Vectorizing double summations using R - r

Question

I am struggling with translating this function into R using via vectorization technique:
Where all I have been able to do so far is this:
c <- matrix(1:9, 3)
z <- 1:3
sum(abs(outer(z, z,"-")) * c)/sum(c)
But I don't think its necessarily correct. I tried a for-loop version but that is too long and my answer is likely wrong anyway. Anyone keen on this? What am I missing (or doing wrong)? Any help would be appreciated.

Answer 1

Here's a double-loop version:
q =
function(z,c){
num = 0
for(i in 1:length(z)){
for(j in 1:length(z)){
num = num + abs(z[i]-z[j]) * c[i,j]
}
}
num/sum(c)
}
Here's your vectorised version, functionised:
q2 =
function(z,c){sum(c*abs(outer(z,z,'-')) /sum(c))}
Not a great difference in timing between them really for a small matrix:
> microbenchmark::microbenchmark(q(z,c), q2(z,c))
Unit: microseconds
expr min lq mean median uq max neval cld
q(z, c) 15.368 15.7505 16.59644 16.0225 16.6290 30.346 100 b
q2(z, c) 12.232 12.8885 13.79178 13.2225 13.6585 44.085 100 a
But for a larger test it's a big win:
> c2 = matrix(runif(100*100),100,100)
> z2 = runif(100)
> microbenchmark::microbenchmark(q(z2,c2), q2(z2,c2))
Unit: microseconds
expr min lq mean median uq max neval cld
q(z2, c2) 7437.031 7588.131 8046.92272 7794.927 8332.104 10729.799 100 b
q2(z2, c2) 74.742 78.647 94.20153 86.113 100.125 188.428 100 a
>
Numeric difference is within floating point tolerance:
> q(z2,c2) - q2(z2,c2)
[1] 6.661338e-16
So unless anyone has faster code, I'd stick with what you've got.

Answer 2

As perfectly explained by #Spacedman, your approach is very efficient, but if you still want to go faster you could try Rcpp :
library(Rcpp)
sourceCpp(code='
#include <Rcpp.h>
// [[Rcpp::export]]
double qRcpp(const Rcpp::NumericVector z, const Rcpp::NumericMatrix cm){
int zlen = z.length();
if(!(zlen == cm.nrow() && cm.nrow() == cm.ncol()))
Rcpp::stop("Invalid sizes");
double num = 0;
for(int i = 0 ; i < zlen ; i++){
for(int j = 0 ; j < zlen ; j++){
num = num + std::abs(z[i]-z[j]) * cm(i,j);
}
}
return num / Rcpp::sum(cm);
}
')
Benchmark :
c2 = matrix(runif(100*100),100,100)
z2 = runif(100)
microbenchmark::microbenchmark(q(z2,c2), q2(z2,c2),qRcpp(z2,c2))
# Unit: microseconds
# expr min lq mean median uq max neval
# q(z2, c2) 10273.035 10976.3050 11680.85554 11348.763 11765.2010 44115.632 100
# q2(z2, c2) 64.292 67.9455 80.56427 75.543 86.3565 244.019 100
# qRcpp(z2, c2) 21.042 21.9180 25.30515 24.256 26.8860 56.403 100