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In this paper, we develop low complexity and stable bootstrap procedures for FastICA estimators. Our bootstrapping techniques allow for performing cost efficient and reliable bootstrap-based statistical inference in the ICA model. Performing statistical inference is needed to quantitatively assess the quality of the estimators and testing hypotheses on mixing coefficients in the ICA model. The developed bootstrap procedures stem from the fast and robust bootstrap (FRB) method [1], which is applicable for estimators that may be found as solutions to fixed-point (FP) equations. We first establish analytical results on the structure of the weighted covariance matrix involved in the FRB formulation. Then, we exploit our analytical results to compute the FRB replicas at drastically reduced cost. The developed enhanced FRB method (EFRB) for FastICA permits using bootstrap-based statistical inference in a variety of applications (e.g., EEG, fMRI) in which ICA is commonly applied. Such an approach has not been possible earlier due to incurred substantial computational efforts of the conventional bootstrap. Our simulation studies compare the complexity and numerical stability of the proposed methods with the conventional bootstrap method. We also provide an example of utilizing the developed bootstrapping techniques in identifying equipotential lines of the brain dipoles from electroencephalogram (EEG) recordings. [ABSTRACT FROM AUTHOR]

In this correspondence, we provide a multiple hypothesis test to detect the number of latent noncircular signals in a complex Gaussian random vector. Our method sequentially tests the results of individual generalized likelihood ratio test (GLRT) statistics with known asymptotic distributions to form the multiple hypothesis detector. Specifically, we are able to set a threshold yielding a precise probability of error. This test can be used to statistically determine if a given complex observation is circular Gaussian, and if not, how many latent signals in the observation are noncircular. Simulations are used to quantify the performance of the detector as compared to a detector based on the minimum description length (MDL) criterion. The utility of the detector is shown by applying it to a beamforming application using independent component analysis (ICA). [ABSTRACT FROM AUTHOR]

This paper provides a rigorous statistical analysis of the deflation-based FastICA estimator, where the independent components (ICs) are extracted sequentially. The focus is on two aspects of the estimator: robustness against outliers as measured by the influence function (IF) and on its asymptotic relative efficiency (ARE) as measured by the ratio of the asymptotic variance of the FastICA w.r.t. the optimal maximum likelihood estimator (MLE). The derived compact closed-form expression of the IF reveals the vulnerability of the FastICA estimator to outliers regardless of the used nonlinearity. A cautionary finding is that even a moderate observation towards certain directions can render the estimator deficient in the sense that its separation performance degrades worse than a plain guess. The IF allows the derivation of a compact closed-form expression for the asymptotic covariance matrix of the FastICA estimator and subsequently its asymptotic relative efficiencies (AREs). The ARE figures calculated for some selected source distributions illustrate the fact that the order which the ICs are found is crucial as the accuracy of the previously extracted components can dominantly affect the accuracy of the successive deflation stages. [ABSTRACT FROM AUTHOR]

Abstract: An extension of the whitening transformation for complex random vectors, called the generalized uncorrelating transformation (GUT), is introduced. GUT is a generalization of the strong-uncorrelating transform [J. Eriksson, V. Koivunen, Complex-valued ICA using 2nd-order statistics, in: Proceedings of the IEEE Workshop on Machine Learning for Signal Processing (MLSP’04), Sao Luis, Brazil, 2004] based upon generalized estimators of the covariance and pseudo-covariance matrix, called the scatter matrix and spatial pseudo-scatter matrix, respectively. Depending on the selected scatter and spatial pseudo-scatter matrix, GUT estimators can have largely different statistical properties. Special emphasis is put on robust GUT estimators. We show that GUT is a separating matrix estimator for complex-valued independent component analysis (ICA) when at most one source random variable possess circularly symmetric distribution and sources do not have identical distribution. In the context of ICA, our approach is computationally attractive as it is based on straightforward matrix computations. Simulations and examples are used to confirm reliable performance of our method. [Copyright &y& Elsevier]

Despite of the increased interest in independent component analysis (ICA) during the past two decades, a simple closed form expression of the Cramér-Rao bound (CRB) for the demixing matrix estimation has not been established in the open literature. In the present paper we fill this gap by deriving a simple closed-form expression for the CRB of the demixing matrix directly from its definition. A simulation study comparing ICA estimators with the CRB is given. [ABSTRACT FROM AUTHOR]

• We formulate graph error models for the adjacency matrix, which help to quantify the deviation from the true matrix using a few parameters. • We study the structural effects of the proposed error models on the adjacency matrix. • The effects of different type of errors in adjacency matrix specification are illustrated in filtering of graph signal and ICA of graph signals. The first step for any graph signal processing (GSP) procedure is to learn the graph signal representation, i.e., to capture the dependence structure of the data into an adjacency matrix. Indeed, the adjacency matrix is typically not known a priori and has to be learned. However, it is learned with errors. A little attention has been paid to modelling such errors in the adjacency matrix, and studying their effects on GSP methods. However, modelling errors in the adjacency matrix will enable both to study the graph error effects in GSP and to develop robust GSP algorithms. In this paper, we therefore introduce practically justifiable graph error models. We also study, both analytically when possible and numerically, the graph error effect on the performance of GSP methods in different types of problems such as filtering of graph signals and independent component analysis of graph signals (graph decorrelation). [ABSTRACT FROM AUTHOR]