UNIT10 PLANE OF REGRESSION

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1 UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons / Answes 0. ITRODUCTIO In Unt 9, you leant the concept of egesson, lnea egesson, lnes of egesson and egesson coeffcent wth ts popetes. Unt 9 was ased on lnea egesson n whch we wee consdeng only two vaales. In Unt 6 of MST-00, you studed the coelaton that measues the lnea elatonshp etween two vaales. In many stuatons, we ae nteested n studyng the elatonshp among moe than two vaales n whch one vaale s nfluenced y many othes. Fo eample, poducton of any cop depends upon sol fetlty, fetlzes used, gaton methods, weathe condtons, etc. Smlaly, maks otaned y students n eam may depend upon the IQ, attendance n class, study at home, etc. In these type of stuatons, we study the multple and patal coelatons. In ths unt you wll study the ascs of multple and patal coelatons that ncludes Yule s notatons, plane of egesson, esduals, popetes of esduals and vaance of esduals. Fo the study of moe than two vaales, thee would e need of much moe notatons n compason to the notatons used n Unt 9. These notatons wee gven y G.U. Yule (897). Yule s notatons and esduals ae desced n Secton 0.. Plane of egesson and nomal equatons ae gven n Secton 0.. Popetes of esduals ae eploed n Secton 0.4 wheeas Secton 0.5 eplans the vaance of esduals. Ojectves Afte eadng ths unt, you would e ale to defne the Yule s notaton; desce the plane of egesson fo thee vaale; eplan the popetes of esduals; desce the vaance of the esduals; fnd out the lnes of egesson; and fnd out the estmates of dependent vaale fom the egesson lnes.

2 Regesson and Multple Coelaton 0. YULE S OTATIO Kal Peason developed the theoy of multple and patal coelaton fo thee vaales and t was genealzed y G.U. Yule (897). Let us consde thee andom vaales, and and, and ae the espectve means. Then egesson equaton of on and s defned as a.. () whee,. and. ae the patal egesson coeffcents of on and on keepng the effect of and fed espectvely. Takng summaton of equaton () and dvdng t y, we get a.. () On sutactng equaton () fom equaton (), we get Suppose.., and ow, plane of egesson on and.. () (equaton ()) can e ewtten as (4) Rght hand sde of equaton () s called the estmate of whch s denoted y.. Thus,... (5) Eo of estmate o esdual s defned as e e..... (6) Ths esdual s of ode. If we ae consdeng n vaales egesson of on,..., n s,,..., n n.4...n... n...(n ) n, the equaton of the plane of (7) and eo of estmate o esdual s...n (.4...n.4...n... n...(n ) n ) (8) 4 ote: In aove epessons we have used suscpts nvolvng dgts,,,, n and dot (.). Suscpts efoe the dot ae known as the pmay suscpts wheeas the suscpts afte the dot ae called seconday suscpts. The nume of seconday suscpts decdes the ode of egesson coeffcent.

3 Fo eample. s the egesson coeffcent of ode,. 4 s of ode and so on n...(n ) of ode (n-). Plane of Regesson Ode n whch seconday suscpts (. 4 o. 4 ) s mmateal ut the ode of pmay suscpts s vey mpotant and decdes the dependent and ndependent vaales. In. 4, s dependent vaale and s ndependent vaale wheeas n. 4, s dependent vaale and s ndependent vaale. Ode of esduals s detemned y the nume of seconday suscpts. Fo eample. s esdual of ode whle as. 4 s of ode. Smlaly,.4...n s esdual of ode (n-). 0. PLAE OF REGRESSIO FOR THREE VARIABLES Let us consde thee vaales, and measued fom the espectve means. The egesson equaton of depends upon and s gven y (fom equaton (4))... (9) If s consdeed as a constant then the gaph of and s a staght lne wth slope., smlaly the gaph of the and would e the staght lne wth slope., f s consdeed as a constant. Accodng to the pncple of least squaes, we have to detemne constants. and. n such a way that sum of squaes of esduals s mnmum.e. (..) U s mnmum. e. hee, e... (0) By the pncple of mama and mnma, we take patal devatves of U wth espect to. and. Thus, U Let us take. U. U 0. 0 (.. )( ) (..) 0 () 0 5

4 Regesson and Multple Coelaton Smlaly, ) 0 ( () U. As we know that 0 (.. ) 0 (). 0 (4). ) ( (fo, and ) Snce,, and ae measued fom the means theefoe 0 then (5) Smlaly, we can wte (fo j =,, ) Cov (, j) j (6) and consequently, usng equatons (5) and (6), coelaton coeffcent etween and can e epessed as j j Cov(, j) j Cov(, j) j j (7) V( )V( ) j j Dvdng equatons () and (4) y povdes.. 0 and (8). 0 (9). Usng equatons (5), (6) and (7) n equatons (8) and (9), we have Cov(, ) Cov(, ) 0 Fom equaton (8) (.. ) 0 6 Smlaly, 0 (0).... Cov(, ) σ Cov(, ) 0 Fom equaton (9)

