(3.3) 6 which partially specifies g as shown in Figure 3. 2. The function approximation problem amounts to finding a function f(ro) by manipulating 0 so that f(re) approximates g as closely as possible We will use this simple data set to illustrate several of the methods we develop in this chapter How do we evaluate how closely a fuzzy system f(ra) approximates the function g(x) for all xe X for a given Notice that sup(x)-f(lo) (34) is a bound on the approximation error (if it exists). However, specification of such a bound requires that the function g be completely known; however, as stated above, we know only a part of g given by the finite set G. Therefore, we are only able to evaluate the accuracy of approximation by evaluating the error between g(x) and f(re) at certain points xe X given by available input-output data. We call this set of input-output data the test set and denote it as r, where FIGURE 3. 2 The training data G generated from the function g0 23 ,1 , ,5 ,6 2 46 G ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞⎛ ⎡⎤ ⎡⎤ ⎡⎤ = ⎨⎜ ⎟ ⎜ ⎟⎜ ⎢⎥ ⎢⎥ ⎢⎥ ⎪ ⎪ ⎩ ⎭ ⎝ ⎠ ⎝ ⎠⎝ ⎣⎦ ⎣⎦ ⎣⎦ ⎞ ⎟⎬ ⎠ (3.3) which partially specifies g as shown in Figure 3.2. The function approximation problem amounts to finding a function f (x θ) by manipulating θ so that f (x θ) approximates g as closely as possible. We will use this simple data set to illustrate several of the methods we develop in this chapter. How do we evaluate how closely a fuzzy system f ( ) x θ approximates the function g (x) for all x∈ X for a givenθ ? Notice that sup{ () ( )} x X gx f x θ ∈ − (3.4) is a bound on the approximation error (if it exists). However, specification of such a bound requires that the function g be completely known; however, as stated above, we know only a part of g given by the finite set G. Therefore, we are only able to evaluate the accuracy of approximation by evaluating the error between g(x) and f (x θ) at certain points x∈ X given by available input-output data. We call this set of input-output data the test set and denote it as Γ , where 0 2 x 1234567 1 x 1 2 3 4 5 6 7 0 1234567 y FIGURE 3.2 The training data G generated from the function g