On the effect of measurement errors in regression-adjusted monitoring of multistage manufacturing processes

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Technical paper

On the effect of measurement errors in regression-adjusted monitoring of multistage manufacturing processes Guoliang Ding a , Li Zeng b,∗ a

Department of Precision Machinery and Instrumentation, University of Science and Technology of China, Hefei, Anhui 230026, China Department of Industrial and Manufacturing Systems Engineering, The University of Texas at Arlington, 500 West First Street, P.O. Box 19017, Arlington, TX 76019, USA b

a r t i c l e

i n f o

Article history: Received 15 August 2012 Received in revised form 24 November 2013 Accepted 24 June 2014 Available online xxx Keywords: Measurement errors Multistage manufacturing process (MMP) Ordinary least squares (OLS) Regression adjustment Total least squares (TLS)

a b s t r a c t A critical challenge in multistage process monitoring is the complex relationships between quality characteristics at different stages. A popular method to deal with this problem is regression adjustment in which each quality characteristic is regressed on its preceding quality characteristics and the resulting residual is monitored to detect changes in local variations. However, the performance of this method depends on the accuracy of the regression coefficient estimation. One source of the estimation errors is measurement errors which commonly exist in practice. To provide guidance on the use of regression-adjusted monitoring methods, this study investigates the effect of measurement errors on the bias of regression estimation theoretically and numerically. Two estimators, the ordinary least squares (OLS) estimator and the total least squares (TLS) estimator, are compared, and insights regarding their performance are obtained. © 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction Multistage manufacturing processes (MMPs) are becoming increasingly common in today’s manufacturing arena [1]. Fig. 1 shows a typical example of such processes which consists of 11 stations and is capable of producing five different types of motor reducers. A defining feature of MMPs is that the outgoing product quality at each single station is determined not only by various local disturbances at that station such as thermal error, cutting-force induced error, and machine geometric error, but also by the propagated variations from upstream stations such as the datum error due to preceding cutting operations. The general model of multistage processes in quality monitoring is shown in Fig. 2, where each node represents a quality characteristic (QC) measured at a certain stage. Due to the variation propagation in the process, these QCs bear complex relationships. This poses significant challenges for process monitoring because the conventional statistical monitoring methods are not able to differentiate local and propagated variations and thus considerable amounts of false alarms could be generated, i.e., the monitoring scheme may mistake the propagated

∗ Corresponding author. Tel.: +1 817 272 3150; fax: +1 817 272 3406. E-mail address: [email protected] (L. Zeng).

variation as local variation and then generate an alarm that is due to other stages. Many efforts have been made to conquer this problem by making use of either the physical models of the processes [e.g., 2–5] or statistical analysis of historical data [e.g., 6,7]. A popular data-driven method for monitoring correlated QCs is the regression adjustment method [e.g., 8,9,10,11,12]. Basically, this method monitors the residual, Zj = Qj − Qˆ j , j = 1, . . ., q, resulted when QC j is regressed on all its preceding QCs instead of monitoring Qj itself. Since the propagated variation, represented by the predictor Qˆ j , is removed, the residual will only contain the information on the local variation of QC j, and thus if Zj is out of control, it means directly that some local faults happened. The idea of regression adjustment has been widely accepted as a simple and effective way to deal with multistage quality control problems and become the basis for many further studies. However, the performance of regression-adjusted monitoring depends closely on the accuracy of coefficient estimation in the regression between each QC and its preceding QCs. There are two sources of estimation errors: sampling uncertainty due to limited sample size and measurement errors in data collection. Shu et al. [13–15] conduct a systematic study of the effects of the first type of estimation error on the performance of regression-adjusted monitoring. It is found that the estimation error will decrease the in-control average run length and increase the out-of-control

http://dx.doi.org/10.1016/j.jmsy.2014.06.013 0278-6125/© 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: Ding G, Zeng L. On the effect of measurement errors in regression-adjusted monitoring of multistage manufacturing processes. J Manuf Syst (2014), http://dx.doi.org/10.1016/j.jmsy.2014.06.013

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Fig. 1. An example of multistage processes.

average run length. Zeng and Zhou [16] consider the effects of the second type of estimation error in the case of large samples. They point out that the existence of measurement errors will cause inaccurate estimation of the coefficients in the regression models which eventually leads to increased false alarms and miss detections in regression-adjusted monitoring. It is worth mentioning that the effects of measurement errors in regression-adjusted monitoring have also been investigated by Li and Huang [17]. However, their study assumes that a preliminary dataset not subject to measurement errors is available from which accurate estimates of the coefficients can be obtained. So their discussion is not on the effect of estimation errors caused by measurement errors in regressionadjusted monitoring. Since measurement errors commonly exist in manufacturing processes [18,19], it will be very useful to investigate the effects of such errors in regression-adjusted monitoring. Essentially, this means to study the estimation error caused by measurement errors. In the study of Zeng and Zhou [16], not much detail on this is provided as the focus of that study is the effects of such estimation errors on the performance of monitoring. To fill the gap, our study concentrates on the effects of measurement errors in coefficient estimation. Moreover, we compare the performance of two popular estimation methods, ordinary least squares (OLS) and total least squares (TLS), in the presence of measurement errors. The results will provide intuitive insights on regression-adjusted process monitoring as well as guidelines on its use in practice. It deserves to point out that to be useful to real-world multistage processes, this study uses an engineering model (a linear state space model) rather than a general regression model as used in many other studies [e.g., 13–15] to characterize the variation flow in the process, which is popular in multistage process research [e.g., 20]. In addition, like in [16], we assume large samples are available; in other words, we are studying the large-sample properties of the estimators. This is often the case in today’s manufacturing processes due to the advancement of information/sensing technologies. Specifically, our study aims to address two concerns on the effect of measurement errors: (1) What is the effect of measurement errors on the coefficient estimation and how is this effect affected

