DEMONSTRATING THE INVARIANCE PROPERTY OF HOTELLING T 2 STATISTIC

Transcription

1 ISSN: Intenational Jounal of Innovative Reseach in Science, Engineeing and echnology Vol., Issue 7, July 013 DEMONSRAING HE INVARIANCE PROPERY OF HOELLING SAISIC Sani Salihu Abubaka 1, Abubaka Yahaya Lectue, Depatment of Statistics, Kano Univesity of Science and echnology, Wudil, Kano State, Nigeia 1 Lectue, Depatment of Mathematics, Ahmadu Bello Univesity, Zaia, Kaduna State, Nigeia Abstact: Numeous contol chat pocedues wee developed and thei widespead applicability as vital tools in statistical pocess contol cannot be oveemphasized. Unde nomal cicumstances, pocess monitoing and contol should include seveal inteelated pocess vaiables. he Hotelling statistic being one of the most popula contol chat pocedues can be utilized to detect whethe the obseved distubance in a given pocess is as a esult of some petubations in the mean vecto of the quality chaacteistics o in thei vaiability. One of the most notable featues of Hotelling statistic is its invaiance popety. his aticle demonstates this impotant popety using a eal dataset obtained fom a well-known cement poduction company in Nigeia. Keywods: Pocess Contol, Hotelling s Statistic, Decomposition, Matix Pemutation. I. INRODUCION Multivaiate statistical pocess contol (MSPC) is of immense impotance and to lage extent being applied in poduction companies as well as manufactuing industies whee data is collected and monitoed on many inteelated pocess vaiables. Pactically, most of the quality chaacteistics to be contolled and monitoed ae dependent. Howeve, ignoing the inteelationship existing between them will be inappopiate. Moe so, the multivaiate contol chating opens up oppotunity in consideing the coelation among pocess vaiables. he complexity in the application of MSPC chating technique in tems of undestanding the multivaiate popeties itself poved to be a challenging task to pactitiones. he application of MSPC equies an undestanding of two distinct phases in contol chating pocedues namely, phase I and phase II opeations. Woodall [8], believes that, it is wothwhile to diffeentiate between the applications of the two phases. Phase I is aimed at detemining whethe the pocess was actually unde contol given that the histoical dataset is popely collected. It is duing this phase that the establishment of the initial contol limits is done in ode to bing the pocess unde contol. In addition, if thee is no infomation about the incontol pocess paametes 0 and 0 both paametes could be estimated fom histoical dataset. Wheeas in phase II, the goal is to investigate whethe the pocess emains in-contol when a futue obsevation is dawn, in this phase, the contol chats ae used fo monitoing and detecting any changes in the pocess mean vecto o covaiance matix which ae the in-contol pocess paametes. Quite numbe of eseaches have discussed in geat detail multivaiate statistical contol pocedues; fo futhe details we efe an inteested eade to the woks of Besimis et al. [1]. he Hotelling chat, being one of the most impotant statistical pocess contol tools aimed at monitoing the mean vecto of quality chaacteistics, has been pesented in the woks of Lowy and Montgomey []. In this aticle ou focus is to demonstate the invaiance popety of Hotelling s statistic and the pape is oganized as follows. In the next section we pesent the basics of Hotelling s contol chats, afte which we show how the statistic can be decomposed. In the same section II we also demonstate the invaiance popety of Hotelling s statistic. In section III, we povide a numeical example to futhe boost the eade s undestanding of the invaiance popety. Discussion of esults and concluding emaks ae pesented in section IV. II. HOELLING S CONROL CHAR Among the popula multivaiate pocess monitoing and contol pocedues; the Hotelling s contol chat is in the foefont. In this section, we intend to pesent the Hotelling s contol chat in elation to the quality chaacteistics of a cetain pocess. Let us assume thee ae two quality chaacteistics ( x 1 and x ) that need to be monitoed in a given pocess and let these chaacteistics be jointly nomally distibuted andom vaiables. Let us also assume that, μ and Copyight to IJIRSE 515

2 ISSN: Intenational Jounal of Innovative Reseach in Science, Engineeing and echnology Vol., Issue 7, July 013 Σ espectively epesent the mean vecto and covaiance matix of the quality chaacteistics unde in-contol conditions; which can be computed using: μ, Σ 1 Given that a pocess is pesumed to be in-contol, the estimates μ and Σ can be obtained fom the peliminay samples taken. Hence, μ x x (.1) 1 1 x 1 1 s1 s 1 Σ S (.) 1 s1 s Accoding to Montgomey [6], S is an unbiased estimato of Σ when the pocess is in-contol. hus, the test statistic fo an individual obsevation becomes; ' -1 x - x Σ x - x (.3) When the chat is used fo monitoing futue poduction, the contol limits to be used is given by: ucl 1n 1 nn 1 p n F p, n1 (.4) F Whee, p, n 1 is the value of F-distibution with p and (n-1) degees of feedom at significance level..1 Decomposition of Statistic Mason et al. [4, 5] as well as Runge et al. [7] have been unaguably among the famous eseaches outlining some well infomed appoaches fo the decomposition of Hotelling statistic. Both eseaches pesented a unique way of decomposing the statistic; fo instance, the pocedue fo the decomposition of Hotelling statistic has been done by Mason et al. [4, 5] using two diffeent tems, namely, the conditional and unconditional tems as follows. (.5) s. whee ( xi x) (.6) s and s. ( x, xs) ( x ) (.7) On the othe hand, the pocedue employed by Runge et al. [7] involved the decomposition of the statistic into components that eflect the contibution of each individual vaiable; that is d (.8) ( ) Copyight to IJIRSE 516

