manufacturing process control - mit opencourseware · manufacturing 28 mistake #1 – multiple...
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MIT OpenCourseWare ____________http://ocw.mit.edu
2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)Spring 2008
For information about citing these materials or our Terms of Use, visit: ________________http://ocw.mit.edu/terms.
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1Manufacturing
Control of Manufacturing Processes
Subject 2.830/6.780/ESD.63Spring 2008Lecture #9
Advanced and Multivariate SPC
March 6, 2008
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2Manufacturing
Agenda• Conventional Control Charts
– Xbar and S
• Alternative Control Charts– Moving average– EWMA– CUSUM
• Multivariate SPC
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3Manufacturing
Xbar ChartProcess Model: x ~ N(5,1), n = 9
• Is process in control?
4.0
4.5
5.0
5.5
6.0M
ean
of x
1 2 3 4 5 6 7 8 9 10 11 12 13Sample
µ0=5.000
LCL=4.000
UCL=6.000
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4Manufacturing
Run Data (n=9 sample size)
01
345
789
11R
un C
hart
of x
0 9 18 27 36 45 54 63 72 81 90 99Run
Avg=4.91
• Is process in control?
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5Manufacturing
S Chart
0.0
0.5
1.0
1.5
2.0
2.5
3.0S
tand
ard
Dev
iatio
n of
x
1 2 3 4 5 6 7 8 9 10 11 12 13Sample
µ0=0.969
LCL=0.232
UCL=1.707
• Is process in control?
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6Manufacturing
n measurementsat sample j
x R j =1n
xii = j
j + n
∑
SR j2 =
1n − 1
(xii = j
j +n
∑ − x Rj )2
Running Average
Running Variance
• More averages/Data • Can use run data alone and
average for S only• Can use to improve resolution
of mean shift
Alternative Charts: Running Averages
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7Manufacturing
Simplest Case: Moving Average• Pick window size (e.g., w = 9)
1
2
3
4
5
6
7
8
9
Mov
ing
Avg
of x
8 16 24 32 40 48 56 64 72 80 88 96Sample
µ0=5.00
LCL
UCL
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8Manufacturing
yj = a1x j −1 + a2x j −2 + a3 xj −3 + ...
General Case: Weighted Averages
• How should we weight measurements?– All equally? (as with Moving Average)
– Based on how recent?• e.g. Most recent are more relevant than less
recent?
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9Manufacturing
0
0.05
0.1
0.15
0.2
0.25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Consider an Exponential Weighted Average
Define a weighting function
Wt − i = r (1− r )i
Exponential Weights
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10Manufacturing
Exponentially Weighted Moving Average: (EWMA)
Ai = rxi + (1 − r)Ai −1 Recursive EWMA
UCL, LCL = x ± 3σ A
σ A =σ x
2
n⎛ ⎝ ⎜
⎞ ⎠ ⎟
r2 − r
⎛ ⎝
⎞ ⎠ 1 − 1− r( )2 t[ ]
time
σA =σ x
2
nr
2 − r⎛ ⎝
⎞ ⎠
for large t
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11Manufacturing
Effect of r on σ multiplier
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
plot of (r/(2-r)) vs. r
wider control limits
r
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12Manufacturing
SO WHAT?• The variance will be less than with xbar,
• n=1 case is valid• If r=1 we have “unfiltered” data
– Run data stays run data– Sequential averages remain
• If r<<1 we get long weighting and long delays– “Stronger” filter; longer response time
σA =σ x
nr
2 − r⎛ ⎝
⎞ ⎠ = σ x
r2 − r
⎛ ⎝
⎞ ⎠
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13Manufacturing
EWMA vs. Xbar
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200 250 300
xbarEWMAUCL EWMALCL EWMAgrand meanUCLLCL
r=0.3
Δμ = 0.5 σ
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14Manufacturing
Mean Shift SensitivityEWMA and Xbar comparison
0
0.2
0.4
0.6
0.8
1
1.2
1 5 9 13 17 21 25 29 33 37 41 45 49
xbar
EWMA
UCLEWMA
LCLEWMA
GrandMean
UCL
LCL
3/6/03
Mean shift = .5 σ
r=0.1n=5
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15Manufacturing
Effect of r
0
0.1
0.2
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0.4
0.5
0.6
0.7
0.8
0.9
1
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
xbar
EWMA
UCLEWMA
LCLEWMA
GrandMean
UCL
LCL
r=0.3
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16Manufacturing
Small Mean Shifts
• What if Δμx is small with respect to σx ?
