• Korean Journal of Mathematics
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• 제18권2호
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• pp.167-174
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• 2010

In this paper, we investigate the possibility of $2{\pi}$-periodic continuous function approximation by periodic neural networks. Using the Riemann sum and the quadrature formula, we show the capability of a periodic neural network approximation.

• Journal of Applied and Pure Mathematics
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• 제4권3_4호
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• pp.185-209
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• 2022

Here we exhibit multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ^N, N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the generalized symmetrical sigmoid function. The approximations are point-wise and uniform. The related feed-forward neural network is with one hidden layer.

• 한국통신학회논문지
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• 제21권1호
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• pp.251-268
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• 1996

In this paper, a higher-order feedforward neural network called ridge polynomial network (RPN) which shows good approximation capability for nonlnear continuous functions defined on compact subsets in multi-dimensional Euclidean spaces, is presented. This network provides more efficient and regular structure as compared to ordinary higher-order feedforward networks based on Gabor-Kolmogrov polynomial expansions, while maintating their fast learning property. the ridge polynomial network is a generalization of the pi-sigma network (PSN) and uses a specialform of ridge polynomials. It is shown that any multivariate polynomial can be exactly represented in this form, and thus realized by a RPN. The approximation capability of the RPNs for arbitrary continuous functions is shown by this representation theorem and the classical weierstrass polynomial approximation theorem. The RPN provides a natural mechanism for incremental function approximation based on learning algorithm of the PSN. Simulation results on several applications such as multivariate function approximation and pattern classification assert nonlinear approximation capability of the RPN.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.317-324
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• 2008

This paper shows the degree of simultaneous neural network approximation for a target function in $C^r$[-1, 1] and its first derivative. We use the Jackson's theorem for differentiable functions to get a degree of approximation to a target function by algebraic polynomials and trigonometric polynomials. We also make use of the de La Vall$\grave{e}$e Poussin sum to get an approximation order by algebraic polynomials to the derivative of a target function. By showing that the divided difference with a generalized translation network can be arbitrarily closed to algebraic polynomials on [-1, 1], we obtain the degree of simultaneous approximation.

• Journal of applied mathematics & informatics
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• 제27권5_6호
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• pp.1087-1095
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• 2009

In this paper, we study approximation method from scattered data to the derivatives of a function f by a radial basis function $\phi$. For a given function f, we define a nearly interpolating function and discuss its accuracy. In particular, we are interested in using smooth functions $\phi$ which are (conditionally) positive definite. We estimate accuracy of approximation for the Sobolev space while the classical radial basis function interpolation applies to the so-called native space. We observe that our approximant provides spectral convergence order, as the density of the given data is getting smaller.

• 대한기계학회논문집A
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• 제33권8호
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• pp.753-759
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• 2009

Feed-forward neural networks have been widely used as function approximation tools in the context of global approximate optimization. In the present study, a wavelet neural network (WNN) which is based on wavelet transform theory is suggested as an alternative to a traditional back-propagation neural network (BPN). The basic theory of wavelet neural network is briefly described, and approximation performance is tested using a nonlinear multimodal function and a composite rotor blade analysis problem. Laplacian of Gaussian function, Mexican function, and Morlet function are considered during the construction of WNN architectures. In addition, approximation results from WNN are compared with those from BPN.

• Communications for Statistical Applications and Methods
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• 제19권2호
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• pp.267-276
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• 2012

Bagai et al. (1989) proposed a distribution-free test for stochastic ordering in the competing risk model, and recently Murakami (2009) utilized a standard saddlepoint approximation to provide tail probabilities for the Bagai statistic under finite sample sizes. In the present paper, we consider the Gaussian-polynomial approximation proposed in Ha and Provost (2007) and compare it to the saddlepoint approximation in terms of approximating the percentiles of the Bagai statistic. We make numerical comparisons of these approximations for moderate sample sizes as was done in Murakami (2009). From the numerical results, it was observed that the Gaussianpolynomial approximation provides comparable or greater accuracy in the tail probabilities than the saddlepoint approximation. Unlike saddlepoint approximation, the Gaussian-polynomial approximation provides a simple explicit representation of the approximated density function. We also discuss the details of computations.

• 대한기계학회논문집A
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• 제31권4호
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• pp.490-495
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• 2007

This research aims to examine accuracy and efficiency of the first four moments corresponding to mean, standard deviation, skewness, and kurtosis, which are estimated by a function approximation moment method (FAMM). In FAMM, the moments are estimated from an approximating quadratic function of a system response function. The function approximation is performed on a specially selected experimental region for accuracy, and the number of function evaluations is taken equal to that of the unknown coefficients for efficiency. For this purpose, three error-minimizing conditions are utilized and corresponding canonical experimental regions constructed accordingly. An interpolation function is then obtained using a D-optimal design and then the first four moments of it are obtained as the estimates for the system response function. In order to verify accuracy and efficiency of FAMM, several non-linear examples are considered including a polynomial of order 4, an exponential function, and a rational function. The moments calculated from various coefficients of variation show very good accuracy and efficiency in comparison with those from analytic integration or the Monte Carlo simulation and the experimental design technique proposed by Taguchi and updated by D'Errico and Zaino.

• 한국전자통신학회논문지
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• 제6권1호
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• pp.20-26
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• 2011

본 논문에서는 NM(node monitoring) 알고리듬과 NP(Piecewise Linear Function Approximation)를 사용해서 LDPC 코드 복호의 복잡도를 감소시키기 위한 효율적인 알고리듬을 제안한다. 이 NM 알고리듬은 새로운 node-threshold 방법과 message passing 알고리듬에 근거해서 제안되었는데, 이에 NP 방법을 사용해서 알고리듬의 복잡도를 더 줄일 수 있었다. 이 알고리듬의 효율성을 입증하기 위해서 모의 실험을 하였다. 모의실험결과, 기존에 잘 알려진 방법에 비해서 20% 정도 더 효율적이었다.

• 한국CDE학회논문집
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• 제12권1호
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• pp.27-38
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• 2007

This paper describes a multiresidual approximation method for scattered volumetric data modeling. The approximation method employs a volumetric NURBS or VNURBS as a data interpolating function and proposes two multiresidual methods as a data modeling algorithm. One is called as the residual series method that constructs a sequence of VNURBS functions and their algebraic summation produces the desired approximation. The other is the residual merging method that merges all the VNURBS functions mentioned above into one equivalent function. The first one is designed to construct wavelet-type multiresolution models and also to achieve more accurate approximation. And the second is focused on its improvement of computational performance with the save fitting accuracy for more practical applications. The performance results of numerical examples demonstrate the usefulness of VNURBS approximation and the effectiveness of multiresidual methods. In addition, several graphical examples suggest that the VNURBS approximation is applicable to various applications such as surface modeling and fitting problems.