543 In Figure 2 (top), we show the 4D omnidirectional variogram of both principal components. To perform GPR, we need to specify a grid. Because we are in 4D, specifying a regular dense Cartesian grid would be costly; indeed, using 100 points in each dimension results in 100Y grid cells. Luckily, GPR can be performed on an irregular, data-adaptive grid. We draw m=50,000 query locations by perturbing each observed sample within a small bandwidth (0.15) and clipping to the variable bounds. This concentrates evaluation where data exist and avoids empty corners of the hypercube. Figure 2 (bottom) shows a 2-D projection (X1 vs. X3) of the 4-D query set (blue) overlaid with the observations (red). Because GPR predicts at arbitrary locations, this unstructured grid is valid and reduces computational cost by orders of magnitude while preserving resolution in the regions of interest. Figure 2: Top: the omnidirectional semi-variogram of the first and second principal components. Bottom: the blue dots constitute the irregular grid on which GPR will be performed, while the red dots are the data. 4. KEY RESULTS, OUTCOMES, OR INSIGHTS Once an irregular grid is defined, GPR simulation of the principal components PC1 and PC2 can be performed independently. Figure 3 shows GPR simulated realizations of PC1 and PC2 projected onto the X1–X2 space with the original observations overlaid for reference.
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