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The fuzzy arithmetic mean
Fuzzy Sets and Systems 107 (1999) 335–348 www.elsevier.com/locate/fss
The fuzzy arithmetic mean Lun Shan Gao∗ ADTEC Corporation, Megurohigashiyama-Bldg. 1-4-4, Higashiyama, Meguro-ku, Tokyo 153, Japan Received August 1997; received in revised form February 1998
Abstract The fuzzy average and the fuzzy arithmetic mean are represented in this paper. It has been proved that with a theorem the fuzzy arithmetic mean converges to the normal arithmetic mean. It is shown how to use the fuzzy average, which is c 1999 Elsevier Science B.V. All rights reserved. compared with fuzzy control rules with an example. Keywords: Linguistic variable; Triangular fuzzy number; Weighted matrix; Consequence matrix; Fuzzy uniform mapping; Fuzzy arithmetic mean; Fuzzy average
1. Introduction and background 1.1. Introduction Since fuzzy set [14] was proposed by Zadeh, the fuzzy c-mean [1] and the fuzzy weighted averages [2] have been investigated. The fuzzy c-mean is an extension of the c-mean that widely applied to cluster algorithms. The fuzzy weighted averages extends the normal weighted mean by using the concept of fuzzy numbers and the extension principle [15]. In the fuzzy control theory and fuzzy logic, a linguistic variable [15] is usually discussed with a universe of discourse U . A linguistic variable X is characterized by a quintuple (x; T (x); U; G; M˜ ) [15] in which x is the name of the variable; T (x) denotes the term set of x, that is, the set of names of linguistic values of x, with each value being a fuzzy variable denoted generically by x and ranging over a universe of discourse U which is associated with the base variable u; G is a syntactic rule for generating the name; M is a semantic rule for associating with each X its meaning, M˜ (x) is a fuzzy subset of U . When a linguistic variable (x; T (x); U; G; M˜ ) is applied to technology, there always exists a problem of how to decide the semantic rule M for the (x; T (x); U; G; M˜ ), that is to say, the problem is how to structure the fuzzy subset M˜ (x). There might be various M˜ (x) for the linguistic variable, when di erent human beings deÿne the fuzzy subset M˜ (x) of the linguistic variable, because people’s recognition of the semantic rule M is not always the same. The purpose of deÿning the fuzzy arithmetic mean is to describe an average of ∗
Present address: 25-10, Nagatakita 1 chome, Minami-ku, Yokohama, Japan 232-0071. E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved. 0165-0114/99/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 0 5 0 - 5
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these di erent M˜ belonging to the linguistic variable, and deciding a certain output on U , when inputs on U corresponding to each di erent M˜ (x) are given. 1.2. Background In the fuzzy control systems, the defuzziÿcation operator is computed as a weighted mean [10,12,13]. For example, in the case of a two-input–single-output fuzzy control system, the fuzzy control rules have the form: if x1 is A1 and x2 is B1 then y is C1 ; if x1 is A2 and x2 is B2 then y is C2 ; where x1 and x2 are input variables, y is an output variable, Ai ; Bi and Ci (i = 1; 2) are fuzzy sets, Ai ; Bi and Ci is a membership function of fuzzy sets Ai ; Bi and Ci , respectively. Here, the fuzzy reasoning of the second type [10] is cited, the defuzziÿcation operator is represented by R R y1 y dy + supp C y2 y dy supp C 2 R ; (1.1) yo = R 1 y1 dy + supp C y2 dy supp C 1
2
where yi = !i Ci (y); !i = minx1 ; x2 {Ai (x1 ); Bi (x2 )} (i = 1; 2): This defuzziÿcation value is represented by a weighted mean of y1 and y2 . It is clear that the defuzziÿcation value depends on the consequence of the fuzzy control rules, that is, if the fuzzy control rules are not given, the defuzziÿcation value cannot be computed. The fuzzy average is a defuzziÿcation operator. Throughout this paper, you can make out that fuzzy control rules are implicit in the fuzzy average, and the fuzzy control rules are represented by the consequence matrix of the fuzzy average. This paper is organized as follows: Section 2 gives deÿnitions of the fuzzy average and the fuzzy arithmetic mean. Section 3 gives deÿnitions of the fuzzy uniform mapping and the degenerate fuzzy number and a theorem with regard to the fuzzy arithmetic mean which converges to the normal arithmetic mean. Section 4 shows numerical results of the fuzzy arithmetic mean. Section 5 gives an example, and the fuzzy average is compared with fuzzy control rules. Section 6 gives a conclusion. 2. The deÿnitions Let us review two deÿnitions, one is the triangular fuzzy number, another is support of a fuzzy number. Deÿnition 2.1. A fuzzy number A is a triangular fuzzy number (TFN) denoted by (a1 ; a2 ; a3 )(a1 6a2 6a3 ) if its membership function A is given by (u − a1 ) ; u ∈ [a1 ; a2 ]; (a 2 − a1 ) (u − a3 ) (2.1) A (u) = ; u ∈ [a2 ; a3 ]; (a − a ) 2 3 0 otherwise: Deÿnition 2.2. A support of a fuzzy number A is an interval on real-valued R denoted by supp A = {u|A (u)¿0; u ∈ R}; if its membership function A (u) is continuous on real-valued, and u is called a mean value of the fuzzy number, if and only if A (u) = 1.
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For example, a TFN A = (a1 ; a2 ; a3 ) (a1 6a2 6a3 ); supp A is an open interval (a1 ; a3 ), such that, supp A = (a1 ; a3 ): a2 is the mean value of this TFN. Then, I show you the fuzzy average concerning two variables. Deÿnition 2.3. Let a linguistic variable X = (x; T (x); U; G; M˜ ). Suppose there exist two M˜ (xi ) to the X , and denote it as M˜ 1 (xi ) and M˜ 2 (xj ). Let M˜ 1 (xi ) correspond to a TFN Ai = (ai1 ; ai2 ; ai3 ) (i = 1; 2; : : : ; n) deÿned on U , and M˜ 2 (xj ) correspond to a TFN Bj = (b1j ; b2j ; b3j ) (j = 1; 2; : : : ; m) deÿned on U . Ai and Bj satisfy the following: ( 6= ∅ (empty set); ( j = i; i + 1); (2:2) supp Ai ∩ supp Aj = ∅; ( j 6= i; i + 1); (2:3) ( 6= ∅ (empty set); (j = i; i + 1); (2:4) supp B i ∩ supp Bj = ∅; (j 6= i; i + 1); (2:5) Ai ; Bj is the membership function of the TFN Ai ; Bi , respectively. The fuzzy average concerning u1 and u2 is deÿned as uf(n×m) (u1 ; u2 ) =
m n X X
!ij (Ai (u1 ); Bj (u2 ))rij ;
∀u1 ∈ supp Ai ; ∀u2 ∈ supp Bj ;
(2.6)
i=1 j=1
where (!ij (Ai (u1 ); Bj (u2 )))n×m is called a weighted matrix, the following is satisÿed: 06!ij (Ai (u1 ); Bj (u2 ))61; n X m X
(2.7)
!ij (Ai (u1 ); Bj (u2 )) = 1:
(2.8)
i=1 j=1
(rij )n×m is called a consequence matrix, especially, when rij satisÿes the following: rij =
ai2 + b2j ; 2
(i = 1; 2; : : : ; n; j = 1; 2; : : : ; m);
the fuzzy average is called the fuzzy arithmetic mean. In general, !ij is deÿned as T (A ; B ) !ij (Ai ; Bj ) = Pn Pm i j i=1 j=1 T (Ai ; Bj )
(i = 1; 2; : : : ; n; j = 1; 2; : : : ; m);
(2.9)
where T (Ai ; Bj ) means t-norm [11] concerning Ai and Bj . As regards the rij , the indices i; j give us a hint, there is a relation between fuzzy numbers Ai ; Bj and the elements rij (i = 1; 2; : : : ; n; j = 1; 2; : : : ; m) of the consequence matrix, let us describe the relation as n × m fuzzy control rules, such that, if Ai and Bj ; then rij ;
i = 1; 2; : : : ; n; j = 1; 2; : : : ; m;
that is to say, (1) n × m fuzzy control rules are implicit in the fuzzy average. (2) rij depends on the TFNs Ai and Bj , here, rij is a real value on the U . In Deÿnition 2.3, (2.2) and (2.4) mean that a common part on the U must exist to fuzzy numbers being adjacent to one another, that is, there exist fuzziness among names of the X being akin to one another. (2.3)
