mvem.stats.multivariate_hyperbolic

mvem.stats.multivariate_hyperbolic.fit

mvem.stats.multivariate_hyperbolic.fit(x, alpha_bar=1, symmetric=False, standardize=False, nit=2000, reltol=1e-08, abstol=1e-07, 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.

Parameters

• x (np.ndarray) – An array of shape (n, p) containing n observations of some p-variate data with n > p.

• alpha_bar (float, optional) – The initial value of alpha_bar, a positive real number, where alpha_bar = chi = psi.

• symmetric (bool, optional) – Whether to fit a symmetric distribution or not. Default to False.

• standardize (bool, optional) – Whether to standardize the data before fitting or not. Default to False.

• nit (int, optional) – The maximum number of iterations to use in the EM algorithm. Defaults to 2000.

• reltol (float, optional) – The relative convergence criterion for the log-likelihood function. Defaults to 1e-8.

• abstol (float, optional) – The relative convergence criterion for the log-likelihood function. Defaults to 1e-7.

• silent (bool, optional) – Whether to print the log-likelihoods and parameter estimates during fitting or not. Defaults to False.

• fmu (np.ndarray, optional) – If fmu!=None, force mu to fmu. Defaults to None.

• fsigma (np.ndarray, optional) – If fsigma!=None, force sigma to fsigma. Defaults to None.

• fgamma (np.ndarray, optional) – If fgamma!=None, force gamma to fgamma. Defaults to None.

• return_loglike (np.ndarray, optional) – Return a list of log-likelihood values at each iteration. Defaults to False.

Returns

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.

Return type

tuple

mvem.stats.multivariate_hyperbolic.loglike

mvem.stats.multivariate_hyperbolic.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.

Parameters

• x (np.ndarray) – An array of shape (n, p) containing n observations of some p-variate data with n > p.

• chi (float) – Univariate parameter > 0.

• psi (float) – Univariate parameter > 0.

• mu (np.ndarray) – Location parameter with shape (p,).

• sigma (np.ndarray) – A positive semi-definite array with shape (p, p).

• gamma (np.ndarray) – Parameter with shape (p,).

Returns

The log-likelihood for given all observations and parameters.

Return type

float

mvem.stats.multivariate_hyperbolic.logpdf

mvem.stats.multivariate_hyperbolic.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.

Parameters

• x (np.ndarray) – An array of shape (n, p) containing n observations of some p-variate data with n > p.

• chi (float) – Univariate parameter > 0.

• psi (float) – Univariate parameter > 0.

• mu (np.ndarray) – Location parameter with shape (p,).

• sigma (np.ndarray) – A positive semi-definite array with shape (p, p).

• gamma (np.ndarray) – Parameter with shape (p,).

Returns

The log-density at each observation.

Return type

np.ndarray with shape (n,).

mvem.stats.multivariate_hyperbolic.mean

mvem.stats.multivariate_hyperbolic.mean(chi, psi, mu, sigma, gamma)

Mean function of the multivariate hyperbolic distribution. We use the (lmbda, chi, psi, mu, sigma, gamma)-parameterisation.

Parameters

• chi (float) – Univariate parameter > 0.

• psi (float) – Univariate parameter > 0.

• mu (np.ndarray) – Location parameter with shape (p,).

• sigma (np.ndarray) – A positive semi-definite array with shape (p, p).

• gamma (np.ndarray) – Parameter with shape (p,).

Returns

The mean of the specified distribution.

Return type

np.ndarray with shape (p,).

mvem.stats.multivariate_hyperbolic.pdf

mvem.stats.multivariate_hyperbolic.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.

Parameters

• x (np.ndarray) – An array of shape (n, p) containing n observations of some p-variate data with n > p.

• chi (float) – Univariate parameter > 0.

• psi (float) – Univariate parameter > 0.

• mu (np.ndarray) – Location parameter with shape (p,).

• sigma (np.ndarray) – A positive semi-definite array with shape (p, p).

• gamma (np.ndarray) – Parameter with shape (p,).

Returns

The density at each observation.

Return type

np.ndarray with shape (n,).

mvem.stats.multivariate_hyperbolic.rvs

mvem.stats.multivariate_hyperbolic.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.

Parameters

• chi (float) – Univariate parameter > 0.

• psi (float) – Univariate parameter > 0.

• mu (np.ndarray) – Location parameter with shape (p,).

• sigma (np.ndarray) – A positive semi-definite array with shape (p, p).

• gamma (np.ndarray) – Parameter with shape (p,).

• size (int) – The number of samples to draw.

Returns

The random p-variate numbers generated.

Return type

np.ndarray with shape (n, p).

mvem.stats.multivariate_hyperbolic.var

mvem.stats.multivariate_hyperbolic.var(chi, psi, mu, sigma, gamma)

Variance function of the multivariate hyperbolic distribution. We use the (lmbda, chi, psi, mu, sigma, gamma)-parameterisation.

Parameters

• chi (float) – Univariate parameter > 0.

• psi (float) – Univariate parameter > 0.

• mu (np.ndarray) – Location parameter with shape (p,).

• sigma (np.ndarray) – A positive semi-definite array with shape (p, p).

• gamma (np.ndarray) – Parameter with shape (p,).

Returns

The variance of the specified distribution.

Return type

np.ndarray with shape (n,).