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Flexible smoothing with Bsplines and penalties
 STATISTICAL SCIENCE
, 1996
"... Bsplines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots ..."
Abstract

Cited by 395 (6 self)
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Bsplines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent Bsplines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of Bsplines, their construction, and penalized likelihood is presented. We discuss properties of penalized Bsplines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.
Smooth function approximation using. neural networks
 IEEE TRANSACTIONS ON NEURAL NETWORK. (JAN
, 2005
"... An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing function’s input, output, and, possibly, gradient information. The training set ..."
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Cited by 24 (4 self)
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An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing function’s input, output, and, possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input/output and/or gradientbased training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.
Hybrid Fuzzy Convolution Model And Its Application In Predictive Control J. Abonyi, . Bdizs, L. Nagy, F. Szeifert
, 2000
"... In this paper a new method for synthesising nonlinear, controloriented process models is presented. The proposed hybrid fuzzy convolution model (HFCM) consists of a steadystate fuzzy model and a gainindependent impulse response model. The proposed HFCM is applied in model based predictive control ..."
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In this paper a new method for synthesising nonlinear, controloriented process models is presented. The proposed hybrid fuzzy convolution model (HFCM) consists of a steadystate fuzzy model and a gainindependent impulse response model. The proposed HFCM is applied in model based predictive control of a laboratoryscale electrical waterheater. Simulation and realtime studies confirm that the method is capable of controlling this delayed and distributed parameter system with a strong nonlinear feature. Keywords: modelbased control, predictive control, fuzzy modelling, impulse response model
Version 2.0Software Support for Metrology Best Practice Guide No. 4 Discrete Modelling and Experimental Data Analysis
, 2004
"... Metrology, the science of measurement, involves the determination from experiment of estimates of the values of physical quantities, along with the associated uncertainties. In this endeavour, a mathematical model of the measurement system is required in order to extract information from the experim ..."
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Metrology, the science of measurement, involves the determination from experiment of estimates of the values of physical quantities, along with the associated uncertainties. In this endeavour, a mathematical model of the measurement system is required in order to extract information from the experimental data. Modelling involves model building: developing a mathematical model of the measurement system in terms of equations involving parameters that describe all the relevant aspects of the system, and model solving: determining estimates of the model parameters from the measured data by solving the equations constructed as part of the model. This bestpractice guide covers all the main stages in experimental data analysis: construction of candidate models, model parameterisation, uncertainty structure in the data, uncertainty of measurements, choice of parameter estimation algorithms and their implementation in software, with the concepts illustrated by case studies. The Guide looks at validation techniques for the main components of discrete modelling: building the functional and statistical model, model solving and parameter estimation methods, goodness of fit of model solutions and experimental design and measurement strategy. The techniques are illustrated in detailed case studies.
Version 1.1Software Support for Metrology Best Practice Guide No. 4 Discrete Modelling
, 2000
"... Metrology, the science of measurement, involves the determination of physical quantities from experiment, along with estimates of their associated uncertainties. In this endeavour, a mathematical model of the measurement system is required in order to extract information from the experimental data. ..."
Abstract
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Metrology, the science of measurement, involves the determination of physical quantities from experiment, along with estimates of their associated uncertainties. In this endeavour, a mathematical model of the measurement system is required in order to extract information from the experimental data. This involves model building: developing a mathematical model of the experimental system in terms of equations involving parameters that describe all the relevant aspects of the system, and model solving: determining estimates of the model parameters from the measured data by solving the equations constructed as part of the model. This Best Practice Guide for discrete modelling covers all the main stages in experimental data analysis: construction of candidate models, model parametrization, error structure in the data, uncertainty of measurements, choice of parameter estimation algorithms and their implementation in software, with the concepts illustrated by case studies. A www version of the Guide will allow for further sections on models, algorithms and case studies to be added.
PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF
, 2002
"... A systematic approach is developed for designing adaptive and reconfigurable nonlinear control systems that are applicable to plants modeled by ordinary differential equations. The nonlinear controller comprising a network of neural networks is taught using a twophase learning procedure realized th ..."
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A systematic approach is developed for designing adaptive and reconfigurable nonlinear control systems that are applicable to plants modeled by ordinary differential equations. The nonlinear controller comprising a network of neural networks is taught using a twophase learning procedure realized through novel techniques for initialization, online training, and adaptive critic design. A critical observation is that the gradients of the functions defined by the neural networks must equal corresponding linear gain matrices at chosen operating points. Online training is based on a dual heuristic adaptive critic architecture that improves control for large, coupled motions by accounting for actual plant dynamics and nonlinear effects. An action network computes the optimal control law; a critic network predicts the derivative of the costtogo with respect to the state. Both networks are algebraically initialized based on prior knowledge of satisfactory pointwise linear controllers and continue to adapt on line during fullscale simulations of the plant. Online training takes place sequentially over discrete periods of time and involves
Regularisation Theroy Applied to Neurofuzzy Modelling
"... A desirable property of any empirical model is the ability to generalise well throughout the models input space. Recent work has seen the development of neurofuzzy model construction algorithms which identify neurofuzzy models from available empirical data and expert knowledge. By matching the model ..."
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A desirable property of any empirical model is the ability to generalise well throughout the models input space. Recent work has seen the development of neurofuzzy model construction algorithms which identify neurofuzzy models from available empirical data and expert knowledge. By matching the models structure to the underlying process represented by the data, parsimonious models are produced. Consequent parsimonious models do generalise better but due to the structural symmetry required in these models enforced by the need for model transparency, and the often sparse distribution of real data these models are still prone to poor generalisation. This report reviews and develops regularisation techniques that can be applied to identified neurofuzzy models to aid their ability to generalise. Essentially regularisation places a prior probability distribution on the weight values which consequently constrains the model output. One of the major overheads encountered when performing regulari...
An evaluation of spline functions for use in cam design
"... This paper shows how spline functions can be employed for kinematic motion specification in cam design. The polynomial spline is introduced as a special case of a continuous piecewise function. Cubic and quintic splines are derived and their properties are discussed in the cam design context. It is ..."
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This paper shows how spline functions can be employed for kinematic motion specification in cam design. The polynomial spline is introduced as a special case of a continuous piecewise function. Cubic and quintic splines are derived and their properties are discussed in the cam design context. It is shown how standard cam laws can be approximated extremely accurately with a small number of points and appropriate boundary conditions. The modijed sinusoidal acceleration cam law is used as an example. The application of quintic splines to nonstandard and special motions is discussed. The algebraic and Bspline representations of spline functions are compared. The former is considered preferable in this context and a list of useful algorithms is given. The real power of the spline function, in particular the algebraic quintic spline, is its simplicity, ease of computation and adaptability to nonstandard design problems. The use of parametrized, de$cient and exponential splines is proposed for specrjic applications. NOTATION mechanism, a radial cam produces a translating folY