APPROXIMATION BY FUZZY B-SPLINE SERIES

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1 J. Appl. Math. & Computing Vol. 20(2006), No. 1-2, pp Website: APPROXIMATION BY FUZZY B-SPLINE SERIES PETRU BLAGA AND BARNABÁS BEDE Abstract. We study properties concerning approximation of fuzzy-number-valued functions by fuzzy B-spline series. Error bounds in approximation by fuzzy B-spine series are obtained in terms of the modulus of continuity. Particularly simple error bounds are obtained for fuzzy splines of Schoenberg type. We compare fuzzy B-spline series with existing fuzzy concepts of splines. AMS Mathematics Subject Classification : 26E50, 03E72. Key words and phrases : Fuzzy number, fuzzy B-spline series, approximation of fuzzy-number-valued functions. 1. Introduction Fuzzy numbers can model linguistic expressions of the type around, early morning, small, big, etc. In [19], L. Zadeh proposed the problem of interpolating some fuzzy data. This problem was solved by R. Lowen and O. Kaleva in [15] and [11]. Other results on approximation and interpolation of fuzzy-number-valued functions can be found in [4], [8], [7] and [12]. Also, recently in [6] the authors give error estimate in polynomial and trigonometric approximation, and the convergence of fuzzy Lagrange interpolation polynomial is studied. In [11] the notion of interpolating fuzzy spline is introduced. In [1] and [3], complete and natural splines interpolating fuzzy data are considered. For approximation of fuzzy-number-valued functions these kinds of interpolating fuzzy splines are not very practical since their coefficients change sign at each knot, and it is difficult (or impossible) to prove approximation theorems (by lack of distributivity for operations with fuzzy numbers). Fuzzy B-spline series are introduced in [2] (there they are called fuzzy B-splines). Approximation properties of fuzzy B-spline series are not studied in [2], however their applications Received May 14, Revised August 8, Corresponding author. c 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 157

2 158 Petru Blaga and Barnabás Bede motivate an accurate study of them. Recently, in [13], having as starting point fuzzy interpolation, consistent fuzzy surfaces are constructed from fuzzy data. In this paper we study approximation by fuzzy B-spline series. Error bounds for the approximation of a continuous fuzzy-number-valued function by fuzzy B- spline series are obtained in terms of the modulus of continuity. Variation and uncertainty diminishing properties are also obtained. Particularly simple error bounds are obtained for approximation by fuzzy B-spline series of Schoenberg type. Some examples are also presented. After a preliminary section, we introduce in Section 3 the fuzzy B-spline series and fuzzy splines of Schoenberg-type which are spline extensions of Bernstein polynomials and we prove some properties such as continuity, variation and uncertainty diminishing property. Section 4 is concerned with the approximation of fuzzy-number-valued functions by fuzzy B-spline series. Jackson-type estimates and particularly simple error bounds are obtained by using the modulus of continuity. We also provide two numerical examples. Some conclusions and further research topics conclude the paper. 2. Preliminaries Let us denote by R F the class of fuzzy subsets of the real axis R (i.e., u : R [0, 1]), satisfying the following properties: (i) u R F, u is normal i.e., x u R with u (x u )=1; (ii) u R F, u is convex fuzzy set (i.e., u (tx +(1 t) y) min {u (x),u(y)}, t [0, 1],x,y R); (iii) u R F, u is upper semi-continuous on R; (iv) {x R : u (x) > 0} is compact, where A denotes the closure of A. Then R F is called the space of fuzzy real numbers (see e.g. [9]). Remark 1. Obviously R R F because any real number x 0 R can be described as the fuzzy number whose value is 1 for x = x 0 and 0 otherwise. For 0 < α 1 and u R F let [u] α = {x R; u (x) α} and [u] 0 = {x R; u (x) > 0}. Then it is well known that for each α [0, 1], [u] α =[u α, u α ] is a bounded closed interval (u α, u α denote the endpoints of the α level set). For u, v R F and λ R, we have the sum u v and the product λ u defined by [u v] α =[u] α +[v] α,[λ u] α = λ [u] α, α [0, 1], where [u] α +[v] α means the usual addition of two intervals (as subsets of R) and λ [u] α means the usual product between a scalar and a subset of R (see e.g. [9], [18]). A fuzzy number u R F is said to be positive if u 1 0, strict positive if u 1 > 0, negative if u 1 0 and strict negative if u 1 < 0. We say that u and v have the same sign if they are both positive or both negative. If u is positive (negative) then u =( 1) u is negative (positive).

