An Adaptive Multivariate Control Chart for Individual

I nternational J ournal V ol .5, N o .1 (2012)

©

of

I ntelligent T echnologies

and

A pplied S tatistics

pp .57-74

A iriti P ress

An Adaptive Multivariate Control Chart for Individual Observations with Adaptive EWMS Star Glyphs for Diagnosis Tzong-Ru Tsai1, Y. L. Lio2, Shing I Chang3*, Shih-Hsiung Chou3 and Yi-Ting Chen1 1 De partment of Statistics, T amkang U niversity, N ew T ai pei City, T aiwan De partment of M athematical Sciences, U niversity of South Dakota, V ermillion, U S A 3 Qulait y Engineering Laborator y, De partment of I ndustrial and M anuf acturing S ystems Engineering, K ansas State U niversity, M anhattan, U S A 2

ABSTRACT A time-weighted multivariate control chart f or individual observations is introduced to monitor process variability changes in multiple quality characteristics in phase II applications. Because of adaptive adjustments, the proposed control chart is sensitive to total variation increase in a multivariate manufacturing process. A new visualization tool, named adaptive EWMS star glyphs, is proposed for process diagnosis once an out-of-control signal is found. An example is presented to illustrate the application of the proposed charting procedure. K eywords: Cumulative sum control chart; Likelihood ratio test; Shewhart control chart; Trace; Type I error

1. Introduction In many industrial applications, the quality of products typically depends on several correlated quality characteristics. Jointly maintaining several individual control charts f or process monitoring may result in unacceptably large overall type I error. It is beneficial to use a multivariate control chart for simultaneously monitoring two or more related quality characteristics rather than maintaining several univariate control charts independently. With modern data acquisition techniques and computing powers, multivariate control charts have received a renewed research interest in the past few decades. Most of the developments of multivariate control charts were based on the pioneer work of [11] and emphasized * Corresponding author: [email protected]

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58

Tsai, Lio, Chang, Chou and Chen

on the monitoring of mean vector of a process. However, process shif ts in the components of covariance matrix may also reflect nonconformities in the production processes of interest. Hence, it is important to simultaneously detect changes in a covariance matrix. One of the major concerns about the structure change in covariance matrix is the total variation increase. Total variation increase f rom the covariance matrix means quality deterioration in multivariate process quality characteristics. Let the variance-covariance matrix of multivariate quality characteristics from a process be denoted as Σ. When the process is in control, Σ is at its target Σ0. Alt [1], Alt and Bedewi [2] and Alt and Smith [3] developed Shewhart control charts f or detecting dif ferent types of changes of Σ f rom the target Σ0. The Shewhart control charts use information only from the current sampling and ignore any other information from the past sequence of sample points generated from the parameter estimator. This feature makes Shewhart control charts relatively insensitive to small process shif ts and potentially makes Shewhart control charts less usef ul in phase II application. The cumulative sum (CUSUM) control chart and the exponentially weighted moving average (EWMA) control chart are two alternatives that provide more sensitivity to small process shifts. Healy [10] proposed a CUSUM control chart to detect the change of Σ from Σ0 to c ⋅ Σ0, where c is a constant not equal to 1. Hawkins [7-8] studied multiple CUSUM and EWMA control charts to detect the changes of Σ from the target Σ0 based on regression-adjusted variables in conjunction with the control chart developed earlier by [6]. Tang and Barnett [15-16] used independent statistics obtained f rom the decomposition of Σ0, and provided various Shewhart control charts to detect the change of Σ f rom the target Σ0. Yeh et al. [18] used an unbiased likelihood ratio test statistic to develop a multivariate EWMA control chart to detect general changes in Σ f rom Σ0. Traditionally, most aforementioned references assume that the subgroup size n is larger than the dimensionality p so that the sample covariance matrix has f ull rank. However, in many industrial applications, particularly in the chemical and semiconductor industries, the subgroup size is usually small or only individual observations are available due to practical concerns such as cost. Most of the charting methods mentioned above are incapable f or monitoring multivariate process variability when only individual observations are available. To overcome this obstacle, [12] and [19] develop two time-weighted multivariate control charts based on a trace transf ormation and a probability integral transformation method, respectively, for monitoring the process variability changes in multivariate quality characteristics. The multivariate exponentially weighted moving variance control chart (MEWMV) and multivariate exponentially weighted mean squared deviation (MEWMS) control chart proposed by [12] are two excellent charts in detecting increases in variances or changes in correlations when only individual observations are available.