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and (2.5) mean that fuzzy numbers apart from each other never have any common part of its supports, that is, there is non fuzziness among names of the X being alien to each other. Clearly, the fuzzy average is a certain value on the real-valued, when u1 ∈ U; u2 ∈ U and rij ∈ U are given, the integers n; m and the real value !ij are determined. The fuzzy average is a convex conbination of elements of the consequence matrix (rij )n×m . Actually, it is found out from (2.7) and (2.8). 3. The fuzzy arithmetic mean and the normal arithmetic mean In order to explain a relation between the fuzzy arithmetic mean and the normal arithmetic mean, let me introduce two deÿnitions, one is the degenerate fuzzy number, another is the fuzzy uniform mapping. ˙ if supp A is degenerated Deÿnition 3.1. A fuzzy number A is called a degenerate fuzzy number denoted by A, into point a on real-valued, such that supp A = a;
A (a) = 1
(a ∈ R):
Especially, a TFN (a1 ; a2 ; a3 ) is a degenerate TFN if and only if a1 = a2 = a3 : The fuzzy arithmetic mean is an extension of the normal arithmetic mean. Actually, let a linguistic variable (x; T (x); U; G; M˜ ) with U = [0; 1], and with n = m = 3. Suppose that there are two M1 (x); M2 (x) to this linguistic variable, let M˜ i (x) correspond to a set of degenerate fuzzy numbers (i = 1; 2), such that, M˜ 1 corresponding to {A˙i };
(i = 1; 2; 3);
M˜ 2 corresponding to {B˙j };
( j = 1; 2; 3);
where A˙i ; B˙j is a degenerate TFN, respectively. Let us take A˙i and B˙j to be A˙1 = (0; 0; 0);
A˙2 = (0:5; 0; 5; 0; 5);
B˙ 1 = (0:2; 0:2; 0:2);
A˙3 = (1:0; 1:0; 1:0);
B˙ 2 = (0:6; 0:6; 0:6);
B˙ 3 = (0:9; 0:9; 0:9):
According to the fuzzy arithmetic mean, ∀u1 ∈ supp A˙i ; ∀u2 ∈ supp B˙j ; (rij )3×3 is as follows: ! 0:1 0:3 0:45 j (3×3) i a2 + b2 = 0:35 0:55 0:7 : (rij )(3×3) = 2 0:6 0:8 0:95 When u1 ∈ supp A˙k (k = 1; 2; 3) and u2 ∈ supp B˙ l (l = 1; 2; 3), based on properties of t-norm, ( 1; i = k; j = l; !ij = 0 otherwise; such that, col: l = 2 ↓ 0 1 0 ← row k = 1: (!ij )(3×3) = 0 0 0 0 0 0
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Fig. 1. Fuzzy uniform mapping.
P3 P3 uf(3×3) = i=1 j=1 !ij rij = r12 ; i.e., the fuzzy arithmetic mean is equal to the normal arithmetic mean in this case. Based on the deÿnition of linguistic variable [15], it is found out that there exist a relation between T (x) and fuzzy variables ranging over the U , and it was denoted with M˜ (x). The notation M˜ (x) can be considered as a mapping from T (x) to a set of fuzzy numbers deÿned on the U . Let me introduce a deÿnition of the fuzzy uniform mapping. Deÿnition 3.2. A mapping M˜ (x) from T (x) to a set of fuzzy numbers deÿned on the U will be called a fuzzy uniform mapping M (x), if and only if the following conditions are satisÿed. (1) T (x) have n (n ∈ N ) elements, i.e., the number of values of the X is n. (2) U is a bounded subset of real-valued R, and it is divided into n − 1 equal parts. (3) The fuzzy numbers deÿned on U are TFNs (ai1 ; ai2 ; ai3 ) (i = 1; 2; : : : ; n), and ai1 ; ai2 ; ai3 must be on points i−1 i is satisÿed (see divided equally. Shapes of the TFNs are uniform isosceles triangles, and ai+1 1 = a2 = a3 Fig. 1). For example, let X be a linguistic variable with the label “odd number” and with a U = [0; 10], then, the X have six values, x1 = “about 0”; x2 = “about 2”; x3 = “about 4”; x4 = “about 6”; x5 = “about 8”; x6 = “about 10”, such that, T (X ) = {x1 ; x2 ; x3 ; x4 ; x5 ; x6 }: Each value of the X can be described as follows: x1 is denoted by a TFN A1 = (0; 0; 2); x2 is denoted by a TFN A2 = (0; 2; 4); x3 is denoted by a TFN A3 = (2; 4; 6); x4 is denoted by a TFN A4 = (4; 6; 8); x5 is denoted by a TFN A5 = (6; 8; 10); x6 is denoted by a TFN A6 = (8; 10; 10). Clearly, the mapping M˜ (xi ) : T (xi ) 7→ {Ai }
(i = 1; 2; : : : ; 6)
is the fuzzy uniform mapping. Deÿnition 3.2 means that the M (x) of a linguistic variable is unique, that is, if there are two M˜ 1 (x) and ˜ M2 (x) to the linguistic variable, and M˜ 1 (x) is the M (x); M˜ 2 (x) is the M (x), then M˜ 1 (x) = M˜ 2 (x) = M (x). Based on the Deÿnition 2.3, the following theorem can be obtained. Theorem 3.1. If the mapping M˜ (x) is the fuzzy uniform mapping M (x); !ij (Ai (u1 ); Bj (u2 )) is a t-norm concerning the Ai and Bj . Then the fuzzy arithmetic mean converges to the corresponding normal arithmetic mean, when n tends to be inÿnite. Proof. Without loss of generality, let U = [0; 1]. Based on M˜ (x) = M (x), then Ai = Bi , such that, ai1 = bi1 ; ai2 = bi2 ; ai3 = bi3 (i = 1; 2; : : : ; n); where Ai ; Bi (i = 1; 2; : : : ; n) are TFNs on [0; 1].
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From (2.8), it is obtained that
uf(n×n) (u1 ; u2 ) −
n n X X u1 + u2 u1 + u2 !ij (Ai (u1 ); Aj (u2 ))rij − = 2 2 i=1 j=1 n X n X u1 + u2 ; !ij (Ai (u1 ); Aj (u2 )) rij − = 2 i=1 j=1
then, n n u1 + u2 X X u1 + u2 uf 6 : !ij (Ai (u1 ); Aj (u2 )) rij − (n×n) (u1 ; u2 ) − 2 2 i=1 j=1
When u1 and u2 are ÿxed, by using properties of the t-norm, elements of the weighted matrix are 0 except the following 2 × 2 submatrices:
!km !k+1m
!km+1 !k+1m+1 ;
(k; m = 1; 2; : : : ; n − 1);
such that,
(n×n) = (!ij )
col: m ↓ 0
0 ·
· ·
· !km
!km+1
!k+1m
!k+1m+1
· · 0 ← row k: · · 0
Actually, ÿxed u1 ∈ (supp Ak ∩ supp Ak+1 ); u2 ∈ (supp Am ∩ supp Am+1 ); then ( Ai (u1 )
6= 0;
i = k; k + 1
=0
otherwise; (
!ij (Ai (u1 ); Aj (u2 ))
( Aj (u2 )
6= 0;
j = m; m + 1
=0
otherwise;
6= 0;
i = k; k + 1; j = m; m + 1;
=0
otherwise:
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Thus, ∀u1 ∈ (supp Ak ∩ supp Ak+1 ); ∀u2 ∈ (supp Am ∩ supp Am+1 ), n X n X
!ij (Ai (u1 ); Aj (u2 ))|rij − 12 (u1 + u2 )|
i=1 j=1
= !km (Ak (u1 ); Am (u2 ))|rkm − 12 (u1 + u2 )| + !km+1 (Ak (u1 ); Am+1 (u2 ))|rkm+1 − 12 (u1 + u2 )| + !k+1m (Ak+1 (u1 ); Am (u2 ))|rk+1m − 12 (u1 + u2 )| + !k+1m+1 (Ak+1 (u1 ); Am+1 (u2 ))|rk+1m+1 − 12 (u1 + u2 )| + 0 m+1 1 1 k ) − 12 (u1 + u2 )| 6| 12 (ak2 + am 2 ) − 2 (u1 + u2 )| + | 2 (a2 + a2 1 1 k+1 + am + am+1 ) − 12 (u1 + u2 )| +| 12 (ak+1 2 ) − 2 (u1 + u2 )| + | 2 (a2 2 2 k m+1 1 − u2 )| = 12 |(ak2 − u1 ) + (am 2 − u2 )| + 2 |(a2 − u1 ) + (a2 k+1 1 − u1 ) + (am − u1 ) + (am+1 − u2 )| + 12 |(ak+1 2 − u2 )| + 2 |(a2 2 2
6 12 (8=(n − 1)): Clearly, when n → ∞, the fuzzy arithmetic mean converges to the corresponding normal arithmetic mean. 4. Numerical results concerning the fuzzy arithmetic mean In this section, the fuzzy arithmetic mean is investigated with the numerical analysis method, the conditions of Theorem 3.1 are satisÿed, such that, (1) Mapping M˜ (x) is the fuzzy uniform mapping M (x). (2) !ij (Ai (u1 ); Bj (u2 )) is a t-norm concerning Ai and Bj . Here, T (Ai ; Bj ) in formula (2.9) is deÿned as follows: T (Ai (u1 ); Bj (u2 )) = min{Ai (u1 ); Bj (u2 )}
(i = 1; 2; : : : ; n; j = 1; 2; : : : ; n):
In this case, !ij (Ai (u1 ); Bj (u2 )) satisÿes formula (2.7) and (2.8), which is a t-norm of the Ai and Bj . Let U = [0; 1], such that, u1 ∈ [0; 1]; u2 ∈ [0; 1]. I show you computation results of the fuzzy arithmetic mean in the case of n = 3. Based on Deÿnition 3.2, to divide [0; 1] into 2 parts, the TFNs Ai (i = 1; 2; 3) and Bj (j = 1; 2; 3) are taken as A1 = B1 = (0; 0; 0:5);
A2 = B2 = (0; 0:5; 1:0);
such that, a12 = b12 = 0;
a22 = b22 = 0:5;
a32 = b32 = 1:
A3 = B3 = (0:5; 1; 1);
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Fig. 2. The fuzzy arithmetic mean over [0; 1] × [0; 1] (n = m = 3).
