# 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 multivariate-normal 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.

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 likelihood

• 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.

Returns

Log Likelihood

Return type

float

property params

The covariance parameters for the Gaussian copula

static pobs(data, ties='average')

Compute the pseudo-observations for the given data matrix

Parameters
• 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

Returns

matrix or vector of the same dimension as data containing the pseudo observations

Return type

ndarray

pseudo_obs()

The pseudo-observations function

property sigma

The covariance matrix for the elliptical copula

Returns

numpy array Covariance matrix for elliptical copula

summary()[source]

Constructs the summary information about the copula