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An Adaptive Multivariate Control Chart for Individual Observations with Adaptive EWMS Star Glyphs for Diagnosis 59

In recent years some researchers proposed multivariate control charts f or simultaneously monitoring the process mean vector and covariance structure. Zamba and Hawkins [20] explored the multivariate change-point model through sequentially using the generalized likelihood ratio (GLR) test. With the use of the Bartlett correction adjustment, their method dramatically improves the chisquare approximation and is able to monitor short runs and unknown parameter process. Zhang et al. [22] combined an EWMA procedure and a GLR test for jointly monitoring both the multivariate process mean and variability. They called this control chart as ELR control chart. Their multivariate control chart has satisfactory performance in detecting various process shifts, especially in the shift of variance decreases. To improve process diagnosis, Zhang and Chang [21] proposed a combined DEWMA-MEWMD control chart for monitoring the mean vector and covariance structure for individual observations. Their proposed method used a sample size larger than one to estimate the covariance matrix Σ through the use of a moving window f or the DEWMA chart during the phase II monitoring. This practice provides dynamic estimates of Σ when individual observation is observed. Due to this dynamic updating, possible variance-covariance changes in Σ would not be reflected in the DEWMA chart which is designed to detect mean shifts only. Hawkins and Maboudou-Tchao [9] developed a multivariate exponentially weighted moving covariance matrix (MEWMC) control chart based on the multi-standardized data vectors to monitor the covariance structure. Let the raw process reading be X i which have in-control mean µ0 and in-control covariance matrix Σ0. They suggested the use of the transformed data, Ui = A(X i - µ0), i = 1, 2,..., for the plot statistics of MEWMC control chart, where A is the inverse-Cholesky root matrix, which corresponds to the cascade regression adjustment of [8]. Hawkins and Maboudou-Tchao also suggested monitoring the process mean and covariance structure through a combination of the multivariate exponentially weighted moving average (MEWMA) and MEWMC control charts. The multivariate control chart proposed by [20] is quite robust for simultaneously monitoring the process mean and covariance structure in multivariate quality characteristics. But the sequential design and Bartlett correction adjustment make this charting procedure more complicated for practical use. The integrated control chart proposed by [22] is robust and has satisfactory perf ormance in detecting various cases of process shifts. However, the Table 2 of [22] shows that the out-ofcontrol (ARL) values are large when the sample size is small for detecting variance increases. In cases where the detection of variance changes is up most important, simultaneous monitoring of both mean vector and covariance matrix may not be the best strategy of process monitoring. In this paper a new time-weighted multivariate control chart is introduced to monitor the total variation increase in multivariate process quality characteristics based on individual observations. An adaptive adjustment technique is used to generate the time-weighted statistics to make the proposed control chart more

