Bayesian sequential - optimal model-robust designs (original) (raw)
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In industrial experiments, cost considerations will sometimes make it impractical to design experiments so that effects of all the factors can be estimated simultaneously. Therefore experimental designs are frequently constructed to estimate main effects and a few pre-specified interactions. A criticism frequently associated with the use of many optimality criteria is the specific reliance on an assumed statistical model. One way to deal with such a criticism may be to assume that instead the true model is an approximation of an unknown element of a known set of models. In this paper, we consider a class of designs that are robust for change in model specification. This paper is motivated by the belief that appropriate Bayesian approaches may also perform well in constructing model robust designs and by the limitation of such approaches in the literature. I will use the traditional Bayesian design method for parameter estimation and incorporates a discrete prior probability on the set of models of interest. Some examples and comparisons with existing approaches will be provided.
Development of a new robust design methodology based on Bayesian perspectives
International Journal of Quality Engineering and Technology, 2012
Robust design (RD), implemented in statistical and mathematical procedures to simultaneously minimise the process bias and variability, is widely used in many areas of engineering and technology to represent complex real-world industrial settings. For RD modelling and optimisation, response surface methodology (RSM) is often utilised as an estimation method to represent the functional relationship between input factors and their associated output responses. Although conventional RSM-based RD methods may offer significant advantages regarding process design, there is room for improvement. In this context, a new RD methodology is developed in this paper by integrating Bayesian principles into the RD procedure. Numerical examples and comparative studies are conducted by using two conventional RSM-based RD models and the proposed model. The results of two numerical examples demonstrate that the proposed RD method provides significantly better RD solutions in terms of the expected quality loss (EQL) than conventional methods.
Model-robust and model-sensitive designs
Computational Statistics & Data Analysis, 2005
The main drawback of the optimal design approach is that it assumes the statistical model is known. In this paper, a new approach to reduce the dependency on the assumed model is proposed. The approach takes into account the model uncertainty by incorporating the bias in the design criterion and the ability to test for lack-of-fit. Several new designs are derived in the paper and they are compared to the alternatives available from the literature.
Two stage designs robust to model uncertainty
Leuven, KU Leuven, Faculteit Economische en …, 2004
D-optimal designs are known to depend quite critically on the particular model that is assumed. These designs tend to concentrate all the experimental runs on a small number of design points and are ideally suited for estimating the coefficients of the assumed model, but they provide little or no ability for model checking. To address this problem we use the notion of empirical models that have both important and potential terms. We propose within the Bayesian paradigm, a two-stage design strategy for planning experiments in the face of model uncertainty. In the first stage, the experimenter's prime interest is to highlight the uncertainties in the specification of the model in order to refine or modify the model(s) initially entertained. A design criterion is used that accounts for precision of the important terms but also facilitates the improvement of the proposed model(s) by detecting lack of fit. Data from the first stage provide model information enabling the second stage design to be chosen efficiently with reduced model uncertainty. The design in the second stage is obtained using a weighted criterion with weights being posterior model probabilities computed from first stage data. The criterion in the second stage also takes into account precise estimation of important terms as in the first stage but now attempts to minimize bias with respect to potential terms. Results from simulations show that the proposed two-stage strategy performs well. The combined first and second stage design has good properties with respect to precision of important terms, lack of fit and also excellent bias properties with respect to a true assumed model in various simulation studies.
Model-sensitive sequential optimal designs
The increasing number of experimenters using computer-generated experimental designs creates an increasing need to have design procedures that are less sensitive to model misspecification. To address this problem, the notion of empirical models that have both important and potential terms is used. A two-stage design strategy for planning experiments in the face of model uncertainty is proposed. The advantage of this procedure resides in the rearrangement of active potential terms at the end of the first stage using marginal posterior probabilities of different candidate models. The two-stage procedure has better estimation efficiency than its one-stage alternatives available from the literature.
Comparing robustness properties of optimal designs under standard and compound criteria
arXiv: Methodology, 2017
Standard optimality criteria (e.g. A-, D-optimality criterion, etc.) have been commonly used for obtaining optimal designs. For a given statistical model, standard criteria assume the error variance is known at the design stage. However, in practice the error variance is estimated to make inference about the model parameters. Modified criteria are defined as a function of the standard criteria and the corresponding error degrees of freedom, which may lead to extreme optimal design. Compound criteria are defined as the function of different modified criteria and corresponding user specified weights. Standard, modified, and compound criteria based optimal designs are obtained for 333^333 factorial design. Robustness properties of the optimal designs are also compared.
Robust design using Bayesian Monte Carlo
International Journal for Numerical Methods in Engineering, 2008
In this paper, we propose an efficient strategy for robust design based on Bayesian Monte Carlo simulation. Robust design is formulated as a multiobjective problem to allow explicit trade-off between the mean performance and variability. The proposed method is applied to a compressor blade design in the presence of manufacturing uncertainty. Process capability data are utilized in conjunction with a parametric geometry model for manufacturing uncertainty quantification. High-fidelity computational fluid dynamics simulations are used to evaluate the aerodynamic performance of the compressor blade. A probabilistic analysis for estimating the effect of manufacturing variations on the aerodynamic performance of the blade is performed and a case for the application of robust design is established. The proposed approach is applied to robust design of compressor blades and a selected design from the final Pareto set is compared with an optimal design obtained by minimizing the nominal performance. The selected robust blade has substantial improvement in robustness against manufacturing variations in comparison with the deterministic optimal blade. Significant savings in computational effort using the proposed method are also illustrated. Copyright
Comparing robust properties of A, D, E and G-optimal designs
Computational Statistics & Data Analysis, 1994
We examine the A, D, E and G-efficiencies of using the optimal design for the polynomial regression model of degree k when the hypothesized model is of degree j and 1 Q j < k Q 8. The robustness properties of each of these optimal designs with respect to the other optimal&y criteria are also investigated. Relationships among these efficiencies are noted and practical implications of the results are discussed. In particular, our numerical results show E-optimal designs possess several properties not shared by the A, D and G-optimal designs.