statistical tools for multivariate six sigma
DESCRIPTION
Statistical Tools for Multivariate Six Sigma. Dr. Neil W. Polhemus CTO & Director of Development StatPoint, Inc. The Challenge. The quality of an item or service usually depends on more than one characteristic. - PowerPoint PPT PresentationTRANSCRIPT
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Statistical Tools for Multivariate Six Sigma
Dr. Neil W. PolhemusCTO & Director of DevelopmentStatPoint, Inc.
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The Challenge
The quality of an item or service usually depends on more than one characteristic.
When the characteristics are not independent, considering each characteristic separately can give a misleading estimate of overall performance.
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The Solution
Proper analysis of data from such processes requires the use of multivariate statistical techniques.
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Outline Multivariate SPC
Multivariate control charts Multivariate capability analysis
Data exploration and modeling Principal components analysis (PCA)
Partial least squares (PLS) Neural network classifiers
Design of experiments (DOE) Multivariate optimization
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Example #1
Textile fiber
Characteristic #1: tensile strength - 115 ± 1
Characteristic #2: diameter - 1.05 ± 0.05
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Sample Data
n = 100
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Individuals Chart - strength
X Chart for strength
0 20 40 60 80 100
Observation
114
114.3
114.6
114.9
115.2
115.5
115.8
X
CTR = 114.98UCL = 115.69
LCL = 114.27
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Individuals Chart - diameter
X Chart for diameter
0 20 40 60 80 100
Observation
1.04
1.043
1.046
1.049
1.052
1.055
1.058
X
CTR = 1.05UCL = 1.06
LCL = 1.04
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Capability Analysis - strength
NormalMean=114.978Std. Dev.=0.238937
Cp = 1.41Pp = 1.40Cpk = 1.38Ppk = 1.36K = -0.02
Process Capability for strength
LSL = 114.0, Nominal = 115.0, USL = 116.0
114 114.4 114.8 115.2 115.6 116
strength
0
4
8
12
16
20
24
freq
uenc
y
DPM = 30.76
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Capability Analysis - diameter
DPM = 44.59
NormalMean=1.04991Std. Dev.=0.00244799
Cp = 1.41Pp = 1.36Cpk = 1.39Ppk = 1.35K = -0.01
Process Capability for diameter
LSL = 1.04, Nominal = 1.05, USL = 1.06
1.04 1.044 1.048 1.052 1.056 1.06diameter
0
4
8
12
16
20
freq
uenc
y
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Scatterplot
Plot of strength vs diameter
1.04 1.045 1.05 1.055 1.06diameter
114
114.5
115
115.5
116
str
en
gth
correlation = 0.89
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Multivariate Normal Distribution
Multivariate Normal Distribution
114 114.5 115 115.5 116
strength
1.041.045
1.051.055
1.06
diameter
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Control Ellipse
Control Ellipse
1.04 1.043 1.046 1.049 1.052 1.055 1.058diameter
114
114.3
114.6
114.9
115.2
115.5
115.8
stre
ng
th
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Multivariate Capability
Determines joint probability of being within the specification limits on all characteristics
Observed Estimated Estimated Variable Beyond Spec. Beyond Spec. DPM strength 0.0% 0.00307572% 30.7572 diameter 0.0% 0.00445939% 44.5939 Joint 0.0% 0.00703461% 70.3461
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Multivariate Capability
Multivariate Normal DistributionDPM = 70.3461
113.5 114 114.5 115 115.5 116 116.5
strength
1.0351.041.0451.051.0551.061.065
diameter
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Capability Ellipse
99.73% Capability Ellipse
MCP =1.27
113.5 114 114.5 115 115.5 116 116.5strength
1.035
1.04
1.045
1.05
1.055
1.06
1.065
diam
eter
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Mult. Capability Indices
Defined to give the
same DPM as in the
univariate case.
Capability Indices Index Estimate MCP 1.27 MCR 78.80 DPM 70.3461 Z 3.80696 SQL 5.30696
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Test for Normality
Probability Plot
-2.6 -1.6 -0.6 0.4 1.4 2.4 3.4normal distribution
-2.6
-1.6
-0.6
0.4
1.4
2.4
3.4
empi
rical
dat
a
strengthdiameter
P-Values Shapiro-Wilk strength 0.408004 diameter 0.615164
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More than 2 Characteristics
Calculate T-squared:
where
S = sample covariance matrix
= vector of sample means
)()( 12 xxSxxT iii
x
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T-Squared Chart
Multivariate Control Chart
UCL = 11.25
0 20 40 60 80 100 120Observation
0
5
10
15
20
25
30
T-S
quar
ed
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T-Squared Decomposition
Subtracts the value of T-squared if each variable is removed.
