Detrended Fluctuation Analysis of Heart
Rate by Means of Symbolic Series
JF Valencia1,2, M Vallverdú1,2, R Schroeder3, A Voss3, I Cygankiewicz4,
R Vázquez5, A Bayés de Luna6, P Caminal1,2
1Dept ESAII, Universitat Politècnica de Catalunya, Barcelona, Spain
2CIBER de Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Spain
3Dept Medical Engineering and Biotechnology, University of Applied Sciences, Jena, Germany
4Institute of Cardiology, Medical University of Lodz, Lodz, Poland
5Hospital Universitario Puerta del Mar, Cádiz, Spain
6Catalan Institute of Cardiovascular Sciences, Barcelona, Spain
Detrended fluctuation analysis (DFA) has been shown
to be a useful tool for diagnosis of patients with cardiac
diseases. The scaling exponents obtained with DFA are
an indicator of power-law correlations in signal
fluctuation, independently of signal amplitude and
external trends. In this work, an approach based on DFA
was proposed for analyzing heart rate variability (HRV)
by means of RR series. The proposal consisted on
transforming consecutive RR increments to symbols,
according to an adapted symbolic-quantization. Three
scaling exponents were calculated, αHF, αLF and αVLF,
which correspond to the well known VLF, LF and HF
frequency bands in the power spectral of the HRV. This
DFA approach better characterized high and low risk of
cardiac mortality in ischemic cardiomyiopathy patients
than DFA applied to RR time series or RR increment
Fluctuations of time intervals between consecutive
heartbeats exhibit a complex dynamics, which is
influenced by the activity of many regulatory systems
interacting over a wide range of time or space scales
[1,2]. Even though the seemingly erratic behavior of
these fluctuations, the heart rate variability (HRV) can be
approximately described by a power law over time scales
of tens of minutes to days, indicating long-term
correlations between heartbeats .
The study of heart rate time series using detrended
fluctuation analysis (DFA) has been shown to be a useful
tool for diagnostic in patients with cardiac diseases .
One advantage of DFA over conventional methods is that
it permits the detection of long-range correlations
embedded in a seemingly non-stationary time series.
Indeed, DFA considers fluctuations only from local linear
trend, which makes it insensitive to spurious correlations
introduced by external nonstationary trends .
Previous studies [1,5] have indicated that correlation
properties showed a crossover when several time scales
were analyzed. Indeed, for healthy subjects and short-
time scales, fluctuations were quite Brownian noise
indicating random walk-like
fluctuations approached to 1/f behavior for long-time
scales. In contrast, it was indicated that correlation
properties of fluctuations can be altered by certain cardiac
diseases, showing a quite random behavior for short-time
scales, whereas fluctuations becomes smoother (like
Brownian noise) as the time scales become larger.
Variability of heart rate fluctuations has been also
studied by using RR increment series (tRR series) and,
particularly, by using series constructed with the
magnitude and the sign of tRR series . This study has
shown that DFA scaling exponents belonging to sign
series allowed statistically differentiate between healthy
subjects and patients with heart failure, obtaining similar
or inclusive better results in comparison with those
calculated over the original RR series.
In this work, an alternative methodology was
introduced in order to
differentiation between risk groups of suffering cardiac
death, by applying DFA over tRR series. The
methodology considered a symbolic transformation of
tRR series by means of an alphabet with four symbols.
Results were compared with those calculated over
original RR series and, series corresponding with
magnitude and sign of tRR series.
improve the statistical
2.1. Analyzed database
Computers in Cardiology 2009;36:405−408.
Patients from MUSIC (MUerte Subita en Insuficiencia
Cardiaca, Sudden Death in Heart Failure) study were
analyzed in the present work. A total of 222 patients
(63.2±0.56 years, 86.9% male) with ischemic dilated
cardiomyopathy (IDC) were enrolled in the present study.
Patients were followed for three years. The inclusion
criteria were: sinus rhythm, symptomatic chronic heart
failure with New York Heart Association functional class
(NYHA) II or III, and ischemic etiology of heart failure.
Age-matched IDC patients were studied considering
cardiac mortality as end-point. The analysis considers 30
patients that suffered cardiac mortality (CM) due to
sudden cardiac death, progressive heart failure or
myocardial infarction, as a high risk group and 192
survivor (SV) as a low risk group. The MUSIC study was
approved by the Ethical Committee of the institution and
all subjects gave their written informed consent before
The RR series, intervals between consecutive heart
beats, were obtained from 24h ECG-Holter recordings
with a sampling frequency of 200 Hz (Spiderview
recorders, ELA Medical, Sorin Group, Paris). An
adaptive filter  was applied to the RR series in order to
replace ectopic beats and artifacts by interpolated RR
intervals. Indeed, the level of interpolated beats related to
the total number of RR intervals was less than 1.5%.
