An index related to autocorrelation function of RR intervals for the ...

Physiological Measurement, February 2006

An index related to autocorrelation function of RR intervals for the analysis of heart rate variability

Jing-Shiang Hwang1, Tsuey-Hwa Hu2 and Lung Chi Chen3 1

Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan E-mail: [email protected]

2

Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan E-mail: [email protected]

3

Department of Environmental Medicine, New York University School of Medicine, Tuxedo, New York 10987, USA E-mail: [email protected]

Running title: An index related to ACF of RR intervals for HRV analysis

Abstract Heart rate variability (HRV) is concerned with the analysis of the variations in the intervals between heartbeats, known as RR intervals. Commonly used HRV indices may be insensitive to detect some dynamic changes related to complex autocorrelation functions of the RR intervals. For example, indices SD1 and SD2 of Poincaré plot can be expressed by the variance and first autocovariance of the signal. The acceleration change index is related to the autocorrelation functions of the series only at the first 3 lags. We extend the idea of characterizing the sign of differences of a time series to propose a new index called VRL, which is the variance of the run length of the sign of the lagged differentiated time series. The theoretical study shows that VRL is directly related to the autocorrelation functions of the RR series at larger lags. Simulated data are used to validate the theoretical results and assess the power of testing group differences measured with VRL and other HRV indices. The performance of VRL is also evaluated for classifying subjects with normal sinus rhythm and congestive heart failure using the RR intervals taken from the PhysioNet database. We apply the index to RR intervals from an animal study of long-term exposure to particulate matter. The VRL values for the susceptible to atherosclerosis young mice in the control and exposure groups decreased gradually with different slopes after several weeks of exposure. The exposure effect changes in this HRV index estimated by fitting a generalized additive model are significant after seven weeks of exposure.

Keyword: heart rate variability, run length, particulate matter

1. Introduction Heart rate variability (HRV) is concerned with the analysis of the variations in the intervals between heartbeats, known as RR intervals. Although the sensitivity and specificity for discriminating normal and pathological subjects may not always be promising, traditional time and frequency domain methods for HRV analysis are widely used in clinical research (Task Force of the ESC and NASPE 1996). However, there is increasing evidence to suggest that the dynamics of heart rate is nonlinear and even chaotic (Guevara et al 1981, Michaels et al 1989). The Poincaré plot analysis is a method becoming popular for assessing the nonlinear dynamics. It is a graph of each RR interval plotted against the next interval or lagged interval. Analysis of Poincaré plots can be performed by a simple visual inspection of the shape of the attractor (like torpedo or butterfly shape). A quantitative analysis of the HRV attractor displayed by the Poincaré plot includes the standard deviation of the instantaneous beat-to-beat RR interval variability (minor axis of the ellipse or SD1), the standard deviation of the long term RR interval variability (major axis of the ellipse or SD2) and the axes ratio (SD1/SD2). It has been shown that SD1 and SD2 can be expressed as second-order statistical moments or as autocorrelation functions, as long as the stationary condition is met. Brennan et al (2001) showed that SD12 (SD22) is the variance minus (plus) the first autocovariance of the time series. This implies that the Poincaré plot index may not have enough sensitivity when the time series have large autocorrelation functions at later lags. When a lag of four heartbeats is used to reconstruct the Poincaré plot, Lerma et al (2003) showed that the SD1/SD2 ratio captured non-linear dynamic changes in the hypotension-resistant chronic renal failure patients after haemodialysis. Lerma et al (2003) also pointed out that it is necessary to choose the lag, indicating the pairs of beats that would define the coordinates of each point in the Poincaré plot. This is important because adequate unfolding depends on the lag we choose according to Takens theorem (Takens, 1981). Ashkenazy et al (2000) proposed analyzing the sign of differences of successive RR intervals to add additional information for cardiac diagnosis and prognosis in certain conditions. They showed that a sign change series is robust with respect to outliers in handling complex signals which include spikes. Gonzalez et al (2003) proposed an acceleration change index (ACI) to characterize the heart rate variability dynamics. ACI is defined as k/M, where k is the number of times that run length of the sign series of the differentiated intervals equal to 1 and M is the total number of runs in the sign series. ACI increases only when a local maximum is followed immediately by a local minimum or vice versa. It detects the presence of very high frequency content on the HRV time series when the tachogram, a RR interval time series, is analyzed. This index is also robust in the presence of artifacts because of its insensitivity to fast changes due to sign operation. ACI, which characterizes the dynamics of zero crossings, is obtained from the sign of differences of successive RR time series. To improve ACI for its insensitive to the magnitude of changes, Arif and Aziz (2005) proposed the threshold-based acceleration change index (TACI) which characterizes the dynamics of threshold crossings. They have also shown robustness of TACI in the presence of artifacts and better performance of TACI for classifying normal sinus rhythm and congestive heart failure using databases from PhysioNet (Goldberger et al 2000). However, the poor performance of ACI in some situations may be explained by the relationship between the index and the complex autocorrelation function (ACF) of the time series. González et al (2003) pointed out that the expectation of ACI is exactly determined by the first three coefficients of the ACF of the time series. In other words, the ACI and TACI have not included information embedded in the ACF at lags beyond 3, which may be crucial to the dynamics of a practical RR time series. In this study, we extend ACI, focusing on the probability of run length 1 of signs of a differentiated series, to an index of variance of run length of signs of the differentiated series, denoted by VRL. We will show that the proposed index is approximately related to the ACF of the RR intervals series at all time lags. We also extend the use of successive difference of each and next RR intervals in ACI to the lagged difference between each and lag-m intervals, where m is some small positive number. We present the relationships between expected probability of run length and ACF. The relationships between VRL and ACF of RR intervals series are then obtained. To demonstrate this, time series simulated from autoregressive process 2

