Statistical Monitoring of Multivariate Linear Profiles A.Saghaei1, R. Noorossana2, M. Eyvazian2, A. Vaghefi2
1
Department of Industrial Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran 2 Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran (
[email protected]) Further studies on the linear profiles can be found in [1, 2, 9 and 10]. Multivariate regression models have also been considered to characterize the quality of a process or product. References [5] and [11] discussed a case where a calibration experiment is performed to investigate the relationship between six response variables and six explanatory variables. In this case, six electrical responses are produced by the strain gauges that are proportional to the magnitude and direction of their respective aerodynamic components. A calibration experiment is performed to model the relationship between the applied forces and moments (explanatory variables) and the electrical responses (response variables). Ideally, each response signal would respond to its respective component of load, and it would have no response to other components of the load. However, this is not entirely possible and each response is affected by six components of load and interaction effects of these components. Since there are several response variables explained by several explanatory or independent variables, one can use multivariate regression to appropriately model the process quality characteristics. Although there is a great deal of attention to monitor linear and nonlinear profiles, almost no study has addressed the monitoring of multivariate regression profiles. In this paper, we propose the use of a control chart scheme for phase II monitoring of a process or product that can be adequately modeled by multivariate linear regression. This control chart scheme is based on the first approach proposed by [3]. The rest of this paper is structured as follows. In Section 2, a brief review of multivariate linear regression modeling is given. In Section 3 we propose a control chart scheme for monitoring multivariate linear profiles. In section 4, the performance of the proposed control chart is investigated numerically in terms of average run length criterion. Concluding remarks are provided in the final section.
Abstract - In some quality control applications, quality of a product or process can be represented by a relationship between two or more variables. This relationship is referred to as profile. However, in other practical applications, two or more profiles are required in order to model the quality of a product or process effectively. In this paper, we propose the use of a control chart scheme in phase II monitoring of multivariate linear profiles. The statistical performance of the proposed method is evaluated using a numerical example. The results reveal that the proposed method is relatively effective in detecting shifts in the process parameters. Keywords - Multivariate linear profile, Average Run Length (ARL), Phase I and II
I. INTRODUCTION In most statistical process control (SPC) applications, it is assumed that the quality of a process or product can be adequately represented by the distribution of a univariate quality characteristic or by the general multivariate distribution of a vector consisting of several quality characteristics. However, in some applications, the quality of a process or product is effectively characterized by a functional relationship between a response variable and one or more explanatory variables. This relationship is typically referred to as profile. Some of the practical applications of profile monitoring has been reported by researchers including [1, 3, 5, 7, 8, 9, 11, 13, 14, 15 and 16]. There has been several control chart methods developed to monitor linear profiles. Reference [3] proposed two methods for monitoring simple linear profiles. The first method uses a bivariate T2 control chart to monitor parameters of the regression line representing the linear profile. The second method uses EWMA and R control charts to monitor mean and variance of errors, respectively. Reference [4] coded the x-values to change the average to zero, and then applied three separate EWMA control charts. They showed that their proposed method is superior to the methods recommended by [3]. Reference [6] compared the approaches of [3 and 4] proposed a method based on a F-test technique in phase I monitoring with calibration applications. In addition, References [17] and [5] proposed control charts based on the change-point model for monitoring the linear profiles particularly when the process parameters are all unknown.
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II. MULTIVARIATE LINEAR PROFILE MODELING According to [3], a profile data consists of a set of measurements with a response variable y and one or more explanatory variables xi, i=1,2…n, which are used to assess the quality of manufactured items. However, it can be assumed that the quality of a process or product is characterized by a multivariate linear profile where multivariate refers to the dependent variables. In these
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Proceedings of the 2008 IEEE IEEM
where σuv is the uth row of the vth column of the
kinds of models, several response variables (y’s) are measured corresponding to one or more explanatory variables (x’s). For the sake of simplicity, we assume that all response variables are affected by one explanatory variable. One can extend the proposed method for the case of two or more explanatory variable. Assume that for the kth random sample collected over time, we have the observations ( xik , yi1k , yi 2 k ,..., yipk ),
covariance matrix ∑ . Proofs are provided in appendix A. III. MULTIVARIATE LINEAR PROFILE MODELING In this section, we present a control chart scheme based on the first method proposed by [3]. For the kth sample, (βˆ k − β k ) T is a 1× 2 p multivariate normal random vector with mean vector zero and known covariance matrix ∑ β . Therefore, the kth chart statistics is
i = 1,2,..., n , where n is the sample size and p is the number of response variables. The relationship between the observations, when the process is in control, can be represented as follows: y ik = β 0 + xi β1 + ε ik , i = 1,2,..., n , (1) where y ik is a 1×p vector of response variables for the ith observation of the kth sample, β 0 and β1 are 1×p vectors of intercept and slope parameters, respectively, and ε ik is a 1×p vector of errors for the ith observation of the kth sample, which is a multivariate normal random vector with mean vector zero and known covariance matrix ∑ . Equation (1) can be rewritten as follows: Yk = XB + E k , (2)
where Yk = ( y1k , y 2 k ,..., y pk ) T
is a n × p
given by Tk2 = (βˆ k − β k ) T ∑ −β1 (βˆ k − β k ) .
