Academic journal article Geographical Analysis

Anisotropic Variance Functions in Geographically Weighted Regression Models

Academic journal article Geographical Analysis

Anisotropic Variance Functions in Geographically Weighted Regression Models

Article excerpt

Most standard methods of statistical analysis used in the social and environmental sciences are built upon the basic assumptions of independence, homogeneity, and isotropy. A notable exception to this rule is the collection of methods used in geographical analysis, which have been designed to take into account serial dependence often observed in spatial data. In addition, recent developments, in particular the method of geographically weighted regression, have provided the tools to model nonstationary processes, and thus evidence that challenges the assumption of homogeneity. The assumption of isotropy, however, although suspect, has received considerably less attention, and there is thus a need for tools to study anisotropy in a more systematic fashion. In this paper we expand the method of geographically weighted regression in a simple yet effective way to explore the topic of anisotropy in spatial processes. We discuss two different estimation situations and exemplify the proposed technical development by means of a case study. The results suggest that anisotropy issues might be a fairly common occurrence in spatial processes and/or in the statistical modeling of spatial processes.

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1. INTRODUCTION

Most standard methods of statistical analysis used in the social and environmental sciences are built upon the basic assumptions of serial independence, homogeneity, and isotropy. A majority of these methods were originally developed within fields for which said assumptions were reasonable, or at a time when they were needed to make the problems tractable (Hepple 1998). A consequence of this historic development is that over time these basic assumptions were transmitted to, and in some cases unconsciously adopted by, different fields for which statistical methods became important tools of analysis. The assumption of independence, however, has long been recognized to be at odds with certain fundamental premises in a number of fields, among which analytical geography takes a prominent place. Indeed, many important contributions to statistical analysis by geographers have been the result of recognizing that independence is the exception, rather than the rule, when it comes to the study of spatial processes: according to Tobler's first law of geography "everything is related to everything else, but near things are more related than distant things" (Tobler 1970). Hence, an important body of literature in geographical analysis has been advanced that deals with methods to detect violations to the independence assumption (statistics of spatial association), and to develop models that explicitly consider spatial dependence or association (spatial autoregression, kriging). Although this topic has been the subject of considerably interest and research efforts, recent work in the domain of local spatial analysis has called into question the assumptions of homogeneity and isotropy that so far have received considerably less attention.

The basic linear regression model, for example, assumes, in addition to independence, a generating process that is stationary (i.e., homogeneous or spatially invariant) and isotropic (i.e., regular or directionally invariant). Stationarity implies that the process operates identically all over the study area. In terms of a model, this means that a single set of global parameters is adequate to describe the process: there are no local variations in the relationships between variables. Although this might genuinely be the case for some processes, or when small regions and/or datasets are the target of analysis, it is increasingly evident that local variations in relationships can play an important exploratory and explanatory role in the analysis of spatial data. Evidence provided by techniques such as the expansion method (Casetti 1972) and by the method of geographically weighted regression (GWR; Brunsdon et al. 1996; Fotheringham et al. 2002; Paez et al. 2002a,b), shows that nonstationarity, the condition by which geographical relationships vary across space, might be more common than previously thought. …

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