3 Approximation by fuzzy B-spline series 159 A special class of fuzzy numbers is the class of triangular fuzzy numbers. Given a b c, a, b, c R, the triangular fuzzy number u =(a, b, c) determined by a, b, c is given such that u α = a+(b a)α and u α = c (c b)α, for all α [0, 1]. Then u 0 = a, u 1 = u 1 = b and u 0 = c. Define D : R F R F R + {0} by D (u, v) = sup α [0,1] max { u α v α, u α v α }. The following properties are known ([9]): D (u w, v w) =D (u, v), u, v, w R F D (k u, k v) = k D (u, v), u, v R F, k R; D (u v, w e) D (u, w) +D (v, e), u,v,w,e R F and (R F,D)isa complete metric space. Also, the following is known. Theorem 1. ([5]) (i) If we denote 0 =χ {0} then 0 R F is neutral element with respect to, i.e., u 0 = 0 u = u, for all u R F. (ii) With respect to 0, none of u R F \ R has opposite in R F (with respect to ). (iii) For any a, b R with a, b 0 or a, b 0, and any u R F, we have (a + b) u = a u b u. For general a, b R, the above property does not hold. (iv) For any λ R and any u, v R F, we have λ (u v) =λ u λ v. (v) For any λ, µ R and any u R F, we have λ (µ u) =(λ µ) u. (vi) If we denote u F = D ( u, 0 ), u R F, then F has the properties of an usual norm on R F, i.e., u F =0iff u = 0, λ u F = λ u F and u v F u F + v F, u F v F D (u, v). The uniform distance between fuzzy-number-valued functions is defined by D(f,g) = sup{d(f(x),g(x) x [a, b]} for f,g :[a, b] R F. For f :[a, b] R F, the function ω(f, ) :R + R given by ω [a,b] (f,δ) = sup{d(f(x),f(y)) x, y [a, b], x y δ} is called the modulus of continuity of the fuzzy-number-valued function f. The fuzzy splines as introduced by O. Kaleva in [11], are given below.

4 160 Petru Blaga and Barnabás Bede Definition 1. Let S l be the family of all splines of order l with the knots t i, i =0, 1,..., n. Then fs(x) = s i (x) u i, where s i S l is the crisp spline i=0 interpolating the data (x i,f j ),j= 0, 1,..., n, where f j = 1 if i = j and 0 otherwise, and u i R F are fuzzy constants. 3. Fuzzy B-spline series Firstly, let us recall the definitions of the crisp B-splines. Let t 0 t 1... t r be points in R, with t r t 0. The B-spline M is given by M(x) =M(x; t 0,..., t r )=r[t 0,..., t r ]( x) r 1 +, (1) where [t 0,..., t r ]f denotes the divided difference of f (see e.g. [14, p. 137]). The B-spline N is defined by N(x; t 0,..., t r )= 1 r (t r t 0 )M(x; t 0,..., t r ). (2) The fuzzy B-spline series are defined as follows (see [2]). Let A =[a, b] ora = R. Let T =(t i ) be a sequence of points in A called basic knots satisfying t i t i+1 and t i <t i+r, for any t i A, i =0,..., n if A =[a, b] and i Z if A = R. If A =[a, b] we need some auxiliary knots t r+1... t 0 = a and b = t n+1... t n+r. To a given sequence of knots corresponds a sequence of crisp B-splines N j (x) =N(x; t j,..., t j+r ), for j Λ, where Λ = Z if A = R and Λ = { r +1,..., n} if A =[a, b]. Definition 2. A fuzzy B-spline series on A (A = R or A =[a, b]) having knots in T =(t i ),i Λ is a function S : A R F, of the form where c j R F. S(x) = N j (x) c j, First of all we prove the continuity, the variation and uncertainty diminishing property of fuzzy B-spline series. Let us recall two useful properties of crisp B-splines (see e.g. [14]). N j (x) 0 for x [t j,t j+r ] and N j (x) = 0 for x/ [t j,t j+r ] (3) The following identity holds N j (x) =1. (4) Theorem 2. The fuzzy B-spline series S(x) given in Definition 2 is continuous as function of x, the knots t i and the coefficients c j.

