Published online: 02 Sep 2013.

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1 This article was downloaded by: [East China Normal University] On: 17 April 14, At: 1:3 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK International Journal of Production Research Publication details, including instructions for authors and subscription information: A self-starting control chart for high-dimensional short-run processes Yanting Li a, Yukun Liu b, Changliang Zou c & Wei Jiang d a Department of Industrial Engineering and Management, Shanghai Jiao Tong University, Shanghai, China b School of Finance and Statistics, East Normal University of China, Shanghai, China c Department of Statistics, Nankai University, Tianjin, China d Antai School of Management, Shanghai Jiao Tong University, Shanghai, China Published online: Sep 13. To cite this article: Yanting Li, Yukun Liu, Changliang Zou & Wei Jiang (14) A self-starting control chart for high-dimensional short-run processes, International Journal of Production Research, 5:, , DOI: 1.18/ To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 International Journal of Production Research, 14 Vol. 5, No., , A self-starting control chart for high-dimensional short-run processes Yanting Li a, Yukun Liu b, Changliang Zou c and Wei Jiang d a Department of Industrial Engineering and Management, Shanghai Jiao Tong University, Shanghai, China; b School of Finance and Statistics, East Normal University of China, Shanghai, China; c Department of Statistics, Nankai University, Tianjin, China; d Antai School of Management, Shanghai Jiao Tong University, Shanghai, China (Received 15 October 1; accepted 7 July 13) A key challenge in using a traditional Hotelling s T chart with high-dimensionality measurements is that monitoring cannot begin until after the number of observations exceeds the dimensionality of the measurements, and the detection sensitivity to early shifts is reduced after that point until a substantial amount of observations has been accumulated. This is especially important with short-run processes where the measurements have high dimensionality. This article proposes a chart that allows monitoring with the second observation regardless of the dimensionality and reduces the average run length in detecting early shifts in high-dimensionality measurements. The proposed control chart can start monitoring quite early before considerable reference data are collected during the initial stage of production. A change point estimate is also available from our procedure, which is shown consistent for locating the correct change point. Both simulation results and an industry example show the effectiveness of the proposed control chart for monitoring short-run processes with high dimensionality. Keywords: high-dimensional observations; short-run processes; mean shift; control chart; average run length 1. Introduction Modern manufacturing systems concern with quality control in short-run processes, which are often characterised by large variability of manufactured products and small batches of production for each product (Nenes and Tagaras 6). To make high-quality product in small batches, extensive use of sensor technology collects a large number of observations related to the production process over time, which are complex and correlated in nature. For example, modern semi-conductor manufacturing processes are generally under continuous surveillance via the monitoring of variables collected from sensors at different process measurement points. On many occasions, more than hundreds of quality characteristics are monitored simultaneously. Process engineers may then use these measurements to determine whether the process is well under control so that corrective adjustments can be made to maintain high quality of process and product during the short production runs. Therefore, statistical process control (SPC) methods for monitoring high-dimensional processes are becoming increasingly important in modern manufacturing environments (Wang and Jiang 9; Chen and Nembhard 1). SPC methods involve monitoring changes in a process by distinguishing special causes of variations from common causes, which are revealed by deviations from the in-control statistical distribution of process measurements. For example, the Hotelling s T chart is the most widely used multivariate SPC procedure for monitoring changes in the mean vector μ = (μ 1,...,μ p ) of p correlated quality characteristics. Given individual observations X i = (X i1,...x ip ) (i = 1,...,n), the Hotelling s T statistic is defined as T = (X i μ) 1 (X i μ), which follows a χ distribution with p-degrees of freedom when X i is normally distributed with process mean μ and variance matrix. When μ and are unavailable and estimated from a large sample of reference data, the Hotelling s T statistic is then T = (X i X) S 1 (X i X), (1) where X and S are the sample mean vector and sample covariance matrix, respectively, of the observations to estimate the process mean μ and variance matrix. Unlike the case when the mean and variance are known, the T statistic follows (after scaled) an F-distribution with (p, n p)-degrees of freedom (Montgomery 9). The Hotelling s T control chart is simple and possesses many appealing theoretical properties, including invariance under linear transformations of data, finite-sample optimality if the observations are from the normal distribution and asymptotic optimality otherwise (Jiang and Tsui 8). Corresponding author. ytli@sjtu.edu.cn 13 Taylor & Francis