Fig. 3. Contour lines of the fuzzy arithmetic mean.
Because of rij = 12 (ai2 + b2j ) (i; j = 1; 2; 3), the consequence matrix (rij )3×3 is as follows:
0:0
0:25
0:5
(rij )(3×3) = 0:25
0:5
0:75 :
0:5
0:75
1:0
According to Deÿnition 2.3, the fuzzy arithmetic mean on the domain [0; 1] × [0; 1] is computed. Fig. 2 shows a distribution of the fuzzy arithmetic mean over [0; 1] × [0; 1], it is clear that this distribution is approximate to that of the normal arithmetic mean. In order to distinguish the fuzzy arithmetic mean from the normal arithmetic mean, contour lines uf = 0:2; 0:4; 0:6 and 0.8 are drawn, Fig. 3 shows these contour lines of the fuzzy arithmetic mean and the normal arithmetic mean. Let us compare the fuzzy arithmetic mean with the normal arithmetic mean, it is clear that the contour lines of the fuzzy arithmetic mean have some curves, but the contour lines of the normal arithmetic mean are straight lines, that is, the fuzzy arithmetic mean has a nonlinear property with regard to parameters u1 and u2 .
5. An example In this section, I will show you how to use the fuzzy average to estimate a situation of a complex system. Let us assume that a complex system has a self diagnosis subsystem, a situation of the complex system is displayed on real time with the self diagnosis subsystem, at the same time, a situation of the complex system will be judged by an experienced operator based on a variety of conditions. Our purpose is to estimate a situation of the complex system by computing the fuzzy average concerning two input situations, one is shown by the self diagnosis subsystem, another is judged by an operator. Moreover, numerical computation results
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by using the fuzzy average is compared with that by using fuzzy control rules. The details are described as follows. 5.1. How to use the fuzzy average Let X = (x; T (x); U; G; M˜ ) be a linguistic variable with a label “situation” and with U = [0; 1]. It consists of a six step procedure that uses the fuzzy average. 5.1.1. Embodying the linguistic variable Suppose that the self diagnosis subsystem of the complex system has only recognized three situations: “bad”, “not so bad” and “ÿne”, i.e., the X has three values: “bad”, “not so bad” and “ÿne”, then, the X is embodied as follows: Xd = (situation; Td (situation); U; Gd ; M˜ d ); where Td (situation) = {bad d , not so badd , ÿned }. On the other hand, operators make the complex system to have ÿve situations: “very bad”, “bad”, “not so bad”, “ÿne” and “very ÿne”, i.e., the X has ÿve values in this case, then, the X is concretely represented by Xo = (situation; To (situation); U; Go ; M˜ o ); where To (situation) = {very bad o , bado , not so bado , ÿneo , very ÿneo }: 5.1.2. Deciding M˜ d and M˜ o In order to decide each mapping corresponding to the linguistic variable Xd or Xo , we have to deÿne TFNs on the U . Let us deÿne TFNs Ai (i = 1; 2; 3) as follows: A1 = (0; 0; 0:5);
A2 = (0; 0:5; 1:0);
A3 = (0:5; 1:0; 1:0);
then, the mapping M˜ d is deÿned as M˜ d : Td (situation) → {Ai };
(i = 1; 2; 3);
such that, bad d → A1 ;
not so bad d → A2 ;
ÿned → A3 :
Similarly, let us deÿne TFNs Bj ( j = 1; 2; : : : ; 5) as follows: B1 = (0; 0; 0:25);
B2 = (0; 0:25; 0:5);
B3 = (0:25; 0:5; 0:75);
B4 = (0:5; 0:75; 1:0);
then, the mapping M˜ o is deÿned as M˜ o : To (situation) → {Bj }
( j = 1; 2; 3; 4; 5);
such that very bad o → B1 ;
bad o → B2 ;
not so bad o → B3 ;
ÿneo → B4 ;
very ÿneo → B5 :
B5 = (0:75; 1:0; 1:0);
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Table 1 Estimated Model 1 Diagnosis Operator
Badd
Not so badd
Fined
very bado bado not so bado ÿneo very ÿneo
very bado bado bado badd badd
bado not so bado not so bado ÿneo ÿneo
badd badd ÿneo ÿneo very ÿneo
5.1.3. Building an estimated model It is important how to combine three situations stated by the self diagnosis subsystem with ÿve situations judged by an operator to an estimated model. For example, the situation of the complex system is judged as “very bado ”, when the situation shown by the self diagnosis subsystem is “badd ” and the situation judged by an operator is “very bado ”. Suppose that an estimated model is built as in Table 1. It is dicult how to decide the situation of the complex system, when the situation judged by an operator is opposed to the situation which gotten from the self diagnosis subsystem, e.g., the situation judged by an operator is “very bado ”, but at the same time, the situation gotten from the self diagnosis subsystem is “ÿned ”, in this case, the complex system has possibly gotten into danger, therefore, the situation of the system is decided to be “badd ” in this model. In general, an estimated model is decided by experts. 5.1.4. Getting the consequence matrix (rij )(5×3) Elements of the consequence matrix (rij )(5×3) is obtained by replacing elements of the estimated model with the mean values of TFNs Ai or Bj corresponding to these elements in the model (Table 1). For example, the element r11 is taken as the mean value b12 of B1 , because the TFN B1 corresponds to “very bado ” in the mapping M˜ o . The r13 is taken as a12 , because the TFN A1 corresponds to “badd ” in the mapping M˜ d . Then, the consequence matrix depends upon the estimated model and the mappings M˜ d and M˜ o , that is, rij will be determined, when the estimated model is decided and the mappings M˜ d and M˜ o are deÿned. According to this model shown in Table 1, the consequence matrix (rij )(5×3) is the following:
b12
2 b2 (5×3) = b22 (rij ) 1 a2 a12
b22 b32 b32 b42 b42
a12
0:0
0:25
a12 0:25 0:5 b42 = 0:25 0:5 b42 0:0 0:75 0:0 0:75 b52
0:0
0:0 0:75 : 0:75 1:0
5.1.5. Computation results Let us divide U = [0; 1] into 10 equal parts, then, points ((i − 1)=10; (j − 1)=10) (i; j = 1; 2; : : : ; 11) are obtained on the U × U plane. The fuzzy average on these points is computed, Fig. 4 shows a three-dimension graph, which presents a relation between two input variables and the fuzzy average. 5.1.6. New approach to the diagnosis system Based on the numerical computation result, to estimate the situations of the complex system, it is required to decide the boundary values between two situations. Let 0.2, 0.4, 0.6 and 0.8 be a boundary value between
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Fig. 4. Distribution of the fuzzy average (n = 5; m = 3).