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sensitivity in term of catching small increases in total variation in multivariate process quality characteristics. We call the new time-weighted control chart the adaptive multivariate exponentially weighted mean square error (AMEWMS) control chart. In addition, we also propose the use of a visual diagnostic tool in addition to the proposed process monitoring tool AMEWMS. Once an out-of-control signal was triggered, this proposed diagnostic tool called star glyphs is used to identify which variables are responsible for the root causes. The proposed diagnosis tool is similar to the CUSUM star glyphs proposed by [5], but we develop an adaptive exponentially weighted mean square error control chart to replace the CUSUM part in the original star glyphs. We call this diagnosis tool the adaptive EWMS star glyphs. Note that the adaptive EWMS control chart is a special case of the AMEWMS control chart with p = 1. The proposed method has the f ollowing f eatures: (1) the AMEWMS control chart is quick to detect the total variation increase in multivariate process quality characteristics using individual observations; (2) a diagnosis procedure is provided to identif y responsible variables immediately af ter the AMEWMS control chart triggers an out-of-control signal; and (3) the proposed method is easy to implement for practical use. The rest of this paper is organized as follows: in Section 2, a new multivariate control chart named A MEWMS control chart is developed f or individual observations. An operational procedure is given to determine the parameters of the AMEWMS control chart. Some control chart parameter estimates are tabulated. In Section 3, a numerical study is conducted to evaluate the performance of the AMEWMS control chart in terms of out-of-control ARL when monitoring total variation increase in multivariate process quality characteristics is the major concern. A diagnosis procedure using the adaptive EWMS star glyphs is provided in Section 4 to identif y responsible variables for causing an out-of-control signal f rom the proposed AMEWMS chart. The properties of the adaptive EWMS star glyphs diagnosis method are discussed and one example is given to demonstrate the application of the proposed method. Finally concluding remarks are given in Section 5.

2. AMEWMS control charts with diagnosis process 2.1 AMEWMS control chart Let X T = ( X 1, X 2,..., X p) represent a p-dimensional ( p ≥ 1) random vector of quality characteristics f or a manuf acturing process. Assume that X has a p-dimensional normal distribution with mean vector µu and covariance matrix Σu, denoted as N p(µu, Σu). When the process is in control, the mean vector and the covariance matrix are at their targets, respectively, as µu = µ0 and Σu = Σ0.

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Let Z Tt = Σ 0-1/2(X t - µ0) = (Z t1,..., Z t p) It can be shown that Z t's are independent and follow a p-dimensional normal distribution N p(µ, Σ), where µ = Σ 0-1/2 (µt - µ0) and covariance matrix Σ = Σ 0-1/2 ΣuΣ 0-1/2 Accordingly, we have µ = 0 and Σ = I p if the process of interest is in control, where I p is an identity matrix of order p. Following the development of the EWMA series introduced by [14], the multivariate EWMS series can be established. Let Yt = Z tZ Tt and

(1)

where Q0 = Z 1Z T1 and λ is a constant in (0, 1). Using the trace transformation to matrices Qt and Yt in Equation (1), and letting Q't = tr(Qt) and Y't = tr(Yt) = Σ pj=1 Zt2j, t = 1, 2,..., an EWMS series of {Q't, t = 1, 2,...} is given accordingly:

(2)

Let et = Y't - Q't -1(≠ 0) and φ (et) = λ (Y't - Q't -1) Based on the development procedure proposed by [17] and [4], we can treat φ (et) as a score function. The EWMS series in (2) can be rewritten as

(3)

For enhancing the sensitivity of the AMEWMS control chart to process shifts due to the total variation increase in multivariate process quality characteristics, we adjust φ (et) to Huber’s score function φ r(et) which is defined by

(4)

where γ is a positive constant. Let ω t = φ r(et)/et, then Equation (3) can be represented as

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(5)

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62


Note that the weight ω t depends on the residual et and is not a constant. Use Equation (4), the weights ω ’s can be presented as

(6)

An EWMS series with dynamical weights generated f rom (6) is called the adaptive EWMS series. The sampling distribution of the adaptive EWMS series in (5) is difficult to be derived analytically. A compromise adjustment is proposed to establish an approximate control limit using the adaptive EWMS series in (5). From (6) we find that ω t = λ if γ /|et| ≥ 1. That is, the adjustment is taken only when the value of γ /|et| is in the interval (-1, 1). Specifically, if 0 < γ / et < 1, ω t = 1 + γ (1 + λ )/ et, and ω t = 1 + γ (1 - λ )/ et if -1 < γ / et < 0. A compromise adjustment is suggested to replace γ / et by the middle value of (-1, 0) if −1 < γ / et < 0 or by the middle value of (0, 1) if 0 < γ / et < 1. Hence, a compromise adjusted weight can be written as

