import numpy as np
from pytensor_distributions import exgaussian as ptd_exgaussian
from scipy.stats import skew
from preliz.distributions.distributions import Continuous
from preliz.internal.distribution_helper import all_not_none, eps, pytensor_jit, pytensor_rng_jit
from preliz.internal.special import mean_and_std
[docs]
class ExGaussian(Continuous):
r"""
Exponentially modified Gaussian (EMG) Distribution.
Results from the convolution of a normal distribution with an exponential
distribution.
The pdf of this distribution is
.. math::
f(x \mid \mu, \sigma, \nu) =
\frac{1}{\nu}\;
\exp\left\{\frac{\mu-x}{\nu}+\frac{\sigma^2}{2\nu^2}\right\}
\Phi\left(\frac{x-\mu}{\sigma}-\frac{\sigma}{\nu}\right)
where :math:`\Phi` is the cumulative distribution function of the
standard normal distribution.
.. plot::
:context: close-figs
from preliz import ExGaussian, style
style.use('preliz-doc')
mus = [0., 0., -3.]
sigmas = [1., 3., 1.]
nus = [1., 1., 4.]
for mu, sigma, nu in zip(mus, sigmas, nus):
ExGaussian(mu, sigma, nu).plot_pdf(support=(-6,9))
======== ========================
Support :math:`x \in \mathbb{R}`
Mean :math:`\mu + \nu`
Variance :math:`\sigma^2 + \nu^2`
======== ========================
Parameters
----------
mu : float
Mean of the normal distribution.
sigma : float
Standard deviation of the normal distribution (sigma > 0).
nu : float
Mean of the exponential distribution (nu > 0).
"""
def __init__(self, mu=None, sigma=None, nu=None):
super().__init__()
self.support = (-np.inf, np.inf)
self._parametrization(mu, sigma, nu)
def _parametrization(self, mu=None, sigma=None, nu=None):
self.mu = mu
self.sigma = sigma
self.nu = nu
self.param_names = ("mu", "sigma", "nu")
self.params = (mu, sigma, nu)
# if nu is too small we get a non-smooth distribution
self.params_support = ((-np.inf, np.inf), (eps, np.inf), (1e-4, np.inf))
if all_not_none(mu, sigma, nu):
self._update(mu, sigma, nu)
def _update(self, mu, sigma, nu):
self.mu = np.float64(mu)
self.sigma = np.float64(sigma)
self.nu = np.float64(nu)
self.params = (self.mu, self.sigma, self.nu)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.mu, self.sigma, self.nu)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.mu, self.sigma, self.nu)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.mu, self.sigma, self.nu)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.mu, self.sigma, self.nu)
[docs]
def entropy(self):
return ptd_entropy(self.mu, self.sigma, self.nu)
[docs]
def mean(self):
return ptd_mean(self.mu, self.sigma, self.nu)
[docs]
def var(self):
return ptd_var(self.mu, self.sigma, self.nu)
[docs]
def std(self):
return ptd_std(self.mu, self.sigma, self.nu)
[docs]
def skewness(self):
return ptd_skewness(self.mu, self.sigma, self.nu)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.mu, self.sigma, self.nu)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.mu, self.sigma, self.nu)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.mu, self.sigma, self.nu)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.mu, self.sigma, self.nu)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.mu, self.sigma, self.nu)
[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.nu, size=size, rng=random_state)
def _fit_moments(self, mean, sigma):
# Just assume this is approximately Gaussian
self._update(mean, sigma, 1e-4)
def _fit_mle(self, sample):
mean, std = mean_and_std(sample)
skweness = max(1e-4, skew(sample))
nu = std * (skweness / 2) ** (1 / 3)
mu = mean - nu
var = std**2 * (1 - (skweness / 2) ** (2 / 3))
self._update(mu, var**0.5, nu)
@pytensor_jit
def ptd_pdf(x, mu, sigma, nu):
return ptd_exgaussian.pdf(x, mu, sigma, nu)
@pytensor_jit
def ptd_cdf(x, mu, sigma, nu):
return ptd_exgaussian.cdf(x, mu, sigma, nu)
@pytensor_jit
def ptd_ppf(q, mu, sigma, nu):
return ptd_exgaussian.ppf(q, mu, sigma, nu)
@pytensor_jit
def ptd_logpdf(x, mu, sigma, nu):
return ptd_exgaussian.logpdf(x, mu, sigma, nu)
@pytensor_jit
def ptd_entropy(mu, sigma, nu):
return ptd_exgaussian.entropy(mu, sigma, nu)
@pytensor_jit
def ptd_mean(mu, sigma, nu):
return ptd_exgaussian.mean(mu, sigma, nu)
@pytensor_jit
def ptd_mode(mu, sigma, nu):
return ptd_exgaussian.mode(mu, sigma, nu)
@pytensor_jit
def ptd_median(mu, sigma, nu):
return ptd_exgaussian.median(mu, sigma, nu)
@pytensor_jit
def ptd_var(mu, sigma, nu):
return ptd_exgaussian.var(mu, sigma, nu)
@pytensor_jit
def ptd_std(mu, sigma, nu):
return ptd_exgaussian.std(mu, sigma, nu)
@pytensor_jit
def ptd_skewness(mu, sigma, nu):
return ptd_exgaussian.skewness(mu, sigma, nu)
@pytensor_jit
def ptd_kurtosis(mu, sigma, nu):
return ptd_exgaussian.kurtosis(mu, sigma, nu)
@pytensor_jit
def ptd_lmoment1(mu, sigma, nu):
return ptd_exgaussian.lmoment1(mu, sigma, nu)
@pytensor_jit
def ptd_lmoment2(mu, sigma, nu):
return ptd_exgaussian.lmoment2(mu, sigma, nu)
@pytensor_jit
def ptd_lmoment3(mu, sigma, nu):
return ptd_exgaussian.lmoment3(mu, sigma, nu)
@pytensor_jit
def ptd_lmoment4(mu, sigma, nu):
return ptd_exgaussian.lmoment4(mu, sigma, nu)
@pytensor_rng_jit
def ptd_rvs(mu, sigma, nu, size, rng):
return ptd_exgaussian.rvs(mu, sigma, nu, size=size, random_state=rng)
