Chapter Linear Regression

Transcription

1 Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use emples, the deved fomuls fo the costts of le egesso model, d. pove tht the costts of the le egesso model e uque d coespod to mmum. Le egesso s the most popul egesso model. I ths model, we wsh to pedct espose to dt pots (, ),(, ),...,(, ) b egesso model gve b () whee d e the costts of the egesso model. A mesue of goodess of ft, tht s, how well pedcts the espose vble s the mgtude of the esdul ε t ech of the dt pots. E ( ) () Idell, f ll the esduls ε e zeo, oe m hve foud equto whch ll the pots le o the model. Thus, mmzto of the esdul s objectve of obtg egesso coeffcets. The most popul method to mmze the esdul s the lest sques methods, whee the estmtes of the costts of the models e chose such tht the sum of the squed esduls s mmzed, tht s mmze E. Wh mmze the sum of the sque of the esduls? Wh ot, fo stce, mmze the sum of the esdul eos o the sum of the bsolute vlues of the esduls? Altetvel, costts of the model c be chose such tht the vege esdul s zeo wthout mkg dvdul esduls smll. Wll of these cte eld ubsed 6.3.

2 6.3. Chpte 6.3 pmetes wth the smllest vce? All of these questos wll be sweed below. Look t the dt Tble. Tble Dt pots To epl ths dt b stght le egesso model, (3) d usg mmzg E s cte to fd d, we fd tht fo (Fgue ) () 6 3 Fgue Regesso cuve fo vs. dt. the sum of the esduls, E s show the Tble. Tble The esduls t ech dt pot fo egesso model. pedcted ε pedcted ε

3 Le Regesso o does ths gve us the smllest eo? It does s E. But t does ot gve uque vlues fo the pmetes of the model. A stght-le of the model 6 () lso mkes E s show the Tble 3. Tble 3 The esduls t ech dt pot fo egesso model 6 pedcted ε pedcted E 6 Fgue Regesso cuve 6 fo vs. dt. ce ths cteo does ot gve uque egesso model, t cot be used fo fdg the egesso coeffcets. Let us see wh we cot use ths cteo fo geel dt. We wt to mmze E ( ) Dffeettg Equto (6) wth espect to d, we get E () (6)

4 6.3. Chpte 6.3 E _ () Puttg these equtos to zeo, gve but tht s ot possble. Theefoe, uque vlues of d do ot est. You m thk tht the eso the mmzto cteo E egtve esduls ccel wth postve esduls. o s mmzg does ot wok s tht E bette? Let us look t the dt gve the Tble fo equto. It mkes E s show the followg tble. Tble The bsolute esduls t ech dt pot whe emplog. pedcted ε pedcted ε The vlue of E lso ests fo the stght le model 6. No othe stght le model fo ths dt hs E <. Ag, we fd the egesso coeffcets e ot uque, d hece ths cteo lso cot be used fo fdg the egesso model. Let us use the lest sques cteo whee we mmze ( ) E () s clled the sum of the sque of the esduls. To fd d, we mmze wth espect to d. ( )( ) () ( )( ) () gvg ()

5 Le Regesso 6.3. (3) Notg tht... () () Fgue 3 Le egesso of vs. dt showg esduls d sque of esdul t tpcl pot,. olvg the bove Equtos () d () gves (6) () Redefg ( ), ( ) 3, 3 ( ), ), ( ( ), E

6 6.3.6 Chpte 6.3 () _ () _ () _ () we c ewte () (3) Emple The toque T eeded to tu the tosol spg of mousetp though gle, θ s gve below Tble Toque vesus gle fo toso spg. Agle, θ Toque, T Rds N m Fd the costts k d k of the egesso model T k k θ oluto Tble 6 shows the summtos eeded fo the clculto of the costts of the egesso model. Tble 6 Tbulto of dt fo clculto of eeded summtos. θ T θ T θ ds N m ds N m

7 Le Regesso k θ T θ θ θ (.6) (6.3)(.) (.) (6.3).6 T N - m/d T _ T..3 N-m θ _ θ ds k T k θ.3 (.6.6 N - m )(.66)

8 6.3. Chpte 6.3 Fgue Le egesso of toque vs. gle dt Emple To fd the logtudl modulus of composte mtel, the followg dt, s gve Tble, s collected. Tble tess vs. st dt fo composte mtel. t tess (%) ( MP ) Fd the logtudl modulus E usg the egesso model.

