cooperative neighbors in defuzzification

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E L S E V I E R Fuzzy Sets and Systems 78 (1996) 37 49

FUZZY sets and systems

Cooperative neighbors in defuzzification

Shounak Roychowdhury, Bo-Hyeun Wang* Advanced Tech. Lab. #4, Central Research Laborator3', GoldStar Co,, Ltd., 16 Woomveon Dong, Seochu-Gu,

Seoul 137 140, South Korea

Received June 1994: revised October 1994

Abstract

Defuzzification is a problem of optimized selection of an element from a fuzzy set. It is in fact an optimization problem, optimization with respect to the whole system under consideration. We believe that neighbors can contribute and make a better selection in the process of defuzzification. We have proposed here a method that uses a collective decision making of cooperative neighbors. Natural selection process has been the basis of our model. We have also outlined three simple functions to show that adaptiveness can be incorporated in the process of defuzzification without making a possibility-probability transformation. The method further uses the concept of logical distance and the interaction between the neighbors to select an element. The neighbors poll and reach a decision of their most ideal representative.

Kevwords. Defuzzification; Operator; Optimization; Selection

1. Introduction

A set is a collection of elements having a given property. While allowing various degrees of the property, an ordinary set is extended to a fuzzy set. Defuzzification is a process to select a representative element from a fuzzy set. We believe that the selected element by defuzzification should have at least some common properties of all the elements in the fuzzy set. However, the basic idea behind defuzzification might not be so simple as it appears. It is indeed a fundamental problem of a selection in general and an optimal selection in particular from a collection of elements whose membership degrees are different.

* Corresponding author.

The two most widely accepted methods of defuzzification found in the literature are center of gravity (COG) and mean of maxima (MoM) [9-1/ , 18]. Recently, a number of researchers have realized that defuzzification is an important issue in the design of fuzzy systems and hence various attempts have been made to have a systematic procedure for choosing a proper defuzzification strategy. For example, Pfluger et al. [14] have discussed a defuzzification method in connection with fuzzy logic controllers and Park [13] has studied it in the problem of minimization of functions with fuzzy parameters. In 1991, Filev and Yager [5] have proposed basic distribution defuzzification (BADD) approach by transforming a possibility distribution to a probabilistic distribution based on Klir's principle of uncertainty invariance [7]. They have also developed a simple adaptive

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38 S. Royehowdhu~, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37 49

defuzzification strategy, called modified semi-linear defuzzification (MSLIDE) method, to simplify the parameter learning process in the BADD approach. Mabuchi [12] has discussed a defuzzification strategy to determine the most proper crisp value either from a fuzzy set or from an interval value with respect to a predetermined criterion.

Since a proper choice of a defuzzification strategy always influences the performance of the system appreciably, it has become clear and evident that there is a need to address the problem of defuzzification in its proper perspective very sys- tematically [11, 14]. However, there is neither a logically convincing technique nor even a profound thought available to us that explains how to defuzzify a fuzzy set.

In this paper, we at tempt to understand the problem of defuzzification from the perspective of optimal selection and establish a systematic procedure that is consistent and flexible enough to be applicable to most conceivable applications. We understand defuzzification as a process of optimized selection while considering the effects and influences of the surrounding modules. Such an optimization requires free parameters for required adaptation. We will show that with the help of cooperative neighbors in the defuzzifier module it is possible to control free parameters. Thus, we transform from a given possibility distribution to a desired possibility while considering the effects of the surroundings.

In Section 2 we will expound our viewpoints and show some differences between similar concepts found in spatial statistics. In the same section we will identify the problem of defuzzification and review some of the recent works. Section 3 demon- strates how neighbors in a set can help for a better selection. We believe that neighbors must recom- mend a rational selection mechanism. It implies that neighbors are equally important in a selective process. It motivates us to propose a method: cooperative neighbors in defuzzification technique.

2. The problem of defuzzification

In this section we will explain what we understand by the problem of defuzzification. Since it is

an open problem we need to address the problem at its roots. Followed by the review of some of the previous works we will propose an idea for defuzzification and will also put forth some supporting interpretations.

2.1. Defuzzification as the problem of optimal selection

Defuzzification is a process to select a representative element from a fuzzy set. A defuzzification method determines a most appropriate single value to represent a collection of elements whose membership degrees are different:

Oef : Af -~ ,Y, (1)

where Ar is a fuzzy set whose universe of discourse is X and ~ ~ X is a selected element. Now we would like to elaborate our viewpoint on defuzzification as optimal selection while reviewing earlier works. We comprehend that the defuzzification process should be addressed as an optimization process within the context of a complete system. The context of a complete system means the surroundings of a defuzzifier. We will see that there are two possible cases regarding the context's influence upon the defuzzifier: (1) when the defuzzifier is not affected by the context and (2) when the defuzzifier is affected by the context.

