Operations on type2 fuzzy sets


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1 Fuzzy Sets and Systems (00) Operations on type fuzzy sets Nilesh N. Karnik, Jerry M. Mendel Signal and Image Processing Institute, Department of Electrical EngineeringSystems, 3740 McClintock Ave., EEB400, University of Southern California, Los Angeles, CA , USA Received 4 August 999; received in revised form 0 April 000; accepted May 000 Abstract In this paper, we discuss set operations on type fuzzy sets (including join and meet under minimum=product tnorm), algebraic operations, properties ofmembership grades oftype sets, and type relations and their compositions. All this is needed to implement a type fuzzy logic system (FLS). c 00 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy relations; Fuzzy numbers; Type fuzzy sets. Introduction The concept ofa type fuzzy set was introduced by Zadeh [30] as an extension ofthe concept ofan ordinary fuzzy set (henceforth called a type fuzzy set). Such sets are fuzzy sets whose membership grades themselves are type fuzzy sets; they are very useful in circumstances where it is dicult to determine an exact membership function for a fuzzy set; hence, they are useful for incorporating linguistic uncertainties, e.g., the words that are used in linguistic knowledge can mean dierent things to dierent people []. A fuzzy relation of higher type (e.g., type) has been regarded as one way to increase the fuzziness of a relation, and, according to Hisdal, increased fuzziness in a description means increased ability to handle inexact information in a logically correct manner [5]. According to John, Type fuzzy sets allow for linguistic grades ofmembership, thus assisting in knowledge representation, and they also oer improvement on inferencing with type sets [6]. Type sets can be used to convey the uncertainties in membership functions of type sets, due to the dependence of the membership functions on available linguistic and numerical information. Linguistic information (e.g., rules from experts), in general, does not give any information about the shapes of the membership functions. When membership functions are determined or tuned based on numerical data, the uncertainty in the numerical data, e.g., noise, translates into uncertainty in the membership functions. In all such cases, information about the linguistic=numerical uncertainty can be incorporated in the type framework. In [8], Liang and Mendel demonstrated (using real data) that a type fuzzy set, a Gaussian with xed mean and Corresponding author. Tel.: + (3) ; fax: + (3) address: (J.M. Mendel) /0/$  see front matter c 00 Elsevier Science B.V. All rights reserved. PII: S (00)
2 38 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) uncertain standard deviation (std), is more appropriate to model the frame sizes of I=P=B frames in MPEG VBR video trac than is a type Gaussian membership function. In this paper, A is a type fuzzy set, and the membership grade (a synonym for the degree of membership) of x X in A is A (x), which is a crisp number in [0; ]. A type fuzzy set in X is A, and the membership grade of x X in A is A (x), which is a type fuzzy set in [0; ]. The elements ofthe domain of A (x) are called primary memberships of x in A and the memberships ofthe primary memberships in A (x) are called secondary memberships of x in A. The latter dene the possibilities for the primary membership, e.g., A (x) can be represented, for each x X as A (x) =f x(u )=u + f x (u )=u + + f x (u m )=u m = i f x (u i )=u i ; u i J x [0; ]: When the secondary MFs are interval sets, we call them interval type fuzzy sets. The operations of interval type fuzzy sets are studied in [9,6]. Example. Fig. (a) shows a type set in which the primary membership grade for each point in the set is a Gaussian type set. The means ofall these Gaussian type sets, follow a Gaussian and the standard deviations ofthese Gaussians decrease as the mean decreases. Uncertainty in the primary membership grades ofthe type MF consists ofa shaded region (the union ofall primary membership grades), that we call the footprint of uncertainty ofthe type MF (e.g., see Fig. (a)). The principal membership function, i.e., the set ofprimary memberships having secondary membership equal to, is indicated with a thick line. This principal membership function is a Gaussian because ofthe way the set is constructed. Intensity ofthe shading is approximately proportional to secondary membership grades. Darker areas indicate higher secondary memberships. The at portion from about.5 to 3.5 appears because primary memberships cannot be greater than (since primary memberships, themselves, are possible membership values, they have to be in [0; ]) and so the Gaussians have to be clipped. The domain ofthe membership grade corresponding to x =4 is also shown. Fig. (b) shows the secondary membership function for x = 4. Many other examples oftype sets can be found in []. The concept of a principal membership function also illustrates the fact that a type fuzzy set can be thought ofas a special case ofa type fuzzy set. We can think ofa type fuzzy set as a type fuzzy set whose membership grades are type fuzzy singletons, having secondary membership equal to unity for only one primary membership and zero for all others, i.e., we can think of the principal membership function of a type set as an embedded type set. A type fuzzy set can also be thought of as a fuzzy