Scaling properties of excursions in heartbeat
dy-namics
I. Reyes-Ram´ırez1,2 and L. Guzm´an-Vargas1
1
Unidad Profesional Interdisciplinaria en Ingenier´ıa y Tecnolog´ıas Avanzadas, Instituto Polit´ecnico Nacional, Av. IPN No. 2580, L. Ticom´an, M´exico D.F. 07340, M´exico
2 Escuela Superior de F´ısica y Matem´aticas,
Instituto Polit´ecnico Nacional, Edif. No. 9 U.P. Zacatenco, M´exico D. F., 07738, M´exico
PACS 87.19.Hh– Cardiac dynamics
PACS 89.20.-a– Time series
PACS 89.75.Da– Scaling and correlations
Abstract.- In this work we study the excursions, defined as the number of beats to return to a local
mean value, in heartbeat interval time series from healthy subjects and patients with congestive
heart failure (CHF). First, we apply the segmentation procedure proposed by Bernaola-Galv´an
et al. [1], to nonstationary heartbeat time series to identify stationary segments with a local mean value. Next, we identify local excursions around the local mean value and construct the distributions to analyze the time organization and memory in the excursions sequences from the whole time series. We find that the cumulative distributions of excursions are consistent with a stretched exponential
function given by g(x) ∼ e−aτb, with a = 1.09±0.15 (mean value ± SD) andb = 0.91±0.11
for healthy subjects and a= 1.31±0.23 andb= 0.77±0.13 for CHF patients. The cumulative
conditional probability G(τ|τ0) is considered to evaluate ifτ depends on a given intervalτ0, that
is, to evaluate the memory effect in excursion sequences. We find that the memory in excursions sequences under healthy conditions is characterized by the presence of clusters related to the fact that large excursions are more likely to be followed by large ones whereas for CHF data we do not observe this behavior. The presence of correlations in healthy data is confirmed by means of the detrended fluctuation analysis (DFA) while for CHF records the scaling exponent is characterized by a crossover, indicating that for short scales the sequences resemble uncorrelated noise.
methods from statistical mechanics and nonlinear dynamics have revealed that some scaling structures and complex variability observed under healthy conditions are altered by disease and aging [3, 6–11]. A recent study to detect local stationary segments of heartbeat in-terval time series [1] revealed that the distribution of these stationary segments follows a power law behavior and the scaling exponent, which characterizes the distribution, is the same for healthy and heart failure groups. These findings indicate that this scale-invariant structure is not a simple consequence of long range correlations [1] displayed by the system and that the nonstationarity is an intrinsic property related to neuroautonomic control and external perturbations. From a physiologic point of view, the presence of local stationary segments can be understood as the capability of the system to preserve an approximated constant value but for a limited period of time. It is also argued that, according to the homeostasis principle, biological systems tend to maintain a constant output in spite of continual perturbations [12]. Indeed, fluctuations or variations around certain value are a common characteristic of some physiologic signals in agreement with the mentioned home-ostasis principle. Statistical analysis of excursions, which are defined as the period of time employed by a walker to return to its average value, may be important to the assessment of the capability of the system to preserve a mean output value. The problem of the first return time has been studied in contexts like financial index, correlated-disordered chains, intermittency, seismic activity and simulated noise [13–18]. There are also important results of zero crossing probabilities for Gaussian long-term correlated data [19]. In this work we focus our attention in the statistical properties of local excursions within the context of nonstationary heart rate time series. We observed that excursions for stationary segments lead to a class of exponential distributions which are characterized by different parameters for healthy subjects and heart failure patients. The paper is organized as follows: In Section II, we briefly describe the segmentation algorithm and the detrended fluctuation analysis method. In section III, the results of the statistics of excursions are described. Finally, some concluding remarks are given in Section IV.
Methods. –
The segmentation algorithm. We briefly explain the segmentation method proposed by Bernaola-Galv´an et al. [1] to detect stationary segments in a nonstationary signal.
• A slider pointer is considered to calculate the statisticst=µr−µl
SD , whereµrandµlare
the mean of the values on the right and left, respectively. SD is the pooled variance give bySD = ((Nl−1)s2l + (Nr−1)s2r)/(Nl+Nr−2)
1/2
(1/Nl+ 1/Nr)1/2, where
sl andsr are the standard deviation of the two sets, and Nland Nr are the number of points in the two sets.
• The quantityt is used to separate two segments with a statistically different mean. A significance level is applied to cut the series into two new segments (typically set to 0.95), as long as the means of the two new segments are significantly different from the mean of the adjacent segments [1, 20].
