Chung-Kang Peng

Fractal Correlation Properties of R-R Interval Dynamics and Mortality in Patients With Depressed Left Ventricular Function After an Acute Myocardial Infarction

BACKGROUND Preliminary data suggest that the analysis of R-R interval variability by fractal analysis methods may provide clinically useful information on patients with heart failure. The purpose of this study was to compare the prognostic power of new fractal and traditional measures of R-R interval variability as predictors of death after acute myocardial infarction. METHODS AND RESULTS Time and frequency domain heart rate (HR) variability measures, along with short- and long-term correlation (fractal) properties of R-R intervals (exponents alpha(1) and alpha(2)) and power-law scaling of the power spectra (exponent beta), were assessed from 24-hour Holter recordings in 446 survivors of acute myocardial infarction with a depressed left ventricular function (ejection fraction </=35%). During a mean+/-SD follow-up period of 685+/-360 days, 114 patients died (25.6%), with 75 deaths classified as arrhythmic (17.0%) and 28 as nonarrhythmic (6.3%) cardiac deaths. Several traditional and fractal measures of R-R interval variability were significant univariate predictors of all-cause mortality. Reduced short-term scaling exponent alpha(1) was the most powerful R-R interval variability measure as a predictor of all-cause mortality (alpha(1) <0.75, relative risk 3.0, 95% confidence interval 2.5 to 4.2, P<0.001). It remained an independent predictor of death (P<0.001) after adjustment for other postinfarction risk markers, such as age, ejection fraction, NYHA class, and medication. Reduced alpha(1) predicted both arrhythmic death (P<0.001) and nonarrhythmic cardiac death (P<0.001). CONCLUSIONS Analysis of the fractal characteristics of short-term R-R interval dynamics yields more powerful prognostic information than the traditional measures of HR variability among patients with depressed left ventricular function after an acute myocardial infarction.

Statistical properties of the volatility of price fluctuations

We study the statistical properties of volatility, measured by locally averaging over a time window T, the absolute value of price changes over a short time interval deltat. We analyze the S&P 500 stock index for the 13-year period Jan. 1984 to Dec. 1996. We find that the cumulative distribution of the volatility is consistent with a power-law asymptotic behavior, characterized by an exponent mu approximately 3, similar to what is found for the distribution of price changes. The volatility distribution retains the same functional form for a range of values of T. Further, we study the volatility correlations by using the power spectrum analysis. Both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents. Also, both methods show the presence of a crossover at approximately 1.5 days. In addition, we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 U.S. companies over the two-year period Jan. 1994 to Dec. 1995.

