As a workaround for the lack of transitive transformations on linear network structures, which are required to consider different notions of distributional invariance, including stationarity, we introduce the notions of pseudostationarity and intensity reweighted moment pseudostationarity for point processes on linear networks. Moreover, using arbitrary so-called regular linear network distances, e.g. the Euclidean and the shortest-path distance, we further propose geometrically corrected versions of different higher-order summary statistics, including the inhomogeneous empty space function, the inhomogeneous nearest neighbour distance distribution function and the inhomogeneous J-function. Such summary statistics detect interactions of order higher than two. We also discuss their nonparametric estimators and through a simulation study, considering models with different types of spatial interaction and different networks, we study the performance of our proposed summary statistics by means of envelopes. Our summary statistic estimators manage to capture clustering, regularity as well as Poisson process independence. Finally, we make use of our new summary statistics to analyse two different datasets: motor vehicle traffic accidents and spiderwebs.

中文翻译：

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线性网络上点过程的不均匀高阶汇总统计量

作为线性网络结构上缺少传递变换的变通办法，需要考虑分布不变性（包括平稳性）的不同概念，我们引入了线性网络上点过程的伪平稳性和强度重加权矩伪平稳性的概念。此外，使用任意所谓的规则线性网络距离（例如，欧几里得距离和最短路径距离），我们还提出了不同高阶汇总统计量的几何校正版本，包括不均匀的空白空间函数，不均匀的最近邻距离分布函数和不均匀的J-功能。此类摘要统计信息检测高于2的顺序的交互。我们还讨论了它们的非参数估计量，并通过模拟研究，考虑了具有不同类型的空间相互作用和不同网络的模型，我们通过包络研究了建议的汇总统计量的性能。我们的摘要统计估计量设法捕获聚类，规律性和泊松过程独立性。最后，我们利用新的摘要统计数据来分析两个不同的数据集：机动车交通事故和蜘蛛网。