Inernaional Journal of Applied Physics and Mahemaics Robusness of Memory-Type Chars o Skew Processes Saowani Sukparungsee* Deparmen of Applied Saisics, Faculy of Applied Science, King Mongku s Universiy of Technology Norh Bangkok, Bangkok, Thailand. * Corresponding auhor. Email: saowani.s@sci.kmunb.ac.h Manuscrip submied May 0, 06; acceped Augus 9, 06. doi: 0.7706/ijapm.06.6.4.65-7 Absrac: This paper aims o sudy he robusness of Double Exponenially Weighed Moving Average (DEWMA) in order o deec a change in parameer when process are underlying skew disribuions. In general, an Average Run Lengh (ARL) is used as a common measuremen o compare he performance of conrol char in erm of quick deecion. The performance of GWMA char are compared wih Exponenially Weighed Moving Average (EWMA) and Generally Weighed Moving Average Conrol Char (GWMA) chars which he former ouperforms and give a minimal ARL for all magniudes of shif. Key words: Performance, monioring, skewness, conrol char.. Inroducion In manufacuring process, sandards of qualiy and low cos producs are key facors of indusrial success. Achieving he qualiy sandards, he manufacure requires eiher process conrol and quickly faul deecion. In fac, he producion sill has a variaion a any poin of ime even he process is well conrolled. Two ypes of he variaion have been described, which are he naure variaion due o chance causes and he variaion due o assignable causes, including man, machine, mehods, and maerials. The Saisical Qualiy Conrol (SQC) play a vial rule in measuring, conrolling, monioring and improving process qualiy. The SQC is caegorized ino hree pars, which are descripive saisics, saisical process conrol chars, and accepance sampling plans. The mos common ool for deecing and monioring a changing in he process is he conrol chars. Theirs major funcions are he manufacuring sandards seing, he producion goal's achievemen, and he produciviy improvemen. The conrol chars have been classified ino variables and aribues. The conrol char for variables such as x -char, R-char and S-char using for deecing he characerisics are measurable and valued coninuous. Whereas he conrol char for aribues such as p-char, np-char, c-char, and u-char using for deecing he characerisics ha have a discree value and are counable. Ideally, he parameers changing should be deeced as soon as possible, which is he main purpose of a conrol char in manufacure. An in-conrol process seing, he false alarm rae should be sufficien large. Oherwise, he rue alarm rae should be minimum when he process is ou-of-conrol.in lieraure reviews, he conrol char known as he Shewhar char, namely Cumulaive Sum [], Exponenially Weighed Moving Average (EWMA) [], is commonly used and widely applied in many fields. The Shewhar conrol char was discovered by Shewhar in 9 []. This kind of conrol char is accepable o deec he large changes in process, bu i is insensiive o minor change deecion. To overcome his problem, here are many memory-ype effecive alernaives o he Shewhar char, namely 65 Volume 6, Number 4, Ocober 06
