increment {rlfsm}

Higher order increments

Description

Difference of the kth order. Defined as following:

\Delta_{i,k}^{n,r} X:= \sum_{j=0}^k (-1)^j{{k}\choose{j}}X_{(i-rj)/n}, i\geq rk.

Index i here is a coordinate in terms of point_num. Although R uses vector indexes that start from 1, increment has i varying from 0 to N, so that a vector has a length N+1. It is done in order to comply with the notation of the paper. This function doesn't allow for choosing frequency n. The frequency is determined by the path supplied, thus n equals to either the length of the path in high frequency setting or 1 in low frequency setting. increment() gives increments at certain point passed as i, which is a vector here. increments() computes high order increments for the whole sample path. The first function evaluates the formula above, while the second one uses structure diff(diff(...)) because the formula is slower at higher k.

Usage

increment(r, i, k, path)

increments(k, r, path)

Arguments

r

difference step, a natural number

i

index of the point at which the increment is to be computed, a natural number.

k

order of the increment, a natural number

path

sample path for which a kth order increment is computed

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

m<-45; M<-60; N<-2^10-M
alpha<-0.8; H<-0.8; sigma<-0.3

lfsm<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm

tryCatch(
increment(r=1,i=length(lfsm),k=length(lfsm)+100,path=lfsm),
error=function(c) 'An error occures when k is larger then the length of the sample path')

increment(r=3,i=50,k=3,path=lfsm)


path=c(1,4,3,6,8,5,3,5,8,5,1,8,6)

r=2; k=3
n <- length(path) - 1
DeltaX = increment(seq(r*k, n), path = path, k = k, r = r)
DeltaX == increments(k=k,r=r,path)