In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

Geostatistical approaches to spatiotemporal prediction in environmental science, climatology, meteorology, and related fields rely on appropriate covariance models. This article proposes general classes of nonseparable, stationary covariance functions for spatiotemporal random processes. The constructions are directly in the space–time domain and do not depend on closed-form Fourier inversions. The model parameters can be associated with the data's spatial and temporal structures, respectively; and a covariance model with a readily interpretable space–time interaction parameter is fitted to wind data from Ireland.

https://www.stat.purdue.edu/%7Echen418/study_research/Tilmann_spatial_stat.pdf

Nonseparable, Stationary Covariance Functions for Space–Time Data

Summary

Environmental systems are complicated. They include very intricate spatio-temporal processes, interacting on a wide variety of scales. There is increasingly vast amounts of data for such processes from geographical information systems, remote sensing platforms, monitoring networks, and computer models. In addition, often there is a great variety of scientific knowledge available for such systems, from partial differential equations based on first principles to panel surveys. It is argued that it is not generally adequate to consider such processes from a joint perspective. Instead, the processes often must be considered as a coherently linked system of conditional models. This paper provides a brief overview of hierarchical approaches applied to environmental processes. The key elements of such models can be considered in three general stages, the data stage, process stage, and parameter stage. In each stage, complicated dependence structure is mitigated by conditioning. For example, the data stage can incorporate measurement errors as well as multiple datasets with varying supports. The process and parameter stages can allow spatial and spatio-temporal processes as well as the direct inclusion of scientific knowledge. The paper concludes with a discussion of some outstanding problems in hierarchical modelling of environmental systems, including the need for new collaboration approaches.

Resume

Les systemes environnementaux sont complexes. Ils incluent des processus spatio-temporels tres complexes, interagissant sur une large variete d'echelles. II existe des quantites de plus en plus grandes de donnees sur de tels processus, provenant des systemes d'information geographiques, des plateformes de teledetection, des reseaux de surveillance et des modeles informatiques. De plus, il y a souvent une grande variete de connaissance scientifique disponible sur de tels systemes, depuis les equations differentielles partielles jusqu'aux enquetes de panels. II est reconnu qu'il n'est generalement pas correct de considerer de tels processus d'une perspective commune. Au contraire, les processus doivent souvent etre examines comme des systemes de modeles conditionnels lies de maniere coherente. Cet article fournit un bref apercu des approches hierachiques appliquees aux processus environnementaux. Les elements cles de tels modeles peuvent etre examines a trois etapes principales: l'etape des donnees, celle du traitement et celle des parametres. A chaque etape, la structure complexe de dependance est attenuee par le conditionnement. Par exemple, le stade des donnees peut incorporer des erreurs de mesure ainsi que de multiples ensembles de donnees sous divers supports. Les stades du traitement et des parametres peuvent admettre des processus spatiaux et spatio-temporels ainsi que l'inclusion directe du savoir scientifique. L'article conclut par une discussion de quelques problemes en suspens dans la modelisation hierarchique des systemes environnementaux, incluant le besoin de nouvelles approches de collaboration.

https://www.jstor.org/stable/info/1403883

Hierarchical Models in Environmental Science

The term geostatistics identifies the part of spatial statistics which is concerned with continuous spatial variation, in the following sense. The scientific focus is to study a spatial phenomenon, s(x)say, which exists throughout a continuous spatial region A ⊂ ℝ2 and can be treated as if it were a realisation of a stochastic process S(·) = {S(x): x ∈ A}. In general, S(·) is not directly observable. Instead, the available data consist of measurements Y 1,..., Y n taken at locations x 1,..., x n sampled within A, and Y i is a noisy version of S(x i ). We shall assume either that the sampling design for x 1,..., x n is deterministic or that it is stochastic but independent of the process S(·), and all analyses are carried out conditionally on x 1,...,x n .

/pdf/an-introduction-to-model-based-geostatistics-195872gul6.pdf

An Introduction to Model-Based Geostatistics

Statistical space–time modelling has traditionally been concerned with separable covariance functions, meaning that the covariance function is a product of a purely temporal function and a purely spatial function. We draw attention to a physical dispersion model which could model phenomena such as the spread of an air pollutant. We show that this model has a non-separable covariance function. The model is well suited to a wide range of realistic problems which will be poorly fitted by separable models. The model operates successively in time: the spatial field at time t +1 is obtained by 'blurring' the field at time t and adding a spatial random field. The model is first introduced at discrete time steps, and the limit is taken as the length of the time steps goes to 0. This gives a consistent continuous model with parameters that are interpretable in continuous space and independent of sampling intervals. Under certain conditions the blurring must be a Gaussian smoothing kernel. We also show that the model is generated by a stochastic differential equation which has been studied by several researchers previously.

Blur‐generated non‐separable space–time models

Motivated by a specific problem concerning the relationship between radar reflectance and rainfall intensity, the paper develops a space–time model for use in environmental monitoring applications. The model is cast as a high dimensional multivariate state space time series model, in which the cross-covariance structure is derived from the spatial context of the component series, in such a way that its interpretation is essentially independent of the particular set of spatial locations at which the data are recorded. We develop algorithms for estimating the parameters of the model by maximum likelihood, and for making spatial predictions of the radar calibration parameters by using realtime computations. We apply the model to data from a weather radar station in Lancashire, England, and demonstrate through empirical validation the predictive performance of the model.

Martin E. Lord

Papers

Space-time calibration of radar rainfall data