CUSUM Generalized Variance Charts

Transcription

1 Georgia Southern University Digital Southern Electronic Theses & Dissertations Graduate Studies, Jack N. Averitt College of Fall 13 CUSUM Generalized Variance Charts Yuxiang Li Georgia Southern University Follow this and additional works at: Part of the Applied Statistics Commons Recommended Citation Li, Yuxiang, "CUSUM Generalized Variance Charts" 13. Electronic Theses & Dissertations This thesis open access is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Southern. It has been accepted for inclusion in Electronic Theses & Dissertations by an authorized administrator of Digital Southern. For more information, please contact

2 CUSUM GENERALIZED VARIANCE CHARTS by YUXIANG LI Under the Direction of Charles W. Champ ABSTRACT The commonly recommended charts for monitoring the mean vector are affected by a shift in the covariance matrix. As in the univariate case, a chart for monitoring for a change in the covariance matrix should be examined first before examining the chart used to monitor for a change in the mean vector. A variety of charts used to monitor for a shift in the process covariance matrix have been introduced into the literature. A group of these charts are based on the sample generalized variance S, where S is the sample covariance matrix. We examine the multivariate Shewhart and cumulative sum CUSUM charts based on function of the generalized variance S and the ln S.. The performance of these chart is based on an analysis of the chart s run length distribution. We give closed form expressions for the distribution of these statistics. Properties of the run length distribution are given as solutions to various integral equations. A method for obtaining approximate solutions to these integral equaltions is discussed. Key Words: average run length, integral equations, Meijer G function, moments, run length distribution 9 Mathematics Subject Classification:

3 CUSUM GENERALIZED VARIANCE CHARTS by YUXIANG LI B.S. in Electrical Engineering, Wuhan Science and Technology University A Thesis Submitted to the Graduate Faculty of Georgia Southern University in Partial Fulfillment of the Requirement for the Degree MASTER OF SCIENCE STATESBORO, GEORGIA 13

4 c 13 YUXIANG LI All Rights Reserved iii

5 CUSUM GENERALIZED VARIANCE CHARTS by YUXIANG LI Major Professor: Charles W. Champ Committee: Broderick O. Oluyede Hani Samawi Daniel Linder Electronic Version Approved: December 13, 13 iv

6 DEDICATION I dedicate this this to my parents. It is their unconditional love that motivates me to obtain my goals. I also dedicate this thesis to my wife, Ting Peng, who has provided me with a strong love shield that always surrounds me which never lets any sadness to enter my life. v

7 ACKNOWLEDGMENTS I would like to thank my advisor Dr. Charles W. Champ for his insightful guidance and constant encouragement. Without his ideas and help I could not have done this thesis. I would also like to thank my committee members Dr. Broderick Oluyede, Dr. Hani Samawi, and Dr. Daniel Linder for their suggestions and willingness to serve on my committee. Also, I would like to thank Dr. Steven E. Rigdon of St. Louis University and Dr. Alan Genz of Washington State University for providing me with some helpful information about my research. I would also like to thank my parents whose encouragement motivated me to achieve my goals in education and in life. I applaud the faculty, staff, and fellow graduates of the Department of Mathematical Sciences for their support. vi

8 TABLE OF CONTENTS Page ACKNOWLEDGMENTS vi LIST OF TABLES ix LIST OF FIGURES x CHAPTER 1 Introduction Quality Control Charts Model and Sampling Method Types of Shifts in the Process Generalized Variance Thesis Prospectus Distributions of Functions of S Introduction Some Distributional Results Distribution of S Distribution of ln n 1 Σ 1 S Evaluating f S w and f ln n 1Σ 1 S u Numerically Conclusion GENERALIZED VARIANCE SHEWHART CHART Introduction Literature Review vii

9 3.3 The Run Length Distribution The Bivariate Case AN EXAMPLE Conclusion GENERALIZED VARIANCE CUSUM CHART Introduction Run Length Distribution Conclusion CONCLUSION General Conclusions Areas for Further Research REFERENCES viii

10 LIST OF TABLES Table Page 3.1 ARL, SDRL, T 5,,λ,.5,.38,1 γ ARL,n = 5, α =.5, τ = α/, λ = ARLs,m = 1, n = 5, α =.395, τ = α/ ARL, SDRL, for m = 1, n = 5, α =.5, and τ = Phase II Data Summary ix

11 LIST OF FIGURES Figure Page.1 Plot of f χ 5 x Plot of f loge χ 5 y solid curve, f log y dashed curve χ 5.3 Plot of f loge χ 5 y solid curve, f log y dashed curve.. 11 χ Plot of ARL versus λ Phase II Plot of S versus t x

12 CHAPTER 1 INTRODUCTION 1.1 Quality Control Charts The quality control chart introduced by Walter A. Shewhart see Shewhart 1931 in the early 19 s. He described two kinds of variability in a quality measurement X found in a process which he labelled as natural and assignable. Natural causes of variability are inherent to the process whereas assignable ones when removed improve the quality of the process. A process that is operating in which the only natural causes of variability are present is said to be in a in-control state. When an assignable causes is present, then the process is in a out-of-control state. Duncan 1986 discussed the three uses of the control chart with respect to natural and assignable causes of variability. He summarized that a control chart can be used by the practitioner as an aid in 1 bringing a process into a state of statistical in control, defining what is meant by the process being in a state of statistical in control, and 3 monitoring for a change in the process. That is, the control chart is an aid to the practitioner in discovering assignable causes of variability, as an aid in defining what is meant by a process being in an in-control state, and detecting when an assignable cause of variability has changed the process. Cases 1 and are part of the first phase of controlling a process. Charts used in this phase are referred to as Phase I or retrospective control charts. In the monitoring phase, the charts are called Phase II or prospective control charts. When there is a p 1 vector X of quality measurements, generally, the main interest of a practitioner is to control the p 1 mean vector µ of the distribution of X. The most popular charts used for this purpose both in Phase I and Phase II are not only affected by a change in the process mean vector µ but also by a change in the process covariance matrix Σ. For example, Champ, Jones-Farmer, and Rigdon

13 5 show that the Hotelling s T chart can be affected by changes in the covariance matrix. It is then typically recommended that a chart for controlling for a change in Σ be use. This chart should be examined first. If there is no evidence in a change in the covariance matrix Σ, then one can assume that any change indicated by the chart for controlling the mean vector µ is due to a change in µ. 1. Model and Sampling Method The model that we will use in this study is 1 the p 1 quality vector X has a multivariate normal distribution with a p 1 mean vector µ and a p p positive definite covariance matrix Σ. When the process is in a state of statistical in-control, then µ = µ and Σ = Σ, where µ and Σ are fixed and typically unknown. In Phase I, the researcher will have available the quality measurements on m samples of each having n items produced. We represent the measurements on these items produced by the process by X i,1,..., X i,n for i = 1,..., m. These measurements are assumed to be m independent random samples with X i,j N p µ, Σ for j = 1,..., n and i = 1,..., m. We will refer to this model for the data as the independent normal model. where We define estimators µ and Σ for µ and Σ by µ = X = 1 m n mn X i,j and Σ = S = 1 m i=1 j=1 m S i, i=1 S i = 1 n T Xi,j X i Xi,j X i. n 1 j=1 It is shown in Anderson 3 under our independent normal model that µ N p µ, 1 mn Σ and µ and Σ are stochastically independent. and Σ 1 W ishart m n 1, m n 1 Σ ;

