Robust Quadratic Regression and Its Application to Energy-Growth Consumption Problem

Yongzhi Wang,Yuli Zhang,Fuliang Zhang,Jining Yi

doi:10.1155/2013/210510

Abstract

We propose a robust quadratic regression model to handle the statistics inaccuracy. Unlike the traditional robust statistic approaches that mainly focus on eliminating the effect of outliers, the proposed model employs the recently developed robust optimization methodology and tries to minimize the worst-case residual errors. First, we give a solvable equivalent semidefinite programming for the robust least square model with ball uncertainty set. Then the result is generalized to robust models underl1- andl∞-norm critera with general ellipsoid uncertainty sets. In addition, we establish a robust regression model for per capital GDP and energy consumption in the energy-growth problem under the conservation hypothesis. Finally, numerical experiments are carried out to verify the effectiveness of the proposed models and demonstrate the effect of the uncertainty perturbation on the robust models.

Highlights

Traditional regression analysis is a useful tool to model the linear or nonlinear relationship between the observed data
From the subfigures on the left hand side, we can see that in both countries there is a gradual increase in economy while the per capital energy consumption may decrease after reaching a certain level; the subfigures on the right hand side inspire us to establish a nonlinear regression model to characterize the relationship
We studied the multivariate quadratic regression model with imprecise statistic data

Summary

Introduction

Traditional regression analysis is a useful tool to model the linear or nonlinear relationship between the observed data. This weak exogeneity assumption makes the linear regression model very powerful to fit the given data or predict the regressand for given known regressors, it may lead to overfitting or inconsistent estimations [1]. Our proposed method is applied in order to safely eliminate the effect generated from imprecise data Under certain assumption on the uncertainty sets, we can obtain a series of equivalent semidefinite programming formulations for robust quadratic regression under different residual error criteria. We first extend the traditional quadratic regression model by introducing the separable ball (2-norm) uncertainty set and formulate the optimal robust regression problem as a min-max problem that tries to minimize the maximal residual error.

Robust Quadratic Regression Models

Robust Energy-Growth Regression Models

Numerical Experiments

Conclusions and Future Works