• But it is “persistent”
• How could we detect?– ARL for xbar would be too large
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17Manufacturing
Another Approach: Cumulative Sums
• Add up deviations from mean– A Discrete Time Integrator
• Since E{x-μ}=0 this sum should stay near zero when in control
• Any bias (mean shift) in x will show as a trend
C j = (xii=1
j
∑ − x)
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18Manufacturing
Mean Shift Sensitivity: CUSUM
-1
0
1
2
3
4
5
6
7
81 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Mean shift = 1σSlope cause by mean shift Δμ
Ci = (xii =1
t
∑ − x )
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19Manufacturing
Control Limits for CUSUM
• Significance of Slope Changes?– Detecting Mean Shifts
• Use of v-mask– Slope Test with Deadband
d
θ
Upper decision line
Lower decision line
d =2δ
ln1 − β
α⎛ ⎝ ⎜
⎞ ⎠ ⎟
δ = Δx σ x
θ = tan−1 Δx2k
⎛⎝⎜
⎞⎠⎟
where k = horizontal scale factor for plot
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20Manufacturing
Use of Mask
-1
0
1
2
3
4
5
6
7
81 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
θ=tan-1(Δμ/2k)k=4:1; Δμ=0.25 (1σ)tan(θ) = 0.5 as plotted
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21Manufacturing
An Alternative• Define the Normalized Statistic
• And the CUSUM statistic
Zi =Xi − μ x
σ x
Si =Zi
i =1
t
∑t
Which has an expected mean of 0 and variance of 1
Which has an expected mean of 0 and variance of 1
Chart with Centerline =0 and Limits = ±3
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22Manufacturing
Example for Mean Shift = 1σ
-1
0
1
2
3
4
5
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47
Normalized CUSUM
Mean Shift = 1 σ
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23Manufacturing
Tabular CUSUM
• Create Threshold Variables:
Ci+ = max[0, xi − (μ0 + K ) + Ci −1
+ ]Ci
− = max[0,(μ0 − K ) − xi + Ci −1− ]
K= threshold or slack value for accumulation
K =Δμ2
Δμ = mean shift to detect
H : alarm level (typically 5σ)
Accumulates deviations from the mean
typical
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24Manufacturing
Threshold Plot
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
C+
C- C+
C-H threshold
μ 0.495
σ 0.170
k=δμ/2 0.049
h=5σ 0.848
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25Manufacturing
Alternative Charts Summary• Noisy data need some filtering• Sampling strategy can guarantee
independence• Linear discrete filters been proposed
– EWMA– Running Integrator
• Choice depends on nature of process• Noisy data need some filtering, BUT
– Should generally monitor variance too!
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26Manufacturing
Motivation: Multivariate Process Control
• More than one output of concern– many univariate control charts– many false alarms if not designed properly– common mistake #1
• Outputs may be coupled– exhibit covariance– independent probability models may not be
appropriate– common mistake #2
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27Manufacturing
Mistake #1 – Multiple Charts
• Multiple (independent) parameters being monitored at a process step– set control limits based on acceptable
α = Pr(false alarm)• E.g., α = 0.0027 (typical 3σ control limits), so 1/370 runs
will be a false alarm– Consider p separate control charts
• What is aggregate false alarm probability?
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28Manufacturing
Mistake #1 – Multiple Tests for Significant Effects
• Multiple control charts are just a running hypothesis test – is process “in control” or has something statistically significant occurred (i.e., “unlikely to have occurred by chance”)?
• Same common mistake (testing for multiple significant effects and misinterpreting significance) applies to many uses of statistics – such as medical research!
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29Manufacturing
The Economist (Feb. 22, 2007)
Text removed due to copyright restrictions. Please see The Economist, Science and Technology.
“Signs of the times.” February 22, 2007.
