A Comparison of Approaches to Stepwise Regression Analysis for Variables Sensitivity Measurements Used with a Multi-Objective Optimization Problem

Abstract

Global sensitivity analysis can be used to identify and rank variables importance (sensitivities) for design objectives and constraints, where the solution space is sampled and a linear regression model is normally adopted in the stepwise manner. The relative importance of variables can be examined by three indicators: the order of variables entry into the linear regression model; the absolute values of the standardized regression coefficients or their rank transformation coefficients; and the size of the R2 changes (coefficient of determination) attributable to additional variables at each step. However, the robustness of the linear regression model constructed from a stepwise regression is related to the choice of procedure options, e.g. the set of samples and data formulation. Different procedure options could lead to different linear regression models, and therefore influence the indication of variables global sensitivities. Thus, this paper investigates the extent to which the procedure options of a stepwise regression can influence the indication of variables global sensitivities, measured by three different sensitivity indicators, for energy demand, capital costs and solution infeasibility, when using both the randomly generated samples and the biased solutions obtained at the start of a multi-objective optimization process (based on NSGA-II). It concludes that the most important variables are always ranked on the top no matter the choice of procedure options, but it is better to adopt both the entry-orders of variables and their standardized (rank) regression coefficients or the contributions to R2 changes, to provide robust orderings of variables importance, for design objectives and constraints. Moreover, when the sample size is smaller, re-generated separate set of samples for sensitivity analysis can avoid misleading variables importance, especially for the variables ranked in the middle. Finally, to improve computational efficiency, this paper concludes that the first 100 solutions obtained from a multi-objective optimization can be used to perform global sensitivity analysis, to identify the important variables for design objectives.