“very bado ” and “bado ”, “bado ” and “not so bado ”, “not so bado ” and “ÿneo ” and “ÿneo ” and “very ÿneo ”, respectively, that is to say, if uf ∈ [0; 0:2), then the situation of the complex system is very bado ; if uf ∈ [0:2; 0:4), then the situation of the complex system is bado ; if uf ∈ [0:4; 0:6), then the situation of the complex system is not so bado ; if uf ∈ [0:6; 0:8), then the situation of the complex system is ÿneo ; if uf ∈ [0:8; 1:0], then the situation of the complex system is very ÿneo . In order to visibly see each area corresponding to each situation, contour lines uf = 0:2; 0:4; 0:6 and 0.8 are drawn. Fig. 5 shows a drawing of these contour lines. As shown in Fig. 5, the areas with white points mean that the situation is “very bado ”, the area with horizontal lines means that the situation is “bado ”, the area with vertical lines means that the situation is “not so bado ”, the area with slanting lines means that the situation is “ÿneo ”, the area with wavy lines means that the situation is “very ÿneo ”. 5.2. Numerical results by using fuzzy control rules Based on the fuzzy control theory, the estimated model 1 (Table 1) can be represented by 3 × 5 = 15 fuzzy control rules, such that, R1 : if u1 is very bado and u2 is badd , then y is very bado ; R2 : if u1 is very bado and u2 is not so badd , then y is bado ; R3 : if u1 is very bado and u2 is ÿned , then y is badd ; R4 : if u1 is bado and u2 is badd , then y is very bado ; R5 : if u1 is bado and u2 is not so badd , then y is not so bado ; R6 : if u1 is bado and u2 is ÿned , then y is badd ; R7 : if u1 is not so bado and u2 is badd , then y is very bado ; R8 : if u1 is not so bado and u2 is not so badd , then y is not so bado ; R9 : if u1 is not so bado and u2 is ÿned , then y is ÿneo ; R10 : if u1 is ÿneo and u2 is badd , then y is very badd ; R11 : if u1 is ÿneo and u2 is not so badd , then y is ÿneo ;
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Fig. 5. Contour lines of the fuzzy average (Fig. 4).
R12 : if u1 is ÿneo and u2 is ÿned , then y is ÿned ; R13 : if u1 is very ÿneo and u2 is badd , then y is badd ; R14 : if u1 is very ÿneo and u2 is not so badd , then y is ÿneo ; R15 : if u1 is very ÿneo and u2 is ÿned , then y is very ÿneo , where u1 is an input variable, which means situations judged by an operator, u2 is another input variable, which means situations shown by the self diagnosis subsystem, y is an output variable, which means the estimated situations based on the input variables u1 and u2 . Let us make use of the M˜ o ; M˜ d , and TFNs Ai (i = 1; 2; 3) and Bj (j = 1; 2; 3; 4; 5) described in Section 5.1. Here, the defuzziÿcation operator is deÿned as formula (1.1), such that, P15 R i=1 supp C i !i Ci (y)y dy o ; y = P15 R i=1 supp C !i Ci (y) dy i
where !i = min{Ai (u1 ); Bi (u2 )}; the Ci is a TFN corresponding to the consequence of these fuzzy control rules. For example, in the R2 ; the C2 is B2 , because the bado corresponding to the B2 in the mapping M˜ o , in the R6 , the C6 is A1 , because the badd corresponding to the A1 in the mapping M˜ d . Similar to the Section 5.1.5, outputs of these fuzzy control rules on points ((i − 1)=10; (j − 1)=10) (i; j = 1; 2; : : : ; 11) are computed. Fig. 6 shows an input–output relation of these fuzzy control rules R1 ; : : : ; R15 . Similar to Fig. 5, let us draw contour lines uf = 0:2; 0:4; 0:6 and 0.8, Fig. 7 shows a drawing of these contour lines. The areas with white points mean that the situation is “very bado ”, the area with horizontal lines means that the situation is “bado ”, the area with vertical lines means that the situation is “not so bado ”, the area with slanting lines means that the situation is “ ÿneo ”, the area with wavy lines means that the situation is “very ÿneo ”. If we compare Fig. 5 with Fig. 7, it is found that each area keeps its balance in Fig. 5, but the red and green areas are too small in Fig. 7. Moreover, implementation of the fuzzy average is easier than that of the fuzzy control rules in point of software, because by using the fuzzy control rules we have to compute each integration, especially, the number of fuzzy control rules is many more, a number of computing these integrations is many more.
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Fig. 6. Input–output relation of the fuzzy control rules.
Fig. 7. Contour lines of the relation (Fig. 6).
6. Conclusion The fuzzy average and the fuzzy arithmetic mean concerning two input variables are suggested, as regards three input variables, four input variables and so on the fuzzy average and the fuzzy arithmetic mean can be similarly deÿned. If we compare the fuzzy average with fuzzy control rules and existing defuzziÿcation operators, we ÿnd that (1) The fuzzy average is not only a defuzziÿcation method, but also a representation of fuzzy control rules by using the consequence matrix.
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(2) The implementation of the fuzzy average is easier than that of the fuzzy control rules. Moreover, it has been proved that the fuzzy arithmetic mean converges to the normal arithmetic mean. Through an example, the procedure of using the fuzzy average is described. Based on deÿnitions of the fuzzy average, the fuzzy arithmetic mean, the arithmetic mean, the weighted mean and the fuzzy weighted average [2], relations among these concepts can be described as: {fuzzy average} ⊇ {fuzzy arithmetic mean} ⊇ {arithmetic mean}; {fuzzy weighted average} ⊇ {weighted mean} ⊇ {arithmetic mean}; that is, the normal arithmetic mean is an intersection of the fuzzy average set and the fuzzy weighted average set. References [1] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum, New York, 1981. [2] W.M. Dong, F.S. Wong, Fuzzy weighted averages and implementation of the extension principle, Fuzzy Sets and Systems 21 (1987) 183–199. [3] D. Dubois, H. Prade, Additions of interactive fuzzy numbers, IEEE Trans. Autom. Control 26 (1981) 926 – 936. [4] L.S. Gao, The application of fuzzy reason to curvic approximation problems, Japan 8th Fuzzy System Symp., Hiroshima, 26 –28 May, 1992, pp. 405– 408. [5] L.S. Gao, Studies on application of fuzzy averaging in numerical analysis, Ph.D. Thesis, Chiba University, 1995. [6] L.S. Gao, H. Imai, H. Kawarada, Fuzzy control of system governed by elliptic partial di erential equations, Technical Reports of Mathematical Sciences, Chiba Univ. 5, 1989. [7] L.S. Gao, H. Kawarada, Application of fuzzy average to curve and surface ÿtting, Proc. FUZZ-IEEE=IFES’95, 1995. [8] A. Kaufmann, M.M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science, Elsevier, Amsterdam, 1988. [9] C.C. Lee, Fuzzy logic in control systems: fuzzy logic controller – Part 1, IEEE Trans. Systems Man Cybernet. SMC-20 (2) (1990) 405– 418. [10] C.C. Lee, Fuzzy logic in control systems: fuzzy logic controller – Part 2, IEEE Trans. Systems Man Cybernet. SMC-20 (2) (1990) 419 – 435. [11] M. Mizumoto, Pictorial representations of fuzzy connectives, Part I: Cases of t-Norms, t-Conorms and averaging operators, Fuzzy Sets and Systems 31 (1989) 217–242. [12] M. Sugeno, Fuzzy Control, Nikkan Kounyo Shinbunsya, Tokyo, 1989 (in Japanese). [13] T. Takagi, M. Sugeno, Derivation of fuzzy control rules from human operator’s control actions, Proc. IFAC Symp. on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseilles, France, July 1983, pp. 55– 60. [14] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338–353. [15] H.J. Zimmermann, Fuzzy Set Theory and Its Application, Kluwer Academic Publishers, Boston, MA, 1984.
