Modeling Archimedean, Extreme-Value and Archimax Copulas with Neural Networks

Copulas are popular in high-dimensional statistical applications as they allow for dependence modeling with arbitrary margins. They are also used in rare event analysis where tail behaviours are important. Current successful applications of copulas however, generally focus on simplified assumptions, a risk in areas such as healthcare, safety and finance where model misspecification may lead to potential catastrophes. Furthermore, for rare event simulation and prediction, likelihood-based estimation may lead to dependencies in the tail being overlooked by dependencies in the bulk of the data. Moreover, there are still unsolved challenges related to the parameterization, estimation and sampling of copulas.

We propose neural representations to better fit data, and solve challenges related to parameterization, estimation and sampling. We focus on the classes of Archimedean, hierarchical Archimedean, extreme-value and Archimax copulas. Notably, these copulas have stochastic representations as mixture models of latent random variables. In particular, the functions that characterize these copulas may be described by expectations of the latent variables. The first neural representation is motivated by the ability of neural networks to represent expectations and function compositions. The second neural representation is motivated by the impressive ability of generative networks at sampling and the convergence of empirical expectations. To avoid repeated differentiation in computing the likelihood of observations during training, we make use of data transforms to collapse d-dimensional observations into multiple one-dimensional transformed observations. To avoid repeated differentiation in computing the conditional distributions in conditional sampling, we use the sampling frameworks that come naturally with the stochastic representations.

We consider the probabilistic construction of Archimedean copulas as mixture models with the completely monotone Archimedean generator given by the Laplace transform of a latent random variable. We model the latent variable with a generative network and consider the empirical Laplace transform given samples of the latent variable. We compute higher-order derivatives using the properties of the Laplace transform. We modify existing Marshall-Olkin type sampling to our parameterization with generative networks. We extend our training and sampling method to an existing network representation. We also extend our method to hierarchical Archimedean copulas, subsequently recovering a richer class of copulas.

We consider the spectral decomposition of the stable tail dependence function in extreme-value copulas. We propose both network and generative representations for the latent spectral variable. Motivated by the data transform in the Pickands estimator, we transform d-dimensional observations into one-dimensional exponentially distributed transformed observations, then perform maximum likelihood estimation using the transformed observations. We sample using the connection to the spectral representation of max-stable processes.

We build on the stochastic representation of Archimax copulas with latent radial and simplex components. The challenges are estimating a flexible Archimedean generator and sampling the simplex component. We consider the more general d-monotone Archimedean generator given as the Williamson d-transform of the radial component, with connection to the decomposition of the Kendall function. We also consider the correspondence between the spectral component, simplex component and generalized Pareto copulas. We empirically validate our algorithms on high dimensional data, extrapolation to extreme and out-of-distribution detection.