Size-Related Properties of Area1 of Approximate Entropy to Characterize Time-series Organization

Aims: There are several entropy estimators to address the organization of time-series. However, the behavior of a given estimator in relation to the size N of the data is not often studied in terms of improving the analysis. Here, we investigate size-related properties of the estimator a1ApEn (area1 of approximate entropy) in order to establish how such properties can improve time-series analysis.

Study Design/Methodology: We established a set of 14 different generating processes, including deterministic maps and limited and unlimited random distributions. Then, we created several vectors of five different sizes (N = 100, 200, 400, 500, 1000) for each process, and a set of indicators (maximum, minimum and mean a1ApEn values) was taken. The correlation between a given indicator and log10(N) was classified as greater or lower than zero, or non-significant, creating a pattern of correlations for each process. Next, we perform a similar analysis in a resampling procedure from vectors of 2,000 points for the same generating processes. In addition, we analyzed heart rate dynamics and solar wind cycles with this method in order to show the applicability of the technique.

Results: The main result is that the patterns of the correlations between indicators and log10(N) are able to segregate the different generating process.

Conclusion: The use of a resampling procedure along with the size-related correlations of the nonlinear estimator a1ApEn is an effective method to discern different generating processes underlying empirical time-series. The method allows for the use of data sets of different sizes in comparisons among results