Bayesian hierarchical modeling for monitoring optical profiles in low-E glass manufacturing processes

Abstract

Low-emittance (low-E) glass manufacturing has become an important sector of the glass industry for energy efficiency of such glasses. However, the quality control scheme in the current processes is rather primitive and advanced statistical quality control methods need to be developed. As the first attempt for this purpose, this article considers monitoring of optical profiles, which are typical quality measurements in low-E glass manufacturing. A Bayesian hierarchical approach is proposed for modeling the optical profiles, which conducts model selection and estimation in an integrated framework. The effectiveness of the proposed approach is validated in a numerical study, and its use in Phase I analysis of optical profiles is demonstrated in a case study. The proposed approach will lay a foundation for quality control and variation reduction in low-E glass manufacturing.

Keywords:

• Bayes factors
• Gibbs sampling
• hierarchical linear mixed-effect (HLME) model
• Phase I analysis
• polynomial models

Appendices

Appendix A:. The Gibbs sampling procedure in Section 3

Starting step: Specify starting values

A simple method to find starting values is fitting a regular polynomial model to each profile. The coefficient estimates will be used as starting values of β, the sample variances of those estimates will be used as starting values of {σ²_p, …, σ₀²}, and the estimate of random error variance will be used as the starting value of σ²_ϵ. The random effects can simply be set to zero.

• Generate a sample {α^(g)_i, i = 1, …, m} from the conditional posterior in Equation (9)(9) .

• Generate a sample {σ^2(g)_p, …, σ₀^2(g)} from the conditional posterior in Equation (10)(10) .

• Generate a sample from the conditional posterior in Equation (11)(11) .

Then go back to Step 1 and repeat this process for g = 1, … , G.

Appendix B: Proof of Equation (9)(9)

We will first prove a proposition and then prove Equation (9)(9) based on that.

Proposition A1. Let U₁, U₂, … , U_n be n independently normally distributed random variables with a common mean and scaled variances; that is, where c₁, … , c_n are known non-zero constants, σ² is known, and θ is the unknown mean that needs to be estimated. Let u₁, u₂, … , u_n be the corresponding observations of these variables. Under a normal prior for θ; that is, the posterior is where Proof. Under the given prior, the posterior of θ is which is a normal distribution with mean and variance The result in the proposition can be obtained by simple manipulations of the above formulas.▪

Coming back to the HLME model, by Equation (3)(3) : which can also be written as (A1) Let z_i = [z_i1, … , z_in]′, x^k = [x^k₁, …, x_n^k]′, and ϵ_i = [ϵ_i1, … , ϵ_in]′, by Equations (A1) and (1), or, equivalently, We can treat z_ij/x^k_j, j = 1, … , n, as n independently normally distributed variables with a common mean α_i,k and scaled variances σ²_ϵ/(x_j^k)², which falls into the exact situation assumed in Proposition A1. Also, by Equation (2)(2) , the prior of the common mean α_i,k is a normal distribution According to Proposition A1, the corresponding posterior is By expressing the above formula in matrix form, Equation (9)(9) will be obtained.▪

Appendix C: The model selection procedure in Section 4

For each given model M_i:

• Generate posterior samples by the procedure in Appendix A.

• Estimate the log marginal likelihood by Equation (15)(15)

  2.1. Calculate the mean, θ*, of the posterior sample of θ.

  2.2. Calculate the log-prior by Equation (17)(17) .

  2.3. Calculate the log-prior by Equation (18)(18) and treat it as .

  2.4. Calculate the log posterior ordinate by Equation (21)(21) .

  2.5. Calculate by Equation (15)(15) using the results in Sections 4.1 and 4.2.

• Calculate the log BF of M_i vs. M_i-1 by Equation (14)(14) .

• If the log BF is larger than three, conclude that M_i is better than M_j.

Appendix D: Proof of Equation (20)(20)

By the independence of parameters (β,σ²_ϵ) and {σ²_p, …, σ₀²}, (A2) So to find the joint conditional posterior at the left-hand side, we just need to find the three marginal conditional posteriors at the right. Under the uniform prior in Equation (6)(6) : under the uniform prior in Equation (3)(3) : and under the weakly informative prior in Equation (7)(7) : Plugging in these distributions into Equation (A2)(A2) will lead to Equation (20)(20) .▪

Additional information

Notes on contributors

Li Zeng

Li Zeng is an Assistant Professor in the Department of Industrial and Manufacturing Systems Engineering at the University of Texas at Arlington. She received her B.S. degree in Precision Instruments and M.S. degree in Optical Engineering from Tsinghua University and Ph.D. in Industrial Engineering and M.S. degree in Statistics from the University of Wisconsin–Madison. Her research interest is process monitoring and control in complex manufacturing and healthcare delivery systems. She is a member of INFORMS and IIE.

Nan Chen

Nan Chen is an Assistant Professor in the Department of Industrial and Systems Engineering at the National University of Singapore. He obtained his B.S. degree in Automation from Tsinghua University and his M.S. degree in Computer Science, M.S. degree in Statistics, and Ph.D. degree in Industrial Engineering from the University of Wisconsin–Madison. His research interests include data analytics in manufacturing systems, prognostics and health management, data-driven modeling, and control of service systems. He is a member of INFORMS, IIE, and IEEE.