When the context does not influence the defuzzification module then we have an optimized selection called the center of gravity (COG). The CoG suggested by Zadeh [18] generates a value which is the center of area of the fuzzy set. It actually minim- izes the membership graded weighted mean of the squared distance [12 3. Consider a fuzzy set that is defuzzified,

, . . . ~ , . . . 7 ,

X i

where ~i =/~(xl) and # ( ' ) is the membership function. Let us consider a cost function that is weighted by its membership function,

K(~) = ~ (xl - 2) 2 p(xl), (3a) i - 1

S. Roychowdhu~, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37 49 39

y Plant t

Defuzzifler

Fuzzy Controller inference

Fig. 1. The fuzzy controlled system with optimization in rulebase and defuzzifier module.

where 2 e X. Minimizing K(£) by differentiation,

dK(2~) 2 Z (xi -- 2 ) # ( x , ) = O, (3b) d?~ i = l

provides the CoG:

n

2 = Y~i= t x i l l (x i ) (3c)

It is clear that although the CoG involves an optimization process, it is not related to the optimal design of overall fuzzy systems at all. Since a choice of defuzzification always influences the performance of fuzzy systems appreciably, it is necessary to deal with the problem of defuzzification as a part of design of fuzzy systems.

Let us consider a fuzzy logic controller as a special case of fuzzy systems. As in Fig. 1, a fuzzy logic controller consists of (1) fuzzifier, (2) inference engine, (3) fuzzy rulebase, and (4) defuzzifier. There are a couple of ways to optimize the fuzzy logic controller. The first approach is to optimize the fuzzy rulebase assuming that the defuzzification module is fixed. In this case, we expect that the given defuzzifier performs well in some sense just as the CoG does. As a matter of fact, most of the available designs belong to this approach. The second approach, when the context influences the defuzzifier, is to optimize the fuzzy rulebase and the defuzzification module as well. Such an optimization of the defuzzifier consequently leads us to the problem of selecting an optimal element by sharing data. Furthermore, the local optimization of the

rules in the rulebase and the strategies in the defuzzifier can be done in either an independent or interactive manner. The key to realize this approach is to add additional flexibilities to adapt the defuzzification module. In our present discussion, it has been assumed, for simplicity, that both defuzzi- tier and rulebase can be optimized while the others are fixed.

Considering the influences of the context on a defuzzifier Filev and Yager proposed methods based on the basic defuzzification distribution (BADD) approach [5]. Their main idea was to transform a possibility distribution to a probabilistic distribution based on Klir's principle of uncertainty invariance [7]. BADD transforms as concentration or dilation operator [18] to a desired degree depend- ing on 6. The BADD transformation is F z ( x i ) =

(Fl(Xi)) a where 6 e [0, ~) and F1 and F2 are the original and transformed fuzzy sets, respectively. They [16] also proposed SLIDE (semi-linear defuzzification) and an adaptive method that used a linear transformation to transform an original fuzzy set to another fuzzy set that is given by

Ta.t~:F 1 -"* F2, (4a)

where

{r/i if rh >/~,

F~i = (1 - fl)qi if ?/i < '~ (4b)

and r/i, and/~i are the membership values of fuzzy sets F1 and F2, respectively; see [5, 16, 17] for

40 S. Roychowdhu~, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37 49

details. In Yager and Filev's work it can be felt that they have considered the problem from a global viewpoint of adaptive system optimization. In their adaptive method, called modified SLIDE, they have used free parameters like ~ and/~ to control the effect of the context's influence of the defuzzifier. They have also used the gradient descent method to find a desired value.

With the advent of fuzzy sets there has been a debate between the Bayesian scientists and the fuzzy theorists regarding the possibility and probability transformation and other various types of uncertainty [6-8] . To mention in brevity, Klir [-7] firmly believes neither of them is stronger nor weaker than the other. He has proved that entropy in probability and non-specificity in possibility have similar valid roles: measures of information and uncertainty. He appreciates transformations from possibility to probability and vice versa. On the other hand, there are supporters of a viewpoint that ignorance and randomness contradict the structural homomorphism because they have different characteristics [4]. Dubois and Prade [4] have presented a well-written survey paper on this debatable topic.

Since we do not intend to use the controversial dilemma in our defuzzification strategy, we would like to use the shared information that could be made available to us from the fuzzy set itself by using cooperative neighbors. In the next subsection we propose our method.

