This function calculates Moran's eigenvectors and their corresponding eigenvalues.
meigen( coords, model = "exp", threshold = 0, enum = NULL, cmat = NULL )Matrix of spatial point coordinates (\(N\) x 2)
Type of kernel to model spatial dependence. The currently available options are "exp" for the exponential kernel, "gau" for the Gaussian kernel, and "sph" for the spherical kernel
Threshold for eigenvalues (scalar). Suppose that \(\lambda\)_1 is the first eigenvalue, eigenvectors whose eigenvalues that are equal or greater than [\(threshold\) x \(\lambda\)_1] are extracted. The value must lie between 0 and 1. Default is zero (see Details)
Optional. A spatial connectivity matrix (\(N\) x \(N\)). It must be provided when the user wants to apply a spatial connectivity matrix other than the default exponential decay matrix (e.g. k-nearest neighbor, adjacency; see Detail)
Optional. The muxmum acceptable mumber of eigenvectors to be extracted (scalar)
Matrix of the first \(L\) eigenvectors (\(N\) x \(L\))
Vector of the first \(L\) eigenvalues (\(L\) x 1)
Vector of all eigenvalues (\(N\) x 1)
List of other outcomes, which are internally used
If \(cmat\) is not provided (default), this function extracts Moran's eigenvectors from MCM, where M = I - 11'/\(N\) is a centering operator. C is a \(N\) x \(N\) connectivity matrix whose (\(i\), \(j\))-th element equals exp(\(-d_i,j / h\)), where \(d_i,j\) is the Euclidean distance between sample sites \(i\) and \(j\), and \(h\) is a range parameter given by the maximum length of the minimum spanning tree connecting sample sites (see Dray et al., 2006). If \(cmat\) is provided, this function performs the same calculation after C is replaced with \(cmat\).
If \(threshold\) is not provided (default), all eigenvectors corresponding to positive eigenvalues are extracted. It implies to consider all elements describing positive spatial dependence. If \(threshold\) is provided, eigenvectors whose corresponding eigenvalues are equal to or greater than [\(threshold\) x \(\lambda\)_1] are extracted. \(threshold\) = 0.00 or 0.25 are standard assumptions (see Griffith, 2003; Murakami and Griffith, 2015).
Dray, S., Legendre, P., and Peres-Neto, P.R. (2006) Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM). Ecological Modelling, 196 (3), 483-493.
Griffith, D.A. (2003) Spatial autocorrelation and spatial filtering: gaining understanding through theory and scientific visualization. Springer Science & Business Media.
Murakami, D. and Griffith, D.A. (2015) Random effects specifications in eigenvector spatial filtering: a simulation study. Journal of Geographical Systems, 17 (4), 311-331.
meigen_f for fast eigen-decomposition