• The process is applied recursively until the significance value is smaller than the thresh-old, or the length of the new segment is smaller than a minimumℓ0. For details in the implementation of the algorithm see [20].
the spectral exponent and is related to the correlation exponent byγ= 1−β. One method which is particularly suited to analyze nonstationary signals is the detrended fluctuation analysis (DFA), which was introduced to quantify long-range correlations in the heartbeat interval time series and DNA sequences [21, 22]. The DFA is described as follows: First, we integrate the original time series to get, y(k) = Pk
i=1[x(i)−xave], where xave is the global mean; the resulting series is divided into boxes of size n. For each box, a straight line is fitted to the points, yn(k). Next, the line points are subtracted from the integrated series,y(k), in each box. The root mean square fluctuation of the integrated and detrended
series is calculated by means ofF(n) =
q
1 N
PN
k=1[y(k)−yn(k)] 2
, this process is taken over several scales (box sizes) to obtain a power law behavior F(n)∼nα, withαan exponent, which reflects self-similar and correlation properties of the signal. It is known thatα= 0.5 corresponds to white noise (non correlated signal), α = 1 means 1/f noise and α = 1.5 represents a Brownian motion.
Results. – We analyze heartbeat interval time series from two groups: 16 healthy subjects and 11 patients with congestive heart failure [23]. We considered RR interval sequences with approximately 3×104beats corresponding to 6 diurnal hours of ECG records. This database is an extended set of records which have been used in previous studies [10, 11, 23].
Distributions of local excursions. First, we apply the segmentation procedure to inter-beat interval time series to detect stationary segments. Fig. 1a shows a representation of the segmentation procedure to the data from one healthy subject. In our study we usedℓ0= 50 as the minimum segment length. As it was reported by Bernaola-Galv´an and coworkers [1], the cumulative distribution of stationary segments with local mean follows a power law,
G(> ℓ)∼ℓ−δ, (1)
with δ ≈ 2.2 for healthy and heart failure groups [1]. Next, we calculate the excursions return times for each stationary segment with respect to the local mean. More specifically, we identify an excursion with sizeτ ifxj > xandxj+τ > xwhilexi> xforj < i < j+τ or conversely xj < x and xj+τ < x while xi < x forj < i < j+τ (see Fig. 1c). Before constructing the distributions from all the segments, we verify if the distribution from one segment corresponds to the same distribution of any other segment. In Fig. 2a representative cases of excursion distributions from segments with small, intermediate and large length
ℓ are depicted. We observe that for the large segment the statistics is better than for distributions from intermediate or small length, but in the three cases a similar functional form is identified. We formulate the question whether any pair of stationary segments, regardless of their length, lead to distributions with the same functional behavior. The Kolmogorov-Smirnov (K-S) test is used to accept or reject the null hypothesis that any pair of segments have the same distribution. We computed the p−valuebetween distributions from all the pairs of stationary segments detected in the signal.
Fig. 1: Representative case of the segmentation procedure for a (a) nonstationary signal from a healthy subject. (b) As in (a) but for a magnification of the interval in (a). (c) Magnification of (b) to illustrate the excursion identification.
by
g(τ)∼e−aτb, (2)
whereaand b are constants. Specifically, we geta= 1.09±0.15 (mean value±SD) and
b= 0.91±0.11. In Fig. 3d, the results for the CHF patients are presented. As in the cases of the healthy group, we observe that the distributions show a slightly different decaying rate. After normalizing the excursions (Fig. 3e), the distributions are very similar for large values whereas for short excursions they are slightly different. The best fit is also consistent with a stretched exponential form, witha= 1.30±0.24 andb= 0.77±0.13. Figs. 3c and 3f show the log-linear plot of the data in Figs. 3b and 3e and the distributions of shuffled records. As we can see, the original data look different than the shuffled cases which are consistent with an exponential distribution. Moreover, we calculate the characteristic scale associated to the stretched exponential distribution given by hτi= a−1/b
b Γ(1/b), with Γ the Gamma function. Note that for b = 1, the mean value of an exponential function is recovered. For healthy data we get hτi = 0.95±0.11, while for CHF hτi= 0.84±0.12, indicating a fast decay under pathologic conditions. An important difference is observed between the in-dividual decaying rates when both groups are compared (p−value= 0.02 by Student’s test).
Correlations of excursions. In order to get a better answer to the question whether the observed distributions of excursions are related to the presence of correlations, we study the memory in the time organization of excursions. Fig. 4 shows representative time evolution of excursions from one healthy subject (Fig. 4a) and one CHF patient (Fig. 4b). We observe that both sequences look different; particularly because of the presence of clusters in healthy data as an indication of memory. In contrast, CHF data is characterized by the presence of many periods with small size excursions and low density of large excursions.
