Chapter 4 다중회귀 - wolfpack.hnu.ac.kr

1 WOLFPACKREGRESSION / 4 . 89 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring 3 .( , ) (1) (2) . 2 .. 1 2 (Polynomial Regression) . 1 2 . )2( p n . ipipiiieXXXY+++++= ..22110, ni,..,2,1= --- ),0(~2 iidNei: ie (1) (2) (3) ( ) p ,..,,10 (parameter) . i i- . piiiXXX,..,,21 deterministic ( , ) . ( ) (ANOVA) . (indicator variable) (dummy variable).

2 EXyeeexxxxxxxxxyyynpnpnnppn+= + = LLLLLLL211021222211121121 1 1 1, ),0(~2 INe , Chapter 4 WOLFPACKREGRESSION / 4 . 90 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring npnnppxxxyxxxyxxxy 212222212112111 LLLL y: ( 1 n), X: ( )1(+ pn) : ( 1)1( +p), e: ( 1 n) XyE=)(, nIyV2)( = . (1) (2) .. ( ), ( ) . p 2)1(21+=+ppCp . (scatter plot matrix) . EXAMPLE 4-1 , IQ(3 , FSIQ(Full scale IQ), VIQ(Verbal, ), PIQ(Performance, ) ( , ) (MRI ) .( ) FSIQ VIQ, PIQ, HEIGHT, WEIGHT, MRI.)

3 (38=n) SAS SAS/INSIGHT . LOG . ? . WOLFPACKREGRESSION / 4 . 91 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring .. SAS MRI (SAS/INSIGHT) . (CRTL ) .. ( ) ( ) . VIQ PIQ FSIQ ( , , ) .. ( ) WOLFPACKREGRESSION / 4 . 92 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring ( ) ( . ) .. (PIQ, VIQ), ( , , ) . SPSS.

4 WOLFPACKREGRESSION / 4 . 93 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring .. HOMEWORK #6-1 DUE 4 13 ( ) ( ) , , , , .. Country: ( , Japan) Car: MPG: Miles per gallon( ) Weight: Drive_Ratio: Lead-screw( ) WOLFPACKREGRESSION / 4 . 94 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring Horsepower: Displacement: Cylinder: , ( ) OLS OLS ( =niiep12,..,10min ) . ipipiiieXXXY+++++= ..22110, OLS . 222110,..,.)..(min10 pipiiixxxyp ),..,,,(210p OLS 0 )1(+p . OLS.

5 EXy+= . )()(minminmin12,..,,21 XYXY eeeniip = = = OLS yXXX = 1)( Gauss Markov Theorem yXXX = 1)( BLUE(Best Linear Unbiased Estimator) =) (E 12)() ( =XXV . ) ( V 12)() ( =XXMSEs . About Matrix pn pnX (matrix X of order pn ) . i , j }{ijpnxX= . ijx (element) . = npnnijpppn .. x .. x xx .. x xxX212222111211 WOLFPACKREGRESSION / 4 . 95 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring : ( , pn=) = 1 1 23 2 42 3 233A : (diagonal element) 0 : (trace) ==niiiAAtr1)( . : 1 0 (Identity Matrix) nI . (Linear Algebra) 1 . (matrix algebra) (inverse matrix) A IAAAA== 11 1 A.

6 =1 0 00 1 00 0 13I : (1) (2) (equal) . BA= ijijba=, jiallfor, . EXAMPLE = = =11322 1 ,1321 ,1321 CBA BA= CA . : (transpose) . pnX npX (np ) .. }{jixX= EXAMPLE = 2 2 51 1 23 2 42 3 234X 43 X = 2 1 3 22 1 2 35 2 4 243X : AA= )( BABA + =+) ABAB = ) : , 'AA=(}{}{jiijaa= ) A (Systematic Matrix) .. : (conformable for addition: ) . : )()()(BtrAtrBAtr+=+ (associate law): )()(CBACBA++=++ WOLFPACKREGRESSION / 4 . 96 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring ( )x( ).

