THE ALGORITHM AND DESIGN FOR REAL-TIME HOTELLING’S T2 AND MEWMA CONTROL CHART IN MSPC Lau Meng Cheng, Yuwaldi Away, Mohammad Khatim Hasan Department of Industrial Computing Faculty of Information Science and Technology National University of Malaysia Email: [email protected], yuwaldi, [email protected]

Abstract This paper deals with the area of the process monitoring and control in multivariate quality with the proposed algorithm and design. The proposed architecture design unifies two existing MSPC techniques which are Hotelling’s T2 and MEWMA control charts that features a helpful graphical user interface (GUI). An architecture of real-time multivariate statistical process control (MSPC) system and a algorithm is also propose in this paper for subgroup data and individual observation scenarios. With the real-time multi-channel hardware sensors (Vernier LabPro) that interfaced directly to a client platform via universal serial bus (USB) or RS232 serial, it is dedicated to monitoring and controlling the multivariate process with this real-time data collecting. Key Words: MSPC (multivariate statistical process control), Hotelling’s T2, MEWMA (multivariate exponentially weighted moving average, real-time.

1. Introduction Generally there are two groups of statistical process control (SPC), i.e. univariate statistical process control (USPC) and multivariate statistical process control (MSPC) which are used for different scenarios. The process monitoring and control primarily apply to the systems or processes from the univariate perspective, which has only one process output variable or quality characteristic measured and tested. One of the disadvantages of the USPC scheme is that for a single process, many variables may be monitored and even controlled. MSPC methods overcome this disadvantage by monitoring several variables simultaneously. The first original study in multivariate quality control was introduced by Hotelling [3]. Three of the most popular multivariate control statistics are Hotelling’s T2, the MEWMA (Multivariate Exponentially-Weighted Moving Average) and the MCUSUM (Multivariate Cumulative Sum). This paper covers both the system development for MSPC and a discussion on some of the techniques currently available. The topics covered by this paper are the algorithms for generating of the Hotelling’s T2 and MEWMA for the difference shift size.

2. Literature review Figure 1 and Figure 2 are the good examples of the advantages of MSPC. Figure 1 shows a superimpose univariate control charts on top of each other and all the points of each control chart in an area of space. This figure shows a scatter plot of multivariate data composed of two variables. The individual control limits for each variable's respective univariate chart are shown in the control rectangle. Since the data points fall within the control rectangle, this particular pattern shows that the process is in-control for each individual variable [7], [14]. However, this method is not very accurate for processes monitoring when the variables are correlated, because relationships between the variables are not capitalized upon and the probability of both charts simultaneously plotting in control

is not 1 − α . If a process is in-control, the probability of p means plotting in control is ( 1 − α )p. Thus, the joint probability of type I error is much larger: ( 1 − α )p [1]. In Figure 2, the same data points plotted on Figure 1 are used, but a control ellipse has been superimposed over the data points. The control ellipse is the true control region for the variables. The point in the upper left corner actually falls out of the control ellipse making it either an assignable cause of variation or a false alarm. The control ellipse takes advantage of the relationships between the variables in the process resulting in the tilt shown. The more correlated the variables are with each other, the more tilted the ellipse will be away from the x1 and x2 axes [8].

Out of control point

UCL for X

X1

X1

Control limit

LCL for X1

LCL for X2

X2

X2

UCL for X2

Figure 1 : Univariate model with control rectangle

Figure 2 : Multivariate model with control ellipse

2.1 Hotelling’s T2 chart The Hotelling’s T2 is the most familiar multivariate process monitoring and control procedure for monitoring the mean vector of the process [11]. When the real population value is unidentified, the Hotelling’s T2 control charts will be used [3]. There are two kinds of Hotelling’s T2 control charts, one for subgroup data and another for individual observation. The formulas use to compute the Hotelling’s T2 chart are as in (1). There are two distinct phases of control chart usage, which are phase 1 for obtain an in-control set of observation so that control limit can be established and phase 2 for monitoring of future production (retrospective analysis) [1] as in (2). T2 = n ( x − x )’ S-1 ( x − x ) for subgroup data T2 = ( x − x )’ S-1 ( x − x ) for individual observation With

 x1  x  2 x=  M     x p 

and

 S12   S=    

S12

S13

2 2

S 23

S

S 32

L S1 p   L S2 p   L  O M  S p2 

n: sample size S: covariance matrices

(1)

