Page 1

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

Abstract

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

series.

1. Introduction

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 [3].

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 [1].

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 [4].

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 [2]. 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.

behavior, whereas

improve the statistical

2. Methods

2.1. Analyzed database

ISSN 0276−6574

405

Computers in Cardiology 2009;36:405−408.

Page 2

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

participation.

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 [6] 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.

2.2. Methodology

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):

∑

1

( )y k[ ( )x i] , 1,...,

k

ave

i

xkN

=

=−=

(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):

2

1

1

N

( )[ ( )

y k

( )]

N

n

k

F ny k

=

=−

∑

(2)

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) [7]:

3 if ( + ) < RR <

∆

2 if < RR ( +

µ

)

1,...,

1 if ( -

) < RR

0 if - < RR ( -

µ sd

)

i

i

i

i

i

µsd

µ sd

SiM

µ sdµ

∞

∆≤

==

∆≤

∞∆≤

(3)

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

[5]. 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

1

)(

−

≈

nsfn

was used

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

406

Page 3

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. Indexes

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 [9].

Concerning to the interpretation of VLF power

spectral component, different physiological mechanisms

have been proposed: physical activity, thermoregulation,

p=0.0004, and

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 [9].

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

LFn 57.78±13.3

HFn 27.96±8.9

LF/HF 2.388±1.353

SV, Survivor; CM, Cardiac mortality. Mean ± standard

deviation. n.s.: non significant

CM (n=30)

793±123.8

77.7±38.85

728±818

307±393

165.6±229

96.8±127.9

48.05±16

30.09±8.789

1.852±1.236

p

n.s.

n.s.

n.s.

0.0181

n.s.

n.s.

0.0004

n.s.

0.0428

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)

αHF

1.0607±0.2262 0.9115±0.2342

RR

αLF

1.2148±0.1622 1.1206±0.2314

αVLF 1.0939±0.117

αHF

0.3796±0.1187 0.3154±0.0775

tRR

αLF

0.313±0.1246

αVLF 0.2345±0.0774 0.2185±0.1361

αHF

0.8313±0.1011 0.8563±0.1024

|tRR| αLF

0.7177±0.0964 0.7272±0.082

αVLF 0.7156±0.0967 0.7381±0.0914

Sign

αHF

0.4671±0.0747 0.4309±0.0539

of tRR αLF

0.4579±0.0655 0.4232±0.0668

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

0.0010

0.0243

0.0103

0.0027

0.0017

n.s.

n.s.

n.s.

n.s.

0.0114

0.0076

0.0011

1.155±0.1399

0.2346±0.1315

with the symbolic

407

Page 4

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)

αHF

0.4171±0.105

tRRST αLF

0.4019±0.0977 0.3411±0.1055

αVLF

0.3647±0.0532 0.3931±0.0989

αHF

0.4113±0.1002 0.3548±0.0686

tRRSW αLF

0.407±0.0991

αVLF

0.3685±0.0607 0.4019±0.1076

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 [10].

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

tRR series.

tRRSW series

CM (n=30)

0.3623±0.0696

p

0.0078

0.0019

0.0193

0.0027

0.0009

0.0141

0.3406±0.1091

4. Conclusions

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.

Acknowledgements

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).

References

[1] Peng C-K, Havlin S, Stanley HE, Goldberger AL.

Quantification of scaling exponents and crossover

phenomena in nonstationary heartbeat time series. Chaos

1995;5(1):82-87.

[2] Ashkenazy Y, Ivanov PC, Havlin S, Peng C-K, Goldberger

AL, Stanley HE. Magnitude and sign correlations in

heartbeat fluctuations.

2001;86(9):1900-1903.

[3] Kaplan DT. Nonlinearity and nonstationarity: the use of

surrogate data in interpreting fluctuations. Frontiers of

Blood Pressure and Heart Rate Analysis 1997:1-13.

[4] Jospin M, Caminal P, Jensen E, Litvan H, Vallverdú M,

Struys M, et al. Detrended fluctuation analysis of EEG as a

measure of depth of anesthesia. IEEE Transactions on

Biomedical Engineering 2007;54(5):840-846.

[5] Baumert M, Wessel N, Schirdewan A, Voss A, Abbott D.

Scaling characteristics of heart rate time series before the

onset of ventricular tachycardia. Annals of Biomedical

Engineering 2007;35(2):201-207.

[6] Wessel N, Ziehmann C, Kurths J, Meyerfeldt U,

Schirdewan A, Voss A. Short-term forecasting of life-

threatening cardiac arrythmias based on symbolic

dynamics and finite time growth rates. Physical Review E

2000;61:733-739.

[7] Voss A, Schroeder R, Truebner S. Comparison of

nonlinear methods symbolic

fluctuation, and Poincaré plot analysis in risk stratification

in patients with dilated

2007;17:0151201-07.

[8] Laitio T, Jalonen J, Kuusela T, Scheinin H. The role of

heart rate variability in risk stratification for adverse

postoperative cardiac events. International Anesthesia

Research Society 2007;105(6):1548-1560.

[9] Guzzetti S, Rovere MTL, Pinna GD, Maestri R, Borroni E,

Porta A, et al. Different spectral components of 24 h heart

rate variability are related to different modes of death in

chronic heart failure. European Heart Journal 2005;26:357-

362.

[10] Beckers F, Verheyden B, Aubert AE. Aging and nonlinear

heart rate control in a healthy population. Am J Physiol

Heart Circ Physiol 2006;290:H2560-H2570.

Address for correspondence

José Fernando Valencia M.

UPC, ESAII, c/ Pau Gargallo 5, cp. 08028, Barcelona, Spain.

jose.fernando.valencia@upc.edu

Physical Review Letters

dynamics, detrended

cardiomyopathy. Chaos

408