P-spline smoothed functional ICA of EEG data

We propose a novel functional data framework for artifact extraction and removal to estimate brain electrical activity sources from EEG signals. Our methodology is derived on the basis of event related potential (ERP) analysis, and motivated by mapping adverse artifactual events caused by body movements and physiological activity originated outside the brain. A functional independent component analysis (FICA) based on the use of fourth moments is conducted on the principal component expansion in terms of B-spline basis functions. We extend this model setup by introducing a discrete roughness penalty in the orthonormality constraint of the functional principal component decomposition to later compute estimates of FICA. Compared to other ICA algorithms, our method combines a regularization mechanism stemmed from the principal eigendirections with a discrete penalization given by the $d$-order difference operator. In this regard, it allows to naturally control high-frequency remnants of neural origin overlapping latent artifactual eigenfunctions and thus to preserve this persistent activity at artifact extraction level. Furthermore, we introduce a new cross-validation method for the selection of the penalization parameter which uses shrinkage to asses the performance of the estimators for functional representations with larger basis dimension and excess of roughness. This method is used in combination with a kurtosis measure in order to provide the optimal number of independent components.The FICA model is illustrated at functional and longitudinal dimensions by an example on real EEG data where a subject willingly performs arm gestures and stereotyped physiological artifacts. Our method can be relevant in neurocognitive research and related fields, particularlly in situations where movement can bias the estimation of brain potentials.