• Title, Summary, Keyword: Higher order polynomial

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Selection of Data-adaptive Polynomial Order in Local Polynomial Nonparametric Regression

  • Jo, Jae-Keun
    • Communications for Statistical Applications and Methods
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    • v.4 no.1
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    • pp.177-183
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    • 1997
  • A data-adaptive order selection procedure is proposed for local polynomial nonparametric regression. For each given polynomial order, bias and variance are estimated and the adaptive polynomial order that has the smallest estimated mean squared error is selected locally at each location point. To estimate mean squared error, empirical bias estimate of Ruppert (1995) and local polynomial variance estimate of Ruppert, Wand, Wand, Holst and Hossjer (1995) are used. Since the proposed method does not require fitting polynomial model of order higher than the model order, it is simpler than the order selection method proposed by Fan and Gijbels (1995b).

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Nonlinear Function Approximation Using Efficient Higher-order Feedforward Neural Networks (효율적 고차 신경회로망을 이용한 비선형 함수 근사에 대한 연구)

  • 신요안
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.21 no.1
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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.

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An Accurate Camera Calibration Using Higher-Order Polynomials (고차 polynomial을 이용한 정밀한 카메라 캘리브레이션)

  • Jo, Tae-Hun
    • Proceedings of the Korean Institute of Intelligent Systems Conference
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    • pp.413-416
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    • 2007
  • 카메라 캘리브레이션은 비젼(vision) 시스템의 광학왜곡을 보정하기 위해, 영상 좌표계와 실세계 좌표계간의 변환관계를 정의해 주는 mapping을 구하는 과정으로 카메라를 이용한 측정, 검사, 위치보정 등의 응용에서 매우 중요하다. 카메라 캘리브레이션 방법으로 많이 사용되는 Tsai 알고리즘은 여러 카메라 내부 상수들을 필요로 하며, 적절한 활용을 위해서는 이에 대한 이해와 카메라와 렌즈왜곡의 모델에 대한 사전지식을 요한다. 본 논문에서는 카메라나 렌즈왜곡에 대한 모델이나 가정없이, 영상좌표와 실세계 좌표간의 변환을 고차(higher order) polynomial을 이용하여 구현하여 사용이 손쉬운 카메라 캘리브레이션 방법을 소개하고 성능을 평가하였다. 성능 평가 결과, 3차 polynomial을 이용한 카메라 캘리브레이션 방법이 Tsai알고리즘보다 정밀도에서 우수하였다.

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A Study on the Adaptive Polynomial Neuro-Fuzzy Networks Architecture (적응 다항식 뉴로-퍼지 네트워크 구조에 관한 연구)

  • Oh, Sung-Kwun;Kim, Dong-Won
    • The Transactions of the Korean Institute of Electrical Engineers D
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    • v.50 no.9
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    • pp.430-438
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    • 2001
  • In this study, we introduce the adaptive Polynomial Neuro-Fuzzy Networks(PNFN) architecture generated from the fusion of fuzzy inference system and PNN algorithm. The PNFN dwells on the ideas of fuzzy rule-based computing and neural networks. Fuzzy inference system is applied in the 1st layer of PNFN and PNN algorithm is employed in the 2nd layer or higher. From these the multilayer structure of the PNFN is constructed. In order words, in the Fuzzy Inference System(FIS) used in the nodes of the 1st layer of PNFN, either the simplified or regression polynomial inference method is utilized. And as the premise part of the rules, both triangular and Gaussian like membership function are studied. In the 2nd layer or higher, PNN based on GMDH and regression polynomial is generated in a dynamic way, unlike in the case of the popular multilayer perceptron structure. That is, the PNN is an analytic technique for identifying nonlinear relationships between system's inputs and outputs and is a flexible network structure constructed through the successive generation of layers from nodes represented in partial descriptions of I/O relatio of data. The experiment part of the study involves representative time series such as Box-Jenkins gas furnace data used across various neurofuzzy systems and a comparative analysis is included as well.

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Polynomial Higher Order Neural Network for Shift-invariant Pattern Recognition (위치 변환 패턴 인식을 위한 다항식 고차 뉴럴네트워크)

  • Chung, Jong-Su;Hong, Sung-Chan
    • The Transactions of the Korea Information Processing Society
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    • v.4 no.12
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    • pp.3063-3068
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    • 1997
  • In this paper, we have extended the generalization back-propagation algorithm to multi-layer polynomial higher order neural networks. The purpose of this paper is to describe various pattern recognition using polynomial higher-order neural network. And we have applied shift position T-C test pattern for invariant pattern recognition and measured generalization by mirror symmetry problem. simulation result shows that the ability for invariant pattern recognition increase with the proposed technique. Recognition rate of invariant T-C pattern is 90% effective and of mirror symmetry problem is 70% effective when the proposed technique is utilized. These results are much better than those by the conventional methods.

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Improved Interpolating Equation for Industrial Platinum Resistance Thermometer (산업용 백금저항온도계를 위한 향상된 내삽식)

  • Yang, In-Seok;Kim, Yong-Gyoo;Gam, Kee-Sool;Lee, Young-Hee
    • 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.

