DESCRIPTION

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QUESTION -ext .R -run 'Rscript omsi_answer1.R'

(Code answer.) In our study of Markov chains, the transition matrix P was 
always either given to us, or easily formulated from the physical structure 
of the problem.  In some applications, though, we may need to estimate P 
from data.  

You'll do that here, writing two functions with call forms 

makeStates(x,nc) 
findP(xmc)

Here x are values observed at times 1,2,3,..., with the values typically
being continuous but not necessarily so.

The first function creates states 1,2,...,nc from continuous data x.  It does 
this by breaking the range of x into nc equal intervals.  It is suggested 
that you use R's cut() function for this.

The second function actually estimates P.  Here xmc is the output of
makeStates(), or is x if x is already in state form.

Of course, all this assumes the Markov Property holds, which we will not
be checking.

makeStates <- function(x,nc)
{


}

findP <- function(xmc)
{
   k <- max(xmc)  # number of states
   n1 <- length(xmc) - 1 
   p <- matrix(nrow=k,ncol=k,0)

   for (m in 1:n1) { 


   }

}

# test it!
nmc <- makeStates(Nile,3)  # Nile is built-in dataset in R
nmc
#   [1] 3 3 2 3 3 3 2 3 3 3 2 2 3 2 2 2 3 2 2 3 3 3 3 3 3 3 2
#  [28] 3 2 2 2 1 2 2 1 2 1 2 2 2 2 1 1 2 1 3 3 2 2 2 2 2 2 2
#  [55] 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 1 1 2 2 1 2 2 2 2 2 2 1
#  [82] 1 2 2 2 2 2 2 2 2 2 2 2 3 2 1 2 1 1 1
findP(nmc)
#            [,1]      [,2]       [,3]
#  [1,] 0.2941176 0.6470588 0.05882353
#  [2,] 0.2131148 0.6721311 0.11475410
#  [3,] 0.0000000 0.4285714 0.57142857