Sparse Autoregressive Modeling via the Least Absolute LP-Norm Penalized Solution

The conventional autoregressive (AR) model has been widely applied in the various electroencephalogram (EEG) analyses such as spectrum estimation, waveform fittings, and in classification tasks. Nevertheless, evoked EEG is usually inevitably contaminated by multiple background activities (ongoing EEG) as well as the strong outliers which may distort the AR estimates of various AR estimation methods including LS, Yule–Walker, and Burg. Moreover, current AR approaches perform well only when the length of the time-series is much larger than the number of brain sites studied, which is exactly the reverse of the situation in neuroimaging whereby relatively short time-series are measured over thousands of voxels thus the need for penalized methods to obtain sparse solutions. In this paper, we introduce a novel ADMM-based AR estimator termed LAPPS (Least Absolute LP (0 < p < 1) Penalized Solution) which employs the L1-loss function for the residual error to alleviate the influence of outliers and another Lp-penalty term (p = 0.5) to obtain the sparse AR parameters while suppressing any spurious noise that may be present. Our obtained simulations result quantitatively show that LAPPS-AR performs better than the commonly used AR estimation methods. In addition, we applied the method to real EEG visual oddball recording with ocular artifacts where LAPPS-AR effectively suppressed the outliers and estimated a P300 EEG power spectrum consistent with its physiological basis.