import numpy as np
from pytensor_distributions import negativebinomial as ptd_negativebinomial
from preliz.distributions.distributions import Discrete
from preliz.internal.distribution_helper import (
all_not_none,
any_not_none,
eps,
pytensor_jit,
pytensor_rng_jit,
)
from preliz.internal.optimization import optimize_mean_sigma, optimize_ml
[docs]
class NegativeBinomial(Discrete):
R"""
Negative binomial distribution.
The negative binomial distribution describes a Poisson random variable
whose rate parameter is gamma distributed.
Its pmf, parametrized by the parameters alpha and mu of the gamma distribution, is
.. math::
f(x \mid \mu, \alpha) =
\binom{x + \alpha - 1}{x}
(\alpha/(\mu+\alpha))^\alpha (\mu/(\mu+\alpha))^x
.. plot::
:context: close-figs
from preliz import NegativeBinomial, style
style.use('preliz-doc')
mus = [1, 2, 8]
alphas = [0.9, 2, 4]
for mu, alpha in zip(mus, alphas):
NegativeBinomial(mu, alpha).plot_pdf(support=(0, 20))
======== ==========================
Support :math:`x \in \mathbb{N}_0`
Mean :math:`\mu`
Variance :math:`\frac{\mu (\alpha + \mu)}{\alpha}`
======== ==========================
The negative binomial distribution can be parametrized either in terms of mu and alpha,
or in terms of n and p. The link between the parametrizations is given by
.. math::
p &= \frac{\alpha}{\mu + \alpha} \\
n &= \alpha
If it is parametrized in terms of n and p, the negative binomial describes the probability
to have x failures before the n-th success, given the probability p of success in each trial.
Its pmf is
.. math::
f(x \mid n, p) =
\binom{x + n - 1}{x}
(p)^n (1 - p)^x
Parameters
----------
alpha : float
Gamma distribution shape parameter (alpha > 0).
mu : float
Gamma distribution mean (mu > 0).
p : float
Probability of success in each trial (0 < p < 1).
n : float
Number of target success trials (n > 0)
"""
def __init__(self, mu=None, alpha=None, p=None, n=None):
super().__init__()
self.support = (0, np.inf)
self._parametrization(mu, alpha, p, n)
def _parametrization(self, mu=None, alpha=None, p=None, n=None):
if any_not_none(mu, alpha) and any_not_none(p, n):
raise ValueError("Incompatible parametrization. Either use mu and alpha, or p and n.")
self.param_names = ("mu", "alpha")
self.params_support = ((eps, np.inf), (eps, np.inf))
if any_not_none(p, n):
self.p = p
self.n = n
self.param_names = ("p", "n")
if all_not_none(p, n):
mu, alpha = self._from_n_p(n, p)
self.mu = mu
self.alpha = alpha
if all_not_none(mu, alpha):
self._update(mu, alpha)
def _from_n_p(self, n, p):
mu = n * (1 / p - 1)
return mu, n
def _to_n_p(self, mu, alpha):
p = alpha / (mu + alpha)
return alpha, p
def _update(self, mu, alpha):
self.mu = np.float64(mu)
self.alpha = np.float64(alpha)
self.n, self.p = self._to_n_p(self.mu, self.alpha)
if self.param_names[0] == "mu":
self.params = (self.mu, self.alpha)
elif self.param_names[0] == "p":
self.params = (self.p, self.n)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.n, self.p)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.n, self.p)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.n, self.p)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.n, self.p)
[docs]
def entropy(self):
return ptd_entropy(self.n, self.p)
[docs]
def mean(self):
return ptd_mean(self.n, self.p)
[docs]
def mode(self):
return ptd_mode(self.n, self.p)
[docs]
def var(self):
return ptd_var(self.n, self.p)
[docs]
def std(self):
return ptd_std(self.n, self.p)
[docs]
def skewness(self):
return ptd_skewness(self.n, self.p)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.n, self.p)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.n, self.p)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.n, self.p)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.n, self.p)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.n, self.p)
[docs]
def rvs(self, size=None, random_state=None):
random_state = np.random.default_rng(random_state)
return ptd_rvs(self.n, self.p, size=size, rng=random_state)
def _fit_moments(self, mean, sigma=None):
optimize_mean_sigma(self, mean, sigma)
def _fit_mle(self, sample):
optimize_ml(self, sample)
@pytensor_jit
def ptd_pdf(x, n, p):
return ptd_negativebinomial.pdf(x, n, p)
@pytensor_jit
def ptd_cdf(x, n, p):
return ptd_negativebinomial.cdf(x, n, p)
@pytensor_jit
def ptd_ppf(q, n, p):
return ptd_negativebinomial.ppf(q, n, p)
@pytensor_jit
def ptd_logpdf(x, n, p):
return ptd_negativebinomial.logpdf(x, n, p)
@pytensor_jit
def ptd_entropy(n, p):
return ptd_negativebinomial.entropy(n, p)
@pytensor_jit
def ptd_mean(n, p):
return ptd_negativebinomial.mean(n, p)
@pytensor_jit
def ptd_mode(n, p):
return ptd_negativebinomial.mode(n, p)
@pytensor_jit
def ptd_median(n, p):
return ptd_negativebinomial.median(n, p)
@pytensor_jit
def ptd_var(n, p):
return ptd_negativebinomial.var(n, p)
@pytensor_jit
def ptd_std(n, p):
return ptd_negativebinomial.std(n, p)
@pytensor_jit
def ptd_skewness(n, p):
return ptd_negativebinomial.skewness(n, p)
@pytensor_jit
def ptd_kurtosis(n, p):
return ptd_negativebinomial.kurtosis(n, p)
@pytensor_jit
def ptd_lmoment1(n, p):
return ptd_negativebinomial.lmoment1(n, p)
@pytensor_jit
def ptd_lmoment2(n, p):
return ptd_negativebinomial.lmoment2(n, p)
@pytensor_jit
def ptd_lmoment3(n, p):
return ptd_negativebinomial.lmoment3(n, p)
@pytensor_jit
def ptd_lmoment4(n, p):
return ptd_negativebinomial.lmoment4(n, p)
@pytensor_rng_jit
def ptd_rvs(n, p, size, rng):
return ptd_negativebinomial.rvs(n, p, size=size, random_state=rng)
