Measuring Non-Gaussianity by Phi-Transformed and Fuzzy Histograms

Independent component analysis (ICA) is an essential building block for data analysis in many applications. Selecting the truly meaningful components from the result of an ICA algorithm, or comparing the results of different algorithms, however, is nontrivial problems. We introduce a very general technique for evaluating ICA results rooted in information-theoretic model selection. The basic idea is to exploit the natural link between non-Gaussianity and data compression: the better the data transformation represented by one or several ICs improves the effectiveness of data compression, the higher is the relevance of the ICs. We propose two different methods which allow an efficient data compression of non-Gaussian signals: Phi-transformed histograms and fuzzy histograms. In an extensive experimental evaluation, we demonstrate that our novel information-theoretic measures robustly select non-Gaussian components from data in a fully automatic way, that is, without requiring any restrictive assumptions or thresholds. 1. Introduction Independent component analysis (ICA) is a powerful technique for signal demixing and data analysis in numerous applications. For example, in neuroscience, ICA is essential for the analysis of functional magnetic resonance imaging (fMRI) data and electroencephalograms (EEGs). The function of the human brain is very complex and can be only imaged at a very coarse spatial resolution. Millions of nerve cells are contained in a single voxel of fMRI data. The neural activity is indirectly measured by the so-called BOLD-effect, that is, by the increased supply of active regions with oxygenated blood. In EEG, the brain function can be directly measured by the voltage fluctuations resulting from ionic current flows within the neurons. The spacial resolution of EEG, however, is even much lower than that of fMRI. Usually, an EEG is recorded using an array of 64 electrodes distributed over the scalp. Often, the purpose of acquiring fMRI or EEG data is obtaing a better understanding of brain function while the subject is performing some task. An example for such an experiment is to show subjects images while they are in the scanner to study the processing of visual stimuli, see Section 4.1.4. Recent results in neuroscience, for example [1], confirm the organization of the human brain into distinct functional modules. During task processing, some functional modules are actively contributing to the task. However, many other modules are also active but not involved into task-specific activities. Due to the low resolution of fMRI and EEG data,