The Role of the Time-Arrow in Mean-Square Estimation of Stochastic Processes
The Role of the Time-Arrow in Mean-Square Estimation of Stochastic Processes
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DOI：
10.1109/lcsys.2017.2740957
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发表时间：
2015-07
影响因子：
3
通讯作者：
Yongxin Chen;J. Karlsson;T. Georgiou
中科院分区：
文献类型：
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作者：
Yongxin Chen;J. Karlsson;T. Georgiou
这封信的目的是指出多变量随机过程的过去值和未来值携带的关于现在的信息之间的某种二分法。更具体地说，向量值二阶随机过程可能在一个时间方向上是确定性的，但在另一个方向上不是确定性的。这种现象在标量过程中是不存在的，它深深地植根于移位算子的几何结构中。我们讨论的论述和例子是基于道格拉斯、夏皮罗和希尔兹关于后向移位循环向量的工作，并与维纳和科尔莫戈洛夫的经典思想有关。我们主要研究一阶随机过程，对于它，我们得到了所有正则过程在逆时间方向上是确定性的显式特征。这封信以实例为基础，目的是为对时间序列的估计理论和建模感兴趣的控制工程受众提供见解。
The purpose of this letter is to point out a certain dichotomy between the information that the past and future values of a multivariate stochastic process carry about the present. More specifically, vector-valued, second-order stochastic processes may be deterministic in one time-direction but not in the other. This phenomenon, which is absent in scalar-valued processes, is deeply rooted in the geometry of the shift-operator. The exposition and the examples we discuss are based on the work of Douglas, Shapiro, and Shields on cyclic vectors of the backward shift and relate to classical ideas going back to Wiener and Kolmogorov. We focus on rank-one stochastic processes for which we obtain an explicit characterization of all regular processes that are deterministic in the reverse time-direction. This letter builds on examples and the goal is to provide insights to a control engineering audience with interests in estimation theory and modeling of time-series.