Spatio‐temporal Ornstein–Uhlenbeck Processes: Theory, Simulation and Statistical Inference

Abstract

Spatio-temporal modelling is an increasingly popular topic in Statistics. Our paper contributes to this line of research by developing the theory, simulation and inference for a spatiotemporal Ornstein-Uhlenbeck process. We conduct detailed simulation studies and demonstrate the practical relevance of these processes in an empirical study of radiation anomaly data. Finally, we describe how predictions can be carried out in the Gaussian setting.A s .x/ A t .x/, 8 s < t; and A t .x/ \ .X .t; 1// D ;:(2)This implies that A t .x/ has a temporal component of . 1; t just like the classical case. These conditions on the integrating set also give the OU^process several characteristic properties Scand J Statist 44 Spatio-temporal OU processes 47 of the OU process: stationarity, Markovianity and an exponentially decaying autocorrelation function (ACF).In the literature, there are other definitions of spatio-temporal OU processes. When the driving Lévy noise is restricted to be Gaussian and X D R, a spatio-temporal process can be formed from the product of a temporal OU process with a spatial one (Traulsen et al., 2004). This is equal to a temporal OU process when the spatial component is fixed and a spatial OU process when the temporal component is fixed. Although this model features exponentially decaying temporal and spatial autocorrelation, one limitation is that the spatio-temporal autocorrelation is separable.Alternatively, if we discretise space, we can create a spatio-temporal OU process by considering a multivariate OU process where each vector component of the process corresponds to a spatial location. Such an approach was taken by Brix & Diggle (2001). Spatial information is contained in the covariance matrix of the driving Brownian motion. Although this also results in a separable spatio-temporal autocorrelation, the spatial autocorrelation is comparatively flexible to that obtained by the previous method.A further step in this direction would be to discretize the time domain. In this case, the OU process can be represented as an autoregressive model. A three-stage iterative procedure for the space-time modelling of autoregressive moving averages (ARMA) models was developed by Pfeifer & Deutrch (1980). The spatio-temporal autocorrelation in this case is defined differently and involves spatial neighbours of different 'orders'.The OU^process in (1) has four key advantages over these alternative models. Firstly, it accommodates non-Gaussian driving noise, which may be more appropriate in practice. Secondly, being a model in continuous time and space, it allows us to work with data of varying temporal and spatial scales. Thirdly, it allows for non-separable autocorrelation structures defined in the usual way. Finally, its integration set helps us identify the influence region for a particular time and space location. This could help scientists better understand the spatio-temporal interactions in real-life phenomena.A wide class of OU^processes for X D R, which we shall refer to as the g-class, is given bywhere ...

Bridging between short-range and long-range dependence with mixed spatio-temporal Ornstein–Uhlenbeck processes

“…The distribution of an MSTOU process is determined by its ambit set A t (x) and the CQ of its Lévy basis. A summary of this can be obtained through its generalised cumulant functional (Barndorff-Nielsen et al 2015, Nguyen & Veraart 2017:…”

“…where A = max(|d t |, |d x |/c) and (8) holds from the results in Example 3 of Nguyen & Veraart (2017).…”

“…The mixed spatio-temporal Ornstein-Uhlenbeck (MSTOU) process is an extension of the STOU process studied in Barndorff-Nielsen & Schmiegel (2003) and Nguyen & Veraart (2017). While the highlight of this set up is the ability to encompass both short-range and long-range dependence, the MSTOU process also retains the ability to create non-separable spatio-temporal covariances and flexible spatial covariances.…”

“…After developing the theory for MSTOU processes in Sections 2 and 3, we presented a simulation algorithm in Section 4. Unlike the discrete convolution algorithms for STOU process in Nguyen & Veraart (2017), our algorithm does not suffer from the kernel discretisation errors. Instead, the simulation error depends on the kernel truncation: for g-class processes with compound Poisson Lévy bases, an upper bound for the mean squared error was shown to shrink to zero as the simulation padding extents increase to infinity.…”

“…Further studies have shown that this class of processes exhibits exponential temporal correlation just like the temporal OU process and boosts flexible spatial correlation structures which are determined by the shape of the ambit set (Nguyen & Veraart 2017). In addition, non-separable covariances, which are desirable in practice, can be obtained.…”