Detrended Fluctuation Analysis and Spectral Analysis of Heart Rate Variability
for Sleep Stage and Sleep Apnea Identification
T Penzel, JW Kantelhardt, HF Becker, JH Peter, A Bunde
Hospital of Philipps-University, Marburg, Germany
Abstract
In a systematic study we compared the performance of
spectral analysis and detrended fluctuation analysis
(DFA) to discriminate sleep stages and sleep apnea.
We investigated 14 healthy subjects, 33 patients with
moderate, and 31 patients with severe sleep apnea with
polysomnography.
Discriminance analysis was used on a person and
sleep stage basis to determine the best method for the
separation of sleep stages and sleep apnea severity.
Using spectral parameters 69.7% of the apnea severity
assignments and 54.6% of the sleep stage assignments
were correct, while using scaling analysis these numbers
increased to 74.4% and 85.0%, respectively. Changes in
heart rate variability are better quantified by scaling
analysis than by spectral analysis.
1. Introduction
Sleep as the absence of wakefulness and the missing
ability to react on external stimuli is regarded as a
unbiased test situation for the autonomic nervous system
[1]. Sleep is not just a constant state controlled by
metabolic needs for the body being at rest. Instead sleep
consists of different well defined sleep stages which
follow a well structured temporal order in normal
restorative sleep. Heart rate and heart rate variability vary
with the sleep stages, and their normal variability is
affected in sleep disorders. It has been shown that
autonomic activity changes from waking to sleep. Big
differences were found between non-REM and REM
sleep [2]. Sympathetic tone drops progressively from
wakefulness over sleep stage 1 to 4. In contrast REM
sleep was characterized by increased sympathetic tone
[3]. Parasympathetic tone increases from wakefulness to
non-REM sleep. Periods of wakefulness during sleep
were found to have an intermediate position between
non-REM and REM sleep [4].
Sleep apnea affects heart rate variability during sleep
described as cyclical variation of heart rate [5]. The
recording of cyclical variation of heart rate together with
snoring has been used in order to detect obstructive sleep
apnea with ambulatory recording devices [6]. It can be
assumed that the cyclical variation of heart rate can be
detected by spectral analysis if the appropriate frequency
range is investigated. The pattern of bradycardia and
tachycardia during apnea has been attributed to an
effective parasympathetic control of heart rate during
sleep [7] interrupted by sympathetic activation
accompanying the intermittent apnea-terminating
arousals.
Spectral analysis of heart rate variability is well
established and provides a quantitative evaluation of
sympathetic and parasympathetic activation of the
heartbeat [8]. Three major oscillatory components were
identified. The physiological interpretation of the very-
low-frequency (VLF) component (< 0.04 Hz) is still
discussed, the low-frequency (LF) component (0.04 –
0.15 Hz) reflects baroreflex sympathetic control of blood
pressure, and the high-frequency (HF) component (0.15 –
0.4 Hz) reflects respiratory rhythm and is believed to be
related to parasympathetic control of heart rate [9].
Detrended fluctuation analysis (DFA) method has
become a widely-used technique for the detection of
long-range correlations in noisy, non-stationary time
series. In the DFA method, long-range correlations
between interbeat intervals separated by several beats are
detected by investigating the scaling behavior of the
heartbeat fluctuations on different time scales
disregarding trends and non-stationarities in the data [10].
This study was performed on existing sleep recordings
to compare spectral analysis of heart rate and DFA in
their ability to distinguish sleep stages in normal and
sleep apnea subjects. We also wanted to see whether
sleep apnea severity can be distinguished using
parameters derived from spectral analysis and DFA and
which one performs better.
2. Methods
Sixty-four patients with symptoms of excessive
daytime sleepiness and arterial hypertension were
recruited. Patients had to be free of any cardiovascular
medication. Patients with apparent cardiac arrhythmias
were excluded. 33 patients with mild to moderate
obstructive sleep apnea with an apnea-hypopnea index
AHI < 40 events/hour and 31 patients with severe sleep
apnea AHI > 40 events/hour were selected for this study.
0276−6547/03 $17.00 © 2003 IEEE 307 Computers in Cardiology 2003;30:307−310.
Table 1. Results of sleep stage scoring and evaluation of
breathing are given for healthy persons and the two sleep
apnea groups. Body mass index (BMI), apnea-hypopnea
index (AHI) and total sleep time (TST) are listed.
healthy moderate
sleep apnea
severe
sleep apnea
subjects (n) 14 33 31
age (years) 33.0 ± 6.4 47.9 ± 9.1 50.0 ± 8.0
BMI (kg/m
2
) 21.7 ± 2.4 28.4 ± 4.2 33.7 ± 6.7
AHI (n/h) 0.6 ± 1.4 19.0 ± 8.0 65.1 ± 18.4
TST (min) 393 ± 37 361 ± 42 358 ± 49
wake (min) 64 ± 27 98 ± 45 103 ± 45
light sleep (min) 248 ± 39 235 ± 41 281 ± 40
deep sleep
(min)
58 ± 19 50 ± 28 11 ± 16
REM sleep
(min)
87 ± 23 75 ± 27 66 ± 21
In order to compare our results with normal subjects
14 healthy persons participated in the study. These
normal controls had no symptoms of sleepiness and no
sleep apnea.
