Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates.
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1 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.
2 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.
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4 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 ( )
5 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. %
6 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
7 Le Regesso: Resdul
8 Le Regesso: Questo How to fd d so tht the eo would be mmum?
9 Le Regesso: Cte fo Best Ft m e ( ) e e e -e
10 Le Regesso: Cte fo Best Ft m e
11 Le Regesso: Cte fo Best Ft m m e
12 Le Regesso: Lest ques Ft e ) ( m e ) (,model),mesued ( Yelds uque le fo gve set of dt.
13 Le Regesso: Lest ques Ft e ) ( m The coeffcets d tht mmze must stsf the followg codtos:
14 [ ] ) ( ) ( o o o Le Regesso: Detemto of o d ( ) + + equtos wth ukows, c be solved smulteousl
15 Le Regesso: Detemto of o d ( )
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19 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 )
20 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
21 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.
22 Lest ques Ft of tght Le: Emple Ft stght le to the d vlues the followg Tble:
23 Lest ques Ft of tght Le: Emple (cot d) ( ) Y
24 Lest ques Ft of tght Le: Emple (Eo Alss) ( ) e ( ) ^ e t t ( ). 743 t
25 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
26 Algothm fo le egesso
27 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.
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29 Lezto of Nole Reltoshps. The epoetl equto. b e l l + b * o +
30 Lezto of Nole Reltoshps. The powe equto b log log + b log * o + *
31 Lezto of Nole Reltoshps 3. The stuto-gowth-te equto * / o / 3 b 3 / 3 * /
32 Emple Ft the followg Equto: b to the dt the followg tble: X*log Y*log b log log( ) log log + b let Y * log log * Y +, X, X * log, * b log
33 Emple X Y X* Log(X) Y* Log(Y) X*Y* X*^ um ()
34 Lezto of Nole Fuctos: Emple log log.46.75
35 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.
36 Poloml Regesso (cot d) A pbol s pefeble
37 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
38 Poloml Regesso (cot d) o o o o ) ( ) ( ) ( o 3 o o 3 le equtos wth 3 ukows ( o,, ), c be solved
39 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
40 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
41 Poloml Regesso- Emple Ft secod ode poloml to dt:
42 Poloml Regesso- Emple (cot d) The sstem of smulteous le equtos: , , t ( ) e
43 Poloml Regesso- Emple (cot d) model e ( -`) The stdd eo of estmte: s /. The coeffcet of detemto: ,
44 Cedts: Chp, Cle The Islmc Uvest of Gz, Cvl Egeeg Deptmet
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