import numpy as np
from pytensor_distributions import studentt as ptd_studentt
from preliz.distributions.distributions import Continuous
from preliz.internal.distribution_helper import (
all_not_none,
eps,
from_precision,
pytensor_jit,
pytensor_rng_jit,
to_precision,
)
from preliz.internal.optimization import optimize_ml
[docs]
class StudentT(Continuous):
r"""
StudentT's distribution.
Describes a normal variable whose precision is gamma distributed.
The pdf of this distribution is
.. math::
f(x \mid \nu, \mu, \sigma) =
\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\
\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}
.. plot::
:context: close-figs
from preliz import StudentT, style
style.use('preliz-doc')
nus = [2., 5., 5.]
mus = [0., 0., -4.]
sigmas = [1., 1., 2.]
for nu, mu, sigma in zip(nus, mus, sigmas):
StudentT(nu, mu, sigma).plot_pdf(support=(-10,6))
======== ========================
Support :math:`x \in \mathbb{R}`
Mean :math:`\mu` for :math:`\nu > 1`, otherwise undefined
Variance :math:`\frac{\nu}{\nu-2}` for :math:`\nu > 2`,
:math:`\infty` for :math:`1 < \nu \le 2`, otherwise undefined
======== ========================
StudentT distribution has 2 alternative parameterization. In terms of nu, mu and
sigma (standard deviation as nu increases) or nu, mu and lam (precision as nu increases).
The link between the 2 alternatives is given by
.. math::
\lambda = \frac{1}{\sigma^2}
Parameters
----------
nu : float
Degrees of freedom, also known as normality parameter (nu > 0).
mu : float
Location parameter.
sigma : float
Scale parameter (sigma > 0). Converges to the standard deviation as nu
increases.
lam : float
Scale parameter (lam > 0). Converges to the precision as nu increases.
"""
def __init__(self, nu=None, mu=None, sigma=None, lam=None):
super().__init__()
self.support = (-np.inf, np.inf)
self.params_support = ((eps, np.inf), (-np.inf, np.inf), (eps, np.inf))
self._parametrization(nu, mu, sigma, lam)
def _parametrization(self, nu=None, mu=None, sigma=None, lam=None):
if all_not_none(sigma, lam):
raise ValueError(
"Incompatible parametrization. Either use nu, mu and sigma, or nu, mu and lam."
)
self.param_names = ("nu", "mu", "sigma")
self.params_support = ((eps, np.inf), (-np.inf, np.inf), (eps, np.inf))
if lam is not None:
self.lam = lam
sigma = from_precision(lam)
self.param_names = ("nu", "mu", "lam")
self.nu = nu
self.mu = mu
self.sigma = sigma
if all_not_none(self.nu, self.mu, self.sigma):
self._update(self.nu, self.mu, self.sigma)
def _update(self, nu, mu, sigma):
self.nu = np.float64(nu)
self.mu = np.float64(mu)
self.sigma = np.float64(sigma)
self.lam = to_precision(sigma)
if self.param_names[2] == "sigma":
self.params = (self.nu, self.mu, self.sigma)
elif self.param_names[2] == "lam":
self.params = (self.nu, self.mu, self.lam)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.nu, self.mu, self.sigma)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.nu, self.mu, self.sigma)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.nu, self.mu, self.sigma)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.nu, self.mu, self.sigma)
[docs]
def entropy(self):
return ptd_entropy(self.nu, self.mu, self.sigma)
[docs]
def mean(self):
return ptd_mean(self.nu, self.mu, self.sigma)
[docs]
def mode(self):
return ptd_mode(self.nu, self.mu, self.sigma)
[docs]
def var(self):
return ptd_var(self.nu, self.mu, self.sigma)
[docs]
def std(self):
return ptd_std(self.nu, self.mu, self.sigma)
[docs]
def skewness(self):
return ptd_skewness(self.nu, self.mu, self.sigma)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.nu, self.mu, self.sigma)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.nu, self.mu, self.sigma)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.nu, self.mu, self.sigma)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.nu, self.mu, self.sigma)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.nu, self.mu, self.sigma)
[docs]
def rvs(self, size=None, random_state=None):
random_state = np.random.default_rng(random_state)
return ptd_rvs(self.nu, self.mu, self.sigma, size=size, rng=random_state)
def _fit_moments(self, mean, sigma):
# if nu is smaller than 2 the variance is not defined,
# so if that happens we use 2.1 as an approximation
nu = self.nu
if nu is None:
nu = 100
elif nu <= 2:
nu = 2.1
else:
sigma = sigma / (nu / (nu - 2)) ** 0.5
self._update(nu, mean, sigma)
def _fit_mle(self, sample):
optimize_ml(self, sample)
@pytensor_jit
def ptd_pdf(x, nu, mu, sigma):
return ptd_studentt.pdf(x, nu, mu, sigma)
@pytensor_jit
def ptd_cdf(x, nu, mu, sigma):
return ptd_studentt.cdf(x, nu, mu, sigma)
@pytensor_jit
def ptd_ppf(q, nu, mu, sigma):
return ptd_studentt.ppf(q, nu, mu, sigma)
@pytensor_jit
def ptd_logpdf(x, nu, mu, sigma):
return ptd_studentt.logpdf(x, nu, mu, sigma)
@pytensor_jit
def ptd_entropy(nu, mu, sigma):
return ptd_studentt.entropy(nu, mu, sigma)
@pytensor_jit
def ptd_mean(nu, mu, sigma):
return ptd_studentt.mean(nu, mu, sigma)
@pytensor_jit
def ptd_mode(nu, mu, sigma):
return ptd_studentt.mode(nu, mu, sigma)
@pytensor_jit
def ptd_median(nu, mu, sigma):
return ptd_studentt.median(nu, mu, sigma)
@pytensor_jit
def ptd_var(nu, mu, sigma):
return ptd_studentt.var(nu, mu, sigma)
@pytensor_jit
def ptd_std(nu, mu, sigma):
return ptd_studentt.std(nu, mu, sigma)
@pytensor_jit
def ptd_skewness(nu, mu, sigma):
return ptd_studentt.skewness(nu, mu, sigma)
@pytensor_jit
def ptd_kurtosis(nu, mu, sigma):
return ptd_studentt.kurtosis(nu, mu, sigma)
@pytensor_jit
def ptd_lmoment1(nu, mu, sigma):
return ptd_studentt.lmoment1(nu, mu, sigma)
@pytensor_jit
def ptd_lmoment2(nu, mu, sigma):
return ptd_studentt.lmoment2(nu, mu, sigma)
@pytensor_jit
def ptd_lmoment3(nu, mu, sigma):
return ptd_studentt.lmoment3(nu, mu, sigma)
@pytensor_jit
def ptd_lmoment4(nu, mu, sigma):
return ptd_studentt.lmoment4(nu, mu, sigma)
@pytensor_rng_jit
def ptd_rvs(nu, mu, sigma, size, rng):
return ptd_studentt.rvs(nu, mu, sigma, size=size, random_state=rng)
