Regression for CERs with Multiplicative Errors

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1 PT 05 - Multpltve-Error Regresso v. Regresso for CERs wth Multpltve Errors Moder Tehques A usophstted forester uses sttsts s druke m uses lmp-posts - for support rther th for llumto. Adrew Lg PT ICEAA All rghts reserved. Akowledgmets ICEAA s deted to TASC, I., for the developmet d mtee of the Cost Estmtg Bod of Kowledge (CEBoK ICEAA s lso deted to Tehoms, I., for the depedet revew d mtee of CEBoK ICEAA s lso deted to the followg dvduls who hve mde sgft otrutos to the developmet, revew, d mtee of CostPROF d CEBoK For wht oer ths spef module o moder tehques of regresso for CERs wth multpltve errors, ICEAA s lso deted to: Rmod P. Covert former Tehl Dretor d Chef Prttoer, Cost d Shedule Alss, MCR, LLC, ow Presdet Covrus LLC, d Tmoth P. Aderso, Dretor, NASA Progrm Assessmets, The Aerospe Corporto PT-05 v ICEAA All rghts reserved. ICEAA 04 Professol Developmet & Trg Workshop

2 PT 05 - Multpltve-Error Regresso Akowledgmets v. The suet mtter of ths trg sesso ws frst orgzed tutorl form uder the ttle Sttstl Foudtos of CER Developmet for Mrh 004 presetto to the mgemet d support stff (ludg FFRDC d otrtor of the NRO Cost Group (ow NRO CAIG, Chtll VA. The preseters would lke to thk the NRO CAIG, uder the dreto of Keth Roertso, for fg the orgl preprto of the tutorl. The would lso lke to thk hs support-stff ollegues for the vgorous telletul dete tht trdtoll preedes, follows, d mproves suh dsussos of methodolog t the NRO CAIG. Espell, the would lke to thk the orgl uthor of ths tutorl, the lte Dr. Stephe A. Book. Steve s ledershp ws the drvg fore ehd m of the ostg tehques we use o dl ss tod. Thk ou Steve! We mss ou! ICEAA All rghts reserved. 3 Cotets v. CER Developmet Ordr Lest Squres (OLS Log-trsformed OLS Sttstl Issues Qult Metrs Geerl-Error Regresso Itertvel Reweghted Lest Squres (IRLS Ftor CERs Ler CERs Trd CERs Qult Metrs Zero Peretge Bs, Mmum Peretge Error (ZMPE Summr ICEAA All rghts reserved. ICEAA 04 Professol Developmet & Trg Workshop

3 PT 05 - Multpltve-Error Regresso Cotets v. CER Developmet Ordr Lest Squres (OLS Log-trsformed OLS Sttstl Issues Qult Metrs Geerl-Error Regresso Itertvel Reweghted Lest Squres (IRLS Ftor CERs Ler CERs Trd CERs Qult Metrs Zero Peretge Bs, Mmum Peretge Error (ZMPE Summr ICEAA All rghts reserved. Mthemtl Formulto of CERs v. = Cost = Tehl Prmeter (Cost Drver Ftor CER: = Ler CER: = + Noler CERs: = = = +,, re Costt Coeffets Derved from Hstorl Dt Ths Tutorl Wll Dsuss the Cse of Ol Oe Cost Drver per CER For Multple Cost Drvers, the Coepts re the Sme, ut the Sttsts re More Complted ICEAA All rghts reserved. 3 ICEAA 04 Professol Developmet & Trg Workshop

4 PT 05 - Multpltve-Error Regresso Idel Hstorl Dt (fter Normlzto v. COST TECHNICAL PARAMETER SCATTERGRAM ICEAA All rghts reserved. Cotets v. CER Developmet Ordr Lest Squres (OLS Log-trsformed OLS Sttstl Issues Qult Metrs Geerl-Error Regresso Itertvel Reweghted Lest Squres (IRLS Ftor CERs Ler CERs Trd CERs Qult Metrs Zero Peretge Bs, Mmum Peretge Error (ZMPE Summr ICEAA All rghts reserved. 4 ICEAA 04 Professol Developmet & Trg Workshop