(7)

It follows that the adaptive EWMS series in (5) can be approximated by

(8)

where δ 1 = (1 + λ )/2, δ 2 = λ and δ 3 = (3 + λ )/2. Figure 1 displays two adaptive EWMS series of size 100 each, where all observations are generated from five-dimensional standard normal distribution with weights (6) and (7), respectively, under γ = 1.7 and λ = 0.2. Both series plots are close to each other. However the adjusted weights (7) are easier to operate. The weights generated from (6) or (7) show that the adaptive EWMS statistic (5) or (8) will be imposed a larger weight than the original EWMS statistic without adaptive weights. The consequence is that the plotting statistic (5) or (8) is more likely over the control limit if the residual et is larger than the threshold γ This adaptive adjustment mechanism makes the proposed control chart more sensitive to process parameter changes due to the total variation increase in multivariate process quality characteristics. Figure 1 shows that the adaptive EWMS series {Q't , t = 1, 2,... } in (5) is approximated by the adaptive EWMS of (8) which uses weight δ 1 = (1 + λ )/2 if

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Figure 1. AEWMS series of Q't with two types of weights from ω t (dash line) and ω't (solid line). et < -γ , δ 2 = λ if |et| ≤ γ and δ 3 = (3 + λ )/2 if et > γ . Let c'h1 = (1 - δ h)t-1 and c'hi = δ h(1 - δ h)t-i, h = 1, 2, 3, i = 2, 3,...t, and t = 1, 2,... It can be shown that the f ollowing three conditions are true: t C1: Σ i=1 c'hi = 1, h = 1, 2, 3. t C2: Σ i=1 c'hi2 → δ h/(2 - δ h), t → ∞. t C3: Q'ht ≅ Σ i=1 c'hi (Σ pj=1 zi2j ), h = 1, 2, 3. The approximate expected value and variance of adaptive EWMS series of (8) can be derived accordingly and provided in Theorem 1. Theorem 1. Considering the ada ptive EW MS series of (8), their ex pected value and variance can be a pproximated, respectively, as f ollows: (a) E[Q't ] and Var[Q't ] can be obtained numerically; In particular, (b) E[Q't ] ≅ p and Var[Q't ] ≅ 2p

,

as t → ∞ and the correlation among et's are low and can be ignored. The proof of Theorem 1(b) is given in the Appendix. The upper control limit (UCL) of AMEWMS control chart is given by

where the estimate of parameter L depends on the prescribed in-control ARL value and parameters γ and λ . In this paper, the probabilities of Pr (et < -γ ), Pr (|et| < γ ) and Pr (et > γ ) in Theorem 1(b) are approximated by their empirical probabilities, denoted by pˆ A, pˆB and pˆC respectively, based on 10,000 simulation runs. The approximate UCL of AMEWMS control chart can be formulated as

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64


where

, δ 2 = λ and

(9)

. Note that the UCL of the proposed AMEWMS

control chart is established by using Equation (9) and empirical probabilities, pˆ A, pˆB, and pˆC in the simulation process based on Theorem 1.

2.2 Properties of the proposed AMEWMS chart In practice, it is dif f icult to analytically and simultaneously determine all parameters of AMEWMS control chart. A feasible parameter selection method is to determine the values of L under a given combination of the other parameters (γ, λ) and the vector dimension p for each in-control ARL level of interest, denoted by ARL0. Table 1 lists the values of L and UCL based on (9)-(12), and ARL0 computed based on 10,000 simulation runs under the combination of γ = (1.2, 1.3, 1.5, 1.7), λ = (0.1, 0.2, 0.3) and p = (1, 2, 3, 5, 7, 10), respectively. The in-control ARL0 is fixed at approximately 370 and 500. Because of the dynamical weighting adjustment, the adaptive EWMS series of (5) contains larger f luctuation to increase the sensitivity to out-of-control signals. Values of L > 3 are selected in Table 1 to keep the ARL0 at the desired level against false alarms. Traditional EWMA control charts use L = 3 for monitoring process mean or variance. From Table 1, we observe that the UCL increases with respect to the vector dimension p. The proposed AMEWMS control chart with smaller γ has a narrower band of control limits than the one with larger γ. The proposed AMEWMS control chart uses γ as a threshold to control the sensitivity to total variation increase in multivariate process quality characteristics. According to the simulation experience from this study, a large λ helps increase the weight on the current observation but decrease the weight on the cumulative weighted statistic. When the process variability shif ts to a higher level, the residual et = Y't - Q't -1 increases positively. Once et increases beyond the threshold γ, a larger weight will be imposed on the adaptive EWMS statistic so that current plotting statistic is more likely to be plotted outside the control limit. Hence, an AMEWMS control chart with a small γ and large λ can provide more sensitivity to the total variation increase in multivariate process quality characteristics under the desired ARL 0 without inflating the type I risk.

