相关熵与循环相关熵信号处理研究进展

奥鹏网院作业

相关熵与循环相关熵信号处理研究进展
邱天爽*
(大连理工大学电子信息与电气工程学部 大连 116024)
摘 要：在无线电监测和目标定位等应用中，接收信号经常会受到脉冲噪声和同频带干扰等复杂电磁环境的影响，传统的基于2阶统计量的信号处理方法往往不能正常工作，基于分数低阶统计量的信号处理方法也由于对信号噪声统计先验知识的依赖性而遇到困难。近年来提出并受到信号处理领域普遍关注的相关熵和循环相关熵信号处理理论与方法，是解决复杂电磁环境下信号分析处理、参数估计、目标定位和其他应用问题的有效技术手段，有力促进了非高斯、非平稳信号处理理论方法和应用的发展。该文系统性地综述了相关熵和循环相关熵信号处理的基本理论和基本方法，包括相关熵与循环相关熵的起源背景、定义概念、性质特点，以及所包含的数学物理意义。该文还介绍了相关熵与循环相关熵信号处理在多个领域的应用问题，希望对非高斯、非平稳统计信号处理的研究和应用有所裨益。
关键词：信号处理；相关熵；循环相关熵；非高斯；非平稳







近年来，由于具有通用非线性逼近特性和在再生核希尔伯特空间上线性与凸性等优良特性，核自适应滤波器受到学术界广泛的重视。针对核最小均方(KLMS)算法在非高斯、非线性条件下的性能退化的问题，Zhao等人[50]基于最大相关熵准则，提出了一种核最大相关熵自适应滤波算法。该算法结合了KLMS和MCC准则的优点，性能优于KLMS和基于MCC的常规线性滤波器。
西安交通大学陈霸东教授团队在相关熵与最大相关熵准则的研究与探索中取得显著成果。Chen等人[51]研究了最大相关熵准则下的贝叶斯估计问题，表明最大相关熵(MCC)估计本质上是一个平滑最大后验(MAP)概率估计。在一定条件下，当相关熵中的核长大于某个值时，MCC估计在平滑后验分布的严格凹区域中会有唯一的最优解。Chen等人[52]还研究了最大相关熵准则下自适应滤波的稳态均方误差(EMSE)问题，得到了稳态超量均方误差(EMSE)的精确值。Wu等人[53]提出了一种鲁棒性的自适应滤波器，称为核递归最大相关熵(KRMC)，适用于非线性和非高斯信号处理。Peng等人[28]在MCC中复合了一个线性约束，提出了一种约束最大相关熵(CMCC)自适应滤波器，具有计算复杂度较低，在强尾脉冲噪声下明显优于由MSE约束的自适应算法。Chen和Liu等人[54–56]分别基于相关熵准则对经典的卡尔曼滤波器和无迹卡尔曼滤波器进行了改造，提出了最大相关熵卡尔曼滤波器、状态约束最大相关熵卡尔曼滤波器和最大相关熵无迹卡尔曼滤波器。这些经过相关熵改进后的卡尔曼滤波器，显著提高了卡尔曼滤波器对于非高斯脉冲噪声的鲁棒性，具有很好的性能。Liu等人[57]把最大相关熵无迹卡尔曼滤波器应用于空间通信网络的相对状态估计，对重尾非高斯噪声具有很强的鲁棒性。
(5) 无线定位中的波达方向估计
波达方向(Direction Of Arrival, DOA)估计是阵列信号处理中的基本问题之一，广泛应用于雷达、声呐以及无线通信等领域[58]。针对复杂电磁环境下非高斯脉冲噪声和同频干扰对DOA估计算法的影响，本文作者团队提出了一系列基于相关熵和循环相关熵的DOA估计与波束形成新方法[59–64]，得到较好的结果。
其中，Zhang等人[60]依据相关熵方法，构建了“相关熵的相关(CRCO)”统计量，提出了基于CRCO-MUSIC的DOA估计算法，其波达方向估计结果优于已有的基于分数低阶统计(FLOS)的MUSIC算法，适合于非高斯Alpha稳定分布噪声环境下使用。Wang等人[61–63]构建了新的广义相关熵，在最大相关熵准则下求解相关熵的优化问题来估计信号子空间，提出了基于最小广义相关熵准则的DOA估计方法和脉冲噪声下基于稀疏表示的韧性DOA估计新方法，得到较好结果。Jin等人[42]基于最大相关熵准则，提出了一种能够抵抗循环频率误差(CEF)的波束形成算法，在同频干扰、低快拍数、低信噪比和大范围CFE环境下具有良好的自适应波束形成效果。
(6) 在时间延迟估计中的应用
时间延迟(Time Delay Estimation, TDE)估计[6 4,6 5]又称为到达时差(Time-Difference Of Arrival, TDOA)估计[66]，是依据接收的同源信号来确定不同接收器之间由于传播距离不同而引起的时间延迟，通常用于目标定位，在雷达、声呐和无线电监测等领域得到广泛应用。