5 PT 05 - Multpltve-Error Regresso Trdtol Ler Regresso v. Ler CER Addtve-Error Model = + + (Atul Cost = Estmted Cost + Error of Estmto Ordr Lest-Squres (OLS Regresso Mmzes Sum of Squred Errors Atul ost for dt pot s Estmted ost for dt pot s + Error of estmto for dt pot s = - ( + Choose vlues for d tht mmze ( - - = Resultg estmtes re used OLS Soluto: = = ICEAA All rghts reserved. - ( ( - ( - 9 OLS Stdrd Error of the Estmte v. A Oe-Sgm -tpe Error Boud o Error Implt OLS CERs SEE = ( Dollrs k = + s the CER tht Epresses Cost ( Terms of Cost-Drvg Tehl or Progrmmt Prmeter ( s the Numer of Dt Pots Used to Derve the CER k s the Numer of Prmeters the Alger Epresso for the CER, e.g., k = for the CER = + Stdrd Error s OLS CER Qult Metr ICEAA All rghts reserved. 0 5 ICEAA 04 Professol Developmet & Trg Workshop

Lwler, Corret d Iorret Error Speftos Sttstl Cost Models, Jourl of Cost Alss, Sprg 994, pge 07. 00-03 ICEAA All rghts reserved. Cotets v.

6 PT 05 - Multpltve-Error Regresso But, There re Altertve Error Speftos v. Y Multpltve Error Y Addtve Error X X Referee: H.L. Eskew d K.S. Lwler, Corret d Iorret Error Speftos Sttstl Cost Models, Jourl of Cost Alss, Sprg 994, pge ICEAA All rghts reserved. Cotets v. CER Developmet Ordr Lest Squres (OLS Log-trsformed OLS Sttstl Issues Qult Metrs Geerl-Error Regresso Itertvel Reweghted Lest Squres (IRLS Ftor CERs Ler CERs Trd CERs Qult Metrs Zero Peretge Bs, Mmum Peretge Error (ZMPE Summr ICEAA All rghts reserved. 6 ICEAA 04 Professol Developmet & Trg Workshop

7 PT 05 - Multpltve-Error Regresso Some Hstor: The 8 th Cetur Approh to Noler* Regresso v. Cosder the Noler Power Model = Tke Logrthms of Both Sdes: log = log + log Determe d to Predt log : Assume Addtve-Error Model log = log + log + E, where E = log - (log + log s Error of Estmto Predtg Logrthm of Cost Choose Vlues for d tht Mmze (log - log - log = E Logrthm Trsformto of Noler Form Ito Ler Form Permts Use of OLS Mthemts to Solve Noler Prolem Eel Tred Le Futo d Other Commo Quke Approhes Use Ths Tehque *For the purposes of ths dsusso, o-ler model refers to power futo ICEAA All rghts reserved. 3 Noler OLS-Bsed Soluto v. Predt log = log + log = A + log where (log (log (log log log log A log ( = 0 log = 0 A Stdrd Error of Estmte s reported s (log (log SEE ( log log log E ICEAA All rghts reserved. 7 ICEAA 04 Professol Developmet & Trg Workshop

8 PT 05 - Multpltve-Error Regresso A Word o Error of Estmto v. I Order for Logrthms to Work, Noler CER Model Must e Multpltve-Error Model = Beuse Applg Logrthms Yelds Addtve-Error Model for Predtg Logrthm of Cost log = log + log + log Settg E = log, wht s Atull Mmzed s (log = E = (log - log - log Isted of ICEAA All rghts reserved. 5 Dsoets Betwee OLS d Log- OLS Regresso Bd: Mmzg (log ot Sme s Mmzg (s Trdtol Ler Regresso d Vlues Tur out to e Dfferet Error of Estmtg Logrthm of Cost s Mmzed Error Epressed Megless Uts ( log dollrs Worse: Stdrd Error Ler Cse Cot e Compred wth Stdrd Error Noler Cse to see whh Futol Form s the Better Estmtor Worst: Log-OLS Noler CERs Must Hve Multpltve Error; OLS Ler CERs Must Hve Addtve Error ICEAA All rghts reserved. ( log 6 v. 8 ICEAA 04 Professol Developmet & Trg Workshop

PT 05 - Multpltve-Error Regresso Eve Worse Cplt of Logrthm-Trsformto Method s Lmted the Mthemtl Propertes of Logrthms Mor Csult of Ths Stuto s ot Hvg Aess to the Noler Futol Form = + v.

