Space-time in high dimensions

A02 addresses problems of high-dimensionality and dimension reduction. It concentrates on a flexible framework for vector autoregressive (panel) models and models spatio-temporal extremes to analyze rare events. In the long term, a framework for high-dimensional space-time data analysis is developed that is equipped with thorough statistical theory and meets practical challenges, such as implementations and tuning parameter calibration.

When analyzing spatio-temporal data in high dimensions, one usually pursues one of the following two general statistical goals: either the (dynamic behavior of the) center of each associated distribution is of greatest interest, or it is the extreme values which rarely occur but can have drastic consequences. The two goals require different statistical tools, which is reflected in the project's research agenda: the focus is both on autoregressive (VAR) models and panel data setups to analyze typical behavior, and on the analysis of spatio-temporal extremes to analyze rare events. The former is challenging because the number of parameters is often very large, in particular in the case of VAR models with additional exogenous variables (VARX). The latter is challenging because the focus on extremes usually leads to comparably small sample sizes.

We aim to devise new estimators that account for high-dimensionality, provide a feasible implementation, equip the estimators with statistical guarantees and test them in simulations and on empirical data. A key technique in our research is regularization, which deals with high-dimensionality by complementing classical objective functions with additional terms that formalize prior information about the data or setting. A by now standard type of prior information, called sparsity, is that only a small number of predictors should have a relevant effect. But many applications require more complex prior information: in VARX models, for example, exogenous predictors that are close in space can behave similarly ("fusion sparsity") or form functional groups ("group sparsity"), and the connectivities across time can also depend on the exogenous predictors ("lag selection"). In extremes, similar phenomena may occur for marginal extreme value models at different locations, and certain conditional independence relations have recently been connected to sparse graphical models for extremes. Such complex types of sparsity are called "structured sparsity" and will be the main focus of our work. The inclusion of structured sparsity will lead to more efficient estimation and more accurate prediction in the space-time applications of TRR 391 and beyond.