Contiguity Analysis and Classification

Abstract

Let us consider n vertices of a symmetric graph G whose associated matrix is M (mii'= 1 if vertices i and i' are joined by an edge, mii'= 0 otherwise). These vertices are simultaneously described by p variables (xijis the value of variable j for vertex i). Such situation occurs when vertices represent time-points, geographic areas. We consider here the case of graphs that are not external, but derived from the observations themselves, namely the series of nearest neighbours graphs. Contiguity Analysis simultaneously uses a local covariance matrix C and the global covariance matrix V. The minimization of the ratio: u'Cu/u'Vu (u being a p-vector) provides a visualization tool allowing for the unfolding of some non-linear structures and generalizing linear discriminant analysis in the case of overlapping clusters.