valued function, which assigns to every x X a type fuzzy membership grade. In this sense, we will call X the domain of the type fuzzy set. Mizumoto and Tanaka [3] have studied the settheoretic operations oftype sets, properties ofmembership grades ofsuch sets, and, have examined the operations ofalgebraic product and algebraic sum for them [4]. Nieminen [5] has provided more details about algebraic structure oftype sets. Dubois and Prade [4] and Kaufmann and Gupta [4] have discussed the join and meet operations between fuzzy numbers under minimum tnorm. Dubois and Prade [ 4] have also discussed fuzzy valued logic and have given a formula for the composition oftype relations as an extension ofthe type supstar composition, but this formula is only for minimum tnorm. Karnik and Mendel [,3] have provided a general formula for the extended supstar composition oftype relations. Type fuzzy sets have been used in decision making [,9], solving fuzzy relation equations [7], survey processing [], preprocessing ofdata [8], and modeling uncertain channel states [,3]. In engineering applications of fuzzy logic, minimum and product tnorm are usually used, so we focus on product and minimum tnorm in this paper, and leave the extensions ofour results to other tnorm to others. Recently, a type FLS [,3] has been developed which uses either tnorm; however, in order to implement such a
3 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig.. (a) Pictorial representation ofa Gaussian type set. The standard deviations ofthe secondary Gaussians decrease by design, as x moves away from 3. The secondary memberships in this type fuzzy set are shown in (b), and are also Gaussian. type FLS, one needs algorithms for settheoretical operations on type sets (which are given in Section ), algebraic operations for type sets (which are given in Section 3), an understanding of set properties of membership grades (which is provided in Section 4) and algorithms for type fuzzy relations and their compositions (which are given in Section 5). So, the main purpose ofthis paper is to provide the foundation for type FLSs.. Settheoretic operations on type sets Consider two fuzzy sets of type, A and B, in a universe X. Let A (x) and B (x) be the membership grades (fuzzy sets in J x [0; ]) ofthese two sets, represented, for each x, as A (x)= u f x(u)=u and B (x)= w g x(w)=w, respectively, where u; w J x indicate the primary memberships of x and f x (u); g x (w) [0; ] indicate the secondary memberships (grades) of x. Using Zadeh s Extension Principle [30,4,], the membership grades for union, intersection and complement of type fuzzy sets A and B have been dened as follows [3]: Union: A B A B (x) = A (x) B (x) = (f x (u)?g x (w))=(u w); () u w Intersection: A B A B (x) = A (x) B (x) = (f x (u)?g x (w))=(u?w); () u w Complement: A A (x) A (x) = f x (u)=( u); (3) u
4 330 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) where represents the max tconorm and? represents a tnorm. The integrals indicate logical union. In the sequel, we adhere to these denitions, and, as in [3], we refer to the operations, as join, meet and negation, respectively. If A (x) and B (x) have discrete domains, the integrals in () (3) are replaced by summations. In (), ifmore than one calculation ofu and w gives the same point u w, then in the union we keep the one with the largest membership grade. We do the same in (). Suppose, for example, u w = and u w =. Then within the computation of() we would have f x (u )?g x (w )= + f x (u )?g x (w )= (4) where + denotes union. Combining these two terms for the common is a type computation in which we can use a tconorm. We choose to use the maximum as suggested by Zadeh [30] and Mizumoto and Tanaka [3]. Using the denitions in () (3), we examine the operations of join, meet and negation in more detail, both under minimum ( ) and product tnorms (which are the most widely used tnorms in engineering applications offuzzy sets and logic). We consider only convex, normal membership grades. Our goal is to obtain algorithms that let us compute the join, meet and negation. These algorithms will be needed later to compute type relations. Since these operations are performed on membership grades of type sets, which themselves are type sets, the results discussed next are in terms oftype sets. The type sets, F i, in Sections..4, can be thought ofas membership grades ofsome type sets, A i, for some arbitrary input x 0. For example, (4) in Fig. (b) is a type set associated with x =4... Join and meet under minimum tnorm Our major result for join and meet under minimum tnorm is given in: Theorem. Suppose that we have n convex; normal; type real fuzzy sets F ;:::;F n characterized by membership functions f ;:::;f n ; respectively. Let v ;v ;:::;v n be real numbers such that v 6v 6 6v n and f (v )=f (v )= =f n (v n )=. Then; using max tconorm and min tnorm; n i= f i(); v ; n Fi() = n i= i=k+ f i(); v k 6 v k+ ; 6k6n ; (5) n i= f i(); v n ; and n i= f i(); v ; n Fi() = k i= i= f i(); v k 6 v k+ ; 6k6n ; n i= f i(); v n : (6) See Appendix A for the proofoftheorem. Fig. shows an example ofan application oftheorem for the case n = 4. Observe that Eqs. (5) and (6) are very easy to program. Dubois and Prade present the result in Theorem just for the case n =, in the context offuzzication ofmax and min operations. Though their method ofproofis very similar to ours, they prove the result for a special case, where f and f have at most three points ofintersection and one needs to keep track of the points ofintersection off and f to use their theorem. We reprove this theorem in a general setting in Appendix A. We believe that our statements of F F () and F F () are more amenable to computer
