Chapter 17. Least Square Regression

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1 The Islmc Uvest of Gz Fcult of Egeeg Cvl Egeeg Deptmet Numecl Alss ECIV 336 Chpte 7 Lest que Regesso Assocte Pof. Mze Abultef Cvl Egeeg Deptmet, The Islmc Uvest of Gz

2 Pt 5 - CURVE FITTING Descbes techques to ft cuves cuve fttg to dscete dt to obt temedte estmtes. Thee e two geel ppoches fo cuve fttg: Lest ques egesso: Dt ehbt sgfct degee of sctte [eos]. 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.

3 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.

4

5 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

6 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. %

7 Chpte 7 Lest ques Regesso Le Regesso Fttg stght le to set of ped obsevtos:,,,,,,. = + + e - slope - tecept e - eo, o esdul, betwee the model d the obsevtos

8 Le Regesso: Resdul

9 Le Regesso: Questo How to fd d so tht the eo would be mmum?

10 Le Regesso: Cte fo Best Ft m e e e e = -e

11 Le Regesso: Cte fo Best Ft m e

12 Le Regesso: Cte fo Best Ft m m e

13 Le Regesso: Lest ques Ft e m e,model,mesued Yelds uque le fo gve set of dt.

14 Le Regesso: Lest ques Ft e m The coeffcets d tht mmze must stsf the followg codtos:

15 o o o Le Regesso: Detemto of o d equtos wth ukows, c be solved smulteousl

16 Le Regesso: Detemto of o d

17

18 Dt sped oud Me Dt sped oud best-ft le 8

19 Emples of le egesso wth smll d b lge esdul eos 9

20 Eo Qutfcto of Le Regesso Totl sum of the sques oud the me fo the depedet vble,, s t um of the sques of esduls oud the egesso le s t o e

21 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

22 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.

23 Lest ques Ft of tght Le: Emple Ft stght le to the d vlues the followg Tble:

24 Lest ques Ft of tght Le: Emple cot d Y =

25 Lest ques Ft of tght Le: Emple Eo Alss e Y = e t t. 743 t e o

26 Lest ques Ft of tght Le: Emple Eo Alss The stdd devto qutfes the sped oud the me: s t The stdd eo of estmte qutfes the sped oud the egesso le Becuse s.99 7 / /.7735, the le egesso model hs good ftess

27 Algothm fo le egesso

28 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.

29 The epoetl equto The powe equto tutogowth-te equto

30 Lezto of Nole Reltoshps. The epoetl equto. b e l l b

31 Lezto of Nole Reltoshps. The powe equto b log log b log

32 Lezto of Nole Reltoshps 3. The stuto-gowth-te equto 3 b 3 3 b 3 3

33 Emple Ft the followg Equto: b to the dt the followg tble: X=log Y=log b log log log log b log let Y log log, X, Y X log b,

34 Emple X Y X* =LogX Y* =LogY X*Y* X*^ um

35 Lezto of Nole Fuctos: Emple log = log.46.75

36 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.

37 Poloml Regesso cot d A pbol s pefeble

38 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

39 Poloml Regesso cot d o o o o 4 3 o 3 o o 3 le equtos wth 3 ukows o,,, c be solved

40 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

41 Poloml Regesso cot d Geel: The m th -ode poloml: A sstem of m+m+ le equtos must be solved fo detemg the coeffcets of the m th -ode poloml. The stdd eo: The coeffcet of detemto: e m m o... / m s t t

42 Poloml Regesso- Emple Ft secod ode poloml to dt: ,

43 Poloml Regesso- Emple cot d The sstem of smulteous le equtos: ,.47857,

44 Poloml Regesso- Emple cot d model e -` The stdd eo of estmte: s /. The coeffcet of detemto: , t e e o

45 Nole Regesso Cosde the pevous epoetl egesso: The sum of the sques of the esduls: The cteo fo lest sques egesso s: o f e e o e f o &

46 Nole Regesso f f f f o o o & o e f o e f

47 Nole Regesso f f f f o The ptl devtves e epessed t eve dt pot tems of o d. Thus, the bove leds to equtos ukows whch c be solved tetvel fo o d.

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