International Journal of Production Research
ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: http://www.tandfonline.com/loi/tprs20
An adaptive step-down procedure for fault variable identification Jinho Kim, Myong K. Jeong, Elsayed A. Elsayed, K.N. Al-Khalifa & A.M.S. Hamouda To cite this article: Jinho Kim, Myong K. Jeong, Elsayed A. Elsayed, K.N. Al-Khalifa & A.M.S. Hamouda (2015): An adaptive step-down procedure for fault variable identification, International Journal of Production Research, DOI: 10.1080/00207543.2015.1076948 To link to this article: http://dx.doi.org/10.1080/00207543.2015.1076948
Published online: 25 Aug 2015.
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Date: 24 October 2015, At: 06:53
International Journal of Production Research, 2015 http://dx.doi.org/10.1080/00207543.2015.1076948
An adaptive step-down procedure for fault variable identification Jinho Kima, Myong K. Jeonga*, Elsayed A. Elsayeda, K.N. Al-Khalifab and A.M.S. Hamoudab a
Department of Industrial and Systems Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ, USA; bDepartment of Mechanical and Industrial Engineering, Qatar University, Doha, Qatar
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(Received 10 March 2015; accepted 22 July 2015) In a process with a large number of process variables (high-dimensional process), identifying which variables cause an out-of-control signal is a challenging issue for quality engineers. In this paper, we propose an adaptive step-down procedure using conditional T2 statistic for fault variable identification. While existing procedures focus on selecting variables that have strong evidence of a change, the proposed step-down procedure selects a variable having the weakest evidence of a change at each step based on the variables that are selected in previous steps. The information of selected unchanged variables is effectively utilised in obtaining a powerful conditional T2 test statistic for identifying the changed elements of the mean vector. The proposed procedure is designed to utilise the correlation information between fault and non-fault variables for the efficient fault variables identification. Further, the simulation results show that the proposed procedure has the better diagnostic performance compared with existing methods in terms of fault variable identification and computational complexity, especially when the number of the variables is high and the number of fault variables is small. Keywords: conditional T2 statistic; fault variable identification; multivariate statistical process control; regression adjusted variables
1. Introduction Multivariate statistical process control (MSPC) has received considerable attention for monitoring multiple quality characteristics and/or process parameters. The primary objectives of MSPC are to detect a change in the process mean vector and to identify the variables responsible for the change (we refer to such variables as fault variables). Multivariate control charts such as Hotelling’s T2, multivariate cumulative sum (MCUSUM) and multivariate exponentially weighted moving average (MEWMA) are often used to detect a shift in the process mean vector (see, e.g. Crosier 1988; Pignatiello and Runger 1990; Lowry et al. 1992; Runger, Alt, and Montgomery 1996; Sullivan and Jones 2002; Jiang and Tsui 2008; Hawkins and Maboudou-Tchao 2008). Such control charts alarm an out-of-control (OC) signal when a shift is detected, but have difficulty in identifying the fault variables that cause the OC signal. Identifying the cause of an OC signal is a challenging issue for quality engineers in high-dimensional processes when an MSPC control chart detects a changed process mean. Since control charts based on T2 statistic have difficulty identifying responsible variables for an OC signal, a variety of diagnostic procedures have been developed (see, e.g. Doganaksoy et al. 1991; Hawkins 1991, 1993; Mason, Tracy, and Young 1995; Runger, Alt, and Montgomery 1996; Sullivan et al. 2007; Li, Jin, and Shi 2008). One popular approach for fault diagnosis evaluates an individual test statistic for an individual variable, respectively (Doganaksoy et al. 1991). This approach ignores correlations among variables. Another approach for fault diagnosis is based on evaluating every possible subset of variables (Murphy 1987; Chua and Montgomery 1992; Sullivan et al. 2007). Although this approach incorporates the correlation information among variables, it may not be practical due to the intensive computational requirements for high-dimensional process. Hawkins (1991, 1993) proposes a monitoring and diagnostic procedure under the assumption that only a single variable is shifted in the mean vector. The procedure is based on regression-adjusted variables using the correlations among variables. Hawkins’ regression-adjusted approach is effective in detecting and identifying a shift of single variable in the mean vector. When the maximum of the absolute values of regression-adjusted variables is significant, it signals and identifies the variable associated with the maximum as a changed variable. However, it can result in poor identification performance when the means of several variables are simultaneously shifted or even when the mean of single variable that is highly correlated with other variables is shifted (Das and Prakash 2008).