In this literature review, when the mean and covariance are estimated from a large number of preliminary sample (m > 20), it is customary to use UCL = χ α2 , p as the upper control limit in both phase 1 and phase 2 [11]. Hotelling’s T2 can also be treated as a standardized univariate Shewhart control chart [6]. So a simple way to analyze the result is to use UCL= T 2 + 3S T 2 as used in analyzing traditional normal distribution. Phase 1:

p (m − 1)(n − 1)

UCL =

mn − m − p + 1

n : sample size m: subgroup number p : number of variables

Fα , p ,mn − m − p +1

LCL = 0 Phase 2:

p (m + 1)(n − 1)

UCL =

mn − m − p + 1

Fα , p ,mn − m − p +1

LCL = 0

(2)

2.2 MEWMA chart The Hotelling’s T2 control chart is a Shewhart-type control chart, thus it uses information only from the current sample, so consequently, they are relatively insensitive to small and moderate shifts in the mean vector. EWMA control chart is developed to provide more sensitivity to small shift in the univariate case, and it can be extended to multivariate quality control problems [11]. Lowry et al. [5] have developed a multivariate version of the univariate EWMA, MEWMA control chart which is defined as in (3). In this paper we proposed a simple estimated upper control limit for MEWMA control chart UCL = M + 3S M . for subgroup data:

Z i = λxi + (1 − λ ) Z i −1

for individual observation:

Where : 0 < λ ≤ 1 dan Z0 = 0

Z i = λxi + (1 − λ ) Z i −1

Covariance matrices:

with

∑

Mi = Z

' i

∑

−1 Zi

Zi

Zi

=

λ

[1 − (1 − λ ) ] ∑ (2 − λ ) 2i

(3)

3. Design and methodology 3.1 Proposed system architecture The conceptual model in Figure 3 depicts the MSPC system that we proposed that consist of several modules including real-time data collection module, graphical module and data storage module.

• •

•

Real-time data collection module: A personal computer (client) connected to a real time data collector (hardware interfacing) and multi-channel sensors for multivariate data collecting. Graphical module: A computer system that presents the information using a graphical user interface (GUI) that was designed and developed using Visual Basic programming to allow users to interact with the multivariate control chart (Hotelling’s T2 and MEWMA control charts that computed and generated in client) and online printer (a printing device to print the control charts and reports for further analysis). Data storage module: Consist of a file server station with a MSPC database (an dynamical design Access file format database) for system setting information, multivariate data storing and data retrieving.

1. Real-time data collection module (input) Real-time hardware interface Vernier LabPro Sensor connections

Sensor

Collecting real-time multivariate data (via USB/ or RS232 serial)

3. Data storage module

system coordinator

` File Server

Client Generating the multivariate control chart and report

Printing the multivariate control chart and report

MSPC Database

A file server with a database for real time multivariate data storing

UCL T2

Data

Multivariate Control Chart

Online Printer

2. Graphical module (Output)

Figure 3 : Conceptual model of the system

3.2 System flow and methodology In this paper we proposed the algorithms for multivariate control chart computing and generating. Figure 4 depicts several steps for multivariate chart selection (Hotelling’s T2 and MEWMA). The algorithms start with the real time multivariate data collection from any devices which can send and integrated with the client application. The algorithms will start the computing for control chart when the collected shift size m > 10. The shift size will be considered for the chart selection where the Hotelling’s T2 is insensitive to small shift of samples [11]. The n is the sample size that decides the

type of a process either with subgroup data or individual observation.

