Optimum Tuning Parameter Selection in Generalized lasso for Clustering with Spatially Varying Coefficient Models

Spatial clustering with spatially varying coefficient models is useful for determining the region with common effects of variables in spatial data. This study focuses on selecting the optimum tuning parameter of the generalized lasso for clustering with the spatially varying coefficient model. The k-fold cross-validation (CV) may fail to split spatial data into a training set and a testing set, if a region contains only a few observations. Moreover, the k-fold CV is known to give a biased estimate of the out-of-sample prediction error. Therefore, we investigated the performance of approximate leave-one-out cross-validation (ALOCV) in comparison with k-fold CV for selecting the tuning parameter in a simulation study on 2-dimensional grid. The ALOCV yielded smaller error than k-fold CV and could detect edges with differences shrunk by generalized lasso appropriately. Then, the ALOCV for selecting the optimum tuning parameter of the generalized lasso in fitting the spatially varying coefficient model is applied to the Chicago crime data. The result of selection by ALOCV was in accordance with the conclusion suggested in the preceding literature. Clustering into regions in advance for making k-fold CV feasible may lead to a wrong result of clustering with a spatially varying coefficient model.