Journal of Applied Mathematics and Computation (JAMC), 2018, 2(10),

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1 Journal of Appled Mahemacs and Compuaon (JAMC), 08, (0), hp:// ISSN Onlne: ISSN Prn: Parameer Esmaon wh Leas-Squares Mehod for he Inverse Gaussan dsrbuon Model Usng Smplex and Quas-Newon Opmzaon Mehods Khzar Haya Khan Deparmen of Mahemacs, College of Scence and Humanes, Prnce Saam Bn Abdulazz Unversy, Al-Kharj, Kngdom of Saud Araba How o ce hs paper: Khan, K. H. (08) Absrac Parameer Esmaon wh Leas-Squares Mehod for he Inverse Gaussan dsrbuon We fnd Survval rae esmaes; parameer esmaes for he nverse Gaussan dsrbuon model usng leas-squares esmaon mehod. We found hese esmaes for Model Usng Smplex and Quas-Newon Opmzaon Mehods. Journal of Appled Mahemacs and Compuaon, (0), he case when paral dervaves were avalable and for he case when paral dervaves were no avalable. The smplex opmzaon (Nelder and Mead, and Hooke DOI: /jamc *Correspondng auhor: Khzar Haya and Jeeves) mehods were used for he case when frs paral dervaves were no Khan, Deparmen of Mahemacs, College avalable and he Quas Newon opmzaon (Davdon-Flecher-Powel (DFP) and of Scence and Humanes, Prnce Saam he Broyden-Flecher-Goldfarb-Shanno (BFGS) mehods were appled for he case Bn Abdulazz Unversy, Al-Kharj, Kngdom of Saud Araba when frs paral dervaves were avalable. The medcal daa ses of Leukema Emal: drkhzar@gmal.com, cancer paens wh me span of 35 weeks were used. k.khan@psau.edu.sa Keywords Inverse Gaussan dsrbuon model, Nelder and Mead, and Hooke and Jeeves, DFP and BFGS opmzaon mehods, Parameer esmaon, Leas Square mehod, Kaplan-Meer esmaes, Survval rae Esmaes, Varance-Covarance marx. DOI: /jamc Journal of Appled Mahemacs and Compuaon(JAMC)

2 . Inroducon The mehods of Nelder and Mead [] and Hooke and Jeeves [] do no requre frs or hgher paral dervaves and n hese mehods, he funcon values are compared o fnd he opmal value of he objecve funcon. Whereas, he Quas Newon Mehods (DFP and BFGS) requre only frs paral dervaves of he funcon of n varables [3], we fnd he survvor rae esmaes, parameer esmaes, opmal funcon values usng smplex opmzaon mehods and Quas-Newon opmzaon mehods. These opmzaon mehods were appled usng medcal daa ses of cancer paens [4]. The mehod of lnear leas squares requres ha a sragh lne be fed o a se of daa pons such ha he sum of squares of he vercal devaons from he pons o be mnmzed [5], [7]. Adren Mere Legendre (75-833) s generally creded for creang he basc deas of he mehod of leas squares. Some people beleve ha he mehod was dscovered a he same me by Karl F. Gauss ( ), Perre S. Laplace (749-87) and ohers. Furhermore, Markov's name s also ncluded for furher developmen of hese deas. In recen years, [6], [9] an effor have been made o fnd beer mehods of fng curves or equaons o daa, bu he leas-squares mehod remaned domnan, and s used as one of he mporan mehods of esmang he parameers. The leas-squares mehod consss of fndng hose parameers ha mnmze a parcular objecve funcon based on squared devaons. I s o be noed ha for he leas-squares esmaon mehod, we are neresed o mnmze some funcon of he resdual, ha s, we wan o fnd he bes possble agreemen beween he observed and he esmaed values. To defne he objecve funcon F, we se up a vecor of resduals. () Then he objecve funcon s a sum of squared resduals - he erm 'leas-squares' derves from he funcon:. () The objecve funcon s he sum of he squares of he devaons beween he observed values and he correspondng esmaed values [6]. The maxmum absolue dscrepancy beween observed and esmaed values s mnmzed usng opmzaon mehods. We used non-paramerc Kaplan-Meer esmaes, [0], [] as he observed values of he objecve funcon and he survvor rae esmaes of nverse Gaussan [8], [0] model as he esmaed value of he objecve funcon F [0]. We consdered he objecve funcon for he model of he form where s he number of falures a me and m s he number of falure groups [0]. Of course s no so easy o work wh an accoun of he relavely complcaed form of Inverse Gaussan Dsrbuon models for s densy and survvor funcons. We used he followng procedure: Noe ha he Kaplan-Meer mehod [0] s ndependen of parameers, so for a parcular value of me, we fnd he value of he Kaplan-Meer esmae, of he survval funcon. The me T requred o cover a dsance d s a random varable wh pdf for Inverse Gaussan, and for ease of compuaon, we pu f ( ) d, 3 ( dv) dexp v d, hen we have, 0 f ( ) 3 exp ( ) We suppose ha he survvor funcon of nverse Gaussan dsrbuon model a me s S ( ) = e,. (3) DOI: /jamc Journal of Appled Mahemacs and Compuaon

