Martin Charlton

Geographically Weighted Regression

In regression models where the cases are geographical locations, sometimes regression coefficients do not remain fixed over space. A technique for exploring this phenomenon, geographically weighted regression is introduced. A related Monte Carlo significance test for spatial non-stationarity is also considered. Finally, an example of the method is given, using limiting long-term illness data from the 1991 UK census.

Geographically Weighted Regression: A Natural Evolution of the Expansion Method for Spatial Data Analysis

Geographically weighted regression and the expansion method are two statistical techniques which can be used to examine the spatial variability of regression results across a region and so inform on the presence of spatial nonstationarity. Rather than accept one set of ‘global’ regression results, both techniques allow the possibility of producing ‘local’ regression results from any point within the region so that the output from the analysis is a set of mappable statistics which denote local relationships. Within the paper, the application of each technique to a set of health data from northeast England is compared. Geographically weighted regression is shown to produce more informative results regarding parameter variation over space.

A Mark 1 Geographical Analysis Machine for the automated analysis of point data sets

Abstract This paper presents the first of a new generation of spatial analytical technology based on a fusion of statistical, GIS and computational thinking. It describes how to build what is termed a Geographical Analysis Machine (GAM), with high descriptive power. A GAM offers an imaginative new approach to the analysis of point pattern data based on a fully automated process whereby a point data set is explored for evidence of pattern without being unduly affected by predefined areal units or data error. No prior information or specification of particular location-specific hypotheses is required. If geographical data contain strong evidence of pattern in geographical space, then the GAM will find it. This technology is demonstrated by an analysis of data on cancer for northern England.

Spatial Nonstationarity and Autoregressive Models

Until relatively recently, the emphasis of spatial analysis was on the investigation of global models and global processes. Recent research, however, has tended to explore exceptions to general processes, and techniques have been developed which have as their focus the investigation of spatial variations in local relationships. One of these techniques, known as geographically weighted regression (GWR), developed by the authors is used here to investigate spatial variations in spatial association. The particular framework in which spatial association is examined here is the spatial autoregressive model of Ord, although the technique can easily be applied to any form of spatial autocorrelation measurement. The conceptual and theoretical foundations of GWR applied to the Ord model are followed by an empirical example which uses data on owner-occupation in the housing market of Tyne and Wear in northeast England where the problems of relying on global models of spatial association are demonstrated. This empirical investigation of spatial variations in spatial autocorrelation prompts a further discussion of several issues concerning the statistical technique.

Geographically weighted summary statistics — a framework for localised exploratory data analysis

Geographical kernel weighting is proposed as a method for deriving local summary statistics from geographically weighted point data. These local statistics are then used to visualise geographical variation in the statistical distribution of variables of interest. Univariate and bivariate summary statistics are considered, for both moment-based and order-based approaches. Several aspects of visualisation are considered. Finally, an example based on house price data is presented.

Geographically weighted regression with a non-Euclidean distance metric: a case study using hedonic house price data

Geographically weighted regression (GWR) is an important local technique for exploring spatial heterogeneity in data relationships. In fitting with Tobler’s first law of geography, each local regression of GWR is estimated with data whose influence decays with distance, distances that are commonly defined as straight line or Euclidean. However, the complexity of our real world ensures that the scope of possible distance metrics is far larger than the traditional Euclidean choice. Thus in this article, the GWR model is investigated by applying it with alternative, non-Euclidean distance (non-ED) metrics. Here we use as a case study, a London house price data set coupled with hedonic independent variables, where GWR models are calibrated with Euclidean distance (ED), road network distance and travel time metrics. The results indicate that GWR calibrated with a non-Euclidean metric can not only improve model fit, but also provide additional and useful insights into the nature of varying relationships within the house price data set.

Building a prototype Geographical Correlates Exploration Machine

The paper describes a exploratory procedure for data analysis for use within GIS. The objective is to search digital map databases for the presence of geographical relationships that may be useful for descriptive purposes, as a pointer towards areas for further investigation, and as a means of generating hypotheses for subsequent testing. A prototype Geographical Correlates Exploration Machine (GCEM) is demonstrated by searching for possible linkages between children with leukaemia and a selection of environmental coverages. Arc Info is used for the GIS parts and a Cray X-MP/48 supercomputer for the analysis. Ultimately, it is envisaged that GCEM will be able to run entirely within a GIS workstation environment.

Stage dependent variability in tractive force distribution through a riffle–pool sequence

High resolution data on spatial and temporal variability in flow hydraulics and sediment transport within riffle–pool sequences are required to improve understanding of how fluvial processes maintain these meso-scale bedforms. This paper addresses this issue by providing velocity and boundary shear stress data over a range of discharges from base flow (0.07 m3 s−1) to just over bankfull (8.52 m3 s−1), from a sequence of four pools and three riffles in the River Rede, Northumberland. The data supports the reversal hypothesis of Keller [Geol. Soc. Am. Bull. 87 (1971) 753.] as the primary explanation for the maintenance of the riffle–pool sequence, although they also indicate that spatial variability in tractive force is highly stage dependent and complex. Section-averaged velocity data indicate reversal to be evident at four out of six riffle–pool units. An equalisation in velocity was found for the other two riffle–pool units close to bankfull stage. The spatial patterns of tractive force exhibited in the study reach as a result of increased discharge demonstrate that riffle–pool units operate independently of one another. Shear stress reversals were observed in individual riffle–pool units at different river stages during a flood hydrograph, and in some instances, two occurred in the same riffle–pool unit during a single flow event. Pools were characterised by coarser bed sediments and narrower channel widths in comparison to riffles, increasing the likelihood of tractive force reversal in the River Rede. Areas of predicted bed sediment entrainment obtained from τo−τc, matched observed channel changes in the upper part of the study reach, but over-estimated change in the middle portion of the reach.

Geographically weighted principal components analysis

Principal components analysis (PCA) is a widely used technique in the social and physical sciences. However in spatial applications, standard PCA is frequently applied without any adaptation that accounts for important spatial effects. Such a naive application can be problematic as such effects often provide a more complete understanding of a given process. In this respect, standard PCA can be (a) replaced with a geographically weighted PCA (GWPCA), when we want to account for a certain spatial heterogeneity; (b) adapted to account for spatial autocorrelation in the spatial process; or (c) adapted with a specification that represents a mixture of both (a) and (b). In this article, we focus on implementation issues concerning the calibration, testing, interpretation and visualisation of the location-specific principal components from GWPCA. Here we initially consider the basics of (global) principal components, then consider the development of a locally weighted PCA (for the exploration of local subsets in attribute-space) and finally GWPCA. As an illustration of the use of GWPCA (with respect to the implementation issues we investigate), we apply this technique to a study of social structure in Greater Dublin, Ireland.

The GWmodel R package: further topics for exploring spatial heterogeneity using geographically weighted models

In this study, we present a collection of local models, termed geographically weighted (GW) models, which can be found within the GWmodel R package. A GW model suits situations when spatial data are poorly described by the global form, and for some regions the localized fit provides a better description. The approach uses a moving window weighting technique, where a collection of local models are estimated at target locations. Commonly, model parameters or outputs are mapped so that the nature of spatial heterogeneity can be explored and assessed. In particular, we present case studies using: (i) GW summary statistics and a GW principal components analysis; (ii) advanced GW regression fits and diagnostics; (iii) associated Monte Carlo significance tests for non-stationarity; (iv) a GW discriminant analysis; and (v) enhanced kernel bandwidth selection procedures. General Election data-sets from the Republic of Ireland and US are used for demonstration. This study is designed to complement a companion GWmodel study, which focuses on basic and robust GW models.