Independent Component Analysis Involving Autocorrelated Sources With an Application to Functional Magnetic Resonance Imaging

Abstract

Independent component analysis (ICA) is an effective data-driven method for blind source separation. It has been successfully applied to separate source signals of interest from their mixtures. Most existing ICA procedures are carried out by relying solely on the estimation of the marginal density functions, either parametrically or nonparametrically. In many applications, correlation structures within each source also play an important role besides the marginal distributions. One important example is functional magnetic resonance imaging (fMRI) analysis where the brain-function-related signals are temporally correlated. In this article, we consider a novel approach to ICA that fully exploits the correlation structures within the source signals. Specifically, we propose to estimate the spectral density functions of the source signals instead of their marginal density functions. This is made possible by virtue of the intrinsic relationship between the (unobserved) sources and the (observed) mixed signals. Our methodology is described and implemented using spectral density functions from frequently used time series models such as autoregressive moving average (ARMA) processes. The time series parameters and the mixing matrix are estimated via maximizing the Whittle likelihood function. We illustrate the performance of the proposed method through extensive simulation studies and a real fMRI application. The numerical results indicate that our approach outperforms several popular methods including the most widely used fastICA algorithm. This article has supplementary material online.

Application of Polynomial Spline Independent Component Analysis to fMRI Data

“…For temporal ICA, this can be implemented by using time series models to account for the source serial correlations. Indeed, Lee et al (2011) has reported that there is noticeable improvement over the marginal density based ICA procedures. It will be important to see if the same will hold for the above spatial ICA approach using tensor products of splines.…”

“…It is important to point out that we should also provide a sensitivity and specificity analysis of the activated spatial locations as described in Lee et al (2011), where popular methods such as Infomax and fastICA were shown to have a higher false-positive/nagative rate. This implies that brain activation should be studied more carefully, and one should avoid using methods that tend to yield false activation.…”

“…We have found that even greater improvement can be achieved by using informative dimension reduction methods such as singular value decomposition (SVD) by choosing the eigen-vectors to relate to the experimental task paradigm closely. This is being referred to as a supervised SVD dimension reduction procedure (Bai et al, 2008) and has been used effectively in Lee et al (2011).…”

Evaluating model misspecification in independent component analysis

“…For simplicity, we focus on the case M = N for the rest of the manuscript. Many algorithms are available for instantaneous mixtures based on different independence measurements, such as high-order statistics [3], information theoretic measurements [4-6], canonical correlations in a reproducing kernel Hilbert space [7], maximum likelihood [8-11], characteristic function [12, 13] and the Whittle likelihood [14]. …”

“…Although identifying mixing models is critical, our literature review showed little research regarding the effects of misspecified ICA models, excepting cases of instantaneous mixing. For example, [14] has indicated that most marginal independence measure based approaches, which intrinsically assume temporal independence of sources, fail to separate autocorrelated sources. Model identification becomes even more difficult in convolutive mixing cases without proper constraints when the sources are autocorrelated.…”