Time series analysis by higher order crossings
Time series analysis by higher order crossings
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通过高阶交叉进行时间序列分析
DOI：
10.1080/00401706.1995.10484318
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
Michael Frey
中科院分区：
文献类型：
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作者：
Michael Frey
现代过零理论始于s 0。Rice著名的高斯过程公式（Rice 1945）。该公式将滤波后的高斯过程的电平平均交叉率与应用的滤波器联系起来。在此后的五十年中，过零理论得到了长足的发展，并在信号处理和时间序列分析中得到了广泛的应用。本书围绕高阶交叉（HOC）的概念系统化了这一进展的结果。为了介绍HOC，作者要求我们想象一个应用于一组滤波器的中心时间序列。时间序列的HOC是滤波器输出的过零计数。本书地址，与一些显著的例外，HOC从独立的，时不变的，线性滤波器。然而，任何一组滤波器——非线性的、耦合的、自适应的或其他的——都是可以想象的，并且是潜在的有用的HOC来源。HOC对信号处理的实际吸引力在于，首先，HOC对异常值具有极强的抵抗力，其次，简单性。有用的HOC信号处理可能会受到影响，例如，只使用重复差分滤波。过零计数器和差分滤波器都很容易在软件和硬件上实现，它们的运行速度使实时HOC信号处理在许多应用中成为可能。这本书有三个主要部分，每个部分有三章。第一部分包括对本书的介绍和概率论的回顾。第二部分介绍了HOC的基本性质，第三部分将HOC应用于时间序列分析中的几个基本问题。每一章以广泛的参考书目和练习部分结束——总共136个练习。章我。第1章通过各种例子对本书进行概述，从座头鲸的应用发声和无损检测到理论的主导频率原理和连续时间过程的谱矩。这些例子准确而引人注目地宣传了本书其余部分的内容。第2章回顾了概率的基本概念，强调了混合分布、随机排序和正交概率等概念，这些概念将在后面的章节中介绍。第三章延续了第二章的思路，介绍了随机过程。大部分的介绍是通过讨论和例子，很少有正式的证明。重点介绍了广义平稳时间序列及其谱表示。采用谱表示对时间序列进行线性滤波。4 - 5章。第4章介绍了椭球（包括高斯）有限平稳序列的期望过零数与一阶自相关的余弦公式。余弦公式建立了平稳时间序列的过零谱表示。这种表示反过来又激发了主导频率原理。这个原理，正是HOC的效用所依赖的，说明过零计数倾向于时间序列频谱的主导频率的邻域值。第5章正式介绍了HOC滤波器及其基本特性，其中着重介绍了差分滤波器（高通滤波器）和求和滤波器（低通滤波器）。第六章关于HOC的统计性质，研究了HOC的渐近正态性。讨论，主要限于高斯的情况下，必然是复杂的技术涉及混合光谱。虽然给出的证明很少，但示例和数值演示很好地识别了问题，并提供了足够的文献参考。
The modem theory of zero crossings begins with S. 0. Rice’s famous formula for Gaussian processes (Rice 1945). This formula relates a filtered Gaussian process’s mean crossing rate of a level to the applied filter. In the succeeding five decades zero-crossing theory has progressed significantly and has been widely applied in signal processing and time series analysis. This book systematizes the results of this progress around the concept of higher order crossings (HOC). To introduce HOC, the author asks us to visualize a centered time series applied to a bank of filters. The HOC’s of the time series are the zero crossing counts of the filter outputs. The book addresses, with some notable exceptions, HOC from independent, time-invariant, linear filters. Any set of filters, however-nonlinear, coupled, adaptive, or otherwise-is conceivable and a potentially useful source of HOC. The practical appeal of HOC for signal processing is, first, that HOC’s are extremely resistant to outliers and, second, simplicity. Useful HOC signal processing can be affected, for example, using only filtering by repeated differencing. Both zero-crossing counters and differencing filters are easily implemented in software and hardware, and their speed of operation makes real-time HOC signal processing possible in many applications. The book has three major parts, each three chapters in length. The first part contains an introduction to the book and a review of probability. The second part contains a presentation of basic properties of HOC, and the third part applies HOC to several fundamental problems in time series analysis. Each chapter ends with an extensive list of references and a section of exercises-136 exercises in all. Chapters I-3. Chapter 1 gives an overview of the book through a variety of examples ranging from the applied-vocalization of humpback whales and nondestructive testing-to the theoretical-the dominanfrequency principle and spectral moments of continuous-time processes. These examples accurately and appealingly advertise the contents of the rest of the book. Chapter 2 reviews basic ideas of probability emphasizing those ideas-mixed distributions, stochastic ordering, and orthant probabilities-needed in later chapters. Chapter 3, continuing in the same vein as Chapter 2, introduces stochastic processes. Much of the presentation is by way of discussion and example with few formal proofs. Wide-sense stationary time series and their spectral representation are highlighted. The spectral representation is used to present linear filtering of time series.Chapters 4-5. Chapter 4 introduces the cosine formula that relates the expected zero-crossing count to the first-order autocorrelation for ellipsoidal (including Gaussian) finite stationary sequences. The cosine formula establishes a zero-crossing spectral representation for stationary time series. This representation in turn motivates the dominant frequency principle. This principle, on which the utility of HOC squarely rests, states that zero-crossing counts tend toward values in neighborhoods of the dominant frequencies of the time series’ spectrum. HOC’s and their basic properties are formally introduced in Chapter 5, in which differencing (high-pass) and summing (low-pass) filters are emphasized. Chapter 6 on the statistical properties of HOC studies the asymptotic normality of HOC. The discussion, mostly limited to the Gaussian case, is necessarily complicated by technicalities involving mixed spectra. Although few proofs are given, the examples and numerical demonstrations nicely identify the issues, and sufficient references to the literature are provided.