Time-dependent frequency domain principal components analysis of multichannel non-stationary signals

High dimensional multi-channel signals often exhibit multi-collinearities. This suggests that such signals can be decomposed into uncorrelated principal components with possibly lower dimension than that of the original signal. A time-localized frequency domain principal components analysis method is proposed for signals that exhibit locally stationary behavior. The first step is to form a mean square consistent estimate of the time-varying spectrum matrix by smoothing the time-localized periodograms using a kernel defined on the frequency axis whose span is selected automatically using a generalized cross-validation procedure that is based on the asymptotic gamma distribution. The eigenvalues of the spectral density estimate are then computed which are the estimated spectra of the principal components. In addition, one may apply a formal statistical procedure for testing whether the weights (components of an eigenvector) at a particular channel change over time. The proposed method can be easily implemented because it only requires the fast Fourier transform (FFT) and eigenvalue–eigenvector decomposition routines. An illustration is presented using a multi-channel brain waves data set recorded during an epileptic seizure.