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Statistical Modelling of Space-Time Variability

TitleStatistical Modelling of Space-Time Variability
Publication TypeConference Paper
Year of Publication2014
AuthorsHeuvelink GBM
Conference NameDailyMeteo.org/2014
Date Published06/2014
PublisherFaculty of Civil Engineering, University of Belgrade
Conference LocationBelgrade
Abstract

Many environmental variables, such as precipitation, temperature and radiation, vary both in space and time. The space-time variability of these variables is governed by physical laws, which are often characterised by partial differential equations. However, these equations can be very complex and their parameters and initial and boundary conditions are often very poorly known. This makes it extremely difficult to obtain practically useful solutions. In such case, statistical modelling offers an alternative. Statistical models are no replacement for mechanistic models because they give less insight into governing processes and cannot easily be extrapolated, but they are easier to implement, calibrate and run. Provided that the observation density is sufficiently large, they often yield sufficiently accurate predictions of the space-time variable at unobserved points. Geostatistics offers a rich methodology for statistical modelling and prediction of spatially distributed variables. The basic approach is to treat the variable of interest as a sum of a deterministic trend and a zero-mean stochastic residual. The trend is often taken as a linear combination of explanatory variables that must be known spatially exhaustively, while the stochastic residual is usually assumed to be normally distributed and stationary. It will typically also be spatially correlated, as characterised by a semivariogram. With this model, predictions at unobserved locations can be made using kriging, which also quantifies the prediction error variance. Extension of the geostatistical model to the space-time domain can be done in various ways. One is to consider the spatial variable at multiple time points, deriving a geostatistical model at each of these time points and characterising the correlation between variables at different time points through a cokriging approach. However, the disadvantage of this approach is that it only addresses the variable at the selected times and not in between, and that modelling is cumbersome when the number of time points is moderate or large. A more attractive alternative is to include time as a third dimension and model space-time variability by means of a spatio-temporal trend and a space-time stochastic residual. Once this model has been defined and calibrated it can be used to predict and simulate at any point in space and time, hence producing a ‘movie’ of the spatial distribution over time. In recent years many advances have been made in developing theoretically sound space-time statistical models. The difficulty is in the space-time stochastic residual, because the associated covariance model must include zonal and geometric anisotropies. Popular representations of the space-time covariance structure are the sum-product model and the sum-metric model. Fitting of these models to real-world data sets and using these models for space-time prediction and simulation has greatly improved in recent years due to advances in the spacetime and gstat packages in R. The main problem with defining valid space-time covariance structures is that these must be semi-positive definite, which is difficult to prove. If, however, the space-time covariance structure is derived from an explicit model of the space-time variable, such as through a space-time auto-regressive moving average (ARMA) model or a so-called state-space model, then the semi-positive definiteness is guaranteed by construction. In such case, spatio-temporal prediction may be done using the Kalman filter and Kalman smoother, which, as does kriging, calculate the conditional probability distribution of a target variable given conditioning data. The attractive property of space-time ARMA and state-space models is also that these bridge the gap with mechanistic modelling of space-time variability. This is because the ARMA and discrete state-space approach may be interpreted as discrete approximations of stochastic partial differential equations. There is yet a lot to be discovered in this research area, and if software development can go hand in hand with theoretical developments we may see major steps forward in the years to come. All statistical approaches described above are explained in this lecture and illustrated with real-world applications.

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