源信号为AR模型的独立成分分析算法及其应用研究
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摘要
独立成分分析(Independent Component Analysis,ICA)是一种新的数据处理与分析方法,目的在于从观测信号中分离或提取出相互统计独立的未知源信号。用ICA来解决盲源分离(Blind Source Separation,BSS)问题已经引起了广泛的关注,并已成功地应用到语音信号处理、通信、人脸识别、图像特征提取、医学信号处理等众多领域。本论文对源信号为自回归(Autoregressive,AR)模型的独立成分分析算法及其应用进行了研究,主要成果如下:
1.对源信号为AR模型的无噪ICA算法进行了研究。首先详细地推导了源信号为AR(1)时模型的对数似然函数,然后给出了三种情况下的ICA学习算法:(1)当新息的概率密度函数为广义高斯分布时,极大化模型的对数似然函数提出了一个梯度算法;(2)由于前一个算法中的参数较多,运行起来较复杂。为克服此不足,用一个非二次光滑的偶函数来近似代替新息的对数概率密度函数时,提出了一个批算法和一个在线算法,并对这两个算法进行了理论分析。应用这两个算法对人工合成的混合信号进行分离的同时又应用批算法对混合的图像进行了分离,均取得了很好的分离效果;(3)由于批算法的收敛性依赖于学习率的选择且收敛速度较慢,为了克服这个不足,利用近似牛顿法极大化模型的对数似然函数给出了一个不动点算法。计算机仿真实验结果表明不动点算法的收敛速度较快且应用起来更为简单。
2.对源信号为AR模型的有噪ICA算法进行了研究。对于这种有噪模型,分别对噪声协方差矩阵已知和未知两种情况下的算法进行了探讨。在噪声协方差矩阵已知的情况下,通过最大化新息的负熵,提出了一个不动点算法。计算机仿真实验结果表明同已有的梯度算法相比不动点算法具有较快的收敛速度,同时应用该算法对有噪的自然混合图像和纹理图像分别进行分离,均取得了很好的分离效果;在噪声协方差矩阵未知的情况下,通过对已有的复杂寻踪梯度算法的改进,提出了一个改进梯度算法和一个新算法:其中包括了对解混矩阵和噪声协方差矩阵进行估计的迭代公式。计算机仿真实验结果表明这两个算法都能够很好地对人工合成的有噪信号进行分离。由于新算法中只涉及一个学习率的选择,因此算法应用起来较简单且收敛速度较快。
3.对源信号为AR模型、方差非平稳变化的无噪ICA算法进行了研究。2005年,Hyv(?)rinen给出了这种情况下模型的对数似然函数,并提出了一个梯度上升算法,但该算法的收敛性依赖于学习率的正确选择。为克服此不足,利用近似牛顿法极大化模型的对数似然函数,推导出了一个不动点算法。计算机仿真实验结果表明同已有的梯度上升算法相比不动点算法具有较快的收敛速度且应用起来更为简单。
4.对源信号为AR模型、方差非平稳变化的有噪ICA算法进行了研究。对于这种有噪模型,分别对噪声协方差矩阵已知和未知两种情况下的算法进行了讨论。在噪声协方差矩阵已知的情况下,利用高斯矩的性质,给出了有噪模型的对数似然函数,通过极大化对数似然函数提出了一个梯度上升算法。但是由于噪声协方差矩阵已知往往是不现实的,因此又对噪声协方差矩阵未知情况下的算法进行了研究。通过对上述梯度算法的改进,提出了一个新梯度算法:其中包括了对解混矩阵和噪声协方差矩阵进行估计的迭代公式。计算机仿真实验结果表明这两个算法都能对人工合成的有噪信号进行很好的分离。
5.对源信号为AR模型的无噪和有噪两种情况下的胎儿心电信号盲提取算法进行了研究。在无噪模型中,利用近似牛顿法极小化目标函数,给出了一个新梯度算法。计算机仿真实验结果表明该算法不但能够较好地对人工合成信号进行胎儿心电信号的提取,而且对某个孕妇真实的心电信号也进行了较好的胎儿心电信号的提取;在有噪模型中,利用高斯矩的性质给出了胎儿心电信号提取的目标函数,提出了一个梯度下降算法。计算机仿真实验验证了算法的有效性和可行性。
Independent component analysis(ICA) is a new data processing and analysis technique for extracting independent sources given only observed data that are mixtures of unknown sources.Recently,blind source separation by ICA has received great attention due to its potential signal processing applications such as speech signal processing, telecommunications,face recognition,image feature extraction and medical signal processing, ect.This dissertation is devoted to the study of several algorithms for temporal independent component analysis and their applications by using AR source model.The main achievements are as follows:
1.The algorithms of noise-free independent component,analysis with AR source model were studied.The log-likelihood of the model is derived in detail when each source is a first-order AR model,then the ICA learning algorithms are proposed in three cases.(1) A gradient algorithm is given by maximizing log-likelihood of the model when the probability density function of innovation is generalized Gaussian distribution; (2) There are many parameters in the gradient algorithm,and the functions are complexity.So we use a nonquadratic function approximate the logarithm of the probability density function of the innovation process,a batch and an on-line algorithms are introduced and their theoretics analysis are carried out simultaneously.Computer simulations show that the algorithms can separate mixed signals,and the batch algorithm can separate mixed images and.achieve better separation effect;(3) There is a learning rate in the batch algorithm which influenced the convergence speed.In order to overcome drawback:a fixed-point algorithm is derived using approximate Newton method by maximizing log-likelihood function of the model.Computer simulations verify the fixed-point algorithm converges faster than the batch algorithm and it is easy to implement due to it does not need any learning rate.