3 446 Y. Li et al. Controlling short-run processes in practice is challenging since there are not enough knowledge or reference data to learn an accurate baseline model for process control, particularly the variance matrix in the Hotelling s T chart. As pointed out by many researchers, poor estimates of μ and severely deteriorate the performance of the Hotelling s T chart (Champ, Jones-Farmer, and Rigdon 5). In fact, when the sample size is smaller than the dimension, the Hotelling s T statistic is even not defined as S is never invertible. Therefore, limited high-dimensional data in short-run processes put forward critical challenges to the Hotelling s T statistic and other change-point control charts are needed for process monitoring (Wasserman 1995). In this article, by applying a two-sample high-dimensional statistical test in a change-point framework, we present a novel SPC scheme for detecting shifts in the mean vector of high-dimensional multivariate distributions. Its main attraction is that it can start monitoring the high-dimensional process much earlier than other change-point control charts, which increases the responsiveness of the control chart, especially for short-run processes. Simulation results also show that the proposed control chart is more powerful than other alternatives in detecting mean changes at the beginning of a process. Moreover, a change-point estimate is available from the procedure, which is consistent for locating the correct change point. The structure of this article is as follows. In Section, existing change-point control charts are reviewed first. In Section 3, a new change-point method for detecting mean shifts in high-dimensional processes is introduced. Section 4 contains numerical evidence for the efficiency of the new control chart under various scenarios comparing with other change-point control charts. An illustrative example is given in Section 5. Section 6 discusses practical and theoretical issues such as data non-normality and change-point estimation consistency. The article concludes in Section 7.. Change-point control charts For short-run high-dimensional processes, the process mean vector and covariance matrix are usually unknown or not fully known. In the setting of unknown in-control parameters, change-point models have been developed for detecting mean or/and scale changes in univariate/multivariate normal processes, see e.g. Sullivan and Woodall (), Hawkins, Qiu, and Kang (3), Hawkins and Zamba (5a,b), Zamba and Hawkins (6, 9), Maboudou-Tchao and Hawkins (11), etc. These change-point control charts have been genaralised to monitoring linear profiles (Zou, Zhang, and Wang 6; Mahmoud et al. 7) and multistage processes (Zou, Tsung, and Liu 8). Change-point estimation techniques have also been proposed for processes of count data, particularly those adequately modelled by Poisson and binomial distributions. An overview of the change-point methods can be found in Amiri and Allahyari (11). Nonetheless, majority of these studies assume that there is only one change point. Consider a sequence of independent observations X i = (X i1,...x ip ) (i = 1,...,n) from a p-dimensional multivariate normal process with mean μ = (μ 1,...,μ p ) and covariance matrices. Assume, after the τth observation, the mean vector changes to μ 1, i.e. { MN(μ, ), i τ X i MN(μ 1, ), i >τ. () This defines the change-point formulation, where the parameter τ is unknown and called the change point. Other parameters μ, μ 1 and are assumed unknown in short-run production processes. Denote Y k,n =[k(n k)/n] 1/ ( X,k X k,n ) with X j,m = m 1 mi= j j+1 X i being the sample mean of observations {X j+1,...,x m }. In order to detect if a mean shift has occurred after observation k (k = 1,...,,n 1), Zamba and Hawkins (6) proposed a multivariate change-point model (denoted as ZH control chart hereafter) based on the Hotelling s T type of statistics Tk,n = Y k,n U 1 k,n Y k,n, where U k,n defines the pooled covariance structure at k, k U k,n = (X i X,k )(X i X,k ) + i=1 n (X i X k,n )(X i X k,n ) i=k+1 / (n 1), (assuming that the variance matrix does not change). The change-point detection statistic is the maximum over all possible split points k, i.e. Tmax,n = max T k,n, (3) 1 k n 1 which signals when Tmax,n exceeds a threshold. The change point is then estimated by the k value that attains the maximum Tk,n.

4 International Journal of Production Research 447 It is easy to see that Tk,n is essentially the square of a two-sample t-statistic when testing the difference between pre-shift and post-shift samples. A nice property of the above change-point test statistic is that, similar to the Hotelling s T statistic in (1), Tk,n also follows (after scaled) an F-distribution with (p, n p)-degrees of freedom. Unfortunately, the above changepoint control chart cannot start monitoring until n > p + 1 and the matrix to be inverted may be ill-conditioned, greatly increasing the variance of the test statistic. Bai and Saranadasa (1996) pointed out that the power of the two-sample test decreases when the dimension gets larger due to the inverse of the sample covariance matrix in the T statistic. Although standardising by the covariance brings benefits to the testing problem with a fixed dimension, it becomes a liability for highdimensional data. In short-run high-dimensional processes, process improvement opportunities may be lost while waiting for data collection before n p. In the next section, we will propose a different multivariate control chart based on the change-point formulation for monitoring short-run high-dimensional processes. 3. A multivariate change-point control chart for short-run processes We use the same working model as defined in Equation (). The definition of the parameters τ, μ, μ 1,, 1 remains the same as those in Section and none of them is assumed known. As mentioned before, T k,n mimics a two-sample t-test for the difference between the mean vectors of pre-shift and post-shift data. In order to circumvent inverting a singular sample covariance matrix when the sample size is smaller than the dimension, Chen and Qin (1) defined a statistic for testing the difference between the mean vectors of pre-shift and post-shift data at time k as follows, k i, j=1;i = j X i X j n i, j=k+1;i = j X i X j k n i=1 j=k+1 W n,k = + k(k 1) (n k)(n k 1) (4) k(n k) which is shown to asymptotically follow a normal distribution. It has been theoretically proved that when k = τ is the true change point, then the mean of W n,k is μ μ 1, and under some mild conditions (see Lemma 1 in the Appendix), the variance of W n,k is σw tr ( ) k(k 1) + tr 1 (n k)(n k 1) + 4tr( 1 ) (5) k(n k) where and 1 are the covariance matrices of the first k and the last n k observations, respectively. In our case, both variance matrices are actually since the process variance is assumed to be unchanged. Unfortunately, the variance estimate of W n,k developed by Chen and Qin (1) is not transformation invariant, which leads to a biased estimate of the covariance matrix. In order to keep the transformation invariant property, we propose a new estimate of σw, sw = k(k 1) s 1 + (n k)(n k 1) s + 4 k(n k) s 3, (6) where s1 = tr 1 k X j X ( j,l) X j X ( j,l) Xl X ( j,l) Xl X ( j,l) k(k 1), j,l=1; j =l s = tr 1 n X j X 1( j,l) X j X 1( j,l) Xl X 1( j,l) Xl X 1( j,l) (n k)(n k 1), j,l=k+1; j =l s3 = tr 1 k n X j X ( j) X j X ( j) Xk X 1(k) Xk X 1(k) k(n k). j=1 l=k+1 X i X j Here, X i( j,l) is the i-th sample mean after excluding X j and X l, and X i( j) is the i-th sample mean after excluding X j. i = or 1 represents the pre-shift sample of the first k or the post-shift sample of the last n k observations, respectively. According to the proof of Theorem 1, we find sw is a ratio-consistent estimator of σ W, the variance of W n,k. Therefore, if conditions (1) and (11) are valid, it follows immediately from Theorem 1 of Chen and Qin (1) that Z n,k = W n,k μ μ 1 sw N(,1) as p and n. (7) So, a natural test statistic for change point is the maximum over all possible splits,

