Multiscale power analysis for heart rate variability

We first introduce multiscale power (MSP) method to assess the power distribution of physiological signals on multiple time scales. Simulation on synthetic data and experiments on heart rate variability (HRV) are tested to support the approach. Results show that both physical and psychological changes influence power distribution significantly. A quantitative parameter, termed power difference (PD), is introduced to evaluate the degree of power distribution alteration. We find that dynamical correlation of HRV will be destroyed completely when PD>0.7.


I. INTRODUCTION
The physiological systems are complex at multiple levels of temporal and spatial scales and regulated by various couplings and feedback loops.The output signals of these systems often exhibit complex oscillations which are not simply attributable to noise but contain information about the underlying dynamics.][3][4][5] A number of methods have been developed to assess the structural complexity of fundamental biological mechanisms on multiple time scales.Costa et al. 5 introduced the multiscale entropy (MSE) analysis method to evaluate the complexity of biological signals by computing sample entropies on different time scales.They found that sample entropy of healthy subjects is highest compared to those of pathologic subjects on high time scales.Wu et al. 6 improved MSE algorithm by composite multiscale entropy (CMSE) algorithm on the concern of accuracy of MSE.It enhanced the statistical reliability of MSE prominently.Ahmed et al. 7,8 produced the multivariate multiscale entropy (MMSE) to extend MSE to multi-channel signals.All these approaches provide solutions for multiscale analysis of physiological signals but also share some drawbacks.First, the method, generating multiscale signals by a coarse-graining procedure, is actually an artificial rough filtering process 9 which ignores the fluctuation regularity of original signal itself.Second, they use the same complexity quantitative indicator, sample entropy, which is easy affected by noise.Third, these approaches need a long data sequence to calculate the entropy as the number of signal points is reduced to 1/τ of original data length after coarse-graining on scale τ.
Empirical mode decomposition (EMD) 10 is a fully adaptive data-driven method that decomposes time series into a set of scale-dependent components, named intrinsic mode function (IMF), and each of which represents the signal's inherent oscillation mode on a particular scale.2][13] In this paper, base on EMD as a scaling tool, we propose the multiscale power (MSP) analysis method to quantify the complexity change of heart rate variability (HRV) from the perspective of data-driven normalized power distribution.We focus on multiscale power distribution for the sake of finding out how life systems regulate the control strength across different time scales.The power variation of HRV is a symbol of functional change of heart.The most popular tool to research power change of HRV is power spectral density (PSD) which analyzes the power of heart rate as a function of frequency.The spectrum is commonly divided into three parts, namely high frequency (HF), low frequency (LF), and very low frequency (VLF) components.Researchers has found that cardiac diseases, such as heart failure and myocardial infarction, will lead to an increased power in HF and decreased power in LF and VLF in normalized units, 14,15 revealing impairment of sympathetic nervous system.However, the PSD method is not data-driven, it divides power spectrum artificially according to some specific frequency thresholds and ignores the fluctuation characteristics of different HRV signals.The divisions are rough and lack detail information.Furthermore, PSD method is sensitive to noise and signal's trend components. 16The time-frequency analysis method, wavelet transform (WT), could decompose HRV signals into more detailed components, but it is also not data-driven and more suitable for stationary data.Besides, the basic wavelet function of WT is of finite length, thus it causes power leakage.For MSP, proposed in this paper, the power is not divided by frequency, but according to inherent fluctuation levels of data.In fact, there is a correspondent relation between power divisions by PSD and those by EMD, which will be discussed in the section III.
In this paper, we first introduce the MSP method and verify it on simulate data.Then experiments on real HRV data are tested to explore how power distribution changes by physical and psychological regulation.Finally, conclusions are given and results are discussed at the end.

II. MULTISCALE POWER ANALYSIS
The MSP algorithm can be summarized as follows: 1.For a given time series x(t) of length N, decompose it into a finite set of IMFs by EMD algorithm: Where, c i is the number i IMF, representing the i-th scale component, and r is the residual component.

Calculate the power of all IMF components:
3. Normalize power value on each scale: Where, E = n  i=1 E i , p i is the normalized power on scale i, representing the proportion of total signal power.
Unlike the shortening process of data length by coarse-graining for MSE, the scale components of MSP, obtained by EMD algorithm, keep an identical data length for all scales.Therefore, much fewer data points are needed for MSP.
We first test the MSP method on simulated white and 1/f noise as shown in Fig. 1.We find that for scale 1, the normalized power value for white noise time series is much higher than that of 1/f time series.However, the power value for 1/f noise remains almost constant across all scales, but it falls monotonically for white noise.Such that for scales>4, the power value for white noise is smaller than the corresponding value for 1/f noise.This consists with the fact that a long-range correlated signal contains a complex structure on multiple temporal scales. 5IG. 1. MSP analysis of white noise and 1/f noise.The noise series contain 10,000 data points, and the plots represent an average of 100 independent realizations and error bars represent the standard deviation (SD).

