MULTIVARIATE SHEWHART QUALITY CONTROL FOR STANDARD DEVIATION

Transcription

1 MULTIVARIATE SHEWHART QUALITY CONTROL FOR STANDARD DEVIATION M. Jabbari Nooghabi, Deartment of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad-Iran. and H. Jabbari Nooghabi, Ph.D. Deartment of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad-Iran. ABSTRACT In univariate quality control, the S Chart have been useful in determining whether the rocess disersion is in-control or not. It would be very useful to have a similar chart alied to the multivariate case. The existing methods do not rovide all the information that a quality control ractitioner would like to ossess such as the indication of which variables are causing the rocess to be out-of-control. In this aer, we roose a method which allows us to simultaneously control the overall rocess quality characteristics and to identify the resonsible variables leading to an out-of-control condition. This method is based on the adequate selection of the symmetric square root of the correlation matrix. The associated critical region is also discussed. The rocess considered is assumed to be multivariate normal with arameters known from historical data or estimated from a large samle. We call this method, "Multivariate Shewhart Chart (MS Chart", because it reduces to the Shewhart Chart when the rocess involves only one variable. The rocedure has been illustrated with the hel of two examles. Key Words: Multivariate Quality Control, Multivariate Shewhart Chart, Symmetric Square Root, and Critical Region.

2 INTRODUCTION Quality control roblems in industry may involve more than a single quality characteristic, i.e. a vector of characteristics, and when these characteristics are correlated, a more aroriate aroach would be to monitor them simultaneously. This has formed the basis of extensive work erformed in the field of multivariate quality control. Shewhart, who is famous for the develoment of the statistical control chart (Shewhart Charts first recognized the need to consider quality control roblems as multivariate in character. The general multivariate statistical quality control roblem considers a reetitive rocess where each item is characterized by -quality characteristics, X, X,..., X. Because of the chance causes inherent in the rocess, these quality characteristics are random variables. Because of the indeendency between the characteristics, the random variables, are correlated. The roblem thus requires a multivariate aroach. The underlying robability distribution of the quality characteristics is assumed to be multivariate normal with mean vector µ and covariance matrix Σ. The multivariate aroach to quality control was first widely ublicized in 97 and 95 by Hotelling [6] in the testing of bombsights. In a set of related aers, Jacson [7, 8] and Jackson and Morris [] use an ellitical control region and extended Hotelling s rocedure for use the rincial comonents in monitoring a hotograhic rocess. Ghare and Torgersen [], Alt [, 5], and Alt et al. [6] examined the simultaneous control of several related variables when the data is in the form of rational subgrous. Some of the conclusions and results resented in the aers by Jackson [7, 8] and those discussed in Ghare and Torgersen [] require certain alterations. Secific details are in Alt [, 5]. Alt and Deutsch [7] determined the aroriate samle size and control chart constant by extending the univariate scheme of to multivariate data. Alt et al. [8] develoed control charts for when there is correlation across the data vectors as well as within each vector. In the univariate case, the rocess disersion is monitored by sigma charts or range charts. Alt [] and Alt et al. [] develo and resent the multivariate counterarts. There are two distinct hases of control chart ractice. Phase I consists of using the charts for ( retrosectively testing whether the rocess was in control when the first subgrous were being drawn and ( testing whether the rocess remains in control when future subgrous are drawn. These are two searate and distinct stages of analysis. Phase II consist of using the control chart to detect any dearture of the underlying rocess from standard values µ,. In this aer, we consider Phase II. ( Σ Notations and Assumtions µ -Mean vector, σ -standard deviation, Σ -covariance matrix, R -correlation matrix, R - square root of the covariance matrix, X - transose of X, χ n, α -chi square value. The rocess considered, is assumed to be multivariate normal and rocess arameters, µ and Σ known or estimated from a large samle.

