Numerical Analysis Topic 4: Least Squares Curve Fitting

Transcription

1 Numerl Alss Top 4: Lest Squres Curve Fttg Red Chpter 7 of the tetook Alss_Numerk

2 Motvto Gve set of epermetl dt: The reltoshp etwee d m ot e ler. Fd futo f tht est ft the dt 3 Alss_Numerk

3 Motvto I egeerg two tpes of ppltos re eoutered: Tred lss: Predtg vlues of depedet vrle m lude etrpolto eod dt pots or terpolto etwee dt pots. Hpothess testg: Comprg estg mthemtl model wth mesured dt.. Wht s the est mthemtl futo f tht represets the dtset?. Wht s the est rtero to ssess the fttg of the futo f to the dt? Alss_Numerk 3

4 Curve Fttg Gve set of tulted dt fd urve or futo tht est represets the dt. Gve:.The tulted dt.the form of the futo 3.The urve fttg rter Fd the ukow oeffets Alss_Numerk 4

5 Lest Squres Regresso Ler Regresso Fttg strght le to set of pred oservtos:. = 0 + +e -slope. 0 -terept. e-error or resdul etwee the model d the oservtos. Alss_Numerk 5

6 Alss_Numerk 6 Seleto of the Futos kow. re 0 0 g g f Geerl f Poloml f Qudrt f Ler k m k k k k k k

7 Dede o the Crtero. Lest Squres Regresso : mmze f Chpter 7.Et Mthg Iterpolto : f Chpter 8 Alss_Numerk 7

8 Lest Squres Regresso Gve:.. The form of the futo s ssumed to e kow ut the oeffets re ukow. e f f The dfferee s ssumed to e the result of epermetl error. Alss_Numerk 8

9 Alss_Numerk 9 Determe the Ukows? : mmze to d ot we do How : mmze to d fd wt to We

10 Determe the Ukows Neessr odto for the mmum : 0 0 Alss_Numerk 0

11 Alss_Numerk Determg the Ukows 0 0

12 Alss_Numerk Norml Equtos

13 Alss_Numerk 3 Solvg the Norml Equtos

14 Alss_Numerk 4 Emple : Ler Regresso f Equtos : : Assume

15 Emple : Ler Regresso 3 sum Equtos : Solvg: Alss_Numerk 5

16 Multple Ler Regresso Emple: Gve the followg dt: t Determe futo of two vrles: ft = + + t Tht est fts the dt wth the lest sum of the squre of errors. Alss_Numerk 6

17 Soluto of Multple Ler Regresso Costrut the sum t 0 3 of the squre of the error d derve the eessr odtos equtg the prtl dervtves wth respet to the ukow prmeters to zero the solve the equtos. 3 Alss_Numerk 7

18 Alss_Numerk 8 Soluto of Multple Ler Regresso odtos : Neessr t t t t t t t f

19 Alss_Numerk 9 Sstem of Equtos t t t t t t

20 Emple : Multple Ler Regresso 3 4 Sum t t t t Alss_Numerk 0

21 Emple : Sstem of Equtos Solvg : f t t t Alss_Numerk

22 Leture 9 Noler Lest Squres Prolems Emples of Noler Lest Squres Soluto of Iosstet Equtos Cotuous Lest Squre Prolems Alss_Numerk

23 Poloml Regresso The lest squres method e eteded to ft the dt to hgher-order poloml Alss_Numerk odtos : Neessr Mmze f e f

24 Equtos for Qudrt Regresso Alss_Numerk Mmze

25 Norml Equtos Alss_Numerk

26 Emple 3: Poloml Regresso Ft seod-order poloml to the followg dt = = = = = =488.8 Alss_Numerk 6

27 Emple 3: Equtos d Soluto Solvg f Alss_Numerk 7

28 How Do You Judge Futos? Gve two or more futos to ft the dt How do ou selet the est? Aswer : Determe the prmeters for eh the ompute for eh oe. The resultg smller lest sum of of the errors s the est. futo futo the squres Alss_Numerk 8

29 Emple showg tht Qudrt s preferle th Ler Regresso Ler Regresso Qudrt Regresso Alss_Numerk 9

30 Fttg wth Noler Futos It s requred to fd futo of the form : f l os e to ft the dt. f Alss_Numerk 30

31 Alss_Numerk 3 Fttg wth Noler Futos Equtos Norml e mmum : for the odto Neessr os l

32 Alss_Numerk 3 Norml Equtos equtos. orml the solve d sums the Evlute os l os os os os l l l os l l e e e e e e

33 Emple 4: Evlutg Sums = =-.3 l =4.556 l os =-3.36 l * e =5.9 * l = os = os * e =-4.48 *os = e =35.39 * e =-.993 Alss_Numerk 33

34 Emple 4: Equtos & Soluto Solvg the ove equtos : Therefore f l.074 os e Alss_Numerk 34

35 Emple 5 Gve: Fd futo Norml Equtos e f e e re tht oted e e 0 e est fts the dt. Alss_Numerk 35 0 usg : Dffult to Solve

36 Alss_Numerk 36 f g e f l l Defe dt. the fts est tht futo Fd solve Eser to : Mmze : mmzg Isted of l l Let l l Defe z e z d z Lerzto Method

37 Alss_Numerk 37 z z z z z d 0 0 usg : oted re Equtos Norml Emple 5: Equtos

38 Evlutg Sums d Solvg 3 = z =l = =4 z = Equtos : Solvg Equtos : l e e f e e Alss_Numerk 38