Mixed Model Analysis of Highly Correlated Data: Tales from the Dark Side of Forestry Christina Staudhammer, PhD candidate Christina Staudhammer, PhD candidate Valerie LeMay, PhD Valerie LeMay, PhD Thomas Maness, PhD Thomas Maness, PhD Robert Kozak, PhD Robert Kozak, PhD THE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, BRITISH COLUMBIA, CANADA
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Mixed Model Analysis of Highly Correlated Data: Tales from the Dark Side of Forestry
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. Introduction - 1. - PowerPoint PPT Presentation
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Mixed Model Analysis of Highly Correlated Data:
Tales from the Dark Side of Forestry
Christina Staudhammer, PhD candidateChristina Staudhammer, PhD candidateValerie LeMay, PhDValerie LeMay, PhDThomas Maness, PhDThomas Maness, PhDRobert Kozak, PhDRobert Kozak, PhDTHE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, BRITISH COLUMBIA, CANADA
Staudhammer, et al.Staudhammer, et al.
Staudhammer, et al.Staudhammer, et al.
Staudhammer, et al.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, s2b, s2
w are within a target range, e.g.,
xsX 3
–SPC is slow and labour-intensive, but important and effective
Staudhammer, et al.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.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.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.Staudhammer, et al.
Data Collection
Staudhammer, et al.Staudhammer, et al.
Profile Data
Profiles (y1 – y4) are computed using the laser readings and the known distance to the centre of the board.
l4
l2
Laser 3
l1
y3 y4
y1 y2
Laser 2
Laser 4
Laser 1
l3
Staudhammer, et al.Staudhammer, et al.
Sample Data - ProfileBoard 001 (side one)
(reduced data set: 50 observations/laser, side, board)
970
980
990
1000
1010
1020
1030
0 2 4 6 8Distance along Board (ft)
Prof
ile (0
.001
inch
)
laser 1 (bottom)
laser 3 (top)
Staudhammer, et al.Staudhammer, et al.
Simple Modelyijkm = + 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; andijkm = the error associated with the mth measurement.
Staudhammer, et al.Staudhammer, et al.
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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…