Explorative Analysis of fMRI Data: Why Functional PCA should Usually be Preferred to ICA

Independent component analysis (ICA [1]) has been proposed as an exploratory technique for fMRI data. Unfortunately ICA generates many components with little information about how to select from them. Here we propose a new technique, functional PCA, to retrieve the signal associated to the experimental paradigm in an explorative setting. ICA posits a linear model in which the observed images Y (each column being an image) are linear combinations M of independent non-Gaussian signal sources P: Y=MP The distributional nature of ICA's assumptions makes it an excellent explorative technique in general. However, its application to fMRI time series is problematic since the omission of an error term appears inappropriate in this context. A large number of ICA components will be generated out of noise. Furthermore, there is certainly some dependency between fMRI signals sampled at adjacent time points. While this latter point does not invalidate the ICA model, it points to an aspect of the data that is missing from its assumption set. If we can assume that the signal of interest is relatively 'smooth', it is appropriate to attempt to separate the signal from undesired high-frequency variance by carrying out a penalty fit voxel-by-voxel with the amount of penalty estimated through generalized cross-validation [2]. This corresponds to a model of the form Y'=XP' + E' where each column of X is a smooth eigenfunction, each row of P' is a component image, and E' is a rough error term. Functional PCA exploits advanced techniques to carry out the eigenanalysis directly on the fitted functions [3], harnessing the information implicit in the smoothness assumption to pull out the signal of interest and place it in the first few components. Fig. 1, left shows the screenplot of the PCA on an fMRI dataset generated in a two-back working memory task in a 6-blocks design [4], indicating that only one component can be considered a genuine signal. On the right, the image of the component loading on the left DLPFC. Fig. 2 shows the eigenfunctions retrieved in 11 datasets obtained with the same task. In 10 datasets the first eigenfunction retrieved the experimental signal, in 1 dataset the second eigenfunction. References: [1] McKeown et al. Human Br Mapping 1998; 6:160–188. [2] Wahba Spline Models for Observational Data. Philadelphia: SIAM, 1990. [3] Ramsay and Silverman Functional Data Analysis. Berlin: Springer, 1997. [4] Walter et al. Cortex 2003; 39: 897–911