#### Transcript PDFA

A brief PPT-Introduction: Using PDFA, a novel changepoint detection method, to extract sleep stage information from the heart beat statistics during sleep Part of the PhD Thesis by Martin Staudacher 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) non-REM has NO such long time correlations as seen in REM-sleep and wakefulness EEG-Scoring according to Rechtschaffen & Kales, examplary night: Use colour-coding of sleep stages: wake light sleep deep sleep Sleep Stage 1 Sleep Stage 3 Sleep Stage 2 Sleep Stage 4 REM-Schlaf 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 RR-Intervals from digital ECGchannel “home-made” interactive MATLAB routine to retrieve RRintervals RR-Intervals non-stationary time series (with drifts or “trends”) 2 1.5 1 Detrended Fluctuation Analysis (DFA) • C.-K. Peng et al. (Chaos 5 (1995)): introduced to investigate the longrange 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 How to perform a DFA analysis • time series (e.g. RR-intervals in a heart beat recording): • calculate cumulated series by summing values (Interpretation: random walk) histogram of a simulated time series cumulative time series distribution of step sizes in a „random walk“ reached distance in a „random walk“ • split the data points of the cumulative time series into windows of a fixed size n • 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 • 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) Example: DFA-1 for artificially generated data relation between asymptotic behaviour of the autocorrelation function C(s) ~ s-γ and the slope α of the FDA function in a log-log-plot: 30 000 random numbers with Gaussian distribution ~ exp(-x2) Progressive DFA (PDFA) • „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) How to calculated the PDFA: • time series: • cumulative series (Interpretation: random walk) : • distribute first p data points into window of fixed size n: • 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 : Difference of DFA and PDFA schematically: PDFA DFA Change in statistics from here on Change in statistics from here on Data set (length of RR-intervals) Data set (length of RR-intervals) Window size (local trend) ... (Steps in between) ... (Steps in between) ... (Steps in between) ... (Steps in between) PDFA-function (depends on window size n !) P[ n ] ( p ) 1 N p l 1 y (l ) y trend (l , n ) 2 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 Validiation of the Method Slope of PDFA curves (by numerical differentiation): Can differences in correlation time be utilized (by means of the PDFA) to localize transitions from one sleep stage to the next ? Colour coded “sleep map” deep sleep light sleep wake stage 1 stage 2 stage 3 stage 4 REM-sleep Results of applying the new method to sleep data: 1. Detection of sleep transitions from „deeper to lighter“ sleep 2. Detection of short episodes of wakefulness 3. On-line differentiation between REM and NREM sleep examples Transitions to lighter sleep transitions 4 3 32 21 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 ) Section 2 short embedded periods of wake as steps Discriminating REM and NREM Discriminating REM and NREM REM Non-REM (including wake) Discriminating REM and NREM REM Non-REM (including wake) 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 (from 3 to 50 points) more general: „scaling parameter dispersion“ Scaling parameter dispersion: PDFA (scaling parameter = window size) moving wavelet analysis (scaling parameter = wavelet basis width) 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 results by testing on artificially produced data sets with chosen change-points and by comparison with wavelet analysis