Analytically derived fuzzy membership functions

The numerical algorithms typically used for determining the fuzzy membership functions are iterative, might face convergence issues, and lack in the mathematical theory. This study suggests an analytical approach to the determination of fuzzy membership functions via variational optimization. The uncertain parameters of a membership function are modeled by variational-membership-functions. The optimal expressions for variational-membership-functions are derived via maximizing analytically the log-membership of the data samples. The uncertain parameters are then averaged to obtain the optimal membership function. Several different scenarios of the uncertain variables are built up and the membership function is designed in each scenario analytically. Experiments have been made to demonstrate the consistently better performance of the proposed methodology than the typical numerical algorithms used for designing fuzzy systems. The application potential of the methodology is demonstrated by studying the problem of 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.