R: Summing up neighboring matrix elements. How to speed up? - r

Question

I'm working with large matrices of about 2500x2500x50 (lonxlatxtime). The matrix contains only 1 and 0. I need to know for each timestep the sum of the 24 surrounding elements. So far I did it about this way:
xdim <- 2500
ydim <- 2500
tdim <- 50
a <- array(0:1,dim=c(xdim,ydim,tdim))
res <- array(0:1,dim=c(xdim,ydim,tdim))
for (t in 1:tdim){
for (x in 3:(xdim-2)){
for (y in 3:(ydim-2)){
res[x,y,t] <- sum(a[(x-2):(x+2),(y-2):(y+2),t])
}
}
}
This works, but it is much too slow for my needs. Has anybody please an advice how to speed up?

Answer 1

Intro
I have to say, there are so many hidden things behind just the setup of the arrays. The remainder of the problem is trivial though. As a result, there are two ways to go about it really:
Bruteforce given by #Alex (written in C++)
Observing replication patterns
Bruteforce with OpenMP
If we want to 'brute force' it, then we can use the suggestion given by #Alex to employ OpenMP with Armadillo
#include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]
// Add a flag to enable OpenMP at compile time
// [[Rcpp::plugins(openmp)]]
// Protect against compilers without OpenMP
#ifdef _OPENMP
#include <omp.h>
#endif
// [[Rcpp::export]]
arma::cube cube_parallel(arma::cube a, arma::cube res, int cores = 1) {
// Extract the different dimensions
unsigned int tdim = res.n_slices;
unsigned int xdim = res.n_rows;
unsigned int ydim = res.n_cols;
// Same calculation loop
#pragma omp parallel for num_threads(cores)
for (unsigned int t = 0; t < tdim; t++){
// pop the T
arma::mat temp_mat = a.slice(t);
// Subset the rows
for (unsigned int x = 2; x < xdim-2; x++){
arma::mat temp_row_sub = temp_mat.rows(x-2, x+2);
// Iterate over the columns with unit accumulative sum
for (unsigned int y = 2; y < ydim-2; y++){
res(x,y,t) = accu(temp_row_sub.cols(y-2,y+2));
}
}
}
return res;
}
Replication Patterns
However, the smarter approach is understanding how the array(0:1, dims) is being constructed.
Most notably:
Case 1: If xdim is even, then only the rows of a matrix alternate.
Case 2: If xdim is odd and ydim is odd, then rows alternate as well as the matrices alternate.
Case 3: If xdim is odd and ydim is even, then only the rows alternate
Examples
Let's see the cases in action to observe the patterns.
Case 1:
xdim <- 2
ydim <- 3
tdim <- 2
a <- array(0:1,dim=c(xdim,ydim,tdim))
Output:
, , 1
[,1] [,2] [,3]
[1,] 0 0 0
[2,] 1 1 1
, , 2
[,1] [,2] [,3]
[1,] 0 0 0
[2,] 1 1 1
Case 2:
xdim <- 3
ydim <- 3
tdim <- 3
a <- array(0:1,dim=c(xdim,ydim,tdim))
Output:
, , 1
[,1] [,2] [,3]
[1,] 0 1 0
[2,] 1 0 1
[3,] 0 1 0
, , 2
[,1] [,2] [,3]
[1,] 1 0 1
[2,] 0 1 0
[3,] 1 0 1
, , 3
[,1] [,2] [,3]
[1,] 0 1 0
[2,] 1 0 1
[3,] 0 1 0
Case 3:
xdim <- 3
ydim <- 4
tdim <- 2
a <- array(0:1,dim=c(xdim,ydim,tdim))
Output:
, , 1
[,1] [,2] [,3] [,4]
[1,] 0 1 0 1
[2,] 1 0 1 0
[3,] 0 1 0 1
, , 2
[,1] [,2] [,3] [,4]
[1,] 0 1 0 1
[2,] 1 0 1 0
[3,] 0 1 0 1
Pattern Hacking
Alrighty, based on the above discussion, we opt to make a bit of code the exploits this unique pattern.
Creating Alternating Vectors
An alternating vector in this case switches between two different values.
#include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]
// ------- Make Alternating Vectors
arma::vec odd_vec(unsigned int xdim){
// make a temporary vector to create alternating 0-1 effect by row.
arma::vec temp_vec(xdim);
// Alternating vector (anyone have a better solution? )
for (unsigned int i = 0; i < xdim; i++) {
temp_vec(i) = (i % 2 ? 0 : 1);
}
return temp_vec;
}
arma::vec even_vec(unsigned int xdim){
// make a temporary vector to create alternating 0-1 effect by row.
arma::vec temp_vec(xdim);
// Alternating vector (anyone have a better solution? )
