import numpy as np
from pytensor_distributions import skewnormal as ptd_skewnormal
from scipy.stats import skew
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_mean_sigma, optimize_ml
[docs]
class SkewNormal(Continuous):
r"""
SkewNormal distribution.
The pdf of this distribution is
.. math::
f(x \mid \mu, \tau, \alpha) =
2 \Phi((x-\mu)\sqrt{\tau}\alpha) \phi(x,\mu,\tau)
.. plot::
:context: close-figs
from preliz import SkewNormal, style
style.use('preliz-doc')
for alpha in [-6, 0, 6]:
SkewNormal(mu=0, sigma=1, alpha=alpha).plot_pdf()
======== ==========================================
Support :math:`x \in \mathbb{R}`
Mean :math:`\mu + \sigma \sqrt{\frac{2}{\pi}} \frac {\alpha }{{\sqrt {1+\alpha ^{2}}}}`
Variance :math:`\sigma^2 \left( 1-\frac{2\alpha^2}{(\alpha^2+1) \pi} \right)`
======== ==========================================
SkewNormal distribution has 2 alternative parameterizations. In terms of mu, sigma (standard
deviation) and alpha, or mu, tau (precision) and alpha.
The link between the 2 alternatives is given by
.. math::
\tau = \frac{1}{\sigma^2}
Parameters
----------
mu : float
Location parameter.
sigma : float
Scale parameter (sigma > 0).
alpha : float
Skewness parameter.
tau : float
Precision (tau > 0).
Notes
-----
When alpha=0 we recover the Normal distribution and mu becomes the mean,
and sigma the standard deviation. In the limit of alpha approaching
plus/minus infinite we get a half-normal distribution.
"""
def __init__(self, mu=None, sigma=None, alpha=None, tau=None):
super().__init__()
self.support = (-np.inf, np.inf)
self._parametrization(mu, sigma, alpha, tau)
def _parametrization(self, mu=None, sigma=None, alpha=None, tau=None):
if all_not_none(sigma, tau):
raise ValueError(
"Incompatible parametrization. Either use mu, sigma and alpha,"
" or mu, tau and alpha."
)
self.param_names = ("mu", "sigma", "alpha")
self.params_support = ((-np.inf, np.inf), (eps, np.inf), (-np.inf, np.inf))
if tau is not None:
self.tau = tau
sigma = from_precision(tau)
self.param_names = ("mu", "tau", "alpha")
self.mu = mu
self.sigma = sigma
self.alpha = alpha
if all_not_none(self.mu, self.sigma, self.alpha):
self._update(self.mu, self.sigma, self.alpha)
def _update(self, mu, sigma, alpha):
self.mu = np.float64(mu)
self.sigma = np.float64(sigma)
self.alpha = np.float64(alpha)
self.tau = to_precision(sigma)
if self.param_names[1] == "sigma":
self.params = (self.mu, self.sigma, self.alpha)
elif self.param_names[1] == "tau":
self.params = (self.mu, self.tau, self.alpha)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.mu, self.sigma, self.alpha)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.mu, self.sigma, self.alpha)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.mu, self.sigma, self.alpha)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.mu, self.sigma, self.alpha)
[docs]
def entropy(self):
return ptd_entropy(self.mu, self.sigma, self.alpha)
[docs]
def mean(self):
return ptd_mean(self.mu, self.sigma, self.alpha)
[docs]
def var(self):
return ptd_var(self.mu, self.sigma, self.alpha)
[docs]
def std(self):
return ptd_std(self.mu, self.sigma, self.alpha)
[docs]
def skewness(self):
return ptd_skewness(self.mu, self.sigma, self.alpha)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.mu, self.sigma, self.alpha)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.mu, self.sigma, self.alpha)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.mu, self.sigma, self.alpha)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.mu, self.sigma, self.alpha)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.mu, self.sigma, self.alpha)
[docs]
def rvs(self, size=None, random_state=None):
random_state = np.random.default_rng(random_state)
return ptd_rvs(self.mu, self.sigma, self.alpha, size=size, rng=random_state)
def _fit_moments(self, mean, sigma):
if self.alpha is None:
self.alpha = 0
optimize_mean_sigma(self, mean, sigma)
def _fit_mle(self, sample):
skewness = skew(sample)
self.alpha = skewness / (1 - skewness**2) ** 0.5
optimize_ml(self, sample)
@pytensor_jit
def ptd_pdf(x, mu, sigma, alpha):
return ptd_skewnormal.pdf(x, mu, sigma, alpha)
@pytensor_jit
def ptd_cdf(x, mu, sigma, alpha):
return ptd_skewnormal.cdf(x, mu, sigma, alpha)
@pytensor_jit
def ptd_ppf(q, mu, sigma, alpha):
return ptd_skewnormal.ppf(q, mu, sigma, alpha)
@pytensor_jit
def ptd_logpdf(x, mu, sigma, alpha):
return ptd_skewnormal.logpdf(x, mu, sigma, alpha)
@pytensor_jit
def ptd_entropy(mu, sigma, alpha):
return ptd_skewnormal.entropy(mu, sigma, alpha)
@pytensor_jit
def ptd_mean(mu, sigma, alpha):
return ptd_skewnormal.mean(mu, sigma, alpha)
@pytensor_jit
def ptd_mode(mu, sigma, alpha):
return ptd_skewnormal.mode(mu, sigma, alpha)
@pytensor_jit
def ptd_median(mu, sigma, alpha):
return ptd_skewnormal.median(mu, sigma, alpha)
@pytensor_jit
def ptd_var(mu, sigma, alpha):
return ptd_skewnormal.var(mu, sigma, alpha)
@pytensor_jit
def ptd_std(mu, sigma, alpha):
return ptd_skewnormal.std(mu, sigma, alpha)
@pytensor_jit
def ptd_skewness(mu, sigma, alpha):
return ptd_skewnormal.skewness(mu, sigma, alpha)
@pytensor_jit
def ptd_kurtosis(mu, sigma, alpha):
return ptd_skewnormal.kurtosis(mu, sigma, alpha)
@pytensor_jit
def ptd_lmoment1(mu, sigma, alpha):
return ptd_skewnormal.lmoment1(mu, sigma, alpha)
@pytensor_jit
def ptd_lmoment2(mu, sigma, alpha):
return ptd_skewnormal.lmoment2(mu, sigma, alpha)
@pytensor_jit
def ptd_lmoment3(mu, sigma, alpha):
return ptd_skewnormal.lmoment3(mu, sigma, alpha)
@pytensor_jit
def ptd_lmoment4(mu, sigma, alpha):
return ptd_skewnormal.lmoment4(mu, sigma, alpha)
@pytensor_rng_jit
def ptd_rvs(mu, sigma, alpha, size, rng):
return ptd_skewnormal.rvs(mu, sigma, alpha, size=size, random_state=rng)
