0 and t< Tmax, then go to Step A3 with te t+ 1; clsc, if E(t)s0 and t s Tmax, then the algorithm terminates Step A9: If t> Tmax and E(t)>8, then go to Step A2 with n=n+ 1 and t=1 In particular, Algorithm SDM is defined as follows Algorithm SDM(c, b, w) 01: inference error Tnax1: the maximum number of learning time n: the number of rules input: current parameters output: parameters c, b, and w after learning Steps A3 to A8 of Algorithm A arc pcrformcd 2.2. Neural gas method Vector quantization techniques encode a data space cR", utilizing only a finite set C=cil iEZ of reference vectors [18] Let the winner vector ci(o) bc dcfincd for any vector vE Vas gmIn i∈Z By using the finite set C, the space v is partitioned as V-{v∈V|-cl‖s|-c‖forj∈Z} (10) where V-Uiez, Vi and VinI- for ifj Learning Algorithms for Fuzzy Inference Systems Using Vector Quantization 133 nttp: //dx. doi. org/10.5772/intechopen 79925 The evaluation function for the partition is defined by E=∑∑ 1=1∈b where n: =V Let us introduce the neural gas method as follows [18] For any input data vector v, the neighborhood ranking cik for keZ determined, being the reference vector for which there are k vectors Ci with U-c;‖<|-ck‖ (12) Let the number k associated with each vector c, denoted by ki (v, ci). Then, the adaption step for adjusting the parameters is given b △ hx(k;(O,C;)·(乙-C;) (13) ha( ki(v, ci))=exp(-ki(v,ci/ where e∈[,1andA>0 Let the probability of v selected from V be denoted by plo) The flowchart of the conventional neural gas algorithm is shown in Figure 1 [18 where eintr fin, and Tmax are learning constants and the maximum number of learning, respectively. The method is called learning algorithm NG Using the set D, a decision procedure for center and width parameters is given as follows Algorithm Center(c) ( ∈Z, plr: the probability of x selected for xED Step 1: By using p(x) for x E D, NG method of Figure 1 [16, 18] is performed As a result, the set C of reference vectors for D" is determined where C=n Step 2: Each valuc for center paramcters is assigncd to a reference vector. Let ∑( (15) ni where Ci and n: are the set and the number of learning data belonging to the ith cluster Ci and C=U1 Ci and n=∑1 As a result, center and width parameters are determined from algorithm center(c) 134 From Natural to Artificial Intelligence -Algorithms and Applications Given e int tinman Lett=1 Each reference vector Wi is selected randomly. Given v E V with(o) Determine the neighborhood ranking k(vW)fori∈Zr Update wi for i∈Z t←t+1 using Eqs. 14, (15)and t/T Eint f Eint Eint/Efin miax END Fi Learning Algorithm B using Algorithm Center(c) is introduced as follows [16, 17] Learning algorithm b 0: threshold of mse wr: maximum number of learning time for ng Tmax: maximum number of learning timc for SDM M: the size of ranges n the number of rules Step 1: Initialize Step 2: Center and width parameters are determined from Algorithm Center(P)and the set D Step 3: Parameters c, b, and w are updated using Algorithm SDM(c, b, w) Step 4: If E(t)se, then algorithm terminates else go to Step 3 with n+-n+ 1 and t+t+1 Learning Algorithms for Fuzzy Inference Systems Using Vector Quantization 135 nttp: //dx. doi. org/10.5772/intechopen 79925 2. 3. The probability distribution of input data based on the rate of change of output It is known that many rules are needed at or near the places where output data change quickly in fuzzy modeling. Then, how can we find the ratc of output change? The probability pM(x) one method to perform it. As shown in Eqs. (16) and(17), any input data where output hanges quickly is selected with the high probability, and any input data where output changes slowly is selected with the low probability, where M is the size of range considering output change Based on the literature [13], the probability( distribution is defined as follows Algorithm Prob(Pm(x)) Input:D={(x,y)lp∈Zp}andD"={(x2)lp∈Zp} Output: PM(r) Step 1: Give an input data x'ED", we determine the neighborhood ranking(r0, xl,.,,xk x' p-1)of the vector x' with x0=x', x'1 being closest to x and xk(k=0,., P-1) being the vector xfor which there are k vectors xwith x-xi