a fuzzy back propagation algorithm

8/3/2019 A Fuzzy Back Propagation Algorithm

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Fuzzy Sets and Systems 112 (2000) 2739

www.elsevier.com/locate/fss

A fuzzy backpropagation algorithm

Stefka Stoevaa, Alexander Nikovb;

a Bulgarian National Library, V. Levski 88, BG-1504 Soa, BulgariabTechnical University of Soa, P.O. Box 41, BG-1612 Soa, Bulgaria

Received November 1994; revised March 1998

Abstract

This paper presents an extension of the standard backpropagation algorithm (SBP). The proposed learning algorithm is

based on the fuzzy integral of Sugeno and thus called fuzzy backpropagation (FBP) algorithm. Necessary and sucient

conditions for convergence of FBP algorithm for single-output networks in case of single- and multiple-training patterns are

proved. A computer simulation illustrates and conrms the theoretical results. FBP algorithm shows considerably greater

convergence rate compared to SBP algorithm. Other advantages of FBP algorithm are that it reaches forward to the target value

without oscillations, requires no assumptions about probability distribution and independence of input data. The convergence

conditions enable training by automation of weights tuning process (quasi-unsupervised learning) pointing out the interval

where the target value belongs to. This supports acquisition of implicit knowledge and ensures wide application, e.g. forcreation of adaptable user interfaces, assessment of products, intelligent data analysis, etc. c 2000 Elsevier Science B.V.

All rights reserved.

Keywords: Neural networks; Learning algorithm; Fuzzy logic; Multicriteria analysis

1. Introduction

Recently, the interest in neural networks has grown

dramatically: it is expected that neural network mod-els will be useful both as models of real brain func-

tions and as computational devices. One of the most

popular neural networks is the layered feedforward

neural network with a backpropagation (BP) least-

mean-square learning algorithm [17]. Its topology is

shown in Fig. 1. The network edges connect the pro-

cessing units called neurons. With each neuron input

there is associated a weight, representing its relative

Corresponding author.E-mail address: [email protected] (A. Nikov)

importance in the set of the neurons inputs. The in-

puts values to each neuron are accumulated through

the net function to yield the net value: the net value is

a weighted linear combination of the neurons inputsvalues.

For the purpose of multicriteria analysis [13,20,21]

a hierarchy of criteria is used to determine an over-

all pattern evaluation. The hierarchy can be encoded

into a hierarchical neural network where each neuron

corresponds to a criterion. The input neurons of the

network correspond to single criterion. The hidden

and output neurons correspond to complex criteria. As

evaluation function it can be used as the net function

of the neurons. However, the criteria can be combined

linearly when it is assumed that they are independent.

0165-0114/00/$- see front matter c 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 0 7 9 - 7


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28 S. Stoeva, A. Nikov / Fuzzy Sets and Systems 112 (2000) 2739

Fig. 1. Example neural network with single-output neuron and a hidden layer.

But in practice the criteria are correlated to some

degree. The linear evaluation function is unable to

capture relationship between the criteria. In order to

overcome this drawback of standard backpropagation

(SBP) algorithm we propose a fuzzy extension called

fuzzy backpropagation (FBP) algorithm. It determinesthe net value through the fuzzy integral of Sugeno [3]

and thus does not assume independence between the

criteria. Another advantage of FBP algorithm is that

it reaches always forward to the target value without

oscillations and there is no possibility to fall into lo-

cal minimum. Necessary and sucient conditions for

convergence of FBP algorithm for single-output net-

works in case of single- and multiple-training patterns

are proved. The results of computer simulation are

reported and analysed: FBP algorithm shows consid-erably greater convergence rate compared to SBP

algorithm.

2. Description of the algorithm

2.1. Standard backpropagation algorithm

(SBP algorithm)

First we describe the standard backpropagation al-

gorithm [17,18] because FBP algorithm proposed by

us can be viewed as fuzzy-logic-based extension of the

standard one. SBP algorithm is an integrative gradient

algorithm designed to minimise the mean-squared er-

ror between the actual output and the desired output by

modifying network weights. In the following we use

for simplicity a network with one hidden layer but theresults are valid also for networks with more than one

hidden layers. Let us consider a layered feedforward

neural network with n0 inputs, n1 hidden neurons and

single-output neuron (cf. Fig. 1). The inputoutput re-

lation of each neuron of the neural network is dened

as follows:

Hidden neurons

net(1)i =

n0

j=1

w(1)ij xj ;

a(1)i = f(net

(1)i ); i = 1; 2; : : : ; n1:

Output neurons

net(2) =

n1i=1

w(2)1i a

(1)i ;

a(2) = f(net(2));

where X = (x1

; x2

; : : : ; xn0

) is the vector of patterns

inputs.