What is physiologic complexity and how does it change with aging and disease

1. IntroductionA defining but elusive feature of physiologic systems istheir daunting complexity. This complexity arises from theinteraction of a myriad of structural units and regulatoryfeedback loops that operate over a wide range of temporaland spatial scales, enabling the organism to adapt to thestresses of everyday life. Quantifying and modeling theremarkable and often bewildering repertoire of behaviorsexhibited by living organisms is one of the major challengesof contemporary science [4,7]. The combination of nonlin-earity and nonstationarity, more the rule than the exceptionin the output of physiologic systems, poses a major chal-lenge to conventional biostatistical assessments and stan-dard reductionist modeling stratagems. To describe andquantify the mechanisms of these “nonhomeostatic” behav-iors, investigators have employed new techniques derivedfrom complexity theory, including fractal analysis and non-linear dynamics. The appropriate application and interpre-tation of such metrics, however, remains incompletely ex-plored. What is clear is that reliance on any single test maygive a misleading representation of physiological complexity.In this issue, Vaillancourt and Newell critique and sug-gest modifications to a general dynamical model of patho-physiology that we and others have elaborated over the pasttwo decades [5,6,8,10,13,14,16,20,21,27]. The theory ofcomplexity loss in aging and disease, as currently formu-lated, has two central postulates:1. The output of healthy systems, under certain param-eter conditions, reveals a type of complex variabilityassociated with long-range (fractal) correlations,along with distinct classes of nonlinear interactions;2. This type of multi-scale, nonlinear complexity breaksdown with aging and disease, reducing the adaptivecapabilities of the individual.The term nonlinear applies to systems whose compo-nents interact in a non-additive way. Nonlinear couplingmay lead to an extraordinary range of dynamics, includingdifferent classes of abrupt changes, (such as bifurcations),deterministic chaos, nonlinear phase transitions, pacemakerentrainment and resetting, stochastic resonance, wave phe-nomena (including spiral waves, solitons, and scroll waves),emergent phenomena, and certain types of fractal scaling.Understanding the specific classes of nonlinear interactionsseen in healthy physiology and characterizing their pertur-bations with aging and disease is just beginning [4,16,27].The term fractal applies to complex-like objects, whichmay be generated by stochastic or nonlinear deterministicmechanisms. Fractal objects show self-similarity (scale-in-variance), such that the smaller-scale structure resemblesthe larger-scale form [10]. Examples in anatomy include theHis-Purkinje network and the tracheobronchial tree. Thefractal concept also extends to complex processes that lacka characteristic, or a single, time scale. Fractal processesgenerate fluctuations over multiple time scales, and theirfrequency spectra typically show an inverse power-law (1/f-like) scaling pattern. Of particular interest is a class offractal processes that demonstrates long-range correlations.This type of “memory” effect has been identified in thefluctuations of the healthy heartbeat, as well as in the inter-stride interval fluctuations in the walking patterns of healthyadults [14,15,21,22].A central caveat when applying concepts and techniquesfrom complexity theory to biomedicine is the recognition

On the trend, detrending, and variability of nonlinear and nonstationary time series

Determining trend and implementing detrending operations are important steps in data analysis. Yet there is no precise definition of “trend” nor any logical algorithm for extracting it. As a result, various ad hoc extrinsic methods have been used to determine trend and to facilitate a detrending operation. In this article, a simple and logical definition of trend is given for any nonlinear and nonstationary time series as an intrinsically determined monotonic function within a certain temporal span (most often that of the data span), or a function in which there can be at most one extremum within that temporal span. Being intrinsic, the method to derive the trend has to be adaptive. This definition of trend also presumes the existence of a natural time scale. All these requirements suggest the Empirical Mode Decomposition (EMD) method as the logical choice of algorithm for extracting various trends from a data set. Once the trend is determined, the corresponding detrending operation can be implemented. With this definition of trend, the variability of the data on various time scales also can be derived naturally. Climate data are used to illustrate the determination of the intrinsic trend and natural variability.

Predicting Survival in Heart Failure Case and Control Subjects by Use of Fully Automated Methods for Deriving Nonlinear and Conventional Indices of Heart Rate Dynamics

BACKGROUND Despite much recent interest in quantification of heart rate variability (HRV), the prognostic value of conventional measures of HRV and of newer indices based on nonlinear dynamics is not universally accepted. METHODS AND RESULTS We have designed algorithms for analyzing ambulatory ECG recordings and measuring HRV without human intervention, using robust methods for obtaining time-domain measures (mean and SD of heart rate), frequency-domain measures (power in the bands of 0.001 to 0.01 Hz [VLF], 0.01 to 0.15 Hz [LF], and 0.15 to 0.5 Hz [HF] and total spectral power [TP] over all three of these bands), and measures based on nonlinear dynamics (approximate entropy [ApEn], a measure of complexity, and detrended fluctuation analysis [DFA], a measure of long-term correlations). The study population consisted of chronic congestive heart failure (CHF) case patients and sex- and age-matched control subjects in the Framingham Heart Study. After exclusion of technically inadequate studies and those with atrial fibrillation, we used these algorithms to study HRV in 2-hour ambulatory ECG recordings of 69 participants (mean age, 71.7+/-8.1 years). By use of separate Cox proportional-hazards models, the conventional measures SD (P<.01), LF (P<.01), VLF (P<.05), and TP (P<.01) and the nonlinear measure DFA (P<.05) were predictors of survival over a mean follow-up period of 1.9 years; other measures, including ApEn (P>.3), were not. In multivariable models, DFA was of borderline predictive significance (P=.06) after adjustment for the diagnosis of CHF and SD. CONCLUSIONS These results demonstrate that HRV analysis of ambulatory ECG recordings based on fully automated methods can have prognostic value in a population-based study and that nonlinear HRV indices may contribute prognostic value to complement traditional HRV measures.