5 448 Y. Li et al. Wn,k Z max,n = max 1 k n 1 sw, (8) which signals when Z max,n exceed a threshold value. A straightforward estimate of τ is then the k value at which Z n,k attains its maximum, i.e. ˆτ = arg max Z n,k. (9) k 1 k n 1 Given a desired in-control average run length (ARL ), which is defined as the average number of samples needed for the control chart to signal when the process is in control, it is necessary to determine the corresponding control limits for the above change-point control chart. Theoretically, the conventional way to determine upper control limits of Z n, ˆτ = max k Z n,k is based on its limiting distribution, which is an extreme-value distribution (Csorgo and Horvath 1997). However, in changepoint problems, the rate of convergence of the distribution of the test statistic based on binary segmentation is usually believed to be slow (see Section 1.3 of Csorgo and Horvath (1997) for some discussions). Consequently, when the sample size is not large enough, the approximation based on limiting distribution yields somewhat conservative results. Therefore, given a nominal ARL, we determine the control limit using numerical simulations. It is important to note that, as Z n,k is asymptotically univariate normal whenever no mean shift occurs, its distribution does not depend on the common mean or the covariance matrix, but is dependent on the size of n and p. So does the control limit, h n,p. Therefore, the determination of the control limit h n,p for the above change-point control chart can be obtained from simple simulations with sufficient repetitions. In this article, we estimate h n,p based on 1,, random samples from MN(μ, I). In general, we formally summarise the above change-point control chart procedure as follows. (1) At time n, based on X 1,...,X n, compute the values of statistics Z n,k (k = 1,...,n 1). () Calculate maxz n,k, the maximum of Z n,k. (3) Check whether Z n, ˆτ > h n,p. If yes, an alarming signal is given. Otherwise, keep on sampling. (4) In case of an alarm, the change point is estimated by ˆτ = arg k max 1 k n 1 Z n,k. 4. Performance of the change-point approach A full-scale performance evaluation of our proposed method is challenging as the performance is affected by a large number of factors, including underlying distribution of the observations, dimensionality, pre-specified significance level, mean shift magnitude and the location of the change point as well. In order to explore the impact of these factors, we conducted an extensive simulation study and compared with the ZH control chart for short-run processes. We take the multivariate changepoint detection method (Zamba and Hawkins 6) as a benchmark to evaluate the performance of the high-dimensional change-point detection method (HC) proposed in this paper under various scenarios. To investigate the control chart performance, pseudo-observations have been simulated. Settings of the Monte Carlo simulation are as follows: (1) The underlying distribution: Multivariate normal distribution MN(μ, I). As discussed in the previous section, when the process is in control, the distribution of the control statistics is indifferent to the covariance matrix. Therefore, for the sake of simplicity, we use the identity matrix as the covariance matrix. () The dimension of the observations, p = 1,, 3, 5. (3) The location of the single change point: (a) for p = 1, τ = 5, 15, 5. (b) for p =, τ = 5, 15, 5. (c) for p = 3, τ = 5,, 4. (d) for p = 5, τ = 1, 4, 6. (4) The mean shift μ 1 = (,...,,δ,...,δ), (a) δ =.5, 1., 1.5,.,.5, 3, (b) the percentage of the out-of-control variables r = 5%, 5%, 75% and 1% and the number of the out-of-control variables is the nearest integer to pr. In case there are two integers that are nearest to pr,the bigger integer is chosen. (5) In-control average run length (a) The in-control average run length ARL is fixed at 1.

