import numpy as np
from pytensor_distributions import vonmises as ptd_vonmises
from scipy.stats import circmean
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 find_kappa, optimize_mean_sigma
[docs]
class VonMises(Continuous):
r"""
Univariate VonMises distribution.
The pdf of this distribution is
.. math::
f(x \mid \mu, \kappa) =
\frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}
where :math:`I_0` is the modified Bessel function of order 0.
.. plot::
:context: close-figs
from preliz import VonMises, style
style.use('preliz-doc')
mus = [0., 0., 0., -2.5]
kappas = [.01, 0.5, 4., 2.]
for mu, kappa in zip(mus, kappas):
VonMises(mu, kappa).plot_pdf(support=(-np.pi,np.pi))
======== ==========================================
Support :math:`x \in [-\pi, \pi]`
Mean :math:`\mu`
Variance :math:`1-\frac{I_1(\kappa)}{I_0(\kappa)}`
======== ==========================================
Parameters
----------
mu : float
Mean.
kappa : float
Concentration (:math:`\frac{1}{\kappa}` is analogous to :math:`\kappa^2`).
"""
def __init__(self, mu=None, kappa=None):
super().__init__()
self._parametrization(mu, kappa)
def _parametrization(self, mu=None, kappa=None):
self.mu = mu
self.kappa = kappa
self.param_names = ("mu", "kappa")
self.params_support = ((-np.pi, np.pi), (eps, np.inf))
self.support = (-np.pi, np.pi)
if all_not_none(mu, kappa):
self._update(mu, kappa)
def _update(self, mu, kappa):
self.mu = np.float64(mu)
self.kappa = np.float64(kappa)
self.params = (self.mu, self.kappa)
self.is_frozen = True
[docs]
def pdf(self, x):
return ptd_pdf(x, self.mu, self.kappa)
[docs]
def cdf(self, x):
return ptd_cdf(x, self.mu, self.kappa)
[docs]
def ppf(self, q):
return ptd_ppf(q, self.mu, self.kappa)
[docs]
def logpdf(self, x):
return ptd_logpdf(x, self.mu, self.kappa)
[docs]
def entropy(self):
return ptd_entropy(self.mu, self.kappa)
[docs]
def mean(self):
return ptd_mean(self.mu, self.kappa)
[docs]
def mode(self):
return ptd_mode(self.mu, self.kappa)
[docs]
def var(self):
return ptd_var(self.mu, self.kappa)
[docs]
def std(self):
return ptd_std(self.mu, self.kappa)
[docs]
def skewness(self):
return ptd_skewness(self.mu, self.kappa)
[docs]
def kurtosis(self):
return ptd_kurtosis(self.mu, self.kappa)
[docs]
def lmoment1(self):
return ptd_lmoment1(self.mu, self.kappa)
[docs]
def lmoment2(self):
return ptd_lmoment2(self.mu, self.kappa)
[docs]
def lmoment3(self):
return ptd_lmoment3(self.mu, self.kappa)
[docs]
def lmoment4(self):
return ptd_lmoment4(self.mu, self.kappa)
[docs]
def rvs(self, size=None, random_state=None):
random_state = np.random.default_rng(random_state)
return ptd_rvs(self.mu, self.kappa, size=size, rng=random_state)
def _fit_moments(self, mean, sigma):
params = mean, 1 / sigma**1.8
optimize_mean_sigma(self, mean, sigma, params)
def _fit_mle(self, sample):
data = np.mod(sample, 2 * np.pi)
mu = circmean(data)
kappa = find_kappa(data, mu)
mu = np.mod(mu + np.pi, 2 * np.pi) - np.pi
self._update(mu, kappa)
[docs]
def eti(self, mass=None, fmt=".2f"):
mean = self.mu
self.mu = 0
hdi_min, hdi_max = super().eti(mass=mass, fmt=fmt)
self.mu = mean
return _warp_interval(hdi_min, hdi_max, self.mu, fmt)
[docs]
def hdi(self, mass=None, fmt=".2f"):
mean = self.mu
self.mu = 0
hdi_min, hdi_max = super().hdi(mass=mass, fmt=fmt)
self.mu = mean
return _warp_interval(hdi_min, hdi_max, self.mu, fmt)
def _warp_interval(hdi_min, hdi_max, mu, fmt):
hdi_min = hdi_min + mu
hdi_max = hdi_max + mu
lower_tail = np.arctan2(np.sin(hdi_min), np.cos(hdi_min))
upper_tail = np.arctan2(np.sin(hdi_max), np.cos(hdi_max))
if fmt != "none":
lower_tail = float(f"{lower_tail:{fmt}}")
upper_tail = float(f"{upper_tail:{fmt}}")
return (lower_tail, upper_tail)
@pytensor_jit
def ptd_pdf(x, mu, kappa):
return ptd_vonmises.pdf(x, mu, kappa)
@pytensor_jit
def ptd_cdf(x, mu, kappa):
return ptd_vonmises.cdf(x, mu, kappa)
@pytensor_jit
def ptd_ppf(q, mu, kappa):
return ptd_vonmises.ppf(q, mu, kappa)
@pytensor_jit
def ptd_logpdf(x, mu, kappa):
return ptd_vonmises.logpdf(x, mu, kappa)
@pytensor_jit
def ptd_entropy(mu, kappa):
return ptd_vonmises.entropy(mu, kappa)
@pytensor_jit
def ptd_mean(mu, kappa):
return ptd_vonmises.mean(mu, kappa)
@pytensor_jit
def ptd_mode(mu, kappa):
return ptd_vonmises.mode(mu, kappa)
@pytensor_jit
def ptd_median(mu, kappa):
return ptd_vonmises.median(mu, kappa)
@pytensor_jit
def ptd_var(mu, kappa):
return ptd_vonmises.var(mu, kappa)
@pytensor_jit
def ptd_std(mu, kappa):
return ptd_vonmises.std(mu, kappa)
@pytensor_jit
def ptd_skewness(mu, kappa):
return ptd_vonmises.skewness(mu, kappa)
@pytensor_jit
def ptd_kurtosis(mu, kappa):
return ptd_vonmises.kurtosis(mu, kappa)
@pytensor_jit
def ptd_lmoment1(mu, kappa):
return ptd_vonmises.lmoment1(mu, kappa)
@pytensor_jit
def ptd_lmoment2(mu, kappa):
return ptd_vonmises.lmoment2(mu, kappa)
@pytensor_jit
def ptd_lmoment3(mu, kappa):
return ptd_vonmises.lmoment3(mu, kappa)
@pytensor_jit
def ptd_lmoment4(mu, kappa):
return ptd_vonmises.lmoment4(mu, kappa)
@pytensor_rng_jit
def ptd_rvs(mu, kappa, size, rng):
return ptd_vonmises.rvs(mu, kappa, size=size, random_state=rng)