of order 1 and fractionally differenced noise are used to show the relationships between expected probability of run length and ACF, and basic characteristics of VRL. Time series of the logistic map are used to study the behavior of the proposed index. We demonstrate that the VRL outperforms ACI and the Poincaré plot index (SD1/SD2) in terms of sensitivity and specificity in classifying normal sinus rhythm and congestive heart failure subjects taken from the PhysioNet databases. Finally, we have applied VRL to examine the RR intervals series obtained from an animal study of long-term repeated exposure of particulate matter (PM).

2. Methods 2.1 The VRL index The proposed index is constructed on a series of signs of differentiated RR intervals. In the sign series of 1 and 0, each maximal subsequence of elements of either 1 or 0 is called a run. The number of elements in this maximal subsequence is called the run length of this run. For example, the sequence 1,1,1,0,0,1,1,0,1 has three runs of 1 with run length 3, 2 and 1 and two runs of 0 with run length 2 and 1. Note that the total number of runs is always one plus the number of sign changes in the sign series. The proposed index can be obtained from the following calculations. Let the lag-m differentiated RR time series be obtained as ∆ m RRn = RRn + m − RRn for n ∈ [1, N − m] where RR n is the RR interval from beat n to beat n+1, and N is the total number of RR intervals. The signs of the differentiated series are represented by { X m, n , n = 1, 2,K, N − m} the elements of which are

either 1 (if ∆ m RRn ≥ 0 ) or 0 (if ∆ m RRn < 0 ).

For notation simplicity, the subscript m of X m,n is dropped out and let x n be a realization of X m,n . The total number of runs, denoted as k m,0 , is the number of pairs (1,0) or (0,1) in the complete sequence x1 , x 2 , L, x N −m , and can be calculated as

N −m −1

k m,0 = ∑i =1

[ xi (1 − xi +1 ) + (1 − xi ) xi +1 ] .

(1)

Let k m, h be the number of runs with run length equal to h ≥ 1 in the complete sequence, x1 , x 2 , L, x N −m . We then can calculate k m, h similarly by the equation: N − m −h −1

k m,h = ∑i =1

[ xi xi + h+1 ∏ j =1 (1 − xi + j ) + (1 − xi )(1 − xi + h+1 )∏ j =1 xi + j ] . h

h

(2)

The value of k m, h increases by one when a sub-sequence xi , xi +1 , L xi + h , xi + h+1 in which xi and xi + h +1 are one kind and the others are the other kind present.

