Eigenfunction properties and approximations of selected incidence matrices employed in spatial analyses
By Daniel A. Griffith. 2000
123 citations recorded by the source corpus.
Abstract Mathematical properties of eigenfunctions of selected incidence matrices appearing in spatial statistics formulae are summarized. Seven theorems are proposed and proved, and three conjectures are posited. Results summarized here allow the determinant of massively large n×n geographic weights matrices to be accurately approximated. In addition, the behavior of eigenfunctions for graphs affiliated with a linear configuration of connected nodes are better understood.
Subfields: Economics and Econometrics, Environmental Engineering
View Paper: https://www.semanticscholar.org/paper/37952948c1d661b10d7066968ecc62df082f6c6c
How researchers see this paper
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In Economics and Econometrics
- Carsten F. Dormann
- Stéphane Dray
- Daniel A. Griffith
- Pierre Legendre
- P. Peres‐Neto
- I. Kühn
- S. Saïd
- François Débias
In Environmental Engineering
- Stéphane Dray
- Pierre Legendre
- P. Peres‐Neto
- S. Saïd
- François Débias
- Arthur Getis
- Roger S. Bivand
- H. Wagner
Citations in the corpus
- Methods to account for spatial autocorrelation in the analysis of species distributional data : a review (2007, methodology): “Due to numerical precision regarding the eigenvector extraction of large matrices (Bai et al. 1996) the method is limited to ca 7000 observations depending on platform and software (but see Griffith 2000a, for solutions based on large binary connectivity matrices).”
- Spatial ordination of vegetation data using a generalization of Wartenberg's multivariate spatial correlation (2008, methodology): “…1 (where = – is a centering operator, 1t = [1,…,1] a (1 by n) row vector and I the identity matrix), for the B and W weighting options respectively, while the eigenvectors of these matrices (Griffith 1996, 2000a) can be used for spatial filtering purposes (Griffith 2000b; Getis & Griffith 2002).”
- Comparative Spatial Filtering in Regression Analysis (2002, background): “Griffith (2000a) outlines his filtering procedure in considerable detail, and includes selected empirical applications to illustrate its use and effectiveness.”
- Generating spatially constrained null models for irregularly spaced data using Moran spectral randomization methods (2015, background): “A set of n observations with a full-rank spatial weights matrixW of size n 9 n will result in n – 1 orthogonal and uncorrelated (Griffith 2000) eigenvectors Vk associated with eigenvalues kk, while a single eigenvector with zero eigenvalue is dropped.”
- Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM) (2006, methodology): “Mathematical formalism extends the original PCNM approach: ⇒ various definitions of spatial weighting matrices ⇒ negative spatial autocorrelation ⇒ analytical formulation (cos and sin) for regular sampling (GRIFFITH 2000) ⇒ very large datasets (e.”