Implicit Copulas: An Overview

Deep distributional time series models and the probabilistic forecasting of intraday electricity prices

“…Those defined in Section 3 have additive homoscedastic errors and are stationary. Moreover, it is straightforward to show (e.g., Smith, 2022) that the implicit copula of a stationary process is also stationary. When this copula is combined with an invariant margin $$$ {p}_Y $$$, as in Section 4, so is the resulting copula time series model.…”

“…The implicit copula of a Gaussian distribution is called a Gaussian copula, and is constructed for () by standardizing the distribution; see, for example, Smith (2022). Let $$$ {\boldsymbol{Z}}_{(t)}={\left({Z}_1,\dots, {Z}_t\right)}^{\prime }={\sigma}^{-1}S{\tilde{\boldsymbol{Z}}}_{(t)} $$$, where $$$ S=\operatorname{diag}\left({\psi}_1,\dots, {\psi}_t\right) $$$ is a diagonal scaling matrix with elements $$$ {\psi}_s={\left(1+{\boldsymbol{b}}_s^{\prime }{\boldsymbol{b}}_s/{\tau}^2\right)}^{-1/2} $$$, and $$$ {\boldsymbol{b}}_s $$$ is the $$$ s $$$‐th row of $$$ {B}_{\xi}\left({X}_{(t)}\right) $$$.…”

“…The implicit copula of a Gaussian distribution is called a Gaussian copula, and is constructed for (9) by standardizing the distribution; see, for example, Smith (2022). Let…”

“…It uses the implicit copula of the time series vector from a Gaussian probabilistic ESN of the type described above. By an 'implicit copula' we mean the copula that is implicit in a multivariate distribution and obtained by inverting Sklar's theorem; see Smith (2022) for an introduction to implicit copulas. This copula is both a deep function of the feature vector and also a 'Gaussian copula process'.…”

Implicit copula variational inference

“…Different choices for F (ψ; π) produce different copula families. These include Gaussian copulas, t copulas, elliptical copulas, skew t copulas, factor copulas and copula processes; Smith (2021) provides an overview of the broad class of implicit copulas.…”