- Email: [email protected]
Uninorms in fuzzy systems modeling
Fuzzy Sets and Systems 122 (2001) 167–175
www.elsevier.com/locate/fss
Uninorms in fuzzy systems modeling Ronald R. Yager ∗ Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA Received 22 February 1999; received in revised form 17 December 1999; accepted 18 January 2000
Abstract Fuzzy systems modeling provides a framework in which input–output relationships can be modeled by partitioning the input into fuzzy regions for which we can describe the output. The fuzzy modeling inference process, the procedure used to obtain the output of a fuzzy systems model for a given input, is shown to involve an aggregation step in which the contributions of the di/erent components of the fuzzy systems model are combined. It is shown that the uninorm operator provides a general class of operators to implement this step. The uninorm it is recalled, is an operator that provides a generalization of the t-norm and t-conorm by allowing the identity element to be any value in the unit interval. Another step in the inference process is the determination of the contribution of each component of the model based on its relevancy to the current input. The performance of this step requires the use of an operator, related to the implication operator, which we call a relevancy transformation (RET) operator. It is shown that the form of this RET operator is dependent upon the identity value of the uninorm used in the aggregation step. We provide some general formulations for this class of operators. We show how the c 2001 Elsevier Science B.V. All well-known forms of fuzzy inference are special cases of our uninorm-based approach. rights reserved. Keywords: Uninorms; Fuzzy modeling; Systems; Inference
1. Introduction Many applications of fuzzy set theory involve the use of fuzzy systems modeling [13] to model complex and perhaps ill-de=ned systems. These applications include fuzzy logic control and fuzzy expert systems. Assume V and U are variables taking their values on the spaces X and Y , respectively. Fuzzy systems modeling involves the use of a fuzzy rule base to model a complex function or system, V = f(U ), by partitioning the input space X into fuzzy regions in which the ∗
Tel.: + 1-212-249-2047; fax: + 1-212-249-1689. E-mail address: [email protected] (R.R. Yager).
output can be e/ectively determined by either an expert or as a result of some data driven process such as clustering [14]. This formulation can be seen as a type of granulation in the spirit described by Zadeh [20]. Typical of these formulations are set of n rules of the form if V is Ai then U is Bi ; where Ai and Bi are fuzzy subsets of the input and output spaces X and Y . These rules are used to provide a model of a system by partitioning the input space X into fuzzy subsets (regions), the Ai such that the value of the consequent U is known in each of the regions.
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 0 2 7 - 0
168
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
The problem of =nding the value of the output U associated with a particular value for the input variable V is called the fuzzy modeling inference process. In fuzzy control the basic paradigm for calculating the output of the fuzzy model is based upon the pioneering work of Mamdani and his colleagues [5–7]. While most applications use a modi=cation of this method based on the work of Sugeno and his colleagues [8,9], both these approaches can be seen as fundamentally based on an inductive or what Yager and Filev [13] call a constructive mechanism. Here the output of the model is built up, starting from the null set, by disjuncting weighted consequents of each of the rules. The weights are related to the relevancy of the rule to the particular input value being considered. The relevancy of a rule is determined by membership grade of the input value in antecedent fuzzy set of the rule, Ai . As discussed by Zadeh [19], in this approach, the rules are e/ectively seen as fuzzy points associated with relationship f. Using the Max and Min to implement operations the output fuzzy set E of this model for a given input x∗ is expressed as E(y) = Max[Min[Ai (x∗ ); Bi (y)]]: i
We note that a defuzzi=cation step [12] is applied to the fuzzy subset E to get a scalar output. In fuzzy expert systems the basic paradigm used for calculating the output of the fuzzy model is based upon Zadeh’s theory of approximate reasoning [17]. Here the output of the model is “built down” by removing portions of the output space that are not supported by relevant rules. Here we form a conjunction of weighted consequents of each of the rules. Again the weights are related to the relevancy of the rule to the particular input value being considered. The relevancy of a rule is also determined by membership grade of the input value in antecedent fuzzy set of the rule, Ai . In this approach, the rules are e/ectively seen in the form of logical implications. Again using the Max and Min to implement operations the output fuzzy set F of this model, for a given input x∗ , is expressed as F(y) = Min[Max[(1 − Ai (x∗ )); Bi (y)]]: i
Our discussion has assumed a crisp input rather then a fuzzy one; this is an assumption we shall carry through out this work. Besides the fact that most applications of fuzzy modeling have crisp inputs, the
consideration of fuzzy inputs introduces an additional level of complexity which serves to obsure the main ideas to be developed in this work. Our purpose here is to provide a general framework for the fuzzy modeling inference process. As we shall see a general framework will result in the above two forms as special cases. A central role in this generalization will be played by the uninorm operators. 2. Uninorms In [15] Yager and Rybalov introduced a generalization of the t-norm and t-conorm which they called uninorms. Fodor and DeBaets [2 – 4] among others have studied the structure of these operators in considerable detail. Denition. A uninorm R is a mapping R : I × I → I having the following properties: (1) Commutativity: R(x; y) = R(y; x). (2) Monotonicity: R(x; y)¿R(u; v) for x¿u; y¿v. (3) Associativity: R(x; R(y; z)) = R(R(x; y); z)). (4) Identity: There exists some =xed element g ∈ [0; 1], called the identity, such we have R(x; g) = x. If Aj ; j = 1 to n, are a collection of fuzzy subsets we can implement a uninorm-based aggregation of these fuzzy subsets as A = Uni(A1 ; A2 ; : : : ; An ) such that A(x) = R(A1 (x); A2 (x); : : : ; An (x)); where R is a uninorm operator. The uninorm operators have the same =rst three properties of t-norm and t-conorm but the fourth condition is more general in that it allows for any identity. In particular, it is recalled that for the t-norm the identity is one while for the t-conorm the identity is zero. In introducing the uninorm we are essentially allowing the identity to be anywhere in the unit interval rather than just at the extremes of one and zero as in the cases of the t-norm and t-conorms. Since R(a; g) = a then the monotonicity condition implies that R(a; b)6a for b¡g and R(a; b)¿a for b¿g. Thus, g determines a boundary which separates those values which can cause a decrease in the aggregated value from those which can cause an increase in the aggregated value. With this property, we see that the inclusion of an additional fuzzy subset to a uninorm-based
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
aggregation can cause either an increase or decrease in overall membership grade. Having introduced the concept of a uninorm we next turn to the question of the existence or construction of such operators. For g = 1 or 0 there exists a large class of such uninorms corresponding to the t-norms and tconorms, respectively. In [4] Fodor et al. introduced a general class of uninorms for any g. In the following we describe this class. Let T and S be any t-norm and t-conorm, respectively. The mappings R de=ned by I and II below are two general classes of uninorm operators with identity g. I.
x y (a) R(x; y) = gT ; g g
if 06x; y6g,
x−g y−g (b) R(x; y) = g + (1 − g)S ; 1−g 1−g if g6x; y61;
II.
x y (a) R(x; y) = gT ; g g
if 06x; y6g,
if g6x; y61,
x−g y−g ; 1−g 1−g
The di/erence between the two classes are in item (c), in the =rst case when one of the arguments is above the identity and the other below the identity we take the Min while in the second case we take the Max. In particular, we note that for the =rst class R(0; 1) = 0 and for the second class R(0; 1) = 1. Fig. 1 provides visualization of the above structure. In the above
x y T (x; y) = gT ; g g
T ∗ (x; y) = Min[x; y];
S(x; y) = Max[x; y];
S ∗ (x; y) = Max[x; y];
T ∗ (x; y) =
xy ; g
S ∗ (x; y) =
x + y − xy − g ; 1−g
T (x; y) = Max[0; x + y − 1]; T ∗ (x; y) = Max[0; x + y − g];
S ∗ (x; y) = Min[1; x + y − g]: In [15] we provided an example of a uninorm not of the above class, called the three operator. For this operator for all x and y in [0; 1]R(x; y) = xy=( xM y+xy) M where xM =1 − x. In this uninorm the identity g = 0:5. Using the associativity property we get n j=1 xj n : R(x1 ; x2 ; : : : ; xn ) = n x j=1 j + j=1 xj
;
x−g y−g S (x; y) = g + (1 − g)S ; 1−g 1−g ∗
T (x; y) = Min[x; y];
S(x; y) = Min[1; x + y];
(c) R(x; y) = Max(x; y) if Min(x; y)6g6Max(x; y).
