Mixtures Experiments with Mixing Errors

Transcription

1 Mures Epermens h Mng Errors Aaa Ahuba & Aeander N. Donev Frs verson: June 009 Research Repor No. 0, 009, Probaby and Sascs Group Schoo o Mahemacs, he Unversy o Mancheser

2 Mures Epermens h Mng Errors Aaa Ahuba and Aeander N. Donev Schoo o Mahemacs, Unversy o Mancheser, UK Absrac. Varous properes o mure producs depend on he proporons o he componens o he mure, bu no on s amoun. In a mure epermen he mures are prepared by mng he reured amouns o mure componens. hs paper consders he mpac o mng errors on he sasca anayss o such daa. Keyords: errors n varabes, regresson cabraon, Scheé poynoma, smpe ace desgn.

3 . Inroducon he man properes o many producs produced by mng severa componens depend on her proporons n he mure bu no on he amoun o he mure. ypca eampes are he ase o a bend o ru uces obaned rom deren rus and he srengh o an aoy made by mng deren meas. here are aso cases here oher acors may aso aec he properes o he mure. For eampe, he eec o a drug combnaon or a erzer made o severa ngredens oud depend no ony on he proporons o componens o he mure bu aso on s amoun and he ay s manuacured and apped. Mures h reured proporons o he ngredens are usuay obaned by mng approprae amouns o hem. hs s oen done h errors hch haever sma can nuence he properes o he mure. e nvesgae he mpac o such errors on he sasca anayss o daa coeced n epermens h mures and sugges mprovemens o he anayss ha gnores he mng errors. e sho ha hese resus aso ao or evauang and conrong he cos o manuacurng mure producs h mng errors. he probem o errors n varabes has receved consderabe aenon n he eraure. Monographs summarzng he avaabe mehods o handng such daa ncude Fuer 987, Cheng and Van Ness 999 and Carro, Rupper and Seansk 006. An mporan dsncon beeen deren cases s based on ho he errors occur and ha her dsrbuon s. For eampe, an epanaory varabe canno be measured drecy or precsey, he vaues used n he sasca anayss are

4 u,,,, n, here n s he number o observaons. Deren assumpons can be made abou he so caed cassca measuremen errors u ; or eampe, hey are oen assumed o be ndependen and normay dsrbued h mean zero and varance,.e. u N 0,. u ~ u Hoever, hen he daa are coeced n a desgned epermen and he speced by he epermena desgn vaues o he h epanaory varabe are se h errors,.e. u,,,, n, he naccurae vaues are usuay used n he esmaon o he sasca mode o he daa because he vaues are no knon. he errors n hs case are knon as Berkson s errors, named aer Berkson 950 ho rs suded hs error srucure. In addon o he es noed above, here are aso many papers addressng ssues specc o he areas here he errors n varabes occur. he Fuer 987, p.79 dened he mng errors as an neresng probem o errors n varabes, her sudy appears med ony o hose presened by Sener and Hamada 997 and Hamada, Marz and Sener 005. hese researchers poned ou ha gnorng he mng errors eads o resus ha overesmae he varance o he response o neres and hence eads o ncreased condence nervas or he esmaes o he mode parameers and reduced poer o he sasca ess or her sgncance. hey proposed a Bayesan approach o esmae he parameers o he reured sasca modes or he daa, based on MCMC smuaons usng pror normaon abou he on dsrbuons o a mode parameers. hs emprca approach provdes a useu pracca oo or a beer sasca anayss han ha gnorng he mng errors, bu does no ao or esabshng a mporan eaures o he mpac o hese errors. 3

5 In hs paper e sudy he mpac o mng errors on he sasca anayss o he resus. ha makes he probem dcu s ha n desgned epermens h mures, hen here are mng errors, he errors made n seng he correc amouns o any ngreden, propagaes o he proporons reured by he epermena desgn or a ngredens. hereore he mng errors ead o a compe error srucure or he proporons o he mures. In Secon, e reve he man eaures o daa coeced n mure epermens and sho ho mng errors aec he resus he mng errors are gnored. As smpe ace desgns are reueny used n epermens h mures, e derve he bas n he esmaes o he mode parameers resung rom anayss ha gnores mng errors hen such desgns are used. he repor concudes h a dscusson abou he useuness o he presened resus.. Mure epermens h mng errors In a ypca sudy o a mure h componens, he epermener s neresed n he ay a response o neres, say y, depends on he proporons o he componens o he mure,,,,. Hence, 0 u,, here andu are oer and upper bounds or he proporon o he h componen n he mure. A mure h he desred proporons s obaned by mng approprae amouns,,,,, o he ngredens. In he absence o errors n seng he amouns o he mure componens, here s he oa amoun o / 4

