A spatial data set consists of a collection of measurements or observations on one or more attributes taken at specified locations. Data sites are referenced so that the relative positions of sites are recorded, for the spatial organisation of the data is important whether the purpose of data analysis is to build a model for the data or to assess the relative merits of different hypotheses concerning some arrangement property of the data or some other (non spatial) characteristic of the data.

The principal purpose of this book is to describe and evaluate methods for spatial data analysis in order to show what is available, how the different techniques relate to one another and what can be achieved – in short, to contribute to the development of a sound inductive methodology for research areas that deal with data in their spatial context. In doing so, the book is aimed primarily at the social and environmental sciences and most of the examples are drawn from those areas. Apart from the fact that there are important links between social and environmental systems so that the study of one may draw in theory and data from the other, there are two other reasons for a methodological book that takes in both areas of research. First, both deal with observational rather than experimental data.

This chapter introduces parametric statistical models for describing spatial variation. The approach followed is rather different from the presentation usually given of models for describing temporal variation. First, time is one-dimensional and the flow of time from past to present to future is usually assumed to impose a natural ordering or direction on patterns of interaction. Space on the other hand is two-dimensional and usually possesses no such equivalent ordering so that spatial models must allow for a wider range and more complex structures of interaction. Second, time series models such as autoregressive models can be specified in terms of covariances or joint probabilities or conditional probabilities. The three formulations are equivalent, and each gives rise to simple functional expressions. But spatial models with simple autoregressive representations in terms of variate relationships generally have covariance and correlation functions that are far from elementary; conditional and joint probability specifications of nearest neighbour spatial models are not equivalent and give rise to different orders of autoregressive model. Third, if the border value of a first order temporal autoregressive series is its initial value, the border or edge effects are of order 1/n where n is the length of the observed series. In the case of spatial data the boundary encircles the region. For an n = N×M rectangular lattice there are usually at least 2N+2M – 4 border sites, so specifying what happens at the edges is likely to be of more importance in the case of spatial models.

This chapter deals with preliminary forms of spatial data analysis, the purpose of which is to make an initial identification of data properties. The emphasis is on data description which should also be useful in developing hypotheses and shaping subsequent statistical analysis. The methods may also be useful for model assessment.

Section 6.1 describes methods for identifying distributional properties of observations. Such summaries may be of substantive interest but also help to indicate whether special data transformations will be required for subsequent statistical analysis. Ways of identifying trend and other forms of spatial arrangement or structure are also considered. This section draws on the methods of EDA (exploratory data analysis: Tukey, 1977; Hoaglin et al., 1983, 1985) adapted to the needs of spatial data analysis. We assume familiarity with standard methods of EDA and the interested reader should refer to these basic sources for more information. Wetherill et al. (1986; Chapter 2) provide a good introduction and many of the methods described below can be implemented using MINITAB. Section 6.2 overviews methods for testing for pattern in spatial data (spatial autocorrelation tests). This is an area already well covered in the literature and the section places these methods in context and discusses when different tests are appropriate. This part of the book draws on the work of Cliff and Ord (1981) and that of Hubert, Golledge and colleagues in the use of the generalised cross product statistic for spatial data analysis.

This chapter considers problems arising in handling and analysing spatial data. The first section deals with the nature of spatial data, the sources and quality of such data with particular reference to the ‘new’ data sources and the problems they may create for analysis. Subsequently, we discuss forms of spatial data and the problems of computerised spatial data storage. The second section examines why the spatial referencing of data is important in the social and environmental sciences and how spatial structure in data comes about. This leads to a discussion of the types of problems that are addressed and for which methods of data analysis are required. The third section is concerned explicitly with the distinctive problems of spatial data analysis. These are problems associated with the observed values, their spatial configuration and the areal system across which the values are observed.

The final section describes a framework for data analysis which summarises the issues of the first two chapters. Exploratory and confirmatory data analysis provide a framework for statistical analysis (Table 2.1). These two elements are concerned with the identification of structure in general data sets and the development of models for such data. The table distinguishes different approaches to confirmatory data analysis. If strict distributional assumptions are known to hold, parametric methods are very efficient (estimators have small sampling variances and hypothesis tests have high power), but otherwise they may be subject to serious error.