∗
Let us look at the form of T ∗ and S ∗ for some special cases of T and S.
S(x; y) = x + y − xy;
(b) R(x; y) = g + (1 − g)S
Fig. 1. Structure of uninorm class in [0; 1] × [0; 1].
T (x; y) = xy;
(c) R(x; y) = Min(x; y) if Min(x; y)6g6Max(x; y).
169
and N (x; y) is either the Max or the Min.
3. Uninorms as aggregation operations for fuzzy modeling As we have noted, a fuzzy systems model provides a representation of a relationship between two variables,
170
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
V and U , called the input and output variables. 1 It consists of a collection of n rules of the form if V is Ai then U is Bi ; where Ai and Bi are fuzzy subsets of X and Y , the domains of the input and output variables, respectively. Each rule provides partial information about the relationship between V and U . As discussed by Yager and Filev [13] the determination of the value of U for a given value of V , the fuzzy modeling inference process (FUMIP), consists of the following four step algorithm: 1. Obtain the relevance of each rule for the current input value. 2. Determination of the e/ective output of each rule for the given input value, expressed as a fuzzy subset of output space. We shall denote these individual rule outputs as U is Fi : 3. Aggregation of the individual rule outputs to obtain the overall fuzzy system output as a fuzzy subset of the output space. We shall denote this overall output as U is F: 4. Selection of some action based upon the output set F. We shall =rst focus on the requirements for the operations that can be used to implement this process. We are particularly concerned with the third step, the rule output aggregation. Because of the strong interrelationship between all the steps in the process we look at all the steps. A particular close connection exists between the second and third step. A fundamental aspect of this connection is the relationship between the choice of identity used in the rule aggregation process and the type of interpretation one gives to the rules in step two. Let us brieNy consider the last step =rst. All we want to do here is make some basic observation regarding the role that F, the system output, plays in the solution selection step [12]. With each y of the output space we can associate some solution, normally it is V = y. Furthermore, for any y ∈ Y; F(y) can be seen as some measure of strength to which the system model, un1 The input can be multiple variables; however for our purposes considering it a single variable is suOcient.
der the current input, outputs the solution y. The key observation is that the action selection process uses F in such a way that if F(y1 )¿F(y2 ) we are more inclined to implement the solution suggested by y1 than that of y2 . Fundamentally, we can say that there is a positive association between the membership grade of an element and the action recommended by that element. One very special case of the solution selection process is the defuzzi=cation step used in fuzzy logic controllers [16]. In this special situation the action is to decide which of the elements from the output space Y to use as the input to the plant. A typical example of the defuzzi=cation step is the center of area method. In this method the output value y∈Y yF(y) ∗ y = : y∈Y F(y) Let us now look at the =rst step of the FUMIP. First consider an individual rule, if V is Ai then U is Bi : The intent of this rule is to say that if the input to the system is “consistent with” or “matches” the antecedent then this rule is relevant and the system output is Bi . In indicating that Bi is the fuzzy set output, this is saying that if Bi (y1 )¿Bi (y2 ) there is a preference for having y1 as the solution rather than y2 . The =rst step in the algorithm is the matching step. In this step, we determine some measure of the degree to which the input to the system and the antecedent of the individual rules are compatible. We shall use i to indicate this value and call it the relevance or weight of the ith rule. In the case where the input is a singleton, V = x∗ , and the antecedent is of the form V is Ai then the usual case is to use i = Ai (x∗ ), the membership grade of x∗ in Ai . If the input is a fuzzy subset and=or the antecedent is more complex the process of matching becomes more complex. We shall not focus on this problem at this point. All that we shall observe about the i is that the larger the i the better the matching, and the more relevant the rule. In particular, we observe that if i = 1 the matching is perfect, and the rule is completely relevant. If i = 0 the rule has no relevance to the current input. Having obtained this =ring level we then use it to obtain the e/ective rule output U is Fi . The individual rule output Fi is seen to be completely determined by
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
i , the rule relevance, and the rule consequent Bi . We denote this as Fi = i ◦ Bi . Having obtained the output of each rule we must now combine these individual rule outputs to obtain the overall system output, F. We denote this process F = F1 F2 F3 · · · Fn : As we shall subsequently see, the process used to obtain the overall system output from the individual Fi and the process used to obtain the e/ective rule outputs from the rule consequent and rule relevance are not independent, that is and ◦ are related. Let us look at the process used to combine the individual rule outputs. A basic assumption we shall make is that the operation is pointwise and likewise. By pointwise we mean that for every y ∈ Y; F(y) just depends upon Fi (y) for i = 1 to n. By likewise we shall mean that the process used to combine the Fi is the same for all of the y. The likewise allows us to focus on the aggregation process for any of the elements y in Y . Let us denote the pointwise process we use to combine the individual rule outputs as F(y) = Agg(F1 (y); F2 (y); : : : ; Fn (y)): In the above Agg is called the aggregation operator and the Fi (y) are the arguments. More generally, we can consider this as an operator a = Agg(a1 ; : : : ; an ); where the ai and a are values from the membership grade space, normally the unit interval. A central focus of our interest here is to study the class of operators that are useful to implement this aggregation process used in fuzzy system modeling. Let us look at the minimal requirements associated with the Agg operator. A =rst observation we can make is that the order in which we list the rules in a fuzzy systems model should not e/ect the output of the model. As a result of this we see that the aggregation of the individual rule outputs should be independent of the choice of indexing of the rules. This implies that a required property that we must associate with the Agg operator is that of commutativity, the indexing of the arguments does not matter. We note that the commutativity property allows us to represent the argument of the Agg operator, (F1 (y); F2 (y); : : : ; Fn (y)); as an unordered collection of values; such an object is a bag.
