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

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

Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering

This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive system of ordinary differential equations (ODEs) for the relevant conditional distribution. Thanks to the simplicity of the ODE system, higher order calculation can be performed easily. Furthermore, solving ODEs sequentially with small sub-periods with updated initial conditions makes it possible to implement a substepping method for asymptotic expansion in a numerically efficient way. This is found to improve the performance significantly where otherwise the approximation fails badly. The method is expected to provide a useful tool for more realistic financial modeling with unobserved parameters, and also for problems involving nonlinear measure-valued processes.Comment: revised version for publication in Ann Inst Stat Mat

On an application of extended kalman filtering to activated sludge processes: a benchmark study

The growing demand for performance improvements of urban wastewater system operation coupled with the lack of instrumentation in most wastewater treatment plants motivates the need for non-linear observers to be used as virtual sensors for estimation and control of effluent quality. This paper is focused on the development of a general procedure for on-line monitoring of activated sludge processes, using an extended Kalman filter (EKF) approach. The Activated Sludge Model no.1 (ASM1) is selected to describe the biological processes in the reactor. On-line measurements are corrupted by additive white noise and unknown inputs are modelled using fast Fourier transform (FFT) and spectrum analyses. The given procedure aims at reducing the original ASM1 model to an observable and identifiable model, which can be used for joint non-linear state and parameter estimations. Simulation results are presented to demonstrate the effectiveness of the proposed methods and show that on-line monitoring of SND and XND concentrations is achieved when dynamic input data are used tocharacterize the influent wastewater for the model