变化检测：一种泛函分析视角
Change Detection: A functional analysis perspective
AI总结：
本文提出一种基于张量积表示和嵌套特征空间的新方法，用于检测随机过程与随机场中的全局或局部变化信号，并能在一般拓扑上扩展应用。
摘要：
我们开发了一种新方法，用于检测随机过程和随机场行为的变化，该方法基于张量积表示，如Karhunen-Loève展开。从协方差算子的相关特征空间出发，构建了一系列嵌套函数空间，从而能够检测位于正交子空间中的信号。特别是，即使信号在全局或局部意义上发生变化，这种方法也能成功检测。我们发展了一种数学方法，基于多层嵌套子空间的构建来定位和度量额外成分的大小。我们在$\nmathbb{R}$和球面域$\nmathbb{S}^{2}$上展示了示例。然而，该方法是灵活的，允许在一般拓扑结构（包括时空域）上检测正交信号。
英文摘要：
We develop a new approach for detecting changes in the behavior of stochastic processes and random fields based on tensor product representations such as the Karhunen-Loève expansion. From the associated eigenspaces of the covariance operator a series of nested function spaces are constructed, allowing detection of signals lying in orthogonal subspaces. In particular this can succeed even if the stochastic behavior of the signal changes either in a global or local sense. A mathematical approach is developed to locate and measure sizes of extraneous components based on construction of multilevel nested subspaces. We show examples in $\mathbb{R}$ and on a spherical domain $\mathbb{S}^{2}$. However, the method is flexible, allowing the detection of orthogonal signals on general topologies, including spatio-temporal domains.
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