Local and weighted regression. Bias reduction and model validation
Nonlinear systems might be estimated, using local linear models. If the estimation data is corrupted by strongly colored noise the local model will have a bias error. In linear system identification the bias error can be reduced by using instrumentalvariable methods. In this thesis, the problem with bias error in local models have been addressed, by adapting linear methods to local models. Two different two-step estimation methods are presented. Both use the fact that the simulated output of a high-order ARX-model, is an approximation of the noise-free output from the system. The first method is a local version of a twostep ARX technique, where the second step uses an ARX-model with a shorter regressor than in the first step. The second method uses the simulated output from the first step as an instrumental-variable in the second step. The impact of the two methods on the bias error is discussed. Expressions for the variance error for both the local two-step ARX-estimate and the local IV-method have been derived. Estimation examples using both simulated data, demonstrating the noise influence and the variance expressions, and laboratory data are included. The thesis also addresses model validation by showing how residual analysis for nonlinear systems can be simplified by using a parametric model of the covariance. A chi-square distributed test variable is derived for testing for correlation of the residuals.