import numpy as np
from mvem.stats import multivariate_genhyperbolic as ghypmv
[docs]def logpdf(x, chi, psi, mu, sigma, gamma):
"""
Log-probability density function of the multivariate hyperbolic distribution.
We use the (lmbda, chi, psi, mu, sigma, gamma)-parameterisation.
:param x: An array of shape (n, p) containing n observations of some
p-variate data with n > p.
:type x: np.ndarray
:param chi: Univariate parameter > 0.
:type chi: float
:param psi: Univariate parameter > 0.
:type psi: float
:param mu: Location parameter with shape (p,).
:type mu: np.ndarray
:param sigma: A positive semi-definite array with shape (p, p).
:type sigma: np.ndarray
:param gamma: Parameter with shape (p,).
:type gamma: np.ndarray
:return: The log-density at each observation.
:rtype: np.ndarray with shape (n,).
"""
lmbda = (len(mu)+1)/2
return ghypmv.logpdf(x, lmbda, chi, psi, mu, sigma, gamma)
[docs]def pdf(x, chi, psi, mu, sigma, gamma):
"""
Probability density function of the multivariate hyperbolic distribution.
We use the (lmbda, chi, psi, mu, sigma, gamma)-parameterisation.
:param x: An array of shape (n, p) containing n observations of some
p-variate data with n > p.
:type x: np.ndarray
:param chi: Univariate parameter > 0.
:type chi: float
:param psi: Univariate parameter > 0.
:type psi: float
:param mu: Location parameter with shape (p,).
:type mu: np.ndarray
:param sigma: A positive semi-definite array with shape (p, p).
:type sigma: np.ndarray
:param gamma: Parameter with shape (p,).
:type gamma: np.ndarray
:return: The density at each observation.
:rtype: np.ndarray with shape (n,).
"""
lmbda = (len(mu)+1)/2
return ghypmv.pdf(x, lmbda, chi, psi, mu, sigma, gamma)
[docs]def loglike(x, chi, psi, mu, sigma, gamma):
"""
Log-likelihood function of the multivariate hyperbolic distribution.
We use the (lmbda, chi, psi, mu, sigma, gamma)-parameterisation.
:param x: An array of shape (n, p) containing n observations of some
p-variate data with n > p.
:type x: np.ndarray
:param chi: Univariate parameter > 0.
:type chi: float
:param psi: Univariate parameter > 0.
:type psi: float
:param mu: Location parameter with shape (p,).
:type mu: np.ndarray
:param sigma: A positive semi-definite array with shape (p, p).
:type sigma: np.ndarray
:param gamma: Parameter with shape (p,).
:type gamma: np.ndarray
:return: The log-likelihood for given all observations and parameters.
:rtype: float
"""
return np.sum(logpdf(x, chi, psi, mu, sigma, gamma))
[docs]def rvs(chi, psi, mu, sigma, gamma, size):
"""
Random number generator of the multivariate hyperbolic distribution.
We use the (lmbda, chi, psi, mu, sigma, gamma)-parameterisation.
:param chi: Univariate parameter > 0.
:type chi: float
:param psi: Univariate parameter > 0.
:type psi: float
:param mu: Location parameter with shape (p,).
:type mu: np.ndarray
:param sigma: A positive semi-definite array with shape (p, p).
:type sigma: np.ndarray
:param gamma: Parameter with shape (p,).
:type gamma: np.ndarray
:param size: The number of samples to draw.
:type size: int
:return: The random p-variate numbers generated.
:rtype: np.ndarray with shape (n, p).
"""
lmbda = (len(mu)+1)/2
return ghypmv.rvs(lmbda, chi, psi, mu, sigma, gamma, size)
[docs]def mean(chi, psi, mu, sigma, gamma):
"""
Mean function of the multivariate hyperbolic distribution. We use
the (lmbda, chi, psi, mu, sigma, gamma)-parameterisation.
:param chi: Univariate parameter > 0.
:type chi: float
:param psi: Univariate parameter > 0.
:type psi: float
:param mu: Location parameter with shape (p,).
:type mu: np.ndarray
:param sigma: A positive semi-definite array with shape (p, p).