171
For an individual e/ective rule output, Fi ; the membership grade Fi (y) indicates the strength to which this rule suggests that y is the appropriate solution. In particular, if for a pair of elements y1 and y2 ∈ Y it is the case that Fi (y1 )¿Fi (y2 ); then we are saying that rule i is preferring y1 as the system output over y2 . From this we can conclude that if all rules prefer y1 over y2 as output then the overall system output should prefer y1 over y2 . This observation requires us to impose a monotonicity condition on the Agg operation. In particular, if Fi (y1 )¿Fi (y2 ) for all i, then F(y1 )¿F(y2 ). This of course means that in calculating F(y) from the individual components that if Fi (y)¿Fˆ i (y) for all ˆ i then F(y)¿F(y). In particular, if ai ¿bi for all the Agg(a1 ; a2 ; : : : ; an )¿Agg(b1 ; b2 ; : : : ; bn ). There is one other condition we need to impose upon the aggregation operator. Assume that there exists some rule whose relevancy is zero. The implication of this is that the rule provides no information regarding what should be the output of the system. It should not a/ect the =nal F. Without loss of generality, we shall assume the nth rule has relevancy zero. One observation we can make is that whatever e/ective output this rule provides it should not make any distinction between the y values. One way to assure this for rule n; the one with zero relevancy, is to have Fn (yi ) = Fn (yj ) for all i and j. Let us denote this value as g. We note of course that g must be a value in the unit interval. In addition to having Fn (yi ) the same for all yi there is a second requirement necessary. Consider any two elements y1 and y2 in Y . Assume that Agg(F1 (y1 ); F2 (y1 ) : : : Fn−1 (y1 )) = a1 ; Agg(F1 (y2 ); F2 (y2 ) : : : Fn−1 (y2 )) = a2 : Since a rule having zero relevancy should play no role in determining which is the preferred output, the inclusion of its output, Fn ; in the aggregation should not a/ect the relationship between the system output value for y1 and y2 . This observation requires that Agg(F1 (yi ); F2 (yi ) · · · Fn−1 (yi ); g) = Agg(F1 (yi ); F2 (yi ) : : : Fn−1 (yi )): That is, g should be an identity element for the aggregation process. Thus, we must require that the aggregation operator has an identity element. In summary,
172
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
we see that the aggregation operator, Agg must satisfy three conditions: 1. Commutativity. 2. Monotonicity. 3. Must contain an identity. These conditions are based on three observations: 1. The indexing of the rules be unimportant. 2. A positive association between e/ective rule output and total system output. 3. Non-relevant rules play no role in the process. A class of operators satisfying these properties, called MICA operators have been identi=ed and studied by Yager [10]. Thus, the class of MICA operators are the prototypical aggregation operators for the combination of individual rule output. While a most general class of operators implementing the Agg operator are MICA, we observe that a special class of these operators are the uninorms which satisfy the three conditions. Thus, the uninorms can be used to implement the aggregation operator in fuzzy systems modeling. With the uninorms we invest our aggregation operator with the additional property of associativity. It should be pointed out that no requirement was made on the choice of the identity g. As we shall subsequently see the only requirement is that it be consistent with the choice of operation used to obtain the e/ective rule output. 4. RET operators and e!ective rule output calculations We now look at the requirements for the operation used to obtain the e/ective rule output from the rule relevancy and the rule consequent, Fi = i ◦Bi . We shall assume that ◦ is a pointwise and likewise operation. Thus, there exists some function which we shall denote as h such that
A =rst requirement on h is that if the antecedent of the rule is completely satis=ed, the rule relevancy is one, our e/ective output should be the rule consequent, Bi . This characteristic requires that h satisfy the condition that for all b h(1; b) = b: A second requirement on h is that if the rule relevancy is zero the e/ective output should not inNuence the aggregation process used in step three. This characteristic imposes a condition on h whereby if the rule relevancy is zero then the output value should be the identity under the aggregation operation used in step three. Formally, this requires that for all b h(0; b) = g; where g is the identity under the aggregation operator Agg. A third condition we require is a kind of monotonicity in the second argument. We require that for any a and any b¿b it is the case that h(a; b)¿h(a; b ): This condition is a reNection of the requirement that we should not inverse the preference ordering between output values. That is no value for relevancy should allow us to reverse the preferences speci=ed in Bi . A fourth condition we shall require is a consistency in the antecedent argument. In particular, we desire that for a =xed consequent value the output of h go monotonically from the value for zero =ring, the identity g, to the value for complete =ring, b. In particular, for any level of relevancy ; Sgn(@h( ; b)=@ ) = Sgn(b − g) or zero. That is if ¿ then if b¿g then h( ; b)¿h( ; b);
Fi (y) = h( i ; Bi (y)):
if b6g then h( ; b)6h( ; b):
More generally, h is a binary operator taking values from I 2 into I; h(a; b) = c; for a; b; c ∈ I . The operation h is closely related to both the implication operator [1] and the weighting operation [11]. We shall call h the relevancy transformation (RET) function. Let us investigate the minimal requirements we desire the operator h to have.
We note that in the special case when g = 1 it is always the case that b6g and hence we get for @h( ; b)=@ 60. At the other extreme is the case when g = 0. In this situation since b¿g for all b we have @h( ; b)=@ ¿0. It is interesting to see that the above requirement imposes the following condition, if b = g then h( ; g) = g for all .
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
173
When using a uninorm with =xed identity g as our aggregation operator we shall say that h is an appropriate RET operator if it satis=es the four conditions speci=ed where h(0; b) = g. For any inference implementation using a uninorm U with identity g and appropriate RET h if i = 0 for all i the system overall output F has F(y) = g for all y. We see this as follows. If i = 0 then Fi (y) = h(0; Bi (y)) = g for all y and all i therefore F(y) = U (g; g; g; g; g; : : : ; g). From associativity and the fact that U (g; g) = g we get the result. This situation of having F(y) the same for all y when all rules have no relevancy is intuitively appealing for it essentially says that our model has provided no information that allows us to distinguish between any of the possible solutions to our model. The use of uninorms as our aggregation operator brings with it an additional property which Yager called 1delity [10]. By =delity we mean to require that when the input to the system is such that one rule is completely relevant and all other rules are completely not relevant then the system output should be exactly the consequent of the relevant rule. Without loss of generality, =delity requires that if 1 = 1 and j = 0 for j = 1 then F = F1 . We see that the associativity of the uninorm guarantees =delity. The choice of an appropriate h means that for any y, B1 (y) = F1 (y) and Fj (y) = g for all j = 1. From this F(y) = U (F1 (y); g; g; : : : ; g). The associativity of U and the identity of g implies that F(y) = F1 (y). A fundamental issue here becomes the selection of an relevancy transformation function given the selection of a uninorm U . The following theorem, which has the Navor of an existence theorem, provides a RET operator that is valid for all choices of g.
However, since aM − aM = − (a − a ) then we get h(a; b) − h(a ; b) = (a − a )(b − g). Thus if b¿g then since a¿a we get h(a; b)¿b(a ; b) and if b6g we get h(a; b)6h(a ; b). Thus, we see that the above form is an appropriate transformation operator for any uninorm with the corresponding g.