Large values indicate that a variable has an important contribution.
T-Squared Decomposition Relative Contribution to T-Squared Signal Observation T-Squared diameter strength 17 26.3659 22.9655 25.951
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Control Ellipsoid
Control Ellipsoid
1.04 1.044 1.048 1.052 1.056 1.06
diameter
114114.4114.8
115.2115.6
116
strength
6.8
7.8
8.8
9.8
10.8
11.8
12.8
rnor
mal
(100
,10,
1)
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Multivariate EWMA Chart
Multivariate EWMA Control Chart
UCL = 11.25, lambda = 0.2
0 20 40 60 80 100 120
Observation
0
3
6
9
12
15
T-S
quar
ed
Largeststrengthdiameter
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Generalized Variance Chart
Plots the determinant of the variance-covariance matrix for data that is sampled in subgroups.
Generalized Variance Chart
0 4 8 12 16 20 24
Subgroup
0
1
2
3
4
5
6(X 1.E-7)
Gen
. Var
ianc
e
UCL = 3.281E-7CL = 7.01937E-8LCL = 0.0
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Data Exploration and Modeling
When the number of variables is large, the dimensionality of the problem often makes it difficult to determine the underlying relationships.
Reduction of dimensionality can be very helpful.
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Example #2
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Matrix PlotMPG City
MPG Highway
Engine Size
Horsepower
Fueltank
Passengers
Length
Wheelbase
Width
U Turn Space
Weight
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Analysis Methods
Predicting certain characteristics based on others (regression and ANOVA)
Separating items into groups (classification)
Detecting unusual items
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Multiple RegressionMPG City = 29.6315 + 0.28816*Engine Size - 0.00688362*Horsepower - 0.297446*Passengers - 0.0365723*Length + 0.280224*Wheelbase + 0.111526*Width - 0.139763*U Turn Space - 0.00984486*Weight Standard T Parameter Estimate Error Statistic P-Value CONSTANT 29.6315 12.9763 2.28351 0.0249 Engine Size 0.28816 0.722918 0.398607 0.6912 Horsepower -0.00688362 0.0134153 -0.513119 0.6092 Passengers -0.297446 0.54754 -0.543241 0.5884 Length -0.0365723 0.0447211 -0.817786 0.4158 Wheelbase 0.280224 0.124837 2.24472 0.0274 Width 0.111526 0.218893 0.5095 0.6117 U Turn Space -0.139763 0.17926 -0.779668 0.4378 Weight -0.00984486 0.00192619 -5.11104 0.0000 R-squared = 73.544 percent R-squared (adjusted for d.f.) = 71.0244 percent Standard Error of Est. = 3.02509 Mean absolute error = 1.99256
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Principal Components
The goal of a principal components analysis (PCA) is to construct k linear combinations of the p variables X that contain the greatest variance.
pp XaXaXaC 12121111 ...
pp XaXaXaC 22221212 ...
…
pkpkkk XaXaXaC ...2211
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Scree Plot
Shows the number of significant components.
Scree Plot
Component
Eig
en
valu
e
0 2 4 6 80
1
2
3
4
5
6
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Percentage Explained
Principal Components Analysis Component Percent of Cumulative Number Eigenvalue Variance Percentage 1 5.8263 72.829 72.829 2 1.09626 13.703 86.532 3 0.339796 4.247 90.779 4 0.270321 3.379 94.158 5 0.179286 2.241 96.400 6 0.12342 1.543 97.942 7 0.109412 1.368 99.310 8 0.0552072 0.690 100.000
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ComponentsTable of Component Weights Component Component 1 2 Engine Size 0.376856 -0.205144 Horsepower 0.292144 -0.592729 Passengers 0.239193 0.730749 Length 0.369908 0.0429221 Wheelbase 0.374826 0.259648 Width 0.38949 -0.0422083 U Turn Space 0.359702 -0.0256716 Weight 0.396236 -0.0298902
First component 0.376856*Engine Size + 0.292144*Horsepower + 0.239193*Passengers + 0.369908*Length + 0.374826*Wheelbase + 0.38949*Width + 0.359702*U Turn Space + 0.396236*Weight Second component -0.205144*Engine Size – 0.592729*Horsepower + 0.730749*Passengers + 0.0429221*Length + 0.259648*Wheelbase - 0.0422083*Width - 0.0256716*U Turn Space – 0.0298902*Weight
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Interpretation
Plot of C_2 vs C_1
C_1
C_2
TypeCompactLarge MidsizeSmall Sporty Van
-6 -4 -2 0 2 4 6-5
-3
-1
1
3
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Principal Component RegressionMPG City = 22.3656 - 1.84685*size + 0.567176*unsportiness Standard T Parameter Estimate Error Statistic P-Value CONSTANT 22.3656 0.353316 63.302 0.0000 size -1.84685 0.147168 -12.5492 0.0000 unsportiness 0.567176 0.339277 1.67172 0.0981 R-squared = 64.0399 percent R-squared (adjusted for d.f.) = 63.2408 percent Standard Error of Est. = 3.40726 Mean absolute error = 2.26553
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Partial Least Squares (PLS)
Similar to PCA, except that it finds components that minimize the variance in both the X’s and the Y’s.