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S. Stoeva, A. Nikov / Fuzzy Sets and Systems 112 (2000) 2739 29

The activation functions f(net) can be linear ones

or Fermi functions of type

f(net) =1

1 + e4(net):

Analogous to the writing and reading phases, there are

also two phases in the supervised learning BP network.

There is a learning (training) phase when a training

data set is used to determine the weights that dene the

neural model. So the task of the BP algorithm is to nd

the optimal weights to minimise the error between the

target value and the actual response. Then the trained

neural model will be used later in the retrieving phase

to process and evaluate real patterns.

Let the patterns output corresponding to the vector

of patterns inputs X be called the target value t.

Then the learning of the neural network for train-

ing pattern (X;t) is performed in order to minimise

the squared-error between the target and the actual

response

E = 12

(t a(2))2:

The weights are changed according to the following

formula:

wnewij = woldij + wij;

where

wij = (l)i a

(l)j :

[0; 1] denotes the learning rate, (l)i is the error

signal relevant to the ith neuron, and a(l)

j the signal at

jth neuron input.

The error signal is obtained recursively by back-

propagation.

Output layer

(2) = f(net(2))(t a(2)); (1)

where the derivation of the activation function

f(net) = 4net(1 net):

Hidden layer

(1)i = f

(net(1)i )

(2)w(2)1i ;

where (2) is determined by formula (1).

When there are more than one training pattern

(Xm; tm); m = 1; 2; : : : ; M , the sum-squared error is

dened as

E=

1

2

M

m=1

(tm

a(2)m )

2

:

2.2. Fuzzy backpropagation algorithm

(FBP algorithm)

Recently, many neuro-fuzzy models have been in-

troduced [2,6,8,10,11]. The following extension of

the standard BP algorithm to fuzzy BP algorithm

is proposed. For yielding the net value neti the in-

puts values of the ith neuron are aggregated. The

mapping is mathematically described by the fuzzy

integral of Sugeno that relies on psychological back-

ground.

Let xj [0; 1] R; j = 1; : : : ; n0, and all network

weights w(l)ij [0; 1] R, where l = 1; 2. Let P(J) be

the power set of the set J of the network inputs (in-

dices), where Jij is the jth input of the ith neuron.

Further on for simplicity, network inputs with their in-

dices are identied. Let g :P(J) [0; 1] be a functiondened as follows:

g() = 0;

...

g({Jij}) = w(1)ij ; 16j6 n0;

...

g({Jij1 ; Jij2 ; : : : ; J ijr}) = w(1)ij1

w(1)ij2 w(1)ijr

;

where {j1; j2; : : : ; jr} {1; 2; : : : ; n0};

...

g({Ji1; Ji2; : : : J ij; : : : ; J in0 }) = 1;

where the notations and stand for the operationsMAX and MIN, respectively, in the unit real inter-

val [0; 1] R. It is proved [20] that the function g isa fuzzy measure on P(J). Therefore, the functional

assumes the form of the Sugeno integral over the -

nite reference set J.

Let h :P(I) [0; 1] be a function, dened in a sim-ilar way.


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Hidden neurons

net(1)i =

GP(J)

pG

xp

g(G)

;

a(1)i = f(net

(1)i ); i = 1; 2; : : : ; n1:

Output neurons

net(2) =

GP(I)

pG

a(1)p

h(G)

;

a(2) = f(net(2)):

The activation functions f(net) are linear ones or

equal to

f(net) =1

1 + e4(net0:5);

in order to obtain f(net) [0; 1].Therefore, the activation values a

(l)i [0; 1], where

l = 1; 2.

The error signal is obtained by back propagation.

Output layer

(2) = |t a(2)|: (2)

Hidden layer

(1)i = |t a

(1)i |; i = 1; : : : ; n1: (3)

In order to obtain wnewij [0; 1] the weights are handledas follows:

Case t= a(2): No adjustment of the weights is

necessary.

