WORKING PAPER SERIES

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1 College of Busness Admnstaton Unvesty of Rhode Island Wllam A. Ome WORKING PAPER SERIES encouagng ceatve eseach 8/9 No. 6 Ths wokng ae sees s ntended to facltate dscusson and encouage the exchange of deas. Incluson hee does not eclude ublcaton elsewhee. It s the ognal wok of the autho(s) and subject to coyght egulatons. Offce of the Dean College of Busness Admnstaton Ballentne Hall 7 Ltt Road Kngston, RI

2 Senstvty atos as a measue of the effects of the mean shft and dseson shft fo Multvaate EWMA montong Xa Pan, Ph.D. Lngnan College, Sun Yat-sen Unvesty Guanghou, Chna anxa@gmal.com Jeffey Jaett College of Busness Admnstaton Unvesty of Rhode Island Kngston, RI 88 jejaett@mal.u.edu * * Coesondng autho /8

3 ABSTRACT Pevously, studes of the multvaate exonentally weghted movng aveage (EWMA) contol ocesses centeed on methods fo constuctng qualty contol chats fo aveage un length (ARL). Smulaton and Makov Chan analyss oduced methods fo constuctng such qualty contol technques fo seally coelated ocesses. In ths ae, we focus on asects of the dstbuton of the chat statstc. Based on the dstbuton of the chat statstcs fo n-contol and out-of-contol stuatons, we oose to use senstvty atos to measue the effects of shfts n both mean and dseson. Usng senstvty measue, we nvestgate the motance of seal coelaton n qualty montong and ts mact on the senstvty of efomance. Ths allows fo adjustments n the otmal exonental weght facto as t elates to seal coelaton. Keywods: Multvaate EWMA, Qualty, Senstvty Analyss. /8

4 Intoducton Multvaate ocess contol smultaneously montos seveal ocesses n a combnaton (e.g., Jackson, 959 and 985; Montgomey and Wadswoth, 97; Alt, 985; Cose, 988; Pgnatello and Runge, 99; Hawkns, 99; Tacy et al, 99; Lowy, Woodall, Cham and Rgdon (99) [heeafte, LWCR]; Lowy and Montgomey, 995; Sullvan and Woodall, 996; Djauha, 5; Khoo and Quah, 3; Kuegel, Value and Vgna, 5; Ye and Chen, ; Ye, Chen and Boo, 4; Ye, Gadano and Feldman, ; Ye, Vlbet and Chen, 3; Besms, Psakas and Panaetos, 6; Lee and Khoo, 6; Khoo, 3, Yeh, Wang and Wu, 4; Pan and Jaett, 4; Yang and Rahm, 5, and fnally Jaett and Pan, 7a and 7b). Shewhat qualty contol chats fo multvaate ocesses emloy the Hotellng T statstc fo a cuent samle. Snce, unvaate Shewhat contol chats ae not senstve to small and modeate shfts n ocess aametes, one often emloys othe methods. Exonentally weghted movng aveage (EWMA) chats ae moe senstve (Robets, 959; Cowde, 989; Lucas and Suscatc, 99). Hence, LWCR extended the unvaate EWMA contol chat to the multvaate case by smulaton. They consdeed that the multvaate EWMA chat has geate senstvty to shfts n the mean than moe tadtonal Hotellng T contol methods. LWCR oosed to buld the EWMA quantty = x + ( ) fo each of the vaables, then fom the quadatc Hotellng T = ' Σ whee the covaance used n the T s the covaance of the EWMA vecto. We wll call the multvaate EWMA scheme as EWMA-M, n accodance wth the ode of these two stes whee the EWMA s bult befoe the Hotellng T. (An altenatve multvaate EWMA scheme s M- 3/8

5 EWMA Pan (5), whch bulds the Hotellng of the EWMA of the T of the vaables befoe the fomaton T s. Lu (996) esented an movement fo EWMA-M. Runge and Pahu (996) used Makov chan analyss to calculate the ARL fo EWMA-M and Pahu and Runge (997) dscussed the desgn of the same scheme. Howeve, all these studes assumed the ocesses to be seally ndeendent. Othes chose to study the usefulness of multvaate EWMA methods as well. Stoumbus and Sullvan nvestgated the effects of non-nomalty on the efomance of the multvaate EWMA contol chat, and ts secal case, the Hotellng s Ch-Squaed contol chat when aled to ndvdual obsevatons. The uose n ths case was to monto the mean vecto of a multvaate ocess vaable. Khoo studed the senstvty of multvaate EWMA contol chats unde othe ccumstances. In addton, Lee and Khoo (6) exloed a method fo otmally desgnng multvaate EWMA chats based on the measues of aveage un length (ARL) and medan un length (MRL). Anothe aoach to the multvaate EWMA chats, M-EWMA, was evously oosed by Pan (5). The M-EWMA scheme bulds the Hotellng T of the vaables befoe the fomaton of the EWMA of the T s. secfcally, fo -dmensonal multvaate nomal ocesses ~ N(, Σ) x at the th obsevaton, the Hotellng T = x ' Σ x s bult at fst. Then, the EWMA of the T s, denoted as Q, s Q T + ) Q, accodng to the ode of constucton stes s the statstc of M- = ( EWMA chat. Pan (5) used ntegal equaton method to comute the ARL's of M- EWMA chats fo n-contol and out-of-contol stuatons wthout the esence of seal coelaton. Both EWMA-M and M-EWMA ae multvaate EWMA schemes. 4/8

