import numpy as np
from pytensor_distributions import pareto as ptd_pareto
from preliz.distributions.distributions import Continuous
from preliz.internal.distribution_helper import all_not_none, eps, pytensor_jit, pytensor_rng_jit
from preliz.internal.optimization import optimize_ml
[docs]
class Pareto(Continuous):
r"""
Pareto distribution.
The pdf of this distribution is
.. math::
f(x \mid \alpha, m) = \frac{\alpha m^{\alpha}}{x^{\alpha+1}}
.. plot::
:context: close-figs
from preliz import Pareto, style
style.use('preliz-doc')
alphas = [1., 5., 5.]
ms = [1., 1., 2.]
for alpha, m in zip(alphas, ms):
Pareto(alpha, m).plot_pdf(support=(0,4))
======== =============================================================
Support :math:`x \in [m, \infty)`
Mean :math:`\dfrac{\alpha m}{\alpha - 1}` for :math:`\alpha \ge 1`
Variance :math:`\dfrac{m^2 \alpha}{(\alpha - 1)^2 (\alpha - 2)}` for :math:`\alpha > 2`
======== =============================================================
Parameters
----------
alpha : float
Shape parameter (alpha > 0).
m : float
Scale parameter (m > 0).
"""
def __init__(self, alpha=None, m=None):
super().__init__()
self.support = (0, np.inf)
self._parametrization(alpha, m)
def _parametrization(self, alpha=None, m=None):
self.alpha = alpha
self.m = m
self.params = (self.alpha, self.m)
self.param_names = ("alpha", "m")
self.params_support = ((eps, np.inf), (eps, np.inf))
if all_not_none(alpha, m):
self._update(alpha, m)
def _update(self, alpha, m):
self.alpha = np.float64(alpha)
self.m = np.float64(m)
self.support = (self.m, np.inf)
self.params = (self.alpha, self.m)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.alpha, self.m)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.alpha, self.m)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.alpha, self.m)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.alpha, self.m)
[docs]
def entropy(self):
return ptd_entropy(self.alpha, self.m)
[docs]
def mean(self):
return ptd_mean(self.alpha, self.m)
[docs]
def mode(self):
return ptd_mode(self.alpha, self.m)
[docs]
def var(self):
return ptd_var(self.alpha, self.m)
[docs]
def std(self):
return ptd_std(self.alpha, self.m)
[docs]
def skewness(self):
return ptd_skewness(self.alpha, self.m)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.alpha, self.m)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.alpha, self.m)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.alpha, self.m)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.alpha, self.m)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.alpha, self.m)
[docs]
def rvs(self, size=None, random_state=None):
random_state = np.random.default_rng(random_state)
return ptd_rvs(self.alpha, self.m, size=size, rng=random_state)
def _fit_moments(self, mean, sigma):
alpha = 1 + (1 + (mean / sigma) ** 2) ** (1 / 2)
m = (alpha - 1) * mean / alpha
self._update(alpha, m)
def _fit_mle(self, sample):
optimize_ml(self, sample)
@pytensor_jit
def ptd_pdf(x, alpha, m):
return ptd_pareto.pdf(x, alpha, m)
@pytensor_jit
def ptd_cdf(x, alpha, m):
return ptd_pareto.cdf(x, alpha, m)
@pytensor_jit
def ptd_ppf(q, alpha, m):
return ptd_pareto.ppf(q, alpha, m)
@pytensor_jit
def ptd_logpdf(x, alpha, m):
return ptd_pareto.logpdf(x, alpha, m)
@pytensor_jit
def ptd_entropy(alpha, m):
return ptd_pareto.entropy(alpha, m)
@pytensor_jit
def ptd_mean(alpha, m):
return ptd_pareto.mean(alpha, m)
@pytensor_jit
def ptd_mode(alpha, m):
return ptd_pareto.mode(alpha, m)
@pytensor_jit
def ptd_median(alpha, m):
return ptd_pareto.median(alpha, m)
@pytensor_jit
def ptd_var(alpha, m):
return ptd_pareto.var(alpha, m)
@pytensor_jit
def ptd_std(alpha, m):
return ptd_pareto.std(alpha, m)
@pytensor_jit
def ptd_skewness(alpha, m):
return ptd_pareto.skewness(alpha, m)
@pytensor_jit
def ptd_kurtosis(alpha, m):
return ptd_pareto.kurtosis(alpha, m)
@pytensor_jit
def ptd_lmoment1(alpha, m):
return ptd_pareto.lmoment1(alpha, m)
@pytensor_jit
def ptd_lmoment2(alpha, m):
return ptd_pareto.lmoment2(alpha, m)
@pytensor_jit
def ptd_lmoment3(alpha, m):
return ptd_pareto.lmoment3(alpha, m)
@pytensor_jit
def ptd_lmoment4(alpha, m):
return ptd_pareto.lmoment4(alpha, m)
@pytensor_rng_jit
def ptd_rvs(alpha, m, size, rng):
return ptd_pareto.rvs(alpha, m, size=size, random_state=rng)
