A nonlinear adaptive framework for the estimation of causal interactions among multivariate data in brain signal recordings -

Abstract

Modeling of dynamical systems is a challenging task in numerous fields, such as neuroscience, control, and econometrics. For instance, the noisy and complex nature of many biological signals, such as the electroencephalogram recordings, may not be fully portrayed via linear models. To this purpose, many nonlinear modeling techniques have been proposed, some of which suffer from high computational complexity, or the inability to simultaneously deal with non-stationarity and noise. These limitations, along with the ease of linear prediction schemes, have shifted attention to the framework of kernel functions mappings. Kernel functions indirectly transform nonlinear data from the original space to a high dimensional space where they are linearly related. In this context, a stationary approach based on an autoregressive model in the feature space has been introduced to model and predict univariate time series. This approach solves a kernelized version of the yule walker equations, and utilizes a fixed point iterative method to find the preimage. In this thesis, we introduce a novel approach based on the previously mentioned solution to model univariate time series in nonlinear and nonstationary environments. The proposed approach employs a Square-Root Cubature Kalman Filter for adaptively tracking model parameters and predicting time series, and with slight modification of the state space formulation, can be made to account for additive white Gaussian noise. The effectiveness of this method is accentuated in the presence of multiple time series, where the analysis of causal interactions between these series can aid in understanding the dynamical evolution of underlying processes. For this purpose, the proposed methodology is further extended to the multivariate case where linear Granger Causality holds. Simulations include nonlinear benchmark examples, such as the Lorenz attractor, as well as real EEG recordings.