6 The above schemes have a common oblem, that s, they cannot be dectly emloyed when the ocesses ae seally coelated. An ndect way to aly the multvaate EWMA schemes fo seally coelated ocesses s to adot Alwan and Robets (988) aoach. They suggest estmatng the esduals,.e., one-ste-ahead foecastng eos, of the autocoelated ocess. In tun, they aly tadtonal contol chats fo the esduals. Extendng ths aoach to multvaate cases, one can aly the above EWMA-M o M-EWMA scheme to the esduals of the seally coelated multvaate ocesses, untl the ocesses ae modeled oely and the ntal numbe of obsevatons s suffcently lage and the esduals ae asymtotcally ndeendent ove tme. At ths ont, we detemne the senstvty of these aoaches to changes n ocess aametes n the esence of seal coelaton. Snce the ocess aametes ae usually unknown, the aoate estmaton and use of the covaance matx s vtal fo coect executon of multvaate EWMA. Ths may occu f the dect samle vaance s a based estmate of the oulaton vaance fo a seally coelated ocess. We wll n the next secton, consde vaaton n the chat statstc detemned by EWMA-M methods fo the absence of seal coelaton. The senstvty measue wll then ad the choce of the otmal EWMA weghtng facto and we wll bette undestand the effects of seal coelaton n the EWMA-M stuctue. Late, we examne the stuaton fo the esence of seal coelaton. EWMA-M fo Seally Indeendent Pocesses Fo a -dmensonal multvaate..d. ocesses x = x, x,, x )' at tme ont (, constuctng the EWMA quanttes based on the evous obsevatons, we have = x + ( ), =, () 5/8

7 Wthout losng genealty the mean vecto of x s set at eo. We then constuct the quadatc Hotellng T of as the chat statstc: T = ' Σ () whee the covaance used n the T s the covaance of, and was chosen as a scala weghtng aamete. It seems obvous that unde the assumton of..d. nomal dstbuton x ~ N(, Σ) wth known aametes, the EWMA-M statstc T = ' Σ follows ch-squae dstbuton wth ode of, because each s stll nomal. Howeve, omtted hee s that ntal value of the EWMA,. As has to be set as a cetan value (say, ), ths may make the T a lttle dffeent fom the exact χ ( ). To nvestgate the detals of the dstbuton, we denote x = x ', x ',, ']' be [ x an column vecto, be a ow vecto = [, ( ),, ( ) ], and I ae a unty matx. Fo ndeendent obsevatons, we have x ~ N (, V ) wth V ( I Σ). Settng the ntal EWMA vecto =, we have, = = ( I ) x (3) And Σ = E( ') = = ( I )( I ( I ) E( xx')( I )' Σ)( I )' = ( ') Σ = ( ( ) = ) Σ whee E( xx ') ( I Σ) because of the ndeendence fo obsevatons at dffeent tme (4) onts ( I s a unty matx), and that ( ') s scala. The esult n (4) s the same as gven n LWCR. Lettng R be a nomaled matx that = ( ' )( '), the EWMA-M chat statstc s R 6/8

8 T = ' Σ = x'( I )' Σ ( I ) x = x'( ' I )(( ') Σ )( I ) x = x'( R Σ ) x. (5) We deve the dstbuton of (5) by alyng the theoems on quadatc fom dscussed n Box (954) and othes (see Aendx ). Accodng to Aendx, the EWMA-M n (5) s aoxmately dstbuted as T ~ ( χ ) whch was exected. Thus, the dstbuton of (5) s elevant to the exonental weghtng facto. Ths esult s consstent wth the eoted smulaton esults n LWCR. Theefoe, the n-contol ARL's and the contol lmts n the table ae the same fo vayng values of. In addton, LWCR noted that the EWMA weghtng aamete s a dagonal matx wth dffeent elements, although they only eoted the smulaton esults fo scala. If the weghts ae dffeent fo dffeent vaables, the EWMA weghtng aamete s a matx W =. (6) We defne a new matx, R = [ W, W ( I W ),, W ( I W ) ], to elace the evous Konecke oduct I ) n (3), so that (3) becomes = Rx. Then, fom the ( dagonal oety of matx RR', t s not dffcult to see that Σ T = R( I Σ) R' = ( RR' ) Σ (7) = ' Σ = x' R' Σ ( RR') Rx (8) 7/8

