A Simple Method to Extract Fuzzy Rules by Measure of Fuzziness

Jieh-Ren Chang

Nai-Jian Wang

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-boxes are

overlapped, a recursive process are applied to additive activation hyper-box in these uncertainty-overlap regions.

Third, the stop criterion for the recursive process --by measure of fuzziness.

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

Measure of Fuzziness of a Fuzzy Set

To measure uncertainty of vagueness .

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

Degree of fuzziness of fuzzy set

(1) ))](1(log))(1(

)(log)([)(

2

2

Xx AA

AA

xx

xxAf

Ain of membership of grade theis )(

A offunction membership theis (.)

xx

A

A

Normalized measure of fuzziness

(2) )(

)(ˆX

AfAf

Xset universal theofy cardinalit thedenotes X

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,

Rk can be rewrote by the T-norm operator with min operation in the following:

The membership value of this rule Rk represented as:

),....,/()()...()( 2121... 22121 nnAAAnXXXk xxxxxxRnkk

))(),...,(),(min(),....,( 2121 21 nAAAnR xxxxxxnkkkk

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

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.

A Fuzzy-Neural Network Structure

..........

..........

1a 2a na

1c 2c cc

Hyperbox node layer

Input layer..........

Output layer

1B 3B2B

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.

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.

1a 2a na Input layer

Output layer

kB)1(1

1kb )(mblkr…

The kth group in

the second layer

… …

…

kc

Max

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.

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

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

Original Iris Data

Pseudo-Iris data

Sepal length

Sepal width

Petal length

Petal width

Setosa 4.4~5.8 2.9~4.4 1.0~1.9 0.1~0.6

Versicolor 5.0~7.0 2.0~3.4 3.0~5.1 1.0~1.8

Verginica 4.9~7.7 2.5~3.8 4.8~6.9 1.4~2.5

單位： cm

Randomly generated area

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.