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.

潜在异质性下的空间趋势估计是提取空间特征和犯罪活动等危害的重要问题。与常用的汇总统计数据（例如均值）相比，分位数提供了有关分布的大量信息，通过关注分位数，不仅可以估计平均趋势，还可以额外估计高（低）风险趋势。在本文中，我们提出了一种贝叶斯分位数趋势过滤方法来估计图上分位数的非平稳趋势，并将其应用于东京 2013 年至 2017 年间的犯罪数据。通过对多个观察案例进行建模，我们可以估计空间的潜在异质性应用程序中多年来的犯罪趋势。为了诱导对趋势的局部自适应贝叶斯推断，我们引入了图差异的一般收缩先验。引入所谓的具有局部尺度参数多元分布的影子先验和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。通过模拟研究证明了所提出方法的数值性能。