multidim_integrator: Multi-dimensional integration via Monte-Carlo methods

In shill1729/numerics: Various numerical methods

Description

A naive but efficient estimator for multi-dimensional integrals via Monte-Carlo methods. The integrand must take a vector as an argument.

Usage

multidim_integrator(g, lbs, ubs, n = 10^4, sig_lvl = 0.05, ...)

Arguments

g	the known integrand, a function of a vector defined over a 'rectangular' region
lbs	the left-end points of each interval per coordinate
ubs	the right end-points of each interval per coordinate
n	the number of variates to simulate of the IID uniform vector
sig_lvl	the significance level for the confidence intervals
...	additional arguments to pass to the function g

Value

data.frame containing

• estimate the point estimate of the integral,

• lb the lower bound of the confidence interval,

• ub the upper bound of the confidence interval,

• std_error the standard error of the point-estimate.

Examples

# A simple example: integrating f(x,y,z)=xyz over [0,1]^3; exact value 1/8
multidim_integrator(function(x) x[1]*x[2]*x[3], lbs = c(0, 0, 0), ubs = c(1, 1, 1))

# More involved example: approximating E(log(1+w dot R))
# where R is the vector of returns of N assets, w is the vector of weights
# Lower and upper bounds of returns, assuming Uniform distribution
lbs <- c(-1, -0.5, -0.25, 0)
ubs <- c(2, 0.5, 1, 1.5)
w <- c(0.1, 0.1, 0.1, 0.7)
N <- length(lbs) # Number of assets
# Mean's of returns from uniform model
mu <- 0.5*(ubs+lbs)
# Quadratic approximation requires the matrix E(R_iR_j) for all assets i,j
B <- matrix(0, N, N)
for(i in 1:N)
{
  for(j in 1:N)
  {
    B[i, j] <- mu[i]*mu[j]
  }
}
# Quadratic approximation to the expectation:
quad_approx <- as.numeric(t(mu)%*%w-0.5*t(w)%*%B%*%w)
# Function of a vector to pass to the integrator
g <- function(x, w)
{
  log(1+w%*%x)
}
# Since we want to approximate the expectation, we divided by the measure
# of the region we are integrating over.
multidim_integrator(g, lbs, ubs, w = w)/prod(ubs-lbs)
quad_approx

shill1729/numerics documentation built on April 17, 2021, 6:46 p.m.