5 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig.. An illustration oftheorems and : Join and meet operations between Gaussians: (a) Participating Gaussians; (b) join under minimum tnorm (solid line) and product tnorm (thick dashed line); and (c) meet under minimum tnorm. implementations than those ofdubois and Prade. Generalization to the case n is also dicult using Dubois and Prade s result, but is provided by us in Theorem. As a consequence oftheorem, we have the following interesting result: Corollary. If we have n convex; normal; type fuzzy sets F ;:::;F n characterized by membership functions f ;:::;f n ; respectively; such that f i ()=f ( k i ); and 0=k 6k 6 6k n ; then; using max tconorm and min tnorm; n i= F i = F n and l n i= F i = F. The prooffollows by a direct application oftheorem to the n type sets described in the statement of the corollary. Fig. 3 shows an example ofunion and intersection oftype sets, using the results in this section... Join under product tnorm For type fuzzy sets satisfying the requirements of Theorem, join operation under product tnorm gives exactly the same result as (5) with product replacing the minimum. Theorem. Suppose that we have n convex; normal; type real fuzzy sets F ;:::;F n characterized by membership functions f ;:::;f n ; respectively. Let v ;v ;:::;v n be real numbers such that v 6v 6 6v n and f (v )=f (v )= = f n (v n )=. Then; the membership function of n i= F i using max tconorm and
6 33 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig. 3. Union and Intersection of two type fuzzy sets using the pictorial representation introduced in Fig., and, for maximum tconorm and minimum tnorm: (a) Participating sets; (b) union; and (c) intersection. product tnorm; can be expressed as n i= f i(); v ; n i= Fi () = n i=k+ f i(); v k 6 v k+ ; 6k6n ; n i= f i(); v n : (7) The proofproceeds in the same way as the proofofpart (I) in Theorem (Appendix A), with the minimum operation replaced by product. See [] for details. Fig. shows an example ofan application oftheorem for the case n =4..3. Meet under product tnorm The meet operation under the product tnorm between two convex normal type sets F and F can be represented as F F = [f (v)f (w)]=(vw): (8) v w Ref. [4] gives a similar result while discussing multiplication of fuzzy numbers (see Section 3 for algebraic operations on fuzzy sets). If is an element of F F, then the membership grade of can be found by nding all the pairs {v; w} such that v F ; w F and vw = ; multiplying the membership grades of v and w in each pair; and then nding the maximum ofthese products ofmembership grades. The possible admissible {v; w} pairs whose
7 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) product is are {v; =v} (v R; v 0) for 0 and {v; 0} or {0;w}, where v; w R for = 0, i.e., ( F F () = sup f (v)f v R;v 0 v [ ] F F (0) = sup f (v)f (0) v R = [ f (0) sup f (v) v R = f (0) f (0): ] ) ; R; 0; [ ] sup f (0)f (w) w R ] [ f (0) sup f (w) w R (9) Since the meet operation under product tnorm is commutative (see Section 4), we get the same result whether we substitute =w = v or =v = w. As is apparent from (9), the result is very much dependent on functions f and f and does not easily generalize like the join and meet operations considered earlier, and generally, it is very dicult to obtain a closedform expression for the result of the meet operation. Even ifboth the fuzzy sets involved have Gaussian membership functions, it is dicult to obtain a nice closedform expression for the result of the meet operation. To determine the membership ofa particular point in F G, we nd all the pairs {v; w} such that v R; w R and vw = ; and multiply the memberships ofeach pair. The membership grade of is given by the supremum ofthe set ofall these products. For example, if = 0, all the pairs {v; w} that give 0 as their product are v and 0=v (v R;v 0). So, the membership grade of0 is given by the supremum of the set ofall the products f(v)g(0=v) (v R;v 0). Fig. 4(c) shows how f(v)g(0=v) looks for F and G in Fig. 4(a). Clearly, it is no easy matter to represent (9) visually. Because ofthe complexity of(9) for general membership functions (MFs), we concentrate on Gaussian MFs. For a derivation of F F when both F and F are Gaussian type sets, see []. It is very complicated, does not lead to a closedform expression, and does not generalize to more than two MFs. Since the meet operation under product tnorm is heavily used in the development ofa type FLS (because it is widely used in type FLSs), here we develop a Gaussian approximation to this operation that will make it practical. Ifthere are n Gaussian fuzzy sets F ;F ;:::;F n with means m ;m ;:::;m n and standard deviations ; ;:::; n, respectively, then, under the product tnorm, F F F n () e (=)(( mm mn)= ) : (0) where (i =;:::;n) = m i + + j m i + + n m i : () i;i i;i j i; i n For the derivation ofthis approximation, see Appendix B. One important feature ofthis approximation is that it is Gaussian; hence, it is reproducible, i.e., Gaussians remain Gaussians. Fig. 5 shows some examples ofthe meet operation under product tnorm, and the approximation in (0). See [9] for a triangular approximation for triangular type sets, i.e., type sets in which all the secondary membership functions are triangular. A popular type fuzzy set is the interval type set, i.e., the type set all of whose secondary MFs are interval sets. Note that an interval [l; r] can also be represented in terms ofits mean and spread as [m s; m+s],
