Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data

Gaining a better understanding of spatial data and a deeper meaning of spatial statistical results gleaned from geo-referenced data requires a more complete knowledge of properties of the underlying geometry. This paper addresses these two topics by further investigating important geometric features of spatial data. The importance of this work is expressed in the geographic information system literature regarding representations of geographic space, as well as various geographic literatures concerned with spatial statistical modelling. Answers to three questions are obtained here. One asks whether or not those eigenvectors associated with the three largest eigenvalues of a binary geographic weights matrix have natural interpretations. A second question asks whether or not the eigenvalues of a geographic weights matrix prove useful in understanding the sampling distribution for spatial autocorrelation in a given geographic landscape. A third question addresses the issue of spatial autocorrelation components geo-referenced phenomena. The analysis summarized in this paper documents responses to these three questions. Empirical evidence is gleaned from both Canadian urban census tract data and the square tessellations of remotely sensed data.