Case ta(2):

If woldij t then wnewij = 1 (w

oldij + wij).

If woldij t then wnewij = w

oldij .

Case ta(2):

If woldij t then wnewij = 0 (w

oldij wij).

If woldij 6 t then wnewij = w

oldij ,

where wij = (l)i ;

(l)i is determined by formula (2)

resp. (3).

In the following let us denote the iteration steps

of the proposed algorithm by s, the corresponding

weights by w(l)ij (s) and the activation values by a(l)i (s).

2.3. Necessary and sucient conditions for

convergence of FBP algorithm for single-output

networks

The following nontrivial necessary and sucientconditions for convergence of FBP algorithm in case

of single-output neural networks will be formulated

and proved: the interval between the minimum and

maximum of the patterns inputs shall include the pat-

terns output, i.e. the target value.

2.3.1. Convergence condition in case of single

training pattern

Denition 1. FBP algorithm is convergent to the tar-

get value t if there exists a number s0 N+ such thatthe following equalities hold:

t= a(2)(s0) = a(2)(s0 + 1) = a

(2)(s0 + 2) = ;

w(l)ij (s0) = w

(l)ij (s0 + 1 ) = w

(l)ij (s0 + 2 ) = ; l = 1; 2:

The following lemma concerns the outputs of the

neurons at the hidden layer.

Lemma 1.

(i) Ifa

(1)

i (s)t then a

(1)

i (s + 1)

t.(ii) Ifa(1)i (s)t then a

(1)i (s + 1)6 t.

The proof of this lemma is similar to that one of the

lemma in [19].

The following lemma concerns the output of the

output neuron:

Lemma 2.

(i) Ifa(2)(s)t then a(2)(s + 1) t;

(ii) Ifa(2)(s)t then a(2)(s + 1)6 t.

Proof. In the sequel for each set G P(I),

hs(G) =

pG

w(2)1p (s); vs(G) =

pG

a(1)p (s);

a2(s) =

GP(I)

[vs(G) hs(G)]:

(i) Since a(2)(s)t, and P(I) is a nite set, there

exists a set Gs P(I) such that

vs(Gs) hs(Gs) = a(2)(s)t;


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i.e.

vs(Gs) a(2)(s)t and hs(Gs) a

(2)(s)t:

By Lemma 1 it follows that vs+1(Gs) t. Since Gs

is a nite set, there exists an input ps Gs, such thatw

(2)1ps

(s) = hs(Gs). Therefore, the weight w(2)1ps

(s + 1) is

produced through the formula

w(2)1ps

(s + 1) = 0 (w(2)1ps (s) |t a(2)(s)|)

0 (w(2)1ps (s) |t a(2)(s)|)

= 0 (w(2)1ps (s) a(2)(s) + t)

0 (a(2)(s) a(2)(s) + t)

= 0 t

= t:

So, vs+1(Gs) w(2)1ps

(s + 1) t. Let

a(2)(s + 1) = vs+1(Gs+1) hs+1(Gs+1)

for some set Gs+1 P(I). Then

a(2)(s + 1) = vs+1(Gs+1) hs+1(Gs+1)

vs+1(Gs) w(2)1ps (s + 1) t:

Thus, a(2)(s + 1) t.

(ii) Since a(2)(s)t, and P(I) is a nite set, there


vs(Gs) hs(Gs) = a(2)(s)t:

Let

a(2)(s + 1) = vs+1(Gs+1) hs+1(Gs+1)

for some set Gs+1 P(I). Since Gs+1 is a nite set,there exists an input ps+1 Gs+1, such that

w(2)1ps+1

(s + 1) = hs+1(Gs+1):

Then it holds:

ta(2)(s) = vs(Gs) hs(Gs)

vs(Gs+1) w(2)1ps+1

(s):

If vs

(Gs+1

)6 a(2)(s)t, then by Lemma 1 it follows

that vs+1(Gs+1)6 t, and thus a(2)(s + 1)t.

If w(2)1ps+1

(s)6 a(2)(s)t, the weight w(2)1ps+1

(s + 1)

is produced through the formula

w(2)1ps+1

(s + 1) = 1 (w(2)1ps+1 (s) + |t a(2)(s)|)

6 1 (w(2)1ps+1 (s) + |t a(2)(s)|)

= 1 (w(2)1ps+1 (s) + t a(2)(s))

6 1 (a(2)(s) + t a(2)(s))

= 1 t

= t:

Hence a(2)(s + 1)6 t.