2.2. The proposed method

The proposed method also uses weighted quad- ratic optimization similar to (3a) but with a different weight:

K(,2) = ~ (xi - 2) z s(xi), (5a) i = l

where s(x~) is called variation function or variation distribution of a fuzzy set which is generated after interaction of neighbors in the fuzzy set. The variation function is in fact a transformed possibility distribution.

There may be many families of variation functions. However, we would concentrate on a class of

variation functions, called the additive normalized family, which satisfies the following condition:

~ s(x i ) = 1. (5b) i=1

When individual properties of each element are only considered, then we have

f / ( ~ l . . . . . ]/i . . . . . ~n) s(xi) = . , (5c)

Z ~ : 1 L (~1, . - - , U l . . . . . m )

wheref( /~l , ... ,/~i . . . . . /~,) will be called the total contribution of interactions by all neighbors of the ith element including the ith element's own contribution. When neighbors do not interact, we have

f~(/~l . . . . . /l~ . . . . . /~,) =f ( /~) . In Section 3, we will discuss the details of how to compute the total contribution of interactions f ( . ).

Minimization of (5a) gives us

X = E~=IXiS (X i )

27=, s(xi) (5d)

It can be seen from (5d) that the variation functions control the quality of selection. Different variation functions in the class of additive normalized family can give rise to different solutions. This will also be analyzed in the next section.

Element to scalar (ETS) mapping maps a selected element to a scalar. We would like to clarify our views about ETS transformation. The role of ETS is to associate a numerical label to the selected element. In the case of a numerically ordered fuzzy set which is often used in fuzzy controllers, the numerical value is enough to quantify the element as a number. Whereas, in general, a scalar (number) or a numerical label is necessary and helpful to quantify those fuzzy elements that are not numbers but are abstract objects like "green" if they were to be used in computational decision making. ETS is very similar to the procedure of associating a color with its wavelength. If "green" is chosen as an element from a fuzzy set of green colors then it is better to quantify the chosen object with its charac- teristic wavelength, say 5310A. The label adds a precision for numerical evaluation. Thus,

ETS:{2} ~ ~. (6a)

S. Roychowdhury, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37 49 41

Defuzzification

Fig. 2. Defuzzif icat ion t rans forms a fuzzy set to a scalar.

In all numerical cases in this study we will consider

ETS{ . f I = 2. (6b)

Thus, the process of defuzzification is a process of local optimization within the defuzzification module followed by an ETS transformation as shown in Fig. 2. Let us consider a couple of examples of variation functions s(x).

Example 1. Consider (5c) with a linear function f (~ ; ) = k/~i. The selected element is given as

.~ = ~ x i klt; i ~.; k Ft; " (7)

Further we use E T S { 2 } = £ and that leads us to the familiar C o G which is widely used in many fuzzy applications.

Example 2. Consider f (p l ) = ( p y in (5c); then we have

; 2 j ( ~ ) ~. (8)

Using ETSI.~ } = £, we have the familiar BADD method, for ~ ~> 1.

2.3. Interpretations

2.3.1. Evolution as a basis' of our model It appears intuitive and a natural instinct of

a thought processing organism to choose the best, which is easily available, or the most appealing and suitable candidate from a group from which a selection can be made. Behind these apparent surfacing qualitative decisions there has to be a quantitative evaluation in a deterministic way. However, it is still unclear what exactly happens in living organ- isms, and how they make their subtle decisions. Even the process of evolution emphasizes the concept of survival of the fittest which is in a broad

sense a process of selection of the best species which can adapt to survive in changing conditions [3]. We would like to refine the idea of "evolution as selection" with qualitative thoughts.

Let us consider the problem from the classical Darwinian viewpoint. Assume in an isolated island there are some different species of iguanas. Some of them are weak, some of them are strong, and others with various special features. With the changing times only strong species of iguanas survive. They survive because of several factors including the following two primary reasons: (1) they are more competitive (individual attributes) and (2) the others are relatively weak (environmental attributes). It implies that the strongest survives on weaknesses of others. It explains that there is a de- pendency between their relative strength and their survival. Finally the strongest iguana survives as a representative of the iguanas. After all, survival of the fittest is one of nature's complex selection processes.

Thus we may realize that the selection procedure in nature involves a number of other variables which may be equally important, apart from the relative properties of the species. On one hand we try to understand this process as a computational selection. On the other hand, the selection mechanism still appears to us as a subjective and a multi-variable problem, because we have not reached that state of art so as to fully comprehend and understand the principles of intelligent decision making.