Therefore, a possible alteration of the results due to the
filter procedure can be discarded. A common length of
60000 beats was selected for all the RR series.
a) Detrended fluctuation analysis
In order to calculate the scaling exponents with DFA,
a given series x(i), with 1≤i≤N being N the length of the
series, is firstly integrated (1):
( )y k[ ( )x i] , 1,...,
where xave is the average of the series x(i). Then, the
integrate time series is divided into boxes of equal length,
n, and in each box, a least square line is fit to the data.
The y coordinate of the straight line segments is denoted
by yn(k). Next, the integrated time series, y(k), is
detrended by subtracting the local trend, yn(k), in each
box. After that, the root-mean-square fluctuation of the
integrated and detrended time series is obtained (2):
( )[ ( )
F ny k
where F(n) represents the average fluctuation as a
function of the box size, n. A linear relationship on a
double log graph between F(n) vs. n indicates the
presence of scaling in the series. The fluctuation can be
quantified by means of the slope of the line (the scaling
exponent α), relating log F(n) to log n. Values of
0<α<0.5 are associated with anti-correlation where large
and small values of the time series are likely to alternate.
The value α=0.5 is related with Gaussian white noise
indicating an uncorrelated behaviour. Values of 0.5<α≤1
indicate persistent long-range correlations. A special case
is α=1 and corresponds to 1/f noise. Values of 1<α≤1.5
are associated to stronger long-correlations, differing
from power law [1,5].
In the present study, heart rate variability was
analyzed by means of DFA applied to: a) RR series; b)
RR increment series (tRR); c) magnitude of the tRR
series; d) sign of the tRR series; e) symbolic series
obtained from tRR series. The criterion used to
transform tRR series into symbols is given in (3) :
3 if ( + ) < RR <
2 if < RR ( +
1 if ( -
) < RR
0 if - < RR ( -
where M, µ and sd are, respectively, the length, the mean
value and the standard deviation of the series. Two
alternative ways were considered in order to determine
these parameters: a) tRRST, symbolic transformation
applied to the total 24-hour tRR series; b) tRRSW,
tRR series were divided into windows of NW=1000
beats without overlapping
transformation was applied to each one of the windows.
Several regions of scale invariance were considered,
corresponding approximately to the well known VLF
[0.003–0.04Hz], LF [0.04–0.15Hz] and HF [0.15–0.4Hz]
frequency bands in the power spectral of HRV. In order
to relate frequency values fn in Hertz and the segment size
n of DFA, the rough approximation
. The scaling exponents of the DFA taken into account
and related to these regions were: αHF (4≤n<8), αLF
(8≤n<30) and αVLF (30≤n≤100), where s has been
approximated to 0.8s.
and the symbolic
b) Time and frequency domain analysis
Time and frequency domain measures were taken
into account. Mean (MRR) and standard deviation (SRR) of
the RR series were calculated, as time domain measures.
In order to obtain the frequency domain measures, the
series of 60000 beats were analyzed according to frames
of 300 beats with an overlap of 50%. Subsequences were
interpolated and re-sampled at 5 Hz. Power spectrum was
estimated over subsequences using an autoregressive
approach (Burg method). The model order was a-priori
assigned and equal to 12. The following measures were
calculated: total power (Ptot); power in the high frequency
band (HF); power in the low frequency band (LF); power
in the very low frequency band (VLF); LF and HF in
normalized units (LFn and HFn); and the LF/HF ratio.
c) Statistical analysis
A statistical analysis based on ANOVA test was
applied on each defined index α. Indeed, Levene’s test
for homogeneity of variance was used to confirm
homoscedasticity were statistically analyzed by applying
U Mann-Whitney test. A significance level p<0.05 was
considered for comparing statistically the risk groups.
A discriminant linear function was built on each one of
the indexes in order to classify the subjects. The
sensitivity (Sen) and specificity (Spe) were taken into
account in this statistical analysis. To test the association
between the power spectral and DFA exponents, a two-
tailed Pearson correlation coefficient (r) was calculated.
α that no fulfil
3. Results and discussion
3.1. Time and frequency domain analysis
Table 1 contains the mean and standard deviation of
time and frequency domain measures, as well as the
significance level of the statistical classification of
subjects in their respective risk groups: survivor (low risk
group) and cardiac death (high risk group). In time
domain, the indexes MRR and SRR were not able to
statistically differentiate between the two risk groups,
however their mean values tended to be higher in SV than
in CM. In frequency domain, VLF, LFn and LF/HF
showed significant differences when risk groups were
statistically compared (p=0.0181,
p=0.0428, respectively). The mean values of VLF, LFn
and LF/HF were lower in high risk group than in low risk
group, suggesting a reduction in the sympathetic branch
activity of the autonomic nervous system in CM group.
The best diagnostic criteria (sensibility=53.4% and
specificity=63.5%) were obtained with LFn component.
