Neural Networks In Determination Of Defuzzification Functionals

Ordered fuzzy numbers as generalization of convex fuzzy numbers are defined together with four algebraic operations. For defuzzification operators, that play the main role when dealing with fuzzy controllers and fuzzy inference systems, new representation formulae are given. Step ordered fuzzy numbers are considered. Approximation method based on forward neural networks is presented for determining defuzzification functionals when training sets of data are given. Results of approximation are given. FUZZY NUMBERS Fuzzy numbers (Zadeh, 1965) are very special fuzzy sets defined on the universe of all real numbers R. In applications the so-called (L,R)–numbers proposed by Dubois and Prade (Dubois & Prade, 1978) as a restricted class of membership functions, are often in use. In most cases one assumes that membership function of a fuzzy number A satisfies convexity assumptions (Nguyen, 1978). However, even in the case of convex fuzzy numbers (CFN) multiply operations are leading to the large grow of the fuzziness, and depend on the order of operations. This as well as other drawbacks have forced us to think about some generalization. Number of attempts to introduce non-standard operations on fuzzy numbers has been made (Drewniak, 2001; Klir, 1997; Sanschez, 1984; Wagenknecht, 2001). Our main observation made in (Kosiński et.al., 2002a) was: a kind of quasi-invertibility (or quasi-convexity (Martos, 1975)) of membership functions is crucial. Invertibility of membership functions of convex fuzzy number A makes it possible to define two functions a1, a2 on [0, 1] that give lower and upper bounds of each α-cut of the membership function μA of the number A A[α] = {x : μA(x) ≥ α} = [a1(α), a2(α)] with a1(α) = μA| incr(α) and a2(α) = μA| −1 decr(α) , where |incr and |decr denote the restrictions of the function μA to its sub-domains on which is increasing or decreasing, respectively. Both functions a1(α), a2(α) were used for the first time by the authors of (Goetschel & Voxman, 1986) in their parametric representation of fuzzy numbers, they also introduced a linear structure to convex fuzzy numbers. ORDERED FUZZY NUMBERS In the series of papers (Kosiński et.al., 2002a; Kosiński et. al., 2003b,a) we have introduced and then developed main concepts of the space of ordered fuzzy numbers (OFNs). In our approach the concept of membership functions has been weakened by requiring a mere membership relation . Definition 1. Pair (f, g) of continuous functions such that f, g : [0, 1]→R is called ordered fuzzy number A. Notice that f and g need not be inverse functions of some membership function. If, however, f is increasing and g – decreasing, both on the unit interval I , and f ≤ g, then one can attach to this pair a continuous function μ and regard it as a membership function a convex fuzzy number with an extra feature, namely the orientation of the number. This attachment can be done by the formula f−1 = μ|incr and g−1 = μ|decr. Notice that pairs (f, g) and (g, f) represent two different ordered fuzzy numbers, unless f = g . They differ by their orientations. Definition 2. Let A = (fA, gA), B = (fB , gB) and C = (fC , gC) are mathematical objects called ordered fuzzy numbers. The sum C = A+B, subtraction C = A−B, product C = A ·B, and division C = A÷B are defined by formula fC(y) = fA(y) ? fB(y) , gC(y) = gA(y) ? gB(y) (1) where ”?” works for ”+”, ”−”, ”·”, and ”÷”, respectively, and where A ÷ B is defined, if the functions |fB | and |gB | are bigger than zero. Scalar multiplication by real r ∈ R is defined as r · A = (rfA, rgA) . The subtraction of B is the same as the addition of the opposite of B, and consequently B − B = 0, where 0 ∈ R is the crisp zero. It means that subtraction is not compatible with the the extension Proceedings 25th European Conference on Modelling and Simulation ©ECMS Tadeusz Burczynski, Joanna Kolodziej Aleksander Byrski, Marco Carvalho (Editors) ISBN: 978-0-9564944-2-9 / ISBN: 978-0-9564944-3-6 (CD) principle, if we confine OFNs to CFN. However, the addition operation is compatible, if