14 In Phase II, the researcher will periodically take the p 1 quality vector X on n items from the output of the process. We assume that these sets of measurements, X t,1,..., X t,n, are independent random samples with common N p µ, Σ distribution, for t = 1,, 3,.... Further, we assume the measurements to be taken in Phases I and II are independent. The estimators we will use in this phase for µ and Σ are µ = X t = 1 n n j=1 X t,j and Σ = S t = 1 n 1 In Anderson 3, it is shown that µ N p µ, 1 n Σ n j=1 Xt,j X t Xt,j X t T. and Σ W ishart n 1, 1n Σ ; 3 and µ and Σ are stochastically independent. It is not difficult to see that µ, Σ, µ, and Σ are stochastically independent. 1.3 Types of Shifts in the Process Generalized Variance The process generalized variance is the determinant Σ of the covariance matrix Σ. A change in Σ may result in a change in the process parameter Σ. It is our interest to control for a change in this process parameter Σ from its in-control value Σ. Under the assumption the covariance matrix Σ Σ is positive definite, then the associated correlation matrix Ψ Ψ is also positive definite. Note that the covariance matrix can be expressed as Σ = ΓΨΓ T Σ = Γ Ψ Γ T, where Γ = Diagonal σ 1,..., σ p Γ = Diagonal σ 1,,..., σ p,. One can show that the eigenvalues of a positive definite matrix are all positive real numbers. It then follows that Ψ Ψ can be expressed as Ψ = VCV T Ψ = V C V T,

15 4 where C = Diagonal ξ 1,..., ξ p C = Diagonal ξ 1,,..., ξ p, is the diagonal matrix of eigenvalues with of Ψ Ψ with the associated normalized eigenvectors v 1,..., v p v 1,,..., v p, the columns of the matrix V V. Define the matrix Further define The square λ of the determinant of Λ is P = ΓVC 1/ P = Γ V C 1/. Λ = P 1 P. λ = Λ = ΛΛ T = P 1 P P 1 P T = P P T 1 PP T = Σ 1 Σ = Σ Σ = ΓΨΓ T Γ Ψ Γ T. Suppose that Ψ = Ψ. It would follow that λ = Γ Γ = p i=1 σ i p i=1 σ i, = p σ i i=1 σi, = p i=1 λ i, where λ i = σ i /σ i, for i = 1,..., p. We see that λ = 1 if the process is in-control and λ 1 if the process is in an out-of-control state with respect to the process generalized variance. Suppose only one of the variances has shifted, say, σ 1 has shifted from its in-control value of σ 1,. It would follow that λ = Λ = σ 1 σ 1, = λ 1. We can also see that ΓVCV T λ Γ = Γ V C V T Γ = ΓCΓ Γ C Γ = p i=1 σi ξ i σi, ξ. i, If a shift has occurred but not with the variances, then their has been a change in the correlation structure of the process covariance. It would follow that λ = Λ = p i=1 ξ i ξ i, = p i=1 ζ i,

16 5 where ζ i = ξ i /ξ i, for i = 1,..., p. We note that the type of shift Healy 1987 assumed is the special case of the one presented here in which Σ = cσ. That is, the standard deviation of each component of the vector of quality measurements shifts the same proportion of their in-control values. For this type of shift in the covariance matrix, λ = Σ 1 Σ = Σ 1 cσ = c p. 1.4 Thesis Prospectus In this thesis, we will discuss analytical methods for analyzing the performance of the Shewhart and CUSUM S and ln S charts. First we examine closed form expressions presented in the literature for the probability and cumulative density functions describing the distribution of the sample generalized variance S and ln S under the independent multivariate normal model. We provide a new closed form expression for the probability density function of S. It is pointed out that each of these methods suffers from the curse of dimensionality. We discuss the use of the methods found in the literature for dealing with the dimension problem for large values of p to evaluating the probability density function describing the distributions of S and ln S. An outline is given for using these results in the analytical methods for analyzing the run length performance of the Shewhart and CUSUM S and ln S charts. Some analytical results are given in the bivariate case. Examples are given to illustrate the procedure. Some recommendations for further research are given.

17 CHAPTER DISTRIBUTIONS OF FUNCTIONS OF S.1 Introduction A measure of the variability in the distribution of a p 1 multivariate measurement X is the population generalized variance Σ, where Σ is the p p matrix of variances and covariances of the components of X. It can be shown that the sample covariance S based on a random sample independence data model X 1,..., X n from the distribution of X is an unbiased estimator of Σ. An estimator of Σ is the sample generalized variance S. In general, the sample generalized variance is a biased estimator of the population generalized variance. Under the multivariate normal model for the distribution of X, it is shown in Anderson 3 that for n > p S Σ n 1 p p i=1 χ n 1 i 1, where χ n 1,..., χ n p are independent chi square random variables with respective degrees of freedom n 1,..., n p. Thus under the independent multivariate normal model is an unbiased estimator of Σ. p i=1 S n i n 1 Anderson 3 states If p = 1, S has the distribution of Σ χ n 1/ N 1. If p =, S has the distribution of Σ χ N 1 χ N / N 1. It follows from Problem 7.15 or 7.37 that when p =, S has the distribution of Σ χn 4 / N. Here his capital N is our lower case n which is the sample size. It follow that when p = 1 Σ = σ the process variance and S = S the sample variance. We see that n 1 Σ 1 S χ n 1, if p = 1; χ n 4 /4, if p =.

18 7 It also follows that ln n 1 Σ 1 S ln χn 1, if p = 1; ln χ n 4 ln, if p =. In this article, we are interested in the distribution of the sample generalized variance S and ln n 1 Σ 1 S under the independent multivariate normal model.. Some Distributional Results A random variable X is said to have Chi Square distribution with ν > degrees of freedom if the probability density function pdf f χ ν x describing the distribution of X has the form f χ ν x = 1 Γ ν ν/ xν/ 1 e x/ I, x, where I, x is an indicator function having the value 1 if x, and otherwise. A graph of f χ 5 x is given in the following figure.

19 8 Figure.1: Plot of f χ 5 x The kth moment of a Chi Square distribution is given in the following theorem. Theorem.1: If X χ ν, then E X k is for k = 1,, 3,.... E X k = Γ ν + k k Γ, ν Proof of Theorem.1: We see that E X k = x k 1 Γ ν ν/ xν/ 1 e x/ dx = Γ ν+k ν+k/ Γ 1 ν ν/ Γ ν+k ν+k/ xν+k/ 1 e x/ dx = Γ ν + k k Γ ν An interesting general family of transformations of χ ν random variable is defined

20 9 by Y = log b χ ν, where b > and b 1. The cdf describing the distribution of Y is F logb χ y = P log ν b X y = P log b X y = P b log b X b y = P X b y = F χ ν b y. It then follows that the pdf of the distribution of X is given by f logb χ ν y = b y f X b y ; ν = 1 Γ ν lnby/ ν ν/ e by. The following figure displays the plot of f loge χ 5 y and f log χ 5 y.

