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Geary’s c and Spectral Graph Theory

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  • Hiroshi Yamada

    (School of Informatics and Data Science, Hiroshima University, 1-2-1 Kagamiyama, Higashi-Hiroshima 739-8525, Japan)

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.

Suggested Citation

  • Hiroshi Yamada, 2021. "Geary’s c and Spectral Graph Theory," Mathematics, MDPI, vol. 9(19), pages 1-23, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:19:p:2465-:d:649227
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    References listed on IDEAS

    as
    1. Garcia, Damien, 2010. "Robust smoothing of gridded data in one and higher dimensions with missing values," Computational Statistics & Data Analysis, Elsevier, vol. 54(4), pages 1167-1178, April.
    2. Yamada, Hiroshi, 2020. "A Smoothing Method That Looks Like The Hodrick–Prescott Filter," Econometric Theory, Cambridge University Press, vol. 36(5), pages 961-981, October.
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