Inernaional Journal of Applied Physics and Mahemaics Cumulaive Sum (CUSUM) [], Exponenially Weighed Moving Average (EWMA) [], Double Exponenially Weighed Moving Average (DEWMA) [4] and Generally Weighed Moving Average (GWMA) [5] chars have been developed o deec small shifs (abou.5σ or less). In general, he performance of conrol char is measures by Average Run Lengh (ARL). According o he expecaion of he sopping imes, ARL is classified ino ARL0 and ARL. The ARL0 (in - conrol ARL) is he expecaion of he sopping ime when he process is in-conrol. On he oher hand, ARL (ou - of - conrol ARL) is he expecaion of delay of rue alarm imes when he process is ou-of-conrol. From previous sudies, here are many lieraures sudied he abiliy in order o deec a change in process of conrol chars which he assumpion of process normaliy [6]. However, in he pracice, his assumpion always deviaed from normal disribuion such as skew processes. The performance of he EWMA char for non-normal disribuions has been invesigaed by [7]. The effec of non-normaliy and auocorrelaion on he performance of EWMA conrol chars was presened by [8] found ha he EWMA char robus o non-normaliy assumpion for deecing small shifs in a process mean and variance. Then, he robusness of conrol char should be invesigaed o violaion of he normaliy assumpion. There are some lieraures o compare he performance beween DEWMA and EWMA for non-normaliy process [9], [0] Consequenly, he aims of his paper is o sudy he robusness of memory-ype conrol chars as EMWA, DEWMA and GWMA chars in order o deec of a change in parameer of skew processes such as gamma and log-normal disribuions.. Conrol Chars and Theirs Properies.. Exponenially Weighed Moving Average (EWMA) Chars The Exponenially Weighed Moving Average (EWMA) saisics is defined as he following form 0 Z = ( X ) Z, 0, () where is a weighing facor for previous observaions. The arge in-conrol parameer 0 is supposed o be seady and he iniial value Z0 Z. 0 0 If he observaions X i of EWMA saisics Z is is usually chosen o be he process in-conrol parameer, i.e., are independen random variables wih variance Since 0, we have ha ( ) 0 as, variance is ( ). Z x, hen he variance and herefore he asympoic value of he Then, an asympoic sandard deviaion is used o find he conrol limis of he EWMA char is he following: where L when Z UCL / LCL L 0 X is he widh of EWMA s conrol limi. The process will be declared o be in an ou-of-conrol sae UCL or Z LCL. ( )[ ( ) ],,,... Z X () 66 Volume 6, Number 4, Ocober 06
Inernaional Journal of Applied Physics and Mahemaics.. Double Exponenially Weighed Moving Average (DEWMA) Char This conrol char was developed from EWMA char by aking accoun o double EWMA saisics which has been proposed by [5] as following Y Z ( ) Y () where = ( ) Y is Z X 0 Z as Equaion (). If from Eq. () and () hen he DEWMA and and Y Z ( ) Y (4) Z = ( X 0) Z. The expecaion and variance of DEWMA saisics are as follows: E( Y ) E( Z ) E( X ) 0 ( ) ( )( ) ( )( ) ( ) 4 4 ( ) x. VY ( ) (5) If from Eq. (5) hen he asympoical variance of DEWMA saisics is ( ) VY ( ) x ( ) Therefore, he conrol limis of DEMWA can be wrien as ( ) UCL / LCL 0 LX. ( ) where L is he widh of DEWMA s conrol limi... Generally Weighed Moving Average (GWMA) Char The GWMA char was iniial presened by [6] and exensive sudied by [] is weighed moving average of sequenial hisorical observaions. Since, each observaion is differenly weighed ha decreases from he presen period o pas periods hen i could be refleced o he imporan observaions on recen process. This char was exended and developed from EWMA char by adding an adjusmen smoohing consan. If he weighed hisorical observaion consan equal o q and, hen he GWMA char coincides he EWMA char. The GWMA saisic is as following ( i) ( i ) i 0. i G q q X q G (6) Taking geomeric series o Eq. (6) hen can be rewrien as ( q)( q ) ( q ) q( q ) G X q G ( q)( q) i 0 67 Volume 6, Number 4, Ocober 06