The value of k m, h is the number of

times that each of consecutive h observations shifts together in the same direction either upward or downward from its lag-m beat. For example, for the case of m = 1, k1,h is the number of times that the observations increase or decrease in the consecutive h beats. . The ratio Rm, h ≡ k m, h / k m,0 can be also interpreted as the probability of run length equal to h, given a lag-m differentiated RR time series. Note that, R1,1 = k1,1 / k1,0 = ACI defined by González et al (2003) is the probability of run length equal to 1. ACI detects only the presence of very high frequency content in the RR intervals, while Rm, h with larger h and/or m will reflect the lower frequency content in the series. Therefore, we extend ACI to a summary index that is built on the probabilities of more run lengths. Specifically, we propose the variance of the run length of the sign of the lag-m differentiated RR intervals 3

(VRLm) as a new index for HRV analysis, which is calculated as H H VRL m = ∑h=1 (h − MRL m ) 2 × Rm,h , where MRL m = ∑h=1 h × Rm,h ,

(3)

for a not too large number H, say 10, to avoid influence from outliers. 2.2 The relationship between VRL and ACF of the RR intervals

The proposed index VRLm is a function of the probabilities Rm, h . In this section, we will show how the expectations of the probabilities relate to the ACF of the RR intervals. In the case of run length equal to 1, from equations (1) and (2), we can calculate the probability as

Rm ,1

∑ =

N − m− 2

i =1

( xi xi + 2 + xi +1 − xi x i +1 − xi +1 xi + 2 )

∑

N − m −1

i =1

( xi + xi +1 − 2 xi x i +1 )

.

(4)

To demonstrate that Rm,1 is related to the autocorrelation functions of the RR intervals, we assume that the RR intervals are from a stationary Gaussian process. When the binary series X m, n is obtained by clipping from a zero mean stationary Gaussian process at level 0, Kedem (1980) showed that 1 N −k 1 1 E ( X m, n = 1, X m, n + k = 1) = lim xi xi + k = + sin −1 ρ (k ) ≡ S 2 ( k ) , (5) ∑ 1 = i N →∞ N 4 2π where ρ (k ) is the ACF of the Gaussian process at lag k. By substituting the summation in (4) with the expectation in (5) and the assumption of E ( X m, n ) = 1 / 2 , we obtain, as N → ∞ ,

π − 2 sin −1 ρ m (2) cos −1 ρ m (2) 1 / 2 − 2S 2 (1) + S 2 (2) =1− = − . (6) 1 1 + 2 S 2 (1) 2π − 4 sin −1 ρ m (1) 2 cos −1 ρ m (1) where ρ m (k ) is the ACF of the lag-m differentiated time series at lag k. González et al (2003) had used cosine formula in Barnett and Kedem (1998) to derive the same result as shown in equation (6) with m = 1 for ACI. E ( Rm,1 ) =

In general, number of run length equal to h in equation (2) can be further expanded as N − m − h −1

k m, h = ∑i =1

N − m − h −1

+ ∑i =1

h +1

( xi xi + h +1 − ∑ j =1 xi xi + j xi + h +1 + ∑1= j < k xi xi + j xi + k xi + h +1 − L + ∏ j = 0 xi + j ) h

h

h +1

h +1

(∏ j =1 xi + j − ∏ j = 0 xi + j − ∏ j =1 xi + j + ∏ j = 0 xi + j ) h

h

.

