Dual CUSUM Control Schemes for Detecting a Range of Mean Shifts

Zhaojun Wang (Joint work with Yi Zhao and Fugee Tsung)

Department of StatisticsSchool of Mathematical Sciences

Nankai Universityzjwang@nankai.edu.cn

Who am I?

Who Are They? They are Zhaojun WANG

He is 王兆军！ It’s me!She is 王昭君！

OUTLIER

• Background and Motivation

• Description of Dual CUSUM Schemes

• The Design of DCUSUM• Criterion for performa

nce evaluation • The Comparisons• Conclusion and Consid

eration• And then…..

Background and Motivation

• Shewhart X-bar control Chart(Shewhart 1931)

• Cumulative Sum (CUSUM) control chart (Page 1954)

Background and Motivation (Cont'd)

• The FIR CUSUM chart (Lucas & Crosier 1982)

• Exponentially Weighted Moving Average (EWMA) control chart (Roberts 1958)

Background and Motivation (Cont'd)

• The combined Shewhart-CUSUM (SCUSUM) schemes (Westgard, Groth, Aronsson, and de Verdier 1977 by Simulations, and Lucas 1982 by Markov Chain)

Background and Motivation (Cont'd)

• The combined Shewhart-EWMA (SEWMA) schemes (Lin and Adams 1996, Klein 1996 and Fan et al. 2000)

• Dual CUSUM schemes (DCUSUM)

Background and Motivation (Cont'd)

Background and Motivation (Cont'd)

• Problems: How to detect the

interval shifts? What Criteria to be

used? How about the

above charts? Any new good

charts?

Background and Motivation (Cont'd)

• Lorden (1971) firstly provides a control scheme could be used to detect the interval shifts, but it is difficult to apply in practice and its performance is difficult to obtain due to great computational complexity.

Description of Dual CUSUM Schemes

Description of Dual CUSUM Schemes (Cont'd)

Description of Dual CUSUM Schemes (Cont'd)

Description of Dual CUSUM Schemes (Cont'd)

Design of DCUSUM

Design of DCUSUM (Cont'd)

Design of DCUSUM (Cont'd)

Design of DCUSUM (Cont'd)

Design of DCUSUM (Cont'd)

与以前不同的在于……

Criterion for performance evaluation

Criterion for performance evaluation (Cont'd)

Criterion for performance evaluation (Cont'd)

• The scatter plot of ARL via mean shift of the DCUSUM scheme when ARL0=1000 and the shift interval is

(0.25, 5)

1 2 3 4 5

-3

-2

-1

1

2

3

1 2 3 4 5

2

4

6

8

10

12

14

ARL

log(ARL-1)

The Comparisons

The Comparisons (Cont'd)

* In this table, the M1 is the same as .Q

The Comparisons (Cont'd)

The Comparisons (Cont'd)

The Comparisons (Cont'd)

The Comparisons (Cont'd)

The Comparisons (Cont'd)

The ratio of out-of-control ARLs between Q4 design based on range [0.25,3] and the optimal CUSUM scheme with in-control ARL=300

The Comparisons (Cont'd)

The Comparisons (Cont'd)

Conclusion and Consideration

• As the range of the shifts or the value of ARL0 increases, the superiority of DCUSUM schemes increases. But, for small range shifts, the CUSUM performs better than the DCUSUM as shown in Table 1.

• In Table 2, it’s found that there is no much difference between M3 and Q4 design, Q2 (single CUSUM) is worse than Q4 and M3 (but M3 is hard to compute).

• In Table 3, we also compare Q4 designs in range [0.25,3], and [0.25,5]. (almost same)

Conclusion and Consideration(Cont'd)

• In Figure 3, we observed that the performance of Q4 scheme is comparable to the optimal CUSUM design for most of the shifts between 0.25 and 3, except that for some very small shifts.

• DCUSUM schemes overall have superior performance when compared with the Shewhart-CUSUM counterparts in terms of IRARL for small and medium shift ranges.

Conclusion and Consideration(Cont'd)

And then

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