R - Compute increments for use in power variation

Question

I am working with high frequent data, which is a vector X of decimal numbers.

I would like to code the following in R, but am at complete loss where to start:

n decides how big intervals we are looking at. This can be any value, this is not so important, since I need to experiment with different values.

I have tried a lot of things, amongst the following:

X <- runif(10000) # 5000 obs. pr. second, with 20 mill obs in total

for(i in 1:length(X){
sum? * (-1)^??
*(for (n in 0:i) print(choose(n, k = 0:n)))
x <- ??
mat[1:length(X)] <- x
}

But I am miles from anything working. Any help is greatly appreciated. I've edited the equation, missed an equal sign.

I've written something that hopefully should make it more clear:

Answer 1

Maybe this can get you started. I think it is what you expressed. However I think it would make more sense with an initialization like rnorm(nx,1,0.01) - i.e. something smoother. It might also make sense to have xsub interpolate between the generated points (i.e. make it a continuous function).

It could obviously be made more efficient but it should be verified to be correct first.

Also I really don't know how to interpret the results or I would have done a plot.

nx <- 100
secs <- 2
x <- runif(nx)
obspersec <- nx/secs

xsub <- function(t){
   i <- trunc(t*obspersec)+1
   if (i<=0) return(0)
   if (i>nx) return(0)
   return(x[i])
}

x_njk <- function (n,i,k){
  if (i<k){
    return (0)
  } 
  sum <- 0
  for (j in 0:k){
    term <- ((-1)^j)*choose(k,j)*xsub((i-j)/n)
    sum <- sum + term 
  }
  return(sum)
}

n <- 100

for (k in 1:2){
   for (i in k:10){
     print(sprintf("n:%d i:%d k:%d  x_njk:%.5f",n,i,k,x_njk(n,i,k)))
   }
}