(7)

Equation (7) consists of summations of products of more than two binary items. For summation of products of three binary items, Kedem (1980) showed that the expectation can be represented by 1 N −k 1 1 lim xi xi + j xi + k = + (sin −1 ρ ( j ) + sin −1 ρ (k ) + sin −1 ρ ( k − j )) ≡ S3 ( j , k ) . (8) ∑ = 1 i N →∞ N 8 4π Using Theorem 4.1 in Kedem (1980), the expectation of products of four binary items can be approximately calculated by, 1 N 1 1 lim ∑i =1 xi xi + j xi + k xi + l ≈ 16 + 8π (sin −1 ρ ( j ) + sin −1 (l − k )) N →∞ N 1 (9) − 2 (sin −1 ρ (k ) × sin −1 ρ (l − j ) + sin −1 ρ (l ) × sin −1 ρ (k − j )) . 4π ≡ S4 ( j, k , l ) The expectation of products of more than four binary items is relatively small. We propose to approximate it by the product of expectations of products of binary items in two disjoint subsets. For example, we have the following approximation 1 N lim (10) ∑i =1 xi xi + j xi + k xi + l xi + m ≈ S3 ( j, k ) × S 2 (m − l ) ≡ S5 ( j, k , l , m) . N →∞ N

4

As N approaches to infinity, the E ( Rm,h ) can be approximately calculated from equations (1) and (7) with sums of products of binary items replaced by the functions of autocorrelations S 2 , S3 and S 4 defined in equations (5), (8) and (9) and approximate functions S p defined similarly to (10) for p > 4 . That is, E ( Rm, h ) ≈

S 2 (h + 1) − ∑ j =1, h S3 ( j , h + 1) + ∑1= j < k = h S 4 ( j , k , h + 1) − L + S h + 2(1, L , h + 1) 1 + 2 S 2 (1)

. (11) S h (1, L , h − 1) − 2S h +1 (1, L , h) + S h + 2(1, L , h + 1) + 1 + 2 S 2 (1) Although the formula is very complicated, it is clear that E ( Rm,h ) consists of the ACF of the lag-m differentiated time series at lags from 1 to h+1. To get an insight on the formula, we present the case of h = 2 in the following, 4 8 4 1 − sin −1 ρ m (1) − sin −1 ρ m (2) + 2 ((sin −1 ρ m (2)) 2 + sin −1 ρ m (1) × sin −1 ρ m (3)) π π π . (12) E ( Rm, 2 ) ≈ 8 4 − sin −1 ρ m (1)

π

The next step is to derive the relationship between the autocorrelation function of the RR time series and the lag-m differentiated RR time series. Under the assumption that m and k 0 and − 0.5 < d < 0.5 . Actually, the autocorrelation function can be calculated as ρ 0 (k ) = Γ(k + d )Γ(1 − d ) /(Γ( k − d + 1)Γ( d )) for a given d. We repeated the above procedures with AR(1) replaced by fractionally differenced noise. Figure 1(b) showed similar distribution of run lengths as the above short memory AR process. For the time series with longer autocorrelations (d = 0.4), the probability of run length 1 is smaller and the tail of the run length distribution is heavier. These two simulation examples confirm that expected run length probability is related to the ACF of the time series.

The logistic map expressed as yn +1 = r yn (1 − yn ) is a chaotic process given an initial value of y0 which satisfies 0 < y0 < 1 . The yn process has no chaos for 0 < r < 3 . It starts oscillation for r > 3 . More bifurcations appear as r increases and the process becomes chaotic when a critical value (r = 3.57…) is reached. For certain r values between this critical value and 4, predictable dynamics can appear. To test whether the proposed index depends on the r values, we have also used the same setup as González et al 2003 to examine the index VRL on the time series of a logistic map. Figure 2 shows the dependence of the VRL1 and VRL3 on the r parameter of the logistic map. The mean and mean ± standard deviation of the index calculated on 50 logistic maps generated from initial values of y0 ranging from 0.01 to 0.99 for each r are plotted. For 3 < r < 3.68, we have VRLm = 0. For r > 3.68, VRLm show similar patterns and depend on the r value. The plots also show that the variance of run length for the lag-3 differentiated time series is larger than that of lag-1. The ACI is determined by the first three autocorrelation functions of the time series, while SD1 and SD2 are determined by the variance and autocovariance at lag 1 only. When the time series have significant autocorrelation functions at lags beyond 3, the indices ACI and SD1/SD2 may have not enough power to detect it. To get some insight about the powers of these HRV indices in testing group mean differences, we generate artificial time series having the same first three autocorrelation functions and different ACF beyond lag 3 for subjects in two groups. Specifically, time series for subject j in group i are generated from a periodic ARMR process plus an extra noise from a normal distribution with mean 0 and standard deviation σ ij ~ Unif (0.8, 1.2) . The periodic ARMA process {Yij ,t } is given by Yij ,t − φi Yij ,t −1 − Φ i (Yij ,t − si − φi Yij ,t − si −1 ) = Z ij ,t + θ i Z ij ,t −1 ,

