A new Kalman filter approach for the estimation of high-dimensional time-variant multivariate AR models and its application in analysis of laser-evoked brain potentials
Introduction
All higher order sensory and cognitive functions depend on precise interactions between numerous brain areas, i.e. the brain's functionality critically depends on the interplay between segregation and integration. From a large-scale perspective, brain functions can be thought of as being implemented by a structure of local neural networks and global interconnected neuronal networks that are probably hierarchically organised. The underlying coordinated neuronal mass activities span functionally diverse and structurally widely distributed cortical and subcortical brain regions. The neuronal network metaphor is very useful here, as it helps to describe the time-varying localisations and dimensions of activated brain regions as well as the complexity changes induced by tv interaction patterns and synchronisation phenomena.
Functional imaging techniques now available for the recording of brain activity have much improved spatial and temporal resolution and provide the unique opportunity to gain new insights into the structure-function relationships within the human brain. However, interactions between activated brain areas cannot be directly measured, thus sophisticated detection and analysis tools as well as modelling that use measured brain activity as input data are required. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causation, and it can be seen as instantaneous mutual action among two (or more) independent neuronal entities (indicated by i ↔ j).
Important tools for analyzing interactions between neuronal activities can be derived from linear Granger Causality (GC) concepts (Granger, 1969). The underlying idea is to split up the analysis of a “two-way” interaction (i ↔ j) into two “one-way” causality analysis steps (i → j and j → i). To use this methodology, multivariate autoregressive (MVAR) models must be adapted to given time series derived from neural activity (e.g. EEG, MEG, fMRI). On principle, interactions and interaction patterns between neuronal entities are time-varying. Therefore, tvMVAR modelling must be used and appropriate algorithms are needed to estimate model parameters. An algorithm which estimates such parameters always involves a special concept about the time course of the parameters. A recursive least square (RLS) algorithm introduced by Moller et al. (2001) has been used for the computation of tvpGCI which stepwise builds up the Yule-Walker equations for stationary estimations of the AR parameters. A reliable tvpGCI analysis requires a reliable tvMVAR parameter estimation, especially for high-dimensional tvMVAR models (model dimension d) because, e.g. for EEG-based investigations, all relevant EEG channels should be included. The rationale for this requirement is the “hidden-source-dilemma,” because hidden sources, i.e. not-measured or non-measurable brain activity, can cause misleading interaction results. Therefore, at least all measured activities should be included in the calculation (see Discussion). To our knowledge the upper dimension bound for tvpGCI has been until now d = 24 (Weiss et al., 2008a). The problem of a model dimension which is too low can be bypassed by using a reduced number of electrodes (Weiss et al., 2008a) or by using modelled source activity, e.g. by using cortical current density waveforms derived from regions of interest after current density reconstruction based on high-dimensional EEG data (Astolfi et al., 2008). The first approach suffers in particular from the “hidden-source-dilemma”, and the second from the inverse problem and other restrictions. Therefore, a quantum jump with regard to the dimension of tvMVAR models is necessary to improve the reliability of tvpGCI and tvPDC (partial directed coherence) analyses. In this study an estimation method for parameters of high-dimensional tvMVAR models is introduced which provides such a breakthrough. This new method draws on experience with our Kalman filter technique introduced more than a decade ago (Arnold et al., 1998), which has been successfully used for bivariate tvAR models (Miltner et al., 1999). This methodology (Gelb, 1974, Haykin, 1986, Kalman, 1960) uses a random walk process to model the time course of the AR parameters.
The aim of this study is to examine the applicability of the RLS and Kalman approaches for tv interaction analyses which are based on high-dimensional EEG data. More specifically, a previous study (Weiss et al., 2008a) has shown that the RLS algorithm failed for dimensions higher than d = 24. Therefore, it was necessary to reduce the whole data set of 58 electrodes to 24 electrodes. We now provide a method with a general linear Kalman filter (GLKF) allowing use of the whole data set. Thus the aim here is actually twofold; first, to compare the applicability of the two principle algorithms RLS and GLKF and second, to compare results of 24 channel vs. 58 channel recordings of LEPs.
This paper is organised in the following way: (1) The RLS algorithm and the Kalman algorithm are compared regarding relevant differences, i.e. the parameters for tuning the adaptation properties of the algorithm and error function used for tvpGCI computation. (2) Both algorithms are applied to real EEG data using 24 EEG channels (d = 24) as a reduced electrode set out of a set of 58 electrodes, where the results based on RLS estimation were recently published by our group (Weiss et al., 2008a). The results are described. (3) These results are compared with those from a tvMVAR model of the dimension d = 58 (whole electrode set) adapted by the new Kalman technique. (4) Finally, the advantages of the Kalman approach for analysis of dynamic interaction patterns are discussed.
Section snippets
Data acquisition
The first data set was obtained by a simulated tvMVAR process. The model dimension was d = 20 to ensure the functioning of the RLS. The length of the process was N = 1800, the number of trials was K = 43 and the order was p = 10 to create a time series with properties equal to the LEP dataset where p = 10 was the result of an order estimation procedure. The additive d-dimensional white noise has the unit matrix as covariance matrix. The AR matrices are always zero matrices with the following exceptions:
Results
The results of the RLS and GLKF based tvMVAR estimations in the case of the simulated time series are depicted in Fig. 3. Only the computed GCI's showing dimensions 17 to 20 are presented to avoid small graphics. The GCI courses were computed using both tvMVAR estimators. The correct connections can be detected by both approaches and the behaviour of the sine function in the AR matrices has a visible effect on the GCIs. However, the RLS-computed GCI courses decrease faster in the interval
Discussion
The most important result of the study is that the GLKF algorithm allows the estimation tvpGCI of a 58-dimensional tvMVAR model without restrictions. This overcomes limitations of our RLS algorithm for the tvpGCI algorithm (Weiss et al., 2008a).
The comparison between the RLS and the Kalman approaches using a reduced set of 24 electrodes provides somewhat similar, but also somewhat different results. Hence, it can be seen that the choice of the tvMVAR estimator may influence the results of an
Acknowledgments
The preliminary results of this study were presented on the occasion of the sixth Biosignal Interpretation workshop (Yale University, USA, June 24th to 26th 2009). This study was in part supported by the German Research Council (DFG Wi 1166/9-1 Gamma; method's development), the Federal Ministry of Education and Research (Bernstein Group, 01GQ0703; application), and the COST Action BM0601 NEUROMATH. We thank E. Ahrens-Kley for language advice.
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