Locally Adaptive Spatial Quantile Smoothing: Application to Monitoring Crime Density in Tokyo

TL;DR

This paper introduces a Bayesian quantile trend filtering method for spatial data, enabling the estimation of heterogeneous crime risk trends across Tokyo's districts over several years, with a focus on local adaptivity and practical inference.

Contribution

It develops a novel locally adaptive Bayesian quantile smoothing approach on graphs, incorporating shadow priors and a Gibbs sampler for non-stationary trend estimation.

Findings

Effective in capturing spatial heterogeneity in crime trends.

Demonstrates strong performance in simulation studies.

Provides a practical Gibbs sampling algorithm for Bayesian inference.

Abstract

Spatial trend estimation under potential heterogeneity is an important problem to extract spatial characteristics and hazards such as criminal activity. By focusing on quantiles, which provide substantial information on distributions compared with commonly used summary statistics such as means, it is often useful to estimate not only the average trend but also the high (low) risk trend additionally. In this paper, we propose a Bayesian quantile trend filtering method to estimate the non-stationary trend of quantiles on graphs and apply it to crime data in Tokyo between 2013 and 2017. By modeling multiple observation cases, we can estimate the potential heterogeneity of spatial crime trends over multiple years in the application. To induce locally adaptive Bayesian inference on trends, we introduce general shrinkage priors for graph differences. Introducing so-called shadow priors with multivariate distribution for local scale parameters and mixture representation of the asymmetric Laplace distribution, we provide a simple Gibbs sampling algorithm to generate posterior samples. The numerical performance of the proposed method is demonstrated through simulation studies.