# RcppNumerical: Numerical integration and optimization with Rcpp

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# Introduction

I have seen several conversations in Rcpp-devel mailing list asking how to

compute numerical integration or optimization in Rcpp. While R in fact

has the functions `Rdqags`

, `Rdqagi`

, `nmmin`

, `vmmin`

etc. in its API

to accomplish such tasks, it is not so straightforward to use them with Rcpp.

For my own research projects I need to do a lot of numerical integration,

root finding and optimization, so to make my life a little bit easier, I

just created the RcppNumerical

package that simplifies these procedures. I havenâ€™t submitted `RcppNumerical`

to CRAN, since the API may change quickly according to my needs or the

feedbacks from other people.

Basically `RcppNumerical`

includes a number of open source libraries for

numerical computing, so that Rcpp code can link to this package to use

the functions provided by these libraries. Alternatively, `RcppNumerical`

provides some wrapper functions that have less configuration and fewer

arguments, if you just want to use the default and quickly get the results.

`RcppNumerical`

depends on `Rcpp`

(obviously) and `RcppEigen`

,

- To use
`RcppNumerical`

with`Rcpp::sourceCpp()`

, add the following two lines

to the C++ source file:

```
// [[Rcpp::depends(RcppEigen)]]
// [[Rcpp::depends(RcppNumerical)]]
```

- To use
`RcppNumerical`

in your package, add the corresponding fields to the

`DESCRIPTION`

file:

```
Imports: RcppNumerical
LinkingTo: Rcpp, RcppEigen, RcppNumerical
```

# Numerical Integration

(Picture from Wikipedia)

The numerical integration code contained in `RcppNumerical`

is based

on the NumericalIntegration

library developed by Sreekumar Thaithara Balan,

Mark Sauder, and Matt Beall.

To compute integration of a function, first define a functor inherited from

the `Func`

class:

```
class Func
{
public:
virtual double operator()(const double& x) const = 0;
virtual void operator()(double* x, const int n) const
{
for(int i = 0; i < n; i++)
x[i] = this->operator()(x[i]);
}
};
```

The first function evaluates one point at a time, and the second version

overwrites each point in the array by the corresponding function values.

Only the second function will be used by the integration code, but usually it

is easier to implement the first one.

`RcppNumerical`

provides a wrapper function for the `NumericalIntegration`

library with the following interface:

```
inline double integrate(
const Func& f, const double& lower, const double& upper,
double& err_est, int& err_code,
const int subdiv = 100,
const double& eps_abs = 1e-8, const double& eps_rel = 1e-6,
const Integrator<double>::QuadratureRule rule = Integrator<double>::GaussKronrod41
)
```

See the README page for the

explanation of each argument. Below shows an example that calculates the

moment generating function of a $Beta(a,b)$ distribution,

$M(t) = E(e^{tX})$:

```
// [[Rcpp::depends(RcppEigen)]]
// [[Rcpp::depends(RcppNumerical)]]
#include
```
using namespace Numer;
// M(t) = E(exp(t * X)) = int exp(t * x) * f(x) dx, f(x) is the p.d.f.
class Mintegrand: public Func
{
private:
const double a;
const double b;
const double t;
public:
Mintegrand(double a_, double b_, double t_) : a(a_), b(b_), t(t_) {}
double operator()(const double& x) const
{
return std::exp(t * x) * R::dbeta(x, a, b, 0);
}
};
// [[Rcpp::export]]
double beta_mgf(double t, double a, double b)
{
Mintegrand f(a, b, t);
double err_est;
int err_code;
return integrate(f, 0.0, 1.0, err_est, err_code);
}

We can compile and run this code in R and draw the graph:

```
library(Rcpp)
library(ggplot2)
sourceCpp("somefile.cpp")
t0 = seq(-3, 3, by = 0.1)
mt = sapply(t0, beta_mgf, a = 1, b = 1)
qplot(t0, mt, geom = "line", xlab = "t", ylab = "M(t)",
main = "Moment generating function of Beta(1, 1)")
```

# Numerical Optimization

Currently `RcppNumerical`

contains the L-BFGS algorithm for unconstrained

minimization problems based on the

libLBFGS library

developed by Naoaki Okazaki.

