The following p.d.f is given:
f(x)=2x/k^2, where 0 <= x <= k
Let k=10, I'm trying to simulate 100 times for this p.d.f, then print the first 5 results and find the mean for all 100 times results.
If you want to simulate with a uniform distribution between 0 and k, then you can pass runif(n, min, max) into your pdf.
f <- function(x, k) {
return(2*x/k^2)
}
k <- 10
res <- f(runif(100, 0, k), k)
print(res[1:5])
print(mean(res))
Per MrFlick's comments, if you were instead wanting to do inverse transform sampling, this should suffice.
pdf <- function(x, k) {
return(2*x/k^2)
}
cdf <- function(x, k){
return(x^2/k^2)
}
icdf <- function(y, k){
return(sqrt(k^2*y))
}
k <- 10
res <- icdf(runif(100,0,1), k)
print(res[1:5])
print(mean(res))
Related
Following this question: How to get the value of `t` so that my function `h(t)=epsilon` for a fixed `epsilon`?
I first sampling 500 eigenvectors v of a random matrix G and then generate 100 different random vectors initial of dimension 500. I normalized them in mats.
#make this example reproducible
set.seed(100001)
n <- 500
#Sample GOE random matrix
A <- matrix(rnorm(n*n, mean=0, sd=1), n, n)
G <- (A + t(A))/sqrt(2*n)
ev <- eigen(G)
l <- ev$values
v <- ev$vectors
#size of multivariate distribution
mean <- rep(0, n)
var <- diag(n)
#simulate bivariate normal distribution
initial <- MASS::mvrnorm(n=1000, mu=mean, Sigma=var) #ten random vectors
#normalized the first possible initial value, the initial data uniformly distributed on the sphere
xmats <- lapply(1:1000, function(i) initial[i, ]/norm(initial[i, ], type="2"))
Then I compute res
h1t <- function(t,x_0) {
h10 <- c(x_0 %*% v[, n])
denom <- vapply(t, function(.t) {
sum((x_0 %*% v)^2 * exp(-4*(l - l[n]) * .t))
}, numeric(1L))
abs(h10) / sqrt(denom)
}
find_t <- function(x, epsilon = 0.01, range = c(-50, 50)) {
uniroot(function(t) h1t(t, x) - epsilon, range,
tol = .Machine$double.eps)$root
}
I want to get res:
res <- lapply(xmats, find_t)
However, it shows error that Error in uniroot(function(t) h1t(t, x) - epsilon, range, tol = .Machine$double.eps) : f() values at end points not of opposite sign
res is a list. I run hist(unlist(res)) and it worked well.
I wrote some R code for simulating random samples from a Poisson distribution, based on the description of an algorithm (see attached image). But my code does not seem to work correctly, because the generated random samples are of a different pattern compared with those generated by R's built-in rpois() function. Can anybody tell me what I did wrong and how to fix my function?
r.poisson <- function(n, l=0.5)
{
U <- runif(n)
X <- rep(0,n)
p=exp(-l)
F=p
for(i in 1:n)
{
if(U[i] < F)
{
X[i] <- i
} else
{
p=p*l/(i+1)
F=F+p
i=i+1
}
}
return(X)
}
r.poisson(50)
The output is very different from rpois(50, lambda = 0.5). The algorithm I followed is:
(Thank you for your question. Now I know how a Poisson random variable is simulated.)
You had a misunderstanding. The inverse CDF method (with recursive computation) you referenced is used to generate a single Poisson random sample. So you need to fix this function to produce a single number. Here is the correct function, commented to help you follow each step.
rpois1 <- function (lambda) {
## step 1
U <- runif(1)
## step 2
i <- 0
p <- exp(-lambda)
F <- p
## you need an "infinite" loop
## no worry, it will "break" at some time
repeat {
## step 3
if (U < F) {
X <- i
break
}
## step 4
i <- i + 1
p <- lambda * p / i ## I have incremented i, so it is `i` not `i + 1` here
F <- F + p
## back to step 3
}
return(X)
}
Now to get n samples, you need to call this function n times. R has a nice function called replicate to repeat a function many times.
r.poisson <- function (n, lambda) {
## use `replicate()` to call `rpois1` n times
replicate(n, rpois1(lambda))
}
Now we can make a reasonable comparison with R's own rpois.
x1 <- r.poisson(1000, lambda = 0.5)
x2 <- rpois(1000, lambda = 0.5)
## set breaks reasonably when making a histogram
xmax <- max(x1, x2) + 0.5
par(mfrow = c(1, 2))
hist(x1, main = "proof-of-concept-implementation", breaks = seq.int(-0.5, xmax))
hist(x2, main = "R's rpois()", breaks = seq.int(-0.5, xmax))
Remark:
Applaud jblood94 for exemplifying how to seek vectorization opportunity of an R loop, without converting everything to C/C++. R's rpois is coded in C, that is why it is fast.