In this case, when the process is in-control, the sample statistics is a chi-square random variable with 2p degrees of freedom. Thus, a 100(1 − α ) percentile of the chi-square distribution with 2p degrees of freedom can be used to construct an upper control limit represented by UCL = χ 22 p ,α .
matrix of
IV. PERFORMANCE OF THE PROPOSED METHOD
th
response variables for the k sample, and X = [1 x] is a
In this section, ARL performance of the control chart scheme discussed in Section 3 is calculate through the use of a Mont Carlo simulation. The control chart scheme is designed to have the in-control ARL of approximately 200. In our simulation study, 5000 replications have been used to estimate all ARL values. Note that although our analyses for all the cases allow for multivariate implementation, without loss of generality, we consider only bivariate case ( p = 2) . The underlying multivariate profile model, which we use in this paper, is Y1 = 3 + 2 x + ε 1 , Y2 = 2 + 1 x + ε 2 , (17) where xi -values are set equal to 2, 4, 6, and 8 (with x = 5 ). The vector of error terms (ε1 , ε 2 ) is a bivariate normal random variable with entries of the mean vector equal zero and known covariance matrix ⎡ σ2 ρσ 1σ 2 ⎤ ∑=⎢ 1 ⎥. σ 22 ⎥⎦ ⎢⎣ ρσ 1σ 2
n × 2 matrix of explanatory variables. B = (β 0 , β1 ) T is a T
2 × p matrix of parameters and E k = (ε1k , ε 2 k ,..., ε pk ) is a
n × p matrix of error terms. For the kth sample, the least
squares estimator of B = (β 0 , β1 ) T , which minimizes (Yk − XBˆ )T (Yk − XBˆ ) , is given by Bˆ k = (βˆ 0 k , βˆ 1k ) T = (X T X)-1 X T Yk . The elements of Bˆ are defined as follows:
(3)
k
βˆ0 jk = y. jk − βˆ1 jk x and βˆ1 jk = S xy ( j ) / S xx , j = 1, 2, …, p (4) n
n
n
i =1
i =1
where y. jk = 1/ n∑ yijk , x = 1 / n∑ xi , S xy ( j ) = ∑ ( xi − x ) yijk i =1
n
and S xx = ∑ ( xi − x ) . 2
i =1
Matrix Bˆ k can be rewritten as a 1× 2 p vector denoted by βˆ k T , where T βˆ k = ( βˆ01k , βˆ02 k ,..., βˆ0 pk , βˆ11k , βˆ12 k ,...βˆ1 pk )
(5)
is a multivariate normal random vector, with the mean vector T β k = ( β 01k , β 02 k ,..., β 0 pk , β11k , β12 k ,...β1 pk ) (6) and 2 p × 2 p covariance matrix ∑ β . The elements of ∑ β are given by the following formulas: Cov ( βˆ0uk , βˆ0vk ) = σ uv (1 / n + x 2 / S xx ) = ρ uvσ uσ v (1 / n + x 2 / S xx ) Cov ( βˆ1uk , βˆ1vk ) = σ uv / S xx = ρ uvσ uσ v / S xx Cov ( βˆ0uk , βˆ1vk ) = −σ uv x / S xx = − ρ uvσ uσ v x / S xx
,
(8)
and (7)
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In this paper, we assume that σ 12 = 1 and σ 22 = 1 . In order to investigate the effects of correlation between response variables, we use different values of ρ, namely, ρ = 0.1 , 0.5 and 0.9. The upper control limit can be found as UCL = 14.86 to give an in-control ARL of roughly 200. Reference [4] stated that the exact ARL of T 2 control chart could be determined analytically. However, they used simulation to find the ARL performance.
