More Regression Lecture Notes CE 311K - McKinney Introduction to Computer Methods Department of Civil Engineering The University of Texas at Austin

Transcription

1 More Regresso Lecture Notes CE K - McKe Itroducto to Coputer Methods Deprtet of Cvl Egeerg The Uverst of Tes t Aust Polol Regresso Prevousl, we ft strght le to os dt (, ), (, ), (, ) usg the lest-squres crtero As we hve see, soe dt re poorl represeted b strght le d for these cses curve s better suted to ft the dt The ost cool used fucto for ths purpose s the polol such s prbol os cubc or geerl -th degree polol: where,,,, re the costt coeffcets of the polol If the reltoshp betwee d were deed trul -th degree polol d there ws o ose the dt, the the coeffcets could be estted such tht the polol pssed through ll of the dt pots However, ths s hrdl ever the cse As the ler cse, the dscrepc (resdul) betwee the true vlue of d the polol pproto s e ( ) CEK McKe October,

2 CEK McKe October, d f there re such prs of pots (, ), the the su of squred resduls over ll the dt pots s [ ] r e S ) ( I order to detere the vlues of the coeffcets,,,,, we c ze r S The zto s ccoplshed b settg the prtl dervtves of r S wth respect to ech coeffcet equl to zero: ( ) [ ] [ ] [ ] [ ] r ) ( S Now, dvdg b - d sug ter b ter, we hve Slrl, the secod equto s ( ) [ ] [ ] [ ] [ ] r ) ( S Dvdg b - d sug ter b ter, we hve

3 CEK McKe October, It c ow be ferred tht the coplete set of sulteous ler equtos the coeffcets (the orl equtos) of the polol s 4 o o o r o o o Eple: Gve the followg dt, choose the ost sutble low order polol d ft t to ths dt usg the lest-squres crtero Tble Dt for polol fttg eple The dt re plotted the followg Fgure The dt pper to hve u er d u er The lowest order polol whch c reproduce such behvor s cubc The lest-squres equtos (orl equtos) for ths set of dt ( 4) d for re

4 CEK McKe 4 October, Guss elto elds 9 Thus the equtos for the terpoltg polol s 9 - -

5 6 4 tredle dt pots Fgure Plot of dt for polol fttg eple Lerzto of Noler Reltoshps I order to ppl the techques of ler lest-squres regresso, the fucto whose coeffcets re beg pproted ust be ler the coeffcets M reltoshps og depedet d depedet vrbles egeerg re ot ler However, cses trsforto c be ppled to the reltoshps to reder the ler the coeffcets Cosder epoetl reltoshp, b e where the bse s the uber e, d d b re costts If we tke the turl logrth of both sdes of the equto, we hve l () l() b whch s ler reltoshp betwee l() d The coeffcets to be detered ths epresso re l() d b A power lw reltoshp be wrtte s CEK McKe October,

6 b If we tke the turl logrth of both sdes of ths equto, we hve or l () l() bl() log () log () blog () whch s ler reltoshp betwee l() d l() Ag, the coeffcets to be detered ths epresso re l() d b Eple Gve the dt the followg tble, use the lest-squres crtero to ft B fucto of the for A to these dt: The power lw reltoshp s B A Tke the turl logrth of both sdes l () l(a) Bl() whch s ler reltoshp betwee l() d l() The coeffcets to be detered re l(a) d B Aother w to look t ths s Y BX whch s ler reltoshp betwee Yl() d Xl() The coeffcets to be detered re l(a) d B CEK McKe 6 October,

7 X l() Fgure Plot of dt o rthetc d log-log es CEK McKe 7 October,

8 The orl equtos re X X Y B X X Y X l( ) X Y l( ) XY X X Y XY l( l( ) l( ) ) l( ) l( ) 4 Pluggg the uercl vlues fro the dt tble, the orl equtos re B 9 4 Soluto elds? so A ep() B?? CEK McKe 8 October,

9 Eple - Crbo Adsorpto Adsorpto volves the ccuulto of dssolved substces t terfces of d betwee terl phses Adsorpto occur s the result of the ttrcto of surfce or terfce for checl speces, such s the dsorpto of substces fro wter b ctvted crbo (Weber d DGo, 996) s cool used hoe wter flters Crbo s well kow for ts dsorptve propertes Actvted crbo s regulrl used to reove tste d odors fro drkg wter sce crbo hs uque blt to reove sthetc orgc checls fro wter supples Adsorpto s the process where olecules of lqud or gs re ttched to d the held t the surfce of sold Phscl dsorpto s the process whereb surfce teso cuses olecules to be held t the surfce of sold Checl dsorpto occurs whe checl recto occurs to cuse olecules to be held t the surfce b checl bodg Phscl dsorpto occurs o ctvted crbo The lrge surfce re of the crbo kes t ecellet dsorbet terl Mcropores the surfce of the ctvted crbo grules provde etrce to the teror of the grul Adsorpto requres three processes: () dffuso through lqud phse to rech the crbo grule, () dffuso of olecules through cropores the crbo grule to dsorpto ste, d () dsorpto of the olecule to the surfce These processes occur t dfferet rtes for dfferet olecules of dfferet substces Sorpto studes re coducted b equlbrtg kow quttes of sorbet (s, crbo) wth solutos of solute (the pollutt) Plots of the resultg dt reltg the vrto of sold-phse cocetrto, or out of the solute (pollutt) sorbed per ut ss of sold (crbo), to the vrto of the soluto-phse cocetrto re tered sorpto sothers (Weber d DGo, 996) The re referred to s sothers becuse the dt re collected t costt teperture To evlute the effectveess of usg ctvted crbo to reove pollutts fro wter, the frst step s to perfor lqud-phse dsorpto sother test Dt re geerted b ddg kow weghts of crbo to wter cotg kow cocetrto of pollutt The crbo-wter ture s ed t costt teperture, the the crbo s reoved b fltrto The resdul pollutt cocetrto the wter s esured d the out of pollutt dsorbed o to the crbo s clculted Ths vlue f dvded b the weght of crbo to detere the crbo lodg (q) CEK McKe 9 October,

10 Severl odels hve bee developed to represet sorpto sothers thetcll These clude: () Ler sother odel q Kc where q s the ss of pollutt sorbed per ut ss of crbo t equlbru wth soluto of pollutt cocetrto c, d K s clled the dstrbuto coeffcet The dstrbuto coeffcet c be detered b fttg strght le through the org to the dt () Lgur sother odel Qbc q bc where Q s the u dsorpto cpct, d b s rte costt, d () Freudlch sother odel q K() c where, K s clled the specfc cpct, dctor of sorpto cpct t specfc pollutt cocetrto; s esure of the eerg of the sorpto recto Both of the preters c be detered fttg strght le to the logrthc trsforto or l q l K ( ) l c log q log K log c CEK McKe October,

11 Tble Adsorpto dt for pollutt (pheol) c q logc logq q tredle c Fgure Pheol sother (rthetc scles o es) CEK McKe October,

12 log(q) tredle log(c) Fgure Pheol sother (logrthc scles o es) Arthetc es: so Logrthc es: so or d q K() c K 747, d 89 log q log K log c logk 87 K CEK McKe October,