2.The algorithms of noisy independent component analysis with AR source model were studied when the noise covariance is known and the noise covariance is unknown. When the noise covariance matrix is known,a fixed-point algorithm is proposed by maximizing the negentropy of innovation.Computer simulations show that the fixedpoint algorithm converges faster than the existing gradient algorithm,and can separate the mixed images and achieve better separation effect.When the noise covariance is unknown,an improved gradient algorithm and a new noisy algorithm are introduced to estimate the mixing matrix and noise covariance matrix simultaneously.Computer simulations verify the two algorithms can separate the mixed signals.Comparison results show that the new noisy algorithm converges faster than the improved gradient algorithm,and it is easy to implement due to there is only one learning rate to be chosen.
3.The algorithm of noise-free independent component analysis were studied with AR source model and nonstationary variances.In 2005,the log-likelihood of the model was given and a gradient algorithm was proposed by maximizing log-likelihood of the model by Hyv(a|¨)rinen.But the convergence speed is influenced by the choice of the learning rate.In order to overcome this drawback,a fixed-point algorithm is proposed using approximate Newton method by maximizing log-likelihood of the model.Computer simulations show that the fixed-point algorithm converges faster than the gradient algorithm,and it is more implement due to it does not need any learning rate.
4.The algorithms of noisy independent component analysis were studied with AR source model and nonstationary variances when the noise covariance matrix is known and the noise covariance matrix is unknown.When the noise covariance matrix is known, the log-likelihood function of the model is given by using the property of Gaussian moments and a gradient algorithm is introduced by maximizing the log-likelihood function of the model;when the noise covariance matrix is unknown,a new gradient algorithm is introduced by developing the gradient algorithm to estimate unmixed matrix and noise covariance matrix simultaneously.Computer simulations show that the two algorithms can separate the artificial mixed signals and achieve better separation effect.
5.The algorithms of blind extraction of FECG with AR model were studied.A new gradient algorithm is given using approximate Newton method by minimizing objective function when Gaussian noise is not present.Computer simulations show that the algorithm can extract FECG from the artificial signals and the real-world ECG data:When Gaussian noise is present in the model,an objective function is given by utilizing the property of Gaussian moments,and a gradient descent algorithm is proposed.Computer simulations verify the gradient,descent algorithm is effective and feasible.
1.对源信号为AR模型的无噪ICA算法进行了研究。首先详细地推导了源信号为AR(1)时模型的对数似然函数,然后给出了三种情况下的ICA学习算法:(1)当新息的概率密度函数为广义高斯分布时,极大化模型的对数似然函数提出了一个梯度算法;(2)由于前一个算法中的参数较多,运行起来较复杂。为克服此不足,用一个非二次光滑的偶函数来近似代替新息的对数概率密度函数时,提出了一个批算法和一个在线算法,并对这两个算法进行了理论分析。应用这两个算法对人工合成的混合信号进行分离的同时又应用批算法对混合的图像进行了分离,均取得了很好的分离效果;(3)由于批算法的收敛性依赖于学习率的选择且收敛速度较慢,为了克服这个不足,利用近似牛顿法极大化模型的对数似然函数给出了一个不动点算法。计算机仿真实验结果表明不动点算法的收敛速度较快且应用起来更为简单。
2.对源信号为AR模型的有噪ICA算法进行了研究。对于这种有噪模型,分别对噪声协方差矩阵已知和未知两种情况下的算法进行了探讨。在噪声协方差矩阵已知的情况下,通过最大化新息的负熵,提出了一个不动点算法。计算机仿真实验结果表明同已有的梯度算法相比不动点算法具有较快的收敛速度,同时应用该算法对有噪的自然混合图像和纹理图像分别进行分离,均取得了很好的分离效果;在噪声协方差矩阵未知的情况下,通过对已有的复杂寻踪梯度算法的改进,提出了一个改进梯度算法和一个新算法:其中包括了对解混矩阵和噪声协方差矩阵进行估计的迭代公式。计算机仿真实验结果表明这两个算法都能够很好地对人工合成的有噪信号进行分离。由于新算法中只涉及一个学习率的选择,因此算法应用起来较简单且收敛速度较快。
3.对源信号为AR模型、方差非平稳变化的无噪ICA算法进行了研究。2005年,Hyv(?)rinen给出了这种情况下模型的对数似然函数,并提出了一个梯度上升算法,但该算法的收敛性依赖于学习率的正确选择。为克服此不足,利用近似牛顿法极大化模型的对数似然函数,推导出了一个不动点算法。计算机仿真实验结果表明同已有的梯度上升算法相比不动点算法具有较快的收敛速度且应用起来更为简单。
4.对源信号为AR模型、方差非平稳变化的有噪ICA算法进行了研究。对于这种有噪模型,分别对噪声协方差矩阵已知和未知两种情况下的算法进行了讨论。在噪声协方差矩阵已知的情况下,利用高斯矩的性质,给出了有噪模型的对数似然函数,通过极大化对数似然函数提出了一个梯度上升算法。但是由于噪声协方差矩阵已知往往是不现实的,因此又对噪声协方差矩阵未知情况下的算法进行了研究。通过对上述梯度算法的改进,提出了一个新梯度算法:其中包括了对解混矩阵和噪声协方差矩阵进行估计的迭代公式。计算机仿真实验结果表明这两个算法都能对人工合成的有噪信号进行很好的分离。
5.对源信号为AR模型的无噪和有噪两种情况下的胎儿心电信号盲提取算法进行了研究。在无噪模型中,利用近似牛顿法极小化目标函数,给出了一个新梯度算法。计算机仿真实验结果表明该算法不但能够较好地对人工合成信号进行胎儿心电信号的提取,而且对某个孕妇真实的心电信号也进行了较好的胎儿心电信号的提取;在有噪模型中,利用高斯矩的性质给出了胎儿心电信号提取的目标函数,提出了一个梯度下降算法。计算机仿真实验验证了算法的有效性和可行性。