III. EXPERIMENTS ON REAL CARDIAC INTERBEAT INTERVALS
The power distribution, representing the control strength of cardiac system on different temporal cycles, displays certain law and varies according to body's conditions.How it changes with diseases (passive change) and mental controlling (active change) is not clearly defined.In this section, MSP will be applied to analyze cardiac interbeat interval (RRI) signals.

A. MSP analysis of RRI signals for healthy and CHF subjects
We apply MSP method to analyze power distribution of RRI signals for healthy and congestive heart failure (CHF) subjects.The database, obtained from PhysioNet (www.physionet.org),contains 54 healthy subjects (age range 28-76, mean 61yrs) and 29 CHF subjects (age range 34-79, mean 55yrs).First, we decompose RRI data by EMD as shown in Fig. 2. It is clear that the first component (IMF 1 or scale 1 component) has the highest frequency and fluctuates on shortest time scale.With scale increases, IMF components fluctuate on longer time scales and their frequencies decrease.In order to find out the relationship between scale number and frequency, average center frequency of scale component is calculated for both healthy and CHF groups and results are listed in Table I.
With scale increases, the center frequency decreases for both groups.We have known that the classic spectrum divisions for PSD are HF (0.15-0.4 Hz), LF (0.04-0.15 Hz) and VLF (0.0033-0.04 Hz).For scale 1, the frequency is above HF; for scale 2, it is in the range of HF; for scale 3 and 4, it is included in LF; and VLF includes scale 5-7; for scale 8, it is lower than VLF.In addition, we also note that for the same scale, the frequency of CHF patients is slightly higher than that of healthy subjects, revealing a change of fluctuation mode caused by heart disease.
We apply MSP only on scale 2-7 by removing scale 1 and the scales higher than scale 7 and the residual.The scale 1 component could be usually contaminated by noise and brings a big fluctuation (high standard deviation).With frequency higher than HF, the physical meaning is undefined.The components higher than scale 7 contain frequencies lower than VLF and represent undefined long term regulation of heart, which is easily affected by external environments.
We plot MSP curves of these two groups on scale 2-7 in Fig. 3. Significance test is carried out through student's t-test.For small scales (scale 2, 3), higher power value is assigned to CHF patients in comparison with healthy subjects, while for high scales (scale>5), it is quite the reverse.On four scales (scale 2, 3, 6, 7), the power values for these two groups are significant different (p<0.01).The MSP curve for healthy subjects goes smoothly on small scales and increases gradually over high scales.In contrast, the power for CHF subjects decreases monotonically over all scales.Next, we apply MSP to evaluate the surrogate data produced by randomizing data points of original RRI.As Fig. 3 shown, the MSP curves for both surrogate data decrease steeply from scale 2 to 7, and the power values are higher on small scales (scale 2, 3) and lower on high scales (scale 5-7) in comparison to those of their original data.
We introduce power difference (PD) as a quantitative parameter of power imbalance between small scales and high scales.Where, PD = (p 2 + p 3 + p 4 ) − (p 5 + p 6 + p 7 ).A higher PD indicates higher power on small scales and lower power on high scales.We calculate PD values of RRI data and its surrogate data for both groups in Table II.
There is a significant difference of PD value between healthy and CHF groups.The PD value of CHF subjects is much higher than that of healthy subjects.The minus PD value for healthy subjects,  indicating a higher power on high scales, demonstrates a stronger control over long time periods.The PD values for both groups show a distinct difference from their surrogate data, revealing that the original physiologic data is more complex than the randomized surrogate data.

B. MSP analysis of RRI signals for two kinds of meditation training
Chi and yoga, originated in China and ancient India, are two popular meditation trainings around the world.Meditation is an effective tool for relaxing, it changes the balance between sympathetic and parasympathetic systems and especially regulates the cardiovascular system. 2,17n this part, we apply MSP method to analyze RRI before and during meditation and also their randomized surrogate data on scale 2-7 in Fig. 4. and Fig. 5.The data is from meditation database of PhysioNet, 18 which includes 8 chi meditators (age range 26-35, mean 29yrs) and 4 yoga meditators (age range 20-52, mean 33yrs).As the RRI data of yoga meditation is short in length (data length range 437-1126, mean 805), in order to obtain a more accurate power value, we use all data points for MSP analysis.While for chi meditation, the data length is 3000.
Fig. 4 shows that the power value during chi meditation is significantly higher on scale 2 and 3 (p<0.01),but smaller on scale 6 and 7 (p<0.01) in comparison to that of pre-chi.The MSP curve for pre-chi goes smoothly on small scales and incrementally over high scales.While for during chi, high values are assigned on scale 2 and 3, then the power value goes degressively over high scales.For yoga meditation (Fig. 5), the MSP curve for during yoga goes close to that of surrogate data.MSP analysis results for yoga are similar to that of chi except for the significant separation points, which are scale 2 (p<0.05),scale 5 (p<0.01) and scale 6 (p<0.05).We measure the PD values to determine the degree of power change in Table III and Table IV, also, the surrogate data is included.
In Table III, the PD value of time series of pre-chi is markedly smaller than that of during chi and their surrogate data.Analogue results are expressed for yoga meditation (Table IV).However, we cannot see any obvious distinction of PD value between the data during meditation and their   surrogate data both for chi (p=0.1521) and yoga (p=0.6486).In this regard, we suppose that the RRI data during meditation behaves like random series, in other words, chi and yoga meditation destroy the long range correlation completely.

