In the analysis of spatial point patterns on linear networks, a critical statistical objective is estimating the first-order intensity function, representing the expected number of points within specific subsets of the network. Typically, non-parametric approaches employing heating kernels are used for this estimation. However, a significant challenge arises in selecting appropriate bandwidths before conducting the estimation. We study an intensity estimation mechanism that overcomes this limitation using adaptive estimators, where bandwidths adapt to the data points in the pattern. While adaptive estimators have been explored in other contexts, their application in linear networks remains underexplored. We investigate the adaptive intensity estimator within the linear network context and extend a partitioning technique based on bandwidth quantiles to expedite the estimation process significantly. Through simulations, we demonstrate the efficacy of this technique, showing that the partition estimator closely approximates the direct estimator while drastically reducing computation time. As a practical application, we employ our method to estimate the intensity of traffic accidents in a neighbourhood in Medellin, Colombia, showcasing its real-world relevance and efficiency.

翻译：在线性网络上的空间点模式分析中，一个关键的统计目标是估计一阶强度函数，该函数表示网络特定子集内点的期望数量。通常，采用基于加热核的非参数方法进行此估计。然而，在进行估计之前选择合适的带宽是一个重大挑战。我们研究了一种利用自适应估计器克服这一限制的强度估计机制，其中带宽根据模式中的数据点自适应调整。尽管自适应估计器已在其他情境中得到探索，但其在线性网络中的应用仍未被充分研究。我们在线性网络背景下研究了自适应强度估计器，并扩展了一种基于带宽分位数的分区技术，以显著加速估计过程。通过模拟，我们证明了该技术的有效性，表明分区估计器在极大减少计算时间的同时，能紧密逼近直接估计器。作为实际应用，我们采用该方法估算了哥伦比亚麦德林市某社区交通事故的强度，展示了其在真实场景中的相关性和效率。