21 1 Figure.: Plot of f loge χ 5 y solid curve, f log y dashed curve χ 5 Also of interest are transformations of X defined by Y = log b χ ν. The cdf describing the distribution of Y is F logb χ ν y = P log b X y = P log b X y = P X b y = 1 P X b y = 1 F χ ν b y. It then follows that the pdf of the distribution of X is given by f logb χ ν y = b y f X b y ; ν = 1 Γ +ν lnby/ ν ν/ e b y. The following figure is displays the plot of f loge χ 5 y and f log χ 5 y.

22 11 Figure.3: Plot of f loge χ 5 y solid curve, f log y dashed curve χ 5 The following theorem is useful. Theorem.: f logb χ ν y = f logb χ ν y and f logb χ ν y = f logb χ ν y. Proof of Theorem.: We see that 1 f logb χ y = ν Γ +ν lnb y/ ν ν/ e b y 1 = Γ ν lnby/ ν ν/ e by = f logb χ ν y. Further, we see that 1 f logb χ y = ν Γ ν lnb y/ ν ν/ e b y 1 = Γ +ν lnby/ ν ν/ e b y = f logb χ ν y.

23 1 Another useful theorem is as follows. Theorem.3: f χ ν x = x 1 log b e f lnχ ν log b x. Proof of Theorem.3: We see that F χ ν x = P log b χ ν logb x = F logb χ ν log b x. Thus, f χ ν x = d dx F log b χ ν log b x = x 1 log b e f lnχ ν log b x..3 Distribution of S Grigoryan and He 5 derived a closed form expression for the probability density function f S w describing the distribution of S under the independent multivariate normal model using the results in Anderson 3. First they derived a closed form expresssion the probability density function f n 1Σ 1 S w describing the distribution of n 1 Σ 1 S p i=1 χ n i, where χ n 1,..., χ n p are independent chi square random variables with respective degrees of freedom n 1,..., n p. Let X i = χ n i and make the transformation W 1 = X 1, W = X 1 X,..., W p = X 1 X p. The inverse of this transformation is X 1 = W 1, X = W /W 1,..., X p = W p /W p 1

24 with Jacobian J = 1 W 1 W W p 1. It follows that the joint probability density function f W1,W,...,W p w 1, w,..., w p describing the joint distribution of W 1, W,..., W p is given by 1 f W1,W,...,W p w 1, w,..., w p = f X1 w 1 f X w /w 1 f Xp w p /w p 1 p 1 i=1 w i 1 = Γ n 1 n 1/ wn 1/ 1 1 e w 1/ 1 Γ w n n / /w 1 n / 1 e w /w 1 / 1... Γ n p w n p/ p/w p 1 n / 1 e wp/w 1 p 1/ p 1 i=1 w i p 1 = wn / 1 p i=1 w 1/ i p i=1 Γ n i pn p 1/4 e w 1/ e p i= w i/w i 1 / It now follows that the probability density function f S w p as given in Grigoryan and He 5 describing the marginal distribution of W p = n 1 Σ 1 S is f n 1Σ 1 S w p = w p n / 1 p 1 i=1 w 1/ i p i=1 Γ n i pn p 1/4 e w 1/ e p i= w i/w i 1 / dw 1... dw p Observing that F S w = P S w = P n 1 Σ 1 S n 1 p Σ 1 w = F n 1Σ 1 S n 1 p Σ 1 w. The probability density function describing the distribution f S w of S is f S w = n 1 p Σ 1 f n 1Σ 1 S n 1 p Σ 1 w = n 1 p Σ 1 n 1 p Σ 1 w n / 1 p 1 p i=1 Γ n i pn p 1/4 e w 1/ e p 1 i= w i/w i 1 +n 1 p Σ 1 w/w p 1/ dw 1... dw p 1. i=1 w 1/ i

25 Pham-Gia and Turkkan 1 show that the distribution of n 1 Σ 1 S can be expressed in terms of the Meijer G function. The Meijer G function is defined by 14 G m r p,q x a 1,..., a p b 1,..., b q = 1 πi L m j=1 Γ b j s r j=1 Γ 1 a j + s q j=m+1 Γ 1 b j + s p j=r+1 Γ a j s xs ds, where the integral is along the complex contour L of a ratio of products of gamma functions. On page 936, they express the pdf describing the distribution of n 1 Σ 1 S with some modification is given as follows f n 1Σ 1 S w = 1 p 1 p j=1 Γ n j G p,p w p n 1,..., n p n 1,..., n p I, w. It would then follow that the pdf describing the distribution of ln n 1 Σ 1 S would have the form f ln n 1Σ 1 S w = e w f n 1Σ 1 S e w = ew p p j=1 1 Γ n j G p,p ew p n 1,..., n p n 1,..., n p. The Meijer G function has been implimented in both MATLAB and Mathematica. The MATLAB code is G m p,q r a 1,..., a p x b 1,..., b q = meijerg[[a 1,..., a r ], [a r+1,..., a p ]], [[b 1,..., b m ], [b m+1,..., b q ]], x.

26 15 It follows that G p a 1,..., a p,p x b 1,..., b q = 1 p j=1 Γ b j s j=1 Γ 1 a j + s πi p L j=p+1 Γ 1 b j + s j=+1 Γ a j s xs ds = 1 p πi Γ b j s x s ds j=1 L = meijerg[[a 1,..., a ], [a +1,..., a ]], [[b 1,..., b p ], [b p+1,..., b p ]], x = meijerg[[], []], [[b 1,..., b p ], []], x. It follows that f n 1Σ 1 S w = 1 p 1 p j=1 Γ n j G m r p,q meijerg[[], []], [[ n 1 In Mathematica, the Meijer -function is implemented as a 1,..., a p x b 1,..., b q We then have f n 1Σ 1 S w = 1 p 1 p j=1 Γ n j,..., n p ], []], w/ p. = MeijerG[a1,..., ar, ar+1,..., ap, b1,..., bm, bm+1,..., bq, x]. MeijerG[[], []], [[ n 1.4 Distribution of ln n 1 Σ 1 S,..., n p ], []], w/ p. The cumulative distribution function F U u, where U = ln n 1 Σ 1 S is given by F U u = P ln n 1 Σ 1 S u = P S Σ e u / n 1 p = F S Σ e u / n 1 p. Hence, the probability density function describing the distribution of U can be expressed in terms of the probability density function describing the distribution of S