May be used with many X variables, even exceeding n.
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Component Extraction
Starts with number of components equal to the minimum of p and (n-1).
Model Comparison Plot
Number of components
Pe
rce
nt
vari
ati
on
XY
1 2 3 4 5 6 7 80
20
40
60
80
100
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Coefficient Plot
PLS Coefficient Plot
Stn
d.
coe
ffic
ien
t
MPG CityMPG HighwayFueltank
-0.7
-0.5
-0.3
-0.1
0.1
0.3
0.5E
ng
ine
Siz
e
Ho
rse
po
we
r
Pa
sse
ng
ers
Le
ng
th
Wh
ee
lba
se
Wid
th
U T
urn
Sp
ace
We
igh
t
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Model in Original Units
MPG City = 50.0593 – 0.214083*Engine Size - 0.0347708*Horsepower
- 0.884181*Passengers + 0.0294622*Length - 0.0362471*Wheelbase
- 0.0882233*Width - 0.0282326*U Turn Space - 0.00391616*Weight
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Classification
Principal components can also be used to classify new observations.
A useful method for classification is a Bayesian classifier, which can be expressed as a neural network.
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6 Types of Automobiles
Plot of unsportiness vs size
size
un
spo
rtin
ess
TypeCompactLarge MidsizeSmall Sporty Van
-6 -4 -2 0 2 4 6-5
-3
-1
1
3
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Neural Networks
Input layer
(2 variables)
Pattern layer
(93 cases)
Summation layer
(6 neurons)
Output layer
(6 groups)
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Bayesian Classifier Begins with prior probabilities for membership in
each group
Uses a Parzen-like density estimator of the density function for each group
jn
i
i
jj
XX
nXg
12
2
exp1
)(
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Options
The prior probabilities may be determined in several ways.
A training set is usually used to find a good value for .
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OutputNumber of cases in training set: 93 Number of cases in validation set: 0 Spacing parameter used: 0.0109375 (optimized by jackknifing during training) Training Set Percent Correctly Type Members Classified Compact 16 75.0 Large 11 100.0 Midsize 22 77.2727 Small 21 76.1905 Sporty 14 85.7143 Van 9 100.0 Total 93 82.7957
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Classification Regions
Classification Plot
size
unsp
ortin
ess
TypeCompact Large Midsize Small Sporty Van
sigma = 0.0109375
-6 -4 -2 0 2 4 6-5
-3
-1
1
3
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Changing Sigma
Classification Plot
size
unsp
ortin
ess
TypeCompact Large Midsize Small Sporty Van
-6 -4 -2 0 2 4 6-5
-3
-1
1
3
sigma = 0.3
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Overlay Plot
Classification Plot
size
un
spo
rtin
ess
TypeCompact Large Midsize Small Sporty Van
sigma = 0.3
-6 -4 -2 0 2 4 6-5
-3
-1
1
3
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Outlier Detection
Control Ellipse
size
unsp
ortin
ess
-8 -4 0 4 8-5
-3
-1
1
3
5
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Cluster Analysis
Cluster Scatterplot
Method of k-Means,Squared Euclidean
-6 -4 -2 0 2 4 6
size
-5
-3
-1
1
3
unsp
ortin
ess
Cluster 1234Centroids
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Design of Experiments
When more than one characteristic is important, finding the optimal operating conditions usually requires a tradeoff of one characteristic for another.
One approach to finding a single solution is to use desirability functions.
