独立向量分析的高斯拟极大似然法性能分析
Performance Analysis of the Gaussian Quasi-Maximum Likelihood Approach for Independent Vector Analysis
摘要：
最大似然估计（ML）需要对基础统计模型有精确的了解。在Quasi ML（QML）中，假设模型被用作（未知）真实模型的替代品。在独立向量分析（IVA）的背景下，我们考虑的高斯QML估计（QMLE）的混合矩阵集，并提出了一个（近似）分析其渐近分离性能。在高斯QML中，源被假定为高斯的，协方差矩阵由一些“有根据的猜测”指定。由此产生的拟似然方程的分层矩阵采取一种特殊的形式，最近被称为一个扩展的“顺序钻孔”联合同余（SeDJoCo）变换，这是让人想起（虽然本质上不同）经典的联合对角化。我们表明，渐近这个QMLE，即，得到的扩展SeDJoCo变换的解实现了完全分离（在一些温和的条件下），而不管源的真实分布和/或协方差矩阵。此外，基于“小误差”假设，我们提出了一个一阶摄动分析的扩展SeDJoCo解决方案。使用所得到的封闭形式的表达式的解决方案矩阵中的错误，我们提供了封闭形式的表达式所得到的干扰源比（ISR）IVA。此外，我们证明了渐近的ISR只依赖于源的协方差，而不是对他们的具体分布。作为这个结果的直接后果，我们提供了一个渐近可达到的下限所产生的ISR。我们还提出了实证结果，证实了我们的分析推导，三个模拟实验的两种可能的模型误差-不准确的协方差矩阵和源的分布建模错误。
英文摘要：
Maximum Likelihood (ML) estimation requires precise knowledge of the underlying statistical model. In Quasi ML (QML), a presumed model is used as a substitute to the (unknown) true model. In the context of Independent Vector Analysis (IVA), we consider the Gaussian QML Estimate (QMLE) of the demixing matrices set and present an (approximate) analysis of its asymptotic separation performance. In Gaussian QML the sources are presumed to be Gaussian, with covariance matrices specified by some "educated guess". The resulting quasi-likelihood equations of the demixing matrices take a special form, recently termed an extended "Sequentially Drilled" Joint Congruence (SeDJoCo) transformation, which is reminiscent of (though essentially different from) classical joint diagonalization. We show that asymptotically this QMLE, i.e., the solution of the resulting extended SeDJoCo transformation, attains perfect separation (under some mild conditions) regardless of the sources' true distributions and/or covariance matrices. In addition, based on the "small-errors" assumption, we present a first-order perturbation analysis of the extended SeDJoCo solution. Using the resulting closed-form expressions for the errors in the solution matrices, we provide closed-form expressions for the resulting Interference-to-Source Ratios (ISRs) for IVA. Moreover, we prove that asymptotically the ISRs depend only on the sources' covariances, and not on their specific distributions. As an immediate consequence of this result, we provide an asymptotically attainable lower bound on the resulting ISRs. We also present empirical results, corroborating our analytical derivations, of three simulation experiments concerning two possible model errors - inaccurate covariance matrices and sources' distribution mismodeling.
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