a brief ppt-introduction: using pdfa, a novel change- point detection method, to extract sleep stage...
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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
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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
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EEG-Scoring according to Rechtschaffen & Kales, examplary night:
Use colour-coding of sleep stages:
wake light sleep
Sleep Stage 1
Sleep Stage 2
deep sleep
Sleep Stage 3
Sleep Stage 4
REM-Schlaf
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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
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RR-Intervals
from digital ECG-channel
“home-made” interactive MATLAB routine to retrieve RR-intervals
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RR-Intervals
non-stationary time series (with drifts or “trends”)
1
1.5
2
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Detrended Fluctuation Analysis (DFA)
• 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
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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)
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cumulative time serieshistogram of a simulated time series
distribution of step sizes in a „random walk“
reached distance in a „random walk“
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• split the data points of the cumulative time series into windows of a fixed size n
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• 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
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• 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)
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Example: DFA-1 for artificially generated data
30 000 random numbers with Gaussian distribution ~ exp(-x2)
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:
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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)
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How to calculated the PDFA:
• time series:
• cumulative series (Interpretation: random walk) :
• distribute first p data points into window of fixed size n:
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• 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
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Difference of DFA and PDFA schematically:
DFA
D a ta se t (le n g th o f R R -in te r v a ls )
C h a n g e in s ta t is t ic s fro m h e r e o n
W in d o w s iz e ( lo c a l tre n d )
... (Steps in betw een)
... (Steps in betw een)
p
ltrendn nlyly
NpP
1
2
][ ),()(1
)( PDFA-function
(depends on window size n !)
PDFA
... (Steps in be tw een)
... (Steps in be tw een)
D a ta se t (le n g th o f R R - in te r v a ls )
C h a n g e in s ta t is t ic s fro m h e re o n
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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
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Validiation of the Method
Slope of PDFA curves (by numerical differentiation):
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Can differences in correlation time be utilized (by means of the
PDFA) to localize transitions from one sleep stage to the next ?
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Colour coded “sleep map”
wake
lig
ht
slee
p
stage 1
stage 2
dee
p s
leep stage 3
stage 4
REM-sleep
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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
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Transitions to lighter sleep
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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 )
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Section 2
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Discriminating REM and NREM
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REM Non-REM(including wake)
Discriminating REM and NREM
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REM Non-REM(including wake)
Discriminating REM and NREM
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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“
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Scaling parameter dispersion:
PDFA (scaling parameter = window size)
moving wavelet analysis (scaling parameter = wavelet basis width)
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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