Chronic heart failure is characterized by a high
sympathetic drive [8, 9]. Indeed, spectral analysis of the
RR series would be reasonably expected to manifest
predominantly LF component, but our results have
revealed a decreased LFn component in high risk group
compared with low risk group. However, the
interpretation of a reduced LFn in chronic heart failure
patients is still an open question including a depressed
sinus node responsiveness, central abnormality in
autonomic modulation, limitation in responsiveness to
high levels of cardiac sympathetic activation, depressed
baroreflex, and increased chemoreceptor sensitivity .
Concerning to the interpretation of VLF power
spectral component, different physiological mechanisms
have been proposed: physical activity, thermoregulation,
rennin-angiotensin-aldosterone system, slow respiratory
patterns, and parasympathetic mechanism. In this sense,
the obtained VLF behaviour could have been influenced
by a reduced physical activity in the patients who were
more ill. Similar results were reported in .
Table 1. Time and frequency domain measures
Measures SV (n=192)
MRR [ms] 844.3±142.6
SRR [ms] 92.29±37.75
Ptot [ms2] 1269±1471
VLF [ms2] 563±568
LF [ms2] 289.4±340
HF [ms2] 167.8±288.5
SV, Survivor; CM, Cardiac mortality. Mean ± standard
deviation. n.s.: non significant
3.2. Detrended fluctuation analysis
The values of the mean and standard deviation of DFA
scaling exponents measured over RR series, tRR series,
magnitude and sign of tRR series are shown in Table 2.
A better significance level can be observed in αHF and
αLF using tRR series than using RR series, whereas
αVLF scaling exponent showed a reverse behavior in RR
series. Magnitude of tRR series has not presented
scaling exponents able to differentiate both risk groups.
Sign of tRR series exhibited scaling exponents with p
value slightly better than those calculated over the
original RR series or even the tRR series.
Table 2. DFA in RR series
Series Index SV (n=192)
αVLF 0.2345±0.0774 0.2185±0.1361
αVLF 0.7156±0.0967 0.7381±0.0914
of tRR αLF
αVLF 0.4386±0.0530 0.4763±0.0845
Mean ± standard deviation. n.s.: non significant
Table 3 contains the values of mean and standard
deviation of DFA scaling exponents calculated over
tRR series transformed to symbols by applying the two
proposed algorithms: tRRST and tRRSW. Comparing
tRR with tRRST and tRRSW series, a better statistical
difference was obtained
transformation, especially for αVLF that is statistically
significant (p=0.0193 and p=0.0141 for tRRST and
CM (n=30) p
with the symbolic
tRRSW, respectively). The mean values of the scaling
exponents of the symbolic transformed tRR series
(Table 3) have similar tendencies that those observed in
the original RR and tRR series (Table 2). Indeed αHF
and αLF showed lower values in high risk group than in
low risk group, whereas αVLF showed a reverse behavior.
It can be observed in Table 3 that tRRSW obtains better
statistical significances than tRRST. The reason should
be due that parameters µ and sd given in (3) are
adaptively calculated in tRRSW series and therefore the
correlation properties are better characterized by this
transformed series. Furthermore,
presented higher statistical differences than sign of tRR
series, when risk groups were compared. It should be due
that sign of tRR series is equivalent to a symbolic
transformation using two symbols, whereas tRRSW used
a codification with four symbols. A linear combination of
αHF, αLF and αVLF in tRRSW statistically classified with
a sensibility=70.0% and specificity=68.2%, by using
leaving-one-out cross-validation method.
Table 3. DFA measures in symbolic RR series
Series Index SV (n=192)
Mean ± standard deviation.
The scaling exponent αLF of tRRSW highly
correlated with LFn, LF/HF and HF with r=0.894
(p<0.0005), r=0.834 (p<0.0005) and r=0.613 (p<0.0005),
respectively. The last correlation suggests that a
relationship seems to exist between respiration and αLF,
but the highest correlation is between αLF and the
sympathetic and vagal activity done in LFn. On the other
hand, αHF and LFn correlated with r=0.613 (p<0.0005),
whereas non spectral power components correlated with
αVLF. Similar results were obtained in .
The mean values of DFA indicate anticorrelated
behaviour in tRR series with or without transformation
to symbols. However, tRRST and tRRSW series
presented less anticorrelation compared with original
This study uses an approximation of DFA based on
symbolic dynamics for analyzing heart rate variability by
means of RR series which are characterized by long-
range correlations. This approach was compared with
other different proposals that involve RR increment
series, magnitude and sign of those RR increment series.
It seems that an adequate symbolic transformation of the
RR series allowed DFA scaling exponents to better
identify different correlation properties between the
studied cardiac risk groups.
This work was supported within the framework of the
CICYT grant TEC2004-02274, the research fellowship
grant FPI BES-2005-9852 from the Spanish Government.
The MUSIC trial was coordinated by University Hospital
St. Pau, Barcelona (I. Carlos III – Spain network group).
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Address for correspondence
José Fernando Valencia M.
UPC, ESAII, c/ Pau Gargallo 5, cp. 08028, Barcelona, Spain.
Physical Review Letters