3 a. b.. ¹. CONTROL SHEWHART CHARTS FOR DISPERSION ( MS CHARTS It was assumed that the rocess disersion remained constant at Σ. This assumtion must be validated in ractice; methods for investigating it are resented in this section. Several different control charts for rocess disersion will be resented since different statistics can be used to describe variability. To lay the groundwork for the develoment of control charts for multivariate data, first consider the case of one quality characteristic. Let S denote the (unbiased samle variance for a random samle size n from a rocess. When the rocess variance is σ, then ( n S ~ χ n, and an S -chart is σ obtained by ivoting on this exression. The control limits are resented as Casea in Table. For successive random samles of size n, this control chart can also be viewed as reeated tests of significance of the form H : σ = σ vs. H : σ σ. A control chart for S is obtained by taking the square root of these limits (Table, Case b. Both charts are resented in Guttman and Wilks []. Table : Univariate Disersion Control Charts (Phase II Statistics LCL CL UCL n ( X i i X S = ( n = σ χ n, ( α / /( n - σ χ n, α / /( n S χ n, ( α / S = σ /( n - σ /( n.5 S = S max{, σ [ c ( c ]} χ n, α /.5 σ c σ [ c + ( ] c S = S B S S B S. V =.5 n max{, σ [ c n ( n nc n = ( X i i X or max{, σ B} ¹. Refer to Aendix..5 ]} σ c.5 σ [ c + n ( n nc or σ B The next two rocess disersion charts do not utilize the comlete distributional roerties of the samle statistic but only the first two moments. It can be shown that E ( S = σ c and Var ( S = σ ( c, where tables of c for n=,,5 are given in Johnson and Leone []. Since most of the robability distribution of S is contained in the interval E ( S ± Var( S, it seems reasonable that a control chart for S would have control limits corresonding to this interval (Table, Case. The lower control limit is relaced by for n < 6. Since S is not normally distributed, these control limits can not be thought of as robability limits []. The same rationale alies in develoment of what is traditionally referred to as the sigma chart, excet that V is used in lace of S, where V = n i= ( X n X are available in Duncan []. i (Table, Case. Tables of c and the B and B factors.5 ]

4 Although the range chart and the standardized range chart are used widely to monitor univariate rocess disersion, they will not discuss here since the multivariate analog is intractable. For multivariate data, the first chart to be considered is the along of the S -chart (Table, Case a, which is equivalent to reeated tests of significance of the form H : σ = σ vs. H : σ σ. Here H : Σ = Σ vs. H : Σ Σ. Using the asymtotic likelihood ratio test result [9], one would comute the following statistic for each random samle: W = n + nln n n ln( Α Σ + tr( Σ Α Where Α denotes the sum of squares and cross-roducts matrix and tr is the trace oerator. Note that Α = ( n S, where S is the ( samle variance-covariance matrix. If the value of this test statistic lots above the UCL= χ ( + /, α, the rocess is deemed to be out of control. Refer to Table, Case. Korin [] found that the asymtotic chi-square aroximation, slightly modified, is quite good even for moderate n. he rooses an F aroximation that aears to be better. Table : Multivariate Disersion Control Charts (Phase II Statistic LCL CL UCL. W - - χ ( + /, α (Aroximate. S Σ ( χ n, α / [( n ] - Σ ( χ n, α / [( n ].5 ª. Σ Σ ( b b ( Σ.5 b Σ b + b a. Refer to Multivariate Quality Control [] A widely used measure of multivariate disersion is the samle generalized variance, denoted by S, where S is the ( samle variance-covariance matrix. The samlegeneralized variance is the basis for the other multivariate disersion charts to be considered. However, Johnson and Wichern [] resent three samle covariance matrices for bivariate data that all have the same generalized variance and yet have distinctly different correlation, r=.8,., and -.8. "Consequently, it is often desirable to rovide more than the single number S as a summary of S." Thus a control chart for S should be used in conjunction with the univariate disersion charts. Suose there are two quality characteristics. Since ( n S / Σ is distributed as χ n. This yields the control chart limits in Table, Case. When there are more than two quality characteristics, one may emloy Anderson's asymtotic normal aroximation [9] or the aroximation suggested by Gnanadesikan and Guta []. Another S -control chart can be constructed using only the first two moments of S and the roerty that most of the robability distribution of S is contained in the interval E ( S ± Var( S. The control chart limits are resented in Table, Case.

5 The Multivariate Shewhart Charts is resented beginning with the simlest case of a multivariate quality control roblem as follows: Consider a rocess that deends uon two characteristics X and X that are comletely indeendent of each other. Suose these two variables are normally distributed with mean zero and varianceσ = σ =. If one sets an overall tye I errorα, then both of the critical regions deicted in Figure -a & -b can be used to monitor the rocess mean. l r Fig -a: Square Critical Region Fig -b: Circular Critical Region Therefore, we can find a radius r and a length l such that l l l l π r x ex( ( x + x dxdx = ex( ( x + x dxdx = α r r r x For examle, whenα =. 5, the radius and the length will be r = 5.99 =. 77 and l =.7 (See Table for different values of α and. The value 5.99 is recisely χ,. 95. π ( Table : Table of critical values for different values of P andα α To control the rocess using the square as the critical region, we calculate the samle mean ( n X, n X. If the oint lies inside the box with corners at ( ± l, ± l then the rocess is in control. If not, either n X > l which makes the first characteristic to be out-of-control and/or n X > l which, makes the second characteristic out-of-control. In either case, the rocess would be considered to be out-of-control. However, since X i >, i =, then, the lower limit for either case is zero. Now consider the matrix

