import numpy as np
from pytensor_distributions import wald as ptd_wald
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 Wald(Continuous):
r"""
Wald distribution.
The pdf of this distribution is
.. math::
f(x \mid \mu, \lambda) =
\left(\frac{\lambda}{2\pi}\right)^{1/2} x^{-3/2}
\exp\left\{
-\frac{\lambda}{2x}\left(\frac{x-\mu}{\mu}\right)^2
\right\}
.. plot::
:context: close-figs
from preliz import Wald, style
style.use('preliz-doc')
mus = [1., 1.]
lams = [1., 3.]
for mu, lam in zip(mus, lams):
Wald(mu, lam).plot_pdf(support=(0,4))
======== =============================
Support :math:`x \in (0, \infty)`
Mean :math:`\mu`
Variance :math:`\dfrac{\mu^3}{\lambda}`
======== =============================
Wald distribution has 3 alternative parametrizations. In terms of mu and lam,
mu and phi or lam and phi.
The link between the 3 alternatives is given by
.. math::
\phi = \dfrac{\lambda}{\mu}
Parameters
----------
mu : float
Mean of the distribution (mu > 0).
lam : float
Relative precision (lam > 0).
phi : float
Shape parameter (phi > 0).
"""
def __init__(self, mu=None, lam=None, phi=None):
super().__init__()
self.support = (0, np.inf)
self._parametrization(mu, lam, phi)
def _parametrization(self, mu=None, lam=None, phi=None):
if all_not_none(mu, lam, phi):
raise ValueError(
"Incompatible parametrization. Either use mu and lam or mu and phi or lam and phi."
)
self.param_names = ("mu", "lam")
self.params_support = ((eps, np.inf), (eps, np.inf))
if phi is not None:
self.phi = phi
if (mu and phi) is not None:
lam = self._from_mu_phi(mu, phi)
self.param_names = ("mu", "phi")
elif (lam and phi) is not None:
mu = self._from_lam_phi(lam, phi)
self.param_names = ("lam", "phi")
self.mu = mu
self.lam = lam
if all_not_none(self.mu, self.lam):
self._update(self.mu, self.lam)
def _from_mu_phi(self, mu, phi):
lam = mu * phi
return lam
def _from_lam_phi(self, lam, phi):
mu = lam / phi
return mu
def _to_phi(self, mu, lam):
phi = lam / mu
return phi
def _update(self, mu, lam):
self.mu = np.float64(mu)
self.lam = np.float64(lam)
self.phi = self._to_phi(self.mu, self.lam)
if self.param_names == ("mu", "lam"):
self.params = (self.mu, self.lam)
elif self.param_names == ("mu", "phi"):
self.params = (self.mu, self.phi)
elif self.param_names == ("lam", "phi"):
self.params = (self.lam, self.phi)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.mu, self.lam)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.mu, self.lam)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.mu, self.lam)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.mu, self.lam)
[docs]
def entropy(self):
return ptd_entropy(self.mu, self.lam)
[docs]
def mean(self):
return ptd_mean(self.mu, self.lam)
[docs]
def mode(self):
return ptd_mode(self.mu, self.lam)
[docs]
def var(self):
return ptd_var(self.mu, self.lam)
[docs]
def std(self):
return ptd_std(self.mu, self.lam)
[docs]
def skewness(self):
return ptd_skewness(self.mu, self.lam)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.mu, self.lam)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.mu, self.lam)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.mu, self.lam)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.mu, self.lam)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.mu, self.lam)
[docs]
def rvs(self, size=None, random_state=None):
random_state = np.random.default_rng(random_state)
return ptd_rvs(self.mu, self.lam, size=size, rng=random_state)
def _fit_moments(self, mean, sigma):
lam = mean**3 / sigma**2
self._update(mean, lam)
def _fit_mle(self, sample):
optimize_ml(self, sample)
@pytensor_jit
def ptd_pdf(x, mu, lam):
return ptd_wald.pdf(x, mu, lam)
@pytensor_jit
def ptd_cdf(x, mu, lam):
return ptd_wald.cdf(x, mu, lam)
@pytensor_jit
def ptd_ppf(q, mu, lam):
return ptd_wald.ppf(q, mu, lam)
@pytensor_jit
def ptd_logpdf(x, mu, lam):
return ptd_wald.logpdf(x, mu, lam)
@pytensor_jit
def ptd_entropy(mu, lam):
return ptd_wald.entropy(mu, lam)
@pytensor_jit
def ptd_mean(mu, lam):
return ptd_wald.mean(mu, lam)
@pytensor_jit
def ptd_mode(mu, lam):
return ptd_wald.mode(mu, lam)
@pytensor_jit
def ptd_median(mu, lam):
return ptd_wald.median(mu, lam)
@pytensor_jit
def ptd_var(mu, lam):
return ptd_wald.var(mu, lam)
@pytensor_jit
def ptd_std(mu, lam):
return ptd_wald.std(mu, lam)
@pytensor_jit
def ptd_skewness(mu, lam):
return ptd_wald.skewness(mu, lam)
@pytensor_jit
def ptd_kurtosis(mu, lam):
return ptd_wald.kurtosis(mu, lam)
@pytensor_jit
def ptd_lmoment1(mu, lam):
return ptd_wald.lmoment1(mu, lam)
@pytensor_jit
def ptd_lmoment2(mu, lam):
return ptd_wald.lmoment2(mu, lam)
@pytensor_jit
def ptd_lmoment3(mu, lam):
return ptd_wald.lmoment3(mu, lam)
@pytensor_jit
def ptd_lmoment4(mu, lam):
return ptd_wald.lmoment4(mu, lam)
@pytensor_rng_jit
def ptd_rvs(mu, lam, size, rng):
return ptd_wald.rvs(mu, lam, size=size, random_state=rng)
