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

Staudhammer, et al.

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, s 2 b, s 2 w are within a target range, e.g., –SPC is slow and labour-intensive, but important and effective

Staudhammer, et al. 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

Staudhammer, et al. 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.

Staudhammer, et al. 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

Staudhammer, et al. Data Collection

Staudhammer, et al. Profile Data Profiles (y 1 – y 4 ) are computed using the laser readings and the known distance to the centre of the board. l4 l4 l2 l2 Laser 3 l1 l1 y 3 y 4 y 1 y 2 Laser 2 Laser 4 Laser 1 l3 l3

Staudhammer, et al. Sample Data - Profile

Staudhammer, et al. Simple Model y ijkm =  +  i +  j + k +  ijkm [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;  i = the ith board effect;  j = the jth side effect; k = the kth laser position effect; and  ijkm = the error associated with the mth measurement.

Staudhammer, et al. ● ● ● ● ● ● ● ● ● ● ● ● 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…

Staudhammer, et al. 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:

Staudhammer, et al. 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

Staudhammer, et al. 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

Staudhammer, et al. Results Simple Model

Staudhammer, et al. Results Model [1] Plus Exponential Error Covariance Structure

Staudhammer, et al. Results Model [1] plus ARMA(1,1) Error Cov. Structure

Staudhammer, et al. Results Model [1] plus Anisotropic Power Error Cov. Structure

Staudhammer, et al. Model selection based on fit statistics –Lowest AIC indicates [4] with Anisotropic Power Structure –What is indicated by directional variograms? Discussion - 1

Staudhammer, et al. Semivariograms vs. Model [4]

Staudhammer, et al. Semivariograms vs. Model [4]

Staudhammer, et al. 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’? Discussion - 2

Staudhammer, et al. 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

Staudhammer, et al. Acknowledgements National Science and Engineering Research Council British Columbia Science Council Izaak Walton Killam Foundation Canadian Forest Products Weyerhauser Company Forintek Canada