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Approximation of vector-valued functions by polynomials with coefficients in normed spaces applications

Anastassiou G., Gal S.

Demonstratio Mathematica

2006

Vol. 39, nr 3

539-552

In this paper we prove basic results in the approximation of vector-valued functions by polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain : estimates in terms of Ditzian-Totik Lp-moduli of smoothness for approximation by Bernstein-Kantorovich generalized polynomials and by other kinds of operators like the Szasz-Mirakian operators, Baskakov operators, Post-Widder operators and their Kantorovich analogues and inverse theorems for these operators. Applications to approximation of random functions and of fuzzy-number-valued functions are given.

Towards a linguistic description of dependencies in data

Batyrshin I., Wagenknecht M.

International Journal of Applied Mathematics and Computer Science

2002

Vol. 12, no 3

391-401

The problem of a linguistic description of dependencies in data by a set of rules Rk: "If X is Tk then Y is Sk" is considered, where Tk's are linguistic terms like SMALL, BETWEEN 5 AND 7 describing some fuzzy intervals Ak. Sk's are linguistic terms like DECREASING and QUICKLY INCREASING describing the slopes pk of linear functions yk=pkx +qk approximating data on Ak. The decision of this problem is obtained as a result of a fuzzy partition of the domain X on fuzzy intervals Ak, approximation of given data {xi,yi}, i=1,...,n by linear functions yk=pkx+qk on these intervals and by re-translation of the obtained results into linguistic form. The properties of the genetic algorithm used for construction of the optimal partition and several methods of data re-translation are described. The methods are illustrated by examples, and potential applications of the proposed methods are discussed.

Iterative Construction and Optimization of Fuzzy Models

Golea N., Benmahammed K.

International Journal of Applied Mathematics and Computer Science

1999

Vol. 9, no 4

899-919

In this paper, a constructive approach to the fuzzy model selection problem is developed. First, the selection of membership functions is decoupled from parameter calculations using an orthogonalization procedure. Since each membership function depends only on its own parameters, the selection of rules is performed in a sequential manner. At each learning step, a new membership function is created and its parameters are optimized. The resulting parameter calculation boils down to the solution of a triangular system. This approach reduces significantly the computational complexity, and allows for the derivation of a simple optimization algorithm. In addition, optimization of the membership functions is related to the approximation accuracy. Simulation results, when compared with the orthogonal least-squares algorithm, show that this approach is less sensitive to the size of the training data and converges rapidly.