8 334 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig. 4. An example showing how f(v)g(0=v) looks for the membership functions f and g oftype sets F and G, respectively, in (a); (b) f(v), which is the same as that in (a) and g(0=v); and, (c) the product f(v)g(0=v). where m =(l + r)= and s =(r l)=. The meet ofinterval sets under product=minimum tnorm has been given by Schwartz [6] and Kaufmann and Gupta [4], and is discussed in detail in Liang and Mendel [7]..4. Negation From the denition ofthe negation operation, it follows that Theorem 3. If a type real fuzzy set F has a membership function has a membership function f( ). The prooffollows by substituting = u in (3). 3. Algebraic operations Just as the tconorm and tnorm operations have been extended to the membership grades oftype sets using the Extension Principle, algebraic operations between type sets have been dened using the Extension Principle [4,4,]. A binary operation dened for crisp numbers, can be extended to two type sets, F = v f (v)=v and F = w f (w)=w, as follows: F F = v [f (v)?f (w)]=(v w) () w
9 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig. 5. Some examples ofthe meet between Gaussians under product tnorm (solid line) and the approximation in (0) (thick dashed line). Means and standard deviations ofparticipating Gaussians in each case are also shown on the gure. In each case, the solid curve was obtained numerically, using (9). where? indicates the tnorm used. We will mainly be interested in multiplication and addition oftype sets. Observe, from (8) and (), that multiplication of two type sets under product tnorm is the same as the meet ofthe two type sets under product tnorm; therefore, all our earlier discussion about the meet under product tnorm applies to the multiplication oftype sets under product tnorm. Theorem 4. Given n interval type numbers F ;:::;F n ; with means m ;m ;:::;m n and spreads s ;s ;:::;s n ; their ane combination n i= if i + ; where i (i =;:::;n) and are crisp constants; is also an interval type number with mean n i= im i + ; and spread n i= i s i. See Appendix C for the proofoftheorem 4. Theorem 5. Given n type Gaussian fuzzy numbers F ;:::;F n ; with means m ;m ;:::;m n and standard deviations ; ;:::; n ; their ane combination n i= if i + ; where i (i =;:::;n) and are crisp constants; is also a Gaussian fuzzy number with mean n i= im i + ; and standard deviation ; where n i= = i i if product tnorm is used; n i= i i if minimum tnorm is used: See Appendix D for the proofoftheorem 5. (3)
10 336 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Table Properties ofmembership grades. In the type case, we assume convex and normal membership grades; and for type laws,, and replace,? and 6, respectively. For product tnorm, the laws that are satised in the type case, but not in the type case, are highlighted. Results for the minimum tnorm are taken from [3] Settheoretic laws Minimum tnorm Product tnorm Type Type Type Type Reexive A 6 A Yes Yes Yes Yes Antisymmetric A 6 B ; B 6 A Yes Yes Yes Yes A = B Transitive A 6 B ; B 6 C Yes Yes Yes Yes A 6 C Idempotent A A = A Yes Yes Yes No A? A = A Yes Yes No No Commutative A B = B A Yes Yes Yes Yes A? B = B? A Yes Yes Yes Yes Associative ( A B ) C Yes Yes Yes Yes = A ( B C ) ( A? B )? C Yes Yes Yes Yes = A? ( B? C ) Absorption A? ( A B )= A Yes Yes No No A ( A? B )= A Yes Yes Yes No Distributive A? ( B C ) Yes Yes Yes No =( A? B ) ( A? C ) A ( B? C ) Yes Yes No No =( A B )? ( A C ) Involution A = A Yes Yes Yes Yes De Morgan s Laws A B = A? B Yes Yes No No A? B = A B Yes Yes No No Identity A 0= A Yes Yes Yes Yes A? = A Yes Yes Yes Yes A = Yes Yes Yes Yes A? 0 = 0 Yes Yes Yes Yes Complement A A Yes Yes Yes Yes (Failure) A?A 0 Yes Yes Yes Yes 4. Properties of membership grades Mizumoto and Tanaka [3] discuss properties ofmembership grades oftype sets in detail, under the minimum tnorm and maximum tconorm. We examined these properties under product tnorm and maximum tconorm for convex, normal membership grades. Our ndings are summarized in Table. See Chapter 3 of [] for details, which we have omitted here because they are rather straightforward. In proving the identity law, we use the concepts of0 and membership grades for type sets [3]; they are represented as =0 and =, respectively, which is in accordance with our earlier discussion about type sets being a special case oftype sets. An element is said to have a zero membership in a type set ifit has a secondary membership equal to corresponding to the primary membership of0, and ifit has all other secondary memberships equal to 0. Similarly, an element is said to have a membership grade equal to in a type set, ifit has a secondary membership equal to corresponding to the primary membership of and ifall other secondary memberships are zero. All our operations on type sets collapse to their type counterparts, i.e., ifall the type sets are replaced by their principal membership functions (assuming principal membership function can be dened for all of