6 International Journal of Production Research 449 Table 1. for p = 1 under various settings of mean shift and change point of multivariate normal distribution. When τ =, ARL ZH = 99.8andARL HC = 1.. r =.5 r =.5 r =.75 r = 1. τ δ ARL HC ARL ZH ARL HC ARL ZH ARL HC ARL ZH ARL HC ARL ZH In particular, for both HC and s, values of the corresponding control limits are obtained so that the false alarm probability of each sampling point is equal, i.e. 1/ARL. Then, the OC performance is measured in terms of the expected delay in detecting a mean change, i.e. E(RL τ + 1 RL τ), where RL denotes the run length of the control chart. The change-point estimate is obtained by counting the value of k that attains the maximum of Z k,n in Equation (9). For convenience, Ave( ˆτ) ZH and Ave( ˆτ) HC are denoted as the corresponding averages of the change-point estimates of the Monte Carlo simulation, which are similar to the definition of average run length. It is clear that the shorter the delay of detection is, the more efficient the control chart is; and the closer the change-point estimate is to the true change point τ,the better the estimation method is. The simulation results are tabulated in Tables 1 8. We bold the shorter delay of detection and change-point estimates which are closer to the true change point in each setting for comparison. 4.1 Comparison of ARL Table 1 compares the delays of detection of the two methods with various change point τ and r when p = 1. Figures 1 and plot the delay of detection w.r.t. the mean shift magnitude δ and shift mean shift percentage r, respectively. It is easy to see that the detection delays of both methods decrease with the increase of δ and r. This indicates that both methods are effective in detecting mean changes no matter where the change point locates. However, the relative comparison of the HC and s depends on the change-point location τ. When τ is smaller than the dimension p, the is uniformly better than the. In fact, it can be observed that the delay of detection of the is inevitably larger than τ p = 5, since the cannot start monitoring until 1 observations are collected. On the other hand, when τ = 5, the is still very efficient in detecting the mean shift. When the change point τ is larger than p, such as 15, the outperforms the when r is small, such as r =.5 and.5. When the mean shift percentage further increases, the responds quicker than the. The advantage of the becomes more obvious when the change point locates farther from p. For example, when τ = 5, the gives smaller delays of detection. In Figure, we plot the scatter plot of r vs. delays of detection with various change point τ by fixing δ = 3 and p = 1. When the change point occurs before p observations are collected, the performs better than the. When the change point τ is bigger than p = 1, such as 15 and 5, the becomes better at detecting mean shift with large r. Similar pattern can be observed with other combination of δ. The comparison of the two methods for higher dimensions, such as p =, 3, 5, are shown in Tables 4. In order to save space, the comparison results are summarised as follows:

7 45 Y. Li et al. Table. for p = under various settings of mean shift and change point of multivariate normal distribution. When τ =, ARL ZH = 1.5andARL HC = 1.. r =.5 r =.5 r =.75 r = 1. τ δ ARL HC ARL ZH ARL HC ARL ZH ARL HC ARL ZH ARL HC ARL ZH Table 3. for p = 3 under various settings of mean shift and change point of multivariate normal distribution. When τ =, ARL ZH = 11.1andARL HC = 1.. r =.5 r =.5 r =.75 r = 1. τ δ ARL HC ARL ZH ARL HC ARL ZH ARL HC ARL ZH ARL HC ARL ZH

8 International Journal of Production Research 451 Table 4. for p = 5 under various settings of mean shift and change point of multivariate normal distribution. When τ =, ARL ZH = 1.8andARL HC = r =.5 r =.5 r =.75 r = 1. τ δ ARL HC ARL ZH ARL HC ARL ZH ARL HC ARL ZH ARL HC ARL ZH Table 5. Change-point estimate for p = 1 under various settings of mean shift and change point of multivariate normal distribution. When τ =, ARL ZH = 99.8andARL HC = 1.. r =.5 r =.5 r =.75 r = 1. τ δ Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH

9 45 Y. Li et al. Table 6. Change point estimate for p = under various settings of mean shift and change point of multivariate normal distribution. When τ =, ARL ZH = 1.5 andarl HC = 1.. r =.5 r =.5 r =.75 r = 1. τ δ Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Table 7. Change-point estimate for p = 3 under various settings of mean shift and change point of multivariate normal distribution. When τ =, ARL ZH = 11.1 andarl HC = 1.. r =.5 r =.5 r =.75 r = 1. τ δ Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH

10 International Journal of Production Research 453 Table 8. Change-point estimate for p = 5 under various settings of mean shift and change point of multivariate normal distribution. When τ =, ARL ZH = 1.8 andarl HC = r =.5 r =.5 r =.75 r = 1. τ δ Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH Ave( ˆτ) HC Ave( ˆτ) ZH When p = : For τ = 5 and τ = 15, i.e. the change points are smaller than the dimension, the new method has consistently shorter average run lengths. However, once the change point is larger than the dimension, for example, τ = 5, ARL ZH is smaller than ARL HC whenever the shift magnitude and the the number of out-of-control variables are relatively large. For example, δ = 3., r = 1., ARL ZH is.1 while ARL HC is 3.. When p = 3: When τ = 1 or τ =, the change point is smaller than the dimension. In terms of average run length, the ARL HC is uniformly smaller than ARL ZH. On the contrary, whenever the change point is larger than the dimension, τ = 4, ARL ZH is smaller than ARL HC whenever the shift magnitude and the the number of out-of-control variables are relatively large. For example, δ = 1.5, r =.75, ARL ZH is.9 while ARL HC is 3.. When p = 5: When τ = 1 or τ = 4, the change point is smaller than the dimension. In terms of average run length, the ARL HC is uniformly smaller than ARL ZH. On the contrary, whenever the change point is larger than the dimension, τ = 6, ARL ZH is smaller than ARL HC whenever the shift magnitude and the the number of out-of-control variables are relatively large. For example, δ = 1.5, r =.5, ARL ZH is 3. while ARL HC is 3.1. Therefore, we recommend the new method for detecting early shifts and the traditional for detecting large mean shifts after the number of observations exceeds the dimensionality of the measurements. 4. Comparison of diagnosis accuracy The accuracy of the change-point estimate is affected by many factors, including the shift magnitude, the number of out-ofcontrol variables and the data used for estimating the change point. Generally, the longer the out-of-control ARL is, the more data we have for change-point diagnosis. Tables 5 8 contain simulation results for change-point estimate p = 1,, 3 and 5, respectively. The findings in these four tables are summarised as follows.

11 454 Y. Li et al r= r= r= r= r=.5 r=.75 r= r=.5 r=1. r= r= r= Figure 1. Comparison of delay of detection of the and the with p = 1, τ = 5, 15, 5, δ =.5, 1., 1.5,.,.5, 3.

12 International Journal of Production Research τ=5 3 5 τ= τ= r r r Figure. Comparison of delay of detection of the and the with p = 1, δ = 3, r = 5%, 5%, 75%, 1%. p = 1 When τ = 5, the ˆτ HC are significantly much closer to the true value of τ than the ˆτ ZH. When τ = 15, i.e. the change point is bigger than the dimension, the traditional method has higher accuracy in estimating the location of shifts, especially when the shift magnitude is large. When τ = 5, the estimation accuracy of two methods are similar. p = When τ = 5orτ = 15, the ˆτ HC are significantly much closer to the true value of τ than the ˆτ ZH. When τ = 5, i.e. the change point is bigger than the dimension, the traditional method has higher accuracy in estimating the location of shifts, especially when the shift magnitude is large. Under some rare occasions, the ˆτ HC is more precise than ˆτ ZH, for example, r =.75 and δ =.5, ˆτ HC = 5 while ˆτ ZH = 4.. A possible explanation could be that the HC method has a longer average run length and consequently the change point is estimated based on more data. p = 3 When τ = 1 or τ =, the ˆτ HC are significantly much closer to the true value of τ than the ˆτ ZH. When τ = 3, i.e. the change point is bigger than the dimension, the traditional method has higher accuracy in estimating the location of shifts, especially when the shift magnitude is large. Under some occasions, the ˆτ HC is more precise than ˆτ ZH,for example, r =.5 and δ =, ˆτ HC = 39. while ˆτ ZH = p = 5 In the viewpoint of change-point estimation accuracy, the shows priority to the when the change point is larger than the dimension, such as τ = 6. In the rest two cases, τ = 1, where the change point is much smaller than the dimension, the change-point estimates based on the are much closer to the ZH method. When τ = 4, the change point is comparable to the dimension, neither of the two methods dominantly performs better and the estimate provided by two methods are quite close. In summary, when the change-point location is smaller than the dimension of data, because the could start monitoring much earlier than the, it has obvious advantage at giving out much earlier signals. Meanwhile, the promptness in detection also accompanied with more precise estimate the change-point location. When the change-point location is close to the quality dimension, the new method still enjoys shorter response time. Although sometimes the change-point estimate of the new method is not as precise as the, the difference between the estimate of the two methods are trivial. Also, in comparing the performance of alternative monitoring methods, those with the best early detection performance are at a disadvantage with respect to accuracy in estimation of the shift onset since there tend to be fewer shifted observations for analysis. Early detection is usually much more important, so usually any degree of sacrifice in estimation accuracy is acceptable. Whenever mean shifts occurs after sample size is much larger than the dimension of data, the usually enjoys higher computational efficiency and shorter average run length, especially when the mean shift is large.

