空间过程因子及其在环境过程灵敏度分析中的应用
Spatial Process Gradients and Their Use in Sensitivity Analysis for Environmental Processes
摘要：
本文提出了基于空间过程的方向导数的局部灵敏度分析方法。空间过程的形式梯度分析在以前的论文中进行了阐述，重点是与假设遵循高斯过程模型的响应变量相关的方向导数的分布理论。在目前的工作中，这些想法被扩展到另外容纳一个连续的协变量，其方向导数也感兴趣，并将响应面的方向导数的行为与协变量表面的行为相关联。在某种意义上，评估响应的梯度是否遵循解释变量的梯度是有意义的。所有变量（包括方向导数）的联合高斯结构允许使用显式分布理论，因此允许使用多元正态理论进行空间区域的克里金法。在贝叶斯分层建模框架内工作，后验样本使所有梯度分析能够在模型拟合后进行。作为概念证明，我们展示了我们的方法可以应用到一个标准的地质统计建模设置使用模拟的例子。对于真实数据的说明，我们使用点模式数据，将我们的梯度分析推迟到强度表面，采用对数高斯Cox过程模型。特别是，我们与海拔数据点模式与杜克森林中的几个树种。
英文摘要：
This paper develops methodology for local sensitivity analysis based on directional derivatives associated with spatial processes. Formal gradient analysis for spatial processes was elaborated in previous papers, focusing on distribution theory for directional derivatives associated with a response variable assumed to follow a Gaussian process model. In the current work, these ideas are extended to additionally accommodate a continuous covariate whose directional derivatives are also of interest and to relate the behavior of the directional derivatives of the response surface to those of the covariate surface. It is of interest to assess whether, in some sense, the gradients of the response follow those of the explanatory variable. The joint Gaussian structure of all variables, including the directional derivatives, allows for explicit distribution theory and, hence, kriging across the spatial region using multivariate normal theory. Working within a Bayesian hierarchical modeling framework, posterior samples enable all gradient analysis to occur post model fitting. As a proof of concept, we show how our methodology can be applied to a standard geostatistical modeling setting using a simulation example. For a real data illustration, we work with point pattern data, deferring our gradient analysis to the intensity surface, adopting a log-Gaussian Cox process model. In particular, we relate elevation data to point patterns associated with several tree species in Duke Forest.
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