Spatial prediction estimates unobserved values from spatially distributed observations. Although flexible predictive models are increasingly available, their practical use remains limited when interpretability is essential. In this study, we formulate a graph partitioning problem to construct a global surrogate model that approximates black-box predictions with simpler representations. Specifically, we select connected subgraphs to minimize within-cluster residuals in vertex-level predictions. Although this formulation can be viewed as a variant of spatial regionalization, it operates on model-based predictions at individual locations rather than predefined areal units. With many locations, this point-level formulation may produce artificial connections between regions, reducing interpretability. We therefore introduce a constraint that excludes such connections while retaining flexible partition boundaries. We theoretically analyze how this constraint affects the resulting partition. We also develop a computational procedure for adjusting partition boundaries. Experimental results show that the proposed method yields spatially coherent partitions with flexible boundaries.
Figures & tables
Figure 1: Overview of the proposed approach. Vertex colors represent the predicted values at individual locations. The locations are partitioned into spatially connected regions to approximate the predictions subject to a constraint defined using sublabels.
Figure 2: Example of a bridge-like connection. Two spatially separated regions of beige vertices are connected through intermediate vertices consisting of one blue vertex and three light-blue vertices. Such a connection may be introduced solely to decrease the objective value, but it does not reflect a spatially coherent regional structure.
Figure 3: Illustration of Definition 3.2 . The left and right panels show two independent graph partitioning instances with m=2 . Each square represents a sublabel group. The dashed line indicates the boundary between the brown–beige partition V1(ω) and the blue–sky-blue partition V2(ω) . White and gray squares represent core and boundary groups, respectively. In the left panel, let S denote the bottom-left gray square. Since G[V1(ω)∖(S∩V1(ω))] is disconnected, this assignment violates the condition of Definition 3.2 .
Figure 4: Example of label reassignment. Let n=8,m=2 , and l=3 . This table illustrates how optimal labels are reassigned for the first and third sublabel groups. For the second sublabel group, no reassignment is performed because its assignment in the optimal partition has only a single label.
Figure 5: Spatial partitions produced by our approach with l=30 . For each target variable, we show the trial with the largest value of ∥f−g([1,…,1]⊤;f)∥2 . The leftmost panel depicts the individual predictions f . The second panel presents the piecewise-constant approximation fσ . The remaining panels give the cluster-wise projection g(ω;f) for the resulting partitions. Colors indicate the levels of the predicted values. Colors are scaled separately for each target variable.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Differences between our approach and MIQCP for l=30 . For each target variable, we show the trial with the largest value of ∥f−g([1,…,1]⊤;f)∥2 . The panels highlight the vertices whose assignments differ between the two methods after matching the cluster labels to minimize the number of differing assignments. Highlighted vertices are outlined in black and colored according to their prediction values, while the remaining vertices are shown in gray. Colors are scaled separately for each target variable.
Figure 7: Greedily refined feasible partitions in Ω0 with l=30 . For each target variable, we show the trial with the largest value of ∥f−g([1,…,1]⊤;f)∥2 . The panels show the cluster-wise projection g(ω;f) for the resulting partitions. Vertices whose assignments differ from those obtained by our approach are outlined in black after matching the cluster labels to minimize the number of differing assignments. Colors indicate the levels of the projected values and are scaled separately for each target variable.
Figure 8: Spatial partitions produced by our approach with l=15 . The figure follows the same layout as Figure 5 .
Figure 9: Differences between our approach and MIQCP for l=15 . The figure follows the same layout as Figure 6 .
Figure 10: Greedily refined feasible partitions in Ω0 with l=15 . The figure follows the same layout as Figure 7 .
Spatial prediction refers to the estimation of unobserved values from spatially distributed observations. Although recent advances have improved the capacity to model diverse observation types, adoption in practice remains limited in industries that demand interpretability. To mitigate this gap, surrogate models that explain black-box predictors provide a promising path toward interpretable decision making. In this study, we propose a graph partitioning problem to construct spatial segments that minimize the sum of within-segment variances of individual predictions. The assignment of data points to segments can be formulated as a mixed-integer quadratic programming problem. While this formulation potentially enables the identification of exact segments, its computational complexity becomes prohibitive as the number of data points increases. Motivated by this challenge, we develop an approximation scheme that leverages the structural properties of graph partitioning. Experimental results demonstrate the computational efficiency of this approximation in identifying spatial segments.
Yuta Shikuri, Hironori Fujisawa
The Graduate University for Advanced Studies · Tokio Marine Holdings, Inc. · Institute of Statistical Mathematics +1
We consider nonparametric regression when the association between a response and its covariates changes across an unknown partition of a spatial domain. The proposed estimator learns the partition and the cluster-specific regression functions jointly. A neural network depending only on location determines cluster membership, while separate neural networks describe the covariate--response relationship within the clusters. An annealed softmax relaxation permits gradient-based estimation of the otherwise discrete assignments. Graph-Laplacian and occupancy penalties are used to discourage fragmented regions and degenerate solutions. We establish identifiability up to label permutation, bound partition error under a margin condition, and decompose prediction risk into regression and assignment components. The resulting rate agrees with that of an oracle estimator when the partition is estimated sufficiently accurately. Simulations show that joint estimation is useful when regression surfaces change abruptly across spatial boundaries, including settings with nonlinear effects, unequal region sizes, preferential sampling, and spatially correlated errors. Finally, a real data analysis is provided to demonstrate the validity and effectiveness of the proposed method.
Kexuan Li, Weidong Ma
Bristol Myers Squibb · University of Pennsylvania, Department of Biostatistics
We present IOAH3 (Importance-Oriented Adaptive H3 partitioning), a computational method for constructing data-driven spatial partitions of geo-referenced observation domains. Standard approaches to spatial aggregation adopt fixed areal units, such as administrative boundaries or uniform hexagonal grids at a single resolution, without regard to the informational content of the underlying observations in each region. This leads to the well-known modifiable areal unit problem: statistical and inferential results depend on the arbitrary choice of partition, and spatially concentrated phenomena are averaged out in coarse cells that obscure fine-scale structure. IOAH3 addresses this by constructing an adaptive partition in three stages: multi-source feature extraction and importance scoring via principal component analysis over road density, POI density, building density, and terrain roughness signals, with population and flood-hazard data entering as auxiliary inputs to cell filtering and spatial smoothness; spatial cell selection via Markov Random Field graph-cut optimisation, which jointly maximises per-cell importance while enforcing spatial contiguity; and data-driven hierarchical refinement of high-importance regions to finer H3 resolution levels, with neighbour-propagated support to avoid isolated fine-resolution islands. The resulting partitions serve as input to spatial inference pipelines and provide a principled resolution of the partition-sensitivity problem prior to any modelling step.
Ehsaneddin Jalilian
GeoSocial Artificial Intelligence, Interdisciplinary Transformation University Austria