6 ρ R =, ρ < ρ and, let C be such that C C = R. The random variable Y = CX has a bivariate normal Π distribution with mean zero and correlation matrix R. If we let ρ = Sin( θ, θ <, some of the choices for matrix C are C Cos( θ Sin(θ = l C = Sin(θ Cos(θ u C ( Cos( θ + Sin( θ = ( Cos( θ + Sin( θ ( Cos( θ + Sin( θ ( Cos( θ Sin( θ C o ( Sin(θ + = ( Sin(θ + + Sin(θ + Sin(θ ( Sin(θ + ( Sin(θ + + Sin(θ + Sin(θ According to relation between Sinus and Cosinus, we have, Cos( θ Sin( θ C o = Sin( θ Cos( θ Note that among these, the matrix Co is symmetric. The decomositions Cl and Cu are called the lower and uer triangular decomosition of R (Chelovsky decomosition and C is the one used in rincial comonent decomosition. In general any choice of C for the decomosition of R is of the form = R T where, C Cos( t Sin( t T =, π t π Sin( t Cos( t i.e. the general form of C is of the forms Cos( θ Sin( θ Cos( t Sin( t Cos( θ t Sin( θ t C t =. = Sin( θ Cos( θ Sin( t Cos( t Sin( θ + t Cos( θ + t We denote this general form of the decomosition of the correlation matrix R as C t. Note that when t = we obtain the symmetric decomositions C o = R and when t = ±θ we get the triangular decomositions C l andc u. When t = π, we get the decomositionc π = C. The transformations C X (for any choice of C such that C C = R, transforms the circle with radius r to the ellise X + + ρ = X XX r

7 The transformationsc l X, C u X, C X, andc o X = R X will transform the square with corners ( ± l, ± l to the regionsr l, R u, R and Ro resectively. Note that in general, the transformation C is a comosition of a rotation T and a transformation underc o = R. t Fig -a: Critical Region R l Fig -b: Critical Region R u Fig -c: Critical Region R Fig -d: Critical Region R o If X has a bivariate normal distribution with correlation matrix R, the variable Z = Ct X has a standardized normal distribution with correlation matrix I. If we are only interested in the overall control of the rocess mean, that is, we are not concerned with the variables that cause the out-of-control condition, then for any given tye I error α choose a region R such that f ( x, x dxdx = α R Where, f ( x, x is the density function of a bivariate normal distribution with correlation matrix R. Then, S, S (With given n samle (Large samle of size m has a aroximately bivariate normal distribution with: Ε S = Cσ, Var( S = σ ( C, Ε S = Cσ, Var( S = σ ( C, and Cov( S, S = σ ( C, or has a aroximately bivariate normal distribution with correlation matrix R = Corr S, S such that ( = Var ( X σ, Var ( X = σ, Cov ( X, X = σ then, choose a decomosition statistic: C t of R and take a samle of size n and calculate the

8 K K = ( c If the oint ( k, k lies outside the region, R we may conclude that the rocess is out-of-control. An equivalent rocedure would be to find the image of the region R under the transformation made by C t to get Rt and lot the oint ( S, S. If this oint lies outside the region, R t we may conclude that the rocess is out-of-control. But if we are concerned about which variable causes the out-of-control condition, we must choose a region Rt whose image under some transformation C t is the square with corners at ( ± l, ± l. The adequate choice of R t and the adequate choice of Ct is needed in order to draw the right conclusions about the state of the rocess standard deviation. As far as Multivariate Statistical Quality Control (MSQC is concerned, any transformation Ct can be used for the urose of statistical control. The above ideas can be extended to the case of >. The following comutational rocedure can be used to, identify the errant variable(s. PROCEDURE R S S Consider a rocess involving characteristics x, x,..., x that we wish to control. Suose the mean µ = ( µ, µ,..., µ variances σ, σ,..., σ and the correlation matrix R of this rocess, are known. Given n samles of size m taken from a multivariate normal oulation, we can normalize the data by subtracting the mean of each characteristic and dividing by the corresonding standard deviation. Therefore, we can assume that the samle is taken from a multivariate normal oulation with mean and correlation, matrix R. To control the rocess standard deviation, first set an overall tye I error α with individualα 's satisfying the condition i α = ( α ( α...( α Set all α i 's to be equal if all variables are equally imortant. In this case the values for the individual tye I errors is given by α = α i Find the values statistic: b i = z. Find the S = S, S,..., S. calculate the value of the α i ( = ( K, K,..., K = ( C R ( S K where, S = S, S,..., S is the samle standard deviation of the n observations. (

