Spatial weights matrix from point coordinates

The spatial weights matrix formalises "neighbour" for a spatial model: \(w_{ij}\) is non-zero when \(j\) is a neighbour of \(i\) and zero otherwise. Everything downstream — the spatial lag \(Wy\), Moran's I, a spatial autoregressive model — is defined relative to this choice, which is why the choice is an assumption rather than an estimate.

Usage

Wmtwgt(
  coords,
  method = "distance",
  k_or_threshold = 1,
  alpha = 1,
  row_standardize = TRUE
)

Arguments

coords: Numeric matrix of \((n, d)\) coordinates, one row per unit. A plain vector is read as \(n\) points on a line.
method: One of "distance", "knn", "inverse".
k_or_threshold: Distance band for "distance" and "inverse", or the number of neighbours \(k\) for "knn".
alpha: Distance decay exponent for "inverse". Default 1.
row_standardize: Divide each non-empty row by its sum. Default TRUE.

Value

Named list with weights, n, scheme, row_standardized, n_links, n_islands, pct_nonzero, method.

Details

Three coordinate-based schemes are offered.

"distance": Distance band: \(w_{ij} = 1\) when \(0 < d_{ij} \le\) k_or_threshold, else 0. A unit with no neighbour inside the band keeps an all-zero row.
"knn": k nearest neighbours: \(w_{ij} = 1\) for the k_or_threshold units nearest to \(i\). This relation is not symmetric.
"inverse": Inverse distance: \(w_{ij} = d_{ij}^{-\alpha}\) for \(0 < d_{ij} \le\) k_or_threshold, else 0. The threshold may be Inf, which connects every pair.

Distances are Euclidean and the diagonal is always zero: a unit is not its own neighbour. With row_standardize the rows that have at least one neighbour are divided by their sum, so a spatial lag is a weighted average of neighbours; all-zero rows are left alone.

Contiguity weights (rook, queen) are not constructed here. They need a lattice or polygon topology rather than points — two units are rook-contiguous when their boundaries share an edge, which a coordinate pair does not determine.

Mirrors morie.fn.wmtwgt on the Python side.

References

Anselin L (1988). Spatial Econometrics: Methods and Models. Kluwer, Dordrecht, Chapter 3.

Examples

Wmtwgt(cbind(c(0, 1, 2, 3), c(0, 0, 1, 1)), "knn", 2)$n_links
#> [1] 8