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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]

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]

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]

Abstract: Spatial independent components analysis (sICA) has become a widely applied data-driven method for fMRI data, especially for resting-state studies. These sICA approaches are often based on iterative estimation algorithms and there are concerns about accuracy due to noise. Repeatability measures such as ICASSO, RAICAR and ARABICA have been introduced as remedies but information on their effects on estimates is limited. The contribution of this study was to provide more of such information and test if the repeatability analyses are necessary. We compared FastICA-based ordinary and repeatability approaches concerning mixing vector estimates. Comparisons included original FastICA, FSL4 Melodic FastICA and original and modified ICASSO. The effects of bootstrapping and convergence threshold were evaluated. The results show that there is only moderate improvement due to repeatability measures and only in the bootstrapping case. Bootstrapping attenuated power from time courses of resting-state network related ICs at frequencies higher than 0.1Hz and made subsets of low frequency oscillations more emphasized IC-wise. The convergence threshold did not have a significant role concerning the accuracy of estimates. The performance results suggest that repeatability measures or strict converge criteria might not be needed in sICA analyses of fMRI data. Consequently, the results in existing sICA fMRI literature are probably valid in this sense. A decreased accuracy of original bootstrapping ICASSO was observed and corrected by using centrotype mixing estimates but the results warrant for thorough evaluations of data-driven methods in general. Also, given the fMRI-specific considerations, further development of sICA methods is strongly encouraged. [Copyright &y& Elsevier]