3. A numerical study A simulation study is conducted to evaluate the performance of the proposed AMEWMS control chart. Two-dimensional data sets are generated from a bivariate normal distribution with mean vector µ and covariance matrix Σ. Parameters of

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Table 1. The values of L and control limits of the AMEWMS control chart for ARL0 ≈ 370 and 500. ARL0 ≈ 370 γ

λ

1.2

0.1

0.2

0.3

1.3

0.1

0.2

0.3

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ARL0 ≈ 500

p

L

UCL

L

UCL

1

8.30

10.441

8.80

11.070

2

6.00

13.304

6.29

13.976

3

5.08

15.669

5.33

16.266

5

4.19

19.660

4.40

20.425

7

3.77

23.393

3.96

24.143

10

3.44

28.426

3.59

29.141

1

7.70

10.596

8.10

11.141

2

5.46

13.350

5.80

14.071

3

4.67

15.760

4.88

16.381

5

3.90

19.769

4.09

20.502

7

3.52

23.462

3.66

24.094

10

3.20

28.507

3.33

29.282

1

7.00

10.620

7.40

11.208

2

5.07

13.586

5.29

14.147

3

4.30

15.846

4.50

16.501

5

3.57

19.778

3.75

20.59

7

3.28

23.722

3.44

24.469

10

2.97

28.650

3.10

29.476

1

8.70

10.569

9.08

11.096

2

6.26

13.470

6.50

13.996

3

5.17

15.725

5.47

16.439

5

4.29

19.757

4.50

20.556

7

3.80

23.247

4.05

24.223

10

3.49

28.430

3.65

29.276

1

7.70

10.619

8.30

11.145

2

5.67

13.508

5.99

14.099

3

4.82

15.857

5.01

16.472

5

3.99

19.938

4.18

20.701

7

3.61

23.657

3.75

24.31

10

3.30

28.776

3.43

29.499

1

7.30

10.832

7.63

11.389

2

5.19

13.597

5.48

14.281

3

4.44

16.020

4.62

16.651

5

3.70

20.055

3.86

20.781

7

3.35

23.808

3.47

24.569

10

3.03

28.768

3.17

29.606

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Table 1. The values of L and control limits of the AMEWMS control chart for ARL0 ≈ 370 and 500. (continued) ARL0 ≈ 370 γ

