Stochastic Partial Differential Equation Based Modelling of Large Space–Time Data Sets

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 advectiondiffusion 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.

A New Covariance Function and Spatio‐Temporal Prediction (Kriging) for A Stationary Spatio‐Temporal Random Process

“…The covariance function given here is fundamentally different from the other covariance functions defined and used by other authors. We further show that the second‐order spectral density function of the spatio‐temporal random process defined here through the aforementioned Laplace operator on the DFTs includes all the second‐order spectra of the processes so far defined through the stochastic versions of the Laplace equation such as the heat equation (transport–diffusion equation of Jones and Zhang (), Lindgren et al (), and Sigrist et al ()), wave equation, and Helmholtz equation. The second‐order spectral density function obtained here corresponds to a non‐separable random process.…”

“…By substituting the spectral representations for the processes { Y t ( s )} and { e t ( s ) } and equating the integrands, after taking expectations of the squared modulus we can show that the spatio‐temporal spectral density function of the process { Y t ( s )} is given by . We note that no closed form for the covariance function is available in this case (see Sigrist et al ()).…”

“…Now, consider the model where the white noise process { e t ( s ) } is defined as earlier. The models considered by Lindgren et al () and recently by Sigrist et al () are variations of the aforementioned diffusion model. If we set the transport direction vector (see Sigrist et al ()) equal to zero and the diffusion matrix to the identity matrix in the models by Lindgren et al () and Sigrist et al (), we acquire the model defined by Jones and Zhang ().…”

“…The models considered by Lindgren et al () and recently by Sigrist et al () are variations of the aforementioned diffusion model. If we set the transport direction vector (see Sigrist et al ()) equal to zero and the diffusion matrix to the identity matrix in the models by Lindgren et al () and Sigrist et al (), we acquire the model defined by Jones and Zhang (). By substituting the spectral representations for the processes { Y t ( s )} and { e t ( s ) } and equating the integrands, after taking expectations of the squared modulus we can show that the spatio‐temporal spectral density function of the process { Y t ( s )} is given by .…”

Forecasting with Big Data: A Review

“…Sigrist et al [80] utilizes stochastic advection diffusion partial differential equations (SPDEs) to improve the precipitation forecasts for northern Switzerland using Big Data from a numerical weather prediction model. They find that following the application of SPDE, the forecasts are greatly improved in comparison to the raw forecasts attained via the numerical model.…”