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.

翻译：在潜在异质性条件下进行空间趋势估计，是提取空间特征与犯罪活动等风险因素的重要课题。相较于均值等常用汇总统计量，分位数能提供更丰富的分布信息，因此不仅需要估计平均趋势，还需额外估计高（低）风险趋势。本文提出一种贝叶斯分位数趋势过滤方法，用于估计图结构上非平稳的分位数趋势，并将其应用于2013年至2017年东京的犯罪数据。通过对多个观测案例建模，我们能够估计应用中多年空间犯罪趋势的潜在异质性。为诱导趋势的局部自适应贝叶斯推断，我们引入图差分的通用收缩先验。通过引入具有多元分布的局部尺度参数"影子先验"以及非对称拉普拉斯分布的混合表示，我们提供了一种简单的吉布斯采样算法来生成后验样本。数值模拟验证了所提方法的性能。