R help - Dec 2004 - Re: Help : generating correlation matrix with a particula r

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Martin,

Thank you for letting us know about posdefify.  It does do exactly what the
Rebonato paper recommended and gives the same result as our code, but it
will be much better behaved in the wild than ours will.

BTW Troels Ring [tring at gvdnet.dk]  found the Rebonato paper at
http://www.quarchome.com/correlationmatrix.pdf

Thank you Troels.

Best regards,
Herb

Herbert G. Desson, ACAS, MAAA

Actuary
JLT Risk Solutions
6 Crutched Friars
London EC3N 2PH

phone:  +44 (0)20 7528 4702
fax:       +44 (0)20 7558 3785



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From: Martin Maechler [mailto:maechler at stat.math.ethz.ch]
Sent: 13 December 2004 16:04
To: Herbert_Desson at jltgroup.com
Cc: r-help at stat.math.ethz.ch
Subject: Re: [R] Re: Help : generating correlation matrix with a
particular

>>>>> "Herbert" == Herbert Desson <Herbert_Desson at
jltgroup.com>
>>>>>     on Mon, 13 Dec 2004 12:24:10 -0000 writes:

    Herbert> Here is some code we have used.

    Herbert> a<-array(c(1,.9,.7,.9,1,.3,.7,.3,1),dim=c(3,3))
    Herbert> a
    Herbert> s<-eigen(a)$vectors
    Herbert> l<-diag(eigen(a)$values)
    Herbert> l[l<0]<-0
    Herbert> b<-s%*%sqrt(l)
    Herbert> for(i in 1:nrow(b)){b[i,]<-b[i,]/sqrt(sum(b[i,]^2))}
    Herbert> ap<-b%*%t(b)
    Herbert> ap

This code does the same thing as my (simplistic, but slightly more
general) function posdefify()  in package "sfsmisc" :

  a <- matrix(c(1,.9,.7,.9,1,.3,.7,.3,1), 3)

  install.packages("sfsmisc")
  library(sfsmisc)

  posdefify(a)

gives
          [,1]      [,2]      [,3]
[1,] 1.0000000 0.8940242 0.6963190
[2,] 0.8940242 1.0000000 0.3009691
[3,] 0.6963190 0.3009691 1.0000000


    Herbert> It is based on a paper by Rebonato etal that formerly was at
    Herbert> www.rebonato.com/correlationmatrix.pdf.
    Herbert> Unfortunately the website has disappeared.

The idea is very simple and has been re-invented many times as
far as I know.

More sophisticated methods for "posdefiying" a matrix exist in
other places. Given symmetrix matrix A, they try to find the
matrix Ap, positive definite, such  ||A - Ap||  is minimal.
The eigen-value based simple solution that you've used above
and I've also coded in posdefify(), is not the same one would
get for `usual' matrix norms  || . || 

[[NB:  posdefify() also has a 2nd method the implementation of
       which has an embarassing bug. The next version of
       sfsmisc, due in a day or two, will have it fixed.
]]

Does anyone know of rigorous mathematical results in this
regard?

Martin Maechler, ETH Zurich