As Thaler [15] points out, the conventional view of evolution considers the environment that is re- sponsible for the selection step. Cairns et al. [1] proposed a more complete view of evolution that included the organism's perception of environment, genotype and phenotype mutations, and DNA metabolism apart from environmental processes. In either of the cases it is well understood and accepted that the environment has distinct factors that affect the process of selection. Either it takes part directly in the selection step or it plays indirectly through genetic mutations. Thaler E15] comments on Cairn's proposal, "... In this proposal, the environment not only selects among the existing variants, it also interacts with the organism in a sophisticated way to generate the variations on

42 S. Roychowdhury, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37-49

which the selection acts..." The latter view of evolution forms our basis of defuzzification.

In the case of fuzzy sets we can presume that other elements in a set form an environment for a particular element. We will call other elements neighbors. The neighbors collectively form an environment. The factors like organism's preception of environment, phenotype variation (characteristics), genotype variations (internal properties or membership values), and metabolism (coupling and relations with other elements) can be modeled as variation funct ions .

2.3.2. Kriging, s imi lar but another di f ferent topic

In statistics of spatial data, we have a similar prediction technique, on a given set of data present at different spatial locations, called Kriging. Here we would only like to point out that Kriging and defuzzification are rather similar but in different domains. Similar because they are both used for making inferences and prediction. The former pre- dicts a value at a given location based on spatial orientation of observed data based on distance measures, whereas the latter would predict a spatial location instead of a value at the location. In other words, defuzzification looks for an element from the support set and not for a belief value from the membership domain.

The Kriging methods are purely stochastic in nature. Here a limited explanation is rendered to facilitate the basics to highlight the differences. Let us have a set of spatial locations, L = {1l, 12 . . . . ,1,}. Also let Z( . ) be a random process generating values at the respective locations. Sup- pose we have observations on the set of spatial locations L:

Z ~ ( Z ( l , ) , Z ( l z ) , Z ( 1 3 ) . . . . . Z(l , )) . (9a)

The problem of optimal prediction is to estimate an optimal value at a given location based on the spatial orientation of the observed data. For this, two basic assumptions are usually required. The first assumption, called the model assumption, is related to the model of prediction:

z ( l ) = E(Z(1)) + ~(l), (9b)

where E ( Z ( . ) ) is the expected value of the random process. The second assumption is called the pre- dictor assumption:

P(Z( ' ) ; I ) = ~ ~iZ(li), ~ }~i = 1. (9C) i=1 i=1

Furthermore, in Kriging a concept of variogram is the link to spatial prediction. The variogram relates the spatial locations and the values at the spatial locations. A variogram, 27(' ), is given by the following equation:

27(si - s t) = V a r ( Z ( s i ) - Z(sj)) . (9d)

The reader may refer to I-2] for an interesting and complete discussion on various types of Kriging. In defuzzification we try to understand optimal prediction of an element in a deterministic way rather than using stochastic methods.

3. Cooperative neighbors in defuzzification: an implementation

In this section we will give some definitions that we will use in the remaining part of our discussions. In the following subsections we will show how different functions can affect the process, and lead to a better understanding of defuzzification.

3.1. Defini t ions

In this subsection we will address how to con- struct the total contribution using a few definitions. Consider a discrete fuzzy set given in (2). Assuming that it is ordered, we introduce the concept of neighbors: an xi (ith element) and its neighbor xi+j ((i + j ) t h element).

Definition 1. The separat ion distance between xl and its neighbor xi+j or xi_j is defined to be separ- ated by j units.

The term separation distance physically implies how far the interacting neighboring element is from the current element which is processing. It is a measure of the distance between two neighbors.

S. Rovchowdhury, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37 49 43

Moreover , just note that the te rm separa t ion distance is not related to any specific known distances like Euclidean or Maha lanob i s , etc.

Definition 2. The contributing factor at x~ by xj is denoted by Cf( i t i , l t i ) and is defined by

h(pi , l t j ) Cf(t~i,l(i) (I.J -- i l ) ' ' r ~> 1, (10)

where h(/zl, IL./) is called the interact ion function and Ij - il is the separa t ion distance.

In this paper we will consider the following interaction functions.

(1) Addit ive interact ion (h(/ti,/~fl = Pl + Pj):

_ t l i + ~L.i (11) Cf(Izi'IZJ) ([.j -- il) ~"

(2) Mult ipl icat ive interact ion (hOt , /9 ) = Pi lq):

lLi P.i (12) C f ( u i , t~j) - ( I J - i l ) ' "

(3) Min imum interaction (h(tti,/~j) = min(tti, ttfl):

min(itl,/~) Cf(l~'12J) - (IJ - i l)' (13)

We must note that the cont r ibut ing factors are those functions which can be adap ted and changed analytically or by exper imenta t ion . To add as a remark it is trivial to see that a s j increases, the effect of the cont r ibut ing factor decreases. Physically it signifies that the cont r ibut ion of a distant ne ighbor is less impor tant .