Theorem 1. The formulation
Proof. We have shown that T (a; b) satis=es the four required conditions: 1. Assume a = 1 then T (1; b) = b. 2. Assume a = 0 then T (0; b) = 0 = g. 3. Assume b¿b then from the monotonicity of tnorms we get T (a; b)¿T (a; b ). 4. Assume a¿a then since b¿g = 0 for all b then T (a; b)¿T (a ; b).
h(a; b) = ab + ag M satis1es all four conditions for any g. Proof. 1. Assume a = 1; then aM = 0 and we get h(a; b) = b. 2. Assume a = 0 then aM = 1 and we get h(a; b) = g. 3. Assume b¿b then h(a; b)−h(a; b ) = a(b−b )¿0. 4. Assume a¿a . Then M − (a b + aM g) h(a; b) − h(a b) = ab + ag = (a − a )b + (aM − aM )g:
The following theorem provides another class of RET operators that are appropriate for any identity g. Theorem 2. The formulation h(a; b) = (a ∧ b) ∨ (aM ∧ g) ∨ (b ∧ g) satis1es all four conditions for any g. Proof. 1. If a = 0 then h(a; b) = g ∨ (b ∧ g) = g. 2. If a = 1 then h(a; b) = b ∨ (b ∧ g) = b. 3. If b1 ¿b2 then h(a; b1 )¿h(a; b2 ). 4. (i) If b¿g then h(a; b) = (a ∧ b) ∨ g; from this we see that h(a1 ; b)¿h(a2 ; b) if a1 ¿a2 . (ii) If g¿b then h(a; b) = (aM ∧ g) ∨ b; from this we see that h(a1 ; b)6h(a2 ; b) if a1 ¿a2 . (iii) If g = b then h(a; b) = g. If we restrict ourselves the special cases of uninorms with g = 0 or 1; t-conorms and t-norms, we get a large class of transformation operators. Theorem 3. For the case of a uninorm in which g = 0; t-conorm; the operator h(a; b) = T (a; b); where T is any t-norm; is an appropriate RET operator.
Two examples of t-norms are the Min and product h(a; b) = Min(a; b) h(a; b) = ab
Min;
Product:
174
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
Using these to implement the h function we get Fi (y) = Min[ i ; Bi (y)]; Fi (y) = i Bi (y): For the uninorm with g = 0 our aggregation operator is a t-conorm S; examples of this are S(a; b) = Max[a; b] and S(a; b) = a + b − ab. If we select the Max t-conorm as our aggregation uninorm and combine this with the Min interpretation of h we get F(y) = Max[ i ∧ Bi (y)]): i
This is the formulation which was used by Mamdani [5] in his original work on fuzzy modeling of controllers. The following theorem provides a corresponding class of appropriate RET functions for aggregation operators based on uninorms with g = 1; t-norms. Theorem 4. For the case of a uninorm in which g = 1; t-norm; the operator h(a; b) = S(a; M b); where S is any t-norm; is an appropriate RET operator. Two important examples of this occur when S(a; b) = Max(a; b); S(a; b) = a + b − ab: If we use S(a; b) = Max(a; b) to implement the h function we get Fi (y) = Max(M i ; Bi (y)) and if we use S(a; b) = a + b − ab to implement the h function we get Fi (y) = Mi + i Bi (y). If we select the Min as our aggregation uninorm and combine this with the Max interpretation of h we get
Fig. 2. Neural model for the fuzzy system inference.
S(a; b) = Min[1; a + b]. Using associativity this extends to the form n aj : U (a1 ; a2 ; : : : ; an ) = Min 1; j=1
Let us use as the RET the product. Thus in this situation we get n F(y) = Min 1; j Bj (y) : j=1
The operation Min[1; x] can be viewed as a quasilinear function f(x) = x
for x61;
f(x) = 1
for x ¿ 1: n Thus F(y) = f( j=1 j Bj (y)). This formulation leads us to a neural network [18] view of the process (see Fig. 2). In this view the neuron calculates the membership grade F(y) of the element y in the output fuzzy set F. Here i is the degree to which the ith rule has =red, Ai (x∗ ); and Bi (y) is the degree of membership of y in the consequent of the ith rule.
F(y) = Min(Max(M i ; Bi (y)))
5. Conclusion
which is the formulation which we indicated was used by Zadeh [17]. There exists another special case which induces a very interesting formulation for the inference process used in fuzzy modeling. Consider the uninorm aggregation operator U based on the t-conorm
The fuzzy modeling inference process was shown to involve an aggregation step in which the contributions of the di/erent rules of the fuzzy systems model are combined. It was shown that the uninorm operator provides a general class of operators to implement this step. Another step in the inference process is the
i
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
determination of the contribution of each component of the model based on its relevancy to the current input. It was shown that the performance of this step requires the use of an operator, related to the implication operator, which we called a relevancy transformation (RET) operator. It was shown that the form of this RET operator is dependent upon the identity value of the uninorm used in the aggregation step. We provided some general formulations for this class of operators. We showed how the well-known forms of fuzzy inference are special cases of our uninorm-based approach. References [1] J.F. Baldwin, B.W. Pilsworth, Axiomatic approach to implication for approximate reasoning with fuzzy logic, Fuzzy Sets and Systems 3 (1980) 193–219. [2] B. De Baets, Uninorms: the known classes, in: D. Ruan, H.A. Abderrahim, P. D’hondt, E.E. Kerre (Eds.), Fuzzy Logic and Intelligent Technologies for Nuclear Science and Industry, World Scienti=c, Singapore, 1998, pp. 21–28. [3] B. De Baets, J. Fodor, On the structure of uninorms and their residual implicators, Proceedings of the 18th Linz Seminar on Fuzzy Set Theory, Linz, Austria, 1997, pp. 81–87. [4] J.C. Fodor, R.R. Yager, A. Rybalov, Structure of uni-norms, Internat. J. Uncertainty, Fuzziness Knowledge-Based Systems 5 (1997) 411–427. [5] E.H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. IEEE 121 (1974) 1585–1588. [6] E.H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man– Mach. Stud. 7 (1975) 1–13. [7] E.H. Mamdani, Advances in the linguistic synthesis of fuzzy controllers, Internat J. Man–Mach. Stud. 8 (1976) 669– 678.
175
[8] M. Sugeno, T. Takagi, A new approach to design of fuzzy controller, in: P.P. Wang (Ed.), Advances in Fuzzy Sets, Possibility Theory and Applications, Plenum Press, New York, 1983, pp. 325–334. [9] T. Takagi, M. Sugeno, Derivation of fuzzy control rules from human operators actions, Proceedings of the IFAC Symposium on Fuzzy Information, Marseille, 1983, pp. 55 – 60. [10] R.R. Yager, Aggregation operators and fuzzy systems modeling, Fuzzy Sets and Systems 67 (1994) 129–146. [11] R.R. Yager, Structures generated from weighted fuzzy intersection and union, J. Chinese Fuzzy Systems Assoc. 2 (1996) 37–58. [12] R.R. Yager, D.P. Filev, On the issue of defuzzi=cation and selection based on a fuzzy set, Fuzzy Sets and Systems 55 (1993) 255–272. [13] R.R. Yager, D.P. Filev, Essentials of Fuzzy Modeling and Control, Wiley, New York, 1994. [14] R.R. Yager, D.P. Filev, Generation of fuzzy rules by mountain clustering, J. Intell. Fuzzy Systems 2 (1994) 209–219. [15] R.R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996) 111–120. [16] J. Yen, R. Langari, L.A. Zadeh, Industrial Applications of Fuzzy Logic and Intelligent Systems, IEEE Press, New York, 1995. [17] L.A. Zadeh, A theory of approximate reasoning, in: J. Hayes, D. Michie, L.I. Mikulich (Eds.), Machine Intelligence, Vol. 9, Halstead Press, New York, 1979, pp. 149–194. [18] J.M. Zaruda, Introduction to Arti=cial Neural Systems, West Publishing, St Paul, MN, 1992. [19] L.A. Zadeh, The calculus of fuzzy if=then rules, AI Expert March, 1992, pp. 23–27. [20] L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems 90 (1997) 111–127.
