Exploring Process Heterogeneity in Environmental Statistics: Examples and Methodological Advances

Environmental models typically rely on stationarity assumptions. However, environmental systems are complex, and processes change over states or seasons, leading to often overlooked heterogeneity. This paper explores methods to incorporate process heterogeneity into statistical models to improve their performance. It considers problems from natural hazards and earth system sciences, demonstrating the effects of process heterogeneity and proposing methodological advances through model extensions. The first problem addresses flood frequency analysis, where floods are generated by different processes in catchment and atmosphere. Amixture model combining peak-over-threshold distributions of flood types can handle this heterogeneity, especially regarding tailheaviness, making it relevant for flood design. The second problem involves minimum flow frequency analysis, with heterogeneity fromdifferent summer and winter processes. A mixture distribution model for minima and a copula-based estimator can incorporate seasonal distributions and event dependence, showing significant performance gains for extreme events. The third problem examines process heterogeneity in rainfall models. Clustering event characteristics (e.g., duration, intensity) using Gower's distance and a lightning index helps distinguish between convective and stratiform events, showing potential to enhance rainfall generators. The fourth problem deals with parameter variation in temporal models of environmental variables, using daily streamflow series. A tree-based machine learning model shows that prediction performance and model parameters vary with quantile loss optimization, suggesting the need for different or combined models for full time series in the presence of process heterogeneity. The study highlights the importance of considering process heterogeneity in modeling from the outset and encourages a better understanding of statistical assumptions and the enrichment of physical knowledge in environmental statistics.