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# Ordinary Least Squares

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 `cpp4r` and `armadillo4r`
[@Sanderson2016]:

```cpp
#include <cpp4r.hpp>
#include <armadillo4r.hpp>

using namespace arma;
using namespace cpp4r;

[[cpp4r::register]] doubles_matrix<> ols_mat_(const doubles_matrix<>& x) {
  // Convert from R to C++
  Mat<double> Y = as_Mat(y);  // Col<double> Y = as_Col(y); also works
  Mat<double> X = as_Mat(x);

  // Armadillo operations
  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)

  // Convert from C++ to R
  return as_doubles_matrix(beta);
}
```

The previous code includes the `cpp4r` and `armadillo4r` libraries
(armadillo4r 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 `armadillo4r` 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.

`armadillo4r` 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;
}

[[cpp4r::register]] doubles_matrix<> ols_mat_(const doubles_matrix<>& y,
                                              const doubles_matrix<>& x) {
  Mat<double> beta = ols_(y, x);
  return as_doubles_matrix(beta);
}

[[cpp4r::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.

# Additional Examples

The package repository includes the directory `armadillo4rtest`, which
contains an R package that uses Armadillo, and that provides additional examples
for eigenvalues, Cholesky and QR decomposition, and linear models.

# References