经典的TDE方法大多是基于2阶统计量的，例如相关法或自适应滤波法等，这类方法简单易行，且具有较好的估计精度。但是，在非高斯Alpha稳定分布噪声和同频干扰等复杂电磁环境下，这类算法性能显著退化。针对这一问题，提出了基于相关熵和循环相关熵的TDE估计方法[25,27,67–70]，具有很好的抑制非高斯脉冲噪声的能力。特别是基于循环相关熵的TDE估计方法，可以在非高斯脉冲噪声和同频带干扰并存的复杂电磁环境中具有较好的鲁棒性。
(7) 在图像处理中的应用
相关熵的理论和方法在图像处理领域也得到广泛的重视和应用。He等人[71]提出了一个稀疏相关熵框架，用于计算人脸识别中的鲁棒稀疏表示。该方法在处理人脸识别中有遮挡和数据损坏问题方面具有很好的鲁棒性和效率。Wang等人[26]在解决图像分割问题时，提出了一种基于局部相关熵K均值聚类的水平集图像分割方法。通过对合成图像和真实图像的大量实验，表明该方法在噪声敏感度和分割精度方面都有很好的性能。文献[72]提出了一种基于相关熵的水平集方法，用于对医学图像进行分割和偏移场校正。通过对合成图像和真实医学图像的大量实验，表明该方法在分割效率和准确性上都有显著的提高。联合稀疏表示(JSR)是一种常用的高光谱图像分类技术。针对JSR对异常值敏感的问题，Peng等人[73]提出了基于相关熵的鲁棒JSR模型，用基于相关熵的测度代替传统的欧氏距离，性能超越了经典的JSR模型。
(8) 在医学信号分析处理中的应用
随着经济与社会的发展，医学与健康问题越来越成为人们关注的焦点。由于医学信号与影像问题中有不少非线性、非平稳和非高斯方面的问题，因此相关熵理论与方法在医学信号分析处理中也得到重视和应用。
针对慢性心力衰竭患者周期性或非周期性呼吸与健康人群呼吸流量信号的特征分类问题，Garde等人[23]基于相关熵原理，提出了一种基于相关熵谱密度的呼吸参数提取与分类方法。在对冠心病患者进性周期或非周期分类时，准确率为88.9%。当将慢性心力衰竭患者与健康受试者进行区分时，准确率为95.2%。将非周期呼吸患者与健康受试者进行区分时，准确率为94.4%。
研究并确定脑电图(EEG)信号与信源之间的关系，对于理解神经系统的功能和作用具有重要意义。为了评价各电极EEG信号之间的相互依赖性，Hassan等人[74]提出了一种基于相关熵谱密度的非线性连通指数，用于检测EEG信号的非线性以及线性耦合。该方法对噪声具有较强的鲁棒性，对耦合强度的突变具有较高的灵敏度。
将电极记录导管贴放在心腔内某一部位后记录到的心脏电活动，称为心腔内电图。心腔内电图是对体表心电图的补充，可对复杂的心电传导机制有进一步的信息描述。传统的心腔内电图显性频率分析方法，以确定激活率高的心脏部位作为消融靶点。但是，这种方法往往会丢弃频谱中的谐波结构或频谱包络等相关信息，不能提供复杂房颤(AF)信号的完整特征。Pérez等人[75]用相关熵函数来估计心房颤动信号的基频，并把相关熵与傅里叶组织分析(FOA)相结合，不仅可以表征FA信号的周期性，而且可以通过用多个分量建模来研究更复杂的信号。
(9) 循环相关熵在通信和机械领域的应用
在天文、机械、雷达、声呐、通信与电力等自然科学和工程技术领域，存在许多循环平稳现象以及由此而引起的循环平稳随机信号。Napolitanoie[38,76]于2016年在Signal Processing上连续发表2篇综述性论文，系统评述了近10年来循环平稳随机信号分析处理领域的研究进展。指出，在严重的噪声和干扰环境中，基于循环平稳性的信号处理技术明显优于经典的平稳信号处理技术。
针对循环平稳信号的特点和常规循环统计量的弱点，近年来发展起来的循环相关熵[6,7,39]理论和方法受到学术界重视，并在许多领域得到应用[6,40–42,69,77]。
在机械工业和其他相关领域中，对于各种设备中的滚动轴承进行故障诊断，是防止意外事故发生，提高工业效率的必要手段。尽管信号的频谱峰度(kurtosis)已成为一种检测轴承故障的有效技术手段，但是在非高斯脉冲噪声下，其性能显著退化。Zhao等人[77]提出一种基于循环相关熵(CCE)及循环相关熵谱(CCES)的故障诊断方法。通过对模拟信号和真实信号的实验分析，表明基于CCE及CCES的故障诊断方法，在抑制脉冲噪声干扰的能力方面，明显优于另外一种强大的频带选择方法。
在无线电监测和通信技术领域，信号调制方式的自动分类识别(AMC)是对信号进行进一步分析处理的重要环节。但是，由于调制信号种类繁多，且受到各类接收噪声和同频带干扰的影响，设计在复杂电磁环境下可靠有效工作的AMC算法与系统，仍然是一项具有重要意义的任务。针对上述问题，Ma等人[40]提出了一种基于循环相关熵谱(CCES)的AMC方法。该方法对待分类识别信号求取CCES，并根据CCES提取不同周期频率的多个切片作为AMC的原始特征，再经由主成分分析对这些切片进一步优化，采用神经网络(或深度学习)作为分类器进行分类判别。蒙特卡罗仿真结果表明，基于循环相关熵谱的AMC算法具有很好的抵抗脉冲噪声和干扰的能力，尤其在低广义信噪比条件下性能优越。