3 where s he sandard normal dsrbuon funcon and s he survvor funcon [9] a he sarng pon. From he numercal values of he Kaplan-Meer esmaes, and he survvor funcon of he nverse Gaussan dsrbuon model a me, we can evaluae errors. The funcon value wh a suable sarng pon s gven by. We fnd numercal value of he objecve funcon, F a nal pon ( 0, 0) and hs funcon value can be used n numercal opmzaon search (Smplex or Quas-Newon) mehods o fnd he opmal pon and hen parameer esmaes a hs opmal pon. For praccal applcaons of leas square mehod, we consdered some medcal daa ses [4]. The drug 6-mercapopurne (6-MP) was compared o a placebo o manan remsson n acue leukema paens. The followng able gves remsson mes for wo groups of weny-one paens each; one group was gven he placebo and was gven he oher he drug 6-MP. Table. Remsson mes for wo groups of weny-one paens Lengh of remsson (n weeks) of leukema paens 6-MP for paens 6,6,6,,7,,0,,, 3,6,,,,,3,, * Censored observaons Placebo for paens,,,, 3, 4, 4, 5, 5, 8, 8, 8, 8,,,,, 5, 7,, 3 We consdered only he daa of weny-one paens who were gven 6-MP drug and here were 7 falures a mes (weeks) 6,7,0, 3, 6, and 3; and of he paens were censored. The daa of weny-one leukema paens were used for assessng he appropraeness of Inverse Gaussan model o fnd he survvor rae esmaes. The resuls of each Inverse Gaussan model usng Leas-Squares mehods and applyng smplex mehods are presened by gvng he values of he varance-covarance marces, parameer esmaes and oher relaed nformaon n ables, able 3 and able 4.. Inverse Gaussan Dsrbuon model usng Leas-Squares Mehod and Applyng Smplex Mehods (Nelder and Mead and Hooke and Jeeves Search Mehods) For a praccal applcaon of he leas-squares esmaon mehod when assumng ha, he paral dervaves of he objecve funcon F were no avalable. Nelder and Mead [], [3], [4] and Hooke and Jeeves [], [5] are smplex mehods and are useful for opmzng he nonlnear programmng problems. These numercal mehods opmze he objecve funcon whou calculang he dervaves a all. These mehods do no requre frs paral dervaves (gradens) so may converge very slow or even may dverge a all [5], [8]. The numercal resuls of Inverse Gaussan dsrbuon model usng Nelder and Mead and Hooke and Jeeves search mehods have been presened n hs paper. The resuls nclude funcon values, parameer esmaes, survvor-rae esmaes; Kaplan-Meer esmaes [0], [] and oher nformaon have been gven n Table and Table 3 and n Fgure and Fgure. Numercal and Graphcal Resuls for Inverse Gaussan Probably Dsrbuon Model usng Leas-Squares Mehods and Applyng Nelder and Mead (NM) and Hooke and Jeeves (HJ) Mehods. DOI: /jamc Journal of Appled Mahemacs and Compuaon

