Online state space filtering of biosignals using neural network-augmented Kalman filter

Abstract

Since its inception, the Kalman filter, which represents the optimal estimator for linear, Gaussian state space models, has been adopted for a wide array of practical applications due to its efficient, recursive formulation. At the same time, the field of nonlinear time series analysis has produced powerful methods for filtering time series with deterministic dynamics. These nonlinear filters differ significantly from linear filters, because they transform the time series via delay embedding and exploiting geometric features in delay space. However, they are not capable of operating recursively like the Kalman filter. In this paper, we propose a state space filter based on the delay embedding principle, but capable of online estimation. This is achieved by formulating the nonlinear delay space filter as a state estimation problem, which can be solved using the extended Kalman filter. In order for this reformulation to work, it is necessary to approximate the dynamics of the time series. For this purpose, we use a feed-forward neural network. By embedding the neural network weights in the Kalman filter state, we are able to simultaneously estimate the hidden dynamics of the time series and perform online state space filtering. We present preliminary performance estimates of our online state space filtering approach obtained from tests with artificial biomedical time series.