Inernaional Journal of Applied Physics and Mahemaics where h G is he GWMA saisic a ime, where he iniial saisic value G. 0 0 X is he observaions of skew process a he i h,,,... i q is a weighed hisorical observaions consan (0 q ) is an adjusmen smoohing consan ( 0). Mean and variance of GWMA saisic are EG ( ) and Var 0 conrol limis of GWMA char are (G ) Q, G x respecively. Therefore, he UCL LCL L Q / 0 X where ( i) i ( ) and i Q q q L is he widh of GWMA s conrol limi. Table. Comparison When Processes Are Gamma (, ) 0., =0.8 and ARL0 70. EWMA DEWMA GWMA L.79 L 5.0 L.97 0.00 70.7 (.4754) 70.58 (.4) 70.56 (4.7) 0.0 56.69 (.46) 5.74* (.4) 59.968 (4.) 0.05 04.7 (.8) 97.0* (.09) 0.555 (.99) 0. 54.54 (0.9689) 44.7* (0.864) 75.05 (.704) 0. 84.94 (0.658) 75.64* (0.56) 06.9 (.0) 0. 4.6 (0.475) 5.864* (0.89) 55.08 (.747) 0.5 96.5 (0.69) 85.7* (0.) 88.66 (.97) Noe: he sandard error is showed in parenheses. Table. Comparison When Processes Are Gamma (, ) 0., =0.8 and ARL0 500. EWMA DEWMA GWMA L.76 L 5.066 L.00 0.00 500.858 (.04) 500.5 (.97) 500.884 (4.875) 0.0 480.74 (.957) 475.99* (.867) 486.75 (4.86) 0.05 406.49 (.66) 9.45* (.54) 45.69 (4.586) 0. 4.0 (.) 6.09* (.69) 79.678 (4.) 0. 5.85 (0.879) 7.87* (0.74) 85.485 (.765) 0. 75.99 (0.6) 6.96* (0.50) 7.58 (.84) 0.5.54 (0.9) 07.5* (0.6) 7.4 (.448) Noe: he sandard error is showed in parenheses.. Average Run Lengh (ARL) Generally, he performance of conrol char are compared by considering he Average Run Lengh (ARL). The ARL is he expeced number of samples obained before a change in process is deeced. I has wo 68 Volume 6, Number 4, Ocober 06
Inernaional Journal of Applied Physics and Mahemaics values under wo saes- ARL before an ou-of-conrol sae is deeced when he process is in conrol defined as ARL0 and ARL before an ou-of-conrol sae is deeced afer process mean changed defined as ARL. In his research, Mone Carlo simulaion is used o evaluae he ARL which is classical mehod and very useful while he closed-form formula and he explici expression of ARL are no exis. In addiion, he resuls obained from MC use for checking an accuracy he resuls from oher approaches. Table. Comparison When Processes Are Log-Normal (0, ) 0., =0.8 and ARL0 70. EWMA DEWMA GWMA L 4. L.68 L 4.66 0.00 70.85 (.654) 70.87 (.6) 70.05 (4.05) 0.0 50.0 (.559) 48.6* (.59) 5.8 (4.4) 0.05 8.07 (.6) 66.97* (.7) 9.8 (4.0) 0..89 (0.950) 87.09* (0.8).7 (4.087) 0. 6.54 (0.576) 94.9* (0.46) 0.5 (.855) 0. 75.805 (0.5) 46.6* (0.) 68.4 (.747) 0.5 7.76 (0.4).00* (0.00).885 (.4) Noe: he sandard error is showed in parenheses. Table 4. Comparison When Processes Are Log-Normal (0, ) 0., =0.8 and ARL0 500. The approximaion ARL by MC is given by EWMA DEWMA GWMA L 4.678 L.907 L 4.7 0.00 500.666 (.08) 500.9 (.94) 49.0 (4.966) 0.0 474.086 (.098) 464.96* (.048) 47.64 (4.05) 0.05 78.745 (.690) 4.88* (.5) 6.58 (4.005) 0. 86.07 (.8) 5.8* (.044) 4.455 (.99) 0. 67.79 (0.754) 6.47* (0.5) 5.564 (.55) 0. 99.8 (0.449) 58.6* (0.79) 6.484 (.4) 0.5 6.4 (0.77).00* (0.00).478 (.) Noe: he sandard error is showed in parenheses. ARL N RL. N The sandard deviaions of ARL (SDRL) as SDRL RL ARL N N ( ). where RL is he number of observaions used o monioring before ou-of-conrol in simulaion round and N =50,000 runs is he number simulaion each siuaions. h 69 Volume 6, Number 4, Ocober 06
