Three-Dimensional Variance Dispersion Graphs for Mixture-Process Experiments with Control and Noise Variables Heidi B. Goldfarb – The Dial Corporation Douglas C. Montgomery – Arizona State University Connie Borror – Arizona State University
Three-Dimensional Variance Dispersion Graphs for Mixture-Process Experiments with Control and Noise Variables Heidi B. Goldfarb – The Dial Corporation.
Mar 27, 2015
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Three-Dimensional Variance Dispersion Graphs for
Mixture-Process Experiments with Control and Noise Variables
Heidi B. Goldfarb – The Dial Corporation
Douglas C. Montgomery – Arizona State University
Connie Borror – Arizona State University
Christine Anderson-Cook – Virginia Tech
2
Outline
• Background
• Model Development
• Variance Dispersion Graphs
• Three-Dimensional Variance Dispersion Graphs
• Examples
• Conclusions
3
Mixture Designs
• Mixture designs are used when the experimental variables have additional constraints on them and it is the proportions of the variables that is important, not the absolute amounts
• For example, consider a 6 oz. fish patty that is a comprised of three different types of fish
• The proportions of the fish types affect the texture of the patty
• The goal is to find the proportions of the three fish types that makes a patty with the firmest texture
• Cornell (2002) gives a comprehensive treatment of mixture designs
4
Robust Designs
• Robust designs are used when we have noise variables - variables that are uncontrollable, difficult to control, or out of our control in practice
• Example: The fish patties are sold to people who cook them. Although recommended temperatures and times are given, we know that not all people follow them exactly
• The goal is to find patties that will have a firm texture throughout a range of temperature / time conditions
• Concepts of robust design were introduced in the United States in the 1980’s by Taguchi
5
MPV Experiments with Control and Noise Variables
• We could also have controllable processing variables, such as the amount of time the patties are precooked before being packaged and sold
• Standard designs generate blends with simplex or D-optimal designs and look at each blend at all possible combinations of the processing and noise variables (or a carefully chosen subset thereof)
• Steiner and Hamada (1997) address this problem but with different models and without correlation among the noise variables
• Goldfarb, Montgomery, and Borror (2003) uses the model shown here and considers correlation
6
Basic Problem
• What blend of fish and pre-cooking time will be “best” under a wide range of temperature and time?
• What type of experimental designs can help find this combination?
• Which designs will build prediction models with the smallest variation?
7
Mixture-Process-Noise Model
• Mixture Components xi i = 1, 2, …, q
• Controllable Process Variables wp p = 1, 2, …, c
• Noise Variables zt t = 1, 2, …, n
( , , ) i i ij i ji i j
ip i p ijp i j pi p i j p
it i t ijt i j ti t i j t
ipt i p t ijpt i j p ti p t i j p t
Y f x w z x x x
x w x x w
x z x x z
x w z x x w z
( )Y f x,w,z x β + x αw + x δz + x λVz +
where V is a cn x n block diagonal matrix with w’s on the diagonals and 0’s elsewhere
8
Expected Value and Variance
( )f x,w,z 0 x β + x αw
( )f x,w,z 0 x δ + x λV
( ) ( )E Y f x,w,z = 0 x β + x αw
2
2
Var( ) ( ) ( )Y f f
z
z
x,w,z 0 Σ x,w,z 0
[x δ + x λV]Σ [x δ + x λV]
Using the Delta Method:
9
Prediction Variance
• For a given design, the standardized prediction variance at a given point, x0, is:
• The scaled version (SPV) allows for fair comparisons among designs with different numbers of runs:
10 02
ˆvar( )( )
0y
x X X x
10 02
ˆvar( )( ) ( )
Nv N
0
0
yx x X X x
10
Mean and Slope Variance Models
• Borror, Montgomery, and Myers (2002) develop mean and slope prediction models for RSM designs with noise variables
• The mean model variance assesses the prediction error variance taking in to account both model errors and the variation transmitted through the noise variables
• The slope model variance measures the precision of the variance of the full model
11
Prediction Variance for MPV Designs with Control and Noise Variables
• Recall the model:
• The prediction variance for the mean model is:
where C is inverse (X*´X*) matrix and X* is the full model form matrix with x, w, and z terms
Y ( )f x,w,z x β + x αw + x δz + x λVz + ε
,
2 11 22
ˆVar[y( , , )] Var( ( ) E( ))
=Var( ( ) ( ))
[ ] [ ] [ ( ) ( )]
y y
y
z
z
x w z x,w,z
x,w,z x β + x αw
x δ x λV Σ x δ x λV x C x xw C xw
12
C-Matrix
13
Mean Model
• We divide by σ2 and multiply by N to allow for comparisons of designs, including those with different numbers of runs
k2a and k2b represent the interaction between the mixture-noise and mixture-control-noise variables, respectively
it= k2a and ipt= k2b.