*Corresponding author. Email:
[email protected] © 2015 Taylor & Francis
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Mason, Tracy, and Young (1995) propose a decomposition procedure based on all possible partitioning of T2 statistic into independent unconditional and conditional T2 terms. This approach is referred to MYT decomposition. While the MYT decomposition is theoretically sound, it may not be practical when the number of variables p is large, since it needs to examine p! decompositions. To reduce the number of computations in the MYT decomposition, Mason, Tracy, and Young (1997) propose a shortened sequential procedure. In a worst-case scenario, however, the procedure still requires the same computations as the original MYT approach. Furthermore, the MYT approach has a concern about diagnostic capability (Li, Jin, and Shi 2008). A main issue associated with the MYT decomposition is to find a meaningful decomposition containing information on identifying a variable or a subset of variables responsible for the process mean shift among different p! decompositions. Recently, Li, Jin, and Shi (2008) propose a causation-based decomposition by integrating the traditional MYT decomposition with a Bayesian causal network that defines the causal relationship between variables. Tan and Shi (2012) propose a Bayesian approach based on Bayesian hierarchical model to determine the changed means and the directions of the shifts. Based on some prior knowledge and experience to specific processes that have known causal or Bayesian hierarchical property, one can investigate a smaller number of decompositions than the MYT approach. While these approaches are effective in some process control situations, they are often not proper in certain processes that have unknown causal or Bayesian hierarchical property. In high-dimensional processes, it is reasonable to assume that shifts in the mean vector occur in only a few variables, which is called the sparsity property (Zou and Qiu 2009). Wang and Jiang (2009) and Zou and Qiu (2009) propose process monitoring and diagnosis schemes based on variable selection methods and the sparsity assumption. Although both schemes provide diagnosis capability, they basically focus on the monitoring task. Recently, Zou, Jiang, and Tsung (2011) propose a LASSO-based diagnosis procedure, which uses an adaptive LASSO-type penalty function for fault variable identification. The LASSO-based procedure is based on the maximum likelihood estimation (MLE) approach, so its performance depends strongly on the number of available OC observations, which are obtained from the observations only after the estimated change point. When the size of a shifted mean is not sufficiently small, the estimated change point can be considerably close to the OC signalled point, resulting in small number of samples for the MLE-based fault identification. In this case, the LASSO-based procedure may not perform well due to a small number of OC observations. In this paper, we propose an adaptive step-down (ASD) procedure for identifying variables whose means are shifted, under the assumption that a shift in the mean vector occurs in only a few variables and a multivariate SPC chart detects the shift. While existing procedures focus on selecting variables that have strong evidence of a change, the proposed procedure selects a variable that has the strongest evidence of no mean change at each step. The unchanged variable selection is based on the variables that are selected in previous steps. When all remaining variables do not have evidence of no change, the procedure stops and identifies fault variables responsible for the OC signal. The information of selected unchanged variables is effectively utilised in obtaining a powerful conditional T2 test statistic for identifying the changed elements of the mean vector. The conditional T2 statistics are based on the projection scheme proposed by Runger (1996). The proposed procedure yields a less computational complexity in a high-dimensional process, since it is based on a polynomial time algorithm. Further, the proposed procedure is designed to consider the correlation structure within non-fault variables and between fault and non-fault variables, while other existing procedures consider only one of them. Thus, it can be an efficient and practical diagnostic tool for real-time fault variable identification in a high-dimensional process. The paper is organised as follows. In the next section, we present conditional T2 statistics when a group of unchanged variables are known. In Section 3, we propose an ASD procedure for the identification of fault variables using the information of a group of estimated unchanged variables, followed by an example to demonstrate the proposed procedure. In Section 4, we compare the proposed procedure with other existing diagnostic procedures in terms of the capability of fault variables identification. Finally, concluding remarks as well as future research are given. 2. Conditional T2 statistics with known group of unchanged variables Assume that a process has p quality characteristics and the measurement, X ¼ ðX1 ; . . .; XP Þ follows a multivariate normal distribution, NP ðl; RÞ. When the process is in-control (IC), the mean vector is l ¼ l0 , where l0 ¼ ðl1 ; . . .; lp Þ, and the covariance matrix R ¼ R0 , where R0 ¼ ½rij 1 i;j p , is known and fixed over time. When the process is OC, the process mean vector is changed to l1 6¼ l0 . The Hotelling’s T2 statistic is decomposed to identify the fault variables when the control charts generate an OC signal, and it is defined as T 2 ¼ ðX l0 Þ0 R1 0 ðX l0 Þ;
(1)
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where T2 statistic follows a χ2 distribution with p degrees of freedom when the process is in control. With a given false alarm rate α0, it signals when T 2 [ v2p;a0 . Although T2 is the optimal test statistic for a general multivariate shift of the process mean vector, it is not optimal when some of the process variables are known to be unchanged in high-dimensional processes (Lowry and Montgomery 1995). Suppose that X ¼ ðXX ; XC Þ, where X and C are index sets of two partitions of X. Then the mean vector and covariance matrix can be partitioned as l0 ¼ ðl0X ; l0C Þ and RXX RXC R0 ¼ ; RCX RCC respectively. Without loss of generality, a shift of the mean vector occurs only in a subset of variables X and there are no changes in all variables of C. Runger (1996) proposes a projection chart based on the conditional T2 given C defined as
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2 TXjC ¼ T 2 TC2 ;
(2)
2 2 where TC2 ¼ ðXC l0C Þ0 R1 CC ðXC l0C Þ is the T of all variables in C. Under the null hypothesis, TXjC follows the chisquare distribution with jXj degrees of freedom, where jXj denotes the number of elements in X. It is shown that using 2 statistic is more powerful than using overall T2 or even TX2 , which is the T2 of all variables in X (Runger (1996). TXjC We propose a new conditional T2-based diagnosis to specify the changed variables by choosing regression-adjusted variables, which are regression on the variables in C, with significantly different values from the target value. For a variable i in X, the proposed test statistic is based on the square of a regression-adjusted variable TiC , which is adjusted by the mean and standard deviation of the conditional distribution of Xi given XC as
TiC ¼
ðXi li Þ b0iC ðXC l0C Þ ðiÞ
ðrii b0iC RXC Þ1=2
;
(3)
where biC is a column vector of the regression coefficients of Xi on Xj for all j 2 C, which are obtained as ðiÞ ðiÞ biC ¼ R1 CC RXC , where RXC is the ith column vector of RXC . As shown in Mason, Tracy, and Young (1995), 2 2 ¼ TC2 þ TiC , so that we can simply obtain the square of TiC as TC[fig 2 2 TiC ¼ TC[fig TC2 ;
(4)
2 2 is the T2 of all variables in C [ fig. The distribution of TiC follows a chi-square with one degree of freewhere TC[fig 2 dom. With a given significant level α1, we can choose a threshold value v21;a1 for TiC and determine changed variables 2 2 in X if TiC [ v1;a1 for all i 2 X. The following proposition shows that the diagnostic power can increase by utilizing the information of known unchanged variables. Proposition 1. Suppose that C is a subset of unchanged variables and C is known in advance. Then for i 2 X, 2 PrðTiC [ v21;a1 Þ PrðTi2 [ v21;a1 Þ:
(5)
2 when EðXi Þ ¼ l0 , both TiC and Ti2 follow a central χ distribution with one degree of freedom. In this case, 2 2 2 2 2 PrðTiC [ v1;a1 Þ ¼ PrðTi [ v1;a1 Þ. When EðXi Þ ¼ l0 þ di , where δi ≠ 0, the distributions of TiC and Ti2 depend on the 2 2 non-centrality parameters, k1 and k2 , of χ distributions, respectively. Since k1 k2 , PrðTiC [ v21;a1 Þ PrðTi2 [ v21;a1 Þ (see Appendix 1 for details). ðiÞ For instance, when p = 2, the correlation coefficient of two variables is b0iC RSC . For simplicity, we also assume that l0 ¼ 0, and σii = 1 for i ¼ 1; 2. Without loss of generality, if we assume that X ¼ f1g and C ¼ f2g, then 2
X1 qX2 T12 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 q2
pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Since 1 q2 \1, if X2 is close to the target value 0, then PrðT12 T12 jX2 ¼ x2 Þ ! 1 as x2 ! 0. Thus, T12 can be 2 significant, although T1 is not significant. The power function for a given shift δ can be defined as 2 gðdÞ ¼ PðT12 [ v21;a1 Þ;
where α1 is a predefined significance level. The function η(δ) is monotonically increasing, and gðdÞ ¼ a1 when δ = 0. 2 Figure 1 depicts the relationship between the power functions of T12 with various ρ when X1 Nðd; 1Þ and 2 X2 Nð0; 1Þ. When ρ = 0, the distribution of T12 is equivalent to that of T12 , which is a test statistic of individual
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0.9
= 0.25 = 0.5
0.8
= 0.75 = 0.9
0.7
Power
0.6 0.5 0.4 0.3 0.2
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0.1 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Shift ( ) 2 Figure 1. Power functions of T12 with various ρ when X1 N ðd; 1Þ and X2 N ð0; 1Þ.