6 he mure. hen 0 and u,,,,, he desgn regon s a reguar smpe hch has verces. Oen > 0 and u < or some or a o he mure componens. I he resung desgn regon agan has verces, ne varabes hch are near combnaons o he componens, caed pseudocomponens, can be dened n such a ay ha he reaonshps are sased or hese varabes. Hence, he probem can be redened as one or a reguar smpe desgn regon. Hoever, he consrans and u can be such ha he desgn regon s rreguar h a number o verces arger han. A comprehensve reve o resus reaed o epermens h mures s gven by Corne, 00. hen he amouns o he mure componens are se h errors, he acua amouns become X e, here e are he errors o seng he amouns o he h mure componen or he h observaon o he response. e assume ha e, or,,,, are ndependeny dsrbued random varabes h mean zero and varance,,,,, proporona o he dscharged amouns, here denes he ay he varance changes h he amoun. For eampe, can assumed ha. Hoever, hs assumpon has o be vered, usuay by a vadaon epermen. he mng errors can be a resu o echnca maons o he eupmen used n he epermen. hey can aso depend on he physca properes o he componens o he mure. For eampe, he componens are n ud orm, he errors n seng her amouns may depend on her vscosy, so ha he varances o he errors o seng he amouns or deren mure 5

7 componens may be deren, even he same eupmen s used o dscharge he reured amouns. Sandard poynomas canno be used o mode he ay he response depends on he mure proporons because o he consrans. Insead, oen he canonca poynomas proposed by Scheé 958 are used. For eampe, he rs and second order Scheé poynomas are y η, ε β β ε y, ε β η β β ε 3 respecvey, here β s a p vecor o a mode parameers and s a vecor o he proporons o he componens o he mure n he h observaon. e assume ha he response s measured h errors hch are normay dsrbued h zero mean and varance ε. Cusomary, smpe ace desgns are used hen he desgn regon s a reguar smpe, or pseudocomponens can be used o ransorm o an denca probem. In a {d, m} smpe-ace desgn, he proporons o each componen ake d euay spaced vaues rom 0 o, 0, /d, /d,...,, or,,...,. One advanage o hese desgns s ha he anayss o he daa and s nerpreaon s smped. For eampe, d s chosen o be eua o he order o Scheé poynoma ha be ed, he eas suares esmaors or he mode parameers o he rs and he second order Scheé poynomas are: βˆ y,,,, 4 6

8 and β ˆ 4y [ y y ],,,,, <, 5 here y denoes he average o he observaons hen he mure consss o he h componen ony, he y denoes he average o he observaons here he proporons o he mure o he h and he h componen are boh eua o 0.5. hen he desgn regon s rreguar, he desgn consrucon can be carred ou usng any o he desgn agorhms descrbed n Aknson e a. Chaper, 007. I ony he amoun o he h componen n he h observaon s se h error, he acua proporon o hs componen becomes e e, he he acua proporons o he remanng componens become. e e denoe he vecor o acua proporons,,...,. I he amouns o a mure componens are se h errors e k e k,,,...,. he error n he proporon o he h componen resung rom mng errors n a componens o he mure s u e e e k e k k k, 7

9 here k e k s he acua oa amoun. Ceary, he mure consss o a snge componen, as reured or some o he observaons o a smpe ace desgn h no consrans on he proporons o he componens, dschargng he rong amoun no change he reured proporon o hs componen,.e.. hereore he oong resus are concerned ony h cases here he mures conss o o or more componens. he mpac o he mng errors on he anayss o he daa s summarzed n he oong emmas. he resus o Lemmas and are obaned drecy by usng he dea mehod; or deas see Meyer 965, p.8. hereore her proos are no provded. Lemma. he epecaon and he varance o he acua proporon o he h componen o a mure n he h observaon, gven he arge proporon,,,,, are E[ ] var, respecvey. Lemma. he epecaon and he varance o he produc o he acua proporons and o he h and he h componens o a mure n he h observaon, gven he arge proporons and, are E 3 3 8

10 3. var 4 Lemmas and sho ha he means o boh he acua proporons and her producs are deren rom hose speced by he epermena desgn, and her varances ncrease h he vaues o he mode parameers. For eampe, n a smpe ace desgn, mng errors occur n he amouns o he componens o a bnary bend, here 0.5, and E. [ ] 0.5, [ ] 0.5 var 0.5. An mporan speca case s hen, as hen E ] [. hs means ha n hs case on average he arge amoun s acuay dscharged, and under some condons, he eec o mng errors coud be regarded neggbe. Hoever, n genera, he mng errors s aec he second order Scheé modes as [ ] 0.5 E, 9