非高斯、非平稳信号处理是统计信号处理领域的前沿与热点研究问题。鉴于循环相关熵谱具有特定的对称性和稀疏性，使得由压缩感知所得到的压缩谱可以用于非高斯、非平稳信号处理问题。为了在非高斯、非平稳条件下更好地完成信号参数估计的工作，Liu等人[41]提出了一种利用压缩循环相关熵谱来估计信号循环频率的新方法，可以有效地利用稀疏重构中的快速贪婪算法，提高了循环频率估计的准确性和效率，具有计算速度快、存储成本低的优点。
(10) 在其它领域中的应用
天文学中的所谓变星(variable star)，是指亮度与电磁辐射不稳定、经常变化、并且伴随着其它物理变化的恒星。而变化呈周期性的变星，叫做周期变星，通常可分为长期变星和短期变星两种。光曲线(light curves)是反映恒星亮度随时间变化的时间序列，其特点是噪声强，且采样不均匀。传统的估计变星周期的方法是时隙相关法(slotted correlation)、周期图法和方差分析法，但是由于非高斯噪声的影响，这些方法性能不够理想。为了更准确地估计周期变星的周期，Huijse等人[78]提出了一种基于时隙相关熵的方法，即使用时隙滞后直接从不规则采样时间序列估计相关熵，并进一步地采用一种新的信息论度量方法来识别相关熵谱密度的峰值。采用这种基于时隙相关熵的方法对多组光曲线时间序列进行周期估计，结果显示，对于来自MACHO的200组数据的正确率为74%，远优于传统时隙相关法50%的正确率。对于来自造父变星(Cepheid variable star)和天琴座RR型变星(RR Lyrae type variable stars)的400组数据的正确率达到97%。
风力发电是世界上使用最广泛的可再生能源之一，风速预测则是风力发电领域研究的重要方向之一。经典的短期风速预测的方法包括自回归滑动平均、支持向量机回归和人工神经网络等。极限学习机(Extreme Learning Machine, ELM)使基于神经网络的学习具有快速的训练速度和良好的生成性能，而在深度学习中开发的堆叠极限学习机(SELM)则将一个较大的神经网络分割成若干个连续计算的较小的神经网络，实现较小的存储占用。由于天气、温度、海拔等诸多不确定因素的影响，风能往往是不稳定的。这些随机波动会引起数据产生误差。针对上述不确定性问题，Luo等人[33]依据相关熵所具有的非线性测度特点，在SELM框架中加入广义相关熵函数，提出一种基于广义相关熵的风速预报方法。通过对多阶秒级和多阶分钟级风速预报实验，验证了基于广义相关熵方法的优越性。与现有传统和最新模型相比，基于广义相关熵和SELM方法的预测精度更高，时间消耗更少。
电量消耗预测(Forecasting of Electricity Consumption, FoEC)是近年来电力市场十分关注的重要问题。如何科学准确地预测和评估电量消耗，是该领域研究的关键课题之一。针对电量消耗预测中尚存在的问题，文献[79]依据最大相关熵准则改进最小二乘支持向量机(LSSVM)模型，以相关熵函数作为局部相似性评价准则，提出了一种电量消耗预测新方法，数据分析实验表明，这种以最大相关熵准则改进的预测方法，比常规的LSSVM具有更好的预测特性，对电力企业制定购电计划和用户定价具有参考意义。
6 结束语
本文对近十几年来基于相关熵和循环相关熵信号处理理论方法与应用技术的研究进展进行了较为系统的综述。介绍了相关熵和循环相关熵的起源背景、定义概念、性质特点，以及所包含的数学物理意义。相关熵是一种广义的相关函数，它可以将数据空间的非线性问题通过非线性变换，映射为再生核希尔伯特空间的线性问题求解，是一种既能有效描述随机过程统计分布，又能刻画其时间结构的单一测度，同时亦是一种局部化的相似性测度，对于远离分布中心的异常值具有很好的抑制作用，非常适合Alpha稳定分布条件下的信号处理、参数估计与目标定位等应用。循环相关熵则在相关熵的基础上融入了循环统计量的理论与方法，具有同时抑制脉冲噪声和同频带干扰的作用。相关熵和循环相关熵在非高斯、非平稳信号处理中具有广泛的应用价值，并具有进一步深入研究的空间。
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Development in Signal Processing Based on Correntropy and Cyclic Correntropy
QIU Tianshuang
