• Title, Summary, Keyword: approximation function

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APPROXIMATION ORDER TO A FUNCTION IN Lp SPACE BY GENERALIZED TRANSLATION NETWORKS

  • HAHM, NAHMWOO;HONG, BUM IL
    • Honam Mathematical Journal
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    • v.28 no.1
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    • pp.125-133
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    • 2006
  • We investigate the approximation order to a function in $L_p$[-1, 1] for $0{\leq}p<{\infty}$ by generalized translation networks. In most papers related to neural network approximation, sigmoidal functions are adapted as an activation function. In our research, we choose an infinitely many times continuously differentiable function as an activation function. Using the integral modulus of continuity and the divided difference formula, we get the approximation order to a function in $L_p$[-1, 1].

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A SIMULTANEOUS NEURAL NETWORK APPROXIMATION WITH THE SQUASHING FUNCTION

  • Hahm, Nahm-Woo;Hong, Bum-Il
    • Honam Mathematical Journal
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    • v.31 no.2
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    • pp.147-156
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    • 2009
  • In this paper, we actually construct the simultaneous approximation by neural networks to a differentiable function. To do this, we first construct a polynomial approximation using the Fejer sum and then a simultaneous neural network approximation with the squashing activation function. We also give numerical results to support our theory.

DEGREE OF APPROXIMATION TO A SMOOTH FUNCTION BY GENERALIZED TRANSLATION NETWORKS

  • HAHM, NAHMWOO;YANG, MEEHYEA;HONG, BUM IL
    • Honam Mathematical Journal
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    • v.27 no.2
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    • pp.225-232
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    • 2005
  • We obtain the approximation order to a smooth function on a compact subset of $\mathbb{R}$ by generalized translation networks. In our study, the activation function is infinitely many times continuously differentiable function but it does not have special properties around ${\infty}$ and $-{\infty}$ like a sigmoidal activation function. Using the Jackson's Theorem, we get the approximation order. Especially, we obtain the approximation order by a neural network with a fixed threshold.

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Function Approximation Using an Enhanced Two-Point Diagonal Quadratic Approximation (개선된 이점 대각 이차 근사화를 이용한 함수 근사화)

  • Kim, Jong-Rip;Kang, Woo-Jin;Choi, Dong-Hoon
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.28 no.4
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    • pp.475-480
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    • 2004
  • Function approximation is one of the most important and active research fields in design optimization. Accurate function approximations can reduce the repetitive computational effort fur system analysis. So this study presents an enhanced two-point diagonal quadratic approximation method. The proposed method is based on the Two-point Diagonal Quadratic Approximation method. But unlike TDQA, the suggested method has two quadratic terms, the diagonal term and the correction term. Therefore this method overcomes the disadvantage of TDQA when the derivatives of two design points are same signed values. And in the proposed method, both the approximate function and derivative values at two design points are equal to the exact counterparts whether the signs of derivatives at two design points are the same or not. Several numerical examples are presented to show the merits of the proposed method compared to the other forms used in the literature.

APPROXIMATION METHOD FOR SCATTERED DATA FROM SHIFTS OF A RADIAL BASIS FUNCTION

  • Yoon, Jung-Ho
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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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.

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APPROXIMATION ORDER TO A FUNCTION IN $C^1$[0, 1] AND ITS DERIVATIVE BY A FEEDFOWARD NEURAL NETWORK

  • Hahm, Nahm-Woo;Hong, Bum-Il
    • Journal of applied mathematics & informatics
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    • v.27 no.1_2
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    • pp.139-147
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    • 2009
  • We study the neural network approximation to a function in $C^1$[0, 1] and its derivative. In [3], we used even trigonometric polynomials in order to get an approximation order to a function in $L_p$ space. In this paper, we show the simultaneous approximation order to a function in $C^1$[0, 1] using a Bernstein polynomial and a feedforward neural network. Our proofs are constructive.

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THE SIMULTANEOUS APPROXIMATION ORDER BY NEURAL NETWORKS WITH A SQUASHING FUNCTION

  • Hahm, Nahm-Woo
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.4
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    • pp.701-712
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    • 2009
  • In this paper, we study the simultaneous approximation to functions in $C^m$[0, 1] by neural networks with a squashing function and the complexity related to the simultaneous approximation using a Bernstein polynomial and the modulus of continuity. Our proofs are constructive.

Approximation for the Two-Dimensional Gaussian Q-Function and Its Applications

  • Park, Jin-Ah;Park, Seung-Keun
    • ETRI Journal
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    • v.32 no.1
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    • pp.145-147
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    • 2010
  • In this letter, we present a new approximation for the twodimensional (2-D) Gaussian Q-function. The result is represented by only the one-dimensional (1-D) Gaussian Q-function. Unlike the previous 1-D Gaussian-type approximation, the presented approximation can be applied to compute the 2-D Gaussian Q-function with large correlations.

Accuracy Analysis of Optimal Trajectory Planning Methods Based on Function Approximation for a Four-DOF Biped Walking Model

  • Peng Chunye;ONO Kyosuke
    • Journal of Mechanical Science and Technology
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    • v.19 no.spc1
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    • pp.452-460
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    • 2005
  • Based on an introduced optimal trajectory planning method, this paper mainly deals with the accuracy analysis during the function approximation process of the optimal trajectory planning method. The basis functions are composed of Hermit polynomials and Fourier series to improve the approximation accuracy. Since the approximation accuracy is affected by the given orders of each basis function, the accuracy of the optimal solution is examined by changing the combinations of the orders of Hermit polynomials and Fourier series as the approximation basis functions. As a result, it is found that the proper approximation basis functions are the $5^{th}$ order Hermit polynomials and the $7^{th}-10^{th}$ order of Fourier series.

Bayesian Estimation of the Reliability Function of the Burr Type XII Model under Asymmetric Loss Function

  • Kim, Chan-Soo
    • Communications for Statistical Applications and Methods
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    • v.14 no.2
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    • pp.389-399
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    • 2007
  • In this paper, Bayes estimates for the parameters k, c and reliability function of the Burr type XII model based on a type II censored samples under asymmetric loss functions viz., LINEX and SQUAREX loss functions are obtained. An approximation based on the Laplace approximation method (Tierney and Kadane, 1986) is used for obtaining the Bayes estimators of the parameters and reliability function. In order to compare the Bayes estimators under squared error loss, LINEX and SQUAREX loss functions respectively and the maximum likelihood estimator of the parameters and reliability function, Monte Carlo simulations are used.