• The Transactions of the Korean Institute of Electrical Engineers D
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• v.49 no.3
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• pp.145-156
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• 2000

In this paper, we propose a new methodology which includes the optimal design procedure of Polynomial Neural Networks(PNN) structure for model identification of complex and nonlinear system. The proposed PNN algorithm is based on GMDA(Group Method of Data handling) method and its structure is similar to Neural Networks. But the structure of PNN is not fixed like in conventional Neural Networks and can be generated. The each node of PNN structure uses several types of high-order polynomial such as linear, quadratic and cubic, and is connected as various kinds of multi-variable inputs. In other words, the PNN uses high-order polynomial as extended type besides quadratic polynomial used in GMDH, and the number of input of its node in each layer depends on that of variables used in the polynomial. The design procedure to obtain an optimal model structure utilizing PNN algorithm is shown in each stage. The study is illustrated with the aid of pH neutralization process data besides representative time series data for gas furnace process used widely for performance comparison, and shows that the proposed PNN algorithm can produce the model with higher accuracy than previous other works. And performance index related to approximation and prediction capabilities of model is evaluated and also discussed.

• Journal of Sensor Science and Technology
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• v.21 no.2
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• pp.109-113
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• 2012

We propose an improved interpolating equation to express temperature-resistance characteristics for modern industrial platinum resistance thermometers (PRTs). Callendar-van Dusen equation which has been widely used for platinum resistance thermometer fails to fully describe temperature characteristics of high quality PRTs and leaves systematic residual when the calibration point include temperatures above $300^{\circ}C$. Expanding Callendar-van Dusen to higher-order polynomial drastically improves the uncertainty of the fitting even with reduced degrees of freedom of the fitting. We found that in the fourth-order polynomial fitting, the third-order and fourth-order coefficients have a strong correlation. Using the correlation, we suggest an improved interpolating equation in the form of fourth-order polynomial, but with three fitting parameters. Applying this interpolating equation reduced the uncertainty of the fitting to 32 % of that resulted from the traditional Callendar-van Dusen. This improvement was better than that from a simple third-order polynomial despite that the degrees of the freedom of the fitting was the same.

• Communications for Statistical Applications and Methods
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• v.2 no.1
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• pp.1-12
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• 1995

An estimator of coefficients of polynomial measurement error model with vector predictor and first-order interaction terms is derived using Hermite polynomial. Asymptotic normality of estimator is provided and some simulation study is performed to compare the small sample properties of derived estimator with those of OLS estimator.

• Journal of Korea Water Resources Association
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• v.55 no.10
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• pp.775-784
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• 2022

In this study, to determine the optimal order of the full-logged I-D-F polynomial equation, which is mainly used to calculate the probable rainfall over a temporal rainfall duration, the probable rainfall was calculated and the regression coefficients of the full-logged I-D-F polynomial equation was estimated. The optimal variable of the polynomial equation for each station was selected using a stepwise selection method, and statistical significance tests were performed through ANOVA. Using these results, the statistically appropriately calculated rainfall intensity equation for each station was presented. As a result of analyzing the variable selection outputs of the full-logged I-D-F polynomial equation at 9 stations in Gyeongbuk, the 1^st to 3^rd order equations at 6 stations and the incomplete 3^rd order at 1 station were determined as the optimal equations. Since the 1^st order equation is similar to the Sherman type equation and the 2^nd order one is similar to the general type equation, it was presented as a unified form of rainfall intensity equation for convenience of use by increasing the number of independent variables. Therefore, it is judged that there is no statistical problem in considering only the 3^rd order polynomial regression equation for the full-logged I-D-F.