www.elsevier.com/locate/fss
Uninorms in fuzzy systems modeling Ronald R. Yager ∗ Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA Received 22 February 1999; received in revised form 17 December 1999; accepted 18 January 2000
Abstract Fuzzy systems modeling provides a framework in which input–output relationships can be modeled by partitioning the input into fuzzy regions for which we can describe the output. The fuzzy modeling inference process, the procedure used to obtain the output of a fuzzy systems model for a given input, is shown to involve an aggregation step in which the contributions of the di/erent components of the fuzzy systems model are combined. It is shown that the uninorm operator provides a general class of operators to implement this step. The uninorm it is recalled, is an operator that provides a generalization of the t-norm and t-conorm by allowing the identity element to be any value in the unit interval. Another step in the inference process is the determination of the contribution of each component of the model based on its relevancy to the current input. The performance of this step requires the use of an operator, related to the implication operator, which we call a relevancy transformation (RET) operator. It is shown that the form of this RET operator is dependent upon the identity value of the uninorm used in the aggregation step. We provide some general formulations for this class of operators. We show how the c 2001 Elsevier Science B.V. All well-known forms of fuzzy inference are special cases of our uninorm-based approach. rights reserved. Keywords: Uninorms; Fuzzy modeling; Systems; Inference
1. Introduction Many applications of fuzzy set theory involve the use of fuzzy systems modeling [13] to model complex and perhaps ill-de=ned systems. These applications include fuzzy logic control and fuzzy expert systems. Assume V and U are variables taking their values on the spaces X and Y , respectively. Fuzzy systems modeling involves the use of a fuzzy rule base to model a complex function or system, V = f(U ), by partitioning the input space X into fuzzy regions in which the ∗
Tel.: + 1-212-249-2047; fax: + 1-212-249-1689. E-mail address: [email protected] (R.R. Yager).
output can be e/ectively determined by either an expert or as a result of some data driven process such as clustering [14]. This formulation can be seen as a type of granulation in the spirit described by Zadeh [20]. Typical of these formulations are set of n rules of the form if V is Ai then U is Bi ; where Ai and Bi are fuzzy subsets of the input and output spaces X and Y . These rules are used to provide a model of a system by partitioning the input space X into fuzzy subsets (regions), the Ai such that the value of the consequent U is known in each of the regions.
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 0 2 7 - 0
168
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
The problem of =nding the value of the output U associated with a particular value for the input variable V is called the fuzzy modeling inference process. In fuzzy control the basic paradigm for calculating the output of the fuzzy model is based upon the pioneering work of Mamdani and his colleagues [5–7]. While most applications use a modi=cation of this method based on the work of Sugeno and his colleagues [8,9], both these approaches can be seen as fundamentally based on an inductive or what Yager and Filev [13] call a constructive mechanism. Here the output of the model is built up, starting from the null set, by disjuncting weighted consequents of each of the rules. The weights are related to the relevancy of the rule to the particular input value being considered. The relevancy of a rule is determined by membership grade of the input value in antecedent fuzzy set of the rule, Ai . As discussed by Zadeh [19], in this approach, the rules are e/ectively seen as fuzzy points associated with relationship f. Using the Max and Min to implement operations the output fuzzy set E of this model for a given input x∗ is expressed as E(y) = Max[Min[Ai (x∗ ); Bi (y)]]: i
We note that a defuzzi=cation step [12] is applied to the fuzzy subset E to get a scalar output. In fuzzy expert systems the basic paradigm used for calculating the output of the fuzzy model is based upon Zadeh’s theory of approximate reasoning [17]. Here the output of the model is “built down” by removing portions of the output space that are not supported by relevant rules. Here we form a conjunction of weighted consequents of each of the rules. Again the weights are related to the relevancy of the rule to the particular input value being considered. The relevancy of a rule is also determined by membership grade of the input value in antecedent fuzzy set of the rule, Ai . In this approach, the rules are e/ectively seen in the form of logical implications. Again using the Max and Min to implement operations the output fuzzy set F of this model, for a given input x∗ , is expressed as F(y) = Min[Max[(1 − Ai (x∗ )); Bi (y)]]: i
Our discussion has assumed a crisp input rather then a fuzzy one; this is an assumption we shall carry through out this work. Besides the fact that most applications of fuzzy modeling have crisp inputs, the
consideration of fuzzy inputs introduces an additional level of complexity which serves to obsure the main ideas to be developed in this work. Our purpose here is to provide a general framework for the fuzzy modeling inference process. As we shall see a general framework will result in the above two forms as special cases. A central role in this generalization will be played by the uninorm operators. 2. Uninorms In [15] Yager and Rybalov introduced a generalization of the t-norm and t-conorm which they called uninorms. Fodor and DeBaets [2 – 4] among others have studied the structure of these operators in considerable detail. Denition. A uninorm R is a mapping R : I × I → I having the following properties: (1) Commutativity: R(x; y) = R(y; x). (2) Monotonicity: R(x; y)¿R(u; v) for x¿u; y¿v. (3) Associativity: R(x; R(y; z)) = R(R(x; y); z)). (4) Identity: There exists some =xed element g ∈ [0; 1], called the identity, such we have R(x; g) = x. If Aj ; j = 1 to n, are a collection of fuzzy subsets we can implement a uninorm-based aggregation of these fuzzy subsets as A = Uni(A1 ; A2 ; : : : ; An ) such that A(x) = R(A1 (x); A2 (x); : : : ; An (x)); where R is a uninorm operator. The uninorm operators have the same =rst three properties of t-norm and t-conorm but the fourth condition is more general in that it allows for any identity. In particular, it is recalled that for the t-norm the identity is one while for the t-conorm the identity is zero. In introducing the uninorm we are essentially allowing the identity to be anywhere in the unit interval rather than just at the extremes of one and zero as in the cases of the t-norm and t-conorms. Since R(a; g) = a then the monotonicity condition implies that R(a; b)6a for b¡g and R(a; b)¿a for b¿g. Thus, g determines a boundary which separates those values which can cause a decrease in the aggregated value from those which can cause an increase in the aggregated value. With this property, we see that the inclusion of an additional fuzzy subset to a uninorm-based
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
aggregation can cause either an increase or decrease in overall membership grade. Having introduced the concept of a uninorm we next turn to the question of the existence or construction of such operators. For g = 1 or 0 there exists a large class of such uninorms corresponding to the t-norms and tconorms, respectively. In [4] Fodor et al. introduced a general class of uninorms for any g. In the following we describe this class. Let T and S be any t-norm and t-conorm, respectively. The mappings R de=ned by I and II below are two general classes of uninorm operators with identity g. I.
x y (a) R(x; y) = gT ; g g
if 06x; y6g,
x−g y−g (b) R(x; y) = g + (1 − g)S ; 1−g 1−g if g6x; y61;
II.
x y (a) R(x; y) = gT ; g g
if 06x; y6g,
if g6x; y61,
x−g y−g ; 1−g 1−g
The di/erence between the two classes are in item (c), in the =rst case when one of the arguments is above the identity and the other below the identity we take the Min while in the second case we take the Max. In particular, we note that for the =rst class R(0; 1) = 0 and for the second class R(0; 1) = 1. Fig. 1 provides visualization of the above structure. In the above
x y T (x; y) = gT ; g g
T ∗ (x; y) = Min[x; y];
S(x; y) = Max[x; y];
S ∗ (x; y) = Max[x; y];
T ∗ (x; y) =
xy ; g
S ∗ (x; y) =
x + y − xy − g ; 1−g
T (x; y) = Max[0; x + y − 1]; T ∗ (x; y) = Max[0; x + y − g];
S ∗ (x; y) = Min[1; x + y − g]: In [15] we provided an example of a uninorm not of the above class, called the three operator. For this operator for all x and y in [0; 1]R(x; y) = xy=( xM y+xy) M where xM =1 − x. In this uninorm the identity g = 0:5. Using the associativity property we get n j=1 xj n : R(x1 ; x2 ; : : : ; xn ) = n x j=1 j + j=1 xj
;
x−g y−g S (x; y) = g + (1 − g)S ; 1−g 1−g ∗
T (x; y) = Min[x; y];
S(x; y) = Min[1; x + y];
(c) R(x; y) = Max(x; y) if Min(x; y)6g6Max(x; y).
