Bayesian analysis of process capability indices for single and multiple sources of variability

Abstract

English: Process capability index (process performance index) -relates the specification limits to the performance of a process, it reduces complex information about the performance of a process to a single number. A capability index is a dimensionless measure of relative variability. In this thesis, Bayesian statistics is employed to simulate and estimate most of the widely used process capability indices. In Bayesian analysis, we assume that we have prior knowledge or information or opinion about parameters of a statistical distribution and very often in practice we do. We then attach a distribution to this belief. Parameters do not really have a distribution, parameters are constants, and so a prior distribution is a way of expressing our belief or opinion on our parameters. A posterior distribution is the belief distribution of the parameters after the outcomes of experiments (data) have been observed. There is now an updated belief distribution in light of the information from the data. Bayesian inference is shown to have a number of advantages. A full Bayesian analysis provides a natural way of taking into account all sources of uncertainty in the estimation of the parameters. Uncertainty about the true value of the process capability index is incorporated into the analysis through the choice of a prior distribution. The most familiar element of the Bayesian school is the use of the non-informative (objective) prior distribution, designed to be minimally informative in some sense. The most famous of these is the Jeffrey’s-rule prior and is utilised throughout the thesis. Scientists hold up objectivity as the ideal of science. Reference priors are a refinement of the Jeffrey’s-rule priors for higher dimensional problems that have proven to be remarkably successful. The probability matching prior is recommended because it is designed to produce posterior credible intervals which are asymptotically identical to their frequentist counterparts. The Bayesian simulation procedure employs the posterior distribution of the parameters in doing the simulations. The procedure is also shown to be useful and comparable to existing classical statistical procedures in solving the supplier selection problem. Data arising from multiple sources of variability are very common in practice. Virtually all industrial processes exhibit between-batch and within-batch components of variation. In some cases the between-batch (or between subgroup) component is viewed as part of the common-cause-system for the process. A process capability index in more general settings is developed using Cpl as a point of reference. Cpl is a single variance index and is adapted to give 2 and 3 variance components indices. The variance component model proves to be suitable for handling multiple sources of variability capability indices. Again, Bayesian simulation methods prove to be useful in handling these multiple sources of variability indices. Results show that the Bayesian simulation approach is just as good if not better than the standard classical statistics approach in assessing the capability of an industrial process. The added advantage of the Bayesian approach is that, from the posterior distribution of the capability indices, we are in a position to obtain quantiles, credible regions and perform other inferential tasks.

Afrikaans: Prosesgeskiktheidsanalise verwys na die moontlikheid om die Bayes-simulasiebenadering toe te pas op prosesgeskiktheidsindekse soos onder andere Cp , Cpk , Pp en Ppk . In hierdie verhandeling word Bayes-statistiek gebruik om die meeste van die prosesgeskiktheidsindekse te simuleer en te beraam. In Bayes-analise neem ons aan dat ons prior kennis of inligting of ‘n opinie het aangaande parameters van ‘n statistiese verdeling, soos die geval dikwels in die praktyk is. ‘n Verdeling kan dan aan hierdie oortuiging gekoppel word. Parameters is konstantes en het nie regtig ‘n verdeling nie, dus is ‘n priorverdeling ‘n manier om ons opinie of oortuiging aangaande parameters uit te druk. ‘n Posteriorverdeling is ‘n oortuigingsverdeling van die parameters nadat die uitkomste of eksperimente (data) waargeneem is. Daar is nou ‘n opgedateerde oortuigingsverdeling in die lig van die inligting uit die data bekom. Bayes-inferensie het ‘n hele aantal voordele. ‘n Volledige Bayes-analise voorsien ‘n natuurlike manier om alle bronne van onsekerheid met die beraming van die parameters in ag te neem. Onsekerheid oor die werklike waarde van die prosesgeskiktheidsindeks word in die analise ingesluit deur middel van die keuse van ‘n priorverdeling. Die mees bekende element van die Bayesskool is die gebruik van die objektiewe priorverdeling, wat ontwerp is om minimale inligting in ‘n sekere sin te gee. Die mees gewildste een is die Jeffreys-reël prior wat deurgaans in die verhandeling gebruik word. Wetenskaplikes hou objektiwiteit as die ideaal van wetenskap voor. Verwysingspriors is ‘n verfyning van die Jeffreys-reël priors vir hoër dimensionele probleme wat reeds as suksesvol beskou word. Die waarskynlikheidsgepaste prior word aanbeveel omdat dit ontwerp is om posterior kredietwaardigheidsintervalle te lewer wat assimptoties identies is aan hulle frekwentistiese teenpartye. Die Bayes-simulasieprosedure gebruik die posteriorverdeling om die simulasies uit te voer. Die prosedure het getoon dat dit geskik en vergelykbaar is met bestaande klassieke statistiese procedures om die verskaffer-seleksieprobleem op te los. Data wat uit meervoudige bronne van variasie voortspruit is baie algemeen in die praktyk. Letterlik alle industriële prosesse toon tussengroep en binnegroep komponente van variasie. In sommige gevalle word die tussengroepkomponent beskou as deel van die algemeen-oorsaak-sisteem van die proses. ‘n Prosesgeskiktheidsindeks in meer algemene omstandighede is ontwikkel deur Cpl as ‘n puntverwysing te gebruik. Cpl is ‘n enkel variansie-indeks en is aangepas om 2 en 3 variansiekomponentindekse te gee. Daar is bewys dat die variansiekomponentmodel geskik is vir die hantering van meervoudige bronne van variasiegeskiktheidsindekse. Weereens kan bewys word dat Bayessimulasiemetodes geskik is vir die hantering van hierdie meervoudige bronne van variasie-indekse. Resultate toon dat die Bayes-simulasiebenadering net so goed, indien nie beter nie, is as die standaard klassieke statistiekbenadering om die vermoë van die industriële proses te assesseer. ‘n Bykomende voordeel van die Bayesbenadering is dat, vanuit die priorverdeling van die geskiktheidsindekse, die moontlikheid geskep word om kwantiele en kredietwaardigheidsintervalle te bekom, asook om ander inferensiële take uit te voer.