Mixed Model Analysis of Highly Correlated Data: Tales from the Dark Side of Forestry

1. Mixed Model Analysis of Highly Correlated Data: Tales from the Dark Side of Forestry Christina Staudhammer, PhD candidate Valerie LeMay, PhD Thomas Maness, PhD Robert Kozak, PhD THE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, BRITISH COLUMBIA, CANADA

4. Introduction - 1 • Current Statistical Process Control (SPC) in Sawmills • Data Collection: • Periodically, a few boards are pulled from a machine • Thickness measured in 6-10 places with digital calipers • Data Analysis • Control Charts are constructed to ensure that X, s2b, s2w are within a target range, e.g., • SPC is slow and labour-intensive, but important and effective

5. Introduction - 2 • Recent advances in SPC • Laser Range Sensors • Real-time measurements available at up to 1000 meas./sec. • Each and every board (or cant) is measured • Research describing Rigid Body Motion • Removes effect of ‘bouncing boards’ (or cants) • enables board profiles to be analyzed, in addition to thickness • On-line machine diagnostics can be monitored to trace quality problems to specific saws

6. Interesting Issues • A great increase in the amount of information available • the data from these devices is subject to noise • External, e.g., wane • Internal, e.g., measurement errors • The data are closely spaced and highly autocorrelated • Boards are almost censused • Observations are easily predicted from their neighbors. • The process variance is underestimated, leading to too narrow control limits for SPC and false signals of an out of control process. • An adequate statistical model to describe the data has not yet been described in the literature.

7. Objectives • Research Objective • To establish a system for collecting and processing real-time quality control data for automated lumber manufacturing • Presentation Objective • To present methods for estimation of the components of variance so that control charts can be constructed

8. Data Collection

9. Profile Data Laser 3 Laser 4 y3y4 y1y2 l4 l3 Laser 1 Laser 2 l1 l2 Profiles(y1 – y4) are computed using the laser readings and the known distance to the centre of the board.

10. Sample Data - Profile

11. Simple Model yijkm = m + bi + yj + lk + eijkm[1] where: i = 1 to b boards; j = 1 to s sides; k = 1 to r laser positions; m = 1 to n measurements along the board; bi = the ith board effect; yj = the jth side effect; lk = the kth laser position effect; and eijkm = the error associated with the mth measurement.

12. ● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ●● ● Model Details • All effects are random, except sides • Observations on a single side of a board are highly correlated, and thus the error covariance structure should be added to the model…

13. Error Covariance Structures Isotropic spatial covariance structures e.g., Exponential: (Other models include Gaussian,spherical, linear) Autoregressive covariance structures e.g., ARMA(1,1): Anisotropic spatial covariance structures e.g., Power:

14. Model Fitting Methods • Models fit: [1] Simple Model [2] Model [1] plus isotropic spatial error covariance structure [3] Model [1] plus autoregressive error cov. structure [4] Model [1] plus anisotropic spatial error covariance structure • Models were fit with SAS PROC MIXED • A reduced dataset was used with 50 meas. per laser/side/board

15. Model Evaluation • Tests for Maximum Likelihood Estimation (MLE) • e.g., Likelihood Ratio Test, Wald Test, etc. • Are tests appropriate? • Fit Statistics for MLE • Information Criteria, e.g., Akaike’s Information Criteria (AIC) • Do not require setting arbitrary significance levels

16. Results SimpleModel

17. Results Model [1] Plus Exponential Error Covariance Structure

18. Results Model [1] plus ARMA(1,1) Error Cov. Structure

19. Results Model [1] plus Anisotropic Power Error Cov. Structure

20. Discussion - 1 • Model selection based on fit statistics • Lowest AIC indicates [4] with Anisotropic Power Structure • What is indicated by directional variograms?

21. Semivariograms vs. Model [4]

22. Semivariograms vs. Model [4]

23. Discussion - 2 • Model selection based on knowledge of system • Appropriateness of isotropic spatial vs. anisotropic spatial vs. autoregressive models of error covariance structure • Should there be a decrease in between-board variance component? • Will a saw travelling at varying speeds yield a consistent ‘saw signature’?

24. Conclusions • Application of QC to automated processes is an important step toward more efficient lumber processing • Model selection should be based on knowledge of the system as well as fit statistics • Further testing should be done on datasets from different days/saws to ensure widespread applicability

25. Acknowledgements • National Science and Engineering Research Council • British Columbia Science Council • Izaak Walton Killam Foundation • Canadian Forest Products • Weyerhauser Company • Forintek Canada