The fuzzy arithmetic mean Lun Shan Gao∗ ADTEC Corporation, Megurohigashiyama-Bldg. 1-4-4, Higashiyama, Meguro-ku, Tokyo 153, Japan Received August 1997; received in revised form February 1998
Abstract The fuzzy average and the fuzzy arithmetic mean are represented in this paper. It has been proved that with a theorem the fuzzy arithmetic mean converges to the normal arithmetic mean. It is shown how to use the fuzzy average, which is c 1999 Elsevier Science B.V. All rights reserved. compared with fuzzy control rules with an example. Keywords: Linguistic variable; Triangular fuzzy number; Weighted matrix; Consequence matrix; Fuzzy uniform mapping; Fuzzy arithmetic mean; Fuzzy average
1. Introduction and background 1.1. Introduction Since fuzzy set [14] was proposed by Zadeh, the fuzzy c-mean [1] and the fuzzy weighted averages [2] have been investigated. The fuzzy c-mean is an extension of the c-mean that widely applied to cluster algorithms. The fuzzy weighted averages extends the normal weighted mean by using the concept of fuzzy numbers and the extension principle [15]. In the fuzzy control theory and fuzzy logic, a linguistic variable [15] is usually discussed with a universe of discourse U . A linguistic variable X is characterized by a quintuple (x; T (x); U; G; M˜ ) [15] in which x is the name of the variable; T (x) denotes the term set of x, that is, the set of names of linguistic values of x, with each value being a fuzzy variable denoted generically by x and ranging over a universe of discourse U which is associated with the base variable u; G is a syntactic rule for generating the name; M is a semantic rule for associating with each X its meaning, M˜ (x) is a fuzzy subset of U . When a linguistic variable (x; T (x); U; G; M˜ ) is applied to technology, there always exists a problem of how to decide the semantic rule M for the (x; T (x); U; G; M˜ ), that is to say, the problem is how to structure the fuzzy subset M˜ (x). There might be various M˜ (x) for the linguistic variable, when di erent human beings deÿne the fuzzy subset M˜ (x) of the linguistic variable, because people’s recognition of the semantic rule M is not always the same. The purpose of deÿning the fuzzy arithmetic mean is to describe an average of ∗
Present address: 25-10, Nagatakita 1 chome, Minami-ku, Yokohama, Japan 232-0071. E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved. 0165-0114/99/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 0 5 0 - 5
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these di erent M˜ belonging to the linguistic variable, and deciding a certain output on U , when inputs on U corresponding to each di erent M˜ (x) are given. 1.2. Background In the fuzzy control systems, the defuzziÿcation operator is computed as a weighted mean [10,12,13]. For example, in the case of a two-input–single-output fuzzy control system, the fuzzy control rules have the form: if x1 is A1 and x2 is B1 then y is C1 ; if x1 is A2 and x2 is B2 then y is C2 ; where x1 and x2 are input variables, y is an output variable, Ai ; Bi and Ci (i = 1; 2) are fuzzy sets, Ai ; Bi and Ci is a membership function of fuzzy sets Ai ; Bi and Ci , respectively. Here, the fuzzy reasoning of the second type [10] is cited, the defuzziÿcation operator is represented by R R y1 y dy + supp C y2 y dy supp C 2 R ; (1.1) yo = R 1 y1 dy + supp C y2 dy supp C 1
2
where yi = !i Ci (y); !i = minx1 ; x2 {Ai (x1 ); Bi (x2 )} (i = 1; 2): This defuzziÿcation value is represented by a weighted mean of y1 and y2 . It is clear that the defuzziÿcation value depends on the consequence of the fuzzy control rules, that is, if the fuzzy control rules are not given, the defuzziÿcation value cannot be computed. The fuzzy average is a defuzziÿcation operator. Throughout this paper, you can make out that fuzzy control rules are implicit in the fuzzy average, and the fuzzy control rules are represented by the consequence matrix of the fuzzy average. This paper is organized as follows: Section 2 gives deÿnitions of the fuzzy average and the fuzzy arithmetic mean. Section 3 gives deÿnitions of the fuzzy uniform mapping and the degenerate fuzzy number and a theorem with regard to the fuzzy arithmetic mean which converges to the normal arithmetic mean. Section 4 shows numerical results of the fuzzy arithmetic mean. Section 5 gives an example, and the fuzzy average is compared with fuzzy control rules. Section 6 gives a conclusion. 2. The deÿnitions Let us review two deÿnitions, one is the triangular fuzzy number, another is support of a fuzzy number. Deÿnition 2.1. A fuzzy number A is a triangular fuzzy number (TFN) denoted by (a1 ; a2 ; a3 )(a1 6a2 6a3 ) if its membership function A is given by (u − a1 ) ; u ∈ [a1 ; a2 ]; (a 2 − a1 ) (u − a3 ) (2.1) A (u) = ; u ∈ [a2 ; a3 ]; (a − a ) 2 3 0 otherwise: Deÿnition 2.2. A support of a fuzzy number A is an interval on real-valued R denoted by supp A = {u|A (u)¿0; u ∈ R}; if its membership function A (u) is continuous on real-valued, and u is called a mean value of the fuzzy number, if and only if A (u) = 1.
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For example, a TFN A = (a1 ; a2 ; a3 ) (a1 6a2 6a3 ); supp A is an open interval (a1 ; a3 ), such that, supp A = (a1 ; a3 ): a2 is the mean value of this TFN. Then, I show you the fuzzy average concerning two variables. Deÿnition 2.3. Let a linguistic variable X = (x; T (x); U; G; M˜ ). Suppose there exist two M˜ (xi ) to the X , and denote it as M˜ 1 (xi ) and M˜ 2 (xj ). Let M˜ 1 (xi ) correspond to a TFN Ai = (ai1 ; ai2 ; ai3 ) (i = 1; 2; : : : ; n) deÿned on U , and M˜ 2 (xj ) correspond to a TFN Bj = (b1j ; b2j ; b3j ) (j = 1; 2; : : : ; m) deÿned on U . Ai and Bj satisfy the following: ( 6= ∅ (empty set); ( j = i; i + 1); (2:2) supp Ai ∩ supp Aj = ∅; ( j 6= i; i + 1); (2:3) ( 6= ∅ (empty set); (j = i; i + 1); (2:4) supp B i ∩ supp Bj = ∅; (j 6= i; i + 1); (2:5) Ai ; Bj is the membership function of the TFN Ai ; Bi , respectively. The fuzzy average concerning u1 and u2 is deÿned as uf(n×m) (u1 ; u2 ) =
m n X X
!ij (Ai (u1 ); Bj (u2 ))rij ;
∀u1 ∈ supp Ai ; ∀u2 ∈ supp Bj ;
(2.6)
i=1 j=1
where (!ij (Ai (u1 ); Bj (u2 )))n×m is called a weighted matrix, the following is satisÿed: 06!ij (Ai (u1 ); Bj (u2 ))61; n X m X
(2.7)
!ij (Ai (u1 ); Bj (u2 )) = 1:
(2.8)
i=1 j=1
(rij )n×m is called a consequence matrix, especially, when rij satisÿes the following: rij =
ai2 + b2j ; 2
(i = 1; 2; : : : ; n; j = 1; 2; : : : ; m);
the fuzzy average is called the fuzzy arithmetic mean. In general, !ij is deÿned as T (A ; B ) !ij (Ai ; Bj ) = Pn Pm i j i=1 j=1 T (Ai ; Bj )
(i = 1; 2; : : : ; n; j = 1; 2; : : : ; m);
(2.9)
where T (Ai ; Bj ) means t-norm [11] concerning Ai and Bj . As regards the rij , the indices i; j give us a hint, there is a relation between fuzzy numbers Ai ; Bj and the elements rij (i = 1; 2; : : : ; n; j = 1; 2; : : : ; m) of the consequence matrix, let us describe the relation as n × m fuzzy control rules, such that, if Ai and Bj ; then rij ;
i = 1; 2; : : : ; n; j = 1; 2; : : : ; m;
that is to say, (1) n × m fuzzy control rules are implicit in the fuzzy average. (2) rij depends on the TFNs Ai and Bj , here, rij is a real value on the U . In Deÿnition 2.3, (2.2) and (2.4) mean that a common part on the U must exist to fuzzy numbers being adjacent to one another, that is, there exist fuzziness among names of the X being akin to one another. (2.3)
