Spatial Data Analysis

Here in this chapter, we first consider the visualisation of area data before examining a number of exploratory techniques. The focus is on spatial dependence (spatial association). In other words, the techniques we consider aim to describe spatial distributions, discover patterns of spatial clustering, and identify atypical observations (outliers). Techniques and measures of spatial autocorrelation discussed in this chapter are available in a variety of software packages. Perhaps the most comprehensive is GeoDa, a free software program (downloadable from http://​www.​geoda.​uiuc.​edu). This software makes a number of exploratory spatial data analysis (ESDA) procedures available that enable the user to elicit information about spatial patterns in the data given. Graphical and mapping procedures allow for detailed analysis of global and local spatial autocorrelation results. Another valuable open software is the spdep package of the R project (downloadable from http://​cran.​r-project.​org). This package contains a collection of useful functions to create spatial weights matrix objects from polygon contiguities, and various tests for global and spatial autocorrelation (see Bivand et al. 2008).

Exploratory spatial data analysis is often a preliminary step to more formal modelling approaches that seek to establish relationships between the observations of a variable and the observations of other variables, recorded for each areal unit. The focus in this chapter is on spatial regression models in a simple cross-sectional setting, leaving out of consideration the analysis of panel data. We, moreover, assume that the data concerned can be taken to be approximately normally distributed. This assumption is—to varying degrees—involved in most of the spatial regression techniques that we will consider. Note that the assumption of normality is not tenable if the variable of interest is a count or a proportion. In these cases we would expect models for such data to involve probability distributions such as the Poisson or binomial. The chapter consists of five sections, starting with a treatment of the specification of spatial dependence in a regression model. Next, specification tests are considered to detect the presence of spatial dependence. This is followed by a review of the spatial Durbin model (SDM) that nests many of the models widely used in the literature, and by a discussion of spatial regression model estimation based on the maximum likelihood (ML) principle. The chapter closes with some remarks on model parameter interpretation, an issue that had been largely neglected so far. Readers interested in implementing the models, methods and techniques discussed in this chapter find useful MATLAB code which is publicly available at spatial-econometrics.com, LeSage’s spatial econometrics toolbox (downloadable from http://​www.​spatial-econometrics.​com/​), see Liu and LeSage (2010) Journal of Geographical Systems 12(1):69–87 for a brief description. Another useful open software is the spdep package of the R project (downloadable from http://​cran.​r-project.​org).

The phenomenon of interest in this chapter may be described in most general terms as interactions between populations of actors and opportunities distributed over some relevant geographic space. Such interactions may involve movements of individuals from one location to another, such as daily traffic flows in which case the relevant actors are individuals such as commuters (shoppers) and the relevant opportunities are their destinations such as jobs (or stores). Similarly, one may consider migration flows, in which case the relevant actors are migrants (individuals, family units, firms) and the relevant opportunities are their possible new locations. Interactions may also involve flows of information such as telephone calls or electronic messages. Here the callers or message senders may be relevant actors, and the possible receivers of calls or electronic messages may be considered as the relevant opportunities (Sen and Smith 1995, Gravity models of spatial interaction behavior. Springer, Berlin pp. 18–19).

Spatial interaction models of the types discussed in the previous chapter take the view that inclusion of a spatial separation function between origin and destination locations is adequate to capture any spatial dependence in the sample data. LeSage and Pace (J Reg Sci 48(5):941–967, 2008), and Fischer and Griffith (J Reg Sci 48(5):969–989, 2008) provide theoretical as well as an empirical motivation that this may not be adequate to model potentially rich patterns that can arise from spatial dependence. In this chapter we consider three approaches to deal with spatial dependence in origin–destination flows. Two approaches incorporate spatial correlation structures into the independence (log-normal) spatial interaction model. The first specifies a (first order) spatial autoregressive process that governs the spatial interaction variable (see LeSage and Pace (J Reg Sci 48(5):941–967, 2008)). The second approach deals with spatial dependence by specifying a spatial process for the disturbance terms, structured to follow a (first order) spatial autoregressive process. In this framework, the spatial dependence resides in the disturbance process (see Fischer and Griffith (J Reg Sci 48(5):969–989, 2008)). A final approach relies on using a spatial filtering methodology developed by Griffith (Spatial autocorrelation and spatial filtering, Springer, Berlin, Heidelberg and New York, 2003) for area data, and leads to eigenfunction based spatial filtering specifications of both the log-normal and the Poisson spatial interaction model versions (see Fischer and Griffith (J Reg Sci 48(5):969–989, 2008)).