We introduce the transport–transform and the relative transport–transform metrics between finite point patterns on a general space, which provide a unified framework for earlier point pattern metrics, in particular the generalized spike time and the normalized and unnormalized optimal subpattern assignment metrics. Our main focus is on barycenters, i.e., minimizers of a q-th-order Fréchet functional with respect to these metrics. We present a heuristic algorithm that terminates in a local minimum and is shown to be fast and reliable in a simulation study. The algorithm serves as a general plug-in method that can be applied to point patterns on any state space where an appropriate algorithm for solving the location problem for individual points is available. We present applications to geocoded data of crimes in Euclidean space and on a street network, illustrating that barycenters serve as informative summary statistics. Our work is a first step toward statistical inference in covariate-based models of repeated point pattern observations.

中文翻译：

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点模式数据的度量和重心

我们介绍了一般空间上有限点模式之间的传输转换和相对传输转换度量，它们为较早的点模式度量提供了统一的框架，尤其是广义的尖峰时间以及规范化和非规范化的最佳子模式分配度量。我们的主要重点是重心，即q的最小化关于这些指标的二阶Fréchet功能。我们提出了一种启发式算法，该算法以局部最小值终止，并且在仿真研究中显示出快速而可靠的效果。该算法用作一般的插件方法，可应用于任何状态空间上的点模式，在这些状态空间上，可以使用用于解决单个点的位置问题的适当算法。我们目前在欧几里得空间和街道网络中对犯罪的地理编码数据进行应用，说明重心可以作为信息摘要。我们的工作是在基于协变量的重复点模式观测模型中进行统计推断的第一步。