Centroid of an Interval Type-2 Fuzzy Set Re-Formulation of the Problem

Transcription

1 Applied Mathematical Sciences, Vol. 6, 2012, no. 122, Centroid of an Interval Type-2 Fuzzy Set Re-Formulation of the Problem Omar Salazar Morales, Jose Jairo Soriano Mendez and Jose Humberto Serrano Devia Universidad Distrital Francisco José de Caldas, Bogotá, Colombia Abstract In theory of interval type-2 (IT2 fuzzy sets, type-reduction is one of the most difficult parts. Karnik-Mendel (KM algorithms have been the most popular method for computing type-reduction of IT2 fuzzy sets. These algorithms present two independent procedures for computing the generalized centroid of an IT2 fuzzy set: the first one for computing its left part (denoted by c l, and the second one for computing its right part (denoted by c r. We present a discussion where we show that the calculation of c l and c r is the same problem and not two different problems, and they can be calculated with a general expression. Mathematics Subject Classification: 03E72, 03B52, 94D05 Keywords: Interval type-2 fuzzy set, Centroid, Karnik-Mendel algorithms 1 Introduction The Karnik-Mendel (KM algorithm has been the method for computing typereduction of interval type-2 (IT2 fuzzy sets [3]. This algorithm has been studied theoretically and experimentally in order to improve its performance on applications. It gives an exact way to get the generalized centroid, which is a closed interval, of an IT2 fuzzy set. The convergence of the KM algorithm was proved by Mendel and Liu [7]. An enhanced version of this algorithm is known as Enhanced Karnik-Mendel (EKM algorithm [5]. EKM algorithm speeds up the search of the centroid by using an optimal initialisation and a better way to compute the involved arithmetic expressions. Both versions always present two parts (even in recent papers [5]: 1. the first one computing c l, which is the left part of the centroid and,

2 6082 O. Salazar, J. Soriano and H. Serrano 2. the second one computing c r, which is the right part of the centroid. Melgarejo et al. [1, 4] presented an alternative version of KM algorithm re-expressing the expressions for c l and c r. The alternative version proposed by Melgarejo, called RAUL (Recursive Algorithm with Unique Loop [1], uses inside of an unique loop re-expressed expressions for c l and c r, and it finds a minimum for c l and a maximum for c r. Such papers still present two parts for computing the centroid by using KM algorithms: one for c l and one for c r. Two different parts for computing the generalized centroid of an IT2 fuzzy set have direct implications on engineering applications, such as in [2] where c l and c r were calculated by hardware. The aim of this paper is to show that the calculation of c l and c r is really the same problem and not two different problems with the deduction of a general expression that involves the calculation of both. Expressions of c l and c r are not independent, one expression can be deduced from the other expression. This paper is organized as follows: Sect. 2 presents a short description of calculation of the generalized centroid of an IT2 fuzzy set. Sect. 3 presents the proof that the calculation of c l and c r is really the same problem. Sect. 4 gives a re-formulation of problem of the centroid. Sect. 5 gives some final comments. Finally, Sect. 6 gives the conclusion. 2 Centroid of an Interval Type-2 Fuzzy Set Given an IT2 fuzzy set à (for more details see [6] which is defined on an universal set X R, with membership function µã(x, x X, its generalized centroid c(ã is a closed interval, i.e., c(ã = [c l,c r ], where c l and c r are respectively the minimum and maximum of all centroids of the embedded type-1 fuzzy sets in the footprint of uncertainty (FOU of Ã. Karnik and Mendel [3] demonstrated that c l and c r can be computed from the lower and upper membership functions of à as follows: ( L L c l = x i µã(x i + x i µã(x i µã(x i + µã(x i (1 ( R c r = x i µã(x i + i=l+1 i=r+1 R x i µã(x i µã(x i + i=l+1 i=r+1 µã(x i where µã and µã are respectively the upper and lower membership functions of à (Fig. 1(a. L N is the switch point that marks the change from µ à to µã (Fig. 1(b, R N is the switch point that marks the change from µã to µã (Fig. 1(c and N N is the number of discrete points on which the x-domain of à has been discretized. It is true that in (1 and (2 x 1 < x 2 < < x N, in which x 1 denotes the smallest sampled value of x and x N denotes the largest sampled value of x [5]. (2

3 Centroid of an interval type-2 fuzzy set 6083 (a (b (c Figure 1: (a Interval Type-2 fuzzy set. (b c l and its interpretation. (c c r and its interpretation. 3 Deduction of a General Expression Let us rewrite the expression (2. If we let j = N +1 i then we will have: if 1 i R then 1 N +1 j R, and hence N R+1 j N; if R+1 i N then R+1 N+1 j N, and hence 1 j N R; therefore (2 can be written as (by properties of sums N R c r = y j µã(y j + y j µã(y j / N R µã(y j + µã(y j where N R = y j µã(y j + L = y j µã(y j + j=l +1 y j µã(y j / N R µã(y j + y j µã(y j / L µã(y j + j=l +1 µã(y j µã(y j (3 y j = x N+1 j, 1 j N, (4 and L = N R. Equations (1 and (3 have the same form. We can obtain one from the other only with the substitution of x i by y j and L by L (or vice versa. Equations (1 and (3 differ in L and L (switch points and that the values