27 16 as f U u = Σ e u / n 1 p f S Σ e u / n 1 p. Using the expression derived by Grigoryan and He 5, we can express the probability density function f U u as f U u = Σ e u / n 1 p n / 1 p 1 p i=1 Γ n i pn p 1/4 i=1 w 1/ i e w 1/ e p 1 i= w i/w i 1 + Σ e u /n 1 p /w p 1 / dw 1... dw p 1 From the expression by Pham-Gia and Turkkan 1 for the probability density function describing the distribution of n 1 Σ 1 S, we have f ln n 1Σ 1 S w = e w f n 1Σ 1 S e w = ew p p j=1 1 Γ n j G p,p ew p n 1,..., n p n 1,..., n p. Here we give another method for deriving the probability density function f U u = f ln n 1Σ 1 S u describing the distribution of U = ln n 1 Σ 1 S p ln χ n i, i=1 where n > p are positive integers, χ n i is a random variable with a Chi Square distribution with n i degrees of freedom, and χ n 1,..., χ n p are independent. The pdf of U is given in the following theorem. Theorem.4: If χ n 1,..., χ n p are independent Chi Square random variables with degrees of freedom n 1,..., n p, respectively, with n > p, then the probability density function f U u describing the distribution of U = p ln χ n i i=1

28 17 can be expressed as f U u = f lnχ n p x p dx dx p = f χ n p e xp dx dx p. f lnχ n 1 u x... x p f lnχ n x e u f χ n 1 e u x... x p fχ n e x Also, the cumulative distribution function F U u can be expressed as F U u = f lnχ n p x p dx dx p = f χ n p e xp dx dx p. F lnχ n 1 u x... x p f lnχ n x e u F χ n 1 e u x... x p fχ n e x Proof of Theorem.4: For convenience, define Y i = ln χ n i for i = 1,..., p for any integer p < n. Also, we let c i = 1 Γ. n i n i/ Consider the one-to-one transformation U i = i j=1 Y j for i = 1,..., p. We see that U = U p. The inverse transformation is Y 1 = U 1 and Y i = U i U i 1 for i =,..., p with Jacobian J = 1. The joint probability density function of U 1,..., U p for u 1 < u <... < u p

29 18 is given by f U1,...,U p u 1,..., u p = f Y1 u 1 f Y u u 1 f Yp u p u p 1. For p = and n > p, we have f U1,U u 1, u = f Y1 u 1 f Y u u 1 = c 1 e eu 1 n 1u 1 / c e eu u 1 n u u 1 / It follows that = c 1 c e eu 1 / e eu u 1 n u u 1 n 1u 1/ 1 i e iu 1 = c 1 c i= i i! e eu u 1 n u u 1 n 1u 1/ 1 i = c 1 c e eu u 1 n u u 1 n 1u 1 iu 1/ i= i i! = c 1 c e n 1u 1 i e / iu i= i i! e eu u 1 +i+1u u 1 / f U u = u f Y1 u 1 f Y u u 1 du 1 = c 1 c e n 1u / i= 1 i e iu i i! u e eu u 1 +i+1u u 1 / du 1 Making the transformation x = u 1 u with dx = du 1, we have f U u = c 1 c e n 1u / = c 1 c e n 1u / = = = i= 1 i e iu i= i i! e e x i+1x / dx 1 i e iu +x e e x x / dx i i! c 1 e eu +x n 1u +x / c e e x +n x / dx f lnχ n 1 u + x f lnχ n x dx f lnχ n 1 u x f lnχ n x dx Hence, the theorem is true for p = and n > p. Now suppose that the theorem is true for p > and n > p. For n > p + 1 and u 1 < u <... < u p+1, we have f U1,...,U p+1 u 1,..., u p+1 = f Y1 u 1 f Y u u 1 f Yp u p u p 1 f Yp+1 u p+1 u p.

30 19 It follows that f Up+1 u p+1 = We now have up+1 up u f Y1 u 1 f Y u u 1 f Yp u p u p 1 f Yp+1 u p+1 u p du 1 du p 1 du p = up+1 up u du 1 du p 1 f Yp+1 u p+1 u p du p = up+1 f Up+1 u p+1 = f Y1 u 1 f Y u u 1 f Yp u p u p 1 f Up u p f Yp+1 u p+1 u p du p. up+1 f lnχ n 1 u p x... x p f lnχ n x f lnχ n p x p dx dx p f Yp+1 u p+1 u p du p = up+1 f lnχ n 1 u p x... x p f Yp+1 u p+1 u p du p f lnχ n x f lnχ n p x p dx dx p.

31 Observe that up+1 up+1 = f lnχ n 1 u p x... x p f Yp+1 u p+1 u p du p c 1 e eup x... xp n 1u p x... x p/ c p+1 e eu p+1 up n p 1u p+1 u p/ du p = c 1 c p+1 up+1 = c 1 c p+1 up+1 e eup x... xp / e eu p+1 up n p 1u p+1 u p n 1u p x... x p/ du p i= e eu p+1 up +p 1+iu p+1 u p/ du p = c 1 c p+1 e n 1u p+1 x... x p/ i= 1 i e iu p+1 x... x p i i! 1 i e n 1+iu p+1 x... x p/ up+1 i i! e eu p+1 up +p+iu p+1 u p/ du p Making the transformation x p+1 = u p+1 u p with dx p+1 = du p, we have up+1 f lnχ n 1 u p x... x p f Yp+1 u p+1 u p du p = c 1 c p+1 e n 1u p+1 x... x p/ e e x p+1 p+ix p+1/ dx p+1 = c 1 c p+1 e n 1u p+1 x... x p/ e e x p+1 p+ix p+1/ dx p+1 = i= i= 1 i e iu p+1 x... x p i i! 1 i e iu p+1 x... x p+x p+1 i i! c 1 e eu p+1 x... xp x p+1 n 1u p+1 x... x p+x p+1 / c p+1 e e x p+1 +n px p+1/ dx p+1 = = f lnχ n 1 u p+1 x... x p + x p+1 f lnχ n p 1 x p+1 dx p+1 f lnχ n 1 u p+1 x... x p x p+1 f lnχ n p 1 x p+1 dx p+1

32 1 It now follows that f Up+1 u p+1 = up+1 f lnχ n 1 u p x... x p f lnχ n x f lnχ n p x p dx dx p f Yp+1 u p+1 u p du p = f lnχ n 1 u p+1 x... x p x p+1 f lnχ n x f lnχ n p 1 x p+1 dx dx p+1. Hence, the theorem is true for p + 1. By the Axiom of Induction, the results holds for all n and p such that n > p. The cumulative distribution function F U u describing the distribution of U is has the form F U u = = u f U t dt u f lnχ n 1 t x... x p f lnχ n x f lnχ n p x p dx dx p dt u = f lnχ n 1 t x... x p dt f lnχ n x f lnχ n p x p dx dx p = u x... x p f lnχ n p x p dx dx p = f lnχ n p x p dx dx p = f χ n p e xp dx dx p. f lnχ n 1 t dt f lnχ n x F lnχ n 1 u x... x p f lnχ n x e u F χ n 1 e u x... x p fχ n e x