Definition 3. The collection of all cont r ibut ing factors leads to total-neighbor-contribution of all neighbors of xi which is denoted as c~i and is given by

7i = ~, Cf( l l i , it j). (14) i = 1. j ~ i

Definition 4. S e l f evaluation is defined as the contr i- but ion made by an element as it processes of its own evaluat ion and is given by g(l&) where g : [o , 1] -,. [o , 13.

Fuzzy set

Fig. 3. All neighbors (empty circles) interact with the ith element (filled circle) and reach a verdict ~. The verdict is passed to the element i. The defuzzifications strategy selects how the element i should interact with other neighboring elements. The loop on the ith element indicates the self-evaluation ,ql l~J.

In this pape r we will consider ,q(#i) =/11 as the self-evaluation. Certa inly one can have ,q(IL~) = laT, n/> 1. In doing so one only reduces the s trength of self-evaluation.

Definition 5. The complete contribution at x~ is defined as

f , ( l~ . . . . . /~.) = ~i ~ g ( l ~ ) , (15)

where ~} is an interact ion ope ra to r which evaluates the self-evaluation of an ith element and its total- ne ighbor-cont r ibu t ion 7~.

We will limit ourselves to a few operators ; ~) e { + , x ]. ~) = + indicates that the total-neighbor -con t r ibu t ion ~i f rom all the neighbors is added to the self-evaluation. This view models collective vote decision in polling, where vot ing f rom all elements is taken into account equally. On the other hand, ~} = x is used when the idea of using com- m o n propert ies of different votes is taken into account.

We now illustrate how the interact ion of the neighbors leads to a decision. Suppose that an element xi, which is represented as a filled circle in Fig. 3, interacts with all o ther elements {neighbors) which are shown by empty circles in the same figure. After interact ion between xl and its neighbors, we can compu te the to ta l -ne ighbor-cont r ibu- t ion zq by collecting all their Cf's . Evaluat ing :~ and

44 s. Roychowdhu~, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37-49

g(&) for all i, it is s t ra ight forward to obtain the var ia t ion function s(xl). Since we have l imited ourselves to three cont r ibu t ion factors with two different interact ion opera tors , there are six possible combina t ions o f f ' s . In the following sections, we will p ropose coopera t ive ne ighbor defuzzification methods based on these functions,

3.2. Cooperative neighbor defuzzification with additive contributing factors

When we consider addit ive cont r ibut ing functions, the total cont r ibut ion of all neighbors of xi is given as

~i = - - ( 1 6 ) j = l , j # i ( I j - i l ) ~

Case ~} = + : Consider ing the addit ive interac- t ion opera tor , the total cont r ibut ion at xi is given as

f / ( /A1 . . . . , /An) = /Ai + ~ i . (17a)

Consider (5c), (16), and (17a), and we have the following s(x):

s(x)= /Ai + ~ ' = 1.j#i((/Ai + &)/(lJ - i 1 ) 9

E T : l (/Ai + E~= 1, j ¢ i ((/Ai "at-/Aj)/(IJ - il)r)) "

(17b)

Further , using E T S { 2 } = 2,

X = ~ X i i = 1

#i + E~= L j # I((/Ai + #j)/(]J -- ilY)

E T : l (/Ai "~- E~'= 1, j ¢ i((/Ai + &)/ ( l J - - il)D) "

(17C)

Using (5d) and (18a), and E T S { 2 } = 2, we have

z., + xi # i ( ~ = l , j e i ( ( / A i +/Aj)/(Ij -- il)r)) 2 n

, '=1 ~7=l(/Ai(Ej=l.j~i((/Ai +/Afl/(lj - il)r)))

(18b)

E x a m p l e 3 (Nearest neighbor defuzzification). As r goes to infinity in (16), we find that the si tuat ion reduces to the nearest neighbors. Thus,

fi(/Ai 1,/Ai,/Ai+ l) = 3 /A i + /Ai-1 -~- / A i + I " (19a)

The defuzzified value is given as

3/Ai -~ /Ai + 1 -~- /Ai - I Lim(2) = xi - - - - . (19b)

3.3. Cooperative neighbor defuzzification with multiplicative contributing factors

In this case we consider that the contr ibut ing factor cont r ibuted by the xj element is as follows:

/A~ #s (20) Cf(#,,/Aj) = IJ - il ~

The total cont r ibut ion by all the neighboring elements is given by

/Ai /Aj ~=, .~, (I j - 7D' (21)

Case ~ = +: When the interact ion opera to r uses the addi t ion of propert ies and thus we have the total cont r ibut ion at xi as

f / (P l , .-. ,~An) = /Ai + O~i. (22a)


Remark: Note that in c o m p u t a t i o n we use the fact that /A-2, /A-1, ... ,/An+l,/An+2, . . . = O. We can see that such a me thod provides a lot of flexibility.