9 PT 05 - Multpltve-Error Regresso Eve Worse Cplt of Logrthm-Trsformto Method s Lmted the Mthemtl Propertes of Logrthms Mor Csult of Ths Stuto s ot Hvg Aess to the Noler Futol Form = + v. Therefore, Usg the OLS Method for Ler Regresso d the OLS-Bsed Logrthm- Trsformto Method for Noler Regresso Puts Us Aother Self-Cotrdtor Stuto: We Allow Ler CERs to Hve Nozero Fed-Cost Compoet, Nmel, ut We Requre Noler CERs to Pss Through the (0,0 Pot ICEAA All rghts reserved. 7 USCM6 Ler CER wth Sttsts v ICEAA All rghts reserved. 9 ICEAA 04 Professol Developmet & Trg Workshop

Ate Cost (FY99K$ PT 05 - Multpltve-Error Regresso USCM6 Noler CER wth Sttsts v. 00-03 ICEAA All rghts reserved. 9 CER Emple: Ate Cost vs. Refletor Dmeter v.

10 Ate Cost (FY99K$ PT 05 - Multpltve-Error Regresso USCM6 Noler CER wth Sttsts v ICEAA All rghts reserved. 9 CER Emple: Ate Cost vs. Refletor Dmeter v. Dollrs-per-Dmeter-Foot Reltoshp for Groud Ates Refletor Dmeter (Feet ICEAA All rghts reserved. 0 0 ICEAA 04 Professol Developmet & Trg Workshop

11 Ate Cost (FY99$ PT 05 - Multpltve-Error Regresso OLS Noler CER d Its Qult Metrs (Compred wth Dt Set o Prevous Chrt 5 OLS-Bsed Noler Log-log CER for Cost of Groud Ates v. 0 %SEE = 47.8% %BIAS = -6.8% R = 78.7% Refletor Dmeter (feet Note: CER = hs opposte ovt from dt set, due to ts eesst to pss through (0,0 pot ICEAA All rghts reserved. Cotets v. CER Developmet Ordr Lest Squres (OLS Log-trsformed OLS Sttstl Issues Qult Metrs Geerl-Error Regresso Itertvel Reweghted Lest Squres (IRLS Ftor CERs Ler CERs Trd CERs Qult Metrs Zero Peretge Bs, Mmum Peretge Error (ZMPE Summr ICEAA All rghts reserved. ICEAA 04 Professol Developmet & Trg Workshop

12 PT 05 - Multpltve-Error Regresso Geerl Multpltve-Error Model Elmtes All These Prolems No Logrthms Futol Forms Predt Cost, Not Logrthm of Cost Stdrd Errors C Be Compred d Rked Mgtude for All Futol Forms Error Model (Addtve or Multpltve C Be Chose Idepedetl of Futol Form v. Tke Advtge of Moder Computg Cplt Lest-squres Mmzto Prolem Does Not Hve to Be Solved Epltl (to Get Formuls for d s the Ler Addtve-error Cse Sequetl-serh Tehques Bsed o Newto s Method or Smple Method Are Used to Fd Error-mmzg Vlues of d All Futol Forms C Be Cosdered, Eve = ICEAA All rghts reserved. 3 Multpltve-Error Model v. Atul Cost Equls Estmte tmes Error ( f Y Multpltve Error Error s Rto of Atul to Estmte Atul f Estmte ( Mmum Peretge Error (MPE CERs: Choose f( s Coeffets so tht Sum of Squred Peretge Errors f ( ( f ( s s Smll s Possle Atul Cost = Estmte ± Peretge of Estmte X ICEAA All rghts reserved. 4 ICEAA 04 Professol Developmet & Trg Workshop

13 PT 05 - Multpltve-Error Regresso Peretge Stdrd Error of the Estmte v. Oe-Sgm -tpe Error Boud tht Chrterzes Multpltve-Error CERs = f( f ( f ( %SEE = 00% k = f( s the CER tht Epresses Cost ( Terms of Tehl or Progrmmt Cost Drver ( s the Numer of Dt Pots Used to Derve the CER k s the Numer of Prmeters the CER s Alger Epresso, e.g., k = for the CER = Peretge Stdrd Error s CER Qult Metr ICEAA All rghts reserved. 5 Multpltve-Error Regresso v. Usg f( = + for Purposes of Illustrto... Multpltve-Error Model where For Best Results, Should e s Close to Oe s Possle Choose,, so tht s s Smll s Possle ( Appl Computto-Itesve Tehques of Numerl Alss ICEAA All rghts reserved. 6 3 ICEAA 04 Professol Developmet & Trg Workshop