11 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) them), all our results remain valid. We, therefore, conclude that if there are any settheoretic laws that are not satised by type fuzzy sets, we can safely say that type sets will not satisfy those laws either; however, the converse ofthis statement may not be true. If any condition is satised by type sets, it may or may not be satised by type sets. Observe, from Table, that the {max, product} tconorm=tnorm pair, for type sets, does not satisfy idempotent, absorption, distributive, De Morgan s and complement laws; hence, ifthe design ofa type fuzzy logic system (controller) involves the use of these laws, it will be in error. Observe also that some laws that are satised in the type case under {max, product} tconorm=tnorm pair are not satised in the type case. 5. Type fuzzy relations, their compositions, and cartesian product 5.. Composition of type fuzzy relations Let X ;X ;:::;X n be n universes. A crisp relation in X X X n is a crisp subset ofthe product space. Similarly, a type fuzzy relation in X X X n is a type fuzzy subset of the product space and a type fuzzy relation in X X X n is a type fuzzy subset of the product space. We concentrate on binary type fuzzy relations. Unions and intersections oftype relations on the same product space can be obtained exactly in the same manner as unions and intersections ofordinary type sets are obtained. If R(U; V ) and S(U; V ) are two type relations on the same product space U V, their union and intersection are determined as R S (u; v)= R (u; v) S (u; v) and R S (u; v)= R (u; v) S (u; v). Consider two dierent product spaces, U V and V W, that share a common set and let R(U; V ) and S(V; W ) betwocrisp relations on these spaces. The composition ofthese relations is dened [5] as a subset T (U; W )ofu W such that (u; w) T ifand only if(u; v) R and (v; w) S. This can be expressed as a max min, maxproduct or, in general, as the supstar composition R S (u; w) = sup v V [ R (u; v)? S (v; w)] where? indicates any suitable tnorm operation. The validity ofthe supstar composition for crisp sets is shown in [8]. If R and S are two crisp relations on U V and V W, respectively, then the membership for any pair (u; w); u U and w W, is ifand only ifthere exists at least one v V such that R (u; v)= and S (v; w) =. In [8], it is shown that this condition is equivalent to having the supstar composition equal to. When we enter the fuzzy domain, set memberships (which are crisp numbers for type sets, and type fuzzy sets for type sets) belong to the interval [0; ] rather than just being 0 or. So, now we can think ofan element as belonging to a set ifit has a nonzero membership in that set. In this respect, the aforementioned condition on the composition of relations can be rephrased as follows (recall the concept of 0 and membership grades in the case oftype sets discussed in Section 4): IfR ( R)andS( S) are two type (type) fuzzy relations on U V and V W, respectively, then the membership for any pair (u; w), u U and w W, is nonzero ifand only ifthere exists at least one v V such that R (u; v) 0 (R (u; v) 0) and S(v; w) =0 ( S (v; w) =0). It can be easily shown that in the type case, this condition is again equivalent to the supstar composition, R S (u; w) = sup [ R (u; v)? S (v; w)] (4) v V and, in the type case, it is equivalent to the following extended version of the supstar composition. R S (u; w) = v V [ R (u; v) S (v; w)]: (5)
12 338 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Proofs of(4) and (5) are very similar. We just give the proofof(5) next. In the proof, we use the following method. Let A be the statement R S (u; w) =0, and B be the statement there exists at least one v V such that R (u; v) =0 and S (v; w) =0. We prove that A i B by rst proving that B A (which is equivalent to proving A B, i.e., necessity of B) and then proving that A B (which is equivalent to proving B A, i.e., suciency of B). Proof. (Necessity) Ifthere exists no v V such that R (u; v) =0 and S (v; w) =0, then this means that either R (u; v)==0 or S (v; w)==0 (or both are zero) for every v V. So, assuming that R (u; v) and S (v; w) are normal, then from the identity law ( A =0==0), R (u; v) =0==0 or=0 S (v; w)==0, so that we have that R (u; v) S (v; w)==0 for every v V, and therefore, in (5), v V =0==0. (Suciency) Ifthe extended supstar composition is zero, then, from (5), v V [ R (u; v) S (v; w)]==0: For each value of v, the term in the bracket is a type fuzzy set. Suppose, for example, that there are two such terms, say P and Q, so that P Q ==0. Observe, from (), that for two normal membership grades, P = u f(u)=u and Q = v g(v)=v, P Q ==0 u (6) [f(u)?g(v)]=(u v) ==0: (7) v For u v =0; u= v = 0, in which case, P = Q ==0. This means that each ofthe two terms in the bracket of(5) equals /0. The extension ofthis analysis to more than two terms is easy. We conclude, therefore, that R (u; v) S (v; w)==0 for every v V, which means that for every v V, either R (u; v) or S (v; w) (or both) is /0. Consequently, there is no v V such that R (u; v) =0 and S (v; w) =0. Dubois and Prade [3,4] give a formula for the composition of type relations under minimum tnorm as an extension ofthe type supmin composition. Their formula is the same as (5); however, we have demonstrated the validity of(5) for product as well as minimum tnorms. Example. Consider the type relation u is close to v onu V, where U = {u ;u } and V = {v ;v ;v 3 } are given as: U = {; }, V = {; 7; 3} and close (u; v) = u u ( v v v 3 0:9 0:4 0: 0: 0:4 0:9 ) : (8) Now consider another type fuzzy relation v is bigger than w onv W where W = {w ;w } = {4; 8}, with the following membership function: bigger (v; w) = w w v 0 0 v 0:6 0 : v 3 0:7 (9)