Proceedings of the 2008 IEEE IEEM
Three different types of shifts are considered in our simulation study. Table 1 gives the ARL values for shifts in the β 01 (intercept of the first profile) in units of σ 1 . Table 2 gives the ARL values for shifts in β11 (the slope of the first profile) in units of σ 1 , and in Table 3, we consider the case when there is a shift in σ 1 . Simulation studies show that the proposed method is relatively effective in detecting shifts in the process parameters. Notice that the control chart scheme performs better when ρ increases. We found similar results for the sustained shifts in the parameters of the second profile. Compared to the T2 control chart presented by [3], the proposed method performs relatively better when the correlation between responses are high. However, if the correlation is
moderate, both methods have the same performance. In the case that the correlation between responses is low, the performance of the T2 control chart method suggested by [3] is superior to the proposed method. It is also worth mentioning that, the EWMA control chart schemes proposed by [3 and 4] can easily be extended to the case of monitoring multiple response profiles. V. CONCLUSION In this paper, a control chart method was proposed for monitoring multivariate linear profiles. This method is an extension of the first method proposed by [3]. Simulation studies reveal that the proposed method is relatively TABLE I
THE SIMULATED ARL VALUES WHEN
β 01 SHIFTS TO β 01 + λ0σ 1
(IN-CONTROL ARL = 200)
λ0 ρ
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.1
157.0
85.8
41.6
20.8
10.5
5.8
3.5
2.4
1.7
1.4
0.5
145.5
69.6
30.8
13.5
6.7
3.7
2.4
1.7
1.3
1.1
0.9
71.4
14.0
3.8
1.7
1.2
1.0
1.0
1.0
1.0
1.0
TABLE II THE SIMULATED ARL VALUES WHEN
β 11 SHIFTS TO β 11 + λ1σ 1
(IN-CONTROL ARL = 200)
λ1 ρ
0.025
0.025
0.025
0.025
0.025
0.025
0.025
0.025
0.025
0.025
0.1
177.8
177.8
177.8
177.8
177.8
177.8
177.8
177.8
177.8
177.8
0.5
171.7
114.0
67.0
37.9
21.1
12.4
7.7
4.9
3.4
2.5
0.9
114.4
38.1
12.7
5.1
2.5
1.6
1.2
1.1
1.0
1.0
TABLE III THE SIMULATED ARL VALUES WHEN
σ 1 SHIFTS TO γσ 1
(IN-CONTROL ARL = 200)
γ ρ
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
0.1
57.3
21.6
11.1
7.0
4.8
3.6
3.0
2.6
2.3
2.1
0.5
53.8
19.8
10.0
6.1
4.2
3.3
2.7
2.3
2.0
1.9
0.9
33.5
9.5
4.5
2.9
2.2
1.8
1.6
1.5
1.4
1.3
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Proceedings of the 2008 IEEE IEEM
Monitoring of Linear Calibration Profiles.“ Inl. J. Prod. Res., vol. 44, pp. 1927-1942. [2] Jensen, W.A., Birch, J.B., Woodall, W.H. (2008) “Monitoring Correlation within Linear Profiles Using Mixed Models”. To appear in J. Qual. Technol . [3] Kang, L., and Albin, S.L. (2000). “On-Line Monitoring When the Process Yields a Linear Profile.“ J. Qual. Technol., vol. 32, pp. 418-426. [4] Kim, K., Mahmoud, M.A., Woodall, W.H. (2003). “On the Monitoring of Linear Profiles.” J. Qual. Technol., vol. 35, pp. 317-328. [5] Mahmoud, M.A., Parker, P.A., Woodall, W. H., and Hawkins, D. M. (2007). “A Change Point Method for Linear Profile Data.” Qual Reliab Eng Int., vol. 23, pp. 247-268. [6] Mahmoud, M.A., and Woodall, W.H. (2004). “Phase I Monitoring of Linear Profiles with Calibration Application.” Technometrics, vol. 46, pp. 380-391. [7] Mestek, O., Pavlik, J., and Suchánek, M. (1994). “Multivariate Control Charts, pp. Control Charts for Calibration Curves.” Fresenius