Measuring complexity using FuzzyEn, ApEn, and SampEn

This paper compares three related measures of complexity, ApEn, SampEn, and FuzzyEn. Since vectors’ similarity is defined on the basis of the hard and sensitive boundary of Heaviside function in ApEn and SampEn, the two families of statistics show high sensitivity to the parameter selection and may be invalid in case of small parameter. Importing the concept of fuzzy sets, we developed a new measure FuzzyEn, where vectors’ similarity is defined by fuzzy similarity degree based on fuzzy membership functions and vectors’ shapes. The soft and continuous boundaries of fuzzy functions ensure the continuity as well as the validity of FuzzyEn at small parameters. The more details obtained by fuzzy functions also make FuzzyEn a more accurate entropy definition than ApEn and SampEn. In addition, similarity definition based on vectors’ shapes, together with the exclusion of self-matches, earns FuzzyEn stronger relative consistency and less dependence on data length. Both theoretical analysis and experimental results show that FuzzyEn provides an improved evaluation of signal complexity and can be more conveniently and powerfully applied to short time series contaminated by noise.