4 Falure Tme (Weeks) Table : Comparson of Survval Rae esmaes for Inverse-Gaussan Dsrbuon Model Number of Nelder and Mead Mehod Hooke and Jeeves Mehod Falures Inverse-Gaussan Kaplan Meer Inverse-Gaussan Kaplan Meer Table-3: Parameer Esmaes and Opmal Funcon Value for Inverse-Gaussan Dsrbuon Model Parameers Esmaes Opmal Funconal value Nelder and Mead Mehod Hooke and Jeeves Mehod DOI: /jamc Journal of Appled Mahemacs and Compuaon

5 Survvor Raes Survvor Raes K. H. Khan Nelder-Meads Mehod Inverse Gaussan Kaplan Meer Tme Hooke-Jeeves Mehod Inverse Gaussan Kaplan Meer Tme Fgure. Survvor rae esmaes usng Nelder-Meads mehod Fgure. Survvor rae esmaes usng Hooke-Jeeves mehod 3. Inverse Gaussan dsrbuon model usng Leas-Squares Mehods and Applyng Quas-Newon Opmzaon Mehods (DFP and BFGS Mehods) We know ha he survvor funcon, he probably of no falure before me, for he wo-parameer nverse Gaussan dsrbuon model s S ( ) = e where s he usual sandard normal dsrbuon funcon (Chhkara e al (974)). Now he objecve funcon for he nverse Gaussan dsrbuon model usng leas-squares esmaon mehod s, (5) where s he number of falures and s he Kaplan-Meer esmae a he falure me. We used he DFP and he BFGS opmzaon mehods o fnd he parameer esmaes for he objecve funcon. Evaluang he frs paral dervaves of he objecve funcon F w.r.. and we oban and where and S () m F ( ) ( ) f S KM (6) m F ( ) ( ) S ( ) f S KM, (7) S ( ) v 3 exp( v / ) v 4 exp( v / ) exp( ) ( v ) (4) DOI: /jamc Journal of Appled Mahemacs and Compuaon(JAMC)

6 here where S () v exp( v / ) v exp( v / ) V = V V and = V V V, 3 V V V V,, 4 V3 = and V4 =. Usng he objecve funcon eq. (5), and he frs paral dervaves eq. (6) and eq. (7) n he DFP and he BFGS opmzaon mehod, we can fnd he esmaed value of he parameers for whch he leas-squares funcon gves he mnmum value for nverse Gaussan dsrbuon model and he resuls are presened n he able 4.. Table 4. Numercal Resuls for he nverse Gaussan Probably Dsrbuon Model usng Leas-Squares Mehod and Applyng Quas Newon Mehods Quas Mehods DFP Model BFGS Model Parameers Esmaes Opmal Funconal value E E-0 Graden a Opmal E E E E-05 The Varance-Covarance a Opmal Concluson The Survval rae esmaes for he Leukema paens for he perod of 35 weeks under observaons wh drug 6-MP [5], [9] were compared usng paramerc Inverse Gaussan dsrbuon model and non-paramerc Kaplan Meer Model []. We found ha he resuls lke parameer esmaes, opmal funcon value, varance covarance marx, graden * * vecor a he opmal pon, were approxmaely same for boh he cases when he dervaves of an objecve DOI: /jamc Journal of Appled Mahemacs and Compuaon