• Proceedings of the KIEE Conference
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• 2005.10b
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• pp.149-151
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• 2005

In this paper, we propose the Optimal Polynomial Neural Networks(PNN) for nonlinear process. The PNN is based on Group Method of Data Handling(GMDH) method and its structure is similar to feedforward Neural Networks. But the structure of PNN is not fixed like in conventional Neural Networks and can be generated. The each node of PNN structure uses several types of high-order polynomial such as linear, quadratic and modified quadratic, and is connected as various kinds of multi-variable inputs. The conventional PNN depends on experience of a designer that select No. of input variable, input variable and polynomial type. Therefore it is very difficult a organizing of optimized network. The proposed algorithm identified and selected No. of input variable, input variable and polynomial type by using Genetic Algorithms(GAs). In the sequel the proposed model shows not only superior results to the existing models, but also pliability in organizing of optimal network. Medical Imaging System(MIS) data is simulated in order to confirm the efficiency and feasibility of the proposed approach in this paper.

• 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.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.53 no.3
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• pp.135-144
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• 2004

In this paper, we propose competitive fuzzy polynomial neurons-based advanced Self-Organizing Neural Networks(SONN) architecture for optimal model identification and discuss a comprehensive design methodology supporting its development. The proposed SONN dwells on the ideas of fuzzy rule-based computing and neural networks. And it consists of layers with activation nodes based on fuzzy inference rules and regression polynomial. Each activation node is presented as Fuzzy Polynomial Neuron(FPN) which includes either the simplified or regression polynomial fuzzy inference rules. As the form of the conclusion part of the rules, especially the regression polynomial uses several types of high-order polynomials such as linear, quadratic, and modified quadratic. As the premise part of the rules, both triangular and Gaussian-like membership (unction are studied and the number of the premise input variables used in the rules depends on that of the inputs of its node in each layer. We introduce two kinds of SONN architectures, that is, the basic and modified one with both the generic and the advanced type. Here the basic and modified architecture depend on the number of input variables and the order of polynomial in each layer. The number of the layers and the nodes in each layer of the SONN are not predetermined, unlike in the case of the popular multi-layer perceptron structure, but these are generated in a dynamic way. The superiority and effectiveness of the Proposed SONN architecture is demonstrated through two representative numerical examples.

• Asian-Australasian Journal of Animal Sciences
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• v.19 no.7
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• pp.915-921
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• 2006

Various random regression models with different order of Legendre polynomials for permanent environmental and genetic effects were constructed to predict future milk yield of Holstein cows in Korea. A total of 257,908 test-day (TD) milk yield records from a total of 28,135 cows belonging to 1,090 herds were considered for estimating (co)variance of the random covariate coefficients using an expectation-maximization REML algorithm in an animal mixed model. The variances did not change much between the models, having different order of Legendre polynomial, but a decreasing trend was observed with increase in the order of Legendre polynomial in the model. The R-squared value of the model increased and the residual variance reduced with the increase in order of Legendre polynomial in the model. Therefore, a model with $5^{th}$ order of Legendre polynomial was considered for predicting future milk yield. For predicting the future milk yield of cows, 132,771 TD records from 28,135 cows were randomly selected from the above data by way of preceding partial TD record, and then future milk yields were estimated using incomplete records from each cow randomly retained. Results suggested that we could predict the next four months milk yield with an error deviation of 4 kg. The correlation of more than 70% between predicted and observed values was estimated for the next four months milk yield. Even using only 3 TD records of some cows, the average milk yield of Korean Holstein cows would be predicted with high accuracy if compared with observed milk yield. Persistency of each cow was estimated which might be useful for selecting the cows with higher persistency. The results of the present study suggested the use of a $5^{th}$ order Legendre polynomial to predict the future milk yield of each cow.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.4
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• pp.307-314
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• 2009

In this work, we introduce a new quatnary approximating subdivision scheme for curve and deal with its analysis (convergence and regularity) using Laurent polynomials method. We also discuss various properties, such as approximation order and support of basic limit function.

• Communications for Statistical Applications and Methods
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• v.2 no.2
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• pp.376-386
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• 1995

Designs for polynomial approximation to the unknown response function are considered. Optimality criteria are monotone functions of the mean squared error matrix of the least squares estimator. They correspond to the classical A-, D-, G- and Q-optimalities. Optimal first order designs are chosen from the invariant designs and then compared with optimal second order designs.