Case ~ = x: When the interact ion ope ra to r seeks c o m m o n propert ies , we model the total con- t r ibut ion at x~ by the following:

fi(/Al . . . . ,/A,) =/Ai x ~i- (18a)

/Ai + ~i s(x) - (22b)

EL 1(~; + ~,)

Further , using E T S { 2 } = 2, we have

2 ~- ~ X i i = 1

/Ai + ~ 7 = l , j # i ( / A i ~ j / ( [ j -- i l ) 9 n

27=, (/Ai + 2 j=l ,a , i (P i /A j / ( [ j --iD'))

(22C)

S. Rqvchowdhuo,, B.-H. Wang/ ' Fuzzy Sets and Systems 78 (1996) 37 49 45

Case {} = x" This corresponds to a si tuation in which f ( l q . . . . . I(.) = :q x/~. In this case we have

..~= n

~ xi (23) n / •

Example 4 (Nearest neighbor dcfuzzification). When r goes to infinity in (21) we find that (23) reduces to a situation where the nearest neighbors only contribute,

fii( Ill 1- t l i , lli + t ) = ,ai(,lli 11i- 1 Jr- ,ai,u. i + 1 )

= I~2( Pi-~ + tq+ ~), (24a)

and following the earlier steps the defuzzified value is given as

" I t ~ ( I t i + l + l q - 1 ) Lim(2) = ~ xi ,, . (24b)

3.4. Cooperative neighbor defuzzification with minimum contributing factors

I n this case we consider that the contr ibut ing factor contr ibuted by x i to x~ is as follows:

min (l(i,/4J) C/'( IL~, I~) - (25)

IJ - i l ~

The total contr ibut ion by all the neighboring elements is given by

min(lli, lti) ~i = ~ (26)

i :~.. i~z ( I j - i l ) "

Case ~ = + : Again consider the interaction opera tor to be purely additive. The total contr ibut ion is given as

. I } ( P l . . . . . I ( . ) = I t i + :~i. (27a)


s(x) = n

n

Y,":, (F~ + Y i= ~. i ~ ,(min(#,,/~j)/(IJ - iIF))

(27b)

Further, after using E T S {2 ~) = 2,

V xi ~ + Z~=l.j¢~(min(~i,l(i)/([j - il)") 2 i = , Y ' ~ - l ( I t i + Z " ~ - l . ~ ¢ i ( m i n ( l ~ i . l l i ) , / ( [ / - - il)~))"

(27c)

Case ~ = ×: When the interaction opera tor seeks c o m m o n properties, we model the total contr ibut ion at x~ by the following:

f i ( I t l . . . . . /tn) = Ill X gi . (28a)

Using (5d) and (28a), and after using ETS ~'~} = 2, we have

/.+ Xi /~i )_.j:L.i#i(min(lGllj)/{[j -- il)h) 2

(28b)

Belief

°.86 O.

O.

0.

Support Set 2 4 6 8 I0

(a). Fuzzy Set #1 h f - { °~ 04 09 , %, o~ o73 ? (~, 0, - T , 7 , r , ~ , - , - , , - ,-~,~}

Belief

0.8

0.6

0.4

0.2

0 Support Set 2 4 6 8 i0

=/T,T,T,~, V o, ( b ) . r u z z y S e t # 2 A f o, . . . . , o ? , v , ,-,~5 V,~}-

Belief

086 O.

0 4

O. Support Set

2 4 6 8 i0

(c). Fuzzy Set # 3 A I ={° i' o~ o, ~ o~ o4 o~ . . . . . . - - , - , 3 , - , - , ' - , ~ , ~ , ~ }

Fig. 4. Typical fuzzy sets that have been used in the defuzzification comparison.

