Multiple Linear Regression

We see from the output that for the trees data our parameter estimates are b = [ −58.0 4.7 0.3 ] ,and consequently our estimate of the mean response is ˆµ given byˆµ(x 1 , x 2 ) = b 0 + b 1 x 1 + b 2 x 2 , (10)≈ − 58.0 + 4.7x 1 + 0.3x 2 . (11)We could see the entire model matrix X with the model.matrix function.> head(model.matrix(trees.lm))(Intercept) Girth Height1 1 8.3 702 1 8.6 653 1 8.8 634 1 10.5 725 1 10.7 816 1 10.8 832.3 Point Estimates of the Regression SurfaceThe parameter estimates b make it easy to find the fitted values, Ŷ. We write them individually asŶ i , i = 1, 2, . . . , n, and recall that they are defined byŶ i = ˆµ(x 1i , x 2i ), (12)= b 0 + b 1 x 1i + b 2 x 2i , i = 1, 2, . . . , n. (13)They are expressed more compactly by the matrix equationŶ = Xb. (14)From Equation 9 we know that b = ( X T X ) −1X T Y, so we can rewrite[(Ŷ = X X T X ) ]−1X T Y , (15)= HY, (16)where H = X ( X T X ) −1X T is the hat matrix. Some facts about H are• H is a symmetric square matrix, of dimension n × n.• The diagonal entries h ii satisfy 0 ≤ h ii ≤ 1.5