R help - Mar 2004 - multiple summation

Hello,
I have to compute a multiple summation (not an integration because the 
independent variables a
are discrete) for all the values of a function of several variables f
(x_1,...,x_n), that is

sum ... sum f(x_1,...,x_n)
x_1     x_n

have you some suggestion? Is it possible?
I know that for multiple integration there is the function adapt, but it has at 
most n=20. In my case n depends on the dimension of the dataset and, hence, it 
could be bigger.

Thank you,
Paolo

Paolo Bulla wrote:
> Hello,
> I have to compute a multiple summation (not an integration because the 
> independent variables a
> are discrete) for all the values of a function of several variables f
> (x_1,...,x_n), that is
> 
> sum ... sum f(x_1,...,x_n)
> x_1     x_n
> 
> have you some suggestion? Is it possible?
> I know that for multiple integration there is the function adapt, but it
has at
> most n=20. In my case n depends on the dimension of the dataset and, hence,
it
> could be bigger.
> 
> Thank you,
> Paolo

I's try to vectorizing f() and apply it on the set of all permutations.

Uwe Ligges

> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html

sum, I guess, as in 
> x <- seq(0,2*pi,length=10)
> y <- x <- seq(0,2*pi,length=10)
> X <- cbind(x,y)
> sum(sapply(1:NROW(X), function(i) sin(X[i,1])*cos(X[i,2])))
[1] 1.991599e-16

There may be more efficient ways to deal with the `sapply' part,
especially if your function is vectorized. 

HTH,
Giovanni
> Date: Tue, 16 Mar 2004 11:12:03 +0100
> From: Paolo Bulla <paolo.bulla at unibocconi.it>
> Sender: r-help-bounces at stat.math.ethz.ch
> Cc: 
> Precedence: list
> User-Agent: Internet Messaging Program (IMP) 3.2.1
> 
> 
> Hello,
> I have to compute a multiple summation (not an integration because the 
> independent variables a
> are discrete) for all the values of a function of several variables f
> (x_1,...,x_n), that is
> 
> sum ... sum f(x_1,...,x_n)
> x_1     x_n
> 
> have you some suggestion? Is it possible?
> I know that for multiple integration there is the function adapt, but it
has at
> most n=20. In my case n depends on the dimension of the dataset and, hence,
it
> could be bigger.
> 
> Thank you,
> Paolo
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html
> 
> 
-- 

 __________________________________________________
[                                                  ]
[ Giovanni Petris                 GPetris at uark.edu ]
[ Department of Mathematical Sciences              ]
[ University of Arkansas - Fayetteville, AR 72701  ]
[ Ph: (479) 575-6324, 575-8630 (fax)               ]
[ http://definetti.uark.edu/~gpetris/              ]
[__________________________________________________]

On 16-Mar-04 Paolo Bulla wrote:
> 
> Hello,
> I have to compute a multiple summation (not an integration
> because the independent variables are discrete) for all
> the values of a function of several variables f(x_1,...,x_n),
> that is
> 
> sum ... sum f(x_1,...,x_n)
> x_1     x_n
> 
> have you some suggestion? Is it possible?
> I know that for multiple integration there is the function adapt,
> but it has at most n=20. In my case n depends on the dimension of the
> dataset and, hence, it could be bigger.
If you can conveniently encapsulate the values of f(x_1,...,x_n)
as an n-dimensional array (which presumes in effect that the ranges
of x_1,...,x_n form a "product space"), then simple 'sum'
should
do it, e.g.

  A<-array(c(1,2,3,4,5,6,7,8,9,10,11,12),dim=c(2,2,3))

  A
  , , 1
       [,1] [,2]
  [1,]    1    3
  [2,]    2    4

  , , 2
       [,1] [,2]
  [1,]    5    7
  [2,]    6    8

  , , 3
       [,1] [,2]
  [1,]    9   11
  [2,]   10   12

  sum(A)
  [1] 78

However, I am concerned about your possibility that 'n' could be
bigger than 20, since this would generate a huge array (at least
8*2^21 = 16,777,216 bytes even with only 2 distinct x_i values per
dimension; with 3 you get  8*3^21 = 83,682,825,624 or 83GB).

While it does work if you can get it into memory and avoid overflow:

  B<-array((1:2^21)/2^22,dim=rep(2,21))
  sum(B)
  [1] 524288.2

you are clearly pushing hard at machine limits here. What sort
of context requires summing over so many dimensions?

An alternative, if 'f' is given by a formula, could be to simply
and crudely iterate over the dimensions, but in its own way that
is even more daunting!

Another alternative, if you don't need a mathematically exact
result, is to estimate the sum by Monte Carlo simulation: sample
the points in each dimension, evaluate the function, accumulate
these mean of these values, and finally scale up appropriately
according to the number of points in the "box" (stop when the SE
of your result is small enough).

Best wishes,
Ted.


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E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
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Date: 16-Mar-04                                       Time: 14:42:34
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