23,964 research outputs found
A unified wavelet-based modelling framework for non-linear system identification: the WANARX model structure
A new unified modelling framework based on the superposition of additive submodels, functional components, and
wavelet decompositions is proposed for non-linear system identification. A non-linear model, which is often represented
using a multivariate non-linear function, is initially decomposed into a number of functional components via the wellknown
analysis of variance (ANOVA) expression, which can be viewed as a special form of the NARX (non-linear
autoregressive with exogenous inputs) model for representing dynamic input–output systems. By expanding each functional
component using wavelet decompositions including the regular lattice frame decomposition, wavelet series and
multiresolution wavelet decompositions, the multivariate non-linear model can then be converted into a linear-in-theparameters
problem, which can be solved using least-squares type methods. An efficient model structure determination
approach based upon a forward orthogonal least squares (OLS) algorithm, which involves a stepwise orthogonalization
of the regressors and a forward selection of the relevant model terms based on the error reduction ratio (ERR), is
employed to solve the linear-in-the-parameters problem in the present study. The new modelling structure is referred to
as a wavelet-based ANOVA decomposition of the NARX model or simply WANARX model, and can be applied to
represent high-order and high dimensional non-linear systems
Approximation with Random Bases: Pro et Contra
In this work we discuss the problem of selecting suitable approximators from
families of parameterized elementary functions that are known to be dense in a
Hilbert space of functions. We consider and analyze published procedures, both
randomized and deterministic, for selecting elements from these families that
have been shown to ensure the rate of convergence in norm of order
, where is the number of elements. We show that both randomized and
deterministic procedures are successful if additional information about the
families of functions to be approximated is provided. In the absence of such
additional information one may observe exponential growth of the number of
terms needed to approximate the function and/or extreme sensitivity of the
outcome of the approximation to parameters. Implications of our analysis for
applications of neural networks in modeling and control are illustrated with
examples.Comment: arXiv admin note: text overlap with arXiv:0905.067
A new class of wavelet networks for nonlinear system identification
A new class of wavelet networks (WNs) is proposed for nonlinear system identification. In the new networks, the model structure for a high-dimensional system is chosen to be a superimposition of a number of functions with fewer variables. By expanding each function using truncated wavelet decompositions, the multivariate nonlinear networks can be converted into linear-in-the-parameter regressions, which can be solved using least-squares type methods. An efficient model term selection approach based upon a forward orthogonal least squares (OLS) algorithm and the error reduction ratio (ERR) is applied to solve the linear-in-the-parameters problem in the present study. The main advantage of the new WN is that it exploits the attractive features of multiscale wavelet decompositions and the capability of traditional neural networks. By adopting the analysis of variance (ANOVA) expansion, WNs can now handle nonlinear identification problems in high dimensions
Theoretical Properties of Projection Based Multilayer Perceptrons with Functional Inputs
Many real world data are sampled functions. As shown by Functional Data
Analysis (FDA) methods, spectra, time series, images, gesture recognition data,
etc. can be processed more efficiently if their functional nature is taken into
account during the data analysis process. This is done by extending standard
data analysis methods so that they can apply to functional inputs. A general
way to achieve this goal is to compute projections of the functional data onto
a finite dimensional sub-space of the functional space. The coordinates of the
data on a basis of this sub-space provide standard vector representations of
the functions. The obtained vectors can be processed by any standard method. In
our previous work, this general approach has been used to define projection
based Multilayer Perceptrons (MLPs) with functional inputs. We study in this
paper important theoretical properties of the proposed model. We show in
particular that MLPs with functional inputs are universal approximators: they
can approximate to arbitrary accuracy any continuous mapping from a compact
sub-space of a functional space to R. Moreover, we provide a consistency result
that shows that any mapping from a functional space to R can be learned thanks
to examples by a projection based MLP: the generalization mean square error of
the MLP decreases to the smallest possible mean square error on the data when
the number of examples goes to infinity
The wavelet-NARMAX representation : a hybrid model structure combining polynomial models with multiresolution wavelet decompositions
