DESCRIPTION

Students:  Please keep in mind the OMSI rules.  Save your files often,
make sure OMSI fills your entire screen at all times, etc.  Remember
that clicking CopyQtoA will copy the entire question box to the answer
box.

In questions involving code which will PARTIALLY be given to you in the
question specs, you may need add new lines.  There may not be
information given as to where the lines should be inserted.

MAKE SURE TO RUN THE CODE IN PROBLEMS THAT INVOLVE CODE!

QUESTION

(Text answer, 25 points.)  In the past, it had always been thought
that the graph of MAPE or MSPE would be U-shaped, but recently it has
been observed that two U-shaped curve segments may occur.  What is
the name of this phenomenon?

QUESTION

(Text answer, 25 points) Suppose the pair (X,Y) is uniformly
distributed on a certain 2-dimensional geometric figure.  Possible
figures are square, rectangle, diamond, triangle, circle, and ellipse,
with the following properties:

* the square has corners (1,-1), (1,1), (-1,1), (-1,-1)

* the rectangle has corners (1,0), (1,1), (-1,1), (-1,0)

* the diamond has corners (1,0), (0,1), (-1,0), (0,-1)

* the major and minor axes of the ellipse are parallel to the horizontal
and vertical axes, respectively, and have lengths 3 and 2, respectively,
and the ellipse's center is at (0,0)

* the triangle has corners at (-1,0), (1,0) and (0,1).  

For each figure state whether X and Y will be uncorrelated, and whether
X and Y will be independent.  Your answer MUST in the following format,
stating which figures have uncorrelated (X,Y) and which have independent
(X,Y).  For instance, if you think the ellipse and triangle give 0
correlation, while the square, rectangle, triangle and circle give
independence, answer this way:

0 correlation:  ellipse, triangle
independence:  square, rectangle, triangle and circle

QUESTION -ext .R -run 'Rscript ./omsi_answer3.R'

(R code answer, 25 points)  One of the problems with polynomial
regression is that the various powers of a feature become highly
correlated with each other.  The code below is intended to illustrate
this.  Fill in the blanks.

polycorr <- function(x,deg)  # look at powers 2,3,...,deg
{
   d <- matrix(x,ncol=1)
   for (      ) {
      d <- cbind(        )
      crr <- cor(d)
      crr <- crr[row(crr) != col(crr)]  # trick to get nondiag elements
      print(max(crr))
      print(min(crr))
   }
}

library(regtools)
data(mlb)
polycorr(mlb$Age,6)

QUESTION -ext .R -run 'Rscript ./omsi_answer4.R'

(R code answer, 25 points.)  The R built-in function eigen() finds
eigenvalues and eigenvectors of a matrix, storing them in the components
$values and $vectors, respectively.  In the latter, each eigenvector is
in one column.  Write your own "homegrown" alternative to prcomp(),
myPCA(data), which will return an R list with components $sdev and
$rotation, as with prcomp(). 

myPCA <- function(x) 
{


}

library(regtools)
data(mlb)
mlbx <- mlb[,c('Height','Weight','Age')] 
myPCA(mlbx)
pcout <- prcomp(mlbx)
pcout$sdev
pcout$rotation