Collecting spatial data. Optimum design of experiments for random fields.
Spatial data have the distinctive characteristic that, attached to every observation, we have a set of coordinates that identifies the (geographical) position of a respective data collection site. The set of locations of those data collection sites (the so-called design) influences the quality of the results of the statistical analysis. Sampling theory and optimum experimental design theory are two large branches of theoretical statistics that provide methods for efficient site positioning. Whereas sampling theory is a basically model-free methodology essentially oriented towards restoring unobserved data, in optimum design theory the aim is to estimate the structure of the data generating process, e.g., the parameters of an assumed (regression) model or functions of these parameters. In this monograph emphasis is on the latter branch. Principles from optimum design theory are used in adapting and developing methods specific for the solution of the spatial design problem. In Chapter 2, a short overview on the predominant geostatistical techniques is given. Estimation techniques for parameters of measures of both first (mean, trend) and second order characteristics (variogram, spatial covariance function) are reviewed. Universal kriging is presented as a special form of generalized least squares estimation. A well-known nonparametric estimation method, the local regression technique, is shown to be a useful complement to kriging. Some of its properties, that are relevant in the spatial setting, are derived. Kriging requires parametric variogram estimation. It is indicated that the most frequently applied (weighted least squares) estimator is generally inconsistent and remedies for this problem are suggested. Chapter 3 contains a description of the basic principles of optimum experimental design theory. Here optimality refers to a prespecified criterion such as the minimization of the variances of the estimates. Emphasis is again on aspects that are particularly relevant for spatial problems such as localizing extremal values for which a reparameterization approach is given. Designs for random coefficient models, that are particularly useful in the presence of spatio-temporal relationships, are compared with some more traditional approaches. Several well known basic techniques such as regular grids, random and stratified sampling, and central composite designs are reviewed in Chapter 4 with particular emphasis on their properties in a spatial framework. Increasingly popular for higher dimensional sampling problems become the so-called space filling designs that aim to spread out intersite distances. A new construction rule for such designs is given and compared with more traditional approaches (like maximin or minimax distance designs). Another method that has value as an exploratory tool is local regression, for which a design criterion (the minimization of the average variance of prediction) is formulated. An optimum design strategy based upon it is given accordingly. The two most distinguishing features of spatial designs are that observations may not be replicated instantaneously and that they are usually correlated across locations. For the former problem there are a number of solutions based on bounding a design measure from above. In the first section of Chapter 5 an alternative method, that is based upon an extension of the concept of an information matrix, is presented and compared with standard techniques. As for the second problem it is common that spatial phenomena exhibit strong local correlations, and thus classical methods of optimum design are applicable only under quite restrictive conditions. There have been several suggestions for the construction of optimum designs, which are briefly reviewed in the second section of Chapter 5. The main part of Chapter 5 is concerned with adapting concepts from classical optimum design theory, such as design measures, to the correlated case. Here again the solution is based upon an extension or rather an approximation of the usual information matrix, that coincides with the classical one when no correlations arise, and that allows us to introduce design measures under correlation. Chapter 6 discusses the applications of optimum design theory for estimation of second-order characteristics of spatial processes such as the variogram. Here similar problems as in Chapter 5 arise due to correlated observations. Several design strategies are suggested and compared. A method for combining several different purpose designs for good overall performance is proposed. In all chapters the formal descriptions are accompanied by an illustrating example taken from air-pollution monitoring, namely the reconstruction of the Upper-Austrian sulfuric dioxide (\(SO_2\)) monitoring network. This is a typical problem in spatial design and there exists a great number of similar applications in this field. New to this edition is also the inclusion of exercises at the end of each chapter (with suggested solutions to be published on a web site). Most of the calculations in this monograph were carried out within the statistical programming environment GAUSS-386 V.3.2.
scientific article; zbMATH DE number 1537376