variable X1. It is clear that the slope of power functions with larger ρ is steeper than those with smaller ρ when d 2 ð0:5; 5:0Þ. 2 As a special case, when X ¼ fjg and C ¼ f1; . . .; j 1; j þ 1; . . .; pg, the conditional TjC statistic is closely 2 related to Hawkins’ regression-adjusted variable. Hawkins (1991, 1993) shows that the test static TjC is based on the optimal test statistic when a shift occurs only in the jth component of X. However, the conditional statistic is often far from optimal when the number of fault variables is greater than one. 3. ASD procedure In practice, the subset of unchanged variables is often unknown. In this case, the step-down procedure (Sullivan et al. 2007) and the MYT decomposition (Mason, Tracy, and Young (1995, 1997)) have been effective in the interpretation of the OC signal. However, interpretation efforts based on these approaches may require extensive computation time. In this section, we propose an ASD procedure based on conditional T2 statistic of a group of estimated unchanged ^ In each step, the proposed adaptive procedure selects a variable with strong evidence of no change given variables, C. previously selected unchanged variables. The variable at ith step is selected as ci ¼ argminj62C^ Tj2C^ ;
(6)
^ ¼ fc1 ; c2 ; . . .; ci1 g is a group of unchanged variables selected in the previous step. When i = 1, the set C ^ is where C 2 2 2 2 empty. In this case, we set TjC^ ¼ Tj , where Tj is the unconditional T of an individual variable.
The procedure has two stopping rules. First, the procedure stops when Tj2C^ ¼ Tj2 where ci ¼ argminj62C^ Tj2C^ is larger ^ than a threshold value v21;a1 , where α1 is the significance level for fault variables, i.e. if the conditional Tj2C^ for all j 62 C ^ are fault variables. Second, the procedure exceeds v2 . In this case, we can conclude all variables which are not in C 1;a1
stops if TC^ [ fci g2 [ v2i;a2 , where α2 is the significance level for the group of unchanged variables and TC^ [ fci g2 ^ is done and follows a χ2 distribution with i degrees of freedom. In this case, we can conclude the search for the set C ^ we can declare only the variables in C ¼ fc1 ; c2 ; . . .; ci1 g have strong evidence of no change. ^ is Ideally, if there are q changed variables among p variables, at iteration p − q + 1, the cardinality of the set C 2 2 2 2 [ vi;a2 will p − q, which is the number of unchanged variables. Then, at iteration p − q + 1, either TjC^ [ v1;a1 or TC[fc ^ g i
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2 ^ and move to the next step. In this paper, we occur. If both Tc2 C^ \v21;a1 and TC[fc \v2i;a2 are satisfied, then add ci to C ^ g i
set a1 ¼ a2 for simplicity.
i
3.1 Initial variable selection The proposed procedure begins with selecting a variable with statistically strongest evidence of no change. Although different schemes can be used for the initial variable selection, it seems reasonable to select an unchanged variable based on the following conditional value 2 c1 ¼ argminj¼1;...;p Tj1;...;j1;jþ1;...;p ;
(7)
2 is the conditional value of jth variable on the remaining p − 1 variables, where Tj1;...;j1;jþ1;...;p 2 2 ¼ T 2 Tf1;...;j1;jþ1;...;pg where f1; . . .; j 1; j þ 1; . . .; pg. The conditional value can be easily obtained as Tj1;...;j1;jþ1;...;p 2 2 is the T2 of the remaining p–1 variables. When the mean of each variable is on-target, Tj1;...;j1;jþ1;...;p Tf1;...;j1;jþ1;...;pg
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follows a χ2 distribution with one degree of freedom. When Tc21 [ v21;a1 , we stop and conclude that all variables whose Tj2 [ v21;a1 for j 2 f1; . . .; pg are fault variables. 3.2 Implementation of the ASD procedure The implementation of the proposed procedure follows the steps below. The decision threshold values v21;a1 and v2i;a2 for i ¼ 1; . . .; p can be obtained in advance from a v2 distribution with i degrees of freedom with pre-specified significance ^ ¼ ;. Then the procedure is defined as follows: level a1 and a2 . Let S ¼ f1; . . .; pg, and set i ¼ 1 and C 2 Step 1: Compute c1 ¼ argminj2S Tj1;...;j1;jþ1;...;p . 2 [ v2i;a2 . Step 2: Repeat substeps a–c until Tc2 C^ [ v21;a1 or TC[fc ^ g ^ ¼C ^ [ fci g. (a) Set C (b) Increase i by 1, i = i + 1. (c) Compute ci ¼ argminj62C^ Tj2C^ .