9 If Z i < z α for all i's, the rocess will be in-control. If on the other hand Z z i i > α the i rocess, will be out-of-control and the variable x i is resonsible for this out-of-control condition. The above rocedure can be dislayed in a Chart, sulying information about the behavior of the individual characteristics in a multivariate rocess. Note that if =, we have the Shewhart S chart. To illustrate the above rocedure, consider the following examle. EXAMPLE I Consider the data for a Carton industrial, Rctcd and Cctcd (two characteristics for stiffness of carton the standard values, either derived from a large amount of ast data or selected by management to attain certain objectives, are σ x =.88, σ y =.5, ρ =.8 In matrix notation, we have R =.8.8 Where, R is the correlation matrix. Let the overall tye I error be α =.5 with z α = z α =.7. Setting Sin ( θ =.8, θ =.986 and Cos( θ Sin( θ R = Sin( θ Cos( θ Therefore, square root of correlation matrix will be R.878 = R.686 = The values K,, are obtained by the formula ( K.95 R S S x y Cσ x σ x Cσ y σ y

10 Given samles of size 5, the values S, S, and h=,... along with S, S, ( h h ( xh yh K, K (for h =, are resented in the Table. ( xh yh Another multivariate control chart for desersion is S -chart [], that uer and lower limits are ( χ n, α / LCL= Σ ( n ( χ n, α / UCL= Σ ( n 8..6 Σ =, χ 6,.975 = Then, LCL=.789 UCL=6.6 And, S =.9 According to multivariate charts for standard deviation S -chart and recent aroach, standard deviation of the rocess is in control. Table : Samle data for Examle I, along with the corresondence K h values n S S xh yh K h K h

11 EXAMPLE II As an examle of a higher dimensional roblem, consider the data for a carton industrial, grammage( X, cobb( X, burst( X and density( X (four characteristics for stiffness of carton The standard values, either derived from a large amount of ast data or selected by management to attain certain objectives, are. In this examle =, and the variances, and the correlation matrix are given as follows: σ σ = 86.69, R = σ σ The inverse of the square root of the matrix R is given by R..576 = From Table, for = and α =.5, the critical value is.9. Given samles of size 5, the values ( S h, S h, Sh, S h, such that h=,... along with are resented in the Table 5.

12 h Table 5: Samle data for Examle II S h S h S h S h The values of K, K, K, are comuted and resented in Table 6. ( K

13 Table 6: The corresondence h K h values for Examle II K h K h K h K h If, K h, K h, K h and K h are greater than.9, the first, second, third, and fourth variables are out-of control for h-observations, resectively. But, another multivariate control chart for desersion is S -chart [], that uer and lower limits are

14 ( χ n, α / LCL= Σ ( n ( χ n, α / UCL= Σ ( n χ 6,.995 = Then, 7 LCL= UCL= And, S =.9 According to multivariate charts for standard deviation S -chart standard deviation of the rocess is in control, and recent aroach, shows the standard deviation of Cobb, Burst and Density is out of control for some cases that resent in following grahs (Grah-. Grah Multivariate Shewhart Chart for Grammage Z=.8 K Z=-.8 Number

15 Grah Multivariate Shewhart Chart for Cobb Number Z=.8 K Z=-.8 Grah Multivariate Shewhart Chart for Burst Number Z=.8 K Z=-.8

16 Grah Multivariate Shewhart Chart for Density Z=.8 K Z=-.8 Number CONCLUSIONS Many quality control related roblems are multivariate in nature. Treating such roblems as a series of indeendent univariate roblems for monitoring uroses may lead to incorrect conclusions. In this aer, we have suggested a multivariate chart for standard deviation of rocess, MS Chart, which is based on the extension of the univariate Shewhart S Chart. This charting technique has the advantage of directing the investigators to the ossible cause(s of an out-of-control signal. We believe that a good extension of this research would be in develoing corresonding charting techniques for the Range structure of multivariate rocesses.

17 Aendix We can easily roof that: Where, Γ( α = x e dx, α x B C C = n = C, B n Γ n Γ C = + C If, given m samles of size n then: LCL S = B S, CL S = S, UCL S = B S (Shewhart Chart for Standard Deviation S + S S m Where, S =. m Factors for Constructing Variables Control Charts Observations Factors for Center Line Factors for Control Limits in Samle, n C C B B For n > 5: ( n C =, n B =, B C ( n = + C ( n

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