A Max-Min fuzzy system can be regarded as a network of max and min operational elements. Thus, the inverse problem of a fuzzy relational equation is interpreted as an input estimation problem from output values in the corresponding network. An approximate network model of a fuzzy relational system is proposed. An algorithm for obtaining an approximate solution of the system is presented for using a neural network technique

Hirota, Kaoru

Ikoma, Norikazu

English

NASA Technical Reports Server

No -  r86
Solution of Inverse Problem of Fuzzy Relational Equation by
using Perceptron Model
Kaoru HIR OTA *
Norikazu IKOMA **
• Department of Instrument and Control Engineering, College of Engineering, HOSEI University
•* The HOSEI University Graduate School, Engineering Division, System Engineering
ABSTRACT
Max-min fuzzy relational system can be regarded as a network of max
and min operational elements. Thus the inverse problem of fuzzy relational
equation is interpreted as an input estimation problem from output values in
the corresponding network. An approximate network model of fuzzy relational
system is proposed. An algorithm of obtaining an approximate solution of the
system is presented by using a neural network technique. The availability is dis-
cussed with a numerical experiment.
Key words : fuzzy relation, fuzzy inverse problem, neural network, per-
ceptron model
Introduction
Inverse problem of fuzzy relational equation (Fuzzy Inverse Problem) was proposed by
E.Sanchez in 197611]. The solution of fuzzy inverse problem was shown by Tukamoto et al in
197712]. And now, it is used for diagnosis of complicated systems.
Max-rain fuzzy relational system can be regarded as a network which consists of max and
rain operational elements. Thus the fuzzy inverse problem is interpreted as an input value esti-
mation problem from output values in the corresponding network.
In this view point, input of network can be identified when output and the network struc-
ture are given. Assuming the network of fuzzy relational system can be approximately con-
structed by the perceptron, we can regard the input of fuzzy system as the input of perceptron
when output and the perceptron structure are given.
In this paper, an input value estimation algorithm based on perceptron model is pro-
posed, and it is applied to solving the fuzzy inverse problem. Numerical experiment is done to
investigate the availability of this method, and the result of experiments is discussed.
Input Estimation Algorithm of Perceptron Model
Perceptron neural network model[3] is used in this paper. It is summarized as follows;
a. Output value ofj -th neuron in the k -th layer is denoted by
b. Threshold value ofj -th neuron in the k -th layer is denoted by
0y.
c. Connection coefficient from the i -th neuron in (k-l) th layer to j -th one in k -th
layer is denoted by
."183
https://ntrs.nasa.gov/search.jsp?R=19910012473 2020-03-19T19:01:15+00:00Z
d. The relation of above three values are described by (1)-(3).
uJ'--i(4)
k _ k-1 k k-I ,'_k
s)=Lw i i'ui -vj
i
1f(s)=
1+ exp(- s)
Input estimation algorithm of perceptron model is described as follows.
Preparation
[1]
[2]
Algorithm
[1]
[2]
Assume the perceptron model has n -input items and m -output items.
Define the evaluation function as
1 tyj-uJ(s)f
E(s)= 2-j=l
where s= [sl.s_s3.''" s.] and yiis the j-th output value of the perceptron.
Set the initial input s as an arbitrary value.
Calculate output
u= [u_,u2,u3, • • • u,,,]
for the current input value s.
[3] Change the value s according to
ds _E
dr- eos
=-e 3u--'_as--TEau ':-'..... as :_-1 "''. .... , a-u.].? as.!..J
where e is a positive value.
[4] Repeat [2]-[3] until the value E attains a sufficiently small value.
[5] Final value s is the estimated value by the perceptron.
(1)
(2)
(3)
(4)
(5)
Fuzzy Inverse Problem
Fuzzy relation R between set X and set Y is regarded as a fuzzy set on the direct product
of X and Y , and its membership function is denoted by
lXR:XxY= {(x,y)lx_X,yEY } (6).
