Figure 1. 1. Map of Great Britain at two different scale levels. (a) Counties, (b)Regions. '-. " Figure 1. 2. Two alternative aggregations of the Italian provincie in 32 larger areas 4 CHAPTER 1 d . , b) Figure 1. 3 Percentage of votes of the Communist Party in the 1987 Italian political elections (a) and percentage of population over 75 years (b) in 1981 Italian Census in 32 polling districts. The polling districts with values above the average are shaded. Figure 1. 4: First order neighbours (a) and second order neighbours (b) of a reference area. INTRODUCTION 5 While there are several other problems relating to the analysis of areal data, the problem of estimating a spatial correlO!J'am merits special attention. The concept of the correlogram has been borrowed in the spatial literature from the time series analysis. Figure l. 4. a shows the first-order neighbours of a reference area, while Figure 1. 4. b displays the second-order neighbours of the same area. Higher-order neighbours can be defined in a similar fashion. While it is clear that the dependence is strongest between immediate neighbouring areas a certain degree of dependence may be present among higher-order neighbours. This has been shown to be an alternative way of look ing at the sca le problem (Cliff and Ord, 1981, p. l 23). However, unlike the case of a time series where each observation depends only on past observations, here dependence extends in all directions.

In recent years there has been a growing interest in and concern for the development of a sound spatial statistical body of theory. This work has been undertaken by geographers, statisticians, regional scientists, econometricians, and others (e. g. , sociologists). It has led to the publication of a number of books, including Cliff and Ord's Spatial Processes (1981), Bartlett's The Statistical Analysis of Spatial Pattern (1975), Ripley's Spatial Statistics (1981), Paelinck and Klaassen's Spatial Economet~ics (1979), Ahuja and Schachter's Pattern Models (1983), and Upton and Fingleton's Spatial Data Analysis by Example (1985). The first of these books presents a useful introduction to the topic of spatial autocorrelation, focusing on autocorrelation indices and their sampling distributions. The second of these books is quite brief, but nevertheless furnishes an eloquent introduction to the rela tionship between spatial autoregressive and two-dimensional spectral models. Ripley's book virtually ignores autoregressive and trend surface modelling, and focuses almost solely on point pattern analysis. Paelinck and Klaassen's book closely follows an econometric textbook format, and as a result overlooks much of the important material necessary for successful spatial data analy sis. It almost exclusively addresses distance and gravity models, with some treatment of autoregressive modelling. Pattern Models supplements Cliff and Ord's book, which in combination provide a good introduction to spatial data analysis. Its basic limitation is a preoccupation with the geometry of planar patterns, and hence is very narrow in scope.

In September 1977 a "Regional Science Symposium" was held at the Faculty of Economics of the University of Goningen in the Netherlands. The impetus in organizing this symposium was the recent estab lishmen t at the F acuIty of Economics of a group engaged in teaching and research within the field of regional science. The aim of the symposium was to familiarize university members with regional science and to introduce the new group to both the national and international scene. Two separate topics of potential interest to both researchers and policy-makers were selected. The first theme, spatial inequalities and regional development, was chosen because of its central place in regional science. Authors from several disciplines were asked to approach this theme from a general, policy-oriented point of view. This ensured the spotlighting of the various dimensions of spatial inequality and its implications for regional policy. The results of their efforts have been collected in a volume entitled Spatial Inequalities and Regional Development. The second theme focussed on spatial statistical analysis. This branch of statistics is a relatively new one. It is receiving growing attention from researchers in the field of applied regional science. The conference dealing with this topic concentrated on recent research results related to the use of appropriate statistical and econometric methods for analyzing spatial data. The papers con cerned have been collected in another volume, entitled Exploratory and Explanatory Statistical Analysis of Spatial Data.

The field of spatial econometrics has come to include the methods and models that deal with estimation and testing problems encountered when attempting to implement regional economic models. Those problems are often characterized by the difficulties associated with assessing the importance of spatial dependence and spatial heterogeneity. This book includes contributions on spatial proximity, spatial patterning and in particular the spatial association (dependence) contained in local map patterns.

The field of spatial econometrics has come to include the methods and models that deal with estimation and testing problems encountered when attempting to implement regional economic models. Those problems are often characterized by the difficulties associated with assessing the importance of spatial dependence and spatial heterogeneity. This book includes contributions on spatial proximity, spatial patterning and in particular the spatial association (dependence) contained in local map patterns.

Understanding spatial statistics requires tools from applied and mathematical statistics, linear model theory, regression, time series, and stochastic processes. It also requires a mindset that focuses on the unique characteristics of spatial data and the development of specialized analytical tools designed explicitly for spatial data analysis. Statistical Methods for Spatial Data Analysis answers the demand for a text that incorporates all of these factors by presenting a balanced exposition that explores both the theoretical foundations of the field of spatial statistics as well as practical methods for the analysis of spatial data. This book is a comprehensive and illustrative treatment of basic statistical theory and methods for spatial data analysis, employing a model-based and frequentist approach that emphasizes the spatial domain. It introduces essential tools and approaches including: measures of autocorrelation and their role in data analysis; the background and theoretical framework supporting random fields; the analysis of mapped spatial point patterns; estimation and modeling of the covariance function and semivariogram; a comprehensive treatment of spatial analysis in the spectral domain; and spatial prediction and kriging. The volume also delivers a thorough analysis of spatial regression, providing a detailed development of linear models with uncorrelated errors, linear models with spatially-correlated errors and generalized linear mixed models for spatial data. It succinctly discusses Bayesian hierarchical models and concludes with reviews on simulating random fields, non-stationary covariance, and spatio-temporal processes. Additional material on the CRC Press website supplements the content of this book. The site provides data sets used as examples in the text, software code that can be used to implement many of the principal methods described and illustrated, and updates to the text itself.