How do I best simulate an arbitrary univariate ran

Question

In R, what's the best way to simulate an arbitrary univariate random variate if only its probability density function is available?

Answer 1

Use cumulative distribution function http://en.wikipedia.org/wiki/Cumulative_distribution_function

Then just use its inverse. Check here for better picture http://en.wikipedia.org/wiki/Normal_distribution

That mean: pick random number from [0,1] and set as CDF, then check Value

It is also called quantile function.

Answer 2

Here is a (slow) implementation of the inverse cdf method when you are only given a density.

den<-dnorm #replace with your own density

#calculates the cdf by numerical integration
cdf<-function(x) integrate(den,-Inf,x)[[1]]

#inverts the cdf
inverse.cdf<-function(x,cdf,starting.value=0){
 lower.found<-FALSE
 lower<-starting.value
 while(!lower.found){
  if(cdf(lower)>=(x-.000001))
   lower<-lower-(lower-starting.value)^2-1
  else
   lower.found<-TRUE
 }
 upper.found<-FALSE
 upper<-starting.value
 while(!upper.found){
  if(cdf(upper)<=(x+.000001))
   upper<-upper+(upper-starting.value)^2+1
  else
   upper.found<-TRUE
 }
 uniroot(function(y) cdf(y)-x,c(lower,upper))$root
}

#generates 1000 random variables of distribution 'den'
vars<-apply(matrix(runif(1000)),1,function(x) inverse.cdf(x,cdf))
hist(vars)

Answer 3

To clarify the "use Metropolis-Hastings" answer above:

suppose ddist() is your probability density function

something like:

n <- 10000
cand.sd <- 0.1
init <- 0
vals <- numeric(n)
vals[1] <- init 
oldprob <- 0
for (i in 2:n) {
    newval <- rnorm(1,mean=vals[i-1],sd=cand.sd)
    newprob <- ddist(newval)
    if (runif(1)<newprob/oldprob) {
        vals[i] <- newval
    } else vals[i] <- vals[i-1]
   oldprob <- newprob
}

Notes:

1. completely untested
2. efficiency depends on candidate distribution (i.e. value of cand.sd). For maximum efficiency, tune cand.sd to an acceptance rate of 25-40%
3. results will be autocorrelated ... (although I guess you could always sample() the results to scramble them, or thin)
4. may need to discard a "burn-in", if your starting value is weird

The classical approach to this problem is rejection sampling (see e.g. Press et al Numerical Recipes)