Bivariate EWMA Control Charts for Autocorrelated Processes

J ournal of K or ean

D a ta & I nf orma t ion S cience S ocie ty 2 002 , Vol. 13, N o.1 p p . 105～ 112

B iv ari at e E W M A C o n t ro l Ch art s f o r A u t o c o rre lat e d P ro c e s s e s

Gy o - Y ou n g Ch o ¹⁾ an d Y oun g S un A h n ^{2 )}

A b s t ra c t

In t his p aper w e e st ablish biv ar iat e ex pon en tially w eig ht ed m ov in g av er ag e (E W M A ) cont r ol ch ar t s for aut ocor r elat ed pr oces ses u sin g r esidu al v ect or s . W e fir st deriv e t h e r esidu al v ect or s , th eir ex pect ation , v ar ian ce - cov ar ian ce m atr ix , t h en ev alu at e t h e cont r ol ch ar t b a sed on t h e av er ag e r un len gt h (A RL ).

K ey w ords : Biv ariat e EWMA Control Chart , Residual Vect ors , ARL

1. In trodu c tion

In st at istical pr oce s s cont r ol, it is u su ally a s su m ed on t h e pr oces s ou tput at differ en t t im es ar e IID . H ow ev er , for m any pr oces s es t h e ob ser v at ion s ar e cor r elat ed an d con tr ol ch ar t s for m onit or in g t h e se pr oces s es h av e r ecen t ly r eceiv ed m u ch at t ent ion . M any aut h or s su g g est ed fitt in g on appr opr iat e t im e ser ie s m odel t o t h e pr oces s an d u sin g r esidu als a s con tr ol st at istic on a con tr ol ch ar t (A br ah am an d K ar th a (1979 ), A lw an an d Rob er t s (1988 ), Y our st on e an d M ont g om er y (1989 ), H ar r is an d Ros s (1991), B ox an d R am ir ez (1992), L on gn eck er an d Ry an (1992), Y a schin (1993 ), W ar dell et al.(1993 ), an d Run g er et al.(1995 )).

T h er e ar e m an y sit u ation s in w h ich t h e sim ult an eou s con tr ol of t w o or m or e r elat ed qu alit y ch ar act er ist ics is n eces sar y so, m any aut h or s su g g est ed m ult iv ar iat e con tr ol ch ar t s (A lt (1985 ), J ack son (1985 ), L ow r y an d M ont g om er y (1995 ), M a son et al (1997 )). L ow r y et al (1992) stu died m u lt iv ar iat e E W M A ch art un der pr oces s

1. A ssociat e Profes sor , Departm ent of st atistics , Kyun gpook National Univ ersity , Daegu , 702- 701, Korea

E - m ail : gycho@knu .ac .kr

2. P art - tim e In struct or , Departm ent of St atistics , Kyungpook Nation al Univer sity , Daegu ,

702- 701, Kor ea

out pu t at differ ent tim es ar e IID .

A u t ocor r elat ion is quit e com m on in in ch em ical an d pr oces s in du st r ies an d oft en sev er al pr oce s s v ar iable s ar e m ea sur ed an d st or ed sim u lt an eou sly . Run g er (1996 ), M a son et al (1996), Bak shi (1998 ), N egiz an d Cin ar (1998) st u died m u lt iv ar iat e cont r ol ch ar t s for aut ocor r elat ed pr oces s es .

Ou r obj ectiv e is t o ev alu at e th e pr op er ties of m u lt iv ar iat e E W M A ch ar t w h en t h e r esidu al v ect or s ar e u s ed t o m on it or an aut ocor r elat ed pr oces s . A sim ple v ect or tim e s er ies m odel is u sed h er e t o r epr esen t t h e ob ser v at ion s fr om an aut ocor r elat ed pr oces s .

2 . M o de lin g an d Controllin g th e P roc e s s

2 .1 V e c t or T im e S e rie s P ro c e s s M e as u re d w ith R an dom Error V e c t or

In t his section , a v ect or t im e ser ie s pr oces s w it h r an dom er r or v ect or w ill b e dev eloped . L et

x

=

+

, (2.1)

w h er e

is t h e ob s er v at ion v ect or at tim e t,

is th e pr oce s s m ean v ect or at t im e t, an d

is t h e m ea sur em en t er r or v ect or at t im e t.

It is also a s su m ed t h at

can b e descr ib ed a s a v ect or fir st or der aut or eg r es siv e (v ect or A R (1)) pr oces s w it h pr oces s lev el , t h at is (Box an d J en kin s (1976 )),

t

= ( - ) +

+

, (2.2)

w h er e

,, t h e r an dom sh ock v ect or at t im e t, is a s su m ed t o b e m u lt iv ariat e n or m ally dist r ibu t ed m ean v ect or 0 an d v ar ian ce - cov an an ce m at rix , in dep en den t of m ea sur em en t er r or v ect or , an d in dep en den t of t h e r an dom sh ock v ect or at any ot h er t im e.