46 S. Royehowdhury, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37-49

Table 1 A numerical comparison is made for the given fuzzy sets with the three proposed cases, where contributions are additive, multiplicative, and minimal along with CoG, Max, and BADD. As r increases in the given equations each equation stabilizes at a certain value (a) Fuzzy set 1 (see Fig. 5)

Eq. # \ r 1 2 3 4 5 20 200 Eq.(17c) 4.932274 4.740326 4.651106 4.616095 4.602459 4.592308 4.592308 Eq.(18b) 4.470511 4.305732 4.208101 4.160995 4.139171 4.121567 4.121567 Eq.(22c) 4.73639 4.604192 4.559878 4.546218 4.541854 4.539376 4.53976 Eq.(23) 4.172837 3.89789 3.744836 3.67173 3.63791 3.67173 3.67173 Eq.(27c) 4.960577 4.798243 4.723339 4.691746 4.67812 4.6667 4.6667 Eq.(28b) 4.480525 4.227655 4.07607 4.000715 3.965299 3.93334 3.93334 CoG 4.931818 4.931818 4.931818 4.931818 4.931818 4.931818 4.931818 B A D D = 2 4.413369 4.413369 4.413369 4.413369 4.413369 4.413369 4.413369 Max 4 4 4 4 4 4 4

(b) Fuzzy set 2 (see Fig. 6)

Eq. # \ r 1 2 3 4 5 20 200 Eq. (17c) 5.318079 5.230629 5.1883357 5.170846 5.163544 5.15748 5.15748 Eq. (18b) 5.260399 5.165772 5.12428 5.109559 5.104506 5.101351 5.101351 Eq. (22c) 5.351106 5.292346 5.274983 5.270884 5.270104 5.270227 5.270227 Eq. (23) 5.146355 4.926481 4.801303 4.743698 4.718324 4.696641 4.696641 Eq. (27c) 5.404301 5.3471 5.318892 5.3069 5.300782 5.295775 5.295775 Eq. (28b) 5.32209 5.180123 5.090304 5.04391 5.021352 5.00001 5.00001 CoG 5.488372 5.488372 5.488372 5.488372 5.488372 5.488372 5.488372 BADD = 2 5.361868 5.361868 5.361868 5.361868 5.361868 5.361868 5.361868 Max 4 4 4 4 4 4 4

(c) Fuzzy set 3 (see Fig. 7)

Eq. # \ r 1 2 3 4 5 20 200 Eq. (17c) 5.756464 5.776287 5.7618449 5.744952 5.734067 5.721805 5.721805 Eq. (18b) 6.342986 6.375082 6.383756 6.382311 6.3793 6.374257 6.374257 Eq. (22c) 5.984254 6.040595 6.053167 6.054087 6.053101 6.051095 6.051095 Eq. (23) 6.529733 6.61417 6.6544595 6.670045 6.675724 6.67933 6.67933 Eq. (27c) 5.644329 5.687537 5.708557 5.717681 5.721644 5.725 5.725 Eq. (28b) 6.070341 6.0669829 6.070536 6.07071 6.070549 6.07 6.07 CoG 6.04447 6.04447 6.04447 6.04447 6.04447 6.04447 6.04447 BADD = 2 6.656827 6.656827 6.656827 6.656827 6.656827 6.656827 6.656827 Max 8 8 8 8 8 8 8

Example 5 (Nearest neighbor defuzzification). If r goes to infinity in (26) then we observe that (28b) reduces to a situation where the nearest neighbors only contribute. We have

f i ( ]Ai- l , lAi, ]Ai + l )

and similarly following the steps as described earlier the defuzzified value is given as

L im( i )= ~" x pi(min(,ui, lai-1) + min(#i+ l,pi)) ~ n ~ - - • - ~ •

. . . . i=1 ~i=l lA i (mln ( l l i ,~ t i -1 ) - t -mln (12 i+l , ]2 i ) )

= pi(min(pi,/~i- 1) + min(pi+ 1,Pi)), (29a) (29b)

S. Rovchowdhu~, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37-49 4 7

5.S

5.3

5.1

4,9

4,7

~ 4.5 4.3

4.1

3.9

3.7

3,5

_° o

I i i i i i

2 3 4 5 20 200

-~O.~ Eqn(17.3) Eqn(18.2) Eqn(22 3)

X Eqn(23) ~K Eqn(27 3)

Eqn(28 2) COG

0 8ADD=2 O MAX

Fig. 5. Graphical representation for fuzzy set # I. X axis represents the value r and Y represents the defuzzified space. We will consider the same in the remaining figures too.

5.7 f

S.3

s . l

4.9

4.7

2 4.5

4.3

4.1

3.9

3.7

. _= _= •

0 ~ ~ ~ 0

i I I I I I

2 3 4 5 20 200

"~41~--"- Eqn( 17 3) '~11"~ Eqn(18 2) , . ~ . ~ Eqn(22 3)

X Eqn(23) .)i( Eqn(27 3)

Eqn(28.2) COG

O BADO=2 O MAX

Fig. 6. Graphical representation for fuzzy set # 2. For different methods results converge to different stable solutions.