33 The distribution of S expressed in terms of the distribution of ln n 1 Σ 1 S under the independent multivariate normal model is given in the following theorem. Theorem.5: If χ n 1,..., χ n p are independent Chi Square random variables with degrees of freedom n 1,..., n p, respectively, with n > p, then the probability density function f S u and cumulative distribution function F S u describing the distribution of S are n 1 p f S w = w 1 w f ln n 1Σ 1 S ln Σ n 1 p w F S w = F ln n 1Σ 1 S ln. Σ and Proof of Theorem.5: Observe that F S w = P S w = P ln n 1 Σ 1 S n 1 p w ln Σ n 1 p w = F ln n 1Σ 1 S ln. Σ Hence, their probability density function f S w is f S w = d dw F S w = w 1 f ln n 1Σ 1 S ln n 1 p w Σ. Theorem.6: The kth moment of the distribution of S for n > p under the independent normal model is E S k = pk Σ k p Γ n i + k n 1 pk i=1 Γ. n i Proof: Recall that n 1 Σ 1 S p i=1 χ n i, χ n 1,..., χ n p are independent Chi Square random variables with degrees of freedom n 1,..., n p, respectively, with n > p. Thus, n E 1 Σ 1 S k [ p ] k = E = p [ χ E ] k n i, i=1 i=1 χ n i

34 since the random variables χ n 1,..., χ n p are independent. From Theorem.1, we have E [ χ ] k n i = Γ n i + k k Γ. n i Hence, the kth moment of W is expressed as n E 1 Σ 1 S k = pk p It follows that the kth moment of S is i=1 Γ n i + k Γ. n i E S k = pk Σ k p Γ n i + k n 1 pk i=1 Γ. n i 3 Note that the distribution of n 1 Σ 1 S can be expressed in terms of the distribuiton of ln n 1 Σ 1 S by f n 1Σ 1 S w = w 1 f ln n 1Σ 1 S ln w I, w. The kth moment of n 1 Σ 1 S can then be determined by n E 1 Σ 1 S k = = = w k f n 1Σ 1 S w dw w k w 1 f ln n 1Σ 1 S ln w dw w k 1 e lnw f χ n 1 e lnw x... x p fχ n e x f χ n p e xp dx dx p dw = w k f χ n 1 e lnw x... x p dw f χ n e x f χ n p e xp dx dx p. Making the change of variable x 1 = ln w x... x p or w = e x +...+x p e x 1 with dw = e x +...+x p e x 1 dx 1,

35 we have E n 1 Σ 1 S k = 4 e x +...+x p e x 1 k fχ n 1 ex 1 e x +...+x p e x 1 dx 1 f χ n e x f χ n p e xp dx dx p = e x 1 k e x 1 f χ n 1 e x 1 dx 1 e x k e x f χ n e x e xp k e xp f χ n p e xp dx dx p = p e x i k e x i f χ n i e x i dx i. i=1 Making the change of variables t i = e x i with dt = e x i dx i, we have n E 1 Σ 1 S k = p i=1 t k f χ n i t dt. Using the results from Theorem.1, we have n E 1 Σ 1 S k = p Γ n i + k k i=1 Γ n i = pk p Γ n i + k i=1 Γ, and n i E S k = pk Σ k p Γ n i + k n 1 pk i=1 Γ. n i When the moment generating function of the distribution of a random variable exists, then one can show that it uniquely determine the distribution. Bain and Engelhardt 199 state and prove that the moment generating function can be determined from its moments about zero. By comparing moments, we have verified that Theorem.4 holds.

36 5.5 Evaluating f S w and f ln n 1Σ 1 S u Numerically Define h u x = π p 1/ e xt x/ f lnχ n 1 u x... x p f lnχ n x f lnχ n p x p, where n > p with x = [x,..., x p ] T. We can then write the probability density function describing the distribution of f U u = π p 1/ = = 1 π p 1/ I φ x h u x dx dx p, e xt x/ h u x dx dx p 1 e x T I 1 x h 1/ u x dx dx p where p 1 1 vector of zeros and I p 1 p 1 identity matrix. φ x = 1 π p 1/ I 1 e x T I 1 x 1/ with Genz and Monahan 1999 give a numerical method for evaluating I h u, where I h u = π p 1/ e xt x/ h u x dx dx p. FORTRAN code to evaluate I f is available from Dr. Alan Genz, Department of Mathematics, Washington State University, Pullman, WA. According to Dr. Genz, his method can be adapted to evaluate f U u. First we need to substitute x with the vector y = [y,..., y p ] T, where y i = x i for i =,..., p and divide by p 1. The method used by Genz and Monahan 1999 to evaluate f U u = 1 p 1 φ y h u y dy dy p

37 6 make a change of variables to a spherical-radial coordinate system to describe the stochastic spherical-radial rules. Defining y = rz, with z T z = 1, such that y T y = r, r results in f U u = π p 1/ p 1 z T z=1.6 Conclusion e r / r p 1 h u rz drdz. In this chapter, we presented and derived some useful distributional results for describing the distribution of the generalized variance S and the distribution of ln n 1 Σ 1 S.

38 CHAPTER 3 GENERALIZED VARIANCE SHEWHART CHART 3.1 Introduction The Phase I Shewhart quality control chart is a plot of a statistic θ i = θ i X i,1,..., X i,n versus the sample number i for each of the m preliminary samples. The plot usually includes a center line CL along with a lower LCL and an upper UCL control limits. The chart is said to signal a potential out-of-control process if either θ i LCL or θ i UCL. These charts are used in a Phase I study of the process in an attempt to remove what Shewhart 1931 referred to as assignable causes of variability that when detected and removed from the process results in a better quality process. The data collected in this phase in which there is evidence it is from an in-control process is used to estimate the in-control process parameters that are used to design a Phase II chart. Once the process is believed to be in a state of statistical in-control, it is desirable to monitor for a change in the process. The monitoring phase is often referred to as Phase II. In this phase, the practitioner plots a statistic θ t = θ t X t,1,..., X t,n versus the sample number t for t = 1,, 3,.... This plot typically includes a CL center line, a LCL lower control limit, and an U CL upper control limit. The charts signals a potential out-of-control process at time sampling stage t if either θ t LCL or θ t UCL. It is our interest in this chapter to examine Phase II Shewhart chart based on the statistic θ t = S t.