14 PT 05 - Multpltve-Error Regresso Peretge Bs v. Smple Peretge Bs = f ( MPE Proedure Teds to Produe Estmtes tht re Bsed Upwrd Ths s Bd? f ( Bs Seems to Our Beuse wll e Smller f the f( Vlues re Lrger Teolote Reommeded tht MPE e Repled Itertvel Reweghted Lest Squres (IRLS, Stdrd Sttstl Method tht Elmtes Peretge Bs Mmum-Peretge-Error Estmtes Peretge Bs s CER Qult Metr ICEAA All rghts reserved. f ( f ( 7 Cotets v. CER Developmet Ordr Lest Squres (OLS Log-trsformed OLS Sttstl Issues Qult Metrs Geerl-Error Regresso Itertvel Reweghted Lest Squres (IRLS Ftor CERs Ler CERs Trd CERs Qult Metrs Zero Peretge Bs, Mmum Peretge Error (ZMPE Summr ICEAA All rghts reserved. 4 ICEAA 04 Professol Developmet & Trg Workshop

15 PT 05 - Multpltve-Error Regresso Itertvel Reweghted Lest Squres v. USCM Refers to IRLS s Mmum Used Peretge Error (MUPE Solve Geertg Sequees of Coeffets (egg wth tl guess 0, 0, 0,, 3,...,, 3,...,, 3,... Gve,, lulte = +, = +, = + mmzg,, = Respetve Lmts of Sequees f Sequees Coverge IRLS CER s = + Not the sme oeffet vlues s MPE CER ICEAA All rghts reserved. 9 R Betwee Estmtes d Atuls* v. The term R s sometmes used to desre the Correlto Betwee Atuls ( vlues d Estmtes ( vlues, Mesurg the Etet to whh the Reltoshp etwee d s Ler R Does ot Deped o Spef Coeffets of Reltoshp R = Proporto of Vrto Estmtes ( tht s Attrutle, through OLS Ler Reltoshp, to Vrtos Atuls ( Lrger (loser to.00 Vlues of R Idte Better Ler Ft If CER s Good, Estmtes Should e Prett Close to Atuls,.e., the (Atul, Estmte = (, Pots Should Le Alog Strght Le = R ( Perso s Correlto Squred s CER Qult Metr R k k k k k k k k k k k k ICEAA All rghts reserved. k k k *Note: ths s dfferet th the lssl defto of R, whh rses OLS, d s defed s -SSE/SST ICEAA 04 Professol Developmet & Trg Workshop

16 Ate Cost (FY99K$ PT 05 - Multpltve-Error Regresso CER Emple: Ate Cost vs. Refletor Dmeter v. Dollrs-per-Dmeter-Foot Reltoshp for Groud Ates Refletor Dmeter (Feet ICEAA All rghts reserved. 3 Cse : Clultg Multpltve-Error Ftor CER = Usg IRLS Method v. Choose Itl Guess 0 (from lookg t the dollrs-per-poud sttergrm Proeed from 0 Through the Sequee,, 3, Suessvel Mmzg the IRLS Oetve Futo F' d d F Note tht + Does ot Deped o, so the IRLS Ftor CER s =, where s Clulted Usg the Formul for + Aove 0 whe ICEAA All rghts reserved. 3 6 ICEAA 04 Professol Developmet & Trg Workshop

17 PT 05 - Multpltve-Error Regresso Note: IRLS-Derved Ftor CER Hs Zero Peretge Bs v. Peretge Bs = PB We Isert the IRLS Soluto for, mel Ad the we Ot PB ICEAA All rghts reserved. Dollrs-per-Foot Dt Bse for IRLS Ftor CER for Groud Ates v. Ate Cost vs. Refletor Dmeter Dt IRLS Ftor CER d Qult Metrs Dmeter ( Cost ( (Feet FY99$K / EST = %SE %BIAS Sums Totls: EST = Estmted FY = Fsl Yer SE = Squred Error ICEAA All rghts reserved. 7 ICEAA 04 Professol Developmet & Trg Workshop