Fractal analysis of heart rate dynamics as a predictor of mortality in patients with depressed left ventricular function after acute myocardial infarction

A number of new methods have been recently developed to quantify complex heart rate (HR) dynamics based on nonlinear and fractal analysis, but their value in risk stratification has not been evaluated. This study was designed to determine whether selected new dynamic analysis methods of HR variability predict mortality in patients with depressed left ventricular (LV) function after acute myocardial infarction (AMI). Traditional time- and frequency-domain HR variability indexes along with short-term fractal-like correlation properties of RR intervals (exponent alpha) and power-law scaling (exponent beta) were studied in 159 patients with depressed LV function (ejection fraction <35%) after an AMI. By the end of 4-year follow-up, 72 patients (45%) had died and 87 (55%) were still alive. Short-term scaling exponent alpha (1.07 +/- 0.26 vs 0.90 +/- 0.26, p <0.001) and power-law slope beta (-1.35 +/- 0.23 vs -1.44 +/- 0.25, p <0.05) differed between survivors and those who died, but none of the traditional HR variability measures differed between these groups. Among all analyzed variables, reduced scaling exponent alpha (<0.85) was the best univariable predictor of mortality (relative risk 3.17, 95% confidence interval 1.96 to 5.15, p <0.0001), with positive and negative predictive accuracies of 65% and 86%, respectively. In the multivariable Cox proportional hazards analysis, mortality was independently predicted by the reduced exponent alpha (p <0.001) after adjustment for several clinical variables and LV function. A short-term fractal-like scaling exponent was the most powerful HR variability index in predicting mortality in patients with depressed LV function. Reduction in fractal correlation properties implies more random short-term HR dynamics in patients with increased risk of death after AMI.

Magnitude and Sign Correlations in Heartbeat Fluctuations

We propose an approach for analyzing signals with long-range correlations by decomposing the signal increment series into magnitude and sign series and analyzing their scaling properties. We show that signals with identical long-range correlations can exhibit different time organization for the magnitude and sign. We find that the magnitude series relates to the nonlinear properties of the original time series, while the sign series relates to the linear properties. We apply our approach to the heartbeat interval series and find that the magnitude series is long-range correlated, while the sign series is anticorrelated and that both magnitude and sign series may have clinical applications.

Fractal mechanisms and heart rate dynamics: Long-range correlations and their breakdown with disease*

Under healthy conditions, the normal cardiac (sinus) interbeat interval fluctuates in a complex manner. Quantitative analysis using techniques adapted from statistical physics reveals the presence of long-range power-law correlations extending over thousands of heartbeats. This scale-invariant (fractal) behavior suggests that the regulatory system generating these fluctuations is operating far from equilibrium. In contrast, it is found that for subjects at high risk of sudden death (e.g., congestive heart failure patients), these long-range correlations break down. Application of fractal scaling analysis and related techniques provides new approaches to assessing cardiac risk and forecasting sudden cardiac death, as well as motivating development of novel physiologic models of systems that appear to be heterodynamic rather than homeostatic.

Dynamic analysis of heart rate may predict subsequent ventricular tachycardia after myocardial infarction.

Dynamics analysis of RR interval behavior and traditional measures of heart rate variability were compared between postinfarction patients with and without vulnerability to ventricular tachyarrhythmias in a case-control study. Short-term fractal correlation of heart rate dynamics was better than traditional measures of heart rate variability in differentiating patients with and without life-threatening arrhythmias.

Correlations in economic time series

The correlation function of a financial index of the New York stock exchange, the S&P 500, is analyzed at 1 min intervals over the 13-year period, Jan 84 -- Dec 96. We quantify the correlations of the absolute values of the index increment. We find that these correlations can be described by two different power laws with a crossover time t_\times\approx 600 min. Detrended fluctuation analysis gives exponents