2

1 S1

4 5

3

6

S2

S3

q

j

by important characteristics of multistage processes? These characteristics include the relationships between QCs at different stages and magnitudes of local variation sources and measurement errors at each stage. (2) What are the advantages/disadvantages of the TLS estimator compared to the OLS estimator, and under what conditions it can be used to replace the OLS estimator in order to alleviate the effect of measurement errors? Both theoretical analysis and numerical study have been done to answer these questions. The remainder of the paper is organized as follows. Section 2 will give some background information, including the process model, conventional procedure of regression-adjusted monitoring, review of estimation methods in the presence of measurement errors, and basics of the TLS method. Theoretical results on the OLS estimator and the TLS estimator will be presented in Section 3. Section 4 gives the results of a numerical study. Section 5 summarizes our findings and discusses implications of the findings on multistage process monitoring. 2. Background and basics 2.1. Process model in the presence of measurement errors A linear state space model will be used in this study. Assume there are q QCs distributed at n stages in a general multistage process, as shown in Fig. 2. For j = 1, . . ., q, define Pj = {1, 2, . . ., p} as the set of QCs in preceding stages of QC j. For example, in Fig. 2, P5 = {1, 2, 3}, while Pq includes all the QCs except q. In this study, we assume that QC j could only be influenced by QCs in Pj . Also, let Uj be the local variation source of QC j. The local variation source represents the quantities which are related to a specific QC and are often not directly observable. They have different physical meanings in different processes. For simplicity, we assume every QC in the process has a different local variation source. A linear model is assumed for Qj and Qi , i ∈ Pj Qj =



ˇij Qi + Uj

where ˇij is the coefficient of Qi in the model of Qj . In the presence of measurement errors, the observed quantities are (Y, X1 , . . ., Xp ) which satisfy Y = Qj + e Xi = Qi + εi ,

2.2. Regression-adjusted monitoring In regression-adjusted monitoring, QC j, j = 1, . . ., q, is monitored using a univariate control chart to detect changes in its local variation, i.e., Uj . Following a standard procedure in SPC practice, the monitoring scheme includes two steps: In Phase I analysis, the relationship of QC j and its preceding QCs is estimated by the ordinary least squares method ˆ ␤

OLS

= (X X)

ˆ where ␤

Sn- 1

Fig. 2. General model of multistage processes.

Sn

(2)

i = 1, . . ., p

where Y is the observation of QC j, Xi is the observation of QC i, e is the measurement error of QC j and εi is that of QC i. As in many studies on multistage process monitoring [e.g., 1], we assume that all the local variation sources and measurement errors follow normal distribution, have zero mean and are independent of each other.

OLS

......

(1)

i ∈ Pj

−1

X Y





(3)

ˆ 2j , . . ., ˇ ˆ pj , and X = [X1 , X2 , . . ., Xp ] . The ˆ 1j , ˇ = ˇ

resulting residual is ˆ Z = Y − X␤

OLS

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X1=Q1+ε1

Mean and variance of the residual will then be found to set up control limits. In Phase II analysis, whenever a new observation is obtained, the value of the residual will be calculated using the estimate in (3) and compared to the control limits. If the residual is out of control, it means directly that the local variation of QC j is out of control. OLS ˆ Clearly, the accuracy of ␤ in Phase I analysis is very criti-

Q1=U1

X1=Q1+ε1

Q1=U1

X2=Q2+ε2 Q2=β12Q1+U2

β12

X2=Q2+ε2

Q2=β12Q1+U2

β23

X3=Q3+ε3

Q3=β13Q1+β23Q2+U3

β13 Fig. 4. Model of a three-stage process.

for the method of moments and maximum likelihood. Like LISREL, the Total least squares (TLS) method based on singular value decomposition (SVD) is a popular solution for measurement error problems which has its origin from computational mathematics. It was introduced by Golub and Van Loan [32], and has been frequently applied in computational mathematics and engineering literature since then [e.g., 33–35]. One of the main reasons for its popularity lies in the availability of efficient and numerically robust algorithms to perform SVD [36]. Another reason is that this method suits for situations where all data contain noises, which makes it a generic way to deal with measurement error problems in various engineering applications [e.g., 37,38]. Since the TLS method is easy to implement and does not require any parameter to be known, it appears to be the most potential remedy for the OLS method. For this reason, its performance is investigated in this study.

2.3. Estimation in the presence of measurement errors

(i) Statistics: The geometric mean method is one of the easiest to use, but limited to simple regression with one single regressor [24]. The grouping method is another approach to tackle the measurement error problem and could be generalized to regression with multiple regressors [25]. However, it is not able to deal with correlated regressors which are common in multistage process monitoring. (ii) Econometrics: In the absence of information about the measurement error variance, consistent estimation of the regression model is still possible, provided the data contain some instrumental variables that are correlated with the regressors but uncorrelated with the error components in the model [26]. In practice, however, it is very difficult to find acceptable instrumental variables. It is also found that some biased estimators such as James–Stein estimator and its variants dominate the OLS estimator uniformly when measurement errors exist [e.g., 27–30]. But the benefit of these estimators is only significant in finite sample cases; with large samples, they bear a trivial difference from the OLS estimator. (iii) Computational mathematics: The linear structural relations (LISREL) model is a method often used to estimate linear relationships in the presence of measurement errors, which adopts an iterative procedure for parameter estimation through constructing a set of estimation equations [31]. However, to implement this procedure, there must be as many estimating equations as parameters, which ties in with the requirements

β12

Fig. 3. Model of a two-stage process.

cal to the performance of the above monitoring scheme. Ideally, if the estimation is perfect, i.e., without estimation errors, the influence of preceding QCs can be completely removed, and thus the residual will be only related with the local variation of QC j; otherwise, the residual will also be related with preceding QCs, and thus false alarms or miss detections will be caused. More details on the effect of estimation errors on the performance of regressionadjusted monitoring in Phase II analysis can be found in Zeng and Zhou’s work [16]. This study focuses on the effect of measurement errors on the estimation of ␤ in Phase I analysis.