where Z ij ,t is a standard normal distribution, the ARMA parameters (φi , θ i ) , the period si and the periodic AR parameter Φ i are given in Table 1. We simulated time series of length 1000 for 50 subjects in each group and calculated the indices for all subjects. The t statistic is used to test no mean index difference between the two groups. The hypothesis of no mean difference of an index is rejected when the p value is smaller than the significance level of 0.05. We repeated the above procedures 100 times and calculated the proportion of rejecting the hypothesis as an estimate of the power of each index. In the first 6

setting, we generated time series with slight differences of ACF at the first 3 lags between the two groups in Table 1. Simulation results show that all the indices have 100% power of rejecting the hypothesis of no index difference. For the next two settings, the two groups have close ACF at the first 3 lags and very different values at least at lags 5 and 6. The results show that ACI and SD1/SD2 have a little power, while VRL outperforms them. In the last setting, the ACF at the first 6 lags shown in Table 1 are close between the two groups, while they are different at lags 8 to 12. The power of VRL drops a little to 89%.

4. Applications

4.1 Classifying subjects in two groups

In this section, we compare the performance of the proposed index VRLm to other indices in terms of sensitivity and specificity in classifying normal sinus rhythm and congestive heart failure subjects taken from the benchmark PhysioNet databases. RR intervals time series data of normal sinus rhythm (NSR) subjects and congestive heart failure (CHF) subjects were taken from the PhysioNet database (Goldberger et al, 2000). The NSR beat annotation files of long-term ECG recordings of 52 subjects including 24 women (aged 56–73 years) and 28 men (aged 28.5–76 years) were taken. The CHF beat annotation files of long-term ECG recordings of 29 subjects (aged 34–79 years) were taken. The percentage of artifacts (mostly ectopic) of these data was 0.44% (Arif and Aziz, 2005). Scattergrams of ACI and VRL6 with subjects along the x-axis and value of the ACI and VRL6 along the y-axis are shown in figure 3. The high overlap of ACI values of the subjects in the NSR and CHF indicates poor classification power of this index. On the other hand, the VRL6 values of the CHF subjects tend to be smaller than those of the NSR subjects. We use receiver operator curve (ROC) for assessing tests based on the indices VRL6, ACI and SD1/SD2 in classifying subjects between NSR subjects and CHF subjects. The ROC is a line graph that plots the probability of a true result (sensitivity) against the probability of a false positive result (1-specificity) for a range of different cutoff points. The closer the curve is to the upper left-hand corner of the graph, the more accurate the test. The area under the ROC (AUC) is often used as an index of diagnostic accuracy. Figure 4 shows that VRL6 are more accurate than the other two indices in separating subjects of the two groups in these databases. The AUC for these indices are 0.89, 0.69 and 0.86, while Arif and Aziz, 2005, reported a value of 0.82 using TACI. 4.2 Animal study

In a subchronic concentrated ambient particles (CAPs) exposure study of mice at the New York University laboratory in Sterling Forest State Park in Tuxedo, 40 miles northwest of Manhattan, Chen and Hwang (2005) and Hwang et al (2005) have found significant exposure-related alterations of SDNN, RMSSD, HR, body temperature, and physical activity in mice lacking apolipoprotein E (ApoE-/-) which is susceptible to atherosclerosis, with smaller and non-significant changes in normal C57 mice. This study is part of an extension investigating the effects of CAPs on the cardiovascular system in a younger population of 16 ApoE-/- mice, four months old with an average life span > 17 months. The mice were randomly assigned into exposure and control groups, eight for each group, and exposed to US northeastern regional background CAPs (average concentration of 130±102 µg/m3, which was 10 times the ambient PM concentration) and filtered air, respectively, for six hours on weekdays. The exposure experiment lasted for 15 weeks from Feb. 10thto May 7th, 2004. Mice were implanted with ECG transmitters (TA10ETA-F20, Data Sciences, St. Paul, MN) at least three weeks prior to CAPs exposure. Ten seconds of ECG, HR, activity, and body temperature data were sampled every five minutes continuously throughout the experiment except during brief periods to transport animals between animal housing and the exposure facility. To examine whether there exist chronic changes in HRV indices, we focused on the RR intervals 7