(Picture from aria42.com)

Again, one needs to first define a functor to represent the multivariate

function to be minimized.

```
class MFuncGrad
{
public:
virtual double f_grad(Constvec& x, Refvec grad) = 0;
};
```

Here `Constvec`

represents a read-only vector and `Refvec`

a writable

vector. Their definitions are

```
// Reference to a vector
typedef Eigen::Ref<Eigen::VectorXd> Refvec;
typedef const Eigen::Ref<const Eigen::VectorXd> Constvec;
```

(Basically you can treat `Refvec`

as a `Eigen::VectorXd`

and

`Constvec`

the `const`

version. Using `Eigen::Ref`

is mainly to avoid

memory copy. See the explanation

here.)

The `f_grad()`

member function returns the function value on vector `x`

,

and overwrites `grad`

by the gradient.

The wrapper function for libLBFGS is

```
inline int optim_lbfgs(
MFuncGrad& f, Refvec x, double& fx_opt,
const int maxit = 300,
const double& eps_f = 1e-6, const double& eps_g = 1e-5
)
```

Also refer to the README page for

details and see the logistic regression example below.

# Fast Logistic Regression: An Example

Letâ€™s see a realistic example that uses the optimization library to fit a

logistic regression.

Given a data matrix $X$ and a 0-1 valued vector $Y$, we want to find a

coefficient vector $beta$ such that the negative log-likelihood function is

minimized:

The gradient function is

So we can write the code as follows:

```
// [[Rcpp::depends(RcppEigen)]]
// [[Rcpp::depends(RcppNumerical)]]
#include
```
using namespace Numer;
using Rcpp::NumericVector;
using Rcpp::NumericMatrix;
typedef Eigen::Map<Eigen::MatrixXd> MapMat;
typedef Eigen::Map<Eigen::VectorXd> MapVec;
class LogisticReg: public MFuncGrad
{
private:
const MapMat X;
const MapVec Y;
public:
LogisticReg(const MapMat x_, const MapVec y_) : X(x_), Y(y_) {}
double f_grad(Constvec& beta, Refvec grad)
{
// Negative log likelihood
// sum(log(1 + exp(X * beta))) - y' * X * beta
Eigen::VectorXd xbeta = X * beta;
const double yxbeta = Y.dot(xbeta);
// X * beta => exp(X * beta)
xbeta = xbeta.array().exp();
const double f = (xbeta.array() + 1.0).log().sum() - yxbeta;
// Gradient
// X' * (p - y), p = exp(X * beta) / (1 + exp(X * beta))
// exp(X * beta) => p
xbeta.array() /= (xbeta.array() + 1.0);
grad.noalias() = X.transpose() * (xbeta - Y);
return f;
}
};
// [[Rcpp::export]]
NumericVector logistic_reg(NumericMatrix x, NumericVector y)
{
const MapMat xx = Rcpp::as<MapMat>(x);
const MapVec yy = Rcpp::as<MapVec>(y);
// Negative log likelihood
LogisticReg nll(xx, yy);
// Initial guess
Eigen::VectorXd beta(xx.cols());
beta.setZero();
double fopt;
int status = optim_lbfgs(nll, beta, fopt);
if(status < 0)
Rcpp::stop("fail to converge");
return Rcpp::wrap(beta);
}

Now letâ€™s do a quick benchmark:

```
set.seed(123)
n = 5000
p = 100
x = matrix(rnorm(n * p), n)
beta = runif(p)
xb = c(x %*% beta)
p = exp(xb) / (1 + exp(xb))
y = rbinom(n, 1, p)
system.time(res1 <- glm.fit(x, y, family = binomial())$coefficients)
## user system elapsed
## 0.339 0.004 0.342
system.time(res2 <- logistic_reg(x, y))
## user system elapsed
## 0.01 0.00 0.01
max(abs(res1 - res2))
## [1] 1.977189e-07
```

This is not a fair comparison however, since `glm.fit()`

will calculate some

other components besides $beta$, and the precision of two methods are also

different.

`RcppNumerical`

provides a function `fastLR()`

that is a more stable

version of the code above (avoiding `exp()`

overflow) and returns similar

components as `glm.fit()`

. The performance is similar:

```
system.time(res3 <- fastLR(x, y)$coefficients)
## user system elapsed
## 0.01 0.00 0.01
max(abs(res1 - res3))
## [1] 1.977189e-07
```

Its source code can be found

here.

# Final Words

If you think this package may be helpful, feel free to leave comments or

request features in the Github

page. Contribution and pull requests would be great.

**leave a comment**for the author, please follow the link and comment on their blog:

**Yixuan's Blog - R**.

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