Estimation of linear regression models with measurement errors, referred to as measurement error models, or errors-invariables models, is a well studied topic in the literature. Adcock [21] first points out the basic fact that the OLS estimator, which is widely adopted in regression-adjusted monitoring, is biased and inconsistent in the presence of measurement errors. More recently, statistical methods for linear regression with measurement errors such as maximum likelihood estimation and method of moments estimator are comprehensively investigated [22,23]. However, seeking for estimators with optimal statistical properties, these methods have a key limitation, that is, they are built on the assumption that the parameters such as the variance and covariance of the measurement errors are known or can be estimated from historical data, which is not very practical. In addition, when the model contains multiple regressors, the problem becomes very complicated and the estimator can no longer be found analytically. For the above reasons, many efforts have been dedicated to alternative estimators in different research fields including

3

2.4. Total least squares estimator The TLS method aims to solve the following optimization problem ˆ {␤ TLS , X, Y } = arg min ||[X, Y ]||F ˆ ␤,X,Y

(4)

s.t. (X + X)␤ = Y + Y Let [X Y] be the pooled data including both the regressors and the response. The singular value decomposition of the pooled data is [XY ] = UV where U and V are unitary matrices, and  is a diagonal matrix. The diagonal entries of  are called singular values. Let ω be the smallest singular value of . The solution of (4), i.e., the TLS estimator, has a similar form as the OLS estimator in (3) ˆ ␤

TLS

= (X X − ωI)

−1

X Y

(5)

3. Effect of measurement errors on the estimators This section will examine the effect of measurement errors on the OLS estimator in (3) and the TLS estimator in (5). The difficulty of this study lies in that there is no analytical form of ω in (5), and thus it is very challenging, if not impossible, to obtain general results on the TLS estimator in the presence of measurement errors. For this reason, two simple cases have been considered in our study: two-stage processes and three-stage processes as illustrated in Figs. 3 and 4, respectively. For two-stage processes, theoretical results on both estimators are found and compared. The favored conditions of the TLS estimator are identified based on those results. For three-stage processes, theoretical results are found only for the OLS estimator and the effect of measurement errors under some special cases is identified.

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Before presenting the results, some notations and terms that will be frequently used in the discussion are given here: u2j and ε2j denote the variance of the local variation source and measurement error of QC j respectively; in the regression of QC j on its preceding QCs, the stage of QC j will be called as response stage, while the stages of its preceding QCs will be called as regressor stages; when talking about the correlation between QCs, the correlation between a regressor and the response will be referred to as regressor–response correlation, while that between two regressors will be referred to as between-regressor correlation. Typically estimation errors in regression are quantified in terms of bias and standard error. Since standard errors are negligibly small in largesample analysis, bias is used as the performance measure in this study. For convenience in the comparison of estimation accuracy under different scenarios, the relative bias as defined below is used

regression which has an analytical form [34]. Therefore, the bias of the TLS estimator can be obtained based on that: TLS 12 →

3.1. Effect of measurement errors in two-stage processes The model of a two-stage process is illustrated in Fig. 3, where only one QC exists in each stage. On the arrow is the associated coefficient in the model which indicates the strength of the correlation between the two QCs. Here the regression X2 –X1 (i.e., X2 is regressed on X1 ) is considered.



−

1 1 + u21 /ε21

(6)

The derivation of (6) can be found in Appendix I. The ratio of the local variation and measurement error in the denominator essentially represents the signal-to-noise ratio (SNR) at the regressor stage since the local variation can be viewed as the signal at that stage. From (6), the following properties can be summarized: (P1) Affecting factors: The bias only depends on the signal-to-noise ratio at the regressor stage (u21 /ε21 ). It has nothing to do with the local variation and measurement error at the response stage. A larger signal-to-noise ratio will lead to a smaller bias, which is consistent with intuition. ˆ OLS | < |ˇ12 |, (P2) Direction of bias: From (6), it is easy to obtain |ˇ 12 where |•| denotes absolute values. In other words, the OLS estimator underestimates the coefficient. (P3) Bias in absence of measurement errors: When the measurement error variance converges to 0, the bias also converges to 0; when there is no measurement error, i.e., in regular regression free of measurement errors, the OLS estimator is unbiased. (P4) Effect of measurement errors: The measurement error variance has positive influence on the bias; the larger the measurement error, the larger the bias. 3.3. TLS estimator For the TLS estimator, though it is not analytically tractable in general, it has been found that, when there is only a single regressor, this estimator is equivalent to the estimator from the orthogonal

−

u21

−

ε21

ε21 − ε22

− ε22 u21



u21

 2 2 + 4ˇ12

⎫ ⎬ (7)

⎭

ε21 − ε22

(8)

u21

The regressor–response correlation affects the bias in a nonlinear way and the bias approaches 0 when this correlation is large, while the quantity in (8) has a monotone decreasing relationship with the algebraic value of the bias. We can also easily find that the bias approaches 0 when this quantity is 0, and the smaller the magnitude of this quantity, the smaller the magnitude of the bias. Also note that unlike in the OLS estimation, the bias is not only related with the local variation and measurement error at the regressor stage, but also related with those at the response stage. (P2 ) Direction of bias: The following facts can be found from (7)

