Transcript A Simple Method to Extract Fuzzy Rules by Measure of

A Simple Method to Extract Fuzzy Rules by Measure
of Fuzziness
Jieh-Ren Chang
Nai-Jian Wang
Abstract
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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 uncertaintyoverlap 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：
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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
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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
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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.
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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.
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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.
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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.
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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 αcut .
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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.
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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.
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Measure of Fuzziness of a Fuzzy
Set
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To measure uncertainty of vagueness .
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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
f ( A)   
x X
[  A ( x) log 2  A ( x) 
(1   A ( x)) log 2 (1   A ( x))]
 A (.) is the membership function of A
 A ( x ) is the grade of membership of x in A
(1)
Normalized measure of
fuzziness
f ( A)
ˆ
f ( A) 
(2)
X
X denotes the cardinalit y of the universal set X
Measure of Fuzziness of a Fuzzy
Rule in a Fuzzy System
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In this section, we define a classification system by a
sequence of multi-input-single-output fuzzy rules as
follows
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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,
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Rk can be rewrote by the T-norm operator with min
operation in the following:
Rk  
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X 1  X 2 ... X n
 A ( x1 )   A ( x2 )...   A ( xn ) /( x1 , x2 ,....xn )
1k
2k
2n
The membership value of this rule Rk represented as:
 R ( x1 , x2 ,....xn )  min(  A ( x1 ),  A ( x2 ),...,  A ( xn ))
k
1k
2k
nk
We can define the measure of fuzziness of the
rule Rk in the fuzzy rule system as:
According to the formula (3)
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We can decide the rule Rk is worth to exist in
this rule-based system or not necessary.
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If the rule have high measure of fuzziness of
a rule, it means too much uncertain for this
rule.
A Fuzzy-Neural Network Structure
Output layer
c2
..........
cc
B1
B2
..........
B3
Hyperbox node
layer
a1
a2
..........
an
Input layer
c1
A variable structure
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We will leave the rule which is very efficient
and useful, so the number of nodes in the
second layer are variable.
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We will reduce the cost, because the
redundant second layer nodes are eliminated.
Second layer includes two Sub
layer
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the first sub layer is configured by the hyperbox 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.
ck
Output layer
Max
bk11 (1)
a1
…
…
a2
bkrl (m)
…
…
Bk
an
The kth
group in
the second
layer
Input layer
Learning Process
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Step1: set level = 1.
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Step2: Set up the hyper-boxes and
membership function for each class.
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Step3: Find the overlap among the activation
hyper-boxes of level l ,then l=l+1.
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Step4:Extract activation hyper-boxes and set
up feature as in step 1.
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Step5:Calculate the measure of fuzziness for
each extracted fuzzy rule. If it is bigger than
threshold, we discard this rule.
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Step6:If none of hyper-box exist in Step 4,
then stop the process, else go to Step 2.
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Step7:Build up the fuzzy-neural network
structure by these extracted fuzzy rules
Performance Evaluation
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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
Randomly generated area
單位：cm
Conclusions
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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 uncertaintyoverlap region. Thus, the network structure is simple
and easy to implement.
The classifier can be generated even for a large
scale of data pattern.