Stochastic Variational Inference for Vine Copula Distributional Regression

Abstract

Structured additive distributional regression flexibly relates all parameters of a conditional response distribution to covariates, but multivariate extensions remain challenging when dependence is complex. We propose a multivariate structured additive distributional regression model based on regular vine copulas. Different vine edges may use different pair-copula families, while every pair-copula parameter may vary with covariates through a structured additive predictor. The model therefore accommodates heterogeneous marginal distributions together with pair-specific asymmetric, tail-dependent, and covariate-dependent dependence structures. For scalable inference, we develop a tree-wise stochastic variational inference procedure based on component-specific Gaussian variational approximations. Marginal models are estimated first, followed by pair-copula regressions sequentially along the vine trees. We also adapt sequential vine selection by fitting candidate covariate-dependent pair-copula regressions and using information criteria based on effective degrees of freedom for both family and tree selection. In simulations, the tree-wise estimator remains close to an oracle using the true recursive conditional inputs, whereas global refinement yields more concentrated approximations and lower frequentist coverage for upstream components. An application to six-dimensional meteorological data from the Netherlands yields a vine combining Gaussian and non-Gaussian pair copulas, with pronounced nonlinear spatial and temporal variation in dependence.