13 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) As is well known, the statement u is close to v and v is bigger than w indicates the composition ofthese two type relations, which can be found by using (4) and the min tnorm, as follows: close bigger (u i ;w j )=[ close (u i ;v ) bigger (v ;w j )] [ close (u i ;v ) bigger (v ;w j )] [ close (u i ;v 3 ) bigger (v 3 ;w j )] (0) where i =; and j =; ; 3. Using (0), we get close bigger (u; w) = u u ( w w 0:4 0: 0:9 0:7 ) : () Now, let us consider the type relations u is close to v and v is bigger than w which are obtained by adding some uncertainty to the type relations in (8) and (9). Let the membership grades be as follows: close ] (u; v) = u u v 0:3=0:8+=0:9 +0:7= 0:5=0+=0: v 0:7=0:3+=0:4 +0:=0:5 0:7=0:3+=0:4 +0:=0:5 v 3 0:5=0+=0: 0:3=0:8+=0:9 +0:7= () and bigger ] (v; w) = v v v 3 w =0+0:6=0: 0:4=0:5+=0:6 +0:9=0:7 0:7=0:9+= w =0+0:=0: =0+0:8=0: +0:=0: : 0:5=0:6+=0:7 +0:7=0:8 (3) The composition ofthe type relations u is close to v and v is bigger than w can be found by using (5), as follows: close ] bigger ] (u i;w j )=[ close ] (u i;v ) bigger ] (v ;w j )] [ close ] (u i;v ) bigger ] (v ;w j )] [ close ] (u i;v 3 ) bigger ] (v 3;w j )] (4) where i =; and j =;. Using (4), (), (3), () and (), we have (using min tnorm and max tconorm) u close ] bigger ] (u; w) = u w w 0:7=0:3+=0:4 0:3=0:8+=0:9 +0:7= 0:5=0+=0: +0:=0: 0:5=0:6+=0:7 +0:7=0:8 : (5) Comparing (5) and (), we observe that the results in (5) are indeed quite similar to the results of the type supstar composition in (), i.e., in the type results, primary memberships corresponding to the memberships in the type results, have unity secondary memberships. Eq. (5) provides additional uncertainty information to account for the secondary MFs.
14 340 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Next, consider the case where one ofthe relations involved in the composition is just a fuzzy set. The composition ofthe type set R U and type fuzzy relation S(U; V ) is given as [6] R S (v) = sup [ R (u)? S (u; v)]; (6) u U which is a type fuzzy set on V. Similarly, in the type case, the composition ofa type fuzzy set in R U and a type relation S(U; V ) is given by R S (v) = u U [ R (u) S (u; v)]: Eq. (7) plays an important role as the inference mechanism of a rule where antecedents or consequents are type fuzzy sets, and is the fundamental inference mechanism for rules in type FLSs (e.g., [0,,7]). 5.. Cartesian product If U and V are domains oftype fuzzy sets F and G, respectively, characterized by certain membership functions, then the Cartesian product of these membership functions will be supported over U V, and each point {(u; v); u U; v V } in this plane will have a membership grade (a crisp number in [0; ]) which is the result ofa tnorm operation between the membership grade ofu in F and the membership grade of v in G. Now consider the case oftype sets F and G. The membership grades ofevery element in F and G are now type fuzzy sets. So, now the tnorm operation in the type case has to be replaced by the meet operation; therefore, when we nd the Cartesian product, it is still supported over U V as in the type case; but now each point (u; v) U V will have a membership grade which is again a type fuzzy set (result ofa meet operation between membership grades of u and v). (7) 6. Conclusions We have discussed settheoretic operations for type sets, properties ofmembership grades oftype sets, and type relations and their compositions, and cartesian product under minimum and product tnorms. We have also developed easily implementable algorithms to perform settheoretic and algebraic operations oftype sets. When actual results are dicult to generalize or implement, we have developed practical approximations. All our results reduce to the correct type results when all the type uncertainties collapse to type uncertainties. All ofthe results in this paper are needed to implement type FLSs [,7], and should be ofuse to other researchers who explore applications of type fuzzy sets. Software to implement some of the algorithms developed herein are online at the URL: mendel=software. Appendix A. Proof of Theorem In the proofoftheorem, given next, we represent fuzzy sets F i as F i = v f i(v)=v (i =;:::;n), where v, in general, varies over the real line. Ifa real number w 0 is not in F i, f i (w 0 ) is zero. Proof. First we prove (5), and then we prove (6). (I) We rst prove (5) for the case n =. The join operation between F and F can be expressed, as F F = [f (v) f (w)]=(v w): (A.) v w
