The Use of Kalman Filter in Biomedical Signal Processing

Abstract

The Kalman Filter (KF) is a powerful tool in the analysis of the evolution of a dynamical model in time. The filter provides with a flexible manner to obtain recursive estimation of the parameters, which are optimal in the mean square error sense. The properties of KF along with the simplicity of the derived equations make it valuable in the analysis of signals. In this chapter an overview of the Kalman Filter, its properties and its applications is presented. More specifically, we focus on the application of Kalman Filter in the Electroencephalogram (EEG) processing, addressing extensions of Kalman Filter such as the Kalman Smoother (KS) in the time varying autoregressive (TVAR) model. The model can be written in a state-space form and the employment of KF provides with an estimation of the AR parameters which can be used for the estimation of the non-stationary signal. It is also demonstrated how these parameters can be used as input features of the signal in a clustering approach. The Kalman Filter is an estimator with interesting properties like optimality in the Minimum Mean Square Error (MMSE). After its discovery in 1960 (Kalman, 1960), this estimator has been used in many fields of engineering such as control theory, communication systems, speech processing, biomedical signal processing, etc. An analogous estimator has been proposed for the smoothing problem (Rauch et al., 1963), which includes three different types of smoothers, namely fixed-lag, fixed-point and fixed interval (Anderson & Moore, 1979; Brown, 1983). In this chapter we address the fixed interval smoother. The difference between the two estimators, the Kalman Filter and the Kalman Smoother, it is related on how they use the observations to perform estimation. The Kalman Filter uses only the past and the present observations to perform estimation, while the Kalman Smoother uses also the future observations for the estimation. This means that the Kalman Filter is used for on-line processing while the Kalman Smoother for batch processing. The derivations of these two estimators is presented in (Kay, 1993; Grewal & Andrews, 2001; Haykin, 2001). Both estimators are recursive in nature. This means that the estimate of the present state is updated using the previous state only and not the entire past states. The Kalman Filter is not only an estimator but also a learning method (Grewal & Andrews, 2001; Bishop, 2006). The observations are used to learn the states of the model. The Kalman Filter is also a computational tool and some problems may exist due to the finite precision arithmetic of the computers.