A fuzzy adaptive network approach to parameter estimation in cases where independent variables come from an exponential distribution

In a regression analysis, it is assumed that the observations come from a single class in a data cluster and the simple functional relationship between the dependent and independent variables can be expressed using the general model; Y=f(X)+εY=f(X)+ε. However; a data cluster may consist of a combination of observations that have different distributions that are derived from different clusters. When faced with issues of estimating a regression model for fuzzy inputs that have been derived from different distributions, this regression model has been termed the ‘switching regression model’ and it is expressed with YL=fL(X)+εL(L=∏i=1pli). Here lili indicates the class number of each independent variable and pp is indicative of the number of independent variables [J.R. Jang, ANFIS: Adaptive-network-based fuzzy inference system, IEEE Transaction on Systems, Man and Cybernetics 23 (3) (1993) 665–685; M. Michel, Fuzzy clustering and switching regression models using ambiguity and distance rejects, Fuzzy Sets and Systems 122 (2001) 363–399; E.Q. Richard, A new approach to estimating switching regressions, Journal of the American Statistical Association 67 (338) (1972) 306–310].In this study, adaptive networks have been used to construct a model that has been formed by gathering obtained models. There are methods that suggest the class numbers of independent variables heuristically. Alternatively, in defining the optimal class number of independent variables, the use of suggested validity criterion for fuzzy clustering has been aimed. In the case that independent variables have an exponential distribution, an algorithm has been suggested for defining the unknown parameter of the switching regression model and for obtaining the estimated values after obtaining an optimal membership function, which is suitable for exponential distribution.