Theorem 1. FBP algorithm is convergent to the tar-

get value t i for the neurons at the input layer thefollowing condition holds:

jj(a(0)j t & a(0)

j 6 t): (4)

Proof. The if part of Theorem 1 is proved by as-

suming that condition (4) is not satised.

Suppose there is no input j J such that a(0)j t.

Then for each input j J, 16j6 n0, it holds thata

(0)j t. Hence, for each set G P(J) it holds that

v(G) =

pG

a(0)p t:

So, at each iteration step s it holds that

a(1)i (s) =

GP(J)

[v(G) gis(G)]

t

GP(J)

gis(G)

= t 1 = tfor each i, 16 i6 n1, where gis(G) =

pG w

(1)ip (s).

Therefore, at each iteration step s it holds that

a(2)(s) =

GP(I)

[vs(G) hs(G)]

=

GP(I)

pG

a(1)p (s) hs(G)

GP(I)[t hs(G)]


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= t

GP(I)

hs(G)

= t 1 = t:

Thus, in such case FBP algorithm is not convergent

to the target value t.

The assumption that there is no input j J, suchthat a

(0)j 6 t, implies in a similar way that at each

iteration step s, a(2)(s)t holds. Thus, in this case

FBP algorithm is not convergent to the target value t,

either.

The only if part of Theorem 1 is proved by as-

suming that FBP algorithm is not convergent to the

target value t. Hence, there exists no iteration step

s0 N+, such that a(2)(s0) = t, i.e. at each iteration

step s it holds either a(2)(s)t or a(2)(s)t.If a(2)(1)t is fullled, then at each iteration step

s, a(2)(s)t is fullled, too. The proof is performed

through mathematical induction. Let a(2)(1)tbe ful-

lled indeed. Since FBP algorithm is not convergent

by Lemma 2, it follows that a(2)(2)t. Let now at

the sth iteration step a(2)(s)tbe fullled. Since FBP

algorithm is not convergent by Lemma 2, it follows

that a(2)(s + 1)t. According to the mathematical in-

duction axiom at each iteration step s, a(2)(s)t is

satised.Since I is a nite set, at each iteration step s there


vs(Gs) hs(Gs) = a(2)(s)t;

i.e. vs(Gs)t and hs(Gs)t.

Let s0 N+ be the iteration step, at which the pro-cess of adjusting all the weights of the network comes

to an end. Then the above inequalities can be fullled

only ifhs0 (Gs0 ) = 1, i.e. if set Gs0 is equal to I, Gs0 I,

and for each input i, 16 i6 n1

, a(1)

i(s

0)t holds.

Since J is a nite set, for each neuron i, 16 i6 n1,

at the hidden layer there exists a set Gis0 P(J), suchthat

v(Gis0 ) gis0 (Gis0 ) = a(1)i (s0)t;

i.e. v(Gis0 )t and gis0 (Gis0 )t. Since the pro-

cess of adjusting all the weights has already been

stopped, the above inequalities can be fullled only if

gis0 (Gis0 ) = 1, i.e. if set Gis0 is equal to J, Gis0 J, and

for each input j, 16j6 n0

, a(0)

jt holds. Therefore,

condition (4) is not satised.

The assumption that a(2)(s)t holds, implies in a

similar way that condition (4) is not satised in that

case, either.

2.3.2. Convergence condition in case of

multiple-training patterns

Denition 2. In case of multiple-training patterns

(Xm; t); m = 1; : : : ; M , FBP algorithm is convergent to

the target value t if there exists a number of s0 N+

such that the following equalities hold:

t= a(2)m (s0) = a(2)m (s0 + 1) = a

(2)m (s0 + 2) = ;

m = 1; : : : ; M ;

w(l)ij (s0) = w

(l)ij (s0 + 1) = w

(l)ij (s0 + 2) = ;

l = 1; 2:

Theorem 2. Let more than one training pattern

(Xm; t); m = 1; : : : ; M ; be given. Let a(0)mj ; a

(0)mj be in-

puts of the mth pattern; such that Theorem 1 holds;

i.e. a(0)mj t and a

(0)mj6 t. Let

a

(0)

m

j

=

M

m=1 a

(0)

mj

and a

(0)

m

j

=

M

m=1 a

(0)

mj

:

Then FBP algorithm is convergent to the target value

t i the following condition holds:

a(0)mj t & a

(0)mj6 t: (5)

Proof. Let condition (5) be satised. Since the in-

equalities t6 a(0)mj6 a

(0)mj and t a

(0)mj a

(0)mj hold

for each m; m = 1; : : : ; M , according to Theorem 1.

