Fuzzy System

A fuzzy inference system (FIS) is a nonlinear mapping of input feature vectors to a scalar output [Mendel, 1995]. It consists of four components: Fuzzifier, Rules, Inferences and Defuzzifier.

Fuzzifier It converts the crisp input to fuzzy inputs (sets). In general if 'x' is crisp input, we apply a function (membership function) f:x --> y There are various types of membership functions:

1. Triangular Membership Function: Here we have a triangular curve, mapping x, based on three values a,b,c. 'a' and 'c' are the feet of the triangle on the x-axis, and 'b' is the peak of the triangle. Mathematically, the function is defined as:

f(x;a,b,c)= 0                ∀ x ≤ a
                  ^(x-a) / _(b-a)     ∀ a < x ≤ b
                  ^(c-x) / _(c-b)     ∀ b < x < c
                    0                ∀ x ≥ c


2. Trapezoidal Membership Function: The function depends on four parameters a,b,c and d. The parameters 'a' and 'd' determine the feet of the trapezoid, while parameters 'b' and 'c' determine the shoulders of the trapezoid. Mathematically we define it as:
   f(x;a,b,c,d)= 0                ∀ x ≤ a
                     ^(x-a) / _(b-a)     ∀ a < x ≤ b
                       1                ∀ b < x ≤ c
                     ^(d-x) / _(d-c)     ∀ c < x < d
                       0                ∀ x ≥ d
   
3. Symmetric Gaussian Membership Function : Symmetric Gaussian function is defined by two parametrs, s and v. Mathematically it is defined as:
f(x;s,v) = e ^-(x-v)2/2c2