# Robust HotellingT2 Control Chart with Consistent Minimum Vector Variance

Authors: Hazlina Ali; Sharifah Soaad Syed Yahaya; Zurni Omar
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Recently, an alternative robust control chart based on a new robust estimator known as minimum vector variance (MVV) estimator,TMVV2, was introduced in Phase II. TMVV2 was able to detect out-of-control signal and simultaneously control false alarm rate even as the dimension increased. However, the estimated UCLs of TMVV2 are large as compared to the traditional chart. In this study, we improved the MVV estimators in terms of consistency and bias. The result showed great improvement in the control limit values while maintaining its good performance in terms of false alarm and probability of detection.


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## Body

## 1. Introduction

HotellingT2 statistic was the first statistic known to be used in multivariate control chart. The control chart is referred to as Hotelling T2 control chart. This statistic is used to measure the significance of the shifted distance from the out-of-control mean vector, μs, to the nominal mean vector, μ0, with the assumption that the covariance matrix remains constant at Σ0. The purpose of the control chart is to monitor the stability of a multivariate process in Phase I and II. Analysis in Phase I seeks to identify a stable historical data set (HDS). From this dataset, the in-control mean vector and the in-control variance-covariance matrix are estimated and later will be used in the Phase II analysis. A successful process monitoring in Phase II totally depends on the estimates of the parameters obtained from a stable HDS. However, the estimators are easily affected by unstable process, that is, multivariate outliers. The existence of outliers can violate the normality assumption. This violation may lead to the inflation of control limits and reduction of the probability of detection in Phase I, which consequently will cause the level of false alarm to be distorted, and the power to detect changes will be reduced in Phase II process [1]. False alarm rate is the probability of out-of-control signal when a process is in control. The value becomes large if the process is unstable due to the increase in variability. Inflated false alarm rate can lead to unnecessary process adjustments and loss of confidence in the control chart as a monitoring tool [2]. Hence, a method which can control the false alarm rate to the desired (nominal) level is necessary.However, the traditional HotellingT2 control chart is only effective in eliminating extreme outliers in small sample sizes, but it fails to detect moderate outliers particularly when the number of variables is large [3–5]. To overcome the problem, alternative estimation methods have been proposed in the literature. One of the approaches is to calculate the T2 statistic based on successive differences variance-covariance matrix estimator [6–9]. Though this approach is effective in detecting shifts in the mean vector, it fails to detect other outliers as shown in Vargas [3]. Another approach is to use robust estimators in place of the classical estimators (x- and S). Robust estimators are known to be more effective in detecting the deviation of data, or outliers as compared to the classical estimators [10]. A wide range of robust estimators of multivariate location and scatter is available; see [11, 12] for a review. However, MCD estimator is more attractive than others because it has good theoretical properties with affine equivariance, high breakdown value, bounded influence function, and better convergence rate [13, 14]. The study on the significant role of MCD estimators in the construction of robust Hotelling T2 chart can be easily found in the literature. Vargas [3] and Jensen et al. [4] introduced robust control chart based on MCD estimator for multivariate individual observations. They identified and removed the outliers in Phase I analysis by using robust estimator and then calculated the classical estimator using the remaining good data points for Phase II analysis. They noticed some drawbacks when MCD was used in Phase I. Hotelling’s T2 issued from MCD needed a larger sample size if large number of outliers was suspected to ensure that MCD estimator did not breakdown and lose its ability especially when monitoring with more quality characteristics (p). To abate the problems, Chenouri et al. [5] proposed robust Hotelling T2 chart based on reweighted MCD estimator. Besides possessing the nice properties of MCD estimator, the estimator was not unduly influenced by outliers and was more efficient than MCD. However, their approach was different from Vargas [3] and Jensen et al. [4] whereby they used RMCD estimator in place of classical estimators in constructing Hotelling T2 chart for Phase II data directly. Using the same approach as Chenouri et al. [5], Alfaro and Ortega [15] conducted a comparison study on the performance of Hotelling T2 control chart in Phase II process based on MCD, MVE, reweighted MCD, and trimmed estimator. Their finding showed a conflict between the percentage of outliers detection and the ability of the robust control charts in controlling the overall false alarm rate under certain conditions. To alleviate this conflict, Yahaya et al. [16] introduced the MVV estimator in Hotelling T2 chart (TMVV2) in Phase II. In general, the result showed that TMVV2 chart was able to increase the detection of out-of-control signals and simultaneously control false alarm rates even with large number of quality characteristics. In contrast, the MCD charts performed well in detecting out-of-control signals but failed in controlling false alarm rates. The traditional chart, however, was able to control false alarm rates but not effective in detecting out-of-control signals. Despite the good performance of TMVV2, the estimated UCLs for Hotelling T2 chart issued from MVV estimators were large as compared to the traditional and MCD charts. Thus, this study attempts to improve the MVV estimators in achieving the desired UCLs by making the estimators consistent at normal model. Since in practice we always deal with finite samples, therefore the issue of bias in a finite sample is also considered in this study. The advantage of having unbiased estimator for a finite sample is that this estimator remains unbiased even though the sample size becomes larger [17]. With respect to the latter issue, this paper will also seek to improve the performance of MVV by making it unbiased for finite samples.The organization of the remaining part of this paper is as follows. The formal definition ofMVV estimator and the adjustment done on the MVV scatter estimator to ensure that it is consistent and unbiased will be discussed next, followed by the investigation on the improved MVV estimator through simulation study. The discussion continues with the computation of control limits for the traditional, MCD, and the improved Hotelling TMVV2 charts, and the improvement of the proposed chart is revealed in this section. A real data analysis from aircraft industry is presented to illustrate the applicability of the proposed charts before arriving to the conclusion in the last section.


