Bayesian Modeling and Optimization of Functional Responses Affected by Noise Factors

Experiments in systems where each run generates a curve, that is, where the response of interest is a set of observed values of a function, are common in engineering. In this paper, we present a Bayesian predictive modeling approach for functional response systems. The goal is to optimize the shape, or profile, of the functional response. A robust parameter design scenario is assumed where there are controllable factors and noise factors that vary randomly according to some distribution. The approach incorporates the uncertainty in the model parameters in the optimization phase, extending earlier approaches by J. Peterson (in the multivariate regression case) to the functional response case based on a hierarchical two-stage mixed-effects model. The method is illustrated with real examples taken from the literature and with simulated data, and practical aspects related to model building and diagnostics of the assumed mixed-effects model are discussed.

[1]  John J. Peterson,et al.  A Bayesian Reliability Approach to Multiple Response Optimization with Seemingly Unrelated Regression Models , 2009 .

[2]  John M. Olin.  On MCMC sampling in hierarchical longitudinal models , 1999 .

[3]  A. Rukhin.  Bayes and Empirical Bayes Methods for Data Analysis , 1997 .

[4]  Joseph Hilbe,et al.  Data Analysis Using Regression and Multilevel/Hierarchical Models , 2009 .

[5]  N. H. Timm.  Applied Multivariate Analysis , 2002 .

[6]  D. Lindley,et al.  Bayes Estimates for the Linear Model , 1972 .

[7]  C. F. Jeff Wu,et al.  Experiments: Planning, Analysis, and Parameter Design Optimization , 2000 .

[8]  A. Gelfand,et al.  Hierarchical Bayes Models for the Progression of HIV Infection Using Longitudinal CD4 T-Cell Numbers , 1992 .

[9]  Kwok-Leung Tsui,et al.  Analysis of dynamic robust design experiments , 1997 .

[10]  Winson Taam,et al.  Analysis of Functional Responses from Robust Design Studies , 2002 .

[11]  David E. Booth,et al.  Applied Multivariate Analysis , 2003, Technometrics.

[12]  Sung-Min Park,et al.  GLMs for the Analysis of Robust Designs with Dynamic Characteristics , 2003 .

[13]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[14]  Arden Miller,et al.  Analysis of Parameter Design Experiments for Signal-Response Systems , 2002 .

[15]  John J. Peterson,et al.  A Bayesian Approach for Multiple Response Surface Optimization in the Presence of Noise Variables , 2004 .

[16]  John J. Peterson.  A Posterior Predictive Approach to Multiple Response Surface Optimization , 2004 .

[17]  Peter Z. G. Qian,et al.  Bayesian Hierarchical Modeling for Integrating Low-Accuracy and High-Accuracy Experiments , 2008, Technometrics.

[18]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[19]  V. Roshan Joseph,et al.  Robust Parameter Design of Multiple-Target Systems , 2002, Technometrics.

[20]  J. Ware,et al.  Applied Longitudinal Analysis , 2004 .

[21]  Arden Miller,et al.  Parameter Design for Signal-Response Systems: A Different Look at Taguchi's Dynamic Parameter Design , 1996 .

[22]  Bernadette Govaerts,et al.  Analysing the Results of a Designed Experiment when the Response is a Curve: Methodology and Application in Metal Injection Moulding , 2005 .

[23]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.

[24]  William F. Guthrie.  Bayesian Statistical Modeling, 2nd edition , 2009 .

[25]  P. Royston.  A Remark on Algorithm as 181: The W‐Test for Normality , 1995 .

[26]  Runze Li,et al.  Design and Modeling for Computer Experiments , 2005 .

[27]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[28]  Enrique Del Castillo,et al.  Model and Distribution-Robust Process Optimization with Noise Factors , 2005 .

[29]  Mahmoud A. Mahmoud,et al.  On the Monitoring of Linear Profiles , 2003 .

[30]  K. Tsui.  Modeling and analysis of dynamic robust design experiments , 1999 .

[31]  S. R. Searle,et al.  Vec and vech operators for matrices, with some uses in jacobians and multivariate statistics , 1979 .

[32]  J. Ware.  Linear Models for the Analysis of Longitudinal Studies , 1985 .