Spline-functions for investigating and forecasting systems
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SPLINE-FUNCTIONS FOR INVESTIGATING AND FORECASTING SYSTEMS UDC 519.2
S. R. Kostadinova
Abstract. An explicit formula of a multivariate B-spline, some useful investigations in the field of linear transformations of independent exponentially distributed random variables, representation of theirs density functions with the help of multivariate exponential spline functions, and their usage are considered. The consideration is illustrated by an appropriate example. Keywords: spline, exponential spline, multivariate spline, probability density of a random vector.
The theory of multivariate splines arose in 1976 when Charles de Boor [5] proposed his definition of Â-splines of many variables and then W. Dahmen and C. A. Micchelli [4] published first works devoted to this line of investigation. At present, this theory is intensively developed and many problems remain to be solved. In this article, the idea of developing and using multivariate exponential splines for the purpose of approximation is considered. An explicit formula of a multivariate Â-spline is given, investigations of the author in the field of linear transformations of independent exponentially distributed random quantities are presented, and a formula for their densities is obtained with the help of multivariate exponential splines. The results obtained can be used for investigating and forecasting systems of different technical and social kinds. Let t 0 , t 1 , ... , t r represent r + 1 points in a k -dimensional Euclidean space R k with their coordinates, respectively, t i = ( t 1i , t 2i ,... , t ki )' , i = 0,... , r. We assume that r > k and that t 0 , t 1 , ... , t r are points in “general position.” This means that, for each j = 1,... , k and different 0 £ i1 ,... , i j + 1 £ r, the following determinants are distinct from zero: 1
t 1i 1
K
t ij 1
1
t 1i2
K
t ij2
K K K K i i 1 t1 j + 1 K t j j + 1
¹ 0.
(1)
Let us consider a linear transformation h = x1t 1 + K + x r t r ,
(2)
where x i , i = 1,... , r, is a number of independent and equally exponentially distributed random quantities with probability densities f x i ( x ) = e - x , x > 0. The main problem is as follows: give a representation of the probability density of a random vector h as a multivariate divided difference of definite functions. Let us recall the definition of a, i.e., the divided difference (a = ( a1 , K, a k )) of a sufficiently smooth function f ( x1 ,... , x k ) of k variables [1]. The quantity r! [ t 0 ,... , t r ] a f : = Da f a! 0 ò r [ t ,..., t ]
St. Kliment Ohridski University of Sofia, Sofia, Bulgaria, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 146–151, January–February 2011. Original article submitted May 23, 2008. 134
1060-0396/11/4701-0134
©
2011 Springer Science+Business Media, Inc.
is called the a-divided difference of the function f ( x1 ,... , x k ) at points t 0 , ... , t r Î R k , where a ! = a1 ! a 2 !K a k !, a
a
æ ¶ ö 1 æ ¶ ö k ÷÷ K çç ÷÷ f , D a f = çç è ¶ x1 ø è ¶ xk ø and [ t 0 , ... , t r ] is
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