## 2. Minimum Vector Variance (MVV) Estimator

Herwindiati et al. [18] had proved that MVV estimators possess three major properties of a good robust estimator, that is, high breakdown point, affine equivariance, and computational efficiency. The main method used in the estimation of MVV is the Mahalanobis squared distances (MSDs). Let X={x1,x2,…,xn} be a data set of p-variate observations. Denote the MVV estimators for the location parameter and scatter by MMVV, and SMVV respectively. Now let H⊆X, the mMVV and SMVV are determined based on the set H consisting of h=⌊(n+p+1)/2⌋ data such that SMVV has minimum trace of SMVV2, denoted as Tr(SMVV2), among all possible sets of h data. To compute the estimates of MVV, we used the MVV algorithm proposed in Yahaya et al. [16]. The location and scatter estimators are defined as
(1)mMVV=1h∑i=1hxi,(2)SMVV=1h∑i=1h(xi-MMVV)(xi-MMVV)t.


### 2.1. Consistency Factor

The aim of HotellingT2 chart in Phase I is to estimate the in-control parameters of location, μ and scatter, Σ. The usual estimators for these parameters are the normal maximum likelihood estimators (MLE). The estimation of parameters is based on the data set x={x1,x2,…,xn} from multivariate normal distribution with density
(3)fμ,Σ(x)=1(2π)p/2|Σ|1/2e(-(x-μ)tΣ-1(x-μ)/2),
with μ∈ℝp and Σ∈ℤ+. However, the distribution of (3) is only an approximation because a portion of the data may be contaminated by outliers [19]. With the existence of outliers, MLE which are known to be sensitive to outliers will not be able to precisely estimate the parameters. To address this problem, we propose MVV estimators, that is, robust estimators with highest breakdown point (50%) proposed by Herwindianti [20] to replace the MLE. We compute the MVV estimators in Phase I data sets, with location and scatter estimators as defined in (1) and (2), respectively. The MVV estimator has a fixed integer h such that
(4)⌊n+p+12⌋≤h<n.The preferred choice ofh for outlier detection is its lower bound, which yields the breakdown value, BP=(n-2(p-1))/2n. Let mMVV and SMVV be the mean and the scatter matrix calculated from the h observations out of xi, whose classical scatter matrix has the lowest vector variance resulting from h smallest MSD. The SMVV is a scatter p×p matrix which is positive definite, symmetric (PDS), and affine equivariant [20]. However, this estimator is not consistent under normal model. Robust scatter estimator is typically calibrated to be consistent for normal model. Known as Fisher consistency, this is a standard concept in robust statistics which denotes that the functionals evaluated at the model distribution return the true parameter value, Σ [19]. In order to achieve consistency under the normal model, SMVV (in (2)) is multiplied by a consistency factor, c(h), as follows:
(5)c(h)SMVV=c(h)h∑i=1h(xi-mMVV)(xi-mMVV)t.
The approximation of consistency factor can be obtained from elliptical truncation in the multivariate normal distribution based on squared distance. If xi~N(μ,Σ),  c(h) is defined as
(6)c(h)=h/nP(χp+22<χp,h/n2),
where χp,h/n2 is the h/n-quantile of χp2 distribution. This formula is derived by Butler et al. [13] and further discussed in Croux and Haesbroeck [14] based on the functional form of the MCD estimator. Since MVV have the same functional form with the MCD estimator, we used (6) as the consistency factor for SMVV. Albeit guaranteed consistency under normality distribution, Pison et al. [17] cautioned that MCD estimators were biased for small sample sizes. Thus, the consistency factor in (6) only might not be sufficient to make MVV estimator unbiased for small sample sizes. For that reason, we also include the computation of correction factor at any sample size n and dimension p.


### 2.2. Correction Factor

A simulation study on the effect of correction factor on theMVV estimator is carried out for several sample sizes n and dimension p = 2, 5, 10, 15, and 20. We generated data sets X(j)∈ℝn×p from standard multivariate normal distribution. For each data set X(j), j=1,…,m we then determine the c(h)SMVV(j) in (5). If the estimator is unbiased, E[c(h)SMVV]=Ip, therefore the pth root of the determinant of c(h)SMVV equals 1. Given the mean of the pth root of the determinant as mean(|c(h)SMVV|)=(1/m)∑j=1m(|c(h)SMVV(j)|)1/p. To determine the correction factor, we performed m=1000 simulations for different sample sizes n and dimensions p, with α=0.05 such that
(7)ϑp,nα=1mean(|c(h)SMVV|).
The computed values are displayed in Table 1. Then, using ϑp,nα in (7) as the correction factor for c(h)SMVV, we obtain
(8)ϑp,nαc(h)SMVV=ϑp,nαc(h)h∑i=1h(xi-mMVV)(xi-mMVV)t.
Since ϑp,nαc(h)SMVV can be considered consistent and unbiased, the determinant of ϑp,nαc(h)SMVV should approach 1.Table 1
Values of theϑp,nα for c(h)SMVV.
p
=
2
p
=
5
p
=
10
p
=
15
p
=
20
n
ϑ
p
,
n
α
n
ϑ
p
,
n
α
n
ϑ
p
,
n
α
n
ϑ
p
,
n
α
n
ϑ
p
,
n
α
10
5.8276
30
2.9304
50
2.3127
80
1.9801
100
1.8503
25
5.4200
50
2.9598
100
2.2098
100
1.9528
200
1.7762
50
5.7715
100
2.9309
200
2.1524
200
1.8937
300
1.7490
100
5.7206
200
2.8912
500
2.1045
500
1.8476
500
1.7180
200
5.6679
500
2.8579
500
5.5842