∗
Let us look at the form of T ∗ and S ∗ for some special cases of T and S.
S(x; y) = x + y − xy;
(b) R(x; y) = g + (1 − g)S
Fig. 1. Structure of uninorm class in [0; 1] × [0; 1].
T (x; y) = xy;
(c) R(x; y) = Min(x; y) if Min(x; y)6g6Max(x; y).
169
and N (x; y) is either the Max or the Min.
3. Uninorms as aggregation operations for fuzzy modeling As we have noted, a fuzzy systems model provides a representation of a relationship between two variables,
170
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
V and U , called the input and output variables. 1 It consists of a collection of n rules of the form if V is Ai then U is Bi ; where Ai and Bi are fuzzy subsets of X and Y , the domains of the input and output variables, respectively. Each rule provides partial information about the relationship between V and U . As discussed by Yager and Filev [13] the determination of the value of U for a given value of V , the fuzzy modeling inference process (FUMIP), consists of the following four step algorithm: 1. Obtain the relevance of each rule for the current input value. 2. Determination of the e/ective output of each rule for the given input value, expressed as a fuzzy subset of output space. We shall denote these individual rule outputs as U is Fi : 3. Aggregation of the individual rule outputs to obtain the overall fuzzy system output as a fuzzy subset of the output space. We shall denote this overall output as U is F: 4. Selection of some action based upon the output set F. We shall =rst focus on the requirements for the operations that can be used to implement this process. We are particularly concerned with the third step, the rule output aggregation. Because of the strong interrelationship between all the steps in the process we look at all the steps. A particular close connection exists between the second and third step. A fundamental aspect of this connection is the relationship between the choice of identity used in the rule aggregation process and the type of interpretation one gives to the rules in step two. Let us brieNy consider the last step =rst. All we want to do here is make some basic observation regarding the role that F, the system output, plays in the solution selection step [12]. With each y of the output space we can associate some solution, normally it is V = y. Furthermore, for any y ∈ Y; F(y) can be seen as some measure of strength to which the system model, un1 The input can be multiple variables; however for our purposes considering it a single variable is suOcient.
der the current input, outputs the solution y. The key observation is that the action selection process uses F in such a way that if F(y1 )¿F(y2 ) we are more inclined to implement the solution suggested by y1 than that of y2 . Fundamentally, we can say that there is a positive association between the membership grade of an element and the action recommended by that element. One very special case of the solution selection process is the defuzzi=cation step used in fuzzy logic controllers [16]. In this special situation the action is to decide which of the elements from the output space Y to use as the input to the plant. A typical example of the defuzzi=cation step is the center of area method. In this method the output value y∈Y yF(y) ∗ y = : y∈Y F(y) Let us now look at the =rst step of the FUMIP. First consider an individual rule, if V is Ai then U is Bi : The intent of this rule is to say that if the input to the system is “consistent with” or “matches” the antecedent then this rule is relevant and the system output is Bi . In indicating that Bi is the fuzzy set output, this is saying that if Bi (y1 )¿Bi (y2 ) there is a preference for having y1 as the solution rather than y2 . The =rst step in the algorithm is the matching step. In this step, we determine some measure of the degree to which the input to the system and the antecedent of the individual rules are compatible. We shall use i to indicate this value and call it the relevance or weight of the ith rule. In the case where the input is a singleton, V = x∗ , and the antecedent is of the form V is Ai then the usual case is to use i = Ai (x∗ ), the membership grade of x∗ in Ai . If the input is a fuzzy subset and=or the antecedent is more complex the process of matching becomes more complex. We shall not focus on this problem at this point. All that we shall observe about the i is that the larger the i the better the matching, and the more relevant the rule. In particular, we observe that if i = 1 the matching is perfect, and the rule is completely relevant. If i = 0 the rule has no relevance to the current input. Having obtained this =ring level we then use it to obtain the e/ective rule output U is Fi . The individual rule output Fi is seen to be completely determined by
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
i , the rule relevance, and the rule consequent Bi . We denote this as Fi = i ◦ Bi . Having obtained the output of each rule we must now combine these individual rule outputs to obtain the overall system output, F. We denote this process F = F1 F2 F3 · · · Fn : As we shall subsequently see, the process used to obtain the overall system output from the individual Fi and the process used to obtain the e/ective rule outputs from the rule consequent and rule relevance are not independent, that is and ◦ are related. Let us look at the process used to combine the individual rule outputs. A basic assumption we shall make is that the operation is pointwise and likewise. By pointwise we mean that for every y ∈ Y; F(y) just depends upon Fi (y) for i = 1 to n. By likewise we shall mean that the process used to combine the Fi is the same for all of the y. The likewise allows us to focus on the aggregation process for any of the elements y in Y . Let us denote the pointwise process we use to combine the individual rule outputs as F(y) = Agg(F1 (y); F2 (y); : : : ; Fn (y)): In the above Agg is called the aggregation operator and the Fi (y) are the arguments. More generally, we can consider this as an operator a = Agg(a1 ; : : : ; an ); where the ai and a are values from the membership grade space, normally the unit interval. A central focus of our interest here is to study the class of operators that are useful to implement this aggregation process used in fuzzy system modeling. Let us look at the minimal requirements associated with the Agg operator. A =rst observation we can make is that the order in which we list the rules in a fuzzy systems model should not e/ect the output of the model. As a result of this we see that the aggregation of the individual rule outputs should be independent of the choice of indexing of the rules. This implies that a required property that we must associate with the Agg operator is that of commutativity, the indexing of the arguments does not matter. We note that the commutativity property allows us to represent the argument of the Agg operator, (F1 (y); F2 (y); : : : ; Fn (y)); as an unordered collection of values; such an object is a bag.
