This study presents a comparative analysis of three predictive models with an increasing degree of flexibility: hidden dynamic geostatistical models (HDGM), generalised additive mixed models (GAMM), and the random forest spatiotemporal kriging models (RFSTK). These models are evaluated for their effectiveness in predicting $$\text {PM}_{2.5}$$PM 2.5 concentrations in Lombardy (North Italy) from 2016 to 2020. Despite differing methodologies, all models demonstrate proficient capture of spatiotemporal patterns within air pollution data with similar out-of-sample performance. Furthermore, the study delves into station-specific analyses, revealing variable model performance contingent on localised conditions. Model interpretation, facilitated by parametric coefficient analysis and partial dependence plots, unveils consistent associations between predictor variables and $$\text {PM}_{2.5}$$PM 2.5 concentrations. Despite nuanced variations in modelling spatiotemporal correlations, all models effectively accounted for the underlying dependence. In summary, this study underscores the efficacy of conventional techniques in modelling correlated spatiotemporal data, concurrently highlighting the complementary potential of Machine Learning and classical statistical approaches.

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≥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.

We propose a novel model selection algorithm based on a penalized maximum likelihood estimator (PMLE) for functional hidden dynamic geostatistical models (f-HDGM). These models employ a classic mixed-effect regression structure with embedded spatiotemporal dynamics to model georeferenced data observed in a functional domain. Thus, the regression coefficients are functions. The algorithm simultaneously selects the relevant spline basis functions and regressors that are used to model the fixed effects. In this way, it automatically shrinks to zero irrelevant parts of the functional coefficients or the entire function for an irrelevant regressor. The algorithm is based on an adaptive LASSO penalty function, with weights obtained by the unpenalised f-HDGM maximum likelihood estimators. The computational burden of maximisation is drastically reduced by a local quadratic approximation of the log-likelihood. A Monte Carlo simulation study provides insight in prediction ability and parameter estimate precision, considering increasing spatiotemporal dependence and cross-correlations among predictors. Further, the algorithm behaviour is investigated when modelling air quality functional data with several weather and land cover covariates. Within this application, we also explore some scalability properties of our algorithm. Both simulations and empirical results show that the prediction ability of the penalised estimates are equivalent to those provided by the maximum likelihood estimates. However, adopting the so-called one-standard-error rule, we obtain estimates closer to the real ones, as well as simpler and more interpretable models.