J. Anal. Chem., vol. 350, pp. 344-351. [8] Montgomery, D.C. (2005). Introduction to Statistical Quality Control. New Jersey, John Wiley & Sons. [9] Noorossana, R., Amiri, A., Soleimani, P. (2008). “On the Monitoring of Autocorrelated Linear Profiles.” Commun. Statist-Theory Meth., vol. 37, pp. 425.442 [10] Noorossana, R., Amiri, A., Vaghefi, A., and Roghanian, E. (2004). “Monitoring Quality Characteristics Using Linear Profile”. Proc. 3rd Conf. International Industrial Engineering, Tehran, Iran. [11] Parker, P.A., Morton, M., Draper, N. R., Line, W. P. (2001), “ A Single-Vector Force Calibration Method Featuring the Modern Design of Experiments,” paper presented at the American Institute of Aeronautics [12] Stover, F.S., and Brill, R.V. (1998). “Statistical Quality Control Applied to Ion Chromatography Calibrations.” J. Chromatogr. A, vol. 804, pp. 37.43. [13] Wang, K., and Tsung, F. (2005). “Using Profile Monitoring Techniques for a Data-Rich Environment with Huge Sample Size.” Qual Reliab Eng Int., vol. 21, pp. 677.688. [14] Williams, J. D., Woodall, W. H., and Birch, J. B. (2007). “Statistical Monitoring of Nonlinear Product and Process Quality Profiles.” Qual Reliab Eng Int., vol. 23, pp. 925941, [15] Woodall, W.H, Spitzner, D.J., Montgomery, D.C., and Gupta, S. (2004). “Using Control Charts to Monitor Process and Product Quality Profiles.” J. Qual. Technol., vol. 36, pp. 309-320. [16] Woodall, W.H. (2007). “Current Research on Profile Monitoring.” Revista Producao, vol. 17, pp. 420-425. [17] Zou, C., Zhang, Y., and Wang, Z. (2006). “Control Chart Based on Change-Point Model for Monitoring Linear Profiles”. IIE Transactions, vol. 38, pp. 1093-1103.
effective in detecting shifts in the process parameters. In addition, the control scheme performance improves when ρ increases. APPENDIX A: CALCULATING THE ELEMENTS OF Σβ According to Section 2, βˆ0uk and βˆ0vk are the least square estimators of the intercepts for the uth and vth profile, respectively. The covariance of βˆ0uk with βˆ0vk is given as follows: Cov ( βˆ0uk , βˆ0vk ) = Cov ( y.uk − βˆ1uk x , y.vk − βˆ1vk x ) n 1 n 1 (x − x)x (x − x)x ) yiuk , ∑ ( + i ) yivk = Cov(∑ ( + i S xx S xx i =1 n i =1 n
)
n 1 n 1 (x − x)x 2 (x − x)x 2 ) Cov( yiuk , yivk ) = σ uv ∑ ( + i ) = ∑( + i S xx S xx i =1 n i =1 n
1 x2 1 x2 ) = ρ uvσ uσ v ( + ), = σ uv ( + n S xx n S xx
(A1)
where ρuv = σ uv /(σ u .σ v ) and σ uv is the uth row of the vth column of the covariance matrix ∑ . In addition, βˆ1uk and βˆ1vk are the least square estimators of the slopes
for the uth and vth profile, respectively. The covariance of βˆ1uk with βˆ1vk is given as follows: S xy (u ) S xy ( v ) Cov ( βˆ1uk , βˆ1vk ) = Cov ( , )= S xx S xx 1
n
n
i =1
i =1
Cov (∑ ( xi − x ) yiuk , ∑ ( xi − x ) yivk ) 2
( S xx ) n
=
=
∑ ( xi − x ) 2 Cov( yiuk , yivk ) i =1
( S xx ) 2
σ uv
=
S xx
ρ uvσ uσ v S xx
n
=
σ uv ∑ ( xi − x ) 2 i =1
( S xx ) 2
(A2)
,
the covariance of βˆ0uk with βˆ1vk is also given as follows: Cov ( βˆ0uk , βˆ1vk ) = Cov ( y.uk − βˆ1uk x , βˆ1vk ) = Cov ( y.uk , βˆ1vk ) − σ uv
x S xx
n 1 x 1 yiuk , ∑ ( xi − x ) yivk ) − σ uv S xx i =1 n i =1 S xx n
= Cov(∑ n
=
σ uv ∑ ( xi − x ) i =1
n ( S xx )
− σ uv
x x x = 0 − σ uv = −σ uv . S xx S xx S xx
(A3)
REFERENCES [1] Gupta, S., Montgomery, D.C., Woodall, W.H. (2006). “Performance Evaluation of Two Methods for Online
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