On the run length of a Shewhart chart for correlated data - Springer Link

Statistical Papers 36, 111 130 (1995)

Statistical Papers

-

(~) Springer-Verlag1995

On the run length of a Shewhart chart for correlated data W. Schmid Received: Nov. 5, 1993; revised version: Jan. 26, 1994

We consider an extension of the classical Shewhart control chart to correlated d a t a w h i c h w a s i n t r o d u c e d by V a s i l o p o u l o s / S t a m b o u l i s (1978). I n e q u a l i t i e s f o r t h e m o m e n t s o f t h e r u n l e n g t h are given u n d e r w e a k c o n d i t i o n s . I t is p r o v e d a n a l y t i c a l l y t h a t t h e a v e r a g e r u n l e n g t h (ARL) in t h e i n - c o n t r o l s t a t e o f t h e c o r r e l a t e d p r o c e s s is l a r g e r t h a n t h a t in t h e c a s e o f i n d e p e n d e n t v a r i a b l e s . T h e e x a c t ARL is c a l c u l a t e d f o r e x c h a n g e a b l e n o r m a l v a r i a b l e s a n d a u t o r e g r e s s i v e p r o c e s s e s (AR). M o r e o v e r , we c o m p a r e t h i s c h a r t w i t h r e s i d u a l c h a r t s . Especially, in t h e c a s e o f a n A R ( t ) - p r o c e s s w i t h p o s i t i v e c o e f f i c i e n t , it t u r n s o u t t h a t t h e out-of-control ARL o f t h e m o d i f i e d S h e w h a r t c h a r t is s m a l l e r t h a n t h a t o f t h e Shewhart chart for the residuals. ~FW~lL

1.

s t a t i s t i c a l p r o c e s s c o n t r o l , S h e w h a r t c h a r t , r u n l e n g t h , c o r r e l a t e d data.

Znt.rodu.c~tJ.on A l o t o f s t a t i s t i c a l c o n t r o l c h a r t s have b e e n i n t r o d u c e d in t h e l i t e r a t u r e ,

e.g. t h e S h e w h a r t ,

EWMA,

CI/SUM

charts

(cf.

M o n t g o m e r y (1991)). The m o s t

w i d e l y u s e d c o n t r o l c h a r t is w i t h o u t d o u b t t h e s t a n d a r d S h e w h a r t c h a r t . Let {Yt} d e n o t e t h e i n - c o n t r o l

p r o c e s s a n d {Xt} t h e o u t - o f - c o n t r o l

pro-

c e s s . Now a v a r i e t y o f d e p a r t u r e s f r o m t h e i n - c o n t r o l s t a t e are p o s s i b l e . H e r e we c o n s i d e r e x c l u s i v e l y t h e c a s e of a n a b r u p t s t e p - l i k e s h i f t ; i.e. it is a s s u m e d t h a t X t : Y t + a I{q,q+l,..}(t),

(1.1)

w h e r e q c b/ d e n o t e s t h e p o s i t i o n a n d a t h e size of t h e c h a n g e . I A s t a n d s f o r t h e i n d i c a t o r f u n c t i o n of t h e s e t A. In t h e c l a s s i c a l S h e w h a r t c h a r t f o r t h e c a s e of k n o w n p a r a m e t e r s t h e p r o c e s s is out-of-control

a t t i m e t, if J X t - ~ o i

> cd, where {Io::E(Yt) , d2:=Var(Yt ) and

c > 0 is a s u i t a b l e c o n s t a n t u s u a l l y t a k e n t o b e 3.

112 A f u n d a m e n t a l a s s u m p t i o n in m o s t s t a t i s t i c a l process c o n t r o l m e t h o d s is t h a t t h e o b s e r v a t i o n s are independent. However, this c o n d i t i o n is n o t f u l f i l l e d f o r

many d a t a s e t s ( s e e e.g. B e r t h o u e x et al. (1978), N o t o h a r d j o n o / E r m e r

(1986), M a c -

G r e g o r / Harris (1990)). In t h e l a s t years several a u t h o r s have d i s c u s s e d the impact o f a c o r r e l a t i o n s t r u c t u r e on t h e behaviour o f the Shewhart, EWMA and C U S U M charts, e.g. B e r t h o u e x e t al. (1978), B a g s h a w / J o h n s o n (1974/5), Yashchin (1989), H a r r i s / R o s s (1991). If it is a s s u m e d t h a t the r a n d o m variables are independent, when in f a c t t h e y are correlated,

it has b e e n s h o w n

via simulations

(e.g.

