Glol Jonl o P nd Appld Mc. ISSN 97-768 Vol, N 6, pp. 4-5 Rc Ind Plcon p://.plcon.co/gp. Avg Rn Lng o Conol C o AR Poc Exponnl W No Pypon Pc Dpn o Sc, Fcly o Scnc, Slpkon Unvy, Sncnd Plc cp, Nkon Po, 7, Tlnd Ac T ocv o Scl Poc Conol SPC o ono opon o n-conol poc. On o cn ool o SPC Clv S conol c c dly d n vl o pplcon c pcc, ngnng, nd nd n o. Fo ny poc o nd n ovon c cloly pcd n ll cold. T o opon Avg Rn LngARL. T n gol o pp o dv nlycl olon o ARL o CUSUM conol c o Aogv c on Explnoy vl AR xponnl no. Cckng cccy o l, ond o xc xpon Ncl Ingl Eqon y G- Lgnd l copd. An xclln gn n xc xpon nd ncl olon ond. T c n ddonl ndcon xc xpon cnly g cccy. Kyod: Aogv on Explnoy vl, Clv S conol c CUSUM, Avg Rn Lng ARL. Inodcon T SPC collcon o cl cnq d o pov ponc o poc, nd o dc von o podc p. T od o SPC ppld o podcon poc n ndl ng. Toclly, c od no ld o ndl pplcon, ov. Pycn on nd n onong pn' dng ov. Tc ngn y nd n onong c lvl pcl ncon. T n ool n SPC o ono poc conol c. T conol c cl ool cn lp n lng poc cpl, dnyng pol
44 Pypon Pc c o o conol condon nd nn conol o podc nd poc qly. T S c nodcd y S [] pl o ndnd nd d vy xnvly o on-ln poc onong, v ddvng y no cv o qck dcon o ll cng n poc p. T ddvng d c c plod c plng pon ncon o d kn only plng pon. T ly o dc ll p cng cn gncnly povd y ng conol c on c c plod ncopo noon co p pl, n ddon o noon n cn pl. A good lnv o S conol c n nd n dcng ll Exponnl Wgd Movng Avg EWMA nodcd y Ro [] nd Clv CUSUM c ppod y Pg[].Tdonl conol c d on pon poc op ndpndn nd dnclly dd IID. ov, n ncng n ovon o ny poc ocold nd ocolon cn v n dv c on ponc o conol c AlnndRo [4]. A volon gncn pc on ponc o clcl SPC pocd,.g., Jonon nd Bg [5], Bg nd Jonon [6], nd Ro [7].Fo xpl, pov ocolon cn podc v ngv n donl o o pocndd dvon, nd podc conol l c g n dd. Tg conol l, cond ocolon n ovon ng plod, cnl n n vg l l c g n xpcdo dd. A vygll ll c poc ponnl o o n npodcvo cng o pcl c. T cn ld o lo o condnc n conol c. T CUSUM c py d o nn n pov cn conol o poc Dncn [8]. T py dvng o CUSUM c ll dnd ddn o pn cng n poc vg o pdly n S conol c ncopong nl S npon l. Fo, on pol o pnpon xc pl cng n poc occd Wll nd Bon [9]. T ponc o CUSUM conol c n pnc o ocolon n dd n n o conx. S, o xpl, Jonon nd Bg [5], Bg nd Jonon [6], nd Ro [7], cn [], Spvll nd Ad [], Tng nd Ad [], Rng, Wlln, nd P [], Scd [4], Vnd Wl [5], VnBckl nd Rynold [6], nd T, Pgnllo, nd Longnck [7].To, oc o pp o dy Fdol yp ngl qon od o dvd clodo olon o Avg Rn Lng ARL o Aogv on Explnoy vl AR xponnl no. T ky o conol c odolog ll vg n lng ARL, o n-conol ARL nd o o conol ARL. On od o copng nd conng conol c od o cop dcon cpl og ARL. nd Ro [7] dcd y ARL cn clcld: ng Mkov Cn Appoc MCA, olvng o Ingl Eqon IE, o ng Mon Clo lon MC. In 987, Cod [8] d ncl
