ABSTRACT
Approximate entropy (ApEn) has been recently introduced as a quantification of regularity in time-series data, motivated by applications to relatively short, noisy data sets (Pincus, 1991). Mathematically, ApEn is part of a general development as the rate of entropy for an approximating Markov Chain to a process (Pincus, 1992). In applications to heart rate, findings have discrimated groups of subjects via ApEn, in instances where c1assical (mean, SO) statistics did not show c1ear group distinctions (Fleisher, Pincus, & Rosenbaum, 1993; Kaplan et al. , 1991; Pincus, Cummins, & Haddad, 1993; Pincus, Gladstone, & Ehrenkrantz, 1991; Pincus & Viscarello, 1992; Ryan, Goldberger, Pincus, Mietus, & Lipsitz, 1994). In applications to endocrine hormone secretion data based on as few as N = 72 points, ApEn has provided vivid distinctions (p < 10-8) between actively diseased subjects and normals,
We next describe ApEn implementation and interpretation, indicating its utility to distinguish correlated stochastic processes, and composite deterministic/stochastic models. We discuss the key technical idea that motivates ApEn, that one need not fully reconstruct an attractor to discriminate in a statistically valid manner-marginal probability distributions often suffice for this purpose. We discuss why algorithms to compute, for example, correlation dimension and the Kolmogorov-Sinai (K-S) entropy often work weil for true dynamical systems, yet sometimes operationally confound for general models. This contrast indicates the need for a thematically faithful modification of a parameter such as the K-S entropy for general applications so that visual intuition matches numerical results, for broad c\asses of stochastic processes as weil as for dynamical systems. We provide a mechanistic hypothesis suggesting greater regularity in a wide range of evolving, complicated systems. Finally, we illustrate the statistical breadth of ApEn via several disparate time-series applications, both to actual "field" data (growth hormone within endocrinology, and Dow Jones stock index), and to theoretical models (coupled stochastic differential equations, and new rejection criteria for a variety of Li.d. or "random" processes).
