CURVE FITTING LEAST SQUARES METHOD

Transcription

1 Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued for of the sste. LEAST SQUARES METHOD Whle fttg le or curve to set of dt pots, the gol s to ze the devtos of the pots fro the pproted le or curve. Severl possble ws of zg the devtos wll be dscussed Mze the Su of Errors e stes.google.co/ste/zdsoud/uercl 5

2 Nuercl Alss for Egeers Ger Jord Uverst A le pssg through the dpot stsfes the bove crtero. Mze the Su of Errors Mgtudes e but e e e costt stes.google.co/ste/zdsoud/uercl 53

3 Nuercl Alss for Egeers Ger Jord Uverst whch es tht le lg betwee les d b stsfes the bove crtero. Mze the Su of Squres of the Errors I the fgure show below, f the pprote le psses through pot, the devtos becoe The su of squres of the devtos s e e e e d e stes.google.co/ste/zdsoud/uercl 54

4 Nuercl Alss for Egeers Ger Jord Uverst s e e e d For the optl postog of the pot, the su s s zed wth respect to the devto e. ds de e d e d e whch loctes the pot o top of pot, whch s tur the desred pssg pot of the pprote le. POLYNOMIAL REGRESSION Polol regresso s the process of fttg polol of th order to set of dt pot the for stes.google.co/ste/zdsoud/uercl 55

5 Nuercl Alss for Egeers Ger Jord Uverst stes.google.co/ste/zdsoud/uercl 56 The error betwee the ect vlue of d the vlue pproted b the polol bove s e the e It s the requred to detere the coeffcets of the polol whch wll ze the su of squres of the errors wth respect to ll coeffcets. The su of squres of the errors s e s ) ( To ze the su wth respect to the coeffcet, the prtl dervtve of the su wth respect to the coeffcet s set to zero ) )( ( s To ze the su wth respect to the rest of the coeffcets, the prtl dervtve of the su wth respect to ll coeffcets s set to zero s ) )( ( ) )( ( ) )( ( s s s

6 Nuercl Alss for Egeers Ger Jord Uverst stes.google.co/ste/zdsoud/uercl 57 The d ters the bove set of equtos c be rerrged s The ter The bove set of equtos c be preseted tr for s 4 3 3

7 Nuercl Alss for Egeers Ger Jord Uverst stes.google.co/ste/zdsoud/uercl 58 The bove tr geerll ll-codtoed but, up to 4 th or 5 th order polols, d usg double precso clcultos, the soluto c be cceptble. Eple Detere the best le ft to the gve set of dt pots usg lest squres ethod The best le ft s gve b To detere the coeffcets of the bove equto, tble s costructed whch cludes the orgl dt d the su of ll quttes the followg equto Usg 8 sgfct fgures clcultos

8 Nuercl Alss for Egeers Ger Jord Uverst The bove tr equto becoes Solvg the sste, the coeffcets re stes.google.co/ste/zdsoud/uercl 59

9 Nuercl Alss for Egeers Ger Jord Uverst = Therefore, the equto of the best le ft becoes WHAT DEGREE OF POLYNOMIAL SHOULD BE USED? Hgher order polols reduce devtos of pots fro the pproted chrt. Whe the degree of the polol reches (-), where s the uber of dt pots, the polol psses through ll pots (ssug o duplcted pots t the se vlue). Oe crese the degree of polol s log s there s sttstcll sgfct decrese the stdrd error defed s s e ( ) stes.google.co/ste/zdsoud/uercl 6

10 Nuercl Alss for Egeers Ger Jord Uverst The bove qutt s dvded b ( ) becuse degrees re used d hece ( ) re the lost coeffcets or degrees of freedo. Eple Detere the stdrd error of the curve fts of the followg set of dt pots usg st order through 4 th order polols Costruct tble to detere the coeffcets of the tr equto of the st order polol s follows. sgfct fgures clcultos re used. Ol 3 sgfct fgures re show for clrt of the soluto steps. stes.google.co/ste/zdsoud/uercl 6

11 Nuercl Alss for Egeers Ger Jord Uverst The st order tr equto s Substtutg vlues fro the bove tble we get stes.google.co/ste/zdsoud/uercl 6

12 Nuercl Alss for Egeers Ger Jord Uverst Solvg the bove sste for the coeffcets of the st order polol we get The equto of the polol becoes To detere the error of the pproto t ll dt pot, costruct the followg tble whch cludes the ect pots d the pprote vlues of obted fro the st order polol bove. stes.google.co/ste/zdsoud/uercl 63

