PDFA

Report
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
32
21
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

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