In this note, we propose partially linear models with least squares support vector machines (LS-SVMs) for nonlinear ARX models. We illustrate how full black-box models can be improved when prior information about model structure is available. A real-life example, based on the Silverbox benchmark data, shows significant improvements in the generalization ability of the structured model with respect to the full black-box model, reflected also by a reduction in the effective number of parameters.

Kernel based partially linear models and nonlinear identification

In this technical note, we propose a partially-linear least-squares regularized regression (PL-LSRR) method for system identification. This method identifies a general nonlinear function as a sum of two functions which come from a linear and a nonlinear function space respectively. Both the linear and nonlinear functions can involve all regressors. Therefore, the PL-LSRR can make use of the partially-linear structure of a given system to reduce prediction errors more efficiently than exiting partially-linear identification methods. Two examples show that the PL-LSRR can reduce prediction errors and estimate the true linear expansion of the system well.

Partially-Linear Least-Squares Regularized Regression for System Identification

Support vector machine (SVM) is a universal learning method. In this paper, an affine support vector machine (ASVM) for regression is presented for identification and control of input-affine nonlinear models. ASVM is a variant of SVM and so inherits its merits. The solution to ASVM is cast into a convex quadratic programming (QP). Hence ASVM has a unique global solution. In addition, the curse of dimensionality is avoided because ASVM is insensitive to the dimensionality of data. A commonly used model for a nonlinear system is a nonlinear autoregressive exogenous (NARX) model. ASVM could get good performance in both identification and control if a NARX model can be well represented by an input-affine nonlinear model. The experimental results validate the efficiency of ASVM in identification and control of discrete-time nonlinear systems.

Identification and control of discrete-time nonlinear systems using affine support vector machines

This paper formulates the problem of learning from examples as a scattered data interpolation problem, and develops a new method that computes interpolan ts that minimize a uveletbased reproducing kernel Hilbert space (RKHS) norm subject to in terpolatory constraias. In contrast to radial basis function kernels, these k ernels are not translation iwariant. Some computational geometry methods are used to construct spatial-adaptive kernels based on local distribution density of unev enly distributed data examples.

http://ieeexplore.ieee.org/iel5/6910/18592/00856440.pdf

Learning from examples with spatial-adaptive wavelet-based reproducing kernels

A fundamental issue in Data Mining is the development of algorithms to extract useful information from very large databases. One important technique is to estimate a smooth function approximating the data. Such an approximation can for example be used for visualisation, prediction, or classification purposes. However, the number of observations can be of the order of millions and there may be hundreds of variables recorded so one has to deal with the so-called ”curse of dimensionality”. The algorithmic complexity of this process is typically of the order γ3d−2 where γ is the number of grid points in each dimension and d is the number of dimensions. We propose a method for approximating a high dimensional surface by computing a projection onto multi-level spaces of low density and we demonstrate that the algorithmic complexity of this method is proportional to (jd−1(2j+1 − 1))3, where j = ⌊log2 γ⌋ — a substantial reduction in computational work. In addition, we show that the approximation error is proportional to (j+d−1 d−1 ) 2−2j , with the proportionality constant depending on the smoothness of the computed surface.

High Dimensional Smoothing Based on Multilevel Analysis

This paper formulates the nonlinear system modeling problem as a scattered data interpolation problem, and develops a new method that computes interpolants that minimize a wavelet-based norm subject to interpolatory constraints. The norm is that of a Reproducing Kernel Hilbert Space (RKHS) for which the wavelet functions that form an orthonormal basis for L 2 are orthogonal. In contrast to radial basis function kernels, these kernels are not translation invariant and they may be designed to provide spatially varying resolution useful for interpolating from unevenly distributed data samples. Furthermore, the discrete wavelet transform can be exploited to efficiently compute the values of the interpolant on a uniform grid. Modeling of systems in Sobolev spaces using different interpolation methods is discussed in details as a special case of RKHS. Numerical simulations are used to demonstrate the effectiveness of the method.

Wavelet Based Modeling of Nonlinear Systems

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Wavelet Based Modeling of Nonlinear Systems