Spatio-temporal probabilistic forecast using MMAF-guided learning
Authors: Leonardo Bardi, Imma Valentina Curato, Lorenzo Proietti
Organizations: Department of Mathematics TU Chemnitz Reichenhainer Straße 41
Abstract
We present a theory-guided generalized Bayesian methodology for spatio-temporal raster data, which we use to train an ensemble of stochastic feed-forward neural networks with Gaussian-distributed weights. The methodology incorporates the dependence and causal structure of a spatio-temporal Ornstein-Uhlenbeck process into training and inference by enforcing constraints on the design of the data embedding and the related optimization routine. In inference mode, the networks are employed to generate causal ensemble forecasts by applying different initial conditions at different horizons. We call this workflow MMAF-guided learning. Experiments conducted on both synthetic and real data demonstrate that our forecasts remain calibrated across multiple time horizons. Moreover, we show that on such data, shallow feed-forward architectures can achieve performance comparable to, and in some cases better than, convolutional or diffusion deep learning architectures used in probabilistic forecasting tasks.
Accurate and reliable forecasting of epidemic incidences is critical for public health preparedness, yet it remains a challenging task due to complex nonlinear temporal dependencies and heterogeneous spatial interactions. Often, point forecasts generated by spatiotemporal models are unreliable in assigning uncertainty to future epidemic events. Probabilistic forecasting of epidemics is therefore crucial for providing the best or worst-case scenarios rather than a simple, often inaccurate, point estimate. We present deep spatiotemporal engression methods to generate accurate and reliable probabilistic forecasts on low-frequency epidemic datasets. The proposed methods act as distributional lenses, and out-of-sample probabilistic forecasts are generated by sampling from the trained models. Our frameworks encapsulate lightweight deep generative architectures, wherein uncertainty is quantified endogenously, driven by a pre-additive noise component during model construction. We establish geometric ergodicity and asymptotic stationarity of the spatiotemporal engression processes under mild assumptions on the network weights and pre-additive noise process. Comprehensive evaluations across six epidemiological datasets over three forecast horizons demonstrate that the proposal consistently outperforms several temporal and spatiotemporal benchmarks in both point and probabilistic forecasting. Additionally, we explore the explainability of the proposal to enhance the models' practical application for informed, timely public health interventions.
Real-world traffic data exhibit heterogeneous spatial correlations and nonlinear temporal dynamics, posing substantial challenges for accurate spatio-temporal forecasting. Existing approaches have developed increasingly sophisticated graph, attention, and decomposition architectures, while the influence of the underlying nonlinear function approximator has received comparatively less attention. In this work, we propose STKAN, a spatio-temporal forecasting architecture that introduces Taylor-polynomial Kolmogorov--Arnold Network modules into spatial and temporal token mixing. STKAN first constructs high-level spatial representations through a learnable soft node-group assignment mechanism, applies group-wise spatial mixing, and subsequently models temporal dependencies over the compressed sequence. Spatial and temporal self-attention layers are further employed to capture long-range interactions. Experiments on five traffic forecasting benchmarks show that STKAN achieves competitive performance and performs better than the evaluated MLP-based variant in the tested settings. These results suggest that the design of nonlinear function approximators can serve as a useful complement to architectural design in spatio-temporal forecasting.
Reliable forecasting of several interrelated environmental variables - such as regional precipitation and temperature, or other correlated geophysical fields - across many locations calls for accurate predictions accompanied by trustworthy statements of their uncertainty. Modern deep-learning models forecast such variables accurately but usually report no uncertainty, and forcing them to output uncertainty through maximum likelihood tends to degrade their accuracy, especially when the variables are strongly correlated. Motivated by this tension, we develop TSCoNet, a two-stage convolutional-recurrent model coupled with a Gaussian copula that jointly forecasts multiple variables over space and time while quantifying predictive uncertainty. The method first learns accurate mean forecasts and then, holding the mean fixed, refines a shared representation to estimate the predictive variance, yielding calibrated prediction intervals after a standard recalibration, so that uncertainty is added without sacrificing point accuracy. We study the approach on simulated non-stationary spatial fields on the sphere and on a real dataset of monthly precipitation and temperature for fifty cities over 2000-2020. The model matches the accuracy of a strong deterministic forecaster while supplying calibrated prediction intervals that the deterministic model cannot, giving a single tool that provides both accurate point forecasts and reliable uncertainty for multivariate spatio-temporal data.