Approximating models based on fuzzy transforms

Transcription

1 Approximating models based on fuzzy transforms Irina Perfilieva University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, Ostrava 1, Czech Republic Abstract The technique of the direct and inverse fuzzy (F-) transforms of three different types is introduced and approximating properties of the inverse F- transforms are described. A number of theorems establishing best approximation properties have been proved. 1 Introduction In classical mathematics, various kinds of transforms (Fourier, Laplace, integral, wavelet) are used as powerful methods for construction of approximation models and for solution of differential or integral-differential equations. The main idea of them consists in transforming the original model into a special space where the computation is simpler. The transform back to the original space produces an approximate model or an approximate solution. In this contribution, we put a bridge between these well known methods and methods for construction of fuzzy approximation models. We will develop a general method called fuzzy transform (or, shortly, F-transform) that encompasses both classical transforms as well as approximation methods based on elaboration of fuzzy IF-THEN rules studied in fuzzy modelling. In this paper, we will construct approximation models on the basis of three different fuzzy transforms and prove the best approximation properties in the corresponding approximation spaces. The paper has been supported by the Grant IAA of the GA AV ČR. 2 Fuzzy Partition of the Universe The core idea of the technique proposed in this paper is a fuzzy partition of the universe. We claim that for a sufficient representation of a function we may consider its average values within some intervals (or factors, or clusters). The latter may be regarded as similarity classes of some chosen nodes. Then, arbitrary function can be considered as a mapping from thus obtained set of factors to the set of function values. We take an interval [a, b] as a universe. That is, all (real-valued) functions considered in this chapter have this interval as a common domain. The fuzzy partition of the universe is given by fuzzy subsets of the universe [a, b] (determined by their membership functions) which must have properties described in the following definition. Definition 1 Let x 1 <... < x n be fixed nodes within [a, b], such that x 1 = a, x n = b and n 2. We say that fuzzy sets A 1,..., A n, identified with their membership functions A 1 (x),..., A n (x) defined on [a, b], form a fuzzy partition of [a, b] if they fulfil the following conditions for k = 1,..., n: 1. A k : [a, b] [0, 1], A k (x k ) = 1; 2. A k (x) = 0 if x (x k 1, x k+1 ) where for the uniformity of denotation, we put x 0 = a and x n+1 = b; 3. A k (x) is continuous; 4. A k (x), k = 2,..., n, monotonically increases on [x k 1, x k ] and A k (x), k = 1,..., n 1, monotonically decreases on [x k, x k+1 ]; 645

2 5. for all x [a, b] A k (x) = 1. (1) The membership functions A 1,..., A n are called basic functions. Let us remark that basic functions are specified by a set of nodes x 1 <... < x n and the properties 1 5. The shape of basic functions is not predetermined and therefore, it can be chosen additionally. 3 Fuzzy transform In this section, we introduce the fuzzy transform (shortly F-transform) which establishes a correspondence between the set of integrable functions on [a, b] and the set of n-dimensional vectors. The formula to which we will refer as an inverse F-transform (inversion formula) converts an n-dimensional vector into a specially represented continuous function which approximates the original one. The advantage of the F-transform is that the inversion formula produces a simple and unique approximate representation of the original function. Thus, in complex computations we can use the inversion formula instead of a representation of the original function. 3.1 Direct fuzzy transform Suppose that the universe is an interval [a, b] and A 1,..., A n are basic functions which form a fuzzy partition of [a, b]. Let L 2 ([a, b], A 1,..., A n ) be the set of functions which are weighted square integrable on the interval [a, b] with the weights given by A k, k = 1,..., n. Lemma 1 The space of functions L 2 ([a, b], A 1,..., A n ) is a normed space with respect to each of the following norms: xk+1 f k = f 2 (x)a k (x)dx, k = 1,..., n. x k 1 (2) Let us remark that the norm f k turns L 2 ([a, b], A 1,..., A n ) into the metric space with the following metric: d k (f, g) = f g k. The following definition (see also [3, 4]) introduces the fuzzy transform of a function f L 2 ([a, b], A 1,..., A n ). Definition 2 Let A 1,..., A n be basic functions which form a fuzzy partition of [a, b] and f(x) be any function from L 2 ([a, b], A 1,..., A n ). We say that the n- tuple of real numbers [F 1,..., F n ] given by b a F k = f(x)a k(x)dx b a A, k = 1,..., n, (3) k(x)dx is the (integral) F-transform of f with respect to A 1,..., A n. Let us remark that this definition is correct because for each k = 1,... n, the product fa k is an integrable function on [a, b]. Denote the F-transform of a function f with respect to A 1,..., A n by F n [f]. Then according to Definitions 2, we can write F n [f] = [F 1,..., F n ]. (4) The elements F 1,..., F n are called components of the F-transform. It is easy to see that if the fuzzy partition of [a, b] (and therefore, basic functions) is fixed then the F-transform establishes a linear mapping from L 2 ([a, b], A 1,..., A n ) to R n so that F n [αf + βg] = αf n [f] + βf n [g] for α, β R and functions f, g L 2 ([a, b], A 1,..., A n ). This linear mapping is denoted by F n where n is the dimension of the image space. We will investigate the following problem: how well is the original function f represented by its F-transform? First of all, we will show that the F-transform components are the weighted mean values of the given function where the weights are given by basic functions. 646