39 8 The Phase II generalized variance Shewhart chart is a plot of the points t, S t for t = 1,, 3,.... The chart signals a potential change in the process from a in-control state to an out-of-control state at sampling stage t if either S t LCL or S t UCL. 3. Literature Review Montgomery and Wadsworth 197 developed a Shewhart S chart using an asymtotic normal approximation to the distribution of the generalized variance. They assumed that the p 1 vector X of quality measurements has a N p µ, Σ distribution. Further, they assumed that periodically the practitioner would have available independent random samples X t,1,..., X t,n in Phase II from the distribution of X. Recall that the two previous assumptions are referred to as the independent multivariate normal model. Under this model, Anderson 3 shows that n 1 S / Σ 1 is asymptotically normally distributed with mean and variance p. It follows that n 1 S / Σ 1 p has an asymptotically standard normal distribution. To determine the asymtotic control limits, we observed that asymtotically α = 1 P z α/ < n 1 S / Σ 1 < z α/. p Next observe that we can write α = 1 P [ ] zα/ p zα/ p 1 Σ < S < + 1 Σ. n 1 n 1 Hence, the asymptotic center line CL, lower LCL and upper U CL probability

40 9 control limits when Σ is known are given by p LCL = z α/ 1 Σ ; CL = n 1 p UCL = + 1 Σ, n 1 z α/ p i=1 n i Σ ; and n 1 where < α <.5. Replacing z α/ with 3 gives the asymptotic 3-sigma limits suggested by Shewhart If Σ is unknown, then it is recommended that Σ be replaced by S. Alt and Smith 1988 introduced a generalized variance S Shewhart chart. They assume the independent multivariate normal model. At each time t, the point t, S t is plotted for t = 1,, 3,.... On this plot a center line CL, lower control LCL and upper control UCL limits are drawn with { LCL = max, E S Σ = Σ 3 } V S Σ = Σ ; CL = E S Σ = Σ ; and UCL = E S Σ = Σ + 3 V S Σ = Σ. It is shown in Anderson 3 that under the given model E S Σ = Σ = b 1 Σ and V S Σ = Σ = b Σ, where b 1 = p i=1 n i n 1 p and b = p [ i=1 n i p + n i n 1 p ] p n i. i=1 n 1 i=1 n 1 It follows that the center line and control limits can be expressed as { LCL = max, Σ b 1 3 } b ; CL = b 1 Σ ; and UCL = Σ b b. Note that under the independent multivariate normal model and assuming Σ is positive definite, Dykstra 197 proved that the sample covariance matrix S is positive

41 definite with probability one. It then follows that the sample generalized variance S is positive with probability one. Thus the LCL can be set to zero if the three sigma lower control limit is negative. The chart is said to signal at sampling stage t when either S t LCL or S t UCL. Aparisi et al studied the properties of the run length distributon of the S chart assuming and the in-control covariance matrix Σ is known. They considered more general control limits of the form { } LCL = max, Σ b 1 k L b ; CL = b 1 Σ ; and UCL = Σ b 1 + k U b, 3 where k L, k U > and k L is chosen so that the LCL is nonnegative. The control limits of the Shewhart S chart can be expressed in terms percentage points of the distribution of the generalized variance S. Note that the distribution of W = n 1 Σ 1 S p i=1 χ n 1 i 1 only depends on n and p. Letting w n,p,1 γ be the 1 1 γth percentile of the distribution of W, then the 1 1 γth percentile of the distribution of S can be expressed as Σ w n,p,1 γ / n 1 p. Hence, the lower and upper control limits for the Shewhart S chart is given by LCL = Σ w n,p,1 τ / n 1 p and UCL = Σ w n,p,α τ / n 1 p, where < τ < α <.5. Typically, the value of τ is selected to be α/. 3.3 The Run Length Distribution The run length T of the chart is the number of the sample in Phase II in which the chart first signals. For the chart proposed by Montgomery and Wadsworth 197,

42 31 the distribution of the run length based on the asymptotic control limit is p LCL = z α/ 1 Σ ; n 1 p n i CL = Σ ; and i=1 n 1 p UCL = + 1 Σ, n 1 z α/ α = α λ, n, p = P T = 1 = 1 P [LCL < S 1 < UCL] [ n 1 S1 / Σ 1 = 1 P B < p ] < A = 1 Φ A + Φ B, where B = A = zα/ p + n 1 λ n 1 p zα/ p + n 1 λ n 1. p and For t > 1, the event {T = t} can be expressed as t 1 { {T = t} = B < i=1 { B < n 1 Si / Σ 1 p n 1 St / Σ 1 p < A} c. } < A We see that {T = t} has been expressed as the intersection of t independent events. Hence, P T = t = t 1 P i=1 P { B < B < n 1 Si / Σ 1 p n 1 St / Σ 1 p < A } c < A. It then follows that P T = t = t 1 i=1 1 α α = α 1 αt 1.

43 Thus, the asymtotic run length distribution is a geometric distribution with parameter α. The expection E T is commonly referred to as the average run length ARL. It follows for this chart that ARL = ARL λ, n, p = 1 α λ, n, p. The process is in-control when λ = 1. The asymtotic in-control average run length 3 is ARL = ARL 1, n, p = 1 α 1, n, p. In the more general setting when the control limits for the Shewhart S are expressed in as percentage points of the distribution of S when Σ, we see that wn,p,1 τ α = 1 P n 1 p Σ < S < w n,p,α τ n 1 p Σ wn,p,1 τ = 1 P < n 1 Σ 1 S w n,p,α τ < λ λ wn,p,α τ wn,p,1 τ = 1 F W + F λ W. λ The ARL of the chart can then be expressed as ARL = 1 F W wn,p,α τ λ 1 + wn,p,1 τ. FW λ As we have seen, the control limits of the Shewhart S are or can be expressed as a function of Σ. If Σ is replaced with S, then the distribution of the run length is no longer a geometric distribution. However, the distribution of T conditioned on S has a geometric distribution. For the Shewhart S with estimated control limits LCL = S w n,p,1 τ / n 1 p and UCL = S wn,p,α τ / n 1 p

44 33 the parameter for this geometric distribution is where α λ, m, n, p W = w S wn,p,1 τ = 1 P n 1 p < S < w w n,p,1 τ = 1 P m p n 1 p λ < W < w w n,p,α τ = 1 F W m p n 1 p λ S wn,p,α τ n 1 p W = w w w n,p,α τ m p n 1 p λ + F W w w n,p,1 τ m p n 1 p λ, W = n 1 Σ 1 S and W = m n 1 Σ 1 S. The ARL of the chart given W = w is ARL λ 1, m, n, p W = w = 1 F w w n,p,α τ W + F w w n,p,1 τ m p n 1 p λ W m p n 1 p λ = 1 F W 1 w w n,p,α τ m p n 1 p λ The unconditional ARL of the chart is then given by ARL λ, m, n, p = We have that the mean of W is + F W w w n,p,1 τ m p n 1 p λ. 1 f W w dw. 1 F w w n,p,α τ W + F w w n,p,1 τ m p n 1 p λ W m p n 1 p λ E W = p i=1 m n 1 i 1. which coverges in probability to m p n 1 p as m goes to infinity. Thus, as m increases, the unconditional ARL approaches the ARL when Σ is known. A Shewhart chart based on the ln S chart is equivalent to a chart based on S. Observe that at sampling stage t, the event defined by the inequality S wn,p,1 τ n 1 p < S < S wn,p,α τ n 1 p.

