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74
Regularization Theory and Neural Networks Architectures
 Neural Computation
, 1995
"... We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Ba ..."
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Cited by 309 (31 self)
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We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, som...
Nonlinear BlackBox Modeling in System Identification: a Unified Overview
 Automatica
, 1995
"... A nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area with structures based on neural networks, radial basis networks, wavelet networks, hinging hyperplanes, ..."
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Cited by 136 (15 self)
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A nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area with structures based on neural networks, radial basis networks, wavelet networks, hinging hyperplanes, as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules. This paper describes all these approaches in a common framework, from a user's perspective. It focuses on what are the common features in the different approaches, the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques. It is pointed out that the nonlinear structures can be seen as a concatenation of a mapping from observed data to a regression vector and a nonlinear mapping from the regressor space to the output space. These mappings are discussed separately. The latter mapping is usually formed as a basis function e...
Model Predictive Control: Past, Present and Future
 Computers and Chemical Engineering
, 1997
"... More than 15 years after Model Predictive Control (MPC) appeared in industry as an effective means to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge. The issues of feasibility of the online optimization, stability and performance a ..."
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Cited by 96 (4 self)
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More than 15 years after Model Predictive Control (MPC) appeared in industry as an effective means to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge. The issues of feasibility of the online optimization, stability and performance are largely understood for systems described by linear models. Much progress has been made on these issues for nonlinear systems but for practical applications many questions remain, including the reliability and efficiency of the online computation scheme. To deal with model uncertainty "rigorously" an involved dynamic programming problem must be solved. The approximation techniques proposed for this purpose are largely at a conceptual stage. Among the broader research needs the following areas are identified: multivariable system identification, performance monitoring and diagnostics, nonlinear state estimation, and batch system control. Many practical problems like control objective prior...
Perspectives on system identification
 In Plenary talk at the proceedings of the 17th IFAC World Congress, Seoul, South Korea
, 2008
"... System identification is the art and science of building mathematical models of dynamic systems from observed inputoutput data. It can be seen as the interface between the real world of applications and the mathematical world of control theory and model abstractions. As such, it is an ubiquitous ne ..."
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Cited by 77 (2 self)
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System identification is the art and science of building mathematical models of dynamic systems from observed inputoutput data. It can be seen as the interface between the real world of applications and the mathematical world of control theory and model abstractions. As such, it is an ubiquitous necessity for successful applications. System identification is a very large topic, with different techniques that depend on the character of the models to be estimated: linear, nonlinear, hybrid, nonparametric etc. At the same time, the area can be characterized by a small number of leading principles, e.g. to look for sustainable descriptions by proper decisions in the triangle of model complexity, information contents in the data, and effective validation. The area has many facets and there are many approaches and methods. A tutorial or a survey in a few pages is not quite possible. Instead, this presentation aims at giving an overview of the “science ” side, i.e. basic principles and results and at pointing to open problem areas in the practical, “art”, side of how to approach and solve a real problem. 1.
A Clustering Technique for the Identification of Piecewise Affine Systems
, 2001
"... We propose a new technique for the identification of discretetime hybrid systems in the PieceWise Affine (PWA) form. This problem can be formulated as the reconstruction of a possibly discontinuous PWA map with a multidimensional domain. In order to achieve our goal, we provide an algorithm that ..."
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Cited by 49 (7 self)
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We propose a new technique for the identification of discretetime hybrid systems in the PieceWise Affine (PWA) form. This problem can be formulated as the reconstruction of a possibly discontinuous PWA map with a multidimensional domain. In order to achieve our goal, we provide an algorithm that exploits the combined use of clustering, linear identification, and pattern recognition techniques. This allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures. Moreover, the clustering step (used for classifying the datapoints) is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions. Measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure.
Generalization Bounds for Function Approximation from Scattered Noisy Data
, 1998
"... this paper we investigate the problem of providing error bounds for approximation of an unknown function from scattered, noisy data. This problem has particular relevance in the field of machine learning, where the unknown function represents the task that has to be learned and the scattered data re ..."