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Example #3
Myers and Montgomery (2002) describe an experiment on a chemical process:
Response variable Goal
Conversion percentage maximize
Thermal activity Maintain between 55 and 60
Input factor Low High
time 8 minutes 17 minutes
temperature 160˚ C 210˚ C
catalyst 1.5% 3.5%
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Experimentrun time temperature catalyst conversion activity (minutes ) (degrees C ) (percent ) 1 10.0 170.0 2.0 74.0 53.2 2 15.0 170.0 2.0 51.0 62.9 3 10.0 200.0 2.0 88.0 53.4 4 15.0 200.0 2.0 70.0 62.6 5 10.0 170.0 3.0 71.0 57.3 6 15.0 170.0 3.0 90.0 67.9 7 10.0 200.0 3.0 66.0 59.8 8 15.0 200.0 3.0 97.0 67.8 9 8.3 185.0 2.5 76.0 59.1 10 16.7 185.0 2.5 79.0 65.9 11 12.5 160.0 2.5 85.0 60.0 12 12.5 210.0 2.5 97.0 60.7 13 12.5 185.0 1.66 55.0 57.4 14 12.5 185.0 3.35 81.0 63.2 15 12.5 185.0 2.5 81.0 59.2 16 12.5 185.0 2.5 75.0 60.4 17 12.5 185.0 2.5 76.0 59.1 18 12.5 185.0 2.5 83.0 60.6 19 12.5 185.0 2.5 80.0 60.8 20 12.5 185.0 2.5 91.0 58.9
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Step #1: Model Conversion
Standardized Pareto Chart for conversion
0 2 4 6 8
Standardized effect
A:timeABAABCBB
B:temperatureCC
C:catalystAC +
-
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Step #2: Optimize ConversionGoal: maximize conversion Optimum value = 118.174 Factor Low High Optimum time 8.0 17.0 17.0 temperature 160.0 210.0 210.0 catalyst 1.5 3.5 3.48086
Contours of Estimated Response Surfacetemperature=210.0
8 9 10 11 12 13 14 15 16 17
time
1.5
2
2.5
3
3.5
cata
lyst
conversion70.072.575.077.580.082.585.087.590.092.595.097.5100.0
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Step #3: Model Activity
Standardized Pareto Chart for activity
0 2 4 6 8
Standardized effect
ACCCBBBC
B:temperatureABAA
C:catalystA:time +
-
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Step #4: Optimize ActivityGoal: maintain activity at 57.5 Optimum value = 57.5 Factor Low High Optimum time 8.3 16.7 10.297 temperature 209.99 210.01 210.004 catalyst 1.66 3.35 2.31021
Contours of Estimated Response Surface
temperature=210.0
8 9 10 11 12 13 14 15 16 17
time
1.5
2
2.5
3
3.5
cata
lyst
activity55.056.057.058.059.060.0
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Step #5: Select Desirability Fcns.
Maximize
Desirability Function for Maximization
Predicted response
Desir
abili
ty, d
s = 1s = 2
s = 8
s = 0.4
s = 0.2
Low
0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
High
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Desirability Function
Hit Target
Desirability Function for Hitting Target
Predicted response
Desir
abilit
y, d
Low HighTarget
s = 1 t = 1
s = 0.1 t = 0.1
s = 5
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
t = 5
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Combined Desirability
where m = # of factors and 0 ≤ Ij ≤ 5. D ranges from 0 to 1.
m
jjm
IIm
II dddD 121
/1
21 ...
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ExampleOptimum value = 0.949092 Factor Low High Optimum time 8.0 17.0 11.1394 temperature 160.0 210.0 210.0 catalyst 1.5 3.5 2.20119
Weights Weights Response Low High Goal First Second Impact conversion 50.0 100.0 Maximize 1.0 3.0 activity 55.0 60.0 57.5 1.0 1.0 3.0
Response Optimum conversion 95.0388 activity 57.5
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Desirability Contours
Contours of Estimated Response Surfacetemperature=210.0
8 9 10 11 12 13 14 15 16 17
time
1.5
2
2.5
3
3.5
cata
lyst
Desirability0.00.10.20.30.40.50.60.70.80.91.0
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Desirability Surface
Estimated Response Surfacetemperature=210.0
8 9 10 11 12 13 14 15 16 17time
1.52
2.53
3.5
catalyst
0
0.2
0.4
0.6
0.8
1
Des
irab
ility
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Overlaid Contours
Overlay Plottemperature=210.0
conversionactivity
10 11 12 13 14 15
time
2
2.2
2.4
2.6
2.8
3
cata
lyst
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References Johnson, R.A. and Wichern, D.W. (2002). Applied Multivariate
Statistical Analysis. Upper Saddle River: Prentice Hall.Mason, R.L. and Young, J.C. (2002).
Mason and Young (2002). Multivariate Statistical Process Control with Industrial Applications. Philadelphia: SIAM.
Montgomery, D. C. (2005). Introduction to Statistical Quality Control, 5th edition. New York: John Wiley and Sons.
Myers, R. H. and Montgomery, D. C. (2002). Response Surface Methodology: Process and Product optimization Using Designed Experiments, 2nd edition. New York: John Wiley and Sons.