5 Approximation by fuzzy B-spline series 161 Proof. Let S(x) = N j (x) c j and S (x ) = N j (x ) c j with c j F < + where the knots of S are (t i ) i Λ and those of S are (t i ) i Λ. Then by the properties of the metric D, we have: ( D(S(x),S (x )) = D N j (x) c j, ) N j(x ) c j D(N j (x) c j,n j(x ) c j) D(N j (x) c j,n j (x) c j ) + D(N j (x) c j,n j (x ) c j ). By Theorem 1, (v) we have: D(S(x),S (x )) N j (x) D(c j,c j)+ i Λ i Λ By (3) and (4) we obtain D(S(x),S (x )) sup N j (x) sup D(c j,c j ) + sup D(c j,c j) + sup Nj (x) N j(x ) c j F Nj (x) N j (x ) Nj (x) N j(x ) c j F, c j F. inequality which combined with [14, Lemma 2.1, p.139] leads to the continuity of S. We say that a fuzzy-number-valued function changes sign in [x 0,x 1 ]iff(x 0 ) is negative (positive) and f(x 1 ) is positive (negative). Theorem 3. A fuzzy B-spline series S(x) = N j (x) c j has the variation diminishing property, i.e., S changes its sign at most as many times as the sequence (c j ) changes it s sign. Proof. We observe that since N j (x) 0 for any x A and j Λ, we have S(x) 1 = 1 N j (x)c j and S(x) 1 = N j (x)c j 1,

6 162 Petru Blaga and Barnabás Bede where c 1 j and c 1 j are the endpoints of the 1-level set of the fuzzy number c j R F. By variation diminishing property of crisp splines it is easy to see that S(x) 1 changes its sign at most as many times as the sequence c 1 j changes sign. Analogously, S(x) 1 changes its sign as many times as c 1 j does. Now the conclusion of the theorem is immediate if we observe that (c j ) changes its sign if and only if c 1 j and c 1 j change their signs. We denote by len(u) the length of the support of the fuzzy number u R F, i.e. len(u) =u 0 u 0. Theorem 4. A fuzzy B-spline series S(x) = N j (x) c j has the uncertainty diminishing property, i.e., the length of the support of S(x) does not exceed the maximum length of the support of the fuzzy numbers c j. (the length of the support of a fuzzy number can be interpreted as the uncertainty on it). Proof. As in the proof of the previous theorem we have S(x) 0 = N j (x)c j 0, S(x) 0 = 0 N j (x)c j and we get len(s(x)) = S(x) 0 S(x) 0 = N j (x) len(c j ) max len(c j). Remark 2. We observe that the splines defined by [11] cannot be written as fuzzy B-spline series. Indeed, by Curry-Schoenberg theorem (see e.g. [14]), the crisp B-splines are a basis for the Schoenberg space of all splines. Let s i be the splines in Definition 1. Then s i (x) = N j (x)d ij, with d ij R. Let fs be as in Definition 1. Then ( ) fs(x) = s i (x) u i = N j (x)d ij u i, i=0 i=0