i
i
^ as fault variables and other variables as non-fault Step 3: Stop and declare variables whose Tj2C^ [ v21;a1 for all j 62 C variables. The proposed procedure reduces the computation time significantly, compared with the approximation procedure of the MTY decomposition (1997). The procedure does not investigate all decompositions to alleviate fault variable identification. In a worst case, the computational complexity of the proposed procedure is O(p2). When all variables are independent, the proposed test statistic is equivalent to individual T2 test statistic (Doganaksoy et al. 1991). 3.3 Choice of parameters The design of an ASD procedure involves the choice of parameters α1 and α2. The significance level α1 is used to check whether the mean of a variable is changed and α2 is used to check whether we have finished identifying all unchanged variables. Let l0 ¼ 0 and R0 ¼ ½rij 1 i;j p , where σii = 1 and σij = ρ. Diagnostic analysis is executed 10,000 times whenever the Hotelling’s T2 chart with ARL0 = 200, whose Type I error is α0 = 0.005, detects a shift of the mean vector. The changed mean vector is defined as l1 ¼ l0 þ d, where d ¼ ðd1 ; . . .; dp Þ. When the mean of the jth variable is shifted, δj ≠ 0, while δj = 0 when the mean is not changed. The performance measure is the relative frequency that the proposed procedure identifies fault variables exactly, which means no fault variables are missed and all fault variables are identified (Zou, Jiang, and Tsung 2011). It is expected that the corresponding variable with large δi can be easily identified as a fault variable even with smaller α1 and unchanged variables may be correctly identified with high probability. So, smaller α1 may provide higher correct identification rate when the mean of a fault variable is changed with a large amount. However, when the amount of a change is small, smaller α1 can often miss the fault variable, while larger α1 can identify falsely unchanged variables as fault variables with higher probability. The basic strategy of selecting α2, which is a significance level for decision on adding the new variable into the set ^ of unchanged variables, is to choose α2 ≥ α1. However, when α2 is too large (i.e. less aggressive in finding the true C), 2 ^ is the empty set) and then uses p unconditional T for identification. the procedure can stop at the first step (i.e. C Suppose that the procedure is at ith step after selecting γi with Tc2i C v21;a1 . If the selected variable is a fault variable and
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Table 1. Relative frequencies identifying fault variables exactly with different α1 and α2 when ρ = 0.75, 0.5, and 0.25.
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α1 = 0.05
α1 = 0.01
α2
0.05
0.01
0.005
0.05
Shifts δ1 = 1 δ1 = 2 δ1 = 3 δ1 = 1, δ2 = 1 δ1 = 2, δ2 = 2 δ1 = 3, δ2 = 3
0.21 0.50 0.63 0.27 0.62 0.70
0.21 0.51 0.63 0.27 0.62 0.70
0.21 0.51 0.63 0.27 0.62 0.70
δ1 = 1 δ1 = 2 δ1 = 3 δ1 = 1, δ2 = 1 δ1 = 2, δ2 = 2 δ1 = 3, δ2 = 3
0.11 0.33 0.52 0.14 0.46 0.64
0.11 0.33 0.53 0.14 0.46 0.64
0.11 0.33 0.53 0.14 0.46 0.64
δ1 = 1 δ1 = 2 δ1 = 3 δ1 = 1, δ2 = 1 δ1 = 2, δ2 = 2 δ1 = 3, δ2 = 3
0.08 0.24 0.42 0.10 0.35 0.57
0.08 0.25 0.42 0.10 0.35 0.57
0.08 0.25 0.42 0.10 0.35 0.57
0.54 0.84 0.91 0.32 0.78 0.92 ρ = 0.50 0.36 0.70 0.86 0.17 0.55 0.84 ρ = 0.25 0.29 0.62 0.81 0.10 0.42 0.70
α1 = 0.005
0.01
0.005
0.05
0.01
0.005
ρ = 0.75 0.54 0.85 0.91 0.32 0.78 0.92
0.54 0.85 0.91 0.32 0.78 0.92
0.61 0.91 0.95 0.25 0.77 0.96
0.61 0.91 0.95 0.25 0.77 0.96
0.61 0.91 0.95 0.25 0.77 0.96
0.37 0.71 0.86 0.17 0.55 0.84
0.36 0.71 0.86 0.17 0.55 0.84
0.43 0.80 0.92 0.12 0.47 0.83
0.42 0.80 0.92 0.12 0.47 0.83
0.42 0.80 0.92 0.12 0.47 0.83
0.29 0.63 0.81 0.10 0.42 0.70
0.29 0.63 0.81 0.10 0.42 0.70
0.34 0.71 0.88 0.07 0.33 0.64
0.34 0.71 0.88 0.07 0.33 0.64
0.34 0.71 0.88 0.07 0.33 0.64
^ since T 2 α1 > α2, then a fault variable γi can be added into C ¼ TC^2 þ Tc2 C^ \v2i;a2 . However, when α1 ≤ α2, it can be ^ g C[fc i
i
2 ^ the procedure is stopped and [ v2i;a2 even though Tc2 C^ v21;a1 . In this case, instead of adding the variable γi to C, TC[fc ^ g i
i
^ as fault variables. declare variables whose Tj2C^ [ v21;a1 for all j 62 C
Table 1 shows the identification performance with various a1 and α2 for three levels of ρ when p = 10. When a single variable is changed, the procedure with α1 = 0.005 provides the best identification performance, while the procedure with α1 = 0.05 provides worse performance than those with α1 = 0.01 and 0.005. In this case, it is reasonable to select α1 = 0.005, which is equal to the value of the Type I error, a0 , of the Hotelling’s T2 chart. When two variables are changed, the procedure with α1 = 0.01 almost outperforms those with α1 = 0.05 and 0.005. Under the assumption that two variables are changed, it is reasonable to choose α1 = 2α0. In most cases, selecting α2 = α1 provides the best results. Hence, we suggest using α2 = α1 for the group of unchanged variables. 3.4 Relation with MYT decomposition Suppose that the proposed procedure stops at (p − 1)th step. Then, the proposed procedure computes all conditional terms of one MYT decomposition (Mason, Tracy, and Young 1995) as T 2 ¼ Tc21 þ Tc22 c1 þ þ Tc2p c1 ;...;cp1 : 2
(8) 2