## 2.1. Consistency Factor

The aim of HotellingT2 chart in Phase I is to estimate the in-control parameters of location, μ and scatter, Σ. The usual estimators for these parameters are the normal maximum likelihood estimators (MLE). The estimation of parameters is based on the data set x={x1,x2,…,xn} from multivariate normal distribution with density
(3)fμ,Σ(x)=1(2π)p/2|Σ|1/2e(-(x-μ)tΣ-1(x-μ)/2),
with μ∈ℝp and Σ∈ℤ+. However, the distribution of (3) is only an approximation because a portion of the data may be contaminated by outliers [19]. With the existence of outliers, MLE which are known to be sensitive to outliers will not be able to precisely estimate the parameters. To address this problem, we propose MVV estimators, that is, robust estimators with highest breakdown point (50%) proposed by Herwindianti [20] to replace the MLE. We compute the MVV estimators in Phase I data sets, with location and scatter estimators as defined in (1) and (2), respectively. The MVV estimator has a fixed integer h such that
(4)⌊n+p+12⌋≤h<n.The preferred choice ofh for outlier detection is its lower bound, which yields the breakdown value, BP=(n-2(p-1))/2n. Let mMVV and SMVV be the mean and the scatter matrix calculated from the h observations out of xi, whose classical scatter matrix has the lowest vector variance resulting from h smallest MSD. The SMVV is a scatter p×p matrix which is positive definite, symmetric (PDS), and affine equivariant [20]. However, this estimator is not consistent under normal model. Robust scatter estimator is typically calibrated to be consistent for normal model. Known as Fisher consistency, this is a standard concept in robust statistics which denotes that the functionals evaluated at the model distribution return the true parameter value, Σ [19]. In order to achieve consistency under the normal model, SMVV (in (2)) is multiplied by a consistency factor, c(h), as follows:
(5)c(h)SMVV=c(h)h∑i=1h(xi-mMVV)(xi-mMVV)t.
The approximation of consistency factor can be obtained from elliptical truncation in the multivariate normal distribution based on squared distance. If xi~N(μ,Σ),  c(h) is defined as
(6)c(h)=h/nP(χp+22<χp,h/n2),
where χp,h/n2 is the h/n-quantile of χp2 distribution. This formula is derived by Butler et al. [13] and further discussed in Croux and Haesbroeck [14] based on the functional form of the MCD estimator. Since MVV have the same functional form with the MCD estimator, we used (6) as the consistency factor for SMVV. Albeit guaranteed consistency under normality distribution, Pison et al. [17] cautioned that MCD estimators were biased for small sample sizes. Thus, the consistency factor in (6) only might not be sufficient to make MVV estimator unbiased for small sample sizes. For that reason, we also include the computation of correction factor at any sample size n and dimension p.


## 2.2. Correction Factor

A simulation study on the effect of correction factor on theMVV estimator is carried out for several sample sizes n and dimension p = 2, 5, 10, 15, and 20. We generated data sets X(j)∈ℝn×p from standard multivariate normal distribution. For each data set X(j), j=1,…,m we then determine the c(h)SMVV(j) in (5). If the estimator is unbiased, E[c(h)SMVV]=Ip, therefore the pth root of the determinant of c(h)SMVV equals 1. Given the mean of the pth root of the determinant as mean(|c(h)SMVV|)=(1/m)∑j=1m(|c(h)SMVV(j)|)1/p. To determine the correction factor, we performed m=1000 simulations for different sample sizes n and dimensions p, with α=0.05 such that
(7)ϑp,nα=1mean(|c(h)SMVV|).
The computed values are displayed in Table 1. Then, using ϑp,nα in (7) as the correction factor for c(h)SMVV, we obtain
(8)ϑp,nαc(h)SMVV=ϑp,nαc(h)h∑i=1h(xi-mMVV)(xi-mMVV)t.
Since ϑp,nαc(h)SMVV can be considered consistent and unbiased, the determinant of ϑp,nαc(h)SMVV should approach 1.Table 1
Values of theϑp,nα for c(h)SMVV.
p
=
2
p
=
5
p
=
10
p
=
15
p
=
20
n
ϑ
p
,
n
α
n
ϑ
p
,
n
α
n
ϑ
p
,
n
α
n
ϑ
p
,
n
α
n
ϑ
p
,
n
α
10
5.8276
30
2.9304
50
2.3127
80
1.9801
100
1.8503
25
5.4200
50
2.9598
100
2.2098
100
1.9528
200
1.7762
50
5.7715
100
2.9309
200
2.1524
200
1.8937
300
1.7490
100
5.7206
200
2.8912
500
2.1045
500
1.8476
500
1.7180
200
5.6679
500
2.8579
500
5.5842


