| Approximate Entropy (ApEn) is a statistic which has been widely applied in a variety of scientific disciplines, but especially in medicine. If one wishes to make an assessment as to whether one sequence appears to be more or less regular than another, ApEn assigns values to each sequence and a comparison of these values can be used to make such an assessment. In such settings, the method has proven to be of enormous potential. The ideas were presented in the 1990's in a series of PNAS papers by Steve Pincus. A search in PubMed for the use of the idea brings up over 560 articles which have applied the technique in medicine and biology alone.;The strength of the idea is that it does not depend upon there being any underlying parametric model for its calculation. What has been "the missing piece" in the development of the methodology is the inability to calculate an accurate standard error for the ApEn statistic (and its generalization: Cross ApEn). In order to decide if two ApEn values are similar or dissimilar, comparisons had to be made via a relatively crude upper hound that did not utilize any specifics of the empirical distribution function.;The present research establishes the asymptotics and derives a justified method by which to numerically calculate, from the data, their standard errors. The series are assumed to be strictly stationary and to satisfy a strong mixing condition. Moreover, there is a small sample bias in the ApEn statistic, which has been somewhat limiting to the small-sample use of the method. In the present work we develop a jackknife procedure by which to reduce such bias. We also developed a technique for the detection of the feedback and/or feedforward interaction time delays between series. The above methods were then applied to clinical problems involving ACTH-cortisol data and to glucose ICU monitoring. |