0 0.2 0.4 0.6 0.8 1
p-value
0 0.2 0.4 0.6 0.8 1
Cumulative Distribution
0.2 0.4 0.6 0.8 1
p-value
Healthy Heart Failure
100 101 102
Excursion size τ
0.01 0.1 1
Cumulative Distribution
l=120 l=420 l =900
b) c)
a)
Fig. 2: (a) Cumulative distribution of excursion sequences obtained from segments with different
length ℓ. We show the cases for ℓ= 120,420,900 from one healthy subject. (b),(c) Cumulative
distribution ofp−valuesfrom the application of the Kolmogorov-Smirnov test to excursion
distri-butions from all the pairs of stationary segments detected in healthy and CHF subjects. We find that, at 5% level of significance, for most of the cases we cannot reject the null hypothesis that any pair of segments have the same excursion distribution, suggesting that we can pool data from all
the segments and improve the statistics. Note that a rapid decay in the distribution of p−values
would indicate that the null hypothesis is rejected for most of the cases. Here, we show the 5% threshold as the grey zone.
the intervalτ0 was followed by an excursion bigger thanτ. For sequences with no memory,
G(τ|τ0) is expected to be independent ofτ0, that is, the order in the sequence of excursions is not correlated. IfG(τ|τ0) shows changes as function ofτ0, it indicates thatτ0”influences“ the next excursion size. For shuffled data,G(τ|τ0) must be exponential and independent of
τ0. To test the effect of τ0 onG(τ|τ0), we consider two intervals for τ0: τ0s and τ0l which correspond to small and large values ofτobtained from a six equal-size partition of the entire interval in increasing order, that is, each interval contains one-sixth of the total number of excursions. Due to the poor statistics for one subject, all the normalized excursions for different subjects constitute an ensemble of excursions to generate the mentioned partition. The results for G(τ|τs
0) and G(τ|τ0l) are presented in Fig. 5. For healthy subjects, the conditional cumulative probability forτs
0 tends to be similar to the probability forτ0l in the range of small excursions whereas for large excursions G(τ|τs
0) < G(τ|τ0l), indicating that large excursions tend to follow large excursions. For heart failure group, we observe that
G(τ|τs
0) is close toG(τ|τ0l) for small excursions whereas for large ones the probability forτ0s is slightly bigger than the probability forτl
0, indicating that large excursions are more likely to be preceded by a small one.
Detrended fluctuation analysis. We use the DFA method to verify the presence of long range memory in the excursion sequences. As shown in Fig 6, for healthy data the scaling behavior along at least for two decades is characterized by the average exponent
α= 0.64±0.04, while for CHF group the scaling is characterized by two regimes; over short scales (4 ≤n ≤102) the average exponent is α
s = 0.563±0.014 whereas for large scales (102 ≤n≤103) the value isα
[image:5.595.111.464.91.342.2]100 101
τ
10-5 10-4 10-3 10-2 10-1 100
Cumulative Distribution
10-1 100 101
τ/σ
Healthy
0 3 6 9 12
τ/σ
100 101
τ
10-5 10-4 10-3 10-2 10-1 100
Cumulative Distribution
10-1 100 101
τ/σ
Heart Failure
0 3 6 9 12
τ/σ
a)
Shuffled Shuffled
b)
c)
[image:6.595.167.459.121.416.2]d)
e)
f)
Fig. 3: Cumulative distributions of excursions for 16 healthy subjects and 11 CHF patients. (a) and (d) Log-log plot for distributions of excursions. (b) and (e) Log-log plot for distributions of normalized excursions. (c) and (f) Log-linear plot for distributions of normalized excursions. We also show the cases of shuffled records, that is, for each segment we shuffled the interbeat time points and then the distribution of excursions was constructed by pooling the data from all segments. For
clarity, the distributions were scaled by a factor 1/10.
2 4 6 8 10
τ/σ
0 200 400 600 800 1000
2 4 6 8 10
τ/σ
a) Healthy
b) Heart Failure
Normalized Excursions
[image:6.595.231.401.544.662.2]100 101
τ/σ
Small τ0 Large τ0
Heart Failure
10-1 100 101
τ/σ
10-6 10-5 10-4 10-3 10-2 10-1 100
G(
τ|τ0
)
Small τ0 Large τ0
Healthy
[image:7.595.179.376.101.283.2]Shuffled Shuffled
Fig. 5: Conditional cumulative distribution for healthy and CHF groups. In this plotτ0 represents
the interval obtained from a six equal-size partition of the entire interval in increasing order. For
a comparison, we used the smallest and largest interval for τ0 (see text for details). The lower
plots represent the distributions obtained from shuffled data (×and + symbols represent small and
largeτ0, respectively). For both groups, we observe that the data can be fitted by an exponential
function. We have scaled the distributions by a factor 1/10.
