A Simple Method to Extract Fuzzy Rules by Measure of Fuzziness

1. A Simple Method to Extract Fuzzy Rules by Measure of Fuzziness Jieh-Ren Chang Nai-Jian Wang

2. Abstract • Use a variable fuzzy-neural network structure to implement the fuzzy rules system. • First, we extract fuzzy rules from different class region which was named as activation hyper-box. • Second, when the activation hyper-boxesare overlapped, a recursive process are applied to additive activation hyper-boxin these uncertainty-overlap regions. • Third, the stop criterion for the recursive process --by measure of fuzziness.

3. Relation between activation hyper-boxes and overlap regions by 2-dimensional example

4. Contents： • Motivation. • Introduction. • Measure of fuzziness for a fuzzy set. • Measure of fuzziness of a fuzzy rule in a fuzzy system. • Fuzzy-neural network. • Learning algorithm. • Compare our method with other methods. • Conclusions.

5. Motivation • To extract more efficiently fuzzy rules from numerical information data in classification problem. • To save computation cost • To get available rules and cancel redundant rules

6. Introduction • Human can always collect the knowledge to discriminate the uncertainty or ambiguous data by their experience. • But computer still can’t be dealt perfectly in classification problem. • So, many methods are still proposed to improve the performance of classification problem.

7. The methods of classification problem are divided into four groups: 1) Statistical method: It is not practical in solving classification problem in a real world. 2) Neural network: It is a system that is constructed to make use of some organizational principles like human brain. It is good for many application.

8. 3)Fuzzy inference engine: By querying experts’ experience or other techniques directly from training data to build fuzzy rule database. 4)Hybrid neural-fuzzy technique: It combines the fuzzy inference and neural network theory to computer-based pattern recognition.

9. Hong and Lee, proposed a method based on the fuzzy clustering technique to setup the decision tables. But they need to determine the scaling it usually takes more computation time.

10. Hong and Chen, they propose the other method to decrease the computation time, but it still generates many rules and take very much computation process, when the training data increase.

11. Wu and Chen have a fuzzy learning algorithm base on theα-cut, can induce the fuzzy rule and reaches a higher average classification ratio. But we don’t know how to select the α-cut . • P.K. Simpson setup the fuzzy rules by an expansion-contraction, it usually generated too many hyper-box that mean too many rules to be concerned.

12. S. Abe and M-S. Lan extract the fuzzy rules by resolving overlaps, it can decrease the learning process. But there some drawback in following points: 1)It needs more computation time to resolve overlaps when the data include many classes. 2)It can’t be resolved in some critical condition. 3)It generate many meaningless fuzzy rules as the data are chaos.

13. Our propose is to decrease the computation time and to extract more efficient fuzzy rule, the method is described in the following steps: 1)Find the activation hyper-box. 2)Find uncertainty overlap. 3)Extracts fuzzy rules . 4)Construct an easy and efficient neural network by measure of fuzziness.

14. Measure of Fuzziness of a Fuzzy Set • To measure uncertainty of vagueness .

15. Measure of fuzziness is a function ƒ, the function ƒ satisfies the following axioms: Axiom 1: ƒ(A)=0 if only if A is a crisp set. Axiom 2: If A B, then ƒ(A) ƒ(B). Where A B denotes that A is shaper than B. Axiom 3: ƒ(A) assumes the maximum value if and only if A is maximally fuzzy

16. Degree of fuzziness of fuzzy set

17. Normalized measure of fuzziness

18. Measure of Fuzziness of a Fuzzy Rule in a Fuzzy System • In this section, we define a classification system by a sequence of multi-input-single-output fuzzy rules as follows • n is the number of attribute of the classification system • c is the number of class of the system • Ai,k is the linguistic label, i=1,2,…n,

19. Rk can be rewrote by the T-norm operator with min operation in the following: • The membership value of this rule Rk represented as:

20. We can define the measure of fuzziness of the rule Rk in the fuzzy rule system as:

21. According to the formula (3) • We can decide the rule Rk is worth to exist in this rule-based system or not necessary. • If the rule have high measure of fuzziness of a rule, it means too much uncertain for this rule.

22. A Fuzzy-Neural Network Structure

23. A variable structure • We will leave the rule which is very efficient and useful, so the number of nodes in the second layer are variable. • We will reduce the cost, because the redundant second layer nodes are eliminated.

24. Second layer includes two Sub layer • the first sub layer is configured by the hyper-box nodes which are created from our proposed algorithm • the second sub-layer is a maximum-operation node, which takes the maximum values of inputs from the first sub-layer.

27. Learning Process • Step1: set level = 1. • Step2: Set up the hyper-boxes and membership function for each class. • Step3: Find the overlap among the activation hyper-boxes of level l ,then l=l+1.

28. Step4:Extract activation hyper-boxes and set up feature as in step 1. • Step5:Calculate the measure of fuzziness for each extracted fuzzy rule. If it is bigger than threshold, we discard this rule. • Step6:If none of hyper-box exist in Step 4, then stop the process, else go to Step 2. • Step7:Build up the fuzzy-neural network structure by these extracted fuzzy rules

29. Performance Evaluation • We use Fisher’s iris data, there are three kinds of flowers, four kinds of attributes. Three flowers: Setosa Versicolor Verginica Four attributes: Sepal length Sepal width Petal length Petal width

30. Original Iris Data

31. Pseudo-Iris data Randomly generated area 單位：cm

33. Conclusions • By this proposed method, we can find more efficient fuzzy rules. • It generates fewer fuzzy rules than other methods [9][10][11][14]. • It avoids a huge matrix computation [9] so its computation time decreases. • It provides a simple recursive process and stopping criteria to extract the fuzzy rules in the uncertainty-overlap region. Thus, the network structure is simple and easy to implement. • The classifier can be generated even for a large scale of data pattern.