45 34 This inequality is equivalent to the inequality S w n,p,1 τ S w n,p,α τ ln n 1 p < ln S < ln n 1 p Hence, a Shewhart ln S chart with control limits S wn,p,1 τ S wn,p,α τ LCL = ln n 1 p and UCL = ln n 1 p is equivalent to the Shewhart S chart with control limits LCL = S wn,p,1 τ n 1 p and UCL = S wn,p,α τ n 1 p. Other measures of the performance of the chart are the standard deviation SDRL of the run length distribution and percentiles of the run length distribution. The SDRL for the chart with known parameters is. SDRL λ, n, p = 1 α λ, n, p. α λ, n, p where α λ, n, p = 1 F W wn,p,α τ λ + F W wn,p,1 τ λ. In the estimated parameters case, the conditional SDRL given W = w is SDRL 1 α λ λ, m, n, p W = w, m, n, p W = w =, α λ, m, n, p W = w where α λ, m, n, p W = w = 1 FW W w n,p,α τ m p n 1 p λ W = w W w n,p,1 τ + F W m p n 1 p λ W = w. The uncondition SDRL λ, m, n, p is SDRL λ, m, n, p = 1 α λ, m, n, p W = w f α λ W w dw,, m, n, p W = w

46 35 In general, the 1γth is the value T 1 γ of T that satisfies the following inequalities P T T 1 γ γ and P T < T 1 γ < γ. For a random variable T having geometric distribution with parameter α, it can be shown that P T > t = 1 α t. Hence, P T t = 1 1 α t and P T < t = 1 1 α t 1 for t > 1. The 1γth percentile T 1 γ of a geometric distribution is the solution to the set of inequalities 1 1 α T 1 γ γ and 1 1 α T1 γ 1 < γ. Observe that 1 γ 1 α T 1 γ and 1 γ < 1 α T 1 γ 1 implies ln 1 γ T 1 γ ln 1 α and ln 1 γ < T 1 γ 1 ln 1 α implies ln 1 γ ln 1 α T ln 1 γ 1 γ and 1 + ln 1 α > T 1 γ implies ln 1 γ ln 1 α T ln 1 γ 1 γ < 1 + ln 1 α. Thus, it follows that T 1 γ = ln 1 γ, ln 1 α where x is the smallest integer greater than or equal to x. In the known parameters case, we have T 1 γ = T n,p,λ,α,τ,1 γ = ln F W wn,p,α τ λ ln 1 γ FW wn,p,1 τ λ.

47 In the estimated parameters case, we see that the 1γth percentile can be expressed as 36 T 1 γ = T m,n,p,λ,α,τ,1 γ = ln 1 γ f W w dw ln F W w n,p,α τ W W m p n 1 p λ = w F W w n,p,1 τ W W m p n 1 p λ = w. 3.4 The Bivariate Case From the results given by Anderson 3 for p = under the independent multivariate normal model, we have W = n 1 Σ 1 S χ n 4 /4 and W = m n 1 Σ 1 S χ mn 1 /4 When Σ is known, then the ARL is determined by ARL = 1 F χ n 4 /4 χ n 4,α τ /4 1 λ + F χ n 4 /4, χ n 4,1 τ /4 λ where w n,,1 γ = χ n 4,1 γ /4. Note that F χ n 4 /4 y = P χ n 4 /4 y = P = F χ n 4 y. χ n 4 4y Thus, ARL = 1 F χ n 4 χ n 4,α τ /4 λ 1 + F χn 4 χ n 4,1 τ /4 λ.

48 The ARL can be numerically evaluated using the Scientific WorkPlace functions ChiSquareInv and ChiSquareDist with 37 χ ν,1 γ = ChiSquareInv γ; ν and F χ ν y = ChiSquareDist y; ν. A graph of the ARL versus λ is given in the following figure for n = 5, α =.5, and τ =.5 solid line along with the case in which τ =.38.

49 38 Figure 3.1: Plot of ARL versus λ α =.5, τ =.5 solid line and τ =.38 dashed line We can see for the case in which τ is chosen to be α/ there are values for λ in which the out-of-control ARL is greater than the in-control ARL. A chart that has this property is referred to an ARL biased chart. ARL biased charts were first studied by Krumbholz 199. An ARL unbiased chart in-control ARL greater than any out-of-control ARL can be designed by selecting τ such that ARL 1 > ARL λ for λ 1 λ >. As one can see from the Figure 3.1, selecting τ =.38 results in a chart that is close to being an unbiased chart. The method presented in Champ 1 could be used to determine the exact value τ that would result in an unbised Shewhart S chart. The following table gives the average run length, standard deviations of the run length, and various percentiles for a Shewhart S chart with a known value for Σ.

50 39 Table 3.1: ARL, SDRL, T 5,,λ,.5,.38,1 γ λ ARL SDRL When Σ is unknown and is to be estimated with S, then the run length performance of the Shewhart S chart is measured by the unconditional ARL which is given by Note that ARL = Thus, we can write ARL λ, m, n = f /4 w χmn 1 dw. 1 F w w n,,α τ χ n 4 + F w w n,,1 τ /4 m n 1 λ χ n 4 /4 m n 1 λ F χ mn 1 /4 y = F χ mn 1 y ; and f χ mn 1 /4 y = y 1/ f χ mn 1 y. 1 F χ n 4 w 1/ f χ mn 1 w dw + F χn 4 w χ n 4,α τ /4 m n 1 λ w χ n 4,1 τ /4 m n 1 λ.

51 4 The results in Table 3. illustrate that as m increases the unconditional ARL of the Shewhart S chart with estimated parameters increases to the ARL of the chart with parameters known. Table 3.: ARL,n = 5, α =.5, τ = α/, λ = 1 m ARL m ARL m ARL Obtaining a parameters estimated chart having the same in-control ARL as the parameters known chart for a given value of m can be done by decreasing the value α. For example, for m = 1 with n = 5, if one choose α =.395 with τ = α/, then the in-control value of the ARL of the chart is.1. But as we can see from Table 3.3 the chart will not on average detect changes in Σ. Table 3.3: ARLs,m = 1, n = 5, α =.395, τ = α/ λ ARLKnow Parameters ARLEstimated Parameters The following table gives the average run lengths, standard deviations of the run

52 length, and various percentiles for a Shewhart S chart when Σ is estimated with S. 41 Table 3.4: ARL, SDRL, for m = 1, n = 5, α =.5, and τ =.38 λ ARL SDRL Both in-control and out-of-control ARLs are less than their counterparts when Σ is assumed known. 3.5 AN EXAMPLE Montgomery 1 gives an example in which tensile X 1 and diameter X of a textile fiber are the quality measurements of interest. At each sampling stage, random samples of size n = 1 are taken from the output of the process. The practitioner has available m = samples each of size n = 1 for the process when the process was believed to be in-control. The estimates for the in-control mean vector and covariance

53 4 matrix using these data are x = x 1 x = and S = It follows that S = = For α =.435 with τ = α/, the unconditional average run length of the chart is The control limits for the Phase II Shewhart S chart are LCL = S wn,p,1 τ / n 1 p = =.3968 χ1 4,1.435/ / ChiSquareInv.435/; =.4 and UCL = S wn,p,α τ / n 1 p =.3968 χ1 4,1.435/ /4 1 1 = The center line CL for this chart is S.3968 CL = n 1 n = =.353. n The following table gives summary information from the process in Phase II.