Montgomery/Mastrangelo

(1991), M a r a g a h / W o o d a l l (1992)) t h a t the classical 30 c o n t r o l limits in S h e w h a r t c h a r t s are n o t s u i t a b l e due to the high frequency of f a l s e alarms. This is n o t s u r p r i s i n g since e.g. in the case of a linear process, i.e. Yt =

~ V~_~

Y. [Sv] 0 , where 8k2:= d e t C k / ( Y o d e t C k 1) and C O := I. It f o l l o w s for x ~ ~I a n d w i t h

z := r

- r

Eo(N•




particularly Eo(N)

v=l

S (v) v! (F(c) - F ( - c - 0 ) ) v - 1 / ( 1 - ( F ( c ) - F ( - c - 0 ) ) ) v , x

(2.7)

> I/( I - (F(c) - F(-c-O) ) ) ,

where F denotes the distribution of the standardized variable (Yi-[1o)/VV~o. IFroof. Let c a := c i/~o. (2.6) implies that Po (N > k ) = Po ( IY 1-11oI~ ca ..... IY k - 1101~ ca )

~ Po ( IYi-lZo[ c~, since (o)

Eo(inf{n,~l:tXnl>C ~ Thus,

if c is d e t e r m i n e d

(o)

~/~o }) ~ Eo(N ~ via tables

) = ~ = Eo(N~).

for independent

variables,

the

resulting

i n - c o n t r o l ARL is l a r g e r than [, provided t h a t the variables are c o r r e l a t e d . lip

to

now

we

have only discussed

the behaviour

of the

ARL in the

i n - c o n t r o l state. N e x t we give a r e s u l t for the o u t - o f - c o n t r o l

situation. In the

following

means

we

shall

always

assume

that

out-of-control

that

a

c h a n g e - p o i n t is p r e s e n t ( s e e model (1.1)). 2,3.

Let {Yt } be a s t o c h a s t i c

process.

Var(Yt) = ~'o ~ (0, oo) for all t c N. F u r t h e r m o r e

Assume

that

E(Y t) =~o and

l e t all marginal d i s t r i b u t i o n s

of

{Yt-[~o} have a c o n t i n u o u s density f which is s y m m e t r i c a b o u t the origin (i.e. f(x) = f ( - x ) for all x ) and, additionally, let f be unimodal ( c f Tong (1980, p.51)), i.e. t h a t {x: f ( x ) > ) . } is convex for all ),>0. Moreover, let •

Then it f o l l o w s

t h a t f o r alt a, q E a,q ( N x) < Eo( N • and t h a t for q fixed E a,q (N ~) is a nonincreasing function in lal. l~roof. Let c * : = c X~'~o , Zl : = Y I - " O and 7-k := (Z1 ..... Zk)L It f o l l o w s w i t h Tong (1980, T h e o r e m 4.1.1) that for lal ~ I~1 P

a,q ( N > k )

= Po(-C

= P

a,q

k ) = P o ( i Y I + a [ < ~ ..... { Y k + a [ < ~ )

and an application of the

t r a n s f o r m a t i o n r u l e for densities s h o w s t h a t for k z p+l and w i t h c~0 : = - I p p f(Y1 . . . . Vk )(zl . . . . Zk) = f(v1 .... yp, Ep§ k

.. , Ek)(Z I .... ZP ' - v=O ~" •v Zp+l-v ......

-vffiO ~'~ 0Iv Z.K_v)

p

no, c p (Zp) v=p§ ]'[ n((z v - iffil ~ a i zv-I )/0), t h e r e s u l t follows. N o t e t h a t Eo(N) d o e s n o t depend on ~. C o e o l J ~ y . Let {Yt} be as described in Theorem 4.1 and p= 1, t h e n the i n - c o n t r o l average run l e n g t h does not change, if al is replaced by - % . ]Proof. We d i s t i n g u i s h

b e t w e e n k even resp. odd. If k is even, we make t h e

s u b s t i t u t i o n tv = -Zv for v odd; else, if k is odd, we s u b s t i t u t e tv = - z v for v even. Now we c o n f i n e o u r s e l v e s to t h e case of an AR(1)- process. &.2. Let { Y J

be a causal

solution

of Y t = a l Y t _ l +

~t" A s s u m e

that

~1 ~ [0,1) and t h a t the r a n d o m variables Et, t ~ Z , are i n d e p e n d e n t and identically distributed

with

~t~No,o2

for

all

t,

where

O c~(pr'pr'r dr) 3/~O }.