,2
11 22 2 22 2 2 2
ˆ Var[y( , , )]
[ ( ) ( )] 2a a b b
N
N Nk k k N Nk
zx w z
x C x xw C xw x Jx x V x x VV x
14
Slope Model
• For the slope model we need to look at the partial derivative of the model with respect to each of the noise variables. For our model this derivative is:
and( )y
x,w,zx δ x λV
z
Var(Slope) Var( ) x δ x λV
15
General Form for the Slope
• For a quadratic mixture model with linear control and noise variables, the general form is:
t
2 33z ( 1) ,( 1)
1
2 44( 1) ( 1) ,( 1) ( 1)
1 1
133( 1) ,( 1)
1 1
2 44( 1) ( 1) ,( 1) ( 1)
1 1
Var (slope)
( )
2
2
m
i t m i t m ii
m c
i p t mc p m i t mc p m ii p
m m
i i t m i t m ii i i
m c
i i p t mc p q i t qc p q ii i i p
x
x w
x x
x x w
C
C
C
C1
1
34( 1) ,( 1) ( 1)
1 1 1
2
m
m m c
i i p t m i t mc p m ii i p
x x w
C
16
Variance Dispersion Graphs
• Prediction variances can be examined with variance dispersion graphs (VDGs), first introduced by Giovannitti-Jensen, A. and Myers, R. H. (1989)
• They plot contains max. variance, average variance, and min. variance versus the distance from the center of the design space
• Piepel, G., Anderson, C. M, and Redgate, P. E (1993) extended the use of VDGs to irregular regions
• VDGs allow us to compare designs to see which have the best overall variance properties over the entire design space
17
Shrinkage
Cube Points for Shrinkage Values of 1, 0.5, and 0
X1X2
X3
Mixture Triangle Points for Shrinkage Values from 0 to 1 by 0.2
X2 X3
X1
Mixture Points for a Constrained Region for Shrinkage Values of 1, 0.5, and 0
X2 X3
X1
18
VDG - Example
Flare Design VDGs
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1
Shrinkage Factor (Distance from Center)
SP
V
A-Min
A-Max
A-Ave
D-Min
D-Max
D-Ave
19
Three-Dimensional VDGs
• Plot the prediction variance by the distances from the centers of the mixture and process spaces
• Distances of 0 signify the center while distances of 1 represent the edges of the spaces
• These graphs allow comparisons among designs and evaluation of the relative increases in prediction as the experimenter moves along both spaces, mixture and process
• Can be used to assess the optimum placement for additional runs
• For the robust design setting, the plots can be done for varying k2a and k2b levels
20
Fish Patty Example
• Consider the fish patty example with 3 fish types, deep fry time, baking time, and baking temperature
• Initially, consider all of the process variables as controllable, giving the following 28-term model:
1 1 2 2 3 3 12 1 2 13 1 3 23 2 3 123 1 2 3 11 1 1
21 2 1 31 3 1 12 1 2 22 2 2 32 3 2 13 1 3 23 2 3
33 3 3 1
i i ij i j ijk i j k il i l
ijl i j l ijkl i j k l
Y x x x x x x x w
x x w x x x w
x x x x x x x x x x x x x w
x w x w x w x w x w x w x w
x w
21 1 2 1 131 1 3 1 231 2 3 1 122 1 2 2 132 1 3 2
232 2 3 2 123 1 2 3 133 1 3 3 233 2 3 3 1231 1 2 3 1
1232 1 2 3 2 1233 1 2 3 3
x x w x x w x x w x x w x x w
x x w x x w x x w x x w x x x w
x x x w x x x w
21
Fish Patty Designs
• Six designs of sizes 28 to 40 runs are considered
• Two designs are from Cornell and Gorman and were constructed to minimize the size of the confidence intervals of the model coefficients
• The other 4 designs are from Design-Expert and were constructed to be D-efficient
• The DX6 designs with an “A” had the extra degrees of freedom split between lack-of-fit and replication
• The other DX6 designs had all of the degrees of freedom allocated to lack-of-fit
22
Fish Patty 3D VDGs - Surface
4 6 8 10 12 14 16 18 20 above
Average Scaled Prediction Variance by Process and Mixture Shrinkage Values
CG 280.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
6
10
14
18
22
26
30
CG 400.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
6
10
14
18
22
26
30
DX6 380.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
6
10
14
18
22
26
30
DX6 A 380.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
6
10
14
18
22
26
30
DX6 400.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
6
10
14
18
22
26
30
DX6 A 400.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
6
10
14
18
22
26
30
Mixture Mixture Mixture
Mixture Mixture Mixture
Process Process Process
Process Process Process
23
Fish Patty 3D VDGs - Contours
4 6 8 10 12 14 16 18 20
Average Scaled Prediction Variance by Process and Mixture Shrinkage Values
Mixture
Pro
cess
CG 28
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
CG 40
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
D-opt 38
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
D-opt A 38
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
D-opt 40
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
D-opt A 40
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
4
6
66
66
6
10
10 10
10 10
10
14
14 14
14 1414
18 18
18 1818
24
Fish Patty 3D VDGs - Surface
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 above
Average Prediction Variance by Process and Mixture Shrinkage Values
CG 280.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
0.15
0.25
0.35
0.45
0.55
0.65
0.75
