Techniques to determine changing system complexity from data are evaluated. Convergence of a frequently used correlation dimension algorithm to a finite value does not necessarily imply an underlying deterministic model or chaos. Analysis of a recently developed family of formulas and statistics, approximate entropy (ApEn), suggests that ApEn can classify complex systems, given at least 1000 data values in diverse settings that include both deterministic chaotic and stochastic processes. The capability to discern changing complexity from such a relatively small amount of data holds promise for applications of ApEn in a variety of contexts.In an effort to understand complex phenomena, investigators throughout science are considering chaos as a possible underlying model. Formulas have been developed to characterize chaotic behavior, in particular to encapsulate properties of strange attractors that represent long-term system dynamics. Recently it has become apparent that in many settings nonmathematicians are applying new "formulas" and algorithms to experimental time-series data prior to careful statistical examination. One sees numerous papers concluding the existence of deterministic chaos from data analysis (e.g., ref. 1) and including "error estimates" on dimension and entropy calculations (e.g., ref. 2). While mathematical analysis of known deterministic systems is an interesting and deep problem, blind application of algorithms is dangerous, particularly so here. Even for low-dimensional chaotic systems, a huge number of points are needed to achieve convergence in these dimension and entropy algorithms, though they are often applied with an insufficient number of points. Also, most entropy and dimension definitions are discontinuous to system noise. Furthermore, one sees interpretations of dimension calculation values that seem to have no general basis in fact-e.g., number of free variables and/or differential equations needed to model a system.The purpose of this paper is to give a preliminary mathematical development of a family of formulas and statistics, approximate entropy (ApEn), to quantify the concept of changing complexity. We ask three basic questions: (i) Can one certify chaos from a converged dimension (or entropy) calculation? (ii) If not, what are we trying to quantify, and what tools are available? (iii) If we are trying to establish that a measure of system complexity is changing, can we do so with far fewer data points needed, and more robustly than with currently available tools?I demonstrate that one can have a stochastic process with correlation dimension 0, so the answer to i is No. It appears that stochastic processes for which successive terms are correlated can produce finite dimension values. A "phase space plot" of consecutive terms in such instances would then demonstrate correlation and structure. This implies neither a deterministic model nor chaos. Compare this to figures 4 a and b of Babloyantz and Destexhe (1).If one cannot hope to establish chaos, presumably one is...

Approximate entropy (ApEn) is a recently developed statistic quantifying regularity and complexity that appears to have potential application to a wide variety of physiological and clinical time-series data. The focus here is to provide a better understanding of ApEn to facilitate its proper utilization, application, and interpretation. After giving the formal mathematical description of ApEn, we provide a multistep description of the algorithm as applied to two contrasting clinical heart rate data sets. We discuss algorithm implementation and interpretation and introduce a general mathematical hypothesis of the dynamics of a wide class of diseases, indicating the utility of ApEn to test this hypothesis. We indicate the relationship of ApEn to variability measures, the Fourier spectrum, and algorithms motivated by study of chaotic dynamics. We discuss further mathematical properties of ApEn, including the choice of input parameters, statistical issues, and modeling considerations, and we conclude with a section on caveats to ensure correct ApEn utilization.

A new statistic has been developed to quantify the amount of regularity in data. This statistic, ApEn (approximate entropy), appears to have potential application throughout medicine, notably in electrocardiogram and related heart rate data analyses and in the analysis of endocrine hormone release pulsatility. The focus of this article is ApEn. We commence with a simple example of what we are trying to discern. We then discuss exact regularity statistics and practical difficulties of using them in data analysis. The mathematic formula development for ApEn concludes the Solution section. We next discuss the two key input requirements, followed by an account of a pilot study successfully applying ApEn to neonatal heart rate analysis. We conclude with the important topic of ApEn as a relative (not absolute) measure, potential applications, and some caveats about appropriate usage of ApEn. Appendix A provides example ApEn and entropy computations to develop intuition about these measures. Appendix B contains a Fortran program for computing ApEn. This article can be read from at least three viewpoints. The practitioner who wishes to use a "black box" to measure regularity should concentrate on the exact formula, choices for the two input variables, potential applications, and caveats about appropriate usage. The physician who wishes to apply ApEn to heart rate analysis should particularly note the pilot study discussion. The more mathematically inclined reader will benefit from discussions of the relative (comparative) property of ApEn and from Appendix A.

A BSTRACT : Approximate entropy (ApEn) is a recently formulated family of parameters and statistics quantifying regularity (orderliness) in serial data, with developments within theoretical mathematics as well as numerous applications to multiple biological contexts. We discuss the motivation for ApEn development, from the study of inappropriate application of dynamical systems ( complexity ) algorithms to general time-series settings. ApEn is scale invariant and model independent, evaluates both dominant and subordinant patterns in data, and discriminates series for which clear feature recognition is difficult. ApEn is applicable to systems with at least 50 data points and to broad classes of models: it can be applied to discriminate both general classes of correlated stochastic processes, as well as noisy deterministic systems. Moreover, ApEn is complementary to spectral and autocorrelation analyses, providing effective discriminatory capability in instances in which the aforementioned measures exhibit minimal distinctions. Representative ApEn applications to human aging studies, based on both heart rate and endocrinologic (hormonal secretory) time series, are featured. Heart rate (HR) studies include gender-and age-related changes in HR dynamics in older subjects, and analyses of "near-SIDS" infants. Endocrinologic applications establish clear quantitative changes in joint LHtestosterone secretory dynamics in older versus younger men (a "partial male menopause"), via cross-ApEn, a related two-variable asynchrony formulation; a disruption in LH-FSH-NPT (penile tumescence) synchrony in older subjects; and changes in LH-FSH secretory dynamics across menopause. The capability of ApEn to assess relatively subtle disruptions, typically found earlier in the history of a subject than mean and variance changes, holds the potential for enhanced preventative and earlier interventionist strategies.

High frequency heart rate spectral power (associated with parasympathetic activity) and the overall complexity of heart rate dynamics are higher in women than men. These complementary findings indicate the need to account for gender-as well as age-related differences in heart rate dynamics. Whether these gender differences are related to lower cardiovascular disease risk and greater longevity in women requires further study.