vineknockoffs.copulas.GaussianCopula#

class vineknockoffs.copulas.GaussianCopula(par=nan)#

Gaussian copula (bivariate).

Parameters#

parfloat: Parameter of the Gaussian copula. Default is np.nan.

Examples#

# ToDo: add an example here

Notes#

The pdf of the Gaussian copula with parameter \(\theta \in (-1, 1)\) is given by

\[c(u, v; \theta) = \frac{1}{\sqrt{1-\theta^2}} \exp\bigg(- \frac{\theta^2 (x^2 + y^2) - 2 \theta x y}{2 (1-\theta^2)}\bigg),\]

with \(x = \Phi^{-1}(u)\) and \(y = \Phi^{-1}(v)\).

Methods

`aic`(u, v)	Evaluate the copula AIC.
`cdf`(u, v)	Evaluate the copula cdf.
`d_cdf_d_par`(u, v)	Evaluate the derivative of the copula cdf wrt the parameter.
`d_hfun_d_par`(u, v)	Evaluate the derivative of the copula hfun wrt the parameter.
`d_hfun_d_u`(u, v)	Evaluate the derivative of the copula hfun wrt u.
`d_hfun_d_v`(u, v)	Evaluate the derivative of the copula hfun wrt v.
`d_inv_hfun_d_par`(u, v)	Evaluate the derivative of the inverse of the copula hfun wrt the parameter.
`d_inv_hfun_d_u`(u, v)	Evaluate the derivative of the inverse of the copula hfun wrt u.
`d_inv_hfun_d_v`(u, v)	Evaluate the derivative of the inverse of the copula hfun wrt v.
`d_inv_vfun_d_par`(u, v)	Evaluate the derivative of the inverse of the copula vfun wrt the parameter.
`d_inv_vfun_d_u`(u, v)	Evaluate the derivative of the inverse of the copula vfun wrt u.
`d_inv_vfun_d_v`(u, v)	Evaluate the derivative of the inverse of the copula vfun wrt v.
`d_vfun_d_par`(u, v)	Evaluate the derivative of the copula vfun wrt the parameter.
`d_vfun_d_u`(u, v)	Evaluate the derivative of the copula vfun wrt u.
`d_vfun_d_v`(u, v)	Evaluate the derivative of the copula vfun wrt v.
`hfun`(u, v)	Evaluate the copula hfun.
`inv_hfun`(u, v)	Evaluate the inverse of the copula hfun.
`inv_vfun`(u, v)	Evaluate the inverse of the copula vfun.
`ll`(u, v)	Evaluate the copula log-likelihood.
`mle_est`(u, v)	Estimation of the copula parameter with maximum likelihood.
`par2tau`(par)	Compute Kendall's tau associated with the given value for the copula parameter.
`pdf`(u, v)	Evaluate the copula pdf.
`sim`([n_obs])	Simulate random observations from the copula.
`tau2par`(tau)	Compute the copula parameter associated with the given value for Kendall's tau.
`vfun`(u, v[, u_])	Evaluate the copula vfun.

Attributes

`n_pars`	The number of copula parameters.
`par`	Parameter of the copula.
`rotation`	Rotation of the copula (0, 90, 180 or 270).

GaussianCopula.aic(u, v)#

Evaluate the copula AIC.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.cdf(u, v)#

Evaluate the copula cdf.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_cdf_d_par(u, v)#

Evaluate the derivative of the copula cdf wrt the parameter.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_hfun_d_par(u, v)#

Evaluate the derivative of the copula hfun wrt the parameter.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_hfun_d_u(u, v)#

Evaluate the derivative of the copula hfun wrt u.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_hfun_d_v(u, v)#

Evaluate the derivative of the copula hfun wrt v.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_inv_hfun_d_par(u, v)#

Evaluate the derivative of the inverse of the copula hfun wrt the parameter.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_inv_hfun_d_u(u, v)#

Evaluate the derivative of the inverse of the copula hfun wrt u.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_inv_hfun_d_v(u, v)#

Evaluate the derivative of the inverse of the copula hfun wrt v.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_inv_vfun_d_par(u, v)#

Evaluate the derivative of the inverse of the copula vfun wrt the parameter.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_inv_vfun_d_u(u, v)#

Evaluate the derivative of the inverse of the copula vfun wrt u.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_inv_vfun_d_v(u, v)#

Evaluate the derivative of the inverse of the copula vfun wrt v.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_vfun_d_par(u, v)#

Evaluate the derivative of the copula vfun wrt the parameter.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_vfun_d_u(u, v)#

Evaluate the derivative of the copula vfun wrt u.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.d_vfun_d_v(u, v)#

Evaluate the derivative of the copula vfun wrt v.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.hfun(u, v)#

Evaluate the copula hfun.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.inv_hfun(u, v)#

Evaluate the inverse of the copula hfun.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.inv_vfun(u, v)#

Evaluate the inverse of the copula vfun.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.ll(u, v)#

Evaluate the copula log-likelihood.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.mle_est(u, v)#

Estimation of the copula parameter with maximum likelihood.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

self : object

GaussianCopula.par2tau(par)#

Compute Kendall’s tau associated with the given value for the copula parameter.

Parameters#

parfloat: The copula parameter

Returns#

taufloat: Kendall’s tau.

GaussianCopula.pdf(u, v)#

Evaluate the copula pdf.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.sim(n_obs=100)#

Simulate random observations from the copula.

Parameters#

n_obs :: The number of observations to simulate.

Returns#

resnumpy.ndarray: Result array.

GaussianCopula.tau2par(tau)#

Compute the copula parameter associated with the given value for Kendall’s tau.

Parameters#

taufloat: Kendall’s tau.

Returns#

parfloat: The copula parameter.

GaussianCopula.vfun(u, v, u_=None)#

Evaluate the copula vfun.

Parameters#

unumpy.ndarray: Array of observations for the first variable.
vnumpy.ndarray: Array of observations for the second variable.

Returns#

resnumpy.ndarray: Result array.