A State-Space modelling framework in geostatistics with application to environmental data

This thesis presents a framework based on geostatistical models for analysing environmental datasets. Geostatistical models rely on stochastic (random) fields, described by a limited number of parameters, aiming to replicate patterns in observed spatio-temporal variability without necessarily following the underlying true physical equations. Space–time geostatistical models are evolving rapidly to address increasingly complex datasets in fields such as geology, climate science, ecology, epidemiology, and economics. Two main conceptual approaches are distinguished: (i) single spatio-temporal random function models, and (ii) dynamical-statistical approaches involving vectors of spatial functions or time series. Links between these approaches are explored, highlighting advantages and limitations. While the first approach often provides satisfactory solutions, it does not model the underlying dynamics. The dynamical-statistical approach explicitly models the dependence of the current state on previous states through mechanistic or probabilistic relationships. These models are specified by unknown parameters, estimated from data using a state-space framework with the Kalman filter and smoother. When parameters are unknown, the expectation–maximisation (EM) algorithm is employed to obtain maximum-likelihood estimates. This thesis focuses on Gaussian state-space models and extends them to multivariate spatial analysis, improving prediction and capturing cross-dependencies. Applications include modelling fine particulate matter (PM2.5 and PM10) and estimating the impact of ammonia (NH3) emissions on PM2.5 in Lombardy, Italy. With growing dataset sizes from remote-sensing and sensor networks, exact inference becomes computationally infeasible. To address this, a low-rank Gaussian process approximation is proposed, leveraging stochastic partial differential equations (SPDEs) to reduce Kalman filter costs while preserving interpretability. Theoretical properties, error bounds, and EM-based parameter estimation are established, and extensive simulations validate accuracy and efficiency. The approach naturally extends to higher-dimensional manifolds, networks, and graphs, highlighting its flexibility and broad applicability.