How do I best simulate an arbitrary univariate random variate using its probability function? - r

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

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 2

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, tune cand.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)

Answer 3

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 4

This is a comment but I don't have enough reputation to drop a comment to Ben Bolker's answer.
I am new to Metropolis, but IMHO this code is wrong because:
a) the newval is drawn from a normal distribution whereas in other codes it is drawn from a uniform distribution; this value must be drawn from the range covered by the random number. For example, for a gaussian distribution this should be something like runif(1, -5, +5).
b) the prob value must be updated only if acceptance.
Hope this help and hope that someone with reputation could correct this answer (especially mine if I am wrong).
# the distribution
ddist <- dnorm
# number of random number
n <- 100000
# the center of the range is taken as init
init <- 0
# the following should go into a function
vals <- numeric(n)
vals[1] <- init
oldprob <- 0
for (i in 2:n) {
newval <- runif(1, -5, +5)
newprob <- ddist(newval)
if (runif(1) < newprob/oldprob) {
vals[i] <- newval
oldprob <- newprob
} else vals[i] <- vals[i-1]
}
# Final view
hist(vals, breaks = 100)
# and comparison
hist(rnorm(length(vals)), breaks = 100)