Tractable Maximum Likelihood Estimation for Latent Structure Influence Models With Applications to EEG & ECoG Processing

Brain signals are nonlinear and nonstationary time series, which provide information about spatiotemporal patterns of electrical activity in the brain. CHMMs are suitable tools for modeling multi-channel time-series dependent on both time and space, but state-space parameters grow exponentially with the number of channels. To cope with this limitation, we consider the influence model as the interaction of hidden Markov chains called Latent Structure Influence Models (LSIMs). LSIMs are capable of detecting nonlinearity and nonstationarity, making them well suited for multi-channel brain signals. We apply LSIMs to capture the spatial and temporal dynamics in multi-channel EEG/ECoG signals. The current manuscript extends the scope of the re-estimation algorithm from HMMs to LSIMs. We prove that the re-estimation algorithm of LSIMs will converge to stationary points corresponding to Kullback-Leibler divergence. We prove convergence by developing a new auxiliary function using the influence model and a mixture of strictly log-concave or elliptically symmetric densities. The theories that support this proof are derived from previous studies by Baum, Liporace, Dempster, and Juang. We then develop a closed-form expression for re-estimation formulas using tractable marginal forward-backward parameters defined in our previous study. Simulated datasets and EEG/ECoG recordings confirm the practical convergence of the derived re-estimation formulas. We also study the use of LSIMs for modeling and classification on simulated and real EEG/ECoG datasets. Based on AIC and BIC, LSIMs perform better than HMMs and CHMMs in modeling embedded Lorenz systems and ECoG recordings. LSIMs are more reliable and better classifiers than HMMs, SVMs and CHMMs in 2-class simulated CHMMs. EEG biometric verification results indicate that the LSIM-based method improves the area under curve (AUC) values by about 6.8 and decreases the standard deviation of AUC values from 5.4 to 3.3 compared to the existing HMM-based method for all conditions on the BED dataset.