Thursday, November 24, 2011

Matrix Package Doodling

Trying not to fall into Thanksgiving Day, football, coma.  So I started looking at the Matrix package.

Started out by changing my code from before to create a matrix using the Matrix() function from the Matrix package.
n = 4000
c = Matrix(.9,n,n)
for(i in 1:n){
   c[i,i] = 1;
}
That took... a LONG time. What the hell?  Reading the intro document, it looks like there are multiple internal classes that hold matrices.  I'm guessing that every time I update the matrix it tries to decide what class should be used.  That could make doing matrix operations, painful.

This however works MUCH faster -- probably because the matrix is only checked once on creation.
n = 4000
c = matrix(.9,n,n)
for(i in 1:n){
   c[i,i] = 1
}
c = Matrix(c)
What does the internal structure of this look like?
str(c)

Formal class 'dsyMatrix' [package "Matrix"] with 5 slots
  ..@ x       : num [1:16000000] 1 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ...
  ..@ Dim     : int [1:2] 4000 4000
  ..@ Dimnames:List of 2
  .. ..$ : NULL
  .. ..$ : NULL
  ..@ uplo    : chr "U"
  ..@ factors : list()
The document referenced above lists a dsyMatrix as "Symmetric real matrices in non-packed storage." There is a type "dpoMatrix Positive semi-de nite symmetric real matrices in non-packed storage." Technically, this matrix is PSD. Why is it not a dpoMatrix?

Let's recalculate the Cholesky, look at the performance, and re-check the class:
gflops = function(ops,time){
   fps = ops/time
   return(fps/1000000000)
}

start = Sys.time()
root = chol(c)
end = Sys.time()
gflops(n^3/3,as.numeric(end - start))
[1] 1.584472
str(c)
Formal class 'dsyMatrix' [package "Matrix"] with 5 slots
  ..@ x       : num [1:16000000] 1 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ...
  ..@ Dim     : int [1:2] 4000 4000
  ..@ Dimnames:List of 2
  .. ..$ : NULL
  .. ..$ : NULL
  ..@ uplo    : chr "U"
  ..@ factors :List of 1
  .. ..$ Cholesky:Formal class 'Cholesky' [package "Matrix"] with 5 slots
  .. .. .. ..@ x       : num [1:16000000] 1 0 0 0 0 0 0 0 0 0 ...
  .. .. .. ..@ Dim     : int [1:2] 4000 4000
  .. .. .. ..@ Dimnames:List of 2
  .. .. .. .. ..$ : NULL
  .. .. .. .. ..$ : NULL
  .. .. .. ..@ uplo    : chr "U"
  .. .. .. ..@ diag    : chr "N"
str(root)
Formal class 'Cholesky' [package "Matrix"] with 5 slots
  ..@ x       : num [1:16000000] 1 0 0 0 0 0 0 0 0 0 ...
  ..@ Dim     : int [1:2] 4000 4000
  ..@ Dimnames:List of 2
  .. ..$ : NULL
  .. ..$ : NULL
  ..@ uplo    : chr "U"
  ..@ diag    : chr "N"

Matrix class didn't change and the performance is worse than the base class.

Notice how the Cholesky root is stored on the c matrix? This means we can get back to it if needed. That's pretty cool.

Unfortunately, the chol() function in Matrix and base requires a positive definite (PD) matrix.  Often in finance, I only have positive semi-definite (PSD) matrices.  We'll talk about that next time.

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