FBP algorithm is convergent to the target value t for

each m; m = 1; : : : ; M .Let now FBP algorithm be convergent to the tar-

get value t for each m; m = 1; : : : ; M . Then according

to Theorem 1 the inequalities a(0)mj t and a

(0)mj6 t

hold for each m = 1; : : : ; M . Therefore, condition (5)

is satised, too.

3. Simulation results

With the help of a computer simulation we demon-

strate the proposed FBP algorithm using two neural


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Fig. 2. Example neural network with single-output neuron in the case of single- and multiple-training patterns.

networks with single-output neuron in case of single-

training pattern (cf. Fig. 2) and in case of multiple

training patterns (cf. Figs. 2 and 7). The FBP algo-

rithm is compared with SBP algorithm (cf. Fig. 8).

3.1. Single-training pattern

In case of single-training pattern FBP algorithm was

iterated with linear activation function, training ac-

curacy 0.001 (error goal), learning rate = 1:0 and

target value t= 0:8 (cf. Fig. 2). According to The-

orem 1 FBP algorithm is convergent to target value

t belonging to the interval between the minimal and

maximal input values [a(0)

j ; a(0)

j ] = [ 0:1; 0:9]. Fig. 3 il-

lustrates that for target values t= 0:2; 0:5 and 0.8

belonging to the interval [0:1; 0:9] the network was

trained only for 2, respectively, for 3 steps. If the

target value (t= 0:0 and 1.0) is outside the interval

[0:1; 0:9] the network output cannot reach it and the

training process is not convergent. This conrms the


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Fig. 3. Output values in case of single-training pattern for dierent target values.

Fig. 4. Output values in case of multiple-training patterns for dierent target values.


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Fig. 5. Sum-squared error in case of multiple-training patterns for dierent target values.

theoretical results. Here the nal output values are 0:1

and 0:9 for t= 0:0 and for t= 1:0, respectively.

3.2. Multiple training patterns

FBP algorithm was iterated with ve training pat-

terns (cf. Fig. 2), linear activation function, error

goal 0:001 and learning rate = 1:0. The interval

between the maximum of minimal input values and

minimum of maximal input values of the training

patterns is [a(0)mj ; a

(0)mj] = [ 0:4; 0:8]. Thus, according

to Theorem 2 for the target values t= 0:4; 0:6 and 0:8

belonging to the interval [0:4; 0:8] the convergence ofFBP algorithm is guaranteed. As illustrated in Fig. 4,

FBP algorithm trains the network only for 10 steps.

If the target value (t= 0:0 and 1:0) is outside the

interval [0:4; 0:8] the output value oscillates. There is

no convergence outside this interval. This conrms

the theoretical result.

The sum-squared error for the targets t= 0:4; 0:6

and 0:8 reaches the error goal 0:001 only for two

epochs (cf. Fig. 5) where during one epoch all ve

training patterns are learned. For target values outside

the interval [0:4; 0:8] the network cannot be trained

and the nal sum-squared error remains constant equal

to 0:59 for t= 0:0 and 0:26 for t= 1:0.

From the denition of FBP algorithm (cf.Section 2.2) it follows that its convergence speed is

maximal for the learning rate = 1:0. Fig. 6 illus-

trates this theoretical result. In case of single- and

multiple-training patterns only 3 and 10 steps, respec-

tively, are needed for training the network at learning

rate = 1:0. If the learning rate decreases the training

steps needed increase. For example for = 0:01 589

training steps in case of single-training pattern and

870 training steps in case of multiple-training patterns

are needed.

