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 candidateChristina Staudhammer, PhD candidateValerie LeMay, PhDValerie LeMay, PhDThomas Maness, PhDThomas Maness, PhDRobert Kozak, PhDRobert Kozak, PhDTHE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, BRITISH COLUMBIA, CANADA

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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

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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

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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.

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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

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Data Collection

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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

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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)

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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.

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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…

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Error Covariance StructuresIsotropic spatial covariance structurese.g., Exponential:

(Other models include Gaussian, spherical, linear)

Autoregressive covariance structures

e.g., ARMA(1,1):

Anisotropic spatial covariance structurese.g., Power:

)/exp(),cov( 2 ijji d

)(1)(1),cov( 12 jijijiji

),,(),,(2),cov( yjidy

xjidxji

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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

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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

model in the parameters ofnumber theis and ;likelihood log maximized empirical theis )|ˆ(

2)|ˆ(log2

Kx

KxAIC

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Results

Anisotropic PowerLaser Spatial

Model AIC Board Position Range x y

[1] 51,235 188.4 0.73 539.0

[2] 42,331 130.6 0.00 b 671.3 12.6c 1444.0 110.6

[3] 42,531 102.8 0.00 b 710.0 0.923 0.848c 1399.4 0.981 0.976

[4] 41,724 119.3 0.00 b 648.2 0.930 0.581c 1109.2 0.991 0.967

Covariance Parameter Parameters Parameters

Residual

ARMA

Simple Model

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Anisotropic PowerLaser Spatial

Model AIC Board Position Range x y

[1] 51,235 188.4 0.73 539.0

[2] 42,331 130.6 0.00 b 671.3 12.6c 1444.0 110.6

[3] 42,531 102.8 0.00 b 710.0 0.923 0.848c 1399.4 0.981 0.976

[4] 41,724 119.3 0.00 b 648.2 0.930 0.581c 1109.2 0.991 0.967

Covariance Parameter Parameters Parameters

Residual

ARMA

Results

Model [1] Plus Exponential Error Covariance Structure

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Anisotropic PowerLaser Spatial

Model AIC Board Position Range x y

[1] 51,235 188.4 0.73 539.0

[2] 42,331 130.6 0.00 b 671.3 12.6c 1444.0 110.6

[3] 42,531 102.8 0.00 b 710.0 0.923 0.848c 1399.4 0.981 0.976

[4] 41,724 119.3 0.00 b 648.2 0.930 0.581c 1109.2 0.991 0.967

Covariance Parameter Parameters Parameters

Residual

ARMA

Results

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

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Results

Anisotropic PowerLaser Spatial

Model AIC Board Position Range x y

[1] 51,235 188.4 0.73 539.0

[2] 42,331 130.6 0.00 b 671.3 12.6c 1444.0 110.6

[3] 42,531 102.8 0.00 b 710.0 0.923 0.848c 1399.4 0.981 0.976

[4] 41,724 119.3 0.00 b 648.2 0.930 0.581c 1109.2 0.991 0.967

Covariance Parameter Parameters Parameters

Residual

ARMA

Model [1] plus Anisotropic Power Error Cov. Structure

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• Model selection based on fit statistics– Lowest AIC indicates [4] with Anisotropic

Power Structure– What is indicated by directional variograms?

Discussion - 1

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Semivariograms vs. Model [4]

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60 70

Distance (inches)

Sem

ivar

iogr

am

FittedVariogram011

015

029

038

065

081

102

107

109

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Semivariograms vs. Model [4]

0

200

400

600

800

1000

1200

1400

1600

1800

0 10 20 30 40 50 60

Distance (inches)

Sem

ivar

iogr

amFitted Variogram

010

011

014

015

020

022

029

030

034

036

038

043

049

051

053

055

059

065

067

072

075

081

101

102

103

107

109

110

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• 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

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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

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Acknowledgements• National Science and Engineering

Research Council• British Columbia Science Council• Izaak Walton Killam Foundation• Canadian Forest Products• Weyerhauser Company• Forintek Canada