recorded on weekends. The VRLm index was computed for each subject in a day period of Saturday and Sunday, which consists of the RR intervals obtained every five minutes during the 8:30−14:30 period. During this period, the slow wave sleep may be an appropriate recording condition offering a "self-controlled" and undisturbed moment of observation for assessing time and frequency domain HRV indices (Brandenberger et al., 2005). Because mice are nocturnal in nature, their respiratory rate is relatively stable during the daytime while they are sleeping. For demonstration, VRL3 was used and averages of the index on Saturday and Sunday for the weekend before exposure and the next 15 weeks were calculated for each mouse and plotted in figure 5(a). Numbers and letters in the plots are used to represent the weekly average index for each mouse in the control and exposure groups, respectively. The plots show that the average VRL3 values of mice in the exposure and control groups are around 5 for the first eight weeks, and then gradually decreased and separated from each other in the next six weeks and moved slightly upward at the end. To estimate the exposure effect changes in this HRV index over the 16 weeks, we applied generalized additive models to fit the nonlinear trends (Hastie and Tibshirani,1990). Specifically, let yijt be the VRL3 value obtained at the tth week for the jth mouse in the ith group, where i =1, 2 represent control group and exposure group, respectively. The model has the form yijt = s (t ) + d (t ) × I (i = 2) + ε ijt , where s (t ) and d (t ) are smoothing spline functions with 4 degrees of freedom, I (⋅) is an indicator function and ε ijt is an error term assumed to be normally distributed and independent with a constant variance. The smoothing function s (t ) is used to model the average VRL3 changes for all the mice over the 16 weeks, while d (t ) is used to model the VRL3 differences between mice in the exposure group and the control group over the time period. The estimates, with 95% confidence intervals, of the effect function d (t ) obtained from the statistical package S-Plus are plotted in figure 5(b). The results show significant exposure effects in this HRV index after seven weeks of exposure. In contrast, figures 6(a) and 7(a) did not reveal clear differences between the two groups using the HRV indices ACI and SDNN. We used the same generalized additive models to estimate the effect changes in these two HRV indices. The estimates, with 95% confidence intervals, of the effect changes plotted in figures 6(b) and 7(b) show no significant differences over the 16 weeks.

5. Conclusion

The VRLm is the variance of run length of signs of lag-m differentiated RR intervals, which reflects the dynamics of beat to beat accelerations and decelerations. The VRLm detects the presence ranging from very high to very low frequency contents on the RR intervals. The VRLm is shown to be related to the autocorrelation functions of RR intervals at longer lags. The proposed index outperforms commonly used indices in separating subjects in two groups when the RR time series have large ACF at later lags. We have evaluated VRLm in two real data sets and found the potential of the index in clinical applications and scientific studies. Since the VRLm is constructed on the signs of lag-m differentiated RR intervals, the index is insensitive to the level of the RR interval differences. Therefore, the VRLm has the advantage of robustness in the presence of artifacts. The index has no technical limitations such as stationary requirement and linear assumptions in time and frequency domain methods. However, since the VRLm has no direct relationship with the variance of the RR intervals, the index may lose power in classifying subjects in two groups with significant different variances of the RR intervals. In this case, other measures such as the Poincaré plot indices of SD1 and SD2 may have better performances for their direct relationship with the variance and the first autocovariance. Similar to the various HRV indices, the new index has its limitation in detecting some heart autonomic function changes. Some indicators of aggregating various HVR indices may be 8

more sensitive than each single index in some situations, e.g., see Matveev et al., 2003. The results of the animal study showed that the VRLm values of each individual subject across the time are relatively more stable than the other indices. These findings indicate the proposed index can reflect each individual’s HRV change and is also robust with respect to outlier signals which frequently occur in ambulatory recordings of the ECG transmitter implanted mice over long-term experiments. We have found that long-term PM exposure may cause chronic effect change of VRLm values in the animal study. However, like other HRV indices, we can only conclude that there were chronic PM effects on mice’s autonomic nervous system. More evidence is still needed in order to better understand the exact mechanism of the exposure effect on heart function.

