Part of the PhD Thesis by Martin Staudacher

1. A brief PPT-Introduction: Using PDFA, a novel change-point detection method, to extract sleep stage information from the heart beat statistics during sleep Part of the PhD Thesis by Martin Staudacher

2. non-REM has NO suchlong time correlations as seen in REM-sleepand wakefulness Heart beat correlations & sleep stages A. Bunde, S. Havlin, J.W. Kantelhardt, T. Penzel, J.-H. Peter, K. Voigt, Phys. Rev. Lett. 85, 3736 (2000) time series analysis of RR-intervals with the Detrended Fluctuation Analysis (DFA) C.-K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Chaos 5, 82 (1995)

3. Sleep Stage 1 Sleep Stage 2 Sleep Stage 3 Sleep Stage 4 REM-Schlaf EEG-Scoring according to Rechtschaffen & Kales, examplary night: Use colour-coding of sleep stages: wake light sleep deep sleep

4. Data Acquisition (sleep research lab) • 18 data sets analyzed • whole night polysomnographies • from 9 healthy male probands (aged 20 - 30) • as reference: sleep stage scoring according to Rechtschaffen & Kales

5. RR-Intervals from digital ECG-channel “home-made” interactive MATLAB routine to retrieve RR-intervals

6. RR-Intervals non-stationary time series (with drifts or “trends”)

7. C.-K. Peng et al. (Chaos 5 (1995)): introduced to investigate the long-range correlation in DNA-base-pair sequences non-coding regions: long range correlations coding regions: short range correlations more than 100 publications in recent years, in many areas of science: Bioinformatics Meteorology Economy Geology and more Detrended Fluctuation Analysis (DFA)

8. time series (e.g. RR-intervals in a heart beat recording): calculate cumulated series by summing values (Interpretation: random walk) How to perform a DFA analysis

9. histogram of a simulated time series cumulative time series distribution of step sizes in a „random walk“ reached distance in a „random walk“

10. splitthedatapointsofthecumulative time seriesintowindowsof a fixedsizen

11. inside the windows: fit the cumulative series to a polynomial (the order of this polynomial fit is the order “ord” of the DFA) linear fit quadratic fit

12. calculate the deviation of the actual data from the polynomial fit curve and eliminates the „trends“ by subtraction: and finally plot this type of „variance“ as a function of the window size n in a doubly logarithmic scale, DFA-coefficient = slope in log-log-plot (see example next page)

13. Example: DFA-1 for artificially generated data relation between asymptotic behaviour of the autocorrelation function C(s) ~ s-γand the slope α of the FDA function in a log-log-plot: 30 000 random numbers with Gaussian distribution ~ exp(-x2)

14. „Weakness“ of the DFA: there is no time axis, since one analyses ALL data points in the time series simultaneously; thus it is not sensitive to changes in the underlying statistics (variance or correlation time, or both) that might ocurr during recording (example: sleep stage changes during whole night recording) thus modify DFA: progressively enlarge set of data point (from first to last point) difference DFA-PDFA: we now have a „time-axis“ use a fixed window size (but can repeat entire procedure for another) Progressive DFA (PDFA)

15. How to calculated the PDFA: • time series: • cumulative series (Interpretation: random walk) : • distribute first p data points into window of fixed size n:

16. inside each window do a polynomial fit of the cumulative time series : • calculate deviation between data and polynomial fit : • PDFA-coefficient = slope in log-log-plot

17. PDFA PDFA-function (depends on window size n !) Difference of DFA and PDFA schematically: DFA

18. Validation of the Method sensitive to change in correlation time OR to change in width of envelope function in artificially generated data same correlation time

19. Validiation of the Method Slope of PDFA curves (by numerical differentiation):

20. Can differences in correlation time be utilized (by means of the PDFA) to localize transitions from one sleep stage to the next ?

21. stage 1 stage 2 stage 3 stage 4 REM-sleep Colour coded “sleep map” wake light sleep deep sleep

22. examples Results of applying the new method to sleep data: • Detection of sleep transitions from „deeper to lighter“ sleep • Detection of short episodes of wakefulness • On-line differentiation between REM and NREM sleep

23. transitions 4 3 3  2 2  1 Transitions to lighter sleep

24. Section 1 non-gradual transitions from deeper to lighter sleep give rise to PDFA „events“ but NOT vice versa! (irrespective of foward or backward processing of data set )

25. short embedded periods of wake as steps Section 2

26. Discriminating REM and NREM

27. REM Non-REM (including wake) Discriminating REM and NREM

28. REM Non-REM (including wake) Discriminating REM and NREM

29. more general: „scaling parameter dispersion“ Why this difference ? • NREM has short correlation time: • light sleep (stage 1 & 2)~ 6 heartbeats (= points) • deep sleep (stage 3 & 4) ~ 3 heartbeats (= points) have scaled window size ACROSS typical correlation time(from3 to 50 points)

30. Scaling parameter dispersion: PDFA (scalingparameter = windowsize) movingwaveletanalysis (scalingparameter = waveletbasiswidth)

31. Conclusions • Reliable partioning of NREM/REM sleep possible • Abrupt changes from deeper sleep to lighter sleep are manifest as „PDFA events“ (i.e. pronounced steps in the PDFA curves) → interpretation • Validation of resultsby testing on artificially produced data sets with chosen change-points and by comparison with wavelet analysis