Gaussian¶
The implementation of the Gaussian (Normal) copula is such that all the univariate marginal distributions are normal and the multivariate joint distribution is a multivariatenormal distribution.

class
copulae.elliptical.
GaussianCopula
(*args, **kwds)[source]¶ The Gaussian (Normal) copula. It is elliptical and symmetric which gives it nice analytical properties. The Gaussian copula is determined entirely by its correlation matrix.
Gaussian copulas do not model tail dependencies very well, it’s tail is flat. Take not that by symmetry, it gives equal weight to tail scenarios. In English, this means upside scenarios happen as often as downside scenarios.
A Gaussian copula is fined as
\[C_\Sigma (u_1, \dots, u_d) = \Phi_\Sigma (N^{1} (u_1), \dots, N^{1} (u_d))\]where \(\Sigma\) is the covariance matrix which is the parameter of the Gaussian copula and \(N^{1}\) is the quantile (inverse cdf) function

property
bounds
¶ Gets the bounds for the parameters
 Returns
Lower and upper bound of the copula’s parameters
 Return type
(scalar or array_like, scalar or array_like)

property
dim
¶ Number of dimensions in copula

fit
(data, x0=None, method='ml', optim_options=None, ties='average', verbose=1, **kwargs)¶ Fit the copula with specified data
 Parameters
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()
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.
kwargs – Other keyword arguments. See Notes for more details
Notes
Other valid keyword arguments and their purpose
scale
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.
See also
scipy.optimize.minimize
the scipy minimize function use for optimization

log_lik
(data, *, to_pobs=True, ties='average')¶ Returns the log likelihood (LL) of the copula given the data.
The greater the LL (closer to \(\infty\)) the better.
 Parameters
data (
ndarray
) – Data set used to calculate the log likelihoodto_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.
 Returns
Log Likelihood
 Return type
float

property
params
¶ The covariance parameters for the Gaussian copula

static
pobs
(data, ties='average')¶ Compute the pseudoobservations for the given data matrix
 Parameters
data ({ array_like, DataFrame }) – Random variates to be converted to pseudoobservations
ties ({ 'average', 'min', 'max', 'dense', 'ordinal' }, optional) – Specifies how ranks should be computed if there are ties in any of the coordinate samples
 Returns
matrix or vector of the same dimension as data containing the pseudo observations
 Return type
ndarray
See also
pseudo_obs()
The pseudoobservations function

property
sigma
¶ The covariance matrix for the elliptical copula
 Returns
numpy array Covariance matrix for elliptical copula

property