3. The proposed learning method using VQ

Let us explain the detailed algorithm of Figure 2(d'). The method is called Learning Algorithm D'. It is composed of four techniques as follows:


The general scheme of the proposed method is shown as Figure 3, where cmin, bmin, and wmin are the optimal parameters for c, b, and w.

Tmax<sup>1</sup> and Tmax2: The maximum numbers of learning time for NG and SDM.

θ and θ1: Thresholds for MSE and SDM

M0, Mmax: The size of initial and final of ranges

△M: The rate of change of the range

D and D<sup>∗</sup> : Learning data D = {(x<sup>i</sup> , yr )|i∈ZP } and D<sup>∗</sup> = {x<sup>i</sup> |i∈ZP }

n: The number of rules

4. The third learning method using VQ is the one that parameters w are determined using GIM after parameters c and b are determined by VQ using pM (x) and all parameters are updated based on SDM. That is, it is learning method composed of three phases. In the first phase, the center parameters c are determined using the probability pM (x), and b is computed from the result of center parameters. In the second phase, weight parameters w are determined by solving the interpolation problem using GIM. In the third phase, all parameters are updated using SDM for the definite number of learning time. In iterating process, the result of SDM is set to initial ones of the next process based on hill climing. Outer process is repeated until the

Figure 2. Concept of conventional and proposed algorithms: mark 1 means that initial values of w are selected randomly

5. The fourth method is the same to the one as the third method except for using pM (x) in learning process of SDM (see Figure 2(d')). This is a proposed method in this paper.

inference error becomes sufficiently small (see Figure 2(d)).

and parameters w are set to the result of SDM after the second step.

138 From Natural to Artificial Intelligence - Algorithms and Applications

E(t): MSE of inference error at step t

Emin: The minimum MSE of E for the rule number

The proposed method of Figure 3 consists of five phases: In the first phase, all values for algorithm are initialized. In the second phase, the probability pM (x) is determined for the size of range M. In the third phase, parameters c are determined by NG using pM (x), and parameters b are computed from parameters c. In the forth phase, parameters w are determined from algorithm weight(c, b). In the fifth phase, all parameters are updated using pM (x) by SDM. The optimum number n<sup>∗</sup> of rules and the optimum size M <sup>∗</sup> of range are determined in Figure 4. That is, the number M for the fixed number n is adjusted, and the optimum values of n<sup>∗</sup> and M<sup>∗</sup> with the minimum number for MSE are determined. Especially, Learning Algorithm D is same method as Learning Algorithm D' except for the step with the symbol "\*" in Figure 3. In learning steps of SDM for Learning Algorithm D, learning data is selected randomly (see Figure 2(d)).

Likewise, we also propose improved methods for Figure 2(a)–(c). In learning process of SDM for algorithm (a), (b), and (c), any learning data is selected randomly. In the proposed methods, any learning data is selected based on pM (x). These algorithms are defined as (a'), (b'), and (c').

4. Numerical simulations

classification are performed.

4.1. Function approximation

and one output with the range [0, 1];

y ¼

for (b), (c), and (d) and <sup>θ</sup> = 1.0 � <sup>10</sup>�<sup>4</sup>

ability for the regression problem.

4.2. Classification problems for UCI database

In order to compare the ability of Learning Algorithms (a'), (b'), (c'), and (d') with Learning Algorithms (a), (b), (c), and (d), numerical simulations for function approximation and pattern

The systems are identified by fuzzy inference systems. This simulation uses four systems specified by the following functions with two-dimensional input space [0, 1]<sup>2</sup> (Eqs. (25)–(28))

<sup>y</sup> <sup>¼</sup> sin <sup>π</sup>x<sup>3</sup>

1

<sup>y</sup> <sup>¼</sup> sin 2πx<sup>3</sup>

1:9 1:35 þ exp ð Þ x<sup>1</sup>

y ¼

the number of learning data is 200 and the number of test data is 2500.

1

cos ð Þþ <sup>π</sup>x<sup>2</sup> <sup>1</sup>

 sin 13ð Þ <sup>x</sup><sup>1</sup> � <sup>0</sup>:<sup>6</sup> <sup>2</sup> exp ð Þ �x<sup>2</sup> sin 7ð Þ <sup>x</sup><sup>2</sup> 

sin 10ð Þ <sup>x</sup><sup>1</sup> � <sup>0</sup>:<sup>5</sup> <sup>2</sup> <sup>þ</sup> <sup>10</sup>ð Þ <sup>x</sup><sup>2</sup> � <sup>0</sup>:<sup>5</sup> <sup>2</sup> <sup>þ</sup> <sup>1</sup>

In this simulation, Tmax<sup>1</sup> = 100000 and Tmax<sup>2</sup> = 50000 for (a) and Tmax<sup>1</sup> = 10000 and Tmax<sup>2</sup> = 5000

Table 1 shows the results for the simulation. In Table 1, the number of rules, MSEs for learning and test, and learning time (second) are shown, where the number of rules means the one when the threshold θ of inference error is achieved in learning. The result of simulation is the average value from 20 trials. As a result, the results of (a'), (b'), (c'), and (d') are almost same as the cases of (a), (b), (c), and (d) as shown in Table 1. It seems that there is no difference of the

Iris, Wine, Sonar, and BCW data from UCI database shown in Table 2 are used as the second numerical simulation [20]. In this simulation, fivefold cross validation is used. As the initial conditions for classification problem, Kc = 0.001, Kb = 0.001, Kw = 0.05, εinit = 0.1, εfin = 0.01, and <sup>λ</sup> = 0.7 are used. Further, Tmax = 50000, <sup>M</sup> = 100, and <sup>θ</sup> = 1.0 � <sup>10</sup>�<sup>2</sup> for iris and wine. Tmax = 50000, <sup>M</sup> = 200, and <sup>θ</sup> = 2.0 � <sup>10</sup>�<sup>2</sup> for BCW; and Tmax = 5000, <sup>M</sup> = 100, and <sup>θ</sup> = 5.0 � <sup>10</sup>�<sup>2</sup> for sonar are used. Table 3 shows the result of classification problem. In Table 3, the number of rules, RMs for learning, and test data are shown, where RM means the rate of misclassification. As a result, the

x<sup>2</sup> (25)

Learning Algorithms for Fuzzy Inference Systems Using Vector Quantization

http://dx.doi.org/10.5772/intechopen.79925

141

<sup>2</sup> (26)

<sup>2</sup> (27)

<sup>2</sup> (28)

, K<sup>0</sup> = 100, Kmax = 190, K = 10, Kc = 0.01, Kb = 0.01, Kc = 0.1,

Figure 3. Flowchart of Learning Algorithm D' corresponding to Figure 2(d').


Figure 4. The optimum values M <sup>∗</sup> and n<sup>∗</sup> for M and n.
