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Coordinate-spatial structured effects ​

Status — Stable (Gaussian mean, coordinate-direct)

Mirrors drmTMB's Coordinate-spatial structured effects. In DRM.jl today: spatial(1 | site) on the Gaussian mean with site coordinates — an exponential spatial correlation K(ρ) = exp(-d / ρ) whose range ρ is estimated jointly. Closed-form GLS (coordinate-direct; mesh/SPDE is planned).

Nearby sites tend to be similar. spatial(1 | site) adds a random intercept with a distance-based correlation built from the site coordinates, u ~ N(0, σ_s² K(ρ)), and estimates the spatial range ρ along with the spatial SD. Pass the coordinates via coords = (one row per site level):

julia
using DRM, Random, LinearAlgebra
Random.seed!(8)

G = 50
coords = rand(G, 2) .* 10.0                  # site locations in [0,10]²
Ddist = [sqrt(sum(abs2, coords[k, :] .- coords[l, :])) for k in 1:G, l in 1:G]
K = exp.(-Ddist ./ 2.0) + 1e-8 * I           # true correlation, range = 2
u = 0.9 .* (cholesky(Symmetric(K)).L * randn(G))

m = 4; n = G * m
site = repeat(1:G, inner = m)
x = randn(n)
y = 0.3 .+ 0.5 .* x .+ u[site] .+ 0.4 .* randn(n)

fit = drm(bf(@formula(y ~ x + spatial(1 | site)), @formula(sigma ~ 1)),
          Gaussian(); data = (; y, x, site), coords = coords)

re_sd(fit)[:site]      # spatial SD (square it for the spatial variance)
1.1726824599329653

re_sd(fit)[:site] is the spatial SD σ_s; vc(fit) is the matching random-effect (co)variance summary for the structured block.

julia
exp(coef(fit, :range)[1])     # estimated spatial range ρ
4.030356344863789

The spatial range is recovered only weakly from a single realization, so treat its point estimate with care; the spatial variance and the fixed effects are the robust outputs. spatial, phylo, animal, and relmat all share one closed-form structured-GLS engine — only the source of the correlation differs (coordinates / tree / pedigree / supplied matrix).

Gaussian only (for now)

A coordinate spatial effect is currently supported on the Gaussian mean. Routing spatial(1 | site) through the sparse-Laplace engine for the non-Gaussian families (Poisson, NB2, Gamma, Beta, Binomial) — alongside relmat / animal — is tracked as future work in issue #167 (phylo already has its non-Gaussian route).

See also ​