## 3. Investigation through Simulation Experiment

Gather and Becker [21] have emphasized that robust estimators to be used in the method of outliers detection should have sufficient rate of convergence to some true underlying model parameter for consistency and unbiased. A sequence of asymptotically unbiased estimators for parameter θ is called consistent if limn→∞P(|θ^n-θ|≥ε)=0. To illustrate the analysis on the consistency of MVV estimator at multivariate normal, data are randomly generated from N(0,Ip). An experiment is carried out for several values of sample sizes n until convergent for a fixed moderate dimension such that p=10. Figure 1 shows the determinants of ϑp,nαc(h)SMVV corresponding to the sample size, n. As the value of n increases, we can observe that the determinant approaches 1 which implies that the ϑp,nαc(h)SMVV is consistent. Next, the investigation using simulation experiment continues to show that mMVV and ϑp,nαc(h)SMVV which replaced the MLE, μ and Σ, in Hotelling T2 are consistent and unbiased. The squared distances using any affine-equivariant robust location and scatter estimators which are consistent and unbiased under normal model are asymptotically χ2 distributed [21]. Therefore, if mMVV and ϑp,nαc(h)SMVV are consistent and unbiased estimators for μ andΣ, then with observations xi i.i.d in ℝp~Np(μ,Σ), it follows that di2=(xi-mMVV)ϑp,nαc(h)SMVV-1(xi-mMVV)t is asymptotically χp2 distributed. Since di2 is similar to Hotelling T2, the asymptotic distribution of the improved Hotelling TMVV2 when n→∞ should follow χp2 distribution if the estimators are consistent. If we consider a sample of p quality characteristics such that xi={xi1,xi2,…,xip} where i=1,2,…,n as a phase I data set, then the improved TMVV2 statistic for xi can be constructed in the following manner:
(9)TMVV(I)2(i)=(xi-mMVV)ϑp,nαc(h)SMVV-1(xi-mMVV)t.
To check on the distributions of the improved TMVV2, we employed the QQ plots and evaluated the goodness of fit on those plots based on the slope and the R-square of the straight line as shown in Table 2. The hypothetical distribution represents the χp2 without error if all points are in a straight line with slope equals 1 and R-square also equals 1 [22]. Random data were generated from multivariate standard normal distribution MVN(0,Ip). This study is carried out for the sample size of n = 10,000 with dimensions of p = 2, 5, 10, 15, and 20.From this table we observe that the R-square values for all p’s are 0.999. With regard to the slopes, we can see a considerable difference in the values between the Hotelling’s T2 with original MVV(TMVV(o)2) and Hotelling’s T2 with improved MVV(TMVV(I)2) especially when p = 2. The slopes for TMVV(I)2 are consistent and approximately equal to 1 regardless of the dimensions (p). In contrast, the slopes for TMVV(o)2 are quite a distance away from 1 even though the pattern shows a declining in values towards 1 as p increases. We observe that the values for the two measurements (R2 and slopes) are very close to the ideal value, which signify that the χp2 distribution fits well with the simulated TMVV(I)2 values. The result implies that the constant ϑp,nαc(h) fulfills the condition of the multiplicative factors to make the SMVV estimators consistent and unbiased for Σ.Table 2
The slope andR-square for TMVV(o)2 and TMVV(I)2.
n
=
10
000
T
MVV
(
o
)
2
T
MVV
(
I
)
2
p
=
2
R
2
0.999
0.999
slope
3.264
1.001
p
=
5
R
2
0.999
0.999
slope
1.952
1.020
p
=
10
R
2
0.999
0.999
slope
1.538
1.003
p
=
15
R
2
0.999
0.999
slope
1.404
1.001
p
=
20
R
2
0.999
0.999
slope
1.341
1.005Figure 1
Determinant ofϑp,nαc(h)SMVV when n→∞ and p = 10.


## 4. MVV HotellingT2 Control Chart

Letxi={xi1,xi2,…,xip} be the p-variate random sample of n observations of preliminary data set in Phase I. Calculate the MMVV and ϑp,nαc(h)SMVV estimators. Since the estimators are known to be free from outliers due to their estimation process, they could be readily used as in-control estimators in Phase II. By using these estimates, TMVV(I)2 statistic in (9) is computed for Phase II observation, xg={xn+1,xn+2,…} where xg∉xi.


