Adaptive Nonparametric Regression Model via a Global Mixing Parameter for the Multi-Response Problem

Abstract

The modeling stage of Response Surface Methodology (RSM) involves using regression models to estimate the functional relationship between the response variable and explanatory variables, relying on data generated through an appropriate experimental design. Traditionally, Ordinary Least Squares (OLS) is employed to model the data using user-specified low-order polynomials. However, OLS performance deteriorates when the homoscedasticity assumption is violated. In the literature, semiparametric regression models are preferred for RSM as they combine the strengths of parametric and nonparametric approaches, unlike purely nonparametric models, 
which are more sensitive to the idiosyncrasies of RSM data. This paper proposes a novel integration of an adaptive nonparametric regression model with a locally adaptive bandwidth selector derived from the explanatory variables to achieve adequate data smoothing. The adaptive nonparametric regression model incorporates local linear regression (LLR) and a product of the optimal mixing parameter and the LLR residuals, providing a second chance to fit portions of the data not captured by the LLR model. Meanwhile, the locally adaptive bandwidth selector addresses challenges such as dimensionality, sparsity in RSM data, and costefficient design. In applying this approach to three types of RSM data, the novel integrated model demonstrated superior performance in terms of goodness-of-fit statistics, zero residual plots, optimization results, and simulations, when compared to OLS, Model Robust Regression 1 (MRR1), and Model Robust Regression 2 (MRR2).