A new hybrid model structure combing polynomial models with multiresolution wavelet decompositions is introduced for nonlinear system identification. Polynomial models play an important role in approximation theory, and have been extensively used in linear and nonlinear system identification. Wavelet decompositions, in which the basis functions have the property of localization in both time and frequency, outperform many other approximation schemes and offer a flexible solution for approximating arbitrary functions. Although wavelet representations can approximate even severe nonlinearities in a given signal very well, the advantage of these representations can be lost when wavelets are used to capture linear or low-order nonlinear behaviour in a signal. In order to sufficiently utilise the global property of polynomials and the local property of wavelet representations simultaneously, in this study polynomial models and wavelet decompositions are combined together in a parallel structure to represent nonlinear input-output systems. As a special form of the NARMAX model, this hybrid model structure will be referred to as the WAvelet-NARMAX model, or simply WANARMAX. Generally, such a WANARMAX representation for an input-output system might involve a large number of basis functions and therefore a great number of model terms. Experience reveals that only a small number of these model terms are significant to the system output. A new fast orthogonal least squares algorithm, called the matching pursuit orthogonal least squares (MPOLS) algorithm, is also introduced in this study to determine which terms should be included in the final model
Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networks
Nonlinear model predictive control (NMPC) often requires real-time solution
to optimization problems. However, in cases where the mathematical model is of
high dimension in the solution space, e.g. for solution of partial differential
equations (PDEs), black-box optimizers are rarely sufficient to get the
required online computational speed. In such cases one must resort to
customized solvers. This paper present a new solver for nonlinear
time-dependent PDE-constrained optimization problems. It is composed of a
sequential quadratic programming (SQP) scheme to solve the PDE-constrained
problem in an offline phase, a proper orthogonal decomposition (POD) approach
to identify a lower dimensional solution space, and a neural network (NN) for
fast online evaluations. The proposed method is showcased on a regularized
least-square optimal control problem for the viscous Burgers' equation. It is
concluded that significant online speed-up is achieved, compared to
conventional methods using SQP and finite elements, at a cost of a prolonged
offline phase and reduced accuracy.Comment: Accepted for publishing at the 58th IEEE Conference on Decision and
Control, Nice, France, 11-13 December, https://cdc2019.ieeecss.org
Generalised additive multiscale wavelet models constructed using particle swarm optimisation and mutual information for spatio-temporal evolutionary system representation
A new class of generalised additive multiscale wavelet models (GAMWMs) is introduced for high dimensional spatio-temporal evolutionary (STE) system identification. A novel two-stage hybrid learning scheme is developed for constructing such an additive wavelet model. In the first stage, a new orthogonal projection pursuit (OPP) method, implemented using a particle swarm optimisation(PSO) algorithm, is proposed for successively augmenting an initial coarse wavelet model, where relevant parameters of the associated wavelets are optimised using a particle swarm optimiser. The resultant network model, obtained in the first stage, may however be a redundant model. In the second stage, a forward orthogonal regression (FOR) algorithm, implemented using a mutual information method, is then applied to refine and improve the initially constructed wavelet model. The proposed two-stage hybrid method can generally produce a parsimonious wavelet model, where a ranked list of wavelet functions, according to the capability of each wavelet to represent the total variance in the desired system output signal is produced. The proposed new modelling framework is applied to real observed images, relative to a chemical reaction exhibiting a spatio-temporal evolutionary behaviour, and the associated identification results show that the new modelling framework is applicable and effective for handling high dimensional identification problems of spatio-temporal evolution sytems
Representation of Functional Data in Neural Networks
Functional Data Analysis (FDA) is an extension of traditional data analysis
to functional data, for example spectra, temporal series, spatio-temporal
images, gesture recognition data, etc. Functional data are rarely known in
practice; usually a regular or irregular sampling is known. For this reason,
some processing is needed in order to benefit from the smooth character of
functional data in the analysis methods. This paper shows how to extend the
Radial-Basis Function Networks (RBFN) and Multi-Layer Perceptron (MLP) models
to functional data inputs, in particular when the latter are known through
lists of input-output pairs. Various possibilities for functional processing
are discussed, including the projection on smooth bases, Functional Principal
Component Analysis, functional centering and reduction, and the use of
differential operators. It is shown how to incorporate these functional
processing into the RBFN and MLP models. The functional approach is illustrated
on a benchmark of spectrometric data analysis.Comment: Also available online from:
http://www.sciencedirect.com/science/journal/0925231
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