4L3. Suppose t h a t the a s s u m p t i o n s of Theorem r

are satisfied. Let P,

~., and ~ d e n o t e e s t i m a t o r s of p, ~_p and o which are calculated via {Yt}. Then it P

f o l l o w s with c~-:=c~(Pr'Cc-*r v / v - p *' ~ ~ , 'o

that

^ > k J p^ =p r ,^ OCpr= 0Cpr, Pa.l(N r 8=or)

k

p

i"[ n(z v- ~ oclzv_|) dZk.. dz I ,

f ( - c ~r - a ) / ~ . (

co : 1 + ~. k=l

cr~-

_ ~| [ (_cr_a)/~,(

....

_ J cr_a)/o]

vffip+l

T

a)/~] k

,_ ~

n

~ (z~)

O,C~-I 1

1=1

k p 1"[ n ( z - ~.cc.z v=p§

v

1-1 -

n

v-a

.)dz...dz.. g

l

l~NlOf. F o l l o w s with the same a r g u m e n t s as Theorem r A c o m p a r i s o n of Theorem 4.1 and r

shows that in the case ~ ~ c~r the true ARL

128 is u n d e r e s t i m a t e d , o t h e r w i s e o v e r e s t i m a t e d , p r o v i d e d t h a t c is c h o s e n s u c h t h a t Eo(N)

=

~.

In Table 9 - 1 1 we s h o w w h a t h a p p e n s , if t h e p a r a m e t e r s are e s t i m a t e d . ^

^

It

tt

T l l b l l 9, Ea,l( N I ~ = 1, a 1 = a t, 8 = 1) f o r various values o f a and r 1 a n d c as in Table 4 { m . S h e w , q = p = l , etI

a1=0.5,

Etch, M=S0000)

0

0.5

1.0

1.5

2.0

2.5

3.0

0.4

298.59

t61.82

60.18

24.82

1t.72

6.17

3.60

0.45

379.02

197.72

71.19

28.48

13.19

6.83

3.92

0.5

499.90

252.78

87.15

33.70

15.24

7.72

4.35

0.55

709.84

345.93

113.43

42.06

18.39

9.03

4.98

0.6

1213.32

560.48

168.95

59.23

24.43

11.54

6.08

/

Tdde

a

A

A

10, Ea,~( N I p = 1, etI = ct1 , 8 = 1 )

f o r various values o f a and a t

and

c as

T a b l e 4 ( S h e w r e s , q = p = 1, a 1 = 0 . 5 ,

Et~r

a1

1.0

1.5

2.0

2.5

3.0

/

a

0

0.5

in

, M=50000)

0.4

469,98

319.17

150.33

70.31

35.35

19.48

11.63

0.45

494.97

351.84

175.87

86.24

44.23

24.67

14.64

O.S

500.00

374.46

202.23

103.98

55.51

31.38

18.87

0.55

494.88

390.09

227.33

124.58

70.11

40.68

24.99

0.6

470.86

390.40

247.69

146.52

86.85

54.83

33.34

T d b l e tt- Ea,1(bll~=l,~1=~,8=l)

for various values of a and a~ and c as in

T a b l e 4 ( E W M A r e s , X=0.75, q = p = l , ct1

/

a

a1=0.5, ~ t ~ ~ , M = 5 0 0 0 0 )

0

0.5

1.0

1.5

2.0

2.5

3.0

0.4

354.01

211.38

86.17

38.25

19.22

11.13

7.26

0.45

425.65

262.04

109.91

48.71

24.32

13.74

8.69

0.5

498.87

320.14

140.90

63.50

31.62

17.52

10.92

0.55

565.51

381.41

179.78

83.94

42.41

23.31

14.!7

0,6

613.70

442.76

225.22

111.07

57.71

32,26

19.42

The r e s u l t s o f T a b l e 9-11 are a b i t s u r p r i s i n g . It can b e s e e n t h a t t h e e s t i m a t i o n

129 of p a r a m e t e r s may have a s t r o n g influence on the ARE The ARL o f t h e m o d i f i e d S h e w h a r t chart increases, if the t r u e p a r a m e t e r value cr is o v e r e s t i m a t e d , o t h e r w i s e it decreases. A similar behaviour can be o b s e r v e d for an E W M A c h a r t f o r the residuals ( T a b l e 11) and for the S h e w h a r t c h a r t for t h e residuals ( Table 10, e x c e p t the case a = 0 ). It is r e m a r k a b l e t h a t t h e i n - c o n t r o l ARL of m.Shew, changes dramatically, even if t h e c o e f f i c i e n t ~1 is e s t i m a t e d quite precisely. This e f f e c t also o c c u r s f o r E W M A res, b u t it is c o n s i d e r a b l y smaller. In comparison, Shewres r e a c t s nearly r o b u s t to deviations in the i n - c o n t r o l state. In s p i t e o f this behaviour, it m u s t be emphasized t h a t the o u t - o f - c o n t r o l