5 .. 0 Plane of Regesson (.. ) 0 0 ().. whee, s the total coelaton coeffcent etween and, s the total coelaton coeffcent etween and and smlaly, s the total coelaton coeffcent etween and. Thus, we have two equatons (0) and (). Solvng the equatons (0) and () fo. and., we otaned.. () Smlaly, If we wte. () t (4). and. and.. can e wtten as t t t t whee, t j s the cofacto of the element n the th ow and th j column of t. Susttutng the values of. and. n equaton (9), we get the equaton of the plane of egesson of on and as 7

6 Regesson and Multple Coelaton t t whee, a = 0 t t t t t 0 (5) Smlaly, the plane of egesson of on and s gven y t t t 0 (6) and the plane of egesson of on and s t t t 0 (7) In geneal the plane of egesson of on the emanng vaale j ( j,,..., n ) s gven y n t t... t... t n 0;,,...,n (8) σ σ σ σ n Eample : Fom the gven data n the followng tale fnd out () Least squae egesson equaton of on and. () Estmate the value of fo 45 and Soluton: () Hee, and ae thee andom vaales wth the espectve means, and. Let, and Then lnea egesson equaton of on and s.. Fom equaton () and (), we have and,.. The value of σ, σ, σ, and can e otaned though some calculatons gven n the followng tale: 8

7 S. o. = 5 = 6 = 7 ( ) ( ) ( ) Plane of Regesson Total fom equaton (5).608 fom equaton (7) 9

8 Regesson and Multple Coelaton ow, we have Thus, egesson equaton of on and s Afte susttutng the value of, and, we wll get the followng egesson equaton of on and s () Susttutng 45 and 8 n egesson equaton we get estmated value of Let us solve some eecses e E) Fo the data gven n the followng tale fnd out () Regesson equaton of on and. () Estmate the value of the value of fo 6 and 8. 0

9 Plane of Regesson 0.4 PROPERTIES OF RESIDUALS Popety : The sum of the poduct of a vaate and a esdual s zeo f the suscpt of the vaate occus among the seconday suscpts of the esdual,.e. e. 0. Hee, suscpt of the vaate.e. s appeang n the second suscpts of the e.. Poof: If the egesson equaton of on and s.. Hee,, and ae measued fom the espectve means. Usng equaton (0) n equatons () and () we have followng nomal equatons e. 0 = e. Smlaly, nomal equaton fo egesson lnes of on and & on and ae e. 0 = e. e. 0 = e. Popety : The sum of the poduct of two esduals s zeo povded all the suscpts, pmay as well as seconday, ae appeang among the seconday suscpts of second esdual.e. e 0, snce pmay as well as.. seconday suscpts ( and ) of the fst esdual s appeang among the seconday suscpts of the second esdual. Poof: We have.e. ( ) e. ( e. e. ) = 0 (Fom Popety : e. 0 and e. 0 Smlaly,.e. 0 ) Popety : The sum of the poduct of any two esduals s unalteed f all the seconday suscpt of the fst occu among the seconday suscpts of the second and we omt any o all of the seconday suscpts of the fst.

10 Regesson and Multple Coelaton Poof: We have e.. e,..e. Rght hand sde and left hand sde ae equal even.e. ( ) e. e = e. e ) (Fom Popety, e 0 (... ) ow let us do some lttle eecses. E) Show that e 0.. E) Show that e VARIACE OF THE RESIDUALS Let, and e thee andom vaales then plane of egesson of on and s defned as a.. Snce and ae measued fom the espectve means so 0 and we get a = 0 and egesson equaton ecomes.. and eo of the estmate o esdual s (See Secton 0.) e... ow the vaance of the esdual s denoted y and σ. (e. e. ). and defned as 0 ecause 0 e.... e. (.. )(.. ) (.. )...(9)

11 . (..).(.. ). (.. ) We know that and Theefoe, (.... ) (.... ) (see equatons () and (4) of Secton 0.) Plane of Regesson. ( ) (0) 0.6 SUMMARY In ths unt, we have dscussed:. The Yule s notaton fo tvaate dstuton;. The plane of egesson fo tvaate dstuton;. How to get nomal equatons fo the egesson equaton of ; on and 4. The popetes of esduals; 5. The vaance of esduals; and 6. How to fnd the estmates of dependent vaale of egesson equatons of thee vaales.

12 Regesson and Multple Coelaton 0.7 SOLUTIOS / ASWERS E) () Hee, and ae thee andom vaales wth the espectve means, and. Let, and Then lnea egesson equaton of on and s.. Fom equaton () and (), we have. and. σ, σ, σ, and can e otaned though the followng tale. S. o. = = = ( ) ( ) ( ) Total fom equaton (5)

13 fom equaton (7) ow, we have Thus, egesson equaton of on and s.9 Afte susttutng the value of, and, we wll get the followng egesson equaton of on and s.9 Plane of Regesson 5

14 Regesson and Multple Coelaton () Susttutng 6 and 8 n egesson equaton..9 we get estmated value of.e. 8 E) Hnt: Accodng to the popety : e. 0, snce suscpt of the vaate.e. s appeang n the second suscpt of e..e. n.. E) Hnt: Accodng to the popety : e. 0, snce suscpt of the vaate.e. s appeang n the second suscpt of e..e. n. 6

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