R help - Mar 2005 - two-dimensional integration?

I find the one-dimensional "integrate" very helpful,
but often enough I stumble into problems that require
two (or more)-dimensional integrals. I suppose there
are no R functions that can do this for me, "directly"?

The ideal thing would be to be able to define say
f <- function(x)
{
x1 <- x[1]
x2 <- x[2]
sin(x1*x2)*exp(x1-x2)
}
and then write say
integrate(f, xlim=c(0,1), ylim=c(0,1))  .

(a) No such thing exists, as of today, right?
(b) There *are* general numerical routines "out there"
for doing such things, right? (Importance sampling
or adaptive important sampling would often do the
job, but it would be difficult to find something that
"always" works -- at least in higher dimension?
Also, iterated one-dimensional integrations could
be attempted, but I find that messy, also because
things lose the g(many) = many(g) property, and
then R refuses to integrate g.)
(c) Will a thing like the above exist in R before
the Tromsoe Olympics in 2014? For which dimensions? 

Nils Lid Hjort
[Professor of statistics at Oslo, but currently at Duke]

Nils,

For 2D, see package 'adapt' on CRAN. e.g.

adapt(2, c(0,0), c(1,1), functn=function(x) sin(prod(x))*exp(x[1]-x[2]))

Package `adapt' will do larger numbers of dimensions, but numerical 
quadrature is often no more effective than Monte-Carlo methods in more 
than a few dimensions.  For very smooth functions, quasi-random numbers 
can help.

A good reference aimed at statisticians is

@Book{Evans.Swartz.00,
   author =       {Michael Evans and Tim Swartz},
   title =        {Approximating Integrals via Monte Carlo and
                   Deterministic Methods},
   publisher =    {Oxford University Press},
   year =         2000,
   address =      {Oxford},
   ISBN =         "0-19-850278-8",
}

BTW, we are not good are predicting to 2014, but fairly good at the 
present.  In this case I could not guess a good search term on
http://search.r-project.org, but it often gets you there.  It has a 
`complete' list of packages, as does CRAN, and searching those pages for 
`integrate' works.

Brian

On Thu, 10 Mar 2005, Nils-at-Duke Lid Hjort wrote:
> I find the one-dimensional "integrate" very helpful,
> but often enough I stumble into problems that require
> two (or more)-dimensional integrals. I suppose there
> are no R functions that can do this for me, "directly"?
>
> The ideal thing would be to be able to define say
> f <- function(x)
> {
> x1 <- x[1]
> x2 <- x[2]
> sin(x1*x2)*exp(x1-x2)
> }
> and then write say
> integrate(f, xlim=c(0,1), ylim=c(0,1))  .
>
> (a) No such thing exists, as of today, right?
> (b) There *are* general numerical routines "out there"
> for doing such things, right? (Importance sampling
> or adaptive important sampling would often do the
> job, but it would be difficult to find something that
> "always" works -- at least in higher dimension?
> Also, iterated one-dimensional integrations could
> be attempted, but I find that messy, also because
> things lose the g(many) = many(g) property, and
> then R refuses to integrate g.)
> (c) Will a thing like the above exist in R before
> the Tromsoe Olympics in 2014? For which dimensions? 
> Nils Lid Hjort
> [Professor of statistics at Oslo, but currently at Duke]

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

This existed even before the Athens Olympics 2004 :)

look at package "adapt" which integrates a function from 2 to 20 
dimensions, e.g.,

library(adapt)
f <- function(x){
    x1 <- x[1]
    x2 <- x[2]
    sin(x1*x2)*exp(x1-x2)
}

adapt(2, c(0,0), c(1,1), functn=f)

I hope it helps.

Best,
Dimitris

----
Dimitris Rizopoulos
Ph.D. Student
Biostatistical Centre
School of Public Health
Catholic University of Leuven

Address: Kapucijnenvoer 35, Leuven, Belgium
Tel: +32/16/336899
Fax: +32/16/337015
Web: http://www.med.kuleuven.ac.be/biostat/
     http://www.student.kuleuven.ac.be/~m0390867/dimitris.htm


----- Original Message ----- 
From: "Nils-at-Duke Lid Hjort" <hjort at isds.duke.edu>
To: <R-help at stat.math.ethz.ch>; <nils at math.uio.no>
Sent: Thursday, March 10, 2005 6:22 AM
Subject: [R] two-dimensional integration?

>I find the one-dimensional "integrate" very helpful,
> but often enough I stumble into problems that require
> two (or more)-dimensional integrals. I suppose there
> are no R functions that can do this for me, "directly"?
>
> The ideal thing would be to be able to define say
> f <- function(x)
> {
> x1 <- x[1]
> x2 <- x[2]
> sin(x1*x2)*exp(x1-x2)
> }
> and then write say
> integrate(f, xlim=c(0,1), ylim=c(0,1))  .
>
> (a) No such thing exists, as of today, right?
> (b) There *are* general numerical routines "out there"
> for doing such things, right? (Importance sampling
> or adaptive important sampling would often do the
> job, but it would be difficult to find something that
> "always" works -- at least in higher dimension?
> Also, iterated one-dimensional integrations could
> be attempted, but I find that messy, also because
> things lose the g(many) = many(g) property, and
> then R refuses to integrate g.)
> (c) Will a thing like the above exist in R before
> the Tromsoe Olympics in 2014? For which dimensions?
> Nils Lid Hjort
> [Professor of statistics at Oslo, but currently at Duke]
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! 
> http://www.R-project.org/posting-guide.html
>

Nils-at-Duke Lid Hjort <hjort at isds.duke.edu> writes:
> I find the one-dimensional "integrate" very helpful,
> but often enough I stumble into problems that require
> two (or more)-dimensional integrals. I suppose there
...
> (c) Will a thing like the above exist in R before
> the Tromsoe Olympics in 2014? For which dimensions? Nils Lid Hjort
> [Professor of statistics at Oslo, but currently at Duke]
Did you check the CRAN package "adapt"? It's been around at least
since Nagano 1998, I believe.

-- 
   O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907