9 If the,, n W s the same, (7) and (8) educe to (4) and (5) esectvely. Aendx ndcates (8) follows χ ( ), a ch-squae detemned by the numbe of dmensons of the system, and elevant to the EWMA weghtng aametes fo the vaables. Snce the scala weght s just a secal case of the matx weght, (3) s a secal case of (7). We note that the esult T ~ ( χ ) s based on the aoxmaton that t has the same fst two moments as (A-). The exact n-contol dstbuton of the chat statstc, (A-), s detemned by the egenvalues of U. Fom the stuctue of U n (A-3), t s concevable that the egenvalues of U deend on (o W) and. (Late we vefy ths by comute aded comutaton.) Theefoe, the weghtng aametes and the tme length fom the ntal ont nfluence the hghe moments of the chat statstc and make t dffe fom χ ( ). MacGego and Has (993), when studyng exonentally weghted movng squae (EWMS), onted out that the aoxmaton s bette fo lage,.e., the geate the dstance fom the ntal tme ont the bette. Shft Effects and the Measue of Senstvty - Mean Shft If a mean shft δ occus at = k, k, then we wte x = u + δ, whee u s denoted as the n-contol ocesses and δ s a column vecto that has δ n the fst k elements and eos n the last elements. Takng the scala case as examle (the case of (6) can be dscussed n the same way), the statstc of (5) becomes T u'( R Σ ) u + δ '( R Σ ) u + δ '( R Σ ) δ. (9) = 8/8

10 Ths s a noncental ch-squae dstbuton. The fst tem n (9) s the n-contol chsquae (note the mean vecto of n-contol ocesses u s assumed to be eos wthout losng genealty). The second tem s a nomal dstbuton wth mean of eo and vaance of 4δ '( R Σ ) δ. The second and the thd tems ae all detemned by the value of the thd tem whch s a constant scala. Defnng the thd tem as, = δ '( R Σ ) δ = δ ( R k Σ ) δ () whee R, the ue-left k k atton matx s R, attoned accodng to shft k occuence tme. Obvously, () s elevant to though R k. Theefoe, the choce of the value of the exonental weght n EWMA-M does matte fo the out-of-contol dstbutons of the chat statstc. By eexamnng (), t s actually δ tmes the sum of all the elements of R Σ. Hence, t s not dffcult to show that () s also δ Σ δ ' tmes the sum of k all the elements of R. Denotng both = δ Σ ' and the sum of all the elements of R k as S k, we have = S k aamete n LWCR. (o k. Note / δ / / ') = ( δ Σ δ s only the noncentalty ) measues the se of the shft n the ocess tself nstead of ncooatng the contol chat, whle the measue s chat-secfc. It can be shown that the S k s S k k ( )( ( ) ) = () ( ( ) ) Snce the mean of (5) s and the mean of (9) can be vewed as + dffeence of them,, can be vewed as the measue fo the dffeence n dstbuton., the 9/8

11 Comang ths dffeence wth the n-contol dseson measued by the n-contol standad devaton,, we can desgn the senstvty ato S k η = = () to measue the efomance of the contol chat fo detectng mean shft. As S k s the only facto deendng on, we choose fo the otmal senstvty though S k. Fgue shows how the S k changes wth fo dffeent occuence tme of the shft. Fo k=, ndcatng a ecent shft, exonental weghtng wll lowe the senstvty. Late, when k=, weghtng at =.5 gves the best senstvty. Fo k s 3, should be chosen at about.3. Fo k s aound 5, should be.. Fo k s about, should be.. Fo k s as lage as 3, should be as small as.4. In geneal, f the soon detecton afte the shft s wanted (say less than 5 tme ntevals), lage s needed. Ths s consstent wth the esult based on ARL ctea n LCWR and Pabhu and Runge (997). Shft Effects and the Measue of Senstvty - Dseson Shft A shft n the dseson aamete s a change n the ocess covaance matx. Yeh, Huwang, and Wu (4) studed a lkelhood-ato-based EWMA aoach fo multvaate vaablty. We now consde the EWMA-M chat by buldng the Hotellng T on the n-contol covaance matx. Fo out-of-contol ocesses, the theoetcal dstbuton whch s the bass fo constuctng the chat s not clea. Moeove, f the mean of x s stll eo, the theoems on quadatc fom (see Aendox) ae dectly alcable. Suose a shft fom Σ to Σ occus at = k, ( k ). Snce the covaance matx s ostvely defnte, thee exsts a matx C so that x = Cu and /8