Assume that A is a fuzzy set on X and B is another fuzzy set on Y , where these
membership functions are [t A and IxB respectively, then fuzzy relation R satisfies (7) which
means (8).
B =R , A (7)
ktB (y)= max_x {I-ta (x ,y)AI.ta (x) } (8)
If A and B are fuzzy input and output, respectively, then (7) is interpreted as an equa-
tion of the system which has fuzzy input and output.
Thenthefuzzyinverseproblemisthe inverseproblemof fuzzyrelationalequation,i. e.
identifyingA for the given B and R in the equation (7).
Method of Solving Fuzzy Inverse Problem by using Pereeptron Model
This method is divided into two phases, i. e. the learning phase and the solving phase.
The learning phase is summarized as follows;
[1] Let A. and B. be the i -th input and output of fuzzy relation system R ( i = 1,2 ....
l !
M ) , respectwely.
[2] Encode A i and B i to x i and Yi ,respectively, according to
[0,1 ]5 ai I.--->xi= 2.2x a i- 1.1E [- 1.1,1.1] (9)
[0,1 ]_b/I--->yi= 0.8× bi+ 0.IE [0.1,0.9] (10).
[3] Let multi layer perceptron be learned by using input-output pair x I and Yi obtained
at [2].
After finishing this learning phase, we can get the solution as follows;
[4] Let B be a fuzzy set to be solved in the fuzzy inverse problem.
[5] Encode B to y by using eq. (10).
[6] Apply the input estimation algorithm to the learned perceptron and estimate the
input for given output y.
[7] Decode y to A by using eq. (9), then we can get the solution of the problem.
Numerical Experimen t
To investigate the availability of the method discussed in the previous section, a numeri-
cal experiment on a digital computer has been done as follows;
[1] Fuzzy relation, i. e. the solution in the fuzzy inverse problem, is given as (11) in this
[21
[3]
experiment.
i R(xI'yl) gR(x2,Yl)
tXR(X I,Yz) IXR(x2,Y2)
R = l.tR (xl,Y3) I.tR (x2,Y3)
_t (x ,Y4) _tR (x2,Y4)
R (Xl,Y5) I'tR (x2,Ys)
... rt_(xs,yl)l to.6 0.5 0.8 0.3 0.2
... _t_(xs,yz)/ /0.4 0.1 0.9 0.6 0.4
... pR(xs,y3)[=|0.1 0.1 0.9 0.8 0.5
... l.tR(xs,Y4) | |0.9 0.2 0.9 0.1 0.5
•" rtR(xs,ys)J LO.4 0.5 0.3 0.8 0.9
(11)
The learning data for the perceptron is generated as follows;
i) Make fuzzy sets A t - A_77e whose membership values at each element take all com-
bination of values {0, 0.2, 0.4, 0.6, 0.8, 1.0} (c.f. 62 = 7776).
ii) Operate each A i to the fuzzy relation R by using max-min composition, then we
can get the fuzzy set B_
!
iii) Encode fuzzy sets A i and B i by eqs' (9) and (10), then we will obtain the learning
data x i and yi.
Let's move to the learning phase by using xj and y as leaning data. The structure of per-i
ceptron used in this experiment is shown as in table 1.
table 1 : Outline of the Perceptron
number of neurons 5 10 5
Error back propagation algorithm[4] is used for learning. Counting the learning process is
defined as follows : one learning process is a learning operation for a paired input-output
data. The perceptron used in this experiment learns about 500 thousand times. The distri-
bution of evaluation function (4) value for learning data is shown in figure I.
185
[4] Thetest-data in the fuzzy inverse problem is made as follows;
i) Make fuzzy sets A. whose membership values at each element are random values
from [0,1] (where I = 1 - i000 in this experiment). Get B i by compositing A i to
the fuzzy relation R.
[5] Encode B i by eq. (10), and get Yl" Applying input estimation algorithm to the percep-
tron, we can get estimated input value x* r Then we get the solution of fuzzy inverse
problem by decoding x* i to A* i by eq. (9).
