Generating random samples from a custom distribution

I am trying to generate random samples from a custom pdf using R. My pdf is:
fX(x)=32(1x2),0x1

I generated uniform samples and then tried to transform it to my custom distribution. I did this by finding the cdf of my distribution (FX(x)) and setting it to the uniform sample (u) and solving for x.

FX(x)=Pr

To generate a random sample with the above distribution, get a uniform sample u \in[0,1] and solve for x in \frac{3}{2} (x – \frac{x^3}{3}) = u

I implemented it in R and I don’t get the expected distribution. Can anyone point out the flaw in my understanding?

nsamples <- 1000;
x <- runif(nsamples);

f <- function(x, u) { 
  return(3/2*(x-x^3/3) - u);
}

z <- c();
for (i in 1:nsamples) {
  # find the root within (0,1) 
  r <- uniroot(f, c(0,1), tol = 0.0001, u = x[i])$root;
  z <- c(z, r);
}

Answer

It looks like you figured out that your code works, but @Aniko pointed out that you could improve its efficiency. Your biggest speed gain would probably come from pre-allocating memory for z so that you’re not growing it inside a loop. Something like z <- rep(NA, nsamples) should do the trick. You may get a small speed gain from using vapply() (which specifies the returned variable type) instead of an explicit loop (there’s a great SO question on the apply family).

> nsamples <- 1E5
> x <- runif(nsamples)
> f <- function(x, u) 1.5 * (x - (x^3) / 3) - u
> z <- c()
> 
> # original version
> system.time({
+ for (i in 1:nsamples) {
+   # find the root within (0,1) 
+   r <- uniroot(f, c(0,1), tol = 0.0001, u = x[i])$root
+   z <- c(z, r)
+ }
+ })
   user  system elapsed 
  49.88    0.00   50.54 
> 
> # original version with pre-allocation
> z.pre <- rep(NA, nsamples)
> system.time({
+ for (i in 1:nsamples) {
+   # find the root within (0,1) 
+   z.pre[i] <- uniroot(f, c(0,1), tol = 0.0001, u = x[i])$root
+   }
+ })
   user  system elapsed 
   7.55    0.01    7.78 
> 
> 
> 
> # my version with sapply
> my.uniroot <- function(x) uniroot(f, c(0, 1), tol = 0.0001, u = x)$root
> system.time({
+   r <- vapply(x, my.uniroot, numeric(1))
+ })
   user  system elapsed 
   6.61    0.02    6.74 
> 
> # same results
> head(z)
[1] 0.7803198 0.2860108 0.5153724 0.2479611 0.3451658 0.4682738
> head(z.pre)
[1] 0.7803198 0.2860108 0.5153724 0.2479611 0.3451658 0.4682738
> head(r)
[1] 0.7803198 0.2860108 0.5153724 0.2479611 0.3451658 0.4682738

And you don’t need the ; at the end of each line (are you a MATLAB convert?).

Attribution
Source : Link , Question Author : Anand , Answer Author : Community

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