Note that the diagnostic procedure starts after Hotelling’s T chart signals since the value of T statistic is significantly large. In this case, since a group of variables {γ1, γ2, …, γp−1} has no evidence of a change, the final variable γp is a fault variable with high probability. 4. An illustrative example To illustrate the proposed procedure, we use a data-set of switch drums from Flury and Riedwyl (1988) and Hawkins (1991). The data-set contains five variables (p = 5): X1 is the inside diameter of the drum, and X2, X3, X4, X5 are distances from the head to the edges of four sectors cut in the drum. The target mean of X is l0 ¼ ð17:96; 10:30; 13:76; 11:08; 8:26Þ and the standard deviation is (1.8622, 1.7053, 1.7090, 1.8718, 2.2114). The covariance matrix of standardised variables is defined as
International Journal of Production Research
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0:1388 1 0:7324 0:9130 0:6932
0:3496 0:7324 1 0:6824 0:8214
0:0829 0:9130 0:6824 1 0:7640
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0:2652 0:6932 0:8214 0:7640 1
A sequence of 50 observations is presented in the data-set. After the first 35 observations, the upward shifts of 0.5 and 0.25 standard deviation are induced to the variables X1 and X5, respectively. The means of all other variables are left unchanged. At the 48th observation, X = (13.065, 11.625, 14.923, 12.589, 12.446), Hotelling’s T2 control chart signals as T2 ¼ 18:19 [ v25;a0 ¼ 15:09, where α0 = 0.01 is the significance level. We set both α1 and α2 as 0.01. In step 1, the third ^ ¼ f3g in step 1. In step 2, variable provides the strongest evidence of no change, where T 2 ¼ 0:80. Then γ1 = 3 and C
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31;2;4;5
2 2 2 2 the conditional T2 values given X3 are evaluated, and select γ2 = 2 since T23 has the smallest value among fT23 ; T43 ; T53 g ^ ^ so C ¼ f2; 3g. Similarly, we select γ3 = 4 so C ¼ f2; 3; 4g in step 3. After step 3, the step-down procedure stops since both 2 2 and T52;3;4 are larger than the predefined threshold values, v21;a1 ¼ v21;a2 ¼ 3:84, and conclude that both X1 and X5 T12;3;4 are fault variables. The values of conditional and unconditional T2 of all steps are shown in Table 2. The bold values in Table 2 indicate the smallest values in each step. The identification of X1 and X5 as the fault variables is similar to the conclusions reached by Hawkins (1991). Mason, Tracy, and Young (1995) shows that 31conditional T2 terms are significant, and X1, X4, X5 are identified as changed variables when the shortened sequential procedure (Mason, Tracy, and Young (1997)) is used. When we test only unconditional T2 for individual variables, X1 is identified as changed variable.
5. Performance comparisons We now compare the proposed diagnostic procedure with the existing procedures such as MYT procedures (Mason, Tracy, and Young 1995), step-down procedure (Sullivan et al. 2007) and LASSO-based procedure (Zou, Jiang, and Tsung 2011). For performance comparisons, the relative frequency that the diagnostic procedures identify fault variables correctly (CR) is used to evaluate the performance of a diagnostic procedure and it is defined by (Zou, Jiang, and Tsung 2011) Pn I ^ CR ¼ i¼1 X¼X ; n where n is the number of repeated identifications and I is the indicator function. A diagnostic procedure performs better if relative frequency is larger. We assume that the diagnosis procedures start after a Hotelling’s SPC detects a shift. The diagnostic procedures are executed 10,000 times after Hotelling’s T2 chart with ARL0 = 200 generates an OC signal. A significance level for a step-down procedure is set to 0.05 (Sullivan et al. 2007). For comparisons, α1 and α2 are set to 0.005 for the proposed ASD procedure. Let the distribution of IC measurements be the multivariate normal distribution with l0 ¼ 0 and R0 ¼ ½rij 1 i;j p , where σii = 1 and σij = ρ|i-j|. We choose ρ = 0.5 and p = 10 in this simulations. The changed mean vector is l1 ¼ l0 þ d, where d ¼ ðd1 ; . . .; dp Þ. We compare the performance of different procedures for two cases: perform the fault identification based on a single observation at the time of an OC signal and all observations from the estimated change point to the time of an OC signal. Note that all procedures except the LASSO-based approach by (Zou, Jiang, and Tsung 2011) uses an observation at the time of an OC signal. 5.1 Fault identification with a single observation at the time of an OC signal For performance comparisons of fault identification procedures, two fault scenarios are used: (a) X ¼ f5g and (b) X ¼ f1; 5g. Figure 2 shows CRs of the ASD and MYT procedures for these two scenarios using only the last point XT Table 2. Conditional and unconditional T2 of all steps in the proposed procedure. Step 1 2 3 4
Test statistics 2 ¼ 10:77 T12;3;4;5 2 ¼ 9:36 T13 2 ¼ 9:22 T12;3 2 ¼ 9:20 T12;3;4
2 T21;3;4;5 ¼ 1:44 T 223 ¼ 0:17 T 242;3 ¼ 0:05 2 T52;3;4 ¼ 6:73
T 231;2;4;5 ¼ 0:80 2 T43 ¼ 0:22 2 T52;3 ¼ 5:32
2 T41;2;3;5 ¼ 2:98 2 T53 ¼ 5:47
2 T51;2;3;4 ¼ 8:30
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(a)
1 ASD MYT
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1 ASD MYT
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0.1
0.5 0.4 0.3 0.2 0.1 0
1
1.5
2
2.5
3 , 1 5
Figure 2. Performance comparisons when only the last observation is shifted for (a) X ¼ f5g and (b) X ¼ f1; 5g, respectively.