[6] Composite the obtained solution A*. to fuzzy relation R , and we get B*. The correct-
ness of solution is considered as th_ closeness of membership value by' each element
between B*. an d B.
i !
[7] Distribution of the values of evaluation functions (12)-(14) for all test-data are shown in
figure 2 - figure 4.
E.,,_=_Ib_-bjl
Em,,,=Max Ibm- b) l
Emi_=Min Ibm- bjl
(12)
(13)
(14)
for allj. biEBi.b_EB*L
Discussion
The availability of using perceptron for fuzzy inverse problem was shown trough a
numerical experiment in previous chapter. The approximate solution of the fuzzy inverse prob-
lem is obtained , but its precision is not enough. This is mainly because that the error of
approximation of fuzzy relation by perceptron is not small enough.
Distribution of approximation error of fuzzy relation was shown in figure 1 in the previ-
ous chapter, but the inputs for error measuring are the same one as learning inputs, precisely.
It is necessary to measure the error for no learning inputs. So the distribution of evaluation
function (4) for no learning inputs is shown in figure 5. By comparing error in learning and no
learning cases, it should be noted that the distributions have the same shape. It depends on the
generalization factor of neural networks. Hence it could be considered that the differences
between learning and no learning are independent with the precision of solution.
The distribution of figure 1 and figure 5 are bell shaped but not exponential. This is con-
sidered that the approximation error is not less than a certain threshold value. This is because a
conflicti0n occurs between one learning input-output pair and others. Therefore, if we can
increase the learning times or use more input-output pairs for learning, the improvement of
precision for approximating fuzzy relation is not expected.
Now let's discuss how small the approximation error of fuzzy relation is. The measure-
ment was done as follows; Let change membership value of fuzzy set A at one element in the
interval [0,1], and compare the membership value of fuzzy set B obtained by compositing
fuzzy relation R in eq. (11) and another one obtained from the output of perceptron. The
membership values used for comparison are shown in tabie-2 and the results are shown in
figure 6 - figure 10. In these figures, the horizontal axis represents membership value of a vari-
able element of fuzzy set A , and vertical axes represent membership values of each element of
fuzzy set B. There exist two lines in these graphs, where one (which has linear shape)
represents the characteristics of approximated fuzzy relation, another (which has not linear
shape) indicates the characteristic of perceptron which approximates the fuzzy relation.
186
table 2 : Conditions of membership value for fuzzy set A
element number
NO.
I 2 3 4 5
I [0,1] 0.3 0.5 0.2 0.7
2 0.3 [0,1] 0.2 0.6 0.4
3 0.2 0.6 [0,1] 0.4 0.7
4 0.8 0.7 0.2 [0,1] 0.5
5 0.2 0.9 0.3 0.4 [0,1]
figure number
fig.6
fig.7
fig.8
fig.9
fig.lO
Concl usion
Input estimation algorithm of perceptron model has been proposed and applied to the
fuzzy inverse problem. Numerical experiments have been done in order to get the solution of
fuzzy inverse problem. The precision of approximate solution obtained by this method is dis-
cussed, and the approximation error distribution is investigated. From the results of these
numerical experiments, it is concluded that this method is available to obtain the approximate
solution of fuzzy inverse problem.
Referen ces
1. E.Sanchez, Resolution of Composite Fuzzy Relation Equations, Information and Control,
30(1), pp.38-48(1976).
2. Tukamoto Y., Tasiro T., Method of Solution to Fuzzy Inverse Problem, Transactions of the
Society of Instrument and Control Engineers, vol 15(1), pp. 21-25(1977) (in Japanese)
3. F.Rosenblatt, The perceptron : A probabilistic model for information sorage and organization in
the brain. Psychol. Rev. 65(6), pp. 386-408(1958).