All subjects underwent two subsequent nights of
polysomnography with EEG, EOG, EMG, recording of
oro-nasal airflow, respiratory movements, snoring,
oxygen saturation, and ECG as required for sleep studies
[11]. Sleep was evaluated according to Rechtschaffen and
Kales. For subsequent analysis some sleep stages were
grouped together. We distinguished 'light sleep' (stage 1
and 2), 'deep sleep' (stage 3 and 4), 'REM sleep', and
'wakefulness'.
Together with the other signals ECG lead II had been
digitized at 100 Hz for patients and 200 Hz for normal
subjects. The interbeat intervals were derived from the
ECG as RR intervals using an R-wave detector. The time
series were obtained for the entire duration of the sleep
recording. All annotated periods of wakefulness, light
sleep, deep sleep and REM sleep were analyzed
separately.
Based on discussions with our cardiologist on
arrhythmia related artifacts in interbeat time series we
chose the following practical criteria for automatic
preprocessing: sleep recordings from our patients were
excluded from our retrospective analysis, if more than
one percent of the interbeat intervals failed to meet the
following criteria: 0.33 s < interbeat interval < 1.5 s and
0.66 s maximum difference from the previous interbeat
interval. All recording epochs, where one sleep stage
persisted shorter than 3 minutes or had more than one
percent of RR intervals violating the criteria were
excluded. In addition, the violating intervals in accepted
epochs were also excluded, concatenating the remaining
parts of the series.
Fig. 1. The calculation of the spectral parameters using
FFT in 5 minute segments was performed separately for
the sleep stages. The summed spectra for each sleep stage
were plotted and used to calculate the spectral bands. The
figure depicts the heart rate and the sleep stage records as
well as the power spectra for light sleep, deep sleep,
REM sleep, and wake for a patient with moderate sleep
apnea. For the spectra the y-axis for light sleep has a
different scale in order to show the pronounced VLF peak
being characteristic for sleep apnea during light sleep.
In order to investigate 'clean' sleep stage effects on
heart rate variability without sleep stage transition effects
and non-stationarities associated with them we removed
the initial and the final 45 seconds of each sleep stage
period. Time domain and frequency domain measures
were calculated according to standard definitions [9].
Mean RR intervals and the standard deviation of all RR
intervals (SDNN) were calculated in the time domain.
For the calculation of the power spectra, the RR
interval time series was resampled at 3.41 Hz using linear
interpolation. Consecutive segments of 5 minutes (1024
points) inside each sleep stage were analyzed by spectral
analysis (FFT) separately. We calculated total power,
VLF (~ 0.04 Hz), LF (0.04 – 0.15 Hz), HF (0.15 –
0.4 Hz) and the ratio LF/HF for the individual sleep
stages separately.
The detrended fluctuation analysis is calculated as the
average over all segments and takes the square root to
obtain the fluctuation function F(t):
308
Ft
N
F
t
t
N
t
() ( ) .»
Ç
É
È
Ú
Ù
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Â
1
2
2
1
2
12
p
p
It is apparent that F(t) will increase with increasing t,
since the deviations from the fits will become larger for
larger segments. If the data are long-range power-law
correlated, F(t) increases, for large values of t, as a
power-law,
Ft t()~ .
c
When the fluctuation function F(t) is plotted as a
function of t on double logarithmic scales, the fluctuation
scaling exponent c can be determined by a linear fit. For
uncorrelated data, the scaling exponent is c = 0.5. For
short-range correlated data c is larger than 0.5 on small
scales t, but a crossover to c = 0.5 is observed on large
scales t. Power-law behavior with c @ 0.5 on large scales
t indicates long-range correlations in the data.
Fig 2. The top part illustrates the disturbed heart rate
together with the sleep stages of a patient with sleep
apnea. The lower part depicts the DFA in double-
logarithmic plot as a mean value for 20 subjects with
sleep apnea. The different slopes for the sleep stages can
be observed.
Differences between sleep stages with the classes 'light
sleep', 'deep sleep', 'REM sleep', 'wake' and differences in
sleep apnea severity with the classes 'normal', 'mild to
moderate, AHI < 40', 'severe, AHI > 40 events/hour',
were tested for two sets of dependent variables. The
dependent variables were mean RR intervals, SDNN,
VLF, LF, HF, and LF/HF in the first set and mean RR
intervals, SDNN, c
1
, c
2
calculated with DFA2 in the
second set. A multiple analysis of variance (MANOVA)
was applied for both sets. In order to check the
differences between the individual groups Bonferroni
tests were applied afterwards for both sets. Statistical
significance was stated for p < 0.05. The statistical test
was performed by SPSS version 10 (SPSS Inc, Chicago
Il. USA).