that generates an OC signal. Because we assume only mean shifts, unconditional T2 statistic for individual variables in the MYT procedure is evaluated for identifying mean shifts as suggested by Mason, Tracy, and Young (1995, 1997). As shown in Figure 2, our procedure shows better performance for different shift levels for two cases: with single and two fault variables. Especially, the CR of ASD increases fast with the shift level increases by 4.0. 5.2 Fault identification with all OC observations after the estimated change point As shown in Zou, Jiang, and Tsung (2011), the performance of the LASSO-based approach strongly depends on the number of available OC observations. The LASSO-based procedure (Zou and Qiu 2009; Zou, Jiang, and Tsung 2011) estimates the change (or shift) point first, then it performs diagnostic procedures using all observations from the estimated change point to the time of an OC signal. For fair comparisons, we use the change point estimation procedure
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that is used in the LASSO-based procedure. This procedure is based on the MLE-based multivariate change point estimator (Zamba and Hawkins 2006). The location of change point τ is defined as t l0 ÞR1 ðX t l0 Þ; ^s ¼ argmin0 t\T ðT tÞðX 0 PT t ¼ 1 where T is the time that an SPC chart detects a shift and X i¼tþ1 Xi . T t Figure 3 shows the CRs of the ASD, the step-down, and LASSO-based procedures with various shift levels when (a) X ¼ f5g and (b) X ¼ f1; 5g. When a shift level is large, SPC charts detect the shift immediately. In this case, only a few recent observations can be used for the fault identification resulting in poor identification performance of the
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Figure 3. Performance comparisons of ASD, step-down and LASSO-based procedures with (a) X ¼ f5g and (b) X ¼ f1; 5g.
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LASSO-based procedure. As expected, the performance of the LASSO-based procedure is good when the shifts are detected later such as the case of smaller shifts, but the performance is poor when the shifts are detected early such as the cases of intermediate or large shifts. Note that the performance of the LASSO-based approach can be affected by the accuracy of the estimation of a change point, especially when the number of available observations is small. The step-down and the ASD procedure are less sensitive to the number of available OC observations used for the fault identification than the LASSO-based procedure. The ASD outperforms the step-down procedure in most cases. Specifically, when the shift levels are between 2.0 and 5.0, the relative frequencies of the ASD for identifying shifted variables correctly are considerably larger than those of the step-down procedure in both (a) and (b) of Figure 3. Compared with the case of fault identification with an observation at the time of an OC signal in Figure 2, the CRs of ASD using all observations from the estimated change point to the time of an OC signal dramatically improve when the shift levels are small, but are slightly worse than those of Figure 2 when the shift levels are large. We also compare the expected error rates in mean shift decisions (EER) defined as (Zou, Jiang, and Tsung 2011) ! p X Number of errors EER ¼ E jIfi2Xg Ifi2Xg ¼E ^ j =p; Number of variables i¼1 where the number of errors is the number of missed fault variables and falsely identified variables. When a diagnostic procedure identifies fault variables exactly, CR = 1 and EER = 0, while CR = 0 and EER = 1 when all identified fault variables are false and all fault variables are missed. Thus, a diagnostic procedure performs better if its value in column ‘CR’ is larger and its value in column ‘EER’ is smaller. The results in Tables 3 and 4 show that the proposed approach has a better diagnostic performance comparable to other procedures. Table 3 reveals the effects of the locations and the number of the fault variables when p = 10. The overall performance of all procedures deteriorates as the number of fault variables increases. The performance of the step-down and ASD procedures is much more improved than the LASSO-based procedure when the locations of the fault variables are closer, that is, when correlations between two fault variables are larger. The values in the brackets represent the variance in Table 3. Table 4 shows the effects of ρ when δi = 3.0, for i 2 X ¼ f1; 5; 6g and p = 10. As shown in, the ASD procedure outperforms the LASSO procedures by a large margin when ρ = 0.75. CRs of ASD tend to be superior to those of stepdown and LASSO procedures as ρ increases. However, when ρ is small, CRs of the LASSO procedures are better than those of the ASD and step-down procedures. This is because the penalty function of the LASSO-based procedure does not consider the correlation information appropriately.
