Hands-on Time Series Experiments for Students

Introduction

Analysis of physiologic signals often tends to focus on average quantities, with comparisons of means and variances (so-called time domain statistics). Additional analyses based on frequency domain techniques involving spectral analysis are also sometimes applied. The utility of newer measures based on fractal analysis and nonlinear dynamics (chaos theory and complexity) remains uncertain, however. Further, these techniques are largely unknown to most investigators involved in physiologic data analysis. A key question, therefore, is whether techniques based on dynamical analysis add information to conventional statistics. If so, what are the appropriate applications and limitations of such techniques? For example, does the use of metrics based on chaos theory add diagnostic or prognostic information to biomedical time series analysis? Does such analysis provide insight into the underlying physiologic control mechanisms?

In this self-guided exploration of the dynamics of physiologic time series, we encourage you to focus on the analysis of actual time series derived from human subjects (1-16). Two classes of ``real world'' physiologic signals are available here, one related to human gait, and the other related to human heartbeat. The accompanying datasets are provided to allow you to begin to explore these and related questions. The datasets are described below. As a starting point, we suggest the following general approach:

• First, the systematic exploration of any time series should begin with a visual inspection of the data, obtained by looking at the time series graphically. The "eyeball analysis" of such data is of great importance because it allows you to develop certain intuitions about the nature of the fluctuations, the degree of stationarity or non-stationarity, and to guide the selection of appropriate analytic techniques. For example, if visual assessment reveals an abrupt change midway through the time series, then spectral analysis across the entire time series would be inappropriate because of this non-stationarity. Indeed, the misapplication of this, or other techniques requiring stationarity, would lead to potentially misleading conclusions about the underlying dynamics. Visual inspection of the data also may reveal the presence of ``outlier'' points, which may either be due to artifact or to some important physiologic event. Sometimes such apparent ``outliers'' may be part of the intrinsic dynamics of the system.
• As a second step, conventional analyses based on the computation of mean and variance, histograms, as well as spectral analysis, should be done. (You can find a variety of software for spectral analysis in the WFDB software package.)
• The third step will be the application of techniques derived from nonlinear dynamics, including complexity and fractal measures. A major question is which of these measures should be applied, and how to implement the possible analytic algorithms. Two such analytic techniques are described here. One is approximate entropy (ApEn), a measurement designed to quantify the degree of regularity versus unpredictability in a given dataset, and the second is detrended fluctuation analysis (DFA), a fractal-related method that provides for estimation of scaling exponents.

These questions may help to guide your explorations:

1. Which measures provide the most robust separation among different subsets?
2. What other dynamic changes that occur in gait and heart rate are associated with age and/or disease?
3. Is there a loss or increase in complexity? How can you quantitatively define "complexity?"
4. What types of control systems might produce the complex fluctuations seen under healthy conditions, and how might these mechanisms be perturbed with aging or disease? For example, would you anticipate that the dynamics would become more regular or more disordered? Might there be an increase or breakdown of long-range (fractal) correlations? Should the overall variance increase or decrease?

As noted, there are many different ways of approaching such datasets, and this short list of techniques only begins to scratch the surface of the analytic possibilities. You may also wish to test other algorithms based directly on chaos theory, including assessment of fractal dimensions, plots of phase-space trajectories, and so forth, using any of a variety of suggested approaches. A key issue, and perhaps the most important one, is how any of these algorithms are limited in their applications, and what the potential pitfalls may be when these tests are applied. Does the statistical test actually yield meaningful information? Are the datasets of sufficient length, and do they meet criteria for stationarity, if that is required by the given statistical test? If the data are non-stationary, what is the nature of the non-stationarity? How does one quantify it, and how does one analyze datasets for which one or more statistical properties are changing over time?

Databases

1. Human Gait Data

Gait is a complex process with multiple inputs and numerous outputs. One of the final outputs of this highly integrated, multi-layered system is the gait cycle duration. Also known as the stride interval, this quantity reflects the rhythm of the locomotor system, and study of the temporal fluctuations in the stride interval can provide a non-invasive, quantitative window into neural control of locomotion and its changes with aging and disease.

The Gait in Aging and Disease Database contains stride interval time series collected from healthy young and old volunteers, and patients with Parkinson's disease.

