• Title/Summary/Keyword: function approximations

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Development of Global Function Approximations of Desgin optimization Using Evolutionary Fuzzy Modeling

  • Kim, Seungjin;Lee, Jongsoo
    • Journal of Mechanical Science and Technology
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    • v.14 no.11
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    • pp.1206-1215
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    • 2000
  • This paper introduces the application of evolutionary fuzzy modeling (EFM) in constructing global function approximations to subsequent use in non-gradient based optimizations strategies. The fuzzy logic is employed for express the relationship between input training pattern in form of linguistic fuzzy rules. EFM is used to determine the optimal values of membership function parameters by adapting fuzzy rules available. In the study, genetic algorithms (GA's) treat a set of membership function parameters as design variables and evolve them until the mean square error between defuzzified outputs and actual target values are minimized. We also discuss the enhanced accuracy of function approximations, comparing with traditional response surface methods by using polynomial interpolation and back propagation neural networks in its ability to handle the typical benchmark problems.

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PARAMETRIZED GUDERMANNIAN FUNCTION RELIED BANACH SPACE VALUED NEURAL NETWORK MULTIVARIATE APPROXIMATIONS

  • GEORGE A. ANASTASSIOU
    • Journal of Applied and Pure Mathematics
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    • v.5 no.1_2
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    • pp.69-93
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    • 2023
  • Here we give multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝN, 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 derived 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 a parametrized Gudermannian sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Cubic Equations in General Saddlepoint Approximations

  • Lee, Young-Hoon
    • Communications for Statistical Applications and Methods
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    • v.9 no.2
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    • pp.555-563
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    • 2002
  • This paper discusses cubic equations in general saddlepoint approximations. Exact roots are found for various cases by trigonometric identities, the root which is appropriate for the general saddlepoint approximations is selected and discussed, and the defective cases in which the general saddlepoint approximations cannot be used are found.

GENERALIZED SYMMETRICAL SIGMOID FUNCTION ACTIVATED NEURAL NETWORK MULTIVARIATE APPROXIMATION

  • ANASTASSIOU, GEORGE A.
    • Journal of Applied and Pure Mathematics
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    • v.4 no.3_4
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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 ℝN, 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.

Saddlepoint approximations for the ratio of two independent sequences of random variables

  • Cho, Dae-Hyeon
    • Journal of the Korean Data and Information Science Society
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    • v.9 no.2
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    • pp.255-262
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    • 1998
  • In this paper, we study the saddlepoint approximations for the ratio of independent random variables. In Section 2, we derive the saddlepoint approximation to the probability density function. In Section 3, we represent a numerical example which shows that the errors are small even for small sample size.

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Saddlepoint Approximations to the Distribution Function of Non-homogeneous Quadratic Forms (비동차 이차형식의 분포함수에 대한 안장점근사)

  • Na Jong-Hwa;Kim Jeong-Soak
    • The Korean Journal of Applied Statistics
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    • v.18 no.1
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    • pp.183-196
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    • 2005
  • In this paper we studied the saddlepoint approximations to the distribution of non-homogeneous quadratic forms in normal variables. The results are the extension of Kuonen's which provide the same approximations to homogeneous quadratic forms. The CGF of interested statistics and related properties are derived for applications of saddlepoint techniques. Simulation results are also provided to show the accuracy of saddlepoint approximations.

Saddlepoint Approximation to Quadratic Form and Application to Intraclass Correlation Coefficient

  • Na, Jong-Hwa
    • Journal of the Korean Data and Information Science Society
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    • v.19 no.2
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    • pp.497-504
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    • 2008
  • In this paper we studied the saddlepoint approximations to the distribution of quadratic forms in normal variables. We derived the approximations as a special case of Na & Kim (2005). Also applications to a statistic which concerns intraclass correlation coefficient are presented. Simulations show the accuracy and availability of the suggested approximations.

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Tail Probability Approximations for the Ratio of two Independent Sequences of Random Variables

  • Cho, Dae-Hyeon
    • Journal of the Korean Data and Information Science Society
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    • v.10 no.2
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    • pp.415-428
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    • 1999
  • In this paper, we study the saddlepoint approximations for the ratio of two independent sequences of random variables. In Section 2, we review the saddlepoint approximation to the probability density function. In section 3, we derive an saddlepoint approximation formular for the tail probability by following Daniels'(1987) method. In Section 4, we represent a numerical example which shows that the errors are small even for small sample size.

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Development of Polynomial Based Response Surface Approximations Using Classifier Systems (분류시스템을 이용한 다항식기반 반응표면 근사화 모델링)

  • 이종수
    • Korean Journal of Computational Design and Engineering
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    • v.5 no.2
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    • pp.127-135
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    • 2000
  • Emergent computing paradigms such as genetic algorithms have found increased use in problems in engineering design. These computational tools have been shown to be applicable in the solution of generically difficult design optimization problems characterized by nonconvexities in the design space and the presence of discrete and integer design variables. Another aspect of these computational paradigms that have been lumped under the bread subject category of soft computing, is the domain of artificial intelligence, knowledge-based expert system, and machine learning. The paper explores a machine learning paradigm referred to as teaming classifier systems to construct the high-quality global function approximations between the design variables and a response function for subsequent use in design optimization. A classifier system is a machine teaming system which learns syntactically simple string rules, called classifiers for guiding the system's performance in an arbitrary environment. The capability of a learning classifier system facilitates the adaptive selection of the optimal number of training data according to the noise and multimodality in the design space of interest. The present study used the polynomial based response surface as global function approximation tools and showed its effectiveness in the improvement on the approximation performance.

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Finite Element Analysis with Paraxial Boundary Condition (파진행 문제를 위한 Paraxial 경계조건의 유한요소해석)

  • Kim, Hee-Seok;Lee, Jong-Seh
    • Proceedings of the Computational Structural Engineering Institute Conference
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    • 2007.04a
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    • pp.475-480
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    • 2007
  • For the propagation of elastic waves in unbounded domains, absorbing boundary conditions at the fictitious numerical boundaries have been proposed. In this paper we focus on both first- and second-order paraxial boundary conditions(PBCs) in the framework of variational approximations which are based on paraxial approximations of the scalar and elastic wave equations- We propose a penalty function method for the treatment of PBCs and apply these into finite element analysis. The numerical verification of the efficiency is carried out through comparing PBCs with Lysmer-Kuhlemeyer' s boundary conditions.

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