12 Σ = CΣC' afte the tme ont. The chat statstc s stll exessed n the fom of (4) and (5) o (7) and (8) excet x s actually shfted. Snce the covaance of x s I k Σ E( xx ') =. (3) I Σ Refeng to Aendx, the matx U s (fo common weght,.e. (4) and (5)) U = E( xx ')( R Σ ) Rk ΣΣ = R k I R R ko I I (4) whee R k, R, R k, and R k ae atton matces of R whch s attoned accodng to shft occuence tme. The egenvalues of U deends on R k, R, and Σ Σ, and the egenvalues of UU deends on all the fou attoned matces of R. We can easly vefy though smulaton that the egenvalues of the attoned R does dffe fo vayng values of. Theefoe, the dstbuton of the chat statstc s stll chsquae but wth aamete shft that deend on the exonental weght (.e., g and v wll be deendent on nstead of beng and ). The deatue fom the n-contol statstc s dstbuton detemnes how quckly on aveage the contol chat detects the shft. To desgn a senstvty measue fo the detecton of the out-of-contol status due to dseson change, we suggest the use of the ato between the vaance fo the out-ofcontol and the vaance fo the n-contol chat statstcs. Snce the n-contol chat statstc s χ ( ), ts vaance s. The vaance of the out-of-contol chat statstc s tace ( UU ), whee U s gven n (4). We have the senstvty η = tace ( UU ) (5) /8

13 Altenatvely, we use the ato of the standad devatons, whch s the squae oot of (5), to measue senstvty. To show the effect of on senstvty η, we consde some atcula cases that the 3-dmensonal ocesses covaance matx shfts fom Σ to Σ at tme ont = k, ( k ). One of the cases s the dseson shft s n such a way that.5.5 Σ Σ =.5.75 (6) Substtutng (6) n (4), we calculate η fo dffeent and k. The esult, shown n Fgue, ndcates that lage yelds hghe senstvtyη. We also examned seveal cases of Σ Σ. As long as the vaance fo each ocess nceases (.e. the dagonal elements of Σ Σ ae geat than one), lage s efeed. Ths ndcates that the cuent samle nfomaton s moe motant. If at least one of the ocess vaances deceases athe than nceases, smalle s efeed (the decease n η s monotone when nceases). In geneal, f not-vey-small k s allowed (k>), values geate than. do not make much dffeence n η. In othe wods, lage ove. only gves bette η fo k<,.e. when the shft just haened not long ago. EWMA-M and Seally Coelated Pocesses When the above EWMA-M scheme s aled to seally coelated ocesses, we note the followng. Fst we focus on estmaton, snce n actce the ocess aametes ae usually unknown. Second, we moe comletely comehend the mact of seal coelaton n the ocesses on the contol chats. /8

14 Fo estmaton, we note that one cannot smly obtan an estmate of a covaance matx even f s estmated coectly, and, n tun constuct the EWMA-M statstc and set contol lmts accodng to ARL. The estmaton ssue s elated to dffcultes concenng the dstbuton. Fom the evous dscusson, the EWMA-M statstc T = ' Σ s χ ( ) when the - dmensonal ocesses ae seally coelated as long as Σ s known. Ths can be also seen fom ( I ) x, = Σ = I ) E( xx')( ' I ) ( and T = ' Σ = x'( ' I )[( I ) E( xx')( ' I )] ( I ) x ~ χ ( ) (7) In the devaton, the seal coelaton n x = x ', x ',, ']' only affects E( x x' ), ths [ x s fnally deleted when calculatng U fo quadatc fom. s obtaned fom the ocess aamete, Σ s not dectly known and t Γ() Γ()' Γ( )' Γ() Γ() Γ( )' E( xx') = Γ =. (8) Γ( ) Γ( ) Γ() whee the Γ(l) s ae the autocovaance matces of the ocesses fo lag l. When seal coelaton exsts, (4) o (7) ae not avalable anymoe, because the Γ(l) s fo l ae not eo. 3/8

15 When the ocess aametes ae unknown, the estmaton can be though ethe of the followng two ways n the ntal hase of constuctng contol chats. Fst, Γ (l) s estmated though to fom Γ. Second, constuct = x + ( ) fom the ntal hase of constucton of x data fom obsevaton to obsevaton, and calculate the samle vaance-covaance based on the values of,,, and to estmate Σ. A second aoach called the Z aoach aeas omsng,,, and ae not andom, snce lmtaton that the ntal value s cetan (fo examle, = ). Altenatvely, the autocovaance aoach s omsng when the ocess model s movng-aveage. Secfcally, f the undelyng ocess s a vecto movng aveage of ode q (denoted as VMA(q)), the autocovaance matces become eo afte q lags. That s, Γ( l ) = fo l >q, and, n tun, Γ() Γ()' Γ( q)' Γ() Γ() Γ()' Γ() Γ( q)' Γ = (9) Γ( q) Γ()' Γ( q) Γ() Γ() If the ode q s small, we need only a few estmates, ˆΓ (), ˆΓ (),, Γ ˆ ( q) examle, VMA(), we have Σ = I ) Γ ( ' I ) ( n n. Fo = ( ( ) ( ) ) Γ() + ( ( ) )( Γ() + Γ()'). () We need only ˆΓ () and ˆΓ () to calculate fom the ntal hase of qualty contol chat constucton. 4/8

