Functional-coefficient models for multivariate time series in designed experiments: With applications to brain signals

To study the neurophysiological basis of attention deficit hyperactivity disorder (ADHD), clinicians use electroencephalography (EEG) which records neuronal electrical activity on the cortex. Instead of focusing on single-channel spectral power, a novel framework for investigating interactions (dependence) between channels in the entire network is proposed. Although dependence measures, such as coherence and partial directed coherence (PDC), are well explored in studying brain connectivity, these measures only capture linear dependence. Moreover, in designed clinical experiments, these dependence measures are observed to vary across subjects, even within a homogeneous group. To address these limitations, we propose the mixed-effects functional-coefficient autoregressive (MXFAR) model which captures between-subject variation by incorporating subject-specific random effects. The advantages of the MXFAR model are the following: (i) it captures potential nonlinear dependence between channels; (ii) it is nonparametric and hence flexible and robust to model misspecification; (iii) it can capture differences between groups when they exist; (iv) it accounts for variation across subjects; (v) the framework easily incorporates well-known inference methods from mixed-effects models; (vi) it can be generalized to accommodate various covariates and factors. Then we formulate a novel nonlinear spectral measure, the functional partial directed coherence (fPDC), to extract dynamic cross-dependence patterns at different frequency oscillations. Finally, we apply the proposed MXFAR-fPDC framework to analyze multichannel EEG signals and report novel findings on altered brain functional networks in ADHD patients.