λ

1.5

0.1

0.2

0.3

1.7

0.1

0.2

0.3

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ARL0 ≈ 500

L

UCL

1

9.19

10.849

9.70

11.328

2

6.37

13.556

6.70

14.041

p

L

UCL

3

5.52

16.022

5.77

16.711

5

4.50

20.041

4.67

20.646

7

4.07

23.838

4.25

24.465

10

3.66

28.799

3.82

29.591

1

8.40

10.889

8.80

11.427

2

6.07

13.889

6.28

14.379

3

5.02

16.078

5.32

16.709

5

4.17

20.257

4.35

20.798

7

3.74

23.878

3.92

24.577

10

3.36

28.789

3.52

29.685

1

7.78

11.142

8.16

11.640

2

5.50

13.992

5.77

14.544

3

4.65

16.293

4.86

16.875

5

3.85

20.287

4.03

21.023

7

3.45

23.876

3.58

24.581

10

3.15

29.182

3.26

29.748

1

9.50

10.952

10.00

11.566

2

6.80

13.896

7.15

14.524

3

5.79

16.265

6.04

16.894

5

4.68

20.128

4.96

21.041

7

4.20

23.846

4.36

24.636

10

3.76

28.968

3.91

29.768

1

8.80

11.142

9.30

11.738

2

6.16

13.879

6.58

14.56

3

5.29

16.325

5.55

17.054

5

4.34

20.371

4.60

21.2

7

3.86

24.010

4.06

24.777

10

3.52

29.183

3.66

29.949

1

8.17

11.400

8.50

11.941

2

5.90

14.238

6.08

14.769

3

4.90

16.510

5.10

17.065

5

4.01

20.544

4.23

21.280

7

3.60

24.179

3.75

24.956

10

3.24

29.336

3.39

30.163

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T µ 0 = (0, 0) and Σ0 =

are set up for in-control process. Since the main focus is

on the total process variation increase, the mean vector is assumed unchanged over time, but the covariance matrix changes to Σ1 =

from Σ0 due to a

60

OC1 OC2 OC3

0

20

40

ARL1

80

100

root cause. Four out-of-control scenarios, denoted as OC1 to OC3 shown as follows, are considered with γ = 1.6 and λ = 0.2 Figure 2 shows ARL1s obtained using AMEWMS based on 20,000 simulation runs to detect the following out-of-control cases. OC1. Let (σ12, σ22, ρ ) = (3, 3, 0), (2, 2, 0), (3, 1, 0), (1.75, 1.75, 0), (1.5, 1.5, 0), (2, 1, 0), (1.75, 1, 0), (1.25, 1.25, 0), (1.5, 1, 0), (1.25, 1, 0). These cases are labeled by parameter shift numbers 1 to 10, respectively. OC2. Let (σ12, σ22, ρ ) = (3, 3, 0.25), (2, 2, 0.25), (1.75, 1.75, 0.25), (1.50, 1.50, 0.25), (1.25, 1.25, 0.25). These cases are labeled by parameter shift numbers 1 to 5, respectively. OC3. Let (σ12, σ22, ρ ) = (3, 3, 0.75), (2, 2, 0.75), (1.75, 1.75, 0.75), (1.50, 1.50, 0.75), (1.25, 1.25, 0.75). These cases are labeled by parameter shift numbers 1 to 5, respectively.

1

2

3

4

5

Parameters shift no.

Figure 2. Out-of-control ARLs obtained using the AMEWMS chart for variability changes due to the total variation increase in multivariate process quality characteristics.

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4. Diagnosis with adaptive EWMS star glyphs 4.1 The adaptive EWMS star glyphs Once AMEWMS control chart provides an out-of-control signal at location or time τ, a visualization tool, named adaptive EWMS star glyphs, is proposed f or process diagnoses. Our strategy is to identify which variables are out of control due to variance increases through the assistance of AMEWMS control chart with p = 1 and star glyphs once the multivariate version of the proposed AMEWMS chart generates an out-of-control point. The use of the multivariate AMEWMS provides a given desirable type I error while the use of univariate version of this chart enables user to quickly identif y responsible variables. The proposed diagnosis procedure similar to [5] star glyph procedure is given as follows: Step 1: Normalize all random variables based on the parameter estimates obtained from phase I application. Step 2: Find an adaptive EWMS series according to (8) for all dimensions until the time, τ, viz calculating {Q'it, i = 1, 2,..., p, t = 1, 2,..., τ}, respectively. Step 3: Construct star glyphs starting from sample period 1. Each glyph consists of a circle with radius of UCL computed using (9) and spikes with length of Q'it from the center of a circle. We choose to start the spikes from the three o’clock position and move the spikes in a counterclockwise direction with identical angle between any adjacent spikes as described in [5]. Step 4: Display the adaptive EWMS star glyphs. Out-of-control signals indicate which variables contribute to the out-of-control statistics of the AMEWMS control chart.

