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.
Figures & tables
Figure 1: Overview of our approach. Data points are grouped through graph partitioning, which is formulated as an MIQP problem. This formulation is approximated, with a performance guarantee, by aggregating nearby data points in advance. The color at each vertex represents the magnitude of the individual predictions. The dashed boxes in the middle figure indicate the aggregation units.
Figure 2: Example of constraint 4.1 . Let V be the collection of vertex sets that induce connected subgraphs whose vertices share the same color. Without imposing the constraint characterized by V , the beige vertex sets remain connected via one blue and three light blue vertices. However, once a blue or light blue vertex set in V is removed, this connection is broken.
Figure 3: Example of prior aggregation. Let n=8,m=2 , and l=3 . The upper table, which corresponds to assumption 5.2 , illustrates how labels are reassigned for the first and third groups of data points sharing the same sublabel. For the second group, no replacement is performed because its assignment in the optimal solution has only a single label. The lower table shows the relationship between the entries of η and η~ .
Figure 4: Examples of violations of assumption 5.7 and assumption 5.8 . Each square represents a group of data points that share the same sublabel. Gray regions highlight the violation of the assumptions. Unconnected vertices in the original undirected graph violate assumption 5.7 . Additionally, assumption 5.8 is violated when the removal of vertices in the gray square with multiple labels causes a connected component in the optimal solution to become disconnected.
Figure 5: Spatial segments obtained through connected graph partitioning using the MIQP formulation. For each target variable, the result of the trial with the largest total sum of squares is displayed. The leftmost panel presents individual predictions η from Gaussian process regression. The second panel displays approximate predictions η~ from the prior aggregation. The remaining panels show segmentations with m∈{2,3,4} clusters. Colors indicate the levels of the predicted values.
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.
Yuta Shikuri
The Graduate University for Advanced Studies, Tokyo, Japan · Tokio Marine Holdings, Inc., Tokyo, Japan
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