Spatial Filtering

ESFModel

An eigenvector spatial filtering fit, returned by fit_esf.

Spatial structure is absorbed by adding selected eigenvectors of the doubly-centred weight matrix to the design as synthetic regressors, chosen stepwise. Because they are orthogonal to the intercept and to each other, the covariate coefficients stay interpretable as ordinary regression coefficients — the appeal of the method over a lag model.

Inspect the chosen basis with selected_eigenvectors.

REESFModel

A random-effects eigenvector spatial filtering fit, returned by fit_re_esf.

Instead of selecting eigenvectors stepwise as fit_esf does, this keeps the whole screened pool and shrinks every coefficient towards zero, governed by one variance ratio theta = tau2/sigma2 estimated by profile REML. That avoids the model-selection variance stepwise search introduces.

SpatialEigen

A screened eigenvector basis for spatial filtering, as returned by spatial_eigenvectors.

Fields

• values::Vector{T}: eigenvalues of Ω = M Wᵤ M, in descending order.
• vectors::Matrix{T}: the corresponding eigenvectors, one per column. Columns are orthonormal and each sums to zero.
• moran::Vector{T}: each eigenvector's Moran coefficient, (N/S₀)λᵢ.
• S0::T: the sum of all weights, Σᵢⱼ wᵢⱼ.
• N::Int: the number of spatial units.

Pass an instance as the eigen_basis keyword of fit_esf or fit_re_esf to reuse one decomposition across several fits.

fit_esf(X, y, W; family=:gaussian, max_eigenvectors=20, criterion=:moran, ...)

Eigenvector spatial filtering (Griffith, 2003). Selected eigenvectors of the doubly-centred W enter as extra regressors, absorbing the spatial autocorrelation so the remaining coefficients can be read as aspatial:

\[g(\mu) = \alpha + X\beta + E_{sel}\gamma\]

family is :gaussian (OLS), :poisson, :logit, or :negbin; the last three are fit by IRLS, with the negative binomial's dispersion updated by method of moments between IRLS passes (Chun et al., 2016). This makes ESF usable wherever a GLM is, which parametric SAR/SEM cannot match.

Eigenvectors are added greedily. criterion=:moran picks, at each step, the candidate that most reduces the absolute Moran coefficient of the residuals (Pearson residuals for GLM families); criterion=:aic picks the one that most reduces AIC. Selection stops at max_eigenvectors, when no candidate improves the criterion, or — under :moran — once |MC| falls below moran_tol.

min_moran_ratio screens the candidate pool to eigenvectors with MCᵢ/max(MC) ≥ ratio (default 0.25, the usual positive-SA screen). Pass a precomputed SpatialEigen as eigen_basis to reuse one decomposition across several fits.

Cost is O(max_eigenvectors × n_candidates) model fits, so keep the pool bounded for GLM families.

fit_re_esf(X, y, W; min_moran_ratio=0.25, n_candidates=nothing, varnames=nothing)

Random-effects eigenvector spatial filtering (Murakami & Griffith, 2015):

\[y = X\beta + E\gamma + \varepsilon, \quad \gamma \sim N(0, \tau^2\Lambda), \quad \varepsilon \sim N(0, \sigma^2 I)\]

Rather than selecting eigenvectors discretely, every positive-eigenvalue eigenvector is retained and its coefficient shrunk toward zero, with the shrinkage governed by the single variance ratio θ = τ²/σ² estimated by profile REML. This sidesteps the model-selection variance of stepwise ESF.

Because E has orthonormal columns, V = σ²(I + θEΛE') inverts in closed form — its eigenvalues are 1 + θλₗ inside span(E) and 1 outside — so each REML evaluation is O(NL), not O(N³). The decomposition itself is still O(N³).

Only positive-eigenvalue eigenvectors are used, the standard choice for positive spatial autocorrelation, and min_moran_ratio screens them further to MCᵢ/max(MC) ≥ ratio.

moran_coefficient(y, W)

Moran coefficient (N/S₀) · (z'Wz)/(z'z) for z = y - ȳ, where S₀ is the sum of all weights. The bare statistic with no inference — use SpatialDiagnostics.moran_global when you need a p-value.

selected_eigenvectors(m::ESFModel)

Indices of the selected eigenvectors (into the candidate pool) with their Moran coefficients, plus the residual Moran coefficient before and after filtering.

spatial_eigenvectors(W; n_candidates=nothing, min_moran_ratio=0.0)

Eigendecomposition of the doubly-centred weight matrix Ω = (I - 11'/N) Wsym (I - 11'/N), following Griffith (2003).

The eigenvectors are mutually orthogonal and uncorrelated, and each carries Moran coefficient MCᵢ = (N/S₀)λᵢ. Eigenvectors with large positive eigenvalues describe positive spatial autocorrelation patterns; large negative eigenvalues describe negative ones. Results are sorted by eigenvalue, descending.

An asymmetric W (e.g. row-standardised) is symmetrised as (W + W')/2 so the eigenvalues are real.

The constant vector is removed by construction rather than by discarding an eigenvector afterwards: with Q an orthonormal basis of 1's orthogonal complement, Q'ΩQ = Q'WsymQ, so eigendecomposing the reduced (N-1)×(N-1) matrix and mapping back as E = QV yields vectors that are orthonormal and exactly mean-zero. Decomposing Ω directly would not: its null space is multi-dimensional, the solver may return any rotation within it, and the constant direction then smears across several near-zero eigenvectors that are each partly collinear with the intercept.

n_candidates keeps only the leading eigenvectors; min_moran_ratio keeps those with MCᵢ / max(MC) ≥ ratio, the usual screen for positive-SA filtering (0.25 is a common choice). Both are applied after sorting.

Dense eigen is O(N³) — intended for moderate N.