171
For an individual e/ective rule output, Fi ; the membership grade Fi (y) indicates the strength to which this rule suggests that y is the appropriate solution. In particular, if for a pair of elements y1 and y2 ∈ Y it is the case that Fi (y1 )¿Fi (y2 ); then we are saying that rule i is preferring y1 as the system output over y2 . From this we can conclude that if all rules prefer y1 over y2 as output then the overall system output should prefer y1 over y2 . This observation requires us to impose a monotonicity condition on the Agg operation. In particular, if Fi (y1 )¿Fi (y2 ) for all i, then F(y1 )¿F(y2 ). This of course means that in calculating F(y) from the individual components that if Fi (y)¿Fˆ i (y) for all ˆ i then F(y)¿F(y). In particular, if ai ¿bi for all the Agg(a1 ; a2 ; : : : ; an )¿Agg(b1 ; b2 ; : : : ; bn ). There is one other condition we need to impose upon the aggregation operator. Assume that there exists some rule whose relevancy is zero. The implication of this is that the rule provides no information regarding what should be the output of the system. It should not a/ect the =nal F. Without loss of generality, we shall assume the nth rule has relevancy zero. One observation we can make is that whatever e/ective output this rule provides it should not make any distinction between the y values. One way to assure this for rule n; the one with zero relevancy, is to have Fn (yi ) = Fn (yj ) for all i and j. Let us denote this value as g. We note of course that g must be a value in the unit interval. In addition to having Fn (yi ) the same for all yi there is a second requirement necessary. Consider any two elements y1 and y2 in Y . Assume that Agg(F1 (y1 ); F2 (y1 ) : : : Fn−1 (y1 )) = a1 ; Agg(F1 (y2 ); F2 (y2 ) : : : Fn−1 (y2 )) = a2 : Since a rule having zero relevancy should play no role in determining which is the preferred output, the inclusion of its output, Fn ; in the aggregation should not a/ect the relationship between the system output value for y1 and y2 . This observation requires that Agg(F1 (yi ); F2 (yi ) · · · Fn−1 (yi ); g) = Agg(F1 (yi ); F2 (yi ) : : : Fn−1 (yi )): That is, g should be an identity element for the aggregation process. Thus, we must require that the aggregation operator has an identity element. In summary,
172
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
we see that the aggregation operator, Agg must satisfy three conditions: 1. Commutativity. 2. Monotonicity. 3. Must contain an identity. These conditions are based on three observations: 1. The indexing of the rules be unimportant. 2. A positive association between e/ective rule output and total system output. 3. Non-relevant rules play no role in the process. A class of operators satisfying these properties, called MICA operators have been identi=ed and studied by Yager [10]. Thus, the class of MICA operators are the prototypical aggregation operators for the combination of individual rule output. While a most general class of operators implementing the Agg operator are MICA, we observe that a special class of these operators are the uninorms which satisfy the three conditions. Thus, the uninorms can be used to implement the aggregation operator in fuzzy systems modeling. With the uninorms we invest our aggregation operator with the additional property of associativity. It should be pointed out that no requirement was made on the choice of the identity g. As we shall subsequently see the only requirement is that it be consistent with the choice of operation used to obtain the e/ective rule output. 4. RET operators and e!ective rule output calculations We now look at the requirements for the operation used to obtain the e/ective rule output from the rule relevancy and the rule consequent, Fi = i ◦Bi . We shall assume that ◦ is a pointwise and likewise operation. Thus, there exists some function which we shall denote as h such that
A =rst requirement on h is that if the antecedent of the rule is completely satis=ed, the rule relevancy is one, our e/ective output should be the rule consequent, Bi . This characteristic requires that h satisfy the condition that for all b h(1; b) = b: A second requirement on h is that if the rule relevancy is zero the e/ective output should not inNuence the aggregation process used in step three. This characteristic imposes a condition on h whereby if the rule relevancy is zero then the output value should be the identity under the aggregation operation used in step three. Formally, this requires that for all b h(0; b) = g; where g is the identity under the aggregation operator Agg. A third condition we require is a kind of monotonicity in the second argument. We require that for any a and any b¿b it is the case that h(a; b)¿h(a; b ): This condition is a reNection of the requirement that we should not inverse the preference ordering between output values. That is no value for relevancy should allow us to reverse the preferences speci=ed in Bi . A fourth condition we shall require is a consistency in the antecedent argument. In particular, we desire that for a =xed consequent value the output of h go monotonically from the value for zero =ring, the identity g, to the value for complete =ring, b. In particular, for any level of relevancy ; Sgn(@h( ; b)=@ ) = Sgn(b − g) or zero. That is if ¿ then if b¿g then h( ; b)¿h( ; b);
Fi (y) = h( i ; Bi (y)):
if b6g then h( ; b)6h( ; b):
More generally, h is a binary operator taking values from I 2 into I; h(a; b) = c; for a; b; c ∈ I . The operation h is closely related to both the implication operator [1] and the weighting operation [11]. We shall call h the relevancy transformation (RET) function. Let us investigate the minimal requirements we desire the operator h to have.
We note that in the special case when g = 1 it is always the case that b6g and hence we get for @h( ; b)=@ 60. At the other extreme is the case when g = 0. In this situation since b¿g for all b we have @h( ; b)=@ ¿0. It is interesting to see that the above requirement imposes the following condition, if b = g then h( ; g) = g for all .
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
173
When using a uninorm with =xed identity g as our aggregation operator we shall say that h is an appropriate RET operator if it satis=es the four conditions speci=ed where h(0; b) = g. For any inference implementation using a uninorm U with identity g and appropriate RET h if i = 0 for all i the system overall output F has F(y) = g for all y. We see this as follows. If i = 0 then Fi (y) = h(0; Bi (y)) = g for all y and all i therefore F(y) = U (g; g; g; g; g; : : : ; g). From associativity and the fact that U (g; g) = g we get the result. This situation of having F(y) the same for all y when all rules have no relevancy is intuitively appealing for it essentially says that our model has provided no information that allows us to distinguish between any of the possible solutions to our model. The use of uninorms as our aggregation operator brings with it an additional property which Yager called 1delity [10]. By =delity we mean to require that when the input to the system is such that one rule is completely relevant and all other rules are completely not relevant then the system output should be exactly the consequent of the relevant rule. Without loss of generality, =delity requires that if 1 = 1 and j = 0 for j = 1 then F = F1 . We see that the associativity of the uninorm guarantees =delity. The choice of an appropriate h means that for any y, B1 (y) = F1 (y) and Fj (y) = g for all j = 1. From this F(y) = U (F1 (y); g; g; : : : ; g). The associativity of U and the identity of g implies that F(y) = F1 (y). A fundamental issue here becomes the selection of an relevancy transformation function given the selection of a uninorm U . The following theorem, which has the Navor of an existence theorem, provides a RET operator that is valid for all choices of g.
However, since aM − aM = − (a − a ) then we get h(a; b) − h(a ; b) = (a − a )(b − g). Thus if b¿g then since a¿a we get h(a; b)¿b(a ; b) and if b6g we get h(a; b)6h(a ; b). Thus, we see that the above form is an appropriate transformation operator for any uninorm with the corresponding g.