Acknowledgements

This work is partially supported by the National Science Council of Taiwan. This research was performed as part of a Center Grant from the U.S. Environmental Protection Agency (R827351), and utilized facility core services supported by a Center Grant from the National Institute of Environmental Health Sciences (ES 00260). References

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Setting 1 i=1 i=2 2 i=1 i=2 3 i=1 i=2 4 i=1 i=2

(φi , θ i ) (0.8,0.2) (0.8,0.2) (0.5,0.0) (0.5,0.0) (0.2,0.8) (0.2,0.8) (0.8,0.2) (0.8,0.2)

si 6 12 6 12 6 12 12 12

Φi 0.9 0.1 0.9 0.1 0.9 0.1 0.9 0.1

ACF estimates at first 6 lags 0.46, 0.27, 0.19, 0.17, 0.18, 0.20 0.43, 0.20, 0.09, 0.04, 0.03, 0.02 0.26, 0.08, 0.06, 0.09, 0.18, 0.38 0.25, 0.07, 0.02, 0.00, 0.00, 0.00 0.29, −.04, 0.01, 0.05, 0.25, 0.35 0.30, −.03, 0.00, 0.00, 0.00, 0.00 0.44, 0.21, 0.11, 0.07, 0.06, 0.06 0.43, 0.19, 0.09, 0.05, 0.03, 0.02

VRL1 100

ACI 100

SD1/SD2 100

100

10

21

96

13

12

89

9

15

Table 1. Simulation results for the power of the three HRV indices. The numbers of rejecting the hypothesis of no group mean index difference (out of 100 replicates) in four settings are shown.

10

0.6

Probability

0.5 0.4 0.3 0.2 0.1 0.0 1

2

3

4

5

Run length (a)

0.7

Probability

0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

2

3

4

5

Run length (b) Figure 1. The distributions of run length of signs of differentiated time series simulated from (a) AR(1) processes with parameter φ = 0.9 (solid line) and 0 (dashed line), and (b) fractionally differenced noise with parameter d = 0.4 (solid line) and -0.4(dashed line). The vertical segments represent sample mean ± standard deviation of the probabilities of run length. The symbol × is the theoretical value of the expectation of the run length probability.

11

1.2

VRL1

1.0 0.8 0.6 0.4 0.2 0.0 3.2

3.4

3.6

3.8

4.0

3.6

3.8

4.0

r (a)

VRL3

1.5

1.0

0.5

0.0 3.2

3.4

r (b) Figure 2. Sample mean ± standard deviation of (a) VRL1 and (b) VRL3 values for data simulated from the logistic map are plotted for each r.

12

Value

0.7

NSR CHF

8

6 0.6 4 0.5 2 0.4

VRL6

ACI

Figure 3. Scattergrams of ACI and VRL6 for NSR and CHF subjects.

1.0

Sensitivity

0.8 0.6 0.4

VRL6 ACI SD1 SD2

0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1-Specificity Figure 4. Receiver operator curve (ROC) for assessing tests based on the indices VRL6, ACI and SD1/SD2 in classifying NSR subjects from CHF subjects.

