[R] matrix of higher order differences
Ravi Varadhan
rvaradhan at jhmi.edu
Wed Apr 27 21:34:09 CEST 2011
My apologies in advance for being a bit off-topic, but I could not quell my curiosity.
What might one do with a matrix of all order finite differences? It seems that such a matrix might be related to the Wronskian (its discrete analogue, perhaps).
http://en.wikipedia.org/wiki/Wronskian
Ravi.
-------------------------------------------------------
Ravi Varadhan, Ph.D.
Assistant Professor,
Division of Geriatric Medicine and Gerontology School of Medicine Johns Hopkins University
Ph. (410) 502-2619
email: rvaradhan at jhmi.edu
-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of Petr Savicky
Sent: Wednesday, April 27, 2011 11:01 AM
To: r-help at r-project.org
Subject: Re: [R] matrix of higher order differences
On Wed, Apr 27, 2011 at 11:25:42AM +0000, Hans W Borchers wrote:
> Jeroen Ooms <jeroenooms <at> gmail.com> writes:
>
> >
> > Is there an easy way to turn a vector of length n into an n by n matrix, in
> > which the diagonal equals the vector, the first off diagonal equals the
> > first order differences, the second... etc. I.e. to do this more
> > efficiently:
> >
> > diffmatrix <- function(x){
> > n <- length(x);
> > M <- diag(x);
> > for(i in 1:(n-1)){
> > differences <- diff(x, dif=i);
> > for(j in 1:length(differences)){
> > M[j,i+j] <- differences[j]
> > }
> > }
> > M[lower.tri(M)] <- t(M)[lower.tri(M)];
> > return(M);
> > }
> >
> > x <- c(1,2,3,5,7,11,13,17,19);
> > diffmatrix(x);
> >
>
> I do not know whether you will call the appended version more elegant,
> but at least it is much faster -- up to ten times for length(x) = 1000,
> i.e. less than 2 secs for generating and filling a 1000-by-1000 matrix.
> I also considered col(), row() indexing:
>
> M[col(M) == row(M) + k] <- x
>
> Surprisingly (for me), this makes it even slower than your version with
> a double 'for' loop.
>
> -- Hans Werner
>
> # ----
> diffmatrix <- function(x){
> n <- length(x)
> if (n == 1) return(x)
>
> M <- diag(x)
> for(i in 1:(n-1)){
> x <- diff(x) # use 'diff' in a loop
> for(j in 1:(n-i)){ # length is known
> M[j, i+j] <- x[j] # and reuse x
> }
> }
> M[lower.tri(M)] <- t(M)[lower.tri(M)]
> return(M)
> }
> # ----
Hi.
The following avoids the inner loop and it was faster
for x of length 100 and 1000.
diffmatrix2 <- function(x){
n <- length(x)
if (n == 1) return(x)
A <- matrix(nrow=n+1, ncol=n)
for(i in 1:n){
A[i, seq.int(along=x)] <- x
x <- diff(x)
}
M <- matrix(A, nrow=n, ncol=n)
M[upper.tri(M)] <- t(M)[upper.tri(M)]
return(M)
}
Reorganizing an (n+1) x n matrix into an n x n matrix
shifts i-th column by (i-1) downwards. In particular,
the first row becomes the main diagonal. The initial
part of each of the remaining rows becomes a diagonal
starting at the first component of the original row.
Petr Savicky.
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