### 4.1. Estimation of Control Limits

In this section, we present the control limit of the improvedTMVV(I)2 control chart by using simulated data with different combinations of sample sizes, n, and number of dimensions, p. The control limit of TMVV(I)2 chart is then compared with the control limit of TMVV(o)2 chart, robust Hotelling T2 chart using MCD (TMCD2) and the traditional Hotelling T2 charts. The application of robust estimators in place of the mean and covariance structure in traditional Phase II Hotelling T2 statistic will cause the distributional properties of the traditional chart to change [9]. To demonstrate the performance of TMVV(o)2 and TMVV(I)2 control charts, we need to identify the distribution of each method in order to obtain appropriate control limits, that is, UCL. Since the exact distribution of TMVV(o)2 is unknown, we apply Monte Carlo method to estimate the quantiles of the TMVV(o)2 and TMVV(I)2, for several combinations of sample sizes and dimensions. In order to estimate the 95% quantile of TMVV(I)2 for a given Phase I of sample size n and dimension p, we generate K = 5000 samples of size n from a standard multivariate normal distribution, MVNp(0,Ip). For each data set of size n, we compute the MVV mean vector and the modified covariance matrix estimates, MMVV(k) and ϑp,nαc(h)SMVV(k), respectively, from k=1,…,K. In addition, for each data set, we randomly generate a new observation xg,k treated as a Phase II observation from MVNp(0,Ip) and calculate the corresponding TMVV(I)2(g,k) values. The empirical distribution function of TMVV(I)2(g) is based on the simulated values
(10)TMVV(I)2(g,1),TMVV2(g,2),…,TMVV2(g,K).
We sort TMVV(I)2(g,k) values in ascending order, and the UCL is the 95% quantile of the 5000 statistics. The results of the investigation are presented in Table 3. We observe that the estimated UCLs for TMVV(o)2 are large as compared to the traditional control charts TS2, TMCD2, and TRMCD2. However, after making the MVV scatter estimator consistent and unbiased as shown in (8), the results improved immensely. As we can see here, the UCLs are closer to the traditional UCLs.Table 3
Control limits of the investigated control charts for various combinations of sample sizes and dimensions.
p
n
T
2
T
MCD
2
T
MVV
(
o
)
2
T
MVV
(
I
)
2
2
10
11.0360
30.6250
76.0122
19.7067
25
7.4275
12.1798
32.4008
9.4443
50
6.6447
8.2762
28.1107
8.0556
100
6.3039
7.4463
24.6037
7.1969
200
6.1443
6.4127
21.7088
6.4469
500
6.0518
6.1558
20.4264
6.1788
5
30
15.6006
27.6404
41.9567
19.5315
50
13.4506
18.3456
33.5214
15.9398
100
12.1579
14.7736
28.4822
14.0082
200
11.5915
12.5765
25.6204
12.9297
500
11.2738
11.4941
22.5296
11.5868
10
50
25.9552
39.8024
62.9323
34.5417
100
21.5264
26.0646
43.1889
25.5450
200
19.7975
20.9145
34.8509
21.4812
500
18.8777
19.5618
31.1418
19.8112
15
80
33.6517
42.7942
69.0937
42.6982
100
31.5083
37.4367
61.0544
38.6310
200
27.9034
29.4809
45.9981
30.6107
500
26.0882
26.6016
39.7181
26.1456
20
100
42.5747
52.7273
83.5238
53.9627
200
36.2033
38.8163
57.2303
39.3577
300
34.4609
36.0595
52.3438
36.8230
500
33.1766
33.6574
47.7808
34.4221


