A Spatiotemporal Approach for Drought Monitoring Using Empirical Orthogonal Function (EOF) Analysis and Neural Networks

Abstract: The current study treats the monthly Standardized Precipitation Index (SPI) and Standardized Precipitation Evapotranspiration Index (SPEI) measured at 32 meteorological stations in Punjab (1981 – 2021) as square-integrable random fields on the product space of time and site. Centering each field and invoking the Karhunen–Loève theorem, we perform an empirical orthogonal function (EOF) expansion: an orthonormal set of temporal basis functions in provides a spectral decomposition. At the same time, the associated spatial coefficient vectors lie in the finite-dimensional Euclidean space of stations. Normalising these coefficients by their singular values through the singular-value decomposition of the empirical covariance operator yields a dimension-reduced representation that preserves 95% of total variance (12 dominant modes for SPI, 8 for SPEI). A multi-output feedforward neural network is trained, under a mean-square risk functional, to approximate the mapping that assigns each station its vector of leading normalised coefficients; reconstruction of the full spatiotemporal field follows by recomposition with the fixed temporal basis. This operator-theoretic deep-learning pipeline is therefore end-to-end differentiable and statistically consistent. The hydro-climatic heterogeneity of the study area, monsoon peaks in Bhakkar, prolonged aridity in Okara, large thermal amplitudes in Multan and Bahawalpur, and subdued seasonality in Murree, is effectively reproduced. In cross-validation, the network attains mean-absolute, mean-squared, and root-mean-square errors of 0.20, 0.17, and 0.27, respectively, for SPEI, outperforming single-output baselines by more than half. Even under aggressive rank truncation, the model conserves climatological zoning, coherently propagates spatiotemporal variability, and uncovers quasi-periodic climate cycles. The proposed framework thus furnishes a mathematically rigorous, computationally efficient instrument for high-resolution drought diagnosis and forecasting in data-scarce, heterogeneous environments.