7 = npnnijppn .. a .. a .. , = pqppijqqqp .. b .. b bb .. ++++++++++++= pqnpqnqnpnpnnpqpqqppqnbababababababababa bababaABLLLMMMLLL22111212111121211111211 21111 : qppnBA , (1)BA BAAB . (2)'')'(ABAB= . ( ) (3)A, B BAABAB= = )( (4))()(BAtrABtr= , AB . (5) (Associate law): )()(CBACBA++=++, ABCBCACAB==)()( (6) (Distribution law): ACABCBA+=+)( (7) (Communication law): )()(ABBA+=+ : MMMM==2 M (Idempotent matrix) . M MMk= (k ) . : IAAAA=='' A (Orthogonal matrix) . 2 : =6 43 7A (Determinant) 304367||= =A(scalar .) 3 : =10 9 87 5 43 2 1A 39 85 4)1(310 87 4)1(210 76 5)1(1||312111 = + + =+++A 39 82 1)1(710 83 1)1(510 93 2)1(4||322212 = + + =+++A(2 ) WOLFPACKREGRESSION / 4.

8 97 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring 37 53 2)1(810 93 2)1(410 97 5)1(1||131211 = + + =+++A(1 ) . n ||)1(||)1(||11ijjinjijijjiniijMaMaA+=+= = = . ||ijM minor ||)1(ijjiM+ cofactor . (1)||||AA= , ||||||BAAB=, ||||BAAB= (2) A 0 . (3) ( ) . (4) ( ) 0 . ( : ) : A IBAAB== B A 1 A . ||1||11 AadjAAA== [A cofactor ] (1) unique . (2)||/1||1AA= , AA= 11)(, )()A(11 = A, 111)( =ABAB (LIN: linearly independent vector): +++ppxaxaxa 0=ia ,,21 (linearly independent) , 0 ia (linearly dependent) .. (full rank): (nxn) ( ) ()(Arank) n full-rank . nAranknn= )( full-rank.

9 WOLFPACKREGRESSION / 4 . 98 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring . full-rank . rank(A)=n A non-singular . |A| 0 Ax=b . full-rank . rank(A)<n A singular . |A|=0 Ax=b . =paaaaM21, =pxxxxM21 (1)axax= )( (2)aaxx= )( (3)xAxAxAxx += )( (A ) A xAxAxx2)(= END of Matrix HOMEWORK #6-2 DUE 4 13 ( ) )()(12 XYXY eeenii = = = . XX . XXyXXyyyXyXy + = )()( . OLS yXXX = 1)( . aXVaXaV = )()( =) (E, 12)() ( =XXV . XXXXH = 1)( HAT . H, )(HI . yHy= yHIer)( == . WOLFPACKREGRESSION / 4 . 99 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring yHyXXXXXy= == 1)( (H Hat ) yHIyyre)( = ==.

10 0)(=rE )()(2 HIrV = . )(rV )(HIMSE . HOMEWORK #6-3 DUE 4 13 ( ) (term paper) 1 , , ( ) ( ) ( ) ( =2)(yySSTi), ( =2) (iiyySSE), ( =2) (yySSRi) (ANOVA) . (source) SS( ) MS ( ) F- Regression ( )JnHySSR])1([ = p pSSRMSR= )1,(~ =npFMSEMSRF Error ( ) yHIySSE][ =1 pn 2 =nSSEMSE To t a l ( ) yJnIySST])1([ = 1 n : SSTSSRR=2 yJynyynYYYYSSTiii'1')()(222 = = = , J 1 yXyyXyXyeeSSE = = = ) () ( JXXnyXSSESSTSSR = =1 WOLFPACKREGRESSION / 4 . 100 Prof. Sehyug Kwon, Dept. of Statistics, HANNAM University @2005 Spring F- F- . :210====pH ( ) 0 : iaallH ( .) F- t- . (coefficient of multiple determination) SSTSSESSTSSRR ==12 ( ) (SST ).