ST790R -- Fall 2009  

  Homework #3 -- due Thursday, 24 September 2009
  (** just turn in 5.13(in text, as modified), and 4, 5, 6 below) 

  *5.13 *modified*  Compare either the algorithm in 5.11 (code
     with a loop) or (2.5.1)(*correct it*) with the 'usual'
     calculations of SUM(x(i)^2) - (SUM(x(i)))^2/N, etc. and
     with R's 'var' function.  Use x(i) = (2^k) + i, for i=1, ..., 48 
     and k=12, 25, 27, 30.  Just compare the variance calculations.

  0)  Centering, or subtracting the mean from a covariate,
     is a common practice in regression, and it has its advantages
     in computation.  Writing Xb = X*S*inv(S)*b = Wc where X and W
     have the same column space, the 2nd through p-th cols of W are
     centered cols of X, write the p*p matrix S, and construct its
     inverse.  ( W = X*S and c = inv(S)*b )

  R Exercises:(from Homework #2)

  1) Let y = (225, 215, 209, 175, 163, 135, 153, 125)' and let
     x = (37, 41, 41, 45, 45, 49, 49, 53)' be the covariate of
     a simple linear regression model, where the first column of
     the design matrix X is one, and the second column is x. 
     Compute y'Pone y, y'(Px-Pone)y, and y'(I-Px)y, where
     Pone is a matrix with each element equal to 1/n (here n=8),
     and Px is X*inv(X'X)X' the usual symmetric projection matrix.

  *** this time, do this using the qr() function ***

   2) Read in the Cork Data from the file 'cork.dat' in the course
     'rfiles' directory.  My old-fashioned code is
       Cork <- t( matrix( scan("cork.dat"), 4, 28) )
     which will make a 28x4 matrix.
   3) Compute the sample covariance matrix using var(Cork)
     and call this matrix V
  *4) Partition the matrix V into the first two and last two rows/
     columns and compute V11 - V12*inv(V22)*V21
  *** For Homework #2, I wanted you to use chol() and forwardsolve(),
      (and that's all!) so redo (4).
  *** this time, do this using either qr() function or sweep ***
     (you can use the original matrix 'Cork' instead of V)

  *5)  (Variation on 5.31)
     a) Write the ridge regression estimator as the solution to a 
     linear least squares problem by adding observations.
     b) In the 'rfiles' directory find a file 'longley.dat' with 
     the response (y) in the first column and six explanatory variables
     in the other columns.  Include an intercept in your model and
     compute the ridge regression estimates for two different values
     of lambda using two methods -- one orthogonalization method and
     one that is not (e.g. Cholesky, sweep).

  *6) Another variation on Cholesky starts at the lower right hand
     corner and constructs an upper triangular matrix R such that
     a positive definite matrix A can be factored as A = RR'.  
     Show the algebra for the induction step and apply this method
     to the 3x3 matrix below:
           9  2 -2
           2  1  0
          -2  0  4