Robust Regression

Journal of Signal Processing Systems, 2014

Polynomial nonlinear system identification suffers from numerical instability related to the ill-conditioning of the involved matrices. Orthogonal methods consist in conditioning the input signal in order to reduce the eigenvalues spread of the correlation matrix. The selection of an appropriate orthogonalization procedure, for robustness improvement, resort to signal statistics. Consequently, several orthogonal polynomial bases are proposed in the literature. Most of them use iterative processes and imply a considerable computational cost. Actually, with the growth of real-time applications, it is important to generate noniterative orthogonal procedures, allowing optimization of algorithm-architecture adequacy. Our paper's motivation is based on the complexity reduction aspect related to the orthogonalization process. Therefore we propose to focus on closed-form expressions of commonly used orthogonal polynomials bases namely the Shifted-Legendre orthogonal polynomials and the Hermite polynomials. A robustness study in terms of numerical stability enhancement of the two bases is carried on. Through comparative simulations results, the basis allowing the best matrix conditioning and an ease generalization for real-time applications with less restrictive hypothesis is selected in order to study the robustness of a polynomial nonlinear system identification

When developing efficient and reliable computer-aided control system design (CACSD) tools for low-order robust control systems analysis and synthesis, the main issue faced by theoreticians and practitioners is the non-convexity of the stability domain in the space of polynomial coefficients, or equivalently, in the space of design parameters. In this paper, we survey some of the recently developed techniques to overcome this non-convexity, underlining their respective pros and cons. We also enumerate some related open research problems which, in our opinion, deserve particular attention.

IEEE Transactions on Automatic Control, 1998

Dhaka University Journal of Science, 2015

For building a linear prediction model, Backward Elimination (BE) is a computationally suitable stepwise procedure for sequencing the candidate predictors. This method yields poor results when data contain outliers and other contaminations. Robust model selection procedures, on the other hand, are not computationally efficient or scalable to large dimensions, because they require the fitting of a large number of submodels. Robust version of BE is proposed in this study, which is computationally very suitable and scalable to large high-dimensional data sets. Since BE can be expressed in terms of sample correlations, simple robustifications are obtained by replacing these correlations by their robust counterparts. A pairwise approach is used to construct the robust correlation matrix -- not only because of its computational advantages over the d-dimensional approach, but also because the pairwise approach is more consistent with the idea of step-by-step algorithms. The performance of ...

2007

Partial least squares regression (PLS) is a linear regression technique developed to relate many regressors to one or several response variables. Robust methods are introduced to reduce or remove the effect of outlying data points. In this paper we show that if the sample covariance matrix is properly robustified further robustification of the linear regression steps of the PLS algorithm

Journal of Mathematical Problems, Equations and Statistics, 2025

An arbitrary order polynomial regression analysis is presented. Under the assumption of satisfaction of some reference points exactly by the polynomial, the general theory is given for the analysis. The polynomials are expressed in a suitable form so that the lower order coefficients can be calculated easily from the restrictions. The remaining coefficients are calculated from the error minimization. The theory is then applied to linear, quadratic and cubic polynomial regression sample problems having constrained data. Compared to the unrestrained regression analysis, the method reduces the computational cost of calculating coefficients.

Journal of Chemometrics, 2009

Partial least squares regression (PLS) is a linear regression technique developed to relate many regressors to one or several response variables. Robust methods are introduced to reduce or remove the effect of outlying data points. In this paper we show that if the sample covariance ma- trix is properly robustified further robustification of the linear regression steps of the PLS

ACM Communications in Computer Algebra, 2010

A new Maple package for solving parametric systems of polynomial equations and inequalities is described. The main idea for solving such a system is as follows. The parameter space Rd is divided into two parts: the discriminant variety W and its complement R d nW . The discriminant variety is a generalization of the well-known discriminant of a univariate polynomial and contains all those parameter values leading to non-generic solutions of the system. The complement R d nW can be expressed as a finite union of open cells such that the number of real solutions of the input system is constant on each cell. In this way, all parameter values leading to generic solutions of the system can be described systematically. The underlying techniques used are Gröbner bases, polynomial real root finding, and cylindrical algebraic decomposition. This package offers a friendly interface for scientists and engineers to solve parametric problems, as illustrated by an example from control theory.

2001

Let (Xj ,Y j) n=1 be a realization of a bivariate jointly strictly station- ary process. We consider a robust estimator of the regression function m(x )= E(Y |X = x) by using local polynomial regression techniques. The estimator is a local M-estimator weighted by a kernel function. Under mixing conditions satisfied by many time series models, together with other appropriate conditions, consistency and asymptotic normality results are established. One-step local M-estimators are introduced to reduce computational burden. In addition, we give a data-driven choice for minimizing the scale factor involving the ψ-function in the asymptotic covariance expression, by drawing a parallel with the class of Huber's ψ-functions. The method is illustrated via two examples.

1999

The paper considers the robust Schur stability veri cation of polynomials with coe cients depending polynomially on parameters varying in given intervals. A new algorithm is presented which relies on the expansion of a multivariate polynomial into Bernstein polynomials and is based on the decomposition of the family of polynomials into its symmetric and antisymmetric parts. It is shown how the inspection of both polynomial families on the upper half of the unit circle can be reduced to the analysis of two related polynomial families on the real interval ?1;1]. Then the Bernstein expansion can be applied in order to check whether both polynomial families have a zero in this interval in common.