Importance sampling

Note: the function \(h\) was to be defined as the of the function given in the exercise. As it stands, the question doesn’t really make sense since it has approximately equal positive and negative parts. See below for a solution to the square.

First we define the function \[h(x) = \left[ \cos(50x) + \sin(20x) \right]^2.\]

h <- function(x) {
    z <- (x > 0) * (x < 1) * (cos(50 * x) + sin(20 * x))
    z^2
}

qplot(x = xs, y = h(xs), geom = "line")

\(h\) can be integrated analytically. Expanding \[h(x) = \cos(50x)^2 + 2 \cos(50x) \sin(20x) + \sin(20x)^2,\] the antiderivative in \([0, 1]\) is \[H(x) = \frac{x}{2} + \frac{\sin(100x)}{200} + 2 \left[ \frac{\cos(30x)}{60} - \frac{\cos(70x)}{140} \right] + \frac{x}{2} - \frac{\sin(40x)}{80} \] and integrated over \([0, 1]\) gives

H <- function(x) {
    x/2 + sin(100 * x)/200 + 2 * (cos(30 * x)/60 - cos(70 * x)/140) + x/2 - 
        sin(40 * x)/80
}
trueValue <- H(1) - H(0)
cat("The true value of I is", trueValue)
## The true value of I is 0.9652009

Uniform Proposal

First, rejection sampling with uniform proposal amounts here to crude Monte Carlo here, as the integral is with respect to the uniform measure on \([0,1]\).

numRepeats <- 1000
numPoints <- 1000
mcErrors <- rep(NA, numRepeats)
for (i in 1:numRepeats) {
    u <- runif(n = numPoints)
    weights <- h(u)
    mcErrors[i] <- abs(mean(weights) - trueValue)/trueValue
}

qplot(x = mcErrors, y = ..density.., geom = "histogram", binwidth = 0.01) + 
    xlab("Relative Error")

Gaussian Mixture Proposal

From our plot of \(h\) we realize it is multimodal. We expect that we should be able to reduce the variance of the Monte Carlo estimator by using importance sampling (IS) with a Gaussian mixture proposal. We roughly place some mixture components by eye here, but feel free to try to automate this procedure using an optimizer, for example.

numComponents <- 13
omega <- c(1, 2, 2, 1, 4, 0.2, 0.2, 4, 1, 2, 2, 1, 2)
omega <- omega/sum(omega)
mu <- c(0.01, 0.12, 0.195, 0.305, 0.38, 0.45, 0.495, 0.565, 0.635, 0.745, 0.825, 
    0.935, 1)
sigma <- c(0.02, 0.02, 0.02, 0.02, 0.02, 0.01, 0.01, 0.02, 0.02, 0.02, 0.02, 
    0.02, 0.02)


# Use a builtin library for sampling from normal mixtures.
library(nor1mix)
mixture <- norMix(m = mu, sigma = sigma, w = omega)

Let’s visually compare the target function with our importance sampling density.

normalized_hs <- abs(h(xs))/sum(abs(h(xs))) * 500/1.4

df <- data.frame(xs = xs, hs = abs(h(xs)), ahs = normalized_hs, qs = dnorMix(xs, 
    mixture))
ggplot(df) + geom_line(aes(x = xs, y = ahs), colour = "black") + geom_line(aes(x = xs, 
    y = qs), colour = "red")

qplot(x = xs, y = h(xs)/dnorMix(xs, mixture), col = "blue")