its components have the same orientations. Notice, however, that addition, as well as subtraction, of two OFNs that are represented by affine functions and possess classical membership functions may lead to result which may not possess its membership functions (in general f(1) needs not be less than g(1)). Relation of partial ordering in the spaceR of all OFN, can be introduced by defining the subset of positive ordered fuzzy numbers: a number A = (f, g) is not less than zero, and write A ≥ 0 if f ≥ 0, g ≥ 0 , andA ≥ B ifA−B ≥ 0 . (2) In this way the spaceR becomes a partially ordered ring. Neutral element of addition in R is a pair of constant function equal to crisp zero. Operations introduced in the space R of all ordered fuzzy numbers (OFN) make it an algebra, which can be equipped with a sup norm ||A|| = max(sup s∈I |fA(s)|, sup s∈I |gA(s)|) if A = (fA, gA) . In R any algebraic equationA+X = C forX , with arbitrarily given fuzzy numbers A and C, can be solved. Moreover, R becomes a Banach space, isomorphic to a Cartesian product of C(0, 1) the space of continuous functions on [0, 1]. It is also a Banach algebra with unity: the multiplication has a neutral element the pair of two constant functions equal to one, i.e. the crisp one. Some interpretations of the concepts of OFN have been given in (Kosiński et.al., 2009a). Fuzzy implications within OFN are presented in (Kosiński et. al., 2009b). STEP NUMBERS It is worthwhile to point out that the class of ordered fuzzy numbers (OFNs) represents the whole class of convex fuzzy numbers with continuous membership functions. To include all CFN with piecewise continuous membership functions more general class of functions f and g in Def.1 is needed. This has been already done by the first author who in (Kosiński, 2006) assumed they are functions of bounded variation. The new space is denoted by RBV . Then operations on elements of RBV are defined in the similar way, the norm, however, will change into the norm of the Cartesian product of the space of functions of bounded variations (BV). Then all convex fuzzy numbers are contained in this new space RBV of OFN. Notice that functions from BV (Łojasiewicz, 1973) are continuous except for a countable numbers of points. Important consequence of this generalization is the possibility of introducing the subspace of OFN composed of pairs of step functions. It will be done as follows. If we fix a natural number K and split [0, 1) into K − 1 subintervals [ai, ai+1) , i = 1, 2, ...,K, i.e. K−1 ⋃ i=1 [ai, ai+1) = [0, 1), where 0 = a1 < a2 < ... < aK = 1, we may define step function f of resolution K by putting value f(s) = ui ∈ R, for s ∈ [ai, ai+1), then each such function f can be identified with a Kdimensional vector f ∼ u = (u1, u2...uK) ∈ R , the K-th value uK corresponds to s = 1, i.e. f(1) = uK . Taking pair of such functions we have an ordered fuzzy number fromRBV . Now we introduce Definition 3. By step ordered fuzzy number A of resolution K we mean ordered pair (f, g) of functions such that f, g : [0, 1]→R are K-step function. We use RK for denotation the set of elements satisfying Def. 3. The set RK ⊂ RBV has been extensively elaborated by our students in (Gruszczyńska & Krejewska, 2008) and (Kościeński, 2010). We can identify RK with the Cartesian product of R × R since each Kstep function is represented by its K values. It is obvious that each element of the space RK may be regarded as approximation of elements fromRBV , by increasing the number K of steps we are getting the better approximation. The norm of RK is assumed to be the Euclidean one of R , then we have a inner-product structure for our disposal. DEFUZZIFICATION FUNCTIONALS In the course of defuzzification operation in CFN to a membership function a real, crisp number is attached. We know number of defuzzification procedures from the literature (Van Leekwijck & Kerre, 1999). Continuous, linear functionals on R give the class of defuzzification functionals . Each of them, say φ, has the representation by the sum of two Stieltjes integrals with respect to two functions h1, h2 of bounded variation,