Chapter03 - Multiple Regression STAT 563 Spring 2007 Design...

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Multiple Regression STAT 563 Spring 2007
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Design Matrix Define the n x p matrix And the column vectors X j =[X 1j ,….., X nj ] Model now can be written as Y = X β + ε = nk n n k k x x x x x x x x x X .. 1 . . . . .. 1 .. 1 2 1 2 22 21 1 12 11
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Model Where = = = k k n y y y Y ε ε ε ε β β β β . , . , . 2 1 1 0 2 1
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Model With p=k+1 predictors, the model can also be written as n i X X X X y i ij k j j i ik k i i i , .... , 2 , 1 ..... 1 0 2 2 1 1 0 = + + = + + + + + = = ε β β ε β β β β
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Fitting the Model Like in simple linear regression, define the SSE as [ ] [ ] β β ε ε ε β X y X y S n i i - - = = = = ' ) ( 1 2
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Criteria Goal is to make all residuals as small as possible All Residuals Zero n parameters with Average residuals zero and many others Minimize average absolute residual L 1 norm Robust with respect to outliers Computational problems Theoretical difficulties 0 ˆ ˆ , ˆ 1 = - = - = - β β X Y Y Y Y X 0 . . ) 1 ( 1 : 1 ˆ = - = - = = Y Y Y Y r Y Y
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Criteria Minimize average squared residuals (or sum of a squared residuals) Least Squares Minimize a bounded, convex function of residuals Robust Estimation
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Least Squares Find that minimizes Determine β ˆ ( 29 ( 29 β β β β β β β β β β X X y X y y X X X y y X y y X y X y S ' ' ' ' 2 ' ' ' ' ' ' ' ' ) ( + - = + - - = - - = Why? ( 29 y X X X y X X X implies which X X y X S ' ' ˆ ' ˆ ' 0 ˆ ' 2 ' 2 1 - = = = + - = β β β β β Normal equations LS Estimate
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Least Squares Provided that (X’X) -1 exists. A linear independence between the regressors (ie, no column of X is a linear combination of the other columns) will guarantee this. Dependence between columns of X will lead to multicollinearity (more later) The fitted value is given by ( 29 ( 29 Matrix Hat ' ' ' ' ˆ ˆ 1 1 the called is X X X X H where y H y X X X X X y - - = = = = β
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Unit length scaling With the unit length scaling, we can define Note that each regressor w j has The regression now is 2 1 0 ) ( , ...... , 2 , 1 ; , .... , 2 , 1 , i n i ij jj Y i i jj j ij ij x x S where SS y y y and k j n i S x x w - = - = = = - = = 1 ) ( 0 1 2 = - = = n i j ij j w w length and w mean 0 1 2 2 1 1 0 ' ) ' ( ˆ ...... y W W W b is estimate squares least the and w b w b w b y i ik k i i i - = + + + + = ε
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Unit length scaling Let us take a look at an off-diagonal element in W’W We can write ) ( . 2 1 12 22 21 11 12 21 12 x and x between n correlatio r S x x S x x w w = - - = = 1 ... .. 1 .. .. ... 1 ... 1 ' 2 1 2 12 1 12 k k k k r r r r r r W W Correlation matrix = ky y y r r r y W .. ' 2 1 0
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Unit length scaling Where j jj jy jj n u u j uj jy j i jj ii ij jj ii n u j uj i ui ij x and y between n correlatio SST S S SST S y y x x r x and x between n correlatio S S S S S x x x x r = - - = = - - = = = 1 2 / 1 1 ) )( ( ) ( ) )( (
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Unit length scaling If unit normal scaling (z-score) is used, then you can show that Z’Z=(n-1)W’W Estimates obtained from both scaling are
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