15 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Let us see what operations are involved here. For every pair ofpoints {v; w}, such that v F and w F, we nd the maximum of v and w and the minimum oftheir memberships, so that v w is an element of F F and f (v) f (w) is the corresponding membership grade. Ifmore than one {v; w} pair gives the same maximum (i.e., the same element in F F ), we use the maximum ofall the corresponding membership grades as the membership ofthis element. So, every element ofthe resulting set is obtained as a result of the max operation on one or more {v; w} pairs, and its membership is the maximum ofall the results ofthe min operation on memberships of v and w. If F F, the possible {v; w} pairs that can give as the result ofthe maximum operation are {v; } where v ( ;] and {; w} where w ( ;]. The process ofnding the membership of in F F can be broken into three steps: () nd the minima between the memberships ofall the pairs {v; } such that v ( ;] and then nd their supremum; () do the same with all the pairs {; w} such that w ( ;]; and, (3) nd the maximum ofthe two suprema, i.e., (F F )() = () () (A.) where and () = sup {f (v) f ()} = f () sup f (v) v ( ;] v ( ;] (A.3) () = sup {f () f (w)} = f () sup f (w): (A.4) w ( ;] w ( ;] We break into the following three ranges: v, v 6 v and v (see Fig. 6). Recall that f (v )= and f (v ) =. Also, observe that convexity of F i (i =; ) is equivalent to the condition that f i is monotonic nondecreasing in ( ;v i ] and monotonic nonincreasing in [v i ; ). = v : See Fig. 6(a). Since f and f both are monotonic nondecreasing in ( ;v ], sup v ( ;] f (v)=f () and sup w ( ;] f (w)=f (); therefore, from (A.) (A.4), we have (F F )() =f () f (); v : (A.5) v 6 = v : See Fig. 6(a). Recall that f (v )= and that f is monotonic nondecreasing in ( ;v ]. Therefore, sup v ( ;] f (v) = and sup w ( ;] f (w)=f (). Using these facts in (A.) (A.4), we have (F F )() =f () [f () f ()] = f (); v 6 v (A.6) where we have made use ofthe fact that a [a b]=a, for real a and b. = 3 v : For in this range [see Fig. 6(a)], both f and f have already attained their maximum values; therefore, sup v ( ;] f (v) = sup w ( ;] f (w) =. Consequently, from (A.) (A.4), we have (F F )() =f () f (); v : (A.7) From (A.5) (A.7), we get f () f (); v ; (F F )() = f (); v 6 v ; f () f (); v : (A.8) Observe that, since F and F both are convex and normal with f (v )=f (v )=: () f f is monotonic nondecreasing in ( ;v ]; () f (v ) f (v )=f (v ); (3) f is monotonic nondecreasing in [v ;v ]; and, (4) f f is monotonic nonincreasing in [v ; ). From these three observations, we see that (F F ) is
16 34 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig. 6. An example oftwo general membership functions, f and f, that satisfy the requirements of Theorem. Observe that for set F, any ofthe points at which f attains its maximum value ofunity may be chosen as v. We arbitrarily chose v =:8: (a) The three possibilities: v, v 6 v, 3 v. (b) Result ofthe join operation. (c) Result ofthe meet operation. The tnorm used is min. monotonic nondecreasing in ( ;v ] and monotonic nonincreasing in [v ; ), which implies that F F is convex [see Fig. 6(b)]. Also, F F is normal with (F F )(v )=. Next, we prove (5) for n. Using the associative law [3], we have F F F 3 =(F F ) F 3. Since, as explained in the previous paragraph, F F is also convex and normal, from (A.8) we have [recall that f 3 (v 3 ) = and that v 3 v ] (F F )() f 3 (); v ; (F F F 3)() = f 3 (); v 6 v 3 ; (A.9) (F F )() f 3 (); v 3 : Since v 6v 3, (A.8) and (A.9) can be rewritten as f () f (); v ; f (F F )() = (); v 6 v ; f () f (); v 6 v 3 ; f () f (); v 3 ; (F F )() f 3 (); v ; (F F )() f 3 (); v 6 v ; (F F F 3)() = f 3 (); v 6 v 3 ; (F F )() f 3 (); v 3 : (A.0) (A.)
17 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Substituting for (F F )() from (A.0) into (A.), we have f () f () f 3 (); v ; f (F F F 3)() = () f 3 (); v 6 v ; f 3 (); v 6 v 3 ; f () f () f 3 (); v 3 ; (A.) 3 i= f i(); v ; 3 = i=k+ f i(); v k 6 v k+ ; 6k6; 3 f i (); v 3 : i= (A.3) It is straightforward to show that F F F 3 is also a convex and normal set. Therefore, (A.8) can be applied again to obtain (F F F 3 F 4). Continuing in this fashion, we get (5). (II) For the case, n =, the meet operation between F and F can be represented as F F = v [f (v) f (w)]=(v w): (A.4) w The only dierence between this operation and the join operation in (A.) is that every F F is the result ofthe minimum operation on some pair {v; w}, such that v F and w F ; the possible such pairs being {v; } where v [; ) and {; w} where w [; ). The process ofnding the membership of in F F can be broken into three steps: () nd the minima between the memberships ofall the pairs {v; } such that v [; ) and then nd their supremum; () do the same with all the pairs {; w} such that w [; ); and, (3) nd the maximum ofthe two suprema. By proceeding in a manner very similar to that in part (I) ofthe proof, we get the result in (6). For details, see []. Appendix B. Solving for the Gaussian meet approximation Consider two Gaussian type sets F and F characterized by membership functions f and f, having means m and m and standard deviations and. Since we will perform the join and meet operations on the membership grades oftype sets when computing unions and intersections oftype sets, the type sets involved in the operations will have [0; ] as their domains. The membership functions of these type sets may, therefore, be clipped (i.e., any portion of the membership function lying outside [0; ] is cut o); and, this clipping may aect the result ofthe meet operation. This constraint needs to be considered in the derivation ofthe actual result ofthe meet between two Gaussian type sets under the product tnorm; however, we will ignore it for simplicity, while deriving the Gaussian approximation. For a detailed discussion about the eects ofthis constraint on the actual result, as well as on the Gaussian approximation and its upper and lower bounds, see []. Applying (9) for Gaussian membership functions, we have F F () = sup v R ;v 0 { [ (v exp ) ( ) ]} m =v m + ; R ; 0; (B.)
