A review of regularised estimation methods and cross-validation in spatiotemporal statistics

This review article focuses on regularised estimation procedures applicable to geostatistical and spatial econometric models and suitable cross-validation techniques for spatiotemporal data. These methods are particularly relevant in the case of big geospatial data for dimension reduction or model selection. To structure the review, we initially consider the most general case of multivariate spatiotemporal processes (i.e., $g>1$ dimensions of the spatial domain, a one-dimensional temporal domain, and $q\ge 1$ random variables). Then, the idea of regularised/penalised estimation procedures and different choices of shrinkage targets are discussed. Guided by the elements of a mixed-effects model setup, which allows for a variety of spatiotemporal models, we show different regularisation procedures and how they can be used for the analysis of geo-referenced data, e.g. for selection of relevant regressors, dimension reduction of the covariance matrices, detection of conditionally independent locations, or the estimation of a full spatial interaction matrix. The second part is dedicated to cross-validation strategies, which are important for evaluating the model performance and selecting regularisation parameters. We outline the three key assumptions a cross-validation partitioning needs to fulfil and discuss how this can be achieved for time series, spatial data and spatiotemporal data. Additionally, software implementations for these techniques are discussed.