import numpy as np
from pytensor_distributions import halfstudentt as ptd_halfstudentt
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
from preliz.internal.special import (
gamma,
)
[docs]
class HalfStudentT(Continuous):
r"""
HalfStudentT Distribution.
The pdf of this distribution is
.. math::
f(x \mid \sigma,\nu) =
\frac{2\;\Gamma\left(\frac{\nu+1}{2}\right)}
{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\nu\pi\sigma^2}}
\left(1+\frac{1}{\nu}\frac{x^2}{\sigma^2}\right)^{-\frac{\nu+1}{2}}
.. plot::
:context: close-figs
from preliz import HalfStudentT, style
style.use('preliz-doc')
sigmas = [1., 2., 2.]
nus = [3, 3., 10.]
for sigma, nu in zip(sigmas, nus):
HalfStudentT(nu, sigma).plot_pdf(support=(0,10))
======== ==========================================
Support :math:`x \in [0, \infty)`
Mean .. math::
2\sigma\sqrt{\frac{\nu}{\pi}}\
\frac{\Gamma\left(\frac{\nu+1}{2}\right)}
{\Gamma\left(\frac{\nu}{2}\right)(\nu-1)}\, \text{for } \nu > 2
Variance .. math::
\sigma^2\left(\frac{\nu}{\nu - 2}-\
\frac{4\nu}{\pi(\nu-1)^2}\left(\frac{\Gamma\left(\frac{\nu+1}{2}\right)}
{\Gamma\left(\frac{\nu}{2}\right)}\right)^2\right) \text{for } \nu > 2\, \infty\
\text{for } 1 < \nu \le 2\, \text{otherwise undefined}
======== ==========================================
HalfStudentT distribution has 2 alternative parameterizations. In terms of nu and
sigma (standard deviation as nu increases) or nu 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).
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, sigma=None, lam=None):
super().__init__()
self.support = (0, np.inf)
self._parametrization(nu, sigma, lam)
def _parametrization(self, nu=None, sigma=None, lam=None):
if all_not_none(sigma, lam):
raise ValueError(
"Incompatible parametrization. Either use nu and sigma, or nu and lam."
)
self.param_names = ("nu", "sigma")
self.params_support = ((eps, np.inf), (eps, np.inf))
if lam is not None:
self.lam = lam
sigma = from_precision(lam)
self.param_names = ("nu", "lam")
self.nu = nu
self.sigma = sigma
if all_not_none(self.nu, self.sigma):
self._update(self.nu, self.sigma)
def _update(self, nu, sigma):
self.nu = np.float64(nu)
self.sigma = np.float64(sigma)
self.lam = to_precision(self.sigma)
if self.param_names[1] == "sigma":
self.params = (self.nu, self.sigma)
elif self.param_names[1] == "lam":
self.params = (self.nu, self.lam)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.nu, self.sigma)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.nu, self.sigma)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.nu, self.sigma)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.nu, self.sigma)
[docs]
def entropy(self):
return ptd_entropy(self.nu, self.sigma)
[docs]
def mean(self):
return ptd_mean(self.nu, self.sigma)
[docs]
def mode(self):
return ptd_mode(self.nu, self.sigma)
[docs]
def var(self):
return ptd_var(self.nu, self.sigma)
[docs]
def std(self):
return ptd_std(self.nu, self.sigma)
[docs]
def skewness(self):
return ptd_skewness(self.nu, self.sigma)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.nu, self.sigma)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.nu, self.sigma)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.nu, self.sigma)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.nu, self.sigma)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.nu, 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.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
gamma0 = gamma((nu + 1) / 2)
gamma1 = gamma(nu / 2)
if np.isfinite(gamma0) and np.isfinite(gamma1):
sigma = (
sigma**2
/ ((nu / (nu - 2)) - ((4 * nu) / (np.pi * (nu - 1) ** 2)) * (gamma0 / gamma1) ** 2)
) ** 0.5
else:
# we assume a Gaussian for large nu
sigma = sigma / (1 - 2 / np.pi) ** 0.5
self._update(nu, sigma)
def _fit_mle(self, sample):
optimize_ml(self, sample)
@pytensor_jit
def ptd_pdf(x, nu, sigma):
return ptd_halfstudentt.pdf(x, nu, sigma)
@pytensor_jit
def ptd_cdf(x, nu, sigma):
return ptd_halfstudentt.cdf(x, nu, sigma)
@pytensor_jit
def ptd_ppf(q, nu, sigma):
return ptd_halfstudentt.ppf(q, nu, sigma)
@pytensor_jit
def ptd_logpdf(x, nu, sigma):
return ptd_halfstudentt.logpdf(x, nu, sigma)
@pytensor_jit
def ptd_entropy(nu, sigma):
return ptd_halfstudentt.entropy(nu, sigma)
@pytensor_jit
def ptd_mean(nu, sigma):
return ptd_halfstudentt.mean(nu, sigma)
@pytensor_jit
def ptd_mode(nu, sigma):
return ptd_halfstudentt.mode(nu, sigma)
@pytensor_jit
def ptd_median(nu, sigma):
return ptd_halfstudentt.median(nu, sigma)
@pytensor_jit
def ptd_var(nu, sigma):
return ptd_halfstudentt.var(nu, sigma)
@pytensor_jit
def ptd_std(nu, sigma):
return ptd_halfstudentt.std(nu, sigma)
@pytensor_jit
def ptd_skewness(nu, sigma):
return ptd_halfstudentt.skewness(nu, sigma)
@pytensor_jit
def ptd_kurtosis(nu, sigma):
return ptd_halfstudentt.kurtosis(nu, sigma)
@pytensor_jit
def ptd_lmoment1(nu, sigma):
return ptd_halfstudentt.lmoment1(nu, sigma)
@pytensor_jit
def ptd_lmoment2(nu, sigma):
return ptd_halfstudentt.lmoment2(nu, sigma)
@pytensor_jit
def ptd_lmoment3(nu, sigma):
return ptd_halfstudentt.lmoment3(nu, sigma)
@pytensor_jit
def ptd_lmoment4(nu, sigma):
return ptd_halfstudentt.lmoment4(nu, sigma)
@pytensor_rng_jit
def ptd_rvs(nu, sigma, size, rng):
return ptd_halfstudentt.rvs(nu, sigma, size=size, random_state=rng)