3 Theorem 1 Let f(x) be a continuous function on [a, b] and A 1,..., A n be basic functions which form a fuzzy partition of [a, b]. Then the k-th component of the integral F-transform gives minimum to the function Φ(y) = b with the domain [f(a), f(b)]. a (f(x) y) 2 A k (x)dx (5) In terms of distances introduced above Theorem1 can be reformulated as follows: Corollary 1 Let f be a continuous function on [a, b]. Then components F 1,..., F n of the F-transform minimize the following sum of squared distances d 2 k (f, c k) (6) with respect to parameters c 1,..., c n. 3.2 Inverse F-transform A reasonable question is the following: can we reconstruct the function by its F-transform? The answer is clear: not precisely in general, because we are losing information when passing to the F- transform. However, the function that can be reconstructed (by the inversion formula) approximates the original one in such a way that a universal convergence can be established. Moreover, the inverse F-transform fulfils the best approximation criterion which can be called the piecewise integral least square criterion. Definition 3 Let A 1 (x),..., A n (x) be basic functions which form a fuzzy partition of [a, b] and f(x) be a function from L 2 ([a, b], A 1,..., A n ). Let F n [f] = [F 1,..., F n ] be the integral F-transform of f with respect to A 1,..., A n. Then the function f F,n (x) = F k A k (x) (7) is called the inverse F-transform. The theorem below shows that the inverse F- transform f F,n can approximate the original continuous function f with an arbitrary precision. Theorem 2 Let f(x) be a continuous function on [a, b]. Then for any ε > 0 there exist n ε and a fuzzy partition A 1,..., A nε of [a, b] such that for all x [a, b] f(x) f F,nε (x) ε (8) where f F,nε is the inverse F-transform of f with respect to the fuzzy partition A 1,..., A nε. 4 Discrete F-Transform Let us specially consider the discrete case, when an original function f(x) is known (may be computed) only at some nodes p 1,..., p l [a, b]. Then the (discrete) F-transform of f is introduced as follows. Definition 4 Let a function f(x) be given at nodes p 1,..., p l [a, b] and A 1,..., A n, n l, be basic functions which form a fuzzy partition of [a, b]. We say that the n-tuple of real numbers [F 1,..., F n ] is the discrete F-transform of f with respect to A 1,..., A n if l F k = f(p j)a k (p j ) l A. (9) k(p j ) Similarly to the integral F-transform, we may show that the components of the discrete F- transform are the weighted mean values of the given function where the weights are given by the basic functions. Lemma 2 Let function f(x) be given at nodes p 1,..., p l [a, b] and A 1,..., A n be basic functions which form a fuzzy partition of [a, b]. Then the k-th component of the discrete F-transform gives minimum of the function Φ(y) = l (f(p j ) y) 2 A k (p j ) (10) with the domain [f(a), f(b)]. 647