18 PT 05 - Multpltve-Error Regresso Dollrs-per-Foot IRLS Ftor CER v. Bsed o Computtos o the Hstorl Dt ( Multpltve-Error CER = (= 637 $/dm-ft Stdrd Error of the Estmte (%SEE Stdrd Error 7 ( drd Error ( (Averge 46.% Aross Dt Rge ICEAA All rghts reserved. Clulto of R Qult Metr for IRLS Ftor CER v Atul EST Sums R-Squred = 0.86 R Numer Term = R Deom Term = R Deom Term = ICEAA All rghts reserved. R Numer R Deom R Deom 36 8 ICEAA 04 Professol Developmet & Trg Workshop

19 Ate Cost (FY99K$ PT 05 - Multpltve-Error Regresso R Betwee Atuls d Estmtes v. Atul Cost ( Estmted Cost ( R k k k k k k k k k k k k k k k ( % Therefore the R Qult Metr for ths CER s 86.% (Perfet Ft s 00% ICEAA All rghts reserved. IRLS Ftor CER d Its Qult Metrs Supermposed o Dt Bse v. Dollrs-per-Dmeter-Foot Reltoshp for Groud Ates %SEE = 46.% %BIAS = 0.0% R = 86.% Refletor Dmeter (Feet ICEAA All rghts reserved. 9 ICEAA 04 Professol Developmet & Trg Workshop

20 PT 05 - Multpltve-Error Regresso ICEAA 04 Professol Developmet & Trg Workshop ICEAA All rghts reserved. v. 39 Cse : Determg Multpltve-Error Ler CER = (+ Usg IRLS Method Choose Itl Guess ( lookg t the dollrs-per-poud sttergrm Proeed from 0 d 0 Through the Sequees,, 3, d,, 3, Suessvel Mmzg the IRLS Oetve Futo, F ICEAA All rghts reserved. v. 40 Clultg the IRLS Coeffets +, + Use Prtl Dervtves to Mmze the Oetve Futo d Fd the Optml Vlues of + d +, F, F

21 PT 05 - Multpltve-Error Regresso ICEAA 04 Professol Developmet & Trg Workshop ICEAA All rghts reserved. v. 4 Solvg the Equtos for + d + Set Both Prtl Dervtves to Zero d Solve the Resultg Smulteous Equtos Solvg these Equtos Yelds the IRLS- Optml Vlues of the Coeffets t Stge ICEAA All rghts reserved. v. 4 The Resultg Vlues of + d +

22 PT 05 - Multpltve-Error Regresso Note: IRLS-Derved Ler CER Hs Zero Peretge Bs v. Peretge Bs = PB, Ulke the Ftor CER Cse, the Alger Cot e Worked Out, euse there re o Closed Epressos for d However, Peretge Bs C e Clulted Numerll Prtulr Cse d Turs Out to e Zero ICEAA All rghts reserved. 43 Dmeter-vs.-Cost Dt Bse d Computtos for IRLS Ler CER v. Ate Cost vs. Refletor Dmeter Dt Dmeter ( Cost ( (Feet FY99$K + (+ /(+ /(+ /(+ /(+ /( Sums EST = Estmted FY = Fsl Yer SE = Squred Error ICEAA All rghts reserved. 44 ICEAA 04 Professol Developmet & Trg Workshop

23 PT 05 - Multpltve-Error Regresso Dmeter-vs.-Cost Dt Bse d IRLS Ler CER Sttsts v. Ate Cost vs. Refletor Dmeter Dt IRLS Ler CER d Qult Metrs Dmeter ( Cost ( (Feet FY99$K EST = + %SE %BIAS Sums Totls: EST = Estmted FY = Fsl Yer SE = Squred Error ICEAA All rghts reserved. 45 Dmeter-vs.-Cost IRLS Ler CER v. Bsed o Computtos o the Hstorl Dt 7.058;.36 Multpltve-Error CER = Stdrd Error of the Estmte (%SEE Stdrd Error 7 ( drd Error 7 ( (Averge 38.% Aross Dt Rge ICEAA All rghts reserved. 3 ICEAA 04 Professol Developmet & Trg Workshop

