Chapter 4: Model Adequacy Checking

Transcription

1 Catr : Modl Adquacy Cckg I t catr, w dcu om troductory act of modl adquacy cckg, cludg: Rdual Aaly, Rdual lot, Dtcto ad tratmt of outlr, T PRE tattc Ttg for lack of ft. T major aumto tat w av mad rgro aaly ar: T rlato btw t ro Y ad t rgror lar, at lat aroxmatly. T rror trm ε a zro ma. T rror trm ε a cotat varacσ. T rror ar ucorrlatd. T rror ar ormally dtrbutd. Aumto ad togtr mly tat t rror ar ddt. Rcall tat aumto rqurd for yot ttg ad trval tmato.

2 Rdual Aaly: T rdual,,, av t followg mortat rort: L a T ma of 0. b T tmat of oulato varac comutd from t rdual : σ R R c c t um of zro, ty ar ot ddt. Howvr, f t umbr of rdual larg rlatv to t umbr of aramtr, t ddcy ffct ca b gord a aaly of rdual. tadardzd Rdual: T quatty d,,, L,, calld tadardzd rdual. T tadardzd rdual av ma zro ad aroxmatly ut varac. A larg tadardzd rdual > ottally dcat a outlr. d R Rcall tat β + ε ε I H Y I H I H Trfor, Var / [ ] var I H ε I H ε I H σ I var. H

3 tudtzd Rdual: T quatty t,,, L,, calld t tudtzd rdual. T tudtzd rdual av aroxmatly a tudt' t dtrbuto wt dgr of frdom. R If w dlt t t obrvato, ft t rgro modl to t rmag obrvato, ad calculat t rdctd valu of corrodg to t dltd y obrvato, t corrodg rdctor rror y y. Grally a larg dffrc btw t ordary rdual ad t PRE rdual wll dcat a ot wr t modl ft t data wll, but a modl wtout tat ot rdct oorly. T rdcto rror ar uually calld PRE rdual or dltd rdual. It ca b ow tat Var. Trfor, Var Var [ σ ] σ Not tat a tadardzd PRE rdual Var σ σ wc, f w u to tmat R σ jut t tudtzd rdual.

4 R-tudt Rdual: T quatty r tudt rdual or jackkf rdual, wr t quatty comutd wt t t obrvato rmovd. It ca b ow tat,,, L,, calld t R- t rdual varac R If t uual aumto rgro aaly ar mt, t jackkf rdual follow xactly a t -dtrbuto wt dgr of frdom. Examl : Codr t followg data: y x x y, /

5 / / / H H , 0.700, 0.8,.9, 0

6 y H I ' R R d d d d d R R R R R t t t t t

7 R R R R R r r r r r

8 A Outut: Rdual, tudtzd Rdual ad R-tudt Rdual Ob Rdual tudt Rtudt cat t r l ot of vr u x x

9 Gracal Aaly of Rdual: a Normal robablty lot: If t ormalty aumto ot badly volatd, t cocluo racd by a rgro aaly wc ormalty aumd wll grally b rlabl ad accurat. A vry ml mtod of cckg t ormalty aumto to cotruct a ormal robablty lot of rdual. Lt,, L, Φ b t rdual rakd crag ordr. Not tat E Φ wr dot t tadard ormal cumulatv dtrbuto. Normal robablty lot ar cotructd by lottg t rakd rdual agat t xctd ormal valu Φ. T rultg ot ould l aroxmatly o a tragt l. ubtatal dartur from a tragt l dcat tat t dtrbuto ot ormal. If ormalty dmd uatfactory, t Y valu may b traformd by ug a Log, quar root, tc. to wtr t w t of obrvato aroxmatly ormal.

10 b Plot of Rdual vru t Fttd valu: A lot of t rdual cald rdual d t r or t, or vru t corrodg fttd valu uful for dtctg vral commo ty of modl adquac. y If t lot of rdual vru t fttd valu ca b cotad a orzotal bad, t tr ar o obvou modl dfct. T outward-og ful attr ml tat t varac of ε a crag fucto of Y. A ward-og ful dcat tat t varac of ε dcra a Y cra. T doubl-bow oft occur w Y a roorto btw zro ad o. T uual aroac for dalg wt qualty of varac to aly a utabl traformato to tr t rgror or t ro varabl. A curvd lot dcat olarty. T could ma tat otr rgror varabl ar dd t modl. For xaml a quard trm may b cary. Traformato o t rgror ad/or t ro varabl may b lful t ca. A lot of rdual vru t rdctd valu may alo rval o or mor uuually larg rdual. T ot ar ottal rdual. Extrm rdctd valu wt larg rdual could alo dcat tr t varac ot cotat or t tru rlato btw Y ad ot lar. T oblt ould b vtgatd bfor t ot ar codrd outlr.