16 The autocovaance aoach s then sutable fo VMA ocesses wth known coeffcent matces Θ and eo tem covaance matx Ω, and Γ (l) s calculated by k q l l h h+ l h= Γ( ) = Θ ΩΘ '. Fo ocesses wth autoegessve tems, Γ (l) s non eo even when the lag l s lage, (.e., Γ( l) = j= Γ( l j) Φ j ', see Rensel, 993). Hence the autocovaance aoach s not sutable. (Note, Aendx gves Σ fo VMA(q) of hghe odes of q.) The esence of seal coelaton n EWMA-M chat To consde the choce of the weghtng facto o R fo EWMA-M on seally coelated ocess, we note that the EWMA-M statstc s seally coelated by the weghtng facto o R. Thus, unnng the chat on seally coelated ocesses s equvalent to vayng the weghtng facto s value(s). Fo examle, consde a smlfed examle of senstvty fo mean shft whee the ocess s a VMA() wth a scala coeffcent θ, and the weghtng facto s also a scala. In ths case, t s not dffcult to see (n Aendx 3) that the ognal vecto = [, ( ),, ( ) ] that foms = ( I ) x and R = ( ' )( ') n (5) becomes θ elated, ( θ ) = [, ( ) + θ,, ( ) + θ ( ) ]. () So R becomes θ elated R (θ ). The othe stes to deve EWMA-M do not change. The n-contol chat statstc s stll χ ( ). Hence, the senstvty η n () becomes S k ( θ ) η ( θ ) = () 5/8

17 whee the S (θ ) s the sum of all the elements of the k-ode ue-left atton matx of R (θ ). k Fgue 3 shows how the θ affects S k (θ ) at dffeent (hence, how θ affects η ). In Fgue 3, the lots dawn n dots coesond to the case of θ beng eo, whch can be a efeence and comaed wth the lots at othe θ. We see fom the fgues that ) lage ostve θ yelds lage senstvty ( S (θ ), actually, same as bellow), and negatve θ wth smalle absolute value yelds lowe senstvty. Ths s geneally tue egadless of whethe the chat s at statng state (small ) o at steady state (lage ), o whethe the shft occued long ago (lage k) o just occued (small k). ) A ostve θ has elatvely less mact on the senstvty than a negatve θ. Hence, the ostve θ changes the senstvty n smalle amounts than negatve θ. Ths s esecally tue fo the steady state. Also, 3) the mact of θ s lage fo the statng state (small ) than fo the steady state (lage ). It s lage fo shfts occued ecently (small k) than fo eale tmes (lage k). Last, 4) ostve θ eques the use of lage to acheve the otmal senstvty, whle negatve θ eques to use a smalle to acheve the otmal senstvty. Thee s lttle mact when ealy tme eods (see Fgue 3(d)). Fnally, the fndngs n 4) may allow us to adjust the otmal weghtng facto fo the EWMA-M chat f the nfomaton on the seal coelaton n the ocess s avalable. k Summay In ths ae, we studed multvaate EWMA contol chats constucted by the method of EWMA-M. Alyng Box quadatc fom, we nvestgated n detal the dstbuton of the chat statstc. Based on the dstbuton of the chat statstcs fo n- 6/8

18 contol and out-of-contol stuatons, we oosed senstvty atos as a measue of the effects of the mean shft and dseson shft. Usng ths senstvty measue, we desgned the otmal exonental weghtng facto, whch s consstent to esults eoted befoe. Although ARL s the usual measue fo SPC chat efomance, t s by no means the only cteon, and t has shotcomngs. Ou oosed senstvty measue has cetan advantages. It s dectly deved fom the dstbutons of the chat statstc, hence, t s not constaned to whee the contol lmt s located. Ths makes the senstvty measue have a boade ange to ft vayng stuatons. We dscussed the EWMA-M chat on ocesses n esence of seal coelaton. Based on the n deth knowledge on the dstbuton of the chat statstc, we suggest a secal way of constuctng the vaance-covaance matx fo the EWMA-M scheme on multvaate MA ocesses. Usng the senstvty measue, we nvestgated the ole of seal coelaton of the ocess n the stuctue of the chat statstc, and ts mact on the senstvty efomance fo a secal ocess atten (VMA ()). Ths allows us to consde adjustng the otmal exonental weghtng facto accodng to the nfomaton on seal coelaton. 7/8

