - 1.0
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mu = n*P
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var = n*P*Q
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g1 = (Q+P)/sqrt(n*P*Q)
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g2 = (1.0 + 6*P*Q) / (n*P*Q)
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return mu, var, g1, g2
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nbinom = nbinom_gen(name='nbinom')
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class geom_gen(rv_discrete):
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r"""A geometric discrete random variable.
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%(before_notes)s
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Notes
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-----
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The probability mass function for `geom` is:
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.. math::
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f(k) = (1-p)^{k-1} p
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for :math:`k \ge 1`, :math:`0 < p \leq 1`
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`geom` takes :math:`p` as shape parameter,
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where :math:`p` is the probability of a single success
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and :math:`1-p` is the probability of a single failure.
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%(after_notes)s
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See Also
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--------
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planck
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%(example)s
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"""
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def _rvs(self, p, size=None, random_state=None):
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return random_state.geometric(p, size=size)
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def _argcheck(self, p):
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return (p <= 1) & (p > 0)
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def _pmf(self, k, p):
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return np.power(1-p, k-1) * p
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def _logpmf(self, k, p):
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return special.xlog1py(k - 1, -p) + log(p)
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def _cdf(self, x, p):
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k = floor(x)
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return -expm1(log1p(-p)*k)
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def _sf(self, x, p):
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return np.exp(self._logsf(x, p))
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def _logsf(self, x, p):
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k = floor(x)
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return k*log1p(-p)
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def _ppf(self, q, p):
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vals = ceil(log1p(-q) / log1p(-p))
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temp = self._cdf(vals-1, p)
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return np.where((temp >= q) & (vals > 0), vals-1, vals)
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def _stats(self, p):
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mu = 1.0/p
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qr = 1.0-p
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var = qr / p / p
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g1 = (2.0-p) / sqrt(qr)
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g2 = np.polyval([1, -6, 6], p)/(1.0-p)
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return mu, var, g1, g2
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geom = geom_gen(a=1, name='geom', longname="A geometric")
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class hypergeom_gen(rv_discrete):
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r"""A hypergeometric discrete random variable.
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The hypergeometric distribution models drawing objects from a bin.
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`M` is the total number of objects, `n` is total number of Type I objects.
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The random variate represents the number of Type I objects in `N` drawn
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without replacement from the total population.
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%(before_notes)s
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Notes
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-----
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The symbols used to denote the shape parameters (`M`, `n`, and `N`) are not
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universally accepted. See the Examples for a clarification of the
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definitions used here.
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The probability mass function is defined as,
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.. math:: p(k, M, n, N) = \frac{\binom{n}{k} \binom{M - n}{N - k}}
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{\binom{M}{N}}
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for :math:`k \in [\max(0, N - M + n), \min(n, N)]`, where the binomial
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coefficients are defined as,
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.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
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%(after_notes)s
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Examples
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--------
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>>> from scipy.stats import hypergeom
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>>> import matplotlib.pyplot as plt
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Suppose we have a collection of 20 animals, of which 7 are dogs. Then if
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we want to know the probability of finding a given number of dogs if we
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choose at random 12 of the 20 animals, we can initialize a frozen
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distribution and plot the probability mass function:
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>>> [M, n, N] = [20, 7, 12]
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>>> rv = hypergeom(M, n, N)
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>>> x = np.arange(0, n+1)
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>>> pmf_dogs = rv.pmf(x)
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>>> fig = plt.figure()
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>>> ax = fig.add_subplot(111)
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>>> ax.plot(x, pmf_dogs, 'bo')
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>>> ax.vlines(x, 0, pmf_dogs, lw=2)
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>>> ax.set_xlabel('# of dogs in our group of chosen animals')
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>>> ax.set_ylabel('hypergeom PMF')
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>>> plt.show()
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Instead of using a frozen distribution we can also use `hypergeom`
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methods directly. To for example obtain the cumulative distribution
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function, use:
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>>> prb = hypergeom.cdf(x, M, n, N)
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And to generate random numbers:
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>>> R = hypergeom.rvs(M, n, N, size=10)
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See Also
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--------
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nhypergeom, binom, nbinom
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"""
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def _rvs(self, M, n, N, size=None, random_state=None):
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return random_state.hypergeometric(n, M-n, N, size=size)
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def _get_support(self, M, n, N):
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return np.maximum(N-(M-n), 0), np.minimum(n, N)
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def _argcheck(self, M, n, N):
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cond = (M > 0) & (n >= 0) & (N >= 0)
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cond &= (n <= M) & (N <= M)
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return cond
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def _logpmf(self, k, M, n, N):
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tot, good = M, n
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bad = tot - good
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result = (betaln(good+1, 1) + betaln(bad+1, 1) + betaln(tot-N+1, N+1) -
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betaln(k+1, good-k+1) - betaln(N-k+1, bad-N+k+1) -
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betaln(tot+1, 1))
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return result
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def _pmf(self, k, M, n, N):
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# same as the following but numerically more precise
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# return comb(good, k) * comb(bad, N-k) / comb(tot, N)
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return exp(self._logpmf(k, M, n, N))
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def _stats(self, M, n, N):
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# tot, good, sample_size = M, n, N
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# "wikipedia".replace('N', 'M').replace('n', 'N').replace('K', 'n')
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M, n, N = 1.*M, 1.*n, 1.*N
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m = M - n
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p = n/M
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mu = N*p
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var = m*n*N*(M - N)*1.0/(M*M*(M-1))
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g1 = (m - n)*(M-2*N) / (M-2.0) * sqrt((M-1.0) / (m*n*N*(M-N)))
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g2 = M*(M+1) - 6.*N*(M-N) - 6.*n*m
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g2 *= (M-1)*M*M
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g2 += 6.*n*N*(M-N)*m*(5.*M-6)
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g2 /= n * N * (M-N) * m * (M-2.) * (M-3.)
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return mu, var, g1, g2
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def _entropy(self, M, n, N):
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k = np.r_[N - (M - n):min(n, N) + 1]
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vals = self.pmf(k, M, n, N)
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return np.sum(entr(vals), axis=0)
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def _sf(self, k, M, n, N):
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# This for loop is needed because `k` can be an array. If that's the
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# case, the sf() method makes M, n and N arrays of the same shape. We
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# therefore unpack all inputs args, so we can do the manual
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# integration.
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res = []
|
|
for quant, tot, good, draw in zip(k, M, n, N):
|
|
# Manual integration over probability mass function. More accurate
|
|
# than integrate.quad.
|
|
k2 = np.arange(quant + 1, draw + 1)
|
|
res.append(np.sum(self._pmf(k2, tot, good, draw)))
|
|
return np.asarray(res)
|
|
|
|
def _logsf(self, k, M, n, N):
|
|
res = []
|
|
for quant, tot, good, draw in zip(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(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
|
|
--------
|
|
>>> 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 _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)
|
|
return cond
|
|
|
|
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 _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` as shape parameter.
|
|
When mu = 0 then at quantile k = 0, ``pmf`` method
|
|
returns `1.0`.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
|
|
# 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 _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 _argcheck(self, lambda_, N):
|
|
return (lambda_ > 0) & (N > 0)
|
|
|
|
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}{high - low}
|
|
|
|
for ``k = low, ..., high - 1``.
|
|
|
|
`randint` takes ``low`` and ``high`` as shape parameters.
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
def _argcheck(self, low, high):
|
|
return (high > low)
|
|
|
|
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 discrete random variable.
|
|
|
|
%(before_notes)s
|
|
|
|
Notes
|
|
-----
|
|
The probability mass function for `zipf` is:
|
|
|
|
.. math::
|
|
|
|
f(k, a) = \frac{1}{\zeta(a) k^a}
|
|
|
|
for :math:`k \ge 1`.
|
|
|
|
`zipf` takes :math:`a` as shape parameter. :math:`\zeta` is the
|
|
Riemann zeta function (`scipy.special.zeta`)
|
|
|
|
%(after_notes)s
|
|
|
|
%(example)s
|
|
|
|
"""
|
|
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')
|
|
|
|
|
|
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 _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 _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):
|
|
px = np.where(x < 0,
|
|
_ncx2_pdf(2*mu2, 2*(1-x), 2*mu1)*2,
|
|
_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,
|
|
_ncx2_cdf(2*mu2, -2*x, 2*mu1),
|
|
1 - _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 _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)
|
|
|
|
|
|
# Collect names of classes and objects in this module.
|
|
pairs = list(globals().items())
|
|
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_discrete)
|
|
|
|
__all__ = _distn_names + _distn_gen_names
|