11 c Plot of Rdual vru t Rgror: Plottg t rdual vru corrodg valu of ac rgror varabl ca alo b lful. Oc aga a orzotal bad cotag t rdual drabl. T ful ad doubl-bow attr dcat ocotat varac. T curvd bad or a olar attr gral dcat tat t aumd rlato btw Y ad t rgror j ot corrct. Tu, tr gr-ordr trm j uc a j or a traformato ould b codrd. Not tat t ml lar rgro t ot cary to lot rdual vru bot rdctd valu ad t rgror varabl c t rdctd valu ar lar combato of t rgror valu. d Plot of Rdual Tm quc: It a good da to lot t rdual agat tm ordr, f t tm quc wc t data wr collctd kow. If a orzotal bad wll clo all of t rdual ad t rdual wll fluctuat a mor or l radom fao wt t bad, t tr ar o autocorrlato.

12 Partal Rgro lot: A lmtato of t lot of rdual vru rgror varabl tat ty may ot comltly ow t corrct or comlt margal ffct of a rgror, gv t otr rgror t modl. T artal rgro lot codr t margal rol of t rgror gv otr rgror tat ar alrady t modl. I t lot, t j ro varabl Y ad t rgror ar bot rgrd agat t otr j rgror t modl ad t rdual obtad for ac rgro. T lot of t rdual agat ac otr rovd formato about t atur of t margal rlato for rgror udr codrato. j If t rgror tr t modl larly, t artal rgro lot ould ow a j lar rlato wt a lo qual to β j t multl lar rgro modl. Not tat: Partal rgro lot oly uggt obl rlato btw rgror ad t ro. T lot may ot gv formato about t ror form of t rlato f vral varabl alrady t modl ar corrctly cfd. It wll uually b cary to vtgat vral altratv form for t rlato btw t rgror ad Y or vral traformato. Rdual lot for t ubqut modl ould b xamd to dtfy t bt rlato or traformato. Partal rgro lot wll ot, gral, dtct tracto ffct amog t rgror. T rc of trog collarty ca cau artal rgro lot to gv corrct formato about t rlato btw t ro ad t rgror varabl.

13 f Partal Rdual lot: uo tat t modl cota t rgror,, L,. T artal rdual for rgror k ar dfd j * Y + j β,,, L, wr t ar t rdual from j a x x j t modl wt all k rgror cludd. T artal rdual ar lottd vru ad t trrtato of t artal rdual lot vry mlar to xj tat of t artal rgro lot. Examl Dlvry Tm Data: A oft drk bottlr aalyzg t vdg mac rvc rout dtrbuto ytm. H trtd rdctg t amout of tm rqurd by t rout drvr to rvc t vdg mac a outlt. T rvc actvty clud tockg t mac wt bvrag roduct ad mor matac or oukg. T dutral gr robl for t tudy a uggtd tat t two mot mortat varabl affctg t dlvr tm Y ar t umbr of ca of roduct tockd ad t dtac walkd by t rout drvr. T gr a collctd 0 obrvato o dlvr tm. A Outut: Rgr o Modl Y o ad y x x N 0 Rq 0. 9 Adj Rq RE CDF of RTUDENT

14 Q-Q-lot of Rt udt Rdual Nor mal Quat l Rgr o Modl Y o ad y x x N 0 Rq 0. 9 Adj Rq RE Pr d ct d Val u

15 Rgr o Modl Y o ad y x x N 0 Rq 0. 9 Adj Rq RE x Rgr o Modl Y o ad y x x N 0 Rq 0. 9 Adj Rq RE x

16 Part al R dual l ot r x Part al R dual l ot r x

17 PRE tattc: PRE rdual ar dfd by y y, wr y t rdctd valu of t t obrvd ro bad o a ft to t rmag aml ot. Larg PRE rdual ar ottally uful dtfyg obrvato wr t modl do ot ft t data wll or obrvato for wc t modl lkly to rovd oor futur rdcto. T PREE tattc dfd by PRE y y PRE grally rgardd a a maur of ow wll a rgro modl wll rform rdctg w data. O vry mortat of t PRE tattc comarg rgro modl. Grally, a modl wt a mall valu of PRE drd. T PRE tattc ca b alo ud to comut a R Pr dcto R PRE -lk tattc for rdcto, ay T T tattc gv om dcato of t rdctv caablty of t rgro modl. Examl Cot.: R RE T R PRE Pr dcto T Trfor, w could xct t modl to xla about 89.0% of t varato rdctg w obrvato, a comard to aroxmatly 9.% of t varablty t orgal data xlad by t lat-quar ft. Lack of Ft of t Rgro Modl: