Fuzzy Entropy Metrics for the Analysis of Biomedical Signals: Assessment and Comparison

Fuzzy entropy (FuzEn) was introduced to alleviate limitations associated with sample entropy (SampEn) in the analysis of physiological signals. Over the past decade, FuzEn-based methods have been widely used in various real-world biomedical applications. Several fuzzy membership functions (MFs), including triangular, trapezoidal, Z-shaped, bell-shaped, Gaussian, constant-Gaussian, and exponential functions have been employed in FuzEn. However, these FuzEn-based metrics have not been systematically compared yet. Since the threshold value r used in FuzEn is not directly comparable across different MFs, we here propose to apply a defuzzification approach using a surrogate parameter called 'center of gravity' to re-enable a fair and direct comparison. To evaluate these MFs, we analyze several synthetic and three clinical datasets. FuzEn using the triangular, trapezoidal, and Z-shaped MFs may lead to undefined entropy values for short signals, thus providing a very limited advantage over SampEn. When dealing with an equal value of the center of gravity, the Gaussian MF, as the fastest algorithm, results in the highest Hedges' g effect size for long signals. Our results also indicate that the FuzEn based on exponential MF of order four better distinguishes short white, pink, and brown noises, and yields more significant differences for the short real signals based on Hedges' g effect size. The triangular, trapezoidal, and Z-shaped MFs are not recommended for short signals. We propose to use FuzEn with Gaussian and exponential MF of order four for characterization of short (around 50-400 sample points) and long data (longer than 500 sample points), respectively. We expect FuzEn with Gaussian and exponential MF as well as the concept of defuzzification to play prominent roles in the irregularity analysis of biomedical signals. The MATLAB codes for the FuzEn with different MFs are available at https://github.com/HamedAzami/FuzzyEntropy_Matlab.