A vectorized version will run much faster than a non-vectorized function using replicate. The idea is to iteratively drop the uniform random samples as i is incremented.
r.poisson1 <- function(n, l = 0.5) {
U <- runif(n)
i <- 0L
X <- integer(n)
p <- exp(-l)
F <- p
idx <- 1:n
while (length(idx)) {
bln <- U < F
X[idx[bln]] <- i
p <- l*p/(i <- i + 1L)
F <- F + p
idx <- idx[!bln]
U <- U[!bln]
}
X
}
#Zheyuan Li's non-vectorized functions:
rpois1 <- function (lambda) {
## step 1
U <- runif(1)
## step 2
i <- 0
p <- exp(-lambda)
F <- p
## you need an "infinite" loop
## no worry, it will "break" at some time
repeat {
## step 3
if (U < F) {
X <- i
break
}
## step 4
i <- i + 1
p <- lambda * p * i
F <- F + p
## back to step 3
}
return(X)
}
r.poisson2 <- function (n, lambda) {
## use `replicate()` to call `rpois1` n times
replicate(n, rpois1(lambda))
}
Benchmark:
microbenchmark::microbenchmark(r.poisson1(1e5),
r.poisson2(1e5, 0.5),
rpois(1e5, 0.5))
#> Unit: milliseconds
#> expr min lq mean median uq max neval
#> r.poisson1(1e+05) 3.063202 3.129151 3.782200 3.225402 3.734600 18.377700 100
#> r.poisson2(1e+05, 0.5) 217.631002 244.816601 269.692648 267.977001 287.599251 375.910601 100
#> rpois(1e+05, 0.5) 1.519901 1.552300 1.649026 1.579551 1.620451 7.531401 100
I have a code that looks that this, What I am trying to do is to simulate stock path, however, obviously I was using the same random number for the same row vector (a particular stock path) How do i modify this such that I could use a different random number for a stock price. Thanks for any comments.
options_monte_DO <- function(r, q, sigma, T, n, m, k, s_0, B) {
x <- matrix(NA, nrow=n, ncol=m)
y <- matrix(NA, nrow=n, ncol=m)
dt <- T/m
for (i in 1:n) {
x[i, ] <- s_0*exp(cumsum((r - q - 0.5*sigma^2)*dt +
sigma*sqrt(dt)*rnorm(n, 0, 1)))
}
y <- x
y[rowSums(x < B) > 0, ] <- NA
option_payoff <- exp(-r*T)*pmax(k - y[, m], 0)
z <- mean(option_payoff, na.rm=T)
hist(option_payoff)
return(z)
}
I have an exercise, in which i have to create an algorithm as follows:
ratio of Uniforms is based on the fact that for a random variable X with density f(x) we can generate X from the desired density by calculating X = U/V for a pair (U, V ) uniformly distributed in the set
Af = {(u,v):0 < v ≤ f(u/v)}
Random points can be sampled uniformly in Af by rejection from the min- imal bounding rectangle, i.e., the smallest possible rectangle that contains Af .
It is given by (u−, u+) × (0, v+) where
v+ = max f(x), x
u− = minx f(x), x
u+ = maxx f(x)
Then the Ratio-of-Uniforms method consists of the following simple steps:
Generate random number U uniformly in (u−, u+).
Generate random number V uniformly in (0, v+).
Set X ← U/V .
If V 2 ≤ f(X) accept and return X.
Else try again.
My code so far:
x <- cnorm(1, mean = 0, sd=1)
myrnorm <- function(pdf){
## call rou() n times
pdf <- function(x) {exp(-x^2/2)}
}
rou <- function(u, v) {
uplus <- 1
vplus <- 1
n <- 100
u <- runif(n, min=0, max=uplus)
v <- runif(n, min=0, max=vplus)
xi <- v/u
while(v < sqrt(xi)) {
if(v^2 <= xi)
return(xi)
}
}
myx <- myrnorm(1000)
hist(myx)
But I really dont know how to go on. Im ´lost with this exercise. I would be really grateful for any advise.
Following example 1 in page 8 of this link and your sample code, I came up this solution:
ratioU <- function(nvals)
{
h_x = function(x) exp(-x)
# u- is b-, u+ is b+ and v+ is a in the example:
uminus = 0
uplus = 2/exp(1)
vplus = 1
X.vals <- NULL
i <- 0
repeat {
i <- i+1
u <- runif(1,0,vplus)
v <- runif(1,uminus,uplus)
X <- u/v
if(v^2 <= h_x(X)) {
tmp <- X
}
else {
next
}
X.vals <- c(X.vals,tmp)
if(length(X.vals) >= nvals) break
}
answer <- X.vals
answer
}
sol = ratioU(1000)
par(mfrow=c(1,2))
hist(sol,breaks=50, main= "using ratioU",freq=F)
hist(rexp(1000),breaks = 50, main="using rexp from R",freq=F)
par(mfrow=c(1,1))
par(mfrow=c(1,2))
plot(density(sol))
plot(density(rexp(1000)))
par(mfrow=c(1,1))
A lot of the code may be optimized but I think it is good enough like this for this purpose. I hope this helps.
Suppose I have a vector x which I want to convolve with itself n times. What is the good way to do this in R?
Suppose that we already have a function conv(u,v) that convolves two vectors.
I can do this:
autoconv<-function(x,n){
r<-1;
for(i in 1:n){
r<-conv(r,x);
}
return(r);
}
is there a more efficient way?
Take the Fast Fourier Transform (fft) of x, raise it to the kth power and take the inverse fft. Then compare that to performing convolutions of k copies of x. No packages are used.
# set up test data
set.seed(123)
k <- 3 # no of vectors to convolve
n <- 32 # length of x
x <- rnorm(n)
# method 1 using fft and inverse fft
yy <- Re(fft(fft(x)^k, inverse = TRUE) / n)
# method 2 using repeated convolutions
y <- x
if (k >= 2) for(i in 2:k) y <- convolve(x, y, FALSE)
# check that the two methods give the same result
all.equal(y, yy)
## TRUE
autoconv <- function(x, n){
if(n == 0){
return(1)
} else if(n == 1){
return(x)
} else {
i <- 2
xi <- conv(x,x)
while(n %% i != 0){
i <- i + 1
xi <- conv(xi,x)
}
return(autoconv(xi,n/i))
}
}
This will call conv() once for each prime factor of n, rather than n times.