:type sigma: np.ndarray
:param gamma: Parameter with shape (p,).
:type gamma: np.ndarray
:return: The mean of the specified distribution.
:rtype: np.ndarray with shape (p,).
"""
lmbda = (len(mu)+1)/2
return ghypmv.mean(lmbda, chi, psi, mu, sigma, gamma)
[docs]def var(chi, psi, mu, sigma, gamma):
"""
Variance function of the multivariate hyperbolic distribution. We use
the (lmbda, chi, psi, mu, sigma, gamma)-parameterisation.
:param chi: Univariate parameter > 0.
:type chi: float
:param psi: Univariate parameter > 0.
:type psi: float
:param mu: Location parameter with shape (p,).
:type mu: np.ndarray
:param sigma: A positive semi-definite array with shape (p, p).
:type sigma: np.ndarray
:param gamma: Parameter with shape (p,).
:type gamma: np.ndarray
:return: The variance of the specified distribution.
:rtype: np.ndarray with shape (n,).
"""
lmbda = (len(mu)+1)/2
return ghypmv.var(lmbda, chi, psi, mu, sigma, gamma)
[docs]def fit(x, alpha_bar=1, symmetric=False, standardize=False, nit=2000, reltol=1e-8,
abstol=1e-7, silent=False, fmu=None, fsigma=None, fgamma=None, return_loglike=False):
"""
Estimate the parameters of the hyperbolic distribution. We use the
(lmbda, chi, psi, mu, sigma, gamma)-parameterisation.
:param x: An array of shape (n, p) containing n observations of some
p-variate data with n > p.
:type x: np.ndarray
:param alpha_bar: The initial value of alpha_bar, a positive real number,
where alpha_bar = chi = psi.
:type alpha_bar: float, optional
:param symmetric: Whether to fit a symmetric distribution or not. Default
to False.
:type symmetric: bool, optional
:param standardize: Whether to standardize the data before fitting or not.
Default to False.
:type standardize: bool, optional
:param nit: The maximum number of iterations to use in the EM algorithm.
Defaults to 2000.
:type nit: int, optional
:param reltol: The relative convergence criterion for the log-likelihood
function. Defaults to 1e-8.
:type reltol: float, optional
:param abstol: The relative convergence criterion for the log-likelihood
function. Defaults to 1e-7.
:type abstol: float, optional
:param silent: Whether to print the log-likelihoods and parameter estimates
during fitting or not. Defaults to False.
:type silent: bool, optional
:param fmu: If fmu!=None, force mu to fmu. Defaults to None.
:type fmu: np.ndarray, optional
:param fsigma: If fsigma!=None, force sigma to fsigma. Defaults to None.
:type fsigma: np.ndarray, optional
:param fgamma: If fgamma!=None, force gamma to fgamma. Defaults to None.
:type fgamma: np.ndarray, optional
:param return_loglike: Return a list of log-likelihood values at each iteration.
Defaults to False.
:type return_loglike: np.ndarray, optional
:return: The fitted parameters (<float> chi, <float> psi, <array> mu, <array> sigma,
<array> gamma). Also returns a list of log-likelihood values at each iteration
of the EM algorithm if ``return_loglike=True``.
:rtype: tuple
"""
opt_pars = {"lmbda": False, "alpha_bar": True, "mu": fmu is None,
"sigma": fsigma is None, "gamma": fgamma is None}
lmbda = (x.shape[1]+1)/2
fit = ghypmv.fitghypmv(
x, lmbda=lmbda, alpha_bar=alpha_bar, mu=fmu, sigma=fsigma, gamma=fgamma,
symmetric=symmetric, standardize=standardize, nit=nit, reltol=reltol,
abstol=abstol, silent=silent, opt_pars=opt_pars)
chi, psi = ghypmv._alphabar2chipsi(fit["alpha_bar"], fit["lmbda"])
if return_loglike:
return fit["lmbda"], chi, psi, fit["mu"], fit["sigma"], fit["gamma"], fit["ll"]
return fit["lmbda"], chi, psi, fit["mu"], fit["sigma"], fit["gamma"]