−

⎧ TLS < 04 (overestimation) > 0 12 ⎪ ⎨

ε21

− ε22 u21

⎪ ⎩

In this model, the bias of the OLS estimator is OLS → 12

u22 u21

u22

The derivation of (7) can be found in Appendix II. This estimator bears the following properties: (P1 ) Affecting factors: The bias depends on the regressor–response correlation (ˇ12 ), and the quantity related with the local variations and measurement errors,

u22 u21

3.2. OLS estimator

 2 (ˇ12 + 1) −

2 − 1) + (ˇ12

−

u21

which may be simply referred to as bias sometimes in the following discussion. To enable an easy and smooth reading, only the concluding results are given in the main text, while all details of the derivation are put in Appendices I and II.

2 2ˇ12

 ·



u22

ˆ ˆ ˇ−ˇ ˇ = = ˇ ˇ

1

TLS → 0 (unbiased estimation) = 0 12

TLS > 0 (underestimation) < 0 12

(P3 ) Bias in absence of measurement errors: When there is no measurement error, the quantity in (8) is larger than 0, indicating that overestimation will be caused. This means that the TLS estimator is not unbiased even when measurement errors do not exist. (P4 ) Effect of measurement errors: It is easy to see that the bias only depends on the difference between the measurement errors in the regressor and response, i.e. ε21 − ε22 , rather than their individual magnitudes, and the larger the gap between this difference and local variation in the response, i.e., u22 − (ε21 − ε22 ), the larger the magnitude of the bias. In the special case with equal measurement errors, the bias is

TLS 12 →

1 2 2ˇ12

·

⎧ ⎨

2 (ˇ12 + 1) −

⎩

u22 u21

 −

2 − 1) + (ˇ12

u22 u21

2 2 + 4ˇ12

⎫ ⎬ ⎭

(9) which only depends on the regressor–response correlation and the ratio of local variations. This is a very insightful fact which implies that in this case the bias will be the same regardless of the magnitude of measurement errors. In other words, there is a possibility that the bias may be very small when measurement errors are very large as long as equal measurement errors are present in the regressor and response. We can see that properties (P1 )–(P4 ) of the TLS estimator are very different from properties (P1)–(P4) of the OLS estimator. The most critical difference is that the OLS estimator can never be unbiased if only measurement errors exist, while the TLS estimator can

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be unbiased in the presence of measurement errors under certain conditions. Two such conditions can be identified from (7) Condition1 : u22 = ε21 − ε22 Condition2 :

ε21

=

ε22 ,

u21

=

u22 ,

ˇ12 >> 2

(10)

The second condition represents equal measurement errors, equal local variations, and large regressor–response correlation, which is more common in practice than the first condition. 3.4. Effect of measurement errors in three-stage processes The model of a three-stage process is shown in Fig. 4. Here the regression X3 ∼ X1 + X2 is considered. The bias of the OLS estimator for this regression can be obtained similarly OLS → 13 OLS → 23

2  2 / 2 +  2 / 2 + 1 − (ˇ ˇ /ˇ ) 2 / 2 ˇ12 23 12 13 u1 u1 ε2 u2 ε2 ε1 2  2 / 2 + ( 2 / 2 + 1) · ( 2 / 2 + 1) ˇ12 u1 ε2 u1 ε1 u2 ε2

u21 /ε21 + 1 − ˇ12 ˇ13 /ˇ23 u21 /ε22 2 ˇ12 u21 /ε22 + (u21 /ε21 + 1) · (u22 /ε22 + 1)

(11)

OLS → 23

1 1 + u21 /ε21 1

(12)

OLS 23

measurement

error

in

regressor

2,

i.e.,

2 + ( 2 / 2 ) ˇ12 u2 u1 2 ˇ12

+ (u22 /u21 ) · (1 + u21 /ε21 ) ˇ12 ˇ13 1 →− · 2 ˇ23 ˇ12 + (u22 /u21 ) · (1 + u21 /ε21 )

(14)

Similar to Case III, the measurement error in regressor 1 will also affect the bias of the coefficient of regressor 2. Compared to Case III, there is an additional layer of complexity in the formula of Case IV, that is, the bias not only depends on the SNR of the regressor with measurement error, but also depends on the ratio of local variations of the regressors. This is due to the upstream/downstream relationships of the regressors in the physical layout of the process. Case V: Equal local variations and measurement errors, i.e., ε21 = ε22 = ε2 , u21 = u22 = u2 ,

OLS → 23

2 − ˇ ˇ /ˇ (ˇ12 23 12 13 + 1) + 1 2 ) + 1 2 + (2 + ˇ12 (1 − ˇ12 ˇ13 /ˇ23 ) + 1

(15)

2 ) + 1 2 + (2 + ˇ12

where  = u2 /ε2 is the common SNR of the regressors. Clearly, the bias depends on the SNR and between-stage correlations. 4. Numerical study Extensive simulations have been done to validate the findings in Section 3 as well as show more details on the performance of the two estimators that cannot be seen clearly from the theoretical results. As in the analysis in Section 3, the two-stage and three-stage processes in Figs. 3 and 4 are used as the models to generate data. In the following, the setting of parameters in the simulations is first presented, and then the performance of the two estimators is compared. A summary is given at the end of this section to summarize our findings in theoretical and numerical studies.

