Analytically Derived Fuzzy Membership Functions
The numerical algorithms typically used for determining the fuzzy membership functions are inexact, slow, and lack in the mathematical theory. This study suggests an analytical approach to the determination of fuzzy membership functions by using the variational optimization method. The uncertain parameters of a membership function are modeled by variational-membership-functions. The optimal expressions for variational-membership-functions are derived by maximizing analytically the log-membership of the data samples. The uncertain parameters are then averaged over the derived optimal variational-membership-functions leading to the so-called optimal determination of the membership function. Several different scenarios of the uncertain variables are built up and the membership function is designed in each scenario analytically. The application potential of the methodology is demonstrated by studying a biomedical signal analysis problem. Another practical application is concerned with the image matching and imaging based personal identification. This study and more studies in this direction will pave the way for the fuzzy researchers to reduce their dependance on numerical algorithms by designing the fuzzy systems in a more analytical manner.