Inernaional Journal of Applied Physics and Mahemaics 4. Numerical Resuls In his secion, he performance of EWMA, DEMWA and GWMA chars in order o deec a change in parameer are compare by considering an ou-of-conrol average run lengh ARL when observaions are underlying gamma (, ) and log-normal (0, ). The weighed parameer of EWMA, DEWMA and GWMA char is given o equal 0., adjusmen smoohing consan is 0.8 and he magniudes of shif are given o be = 0.0, 0.05, 0., 0., 0. and 0.5. The comparison of ARL for gamma disribuion when ARL 0 = 70 and 500 are shown on Table and, respecively. When processes are log-normal disribued, ARL he numerical comparison of are presened on Table and 4. 5. Conclusions According o he numerical resuls, he performance of memory-ype conrol chars as EWMA, DEWMA and GWMA are invesigaed o sudy he robusness o skew process such gamma and log-normal disribuions. The hisorical weighed of hose conrol chars are given equally o 0. ha means he pas ARL informaion are concerned similarly of all hree conrol chars. Since, he value of obained from DEWMA is minimum i can explain ha he DEWMA ouperforms o deec a change in parameer when observaions are boh gamma and log-normal disribuions. Therefore, DEWMA s performance robus o he skew processes. Acknowledgmen The auhor would like o express my graiude o Deparmen of Applied Saisics, Faculy of Sciences, King Mongku s Universiy of Technology, Norh Bangkok, Thailand for supporing he gran. References [] Page, E. S. (954). Coninuous inspecion schemes. Biomerika, 4(), 00-4. [] Robers, S. W. (959). Conrol char ess based on geomeric moving average. Technomerics,, 9-50. [] Shewhar, W. A. (9). Economic Conrol of Qualiy Manufacured Produc. MacMillan, London. [4] Buler, S. W., & Sefani, J. A. (994). Supervisory run-o-run conrol of a polysilicon gae ech using in siu ellipsomery. IEEE Transacions on Semiconducor Manufacuring, 7(4), 9-0. [5] Sheu, S. H., & Yang, L. (00). The generally weighed moving average conrol char for deecing small shifs in he process mean. Qualiy Engineering, 6(), 09-. [6] Mongomery, D. C. (005). Inroducion o Saisical Qualiy Conrol. New York, Chicheser: Wiley. [7] Borror, C. M., Mongomery, D. C., & Runger, G. C. (999). Robusness of he EWMA conrol char o non-normaliy. Journal of Qualiy Technology, (), 09-6. [8] Soumbos, Z. G., & Reynolds, M. R. (000). Robusness o non-normaliy and auocorrelaion of individuals conrol chars. Journal of Saisical Compuaion and Simulaion, 66(), 45-87. [9] Alkahani, S. S. (0). Robusness of DEWMA versus EWMA conrol chars o non-normal processes. Journal of Modern Applied Saisical Mehods, (), 48-6. [0] Shamma, S. E., & Shamma, A. K. (99). Developmen and evaluaion of conrol chars using double exponenially weighed moving averages. Inernaional Journal of Qualiy & Reliabiliy Managemen, 9(6), 8-5. [] Chiu, W. C. (009). Generally weighed moving average conrol chars wih fas iniial response feaures. Journal of Applied Saisics, 6(), 55-75. 70 Volume 6, Number 4, Ocober 06
Inernaional Journal of Applied Physics and Mahemaics Saowani Sukparungsee was born in Chonburi, Thailand since April 0, 976 and graduaed he Ph.D. in mahemaical science from Universiy of Technology, Sydney, Ausralia since 009, he M.S. in saisics from Chulalongkorn Universiy, Bangkok, Thailand since 000 and B.S. in applied saisics from King Mongku s Universiy of Technology, Norh Bangkok (KMUTNB), Thailand since 997. She currenly works for KMUTNB a he Deparmen of Applied Saisics, Faculy of Applied Science as he head of he Deparmen and Academic Posiion is associae professor. The major fields of sudy are parameric and non-parameric conrol chars, saisical qualiy conrol, forecasing and ime series analysis, sochasic processes and applied saisics areas. 7 Volume 6, Number 4, Ocober 06