ST790R -- Fall 2010

  Homework #3 -- due Tuesday, 28 September 2010

  Exercises in Textbook: (*just turn in these)
  *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 (2^k) + i, for i=1, ...,32 
     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:

  1) Let y = (4.82, 4.88, 4.70, 4.72, 4.88, 4.66, 4.94, 4.80)' and let
     x = (38, 40, 42, 44, 36, 42, 38, 40)' 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 Ramus Data from the file 'ramus.dat' in the course
     'rfiles' directory.  My old-fashioned code is
       Ramus <- t( matrix( scan("ramus.dat"), 4, 20) )
     which will make a 20x4 matrix.
   3) Compute the sample covariance matrix using var(Ramus)
     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 'Ramus' instead of V)

  *5)(similar to *New *1* in 'Update' for Chapter 5)
     In the file "uspop.dat" are values of the US population for 
     the years 1948-1971.  The first column is the year and the 
     second column is the population in millions.  Compute the
     condition number for a quadratic regression (three columns:
     intecept, linear, quadratic).  Center the year either by 1900
     or by its mean and recompute the condition number.