In gvegayon/numint:

Definition

From Wolfram MathWorld:

Picking $N$ randomly distributed points $x_1, x_2, ..., x_N$ in a multidimensional volume $V$ to determine the integral of a function $f$ in this volume gives a result

$$ \tag{1} \int f dV \approx V \langle f\rangle \pm V\sqrt{\frac{\langle f^2 \rangle- \langle f \rangle^2}{N}} $$ Where

$$ \begin{align} \langle f \rangle & = \frac{1}{N}\sum^N{i=1}f(x_i)\tag{2}\ \langle f \rangle & = \frac{1}{N}\sum^N{i=1}f(x_i)^2\tag{3} \end{align} $$

Example with 2D Multivariate Normal

Integrate between a = {-1, -10} and b = {1, 1}

set.seed(1231)
library(mvtnorm)
library(numint)

ans <- num_int(
  dmvnorm, a = c(-1, -10), b = c(1, 1),
  mean = c(0,0), sigma = diag(2),
  N = 1e7, ncores = 2
)
ans$val
pmvnorm(lower = c(-1,-10), upper = c(1,1), mean = c(0,0), sigma = diag(2))

We can visualize a single dim of the area

plot(ans)

References

Weisstein, Eric W. "Monte Carlo Integration." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MonteCarloIntegration.html

gvegayon/numint documentation built on May 24, 2019, 4:05 a.m.