24 PT 05 - Multpltve-Error Regresso Clulto of R Qult Metr for IRLS Ler CER v. Ate Cost vs. Refletor Dmeter Dt EST ( Cost ( FY99$K FY99$K Sums Num R = De R = De R = R = ICEAA All rghts reserved. 47 R Betwee Atuls d Estmtes v. EST ( Cost ( R k 86. % k k k k k k k k k k k k k ( Therefore the R Qult Metr for ths CER s 86.% (Perfet Ft s 00% k ICEAA All rghts reserved. 4 ICEAA 04 Professol Developmet & Trg Workshop

25 Ate Cost (FY99K$ PT 05 - Multpltve-Error Regresso IRLS Ler CER d Its Qult Metrs Supermposed o Dt Bse v. Dmeter vs. Cost IRLS Ler Reltoshp for Groud Ates %SEE = 38.% %BIAS = 0.0% R = 86.% Refletor Dmeter (Feet ICEAA All rghts reserved. Cse 3: Determg Multpltve-Error Trd CER = (+ usg IRLS Method v. Choose Itl Guess 0 0 (from lookg t the dollrs-per-poud stter-grm Proeed from 0, 0, d 0 Through the Sequees,, 3,,,, 3,, d,, 3, Suessvel Mmzg the IRLS Oetve Futo F,, ICEAA All rghts reserved ICEAA 04 Professol Developmet & Trg Workshop

26 PT 05 - Multpltve-Error Regresso Attempt to Clulte IRLS Trd Coeffets +, +, + v. Use Prtl Dervtves (wth respet to +, +, d + s Before to Mmze Oetve Futo d Estlsh Smulteous Equtos for Optml Vlues of +, +, + Ufortutel, These Smulteous Equtos re ot Ler d Cot e Solved Epltl to Get Alger Epressos for +, +, + Optml Numerl Vlues of +, +, d + Must e Foud Itertve Tehques, suh s those Used Eel s Solver Route ICEAA All rghts reserved. 5 PB, Note: IRLS-Derved Trd CER Hs Zero Peretge Bs Peretge Bs = PB, Alger g ot e worked out s the Ftor CER Cse, euse there re o losed Epressos for,, d v. Ag, though, Peretge Bs wll tur out to e Zero spef se (whe ou mke the spef omputtos ICEAA All rghts reserved. 5 6 ICEAA 04 Professol Developmet & Trg Workshop

27 PT 05 - Multpltve-Error Regresso Trd CER Suessve Itertve Prmeter Solutos v. Itl Guesses IRLS Sol. #0 # # #3 #4 #5 #6 #7 = = = Note: Itertve Proess Coverges to Soluto 7 Steps, Strtg from Itl Guess # ICEAA All rghts reserved. 53 Dmeter-vs.-Cost Dt Bse, wth IRLS Trd CER Sttsts v. Ate Cost vs. Refletor Dmeter Dt IRLS Trd CER d Qult Metrs Dmeter ( Cost ( (Feet FY99$K EST = +^ %SE %BIAS Sums Totls: EST = Estmted FY = Fsl Yer SE = Squred Error MCR, LLC ICEAA All rghts reserved. 7 ICEAA 04 Professol Developmet & Trg Workshop

28 PT 05 - Multpltve-Error Regresso Emple: Dmeter vs. Cost Trd CER Bsed o Computtos usg the Hstorl Dt 8.484; Multpltve-Error CER ; v. = Stdrd Error of the Estmte (%SEE Stdrd Error ( drd Error ( (Averge 40.5% Aross Dt Rge ICEAA All rghts reserved. Clulto of R Qult Metr for IRLS Trd CER v. Ate Cost vs. Refletor Dmeter Dt EST ( Cost ( FY99$K FY99$K Nu De De Sums Num R = De R = De R = R = MCR, LLC ICEAA All rghts reserved. 8 ICEAA 04 Professol Developmet & Trg Workshop

Ate Cost (FY99K$ PT 05 - Multpltve-Error Regresso R Betwee Atuls d Estmtes v. R k k k k k k k k k k k k k k ( 85.6 ( 85.6 89.8% 89.8% 95.74 899.349 95.74 899.349 k 0.898 0.