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and (2.5) mean that fuzzy numbers apart from each other never have any common part of its supports, that is, there is non fuzziness among names of the X being alien to each other. Clearly, the fuzzy average is a certain value on the real-valued, when u1 ∈ U; u2 ∈ U and rij ∈ U are given, the integers n; m and the real value !ij are determined. The fuzzy average is a convex conbination of elements of the consequence matrix (rij )n×m . Actually, it is found out from (2.7) and (2.8). 3. The fuzzy arithmetic mean and the normal arithmetic mean In order to explain a relation between the fuzzy arithmetic mean and the normal arithmetic mean, let me introduce two deÿnitions, one is the degenerate fuzzy number, another is the fuzzy uniform mapping. ˙ if supp A is degenerated Deÿnition 3.1. A fuzzy number A is called a degenerate fuzzy number denoted by A, into point a on real-valued, such that supp A = a;
A (a) = 1
(a ∈ R):
Especially, a TFN (a1 ; a2 ; a3 ) is a degenerate TFN if and only if a1 = a2 = a3 : The fuzzy arithmetic mean is an extension of the normal arithmetic mean. Actually, let a linguistic variable (x; T (x); U; G; M˜ ) with U = [0; 1], and with n = m = 3. Suppose that there are two M1 (x); M2 (x) to this linguistic variable, let M˜ i (x) correspond to a set of degenerate fuzzy numbers (i = 1; 2), such that, M˜ 1 corresponding to {A˙i };
(i = 1; 2; 3);
M˜ 2 corresponding to {B˙j };
( j = 1; 2; 3);
where A˙i ; B˙j is a degenerate TFN, respectively. Let us take A˙i and B˙j to be A˙1 = (0; 0; 0);
A˙2 = (0:5; 0; 5; 0; 5);
B˙ 1 = (0:2; 0:2; 0:2);
A˙3 = (1:0; 1:0; 1:0);
B˙ 2 = (0:6; 0:6; 0:6);
B˙ 3 = (0:9; 0:9; 0:9):
According to the fuzzy arithmetic mean, ∀u1 ∈ supp A˙i ; ∀u2 ∈ supp B˙j ; (rij )3×3 is as follows: ! 0:1 0:3 0:45 j (3×3) i a2 + b2 = 0:35 0:55 0:7 : (rij )(3×3) = 2 0:6 0:8 0:95 When u1 ∈ supp A˙k (k = 1; 2; 3) and u2 ∈ supp B˙ l (l = 1; 2; 3), based on properties of t-norm, ( 1; i = k; j = l; !ij = 0 otherwise; such that, col: l = 2 ↓ 0 1 0 ← row k = 1: (!ij )(3×3) = 0 0 0 0 0 0
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Fig. 1. Fuzzy uniform mapping.
P3 P3 uf(3×3) = i=1 j=1 !ij rij = r12 ; i.e., the fuzzy arithmetic mean is equal to the normal arithmetic mean in this case. Based on the deÿnition of linguistic variable [15], it is found out that there exist a relation between T (x) and fuzzy variables ranging over the U , and it was denoted with M˜ (x). The notation M˜ (x) can be considered as a mapping from T (x) to a set of fuzzy numbers deÿned on the U . Let me introduce a deÿnition of the fuzzy uniform mapping. Deÿnition 3.2. A mapping M˜ (x) from T (x) to a set of fuzzy numbers deÿned on the U will be called a fuzzy uniform mapping M (x), if and only if the following conditions are satisÿed. (1) T (x) have n (n ∈ N ) elements, i.e., the number of values of the X is n. (2) U is a bounded subset of real-valued R, and it is divided into n − 1 equal parts. (3) The fuzzy numbers deÿned on U are TFNs (ai1 ; ai2 ; ai3 ) (i = 1; 2; : : : ; n), and ai1 ; ai2 ; ai3 must be on points i−1 i is satisÿed (see divided equally. Shapes of the TFNs are uniform isosceles triangles, and ai+1 1 = a2 = a3 Fig. 1). For example, let X be a linguistic variable with the label “odd number” and with a U = [0; 10], then, the X have six values, x1 = “about 0”; x2 = “about 2”; x3 = “about 4”; x4 = “about 6”; x5 = “about 8”; x6 = “about 10”, such that, T (X ) = {x1 ; x2 ; x3 ; x4 ; x5 ; x6 }: Each value of the X can be described as follows: x1 is denoted by a TFN A1 = (0; 0; 2); x2 is denoted by a TFN A2 = (0; 2; 4); x3 is denoted by a TFN A3 = (2; 4; 6); x4 is denoted by a TFN A4 = (4; 6; 8); x5 is denoted by a TFN A5 = (6; 8; 10); x6 is denoted by a TFN A6 = (8; 10; 10). Clearly, the mapping M˜ (xi ) : T (xi ) 7→ {Ai }
(i = 1; 2; : : : ; 6)
is the fuzzy uniform mapping. Deÿnition 3.2 means that the M (x) of a linguistic variable is unique, that is, if there are two M˜ 1 (x) and ˜ M2 (x) to the linguistic variable, and M˜ 1 (x) is the M (x); M˜ 2 (x) is the M (x), then M˜ 1 (x) = M˜ 2 (x) = M (x). Based on the Deÿnition 2.3, the following theorem can be obtained. Theorem 3.1. If the mapping M˜ (x) is the fuzzy uniform mapping M (x); !ij (Ai (u1 ); Bj (u2 )) is a t-norm concerning the Ai and Bj . Then the fuzzy arithmetic mean converges to the corresponding normal arithmetic mean, when n tends to be inÿnite. Proof. Without loss of generality, let U = [0; 1]. Based on M˜ (x) = M (x), then Ai = Bi , such that, ai1 = bi1 ; ai2 = bi2 ; ai3 = bi3 (i = 1; 2; : : : ; n); where Ai ; Bi (i = 1; 2; : : : ; n) are TFNs on [0; 1].
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From (2.8), it is obtained that
uf(n×n) (u1 ; u2 ) −
n n X X u1 + u2 u1 + u2 !ij (Ai (u1 ); Aj (u2 ))rij − = 2 2 i=1 j=1 n X n X u1 + u2 ; !ij (Ai (u1 ); Aj (u2 )) rij − = 2 i=1 j=1
then, n n u1 + u2 X X u1 + u2 uf 6 : !ij (Ai (u1 ); Aj (u2 )) rij − (n×n) (u1 ; u2 ) − 2 2 i=1 j=1
When u1 and u2 are ÿxed, by using properties of the t-norm, elements of the weighted matrix are 0 except the following 2 × 2 submatrices:
!km !k+1m
!km+1 !k+1m+1 ;
(k; m = 1; 2; : : : ; n − 1);
such that,
(n×n) = (!ij )
col: m ↓ 0
0 ·
· ·
· !km
!km+1
!k+1m
!k+1m+1
· · 0 ← row k: · · 0
Actually, ÿxed u1 ∈ (supp Ak ∩ supp Ak+1 ); u2 ∈ (supp Am ∩ supp Am+1 ); then ( Ai (u1 )
6= 0;
i = k; k + 1
=0
otherwise; (
!ij (Ai (u1 ); Aj (u2 ))
( Aj (u2 )
6= 0;
j = m; m + 1
=0
otherwise;
6= 0;
i = k; k + 1; j = m; m + 1;
=0
otherwise:
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Thus, ∀u1 ∈ (supp Ak ∩ supp Ak+1 ); ∀u2 ∈ (supp Am ∩ supp Am+1 ), n X n X
!ij (Ai (u1 ); Aj (u2 ))|rij − 12 (u1 + u2 )|
i=1 j=1
= !km (Ak (u1 ); Am (u2 ))|rkm − 12 (u1 + u2 )| + !km+1 (Ak (u1 ); Am+1 (u2 ))|rkm+1 − 12 (u1 + u2 )| + !k+1m (Ak+1 (u1 ); Am (u2 ))|rk+1m − 12 (u1 + u2 )| + !k+1m+1 (Ak+1 (u1 ); Am+1 (u2 ))|rk+1m+1 − 12 (u1 + u2 )| + 0 m+1 1 1 k ) − 12 (u1 + u2 )| 6| 12 (ak2 + am 2 ) − 2 (u1 + u2 )| + | 2 (a2 + a2 1 1 k+1 + am + am+1 ) − 12 (u1 + u2 )| +| 12 (ak+1 2 ) − 2 (u1 + u2 )| + | 2 (a2 2 2 k m+1 1 − u2 )| = 12 |(ak2 − u1 ) + (am 2 − u2 )| + 2 |(a2 − u1 ) + (a2 k+1 1 − u1 ) + (am − u1 ) + (am+1 − u2 )| + 12 |(ak+1 2 − u2 )| + 2 |(a2 2 2
6 12 (8=(n − 1)): Clearly, when n → ∞, the fuzzy arithmetic mean converges to the corresponding normal arithmetic mean. 4. Numerical results concerning the fuzzy arithmetic mean In this section, the fuzzy arithmetic mean is investigated with the numerical analysis method, the conditions of Theorem 3.1 are satisÿed, such that, (1) Mapping M˜ (x) is the fuzzy uniform mapping M (x). (2) !ij (Ai (u1 ); Bj (u2 )) is a t-norm concerning Ai and Bj . Here, T (Ai ; Bj ) in formula (2.9) is deÿned as follows: T (Ai (u1 ); Bj (u2 )) = min{Ai (u1 ); Bj (u2 )}
(i = 1; 2; : : : ; n; j = 1; 2; : : : ; n):
In this case, !ij (Ai (u1 ); Bj (u2 )) satisÿes formula (2.7) and (2.8), which is a t-norm of the Ai and Bj . Let U = [0; 1], such that, u1 ∈ [0; 1]; u2 ∈ [0; 1]. I show you computation results of the fuzzy arithmetic mean in the case of n = 3. Based on Deÿnition 3.2, to divide [0; 1] into 2 parts, the TFNs Ai (i = 1; 2; 3) and Bj (j = 1; 2; 3) are taken as A1 = B1 = (0; 0; 0:5);
A2 = B2 = (0; 0:5; 1:0);
such that, a12 = b12 = 0;
a22 = b22 = 0:5;
a32 = b32 = 1:
A3 = B3 = (0:5; 1; 1);
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Fig. 2. The fuzzy arithmetic mean over [0; 1] × [0; 1] (n = m = 3).
