This function performs a fast approximation of Moran's eigenvectors and their corresponding eigenvalues.
meigen_f( coords, model = "exp", enum = 200 )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
Number of eigenvectors and eigenvalues to be extracted (scalar). Default is 200
Matrix of the first \(L\) approximated eigenvectors (\(N\) x \(L\))
Vector of the first \(L\) approximated eigenvalues (\(L\) x 1)
Vector of all approximated eigenvalues (\(enum\) x 1)
List of other outcomes, which are internally used
This function extracts approximated Moran's eigenvectors from MCM. M = I - 11'/\(N\) is a centering operator, and C is a spatial connectivity matrix whose (\(i\), \(j\))-th element is given by 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).
Following a simulation result that 200 eigenvectors are sufficient for accurate approximation of ESF models (Murakami and Griffith, 2017), this function approximates the first 200 eigenvectors by default (i.e., \(enum\) = 200). If \(enum\) is given by a smaller value like 100, the computation time will be shorter, but with greater approximation error. Following meigen, eigenvectors corresponding to negative eigenvalues are omitted among the 200 eigenvectors.
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.
Murakami, D. and Griffith, D.A. (2018) Eigenvector spatial filtering for large data sets: fixed and random effects approaches. Geographical Analysis, doi: 10.1111/gean.12156.