3 Whee the statistic, ( ) ISSN: Intenational Jounal of Innovative Reseach in Science, Engineeing and echnology Vol., Issue 7, July 013 d is an indicato of the elative contibution of the th vaiable to the oveall statistics; wheeas, is the value of statistic fo all othe quality vaiables except the th one. hus, ' n x x x x 1 ( ) (.9). he Invaiance Popety of Hotelling s statistic he invaiance pinciple seeks what always emains the same o invaiant. he invaiance popety of pemuting components guaantees that each odeing of an obsevation vecto will poduce the same oveall value, since thee pemutations of the components of vectos x1, x, xp ae p! p p 1 p... 1 implies that we can patition a value in p! diffeent ways. o illustate, suppose p = 3, then can be decomposed in 6 possible ways as follows: this Accoding to Mason and Young [3], value cannot change by pemuting the component of the obsevation vecto, hence: x1 x xp x x1 xp (,,, ) (,,, ) III. ILLUSRAIVE EXAMPLE In this section we pesent an illustative example aimed at demonstating the invaiance popety of Hotelling s statistic. he data fo this eseach (see APPENDIX 1) was collected fom the poduction line of West Afican Potland Cement Company (WAPCO) a cement poduction company in Nothen Nigeia. hee vaiables (X 1, X, and X 3 ) epesenting tempeatue eadings fom thee boile machines wee monitoed and unde each twenty five obsevations wee ecoded at diffeent time peiods. he values of Hotelling s statistic obtained fom the obsevations on APPENDIX 1 can be found on APPENDIX. hus: S , ' X and 55 X In ode to demonstate the invaiance popety of, we conside 1 x x s x x ( )' ( ) Now, to see the contibution of each component, the above is decomposed as follows: Copyight to IJIRSE 517

4 ISSN: Intenational Jounal of Innovative Reseach in Science, Engineeing and echnology Vol., Issue 7, July IV. CONCLUSION In this pape, we pesented one of the novel tools used in MSPC, which is the Hotelling s statistic. We also eviewed diffeent popula Hotelling statistic decomposition techniques. By making use of eal life dataset obtained fom a cement poduction company, we wee able to demonstate the invaiance popety of the statistic which guaantees that each odeing of an obsevation vecto will poduce the same oveall value. REFERENCES [1] Besimis, S., Psaakis, S. and Panaetos, J., Multivaiate Statistical Pocess Contol Chats: an Oveview, Quality and Reliability Engineeing Intenational, Vol. 3, pp , 007. [] Lowy, C. A. and Montgomey, D. C., A Review of Multivaiate Contol Chat, IIE ansactions, Vol. 7, pp , [3] Mason, R. L. and Young, J. C., Multivaiate Statistical Pocess Contol with Industial Applications, ASA-SIAM, Philadelphia, 00. [4] Mason, R. L., acy, N. D. and Young, J. C., Decomposition of fo Multivaiate Contol Chat Intepetation, Jounal of Quality echnology, Vol. 7, pp , [5] Mason, R. L., acy, N. D. and Young, J. C., Pactical Appoach fo Intepeting Multivaiate Contol Chat Signals, Jounal of Quality echnology, Vol. 9, pp , [6] Montgomey, D. C. Intoduction to Statistical Quality Contol, New Yok: John Wiley & Sons, 005. [7] Runge, G. C., Fank, A. B. and Montgomey, D. C., Contibutos to a Multivaiate Statistical Pocess Contol Chat Signals, Communication in Statistics heoy and Methods, Vol. 33, pp , 1996 [8] Woodall, W. H., Contovesies and Contadictions in Statistical Pocess Contol, Jounal of Quality echnology, Vol. 3, pp , 000. Biogaphy M Sani Salihu Abubaka is a Lectue in the Depatment of Statistics, Kano Univesity of Science and echnology, Wudil, Nigeia. In Decembe 005, he obtained his Bachelo s Degee in Statistics fom Kano Univesity of Science and echnology, Wudil, Nigeia with Second Class Uppe. He is cuently pusuing his Mastes degee in Statistics at the Depatment of Mathematics, Ahmadu Bello Univesity, Zaia, Nigeia. His main eseach inteest is in Multivaiate Statistical Analysis D. Abubaka Yahaya is a Lectue in Statistics and the Postgaduate Coodinato in the Depatment of Mathematics, Ahmadu Bello Univesity, Zaia, Nigeia. In 000, he obtained his Bachelo s Degee in Statistics fom Usmanu Danfodiyo Univesity, Sokoto, Nigeia with Fist Class. In 004, he also obtained his Mastes Degee in Decision Modelling & Infomation Systems fom Bunel Univesity, Uxbidge, United Kingdom unde the sponsoship of Commonwealth Scholaship Commission. Futhemoe, in Januay 011, D. Abubaka successfully completed his Doctoate Degee pogamme in Management Science (Opeations Reseach) fom Lancaste Univesity Management School, Lancaste, UK unde the sponsoship of Petoleum echnology Development Fund (OSS-PDF), Abuja, Nigeia. he eseach inteests of D. Abubaka cut acoss divese aeas in Statistics and Opeations Reseach. Howeve, his main eseach inteest lies in the Application of Metaheuistic Optimization Algoithms in solving eal life poblems elated to Finance and Engineeing. Copyight to IJIRSE 518

5 ISSN: Intenational Jounal of Innovative Reseach in Science, Engineeing and echnology Vol., Issue 7, July 013 APPENDIX 1 Boile empeatue Data Peiods of Obsevations X 1 X X otal Aveage APPENDIX Computation of Hotelling s Peiods of Obsevations Hotelling s Value F- Citical Value ** ** Copyight to IJIRSE 519