Fig. 3. Contour lines of the fuzzy arithmetic mean.
Because of rij = 12 (ai2 + b2j ) (i; j = 1; 2; 3), the consequence matrix (rij )3×3 is as follows:
0:0
0:25
0:5
(rij )(3×3) = 0:25
0:5
0:75 :
0:5
0:75
1:0
According to Deÿnition 2.3, the fuzzy arithmetic mean on the domain [0; 1] × [0; 1] is computed. Fig. 2 shows a distribution of the fuzzy arithmetic mean over [0; 1] × [0; 1], it is clear that this distribution is approximate to that of the normal arithmetic mean. In order to distinguish the fuzzy arithmetic mean from the normal arithmetic mean, contour lines uf = 0:2; 0:4; 0:6 and 0.8 are drawn, Fig. 3 shows these contour lines of the fuzzy arithmetic mean and the normal arithmetic mean. Let us compare the fuzzy arithmetic mean with the normal arithmetic mean, it is clear that the contour lines of the fuzzy arithmetic mean have some curves, but the contour lines of the normal arithmetic mean are straight lines, that is, the fuzzy arithmetic mean has a nonlinear property with regard to parameters u1 and u2 .
5. An example In this section, I will show you how to use the fuzzy average to estimate a situation of a complex system. Let us assume that a complex system has a self diagnosis subsystem, a situation of the complex system is displayed on real time with the self diagnosis subsystem, at the same time, a situation of the complex system will be judged by an experienced operator based on a variety of conditions. Our purpose is to estimate a situation of the complex system by computing the fuzzy average concerning two input situations, one is shown by the self diagnosis subsystem, another is judged by an operator. Moreover, numerical computation results
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by using the fuzzy average is compared with that by using fuzzy control rules. The details are described as follows. 5.1. How to use the fuzzy average Let X = (x; T (x); U; G; M˜ ) be a linguistic variable with a label “situation” and with U = [0; 1]. It consists of a six step procedure that uses the fuzzy average. 5.1.1. Embodying the linguistic variable Suppose that the self diagnosis subsystem of the complex system has only recognized three situations: “bad”, “not so bad” and “ÿne”, i.e., the X has three values: “bad”, “not so bad” and “ÿne”, then, the X is embodied as follows: Xd = (situation; Td (situation); U; Gd ; M˜ d ); where Td (situation) = {bad d , not so badd , ÿned }. On the other hand, operators make the complex system to have ÿve situations: “very bad”, “bad”, “not so bad”, “ÿne” and “very ÿne”, i.e., the X has ÿve values in this case, then, the X is concretely represented by Xo = (situation; To (situation); U; Go ; M˜ o ); where To (situation) = {very bad o , bado , not so bado , ÿneo , very ÿneo }: 5.1.2. Deciding M˜ d and M˜ o In order to decide each mapping corresponding to the linguistic variable Xd or Xo , we have to deÿne TFNs on the U . Let us deÿne TFNs Ai (i = 1; 2; 3) as follows: A1 = (0; 0; 0:5);
A2 = (0; 0:5; 1:0);
A3 = (0:5; 1:0; 1:0);
then, the mapping M˜ d is deÿned as M˜ d : Td (situation) → {Ai };
(i = 1; 2; 3);
such that, bad d → A1 ;
not so bad d → A2 ;
ÿned → A3 :
Similarly, let us deÿne TFNs Bj ( j = 1; 2; : : : ; 5) as follows: B1 = (0; 0; 0:25);
B2 = (0; 0:25; 0:5);
B3 = (0:25; 0:5; 0:75);
B4 = (0:5; 0:75; 1:0);
then, the mapping M˜ o is deÿned as M˜ o : To (situation) → {Bj }
( j = 1; 2; 3; 4; 5);
such that very bad o → B1 ;
bad o → B2 ;
not so bad o → B3 ;
ÿneo → B4 ;
very ÿneo → B5 :
B5 = (0:75; 1:0; 1:0);
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Table 1 Estimated Model 1 Diagnosis Operator
Badd
Not so badd
Fined
very bado bado not so bado ÿneo very ÿneo
very bado bado bado badd badd
bado not so bado not so bado ÿneo ÿneo
badd badd ÿneo ÿneo very ÿneo
5.1.3. Building an estimated model It is important how to combine three situations stated by the self diagnosis subsystem with ÿve situations judged by an operator to an estimated model. For example, the situation of the complex system is judged as “very bado ”, when the situation shown by the self diagnosis subsystem is “badd ” and the situation judged by an operator is “very bado ”. Suppose that an estimated model is built as in Table 1. It is dicult how to decide the situation of the complex system, when the situation judged by an operator is opposed to the situation which gotten from the self diagnosis subsystem, e.g., the situation judged by an operator is “very bado ”, but at the same time, the situation gotten from the self diagnosis subsystem is “ÿned ”, in this case, the complex system has possibly gotten into danger, therefore, the situation of the system is decided to be “badd ” in this model. In general, an estimated model is decided by experts. 5.1.4. Getting the consequence matrix (rij )(5×3) Elements of the consequence matrix (rij )(5×3) is obtained by replacing elements of the estimated model with the mean values of TFNs Ai or Bj corresponding to these elements in the model (Table 1). For example, the element r11 is taken as the mean value b12 of B1 , because the TFN B1 corresponds to “very bado ” in the mapping M˜ o . The r13 is taken as a12 , because the TFN A1 corresponds to “badd ” in the mapping M˜ d . Then, the consequence matrix depends upon the estimated model and the mappings M˜ d and M˜ o , that is, rij will be determined, when the estimated model is decided and the mappings M˜ d and M˜ o are deÿned. According to this model shown in Table 1, the consequence matrix (rij )(5×3) is the following:
b12
2 b2 (5×3) = b22 (rij ) 1 a2 a12
b22 b32 b32 b42 b42
a12
0:0
0:25
a12 0:25 0:5 b42 = 0:25 0:5 b42 0:0 0:75 0:0 0:75 b52
0:0
0:0 0:75 : 0:75 1:0
5.1.5. Computation results Let us divide U = [0; 1] into 10 equal parts, then, points ((i − 1)=10; (j − 1)=10) (i; j = 1; 2; : : : ; 11) are obtained on the U × U plane. The fuzzy average on these points is computed, Fig. 4 shows a three-dimension graph, which presents a relation between two input variables and the fuzzy average. 5.1.6. New approach to the diagnosis system Based on the numerical computation result, to estimate the situations of the complex system, it is required to decide the boundary values between two situations. Let 0.2, 0.4, 0.6 and 0.8 be a boundary value between
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Fig. 4. Distribution of the fuzzy average (n = 5; m = 3).