19 APPENDIX. The dstbuton of EMWA-M To deve the dstbuton of EWMA-M n (3), we aly the theoems quadatc fom and ts aoxmaton n Box (954). Accodng to Box (954), f x = x, x )' s ( m multvaate nomal N (, V ) wth ode of m, then the quantty Q = x ' Ax wth ank m s dstbuted as Q = j= λ χ () (A-) j j whee λ j 's ae the latent oots of gves the same fst two moments s whee U = VA. An aoxmaton to ths dstbuton that Q ~ gχ ( v), (A-a) and g = j= j= j λ λ = tace( UU ) tace( U ) j= j v = ( λ ) λ = [ tace( U )] / tace( UU ). (A-b) j j= j An alcaton of ths quadatc fom n unvaate SPC can be found n MacGego and Has (993), whee they examned exonentally weghted movng squae (EWMS) and exonentally weghted movng vaance (EWMV). Alyng the above quadatc fom nto (3), we teat U = ( I Σ)( R Σ ) = R I, and UU = ( R I )( R I ) = R I. Snce R = = ' ' ( ') = ' ( ') R, then UU = R I. Snce tace ( ) = and R tace( I ) =, and snce the egenvalues of a Konecke oduct ae the oducts of the egenvalues of the esectve matces that fom the Konecke oduct (also, note that 8/8

20 the oduct of egenvalues of a matx s the tace of the matx), we have tace( U ) = tace( R ) =, and tace ( UU ) =. Fom (A-b), t s easy to see that g = and v =. Theefoe, the quadatc fom n (3) follows T ~ ( χ ). When the exonental weghtng facto s matx gven n (6), We can use the matx R = [ W, W ( I W ),, W ( I W ) ], and aly the Box quadatc fom wth (7) and (8). Then, U = ( I Σ)( R' Σ ( RR' ) R) = R' ( RR' ) R (A-3) UU ( '( ') = R RR R) ( '( ') R RR R) = R' ( RR' ) R =U (A-4) Snce tace( R'( RR') R) = tace( RR'( RR') ) = tace( I ) =, (A-5) we have agan, tace ( U ) = tace( UU ) =, hence g = and v =. The chat statstc s then stll T ~ χ ( ), accodng to (A-). APPENDIX. The Covaance Matx of the EWMA Vecto Gven the fom of R (agan, wth scala ), followng the algothm ules of Konecke matx, t s easy to get the esults of VMA() ocesses the Σ s, Σ esults fo hghe VNA(q) odes. Fo ( ) ( ) Σ = ( ( ) ) Γ() + ( ( ) )( Γ() + Γ()') + ) + ( ( ) ( ( ) )( Γ() + Γ()'), (A-) 9/8

21 and fo VMA(3), ( ) ( ) Σ = ( ( ) ) Γ() + ( ( ) )( Γ() + Γ()') + ) + ( ( ) ) + Fnally fo VMA(q), ( ( ) 3 ( ( 3) ( ( ) )( Γ() + Γ()') + )( Γ(3) + Γ(3)'). (A-) Σ = ( ( ) ( ) ) Γ() + ( ( ) ( ) )( Γ() + Γ()') + q ( ) ( q) + + ( ( ) )( Γ( q) + Γ( q)' ). (A-3) When the weghtng facto s a matx W, t s also not dffcult to deve the elatonsh between Σ and APPENDIX 3. The ole of MA() tem fo EWMA. As we ae talkng about a secal examle that the MA() coeffcent s a scala, we teat the multvaate ocesses just lke a unvaate ocess. Fo the ocess xt ε, the EWMA s t = xt + ( ) t. Hence, = t + θε t t ε θε = t + t + ( ) t ε θε + ( )( xt + ( ) ) = t + t t = ε t + θε t t + ( )( ε t + θε t + ( ) ) = ε t + θε t + ( ) ε t + ( ) θε t + ( ) t = t ( t t t t 3 ε + θ + ( )) ε + ( ) θε + ( ) ( x + ( ) ) (A3-) /8

22 Theefoe, settng =, we have t t, ( ) + θ,... ( ) + θ ( ) ][ ε, ε,... ε ]', (A3-) t = [ t t whee ε, ε,... ε t t ae ndeendent. Comang wth the case that seal coelaton s absent n the ocess x t, what s dffeent fo the stuctue of EWMA-M statstc s just n the vecto ( θ ) = [, ( ) + θ,..., ( ) + θ ( ) ] (A3-3) whch becomes θ -elated (hee n (A3-) we change denotaton t n (A3-) back to as we used n ths ae. /8