7 Approximation by fuzzy B-spline series 163 where u i R F,i=0,..., n. By Theorem 1, (iii) the two sums cannot be interchanged, because the splines s i (and so also the coefficients d ij ) change their sign at each knot. Changing the order of the sums is possible if all d ij have the same sign for j Λ. The same remark is true for fuzzy splines defined in [1] and [3]. Remark 3. Let us observe that the fuzzy B-spline series S(x) = N j (x) c j can be easily computed by using fuzzy arithmetic (i.e., addition of fuzzy numbers and multiplication of a fuzzy number by a crisp real). Remark 4. The fuzzy B-spline series in Definition 2 can be extended easily to the bivariate case. Indeed, if we have a rectangular grid of knots we consider S(x, y) = N j,r1 (x)n k,r2 (y) c j.k, 1 k Λ 2 where N j,r1 (x) is the crisp B-spline with respect to variable x, N k,r2 (y) is the crisp B-spline with respect to y and c j.k are fuzzy constants. Then S(x, y) is called bivariate fuzzy B-spline series or fuzzy B-spline surface (see also [2], [13]). Since S(x, y) α = α N j,r1 (x)n k,r2 (y)c j,k k Λ 2 and 1 S(x, y) α = 1 k Λ 2 N j,r1 (x)n k,r2 (y)c j,k for any α [0, 1], a fuzzy B-spline surface is a consistent fuzzy surface in the sense of [13], i.e., (1) if α 1 α 2 S(x, y) α1 S(x, y) α2 S(x, y) α2 S(x, y) α1, α i.e., surfaces with larger α-cut values are contained in surfaces with lower α-cut values, and (2) S(x, y) α and S(x, y) α are crisp B-spline series, so they posses the underlying smoothness and continuity properties of the approximation method which is used. 4. Approximation by fuzzy B-spline series We approximate a continuous fuzzy-number-valued function f :[0, 1] R F by some fuzzy B-spline series S :[0, 1] R F. For this aim we consider the sequence of knots 0 <t 1... t n < 1, and auxiliary knots t r+1...

8 164 Petru Blaga and Barnabás Bede t 0 =0, 1=t n+1... t n+r. Let ξ j [0, 1] supp N j,j= r +1,..., n (where supp N j = {x [0, 1] : N j (x) 0}). Then we consider the fuzzy B-spline series S(f,x) = N j (x) f(ξ j ). (5) Let also δ = max 0 j n (t j+1 t j ). We observe that everything is also valid for f :[a, b] R F. Approximation properties of fuzzy B-spline series are given in the following theorem. Theorem 5. For f :[0, 1] R F continuous we have: D(f(x),S(f,x)) rω(f,δ), where ω(f,δ) is the modulus of continuity of the function f. Proof. By (3) and (4) we have: ( D(f(x),S(f,x)) = D f(x), ( n = D ) N j (x) f(ξ j ) N j (x) f(x), By the properties of the distance D we have: D(f(x),S(f,x)) N j (x)d(f(x),f(ξ j )). ) N j (x) f(ξ j ). It is well known that supp N j =(t j,t j+r ). For x supp N j by some well-known properties of the modulus of continuity (see e.g. [14]) we get D(f(x),S(f,x)) ω(f,t j+r t j ) ω(f,rδ). Finally we obtain D(f(x),S(f,x)) rω(f,δ). Better estimates can be obtained for fuzzy splines of Schoenberg type (for crisp Schoenberg splines see e.g. [17], [16]). Let the knots and auxiliary knots given as above, and ξ j = t j t j+r 1,j= r +1,..., n. We define the r 1 fuzzy spline of Schoenberg type S(f,x) = N j (x) f(ξ j ). If we have given fuzzy data (ξ j,f(ξ j )), j= r +1,..., n, ξ j being the nodes, then the sequence of knots t j,j= r +1,..., n considered for the fuzzy Schoenberg

9 Approximation by fuzzy B-spline series 165 splines is as in the crisp case a linear functional of the nodes (see [17], [16]). If there are no basic knots in the interior of the interval [0, 1] then the fuzzy Schoenberg spline reduces to the fuzzy Bernstein polynomial similar to the crisp case, so the results of this paper extend the results in [10] and [12]. As in [16] we obtain the error bound in approximation by fuzzy splines of Schoenberg type: Theorem 6. Concerning the error in approximation by fuzzy Schoenberg splines we have: D(f(x),S(f,x)) (1 + h(r, δ)) ω(f,δ), (6) where { } 1 r h(r, δ) = min, 2r 2 12 δ. (7) Proof. By (4) we have: ( n D(f(x),S(f,x)) = D N j (x) f(x), N j (x)ω(f, ξ j x ). ) N j (x) f(ξ j ) By the properties of the modulus of continuity we get: ( D(f(x),S(f,x)) N j (x)ω f, ξ j x δ ) δ ( N j (x) 1+ ξ ) j x ω(f,δ) δ ( = 1+ 1 ) N j (x) ξ j x ω(f,δ). δ By [16] we have: where 0 E(x) = Finally we obtain ( n N j (x) ξ j x ) 2 = E(x), ξ j N j (x) x 2 h 2 (r, δ), with { } 1 r h(r, δ) = min, 2r 2 12 δ. D(f(x),S(f,x)) (1 + h(r, δ)) ω(f,δ).