(Faculty of Electronic Information and Electrical Engineering,Dalian University of Technology, Dalian 116024, China)
Abstract: In radio monitoring and target location applications, the received signals are often affected by complex electromagnetic environment, such as impulsive noise and cochannel interference. Traditional signal processing methods based on second-order statistics often fail to work properly. The signal processing methods based on fractional lower order statistics also encounter difficulties due to their dependence on prior knowledge of signals and noises. In recent years, the theory and method of correntropy and cyclic correntropy signal processing, which are widely concerned in the field of signal processing, are put forward. They are effective technical means to solve the problems of signal analysis and processing, parameter estimation, target location and other applications to complex electromagnetic environment. They promote greatly the development of the theory and application of non-Gaussian and non-stationary signal processing. This paper reviews systematically the basic theory and methods of correntropy and cyclic correntropy signal processing, including the background,definition, properties and characteristics of correntropy and cyclic correntropy, as well as their mathematical and physical meanings. This paper introduces also the applications of correntropy and cyclic correntropy signal processing to many fields, hoping to benefit the research and application of non-Gaussian and non-stationary statistical signal processing.
Key words: Signal processing; Correntropy; Cyclic correntropy; Non-Gaussian; Non-stationary
中图分类号：TN911.7
文献标识码：A
文章编号：1009-5896(2020)01-0105-14
DOI: 10.11999/JEIT190646
收稿日期：2019-08-28；改回日期：2019-11-05；网络出版：2019-11-12
*通信作者： 邱天爽 qiutsh@dlut.edu.cn
基金项目：国家自然科学基金(61671105, 61172108, 61139001, 81241059)
Foundation Items: The National Natural Science Foundation of China (61671105, 61172108, 61139001, 81241059)
邱天爽：男，1954年生，教授，博士生导师，主要研究方向为非高斯、非平稳统计信号处理.