Dear colleagues
When computing the Fuzzy Entropy measure, most works use an Exponential fuzzy function. Yet, there are two deviations between the works.
Some use the term
exp(-dij^n/r)
so the power 'n' goes to the numerator only, while other works use
exp(-(dij/r)^n)
so the power 'n' goes to the whole fraction.
Which one is the correct?
Relevant works are the following, where the issue 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.
Fuzzy entropy is a type of entropy used in fuzzy systems and fuzzy clustering analysis. It is a measure of the complexity or randomness of a fuzzy set, which is a set in which each element has a degree of membership between 0 and 1.
There are several definitions of fuzzy entropy, and the choice of exponential term depends on the specific definition being used. Some common choices include:
The natural logarithm (base e): This is used in the definition of Tsallis entropy, which is a generalization of Shannon entropy that includes a parameter q that can take on values other than 1. The fuzzy Tsallis entropy is defined as:
Hq(X) = [1 - ∑μi^q]/(q - 1)
where X is the fuzzy set, μi is the degree of membership of the i-th element of X, and q is the Tsallis entropy parameter.
The base 2 logarithm: This is used in the definition of fuzzy Shannon entropy, which is a measure of the uncertainty or randomness of a fuzzy set. The fuzzy Shannon entropy is defined as:
H(X) = - ∑μi log2(μi)
where X is the fuzzy set and μi is the degree of membership of the i-th element of X.
Other exponential terms: Depending on the specific application and definition, other exponential terms such as the base 10 logarithm or a power law function may be used.
It is important to carefully choose the appropriate definition and exponential term for the specific application and context, as different choices may lead to different results and interpretations
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