23 REFERENCE Alt, F. B. (985), "Multvaate Qualty contol," n The Encycloeda of Statstcal Scences, eds. S. Kot, N. L. Johnson, and C. R. Read, John New Yok Wley, Alwan, B.M., and H.V. Robets (988). Tme-Sees Modelng fo Detectng Level Shfts of Autocoelated Pocesses, Jounal of Busness and Economcs Statstcs, 6,, Besms, S., Psaaks, S. and Paneetos, J. (6) Multvaate Statstcal Pocess Contol Chats: An Ovevew Qualty& Relablty Engneeng Intenatonal, 3, 5, Box, G. E. P, (954) Some Theoems on Quadatc foms Aled n the Study of Analyss of Vaance Poblems: Effect of Inequalty of Vaance n One-Way Classfcaton, Annals of Mathematcal Statstcal Statstcs, 5, 9-3 Cose, R.B. (988). Multvaate Genealatons of Communcatve Sum Qualty Contol Schemes," Technometcs, 3, 3, 9-33 Cowde, S.V. (989), Desgn of Exonentally Weghted Movng Aveage Schemes, Jounal of Qualty Technology,,, 55-6 Djauha, M.A. (5). Imoved Montong of Multvaate Pocess Vaablty, Jounal of Qualty Technology, 39,, 3-39 Hawkns, D.M.(99). Multvaate Qualty Contol based on Regesson Adjustment, Technometcs,, 33,, 6-75 Jackson, J.E. (959) Qualty Contol Methods fo Seveal Related Vaables, Technometcs,, 4, Jackson, J.E. (985) Multvaate Qualty Contol, Communcatons n Satstcs-Theoy and Methods, 4,, Jaett, J.E. and Pan, Xa (7a) The Qualty Contol Chat fo Montong Multvaate Autocoelated Pocesses, Comutatonal Statstcs and Data Analyss, 5, Jaett, J.E. and Pan, Xa (7b) Usng Vecto Autoegessve Resduals to Monto Multvaate Pocesses n the Pesence of Seal Coelaton Intenatonal Jounal of Poducton Economcs, 6, 4-6 Khoo, M.B.C., and Quah, S.H. (3). Multvaate Qualty Chat fo Pocess Dseson Based on Indvdual Obsevatons, Qualty Engneeng, 5,4, /8

24 Kuegel, C. Value, F. and Vgna G. (5) Intuson Detecton and Coelaton, Challenges and Solutons, Snge Scence + Busness Meda Inc.Boston Lowy, C. A. & Montgomey, D. C. (995). A Revew of Multvaate Contol Chats. IIE Tansactons, Lowy, C. A. W. Woodall, CW. Cham and S.E. Rgdon (99) A Multvaate Exonentally Weghted Movng Aveage Contol Chat, Technometcs, vol. 34, Lucas, J.M., and M. S. Saccucc (99). Exonentally weghted Movng Aveage Contol Schemes: Poetes and Enhancements, Technometcs, vol. 3, -. Lu, Y. (996) An movement fo MEWMA n Multvaate ocess Contol, Comute & Industal Engneeng 3, MacGego, J. F. and T.J. Has (993) "The Exonentally Weghted Movng Vaance", Jounal of Qualty Technology Vol. 5, No., 6-8 Montgomey, D.C. and H.W. Wadswoth, J. (97) Some Technques fo Multvaate Qualty Contol Alcatons, ASQC Techncal Confeence Tansactons, Washngton, D. C. Pan, Xa. (5) An Altenatve Aoach to Multvaate EWMA Chat, Jounal of Aled Statstcs, Vol. 3, Pan, Xa, and Jaett, J.E. (4) Alyng State Sace to SPC: Montong Multvaate Tme Sees, Jounal of Aled Statstcs, 3, 4, Pgnatello, J. J. J., and G.C. Runge (99) Comason of Multvaate CUSUM Chats, Jounal of Qualty Technology,, 3, Pabhu, S.S. and G.C. Runge (997) Desgnng a Multvaate EWMA Contol Chat, Jounal of Qualty Technology, 9, 3-4 Rensel, G.C. (993). Multvaate Tme Sees Analyss, John Wley & Sons Robets, S.W. (959) "Contol Chat Tests Based on Geometc Movng Aveages," Technometcs,, 39-5 Runge, G.C. and S.S. Pabhu (996) A Makov Chan Model fo the Multvaate Exonentally Weghted Movng Aveages Contol Chat, Jounal of Amecan Statstcal Assocaton, 9, 7-76 Stoumbos, Z. and Sullvan, J.H. () Robustness to non-nomalty of the multvaate EWMA Contol Chat, Jounal of Qualty Technology, 34, 3, /8

25 Sullvan, J.H. and W.H. Woodall (996). A Comason of Multvaate Qualty Contol Chats fo Indvdual Obsevatons, Jounal of Qualty Technology 8, 4, Tacy, N.D., J.C. Young and R.L. Mason (99). Multvaate Qualty Contol Chats fo Indvdual Obsevatons, Jounal of Qualty Technology, vol.4 Yang, S.F. and Rahm, R.A. (5) Economc Statstcal Pocess Contol fo Multvaate Qualty Chaactestcs Unde Webull shock model, Intenatonal Jounal of Poducton Economcs, 98, 5-6 Ye, N. and Chen, Q. (). An Anomaly Detecton Technque Based on a Ch Squae Statstcs fo Detectng Intuson nto Infomaton System, Qualty and Relablty Engneeng, 7, 5- Ye, N., Chen, Q. and Boo, C.M. (4). EWMA Foecast of Nomal System Actvty fo Comute Intuson Detecton, IEEE Tansactons on Relablty, 53,4, Ye. N, Gadano, J. and Feldman, J. (). A Pocess Contol Aoach to Cybe Attack Detecton, Communcaton of the ACM, 44, 8, 76-8 Ye, N, Vlbet, S and Chen., Q (3) Comute Intuson Detecton Though EWMA fo Autocoelated and Uncoelated Data, IEEE Tansactons on Relablty, 5, 75-8 Yeh, A.B., Huwang, L.C. Wu, Y.F (4). A lkelhood-ato-based EWMA contol chat fo montong vaablty of multvaate nomal ocesses. IIE Tansactons 36 (9): /8