### 4.2. Real Data Analysis

The application of the improved methodTMVV(I)2 on real data is illustrated using data furnished by Asian Composites Manufacturing Sdn. Bhd. (ACM) which involves in the production of advanced composite panels for the aircraft industry. ACM produces flat and contoured primary (Aileron Skins, Spoilers and Spars) and secondary (Flat Panels, Leading Edges and MISC: Components) structure composite bond assemblies and sub-assemblies for aerospace industries. The company provided us with data on spoilers as shown in Table 4. For the purpose of this study, a sample of 47 spoilers (n=47) which consists of several features, namely, trim edge (X1), trim edge spar (X2), and drill hole (X3) were furnished to us by the company. Out of the total, 21 spoilers were collected from 2009, while the rest were from 2010. Hence, we decided to use the 2009 spoilers as Phase I historical data and considered the spoilers from 2010 as future data in this study. Estimates for the location vector (X-) and scatter matrix (S) are presented in Table 5. In the last column of Table 5, we could clearly observe that the upper control limit (UCL) for TMVV(I)2 is the closest to the traditional Hotelling T2 with values of 11.5513 and 11.035, respectively, whereas the other control charts produce large UCL values especially the original TMVV2(TMVV(o)2). When we compare the improved with the original TMVV2, we observe a large disparity between the two values such that TMVV(o)2 = 41.298, and TMVV(I)2 = 11.5513. The result indicates great improvement in the UCL values from TMVV(o)2 to TMVV(I)2. Table 6 identifies the out-of-control data (bold font) using the different T2 statistics. Among the four statistics, TMVV(o)2, TMVV(I)2 and TMCD2 consider signal observations 20, 22, and 25 as out-of-control but T2 fails to signal observation 22 and only consider, observations 20 and 25 as out-of-control. This is expected due to the low probability of detection in the traditional control chart [16]. For a clearer visualisation on the performance of the control charts in detecting out or control observations, graphical presentation of the corresponding control charts are put on view in Figure 2.Table 4
List of Phase I and Phase II real data.
No. of product
Phase I data
Phase II data
X
1
X
2
X
3
X
1
X
2
X
3
1
−0. 0011
0.0003
0.0128
0.0041
0.0087
0.0129
2
0.0011
0.0021
0.0246
0.0047
0.0109
0.0124
3
0.0252
0.0308
0.0378
0.0031
0.0057
0.0096
4
−0.0017
0.0109
0.0177
0.0035
−0.0020
0.0101
5
−0.0005
−0.0010
0.0106
0.0040
−0.0028
0.0125
6
0.0016
−0.0059
0.0128
0.0031
0.0008
0.0061
7
0.0004
0.0001
0.0062
−0.0019
0.0101
0.0112
8
0.0078
0.0003
0.0159
0.0009
0.0039
0.0082
9
0.0076
0.0089
0.0097
−0.0052
0.0090
0.0203
10
0.0020
0.0005
0.0071
−0.0008
0.0110
0.0184
11
0.0108
0.0011
0.0092
−0.0021
0.0139
0.0170
12
0.0039
0.0034
0.0425
−0.0017
0.0092
0.0061
13
0.0060
−0.0033
0.0160
−0.0010
0.0133
0.0138
14
0.0066
0.0100
0.0056
−0.0030
0.0002
0.0053
15
0.0045
−0.0067
0.0147
0.0016
0.0134
0.0151
16
0.0110
−0.0207
0.0337
0.0027
0.0086
0.0070
17
0.0047
0.0059
0.0065
0.0004
0.0086
0.0087
18
0.0077
0.0003
0.0191
−0.0036
0.0136
0.0129
19
0.0015
0.0123
0.0124
−0.0028
0.0003
0.0078
20
0.0011
0.0038
0.0104
0.0120
0.0123
0.0768
21
0.0056
0.0065
0.0063
−0.0015
0.0004
0.0115
22
0.0009
0.0232
0.0202
23
−0.0035
0.0088
0.0107
24
0.0016
0.0061
0.0066
25
−0.0228
−0.0466
0.0231
26
0.0037
−0.0038
0.0147Table 5
Estimates of location vector, covariance matrix, and UCL.
Types of control chart
Location Vector (X-)
Scatter Matrix (S)
Upper Control Limit (UCL)
T
2
[
0.00504
0.00284
0.01579
]
[
0.00004
0.00002
0.00003
0.00002
0.00009
0.00001
0.00003
0.00001
0.00011
]
11.035
T
MCD
2
[
0.00414
0.00207
0.01096
]
[
0.00002
0.00000
0.00000
0.00002
0.00009
-
0.00002
0.00000
-
0.00002
0.00003
]
21.946
T
MVV
(
o
)
2
[
0.00336
0.00354
0.00913
]
[
0.00001
0.00001
0.00000
0.00001
0.00003
0.00000
0.00000
0.00000
0.00001
]
41.298
T
MVV
(
I
)
2
[
0.00336
0.00354
0.00913
]
[
0.00003
0.00002
-
0.00001
0.00002
0.00007
-
0.00001
-
0.00001
-
0.00001
0.00002
]
11.5513Table 6
The HotellingT2 values for the future (Phase II) data.
No.
T
2
T
MCD
2
T
MVV
(
o
)
2
T
MVV
(
I
)
2
1
0.55822
1.76591
4.39082
1.56608
2
0.90026
2.46944
5.16947
1.84380
3
0.49916
0.34367
0.29917
0.10671
4
0.54633
0.54563
1.50640
0.53729
5
0.45922
0.45797
3.78687
1.35067
6
0.90130
1.25274
2.24213
0.79971
7
3.09329
4.44043
6.53612
2.33125
8
0.80608
0.68370
1.05554
0.37648
9
7.36021
14.97663
26.04990
9.29127
10
3.61976
9.74168
19.17603
6.83955
11
5.38392
11.87166
19.63128
7.00193
12
2.73870
2.97882
8.13879
2.90288
13
3.80577
7.40398
11.38954
4.06233
14
2.05480
3.30863
9.14983
3.26349
15
2.50731
6.80538
12.38812
4.41850
16
1.19755
1.06789
2.05633
0.73344
17
1.57979
1.75966
2.87650
1.02597
18
5.79103
9.28168
13.92929
4.96819
19
1.83044
2.41775
4.87909
1.74023
20
38.13972
214.92329
894.51844
319.04971
21
1.26507
1.54862
2.06405
0.73619
22
8.41812
24.65515
45.24620
16.13805
23
3.75884
4.87934
7.53275
2.68672
24
1.06020
0.93200
2.23575
0.79743
25
42.84468
68.63065
116.02933
41.38442
26
0.48316
0.77959
7.32655
2.61318HotellingT2 Control Charts.
(a)
(b)
(c)
(d)


