A sliding window approach to detrended fluctuationanalysis of heart rate variability
Daniel Lucas Ferreira e Almeida ([email protected])Fabiano Araujo Soares ([email protected])
Joao Luiz Azevedo de Carvalho ([email protected])
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 [email protected] http://www.pgea.unb.br/∼joaoluiz/