1 + u22 /ε22

which is similar to (6). This means that when the regressors are not correlated, the measurement error at each stage will only affect the bias of that particular stage. Case II: No measurement errors in both regressors, i.e., ε21 = 0, ε22 = 0, OLS → 0 13 OLS → 0 23

This means that the OLS estimator is unbiased when measurement errors do not exist. Case III: No measurement error in regressor 1, i.e., ε21 =

/ 0, 0, ε22 = OLS → − 13 OLS → 23

OLS → 13

OLS → 13

The derivation can be found in Appendix II. We can see that the bias has a complex relationship with the between-stage correlations and the local variations and measurement errors at the regressor stages, but has nothing to do with the response stage as in the two-stage case. A facet of the complexity is that the bias not only depends on the SNR at each stage, but also depends on the ratio of the local variation at one stage and the measurement error at the other stage (i.e., the term u21 /ε22 in (11)), which is due to the between-regressor correlation (i.e., ˇ12 ). To obtain some insights on the effect of measurement errors, the biases in a set of special cases are found from the general formula in (11): Case I: Independent regressors, i.e., ˇ12 = 0, OLS → 13

Case IV: No ε21 = / 0, ε22 = 0,

5

ˇ23 ˇ12 1 · ˇ13 1 + u22 /ε22 1

4.1. Parameter setting For the two-stage model, we consider the effect of two factors on the resulting bias of estimation: the true coefficient ˇ12 and the measurement error variance. The purpose is to validate the properties (P1–P4) and (P1 –P4 ) in Section 3.1. To find the effect of ˇ12 , biases under different values of ˇ12 are obtained, with other factors being constant; for the effect of the measurement error variance, equal variance (ε2 ) is assumed at both stages and biases under different values of the variance are obtained. These results are shown in Fig. 5. For the three-stage model, the 5 special cases in Section 3.2 are generated. The performance of the OLS and the TLS estimator in each case is evaluated and shown in Figs. 6–10. The performance in a general case is shown in Fig. 11 to provide additional information. Specific settings of parameters in the simulations can be found in Appendix III.

(13)

1 + u22 /ε22

Both estimates are subject to the measurement error in regressor 2. The bias of ˇ23 is the same as in (6), which is equivalent to conducting regression on regressor 2 alone; the bias of ˇ13 depends on the bias of ˇ23 and all the between-stage correlations. This implies that even if no measurement error is present in one regressor, the estimate of its associated coefficient will still be affected by measurement errors in other regressors due to the betweenregressor relationships.

5. Results Fig. 5 shows the performance of the two estimators in the twostage process. From Fig. 5(a), we can see that the relative bias of the OLS estimator does not depend on the regressor–response correlation (i.e., the true coefficient ˇ12 ), which is consistent with (6); the bias of the TLS estimator decreases as the regressor–response correlation increases, and approaches 0 when the correlation is very large, which is consistent with (8). In terms of the direction of bias, the OLS estimator underestimates the coefficient (indicated by the

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6 1

1

ζ 12 by OLS

0.8

0.8

ζ 12 by TLS

0.6

0.6

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

1

2

3

4

5

6

7

8

9

10

β12

-1

0

0.5

1

1.5

2

2.5

3

σε2

(a)

(b)

Fig. 5. Results of the two-stage process: effect of the true coefficient (a) and variance of measurement errors (b).

Fig. 6. Results of the three-stage process in Case I: effect of measurement errors at Stage 1 (a) and Stage 2(b).

1

ζ13 by OLS

0.8

ζ23 by OLS

0.6

ζ

0.4

ζ23 by TLS

13

by TLS

0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

0.5

1

1.5

2

2.5

3

3

Fig. 7. Results of the three-stage process in Case II.

3.5

4

positive biases), while the TLS estimator overestimates the coefficient. From Fig. 5(b), we can see that when measurement errors do not exist (i.e., ε2 = 0), the OLS estimator has no bias and the TLS estimator has a small bias; the bias of the OLS estimator increases as the measurement error variance increases, while that of the TLS estimator does not change with the variance and large measurement errors lead to the same amount of bias as small measurement errors. Comparing the magnitude of bias in both figures, we can conclude that the TLS estimator outperforms the OLS estimator when the regression–response correlation is large or the measurement errors at the two stages are of equal magnitude, which is consistent with properties (P3–P4) and (P3 –P4 ) in Section 3.1. Fig. 6 shows the effect of measurement errors in the three-stage process in Case I, i.e., with independent regressors. For the OLS estimator, its performance in estimating ˇ23 does not depend on the measurement error at Stage 1 (Fig. 6(a)), and that in estimating ˇ13 does not depend on the measurement error at Stage 2 (Fig. 6(b)). Also, it always causes underestimation. This validates the results in (12). In contrast, the TLS estimator is affected by measurement errors at both stages, and it may cause overestimation or underestimation. In terms of the magnitude of bias, there is no uniform conclusion on the two estimators. The TLS estimator may produce a smaller or larger bias than the OLS estimator.

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Fig. 8. Results of the three-stage process in Case III: effect of measurement errors at Stage 2 (a) and Stage 3(b).

Fig. 9. Results of the three-stage process in Case IV: effect of measurement errors at Stage 1 (a) and Stage 3 (b).

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

ζ13 by OLS

-0.6

ζ23 by OLS ζ

-0.8 -1

ζ 0

0.5

1

1.5

2

13

by TLS

23

by TLS

2.5

Fig. 10. Results of the three-stage process in Case V.