“very bado ” and “bado ”, “bado ” and “not so bado ”, “not so bado ” and “ÿneo ” and “ÿneo ” and “very ÿneo ”, respectively, that is to say, if uf ∈ [0; 0:2), then the situation of the complex system is very bado ; if uf ∈ [0:2; 0:4), then the situation of the complex system is bado ; if uf ∈ [0:4; 0:6), then the situation of the complex system is not so bado ; if uf ∈ [0:6; 0:8), then the situation of the complex system is ÿneo ; if uf ∈ [0:8; 1:0], then the situation of the complex system is very ÿneo . In order to visibly see each area corresponding to each situation, contour lines uf = 0:2; 0:4; 0:6 and 0.8 are drawn. Fig. 5 shows a drawing of these contour lines. As shown in Fig. 5, the areas with white points mean that the situation is “very bado ”, the area with horizontal lines means that the situation is “bado ”, the area with vertical lines means that the situation is “not so bado ”, the area with slanting lines means that the situation is “ÿneo ”, the area with wavy lines means that the situation is “very ÿneo ”. 5.2. Numerical results by using fuzzy control rules Based on the fuzzy control theory, the estimated model 1 (Table 1) can be represented by 3 × 5 = 15 fuzzy control rules, such that, R1 : if u1 is very bado and u2 is badd , then y is very bado ; R2 : if u1 is very bado and u2 is not so badd , then y is bado ; R3 : if u1 is very bado and u2 is ÿned , then y is badd ; R4 : if u1 is bado and u2 is badd , then y is very bado ; R5 : if u1 is bado and u2 is not so badd , then y is not so bado ; R6 : if u1 is bado and u2 is ÿned , then y is badd ; R7 : if u1 is not so bado and u2 is badd , then y is very bado ; R8 : if u1 is not so bado and u2 is not so badd , then y is not so bado ; R9 : if u1 is not so bado and u2 is ÿned , then y is ÿneo ; R10 : if u1 is ÿneo and u2 is badd , then y is very badd ; R11 : if u1 is ÿneo and u2 is not so badd , then y is ÿneo ;
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Fig. 5. Contour lines of the fuzzy average (Fig. 4).
R12 : if u1 is ÿneo and u2 is ÿned , then y is ÿned ; R13 : if u1 is very ÿneo and u2 is badd , then y is badd ; R14 : if u1 is very ÿneo and u2 is not so badd , then y is ÿneo ; R15 : if u1 is very ÿneo and u2 is ÿned , then y is very ÿneo , where u1 is an input variable, which means situations judged by an operator, u2 is another input variable, which means situations shown by the self diagnosis subsystem, y is an output variable, which means the estimated situations based on the input variables u1 and u2 . Let us make use of the M˜ o ; M˜ d , and TFNs Ai (i = 1; 2; 3) and Bj (j = 1; 2; 3; 4; 5) described in Section 5.1. Here, the defuzziÿcation operator is deÿned as formula (1.1), such that, P15 R i=1 supp C i !i Ci (y)y dy o ; y = P15 R i=1 supp C !i Ci (y) dy i
where !i = min{Ai (u1 ); Bi (u2 )}; the Ci is a TFN corresponding to the consequence of these fuzzy control rules. For example, in the R2 ; the C2 is B2 , because the bado corresponding to the B2 in the mapping M˜ o , in the R6 , the C6 is A1 , because the badd corresponding to the A1 in the mapping M˜ d . Similar to the Section 5.1.5, outputs of these fuzzy control rules on points ((i − 1)=10; (j − 1)=10) (i; j = 1; 2; : : : ; 11) are computed. Fig. 6 shows an input–output relation of these fuzzy control rules R1 ; : : : ; R15 . Similar to Fig. 5, let us draw contour lines uf = 0:2; 0:4; 0:6 and 0.8, Fig. 7 shows a drawing of these contour lines. The areas with white points mean that the situation is “very bado ”, the area with horizontal lines means that the situation is “bado ”, the area with vertical lines means that the situation is “not so bado ”, the area with slanting lines means that the situation is “ ÿneo ”, the area with wavy lines means that the situation is “very ÿneo ”. If we compare Fig. 5 with Fig. 7, it is found that each area keeps its balance in Fig. 5, but the red and green areas are too small in Fig. 7. Moreover, implementation of the fuzzy average is easier than that of the fuzzy control rules in point of software, because by using the fuzzy control rules we have to compute each integration, especially, the number of fuzzy control rules is many more, a number of computing these integrations is many more.
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Fig. 6. Input–output relation of the fuzzy control rules.
Fig. 7. Contour lines of the relation (Fig. 6).
6. Conclusion The fuzzy average and the fuzzy arithmetic mean concerning two input variables are suggested, as regards three input variables, four input variables and so on the fuzzy average and the fuzzy arithmetic mean can be similarly deÿned. If we compare the fuzzy average with fuzzy control rules and existing defuzziÿcation operators, we ÿnd that (1) The fuzzy average is not only a defuzziÿcation method, but also a representation of fuzzy control rules by using the consequence matrix.
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(2) The implementation of the fuzzy average is easier than that of the fuzzy control rules. Moreover, it has been proved that the fuzzy arithmetic mean converges to the normal arithmetic mean. Through an example, the procedure of using the fuzzy average is described. Based on deÿnitions of the fuzzy average, the fuzzy arithmetic mean, the arithmetic mean, the weighted mean and the fuzzy weighted average [2], relations among these concepts can be described as: {fuzzy average} ⊇ {fuzzy arithmetic mean} ⊇ {arithmetic mean}; {fuzzy weighted average} ⊇ {weighted mean} ⊇ {arithmetic mean}; that is, the normal arithmetic mean is an intersection of the fuzzy average set and the fuzzy weighted average set. References [1] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum, New York, 1981. [2] W.M. Dong, F.S. Wong, Fuzzy weighted averages and implementation of the extension principle, Fuzzy Sets and Systems 21 (1987) 183–199. [3] D. Dubois, H. Prade, Additions of interactive fuzzy numbers, IEEE Trans. Autom. Control 26 (1981) 926 – 936. [4] L.S. Gao, The application of fuzzy reason to curvic approximation problems, Japan 8th Fuzzy System Symp., Hiroshima, 26 –28 May, 1992, pp. 405– 408. [5] L.S. Gao, Studies on application of fuzzy averaging in numerical analysis, Ph.D. Thesis, Chiba University, 1995. [6] L.S. Gao, H. Imai, H. Kawarada, Fuzzy control of system governed by elliptic partial di erential equations, Technical Reports of Mathematical Sciences, Chiba Univ. 5, 1989. [7] L.S. Gao, H. Kawarada, Application of fuzzy average to curve and surface ÿtting, Proc. FUZZ-IEEE=IFES’95, 1995. [8] A. Kaufmann, M.M. Gupta, Fuzzy Mathematical Models in Engineering and Management Science, Elsevier, Amsterdam, 1988. [9] C.C. Lee, Fuzzy logic in control systems: fuzzy logic controller – Part 1, IEEE Trans. Systems Man Cybernet. SMC-20 (2) (1990) 405– 418. [10] C.C. Lee, Fuzzy logic in control systems: fuzzy logic controller – Part 2, IEEE Trans. Systems Man Cybernet. SMC-20 (2) (1990) 419 – 435. [11] M. Mizumoto, Pictorial representations of fuzzy connectives, Part I: Cases of t-Norms, t-Conorms and averaging operators, Fuzzy Sets and Systems 31 (1989) 217–242. [12] M. Sugeno, Fuzzy Control, Nikkan Kounyo Shinbunsya, Tokyo, 1989 (in Japanese). [13] T. Takagi, M. Sugeno, Derivation of fuzzy control rules from human operator’s control actions, Proc. IFAC Symp. on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseilles, France, July 1983, pp. 55– 60. [14] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338–353. [15] H.J. Zimmermann, Fuzzy Set Theory and Its Application, Kluwer Academic Publishers, Boston, MA, 1984.