Duangporn Jearkpaporn, Connie M. Borror Douglas C. Montgomery and George C. Runger Arizona State University Tempe, AZ

Transcription

1 Process Monitoring for Correlated Gamma Distributed Data Using Generalized Linear Model Based Control Charts Duangporn Jearkpaporn, Connie M. Borror Douglas C. Montgomery and George C. Runger Arizona State University Tempe, AZ

2 Outline Introduction Problem Statement Review of Model-Based Control Chart Review of Generalized Linear Models Recent Researches Development of Proposed Method Data Model for Model-Based Control Chart Generalized Likelihood Ratio Monitoring Scheme Simulation Results (ARL Performance) Summary and Conclusions Fall Technical Conference

3 Problem Statement A need to simultaneously monitor several related process variables. All characteristics ointly determine the usefulness of the product. Mixtures of normally and nonnormally distributed variables. Fall Technical Conference 2002 duang@asu.edu 3

4 Gamma Distribution f ( y; θ, κ) = θ κ 1 Γ( κ) y κ 1 y / θ, y > e and 0 zero otherwise κ is a shape parameter and > 0 θ is scale parameter and > 0 E(y) = µ = κθ and Var(y) = κθ 2 Fall Technical Conference 2002 duang@asu.edu 4

5 Model-Based Control Chart A model-based control strategy that relates inputs to outputs [Mandel (1969), Zhang (1984), Hawkins (1993)] Residuals are control statistics Model need not be linear Benefits Simple interpretations Easy to perform. No correlation over time Fall Technical Conference 2002 duang@asu.edu 5

6 Generalized Linear Models (GLM) Generalized linear models have three components Random Component Systematic Component, η = Xβ Link Function, g( ) i.e. Ordinary Least Squares (OLS) Normal Distribution Linear combination of inputs, Xβ Identity link, µ = g -1 (Xβ) = Xβ Residual analysis of a GLM is similar to OLS Available from most software packages (Asymptotically) Normally distributed Fall Technical Conference 2002 duang@asu.edu 6

7 Recent Research Skinner, Montgomery, and Runger s (2001) modelbased control chart for counted data Using Generalized Linear Model Based Control Charts. Hauck, Runger, and Montgomery (1999) used process knowledge to group variables into one of three charts (model-void, model fixed and cause-selecting). Wade and Woodall s (1993) cause-selecting control charts regress input variables on output variables. Various charts of standardized residuals are constructed and used to interpret the signal. Fall Technical Conference 2002 duang@asu.edu 7

8 Data Model for Model-Based Control Chart Consider p Gamma variables, y y is a function of k inputs, x i with the link g( ) E(y ) = µ = g -1 (β 0 + β 1 x 1 + β 2 x β k x k ) y ~ gamma κ, θ = g 1 ( β 0 + β 1 x 1 + β κ 2 x 2 + K+ β k x k ) Fall Technical Conference 2002 duang@asu.edu 8

9 Data Model for Model-Based Control Chart (Con t) Assume log link, one x and py s All y s have different means µ = exp(β 0 + β 1 x) Variance of y is µ 2 /κ = exp[2 (β 0 + β 1 x)]/κ y ~ gamma κ, θ = exp( β 0 κ + β 1 x) Fall Technical Conference 2002 duang@asu.edu 9

10 Generalized Likelihood Ratio Likelihood ratio statistics for detecting mean changes for models given above were developed. Detecting the mean change is equivalent to testing H 0 : µ = µ 0 vs. H 1 : µ = µ 1. The GLR for testing this is Γ(κ -2 ln Γ(κˆ ) ) + κˆ y + ln κ where ˆµ 1and ˆκ 1 are the MLE of µ and κ under H 1. Under H 0, test statistic ~ χ 2 (m) where m = # parameter we wish to test. Reect H 0 if this statistic > c µ + ln µ ˆ κˆ 1 κ yκ 0 ( κˆ 1 κ 0 ) ln + κ0 κˆ µ yκˆ µ ˆ Fall Technical Conference 2002 duang@asu.edu 10

11 Generalized Likelihood Ratio (Con t) Consider the case where A mean shift is due to the scale parameter changes. Assume κ is known and equal κ 0 = κ 1 = κ A univariate case p = 1. The test statistic simplifies to µ ˆ = g = sign where. r ( ' βˆ ) 1 x ( y µ ˆ ) y 2 y ln µ ˆ + y µ ˆ µ ˆ A likelihood ratio statistic is the deviance residual 1 2 Fall Technical Conference 2002 duang@asu.edu 11

