[Paper Review] Spatially Constrained Spectral Clustering Algorithms for Region Delineation

Shuai Yuan, Pang‐Ning Tan+3 more|arXiv (Cornell University)|May 21, 2019

Land Use and Ecosystem Services49 references4 citations

TL;DR

This paper proposes a spatially constrained spectral clustering framework, BSSC (Balanced Spatial Spectral Clustering), that integrates landscape feature similarity with spatial contiguity via a truncated exponential kernel to produce homogeneous, contiguous, and size-balanced regions. The method extends to hierarchical clustering using a greedy recursive bisection strategy, outperforming baseline constrained clustering methods in balancing homogeneity, contiguity, and region size uniformity, especially in multi-scale regionalization tasks.

ABSTRACT

Regionalization is the task of dividing up a landscape into homogeneous patches with similar properties. Although this task has a wide range of applications, it has two notable challenges. First, it is assumed that the resulting regions are both homogeneous and spatially contiguous. Second, it is well-recognized that landscapes are hierarchical such that fine-scale regions are nested wholly within broader-scale regions. To address these two challenges, first, we develop a spatially constrained spectral clustering framework for region delineation that incorporates the tradeoff between region homogeneity and spatial contiguity. The framework uses a flexible, truncated exponential kernel to represent the spatial contiguity constraints, which is integrated with the landscape feature similarity matrix for region delineation. To address the second challenge, we extend the framework to create fine-scale regions that are nested within broader-scaled regions using a greedy, recursive bisection approach. We present a case study of a terrestrial ecology data set in the United States that compares the proposed framework with several baseline methods for regionalization. Experimental results suggest that the proposed framework for regionalization outperforms the baseline methods, especially in terms of balancing region contiguity and homogeneity, as well as creating regions of more similar size, which is often a desired trait of regions.

Motivation & Objective

• To address the challenge of creating spatially contiguous and homogeneous regions in landscape regionalization.
• To overcome the limitation of standard clustering methods that often produce non-contiguous regions.
• To develop a hierarchical regionalization framework that generates nested, fine-scale regions within broader-scale regions.
• To balance region homogeneity, spatial contiguity, and region size uniformity—key criteria for ecological and management applications.
• To reduce sensitivity to parameter tuning and improve stability across varying spatial constraints.

Proposed method

• Uses a truncated exponential kernel to model spatial contiguity constraints, integrated via Hadamard product with the feature similarity matrix.
• Applies spectral clustering with a normalized cut objective function to optimize region homogeneity and contiguity simultaneously.
• Employs a greedy recursive bisection approach to extend the non-hierarchical BSSC into a hierarchical framework (HSSC).
• Introduces a flexible parameter δ to control the spatial neighborhood size, enabling scalable and stable regionalization.
• Uses the adjusted Rand index to evaluate clustering stability across different δ values.
• Optimizes the objective function to inherently favor more uniformly sized clusters, avoiding the imbalance seen in Ward’s and single-link methods.

Experimental results

Research questions

• RQ1 Can a spectral clustering framework effectively balance landscape homogeneity and spatial contiguity in regionalization?
• RQ2 How does the proposed BSSC method compare to existing constrained clustering methods in terms of region size balance and contiguity?
• RQ3 Does the recursive bisection extension (HSSC) preserve the quality of non-hierarchical clustering while enabling hierarchical regionalization?
• RQ4 How sensitive are the resulting regions to changes in the spatial constraint parameter δ?
• RQ5 Can the proposed method produce stable, consistent regionalizations across different spatial scales?

Key findings

• BSSC achieved the lowest objective function value (13,458.13) at K=10 for Michigan, outperforming HSSC (14,284.21) and Ward’s method in normalized cut optimization.
• BSSC produced regions with higher spatial contiguity (c metric) and better cluster balance (Cbalance) than Ward’s and UPGMA methods across all cluster counts.
• The adjusted Rand index for BSSC (0.92) was significantly higher than Ward’s (0.66), indicating superior stability across varying δ values.
• HSSC achieved slightly better SSW and Cbalance than BSSC, but at the cost of slightly worse PctML preservation, indicating trade-offs in objective function trade-offs.
• The spatially constrained single-link method produced highly imbalanced regions, requiring ad-hoc tuning of maximum region size, unlike BSSC which avoids this via inherent size bias.
• BSSC and HSSC maintained high contiguity and homogeneity even as δ increased from 1/41 to 4/41, while Ward’s method failed to maintain contiguity at higher δ.