非齐次随机过程的拓扑引导抽样
Topology-guided sampling of nonhomogeneous random processes
摘要：
拓扑测量作为量化复结构的重要工具正日益被接受。在许多应用中，这些结构可以表示为实值函数的节点域，并且仅通过实验观察或数值模拟获得。在这两种情况下，拓扑测量所基于的数据都是通过某种形式的有限采样或离散化得到的。本文提出了一种通过有限离散化来定量实直线上非齐次随机过程广义节点域分量个数的概率方法，即考虑随机过程相对于非常数确定性门限函数的漂移集.我们的研究结果提供了明确的概率先验界的某些离散化大小的适合性，也提供了信息的采样点的位置的选择，以尽量减少错误概率。我们说明了我们的各种随机过程的结果，演示了它们如何可以被用来采样的经典节点域的确定性函数扰动的加性噪声，并讨论它们的关系的密度为零。
英文摘要：
Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications, these structures can be expressed as nodal domains of real-valued functions and are obtained only through experimental observation or numerical simulations. In both cases, the data on which the topological measurements are based are derived via some form of finite sampling or discretization. In this paper, we present a probabilistic approach to quantifying the number of components of generalized nodal domains of nonhomogeneous random processes on the real line via finite discretizations, that is, we consider excursion sets of a random process relative to a nonconstant deterministic threshold function. Our results furnish explicit probabilistic a priori bounds for the suitability of certain discretization sizes and also provide information for the choice of location of the sampling points in order to minimize the error probability. We illustrate our results for a variety of random processes, demonstrate how they can be used to sample the classical nodal domains of deterministic functions perturbed by additive noise and discuss their relation to the density of zeros.
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