Stochastic partial differential equation based modelling of large space-time data sets

TL;DR

This paper introduces a computationally efficient stochastic PDE-based model for large-scale spatio-temporal data, enabling better calibration and uncertainty quantification in environmental forecasts.

Contribution

It develops a spectral method for solving stochastic PDEs that scales linearly with data dimension, improving inference for large spatio-temporal datasets.

Findings

Model outperforms raw weather forecasts in accuracy.

Provides calibrated uncertainty quantification.

Efficient spectral solution avoids error accumulation.

Abstract

Increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection-diffusion partial differential equation provides a flexible model class for spatio-temporal processes which is computationally feasible also for large data sets. The Gaussian process defined through the stochastic partial differential equation has in general a nonseparable covariance structure. Furthermore, its parameters can be physically interpreted as explicitly modeling phenomena such as transport and diffusion that occur in many natural processes in diverse fields ranging from environmental sciences to ecology. In order to obtain computationally efficient statistical algorithms we use spectral methods to solve the stochastic partial differential equation. This has the advantage that approximation errors do not accumulate over time, and that in the spectral space the computational cost grows linearly with the dimension, the total computational costs of Bayesian or frequentist inference being dominated by the fast Fourier transform. The proposed model is applied to postprocessing of precipitation forecasts from a numerical weather prediction model for northern Switzerland. In contrast to the raw forecasts from the numerical model, the postprocessed forecasts are calibrated and quantify prediction uncertainty. Moreover, they outperform the raw forecasts, in the sense that they have a lower mean absolute error.