3.5. Observations

We have considered three typical fuzzy sets as they are the c o m m o n fuzzy output held for defuzzification as shown in Fig. 4. The first fuzzy set shown in Fig. 4(a) is a typical example of prohibitive learning. Pfluger et al. [14] had confronted such a fuzzy set. The second fuzzy set is a more constrained example. The random generation defuzzification (RAGE) method proposed by Yager [17] used random number generation to select one of the probable defuzzified values in such a constrained situation. The third fuzzy set is a typical output in Mamdanrs inferencing method.

When we have applied the cooperative neighbor defuzzification methods to the above three typical fuzzy sets, we obtain Table 1 which is graphically illustrated in Figs. 5-7. From Table 1, we have observed a couple of important points. It has been observed that with the increase in r the variations are less felt in the defuzzified outputs. In other words, the defuzzified values of each method converge to a certain value since h(~q,l~j)/Ij- ilr~ h(#i,t~j)/lj- il r+t for a sufficiently large r. This suggests that in practice we can limit r to relatively small values such as 4 or 5. The second observation is that the defuzzified outputs using neighbors lie between C o G and Max in general. This implies that

48 S. Roychowdhury, B.-H. Wang / Fuzzy Sets and Systems 78 (1996) 37 49

7.5

7

~o > 6.5

4.5

O 0 O C 0 -0

0 C ~ 0 --_ = -_ =_ .~ ~=

_ A =_ ~= =_ =_ =

~ , =!t====-~. x= x x

---- .~ Eqn(17.3) ' - -~ '~ Eqn(l 8.2) '--t~--'- Eqn(22 3)

X Eqn(23] --~'1~-- Eqn(27 3)

Eqn(28 2) COG BADD=2

O MAX

I I I I I I

2 3 4 5 20 200

Fig. 7. Graphical representation for fuzzy set #3. We can note an increasing trend. Eqs. (18b) and (28b) give results less than CoG, whereas results from BADD at 2 and (23) are quite close.

10

9

8

7

6

5

4

3

2

1

---~----FS#1 S O -- ~ FS~2

~ F S # 3

5 10 15 20

12

Fig. 8. Defuzzified values as computed by BADD for different values of ~ for all the three fuzzy sets.

cooperative neighbor defuzzification can give a more flexible solution of which the designers can take effective advantage.

In the proposed method we may consider ~i in (14) as the free parameters which can be used for adaptation in order to realize the optimal defuzzi- fled scalar with respect to a given environment or context.

At the same time we should be careful about the fact that any movement of the defuzzified value closer towards the maximum value loses information of the entire fuzzy set. As Fig. 8 shows B A D D reaches the maximum value very fast because of the exponential factor. In that respect, B A D D provides

very limited flexibility to the designer. Fine tuning might not be very easy in it.

4. Conclusion

Selection of a representative element from a var- iety of elements is a complex problem in nature as well as in engineering. Defuzzification is a process of optimal selection of an element from a fuzzy set. It is in fact an optimization process. We understand that such an optimization should be understood by including the effects and influences of the surroundings and context. In that framework, without

S. Roychowdhur),, B.-H. Wang / Fuzzv Sets and Systems 78 (1996) 37 49 49

referring to possibility and probability transformation we undertake possibility to possibility (variation function) transformation using the information available to us from the neighboring elements. We refer to them as neighbors. Along with the optimization process the neighbors also certainly play an important role in the selection process.

We have proposed an idea for defuzzification based on the concept of neighborhood interaction called cooperative neighbors in defuzzification. The method of defuzzification presented here uses three functional classes indicating three types of interaction functions. These functional classes are: (1) the additive interaction, (2) the multiplicative interaction, and (3) the minimum interaction. We may also see that there may be quite a few more functional forms by which neighbors can interact. Along with these we have also considered here two different interaction operators to get the complete contribution. The numerical data show that the additive interaction operator gives output that is less closer to the maximum value when compared with the product interaction operator.

Regarding defuzzification as an optimal selection with respect to environment, the proposed defuzzification method is able to provide free parameters which are adapted for realizing an optimal defuzzi- fled value. The implementation of the issue of adaptation is currently in progress.

Although the procedures discussed here may be computationally expensive, yet there is a rational intellect to consider neighbors in a process where selection of the representative elements is intended. The neighbors poll and reach a decision of their most ideal representative. Our methods for defuzzification are based on these simple ideas of sharing, environment, and evolution.

Acknowledgements

The authors would like to thank the reviewers and the principal editor for their careful reading of the paper and their invaluable suggestions.

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