Figure 5 and Figure 6 are the continuous flowchart from Figure 4 (A and B). Figure 5 flowchart depicts the algorithm to compute the Hotelling’s T2 chart module in subgroup data case and Figure 6 for individual observation case. The inverse of covariance matrix, S-1; ( x − x ) matrices (A); transpose of matrices ( x − x ) , ( x − x ) matrices (B), transpose of ( x − x ) and the upper control limit are computed to finalize a Hotelling’s chart.

Start

Real time multivariate data No m > 10

Yes

Compute the Hotelling’s T 2

A

Yes

Sample size, n>1

No

No

Small shift size

B

C

Compute The MEWMA

Yes

Sample size, n >1

Yes

No

D

Figure 4 : Flowchart for multivariate charts selection

Set the matrices, A

Compute the sample mean, x ik

Inverse the covariance matrix, S-1

Hotelling’s T2 Statistic,

n( x − x ) ' S −1 ( x − x )

Compute the sample mean average,

xi

Set the covariance matrix, S

Set the sample mean and sample mean average matrices, &

x

x

Compute the average of variance & covarance, s & s i

ij

Compute the Hotelling’s T2 mean and standard deviation, µ &σ

Finish

(x − x )

and the transpose matrices

( x − x )'

Compute the sample variance & covariace, sik & sijk

Compute the Upper Control Limit,

UCLT 2 = µ + 3σ Hotelling’s T2 control chart

Figure 5 : Flowchart for computation the Hotelling’s T2 module with sample size, n > 1

Set the matrices, B

( x − x)

Set the sample data and sample mean matrices, x& x

Compute the sample mean,

xi

Compute the Hotelling’s T2 mean and standard deviation, µ &σ

( x − x) '

Hotelling’s T2 Statistic, −1

n(x − x) S (x − x) '

Compute the Upper Control Limit,

Inverse the covariance matrix, S-1

Hotelling’s T2 control chart

UCLT 2 = µ + 3σ

Compute the sample variance & covariace, sik & sijk

and the transpose matrices

Set the covariance matrix, S

Finish

Figure 6 : Flowchart for computation the Hotelling’s T2 module with sample size, n = 1

Figure 7 and Figure 8 are the continuation of the flowchart from Figure 4 which consist C and D for the MEWMA chart computing algorithms. Both of these algorithms need to compute the inverse of the covariance matrices from the covariance matrix of the input data and the composition matrices.

C

C om pute the sam ple m ean,

Com pute and set the M EW MA vector m atrice,

x ip

C om pute the sam ple variance & covariace, Σ ip & Σ ijp

Com pute the average of variance & covarance, Σ & Σ i

Z i = λ x ip + (1 − λ ) Z i −1

Compute the U pper C ontrol Limit,

U CL M EW M A = M + 3 S

MEW M A control chart

Z i' Compute the composition value,

Set the Com position Matrices,C

λ

∑i

Com pute the M EW MA m ean and standard deviation , M & S

[ 1 − (1 − λ ) ] 2i

(2 − λ )

Set the covariance m atrices,

ij

Transpose the M EW MA vector matrices,

Inverse the covariance m atrices,

Σ i−1

MEW M A Statistic,

M i = Z i' Σ −Z1i Z i

Finish

Figure 7 : Flowchart for computation the MEWMA module with sample size, n > 1

D

Compute and set the MEWMA vector matrice,

Set the data matrice,

xip

Zi = λxip +(1 − λ)Zi−1

Compute the data variance & covariace, Σip & Σijp

Set the Composition Matrices,C

Inverse the covariance matrices,

Set the covariance matrices,

Σi−1

∑i

Finish

MEWMA control chart

Compute the Upper Control Limit,

UCLMEWMA = M + 3S

Transpose the MEWMA vector matrices,

Zi' Compute the composition value,

λ

[ 1− (1− λ) ] 2i

(2 − λ)

MEWMA Statistic,

M i = Zi' Σ−Z1i Zi

Compute the MEWMA mean and standard deviation, M & S

Figure 8 : Flowchart for computation the MEWMA module with sample size, n = 1

The MEWMA vector matrices are computed in both of the algorithms (C and D) but for the individual observation, individual data are used but in the subgroup data case the computed data mean is used. Last, the MEWMA statistic and the upper control limit are computed.