In order to compare the set of parameters derived by
spectral analysis with the set of parameters given by DFA
to determine their ability to discriminate between sleep
stages and between differences in severity of sleep apnea
we choose discriminance analysis. As parameters derived
by DFA we choose c
1
and c
2
calculated with DFA2 as
used in the MANOVA. From the spectral analysis we
choose the variables VLF, LF, HF, and LF/HF as used in
the MANOVA. The target variables for sleep were 'light
sleep', 'deep sleep', 'REM sleep', and 'wake' derived for
each subject and for apnea were 'normal', AHI < 40, and
AHI > 40 events/hour. The model derived by
discriminance analysis creates hyperplanes in the
hyperspace. The hyperplanes for the independent
variables were applied to predict the correct assignment
of each single subject into the corresponding class of
sleep and apnea – corresponding to the segment in the
hyperspace. The numbers of correct assignments were
calculated in percent.
3. Results
By applying discriminance analysis which separates
the hyperspace created by the dependent parameters with
hyperplanes we could prove that separation of sleep
stages was performed best using c
1
and c
2
derived by
DFA. 78.4% of sleep assignments were correct. If mean
RR intervals and SDNN were added, the correct
assignments increased to 85.0%. The assignments of
sleep stages based on spectral analysis parameters
resulted in 51.4%. If mean RR interval and SDNN were
added 54.6% of correct assignments were reached for
sleep stages.
Separation of apnea severity based on spectral
parameters performed better than based on DFA
parameters. 63.6% of apnea severity assignments were
correct. If mean RR intervals and SDNN were added to
the discriminance analysis model, the correct assignments
increased to 69.7%. The assignments of apnea severity
based on DFA parameters resulted in 60.1%. If mean RR
interval and SDNN were added 74.4% of assignments
were correct. This was slightly better than the spectral
parameter set together with time domain parameters.
309
If both classes were separated at the same time, and the
corresponding discriminant model was applied, DFA
analysis was better with 54.9% of correct assignments
compared to spectral parameters with 36.3% correct
assignments. In both cases, mean RR intervals and SDNN
were included in the model.
As a last test, all variables, derived by DFA, the
spectral parameters, mean RR intervals, and SDNN were
taken together. Then we achieved 84.1% correct
assignments for sleep, 72.9% for apnea and 56.1% for
separating both classes at the same time.
4. Discussion
This is the first study which systematically compared
the method of spectral analysis of heart rate variability
and DFA in a group with a defined disorder of high
interest. We used both methods to compare the ability to
discriminate sleep stages and sleep apnea severity. The
separation of sleep stages was performed best using the
two parameters derived by DFA together with time
domain measures. The separation of apnea severity was
also better using the parameters derived by DFA taken
together with time domain measures. If only spectral
parameters were compared to DFA parameters, they were
better in the case of apnea severity. The results indicate
that DFA derived parameters reflect heart rate regulation
properties which complement time domain measures and
their combination performs better than spectral measures
when we want to distinguish sleep stage and apnea
severity.
A limitation of our study is, that the age and body mass
index of our healthy control subjects (employees of the
hospital) is considerably lower than age and body mass
index of our patients. Both factors play a role in heart rate
regulation. Our patients had no other cardiac or
pulmonary disorder beside sleep apnea. These other
disorders were excluded prior to the study. The patients
with sleep apnea had an elevated office blood pressure at
the time of being recruited for this study.
Age and body mass index are very typical for sleep
apnea patients. As the influence of age and body mass
index on our results cannot be completely excluded this
presents a limitation of our study. This specific limitation
is a very common limitation to most studies on sleep
disordered breathing.
Our results do indicate that it might be possible to
improve heart rate analysis in such a way that it is
possible to recognize the severity of sleep apnea in rough
classes as had been used here and sleep stages in a
general way which distinguishes wake, light sleep, deep
sleep and REM sleep. In order to prove these hypotheses
prospective studies with implementations of the
discriminance functions must be performed on subjects
with sleep disordered breathing as well as in subjects
which suffer from other disorders affecting the autonomic
system.
Acknowledgements
Sleep recordings of healthy subjects were performed
as part of the Siesta project funded by the European
Commission (Biomed 2 BMH-97-2040). Sleep
recordings of patients were baseline studies with a study
supported by Roche (G-5001 and M-21222). We thank
Wilma Althaus and Dr. Regina Conradt for sleep stage
and respiratory event scoring. We thank Thomas Ploch
for statistical advice and providing the statistical tests.
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Address for correspondence.
Dr. Thomas Penzel
Hospital of Philipps-University
Depart. Internal Medicine, Div. of Pulmonary Diseases
Baldingerstr. 1, D-35033 Marburg, Germany
E-mail: Penzel@mailer.uni-marburg.de
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