ME 310 Numerical Methods. Least Squares Regression

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1 ME 3 Numecl Mehods Les ques Regesso These peseos e peped b D. Cue e Mechcl Egeeg Depme Mddle Es Techcl Uves Ak, Tuke cse@meu.edu. The c o be used whou he pemsso of he uho

2 Abou Cuve Fg Cuve fg s epessg dscee se of d pos s couous fuco. I s fequel used egeeg. Fo emple he empcl elos h we use he sfe d flud mechcs e fucos fed o epemel d. Regesso: Ml used wh epemel d, whch mgh hve sgfc mou of eo (ose). No eed o fd fuco h psses hough ll dscee pos. f() Le Regesso f() Poloml Regesso Iepolo: Used f he d s kow o be ve pecse. Fd fuco (o sees of fucos) h psses hough ll dscee pos. f() A sgle fuco f() Fou dffee fucos Poloml Iepolo ple Iepolo

3 Les ques Regesso (Red he sscs evew fom he book.) Fg sgh le o se of d se (ped d pos). (, ), (, ), ( 3, 3 ),..., (, ) e + + : -ecep (ukow) : slope (ukow) e = - - Eo (devo) fo he h d po Mmze he eo (devo) o ge bes-f le (o fd d ). evel posbles e: Mmze he sum of dvdul eos. Mmze he sum of bsolue vlues of dvdul eos. Mmze he mmum eo. Mmze he sum of sques of dvdul eos. Ths s he pefeed seg (Check he booko see wh ohes fl). 3

4 4 - ) ( ( ) Mmzg he que of Idvdul eos Deeme he ukows d b mmzg. To do hs se he devves of w d o zeo. ) ( e um of sques of he esduls ] ) [( - ) ( - equos. e clled he oml These o olve hese fo d. The esuls e whee -b d -b e he mes of d, especvel.

5 Emple 4: Use les-sques egesso o f sgh le o () ( ) * *96 5 * 73 * Eecse 4: I s lws good de o plo he d pos d he egesso le o see how well he le epeses he pos. You c do hs wh Ecel. Ecel wll clcule d fo ou. 5

6 6 Eo of Le Regesso (How good s he bes le?) esme of eo sd. s ) ( e / The mpoveme obed b usg egesso le sed of he me gves mesue of how good he egesso f s. coeffce of deemo coelo coeffce devo sd. s ) ( + ped of d oud he me ped of d oud he egesso le ) ( ) ( ( )

7 How o epe he coelo coeffce? Two eeme cses e = = descbes pefec f (sgh le pssg hough ll pos). = = descbes cse wh o mpoveme. Usull vlue close o epeses good f. Bu be ceful d lws plo he d pos d he egesso le ogehe o see wh s gog o. Emple 4 (co d): Clcule he coelo coeffce ( ( ) 64. )

8 Emple 4 (co d): Revese d. Fd he le egesso le d clcule. = = 374.5, = 7.9 (dffee h befoe). =.87, =.9 (sme s befoe). Eecse 5: Whe wokg wh epemel d we usull ke he vble h s coolled b us pecse w s. The mesued o clculed ques e. ee Mdem II of Fll 3 fo emple. Lezo of Nole Behvo Le egesso s useful o epese le eloshp. If he elo s ole ehe ohe echque c be used o he d c be sfomed so h le egesso c sll be used. The le echque s fequel used o f he he followg ole equos o se of d. Epoel equo ( =A e B ) Powe equo ( =A B ) uo-gowh e equo ( =A 3 / (B 3 +) ) 8

9 Lezo of Nole Behvo (co d) () Epoel Equo ( = A e B ) = A e B Lezo l l = l A + B o l = + Emple 5: F epoel model o he followg d se Cee he followg ble l F sgh le o hs ew d se. Be ceful wh he oo. You c defe z = l Clcule = 6.5 d =.84. gh le s l = wch bck o he ogl equo. A = e = 58, B = =.84. Theefoe he epoel equo s = 58 e.84. Check hs soluo wh couple of d pos. Fo emple (.) = 58 e.84 (.) = 4 o (.3) = 58 e.84 (.3) = OK. 9

10 Lezo of Nole Behvo (co d) () Powe Equo ( = A B ) = A B Lezo log log = log A + B log o log = + log log Emple 6: F powe equo o he followg d se log log F sgh le o hs ew d se. Be ceful wh he oo. Clcule =. d = gh le s log =..53 log wch bck o he ogl equo. A = =.5, B = = Theefoe he powe equo s = Check hs soluo wh couple of d pos. Fo emple (5) =.5 * = 4.8 o (5) =.5 * =.39. OK.

11 Lezo of Nole Behvo (co d) (3) uo-gowh e Equo ( = A 3 / (B 3 +) ) = =A 3 / (B 3 +) Lezo / / = /A 3 + B 3 /A 3 o / = + / Emple 7: F suo-gowh-e equo o he followg d se / / F sgh le o hs ew d se. Be ceful wh he oo. Clcule =.5 d =.56. gh le s / = (/) wch bck o he ogl equo. A 3 = / =.953, B 3 = A 3 =.97. Theefoe he suo-gowh e equo s / =.953 /(.97+). Check hs soluo wh couple of d pos. Fo emple () =.953*/(.97+) =.6 OK.

12 Poloml Regesso (Eeso of Le Les ques) Used o fd bes-f le fo ole behvo. Ths s o ole egesso descbed pge 468 of he book. Th seco s omed Emple fo secod ode poloml egesso e = Eo (devo) fo he h d po e ( ) Mmze hs sum o ge he oml equos um of sques of he esduls,, olve hese equos wh oe of he echques h we leed o ge, d.

13 3 Fd he les-sques pbol h fs o he followg d se. Poloml Regesso Emple Noml equos o fd les-sques pbol e ,.479,

14 4 = (, ) Idvdul eos e e = um of sques of he esduls s Mmze hs sum o ge he oml equos Mulple Le Regesso olve hese equos o ge, d.,, ) ( e

15 Emple 8: Use mulple le egesso o f z z 4.7 z z , 9.3, z Eecse 6: Clcule he sdd eo of he esme (s / ) d he coelo coeffce (). 5

16. Mean Square Estimation

16. Mean Square Estimation 6 Me Sque stmto Gve some fomto tht s elted to uow qutty of teest the poblem s to obt good estmte fo the uow tems of the obseved dt Suppose epeset sequece of dom vbles bout whom oe set of obsevtos e vlble