3.3. Comparison of FBP algorithm with SBP

algorithm

The convergence speed of FBP algorithm was

compared with that one of SBP algorithm (cf. Fig. 8)

for a network with four hidden layers (cf. Fig. 7),

100 randomly generated training patterns, randomly

generated initial network weights, error goal 0:02

and target value t= 0:8 belonging to the interval

[a(0)j ; a(0)

j ] = [ 0:4; 0:8]: FBP algorithm is signicantly


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Fig. 6. Dependence between learning rate and training steps needed for single-training pattern (STP) and multiple-training patterns (MTP).

Fig. 7. Example neural network with single-output neuron and four hidden layers.


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Fig. 8. Comparison of FBP and SBP algorithms in terms of steps needed for convergence in case of multiple-training patterns for two

learning rates (LR).

faster (seven epochs) than SBP algorithm (5500epochs) for learning rate = 0:01. SBP algorithm is

not convergent for learning rate 0:01. In Fig. 8

this is shown for = 1:0. For the same learning rate

FBP algorithm is convergent only for two epochs.

Numerous other experiments with dierent neu-

ral networks, learning rates and training patterns

conrmed the greater convergence speed of FBP al-

gorithm over SBP algorithm. Of course, this speed

depends also on the initial network weights.

4. Conclusions

In this paper we propose a fuzzy extension of the

backpropagation algorithm: fuzzy backpropagation al-

gorithm. This learning algorithm uses as net function

the fuzzy integral of Sugeno. Necessary and sucient

conditions for its convergence for single-output net-

works in case of single- and multiple-training patterns

are dened and proved. A computer simulation con-

rmed these theoretical results.

FBP algorithm has a number of advantages com-pared to SBP algorithm:

(1) greater convergence speed implying signicant

reduction in computation time what is important

in case of large sized neural networks,

(2) reaches always forward to the target value without

oscillations and there is no possibility to fall into

local minimum,

(3) requires no assumptions about probability distri-

butions and independence of input data (patterns

inputs),(4) does not require initial weights to be dierent from

each other,

(5) enables the automation of the weights tuning pro-

cess (quasi-unsupervised learning) by pointing

out the interval where the target value belongs to

[14,15]. The target values are determined:

automatically by computer in the interval,specied by input data,

semi-automatically by experts that dene thedirection and degree of target value changes.

Thus the actual target value is determined by


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fuzzy sets relevant to predened linguistic

expressions.

Another advantage of FBP algorithm is that it presents

a generalisation of the fuzzy perceptron algorithm

[19].FBP algorithm can support knowledge acquisition

and presentation especially in cases of uncertain im-

plicit knowledge, e.g. for construction of Common

KADS models in creative design process [5]; for in-

telligent data analysis [1]; for creating adaptive and

adaptable interfaces [4], for ergonomic assessment of

products [9], etc. It can be implemented as a module of

a CAKE-tool like CoMo-Kit [12] or as a stand-alone

tool for knowledge acquisition in systems for analysis

and design like ERGON EXPERT [7] and ISO 9241

evaluator [16].The ability of FBP algorithm for quasi-unsupervised

learning was successfully implemented for the cre-

ation of adaptable user interface of the information

system of Bulgarian parliament [15] and usability

evaluation of hypermedia user interfaces [17].

Finally, we should say that this study is not yet ap-

plied to many complex problems and the results are

not compared with dierent modications of SBP al-

gorithm. However, the current results are encouraging

for continuation of our work. We will have to make

many experiments and practical implementations of

FBP algorithm.

Appendix 1: Notation list

xj the jth patterns input

net(l)i net value of the ith neuron at the lth layer

a(l)i activation value of the ith neuron at the

lth layer

w(l)ij weight of the jth input of the ith neuron

at the lth layer

f(net) activation function

t the patterns output, i.e. target value

(l)i error signal relevant to the ith neuron at

the lth layer

s the sth iteration step

w(l)ij (s) weight w

(l)ij at the sth step

a(l)

i(s) activation value a

(l)

iat the sth step

wij weight change

woldij weight w(l)ij (s)

wnewij weight w(l)ij (s + 1)

the empty set

R the set of real numbers

N+

the set of positive integersJ the set of (indices of) network inputs

I the set of (indices of) neurons at the

hidden layer

P(I) the power set of the set I

max operation in unit interval [0; 1] min operation in unit interval [0; 1]

a(l)m activation value a

(l) relevant to the mth

pattern

tm the patterns output of the mth pattern

g fuzzy measure on the set P(J)

h fuzzy measure on the set P(I)

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