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 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
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
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
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
Dede o the Crtero. Lest Squres Regresso : mmze f Chpter 7.Et Mthg Iterpolto : f Chpter 8 Alss_Numerk 7
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
Alss_Numerk 9 Determe the Ukows? : mmze to d ot we do How : mmze to d fd wt to We
Determe the Ukows Neessr odto for the mmum : 0 0 Alss_Numerk 0
Alss_Numerk Determg the Ukows 0 0
Alss_Numerk Norml Equtos
Alss_Numerk 3 Solvg the Norml Equtos
Alss_Numerk 4 Emple : Ler Regresso f Equtos : : Assume 3 5. 5.9 6.3
Emple : Ler Regresso 3 sum 3 6 5. 5.9 6.3 7.3 4 9 4 5..8 8.9 35.8 Equtos : 3 6 6 4 7.3 35.8 Solvg: 4.5667 0.60 Alss_Numerk 5
Multple Ler Regresso Emple: Gve the followg dt: t 0 3 0. 0.4 0. 0. 3 Determe futo of two vrles: ft = + + t Tht est fts the dt wth the lest sum of the squre of errors. Alss_Numerk 6
Soluto of Multple Ler Regresso Costrut the sum t 0 3 of the squre of the error d derve the 0. 0.4 0. 0. eessr odtos equtg the prtl dervtves wth respet to the ukow prmeters to zero the solve the equtos. 3 Alss_Numerk 7
Alss_Numerk 8 Soluto of Multple Ler Regresso 0 0 0 odtos : Neessr t t t t t t t f
Alss_Numerk 9 Sstem of Equtos t t t t t t
Emple : Multple Ler Regresso 3 4 Sum t 0 3 6 0. 0.4 0. 0. 0.9 3 8 0.0 0.6 0.04 0.04 0.5 t 0 0.4 0.4 0.6.4 0.3 0.8 0. 0.4.7 t 0 4 9 4 t 0 6 0 Alss_Numerk 0
Emple : Sstem of Equtos 4 0.9 6 8 0.9 0.5.4 6.4 4 0.7 Solvg :.9574.70 0.3898 f t t.9574.70 0.3898 t Alss_Numerk
Leture 9 Noler Lest Squres Prolems Emples of Noler Lest Squres Soluto of Iosstet Equtos Cotuous Lest Squre Prolems Alss_Numerk
Poloml Regresso The lest squres method e eteded to ft the dt to hgher-order poloml Alss_Numerk 3 0 0 0 odtos : Neessr Mmze f e f
Equtos for Qudrt Regresso Alss_Numerk 4 0 0 0 Mmze
Norml Equtos Alss_Numerk 5 4 3 3
Emple 3: Poloml Regresso Ft seod-order poloml to the followg dt 0 3 4 5 =5. 7.7 3.6 7. 40.9 6. =5.6 0 4 9 6 5 =55 3 0 8 7 64 5 5 4 0 6 8 56 65 =979 0 7.7 7. 8.6 63.6 305.5 =585.6 0 7.7 54.4 44.8 654.4 57.5 =488.8 Alss_Numerk 6
Emple 3: Equtos d Soluto 6 5 55 5.6 5 55 5 585.6 55 5 979 488.8 Solvg....4786.3593.8607 f.4786.3593.8607 Alss_Numerk 7
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
Emple showg tht Qudrt s preferle th Ler Regresso Ler Regresso Qudrt Regresso Alss_Numerk 9
Fttg wth Noler Futos 0.4 0.65 0.95.4.73.0.3.5 0.3-0.3 -. -0.45 0.7 0. -0.9 0.4 It s requred to fd futo of the form : f l os e to ft the dt. f Alss_Numerk 30
Alss_Numerk 3 Fttg wth Noler Futos Equtos Norml e 0 0 0 mmum : for the odto Neessr os l
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
Emple 4: Evlutg Sums 0.4 0.65 0.95.4.73.0.3.5 =.57 0.3-0.3 -. -0.45 0.7 0. -0.9 0.4 =-.3 l.036 0.856 0.006 0.0463 0.3004 0.4874 0.643 0.8543 =4.556 l os -.386-0.349-0.098 0.0699-0.0869-0.969-0.49-0.754 =-3.36 l * e -.84-0.85-0.36 0.7433 3.098 5.04 7.4585.487 =5.9 * l -0.38 0.099 0.0564-0.0968 0.480 0.0698-0.36 0.8 =-0.065 os 0.943 0.6337 0.3384 0.055 0.05 0.808 0.375 0.6609 =3.6307 os * e.35.549.504.4-0.894-3.735-5.696-0.04 =-4.48 *os 0.3-0.83-0.6399-0.46-0.048-0.045 0.776-0.95 =-0.8485 e.66 3.6693 6.6859.94 3.87 55.70 86.488 54.47 =35.39 * e 0.94-0.4406 -.844 -.555.53 0.7463 -.697.989 =-.993 Alss_Numerk 33
Emple 4: Equtos & Soluto 4.55643 3.3547 5.9 0.06486 3.3547 3.6307 4.485 0.84854 5.9 4.485 35.388.9983 Solvg the ove equtos : 0.8885.074 Therefore f 0.8885 l.074 os 0.0398 0.0398 e Alss_Numerk 34
Emple 5 Gve: 3.4 5 9 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
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
Alss_Numerk 37 z z z z z d 0 0 usg : oted re Equtos Norml Emple 5: Equtos
Evlutg Sums d Solvg 3 =6.4 5 9 z =l 0.875469.609438.975 =4.683 4 9 =4 z 0.875469 3.8876 6.59674 =0.6860 Equtos : 3 6 4.683 6 4 0.686 Solvg Equtos : 0.3897 0.66087 l e e f 0.3897 e.6994.6994 e 0.66087 Alss_Numerk 38