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
1
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart • CASE II(a): when n = 1. - Test statistic
- Its distribution and UCL
2
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart • Derivation of the distribution of T2 under CASE II(a)
3
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart • CASE II(b): when n > 1.
4
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart • CASE II(b): when n > 1. - Test statistic
- Its distribution and UCL
5
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart • Derivation of the distribution of T2 under CASE II(b).
6
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart • 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
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
7
Dr. Yu Ding
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 incontrol 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. ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
8
Dr. Yu Ding
Hotelling T2 control chart: Phase I analysis • Phase I analysis: when n > 1
9
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart: Phase I analysis • Derivation of the distribution of T2 for Phase I analysis
10
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart: Phase I analysis • Derivation of the distribution of T2 for Phase I analysis
11
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart: Phase I analysis • 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.
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
12
Dr. Yu Ding
Hotelling T2 control chart: Summary • Summary table for Hotelling T2 control charts
13
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
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
14
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart: Example 3.3 •
(Example 3.3)
T2 chart
95% probability contour 15
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Hotelling T2 control chart: Example 3.3 •
(Example 3.3)
revised T2 chart ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
16
Dr. Yu Ding
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.
17
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate CUSUM chart • Procedure for m-CUSUM
18
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate CUSUM chart • 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
19
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate CUSUM chart • 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. 20
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
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:
21
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate EWMA chart • 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
22
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate EWMA chart Note that here in the table is the statistical distance between µ0 and the shifted mean we try to detect.
23
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
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 (μ ) (μ μ ) Σ (μ μ ) 2 is the mean shift to be detected. T
1
1
1
0
1
0
• 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.
UCL
10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4
UCL for p = 5 is estimated as 6.6.
0
1
2
3
4
5
6
7
8
9
10 11
p 24
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate EWMA chart: Example 3.4 • 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.
25
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate EWMA chart: Example 3.4 • 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
26
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate EWMA chart: Example 3.4 • 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 10 9 8
UCL=5.99 for α=0.05
7 6 5 4 3 2 1 0
0
2
4
6
8
10
12
14
16
27
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate EWMA chart: Example 3.4 • 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 onetenth of T2 chart). That intuitively explains why when we saw an out-ofcontrol process indicated in the T2 chart but was not detected by the mEWMA.
28
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate EWMA chart: Example 3.4 • 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. 12
UCL=10.6 for alpha=0.005 10
8
UCL=5.99 for alpha=0.05 6
4
2
0
0
2
4
6
8
10
12
14
16
29
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
Multivariate EWMA chart: Example 3.4 • 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. 12
UCL=10.6 for alpha=0.005 h4=9.35 for m-EWMA
10
8
UCL=5.99 for alpha=0.05
6
4
2
0
0
2
4
6
8
10
12
14
16
30
ISEN 614 Advanced Quality Control (Anomaly and Change Detection)
Dr. Yu Ding
