An Optimization Framework for Spatial Component Analysis and Its Extension to Two-Scale Discriminant Function Analysis

Abstract

This study formulates spatial component analysis as a ratio-of-trace optimization problem. Under this framework, the one-scale spatial component analysis is extended to a two-scale analysis, which is also essentially an adaptation of linear discriminant analysis (LDA). It does so by including spatial effects with respect to spatial autocorrelation at either one or both observation scales. The spatialized LDA (SLDA) seeks the projection that minimizes the similarity between neighboring regions and maximizes the similarity within regions or neighboring units within regions. Regionalization results for the AZP and Max-p methods are evaluated for three different heuristic algorithms. The evaluation is designed specifically to rank the ability of regionalization methods to separate classified neighbor regions in a one-dimensional subspace, which is constructed by the SLDA. The process is mainly based on the first pseudo eigenvalue of the SLDA.