Hotelling T control chart - Industrial & Systems Engineering | …ise.tamu.edu/inen614/chapter 3_p2.pdf · · 2010-04-08Hotelling T2 control chart ... that you have a p that is

ISEN 614 Advanced Quality Control (Anomaly and Change Detection) Dr. Yu Ding

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Hotelling T2 control chart

• CASE II: the in-control µ0 and 0 are NOT known. We need to

estimate them from training data.

• CASE II(a), when n = 1


2


• CASE II(a): when n = 1.

- Test statistic

- Its distribution and UCL


3


• Derivation of the distribution of T2 under CASE II(a)


4


• CASE II(b): when n > 1.


5


• CASE II(b): when n > 1.

- Test statistic

- Its distribution and UCL


6


• Derivation of the distribution of T2 under CASE II(b).


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• Relation of UCL under CASE II(b) to UCL under CASE I

• Be aware that how large m (and n) should be is relative to the value of

p. For example, for n=5, in order for 2 distribution to approximate F

distribution,p m required

2 > 50

10 >75

20 > 100


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Hotelling T2 control chart: Phase I analysis

• In change detection literature, the monitoring and detection

process is divided into two phases:

- Phase I: identify the in-control training data (which are used to

estimate the distribution parameters). Typically, apply a chart to

the training data to see if the training data are really in control.

Remove all out-of-control data and iterate until all training data

are in control.

- Phase II: apply the control charts established from the in-

control training data to future observations.

• This Phase I & II analysis should also be performed in the

univariate detection, even though we did not explicitly mention it.

• For T2 charts, so far we only discuss the Phase II analysis by

assuming that in-control training data have been already

identified.


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• Phase I analysis: when n > 1


10


• Derivation of the distribution of T2 for Phase I analysis


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• Derivation of the distribution of T2 for Phase I analysis


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• Phase I analysis: when n = 1

How good is this

approximation?

dist UCL

F 35.72

2 17.6

exact 12.0

for one chosen set of

p, n, m, and α.


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Hotelling T2 control chart: Summary

• Summary table for Hotelling T2 control charts


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Hotelling T2 control chart: Example 3.3

• Example 3.3 T2 chart. Study the Madison, Wisconsin, Police

Department data shown in the following table.

- Use data on x1 = Legal Appearances Hours and x2= Extraordinary Event

Hours, construct a T2 chart. Does the process represented by the

bivariate observations appear to be in control? set =0.05.

- Using the data of x1 = Legal Appearances Hours and x2= Extraordinary

Event Hours to estimate 0 and 0 for future observation x=(x1, x2)T


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• (Example 3.3)

T2 chart 95% probability contour


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• (Example 3.3)

revised T2 chart


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Multivariate CUSUM chart

• As we mentioned before, Hotelling T2 chart does not perform

well when the magnitude of change is small, just like what a

Shewhart suffers. We can increase the sample size, or we can

implement a multivariate version of CUSUM or EWMA to help

enhance the sensitivity of detection for small changes.

• Basic setting.


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• Procedure for m-CUSUM


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• m-CUSUM chart signals when MCi > UCL.

• The parameters to be chosen for m-CUSUM chart design are k

and UCL.

• The selection of k


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• The selection of UCL

- We have to utilize the Monte Carl simulation method to

evaluate the ARL under a choose UCL.

- Some results have been documented in literature.

- The above figure only list UCL's for p = 2, 3, and 10. In case

that you have a p that is not tabulated, you can find the UCL

by interpolation.


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Multivariate EWMA chart

• Same set up as in m-CUSUM, and still n = 1. After collecting the

ith observation, define:

• An m-EWMA chart signals when:


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• Selection of r and UCL

- Selection of r is very similar to the selection of in the

univariate case. A small r gives longer memory and is good

for detection small changes, while a large r gives shorter

memory and is good for detecting large changes. When

r = 1, an m-EWMA becomes a T2 chart.