18 344 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) { F F (0) = exp ( m ) } { exp ( ) } m : (B.) Finding F F () for 0, requires minimization of ( ) ( ) ( ) ( ) v m =v m v m m v J (v) = + = + : (B.3) v Observe that J is nonconvex and dicult to minimize. Therefore, we simplify the problem by replacing the v in the denominator ofthe second term ofj (v) by a constant k. This gives us the following objective function to minimize: ( ) ( ) v m m v H(v) = + : (B.4) k Observe that H is convex (H == +m = 0). Therefore, equating H to zero and assuming that the inmum is obtained at v = v,weget ( v )( m ) + ( m v )( ) m k Substituting (B.5) into (B.4), we get inf v H(v) = ( k =0 v = m + m k m + k : (B.5) m m m + k ) : (B.6) Recall that the type sets involved in the meet operation have [0; ] as their domains. Since the constant k in (B.4) replaces v that varies between 0 and, it seems apparent that some value of k between 0 and will give a good approximation to the actual result ofthe meet operation. Observe, from (B.5), that replacing k by m, makes the Gaussian approximation commutative [i.e., ifwe interchange {m ; } and {m ; },we still get the same result] just as the actual meet operation is (see Section 4). Therefore, by replacing k with m, we obtain the Gaussian approximation. ( F F () exp m m m + m ) : (B.7) Generalization ofthis result to the meet of n Gaussian type sets is straightforward and gives the result in (0). For a detailed derivation ofbounds on the approximation error between (0) and the actual value of F F F n (), see []. Appendix C. Proof of Theorem 4 Consider F i =[m i s i ;m i + s i ]. Multiplying F i by a crisp constant i (== i ) yields (see ()) i F i = =( i v); v [m i s i ;m i + s i ]: (C.) v
19 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Adding a crisp constant (==) to i F i yields i F i + = =( i v + ); v [m i s i ;m i + s i ]: (C.) v Substituting w = i v +, (C.) gives us i F i + = =w; w [ i m i + i s i ; i m i + + i s i ]: (C.3) w Since F i can be represented as [l i ;r i ], where l i = m i s i and r i = m i + s i, then [ n n ] n n n F i = m i s i ; m i + s i : (C.4) i= i= i= i= i= Using (C.3) and (C.4), we get the result in Theorem 4. Appendix D. Proof of Theorem 5 We follow a method similar to the one used in the proofoftheorem 4 in Appendix C. We prove the theorem in two parts: (a) we prove that i F i + is a Gaussian fuzzy number with mean i m i + and standard deviation i i ; and (b) we prove that n i= F i is a Gaussian fuzzy number with mean n i= m i and standard deviation, where n i= = i ifproduct tnorm is used; (D.) n i= i ifminimum tnorm is used: (a) Consider { F i = exp ( ) }/ v mi v: (D.) v i Multiplying F i by a constant i (== i ) and adding a constant (==) to i F i, yields [see ()] [ { i F i + = exp ( ) } ]/ v mi? ( i v + ) v i { = exp ( ) }/ v mi ( i v + ): (D.3) v i Let i v + = v ; this gives v =(v )= i, which when substituted into (D.3), leads to { i F i + = exp [ (v ] }/ )= i m i v v i { = exp [ v ] }/ ( i m i + ) v i i v (D.4)
20 346 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) which shows that i F i + is a Gaussian fuzzy number with mean i m i + and standard deviation i i. Note that this result does not depend on the kind oftnorm used, since i and are crisp numbers. (b) Consider F and F, with means m and m and standard deviations and, respectively. The sum ofthese two fuzzy numbers can be expressed as [see ()] { F + F = ( ) } { v m? exp ( ) }/ w m [v + w] (D.5) v F w F exp where? indicates the chosen tnorm. (i) Product tnorm: In this case, (D.5) reduces to { F + F = ( ) } { v m exp ( ) }/ w m [v + w]: (D.6) v F w F exp If is an element of F + F, the membership grade of in F + F can be obtained by considering all the {v; w} pairs such that v F and w F and v + w =, multiplying the memberships of v and w in every pair, and, choosing the maximum ofall these membership products. In other words, { [ (v (F+F )() = sup exp ) ( ) ]} m ( v) m + : (D.7) v Let v maximize the expression on the RHS of(d.7). v is obtained by minimizing ( ) [ ] v m ( v) m J (v) = + : (D.8) Since J is convex (J == += 0), equating the rst derivative of J to zero, we get ( v )( ) [ m ( v ) ) m + ]( =0; ( v + ) = m + m ; v = m +( m ) + : Substituting (D.9) into (D.7), we get [ ] (F+F )() = exp (m + m ) + : (D.9) (D.0) This result generalizes easily to the sum of n Gaussian type sets, F ;:::;F n ; hence, n i= F i, under product tnorm, is a Gaussian fuzzy number with mean n i= m n i and standard deviation i= i. (ii) Minimum tnorm: In this case, (D.5) reduces to { F + F = ( ) } { v m exp ( ) }/ w m [v + w]: (D.) v F w F exp
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