11 he 0.5 and he modes become based. he eac mpac o he mng errors on he sasca anayss o he resus depends on he epermena desgn ha has been used and he sasca mode ha s esmaed. heorem. he mng errors make he mode based, ecep n he case hen he mure consss o a snge componen,.e. E[ y ] η β, B, here he bas B depends on he orm o he rue mode. he resu o hs heorem oos drecy rom Lemmas and. As usraon o hs heorem, he dervaon o he bas or he rs and second order Scheé poynoma modes s gven n he Append. heorem. he mng errors make he varance o he mode heerogeneous and arger han,.e. var y >. ε ε he resu o hs heorem oos by nong ha var y ε var B. he epresson or he varance o he response s very compe. hereore e ony sho s dervaon or he rs order Scheé poynoma mode n he Append. he mpac o he mng errors on he esmaes o he mode parameers hen he mng errors are gnored naïve esmaes depends on ha epermena desgn has been used n he sudy. Neverheess, s mporan o pon ou ha he esmaes o mode parameers are based, hs s because he epecaon o he response gven he 0

12 arge desgn s based. So, by anayzng he daa usng he proporons speced by he epermena desgn, he errors n proporons are ransmed o he error erm ε n he mode, causng E ε 0. e assume hou oss o generay ha a smpe ace desgn has been used. Lemmas and ao or ndng he correspondng resus hen oher epermena desgns are used. heorem 3. I daa coeced h a smpe ace desgn, he eas suares esmaors or he mode parameers are based, so ha E ˆ β β E ˆ β β [ ] β [ ] β β,,,,, <. he proo o hs heorem reures edous bu sraghorard agebra appyng he resu o heorem o mody he eas suares esmaors 4 and 5 and s omed. 3. Dscusson he resus presened n hs repor sho ha gnorng he mng errors usuay eads o aenuaon n he esmaes o mode parameers so ha he ordnary eas suares esmaes are based. he sze o he bas depends on many acors, bu many on he sze o he mng error varances. As a resu o he mng errors, he mode, hch may oherse be chosen correcy, becomes based and h ncreased heerogeneous varance. he speccaon o he orm o he bas s very mporan because aos or correcng or usng he regresson cabraon approach Carro and Seansk, 990.

13 An neresng appcaon o he presened resus be o use hem o devse suabe correcons o manuacurng sengs o mure producs hen errors n he dscharged amouns o he mure componens canno be avoded. hs s been subec o ongong research. Append Proo o heorem or he rs order Scheé poynoma mode I he amouns o a mure componens are dscharged h errors, he rue mode s y β ε, From Lemma, oos ha a naïve anayss o he daa gnorng he mng errors usng mode eads o predcons E[ y ] β β η β, Hence, he epeced response s based. B. Proo o heorem or he second order Scheé poynoma mode I a componens are measured h errors, he rue mode s gven by y β β ε. I oos rom Lemmas and ha E[ y ] β β B η β, B, here he bas s

14 B β β Proo o heorem or he rs order Scheé poynoma mode he varance o he response gven he desgn pons s, cov var var y β β β ε. Snce, appyng he resus o Lemmas and, gves ] [ ] [ ] [, cov E E E y var β ε 3 3 β β 3. Ceary he varance s arger han and depends on he proporon o he mure. ε Reerences Berkson, J Are here o Regressons? Journa o he Amercan Sasca Assocaon, 45, 50,

15 Carro, R.J., Seansk, L.A Appromae uas-kehood esmaon n modes h surrogae predcors. J. Amer. Sas. Assoc. 85, Carro, R., Rupper, D. and Seansk, L. A Measuremen Error n Nonnear Modes. nd edon, Chapman and Ha. Cheng, C. and Van Ness, J Sasca Regresson h Measuremen Errors. John ey & Sons, Ne York. Corne, J.A. 00 Epermens h mures: Desgn, Modes, and he Anayss o Mure Daa. 3rd Edon. ey, Ne York. Fuer,.A Measuremen Error Modes. John ey & Sons, Ne York. Meyer, P.L.965. Inroducory probaby and sasca appcaons. Addson- esey, Ne York. Scheé, H. 958 Epermens h mures. J.R.Sas.Soc. B, 0, Sener, S.H., Hamada, M Makng mures robus o nose and mng measuremen errors. Journa o Quay echnoogy, 9, 4, Hamada, M., Marz, H F., Sener S Accounng or Mure Errors n Anayzng Mure Epermens. Journa o Quay echnoogy, 37,,