CURVE FITTING Descbes techques to ft cuves (cuve fttg) to dscete dt to obt temedte estmtes. Thee e two geel ppoches fo cuve fttg: Regesso: Dt ehbt sgfct degee of sctte. The stteg s to deve sgle cuve tht epesets the geel ted of the dt. Itepolto: Dt s ve pecse. The stteg s to pss cuve o sees of cuves though ech of the pots.
Itoducto I egeeg, two tpes of pplctos e ecouteed: Ted lss. Pedctg vlues of depedet vble, m clude etpolto beod dt pots o tepolto betwee dt pots. Hpothess testg. Compg estg mthemtcl model wth mesued dt.
Mthemtcl Bckgoud Athmetc me. The sum of the dvdul dt pots () dvded b the umbe of pots ().,,, tdd devto. The most commo mesue of sped fo smple. t, t ( )
Mthemtcl Bckgoud (cot d) Vce. Repesetto of sped b the sque of the stdd devto. ( ) o ( ) / Coeffcet of vto. Hs the utlt to qutf the sped of dt. c. v. %
Lest ques Regesso Chpte 7 Le Regesso Fttg stght le to set of ped obsevtos: (, ), (, ),,(, ). + + e - slope - tecept e - eo, o esdul, betwee the model d the obsevtos
Le Regesso: Resdul
Le Regesso: Questo How to fd d so tht the eo would be mmum?
Le Regesso: Cte fo Best Ft m e ( ) e e e -e
Le Regesso: Cte fo Best Ft m e
Le Regesso: Cte fo Best Ft m m e
Le Regesso: Lest ques Ft e ) ( m e ) (,model),mesued ( Yelds uque le fo gve set of dt.
Le Regesso: Lest ques Ft e ) ( m The coeffcets d tht mmze must stsf the followg codtos:
[ ] ) ( ) ( o o o Le Regesso: Detemto of o d ( ) + + equtos wth ukows, c be solved smulteousl
Le Regesso: Detemto of o d ( )
Eo Qutfcto of Le Regesso Totl sum of the sques oud the me fo the depedet vble,, s t t ( ) um of the sques of esduls oud the egesso le s e ( o )
Eo Qutfcto of Le Regesso t - qutfes the mpovemet o eo educto due to descbg dt tems of stght le the th s vege vlue. t t : coeffcet of detemto : coelto coeffcet
Eo Qutfcto of Le Regesso Fo pefect ft: d, sgfg tht the le epls pecet of the vblt of the dt. Fo, t, the ft epesets o mpovemet.
Lest ques Ft of tght Le: Emple Ft stght le to the d vlues the followg Tble:.5.5.5 5 4 3 6 9 4 4 6 6 5 3.5 7.5 5 6 6 36 36 7 5.5 38.5 49 8 4 9.5 4 8 4. 4 9.5
Lest ques Ft of tght Le: Emple (cot d) ( 7 9.5 8 4.839857 7 4 8 ) 3.4857.839857 4.74857 Y.74857 +.839857
Lest ques Ft of tght Le: Emple (Eo Alss).5.5 3. 4 4. 5 3.5 6 6. 7 5.5 ( ) e ( ) 8.5765.687.86.565.48.3473.365.365.5.5896 6.6.797 4.98.993 8 4..743.99 ^ e t t ( ). 743 t.868.99.868.93
Lest ques Ft of tght Le: Emple (Eo Alss) The stdd devto (qutfes the sped oud the me): s t.743 7.9457 The stdd eo of estmte (qutfes the sped oud the egesso le) Becuse s.99 7 / < /.7735, the le egesso model hs good ftess
Algothm fo le egesso
Lezto of Nole Reltoshps The eltoshp betwee the depedet d depedet vbles s le. Howeve, few tpes of ole fuctos c be tsfomed to le egesso poblems. The epoetl equto. The powe equto. The stuto-gowth-te equto.
Lezto of Nole Reltoshps. The epoetl equto. b e l l + b * o +
Lezto of Nole Reltoshps. The powe equto b log log + b log * o + *
Lezto of Nole Reltoshps 3. The stuto-gowth-te equto * / o / 3 b 3 / 3 * /
Emple Ft the followg Equto: b to the dt the followg tble:.5.7 3 3.4 4 5.7 5 8.4 5 9.7 X*log Y*log -.3.3.6.477.534.6.753.699.9.79.4 b log log( ) log log + b let Y * log log * Y +, X, X * log, * b log
Emple X Y X* Log(X) Y* Log(Y) X*Y* X*^.5. -.3...7.3.34.694.96 3 3.4.477.535.536.76 4 5.7.6.7559.455.365 5 8.4.699.943.646.4886 um 5 9.7.79.4.44.69 5.44.79.4.75 () 5.69.79.48.75.4584.334
Lezto of Nole Fuctos: Emple log -.334+.75log.46.75
Poloml Regesso ome egeeg dt s pool epeseted b stght le. Fo these cses cuve s bette suted to ft the dt. The lest sques method c edl be eteded to ft the dt to hghe ode polomls.
Poloml Regesso (cot d) A pbol s pefeble
Poloml Regesso (cot d) A d ode poloml (qudtc) s defed b: The esduls betwee the model d the dt: The sum of sques of the esdul: e o + + + o e ( ) o e
Poloml Regesso (cot d) o o o o ) ( ) ( ) ( + + + + + + 4 3 o 3 o o 3 le equtos wth 3 ukows ( o,, ), c be solved
Poloml Regesso (cot d) A sstem of 33 equtos eeds to be solved to deteme the coeffcets of the poloml. The stdd eo & the coeffcet of detemto 3 / s t t 4 3 3
Poloml Regesso (cot d) Geel: The mth-ode poloml: A sstem of (m+)(m+) le equtos must be solved fo detemg the coeffcets of the mth-ode poloml. The stdd eo: The coeffcet of detemto: e m m o + + + + +... ( ) / + m s t t
Poloml Regesso- Emple Ft secod ode poloml to dt: 3 4. 7.7 7.7 7.7 3.6 4 8 6 7. 54.4 3 7. 9 7 8 8.6 44.8 4 4.9 6 64 56 63.6 654.4 5 6. 5 5 65 35.5 57.5 5 5.6 55 5 979 585.6 489 3 5 4 979 5 5.6 55 585.6 488.8
Poloml Regesso- Emple (cot d) The sstem of smulteous le equtos: 6 5 55 5 55 5 55 5 979 5.6 585.6 488.8.47857,.47857 +.3599,.3599 +.867.867 t ( ) 53. 39 e 3.74657
Poloml Regesso- Emple (cot d) model e ( -`)..4786.433 544.4889 7.7 6.6986.86 34.4599 3.6 4.64.858 4.989 3 7. 6.33.849 3.9 4 4.9 4.687.695 39.89 5 6. 6.793.9439 7.3489 5 5.6 3.74657 53.39333 The stdd eo of estmte: 3.74657 6 3 s /. The coeffcet of detemto: 53.39 3.74657.9985, 53.39.9995
Cedts: Chp, Cle The Islmc Uvest of Gz, Cvl Egeeg Deptmet