T h e v ect or A R (1) pr oce s s ob s er v ed w it h m ea su r em ent er r or v ect or is equ iv alen t t o a v ect or fir st - or der an t or eg r e s siv e fir st - or der m ov in g av er ag e (v ect or A RM A (1,1)) pr oces s (Box an d J en kin s (1976 )). T h e A RM A (1, 1) pr oces s can b e w r itt en a s

( - )

= ( - ) + ( - ) a

(2.3)

w h er e B is a b a ck shift op er at or su ch t h at B x

= x

，an d a

is u n cor r elat ed

an d m ult iv ar iat e n or m ally distr ibut ed w ith m ean v ect or 0 an d v ar ian ce - cov ar ian ce m at r ix a an d is t h e m ov in g - av er a g e par am et er m at r ix an d is t h e au t or egr es siv e par am et er m atr ix .

2 .2 S t e p Ch an g e in th e P ro c e s s Lev e l

W h en t h e pr oces s is in - con tr ol ( =

) , m in im um M S E for eca st v ect or m ade at tim e t for t + 1 is

x

=

+ ( x

-

) - e

w h er e e

= x

- x

= x

-

- ( x

-

) + e

an d e

is th e r e sidu al v ect or at tim e t. S u ppose t h at th er e is a st ep ch an g e fr om

t o

in t h e pr oces s lev el b et w een t im e t = k - 1 an d k . T h en th e pr oces s can b e w r it t en a s

x

= {

^-

^{+ x}

^{+ a}

^{- a}

^, t = k - 1, k - 2 ,

1

-

+ x

+ a

- a

, t = k

1

-

+ x

+ a

- a

, t = k + 1, k + 2 ,

an d t h e ex pect ation of th e ob s er v at ion v ect or x

is

E ( x

) = {

t = k - 1, k - 2 ,

1 ,

t = k , k + 1, k + 2 , T h er efor e t h e r esidu al v ect or s ar e

e

= { ^{( [ ( I + a}

^,

^{- +}

^{) +}

^a

⁺

^{)( I - ) +}

^](

-

) + a

^{t< k t = k} t = k + l, l = 1, 2 , 3 , S in ce a

is w hit e n oise, w e h av e

E ( e

) = { ^{0 ,} t = k - 1, k - 2 ,

1

-

, t = k

[ ( I + +

+ +

)( I - ) +

](

-

) , t = k + 1, k + 2 ,

3 . M ultiv ari at e EW M A Ch art of R e s i du al V e c t ors

T h e M ultiv ar iat e E W M A con t r ol st at ist ic u sin g r esidu al v ect or s is Y

= ( 1 - ) Y

+ W

0 < 1

w h er e W

= [ e

- E ( e

) ] '

[ e

- E ( e

) ] .

T h e con tinu ou s - st at e M ar k ov ch ain is discr et ized by div idin g th e in t erv al b et w een 0 an d t h e cont r ol lim it L int o n su bin t erv als of w idth w = L / n . L et S

b e t h e m idp oint of th e j th in t er v al, j = 1, 2,・・・, n . S o, Y

is in t r an sien t st at e j at t im e t if S

- w

2 < Y

< S

+ w

2 . Br ook an d E v an s (1972) h a s sh ow n t h at t h e A RL v ect or N w h en t h e pr oces s is in cont r ol is g iv en by

N = ( I - R )

1 ,

w h er e th e jt h elem ent of N r epr esen t s t h e A RL for t h e pr oce s s st ar t s fr om st at e j . T r an sition pr ob abilit y m at r ix , r epr esent ed in par t it ion ed m atr ix form , for th is M ark ov ch ain can b e w r itt en a s

P = ( p

) = P (g oin g t o st at e k | in st at e j )

=

R ( I - R ) 1

0

1

,

w h er e t h e subm at r ix R is a ( n n ) m at rix w hich con t ain s t h e tr an sient pr ob ab ilit y of g oin g fr om on e t r an sien t st at e t o an ot h er , I is a ( n n ) ident it y m at r ix , 0 is a colu m n v ect or of zer os an d 1 is th e colum n v ect or of on e s .

4 . N u m e ric al R e s u lt s an d Con c lu s ion s

T h e par am et er s , , an d in t h e t im e s er ies m odel can b e est im at ed fr om t h e pr elim in ar y dat a . F or sim plicity , supp os e th at th ose par am et er s ar e k n ow n . T h e follow in g con t r ol pr ocedu r es w ill b e com par ed on t h e b a sis of th eir A RL per for m an ces .