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30Manufacturing
Approximate Corrections for Multiple (Independent) Charts
• Approach: fixed α’– Decide aggregate acceptable false alarm rate, α’– Set individual chart α to compensate
– Expand individual control chart limits to match
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31Manufacturing
Mistake #2: Assuming Independent Parameters
• Performance related to many variables • Outputs are often interrelated
– e.g., two dimensions that make up a fit– thickness and strength– depth and width of a feature (e.g.. micro embossing)– multiple dimensions of body in white (BIW)– multiple characteristics on a wafer
• Why are independent charts deceiving?
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32Manufacturing
Examples
• Body in White (BIW) assembly– Multiple individual dimensions measured– All could be OK and yet BIW be out of spec
• Injection molding part with multiple key dimensions
• Numerous critical dimensions on a semiconductor wafer or microfluidic chip
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33Manufacturing
LFM Application
Rob York ‘95
“Distributed Gaging Methodologies for Variation Reduction in and Automotive Body Shop”
Rob York ‘95
“Distributed Gaging Methodologies for Variation Reduction in and Automotive Body Shop”
Body in White “Indicator”
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34Manufacturing
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35Manufacturing
Independent Random Variables
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-6
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-2
0
2
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
-5
-4
-3
-2
-1
0
1
2
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4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
x1
x1
x2
Proper Limits?x2
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36Manufacturing
Correlated Random Variables
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0
1
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3
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
-4
-3
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
x2
x2
x1
x1
Proper Limits?
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37Manufacturing
Outliers?
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-3
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
-4
-3
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
x2
x1
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38Manufacturing
Multivariate Charts
• Create a single chart based on joint probability distribution– Using sample statistics: Hotelling T2
• Set limits to detect mean shift based on α• Find a way to back out the underlying
causes• EWMA and CUSUM extensions
– MEWMA and MCUSUM
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39Manufacturing
Background• Joint Probability Distributions• Development of a single scale control chart
– Hotelling T2
• Causality Detection– Which characteristic likely caused a problem
• Reduction of Large Dimension Problems– Principal Component Analysis (PCA)
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40Manufacturing
Multivariate Elements• Given a vector of measurements
• We can define vector of means:
• and covariance matrix:where p = # parameters
variance
covariance
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41Manufacturing
Joint Probability Distributions
• Single Variable Normal Distribution
• Multivariable Normal Distribution
squared standardizeddistance from
mean
squared standardizeddistance from mean
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42Manufacturing
Sample Statistics• For a set of samples of the vector x
• Sample Mean
• Sample Covariance
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43Manufacturing
Chi-Squared Example - True Distributions Known, Two Variables
• If we know μ and Σ a priori:
will be distributed as
– sum of squares of two unit normals
• More generally, for p variables:
distributed as (and n = # samples)
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44Manufacturing
Control Chart for χ2? • Assume an acceptable probability of Type I
errors (upper α)• UCL = χ2
α, p– where p = order of the system
• If process means are µ1 and µ2 then χ20 < UCL
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45Manufacturing
Univariate vs. χ2 Chart
Montgomery, ed. 5e
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
LCLx1
UCLx1
x1
12345678910111213141516
LCLx2
UC
Lx2
x2
Joint control region for x1 and x2
10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
UCL = xa.2
x02
2
Sample number
Figure by MIT OpenCourseWare.
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46Manufacturing
Multivariate Chart with No Prior Statistics: T2
• If we must use data to get• Define a new statistic, Hotelling T2
• Where is the vector of the averages for each variable over all measurements
• S is the matrix of sample covariance over all data
x and S
x
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47Manufacturing
Similarity of T2 and t2
vs.