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Cited by 31 (1 self)
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this paper we investigate the problem of providing error bounds for approximation of an unknown function from scattered, noisy data. This problem has particular relevance in the field of machine learning, where the unknown function represents the task that has to be learned and the scattered data represents the examples of this task. An obvious quantity of interest for us is the generalization error  a measure of how much the result of the approximation scheme differs from the unknown function  typically studied as a function of the number of data points. Since the data are randomly generated and noisy, the analysis of the generalization error necessarily involves statistical considerations in addition to the traditional
A.: Functional aggregation for nonparametric regression
 Ann. Stat
, 2000
"... We consider the problem of estimating an unknown function f from N noisy observations on a random grid. In this paper we address the following aggregation problem: given M functions f 1�����f M find an “aggregated” estimator which approximates f nearly as well as the best convex combination f ∗ of f ..."
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Cited by 30 (4 self)
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We consider the problem of estimating an unknown function f from N noisy observations on a random grid. In this paper we address the following aggregation problem: given M functions f 1�����f M find an “aggregated” estimator which approximates f nearly as well as the best convex combination f ∗ of f 1�����f M. We propose algorithms which provide approximations of f ∗ with expected L 2 accuracy O�N −1/4 ln 1/4 M�. We show that this approximation rate cannot be significantly improved. We discuss two specific applications: nonparametric prediction for a dynamic system with output nonlinearity and reconstruction in the Jones– Barron class. 1. Introduction. Consider
Nonlinear BlackBox Models in System Identification: Mathematical Foundations
, 1995
"... In this paper we discuss several aspects of the mathematical foundations of nonlinear blackbox identification problem. As we shall see that the quality of the identification procedure is always a result of a certain tradeoff between the expressive power of the model we try to identify (the larger ..."
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Cited by 29 (5 self)
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In this paper we discuss several aspects of the mathematical foundations of nonlinear blackbox identification problem. As we shall see that the quality of the identification procedure is always a result of a certain tradeoff between the expressive power of the model we try to identify (the larger is the number of parameters used to describe the model, more flexible would be the approximation), and the stochastic error (which is proportional to the number of parameters). A consequence of this tradeoff is a simple fact that good approximation technique can be a basis of good identification algorithm. From this point of view we consider different approximation methods, and pay special attention to spatially adaptive approximants. We introduce wavelet and "neuron" approximations and show that they are spatially adaptive. Then we apply the acquired approximation experience to estimation problems. Finally, we consider some implications of these theoretic developments for the practically...
Identification of piecewise affine systems via mixedinteger programming
 Automatica
, 2004
"... This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes (HHARX) and Wiener piecewise affine (WPWARX) autoregressive exogenous models. In particular, we provide algorithms based on mixedinteger linear or quadratic programming ..."
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Cited by 22 (4 self)
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This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes (HHARX) and Wiener piecewise affine (WPWARX) autoregressive exogenous models. In particular, we provide algorithms based on mixedinteger linear or quadratic programming which are guaranteed to converge to a global optimum. For the special case where switches occur only seldom in the estimation data, we also suggest a way of trading off between optimality and complexity by using a change detection approach. 1
Low Entropy Coding with Unsupervised Neural Networks
"... ed on visual and speech data. The ability of the network to automatically generate wavelet codes from natural images is demonstrated. These bear a close resemblance to 2D Gabor functions, which have previously been used to describe physiological receptive fields, and as a means of producing compact ..."
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Cited by 20 (0 self)
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ed on visual and speech data. The ability of the network to automatically generate wavelet codes from natural images is demonstrated. These bear a close resemblance to 2D Gabor functions, which have previously been used to describe physiological receptive fields, and as a means of producing compact image representations. Keywords: neural networks, unsupervised learning, selforganisation, feature extraction, information theory, redundancy reduction, sparse coding, imaging models, occlusion, image coding, speech coding. Declaration This dissertation is the result of my own original work, except where reference is made to the work of others. No part of it has been submitted for any other university degree or diploma. Its length, including captions, footnotes, appendix and bibliography, is approximately 58000 words. Acknowledgements I would like first and foremost to thank Richard Prager, my supervisor, fo