13 456 Y. Li et al. 5. Discussions of the change-point control chart 5.1 Robustness to non-normality Univariate normality of course can be verified using normal probability plots. However, for multivariate observations, drawing inferences from univariate statistics could be misleading since marginal normality does not insure multivariate normality. Generally, it is a quite difficult task to assessing multivariate normality in multivariate dimensions, especially when the dimension is high. Fortunately, as shown by Theorem 1 in the Appendix, the proposed test statistic in Equation (8) still follows an asymptotic normal distribution even for non-normal data with heterogeneous variances. We have performed an extensive simulation and found that the the probability Pr(Z n, ˆτ > c) obtained by Monte Carlo simulation are quite robust to moderate violation of the normality model assumption. Therefore, the control limits h n,p obtained from the multivariate normal distribution can still be used for monitoring non-normal data. 5. Consistency of the estimate of change point The consistency of the change-point estimate is an attractive property. Here, consistency means that having a sufficiently large number of observations, the estimator converges in probability to its true value; that is, it is possible to estimate the true value of the parameter with arbitrary precision. Many works on change-point control charts studied the consistency of their change-point estimators, such as Zou, Zhang, and Wang (6) and Zou, Tsung, and Liu (8). The consistency of ˆτ is proved in the Appendix. We prove that ˆτ τ = O p (1) which means that the gap between the estimator ˆτ and τ is negligible compared with the sample size when the sample size increases to infinity. In other words, if we have enough in-control and out-of-control observations, the estimate ˆτ can be as relatively close to the true change point as possible. When the sample size tends to infinity, the estimate will converge to the true value of the change point in probability. Meanwhile, the newly proposed change-point estimator ˆτ is also consistent in presence of non-normality. This implies that the proposed high-dimensional change-point control chart works even when the normality assumption fails in practice, which is another attraction of the proposed method. 6. An industrial example A large amount of sensors are installed at a semi-conductor manufacturing process to collect process measurements, which can be used for quality monitoring and root cause identification. A data-set can be found at datasets/secom. The data-set file consists nearly 6 variables that could impact the quality level of the final product. Whenever an out-of-control is detected, it is necessary to locate the change point which will enable an increase in process throughput and decreased time to respond to the faults occurred in the process. Figure 3. Difference in the mean vector of qualified units and defective units in a semi-conductor manufacturing process.

14 International Journal of Production Research Change point τ Change point estimate Change point τ Figure 4. Comparison of the and the for the semiconductor example. The data-set file consists of 1567 records each with 591 variables and a label representing a simple pass/fail yield for in-house line testing where 1 corresponds to a pass (1463 records, 93.4%) and 1 corresponds to a fail (14 records, 6.6%). As with any real-life data, this data-set contains null values and constant values. Therefore, pre-process is applied to the data-set. We removed the constant variables and records containing null values, leaving 179 variables (3% of the original), 1387 qualified records (94.8% of the original) and 11 defective units (97.1% of the original). The difference between the mean of the qualified and defective units is shown in Figure 3. Among 179 variables, between the qualified and defective parts, only few variables mean experience significant shifts. In the rest part of this section, we use this example to show the implementation the proposed method. Also, since it is quite difficult to justify the multivariate normality of the dataset, we shall apply the proposed method to this data-set to test and demonstrate the robustness of the proposed method. We assume that the process start from an in-control status and the mean vector of the process has a step shift at time τ. For τ, we consider four values, 5, 1, 15 and. The in-control and out-of-control observations are drawn from the 1387 qualified units and 11 defective units by applying bootstrap method. The control limits of Zamba and Hawkins (6) method (ZH) and the newly proposed method (HC) determined by Monte Carlo simulation by making the in-control average run length 3. The results are shown in Figure 4. From Figure 4, the advantage of the is quite obvious in term of responsiveness to the mean shifts. When the change point τ is smaller than the observation dimension p = 179, the always signals much earlier than the ZH method. Concerning the change-point estimation, the estimate given by the is more closer to the true change point for τ = 5, 1. On the other hand, for τ = 15 and, the average of the change-point estimate given by is around accuracy of the is higher.