4.2

An example to demonstrate the operation of adaptive EWMS star glyphs

A f ive-dimensional normal distribution data set is used to demonstrate the application of the proposed AMEWMS control chart and adaptive EWMS star glyphs with γ = 1.2, λ = 0.3 and ARL ≈ 370. The AMEWMS control chart in Figure 3(a) shows that an out-of-control signal is given at the time τ = 17. This out-of-control signal would trigger the use of the adaptive EWMS star glyphs shown in Figure 3(b). It shows that the third dimension is out-of-control at sample period 15 and the second dimension is alarmed at sample period 17. In this case, the proposed star glyphs are able to identify the correct variables that are responsible for the out-ofcontrol signal.

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10 5

AMEWMA

15

20

17

0

10

20

30

40

index (a)

Figure 3. (a) The AMEWMS control chart and (b) AEWMS star glyphs for Case 3.

4.3 Properties of adaptive EWMS star glyphs and illustrative examples Diagnoses of variance increases are more complicated than those of mean shif ts. Suppose a production process contains multiple quality characteristics and the proposed AMEWMS control chart triggers an alarm at the time τ. This outof-control signal means that the total variation in multivariate process quality characteristics due to individual variances have increased. However, the increases or changes due to individual quality characteristics may not necessarily take place within the time window τ - 10. A numerical study is conducted to demonstrate the

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difficulty of implementing diagnosis of variance shifts. Assume that random vectors Z Ti = (Z 1i , Z 2i ,... Z 5i ), i = 1, 2,... are generated f rom a f ive-dimensional normal distribution, N5(0, Σ), where 0 is a zero mean vector and Σ = (σ i j) is a 5 × 5 variancecovariance matrix, where σ i j = σ2i , if i = j; otherwise σ i j = ρσ iσ j, 0 ≤ ρ < 1, for i ≠ j, and 2 = 1, i = 1, 2, 3, 4, 5 and ρ = ρ0 = 0 are set i, j = 1, 2, 3, 4, 5. Process parameters σ12 = σ10 up for in-control process. Assume that an assignable cause was introduced into the process such that the ith variance shifts to σ' 2i (> 1) and/or the correlation coefficient shift to ρ1 > 0. The AMEWMS control chart with λ = 0.3, γ = 1.2 and ARL0 ≈ 370 is used to monitor the structure of covariance matrix. Once the AMEWMS provides an out-of-control signal at the time τ, the adaptive EWMS star glyphs method with λ = 0.3, γ = 1.2 and ARL0 ≈ 370 is used to search for out-of-control variables. Let ∆i1 = σ' i / σi0 = σ' i denote the shif t magnitude of the ith process variance f rom its target, i = 1, 2,...5. The following six out-of-control scenarios, denoted by Case 1 to Case 6, respectively, are used to evaluate the performance of adaptive EWMS star glyphs method: ∆i1

∆i2

∆i3

∆i4

∆i5

ρ1

Case 1

1

1.5

1.5

1

1

0

Case 2

1

1.5

1.5

1

1

0.2

Case 3

1

1.5

1.5

1

1

0.5

Case 4

1

2

2

1

1

0

Case 5

1

2

2

1

1

0.2

Case 6

1

2

2

1

1

0.5

In this study, 10,000 simulation runs are used f or each case of these six out-ofcontrol scenarios. A sample of size five is considered. Within each simulation run, ten in-control observations are generated before out-of-control samples are simulated. The main perf ormance statistic of the proposed adaptive EWMS star glyphs is the discrimination rate without false alarms (labeled as DR W ). Specif ically the discrimination rate in this case study is def ined as the percentage of successf ul identification of Z 2, Z 3, or both as the out-of-control variables instead of any one of the Z 1, Z 4 or Z 5. The DRW values with γ = 1.2, λ = 0.3, ARL0 ≈ 370 are given in Table 2. We conclude that the adaptive EWMS star glyphs diagnosis method performs satisfactory with its values ranging from 0.750 to 0.918. When the shift magnitude is large, the discrimination rates are high. However, it is difficult for the proposed star glyphs to identif y that both Z 2 and Z 3 are out-of-control variables within the sampling window given especially when the shif t magnitude is small. This phenomenon takes place because the proposed AMEWMS chart already signals that the process of interest is out of control. At this point, there is no new point plotted. Since a diagnosis procedure is conducted retrospectively, the star glyphs could not go beyond this out-of-control point. We will now investigate a second performance statistic to learn more about this issue.

