1. T-61.181 – Biomedical Signal Processing Chapters 3.4 - 3.5.2 14.10.2004

2. Overview Model-based spectral estimation Three methods in more detail Performance and design patterns Spectral parameters EEG segmentation Periodogram and AR-based approaches

3. Model-based spectral analysis Linear stochastic model Autoregressive (AR) model Linear prediction

4. Prediction error filter Estimation of parameters based on minimization of prediction error e p variance

5. Estimation of model parameters Parameter estimation process critical for the successful use of an AR model Three methods presented Autocorrelation/covariance method Modified covariance method Burg’s method The actual model is the same for all methods

6. Straightforward minimization of error variance Linear equations solved with Lagrange multipliers (constraint a p T i=1) Autocorrelation/covariance method

7. Levinson-Durbin recursion Recursive method for solving parameters Exploits symmetry and Toeplitz properties of the correlation matrix Avoids matrix inversion Parameters fully estimated at each recursion step

8. The correlation matrix can be directly estimated with data matrices In covariance method the data matrix does not include zero padding, but the resulting matrix is not Toeplitz In autocorrelation method the data matrix is zero-padded Data matrix

9. Data matrices in detail

10. Modified covariance method Minimization of both backward and forward error variances Parameters from forward and backward predictors are the same Correlation matrix estimate not Toeplitz so the forward and backward estimates will differ from each other

11. Burg’s method Based on intensive use of Levinson- Durbin recursion and minimization of forward and backward errors Prediction error filter formed from a lattice structure

12. Burg’s method recursion steps

13. Performance and design parameters Choosing parameter estimation method Two latter methods preferred over the first Modified covariance method no line splitting might be unstable Burg’s method guaranteed to be stable line splitting Both methods dependant on initial phase

14. Selecting model order Model order affects results significantly A low order results in overly smooth spectrum A high order produces spikes in spectrum Several criteria for finding model order Akaike information criterion (AIC) Minimum description length (MDL) Also other criteria exist Spectral peak count gives a lower limit

15. Sampling rate Sampling rate influences AR parameter estimates and model order Higher sampling rate results in higher resolution in correlation matrix Higher model order needed for higher sampling rate

16. Spectral parameters Power, peak frequency and bandwidth Complex power spectrum Poles have a complex conjugate pair

17. Partial fraction expansion Assumption of even-valued model order Divide the transfer function H(z) into second-order transfer functions H i (z) No overlap between transfer functions

18. Partial fraction expansion, example

19. Power, frequency and bandwidth

20. EEG segmentation Assumption of stationarity does not hold for long time intervals Segmentation can be done manually or with segmentation methods Automated segmentation helpful in identifying important changes in signal

21. EEG segmentation principles A reference window and a test window Dissimilarity measure Segment boundary where dissimilarity exceeds a predefined threshold

22. Design aspects Activity should be stationary for at least a second Transient waveforms should be eliminated Changes should be abrupt to be detected Backtracking may be needed Performance should be studied in theoretical terms and with simulations

23. The periodogram approach Calculate a running periodogram from test and reference window Dissimilarity defined as normalized squared spectral error Can be implemented in time domain

24. The whitening approach Based on AR model Linear predictor filter “whitens” signal When the spectral characteristics change, the output is no longer a white process Dissimilarity defined similarly to periodogram approach The normalization factor differs Can also be calculated in time domain

25. Dissimilarity measure for whitening approach Dissimilarity measure asymmetric Can be improved by including a reverse test by adding the prediction error also from reference window (clinical value not established)

26. Summary Model-based spectral analysis Stochastic modeling of the signal Is the signal an AR process? Spectral parameters Quantitative information about the spectrum EEG segmentation Detect changes in signal