4. Runmelhart et al., Learning Representations by Back-propagating errors, Nature 323-9,
pp.533-536(1986)
187
Table 1. Outline of the Perceptron
Outline of the Perceptron [
number of neurons 5 I 10 /10 } 51
Table 2. Conditions of membership value for fuzzy set A
Conditions of membership value for fuzzy set A
element number figure number
No.
1 2 3 4 5
1 [0,1] 0.3 0.5 0.2 0.7 fig.6
2 0.3 [0,1] 0.2 0.6 0.4 fig.7
3 0.2 0.6 [0,1] 0.4 0.7 fig.8
4 0.8 0.7 0.2 [0,1] 0.5 fig.9
5 0.2 0.9 0.3 0.4 [0,1] fig. 10
188
1500
1000
5OO
0
0.0 0_02
R
1500
1000
500
Figure 1. Distribution of evaluation function E for learning data
189
100
60
! I
0 0.05 0.1 0.15
100
5O
Figure 2. Distribution of evaluation function E,.ea" for meaning evaluation
190
nmbex of
samples
I00
5O
0
i
m
I l i
I00
5O
0
O.2
Em_
Figure 3. Distribution of evaluation fimction Emax for maximum evaluation
191
numbe.rof
100
50
I l I "
100
5O
0
Figure 4. Distribution of evaluation function E . for minimum evaluation
mtn
192
200
150
100
50
0
O.0 O.01 -"u.'oz
250
200
150
I00
50
0
0.O3
E
Figure 5. Distribution of evaluation function E for no learning data
193
1.0 -
_sk_
val_ of b5
0.0
1.0-
valueof b4
1.
membership
value of b3
i° -
me_._rsl_
vahm of b2
a8-1.
_bership
value of bl
0.0.
0.0
-1.0
0.0
-I.0
_0.0
-I,0
0.0
1,0
0.0
1.0
, .0.0
I.I
m_e_rship value of a
!
O.5
Figure 6. Comparison of fuzzy set B and output of perceptron (No.l)
194
1.0 -
mmbersh.i_
value of b5
O.
1.8-
me_ership
valtm of b4
1, -
value of b3
1, -
_e_ersh_
value of b2
o.8-1°
_rbersl_ --
value of b 1
_0
0.0
x
/
I I !
0.5
-1.0
_0.0
-I. 0
0.0
-I. 0
0.0
1.0
_0.0
-1.0
0.0
1.0
membershipval,._of a
Figure 7. Comparison of fuzzy set B and output of perceptron (No.2)
195
1.0
memberslzi_
valueof b5
a8_L
_e_xship
valueof b4
a 8-1°
_sh_
value of b3
0.0_
l.O-
value of b2
0.0_
l,O-
value of bl
0.0
1.0
_0
1.0
0.0
1.0
0.0
1.0
_0
1.0
, , , _0
_0 0.5 1.0
membershipvalueof a
Figure 8. Comparison of fuzzy set B and output of perceptron (No.3)
196
i, 0 -
_sh£p
value of b5
va_ of b4
_ership
value of b3
1.
_be.mh._
value of b2
(_0
1.0
m_be_sh_
v_tm of bl
0.0
| i z
-1.0
0.0
-i.0
0.0
-I.0
_0
-I.0
, _0
1.0
me_be_ va]m of a
-- ! !
0.0 0.5
Figure 9. Comparison of fuzzy set B and output of perceptron (No.4)
197
1. O-
me_e_sh_
valu_of b5
O.1.8
m_exsl_
va].ueof b t
O.
valm of b3 --
O.1.8
a_exsl_
wlue of b2
1.8-
value of bl
0.0-
0.0
-_I. 0
o.o
1.0
_0.0
-1.0
_0.0
-1.0
0.0
1.0
0.0
1.0
membershipvalueof a
I I
O.5
Figure 10. Comparison of fuzzy set B and output of perceptron (No.5)
198
Overview of LIFE (Laboratory for International Fuzzy Engineering)
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Solution of inverse problem of fuzzy relational equation by using perceptron model

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