Geary’s c and Spectral Graph Theory

Hiroshi Yamada
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Hiroshi Yamada: School of Informatics and Data Science, Hiroshima University, 1-2-1 Kagamiyama, Higashi-Hiroshima 739-8525, Japan

Mathematics, 2021, vol. 9, issue 19, 1-23

Abstract: Spatial autocorrelation, of which Geary’s c has traditionally been a popular measure, is fundamental to spatial science. This paper provides a new perspective on Geary’s c . We discuss this using concepts from spectral graph theory/linear algebraic graph theory. More precisely, we provide three types of representations for it: (a) graph Laplacian representation, (b) graph Fourier transform representation, and (c) Pearson’s correlation coefficient representation. Subsequently, we illustrate that the spatial autocorrelation measured by Geary’s c is positive (resp. negative) if spatially smoother (resp. less smooth) graph Laplacian eigenvectors are dominant. Finally, based on our analysis, we provide a recommendation for applied studies.

Keywords: spatial autocorrelation; Geary’s c; spectral graph theory; graph Laplacian; graph Laplacian quadratic form; graph Fourier transform; graph Laplacian eigenvalues; graph Laplacian eigenvectors; Pearson’s correlation coefficient; path graph; discrete cosine transform (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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