Avg Rn Lng o Conol C o Ax Poc 45 cnq o dn ARL o conol c. d n IE knon Fdol ngl qon o cond knd o n lng nd vnc nd ppoxd olon ng y o ln qon gvng xc xpon o n nd vnc. L, Lc nd Scccc [9] dd n lng don o conol c y ng MCA. Vnckl nd Rynold [6] d ARL y ng n IE cnq nd MCA o vl EWMA nd CUSUM conol c n c o od Aogv, AR poc ddonl ndo o. T ppo o pp o dv nlycl ol nd ncl od o nd ARL o CUSUM conol c o AR poc xponnl no o dcng o cng n poc n. W d ngl qon o dv xc xpon o ARL. T opon o pp ollong: n con nodcd ccc o CUSUM conol c o AR poc. T dvon o clod-o xpon o ARL xpd n con, ncl od o olvng ngl qon o on ppoxon o ARL pnd n con 4, copon o l ddd n Scon 5 nd conclon pnd n con 6.. T AR Poc o CUSUM Conol C T CUSUM conol c conologcl plo o clv o dvon o poc n. I ll knon CUSUM c dn ll n poc c n ndd x conol c Jn nd Gyn [].Gvn qnc o o-gv Explnoy vl: AR ndo poc. T CUSUM poc g cnvl on p vl o l,,,, nd p ndo o occd n p pod,,,,. T, cn vl no o. T dnon o CUSUM c d on AR poc ollong con: C x C, ;,, C CUSUM c, qnc o ndpndn nd dnclly don ndo vl. T vl oc n nl vl o CUSUM c, C nd non-zo conn. T gnl Aogv Explnoy vl: AR poc cn n :
46 Pypon Pc o no poc d xponnl don. T nl vl nolly o poc n, n ogv cocn < <. T p < < nd < < cocn o n ncon. I d nl vl o AR poc,,,, nd,,., p, In pp, c o pov cng n don c cong pp conol l l nly dcd. Gvn,,, qnc o ndpndn dnclly don ndo vl xponnl p. I nolly d nd n conol, p knon n-conol vl. T p cold cngd o o-o-conol vl, n θ, cng-pon. T pg o CUSUM cn n : τ n > : C >, > τ oppng conn p knon Upp conol L UCL. T ARL xpcon vl oτ. Mo o ccc d ponc o conol c ARL nd ARL ollong: ARL E τ 4 ARL Eθ τ θ τ θ 5 W E. xpcon copondng o g vl nd d o lg nog. Eθ. xpcon nd pon cng-pon occ θ.. Exc xpon o CUSUM conol c oar Poc T noon P c dno poly nd E c dno xpon copondng c [,]. T olon o ngl qon ollong E c [I{ < C < }C ]P c {C } 6. To, ngl qon o CUSUM conol c. So, [
Avg Rn Lng o Conol C o Ax Poc 47 y dy y ] 7. L k y dy y. Conqnly, cn n [ k ] 8. In pcl, on ollong o [ k ].. k Sng no Eqon, n ollong o [ k ]. k Conqnly, k. 9. To nd conn k ollong o y dy y k
48 Pypon Pc. Sng conn k no Eqon 9 ollo. T, g xplc ol o ARL o CUSUM conol c ollo. Snc poc n-conol xponnl p, on xplc ol o ARL ollo ARL. Snc poc o-o-conol xponnl p,t xplc ol o ARL cn n ollo ARL. < < Aogv cocn, < < nd < < p o n ncon,,,, Explnoy vl, <, < λ,,,,, p nd,,, oong p nd nl vl pcvly. 4. Ncl Ingl Eqon Gnlly, Ingl Eqon cold no nlyclly olvd nd ncy o ncl od o olv. Knoovc nd kylov []; Aknon nd n [] v n dvlopd ncl c o olvng ngl qon. W ll qd l o ppox ngl y n o o cngl d on / gnnng zo. Pclly, onc coc o qd l d, nvl [,] dvdd no pon nd o conn gd. / T ppoxon o n ngl o o:
Avg Rn Lng o Conol C o Ax Poc 49 dx x x W nd,, ;,. L dno o ncl ppoxon o ngl qon, c cn ond olon o ln qon ollo: F. T ov qon y o ln qon n nknon,,,, c cn ngd F M F
5 Pypon Pc. Solvng o qon o ppox vl,, n, ncl ngon o ncon F, nd,, ;,. 5. Copon l o CUSUM Conol C y Exc Expon nd Ncl Ingl Eqon od In con, l o ARL nd ARL o AR p poc, c ond o xc xpon ncl olon o ngl qon od copd. T l o ARL xpd n Tl o Tl. T p vl o n-conol p β nd p o o-o-conol β.,.,.,.4,.5,.6,.7,.8,.9,.,.,.5,, nd 5 pcvly. T ponc o ppod xc xpon condd y coponl nd ol pcng dnc. %. % L L L D T l o Tl nd Tl pn od n good gn. T nlycl l g ncl ppoxon n ol pcng dnc l n.5% o,5 on nd o coponl o ppoxly 5 cond. T coponl o popod nlycl xc xpon l n cond.