(1) =

1 1

, = 0.2, 0.5, 0.8

(2) =

0 . 2 0 . 1 0 . 1 0 .2

, =

0 . 3 0 . 1 0 . 1 0 . 3

, =

0 . 8 0 . 5 0 . 5 0 . 8

,

=

0 . 9 0 . 8 0 . 8 0 . 9

(3)

= ( ) 0 ¹ ,

= ( ) ⁰ 1 ,

= ( ) ^{- 1} 1 ,

= ( ) ² 0

W h en com par in g con tr ol ch ar t s , som e k in ds of st an dar d for com p ar ison is n eces sar y . T h e ch ar t s ar e m at ch ed for A RL w h en th e pr oces s is in con t r ol. T his en ables th e per for m an ce t o b e ev alu at ed w h en th e pr oces s h a s shift ed aw ay fr om it s t ar g et v alu e. In our com put at ion , t h e A RL in cont r ol w a s fix ed t o b e 200.

T ables 1- 4 sh ow t h at A RL of th e biv ar iat e E W M A cont r ol ch ar t s decr ea se a s in cr ea s es , an d decr ea s es a s an d in cr ea se.

In con clu sion , w e dr iv e r esidu al v ect or , v ar ian ce - cov arian ce m atr ix an d t h en con st ru ct m u lt iv ar iat e E W M A con tr ol st at istic for aut ocor r elat ed pr oces s . A ls o w e ev alu at e biv ar iat e E W M A con t r ol ch ar t s b a s ed on t h e A RL an d ident ify t h e r elat ion ship am on g , , an d A RL .

(T able 1) A RL v alu es for m ultiv ar iat e E W M A ch ar t s b a s ed on r esidu al v ect or s w it h =

0 .2 0 . 1 0 . 1 0 . 2

an d =

0 . 3 0 . 2 0 . 2 0 . 3

' =0.05 =0.1 =0.3

( 1 0 )

0.2 0.5 0.8

38.458 30.265 30.840

26.303 19.002 16.663

11.602 7.172 4.014

( 0 1 )

0.2 0.5 0.8

25.249 18.105 16.022

17.270 11.622 8.681

7.619 4.740 2.634

(- 1 1 )

0.2 0.5 0.8

9.362 5.794 3.192

5.778 3.317 1.595

1.842 1.077 1.001

( 2 0 )

0.2 0.5 0.8

10.957 6.706 3.672

6.843 3.786 1.719

2.002 1.093 1.001

(T able 2) A RL v alu es for m ultiv ar iat e E W M A ch ar t s b a s ed on r esidu al v ect or s w it h =

0 . 3 0 . 1 0 . 1 0 .3

an d =

0 . 3 0 . 2 0 . 2 0 . 3

' =0.05 =0.1 =0.3

( 1 0 )

0.2 0.5 0.8

72.258 64.940 71.774

8.682 5.926 4.013

26.107 18.670 15.979

( 0 1 )

0.2 0.5 0.8

50.372 42.186 45.680

8.202 5.629 3.792

17.419 11.636 8.568

(- 1 1 )

0.2 0.5 0.8

21.365 14.729 11.673

6.532 4.278 2.615

5.247 2.748 1.276

( 2 0 )

0.2 0.5 0.8

24.354 17.044 14,059

6.298 4.072 2.465

6.191 3.079 1.347

(T able 3 ) A RL v alu es for m ultiv ar iat e E W M A ch ar t s b a s ed on r esidu al v ect or s w it h =

0 . 8 0 .5 0 . 5 0 .8

an d =

0 . 3 0 . 2 0 . 3 0 . 2

' =0.05 =0.1 =0.3

( 1 0 )

0.2 0.5 0.8

77.216 70.730 77.999

70.593 63.350 70.376

35.140 26.912 26.482

( 0 1 )

0.2 0.5 0.8

88.769 82.840 91.036

86.426 80.816 89.249

44.801 36.334 37.736

(- 1 1 )

0.2 0.5 0.8

35.255 27.084 26.465

25.545 18.019 15.231

10.343 5.945 2.800

( 2 0 )

0.2 0.5 0.8

31.204 23.363 22.042

22.625 15.638 12.412

9.214 5.296 2.524

(T able 4 ) A RL v alu es for m ultiv ar iat e E W M A ch ar t s b a s ed od r esidu al v ect or s w it h =

0 .9 0 . 8 0 .8 0 . 9

an d =

0 . 3 0 .2 0 . 2 0 .3

' =0.05 =0.1 =0.3

( 1 0 )

0.2 0.5 0.8

25.953 18.799 16.696

17.675 12.037 9.150

7.648 4.818 2.735

( 0 1 )

0.2 0.5 0.8

38.794 30.267 30.665

26.355 18.931 16.480

10.648 6.377 3.370

(- 1 1 )

0.2 0.5 0.8

9.280 5.741 3.163

5.760 3.308 1.588

1.818 1.072 1.001

( 2 0 )

0.2 0.5 0.8

7.783 4.921 2.788

5.008 3.068 1.601

1.935 1.091 1.001

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