Theorem 1. The formulation
Proof. We have shown that T (a; b) satis=es the four required conditions: 1. Assume a = 1 then T (1; b) = b. 2. Assume a = 0 then T (0; b) = 0 = g. 3. Assume b¿b then from the monotonicity of tnorms we get T (a; b)¿T (a; b ). 4. Assume a¿a then since b¿g = 0 for all b then T (a; b)¿T (a ; b).
h(a; b) = ab + ag M satis1es all four conditions for any g. Proof. 1. Assume a = 1; then aM = 0 and we get h(a; b) = b. 2. Assume a = 0 then aM = 1 and we get h(a; b) = g. 3. Assume b¿b then h(a; b)−h(a; b ) = a(b−b )¿0. 4. Assume a¿a . Then M − (a b + aM g) h(a; b) − h(a b) = ab + ag = (a − a )b + (aM − aM )g:
The following theorem provides another class of RET operators that are appropriate for any identity g. Theorem 2. The formulation h(a; b) = (a ∧ b) ∨ (aM ∧ g) ∨ (b ∧ g) satis1es all four conditions for any g. Proof. 1. If a = 0 then h(a; b) = g ∨ (b ∧ g) = g. 2. If a = 1 then h(a; b) = b ∨ (b ∧ g) = b. 3. If b1 ¿b2 then h(a; b1 )¿h(a; b2 ). 4. (i) If b¿g then h(a; b) = (a ∧ b) ∨ g; from this we see that h(a1 ; b)¿h(a2 ; b) if a1 ¿a2 . (ii) If g¿b then h(a; b) = (aM ∧ g) ∨ b; from this we see that h(a1 ; b)6h(a2 ; b) if a1 ¿a2 . (iii) If g = b then h(a; b) = g. If we restrict ourselves the special cases of uninorms with g = 0 or 1; t-conorms and t-norms, we get a large class of transformation operators. Theorem 3. For the case of a uninorm in which g = 0; t-conorm; the operator h(a; b) = T (a; b); where T is any t-norm; is an appropriate RET operator.
Two examples of t-norms are the Min and product h(a; b) = Min(a; b) h(a; b) = ab
Min;
Product:
174
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
Using these to implement the h function we get Fi (y) = Min[ i ; Bi (y)]; Fi (y) = i Bi (y): For the uninorm with g = 0 our aggregation operator is a t-conorm S; examples of this are S(a; b) = Max[a; b] and S(a; b) = a + b − ab. If we select the Max t-conorm as our aggregation uninorm and combine this with the Min interpretation of h we get F(y) = Max[ i ∧ Bi (y)]): i
This is the formulation which was used by Mamdani [5] in his original work on fuzzy modeling of controllers. The following theorem provides a corresponding class of appropriate RET functions for aggregation operators based on uninorms with g = 1; t-norms. Theorem 4. For the case of a uninorm in which g = 1; t-norm; the operator h(a; b) = S(a; M b); where S is any t-norm; is an appropriate RET operator. Two important examples of this occur when S(a; b) = Max(a; b); S(a; b) = a + b − ab: If we use S(a; b) = Max(a; b) to implement the h function we get Fi (y) = Max(M i ; Bi (y)) and if we use S(a; b) = a + b − ab to implement the h function we get Fi (y) = Mi + i Bi (y). If we select the Min as our aggregation uninorm and combine this with the Max interpretation of h we get
Fig. 2. Neural model for the fuzzy system inference.
S(a; b) = Min[1; a + b]. Using associativity this extends to the form n aj : U (a1 ; a2 ; : : : ; an ) = Min 1; j=1
Let us use as the RET the product. Thus in this situation we get n F(y) = Min 1; j Bj (y) : j=1
The operation Min[1; x] can be viewed as a quasilinear function f(x) = x
for x61;
f(x) = 1
for x ¿ 1: n Thus F(y) = f( j=1 j Bj (y)). This formulation leads us to a neural network [18] view of the process (see Fig. 2). In this view the neuron calculates the membership grade F(y) of the element y in the output fuzzy set F. Here i is the degree to which the ith rule has =red, Ai (x∗ ); and Bi (y) is the degree of membership of y in the consequent of the ith rule.
F(y) = Min(Max(M i ; Bi (y)))
5. Conclusion
which is the formulation which we indicated was used by Zadeh [17]. There exists another special case which induces a very interesting formulation for the inference process used in fuzzy modeling. Consider the uninorm aggregation operator U based on the t-conorm
The fuzzy modeling inference process was shown to involve an aggregation step in which the contributions of the di/erent rules of the fuzzy systems model are combined. It was shown that the uninorm operator provides a general class of operators to implement this step. Another step in the inference process is the
i
R.R. Yager / Fuzzy Sets and Systems 122 (2001) 167–175
determination of the contribution of each component of the model based on its relevancy to the current input. It was shown that the performance of this step requires the use of an operator, related to the implication operator, which we called a relevancy transformation (RET) operator. It was shown that the form of this RET operator is dependent upon the identity value of the uninorm used in the aggregation step. We provided some general formulations for this class of operators. We showed how the well-known forms of fuzzy inference are special cases of our uninorm-based approach. References [1] J.F. Baldwin, B.W. Pilsworth, Axiomatic approach to implication for approximate reasoning with fuzzy logic, Fuzzy Sets and Systems 3 (1980) 193–219. [2] B. De Baets, Uninorms: the known classes, in: D. Ruan, H.A. Abderrahim, P. D’hondt, E.E. Kerre (Eds.), Fuzzy Logic and Intelligent Technologies for Nuclear Science and Industry, World Scienti=c, Singapore, 1998, pp. 21–28. [3] B. De Baets, J. Fodor, On the structure of uninorms and their residual implicators, Proceedings of the 18th Linz Seminar on Fuzzy Set Theory, Linz, Austria, 1997, pp. 81–87. [4] J.C. Fodor, R.R. Yager, A. Rybalov, Structure of uni-norms, Internat. J. Uncertainty, Fuzziness Knowledge-Based Systems 5 (1997) 411–427. [5] E.H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. IEEE 121 (1974) 1585–1588. [6] E.H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man– Mach. Stud. 7 (1975) 1–13. [7] E.H. Mamdani, Advances in the linguistic synthesis of fuzzy controllers, Internat J. Man–Mach. Stud. 8 (1976) 669– 678.
175
[8] M. Sugeno, T. Takagi, A new approach to design of fuzzy controller, in: P.P. Wang (Ed.), Advances in Fuzzy Sets, Possibility Theory and Applications, Plenum Press, New York, 1983, pp. 325–334. [9] T. Takagi, M. Sugeno, Derivation of fuzzy control rules from human operators actions, Proceedings of the IFAC Symposium on Fuzzy Information, Marseille, 1983, pp. 55 – 60. [10] R.R. Yager, Aggregation operators and fuzzy systems modeling, Fuzzy Sets and Systems 67 (1994) 129–146. [11] R.R. Yager, Structures generated from weighted fuzzy intersection and union, J. Chinese Fuzzy Systems Assoc. 2 (1996) 37–58. [12] R.R. Yager, D.P. Filev, On the issue of defuzzi=cation and selection based on a fuzzy set, Fuzzy Sets and Systems 55 (1993) 255–272. [13] R.R. Yager, D.P. Filev, Essentials of Fuzzy Modeling and Control, Wiley, New York, 1994. [14] R.R. Yager, D.P. Filev, Generation of fuzzy rules by mountain clustering, J. Intell. Fuzzy Systems 2 (1994) 209–219. [15] R.R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996) 111–120. [16] J. Yen, R. Langari, L.A. Zadeh, Industrial Applications of Fuzzy Logic and Intelligent Systems, IEEE Press, New York, 1995. [17] L.A. Zadeh, A theory of approximate reasoning, in: J. Hayes, D. Michie, L.I. Mikulich (Eds.), Machine Intelligence, Vol. 9, Halstead Press, New York, 1979, pp. 149–194. [18] J.M. Zaruda, Introduction to Arti=cial Neural Systems, West Publishing, St Paul, MN, 1992. [19] L.A. Zadeh, The calculus of fuzzy if=then rules, AI Expert March, 1992, pp. 23–27. [20] L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems 90 (1997) 111–127.