4 In the discrete case, we define the inverse F- transform only at nodes where the original function is given. Definition 5 Let function f(x) be given at nodes p 1,..., p l [a, b] and F n [f] = [F 1,..., F n ] be the discrete F- transform of f w.r.t. A 1,..., A n. Then the function f F,n (p j ) = F k A k (p j ), defined at the same nodes, is the inverse discrete F-transform. Analogously, the inverse discrete F-transform f F,n approximates the original function f at common nodes with an arbitrary precision. Theorem 3 Let function f(x) be given at nodes p 1,..., p l [a, b]. Then, for any ε > 0 there exist n ε and a fuzzy partition A 1,..., A nε of [a, b] such that for all p {p 1,..., p l } f(p) f F,nε (p) ε (11) where f F,nε is the inverse discrete F-transform of f with respect to the fuzzy partition A 1,..., A n of [a, b] and the chosen nodes. 5 F-Transforms Expressed By Residuated Lattice Operations Our purpose is to introduce two new fuzzy transforms with the help of operations of a residuated lattice on [0, 1]. These new transforms lead to new approximation models which are expressed using weaker operations than the arithmetic ones used above in the case of the F-transform. However, these operations are successfully used in modelling dependencies expressed by natural language words (e.g. fuzzy IF THEN rules) and also, in modelling continuous functions. Therefore, two new F-transforms which we are going to introduce in this section, extend and generalize the F-transform considered above. Formally, both new F-transforms taken as transformations, map a space of the original functions onto the space of n-dimensional vectors. Moreover, two inverse F-transforms from different image spaces give lower and upper approximations of the original function. Let us refer to [2] for the notion of residuated lattice which will be a basic algebra of operations in the sequel. 5.1 Direct F (F )-transform Let us fix some residuated lattice L on [0, 1]: L = [0, 1],,,,, 0, 1 and define the operations of negation and biresiduation a = a 0 a b = (a b) (b a). Suppose that the universe is the interval [0, 1], so that all functions in this section are defined on this interval and take values from it. We redefine here a fuzzy partition of [0, 1] assuming that it is given by fuzzy sets A 1,..., A n identified with their membership functions A 1 (x),..., A n (x) fulfilling the following covering property ( x)( i) A i (x) > 0. (12) As above, A 1,..., A n are called the basic functions. Definition 6 Let a function f(x) be defined at nodes p 1,..., p l [0, 1] and A 1,..., A n, n l, be basic functions which form a fuzzy partition of [a, b]. We say that the n-tuple of real numbers F n[f] = [F 1,..., F n] is a F -transform of f w.r.t. A 1,..., A n if F k = l (A k (p j ) f(p j )). (13) Moreover, we say that the n-tuple of real numbers F n[f] = [F 1,..., F n] is the F -transform of f w.r.t. A 1,..., A n if F k = l (A k (p j ) f(p j )). (14) 648