12 Generalized Likelihood Ratio (Con t) For the shift in p variables, the test statistic is 2 y ln µ ˆ y + µ ˆ p κ = 1 µ ˆ ( ) 1 where µ ˆ = g x β ˆ the fitted mean of the th output under H 0 (the mean before the shift). Fall Technical Conference 2002 duang@asu.edu 12

13 Monitoring Scheme Consider a process with p output variables. An initial set of n observations is used to fit GLM for each of p variables. Construct control limits based on deviance residuals obtained from GLM. Observe new data. Compute deviance residuals for new observations. Plot deviance residuals on p individual Charts. A signal on any of the p charts indicates that the variable it monitors is out of control. Fall Technical Conference 2002 duang@asu.edu 13

14 Results: Comparing ARL Performance Simulation scenarios: Univariate Case. Bivariate Case with µ 1 = µ 2 and κ 1 = κ 2 Bivariate Case with µ 1 µ 2 and κ 1 = κ 2 Two types of mean shift are introduced in each scenario: Additive shift µ = g -1 (β 0 + β 1 x) +δ Multiplicative Shift µ = g -1 (β 0 + β 1 x +δ ) µ = g -1 (β 0 + (β 1 +δ ) x) Fall Technical Conference 2002 duang@asu.edu 14

15 Univariate Case 40 AddOneSigma b0onesigma b1onesigma Add3Sigma b03sigma b13sigma ARL y-chart ARL DR-chart Kappa Fall Technical Conference 2002 duang@asu.edu 15

16 Bivariate Case: Equal Means y 1 Shifted 40 AddOneSigma b0onesigma b1onesigma Add3Sigma b03sigma b13sigma ARL y-chart ARL DR-chart Kappa Fall Technical Conference 2002 duang@asu.edu 16

17 Bivariate Case: Equal Means y 1 & y 2 Shifted AddOneSigma b0onesigma b1onesigma Add3Sigma b03sigma b13sigma ARL y-chart ARL DR-chart Kappa Fall Technical Conference 2002 duang@asu.edu 17

18 Bivariate Case: Unequal Means y 1 Shifted AddOneSigma b0onesigma b1onesigma Add3Sigma b03sigma b13sigma ARL y-chart ARL DR-chart Kappa Fall Technical Conference 2002 duang@asu.edu 18

19 Bivariate Case: Unequal Means y 2 Shifted 30 AddOneSigma b0onesigma b1onesigma Add3Sigma b03sigma b13sigma ARL y-chart ARL DR-chart Fall Technical Conference 2002 duang@asu.edu 19

20 Bivariate Case: Unequal Means y 1 &y 2 Shifted AddOneSigma b0onesigma b1onesigma Add3Sigma b03sigma b13sigma ARL y-chart ARL DR-chart Fall Technical Conference 2002 duang@asu.edu 20

21 Summary and Conclusions This study shows: Overall, deviance residuals based control chart outperform the individual Shewhart charts. Deviance chart is best to detect the additive shift. detect the shift quicker than individual Shewhart charts when outputs are correlated and non-normal. Ability of model-based control chart to handle relationships between the responses. Application of the GLM in process monitoring Fall Technical Conference

22 What Else? Apply the same idea to CUSUM for detecting the drift in mean Cascade Processes Using other methods in model fitting GEE Robust fitting technique Fall Technical Conference

23 Question? Fall Technical Conference

24 References Engelhardt, B. (1992), Introduction to Probability and Mathematical Statistics, 2 nd ed., Duxbury Press, Belmont, California, pp.418. Hauck D.J., G.C. Runger, and D.C. Montgomery (1999), Multivariate statistical process monitoring and diagnosis with grouped regression-adusted variables, Commun Stat-Simul 28(2) pp Hawkins, D.M., Regression Adustment for Variables in Multivariate Quality Control, JQT, pp Mendel, B.J. (1969), The Regression control Chart, JQT, pp.1-9. Skinner, K. R., Runger, G. C., and Montgomery, D. C. (2001), Process Monitoring for Multiple Count Data Using Generalized Linear Model Based control Charts, submitted to IJPR. Wade, M.R. and W.H. Woodall (1993), A review and analysis of cause-selecting control charts, J. of Quality Technology 25(3) pp Zhang, G.X. (1984), A New Type of Control Charts and a Theory of Diagnosis with Control Charts, World Quality congress Transactions. American society for Quality Control, pp Fall Technical Conference 2002 duang@asu.edu 24