On Consistency of Cross-Approximate Entropy in Cardiovascular and Artificial Environments

Abstract

Cross-approximate entropy (XApEn) quantifies the mutual orderliness of simultaneously recorded time series. Despite being derived from the firmly established solitary entropies, it has never reached their reputation and deployment. The aim of this study is to identify the problems that preclude wider XApEn implementation and to develop a set of solutions. Exact expressions for XApEn and its constitutive parts are derived and compared to values estimated from artificial data. This comparison revealed vast regions within the parameter space that do not guarantee reliable probability estimation, making XApEn estimates inconsistent. A simple correction to one of the XApEn procedural steps is proposed. Three sets of formulae for joint parameter selection are derived. The first one is intended to maximize threshold profile. The remaining ones minimize XApEn instability according to the strong and weak criteria. The derived expressions are verified using cardiovascular signals recorded from rats (long signals) and healthy volunteers (short clinical signals), proposing a change of traditional parameter guidelines.

Sequences and antisequences in hypertensive patients under therapyPlatform

“…For this reason, the probability zero is omitted and the averaging performed without the omitted probability. Different ways of correction exist, systematized in [10], but the comparison to the theoretical (and therefore correct) values in artificial signals with known distributions has shown that this simple correction outperforms the more complex ones.…”

“…In addition to the method for the quantification of baroreflex sensitivity [4,5], classical contributions rely on power spectrum density approach, as well as on multivariate autoregressive analysis [6]. More recent analyses implement tools designed specifically to assess the mutual orderliness of simultaneously recorded time series and the level of their (a)synchrony, such as cross-entropy [7,8] and its variations such as [9], [10], [3], including the multi-scale cross-entropy [11,12,13,14,16].…”

“…In respect to the previous works, our analysis implement an original method that increases the reliability of cross-entropy [10], as well as the analysis of SBP-RRI dependency structures performed by copula density [12]. The aim of this study is to apply the modified cross-entropy and copula density methods in patients before and immediately after Verapamil administration and to test the possibilities of the methods to capture the changes in cardiovascular parameters methods in respect to the statistics of baroreflex sequences and antisequences.…”

Artifacts in dependency structures of Portapres® signals

“…The most implemented types of entropy estimates used for measuring the similarity and self-similarity of biomedical data are Approximate entropy (ApEn) and Sample entropy (SampEn). Approximate entropy was introduced in [19] and well explained in [20,21]. It initiated many variations (FuzzyEn, BinEn) [22,23] but the SampEn [24] is the method that, considering its quotation rate [25], almost equals its predecessor ApEn.…”

“…Finding the threshold level appropriate for the observed time series is a critical issue which has been elaborated in numerous contributions [27]- [29]. For this paper we used the method for cross-entropy proposed in [20].…”