How do I best simulate an arbitrary univariate random variate using its probability function?
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)
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:
- completely untested
- efficiency depends on candidate distribution (i.e. value of
cand.sd
). For maximum efficiency, tunecand.sd
to an acceptance rate of 25-40% - results will be autocorrelated ... (although I guess you could always
sample()
the results to scramble them, or thin) - 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)