bigger than 0.5, indicating the presence of positive correlations. A significant difference is observed between αsand αl for CHF data (p-value<10−3 by the Student’s test). We also remark that for short scales, the average DFA exponent for the healthy group is slightly bigger than the corresponding scaling DFA exponent of CHF group, confirming that the excursion sequences under healthy conditions are more correlated than for pathologic data. In order to get a better evaluation of the presence of the crossover in the scaling exponent, we extract both scaling exponents (αs and αl) for healthy data. Fig. 7 shows the scatter plot of the scaling exponents αs vs αl from healthy and CHF subjects. We observe that both groups are segregated.
1 2 3
log n
00.5 1 1.5 2 2.5
log F(n)
Healthy Heart Failure
αs=0.55
αl=0.71
[image:7.595.192.365.519.663.2]α=0.65
Fig. 6: DFA analysis of excursion sequences from a healthy subject and a patient with CHF. For the healthy subject, a single scaling exponentα≈0.65 is identified for time scales 10< n <100. In contrast, a crossover pattern is observed in the CHF patient. For short scales the scaling exponent
is close to the white noise value (αs≈0.55) whereas for large scales, the excursion sequences display
0.5 0.55 0.6 0.65 0.7
αs
0.4 0.5 0.6 0.7 0.8 0.9 1
α l
[image:8.595.236.403.107.238.2]Healthy Heart Failure
Fig. 7: Scatter plot ofαsvsαlfor excursion sequences from healthy and CHF patients. We estimate
αsover short scales 4≤n≤120 andαlover large ones 120≤n≤1000. A clear separation between
the two groups is observed.
Finally, we verify if the observed stretched exponential distribution of excursions is re-lated to the presence of correlations in the heartbeat interval time series. To this end, we generate Gaussian correlated 1/fβ-noise by using the Fourier filtering method with 0 ≤ β ≤ 1 [24]. The case β ≈ 1 roughly corresponds to the observed average spectral exponents in healthy data [3]. The simulated data consists of 32000 values with mean zero and standard deviation equal to one.
0 0.2 0.4 0.6 0.8 1
Spectral exponent (β)
0.5 1
DFA exponent (
α)
0 0.2 0.4 0.6 0.8 1
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
CDF exponent (b)
b (Unsegmented) α (Unsegmented)
Segmented
Segmented
Fig. 8: Statistics ofband αfor excursion sequences from unsegmented and segmented correlated
1/fβ-noise.
[image:8.595.232.403.404.517.2]almost do not show significant changes for different values of β. In particular, we observe that b remains close to one and α≈0.5, indicating that the fluctuations of excursions are close to the uncorrelated case and the segmentation procedure almost destroyed long range correlations in excursion sequences. From these findings, we can conclude that the observed scaling values in excursion sequences from real data are not only related to the presence of long range correlations in heartbeat interval series.
Conclusions. – We have analyzed excursion sequences from stationary segments de-tected in hearbeat interval series. Our study reveals that healthy and heart failure excur-sion sequences are characterized by stretched distributions with different fitting parameters. Specifically, we observed that under healthy conditions the characteristic scale associated to excursions, is bigger than the corresponding average value for the heart failure group, indicating a fast decay under pathologic conditions. Our results about the similar behav-ior of the excursion distributions observed in healthy data are in concordance with previous studies which report common scaling properties in the distributions of beat-to-beat variation amplitudes obtained by means of wavelet and Hilbert-transform analyzes under healthy con-ditions [26]. In our case, nonstationarities related with constant values are removed when the original signal is segmented. This process is related to the wavelet transform where masking effects of nonstationarities are reduced due to the removal of local trends. When the presence of correlations is explored, we find changes in the conditional cumulative prob-ability for healthy and CHF data when τ0 is large. We remark that this difference in the conditional probability can be understood as a degradation of the memory over very short scales. By means of DFA analysis, we confirm the presence of long-term correlations in excursion sequences from healthy data whereas under pathologic conditions correlations are described by a crossover, indicating that over short scales excursions are close to uncorre-lated fluctuations whereas long-range correlations are present for large scales. Finally, we find that the positive correlations observed in excursion sequences from real data are not a simple consequence of the presence of long-term correlations in hearbeat interval series.
∗ ∗ ∗
We thank F. Angulo, R. Hern´andez, A. A. Moreira and D. B. Stouffer for fruitful com-ments and suggestions. This work was partially supported by US-Mexico Foundation for Science (FUMEC), EDI-IPN, COFAA-IPN and Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT, J49128F-26020), M´exico.
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