Independent component analysis(ICA) is a new data processing and analysis technique for extracting independent sources given only observed data that are mixtures of unknown sources.Recently,blind source separation by ICA has received great attention due to its potential signal processing applications such as speech signal processing, telecommunications,face recognition,image feature extraction and medical signal processing, ect.This dissertation is devoted to the study of several algorithms for temporal independent component analysis and their applications by using AR source model.The main achievements are as follows:
1.The algorithms of noise-free independent component,analysis with AR source model were studied.The log-likelihood of the model is derived in detail when each source is a first-order AR model,then the ICA learning algorithms are proposed in three cases.(1) A gradient algorithm is given by maximizing log-likelihood of the model when the probability density function of innovation is generalized Gaussian distribution; (2) There are many parameters in the gradient algorithm,and the functions are complexity.So we use a nonquadratic function approximate the logarithm of the probability density function of the innovation process,a batch and an on-line algorithms are introduced and their theoretics analysis are carried out simultaneously.Computer simulations show that the algorithms can separate mixed signals,and the batch algorithm can separate mixed images and.achieve better separation effect;(3) There is a learning rate in the batch algorithm which influenced the convergence speed.In order to overcome drawback:a fixed-point algorithm is derived using approximate Newton method by maximizing log-likelihood function of the model.Computer simulations verify the fixed-point algorithm converges faster than the batch algorithm and it is easy to implement due to it does not need any learning rate.
2.The algorithms of noisy independent component analysis with AR source model were studied when the noise covariance is known and the noise covariance is unknown. When the noise covariance matrix is known,a fixed-point algorithm is proposed by maximizing the negentropy of innovation.Computer simulations show that the fixedpoint algorithm converges faster than the existing gradient algorithm,and can separate the mixed images and achieve better separation effect.When the noise covariance is unknown,an improved gradient algorithm and a new noisy algorithm are introduced to estimate the mixing matrix and noise covariance matrix simultaneously.Computer simulations verify the two algorithms can separate the mixed signals.Comparison results show that the new noisy algorithm converges faster than the improved gradient algorithm,and it is easy to implement due to there is only one learning rate to be chosen.
3.The algorithm of noise-free independent component analysis were studied with AR source model and nonstationary variances.In 2005,the log-likelihood of the model was given and a gradient algorithm was proposed by maximizing log-likelihood of the model by Hyv(a|¨)rinen.But the convergence speed is influenced by the choice of the learning rate.In order to overcome this drawback,a fixed-point algorithm is proposed using approximate Newton method by maximizing log-likelihood of the model.Computer simulations show that the fixed-point algorithm converges faster than the gradient algorithm,and it is more implement due to it does not need any learning rate.
4.The algorithms of noisy independent component analysis were studied with AR source model and nonstationary variances when the noise covariance matrix is known and the noise covariance matrix is unknown.When the noise covariance matrix is known, the log-likelihood function of the model is given by using the property of Gaussian moments and a gradient algorithm is introduced by maximizing the log-likelihood function of the model;when the noise covariance matrix is unknown,a new gradient algorithm is introduced by developing the gradient algorithm to estimate unmixed matrix and noise covariance matrix simultaneously.Computer simulations show that the two algorithms can separate the artificial mixed signals and achieve better separation effect.
5.The algorithms of blind extraction of FECG with AR model were studied.A new gradient algorithm is given using approximate Newton method by minimizing objective function when Gaussian noise is not present.Computer simulations show that the algorithm can extract FECG from the artificial signals and the real-world ECG data:When Gaussian noise is present in the model,an objective function is given by utilizing the property of Gaussian moments,and a gradient descent algorithm is proposed.Computer simulations verify the gradient,descent algorithm is effective and feasible.
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