Spatially clustered regression

Abstract

Spatial regression and geographically weighted regression models have been widely adopted to capture the effects of auxiliary information on a response variable of interest over a region. In contrast, relationships between response and auxiliary variables are expected to exhibit complex spatial patterns in many applications. This paper proposes a new approach for spatial regression, called spatially clustered regression, to estimate possibly clustered spatial patterns of the relationships. We combine K-means-based clustering formulation and penalty function motivated from a spatial process known as Potts model for encouraging similar clustering in neighboring locations. We provide a simple iterative algorithm to fit the proposed method, scalable for large spatial datasets. Through simulation studies, the proposed method demonstrates its superior performance to existing methods even under the true structure does not admit spatial clustering. Finally, the proposed method is applied to crime event data in Tokyo and produces interpretable results for spatial patterns. The R code is available at https://github.com/sshonosuke/SCR.

Introduction

Spatial heterogeneity, which is often referred to as the Second Law of Geography (Goodchild, 2004), is ubiquitous in spatial science. Geographically weighted regression (GWR; Brunsdon et al., 1998, Fotheringham et al., 2002), which is a representative approach for modeling spatial heterogeneity, has widely been adopted for modeling spatially varying regression coefficients; its applications cover social science (e.g. Hu et al., 2016), epidemiology (e.g. Nakaya et al., 2005) and environmental science (e.g. Zhou et al., 2019).

Despite the success, GWR is known to be numerically unstable and may produce extreme estimates of coefficients (e.g. Wheeler and Tiefelsdorf, 2005, Cho et al., 2009). To address the drawback, a wide variety of regularized GWR approaches have been developed (e.g. Wheeler, 2007, Wheeler, 2009, Bárcena et al., 2014, Comber et al., 2016). Still, it is less clear how to regularize GWR to improve stability while maintaining its computational efficiency. Bayesian spatially varying coefficient model (Gelfand et al., 2003, Finley, 2011) is another popular approach for modeling spatial heterogeneity in regression coefficients. While Wheeler and Waller (2009) and Wolf et al. (2018) among others have suggested its stability and estimation accuracy, this approach can be computationally very intensive for large samples, limiting applications of spatial regression techniques to large spatial datasets. Therefore, an alternative method that has stable estimation performance, as well as computational efficiency under large datasets, is strongly required.

This paper proposes a new effective approach for spatial regression with possibly spatially varying coefficients or non-stationarity. Our fundamental idea is a combination of regression modeling and clustering; we assume all the geographical locations can be divided into a finite number of groups, where locations in the same groups share the same regression coefficients. Hence, possibly smoothed surfaces of varying regression coefficients are approximated by step functions. Owing to the clustering technique, the estimation results would be numerically stable and more accessible to interpret than GWR. The idea to incorporate spatial clustering into regression is not new. There have been some two-stage procedures (e.g. Anselin, 1990, Billé et al., 2017, Lee et al., 2017, Nicholson et al., 2019), but they tend to be ad-hoc combinations of clustering and regression. In contrast, the proposed method carries out regression and clustering simultaneously, which can produce reasonable spatial clustering depending on regression structures.

To introduce such a spatial clustering nature, we employ indicators showing the group to which the corresponding location belongs, and we estimate the grouping parameters and group-wise regression models simultaneously. For estimating group memberships, it would be reasonable to impose that the geographically neighboring locations are likely to belong to the same groups. To this end, we introduce a penalty function to encourage such spatially clustered structures motivated from the hidden Potts model (Potts, 1952) that was originally developed for modeling spatially correlated integers. We will demonstrate that the proposed objective function can be easily optimized by a simple iterative algorithm similar to $K$-means clustering. In particular, updating steps in each iteration do not require computationally intensive manipulations, so that the proposed algorithm is much more scalable than GWR. For selecting the number of groups $G$, we employ an information criterion. Moreover, the proposed approach allows substantial extensions to include variable selection or semiparametric additive modeling, which cannot be achieved by existing techniques such as GWR.

Recently, sophisticated statistical methods combining regression modeling and clustering have been studied in the literature. In the context of spatial regression, Li and Sang (2019) and Zhao and Bondell (2020) adopted a fused lasso approach to shrink regression coefficients in neighboring areas toward $0$, which results in spatially clustered regression coefficients. However, the computation cost under large datasets is substantial, and the performance is not necessarily reasonable, possibly because the method does not take account of spatially heterogeneous variances, which will be demonstrated in our numerical studies. On the other hand, in the context of panel data analysis, clustering approaches using grouping indicators like the proposed method have been widely studied (e.g. Bonhomme and Manresa, 2015, Wang et al., 2018, Ito and Sugasawa, 2020). Still, the existing works did not take account of spatial similarities among the grouping indicators.

This paper is organized as follows. In Section 2, we introduce the proposed methods, estimation algorithms and discuss some related issues. In Section 3, we evaluate the numerical performance of the proposed methods together with some existing methods through simulation studies. In Section 4, we demonstrate the proposed method through spatial regression modeling of the number of crimes in the Tokyo metropolitan area. Finally, we give some discussions in Section 5.