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

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

Sleep Stage 1

Sleep Stage 2

deep sleep

Sleep Stage 3

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 ECG-channel

“home-made” interactive MATLAB routine to retrieve RR-intervals

RR-Intervals

non-stationary time series (with drifts or “trends”)

1

1.5

2

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

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)

cumulative time serieshistogram of a simulated 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

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:

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:

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

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”

wake

lig

ht

slee

p

stage 1

stage 2

dee

p s

leep 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

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

Discriminating REM and NREM

REM Non-REM(including wake)

Discriminating REM and NREM

REM Non-REM(including wake)

Discriminating REM and NREM

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