4. Conclusion and further research The proposed algorithms are verified with the data (small shift) from Lowry et. al., 1992 for MEWMA and the data (large shift) from Amitava [2] for Hotelling’s T2 which are depict in Figure 9 and Figure 10. In this paper we used summation of mean and three standard deviations as the control limit for both Hotelling’s T2 and MEWMA control charts. Figure 9 shows that the Hotelling’s T2 control chart is more sensitive to the data from Amitava compares with the MEWMA control chart. However, Figure 10 shows that the MEWMA are more sensitive when the data from Lawry et. al. [5] are used. Currently, we are working on the integration and hardware interfacing of the real time device (Vernier LabPro) that can provides the multivariate data with the system on multi-channel sensors.

5. Acknowledgments This work is sponsored Ministry of Science, Technology and Innovation of Malaysia (MOSTI) under the Intensification of Research in Priority Areas (IRPA) for the project IRPA 04-02-02-0011-EA011.

Figure 9: The computed Hotelling’s T2 and MEWMA chart with Amitava’s data

Figure 10: The computed Hotelling’s T2 and MEWMA chart with Lowry et al.’s data

6. References [1] Alt, F.B. Multivariate Quality Control: State of the Art. Transactions of the 1982 ASQC Quality Congress, 1982, pp. 886-893. [2] Amitava, M. Fundamentals of Quality Control and Improvement. New York: Macmillan Publishing Company, 1993. [3] Hotelling, H.H. Multivariate Quality Control Illustrated by the Air Testing of Sample Bombsights. Techniques of Statistical Analysis, 1947, pp. 111-184. [4] Linderman, K., Love, T. E. Economic and Economic Statistical Designs for MEWMA Control Chart. Journal of Quality Technology, 2000, Vol. 32, No. 4, pp. 410-417. [5] Lowry, C.A., Woodall, W.H., Champ, C.W., Rigdon, S.E. A Multivariate Exponentially Weighted Moving Average Control Chart. Technometrics, 1992, Vol.34, pp.46-53. [6] Mason, R.L., Tracy, N.D., Young, J.C. Decomposition of T2 for Multivariate Control Chart Interpretation. Journal of Quality Technology, 1995, Vol. 27, No.2, pp. 99-108. [7] Mastrangelo, C.M., Runger, G.C., Montgomery, D.C. Statistical process monitoring with principal component. Quality and Reliability Engineering International, 1996, Vol. 29, No. 4, pp. 203210. [8] Mastrangelo, C.M.. Multivariate Statistical Process Control. Department of Systems Engineering University of Virginia. (online) http://www.sys.virginia.edu/mqc/, 18 December 2002. [9] Molnau, W. E., Runger, G. C., Montgomery, D. C., Skinner K. R., Loredo, E. N., Prabhu, S. S. A Program for ARL Calculation for Multivariate EWMA Charts. Journal of Quality Technology, 2001, Vol. 33 No. 4. [11] Montgomery, D.C. Introduction to Statistical Quality Control, Ed. Ke-4. United States of America: John Wiley & Sons, Inc, 2001. [12] RAID A. AL-IMAM. Neuro-Fuzzy Model for Classifying and Monitoring Multivariate Attribute Proceses. Agilent Technologies, 2002. [13] S. ÜMÝT OKTAY FIRAT RES, ÇÝÐDEM ARICIGÝL. Multivariate Statistical Process Control Methods and New Approaches. The 6th TQM World Congress, 2001, pp. 239-246. [14] Tracy, N.D., Young, J.C., Mason, R.L. Multivariate control chart for individual observation. Journal of Quality Technology 1992, Vol. 24, No. 2, pp. 88-95.