29 Ate Cost (FY99K$ PT 05 - Multpltve-Error Regresso R Betwee Atuls d Estmtes v. R k k k k k k k k k k k k k k ( 85.6 ( % 89.8% k Therefore the R Qult Metr for ths CER s 89.8% (Perfet Ft s 00% ICEAA All rghts reserved. Trd CER d Its Qult Metrs (Compre wth Dt Bse o Erler Chrt v Dmeter vs. Cost Trd CER for Groud Ates %SEE = 40.5% %BIAS = 0.0% R = 89.8% Refletor Dmeter (Feet ICEAA All rghts reserved. 9 ICEAA 04 Professol Developmet & Trg Workshop

30 PT 05 - Multpltve-Error Regresso Cotets v. CER Developmet Ordr Lest Squres (OLS Log-trsformed OLS Sttstl Issues Qult Metrs Geerl-Error Regresso Itertvel Reweghted Lest Squres (IRLS Ftor CERs Ler CERs Trd CERs Qult Metrs Zero Peretge Bs, Mmum Peretge Error (ZMPE Summr ICEAA All rghts reserved. Zero-Peretge-Bs, Mmum- Peretge-Error CERs v. I 998, the Zero Peretge Bs Mmum Peretge Error (ZMPE Tehque ws Proposed* to Yeld CERs Gurteed to Hve Mmum Possle Peretge Error mog ll Used CERs for Gve Dt Set tht Hve the Futol Form eg Cosdered ZMPE Pursues the Mmum-Peretge-Error Gol Dretl Computes Mmum-Peretge-Error CER, Suet to Costrt tht Peretge Bs e Etl Zero CERs Derved Usg Costred Optmzto Aother Cplt of Eel Solver But, uto must e tke whe usg Eel * Book, S. d Lo, N, Mmum-Peretge-Error Regresso uder Zero-Bs Costrts, Proeedgs of the Fourth Aul U.S. Arm Coferee o Appled Sttsts, -3 Otoer 998, U.S. Arm Reserh Lortor, Report No. ARL-SR- 84, Novemer 999, pges ICEAA All rghts reserved. 30 ICEAA 04 Professol Developmet & Trg Workshop

31 PT 05 - Multpltve-Error Regresso ZMPE Mthemts v. Usg the Trd Cse s Emple, ZMPE Mmzes F,, Suet to the Costrt k k k k, k %Bs,, k k k ICEAA All rghts reserved. ZMPE Dmeter vs. Cost Trd CER v. Bsed o Computtos o the Hstorl Dt 36.;.4; 0.06 Multpltve-Error CER = Stdrd Error of the Estmte (%SEE Stdrd Error ( ( Averge 39.49% Aross Dt Rge ICEAA All rghts 6 reserved. 3 ICEAA 04 Professol Developmet & Trg Workshop

32 Ate Cost (FY99$ PT 05 - Multpltve-Error Regresso ZMPE Trd CER, Qult Metrs Supermposed o Dt Bse v. 35 Dmeter vs. Cost MPE-ZPB Trd Reltoshp for Groud Ates ( = ^ %SEE = 39.49% %BIAS = 0.00% R = 9.43% Refletor Dmeter (feet ICEAA All rghts 63 reserved. Geerl Commets v. Note tht the ZMPE CER s Qult Metrs re Better th re the IRLS CER s Qult Metrs for Dt Set of the Emple Ths s Geerl Pheomeo A Presetto Dr. S.A. Book to the 006 ISPA Itertol Coferee Bellevue WA (006 Showed tht ZMPE d MUPE CERs Derved from the Sme Dt Set re Dfferet d tht the ZMPE CER hs Smller Peretge Stdrd Error th does the IRLS/MUPE CER A Moogrph (003 Dr. M.S. Golderg d A.E. Tuow, the of IDA, Poted Out tht IRLS (k MUPE CERs re Atull ot Used Whe Bsed o Smll Smples A Pper (998 Dr. S.A. Book d N.Y. Lo Reommeded the Use of Costred Optmzto to Mmze Peretge Stdrd Error, Suet to the Costrt tht the CER s Peretge Smple Bs s Zero ICEAA All rghts reserved ICEAA 04 Professol Developmet & Trg Workshop

