Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron

8/13/2019 Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron

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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.

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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

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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

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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 ))},

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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


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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

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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


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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

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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.

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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


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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.

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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.

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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 }.

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Implementing Fuzzy if-TheN Rules by Trainable Neural Nets. Fuzzy Neuron