A sliding window approach to detrended fluctuationanalysis of heart rate variability

Daniel Lucas Ferreira e Almeida (daniel.lucas@msn.com)Fabiano Araujo Soares (fabianosoares@unb.br)

Joao Luiz Azevedo de Carvalho (joaoluiz@pgea.unb.br)

Department of Electrical EngineeringUniversity of Brasilia, Brasilia–DF, Brasil

Introduction

I Heart rate variability (HRV): study of the nervous systemcontrol over circulatory function [1]

I Traditional analysis:I Time-domain, frequency-domain, geometrical techniquesI Generally require the signal to be stationaryI Analysis of long duration signals not recommeded

I Detrended Fluctuation Analysis (DFA) [2]I Quantifies long–term correlationsI Able to distinguish intrinsic HRV features from non-stationary external

trends [2,3]I Allows analyzing long duration signalsI Two coefficients:

Iα1: short-term correlationsIα2: long-term correlations

I DFA involves segmenting the signal into windowsI Detrended signal: discontinuities at the edges of each windowI At least one window will have fewer samples

I In this paper:I We propose a sliding window approach for DFAI Evaluation using random signals and real HRV signalsI Results are compared with the traditional approach

DFA: Traditional Calculation [2,3]

1) Obtain “integrated signal”: y(n) =n∑η=0

[RR(η)− RR],

where RR is the mean value the HRV signal RR(n)2) y(n) is segmented into windows of length lk. The “trendsignal”, yk(n), is a piecewise-linear approximation of y(n),obtained by replacing the samples of y(n) with the valuesobtained by linear fitting within each window3) Calculate “detrended signal”: ek(n) = y(n)− yk(n)

4) Calculate RMS approximation error: Ek =

√1

N

N−1∑n=0

|ek(n)|2

5) Repeat steps 2 through 4 for multiple values of lk6) Obtain f(x) = 20log10(Ek), where x = 20log10(lk) (Fig. 1)7) α1: angular coefficient of first-order approximation of f(x),for x s.t. 4 ≤ lk ≤ 168) α2: angular coefficient of first-order approximation of f(x),for x s.t. 16 ≤ lk ≤ N, where N is the length of RR(n)

10 15 20 25 30 35 40 45 5020

25

30

35

40

45

50

55

60

window length (dBhb)

det

ren

ded

sig

nal

RM

S va

lue

(dB

ms)

Figure 1: Calculating α1 and α2: RMS approximation error vs. window length

A Sliding Window Approach to DFA

I If N/lk /∈ Z, last window would have fewer samples (Fig. 2a)I Overlapping windows (Fig. 2b):

I Good: Makes all windows exactly lk-sample longI Bad: Does not eliminate discontinuities in detrended signal (Fig. 3a)

I Sliding window approach (Fig. 2c):I Replace step 2 with: ”For η = 1, 2, ...,N, the value of yk(η) is obtained

by: (i) taking a segment of y(n), with lk samples, centered around theη-th sample of y(n); (ii) least-square fitting a first-order polynomial,y′(n), to said segment; (iii) evaluating this polynomial at n = η, andmaking yk(η) = y′(η).”

I Results in a smooth trend signal (Fig. 3b)I No discontinuities in detrended signal (Fig. 3b)

(a)

(b)

lk lk lk lk lk lk-m

lk

lk

lk

lk

lk

lk

N samples

(c)

lk

yk(n)

Figure 2: Different approaches for segmenting the signal:(a) traditional approach; (b) overlapping window approach;

(c) sliding window approach

0 50 100 150 200 250 300−200

0

200

400

600

800

1000

integrated signal, y[n]trend signal, yk[n]detrended signal, ek[n]

Integrated SignalTrend SignalDetrended Signal

0 50 100 150 200 250 300−200

0

200

400

600

800

1000

integrated signal, y[n]trend signal, yk[n]detrended signal, ek[n]

(a)

(b)

heartbeat number (n)

RR in

terv

al d

ura

tio

n (m

s)RR

inte

rval

du

rati

on

(ms)

Figure 3: Integrated, trend, and detrended signals, obtained with:

(a) traditional approach; (b) sliding window approach

Evaluation: Random Signals

I Created with different power-law correlation characteristics:white noise, pink noise (1/f) and Brownian noise (1/f2)I Expected α for each of these is 0.5, 1.0 and 1.5, respectively [2]I 250 signals of each type; 4096 samples each

I Table 1: mean error and standard deviation for eachapproach

I Student’s t-test: α1 and α2 coefficients for sliding windowapproach are different from those for traditional approach(p < 0.05)

Table 1: Mean error (from expected value, α) and standard deviation for

α1 and α2 for traditional and proposed approaches

Evaluation: Real HRV Signals

I Signals obtained from Physiobank: 11 normals, 9 eliteathletes, 8 Chi meditators (during meditation and duringrest), 12 apneics, 7 with mild epileptic seizures [4,5,6]

I Each signal has ≈20,000 interbeat intervalsI Ectopic beats and false +/− were removed and corrected

using ECGLab [7]I Figure 4: α1 and α2 for each group (sliding window

approach)I Statistical tests: nonparametric Friedman ANOVA test, as

some groups tested negative for normality (Table 2)I Table 3: correlation (r) and regression (m) coefficients

between traditional and proposed approaches

alpha 1

alp

ha

2

alpha 1

alp

ha

2

Normal

Subjects with Apnea

Epileptics

0.40.40.4

0.4 0.8

0.8

0.8

0.8

1.2 1.2

1.2 1.2

1.61.6

1.6 1.6NormalAthletes

Meditators (at rest)Meditators (meditating)

NormalSubjects with ApneaEpileptics

Figure 4: α1 and α2 for HRV signals from different groups of subjects

(sliding window approach)

Table 2: ANOVA: traditional vs. sliding window approach (p values)

Table 3: Comparison between traditional and sliding window approaches

Conclusions

I While the results from the two approaches are quantitativelysimilar, extremely correlated, and with similar precision forboth α1 and α2, the proposed approach presented someadvantages, e. g., a better accuracy for white noise signals

I Both approaches seem able to visually differentiate betweenmost of the studied groups

References

[1] Malik M & Camm AJ. Heart Rate Variability, Wiley-Blackwell, 1995[2] Peng CK et al. Chaos 5(1):82–87, 1995[3] Leite FS et al. Proc BIOSIGNALS 3:225–229, 2010[4] Peng CK et al. Int J Cardiol 70(2):101–107, 1999[5] Penzel T et al. Comput Cardiol 27:255–258, 2000[6] Al-Alweel IC et al. Neurology 53(7):1590–1592, 1999[7] Carvalho JLA et al. Proc ICSP 6(2):1488–1491, 2002

Financial Support

PIBIC/CNPq; FAP-DF; PROAP/CAPES; PGEA/ENE/UnB

35th IEEE EMBC — Osaka, Japan — July 5, 2013 joaoluiz@pgea.unb.br http://www.pgea.unb.br/∼joaoluiz/