15 458 Y. Li et al. 7. Conclusion and summary Testing whether the mean vector of a multivariate process is constant has been an important problem in multivariate SPC for decades. Many statistical procedures including the classical Hotelling s T charts, Multivariate CUSUM charts and Multivariate EWMA charts have been proposed and widely used. These testing procedures have been well studied in the conventional low-dimensional setting. Driven by a wide range of contemporary industrial and financial applications, analysis of high-dimensional data is of significant current interest. In the high-dimensional setting, where the dimension can be much larger than the sample size, the conventional testing procedures perform poorly or are not even well defined. For instance, Hotelling s T charts cannot be start until enough sample are collected. In this paper, we combine one efficient existing high-dimensional statistical analysis technique and typical statistical change-point detection formulation together and develop a test that is powerful for of high-dimensional multivariate individual observations. In particular, we provide theoretical and numerical evidence for the desirable efficiency of the proposed test. We prove the asymptotical normality of the proposed test statistic and the consistency of the proposed change-point estimator when process is in control and out or control, respectively. The efficiency of the proposed method is evaluated via Monte Carlo simulation and compared with one representative existing test under various scenarios. The primary attraction of the new method is that SPC using the proposed control chart can start much earlier than existing methods. The same idea can be extended to high-dimensional covariance matrix monitoring. Other possible research topics include developing hybrid control schemes which combine the traditional methods and the newly developed method in this paper, and optimising the algorithm to accelerate the computation speed and maintain the precision meanwhile. Acknowledgements The authors deeply thank the editor and two anonymous referees for their many helpful comments that have resulted in significant improvements in the article. Li and Liu s research was supported by the National Natural Science Foundation of China (7173, , 11183). Zou s research was supported by the National Natural Science Foundation of China (111136, , 79314), Foundation for the Author of National Excellent Doctoral Dissertation of PR China (H5111) and New Century Excellent Talents in University (NCET-1-76). Jiang s research was supported by the National Natural Science Foundation of China ( ), Ministry of Education of China (NCET11-31) and Shanghai Pujiang Programme. References Amiri, A., and S. Allahyari. 11. Change Point Estimation Methods for Control Chart Postsignal Diagnostics: A Literature Review. Published online in Quality and Reliability Engineering International. Bai, Z., and H. Saranadasa Effect of High Dimension: By an Example of a Two Sample Problem. Statistical Sinica 6 : Champ, C. W., L. A. Jones-Farmer, and S. E. Rigdon. 5. Properties of the T Control Chart When Parameters are Estimated. Technometrics 47 (4): Chen, S., and B. H. Nembhard. 1. A High-dimensional Control Chart for Profile Monitoring. Quality and Reliability International 7 : Chen, S., and Y. Qin. 1. A Two-sample Test for High-dimensional Data with Applications to Gene-set Testing. Annals of Statistics 38 : Csorgo, M., and L. Horvath Limit Theorems in Change-Point Analysis. New York: Wiley. Hawkins, D. M., P. Qiu, and C. W. Kang. 3. The Change Point Model for Statistical Process Control. Journal of Quality Technology 35 (4): Hawkins, D. M., and K. D. Zamba. 5a. Statistical Process Control for Shifts in Mean or Variance using a Change Point Formulation. Technometrics 47 (): Hawkins, D. M., and K. D. Zamba. 5b. A Change Point Model for a Shift in Variance. Journal of Quality Technology 37 (1): Jiang, W., and K. L. Tsui. 8. A Theoretical Framework and Efficiency Study of Multivariate Control Charts. IIE Transactions 4 (7): Maboudou-Tchao, E. M., and D. M. Hawkins. 11. Self-starting Multivariate Control Charts for Location and Scale. Journal of Quality Technology 43 (): Mahmoud, M. A., P. A. Parker, W. H. Woodall, and D. M. Hawkins. 7. A Change Point Method for Linear Profile Data. Quality and Reliability Engineering International 3 (): Montgomery, D. C. 9. Introduction to Statistical Quality Control. 6th ed. New York: John Wiley & Sons. Nenes, G., and G. Tagaras. 6. The Economically Designed CUSUM Chart for Monitoring Short Production Runs. International Journal of Production Research 44 (8): Sullivan, J. H., and W. H. Woodall.. Change-point Detection of Mean Vector or Covariance Matrix Shifts using Multivariate Individual Observations. IIE Transactions 3 : Wang, K., and W. Jiang. 9. High-dimensional Process Monitoring and Fault Isolation via Variable Selection. Journal of Quality Technology 41 (3):