for (unsigned int i = 0; i < xdim; i++) {
temp_vec(i) = (i % 2 ? 1 : 0); // changed
}
return temp_vec;
}
Creating the three cases of matrix
As mentioned above, there are three cases of matrix. The even, first odd, and second odd cases.
// --- Handle the different cases
// [[Rcpp::export]]
arma::mat make_even_matrix(unsigned int xdim, unsigned int ydim){
arma::mat temp_mat(xdim,ydim);
temp_mat.each_col() = even_vec(xdim);
return temp_mat;
}
// xdim is odd and ydim is even
// [[Rcpp::export]]
arma::mat make_odd_matrix_case1(unsigned int xdim, unsigned int ydim){
arma::mat temp_mat(xdim,ydim);
arma::vec e_vec = even_vec(xdim);
arma::vec o_vec = odd_vec(xdim);
// Alternating column
for (unsigned int i = 0; i < ydim; i++) {
temp_mat.col(i) = (i % 2 ? o_vec : e_vec);
}
return temp_mat;
}
// xdim is odd and ydim is odd
// [[Rcpp::export]]
arma::mat make_odd_matrix_case2(unsigned int xdim, unsigned int ydim){
arma::mat temp_mat(xdim,ydim);
arma::vec e_vec = even_vec(xdim);
arma::vec o_vec = odd_vec(xdim);
// Alternating column
for (unsigned int i = 0; i < ydim; i++) {
temp_mat.col(i) = (i % 2 ? e_vec : o_vec); // slight change
}
return temp_mat;
}
Calculation Engine
Same as the previous solution, just without the t as we no longer need to repeat calculations.
// --- Calculation engine
// [[Rcpp::export]]
arma::mat calc_matrix(arma::mat temp_mat){
unsigned int xdim = temp_mat.n_rows;
unsigned int ydim = temp_mat.n_cols;
arma::mat res = temp_mat;
// Subset the rows
for (unsigned int x = 2; x < xdim-2; x++){
arma::mat temp_row_sub = temp_mat.rows(x-2, x+2);
// Iterate over the columns with unit accumulative sum
for (unsigned int y = 2; y < ydim-2; y++){
res(x,y) = accu(temp_row_sub.cols(y-2,y+2));
}
}
return res;
}
Call Main Function
Here is the core function that pieces everything together. This gives us the desired distance arrays.
// --- Main Engine
// Create the desired cube information
// [[Rcpp::export]]
arma::cube dim_to_cube(unsigned int xdim = 4, unsigned int ydim = 4, unsigned int tdim = 3) {
// Initialize values in A
arma::cube res(xdim,ydim,tdim);
if(xdim % 2 == 0){
res.each_slice() = calc_matrix(make_even_matrix(xdim, ydim));
}else{
if(ydim % 2 == 0){
res.each_slice() = calc_matrix(make_odd_matrix_case1(xdim, ydim));
}else{
arma::mat first_odd_mat = calc_matrix(make_odd_matrix_case1(xdim, ydim));
arma::mat sec_odd_mat = calc_matrix(make_odd_matrix_case2(xdim, ydim));
for(unsigned int t = 0; t < tdim; t++){
res.slice(t) = (t % 2 ? sec_odd_mat : first_odd_mat);
}
}
}
return res;
}
Timing
Now, the real truth is how well does this perform:
Unit: microseconds
expr min lq mean median uq max neval
r_1core 3538.022 3825.8105 4301.84107 3957.3765 4043.0085 16856.865 100
alex_1core 2790.515 2984.7180 3461.11021 3076.9265 3189.7890 15371.406 100
cpp_1core 174.508 180.7190 197.29728 194.1480 204.8875 338.510 100
cpp_2core 111.960 116.0040 126.34508 122.7375 136.2285 162.279 100
cpp_3core 81.619 88.4485 104.54602 94.8735 108.5515 204.979 100
cpp_cache 40.637 44.3440 55.08915 52.1030 60.2290 302.306 100
Script used for timing:
cpp_parallel = cube_parallel(a,res, 1)
alex_1core = alex(a,res,xdim,ydim,tdim)
cpp_cache = dim_to_cube(xdim,ydim,tdim)
op_answer = cube_r(a,res,xdim,ydim,tdim)
all.equal(cpp_parallel, op_answer)
all.equal(cpp_cache, op_answer)
all.equal(alex_1core, op_answer)
xdim <- 20
ydim <- 20
tdim <- 5
a <- array(0:1,dim=c(xdim,ydim,tdim))
res <- array(0:1,dim=c(xdim,ydim,tdim))
ga = microbenchmark::microbenchmark(r_1core = cube_r(a,res,xdim,ydim,tdim),
alex_1core = alex(a,res,xdim,ydim,tdim),
cpp_1core = cube_parallel(a,res, 1),
cpp_2core = cube_parallel(a,res, 2),
cpp_3core = cube_parallel(a,res, 3),
cpp_cache = dim_to_cube(xdim,ydim,tdim))