## 4.1. Estimation of Control Limits

In this section, we present the control limit of the improvedTMVV(I)2 control chart by using simulated data with different combinations of sample sizes, n, and number of dimensions, p. The control limit of TMVV(I)2 chart is then compared with the control limit of TMVV(o)2 chart, robust Hotelling T2 chart using MCD (TMCD2) and the traditional Hotelling T2 charts. The application of robust estimators in place of the mean and covariance structure in traditional Phase II Hotelling T2 statistic will cause the distributional properties of the traditional chart to change [9]. To demonstrate the performance of TMVV(o)2 and TMVV(I)2 control charts, we need to identify the distribution of each method in order to obtain appropriate control limits, that is, UCL. Since the exact distribution of TMVV(o)2 is unknown, we apply Monte Carlo method to estimate the quantiles of the TMVV(o)2 and TMVV(I)2, for several combinations of sample sizes and dimensions. In order to estimate the 95% quantile of TMVV(I)2 for a given Phase I of sample size n and dimension p, we generate K = 5000 samples of size n from a standard multivariate normal distribution, MVNp(0,Ip). For each data set of size n, we compute the MVV mean vector and the modified covariance matrix estimates, MMVV(k) and ϑp,nαc(h)SMVV(k), respectively, from k=1,…,K. In addition, for each data set, we randomly generate a new observation xg,k treated as a Phase II observation from MVNp(0,Ip) and calculate the corresponding TMVV(I)2(g,k) values. The empirical distribution function of TMVV(I)2(g) is based on the simulated values
(10)TMVV(I)2(g,1),TMVV2(g,2),…,TMVV2(g,K).
We sort TMVV(I)2(g,k) values in ascending order, and the UCL is the 95% quantile of the 5000 statistics. The results of the investigation are presented in Table 3. We observe that the estimated UCLs for TMVV(o)2 are large as compared to the traditional control charts TS2, TMCD2, and TRMCD2. However, after making the MVV scatter estimator consistent and unbiased as shown in (8), the results improved immensely. As we can see here, the UCLs are closer to the traditional UCLs.Table 3
Control limits of the investigated control charts for various combinations of sample sizes and dimensions.
p
n
T
2
T
MCD
2
T
MVV
(
o
)
2
T
MVV
(
I
)
2
2
10
11.0360
30.6250
76.0122
19.7067
25
7.4275
12.1798
32.4008
9.4443
50
6.6447
8.2762
28.1107
8.0556
100
6.3039
7.4463
24.6037
7.1969
200
6.1443
6.4127
21.7088
6.4469
500
6.0518
6.1558
20.4264
6.1788
5
30
15.6006
27.6404
41.9567
19.5315
50
13.4506
18.3456
33.5214
15.9398
100
12.1579
14.7736
28.4822
14.0082
200
11.5915
12.5765
25.6204
12.9297
500
11.2738
11.4941
22.5296
11.5868
10
50
25.9552
39.8024
62.9323
34.5417
100
21.5264
26.0646
43.1889
25.5450
200
19.7975
20.9145
34.8509
21.4812
500
18.8777
19.5618
31.1418
19.8112
15
80
33.6517
42.7942
69.0937
42.6982
100
31.5083
37.4367
61.0544
38.6310
200
27.9034
29.4809
45.9981
30.6107
500
26.0882
26.6016
39.7181
26.1456
20
100
42.5747
52.7273
83.5238
53.9627
200
36.2033
38.8163
57.2303
39.3577
300
34.4609
36.0595
52.3438
36.8230
500
33.1766
33.6574
47.7808
34.4221