CG 400.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
0.15
0.25
0.35
0.45
0.55
0.65
0.75
DX6 380.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
0.15
0.25
0.35
0.45
0.55
0.65
0.75
DX6 A 380.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
0.15
0.25
0.35
0.45
0.55
0.65
0.75
DX6 400.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
0.15
0.25
0.35
0.45
0.55
0.65
0.75
DX6 A 400.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
0.15
0.25
0.35
0.45
0.55
0.65
0.75
Mixture Mixture Mixture
Mixture Mixture Mixture
Process Process Process
Process Process Process
25
3D VDG - Example
• Consider a setting with three mixture components and two controllable process variables (Kowalski, Cornell, and Vining (2000)) with the following 15-term model:
• We consider six competing designs – two with 17 runs, three with 23 runs, and one with 25 runs
• Extra runs beyond those for model fit were allocated differently for each design
2
2 21 1 2 2 3 3 12 1 2 13 1 3 23 2 3 11 1 22 2 12 1 2
11 1 1 21 2 1 31 3 1 12 1 2 22 2 2 32 3 2
( , ) i i ij i j kk k kl k l ik i lY f x x x w w w x w
x x x x x x x x x w w w w
x w x w x w x w x w x w
x w
26
Surface Plot VDGs
4 6 8 10 12 14 16 18 20 above
Average Scaled Prediction Variance by Process and Mixture Shrinkage Values
KCV 230.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
8
14
20
26
DX6 170.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
8
14
20
26
DX6 230.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
8
14
20
26
DX6 A 170.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
8
14
20
26
DX6 A 230.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
8
14
20
26
DX6 A 250.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
8
14
20
26
Mixture Mixture Mixture
Mixture Mixture Mixture
Process Process Process
Process ProcessProcess
27
Contour Plot VDGs
4 6 8 10 12 14 16 18 20
Average Scaled Prediction Variance by Process and Mixture Shrinkage Values
Mixture
Pro
cess
KCV 23
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
DX6 17
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
DX6 23
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
DX6 A 17
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
DX6 A 23
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
DX6 A 25
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
4 4
8
8
8
8
10
10
10
10
10
10
6 6
6
12
1212
1414
14
1412
141618
1212
28
Example with Noise Variables -Fish Patties
• Now consider the fish patty example where baking time and baking temperature are treated as noise variables
• We will fit the following 36-term model
( , , )
i i ij i ji i j
ip i p ijp i j pi p i j p
it i t ijt i j ti t i j t
ipt i p t ijpt i j p ti p t i j p t
Y f x w z x x x
x w x x w
x z x x z
x w z x x w z
29
Fish Patty Designs
• Three designs are being considered
• Design A - The 56-run design from Cornell (2002) which is a 7-run simplex-centroid in the mixture components crossed with an 8-run full factorial in the process and noise variables
• Design B - A 56-run design generated with the D-optimal design generator in Design-Expert 6.0 with 10 lack-of-fit points and 10 pure replicate points
• Design C - A 36-run design generated with the D-optimal design generator in Design-Expert 6.0 with no lack-of-fit points or pure replicate points
• The 3D VDGs show that Design C is the best from a scaled prediction variance standpoint
30
3D VDGs – Mean Model
0 50 100 150 200 250 300 350 400 450 above
VDGs for the Average SPV for the Mean Model for Design A
k2a=0, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
31
3D VDGs – Mean Model
0 50 100 150 200 250 300 350 400 450 above
VDGs for the Average SPV for the Mean Model for Design B
k2a=0, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
32
3D VDGs – Mean Model
0 50 100 150 200 250 300 350 400 450 above
VDGs for the Average SPV for the Mean Model for Design C
k2a=0, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=0.5, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=0
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=0.5
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
k2a=1, k2b=1
MixtureProcess0.0
0.20.4
0.60.8
1.0
0.00.2
0.40.6
0.81.0
50
150
250
350
450
33
Conclusions
• 3D VDG’s allow an experimenter to look at the prediction variance profiles of designs in both the mixture and process spaces simultaneously
• The plots can be used to compare designs and to determine the placement of additional runs
• Noise variables can be handled by looking at a grid of plots
34
References
• Borror, C. M., Montgomery, D. C., and Myers R. H. (2002) “Evaluation of Statistical Designs for Experiments Involving Noise Variables”. Journal of Quality Technology 34, pp. 54-70.
• Cornell, J. A. (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, Third Edition. John Wiley & Sons, New York, NY.
• Giovannitti-Jensen, A. and Myers, R. H. (1989). “Graphical Assessment of the Prediction Capability of Response Surface Designs”. Technometrics 31, pp. 159-171.
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