AN INTRODUCTION TO THE ANALYSIS OF SPATIAL PATTERNS

It was suggested to me that it would be useful in this introductory paper to survey some of the methods and models that have been employed in the statistical analysis of spatial pattern. However, I would emphasize that with such a wide spectrum of research workers interested in this general area, it is difficult to cater for everyone, and I must obviously confine my attention to some of the topics that have arisen from my own interests, which one might perhaps describe as reasonably central in the range from the theoretical to the applied. These topics I have discussed on previous occasions, for example at a conference at Sheffield in 1973, and at a summer school in Melbourne in 1974, so I apologize to anyone who is already familiar with some of the material I shall use. First of all, it is necessary to limit my coverage, and I shall try to concentrate on situations intermediate between the simple detection of nonrandomness on the one hand, and the study of 'specialized situations' of pattern recognition on the other. I shall also for simplicity confine my attention to the univariate case. There is still a considerable range of problems, as may be depicted in Figure 1, which shows some different types of two-dimensional data, with some references (not claimed to be other than illustrative) to relevant authors. I shall also assume that the spatial data may be regarded as 'stationary', using this term in the spatial generalization of 'stationarity' as understood in the theory of time-series. This is not to deny the importance of non-stationary problems; but what is left is still analogous to the extensive field of stationary one-dimensional time-series, with two additional complications. The first arises even for one-dimensional spatial series, and is connected with the transfer of attention from a time-series developing in one direction from past to future to a symmetric two-way spatial process. The second is associated with the problems for two-dimensional processes arising from the lack of the ordering which is possible for a one-dimensional process. With regard to types of data, I shall not consider line processes, nor mosaic patterns (see Figure 1), but otherwise consider two types of variable: (i) quantitative, denoted by X(r), where r denotes the vector spatial co-ordinate, or (ii) 'point', denoted by the cumulative number N(r), or the increment dN(r)