Upload
sushil4056
View
229
Download
0
Embed Size (px)
Citation preview
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
1/21
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
2/21
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
3/21
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
4/21
learning process is relatively slow and analysis of the trained network is dif cult (black box).
Neither is it possible to extract structural knowledge(rules) from the trained neural network, nor can we
integrate special information about the problem intothe neural network in order to simplify the learningprocedure.
Fuzzy systems are more favorable in that their be-havior can be explained based on fuzzy rules and
thus their performance can be adjusted by tuning therules.
But since, in general, knowledge acquisition is dif -cult and also the universe of discourse of each inputvariable needs to be divided into several intervals,
applications of fuzzy systems are restricted to theelds where expert knowledge is available and thenumber of input variables is small.
4
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
5/21
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
6/21
Each rule in (1) can be interpreted as a training pat-tern for a multilayer neural network, where the an-tecedent part of the rule is the input and the conse-quence part of the rule is the desired output of theneural net.
The training set derived from (1) can be written inthe form
{(A1 , B 1 ), . . . , (An , Bn)}
If we are given a two-input-single-output (MISO)fuzzy systems of the form
i : If x is Ai and y is Bi, then z is C iwhere Ai, B i and C i are fuzzy numbers, i = 1, . . . , n .
Then the input/output training pairs for the neuralnet are the following
{(Ai, B i), C i}, 1 i n.If we are given a two-input-two-output (MIMO) fuzzy
6
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
7/21
systems of the form
i : If x is Ai and y is Bi, then r is C i and s is D i
where Ai, Bi, C i and Di are fuzzy numbers, i =1, . . . , n .
Then the input/output training pairs for the neuralnet are the following
{(Ai, B i), (C i, D i)}, 1 i n.
There are two main approaches to implement fuzzyIF-THEN rules (1) by standard error backpropaga-
tion network .
In the method proposed by Umano and Ezawa afuzzy set is represented by a nite number of its
7
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
8/21
membership values.
Let [ 1 , 2 ] contain the support of all the Ai, plusthe support of all the A we might have as input tothe system.
Also, let [ 1 , 2 ] contain the support of all the Bi ,plus the support of all the B we can obtain as out-puts from the system. i = 1, . . . , n .
Let M 2 and N be positive integers. Let
x j = 1 + ( j 1)( 2 1 )/ (N 1)
yi = 1 + (i 1)( 2 1 )/ (M 1)
for 1 i M and 1 j N .
A discrete version of the continuous training set isconsists of the input/output pairs
{(Ai(x1 ), . . . , A i(xN )), (B i(y1 ), . . . , B i(yM ))},
8
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
9/21
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
10/21
x1
y1 yM
xN
A i
Bi
a ij
b ij
y
Multilayer neural network
j
x i
A network trained on membership values fuzzy numbers.Example 1. Assume our fuzzy rule base consists of three rules
1 : If x is small then y is negative ,2 : If x is medium then y is about zero ,3 : If x is big then y is positive ,
where the membership functions of fuzzy terms are10
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
11/21
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
12/21
1
1
negative positiveabout zero
-1
negative (u) = u if 1 u 00 otherwise
about zero (u) =1 2|u| if 1/ 2 u 1/ 20 otherwise
positive (u) =u if 0 u 10 otherwise
The training set derived from this rule base can bewritten in the form
12
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
13/21
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
14/21
A discrete version of the continuous training set isconsists of three input/output pairs
{(a11 , . . . , a 15 ), (b11 , . . . , b15 )}
{(a21 , . . . , a 25 ), (b21 , . . . , b25 )}
{(a31
, . . . , a35
), (b31
, . . . , b35
)}where
a1 j = small (x j ), a2 j = medium (x j ), a3 j = big(x j )
for j = 1, . . . , 5, and
b1 i = negative (yi), b2 i = about zero (yi),
b3 i = positive (yi)for i = 1, . . . , 5.
Plugging into numerical values we obtain the fol-lowing training set for a standard backpropagation
network
{(1, 0.5, 0, 0, 0), (1, 0.5, 0, 0, 0)}
{(0, 0.5, 1, 0.5, 0), (0, 0, 1, 0, 0)}14
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
15/21
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
16/21
the product
pi = wix i, i = 1, 2.
The input information pi is aggregated, by addition,to produce the input
net = p1 + p2 = w1 x1 + w2 x2
to the neuron.
The neuron uses its transfer function f , which couldbe a sigmoidal function,
f (x) = 1
1 + e x,
to compute the output
16
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
17/21
x1
x2
w1
w2
y = f(w 1x1+w2x2)
y = f (net ) = f (w1 x1 + w2 x2 ).
This simple neural net, which employs multiplica-tion, addition, and sigmoidal f , will be called as
regular (or standard) neural net.
Simple neural net.
If we employ other operations like a t-norm, or a t-conorm, to combine the incoming data to a neuronwe obtain what we call a hybrid neural net .
These modi cations lead to a fuzzy neural architec-ture based on fuzzy arithmetic operations.
17
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
18/21
Let us express the inputs (which are usually mem-bership degrees of a fuzzy concept) x1 , x 2 and theweigths w1 , w2 over the unit interval [0, 1].
A hybrid neural net may not use multiplication, ad-
dition, or a sigmoidal function (because the resultsof these operations are not necesserily are in the unitinterval).Denition 1. A hybrid neural net is a neural net with crisp signals and weights and crisp transfer function. However,
we can combine xi and wi using a t-norm, t-conorm, or some other continuous operation,
we can aggregate p1 and p2 with a t-norm, t-conorm, or any other continuous function
f can be any continuous function from input tooutput
We emphasize here that all inputs, outputs and the18
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
19/21
weights of a hybrid neural net are real numbers takenfrom the unit interval [0, 1].
A processing element of a hybrid neural net is called fuzzy neuron .
De nition 2. (AND fuzzy neuron
The signal xi and wi are combined by a triangular conorm S to produce
pi = S (wi, x i), i = 1, 2.
The input information pi is aggregated by a trian-gular norm T to produce the output
y = AND ( p1 , p2 ) = T ( p1 , p2 )
= T (S (w1 , x 1 ), S (w2 , x 2 )),
of the neuron.
19
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
20/21
x1
x2
w1
w2
y = T(S(w 1, x1), S(w 2, x2))
So, if T = min, S = max
then the AND neuron realizes the min-max compo-sition
y = min{w1 x1 , w2 x2 }.
AND fuzzy neuron.
De nition 3. (OR fuzzy neuron
The signal xi and wi are combined by a triangular norm T to produce
pi = T (wi, x i), i = 1, 2.
20
8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron
21/21
x1
x2
w1
w2
y = S(T(w 1, x1), T(w 2, x 2))
The input information pi is aggregated by a trian-gular conorm S to produce the output
y = OR( p1 , p2 ) = S ( p1 , p2 ) = S (T (w1 , x 1 ), T (w2 , x 2 ))
of the neuron.
OR fuzzy neuron.
So, if T = min, S = max
then the OR neuron realizes the max-min composi-tion
y = max{w1 x1 , w2 x2 }.
21