26 6 Senstvty fo and k exonental weght Fgue (a) The Senstvty Facto S k fo Mean Shft, lage (=) ' ': k=; '-.': k=; ' ': k=3; '+': k=5; ' ': k=7 5 Senstvty fo and k exonental weght Fgue (b) The Senstvty Facto S k fo Mean Shft, lage (=) ' ': k=5; '-.': k=; ' ': k=5; '+': k=; ' ': k=3 5/8

27 6 Senstvty fo and k exonental weght Fgue (c). The Senstvty Facto S k fo Mean Shft, small (=) ' ': k=; '-.': k=; ' ': k=3; '+': k=5; ' ': k=7 Senstvty fo and k exonental weght Fgue. The Senstvty η of EWMA-M Chat fo Dseson Shft, (=4) ' ': k=5; '-.': k=; ' ': k=5; '+': k=; ' ': k=3 6/8

28 Senstvty fo and theta exonental weght Fgue 3 (a) S k (θ ) fo Senstvty η of EWMA-M Chat fo Dseson Shft, k= Stang State (=) '-.': θ = -.8; ' ': θ = -.5; + : θ = -.; ' ': θ =; ' ': θ =.; : θ =.5; - - :θ =.8; 4.5 Senstvty fo and theta exonental weght Fgue 3 (b) S k (θ ) fo Senstvty η of EWMA-M Chat fo Dseson Shft, k=5 Stang State (=) '-.': θ = -.8; ' ': θ = -.5; + : θ = -.; ' ': θ =; ' ': θ =.; : θ =.5; - - :θ =.8; 7/8

29 9 Senstvty fo and theta exonental weght Fgue 3 (c) S k (θ ) fo Senstvty η of EWMA-M Chat fo Dseson Shft, k= Steady State (=) '-.': θ = -.8; ' ': θ = -.5; + : θ = -.; ' ': θ =; ' ': θ =.; : θ =.5; - - :θ =.8; 5 Senstvty fo and theta exonental weght Fgue 3 (d) S k (θ ) fo Senstvty η of EWMA-M Chat fo Dseson Shft, k=3 Steady State (=) '-.': θ = -.8; ' ': θ = -.5; + : θ = -.; ' ': θ =; ' ': θ =.; : θ =.5; - - :θ =.8; 8/8

30 Founded n 89, the Unvesty of Rhode Island s one of eght land, uban, and sea gant unvestes n the Unted States. The,-ace ual camus s less than ten mles fom Naagansett Bay and hghlghts ts tadtons of natual esouce, mane and uban elated eseach. Thee ae ove 4, undegaduate and gaduate students enolled n seven degeegantng colleges eesentng 48 states and the Dstct of Columba. Moe than 5 ntenatonal students eesent 59 dffeent countes. Eghteen ecent of the feshman class gaduated n the to ten ecent of the hgh school classes. The teachng and eseach faculty numbes ove 6 and the Unvesty offes undegaduate ogams and 86 advanced degee ogams. URI students have eceved Rhodes, Fulbght, Tuman, Goldwate, and Udall scholashs. Thee ae ove 8, actve alumnae. The Unvesty of Rhode Island stated to offe undegaduate busness admnstaton couses n 93. In 96, the MBA ogam was ntoduced and the PhD ogam began n the md 98s. The College of Busness Admnstaton s accedted by The AACSB Intenatonal - The Assocaton to Advance Collegate Schools of Busness n 969. The College of Busness enolls ove 4 undegaduate students and moe than 3 gaduate students. Msson Ou esonsblty s to ovde stong academc ogams that nstll excellence, confdence and stong leadesh sklls n ou gaduates. Ou am s to () omote ctcal and ndeendent thnkng, () foste esonal esonsblty and (3) develo students whose efomance and commtment mak them as leades contbutng to the busness communty and socety. The College wll seve as a cente fo busness scholash, ceatve eseach and outeach actvtes to the ctens and nsttutons of the State of Rhode Island as well as the egonal, natonal and ntenatonal communtes. The ceaton of ths wokng ae sees has been funded by an endowment establshed by Wllam A. Ome, URI College of Busness Admnstaton, Class of 949 and fome head of the Geneal Electc Foundaton. Ths wokng ae sees s ntended to emt faculty membes to obtan feedback on eseach actvtes befoe the eseach s submtted to academc and ofessonal jounals and ofessonal assocatons fo esentatons. An awad s esented annually fo the most outstandng ae submtted. Ballentne Hall Quadangle Unv. of Rhode Island Kngston, Rhode Island