Answer 2

Here's one solution that's fast for a large array:
res <- apply(a, 3, function(a) t(filter(t(filter(a, rep(1, 5), circular=TRUE)), rep(1, 5), circular=TRUE)))
dim(res) <- c(xdim, ydim, tdim)
I filtered the array using rep(1,5) as the weights (i.e. sum values within a neighborhood of 2) along each dimension. I then modified the dim attribute since it initially comes out as a matrix.
Note that this wraps the sum around at the edges of the array (which might make sense since you're looking at latitude and longitude; if not, I can modify my answer).
For a concrete example:
xdim <- 500
ydim <- 500
tdim <- 15
a <- array(0:1,dim=c(xdim,ydim,tdim))
and here's what you're currently using (with NAs at the edges) and how long this example takes on my laptop:
f1 <- function(a, xdim, ydim, tdim){
res <- array(NA_integer_,dim=c(xdim,ydim,tdim))
for (t in 1:tdim){
for (x in 3:(xdim-2)){
for (y in 3:(ydim-2)){
res[x,y,t] <- sum(a[(x-2):(x+2),(y-2):(y+2),t])
}
}
}
return(res)
}
system.time(res1 <- f1(a, xdim, ydim, tdim))
# user system elapsed
# 14.813 0.005 14.819
And here's a comparison with the version I described:
f2 <- function(a, xdim, ydim, tdim){
res <- apply(a, 3, function(a) t(filter(t(filter(a, rep(1, 5), circular=TRUE)), rep(1, 5), circular=TRUE)))
dim(res) <- c(xdim, ydim, tdim)
return(res)
}
system.time(res2 <- f2(a, xdim, ydim, tdim))
# user system elapsed
# 1.188 0.047 1.236
You can see there's a significant speed boost (for large arrays). And to check that it's giving the correct solution (note that I'm adding NAs so both results match, since the one I gave filters in a circular manner):
## Match NAs
res2NA <- ifelse(is.na(res1), NA, res2)
all.equal(res2NA, res1)
# [1] TRUE
I'll add that your full array (2500x2500x50) took just under a minute (about 55 seconds), although it did use a lot of memory in the process, FYI.

Answer 3

Your current code has a lot of overhead from redundant subsetting and calculation. Clean this up if you want better speed.
At xdim <- ydim <- 20; tdim <- 5, I see a 23% speedup on my machine.
At xdim <- ydim <- 200; tdim <- 10, I see a 25% speedup.
This comes at small cost of additional memory, which is obvious by examining the code below.
xdim <- ydim <- 20; tdim <- 5
a <- array(0:1,dim=c(xdim,ydim,tdim))
res <- array(0:1,dim=c(xdim,ydim,tdim))
microbenchmark(op= {
for (t in 1:tdim){
for (x in 3:(xdim-2)){
for (y in 3:(ydim-2)){
res[x,y,t] <- sum(a[(x-2):(x+2),(y-2):(y+2),t])
}
}
}
},
alex= {
for (t in 1:tdim){
temp <- a[,,t]
for (x in 3:(xdim-2)){
temp2 <- temp[(x-2):(x+2),]
for (y in 3:(ydim-2)){
res[x,y,t] <- sum(temp2[,(y-2):(y+2)])
}
}
}
}, times = 50)
Unit: milliseconds
expr min lq mean median uq max neval cld
op 4.855827 5.134845 5.474327 5.321681 5.626738 7.463923 50 b
alex 3.720368 3.915756 4.213355 4.012120 4.348729 6.320481 50 a
Further improvements:
If you write this in C++, my guess is that recognizing res[x,y,t] = res[x,y-1,t] - sum(a[...,y-2,...]) + sum(a[...,y+2,...]) will save you additional time. In R, it did not in my timing tests.
This problem is also embarrassingly parallel. There's no reason you couldn't split the t dimension to make more use of a multi-core architecture.
Both of these are left to the reader / OP.