2. Heart Rate Data

Five young (21-34 years old) and five elderly (68-81 years old) rigorously-screened healthy subjects underwent 120 minutes of continuous supine resting while continuous electrocardiographic (ECG) signals were collected (7).

All subjects remained in a resting state in sinus rhythm while watching the movie Fantasia (Disney) to help maintain wakefulness. The continuous ECG was digitized at 250 Hz. Each heartbeat was annotated using an automated arrhythmia detection algorithm, and each beat annotation was verified by visual inspection. The R-R interval (interbeat interval) time series for each subject was then computed. The Fantasia Database provides these time series as one-column data files with the interbeat intervals in seconds.

Software

Here are two techniques that can be used to explore these databases:

Approximate entropy (ApEn) measures the unpredictability of fluctuations in a time series. (At the request of S.M. Pincus, who first defined approximate entropy, software for calculation of ApEn has not been posted here. It is not difficult to implement the calculation in about 10 lines of code, however, using the detailed description of the algorithm for computing ApEn given here. If you do this, be sure to check your implementation using the example data provided.)

Detrended Fluctuation Analysis (DFA) can reveal the presence of long-term correlations (self-similarity) even when embedded in non-stationary time series.

Selected References Pertinent to Heart Rate and Gait Dynamics

1. Lipsitz LA, Goldberger AL. Loss of ``complexity'' and aging: potential applications of fractals and chaos theory to senescence. JAMA 1992;267:1806-1809.
2. Hausdorff JM, Peng C-K, Ladin Z, Wei JY, Goldberger AL. Is walking a random walk? Evidence for long-range correlations in stride interval of human gait. J Appl Physiol 1995;78:349-358.
3. Peng C-K, Havlin S, Stanley HE, Goldberger AL. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 1995;5:82-87.
4. Goldberger AL. Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside. Lancet 1996;347:1312-1314.
5. Ivanov PCh, Rosenblum MG, Peng C-K, Mietus J, Havlin S, Stanley HE, Goldberger AL. Scaling behaviour of heartbeat intervals obtained by wavelet-based time series analysis. Nature 1996;383:323-327.
6. Hausdorff JM, Purdon PL, Peng C-K, Ladin Z, Wei JY, Goldberger AL. Fractal dynamics of human gait: stability of long-range correlations in stride interval fluctuations. J Appl Physiol 1996;80:1448-1457.
7. Iyengar N, Peng C-K, Morin R, Goldberger AL, Lipsitz LA. Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics. Am J Physiol 1996;271:R1078-R1084.
8. Goldberger AL. Fractal variability versus pathologic periodicity: complexity loss and stereotypy in disease. Perspect Biol Med 1997;40:543-561.
9. Hausdorff JM, Mitchell SL, Firtion R, Peng C-K, Cudkowicz ME, Wei JY, Goldberger AL. Altered fractal dynamics of gait: reduced stride interval correlations with aging and Huntington’s disease. J Appl Physiol 1997;82:262-269.
10. Ho KKL, Moody GB, Peng C-K, Mietus JE, Larson MG, Levy D, Goldberger AL. 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. Circulation 1997;96:842-848.
11. Amaral LAN, Goldberger AL, Ivanov PCh, Stanley HE. Scale-independent measures and pathologic cardiac dynamics. Phys Rev Lett 1998;81:2388-2391.
12. Ivanov PCh, Amaral LAN, Goldberger AL, Stanley HE. Stochastic feedback and the regulation of biological rhythms. Europhys Lett 1998;43:363-368.
13. Hausdorff JM, Cudkowicz ME, Firtion R, Wei JY, Goldberger AL. Gait variability and basal ganglia disorders: stride-to-stride variations of gait cycle timing in Parkinson's and Huntington's disease. Movement Disorders 1998;13: 428-437.
14. Hausdorff JM, Zemany L, Peng C-K, Goldberger AL. Maturation of gait dynamics: stride-to-stride variability and its temporal organization in children. J Appl Physiol 1999;86:1040-1047.
15. Ivanov PCh, Amaral LAN, Goldberger AL, Havlin S, Rosenblum MG, Struzik Z, Stanley HE. Multifractality in human heartbeat dynamics. Nature 1999;399:461-465.