Weight Matrices

SpatialConnectivity

Summary of a weight matrix's structure, returned by spatial_connectivity. Worth looking at before fitting anything.

Fields

• N, n_nonzero, density: size and sparsity.
• avg_neighbors, min_neighbors, max_neighbors: the degree distribution.
• n_islands: units with no neighbours at all.
• is_symmetric: whether W == W'. Row-standardizing a symmetric contiguity matrix generally destroys symmetry, so false here is normal and not a problem.
• spectral_radius: largest eigenvalue magnitude; 1 for a row-standardized matrix.

band_weights(coords, d_min, d_max; row_standardize=true)

Binary weights: w_ij = 1 if d_min < d_ij ≤ d_max, else 0.

block_diagonal_weights(W_list)

Combine a vector of sub-region weight matrices into one block-diagonal matrix.

distance_weights(coords, cutoff; decay=:inverse, alpha=1.0, row_standardize=true)

Build distance-decay spatial weights. Pairs beyond cutoff get zero weight.

Decay functions:

• :inverse — 1 / d^alpha
• :negative_exponential — exp(-alpha * d)
• :power — d^(-alpha) (same as inverse, explicit name)
• :bisquare — (1 - (d/cutoff)^2)^2

knn_weights(coords, k; binary=false, row_standardize=true, symmetrize=false)

Build a spatial weight matrix from k-nearest neighbors. coords is N×D (rows = observations). Returns a sparse N×N matrix.

minmax_standardize(W) -> SparseMatrixCSC

Copy of W divided by the smaller of its largest row sum and largest column sum. See minmax_standardize! for the in-place version.

A middle course between row_standardize and spectral_standardize: it keeps relative connectedness while being cheaper than an eigenvalue computation, which matters for large W.

minmax_standardize!(W)

MinMax standardization: W / (min(r_max) + max(r_max)) where r_max is the vector of max absolute row sums and max absolute column sums.

read_gal(path; ids=nothing) → (W, ids)

Read a GeoDa GAL contiguity weights file. Returns the row-standardized sparse weight matrix and the observation ID vector, where row i of W corresponds to ids[i].

The ordering comes from the file's observation records, in record order. Pass ids to impose an external table's ordering instead. A neighbour id with no observation record raises an error.

read_gwt(path; ids=nothing) → (W, ids)

Read a GeoDa GWT (general weights) file. Returns the sparse weight matrix and the observation ID vector, where row i of W corresponds to ids[i].

A GWT file is only an edge list and carries no ordering of its own, so the ids found in it are sorted — numerically when they all parse as integers, else lexicographically. Pass ids to impose an external table's ordering instead.

row_standardize(W) -> SparseMatrixCSC

Row-standardized copy of W: every row with at least one neighbour sums to 1. See row_standardize! for the in-place version.

This is the conventional normalization, and most of this package assumes it. It makes W*y a neighbourhood average, bounds the spatial parameter inside (-1, 1), and makes Moran's I equal the slope in a Moran scatter plot. Rows with no neighbours stay all-zero.

row_standardize!(W)

Row-standardize a sparse weight matrix in place: each row sums to 1. Rows with no neighbors are left as zeros.

spatial_connectivity(W)

Compute diagnostic statistics for a spatial weight matrix.

spectral_standardize(W) -> SparseMatrixCSC

Copy of W divided by its spectral radius. See spectral_standardize! for the in-place version.

Bounds the spatial parameter in (-1, 1) as row-standardization does, but without rescaling rows relative to one another, so differences in connectedness between units are preserved. Preferable when the number of neighbours is itself meaningful; the cost is that W*y is no longer a simple neighbourhood average.

spectral_standardize!(W)

Divide all entries by the spectral radius (largest eigenvalue magnitude).

write_gal(path, W, ids)

Write a weight matrix in GeoDa GAL format.

write_gwt(path, W, ids; threshold=0.0)

Write a weight matrix in GeoDa GWT format.