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48Manufacturing
Distribution for T2
– Given by a scaled F distribution
α is type I error probabilityp and (mn – m – p + 1) are d.o.f. for the F distributionn is the size of a given samplem is the number of samples takenp is the number of outputs
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49Manufacturing
F - Distribution
Significance level α
Tabulated Values
Fν1 ,ν2ν1 = 8,ν2 = 16
Fα , p,(mn− m − p+1)
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50Manufacturing
Phase I and II?• Phase I - Establishing Limits
• Phase II - Monitoring the Process
NB if m used in phase 1 is large then they are nearly the same
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51Manufacturing
Example• Fiber production• Outputs are strength and weight• 20 samples of subgroups size 4
– m = 20, n = 4• Compare T2 result to individual control charts
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52Manufacturing
Data SetOutput x1 Output x2
Sample Subgroup n=4 R1 Subgroup n=41 80 82 78 85 7 19 22 20 202 75 78 84 81 9 24 21 18 213 83 86 84 87 4 19 24 21 224 79 84 80 83 5 18 20 17 165 82 81 78 86 8 23 21 18 226 86 84 85 87 3 21 20 23 217 84 88 82 85 6 19 23 19 228 76 84 78 82 8 22 17 19 189 85 88 85 87 3 18 16 20 1610 80 78 81 83 5 18 19 20 1811 86 84 85 86 2 23 20 24 2212 81 81 83 82 2 22 21 23 2113 81 86 82 79 7 16 18 20 1914 75 78 82 80 7 22 21 23 2215 77 84 78 85 8 22 19 21 1816 86 82 84 84 4 19 23 18 2217 84 85 78 79 7 17 22 18 1918 82 86 79 83 7 20 19 23 2119 79 88 85 83 9 21 23 20 1820 80 84 82 85 5 18 22 19 20
Mean Vector Covariance Matrix
82.46 7.51 -0.3520.18 -0.35 3.29
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53Manufacturing
Cross Plot of Data
17.00
18.00
19.00
20.00
21.00
22.00
23.00
78.00 80.00 82.00 84.00 86.00 88.00
X1 bar
74.00
76.00
78.00
80.00
82.00
84.00
86.00
88.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
X1 bar
X2 bar
17.00
18.00
19.00
20.00
21.00
22.00
23.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
X2 bar
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54Manufacturing
T2 Chart
0
2
4
6
8
10
12
14
16
18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T2
UCL
α = 0.0054
(Similar to ±3σ limits ontwo xbar charts combined)
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55Manufacturing
Individual Xbar Charts
17.00
18.00
19.00
20.00
21.00
22.00
23.00
24.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
X2 bar
UCL X2
LCL X2
74.00
76.00
78.00
80.00
82.00
84.00
86.00
88.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
X1 bar
UCL X1
LCL X1
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56Manufacturing
Finding Cause of Alarms• With only one variable to plot, which
variable(s) caused an alarm?• Montgomery
– Compute T2
– Compute T2(i) where the i th variable is not included
– Define the relative contribution of each variable as• di = T2 - T2
(i)
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57Manufacturing
Principal Component Analysis
• Some systems may have many measured variables p– Often, strong correlation among these variables:
actual degrees of freedom are fewer• Approach: reduce order of system to track only
q << p variables– where each z1 … zq is a linear combination of the
measured x1 … xp variables
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58Manufacturing
Principal Component Analysis
• x1 and x2 are highly correlated in the z1direction
• Can define new axes zi in order of decreasing variance in the data
• The zi are independent• May choose to neglect dimensions with
only small contributions to total variance ⇒ dimension reduction
Truncate at q < p
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59Manufacturing
Principal Component Analysis• Finding the cij that define the principal
components:– Find Σ covariance matrix for data x– Let eigenvalues of Σ be – Then constants cij are the elements of the ith
eigenvector associated with eigenvalue λis• Let C be the matrix whose columns are the eigenvectors• Then
where Λ is a p x p diagonal matrix whose diagonals are the eigenvalues
• Can find C efficiently by singular value decomposition (SVD)– The fraction of variability explained by the ith principal
components is
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60Manufacturing
Extension to EWMA and CUSUM
• Define a vector EWMA
• And for the control chart plot
where
Zi = rxi + (1 + r)Zi −1
Ti2 = Zi
T ΣZi
−1Zi
ΣZi i =r
2 − r[1 − (1 − r)2 ]Σ
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61Manufacturing
Conclusions
• Multivariate processes need multivariate statistical methods
• Complexity of approach mitigated by computer codes
• Requires understanding of underlying process to see if necessary– i.e. if there is correlation among the variables of
interest