10 166 Petru Blaga and Barnabás Bede Particularly simple error bounds can be obtained for fuzzy splines of Schoenberg type with equally spaced knots and for dyadic fuzzy splines of Schoenberg type. Corollary 7. For fuzzy splines of Schoenberg type with equally spaced knots we have ( ( D(f(x),S(f,x)) 1+h r, 1 )) ( ω f, 1 ), n n where h(r, δ) as in (7). Corollary 8. For dyadic fuzzy splines of Schoenberg type (i.e., having the knots t j = j 2 n,j=1,..., 2 n 1) we have ( ( )) ( ) 1 1 D(f(x),S(f,x)) 1+h r, 2 n ω f, 2 n, where h(r, δ) as in (7). Remark 5. The above results can be easily extended to the bivariate case. Indeed, let us consider the bivariate fuzzy B-spline series as in Remark 4, S(f,x,y) = n 1 where ξ j = t j t j+r1 1 r 1 1 n 2 j= r 1+1 k= r 2+1 N j,r1 (x)n k,r2 (y) f(ξ j,η k ),, η k = s j s j+r2 1. Then r 2 1 D(S(f,x,y),f(x, y)) (1 + h(r 1,δ 1 ))(1 + h(r 2,δ 2 ))ω(f,δ 1,δ 2 ), where h(r, δ) is as in (7) and ω(f,δ 1,δ 2 ) is the modulus of continuity of a bivariate function f(x, y), defined by ω(f,δ 1,δ 2 ) = sup x 1 x 2 δ 1 y 1 y 2 δ 2 D(f(x 1,y 1 ),f(x 2,y 2 )). This result can be seen as an error estimate for the digital terrain modelling method described in [2]. In what follows we give two examples, and we compare our results with the examples in [1], [3] and [11]. In these examples we use triangular fuzzy numbers.

11 Approximation by fuzzy B-spline series 167 Figure 1 Example 9. We construct the cubic fuzzy B-spline series approximating the triangular fuzzy data ξ 2 =1,f(ξ 2 )=( 2, 0, 1), ξ 1 =1.1,f(ξ 1 )=(4, 5, 7), ξ 0 =1.2,f(ξ 0 )=(1, 1, 4), ξ 1 =3,f(ξ 1 )=(0, 4, 7), ξ 2 =3.5,f(ξ 2 )=( 3, 0, 2), ξ 3 =4,f(ξ 3 )=(0, 1, 2), with the knots t 2 =0.5, t 1 =1.05, t 0 =1.15, t 1 =1.25, t 2 =1.5, t 3 =2.75, t 4 =3.25, t 5 =3.75, t 6 =4.25. The endpoints of the 0 level set and the 1 level set of the fuzzy B-spline series can be seen in Figure 1. Example 10. Also, we construct the cubic fuzzy B-spline series approximating the triangular fuzzy data ξ 2 =1,f(ξ 2 )=( 1, 0, 1), ξ 1 =2,f(ξ 1 )=(3, 4, 5), ξ 0 =3,f(ξ 0 )=( 2, 1, 0), ξ 1 =4,f(ξ 1 )=(0, 1, 2), ξ 2 =5,f(ξ 2 )=(4, 5, 6), ξ 3 =6,f(ξ 3 )=( 1, 0, 1), with the knots t 2 =0,t 1 =0.5, t 0 =1.5, t 1 =2.5, t 2 =3.5, t 3 =4.5, t 4 =5.5, t 5 =6.5, t 6 = 7 (see Figure 2).

