Assume that all that is available to choose the parameters B of the fuzzy system f(rO) is some part of the function g in the form of a finite set of input-output data pairs (i.e, the functional mapping implemented by g is largely unknown The ith input-output data pair from the system g is denoted by(x,y), where x∈X,y∈Y,andy=g(x). We let x'=Lx,, x2,,n] represent the input vector for the ith data pair. Hence, x', is the jth element of the ith data vector(it has a specific value and is not a variable). We call the set of input output data pairs the training data set and denote it by G=(,y),,(",y cXxY (32 where M denotes the number of input-output data pairs contained in g. for convenience. we will sometimes use the notation d(i) for data pair(x,y') 16
16 ◼ Assume that all that is available to choose the parameters of the fuzzy system is some part of the function g in the form of a finite set of input-output data pairs (i.e., the functional mapping implemented by g is largely unknown). The ith input-output data pair from the system g is denoted by ,where , , and . We let represent the input vector for the ith data pair. Hence, is the jth element of the ith data vector (it has a specific value and is not a variable). We call the set of inputoutput data pairs the training data set and denote it by (3.2) ◼ where M denotes the number of input-output data pairs contained in G. For convenience, we will sometimes use the notation d(i) for data pair . f x( ) ( , ) i i x y i x X i y Y ( ) i i y g x = 1 2 [ , ,..., ] i i i i T n x x x x = i j x 1 1 {( , ),...,( , )} M M G x y x y X Y = ( , ) i i x y
a To get a graphical picture of the function approximation problem, see Figure 3. 1. This clearly shows the challenge: it can certainly be hard to come up with a good function f to match the mapping g when we know only a little bit about the association between X and Y in the form of data pairs G. moreover, it may be hard to know when we have a good approximation--that is, when approximates over the whole space of inputs X
17 ◼ To get a graphical picture of the function approximation problem, see Figure 3.1. This clearly shows the challenge; it can certainly be hard to come up with a good function f to match the mapping g when we know only a little bit about the association between X and Y in the form of data pairs G. Moreover, it may be hard to know when we have a good approximation—that is, when approximates over the whole space of inputs X
g G FIGURE 3. 1 Function mapping with three known input-output data pairs 18
18 FIGURE 3.1 Function mapping with three known input-output data pairs
To make the function approximation problem even more concrete, consider a simple example. Suppose that n=2,Xc界2,Y=[0,10]andg:X→>Y.LetM 3 and the training data set (3.3) 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(0)approximates g as closely as possible We will use this simple data set to illustrate several of the methods we develop in this chapter 19
19 ◼ To make the function approximation problem even more concrete, consider a simple example. Suppose that n=2, , Y = [0, 10] and . Let M = 3 and the training data set (3.3) ◼ which partially specifies g as shown in Figure 3.2. The function approximation problem amounts to finding a function by manipulating so that approximates g as closely as possible. We will use this simple data set to illustrate several of the methods we develop in this chapter. 2 X g X Y : → 0 2 3 ,1 , ,5 ,6 2 4 6 G = f x( ) f x( )
How do we evaluate how closely a fuzzy system f(0) approximates the function g() for all xeX for a given 0? Notice that sup(g(x)f(xpe) (3.4) x∈X 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()and f(r0)at certain points xEX given by available input-output data. We call this set of input-output data the test set and denote it as t where 20
20 ◼ How do we evaluate how closely a fuzzy system approximates the function g (x) for all for a given ? Notice that ◼ (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 at certain points given by available input-output data. We call this set of input-output data the test set and denote it as , where f x( ) x X sup ( ) ( ) x X g x f x − f x( ) x X