### Faster creation of binomial matrices

Scott Chamberlain blogged about faster creation of binomial matrices the other day, and even referred to our RcppArmadillo package as a possible solution (though claiming he didn't get it to work, tst tst -- that is what the rcpp-devel list is here to help with).

The post also fell short of a good aggregated timing comparison for which we love the rbenchmark package. So in order to rectify this, and to see what we can do here with Rcpp, a quick post revisiting the issue.

As preliminaries, we need to load three packages: inline to create compiled code on the fly (which, I should mention, is also used together with Rcpp by the Stan / RStan MCMC sampler which is creating some buzz this week), the compiler package included with R to create byte-compiled code and lastly the aforementioned rbenchmark package to do the timings. We also set row and column dimension, and set them a little higher than the original example to actually have something measurable:

```library(inline)
library(compiler)
library(rbenchmark)

n <- 500
k <- 100
```
The first suggestion was the one by Scott himself. We will wrap this one, and all the following ones, in a function so that all approaches are comparable as being in a function of two dimension arguments:
```scott <- function(N, K) {
mm <- matrix(0, N, K)
apply(mm, c(1, 2), function(x) sample(c(0, 1), 1))
}
scottComp <- cmpfun(scott)
```
We also immediatly compute a byte-compiled version (just because we now can) to see if this helps at all with the code. As there are no (explicit !) loops, we do not expect a big pickup. Scott's function works, but sweeps the `sample()` function across all rows and columns which is probably going to be (relatively) expensive.

Next is the first improvement suggested to Scott which came from Ted Hart.

```ted <- function(N, K) {
matrix(rbinom(N * K, 1, 0.5), ncol = K, nrow = N)
}
```
This is quite a bit smarter as it vectorises the approach, generating N times K elements at once which are then reshaped into a matrix.

Another suggestion came from David Smith as well as Rafael Maia. We rewrite it slightly to make it a function with two arguments for the desired dimensions:

```david <- function(m, n) {
matrix(sample(0:1, m * n, replace = TRUE), m, n)
}
```
This is very clever as it uses `sample()` over zero and one rather than making (expensive) draws from random number generator.

Next we have a version from Luis Apiolaza:

```luis <- function(m, n) {
round(matrix(runif(m * n), m, n))
}
```
It draws from a random uniform and rounds to zero and one, rather than deploying the binomial.

Then we have the version using RcppArmadillo hinted at by Scott, but with actual arguments and a correction for row/column dimensions. Thanks to inline we can write the C++ code as an R character string; inline takes care of everything and we end up with C++-based solution directly callable from R:

```arma <- cxxfunction(signature(ns="integer", ks="integer"), plugin = "RcppArmadillo", body='
int n = Rcpp::as<int>(ns);
int k = Rcpp::as<int>(ks);
return wrap(arma::randu(n, k));
')
```
This works, and is pretty fast. The only problem is that it answers the wrong question as it returns U(0,1) draws and not binomials. We need to truncate or round. So a corrected version is
```armaFloor <- cxxfunction(signature(ns="integer", ks="integer"), plugin = "RcppArmadillo", body='
int n = Rcpp::as<int>(ns);
int k = Rcpp::as<int>(ks);
return wrap(arma::floor(arma::randu(n, k) + 0.5));
')
```
which uses the the old rounding approximation of adding 1/2 before truncating.

With Armadillo in the picture, we do wonder how Rcpp sugar would do. Rcpp sugar, described in one of the eight vignettes of the Rcpp package, is using template meta-programming to provide R-like expressiveness (aka "syntactic sugar") at the C++ level. In particular, it gives access to R's RNG functions using the exact same RNGs as R making the results directly substitutable (whereas Armadillo uses its own RNG).

```sugar <- cxxfunction(signature(ns="integer", ks="integer"), plugin = "Rcpp", body='
int n = Rcpp::as<int>(ns);
int k = Rcpp::as<int>(ks);
Rcpp::RNGScope tmp;
Rcpp::NumericVector draws = Rcpp::runif(n*k);
return Rcpp::NumericMatrix(n, k, draws.begin());
')
```
Here `Rcpp::RNGScope` deals with setting/resetting the R RNG state. This draws a vector of N time K uniforms similar to Luis' function -- and just like Luis' R function does so without looping -- and then shapes a matrix of dimension N by K from it.

And it does of course have the same problem as the RcppArmadillo approach earlier and we can use the same solution:

```sugarFloor <- cxxfunction(signature(ns="integer", ks="integer"), plugin = "Rcpp", body='
int n = Rcpp::as<int>(ns);
int k = Rcpp::as<int>(ks);
Rcpp::RNGScope tmp;
Rcpp::NumericVector draws = Rcpp::floor(Rcpp::runif(n*k)+0.5);
return Rcpp::NumericMatrix(n, k, draws.begin());
')
```

Now that we have all the pieces in place, we can compare:

```res <- benchmark(scott(n, k), scottComp(n,k),
ted(n, k), david(n, k), luis(n, k),
arma(n, k), sugar(n,k),
armaFloor(n, k), sugarFloor(n, k),
order="relative", replications=100)
print(res[,1:4])
```
With all the above code example in a small R script we call via littler, we get
```edd@max:~/svn/rcpp/pkg\$ r /tmp/scott.r
test replications elapsed   relative
7      sugar(n, k)          100   0.072   1.000000
9 sugarFloor(n, k)          100   0.088   1.222222
6       arma(n, k)          100   0.126   1.750000
4      david(n, k)          100   0.136   1.888889
8  armaFloor(n, k)          100   0.138   1.916667
3        ted(n, k)          100   0.384   5.333333
5       luis(n, k)          100   0.410   5.694444
1      scott(n, k)          100  33.045 458.958333
2  scottComp(n, k)          100  33.767 468.986111
```
We can see several takeaways:
• Rcpp sugar wins, which is something we have seen in previous posts on this blog. One hundred replication take only 72 milliseconds (or 88 in the corrected version) --- less than one millisecond per matrix creation.
• RcppArmadillo does well too, and I presume that the small difference is due not to code in Armadillo but the fact that we need one more 'mapping' of data types on the way back to R
• The `sample()` idea by David and Rafael is very, very fast too. This proves once again that well-written R code can be competitive. It also suggest how to make the C++ solution by foregoing (expensive) RNG draws in favour of sampling
• The approaches by Ted and Luis are also pretty good. In practice, the are probably good enough.
• Scott's function is not looking so hot (particularly as we increased the problem dimensions) and byte-compilation does not help at all.
Thanks to Scott and everybody for suggesting this interesting problem. Trying the `rbinom()` Rcpp sugar function, or implementing `sample()` at the C++ level is, as the saying goes, left as an exercise to the reader.