Avg Rn Lng o Conol C o Ax Poc 5 Tl : Copon o ARL nd ARL o CUSUM conol c y xc xpon ncl ngl qon o AR poc,,.nd.. P vl o EWMA c.5, nd.959 Exc xpon Ncl IE T d D %. 7.65 7.457.46.59. 46.54 46.456..48. 4.498 4.9757.5.5. 4.45 4.55.47..4 85.99 85.8756.8.868.5 68.86 68.78555..7896.6 5. 5.5456.8.6865.7 8.577 8.5957.8.4.8 5.7 5.657.5.988.9.647.59856.7.4..78.55.4.877. 79. 79.8655.4.87.5. 5. 4.476 5.496.748 4.49457.8 5.489456.58.74456.44.88.8578.6777 Tl : Copon o ARL nd ARL o CUSUM conol c y xc xpon ncl ngl qon o AR poc,,.,. nd.. P vl o EWMA c.5, nd.984 Exc xpon Ncl IE T d D %. 7. 69.98957..895. 47.65 47.4856.45.86. 6.86 6.7456..79. 7.7 7.59857.8.67.4 9.4 89.9856.54.655.5 7.69 7.5455.4.584.6 58.468 58.7856..48.7 44.96 44.455.4.55.8.7.5456.5.557.9 9.76 9.855.47.5. 7.865 7.856.48.789. 85.479 85.44656.9.68.5. 5. 44.66 5.88.8747 44.67457. 5.88656..86956.44.866.5497.959
5 Pypon Pc 6. Conclon Exc xpon o ARL o CUSUM n c o AR poc xponnl no dvd. T ol vy cc, nd y o clcl nd pog. Mo pcclly, xc xpon k coponl c l n ncl ngl qon. Rnc [] S, W. A., 9, Econoc Conol o Qly o Mncd Podc, Vn Nond, N ok. [] Ro, S. W., 959, Conol c d on goc ovng vg, Tcnoc,, pp. 9-5. [] Pg, E. S., 954, Conno npcon c, Bok, 4, pp. - 4. [4] Aln, L. C., nd Ro,. V., 988, T odlng o cl poc conol, Jonl o Bn & Econoc Sc, 6, pp. 87-95. [5] Jonon, R. A., nd Bg, M., 974, T c o l colon on ponc o c, Tcnoc, 6, pp.-. [6] Bg, M., nd Jonon, R. A., 975, T c o l colon on ponc o c II, Tcnoc, 7, pp.7-8. [7], T. J., nd Ro, W.., 99, Scl poc conol pocd o cold ovon, T Cndn Jonl o Ccl Engnng, 69, pp.48-57. [8] Dncn, A. J., 965, Qly Conol nd Indl Sc, Rcd D. In, Inc. oood, Illno. [9] Wll, G. B., nd Bon, D. W., 99, Scl Poc Conol. Toy nd Pcc, Cpn nd ll, London. [] cn, E., 99, Ponc o CUSUM conol c o lly cold ovon, Tcnoc, 5, pp.7-5. [] Spvll, C. R., nd Ad, B. M.,994, An vlon o oc-d qly conol c, Concon n Sc Slon nd Copon,, pp. 645-66. [] Tng, S., nd Ad, B. M., 994, Monong ocold poc n xponnlly gd ovng vg oc, Jonl o Scl Copon nd Slon, 5, pp.87-95. [] Rng, G. C., Wlln, T. R. nd P, S., 995, Avg n lng o CUSUM conol c ppld o dl, Concon n Sc oy nd Mod, 4, pp.7-8. [4] Scd, W., 997, On EWMA c o, In Lnz,. J. nd Wlc, P. T. Ed, Fon n Scl Qly Conol 5, dlg:pyc-vlg, pp.5-7. [5] VndWl, S. A., 996, Monong Poc T Wnd Ung Ingd Movng Avg Modl, Tcnoc, 8, pp.9-5.
Avg Rn Lng o Conol C o Ax Poc 5 [6] VnBckl, L. N. nd Rynold, M. R., J., 997, EWMA nd CUSUM conol c n pnc o colon, Concon n Sc Slon nd Copon, 6, pp. 979-8. [7] T, D.., Pgnllo, J. J., nd Longnck, M.T., 998, T dvlopn nd vlon o CUSUM-d conol c o n AR poc, IIE Tncon,, pp.55-54. [8] Cod, S. V., 987, A pl od o dyng n lng don o xponnlly gd ovng vg c, Tcnoc, 9, pp4-47. [9] Lc, J. M., nd Scccc, M. S.,99, Exponnlly gd ovng vg conol c: pop nd nncn, Tcnoc,, pp9-44. [] Jn, J. M., nd Gyn, F.,99, Qly Plnnng nd Anly, d d., N, McG- ll. [] Knoovc, L.V., nd Kylov V. I.,958, Appox Mod o g Anly, Incnc Pl, Inc., N ok. [] Aknon, K. nd n, W.,, Tocl Ncl Anly: A Fnconl Anly Fok, N ok, Spng-vlg.
54 Pypon Pc