ST758 -- Fall 2011  

  Homework #3 -- due Thursday, 28 September 2011
  (** just turn in 5.13, and 1, 4, 5 below) 

  *5.13 *modified*  Compare either the algorithm in 5.11 (code
     with a loop) or (2.5.1) 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.

  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.  Use 'readtable' or my old-fashioned code 
       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).