Design of Particle Swarm Optimization-based Polynomial Neural Networks (입자 군집 최적화 알고리즘 기반 다항식 신경회로망의 설계)

  • Park, Ho-Sung;Kim, Ki-Sang;Oh, Sung-Kwun
    • The Transactions of The Korean Institute of Electrical Engineers
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    • v.60 no.2
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    • pp.398-406
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    • 2011
  • In this paper, we introduce a new architecture of PSO-based Polynomial Neural Networks (PNN) and discuss its comprehensive design methodology. The conventional PNN is based on a extended Group Method of Data Handling (GMDH) method, and utilized the polynomial order (viz. linear, quadratic, and modified quadratic) as well as the number of node inputs fixed (selected in advance by designer) at Polynomial Neurons located in each layer through a growth process of the network. Moreover it does not guarantee that the conventional PNN generated through learning results in the optimal network architecture. The PSO-based PNN results in a structurally optimized structure and comes with a higher level of flexibility that the one encountered in the conventional PNN. The PSO-based design procedure being applied at each layer of PNN leads to the selection of preferred PNs with specific local characteristics (such as the number of input variables, input variables, and the order of the polynomial) available within the PNN. In the sequel, two general optimization mechanisms of the PSO-based PNN are explored: the structural optimization is realized via PSO whereas in case of the parametric optimization we proceed with a standard least square method-based learning. To evaluate the performance of the PSO-based PNN, the model is experimented with using Gas furnace process data, and pH neutralization process data. For the characteristic analysis of the given entire data with non-linearity and the construction of efficient model, the given entire system data is partitioned into two type such as Division I(Training dataset and Testing dataset) and Division II(Training dataset, Validation dataset, and Testing dataset). A comparative analysis shows that the proposed PSO-based PNN is model with higher accuracy as well as more superb predictive capability than other intelligent models presented previously.

Automatic classification of power quality disturbances using orthogonal polynomial approximation and higher-order spectra (직교 다항식 근사법과 고차 통계를 이용한 전력 외란의 자동식별)

  • 이재상;이철호;남상원
    • 제어로봇시스템학회:학술대회논문집
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    • pp.1436-1439
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    • 1997
  • The objective of this paper is to present an efficient and practical approach to the automatic classification of power quality(PQ) disturbances, where and orthogonal polynomial approximation method is emloyed for the detection and localization of PQ disturbances, and a feature vector, newly extracted form the bispectra of the detected signal, is utilized for the automatic rectgnition of the various types of PQ disturbances. To demonstrae the performance and applicabiliyt of the proposed approach, some simulation results are provided.

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A New Architecture of Genetically Optimized Self-Organizing Fuzzy Polynomial Neural Networks by Means of Information Granulation

  • Park, Ho-Sung;Oh, Sung-Kwun;Ahn, Tae-Chon
    • 제어로봇시스템학회:학술대회논문집
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    • pp.1505-1509
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    • 2005
  • This paper introduces a new architecture of genetically optimized self-organizing fuzzy polynomial neural networks by means of information granulation. The conventional SOFPNNs developed so far are based on mechanisms of self-organization and evolutionary optimization. The augmented genetically optimized SOFPNN using Information Granulation (namely IG_gSOFPNN) results in a structurally and parametrically optimized model and comes with a higher level of flexibility in comparison to the one we encounter in the conventional FPNN. With the aid of the information granulation, we determine the initial location (apexes) of membership functions and initial values of polynomial function being used in the premised and consequence part of the fuzzy rules respectively. The GA-based design procedure being applied at each layer of genetically optimized self-organizing fuzzy polynomial neural networks leads to the selection of preferred nodes with specific local characteristics (such as the number of input variables, the order of the polynomial, a collection of the specific subset of input variables, and the number of membership function) available within the network. To evaluate the performance of the IG_gSOFPNN, the model is experimented with using gas furnace process data. A comparative analysis shows that the proposed IG_gSOFPNN is model with higher accuracy as well as more superb predictive capability than intelligent models presented previously.

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Genetically Optimized Self-Organizing Polynomial Neural Networks (진화론적 최적 자기구성 다항식 뉴럴 네트워크)

  • 박호성;박병준;장성환;오성권
    • The Transactions of the Korean Institute of Electrical Engineers D
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    • v.53 no.1
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    • pp.40-49
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    • 2004
  • In this paper, we propose a new architecture of Genetic Algorithms(GAs)-based Self-Organizing Polynomial Neural Networks(SOPNN), discuss a comprehensive design methodology and carry out a series of numeric experiments. The conventional SOPNN is based on the extended Group Method of Data Handling(GMDH) method and utilized the polynomial order (viz. linear, quadratic, and modified quadratic) as well as the number of node inputs fixed (selected in advance by designer) at Polynomial Neurons (or nodes) located in each layer through a growth process of the network. Moreover it does not guarantee that the SOPNN generated through learning has the optimal network architecture. But the proposed GA-based SOPNN enable the architecture to be a structurally more optimized network, and to be much more flexible and preferable neural network than the conventional SOPNN. In order to generate the structurally optimized SOPNN, GA-based design procedure at each stage (layer) of SOPNN leads to the selection of preferred nodes (or PNs) with optimal parameters- such as the number of input variables, input variables, and the order of the polynomial-available within SOPNN. An aggregate performance index with a weighting factor is proposed in order to achieve a sound balance between approximation and generalization (predictive) abilities of the model. A detailed design procedure is discussed in detail. To evaluate the performance of the GA-based SOPNN, the model is experimented with using two time series data (gas furnace and NOx emission process data of gas turbine power plant). A comparative analysis shows that the proposed GA-based SOPNN is model with higher accuracy as well as more superb predictive capability than other intelligent models presented previously.