12 168 Petru Blaga and Barnabás Bede Figure 2 Compared with the results in [1], [3] and [11], we observe that the fuzzy B- spline series as approximations of fuzzy data have length of their support (i.e. uncertainty) smaller than the fuzzy splines given in [1], [3] and [11] (as it is proved in Theorem 4). 5. Conclusions and further research Approximation of continuous fuzzy-number-valued functions by fuzzy B-spline series have been studied and error estimates are obtained in terms of the modulus of continuity. Since fuzzy B-spline series are already studied from a practical point of view for digital terrain modelling (see [2], [13]) their study from a theoretical point of view is motivated. For further research we propose construction of spline-type functions approximating fuzzy input fuzzy output functions (i.e., functions f : R F R F ). Acknowledgement. The authors would like to thank the referees for their valuable suggestions. References 1. S. Abbasbandy, Interpolation of fuzzy data by complete splines, J. Appl. Math. & Computing (old:kjcam) 8(3) (2001), A. M. Anile, B. Falcidieno, G. Gallo, M. Spagnuolo and S. Spinello, Modeling uncertain data with fuzzy B-splines, Fuzzy Sets and Systems 113 (2000),

13 Approximation by fuzzy B-spline series S. Abbasbandy and E. Babolian, Interpolation of fuzzy data by natural splines, J. Appl. Math. & Computing (old:kjcam) 5(2) (1998), G. A. Anastassiou, Rate of convergence of fuzzy neural network operators, univariate case, J. Fuzzy Math. 10(3) (2002), G. A. Anastassiou and S. G. Gal, On a fuzzy trigonometric approximation theorem of Weierstrass-type, J. Fuzzy Math. 9 (2001), B. Bede and S. G. Gal, Best approximation and Jackson-type estimates for fuzzy-numbervalued functions, Journal of concrete and applicable mathematics, to appear. 7. P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, New Jersey, P. Diamond, P. Kloeden and A. Vladimirov, Spikes, broken planes and the approximation of convex sets, Fuzzy Sets and Systems 99 (1999), D. Dubois and H. Prade, Fuzzy numbers: an overview, in Analysis of fuzzy information, CRC Press, Boca Raton, Fl, 1987, vol 1, S. G. Gal, Approximation Theory in Fuzzy Setting, Chapter 13 in Handbook of Analytic- Computational Methods in Applied Mathematics (ed. G. A. Anastassiou) Chapman & Hall/CRC, Boca Raton-London-New York-Washington D. C., 2000, pp O. Kaleva, Interpolation of fuzzy data, Fuzzy Sets and Systems 61 (1994), Puyin Liu, Analysis of approximation of continuous fuzzy functions by multivariate fuzzy polynomials, Fuzzy Sets and Systems 127 (2002), W. Lodwick and J. Santos, Constructing consistent fuzzy surfaces from fuzzy data, Fuzzy Sets and Systems 135 (2003) G. G. Lorentz and R. A. DeVore, Constructive Approximation, Polynomials and Splines Approximation, Springer-Verlag, New York, Berlin, Heidelberg R. Lowen, A fuzzy Lagrange interpolation theorem, Fuzzy Sets and Systems 34 (1990) M. J. Marsden, An identity for spline functions with applications to variation-diminishing spline approximation, J. Approx. Theory 3 (1970), I. J. Schoenberg, On spline functions, In Inequalities Symposium at Wright-Patterson Air Force Base (O. Shisha, ed.), Academic Press, New York, 1967, Wu Congxin and Gong Zengtai, On Henstock integral of fuzzy-number-valued functions, I, Fuzzy Sets and Systems 120 (2001), L. A. Zadeh, Fuzzy Sets, Inform. and Control 8 (1965), Petru Blaga is a Professor at the Faculty of Mathematics and Computer Science, University Babes-Bolyai, Cluj-Napoca, Romania. His research interests focus on numerical analysis, approximation theory, spline functions. Department of Applied Mathematics, University Babes-Bolyai, Cluj-Napoca, Mihail Koglniceanu no. 1, Cluj-Napoca, Romania blaga@math.ubbcluj.ro Barnabas Bede received his PhD from the Faculty of Mathematics and Computer Science, University Babes-Bolyai, Cluj-Napoca, Romania. He is a Lecturer at the University of Oradea, Oradea, Romania and Budapest Tech, Budapest, Hungary. His research interests focus on: fuzzy numbers, fuzzy differential equations, numerical analysis. Department of Mathematics and Informatics, University of Oradea, Armatei Romane no. 5, Oradea, Romania, and Department of Mechanical and System Engineering, Budapest Tech, Nepszinhaz u. 8, H-1081 Budapest, Hungary bbede@uoradea.ro