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Table 2. The discrimination rates of AEWMS star glyphs for various out-of-control situations. DRW

PUD

ARL1

sd(RL1)

Case 1

0.786

0.239

24.429

23.598

Case 2

0.780

0.256

22.569

21.585

Case 3

0.750

0.302

17.028

15.946

Case 4

0.917

0.331

6.627

5.909

Case 5

0.918

0.331

6.465

5.809

Case 6

0.904

0.351

6.169

5.522

Let PU denote the proportion of simulation runs that only one of the Z 2 or Z 3 is identified as the out-of-control variable before time τ is reached. The sampling window given is τ - 10. The values of PU given in Table 2 ranges from 0.239 to 0.351 for Case 1 to Case 6. The PU values for Case 4 to Case 6 indicate that over 30% of simulation runs only identify one rather both as the out-of-control variable. Table 2 also lists the mean and standard deviation of run lengths denoted as ARL1 and sd(RL1) while maintaining the ARL0 ≈ 370 based on 10,000 simulated runs. We find that the proposed AMEWMS control chart provides short out-of-control ARL1 and sd(RL1) values for all out-of-control scenarios especially for Case 3 to Case 6. For example, the ARL1 value of Case 6 is 6.169. This small number implies that it is difficult for the proposed star glyphs to identify both Z 2 and Z 3 as the out-of-control variables within such a short window. In practice, the DR W statistic is much more important than the P U statistic. Although we only identify one of the Z 2 or Z 3 as the out-of-control variable through the adaptive EWMS star glyphs after the AMEWMS control chart alarms at time τ. The practice of statistical process control will remove the root cause of the outof-control variable that is first found then return to the AMEWMS control chart monitoring. The other out-of-control variable or variables will be identified through the AMEWMS chart and then the adaptive EWMS star glyphs in a recursive manner.

5. Conclusions A new adaptive time-weighted multivariate control chart, named AMEWMS control chart, is introduced in this paper for monitoring variability change due to total variation increase in multivariate process quality characteristics using individual observations. The proposed adaptive adjustment scheme dynamically weights the observations over time to make the proposed AMEWMS control chart more sensitive in detecting total variation increase in multivariate process quality characteristics. To overcome tedious numerical computation ef f orts, the

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upper control limit of AMEWMS control chart is evaluated numerically and tabulated for parameters γ = 1.2, 1.3, 1.5, 1.7, λ = 0.1, 0.2, 0.3 and the vector dimensions p = 2, 3, 5, 7, 10 to produce an in-control average run length approximately 370 and 500, respectively. These tabulated upper control limits provide a guideline in establishing the proposed AMEWMS control chart in practice. Once the AMEWMS control chart provides an out-of-control signal, a visualization tool, named adaptive EWMS star glyphs, is used to search for responsible individual out-of-control variables. The combined use of the AMEWMS control chart and the adaptive EWMS star glyphs diagnosis procedure is recommended to remove root causes f rom a manufacturing process when the process is out of control due to the total variation increase in multivariate process quality characteristics. However, the proposed charting procedure cannot diagnose the changes in correlations, which is a potential topic for future study.

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APPENDIX: PROOF OF THEOREM 1 Assume that the correlation among et's are low and can be ignored. Since Σ pj=1 z 2i j has an approximate chi-square distribution with degrees of f reedom p. Using Condition C1, we have

It follows that

Using Condition C2, we have

Then

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