Table 3. Effects of the locations and the number of the fault variables to the performance of ASD, step-down and LASSO-based procedures when δi = 3.0 for i 2 X. Step-down X {1} {5} {1,10} {5,6} {1,5,6} {4,5,6}
CR 0.881 0.901 0.581 0.794 0.410 0.570
LASSO EER
0.017 0.014 0.051 0.032 0.093 0.079
(0.052) (0.048) (0.070) (0.077) (0.099) (0.113)
CR 0.760 0.756 0.577 0.611 0.463 0.452
ASD EER
0.035 0.035 0.062 0.059 0.087 0.091
(0.074) (0.074) (0.087) (0.089) (0.101) (0.105)
CR 0.906 0.915 0.697 0.839 0.508 0.590
EER 0.013 0.012 0.038 0.026 0.081 0.095
(0.050) (0.046) (0.066) (0.077) (0.098) (0.158)
Table 4. Effects of the correlation between fault variables to the performance of ASD, step-down and LASSO-based procedures when δi = 3.0 for 2 X. Step-down ρ 0.25 0.50 0.75
LASSO
ASD
CR
EER
CR
EER
CR
EER
0.294 0.410 0.774
0.100 (0.087) 0.093 (0.099) 0.038 (0.090)
0.434 0.463 0.534
0.087 (0.095) 0.087 (0.101) 0.082 (0.118)
0.372 0.508 0.854
0.098 (0.094) 0.079 (0.098) 0.029 (0.089)
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Table 5. Average computation time (standard deviations) in seconds. p
Step-down
LASSO
10 15 20 25
0.0810 (0.0379) 2.1800 (0.0459) 75.2758 (0.6705) 2643.5394 (4.6083)
0.2044 0.3053 0.3661 0.4360
ASD
(0.0553) (0.1068) (0.1323) (0.1465)
0.0035 0.0068 0.0112 0.0180
(0.0012) (0.0013) (0.0013) (0.0011)
5.3 Case study: automatic monitoring using bolt image data In this section, we demonstrate the proposed procedure in a real case study conducted in the laboratory to develop an automatic monitoring system using image data. In the system, a bolt travels on a conveyer belt and a high-resolution camera takes a picture of the bolt when infrared sensors detect its presence automatically. The dimensions of bolts such as height, length and diameter are compared with those of IC images. When the measured dimensions of a bolt exceed their corresponding threshold values, the bolt is automatically diverted from the conveyer. We consider a bolt to be acceptable when all the following four dimensions are within the acceptable thresholds. The observation data shown in Appendix 2 contain four variables: the head diameter (X1), the head height (X2), the bolt diameter (X3) and the length of the bolt (X4). When the process is in control, the means and the standard deviations of the four variables are (0.3673, 0.2449, 0.2502, 0.7346) and (0.0018, 0.0063, 0.0032, 0.0075), respectively. The correlations among variables are q1;2 ¼ 0:1853, q1;3 ¼ 0:3231, q1;4 ¼ 0:2026, q2;3 ¼ 0:1025, q2;4 ¼ 0:9511, q3;4 ¼ 0:1516. Figure 4 shows the resulting Hotelling T2 chart along with its control limit, 14.86 (solid horizontal line). The Hotelling chart signals at the 32nd observation, x32 ¼ ð0:36616; 0:24408; 0:24986; 0:72286Þ with T 2 ¼ 31:1652. The proposed diagnostic procedure is used to identify the fault variables immediately after the chart signals. Table 6 shows the conditional T2 values calculated in each step. At the first step, X1 is selected as an unchanged variable, next X3 is selected at 40 35 30 25
T2
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Because the step-down procedure requires 2p−1 computations, it takes significant computation time when p is large. Average computation times and standard deviations of the step-down, LASSO and the proposed ASD procedures for single diagnosis with p = 10, 15, 20 and 25 are shown in Table 5. When p = 25, the average execution time of the step-down procedure is about 44 min, while that of the proposed procedure is about 0.02 s. Therefore, the step-down procedure is not practical for the situation that needs to identify fault variables quickly when p is large. Since the LASSO procedure is based on a polynomial time algorithm, the execution time of LASSO is also practical when p is large. The experiments are executed on MATLAB 2012b in Window 7 (64 bits) desktop with 8 GB RARM and 2.10 GHz Dual-Core CPU.
20 15 10 5 0
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Figure 4. Hotelling’s T2 chart. Shift is detected at the 32nd observation.
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Table 6. Conditional T 2 of all steps in the proposed procedure. Step 1 2 3 4
Test statistics T 212;3;4 ¼ 0:04 2 ¼ 0:49 T21 T 221;3 ¼ 0:08 2 ¼ 30:32 T41;2;3
2 T21;3;4 ¼ 28:24 2 T 31 ¼ 0:44 2 T41;3 ¼ 2:16
2 T31;2;4 ¼ 1:09 2 T41 ¼ 2:55
2 T41;2;3 ¼ 30:32
step 2 and then X2 at step 3. These values indicate that the shift may have occurred at the last variable X4 (the length of 2 the bolt), since T41;2;3 ¼ 30:32 is significantly large. The bold values in Table 6 indicate the smallest values in each step.
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6. Concluding remarks When a shift in the mean vector is detected, identifying variables that cause an OC signal is very crucial. Conventional diagnosis approaches such as step-down and MYT decomposition are theoretically sound for diagnosing root causes of the process changes, but computationally impractical when the process has a large number of variables. In this paper, we develop an ASD procedure based on the conditional T2 statistic for fault variable identification. By utilizing the information of variables that have strong evidence of no change based on our proposed procedure, we can construct the powerful diagnostic test statistic. The proposed procedure is computationally efficient in high-dimensional processes and enhances diagnostic power in identifying the changed variables of the mean vector when a shift occurs only in a few variables. As shown in the simulation results, the proposed procedure outperforms the MYT and step-down procedures. Moreover, the proposed procedure is superior to the LASSO-based procedure when the shift levels are not small. In future work, we may extend our procedure for both monitoring and diagnosis in multistage processes. Acknowledgements The authors thank the anonymous reviewers and editors for their helpful and constructive comments that greatly contributed to improving the paper.
Disclosure statement No potential conflict of interest was reported by the authors.
Funding This work was supported by Qatar National Research Fund (QNRF) [grant number NPRP 5-364-2-142]. The statements made herein are solely the responsibility of the authors.
References Chua, M. K., and D. C. Montgomery. 1992. “Investigation and Characterization of a Control Scheme for Multivariate Quality Control.” Quality and Reliability Engineering International 8: 37–44. Crosier, R. B. 1988. “Multivariate Generalizations of Cumulative Sum Quality-control Schemes.” Technometrics 30: 291–303. Das, N., and V. Prakash. 2008. “Interpreting the Out-of-control Signal in Multivariate Control Chart – A Comparative Study.” The International Journal of Advanced Manufacturing Technology 37: 966–979. Doganaksoy, N., F. W. Faltin, and W. T. Tucker. 1991. “Identification of Out of Control Quality Characteristics in a Multivariate Manufacturing Environment.” Communications in Statistics – Theory and Methods 20: 2775–2790. Flury, B., and H. Riedwyl. 1988. Multivariate Statistics: A Practical Approach. London: Chapman & Hall. Hawkins, D. M. 1991. “Multivariate Quality Control Based on Regression-adjusted Variables.” Technometrics 33: 61–75. Hawkins, D. M. 1993. “Regression Adjustment for Variables in Multivariate Quality Control.” Journal of Quality Technology 25: 170–182.