33 PT 05 - Multpltve-Error Regresso Multpltve-Error CER Fts Mmum-Peretge-Error (MPE Multpltve-Error CERs of Artrr Futol Form CERs tht Mmze Peretge Error of the Estmte CERs tht re Bsed Hgh CERs wth Heurst Sttstl Propertes* Itertvel Reweghted Lest Squres (IRLS Multpltve-Error CERs of Artrr Futol Form CERs tht re Used CERs wth Good Sttstl Propertes* CERs tht Mmze the Qus-Lkelhood Zero-Peretge-Bs/Mmum- Peretge-Error (ZMPE CERs Multpltve-Error CERs of Artrr Futol Form CERs tht Mmze Peretge Error of the Estmte Suet to eg Used CERs tht re Used CERs wth Heurst Sttstl Propertes* * The term sttstl propertes refers to hpothess tests, ofdee tervls, d ther ssoted outremets suh s t d F vlues. v ICEAA All rghts reserved. Cotets v. CER Developmet Ordr Lest Squres (OLS Log-trsformed OLS Sttstl Issues Qult Metrs Geerl-Error Regresso Itertvel Reweghted Lest Squres (IRLS Ftor CERs Ler CERs Trd CERs Qult Metrs Zero Peretge Bs, Mmum Peretge Error (ZMPE Summr ICEAA All rghts reserved. 33 ICEAA 04 Professol Developmet & Trg Workshop

34 PT 05 - Multpltve-Error Regresso Summr CERs re Derved Applg Sttstl Alss to Cost Dt Bses Refletg Hstorl Cost Eperee Multpltve-Error Regresso Frequetl More Approprte th Addtve-Error Regresso for CERs IRLS (k MUPE d ZMPE Allow CERs of All Approprte Forms to e Derved v. CER Qult Metrs Support Credlt of Estmtes Derved from Multpltve-Error CERs Peretge Stdrd Error of the Estmte Peretge Bs Perso s Correlto Squred etwee Estmtes d Atuls MCR, LLC ICEAA All rghts reserved. Referees v. Covert, R. d Aderso,T., Moder Tehques of Regresso for CERs wth Multpltve Errors, 0 SCEA/ISPA Jot Aul Coferee & Trg Workshop Orldo, FL 6-9 Jue 0 Book, S. d Lo, N, Mmum-Peretge-Error Regresso uder Zero-Bs Costrts, Proeedgs of the Fourth Aul U.S. Arm Coferee o Appled Sttsts, -3 Otoer 998, U.S. Arm Reserh Lortor, Report No. ARL- SR-84, Novemer 999, pges Book, S., IRLS/MUPE CERs Are Not MPE-ZPB CERs, Itertol Soet of Prmetr Alsts, 8 th Aul Coferee, Settle WA, 3-6 M 006. Golderg, Mtthew S. d Tuow, Adu E., Sttstl Cosdertos Estmtg Lerg Curves d Multpltve CERs, 3d Aul DoD Cost Alss Smposum, Wllmsurg VA, -5 Ferur 999, 44 hrts. Golderg, Mtthew S. d Touw, Adu E., Sttstl Methods for Lerg Curves d Cost Alss, Isttute for Opertos Reserh d the Mgemet Sees (INFORMS Tops Opertos Reserh Seres, 003, 96 pges. Nelder, J. A., Weghted Regresso, Qutl Respose Dt, d Iverse Polomls, Bometrs, Vol. 4 (968, pges Wedderur, R.W.M., Qus-lkelhood Futos, Geerlzed Ler Models, d the Guss-Newto Method, Bometrk, Vol. 6, Numer 3 (974, pges Uted Sttes Ar Fore Spe d Mssle Sstems Ceter, Umed Spe Vehle Cost Model, Sth Edto (USCM6, ICEAA All rghts reserved. 34 ICEAA 04 Professol Developmet & Trg Workshop

Numerical Analysis Topic 4: Least Squares Curve Fitting

Numerical Analysis Topic 4: Least Squares Curve Fitting Numerl Alss Top 4: Lest Squres Curve Fttg Red Chpter 7 of the tetook Alss_Numerk Motvto Gve set of epermetl dt: 3 5. 5.9 6.3 The reltoshp etwee d m ot e ler. Fd futo f tht est ft the dt 3 Alss_Numerk Motvto