Using geovisual analytics to compare the performance of geographically weighted discriminant analysis versus its global counterpart, linear discriminant analysis

Abstract

Geographically weighted spatial statistical methods are a family of spatial statistical methods developed to address the presence of non-stationarity in geographical processes, the so-called spatial heterogeneity. While these methods have recently become popular for analysis of spatial data, one of their characteristics is that they produce outputs that in themselves form complex multi-dimensional spatial data sets. Interpretation of these outputs is therefore not easy, but is of high importance, since spatial and non-spatial patterns in the results of these methods contain clues to causes of underlying non-stationarity. In this article, we focus on one of the geographically weighted methods, the geographically weighted discriminant analysis (GWDA), which is a method for prediction and analysis of categorical spatial data. It is an extension of linear discriminant analysis (LDA) that allows the relationship between the predictor variables and the categories to vary spatially. This produces a very complex data set of GWDA results, which include on top of the already complex discriminant analysis outputs (e.g. classifications and posterior probabilities) also spatially varying outputs (e.g. classification function parameters). In this article, we suggest using geovisual analytics to visualise results from LDA and GWDA to facilitate comparison between the global and local method results. For this, we develop a bespoke visual methodology that allows us to examine the performance of global and local classification method in terms of quality of classification. Furthermore, we are also interested in identifying the presence (or absence) of non-stationarity through comparison of the outputs of both methods. We do this in two ways. First, we visually explore spatial autocorrelation in both LDA and GWDA misclassifications. Second, we focus on relationships between the classification result and the independent variables and how they vary over space. We describe our visual analytic system for exploration of LDA and GWDA outputs and demonstrate our approach on a case study using a data set linking election results with a selection of socio-economic variables.

Keywords:

• geovisual analytics
• geovisualisation
• geographically weighted discriminant analysis
• spatial statistics

Acknowledgements

This work was supported by a Research Frontiers Programme Grant (09/RFP/CMS2250) awarded to Dr. Urška Demšar by Science Foundation Ireland under the National Development Plan. We thank Prof. Chris Brunsdon from the University of Liverpool, UK; Drs. Frank Hardisty and Anthony Robinson from the Pennsylvania State University, USA; Prof. Stewart Fotheringham from the University of St. Andrews, UK; Mr. Martin Charlton from the National University of Ireland, Ireland; and Profs. Jason Dykes and Jo Wood from the City University, UK, for their suggestions and advice.