08_ICA - Unmixing fMRI with Independent Component Analysis...

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IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE MARCH/APRIL 2006 79 FUNCTIONAL MAGNETIC RESONANCE IMAGING Unmixing fMRI with Independent Component Analysis Using ICA to Characterize High-Dimensional fMRI Data in a Concise Manner. BY VINCE D. CALHOUN AND TÜLAY ADALI ©DIGITAL STOCK 0739-5175/06/$20.00©2006IEEE hat an antithetical mind!—tenderness, rough- ness—delicacy, coarseness—sentiment, sensual- ity—soaring and groveling, dirt and deity—all mixed up in that one compound of inspired clay! —Lord Byron Independent component analysis (ICA) is a statistical method used to discover hidden factors (sources or features) from a set of measurements or observed data such that the sources are maximally independent. Typically, it assumes a generative model where observations are assumed to be linear mixtures of independent sources, and unlike principal component analy- sis (PCA), which uncorrelates the data, ICA works with high- er-order statistics to achieve independence. An intuitive example of ICA can be given by a scatter-plot of two independent signals s 1 and s 2 . Figure 1(a) shows a plot of the two independent signals ( s 1 , s 2 ) in a scatter plot. Figure 1(b) and (c) shows the projections for PCA and ICA, respectively, for a linear mixture of s 1 and s 2 . PCA finds the orthogonal vectors u 1 , u 2 but does not find independent vectors. In con- trast, ICA is able to find the independent vectors a 1 , a 2 of the linear mixed signals ( s 1 , s 2 ) and is thus able to restore the orig- inal sources. A typical ICA model assumes that the source signals are not observable, are statistically independent, and are non- Gaussian, with an unknown but linear mixing process. Consider an observed M- dimensional random vector denoted by x = ( x 1 ,... x M ) T , which is generated by the ICA model: x = As ,( 1 ) where s = [ s 1 , s 2 s N ] T is an N- dimensional vector whose elements are assumed independent sources and A M × N is an unknown mixing matrix. Typically M > = N , so A is usually of full rank. The goal of ICA is to estimate an unmixing matrix W N × M such that y [defined in (2)] is a good approxima- tion to the true sources: s . y = Wx ( 2 ) ICA is hence an approach to solving the blind source separation problem, which traditionally addresses the solu- tion of the cocktail party problem in which several people are speaking simultaneously in the same room. The prob- lem is to separate the voices of the different speakers by using recordings of several microphones in the room [2]. The basic ICA model for blind source separation is shown in Figure 2. Popular approaches for performing ICA include maximiza- tion of information transfer, which is equivalent to maximum likelihood estimation, maximization of non-Gaussianity, mutual information minimization, and tensorial methods. The most commonly used ICA algorithms are Infomax [3], FastICA [4], and joint approximate diagonalization of eigen- matrices (JADE) [5]. The original Infomax algorithm for blind separation by [3] is better suited to estimation of super- Gaussian sources. To overcome this limitation, techniques
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08_ICA - Unmixing fMRI with Independent Component Analysis...

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