ARL of

m.Shew, is the s m a l l e s t of all three charts, if a > 1. This s h o w s t h a t in r e l a t i o n to the other charts, the A R L known We

parameters E W M A r e s

of m.Shew, decreases more rapidly. As in the case of is better than Shewres, provided that a > I.

think that these simulations s h o w that the influence of parameter esti-

mation on the A R L

has to be studied in more detail in future. This is a crucial

point, especially in connection with correlated data, which was not treated up to n o w in literature.

Ac.knowledg~ffi~t, I

w o u l d like to thank Prof. H. W o l f f f o r calling my a t t e n t i o n

to this subject and t h e r e f e r e e for helpful c o m m e n t s on a previous version of this paper.

~

m

A b r a m o w i t z , M.; Stegun, I.A. (1984). Pocketbook of M a t h e m a t i c a l Functions. Harry Deutsch, Frankfurt. Berthouex, P.M.; Hunter, W.G.; Pallesen, L. (1978). Monitoring sewage treatment p l a n t s : s o m e quality c o n t r o l aspects. J. Quality C o n t r o l 10(4), 139-149. Box, G.E.P.;Jenkins, G.M. (1976). Time Series Analysis - F o r e c a s t i n g and C o n t r o l . H o l d e n - Day, San Francisco. Brockwell, P.J.; Davis, R.A. (1991). Time Series Analysis. S p r i n g e r - Verlag. Das

Gupta, S.; Eaton, M.L.; Olkin, i.; Perlman, M.; Savage, L J . (1972). I n e q u a l i t i e s on t h e probability c o n t e n t of convex regions f o r elliptically c o n t o u r e d d i s t r i b u t i o n s . In: Sixth Berkeley Symposium, Vol. II, 241-265. C a m b r i d g e U n i v e r s i t y Press, London.

130 Hansen, E.R. (197S). A Table of Series and Products. Prentice-Hall. Harris, TJ.; Ross, W.H. (1991). Statistical process control procedures for correlated observations. Canadian J. Chemical Engineering 69, 4 8 - $ 7 . Johnson, R.A.;Bagshaw, M. (1974). The effect of serial correlation on the performance of CUSUM tests. Technometrics 16 (1), 103-112. Johnson, R.A.;Bagshaw, M. (197S). The effect of serial correlation on the performance of CUSUM tests II. Technometrics 17 (1), 7 3 - 8 0 . Lu, C.W.; Reynolds, M.R. (1992). An EWMA control chart on the residuals of a time series model. Proc. Sect. Quality and Productivity, ASA, 101-105. Lucas, J.M.;Saccucci, M.S.(1990). Exponentially weighted moving average control schemes: properties and enhancements. Technometrics 32 (1), l - 1 2 . MacGregor, J.F.;Harris, TJ. (1990). Discussion of "Exponentially weighted moving average control schemes: properties and enhancements". Technometrics 32(1), 23-26. MacGregor, J.F.; Harris, T.J. (1993). The exponentially weighted moving variance. J. Quality Technology 2S (4), 106- 118. Maragah, H.D.; Woodall, W.H. (1992). The effect of autocorrelation on the retrospective ~ - chart. J. Statist. Comput. Simula. 40, 29 - 42. Montgomery, D.C. (1991). Statistical Quality Control. Wiley, New York. Montgomery, D.C.; Mastrangelo, C.M. (1991). Some statistical process control methods for autocorrelated data. J. Quality Technology 23(3), 179-193. Notohardjono, D.; Ermer, D.S. (1986). Time series control charts for correlated and contaminated data. J. Engineering for Industry I08, 219-226. Ryan, T.P.(1991). Discussion of "Some statistical process control methods for autocorrelated data". J. Quality Technology 23 (3), 2 0 0 - 202. Tong, Y.L. (1980). Probability Inequalities in Multivariate Distributions. Academic Press. Tong, Y.L. (1990). The Multivariate Normal Distribution. Springer-Verlag. Vasilopoulos, A.V.; Stamboulis, A.P. (1978). Modification of control chart limits in the presence of data correlation. J. Quality Technology I0 (I), 2 0 - 3 0 . Yashchin, E. (1989). Performance of CUSUM control charts for serially correlated observations. IBM Research Report.

Wolfgang Schmid Universit~it Ulm, Abteilung Stochastik D - 89069 Ulm