4 6084 O. Salazar, J. Soriano and H. Serrano of x are indexed in reverse order as (4 establishes. Equation (4 means that y 1 = x N, y 2 = x N 1,..., y N = x 1, asweshowinfig.2. Itisjustapermutation (a bijective function of the N values of x. Equation (4 can be thought as an indexation of the N values of x in reverse order. Then, the problem for computingc l andc r canbereducedtocalculatethemwiththesameprocedure. It is just necessary reverse the order in which the values of x are indexed, and if we are computing c l then we will need to find a minimum, and if we are computing c r then we will need to find a maximum. x1 x2 xn j xn+1 j xn+2 j xn 1 xn y1 y2 yj 1 yj yj+1 yn 1 yn Figure 2: Permutation y j = x N+1 j (1 j N that inverts the order in which the values of x are indexed If we start form (1 by using a similar argument then we will obtain an analogous expression to (2, i.e., there will be an expression R c l = z j µã(z j + j=r +1 z j µã(z j / R µã(z j + j=r +1 µã(z j (5 which is analogous to (2, where z j = x N+1 j, 1 j N, and R = N L. 4 Definition of a General Expression We define a general expression 1 (6 for computing a centroid (c l or c r because of the duality between (1 and (3. It is just necessary to replace appropriate values in order to find c l or c r as we show in Table 1. ( M M c = w i µã(w i + w i µã(w i µã(w i + µã(w i (6 The substitution of M and w i in (6 by L and x i respectively gives the expression (1; and the substitution of M and w i in (6 by L (= N R and y i (= x N+1 i respectively gives the expression (3 (which is the same equation (2 as we showed above. 1 This problem can also be re-formulated with the definition of a general expression by using the duality between (2 and (5

5 Centroid of an interval type-2 fuzzy set 6085 Table 1: Summary for computing a centroid (c l or c r by using (6 c M w i Observation (1 i N c l L x i If we are finding c l, we will have to find L such that (6 is minimum by using x i c r L y i If we are finding c r, we will have to find L (= N R such that (6 is maximum by using y i (= x N+1 i 5 Relationship with the Concept of Convex Combination Zadeh [8, page 345] defined the concept of convex combination of three arbitrary (type-1 fuzzy sets A, B and Λ, with membership functions µ A, µ B and µ Λ, by the relation A,B;Λ = ΛA+Λ B, where Λ is the complement of Λ. Written out in terms of membership functions µ A,B;Λ (w = µ Λ (wµ A (w+(1 µ Λ (wµ B (w, w X. (7 If we apply this concept to an IT2 fuzzy set à (with lower and upper type-1 fuzzy sets A and A whose membership functions are µã and µã: µ A (w i = µã(w i, µ B (w i = µã(w i, and µ Λ (w i = { 1, if i M, 0, if i M +1, with i = 1,...,N, then (7 reduces to µ A,A;Λ (w i = µ Λ (w i µã(w i +(1 µ Λ (w i µã(w i = { µã(w i, if i M, µã(w i, if i M +1. (8 With the aid of (8, the numerator of (6 can be written as M w i µ A,A;Λ (w i = w i µ A,A;Λ (w i + = M w i µã(w i + w i µã(w i, w i µ A,A;Λ (w i and similarly for its denominator. Hence (6 can be written as ( N N c = w i µ A,A;Λ (w i µ A,A;Λ (w i. Melgarejo et al. [1, 4] deduced and used analogous expressions in his work of the RAUL algorithm, but he did not cite the concept of convex combination.

6 6086 O. Salazar, J. Soriano and H. Serrano 6 Conclusion This paper showed that expressions (1 and (2, which were given by Karnik and Mendel in order to calculate c l and c r, have the same form with a simple substitution of its index variable. Then, there is a duality between them. We presented a general expression (6 for computing c l and c r. It is just necessary to replace appropriate values in order to find c l or c r as we showed in Table 1. We also deduced a general expression based on the concept of convex combination given by Zadeh. The calculation of c l or c r can be done with a membership function that is the convex combination of three type- 1 fuzzy sets: µã (lower membership function of Ã, µã (upper membership function of à and µ Λ (a crisp set. References [1] Hector Bernal, Karina Duran, and Miguel Melgarejo. A comparative study between two algorithms for computing the generalized centroid of an interval type-2 fuzzy set. In Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ 2008, pages , [2] Linda K. Durán and Miguel A. Melgarejo. Implementación hardware del algoritmo Karnik-Mendel mejorado basada en operadores CORDIC. Ingeniería y Competitividad, 11(2:21 39, [3] Nilesh N. Karnik and Jerry M. Mendel. Centroid of a type-2 fuzzy set. Information Sciences, 132: , [4] Miguel Melgarejo. A fast recursive method to compute the generalized centroid of an interval type-2 fuzzy set. In Annual Meeting of the North American Fuzzy Information Processing Society NAFIPS 2007, pages , San Diego, California, USA, June [5] Jerry M. Mendel. On centroid calculations for type-2 fuzzy sets. Appl. Comput. Math., 10(1:88 96, [6] Jerry M. Mendel and Robert I. Bob John. Type-2 fuzzy sets made simple. IEEE Transactions on Fuzzy Systems, 10(2: , [7] Jerry M. Mendel and Feilong Liu. Super-exponential convergence of the Karnik-Mendel algorithms for computing the centroid of an interval type-2 fuzzy set. IEEE Transactions on Fuzzy Systems, 15(2: , April [8] Lofti A. Zadeh. Fuzzy sets. Information and Control, 8(3: , Received: June, 2012