---
title: "Basic 'cpp11armadillo' usage"
output: rmarkdown::html_vignette
bibliography: "references.bib"
vignette: >
  %\VignetteIndexEntry{Basic 'cpp11armadillo' usage}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

# Ordinary Least Squares Examples

The Ordinary Least Squares (OLS) estimator is $\hat{\beta} = (X^tX)^{-1}(X^tY)$
for a design matrix $X$ and an outcome vector $Y$ [@Hansen2022].

The following code shows how to compute the OLS estimator using Armadillo and
sending data from R to C++ and viceversa using `cpp11` and `cpp11armadillo`
[@Sanderson2016]:

```cpp
#include <cpp11.hpp>
#include <cpp11armadillo.hpp>

using namespace arma;
using namespace cpp11;

[[cpp11::register]] doubles_matrix<> ols_mat_(const doubles_matrix<>& x) {
  Mat<double> Y = as_Mat(x); // convert from R to C++
  Mat<double> Yinv = inv(Y); // Y^(-1)
  return as_doubles_matrix(Yinv); // convert from C++ to R
}
```

The previous code includes the `cpp11` and `cpp11armadillo` libraries
(cpp11armadillo calls Armadillo) to allow interfacing C++ with R. It also loads
the corresponding namespaces in order to simplify the notation (i.e., using
`Mat` instead of `arma::Mat`), and the function `as_Mat()` and
`as_doubles_mat()` are provided by `cpp11armadillo` to pass a `matrix` object
from R to C++ and that Armadillo can read and then pass it back to R.

The use of `const` and `&` are specific to the C++ language and allow to pass
data from R to C++ without copying the data, and therefore saving time and
memory.

`cpp11armadillo` provides flexibility and in the case of the resulting vector
of OLS coefficients, it can be returned as a matrix or a vector. The following
code shows how to create three functions to compute the OLS estimator and return
the result as a matrix or a vector avoiding repeated code:

```cpp
Mat<double> ols_(const doubles_matrix<>& y, const doubles_matrix<>& x) {
  Mat<double> Y = as_Mat(y);  // Col<double> Y = as_Col(y); also works
  Mat<double> X = as_Mat(x);

  Mat<double> XtX = X.t() * X;             // X'X
  Mat<double> XtX_inv = inv(XtX);          // (X'X)^(-1)
  Mat<double> beta = XtX_inv * X.t() * Y;  // (X'X)^(-1)(X'Y)

  return beta;
}

[[cpp11::register]] doubles_matrix<> ols_mat_(const doubles_matrix<>& y,
                                              const doubles_matrix<>& x) {
  Mat<double> beta = ols_(y, x);
  return as_doubles_matrix(beta);
}

[[cpp11::register]] doubles ols_dbl_(const doubles_matrix<>& y,
                                     const doubles_matrix<>& x) {
  Mat<double> beta = ols_(y, x);
  return as_doubles(beta);
}
```

In the previous code, the `ols_mat_()` function receives inputs from R and calls
`ols_()` to do the computation on C++ side, and `ols_dbl_()` does the same but
it returns a vector instead of a matrix.

# Eigenvalues benchmark

A proper benchmark is to compute eigenvalues for large matrices. Both
`cpp11armadillo` and `RcppArmadillo` use Armadillo as a backend, and the
marginal observed differences are because of how cpp11 and Rcpp pass data from
R to C++ and viceversa. The computation times are identical.

|Input     | Median time cpp11armadillo | Median time RcppArmadillo |
|:---------|---------------------------:|--------------------------:|
|500x500   |                     35.07ms|                     36.4ms|
|1000x1000 |                    260.28ms|                   263.21ms|
|1500x1500 |                    874.62ms|                   857.31ms|
|2000x2000 |                       2.21s|                      2.21s|

|Input     | Memory allocation cpp11armadillo | Memory allocation RcppArmadillo |
|:---------|---------------------------------:|--------------------------------:|
|500x500   |                            17.1KB|                           4.62MB|
|1000x1000 |                              21KB|                           4.62MB|
|1500x1500 |                            24.9KB|                           4.63MB|
|2000x2000 |                            28.8KB|                           4.63MB|

The `cpp11armadillo` computation was obtained with the following function:

```cpp
[[cpp11::register]] doubles_matrix<> eigen_sym_mat(const doubles_matrix<>& x) {
  Mat<double> X = as_Mat(x);
  Mat<double> y = eig_sym(X);
  return as_doubles_matrix(y);
}
```

The `RcppArmadillo` computation was obtained with the following function:

```cpp
#include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]

using namespace Rcpp;

// [[Rcpp::export]]
arma::mat eigen_sym_mat(const arma::mat& x) {
  arma::mat y = eig_sym(x);
  return y;
}
```

In order to get the `RcppArmadillo` function to work, we had to dedicate time
to search online about the error `function 'enterRNGScope' not provided by
package 'Rcpp'`, which required to include `// [[Rcpp::depends(RcppArmadillo)]]`
for the function to work.

# Additional Examples

The package repository includes the directory `cpp11armadillotest`, which
contains an R package that uses Armadillo, and that provides additional examples
for eigenvalues, Cholesky and QR decomposition, and linear models.

# References