When computing the Fuzzy Entropy measure, most works use an Exponential fuzzy function. Yet, there are two deviations between the works. For the exponential membership function, some use the term
$\exp^{-\frac{d_{ij}^n}{r}}$
so the power $n$ goes to the numerator only, while other works use
$\exp^{-\left(\frac{d_{ij}^n}{r}\right)^n}$
so the power $n$ goes to the whole fraction. Here, $r$ denotes the tolerance, and $d_{ij}$ is the maximum absolute difference calculated of the corresponding scalar components $X_i^m$, $X_j^m$, and $m$ is the length of the given window.
Which one is the correct? Relevant works are the following, where the variation in the formula appears: --Chen, W., Wang, Z., Xie, H., & Yu, W. (2007). Characterization of surface EMG signal based on fuzzy entropy. IEEE Transactions on neural systems and rehabilitation engineering, 15(2), 266-272.
--Chen, W., Zhuang, J., Yu, W., & Wang, Z. (2009). Measuring complexity using fuzzyen, apen, and sampen. Medical engineering & physics, 31(1), 61-68.
--Azami, H., Li, P., Arnold, S. E., Escudero, J., & Humeau-Heurtier, A. (2019). Fuzzy entropy metrics for the analysis of biomedical signals: Assessment and comparison. IEEE Access, 7, 104833-104847.
--Xiang, J., Li, C., Li, H., Cao, R., Wang, B., Han, X., & Chen, J. (2015). The detection of epileptic seizure signals based on fuzzy entropy. Journal of neuroscience methods, 243, 18-25.