16 International Journal of Production Research 459 Wasserman, G. S An Adaptation of the EWMA Chart for Short Run SPC. International Journal of Production Research 33 : Zamba, K. D., and D. M. Hawkins. 6. A Multivariate Change-point Model for Statistical Process Control. Technometrics 48 (4): Zamba, K. D., and D. M. Hawkins. 9. A Multivariate Change-point Model for Change in Mean Vector and/or Covariance Structure. Journal of Quality Technology 41 (3): Zou, C., F. Tsung, and Y. Liu. 8. A Change Point Approach for Phase I Analysis in Multistage Processes. Technometrics 5 (3): Zou, C., Y. Zhang, and Z. Wang. 6. A Control Chart Based on a Change-point Model for Monitoring Linear Profiles. IIE Transactions 38 (1): Appendix The appendix consists of two technical results on the proposed test: Lemma 1 formally presents the variance formula (4) and Theorem 1 confirms that the proposed change point estimator ˆτ is consistent in sense ˆτ τ = O(1). Before giving these results, we recall the following two key assumptions that Chen and Qin (1) made to achieve the asymptotical normality of their W n,k, (μ μ 1 ) (μ μ 1 ) ( = o(tr ) ) /n, (1) ( [ ] ) tr( 4 ) = o tr( ), (11) because they are also very important to our proposed test. Assumption (1) implies that the difference between μ and μ 1 is negligible relative to tr( )/n, as a result Chen and Qin (1) s variance estimator and estimator (6) are both ratio consistent. Assumption (11) is used to restrict the increasing rate of p in an implicit way in order to guarantee the validation of the asymptotical normality in Equation (7). Lemma 1 Suppose {X i = (X i1,...x ip ), i = 1,...,n} are n independent p-variate normally distributed observations, with the first τ from N(μ, )and the rest n τ from N(μ 1, ). Suppose there exists a constant θ (, 1) such that τ/n = θ + o(1) as n tends to infinity. If μ = μ 1 and condition (1) holds, then for k = τ, the variance of W n,k defined in (6) is (4). Proof Throughout this proof, we fix k = τ.ase(w n,k ) = μ μ 1, the remaining task is to compute E(Wn,k ). Denote k X i X n j X i X k n j X i X j i, j=1;i = j i, j=k+1;i = j W n,k = + k(k 1) (n k)(n k 1) i=1 j=k+1 I 1 + I + I 3. (1) k(n k) Therefore, E Wn,k = E(I1 + I + I 3 + I 1 I + I 1 I 3 + I I 3 ). To compute E(I1 ), we assume temporarily i = j, r = s and 1 i, j, r, s k, and let S ={(i, j, s, r) : i = r, j = s} and It is clear that and S 1 ={(i, j, s, r) : i = s, j = r}, S ={(i, j, s, r) : i = s, j = r}, S 3 ={(i, j, s, r) : i = s, j = r}, S 4 ={(i, j, s, r) : i = r, j = s}, S 5 ={(i, j, s, r) : i = r, j = s}, S 6 ={(i, j, s, r) : i = r, j = s}. I 1 = 1 k (k 1) k k i, j=1;i = j r,s=1;i = j X i X j X r X s = 1 k (k 1) 6 X i X j X r X s, u= S u E X i X j X r X s = E [ X i X j X ( r X s = k(k 1)tr + μ μ ) ], S S 1 E S X i X j X r X s = E S 3 X i X j X r X s = E S 4 X i X j X r X s = E S 5 X i X j X r X s = k(k 1)(k )tr [( + μ μ ) μ μ ], E [ X i X j X (μ r X s = k(k 1)(k )(k 3)tr μ ) ]. S 6 Here and in what follows, the variance matrices and 1 are actually. This immediately implies E I1 tr = k(k 1) + 4 k μ μ + ( μ μ ).

17 46 Y. Li et al. Similarly, it can be found that E I tr 1 = (n k)(n k 1) + 4 n k μ 1 1μ 1 + ( μ 1 μ ) 1, E(I 1 I ) = tr [ μ μ μ 1μ 1] = (μ μ 1 ), E(I 1 I 3 ) = { } μ k μ 1 + kμ μ μ μ 1 = 4 k μ μ 1 μ μ μ μ 1, E(I I 3 ) = 4 n k μ 1μ 1 μ 1 μ 1μ 1 μ, E I3 = 4tr {( μ μ + /k )( μ 1 μ 1 + 1/(n k) )}. Without loss of generality we assume μ = and, therefore, δ = μ 1 μ = μ 1. Due to the assumption δ δ = o(tr( )/n), itthen follows that E Wn,k tr = k(k 1) + tr( 1 ) (n k)(n k 1) + 4μ 1 1μ 1 + ( μ n k 1 μ ) 4μ μ 1 + 4tr{ 1 } k k(n k) = tr k(k 1) + ( 1 ) tr (n k)(n k 1) + 4tr ( 1 ) k(n k) [1 + o(1)]. This proves Lemma 1. One key concern about the proposed change estimator ˆτ is its consistency under the alternative hypothesis, which is shown in the following theorem. Theorem 1 Under the same conditions of Lemma 1,ifμ = μ 1, then the change-point estimator ˆτ in (9) satisfies ˆτ τ = O p (1). Proof Recall the true change-point τ = nθ(1 + o(1)) for some θ (, 1). Letl = nβ for some β (, 1). To prove this theorem, it sufficient to show that Z n,l with β θ takes its minimum at l = nθ(1 + o(1)), as the case β θ can be proved similarly. To begin with, we need to derive the dominant parts of W n,l and sw.letδ = μ 1 μ and α = τ/l. It is found that The approximation of s W W n,l = α δ δ(1 + o p (1)). is based on the following three approximations s 1 = [tr( ) + α (1 α) (δ δ) ] (1 + o(1)), s = tr( )(1 + o(1)), s 3 = tr( )(1 + o(1)) which clearly indicate that at the true change-point l = nθ(1 + o(1)), or equivalently α = 1, these three estimators are all ratio-consistent estimates of tr( ). With l = nβ(1 + o(1)), the variance estimate can be approximated as { s sw = 1 l(l 1) + s (n l)(n l 1) + 4s 3 { = 1 n = n } (1 + o p (1)) l(n l) [ β tr( ) + α (1 α) (δ δ) ] + { tr( ) β (1 β) + α (1 α) (δ δ) } β (1 + o p (1)). (1 β) tr( ) + } 4 β(1 β) tr( ) (1 + o p (1)) Based on the above two approximations and the fact that θ = αβ(1 + o(1)), Z n,l can be approximated as Z n,l = W n,l s W = nδ δ (1 + o(1)). where ( ) 1 = θ(1 β) 1 tr( ) + (1 θ/β) θ (δ δ).