Area 1 of Approximate Entropy as a Fast and Robust Tool to Address Temporal Organization

Aims: To evaluate the consistency and robustness of an informational entropy analytical tool derived from Approximate Entropy (ApEn).

Study Design: A set of in machina time-series of known properties were generated to test and compare the proposed tool with the standard ApEn and with peak-ApEn.

Place and Duration of Study: Laboratory of Energetics and Theoretical Physiology, Dept. Physiology, Biosciences Institute, University of São Paulo. From April 2014 to May 2015.

Methodology: The proposed tool consists in obtaining a detailed tolerance vector with more than 100 values and, then, to compute ApEn for window m = 1 for each one of these tolerance values. This creates a curve that is numerically integrated using a normalized tolerance vector as the basis, thus obtaining the area under the curve of m = 1 ApEn (a1ApEn). In order to make comparisons, 17 time-series from different generating processes were constructed using Matlab R2013a. Employing the above-cited analytical tools, we approached the following queries: (a) for a given process, how variable is the estimator value? (b) is a1ApEn more consistent than peak-ApEn in classifying different processes?

Results: The answer for (a) is that, in relation to ApEn, the variance of a1ApEn is significantly lower in 16 cases (all P < .01, F-test for sample variance), and we explain why the one exception occurs. In relation to peak-ApEn, the variance is lower for all 17 series (all P < .01). The answer for (b) is that a1ApEn is able to correct inconsistencies found when using peak-ApEn (all P < .01, Student’s t-test).

Conclusion: The proposed tool, the area under the curve for ApEn of window 1 (a1ApEn) is objective and more consistent than both the ApEn and the peak-ApEn estimators.