平稳图过程的遍历性：一个弱大数定律
Ergodicity in Stationary Graph Processes: A Weak Law of Large Numbers
摘要：
对于时间上的平稳信号，弱大数定律（WLLN）指出，当考虑N个信号分量时，系综平均值和实现平均值在e以内，概率为O（1/Ne ^2）。本文中介绍的图WLLN表明，这基本上是真实的支持图上的信号。然而，平稳性，系综平均值和实现平均值的概念是不同的。最近的论文定义图平稳信号为那些满足一种形式的不变性相对于图扩散。图平稳信号的系综均值不是一个常数，而是一个节点变化的信号，其结构依赖于图的谱特性。图信号的实现平均值在此被定义为局部信号值与相邻节点的信号值的连续加权平均值的平均值。图WLLN显示这两个节点变化信号在至少一些节点中以O（1/Ne ^2）阶的概率彼此在e之内。在平稳时间信号中，实现平均不仅是系综平均的一致估计，而且在均方误差（MSE）方面也是最优的。这对于图形信号来说是不正确的。最佳MSE图滤波器的设计。给出了一个关于高斯随机场均值估计的例子。
英文摘要：
For stationary signals in time the weak law of large numbers (WLLN) states that ensemble and realization averages are within e of each other with a probability of order O(1/Ne^2) when considering N signal components. The graph WLLN introduced in this paper shows that the same is essentially true for signals supported on graphs. However, the notions of stationarity, ensemble mean, and realization mean are different. Recent papers have defined graph stationary signals as those that satisfy a form of invariance with respect to graph diffusion. The ensemble mean of a graph stationary signal is not a constant but a node-varying signal whose structure depends on the spectral properties of the graph. The realization average of a graph signal is defined here as an average of successive weighted averages of local signal values with signal values of neighboring nodes. The graph WLLN shows that this two node-varying signals are within e of each other with probability of order O(1/Ne^2) in at least some nodes. In stationary time signals, the realization average is not only a consistent estimator of the ensemble mean but also optimal in terms of mean squared error (MSE). This is not true of graph signals. Optimal MSE graph filter designs are also presented. An example problem concerning the estimation of the mean of a Gaussian random field is presented.
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