54 43 Table 3.5: Phase II Data Summary t x 1 x s 1 s s 1, S Unknown to the practitioner, the variance for measurement X made a substained shift at time t = 6 by a value of 1.5. The following plot of the sample generalized variance versus the sample number indicates there may have been a change in the process at time t = 8, but the chart did not signal.

55 44 Figure 3.: Phase II Plot of S versus t n 1 It would be interesting to see how a CUSUM ln 1 S S chart would n 1 compare to Shewhart ln 1 S chart which is equivalent to Shewhart S with estimated parmaeters. S 3.6 Conclusion Control charting procedures with a center line and control limits were introduced by Walter A. Shewhart see Shewhart 1931 and have been referred to as Shewhart charts. Over time there have bee several Shewhart charts introduced into the literature by a variety of authors. The Shewhart generalized variance chart is one of those. We have show that the Shewhart S and ln S charts are equivalent in the sense that when one of the charts signals the other also signals a potential out-of-control process. Also, we have show how to analyze each of these charts when parameters are estimated.

56 CHAPTER 4 GENERALIZED VARIANCE CUSUM CHART 4.1 Introduction Page 1956 in his seminal work proposed the cumulative sum CUSUM quality control chart for monitoring for a change in the process mean. Ewan and Kemp 196 recommended a tabular form of the CUSUM consisting of a lower and upper sided CUSUM sum procedures. In the design of a CUSUM chart in which the statistic whose values are to be acumulated is always positive, one must adjust their method. For the lower-sided CUSUM chart base on the generalized variance S with head start value c as proposed by Lucas 1985, the CUSUM statistic has the form C = c and C t = min {, C t 1 + S t k }, for t = 1,, 3,..., where k >. The chart is a plot of the points t, C t for t = 1,, 3,.... The chart signals at the first sampling stage t in which C t h. The head start value is selected to be in the interval h, ]. The values c, k, and h are referred to as chart parameters. The upper-sided CUSUM chart based on the generalized variance S with head start value c + is a plot of the points t, C + t, where C + = c + and C + t = max {, C + t 1 + S t k +}, for t = 1,, 3,..., where k + >. The chart is a plot of the points t, C + t for t = 1,, 3,.... The chart signals at the first sampling stage t in which C + t h +. The head start value c + is selected to be in the interval [, h +. The two-sided CUSUM chart based on the generalized variance S is the chart in which the points t, Ct and t, C + t are plotted on the same graph. Note that setting h and h + both to zero in the two-sided CUSUM S results in a Shewhart S chart with LCL = k and UCL = k +. That is the Shewhart chart is a special case of the CUSUM chart.

57 Healy 1987 used what he called rescaling to obtain an equivalent form of the chart in the sense that the charts either both signal or not at each time t. One rescaling of the lower and upper CUSUM S charts is accomplished by defining C t = n 1 Σ 1 Ct and Ct + = n 1 Σ 1 C t The CUSUM statistics and chart parameters for this rescaled chart become with C = c and Ct = min {, Ct 1 + n 1 Σ 1 S } t k and C + = c + and Ct + = max {, Ct n 1 Σ 1 S } t k + c = n 1 Σ 1 c, k = n 1 Σ 1 k, h = n 1 Σ 1 h, c + = n 1 Σ 1 c +, k + = n 1 Σ 1 k +, and h + = n 1 Σ 1 h +. If the in-control covariance matrix Σ is unknown, then the rescaling is done by replacing Σ with S. Note that we can express n 1 Σ 1 S t = Σ 1 Σ n 1 Σ 1 S t = λ n 1 Σ 1 S t = λ W t, when parameters are known, where W t = n 1 Σ 1 S t. In the parameters estimated case, we see that n 1 S 1 S = m p n 1 p Σ 1 Σ m n 1 Σ 1 S 1 n 1 Σ 1 S = m p n 1 p m n 1 Σ 1 S 1 λ n 1 Σ 1 S = m p n 1 p W 1 λ W t 1 W = λ m p n 1 p W t,

58 where W = m n 1 Σ 1 S. We see that replacing m p n 1 p W 1 with 1 in the parameters estimated case gives the parameters known case. Another CUSUM chart that could be used to monitor for a change in the process generalized variance is a CUSUM chart based on the ln S or equivalently the ln n 1 Σ 1 S n 1 if Σ is known or ln 1 S when Σ is unknown. The S lower and upper CUSUM ln S charts are defined by 47 C = c and C t = min {, C t 1 + ln S t k } and C + = c + and C + t = max {, C + t 1 + ln S t k +}. Each of these charts are equivalent to a CUSUM ln n 1 Σ 1 S t chart when n 1 Σ is known and a CUSUM ln 1 S S t when Σ is estimated by S. This can be seen to hold, for example, for the upper one-sided CUSUM ln S t chart by observing that the statistic C + t can be expressed as C + t = max {, C t ln n 1 Σ 1 S t k + + ln n 1 Σ 1 }. = max {, C t ln n 1 Σ 1 S } t k + Defining k + = k + + ln n 1 Σ 1, the upper one-sided CUSUM ln n 1 Σ 1 S t chart is defined by the sequence C + = c + and C + t = max {, C + t 1 + ln n 1 Σ 1 S t k + } for t = 1,, 3,.... This would hold for the lower one-sided chart and the charts with estimated parameters. Note that ln n 1 Σ 1 S t = ln λ n 1 Σ 1 S t

59 Other CUSUM charts for monitoring for a change in the process covariance matrix have been proposed. A chart derived by Healy 1987 after rescaling is based on the CUSUM statistics C i = max {, C i 1 + n } X i,j µ T Σ 1 X i,j, µ k, j=1 where k = pnc ln c / c 1 and C < h. This chart was designed specifically to detect a change in Σ from Σ to cσ for c > given. Pignatiello, Runger, and Korpela 1986 proposed using the CUSUM chart to monitor for a change in the mean vector to monitor for a change in the covariance matrix. Crosier 1986 proposed using the CUSUM chart based on the CUSUM statistics { n } C i = max, C i 1 + X i,j µ T Σ 1 X i,j, µ k, j=1 when n = 1 without the restriction Healy 1987 placed on the chart parameter k. It is not difficult to show that all the CUSUM charts discussed in this section have as a special case a Shewhart chart Run Length Distribution The run length T of a CUSUM chart is the first sampling stage t in which the chart first signals a potential out-of-control process. The three most commonly used methods for evaluating the run length distribution are 1 simulation, approximating the chart as a discrete state Markov chain, and 3 expressing a parameter of the run length distribution, such as the ARL, as the solution to an integral equation. Champ and Rigdon 1991 demonstrated that the Markov chain and the integral equation approach for the CUSUM X chart are equivalent. Champ, Rigdon, and Scharnagl 1 derive various integral equations whose exact solutions are parameters of the run length distribution. Recall that the lower