Abstract
The goal of this paper is to investigate the changes of entropy estimates when the amplitude distribution of the time series is equalized using the probability integral transformation. The data we analyzed were with known properties—pseudo-random signals with known distributions, mutually coupled using statistical or deterministic methods that include generators of statistically dependent distributions, linear and non-linear transforms, and deterministic chaos. The signal pairs were coupled using a correlation coefficient ranging from zero to one. The dependence of the signal samples is achieved by moving average filter and non-linear equations. The applied coupling methods are checked using statistical tests for correlation. The changes in signal regularity are checked by a multifractal spectrum. The probability integral transformation is then applied to cardiovascular time series—systolic blood pressure and pulse interval—acquired from the laboratory animals and represented the results of entropy estimations. We derived an expression for the reference value of entropy in the probability integral transformed signals. We also experimentally evaluated the reliability of entropy estimates concerning the matching probabilities.
Highlights
The sampling theorem [1] paved a way for pervasive signal processing within the scientific fields where it was once inconceivable
This paper aims to investigate the approximate entropy (ApEn), SampEn, and XEn of pulse interval (PI)-transformed cardiovascular time series, and PIT is closely related to the copula density
The aim of this paper was to apply the ApEn-based entropies and cross-entropies to the signals submitted to the probability integral transformation
Summary
The sampling theorem [1] paved a way for pervasive signal processing within the scientific fields where it was once inconceivable. Tools developed for classical thermodynamics or communications engineering found new multidisciplinary implementation. The function developed to estimate the uncertainty of the communication signals—entropy [2]. Other entropy concepts were accepted as well—Kolmogorov–Sinai [3], Grassberger et al [4] and Eckmann et al [5], despite difficult implementation and firm theoretical framework. Pincus [6] proposed the approximate entropy (ApEn) that avoids the rigid mathematical requirements of its theoretical predecessors ( the name—approximate). The researchers readily accepted ApEn and its modification SampEn (sample entropy, [7]), with a commendation of the rapidly growing number of citations [8]
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