The implementation of the Student (T) copula is such that all the univariate marginal distributions are student and the multivariate joint distribution is a multivariate-student distribution.
Note that the degrees-of-freedom parameter is shared by all univariate margins as well as the joint multivariate distribution
copulae.elliptical.
StudentCopula
The Student (T) Copula. It is elliptical and symmetric which gives it nice analytical properties. The Student copula is determined by its correlation matrix and the degrees of freedom. Student copulas have fatter tails as compared to Gaussian copulas.
A Student copula is fined as
where \(\Sigma\) and \(\nu\) are the covariance matrix and degrees of freedom which describes the Student copula and \(t_\nu^{-1}\) is the quantile (inverse cdf) function
bounds
Gets the bounds for the parameters
Lower and upper bound of the copula’s parameters
(scalar or array_like, scalar or array_like)
dim
Number of dimensions in copula
fit
Fit the copula with specified data
data (ndarray) – Array of data used to fit copula. Usually, data should be the pseudo observations
x0 (ndarray) – Initial starting point. If value is None, best starting point will be estimated
method ({ 'ml', 'irho', 'itau' }, optional) – Method of fitting. Supported methods are: ‘ml’ - Maximum Likelihood, ‘irho’ - Inverse Spearman Rho, ‘itau’ - Inverse Kendall Tau
optim_options (dict, optional) – Keyword arguments to pass into scipy.optimize.minimize()
scipy.optimize.minimize()
ties ({ 'average', 'min', 'max', 'dense', 'ordinal' }, optional) – Specifies how ranks should be computed if there are ties in any of the coordinate samples. This is effective only if the data has not been converted to its pseudo observations form
verbose (int, optional) – Log level for the estimator. The higher the number, the more verbose it is. 0 prints nothing.
to_pobs (bool) – If True, casts the input data along the column axis to uniform marginals (i.e. convert variables to values between [0, 1]). Set this to False if the input data are already uniform marginals.
scale (float) – Amount to scale the objective function value of the numerical optimizer. This is helpful in achieving higher accuracy as it increases the sensitivity of the optimizer. The downside is that the optimizer could likely run longer as a result. Defaults to 1.
fix_df (bool, optional) – If True, the degree of freedom specified by the user (param) is fixed and will not change. Otherwise, the degree of freedom is subject to changes during the fitting phase.
See also
scipy.optimize.minimize
the scipy minimize function use for optimization
irho
irho is not implemented for t copula
log_lik
Returns the log likelihood (LL) of the copula given the data.
The greater the LL (closer to \(\infty\)) the better.
data (ndarray) – Data set used to calculate the log likelihood
ndarray
to_pobs – If True, converts the data input to pseudo observations.
ties – Specifies how ranks should be computed if there are ties in any of the coordinate samples. This is effective only if to_pobs is True.
to_pobs
Log Likelihood
float
params
The parameters of the Student copula. A tuple where the first value is the degrees of freedom and the subsequent values are the correlation matrix parameters
A dataclass with 2 properties
Degrees of freedom
Correlation parameters
StudentParams
pobs
Compute the pseudo-observations for the given data matrix
data ({ array_like, DataFrame }) – Random variates to be converted to pseudo-observations
ties ({ 'average', 'min', 'max', 'dense', 'ordinal' }, optional) – Specifies how ranks should be computed if there are ties in any of the coordinate samples
matrix or vector of the same dimension as data containing the pseudo observations
pseudo_obs()
The pseudo-observations function
sigma
The covariance matrix for the elliptical copula
numpy array Covariance matrix for elliptical copula