Avoid two for loops in R - r

Question

I have a R code that can do convolution of two functions...
convolveSlow <- function(x, y) {
nx <- length(x); ny <- length(y)
xy <- numeric(nx + ny - 1)
for(i in seq(length = nx)) {
xi <- x[[i]]
for(j in seq(length = ny)) {
ij <- i+j-1
xy[[ij]] <- xy[[ij]] + xi * y[[j]]
}
}
xy
}
Is there a way to remove the two for loops and make the code run faster?
Thank you
San

Answer 1

Since R is very fast at computing vector operations, the most important thing to keep in mind when programming for performance is to vectorise as many of your operations as possible.
This means thinking hard about replacing loops with vector operations. Here is my solution for fast convolution (50 times faster with input vectors of length 1000 each):
convolveFast <- function(x, y) {
nx <- length(x)
ny <- length(y)
xy <- nx + ny - 1
xy <- rep(0, xy)
for(i in (1:nx)){
j <- 1:ny
ij <- i + j - 1
xy[i+(1:ny)-1] <- xy[ij] + x[i] * y
}
xy
}
You will notice that the inner loop (for j in ...) has disappeared. Instead, I replaced it with a vector operation. j is now defined as a vector (j <- 1:ny). Notice also that I refer to the entire vector y, rather than subsetting it (i.e. y instead of y[j]).
j <- 1:ny
ij <- i + j - 1
xy[i+(1:ny)-1] <- xy[ij] + x[i] * y
I wrote a small function to measure peformance:
measure.time <- function(fun1, fun2, ...){
ptm <- proc.time()
x1 <- fun1(...)
time1 <- proc.time() - ptm
ptm <- proc.time()
x2 <- fun2(...)
time2 <- proc.time() - ptm
ident <- all(x1==x2)
cat("Function 1\n")
cat(time1)
cat("\n\nFunction 2\n")
cat(time2)
if(ident) cat("\n\nFunctions return identical results")
}
For two vectors of length 1000 each, I get a 98% performance improvement:
x <- runif(1000)
y <- runif(1000)
measure.time(convolveSlow, convolveFast, x, y)
Function 1
7.07 0 7.59 NA NA
Function 2
0.14 0 0.16 NA NA
Functions return identical results

Answer 2

For vectors, you index with [], not [[]], so use xy[ij] etc
Convolution doesn't vectorise easily but one common trick is to switch to compiled code. The Writing R Extensions manual uses convolution as a running example and shows several alternative; we also use it a lot in the Rcpp documentation.

Answer 3

As Dirk says, compiled code can be a lot faster. I had to do this for one of my projects and was surprised at the speedup: ~40x faster than Andrie's solution.
> a <- runif(10000)
> b <- runif(10000)
> system.time(convolveFast(a, b))
user system elapsed
7.814 0.001 7.818
> system.time(convolveC(a, b))
user system elapsed
0.188 0.000 0.188
I made several attempts to speed this up in R before I decided that using C code couldn't be that bad (note: it really wasn't). All of mine were slower than Andrie's, and were variants on adding up the cross-product appropriately. A rudimentary version can be done in just three lines.
convolveNotAsSlow <- function(x, y) {
xyt <- x %*% t(y)
ds <- row(xyt)+col(xyt)-1
tapply(xyt, ds, sum)
}
This version only helps a little.
> a <- runif(1000)
> b <- runif(1000)
> system.time(convolveSlow(a, b))
user system elapsed
6.167 0.000 6.170
> system.time(convolveNotAsSlow(a, b))
user system elapsed
5.800 0.018 5.820
My best version was this:
convolveFaster <- function(x,y) {
foo <- if (length(x)<length(y)) {y %*% t(x)} else { x %*% t(y) }
foo.d <- dim(foo)
bar <- matrix(0, sum(foo.d)-1, foo.d[2])
bar.rc <- row(bar)-col(bar)
bar[bar.rc>=0 & bar.rc<foo.d[1]]<-foo
rowSums(bar)
}
This was quite a bit better, but still not nearly as fast as Andrie's
> system.time(convolveFaster(a, b))
user system elapsed
0.280 0.038 0.319

Answer 4

The convolveFast function can be optimized a little by carefully using integer math only and replacing (1:ny)-1L with seq.int(0L, ny-1L):
convolveFaster <- function(x, y) {
nx <- length(x)
ny <- length(y)
xy <- nx + ny - 1L
xy <- rep(0L, xy)
for(i in seq_len(nx)){
j <- seq_len(ny)
ij <- i + j - 1L
xy[i+seq.int(0L, ny-1L)] <- xy[ij] + x[i] * y
}
xy
}

Answer 5

How about convolve(x, rev(y), type = "open") in stats?
> x <- runif(1000)
> y <- runif(1000)
> system.time(a <- convolve(x, rev(y), type = "o"))
user system elapsed
0.032 0.000 0.032
> system.time(b <- convolveSlow(x, y))
user system elapsed
11.417 0.060 11.443
> identical(a,b)
[1] FALSE
> all.equal(a,b)
[1] TRUE

Answer 6

Some say the apply() and sapply() functions are faster than for() loops in R. You could convert the convolution to a function and call it from within apply().
However, there is evidence to the contrary
http://yusung.blogspot.com/2008/04/speed-issue-in-r-computing-apply-vs.html