5 We can prove that the F (F )-transform being a mapping from the set of all functions on [0, 1] to R n has the property analogous to the property of linearity of the F-transform. Lemma 3 Let functions f, g be defined at nodes p 1,..., p l [0, 1] and A 1,..., A n be basic functions which form a fuzzy partition of [0, 1]. Then, for arbitrary α, β [0, 1], the following equalities holds: F n[α f β g] = α F n[f] β F n[g], F n[(α f) (β g)] = (α F n[f]) (β F n[g]). where on the right-hand sides, the operations and are taken componentwise. 6 Inverse F (F )-Transforms All F-transforms convert the space of functions into the space of n-dimensional real vectors. It is possible to define the inverse transform to be able to return to the original space of functions. In the construction of the inverse F (F )-transform we use the fact that the operations and are mutually adjoint in a residuated lattice. Definition 7 Let function f(x) be defined at nodes p 1,..., p l [a, b] and let F n[f] and F n[f] be the F and F - transforms of f w.r.t. basic functions A 1,..., A n. Then the function n f F,n (p j) = (A k (p j ) F k ) is called the inverse F -transform and the function n f F,n (p j) = (A k (p j ) F k ) is called the inverse F -transform. Both inverse F -transform and F -transform are called the inverse lattice transforms. Theorem 4 Let f F,n and f F,n be the inverse lattice transforms of f, computed with respect to basic functions A 1,..., A n. Then for all j = 1,..., l f F,n (p j) f(p j ) f F,n (p j). It is interesting that the inverse F -transform of the function f F,n is again f F,n and the inverse F - transform of f F,n is again f F,n. This easily follows from the theorem given below. Theorem 5 Let function f(x) be defined at nodes p 1,..., p l [a, b] and let F n[f] and F n[f] be the F and F - transforms of f w.r.t. basic functions A 1,..., A n. Then for all k = 1,..., n and F k = F k = l (A k (p j ) f F,n (p j)) l (A k (p j ) f F,n (p j)). 7 Approximation By Inverse F-Transforms As we saw in the previous section, the inverse F - transform f F,n of f can be taken as its upper approximation. On the other hand, the inverse F -transform f F,n of f can be taken as its lower approximation. We may be interested in the estimation of proximity between f and its two mentioned approximations. This can be done with some additional assumptions which, due to the limitation of a space, we will not explain in the details: 1. a residuated lattice on [0, 1] is a BL-algebra ([1]), 2. basic functions A 1,..., A n fulfil (12) and ( i)( j) ( x [0,1] (A i (x) A j (x)) x [0,1] (A i (x) A j (x))), 3. a function f : [0, 1] [0, 1], is extensional on the set {p 1,..., p l } with respect to the similarity E = n i=1 (A i(x) A i (y)), i.e. ( i)( j) (E(p i, p j ) f(p i ) f(p j )). 649

6 Theorem 6 Let the inverse lattice transforms f F,n and f F,n of an original function f be defined with respect to A 1,..., A n at the same nodes as f. Then under the above given assumptions n ( j) (f(p j ) f F,n (p j)) A 2 i (p j )) and ( j) (f(p j ) f F,n (p j)) i=1 n A 2 i (p j )). i=1 8 Comparison of Three F-Transforms This section is aiming to compare three different F-transforms which have been introduced in this paper. For this purpose we will consider [0, 1] as the common universe which enables us to use arithmetic operations as well as operations of a residuated lattice. Moreover, we will choose the Lukasiewicz MV-algebra L L = [0, 1],,,, L, 0, 1 as a basic residuated lattice. Let us consider a partition of [0, 1] in the sense of Definition 1 which, moreover, fulfils the covering property (12). Then, for a given function f : [0, 1] [0, 1], we may construct all three different discrete F-transforms and also, the inverse F-transforms. Theorem 7 Let a function f(x) be defined at nodes p 1,..., p l [0, 1] and A 1,..., A n be basic functions which form a normalized fuzzy partition of [0, 1], i.e. Then l A k (p j ) = 1, k = 1,..., n. Theorem 8 Let function f(x) be defined at nodes p 1,..., p l [0, 1]. Then for any ε > 0 there exist n ε and a fuzzy partition A 1,..., A nε of [0, 1] such that for all p j, j = 1,..., l, f F,n ε (p j ) ε f F,nε (p j ) f F,n ε (p j ) + ε. (15) 9 Conclusion We have introduced a new technique of direct and inverse fuzzy transforms (F-transforms for short) which enables us to construct various approximating models depending on the choice of basic functions. The best approximation property of the inverse F-transform is established within the respective approximating space. Two fuzzy transforms are built with the help of operations of a residuated lattice on [0, 1]. They lead to new approximation models which are expressed with the help of weaker operations than the arithmetic ones used in the case of the first fuzzy transform. References [1] P. Hájek, Metamathematics of fuzzy logic, Kluwer, Dordrecht, [2] U. Höhle, Commutative residuated l- monoids, in: Höhle U., E.P. Klement (Eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer, Dordrecht, 1995, [3] I. Perfilieva, Fuzzy Transform: Application to Reef Growth Problem, in: R. B. Demicco, G. J. Klir (Eds.), Fuzzy Logic in Geology, Academic Press, Amsterdam, 2003, [4] I. Perfilieva, Fuzzy Transforms, Fuzzy Sets and Systems, submitted, F k F k F k, k = 1,..., n. The theorem given below shows that the inverse F-transform lies in between the two other inverse F-transforms, based on residuated lattice operations, which are proved to be a lower and an upper bound of the original function. 650