Quantification of evolution from order to randomness in practical time series analysis

The principal focus of this chapter is the description of a recently developed, readily usable regularity statistic, ApEn, that quantifies the continuum from perfectly orderly to completely random in time series data. Several properties of ApEn facilitate its utility for practical time series analysis: (1) ApEn is nearly unaffected by noise of magnitude below a de facto specified filter level; (2) ApEn is robust to outliers; (3) ApEn can be applied to time series of 100 or more points, with good confidence (established by standard deviation calculations); (4) ApEn is finite for stochastic, noisy deterministic, and composite (mixed) processes, the last of which are likely models for complicated biological systems; (5) increasing ApEn corresponds to intuitively increasing process complexity in the settings of (4). This applicability to medium-sized data sets and general stochastic processes is in marked contrast to capabilities of "chaos" algorithms such as the correlation dimension, which are properly applied to low-dimensional iterated deterministic dynamical systems. The potential uses of ApEn to provide new insights in biological settings are thus myriad, from a perspective complementary to that given by classic statistical methods. The ApEn statistic is typically calculated by a computer program, with a FORTRAN listing for a "basic" code referenced above. It is imperative to view ApEn as a family of statistics, each of which is a relative measure of process regularity. For proper implementation, the two input parameters m (window length) and r (tolerance width, de facto filter) must remain fixed in all calculations, as must N, the data length, to ensure meaningful comparisons. Guidelines for m and r selection are indicated above. We have found normalized regularity to be especially useful; "r" is chosen as a fixed percentage (often 15 or 20%) of the SD of the subject rather than of a group SD. This version of ApEn has the property that it is decorrelated from process SD, in that it remains unchanged under uniform process magnification or reduction; thus we can entirely separate the questions of SD change and regularity change in data analysis. Because regularity questions are thematically orthogonal to the type of information that, for example, moment statistics ascertain, we highly recommend that ApEn be used in conjunction with other such statistics, rather than as a sole indicator of process typicality. Last and yet foremost, we recommend the following order of detail in Practical analysis. First, either visually or algorithmically, eliminate outliers.(ABSTRACT TRUNCATED AT 400 WORDS)