IV. DISCUSSION AND CONCLUSION
In our work, the trend of power to short time scales, caused by CHF (physical factor), is clearly exhibited.But why this happened?The pathogenesis of CHF may give some clues.We have known that CHF is a serious heart disease with inability or failure to pump sufficient blood to meet the body's needs. 19With changing structure and function of the heart, patients can easily decompensate when a change occurs to their body, and the patient's heart may not be able to react to the body's changing environment.Therefore, CHF patients need more energy to control heart beat rhythm on short time period to satisfy body's immediate needs.On the other hand, with damaged myocardium, patients may lose the ability to regulate heart beat rhythm on long time scales.
What about the meditation (psychological factor)?We know that meditation is a complex physiological process of adjusting neural, psychological, behavioral, and autonomic functions. 2It turns people's attention to a single point of reference, such as breath, bodily sensations, or a word or phrase known as a mantra.The heartbeat becomes regular and loses long range correlation during meditation. 20,21Because the heart beat rhythm on long time period is somewhat dependent on external environment.When doing meditation, people are less correlated to the external stimuli. 17onsequently, the power on long time scales decreases.
The experimental results prove that MSP method could effectively evaluate power distribution of heartbeat both for physical and psychological changes.We tested MSP in two forms of uncorrelated signals: white noise and randomized RRI signals, both of which showed similar distributions; the power is highest on shortest scale and diminishes gradually across all scales.So, we assume that any RRI signals with power distribution similar to that of randomized series or white noise lose dynamical complexity or long-range correlation.For correlated series (1/f noise and RRI signals of healthy subjects or meditators before meditation), the MSP plot goes smoothly or incrementally.We believe that MSP of any healthy subjects should behave like this.For correlated data, with uniform distribution of sample entropy on all scales based on MSE, 5 should also have a uniform power distribution on all scales based on MSP.Therefore, for CHF patients or meditators during meditation, with decreased long-range correlation of HRV, the MSP curves trend to that of randomized surrogated series.Consequently, the MSP method is a reliable tool to analyze power distribution and dynamical changes of HRV.
The parameter PD is effective in quantifying the MSP plot.From experiments, It is clear that the PD values for all surrogate signals are larger than 0.7.Therefore, for signals during meditation, with PD value close to 0.7 (for chi) or larger than 0.7 (for yoga), the dynamical correlation disappears.For healthy subjects, the PD value should be lower than zero.Besides, it is noteworthy that CHF patients showed a significantly larger PD than that of healthy controls, and also a significantly smaller PD compared to randomized surrogate signals.In view of this fact, we believe that the dynamical correlation for CHF patients still exists, although it is weak compared with healthy subjects.So, as a result, the PD value could be used to quantify the degree of power distribution variation and dynamical correlation change.The higher it rises, the lower the correlation falls.
FIG. 2. RRI signal of a healthy subject and its IMFs obtained by EMD.From top to bottom: original RRI data, IMF1-8 and the residual.

FIG. 4 .
FIG. 4. MSP analysis of RRI series for chi meditation training.The number of data points is 3,000.Values are given as means ± standard error.Significant difference of t-test between pre-chi and during chi are marked by * (for p<0.01) above the plots.

TABLE I .
Average center frequency of healthy and CHF subjects.
All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/Downloaded to IP: 114.212.123.231On: Mon, 29 Jun 2015 14:37:50 FIG. 3. MSP analysis of RRI series for healthy and CHF subjects on scale 2-7.The data length is 10,000 and values are given as means ± standard error.Significant difference of student's t-test between healthy and CHF groups are marked by * (for p<0.01) above the plots.

TABLE II .
PD values of healthy and CHF patients.
PD (mean±std)-0.2092± 0.2940 0.2642 ± 0.4070 a 0.7306 ± 0.0226 b 0.7230 ± 0.0328 b a represents p<0.0001 for t-test between healthy and CHF subjects.b represents p<0.0001 for t-test between surrogate data and its original data.

TABLE III .
PD values of chi training.

TABLE IV .
PD values of yoga training.