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Hawkins, D. M., and E. M. Maboudou-Tchao. 2008. “Multivariate Exponentially Weighted Moving Covariance Matrix.” Technometrics 50: 155–166. Jiang, W., and K. L. Tsui. 2008. “A Theoretical Framework and Efficiency Study of Multivariate Control Charts.” IIE Transactions on Quality and Reliability 40: 650–663. Li, J., J. H. Jin, and J. J. Shi. 2008. “Causation-Based T2 Decomposition for Multivariate Process Monitoring and Diagnosis.” Journal of Quality Technology 40: 46–58. Lowry, C. A., and D. C. Montgomery. 1995. “A Review of Multivariate Control Charts.” IIE Transactions 27: 800–810. Lowry, C. A., W. H. Woodall, C. W. Champ, and S. E. Rigdon. 1992. “A Multivariate Exponentially Weighted Moving Average Control Chart.” Technometrics 34: 46–53. Mason, R. L., N. D. Tracy, and J. C. Young. 1995. “Decomposition of T2 for Multivariate Control Chart Interpretation.” Journal of Quality Technology 27: 99–108. Mason, R. L., N. D. Tracy, and J. C. Young. 1997. “A Practical Approach for Interpreting Multivariate T2 Control Chart Signals.” Journal of Quality Technology 29: 396–406. Murphy, B. J. 1987. “Selecting Out-of-control Variables with T2 Multivariate Quality Procedures.” The Statistician 36: 571–583. Pignatiello, J. J., and G. C. Runger. 1990. “Comparisons of Multivariate CUSUM Charts.” Journal of Quality Technology 22: 173–186. Runger, G. C. 1996. “Projections and the U2 Multivariate Control Chart.” Journal of Quality Technology 28: 313–319. Runger, G. C., F. B. Alt, and D. C. Montgomery. 1996. “Contributors to a Multivariate Statistical Process Control Chart Signal.” Communications in Statistics – Theory and Methods 25: 2203–2213. Sullivan, J. H., and L. A. Jones. 2002. “A Self-Starting Control Chart for Multivariate Individual Observations.” Technometrics 44: 24–33. Sullivan, J. H., Z. G. Stoumbos, R. L. Mason, and J. C. Young. 2007. “Step-Down Analysis for Changes in the Covariance Matrix and Other Parameters.” Journal of Quality Technology 39: 66–84. Tan, M. H., and J. Shi. 2012. “A Bayesian Approach for Interpreting Mean Shifts in Multivariate Quality Control.” Technometrics 54 (3): 294–307. Wang, K., and W. Jiang. 2009. “High-Dimensional Process Monitoring and Fault Isolation via Variable Selection.” Journal of Quality Technology 41: 247–258. Zamba, K. D., and D. M. Hawkins. 2006. “A Multivariate Change-Point for Statistical Process Control.” Technometrics 48: 539–549. Zou, C., and P. Qiu. 2009. “Multivariate Statistical Process Control Using LASSO.” Journal of the American Statistical Association 104: 1586–1596. Zou, C., W. Jiang, and F. Tsung. 2011. “A LASSO-Based Diagnostic Framework for Multivariate Statistical Process Control.” Technometrics 53: 297–309.
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2 Appendix 1. Derivation of the non-centrality parameter of TiC
For simplicity, we assume that l0 ¼ 0, and σii = 1 and 0 ≤ σij < 1 for i, j ∈ {1, …, p}. Suppose that the mean of Xi, where i ∈ S, is shifted to δi ≠ 0, and the means of all variables in Γ are zero, EðXj Þ ¼ 0 for all j ∈ Γ. Then the expected value of TiC is di EðTiC Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðiÞ 1 bðiÞ RCS ðiÞ0
2 where bðiÞ ¼ RCS R1 CC . Therefore, the non-centrality parameter of TiC is
ki ¼
d2i ðiÞ
1 bðiÞ RCS
: ðiÞ
Let d2i be the non-centrality parameter of Ti2 , which follows a χ2 distribution with one degree of freedom. Since 0 1 bðiÞ R21 \1, it can be shown that ki d2i . 2 Also, for a given t > 0, we can show that the difference between cdf’s of non-central χ2, Ti2 and TiC , is greater than or equal to 0, which means
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2 PrðTiC [ tÞ PrðTi2 [ tÞ:
Appendix 2. Bolt measurements (inch) Sample number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
X1
X2
X3
X4
0.37064 0.36612 0.36612 0.37064 0.37064 0.36838 0.36612 0.36838 0.36612 0.36838 0.36838 0.36838 0.37064 0.36838 0.37064 0.36838 0.36838 0.36612 0.36612 0.36838 0.36838 0.37064 0.36838 0.36838 0.36612 0.36612 0.36612 0.36786 0.36838 0.36712 0.36612 0.36616
0.24634 0.24408 0.24634 0.24634 0.24408 0.24408 0.24634 0.24634 0.24408 0.24408 0.24182 0.24408 0.24182 0.24182 0.24634 0.24182 0.24182 0.24408 0.24408 0.24408 0.24634 0.24634 0.24182 0.24408 0.24634 0.24182 0.24634 0.24408 0.24408 0.24634 0.24408 0.24408
0.24860 0.24860 0.25086 0.25086 0.24634 0.25086 0.24860 0.25086 0.25086 0.24860 0.24860 0.24860 0.24860 0.25086 0.25086 0.25086 0.24860 0.25086 0.24860 0.25312 0.24860 0.25086 0.25086 0.24860 0.25086 0.24934 0.25086 0.24960 0.24860 0.25086 0.24860 0.24986
0.73450 0.73224 0.73450 0.73224 0.73224 0.73676 0.73450 0.73224 0.73902 0.73902 0.74580 0.73224 0.73676 0.73450 0.73902 0.73450 0.73902 0.73676 0.73224 0.73676 0.73450 0.73224 0.73902 0.73676 0.73224 0.73676 0.72772 0.73224 0.72894 0.72772 0.72894 0.72286