13 Nuercl Alss for Egeers Ger Jord Uverst e e The stdrd error s e s. ( ) () where s the uber of pots, d s the order of the polol. 3 stes.google.co/ste/zdsoud/uercl 64

14 Nuercl Alss for Egeers Ger Jord Uverst The bove steps re repeted fro the d, 3 rd, d 4 th order polols. The followg tble shows the obted polols d the stdrd error of ech ft. Degree Polol Stdrd error Mu stdrd error s cheved usg d order polol. The stdrd error strts to crese fter the d order. Ths due to the ll-codto of the curve fttg tr. The degree of lless of the tr creses wth the crese of ts sze. Therefore, sttstcll, d order polol gves the best ft to the gve dt. The ctul dt gve the eple tble ws obted usg the followg d order polol. LINEARIZING NONLINEAR RELATIONSHIPS The respose of phscl sstes s geerll oler, such s dped vbrtos. The pltude of osclltos t te s gve b e t stes.google.co/ste/zdsoud/uercl 65

15 Nuercl Alss for Egeers Ger Jord Uverst Oe of the coeffcets the bove equto ( ) s cluded wth oler fucto. Ths proble c be solved b trsforg the oler reltoshp to ler reltoshp usg vrble trsforto s New vrbles c be defed s Therefore, the ew reltoshp becoes where e l l l l t t e t l e l l t z l t z l e The gve dt s trsfored to the ew vrbles usg the reltoshps z l d t, the best le s obted for the ew dt usg z. The obted coeffcets re the trsfored bck to the orgl for. Eple 3 Ft oler curve the for e to the followg set of dt pots. stes.google.co/ste/zdsoud/uercl 66

16 Nuercl Alss for Egeers Ger Jord Uverst To trsfor the gve oler reltoshp to ler oe, wth costts pperg le wth the depedet vrble, the turl logrth of both sdes of the equto s used s follows Appl the trsfortos The resultg ler reltoshp becoes e l l l l The ler curve fttg tr equto s e l e l l z l l z stes.google.co/ste/zdsoud/uercl 67

17 Nuercl Alss for Egeers Ger Jord Uverst stes.google.co/ste/zdsoud/uercl 68 z z Costruct the lest-squres curve fttg tble s follows z l z Clcultos re crred out usg sgfct fgures. The dspled vlues re rouded to 6 sgfct fgures. The tr equto becoes Solvg the sste, the coeffcets re

18 Nuercl Alss for Egeers Ger Jord Uverst Therefore, The fl for of the ftted curve becoes l e e. The fgure below shows the orgl dt pots d the curve ft GOODNESS OF FIT To qutf the goodess of ft, the totl su of the squres of errors roud the e s copred to the totl su of the squres of errors roud the curve ft. The e vlue of the dt s detered s stes.google.co/ste/zdsoud/uercl 69

19 Nuercl Alss for Egeers Ger Jord Uverst The su of squres roud the e s The su of squres roud the ft s S t S r where s the vlue pproted b the curve ft. 8 6 Curve ft 4 Me The reltve dfferece betwee the two sus qutfes the out of proveet cheved. Ths reltve dfferece s kow s the Coeffcet of Deterto ( r ), d s defed s stes.google.co/ste/zdsoud/uercl 7

20 Nuercl Alss for Egeers Ger Jord Uverst r St S S The squre root of the Coeffcet of Deterto ( r ) s kow s the Correlto Fctor. A Correlto Fctor r correspods to perfect ft (%). A zero Correlto Fctor ( r ) ples o proveet or o dvtge to the curve ft over the e of the dt. Eple 4 I eple, strght le ws ftted to the followg set of dt pots. t r The e vlue of the dt s The equto of the strght le ws stes.google.co/ste/zdsoud/uercl 7

21 Nuercl Alss for Egeers Ger Jord Uverst The followg tble shows the orgl vlues of d the vlues obted fro the le equto. The errors d the squres of the errors re lso cluded the tble e e e The Coeffcet of Deterto s clculted s The Correlto Fctor s r 5 5 e e St Sr S 5 t e r e Ths dctes 99.6% proveet cheved usg the best le ft over the e vlue of the gve dt. stes.google.co/ste/zdsoud/uercl 7