Contributions to variable selection for mean modeling and variance modeling in computer experiments
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This thesis consists of two parts. The first part reviews a Variable Search, a variable selection procedure for mean modeling. The second part deals with variance modeling for robust parameter design in computer experiments. In the first chapter of my thesis, Variable Search (VS) technique developed by Shainin (1988) is reviewed. VS has received quite a bit of attention from experimenters in industry. It uses the experimenters' knowledge about the process, in terms of good and bad settings and their importance. In this technique, a few experiments are conducted first at the best and worst settings of the variables to ascertain that they are indeed different from each other. Experiments are then conducted sequentially in two stages, namely swapping and capping, to determine the significance of variables, one at a time. Finally after all the significant variables have been identified, the model is fit and the best settings are determined. The VS technique has not been analyzed thoroughly. In this report, we analyze each stage of the method mathematically. Each stage is formulated as a hypothesis test, and its performance expressed in terms of the model parameters. The performance of the VS technique is expressed as a function of the performances in each stage. Based on this, it is possible to compare its performance with the traditional techniques. The second and third chapters of my thesis deal with variance modeling for robust parameter design in computer experiments. Computer experiments based on engineering models might be used to explore process behavior if physical experiments (e.g. fabrication of nanoparticles) are costly or time consuming. Robust parameter design (RPD) is a key technique to improve process repeatability. Absence of replicates in computer experiments (e.g. Space Filling Design (SFD)) is a challenge in locating RPD solution. Recently, there have been studies (e.g. Bates et al. (2005), Chen et al. (2006), Dellino et al. (2010 and 2011), Giovagnoli and Romano (2008)) of RPD issues on computer experiments. Transmitted variance model (TVM) proposed by Shoemaker and Tsui. (1993) for physical experiments can be applied in computer simulations. The approaches stated above rely heavily on the estimated mean model because they obtain expressions for variance directly from mean models or by using them for generating replicates. Variance modeling based on some kind of replicates relies on the estimated mean model to a lesser extent. To the best of our knowledge, there is no rigorous research on variance modeling needed for RPD in computer experiments. We develop procedures for identifying variance models. First, we explore procedures to decide groups of pseudo replicates for variance modeling. A formal variance change-point procedure is developed to rigorously determine the replicate groups. Next, variance model is identified and estimated through a three-step variable selection procedure. Properties of the proposed method are investigated under various conditions through analytical and empirical studies. In particular, impact of correlated response on the performance is discussed.