- Selecting UCL


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Note that here in the table is the statistical distance between

µ0 and the shifted mean we try to detect.


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Multivariate EWMA chart: Example 3.4

• Example 3.4: Try to use a CUSUM chart to see if there is any

change in the Madison, Police Department data in Example 3.3.

Select control limits corresponding to ARL0 =200, and use

k=0.5, where is the mean shift to be

detected.

• Here p = 5. But in the data table given in the previous slide, the

UCLs for ARL0=200 and p= 2, 3 and 10 are listed. We need to

utilize an interpolation to decide UCL for p = 5.

2)()()( 01

1

011 μμΣμμμ

T

44.5

55.5

66.5

77.5

88.5

99.510

0 1 2 3 4 5 6 7 8 9 10 11

UCL

p

UCL for p = 5

is estimated as 6.6.


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• Example 3.4: The CUSUM chart: plot the statistic and upper control

limit in the following figure. There is not data point out of control. The

overall process (with five random variables) is stable.


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• Example 3.4: Construct a multivariate EWMA chart for the data of

x1 = Legal Appearances Hours and x2= extraordinary event hours in the

above Table in Example 3.3, with in-control ARL 200. This

multivariate EWMA chart will be used to detect a mean shift of =1

(here means the statistical distance of a mean shift, not the constant

in the univariate EWMA).

• Given that =1, p=2 and ARL0= 200, r is chosen as 0.16 and h4= 9.35.

The MEWMA chart is plotted as follows. There is no signal of out of

control.

m-EWMA

chart


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• Example 3.4: The conclusion appears different from the T2 chart in

Example 3.3. In fact, when we apply the EWMA chart, we first centered

the x's, namely that the mean of x were subtracted. Does that explain

the difference in the charts?

• If we apply T2 to the centered x1 and x2 with an alpha=0.05, the T2

chart looks like

0 2 4 6 8 10 12 14 160

1

2

3

4

5

6

7

8

9

10

UCL=5.99 for α=0.05


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• Example 3.4: One of the reasons for this discrepancy is that we

selected different values of error. We set = 0.05 when using T2

chart. But when we use h4 in the EWMA, it corresponds to ARL0=200.

Although ARL0 does not exactly equal to 1/ for a control chart other an

x-bar chart, we may use 1/ as an approximation to elaborate the

difference.

• When set =0.05, the ARL0 for T2 chart is about 20, which is far smaller

than ARL0=200 as when the EWMA is used. That is, the m-EWMA

chart will see much fewer alarms than the T2 chart (only about one-

tenth of T2 chart). That intuitively explains why when we saw an out-of-

control process indicated in the T2 chart but was not detected by the m-

EWMA.


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• Example 3.4: On the other hand, when use m-EWMA, ARL0=200 can

roughly translate to 0.005. If we use 0.005 for the UCL of the

T2 chart, it will be =10.6. Use this new UCL for the T2 chart for x1 and

x2, the chart looks like as follows, where the process is in control – no

sample point is above the UCL.

0 2 4 6 8 10 12 14 160

2

4

6

8

10

12

UCL=5.99 for alpha=0.05



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• Example 3.4: we plot T2 and EWMA together on the same graph: the

dotted black line is EWMA and the blue solid line is T2. We find that

had we used a much smaller UCL for m-EWMA, it will not identify the

same out-of-control points as the T2 chart did. That is because here we

have a spike-type change (a type of change different from a sustained

mean shift) and EWMA is not sensitive to detecting a spike. Rather,

EWMA is good at detecting a sustained mean shift.

0 2 4 6 8 10 12 14 160

2

4

6

8

10

12


h4=9.35 for m-EWMA


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Hotelling T control chart - Industrial & Systems Engineering | …ise.tamu.edu/inen614/chapter 3_p2.pdf · · 2010-04-08Hotelling T2 control chart ... that you have a p that is