Spatial Point Pattern Analysis Using Composite Likelihood

提出了一种新的基于似然的空间点过程模型拟合方法。复合似然（CL）已成功应用于许多不可行或无法获得完整最大似然的情况。然而，它在空间点过程建模中的应用从未被研究过。本研究旨在证明，对于任何二阶强度函数可以显式定义的空间点过程，都可以形成一个共轭矩阵。所提出的似然方法易于获得，可用于许多不同的空间点图分析。特别是，它可以用于拟合齐次和非齐次空间点过程模型。此外，它还可以用于选择用于估计对相关函数的带宽，这是空间点图分析中极具探索性和模型拟合的工具。这项研究的动机是由阿肯色大学的昆虫学家收集的红橡树蛀虫数据。这些数据包括红橡树蛀虫幼虫在吃树时造成的攻击洞的地图位置。主要目的是了解影响成年红橡树蛀虫决定产卵地点的因素。建议的模型拟合程序将应用于将攻击孔的位置与树的重要特征（如树的侧面和高度）联系起来。这项工作将具有重要的生物学意义和现实意义。特别是，该结果将为成年蛀虫的繁殖习性提供生物学见解。这种生物学见解反过来可以用于实践，指导开发更有效的诱捕技术，用于控制红橡树蛀虫的数量。因此，拟议的研究可能在拯救美国数百万棵受到红橡树蛀虫爆发威胁的红橡树的努力中发挥关键作用。

The proposed research introduces a new likelihood based method in fitting spatial point process models using the idea of composite likelihood. Composite likelihood (CL) has been successfully applied in numerous settings where a full maximum likelihood is not feasible or is not available. Its use in spatial point process modeling, however, has never been studied. This research intends to show that a CL can be formed for any spatial point process whose second-order intensity function can be explicitly defined. The proposed likelihood is easy to obtain and can be used in many different spatial point pattern analyses. In particular, it can be used to fit both homogeneous and inhomogeneous spatial point process models to data. Furthermore, it can also be used to select the bandwidth used in estimating the pair correlation function, which is an extremely exploratory and model fitting tool in spatial point pattern analysis.This research is motivated by the red oak borer data that were collected by entomologists at the University of Arkansas. The data consist of mapped locations of attack holes caused by larvae of red oak borers when they eat their way into the trees. The main objective is to understand what affects adult red oak borers to decide where to lay their eggs. The proposed model fitting procedures will be applied to link the locations of attack holes to important tree characteristics such as side and height of the tree. This work will have both important biological and practical significance. In particular, the results will provide biological insight on the breeding habit of the adult borers. This biological insight can in turn be used in practice to guide the development of more effective trapping techniques that can be used to control the population of the red oak borers. Thus the proposed research may play a critical role in the effort to save millions of red oak trees in the US that are being threatened by the outbreaks of red oak borers.