Fig. 3.

Model designs and configurations of different partitions on $\mathcal{S}$, the 100 × 100 equally spaced grid points in two-dimensional space [0, 1]², in Setting 4 of the simulation study for spatial variable screening: (a) the true coefficient β_j, which takes only three possible values, −3 in blue, 3 in red and 0 in green; (b) one set of simulated predictors X_j over space from a Gaussian random field on $\mathcal{S}$ with covariance kernel exp(−10||s − s′||₂) for s, s′ ∈ $\mathcal{S}$; (c) the partition ${\{{\mathcal{S}}_g\}}_{g=1}^{100}$ in Setting 4 to define the true nonzero coefficients, with β_j ⧧ 0 if and only if $s_j\in {\mathcal{S}}_1$, in yellow. Panels (c), (d), (e) and (f) respectively represent the optimal partition ${\mathcal{G}}^{\text{opt}}$, the size-reduced partition ${\mathcal{G}}^{\text{red}}$, and two misspecified partitions, ${\mathcal{G}}^{{\text{mis}}_1}$ and ${\mathcal{G}}^{{\text{mis}}_2}$, for the corresponding partition-based screening for selecting variables in Setting 4; in panels (c)–(f), ${\mathcal{S}}_1$ is coloured yellow.