## 4.2. Real Data Analysis

The application of the improved methodTMVV(I)2 on real data is illustrated using data furnished by Asian Composites Manufacturing Sdn. Bhd. (ACM) which involves in the production of advanced composite panels for the aircraft industry. ACM produces flat and contoured primary (Aileron Skins, Spoilers and Spars) and secondary (Flat Panels, Leading Edges and MISC: Components) structure composite bond assemblies and sub-assemblies for aerospace industries. The company provided us with data on spoilers as shown in Table 4. For the purpose of this study, a sample of 47 spoilers (n=47) which consists of several features, namely, trim edge (X1), trim edge spar (X2), and drill hole (X3) were furnished to us by the company. Out of the total, 21 spoilers were collected from 2009, while the rest were from 2010. Hence, we decided to use the 2009 spoilers as Phase I historical data and considered the spoilers from 2010 as future data in this study. Estimates for the location vector (X-) and scatter matrix (S) are presented in Table 5. In the last column of Table 5, we could clearly observe that the upper control limit (UCL) for TMVV(I)2 is the closest to the traditional Hotelling T2 with values of 11.5513 and 11.035, respectively, whereas the other control charts produce large UCL values especially the original TMVV2(TMVV(o)2). When we compare the improved with the original TMVV2, we observe a large disparity between the two values such that TMVV(o)2 = 41.298, and TMVV(I)2 = 11.5513. The result indicates great improvement in the UCL values from TMVV(o)2 to TMVV(I)2. Table 6 identifies the out-of-control data (bold font) using the different T2 statistics. Among the four statistics, TMVV(o)2, TMVV(I)2 and TMCD2 consider signal observations 20, 22, and 25 as out-of-control but T2 fails to signal observation 22 and only consider, observations 20 and 25 as out-of-control. This is expected due to the low probability of detection in the traditional control chart [16]. For a clearer visualisation on the performance of the control charts in detecting out or control observations, graphical presentation of the corresponding control charts are put on view in Figure 2.Table 4
List of Phase I and Phase II real data.
No. of product
Phase I data
Phase II data
X
1
X
2
X
3
X
1
X
2
X
3
1
−0. 0011
0.0003
0.0128
0.0041
0.0087
0.0129
2
0.0011
0.0021
0.0246
0.0047
0.0109
0.0124
3
0.0252
0.0308
0.0378
0.0031
0.0057
0.0096
4
−0.0017
0.0109
0.0177
0.0035
−0.0020
0.0101
5
−0.0005
−0.0010
0.0106
0.0040
−0.0028
0.0125
6
0.0016
−0.0059
0.0128
0.0031
0.0008
0.0061
7
0.0004
0.0001
0.0062
−0.0019
0.0101
0.0112
8
0.0078
0.0003
0.0159
0.0009
0.0039
0.0082
9
0.0076
0.0089
0.0097
−0.0052
0.0090
0.0203
10
0.0020
0.0005
0.0071
−0.0008
0.0110
0.0184
11
0.0108
0.0011
0.0092
−0.0021
0.0139
0.0170
12
0.0039
0.0034
0.0425
−0.0017
0.0092
0.0061
13
0.0060
−0.0033
0.0160
−0.0010
0.0133
0.0138
14
0.0066
0.0100
0.0056
−0.0030
0.0002
0.0053
15
0.0045
−0.0067
0.0147
0.0016
0.0134
0.0151
16
0.0110
−0.0207
0.0337
0.0027
0.0086
0.0070
17
0.0047
0.0059
0.0065
0.0004
0.0086
0.0087
18
0.0077
0.0003
0.0191
−0.0036
0.0136
0.0129
19
0.0015
0.0123
0.0124
−0.0028
0.0003
0.0078
20
0.0011
0.0038
0.0104
0.0120
0.0123
0.0768
21
0.0056
0.0065
0.0063
−0.0015
0.0004
0.0115
22
0.0009
0.0232
0.0202
23
−0.0035
0.0088
0.0107
24
0.0016
0.0061
0.0066
25
−0.0228
−0.0466
0.0231
26
0.0037
−0.0038
0.0147Table 5
Estimates of location vector, covariance matrix, and UCL.
Types of control chart
Location Vector (X-)
Scatter Matrix (S)
Upper Control Limit (UCL)
T
2
[
0.00504
0.00284
0.01579
]
[
0.00004
0.00002
0.00003
0.00002
0.00009
0.00001
0.00003
0.00001
0.00011
]
11.035
T
MCD
2
[
0.00414
0.00207
0.01096
]
[
0.00002
0.00000
0.00000
0.00002
0.00009
-
0.00002
0.00000
-
0.00002
0.00003
]
21.946
T
MVV
(
o
)
2
[
0.00336
0.00354
0.00913
]
[
0.00001
0.00001
0.00000
0.00001
0.00003
0.00000
0.00000
0.00000
0.00001
]
41.298
T
MVV
(
I
)
2
[
0.00336
0.00354
0.00913
]
[
0.00003
0.00002
-
0.00001
0.00002
0.00007
-
0.00001
-
0.00001
-
0.00001
0.00002
]
11.5513Table 6
The HotellingT2 values for the future (Phase II) data.
No.
T
2
T
MCD
2
T
MVV
(
o
)
2
T
MVV
(
I
)
2
1
0.55822
1.76591
4.39082
1.56608
2
0.90026
2.46944
5.16947
1.84380
3
0.49916
0.34367
0.29917
0.10671
4
0.54633
0.54563
1.50640
0.53729
5
0.45922
0.45797
3.78687
1.35067
6
0.90130
1.25274
2.24213
0.79971
7
3.09329
4.44043
6.53612
2.33125
8
0.80608
0.68370
1.05554
0.37648
9
7.36021
14.97663
26.04990
9.29127
10
3.61976
9.74168
19.17603
6.83955
11
5.38392
11.87166
19.63128
7.00193
12
2.73870
2.97882
8.13879
2.90288
13
3.80577
7.40398
11.38954
4.06233
14
2.05480
3.30863
9.14983
3.26349
15
2.50731
6.80538
12.38812
4.41850
16
1.19755
1.06789
2.05633
0.73344
17
1.57979
1.75966
2.87650
1.02597
18
5.79103
9.28168
13.92929
4.96819
19
1.83044
2.41775
4.87909
1.74023
20
38.13972
214.92329
894.51844
319.04971
21
1.26507
1.54862
2.06405
0.73619
22
8.41812
24.65515
45.24620
16.13805
23
3.75884
4.87934
7.53275
2.68672
24
1.06020
0.93200
2.23575
0.79743
25
42.84468
68.63065
116.02933
41.38442
26
0.48316
0.77959
7.32655
2.61318HotellingT2 Control Charts.
(a)
(b)
(c)
(d)


## 5. Conclusion

The UCL value for the HotellingT2 control chart using consistent and unbiased MVV estimators seemed to improve significantly from the Hotelling T2 control chart based on the original MVV estimators. The improved control chart (TMVV(I)2) was put to test on real data. Even though the performance of the improved TMVV2 control chart was on par with the original TMVV2 chart, the improved estimators have successfully reduced the inflated UCL of the original TMVV2 close to the UCL of the traditional Hotelling T2 (T2) control chart. However, when the improved control chart was compared with the traditional chart based on their almost equal UCLs, the finding showed that the improved control chart performed better in detecting out-of-control observations. With the good properties and performance, this improved MVV estimators that should be considered as alternative estimators to replace the usual mean and variance vector in the construction of the robust Hotelling T2 control chart as well as other multivariate statistical procedures.



---

*Source: 401350-2013-09-25.xml*