Fokker-Planck formalism and Shannon entropy in forecasting weather extremes

The article presents an integrated approach to the analysis and prediction of extreme weather events by combining Extreme Value Theory (EVT), informational entropy, and the formalism of the Fokker–Planck equation. The starting point is the identification of extreme values of temperature and precipitation—defined as observations beyond the 90th and 10th percentiles, respectively—and their increasing significance in the context of climate change. The authors highlight the limitations of traditional climate models, which primarily focus on mean values and tend to neglect distributional variability, asymmetry, and discontinuities. The proposed methodology employs empirical estimations of Shannon entropy as a nonparametric measure of statistical uncertainty and explores the structural properties of probability density functions of weather extremes. A key element of the framework is the application of the Fokker–Planck equation, which separates deterministic components (drift) from stochastic ones (diffusion), while capturing their influence on the spatial and temporal evolution of the system. A novel aspect of the analysis involves using spatiotemporal derivatives \(\:(\partial\:P/\partial\:t,\:\nabla\:·(vP),\:\nabla\:²(DP)\)) to describe the dynamics of probability distributions and the production of entropy (\(\:dH/dt\)) as an indicator of climatic reorganization. The introduced concept of “entropy flux” allows for the identification of probabilistic sources and sinks, highlighting regions of increased risk for extreme events. The analysis is based on monthly temperature and precipitation data at 0.5° resolution for the period 1901–2010, employing marginal distribution estimation, copula functions, and local heuristic metrics (gradient, Laplacian). Classification of extremes is performed according to their generative mechanism (e.g., drought vs. storm, cold vs. heat), and model performance is evaluated using TPR, PPV, spatial correlation, and chi-squared tests. The results indicate the highest predictive accuracy and precision for the \(\:{P}_{min}\) index (minimal precipitation), while \(\:{T}_{max}\) and \(\:{T}_{min}\)—despite high sensitivity—exhibit lower precision and greater spatial dispersion of false alarms. The Fokker–Planck model demonstrates spatial and temporal coherence, and the observed entropy production reflects the evolution of dynamic complexity in the climate system. This approach integrates statistical physics with information-theoretic classification, offering a comprehensive framework for detecting and forecasting weather extremes. The model shows strong potential for operational applications in weather forecasting, climate risk assessment, disaster management, and adaptive planning in the face of global climate change.