9 Le Regesso 6.3. σ Eε () oluto Rewtg dt fom Tble, stesses vesus st dt Tble Tble tess vs st dt fo composte I sstem of uts t ( m/m ) tess ( P ) Applg the lest sque method, the esduls γ t ech dt pot s γ σ Eε The sum of sque of the esduls s γ ( σ Eε ) Ag, to fd the costt E, we eed to mmze b dffeettg wth espect to E d the equtg to zeo d ( σ Eε )( ε ) de Fom thee, we obt E σ ε ε Note, Equto () ol so f hs show tht t coespods to locl mmum o mmum. C ou show tht t coespods to bsolute mmum. The summtos used Equto () e gve the Tble. ()

10 6.3. Chpte 6.3 E Tble Tbulto fo Emple fo eeded summtos ε σ ε εσ ε.6 3 σ ε.333 σ ε ε GP

11 Le Regesso 6.3. Fgue Le egesso model of stess vs. st fo composte mtel. QUETION: Gve dt ps, (, ),,(, ), do the vlues of the two costts d the lest sques stght-le egesso model coespod to the bsolute mmum of the sum of the sques of the esduls? Ae these costts of egesso uque? ANWER: Gve dt ps ( ),,(, ),, the best ft fo the stght-le egesso model (A.) s foud b the method of lest sques. ttg wth the sum of the sques of the esduls ( ) (A.) d usg gves two smulteous le equtos whose soluto s (A.3) (A.)

12 6.3. Chpte 6.3 (A.) (A.b) But do these vlues of d gve the bsolute mmum of vlue of (Equto (A.))? The fst devtve lss ol tells us tht these vlues gve locl mm o mm of, d ot whethe the gve bsolute mmum o mmum. o, we stll eed to fgue out f the coespod to bsolute mmum. We eed to fst coduct secod devtve test to fd out whethe the pot ), ( fom Equto (A.) gves locl mmum o locl mmum of. Ol the c we poceed to show f ths locl mmum (o mmum) lso coespods to the bsolute mmum (o mmum). Wht s the secod devtve test fo locl mmum of fucto of two vbles? If ou hve fucto ( ) f, d we foud ctcl pot ( ) b, fom the fst devtve test, the ( ) b, s mmum pot f > f f f, d (A.6) > f OR > f (A.) Fom Equto (A.) ( ) ( ) ) ( (A.) ( ) ( ) ) ( (A.) the (A.)

13 Le Regesso (A.) (A.) o, we stsf codto (A.) becuse fom Equto (A.) we see tht s postve umbe. Although ot equed, fom Equto (A.) we see tht s lso postve umbe s ssumg tht ll dt pots e NOT zeo s esoble. Is the othe codto (Equto (A.6)) fo beg mmum met? Yes, we c show (poof ot gve tht the tem s postve) ( ) ( ) j > (A.3) < j o the vlues of d tht we hve Equto (A.) do coespod to locl mmum of. But, s ths locl mmum lso bsolute mmum. Yes, s gve b Equto (A.), the fst devtves of e zeo t ol oe pot. Ths obsevto lso mkes the stght-le egesso model bsed o lest sques to be uque. As sde ote, the deomto Equtos (A.) s ozeo s show b Equto (A.3). Ths shows tht the vlues of d e fte. LINEAR REGREION Topc Le Regesso umm Tetbook otes of Le Regesso Mjo Geel Egeeg Authos Egwu Klu, Aut Kw, Cuog Ngue Dte August 3, Web te