3

Fig. 7 shows the results of the three-stage process in Case II, i.e., without measurement errors at the two regressor stages. Consistent with the theoretical analysis, the OLS estimators are unbiased in this case. However, the TLS estimators are biased and increase with the measurement error variance at the response stage. Particularly, when all the three stages are free of measurement errors (i.e., ε23 = 0), the TLS estimators are still biased. Also, the directions of the bias on the two TLS estimators are different. Fig. 8 shows the results of the three-stage process in Case III, i.e., no measurement error at Stage 1. As predicted by (13), the OLS estimator for each coefficient is affected by measurement errors at Stage 2 and not by measurement errors at the response stage. However, the TLS estimators are affected by measurement errors at both Stage 2 and the response stage. Moreover, the magnitude of the TLS bias is larger than the OLS bias. Fig. 9 shows the results of the three-stage process in Case IV, i.e., no measurement error at Stage 2. Similar to Case III, the OLS estimators depend on measurement errors at Stage 1 but not those at Stage 3, while the TLS estimators depend on both of them. However, in this case, the magnitude of the TLS bias may be smaller than the OLS bias sometimes. Fig. 10 shows the results in Case V, i.e., equal local variations and measurement errors. In this case, the OLS estimators may

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1

0.8

0.8

ζ13 by OLS

0.6

0.6

ζ23 by OLS

0.4

ζ

by TLS

0.4

0.2

ζ23 by TLS

0.2

13

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

-1

0

0.5

1

1.5

2

2.5

3

2

(a)

(b)

Fig. 11. Results of the three-stage process in a general case: effect of measurement errors at Stage 1 (a) and Stage 2 (b).

cause overestimation or underestimation and the magnitude of bias increases with the measurement errors. In contrast, the TLS estimators do not depend on the measurement errors, and keep as a constant when the measurement errors increase. In the presence of large measurement errors, the TLS estimators are uniformly better than the OLS estimators, which is similar to the case in the two-stage process. It is also interesting to find the performance of the two estimators in a general case where measurement errors are present at all the three stages and of different magnitudes. The resulting bias is shown in Fig. 11. We can see that both the two estimators are affected by the measurement errors at each stage; they may cause underestimation when the variance of the measurement errors is in a certain range and cause overestimation when it is beyond that range; sometimes the TLS estimator works better than the OLS estimator and sometimes the contrary. In a word, there is no uniform conclusion on the performance of the two estimators in a general case. 5.1. Summary of findings The results of simulations validate the theoretical analysis in Section 3 and provide more details on the performance of the two estimators, especially the performance of the TLS estimator in the three-stage process. Our findings and implications are summarized in the following. 5.2. Two-stage process Table 1 lists the main differences of the two estimators in the two-stage process. These differences lie in four aspects: (1) In terms of affecting factors, the OLS estimator is only affected by the signalto-noise ratio in the regressor, while the TLS estimator is affected by measurement errors and local variations in both the regressor and response as well as the regressor–response correlation. (2) In terms of direction of bias, the OLS estimator always underestimates the coefficient, while the TLS estimator can underestimate or overestimate the coefficient or have no bias depending on the magnitudes of the variation sources. (3) When there is no measurement error, the OLS estimator is unbiased, while the TLS estimator is biased. (4) The bias of the OLS estimator depends on the measurement error variance in the regressor, and larger variance leads to larger bias, while

the bias of the TLS estimator depends on the difference between the measurement errors in the regressor and that in the response, and larger gap between this difference and the local variation in the response leads to larger bias. The above results suggest that the TLS estimator can be used to replace the OLS estimator in two situations: (i) Equal and large measurement errors in the regressor and response. In the presence of equal measurement errors, the bias of the TLS estimator is a constant however large the variance of the measurement error is. Thus, when serious measurement error is present, the TLS estimator will lead to smaller bias than the OLS estimator. (2) Equal measurement errors, equal local variations and large regressor–response correlation. In this case, the magnitude of the bias of the TLS estimator approaches to zero.

5.3. Three-stage process In general, the performance of the two estimators in the threestage process is complex and no uniform conclusions can be drawn. One source of the complexity lies in the correlation between regressors, which causes the following phenomena: (1) When the regressors are independent, the OLS estimator of their associated coefficients has the same properties as in the two-stage case. In other words, the measurement error in each regressor will only affect its associated coefficient. However, when regressors are not independent, the measurement error in one regressor will affect the estimates of all the coefficients. (2) When there is no measurement error in a regressor, the estimate of its associated coefficient will still be affected by measurement errors in other regressors due to the between-regressor correlation. The performance of the TLS estimator is similar and even more complex, but like in the two-stage case, its resulting bias is stable and smaller than the OLS estimator in the presence of equal and large measurement errors in all the variables. The findings in both the two-stage and three-stage regression suggest that the existence of equal measurement errors is a favored condition for using the TLS estimator especially when the measurement error is large. This is not an uncommon case in practice. For example, if the operations performed and/or equipments used at different stages in a multistage process are similar, the measurement errors are likely to be similar.

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Table 1 Comparison of the OLS estimator and TLS estimator in the two-stage process.

Affecting factors Direction of bias Bias in absence of measurement errors Effect of measurement errors

Ordinary least squares (OLS)

Total least squares (TLS)

Signal-to-noise ratio in regressor

i. Regressor–response correlations ii. Difference of measurement errors in response and regressor iii. Local variations in response and regressor

Underestimate Unbiased

Underestimate, unbiased, or overestimate Overestimate

Larger measurement error leads to larger bias

Larger gap between difference in measurement errors and local variation in response leads to larger bias

Although the above findings are based on two-stage and threestage processes, they are quite useful for practitioners: First, two-stage and three-stage structure is very common in multistage process monitoring. Many multistage manufacturing processes bear a cascade structure, that is, the quality of each stage directly depends on the quality of its preceding stage, which requires a twostage model in monitoring each QC. Examples of such processes can be found in related studies [e.g., 7,8,39,41]. In fact, the well known cause-selecting charts [6,7], which is similar to the regressionadjusted charts, are designed for such processes. For processes with more complex structures, some variable selection methods can be used to identify regressors that have direct influence on the monitored QC among its upstream QCs, and only those regressors will be included in the regression [40]. Such regressors may not be many and thus the two-stage or three-stage model can still be applicable. For example, a case study on a multistage car-body assembly process finds that most QCs have only one directly influential regressor [40]. More importantly, results on the two-stage and three-stage processes provide generic insights on the measurement error problem in multistage process monitoring. Some of those insights include (i) the estimation bias caused by measurement errors depends on the correlations between stages and local variations in a complex way, (ii) even if only one stage is subject to measurement errors, all the correlated stages will be affected, and (iii) The TLS method is only advantageous under some special conditions, and when those conditions are not satisfied, replacing the OLS with it may not work or even cause more serious bias.

6. Conclusion and discussion This study investigates the effect of measurement errors on the estimation of linear regression models which is a built-in element in regression-adjusted multistage process monitoring. The performance of the OLS estimator and the TLS estimator is compared both theoretically and numerically. The findings address the two concerns given in the Introduction: (1) The effect of measurement errors: Measurement errors will lead to bias in the estimation, and the direction and magnitude of the bias depend on correlations between stages, and the local variation and measurement error at each stage in a complex way. The between-stage correlation plays a critical role on this as it propagates the effect of measurement error at each stage to all stages involved in the regression. (2) The OLS vs. the TLS estimator: Compared to the OLS estimator, the effect of measurement errors takes a more complex form for the TLS estimator. An important feature of the TLS estimator is that unlike the OLS estimator, it is not unbiased when there is no measurement error, and can be unbiased when measurement errors exist. Roughly speaking, it can be used to replace the OLS estimator in cases with similar and large measurement errors at different stages of the multistage process. Since it has been found that the estimation bias will degrade the performance of regression-adjusted monitoring [13–16], the findings in this study provide useful guidelines on the

monitoring of multistage manufacturing processes. First, the TLS method should be used under its favored conditions to reduce the estimation bias and consequently the degradation in the performance of monitoring. Second, in cases where those favored conditions are not satisfied, the TLS method may not work better than the OLS method; in other words, whichever method to use, serious estimation bias may exist and thus considerable degradation in the monitoring performance is not avoidable. According to [16], there are two types of degradations: increased false alarming rate and miss detection rate. Specifically, in the presence of measurement errors, there will be a higher chance that an incontrol stage is falsely concluded to be out of control due to changes occurred at its upstream stages (i.e., false alarm) and that an out-ofcontrol stage is mistaken to be in control (i.e., miss detection). Since the defining advantage of regression adjustment is to eliminate the false alarms and miss detections due to the mixing of upstream variations and the local variation at each stage, the value of regression adjustment will be limited. In this case, a computationally simpler alternative, i.e., monitoring each QC directly instead of their residuals, might be considered. Optimal design of such control systems has been provided in the literature [41–43].

Appendix I. Derivation of (6) and (11) In the two-stage model, X = X1 , Y = X2 , the OLS estimator of ˇ12 is ˇ12 u21 ˆ OLS = (X  X)−1 X  Y → Cov(X1 , X2 ) = ˇ 12 Var(X1 ) u21 + ε21 Consequently ˆ OLS → ˇ12 − ˇ 12

ˇ12 u21 u21 + ε21

=

ˇ12 ε21 u21 + ε21

which leads to (6). In the three-stage model, X = [X1 , X2 ] , Y = X3 , the OLS estimator of ␤ = [ˇ13 , ˇ23 ] is ˆ ␤

OLS

=

= (X X)

−1

 X Y =

X1 X2

1 X1 X1 · X2 X2 − X1 X2 · X2 X1





· X1



X2



 −1 X  1 ·

X2

· X3

X2 X2 · X1 X3 − X1 X2 · X2 X3 X1 X1 · X2 X3 − X2 X1 · X1 X3



⎛⎡ var(X ) · cov(X , X ) − cov(X , X ) · cov(X , X ) ⎤⎞ 2 1 3 1 2 2 3 2 (X , X ) var(X ) · var(X ) − cov 1 2 1 2 ⎝ ⎣ ⎦⎠ → var(X1 ) · cov(X2 , X3 ) − cov(X1 , X2 ) · cov(X1 , X3 ) var(X1 ) · var(X2 ) − cov2 (X1 , X2 )

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which means ˆ OLS → var(X2 ) · cov(X1 , X3 ) − cov(X1 , X2 ) · cov(X2 , X3 ) ˇ 13 var(X1 ) · var(X2 ) − cov2 (X1 , X2 )

(A1)

ˆ OLS → var(X1 ) · cov(X2 , X3 ) − cov(X1 , X2 ) · cov(X1 , X3 ) ˇ 23 var(X1 ) · var(X2 ) − cov2 (X1 , X2 ) From the three-stage model in Fig. 4, we can easily find that 2 2 + 2 + 2 var(X2 ) → ˇ12 u1 u2 ε2

var(X1 ) → u21 + ε21 , cov(X1 , X2 ) → ˇ12 u21 ,

cov(X1 , X3 ) → (ˇ13 + ˇ23 ˇ12 )u21

ˇ12 (ˇ13 + ˇ23 ˇ12 )u21

cov(X2 , X3 ) →

(A2)

+ ˇ23 u22

Substituting the above formulas into (A1) yields ˆ OLS → ˇ 13

ˆ OLS → ˇ 23

2  2 +  2 +  2 ) · (ˇ 2 2 2 2 (ˇ12 13 + ˇ23 ˇ12 )u1 − (ˇ12 u1 ) · [ˇ12 (ˇ13 + ˇ23 ˇ12 )u1 + ˇ23 u2 ] u1 u2 ε2 2  2 +  2 +  2 ) − (ˇ  2 ) (u21 + ε21 ) · (ˇ12 12 u1 u1 u2 ε2

2

(u21 + ε21 ) · [ˇ12 (ˇ13 + ˇ23 ˇ12 )u21 + ˇ23 u22 ] − (ˇ12 u21 ) · [(ˇ13 + ˇ23 ˇ12 )u21 ] 2  2 +  2 +  2 ) − (ˇ  2 ) (u21 + ε21 ) · (ˇ12 12 u1 u1 u2 ε2

=

2

=

ˇ13 u21 (u22 + ε22 ) + ˇ23 ˇ12 u21 ε22 2  2  2 + ( 2 +  2 ) · ( 2 +  2 ) ˇ12 u1 ε1 u1 ε1 u2 ε2

ˇ23 u22 (u21 + ε21 ) + ˇ12 (ˇ13 + ˇ23 ˇ12 )u21 ε21 2  2  2 + ( 2 +  2 ) · ( 2 +  2 ) ˇ12 u1 ε1 u1 ε1 u2 ε2

(11) will be obtained through manipulating the terms in the above formulas. Appendix II. Derivation of (7)–(10) The TLS estimator in the two-stage model is [34] ˆ TLS → var(X2 ) − var(X1 ) + ˇ 12



[var(X2 ) − var(X1 )]2 + 4cov(X1 , X2 )2 2cov(X1 , X2 )

Substituting (A2) into the formula, we can obtain



ˆ TLS → ˇ 12

2 − 1) 2 +  2 + ( 2 −  2 ) + (ˇ12 u1 u2 ε2 ε1

2

2 − 1) 2 +  2 + ( 2 −  2 )] + 4ˇ2  4 [(ˇ12 u1 u2 ε2 ε1 12 u1

2ˇ12 u21

Consequently,



ˆ TLS = ˇ12 − ˇ ˆ TLS → ˇ 12 12

=

2 + 1) 2 −  2 − ( 2 −  2 ) − (ˇ12 u1 u2 ε2 ε1

⎧ ⎨

1 · (ˇ2 + 1) − 2ˇ12 ⎩ 12



u22 u21

−

ε21 − ε22

 

u21

2

2 − 1) 2 +  2 + ( 2 −  2 )] + 4ˇ2  4 [(ˇ12 u1 u2 ε2 ε1 12 u1

2ˇ12 u21



2 − 1) + (ˇ12

−

u22 u21

−

ε21 − ε22 u21

 2 2 + 4ˇ12

⎫ ⎬ ⎭

which gives (7). Results under some special cases are as follows: (i)

u2

2 u2 1

−

ε2 −ε2

ˆ TLS ˇ 12 ˇ12

1

u2

2

1

→

= 0, i.e., u22 = ε21 − ε22

1 2 2ˇ12

 ·



 2 (ˇ12 + 1) −

2

2 − 1) + 4ˇ2 (ˇ12 12

=0

(ii) ε21 = ε22 ˆ TLS ˇ 12 ˇ12

→

1 2 2ˇ12

·

⎧ ⎨

2 (ˇ12 + 1) −

⎩

u22 u21

 −

2 − 1) + (ˇ12

u22 u21

2 2 + 4ˇ12

⎫ ⎬ ⎭

which gives (9). (iii) ε21 = ε22 , u21 = u22 ˆ TLS ˇ 12 ˇ12

→

1 2 2ˇ12

 2 · (ˇ12

−

4 ˇ12

2 ) + 4ˇ12

1 = − 2



1 2 ˇ12

1 1 + ≈ − 4 2



1 = 0 when ˇ12 >> 2 4

which gives (10).

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Appendix III. Setting of parameters in the numerical study Two-stage process Fig. 5(a)

Fig. 5(b)

u21 ε21 ε22

ˇ=3 u21 = 3, u22 = 4 ε21 = ε22 = ε2 ∈ [0, 3]

=3 = 0.2, u22 = 4 = 0.4

Three-stage process Fig. 6(a)

Fig. 6(b)

Fig. 7

Fig. 8(a)

Fig. 8(b)

ˇ12 = 0, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 ∈ [0, 5] ε22 = 0.2 ε23 = 0.4

ˇ12 = 0, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 = 0.6 ε22 ∈ [0, 3] ε23 = 0.4

ˇ12 = 2, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 = 0 ε22 = 0 ε23 ∈ [0, 4]

ˇ12 = 2, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 = 0 ε22 ∈ [0, 3] ε23 = 0.4

ˇ12 = 2, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 = 0 ε22 = 0.2 ε23 ∈ [0, 4]

Fig. 9(a)

Fig. 9(b)

Fig. 10

Fig. 11(a)

Fig. 11(b)

ˇ12 = 2, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 ∈ [0, 5] ε22 = 0 ε23 = 0.4

ˇ12 = 2, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 = 0.6 ε22 = 0 ε23 ∈ [0, 4]

ˇ12 = 2, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 = ε22 = ε23 ε21 = ε2 ε2 ∈ [0, 3]

ˇ12 = 2, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 ∈ [0, 5] ε22 = 0.2 ε23 = 0.4

ˇ12 = 2, ˇ13 = 5 ˇ23 = 3, u21 = 5 u22 = 3, u23 = 4 ε21 = 0.6 ε22 ∈ [0, 3] ε23 = 0.4

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