7 funcon were avalable (usng Quas-Newon mehod (DFP and BFGS mehods)) and when frs paral dervaves of he objecve funcon were no avalable (usng he Hooke and Jeeves, and Nelder and Mead mehod). We also noed ha he parameer esmaes for he Nelder and Mead mehod are very close o he Quas-Newon mehods (DFP or BFGS). Usng unconsraned opmzaon mehod (DFP-mehod and BFGS), we presened parameer esmaes, nverse Hessan marces, survvor-rae esmaes and he maxmum lkelhood funcon values for he nverse Gaussan dsrbuon model usng survval me cancer daa ses. Wh he help of DFP-mehod and BFGS mehod varance covarance marx can be found auomacally whch s very dffcul f we apply some oher mehod (e.g. Newon-Raphson mehod) for he nverse Gaussan dsrbuon model. Acknowledgemen The auhor (Khzar H. Khan) hankfully acknowledges he suppor provded by he Deparmen of Mahemacs, College of Scence and Humanes, Prnce Saam Bn Abdulazz Unversy, Al-Kharj, (Ryadh) and Mnsry of Hgher Educaon, Saud Araba, for provdng he facles and an envronmen o perform hs research work. References [] J.A. Nelder, R. Mead, A Smplex mehod for funcon mnmzaon, Compuer Journal 7, (965), [] Rober Hooke and T.A. Jeeves, Wesnghouse Research Laboraores, Psburgh, Pennsylvana, Drec Research Soluon of Numercal and sascal Problem research, Inernaonal Journal of Sascs, Vol.6, pp ,960. [3] Flecher, Roger (987), Praccal mehods of opmzaon (nd ed.), New York: John Wley & Sons, ISBN [4] Frerech e al. (963) The Effec of 6-mercapopurne on he duraon of serod nduced remsson n acue leukema, Blood, pp [5] Placke, R.L. (97). The dscovery of he mehod of leas squares. Bomerka, 59,39 5. [6] J. T. Bes, Solvng he Nonlnear Leas Square Problems: Applcaon of a General Mehod, Journal of Opmzaon Theory and Applcaons, vol. 8, no. 4, 976. [7] J.E. Denns Jr., R.B. Schnabel. Nonlnear leas squares Numercal Mehods for Unconsraned Opmzaon and Nonlnear Equaons, SIAM, Phladelpha, PA (996), pp [8] J. L. Folks and R. S. Chhkara, The Inverse Gaussan Dsrbuon and Is Sascal Applcaon--A Revew Journal of he Royal Sascal Socey. Seres B (Mehodologcal), Vol. 40, No. 3, pp (978). [9] Ledvj, M. "Curve Fng Made Easy." Indusral Physcs 9, 4-7, May 003. [0] Kaplan E.L., Meer P. (958). Nonparamerc esmaon from ncomplee observaons, J. Amer. Sas. Assoc., 53, pp [] Cox D.R.,Oaks D. (984) Analyss of Survval Daa. London Chapman and Hall. [] C.J. Prce, I.D. Coope, D. Bya, A convergen varan of he Nelder-Mead algorhm, Journal of Opmzaon Theory and Applcaons, 3, (00), 5 9. [3] L. Han, M. Newmann, Effec of dmensonaly on he Nelder-Mead smplex mehod, Opmzaon Mehods and Sofware,, (006), -6. [4] Nam P., Bogdan M.Wlamow mproved Nelder Mead s Smplex mehod and Applcaons, Journal Of Compung, Volume 3, Issue 3, ISSN 5-967, March 0. [5] G.S Kra, A.N Surde Revew of Hooke and Jeeves Drec Search Soluon Mehod Analyss Applcable To Mechancal Desgn Engneerng Novaeur Publcaons Inernaonal Journal of Innovaons n Engneerng Research and Technology [IJIERT], ISSN: , Volume, Issue, Dec-04. [6] Lsawad, S., Parameer Esmaon for he Two-Sded BS and IG Dsrbuons, VDM Verlag, Saarbrücken, 009. [7] Zhenln Yang, Maxmum Lkelhood Predcve Denses for The Inverse Gaussan Dsrbuon wh Applcaons o Relably and Lfeme Predcons, Mcroelecroncs Relably, 39, 43-4, 999. [8] Seshard, V. The Inverse Gaussan Dsrbuon. Oxford Scence Publcaons.,993. [9] P. Basak, N. Balakrshnan, Esmaon for he hree-parameer nverse Gaussan dsrbuon under progressve Type-II censorng, Journal of he Sascs Compuaon and Smulaon 8 (7), , 0. [0] M. C. K.Tweede, Sascal properes of nverse Gaussan dsrbuon, Annals of Mahemacal Sascs 8 (), ,957. DOI: /jamc Journal of Appled Mahemacs and Compuaon