13

10

f h

f

VRL3

8 6 4 2

f

a f h f b 7 5 g 2 h c3 6 e d 1 4

cb 3 2 a 5 7 g e d 4 6 1

f

f h

h g c 3 2 a 7 5 d 1 6 4 e

c g 3 6 7 b 5 a 2 d 1 4 e

h 3 g c1 d a 2 5 e 7 b 4 6

f

f h f c b 3 1 d 2 a 5 7 g e

h f c2 e 3 1 b g a 7 d 5 4 6

b h f c2 3 1 7 4 5 6 g d a e

f h

h c3 d 2 a 1 5 7 e b 4 g 6

4 6

b

f h

h c 5

5 c 3 a d 7 1 b e 4 g 2 6

c3 5 d 2 a g e 7 4 1 6

a 2 7 d e g b 4 3

b

6

1

h 5 f

3 5

d 3 c2 a 4 e 7 g b 6

c a d e 4 7 2 g

1

6

b 1

h 5 a c e 7 3 4 b 2 d g 6

h f c d 7 5 3 a e 2 4 6 g

b 1

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week (a)

Effect

1.0 0.5 0.0 -0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week (b) Figure 5. (a) The VRL3 values calculated for each mouse over the 16 weekends are plotted with alphabet and number for exposure and control groups, respectively. The fine and coarse lines are geometric average of VRL3 in exposure and control, respectively. (b) The estimates (solid), with 95% confidence intervals, of the VRL3 differences between mice in exposure and control groups.

14

ACI

0.70

e

0.65

4 1 3

0.60 0.55

3 4 e 6 h g c1 5 d b f 2 a 7

1 6 3 e 4 g c h f a b 2 5 d 7

6 a h cb g f 2 5 7 d

e 3 4 a c 6 1 g 2 d 7 5 b f h

4 6 c 1 3 a 5 e

g

4 6 a 3 g

g 4 6

6 g

c1

2 g h b 7

e 2 h 5 b

f d

d f 7

6 c4 1 a d e b 5 f 3 h 2

3 1 c4 d a g e b h f 2 7 5

1 3 a h cb e d 2 7 f 5

7

a 6 cg 4

6 1

3 6

1 e b 3

c4 a e 3 g

b a c4 1 g e 2 h

2 d h f 5 7

b h f 7 2 d 5

f d 5 7

g a 3 h 6 b c4 1 e 7 2 d f 5

6 g

6

g

a 4 d 2 cb e 1 3 h

3 c4 e a 1 b h d 2 f 5

6 a 1 3 b 4 e

5

c 2 f d h 5

7 f 7 7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week (a)

0.03

Effect

0.02 0.01 0.00 -0.01 -0.02 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week (b)

Figure 6. (a) The ACI values calculated for each mouse over the 16 weekends are plotted with alphabet and number for exposure and control groups, respectively. The fine and coarse lines are the average of ACI in exposure and control, respectively. (b) The estimates (solid), with 95% confidence intervals, of the ACI differences between mice in exposure and control groups.

15

-2.5 1

b

1 1

1

1

b

-3.0

log SDNN

b

-3.5 b

-4.0 -4.5 -5.0 -5.5

1 2 b 6 e f h a 7 d c 5 g 4 3

2

b 6 b h 2 a c3 f 7 e d g 5 4 1

6 h 2 c f 3 e a g 1 7 4 d 5

6 2 cb h a g 7 f 3 e d 4 1 5

6 2 h f g 7 a 3 1 e c4 d 5

6 b h 2 f a e d 7 g c3 1 4 5

6

6 2 h d e 1 g f cb 7 a 3 4 5

2 e h cb 7 g d a 1 f 4 3

b 6 h g a f d 7 c3 e 4 1 5

2 6 b f ch e 7 a 1 d g 3 4 5

2 h cf 6 e a 7 1 d g 3 4 5

6 f 2 h a e b 7 g c d 3 4 5

b f 2 7 e h 6 ca g 4 d 3 5

2 f 6 h 7 c3 e a g d 4

b 2 f h 7 g 6 e a d c3 4

5

2 f 6 7 g a ch e 3 d 4

5

5

5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week (a)

0.2

Effect

0.1 0.0 -0.1 -0.2 -0.3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week (b) Figure 7. (a) The log SDNN values calculated for each mouse over the 16 weekends are plotted with alphabet and number for exposure and control groups, respectively. The fine and coarse lines are the average of log SDNN in exposure and control, respectively. (b) The estimates (solid), with 95% confidence intervals, of the log SDNN differences between mice in exposure and control groups.

16