path {rlfsm}

Generator of linear fractional stable motion

Description

The function creates a 1-dimensional LFSM sample path using the numerical algorithm from the paper by Otryakhin and Mazur. The theoretical foundation of the method comes from the article by Stoev and Taqqu. Linear fractional stable motion is defined as

X_t = \int_{\R} \left\{(t-s)_+^{H-1/\alpha} - (-s)_+^{H-1/\alpha} \right\} dL_s

Usage

path(
  N = NULL,
  m,
  M,
  alpha,
  H,
  sigma,
  freq,
  disable_X = FALSE,
  levy_increments = NULL,
  seed = NULL
)

Arguments

N

a number of points of the lfsm.

m

discretization. A number of points between two nearby motion points

M

truncation parameter. A number of points at which the integral representing the definition of lfsm is calculated. So, after M points back we consider the rest of the integral to be 0.

alpha

self-similarity parameter of alpha stable random motion.

H

Hurst parameter

sigma

Scale parameter of lfsm

freq

Frequency of the motion. It can take two values: "H" for high frequency and "L" for the low frequency setting.

disable_X

is needed to disable computation of X. The default value is FALSE. When it is TRUE, only a levy motion is returned, which in turn reduces the computation time. The feature is particularly useful for reproducibility when combined with seeding.

levy_increments

increments of Levy motion underlying the lfsm.

seed

this parameter performs seeding of path generator

Value

It returns a list containing the motion, the underlying Levy motion, the point number of the motions from 0 to N and the corresponding coordinate (which depends on the frequency), the parameters that were used to generate the lfsm, and the predefined frequency.

References

Mazur S, Otryakhin D (2020). “Linear Fractional Stable Motion with the rlfsm R Package.” The R Journal, 12(1), 386–405. doi:10.32614/RJ-2020-008.

Stoev S, Taqqu MS (2004). “Simulation methods for linear fractional stable motion and FARIMA using the Fast Fourier Transform.” Fractals, 95(1), 95-121. https://doi.org/10.1142/S0218348X04002379.

See Also

paths simulates a number of lfsm sample paths.

Examples

# Path generation

m<-256; M<-600; N<-2^10-M
alpha<-1.8; H<-0.8; sigma<-0.3
seed=2

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

# Normalized paths
Norm_lfsm<-List[['lfsm']]/max(abs(List[['lfsm']]))
Norm_oLm<-List[['levy_motion']]/max(abs(List[['levy_motion']]))

# Visualization of the paths
plot(Norm_lfsm, col=2, type="l", ylab="coordinate")
lines(Norm_oLm, col=3)
leg.txt <- c("lfsm", "oLm")
legend("topright",legend = leg.txt, col =c(2,3), pch=1)


# Creating Levy motion
levyIncrems<-path(N=N, m=m, M=M, alpha, H, sigma, freq='L',
                  disable_X=TRUE, levy_increments=NULL, seed=seed)

# Creating lfsm based on the levy motion
  lfsm_full<-path(m=m, M=M, alpha=alpha,
                  H=H, sigma=sigma, freq='L',
                  disable_X=FALSE,
                  levy_increments=levyIncrems$levy_increments,
                  seed=seed)

sum(levyIncrems$levy_increments==
    lfsm_full$levy_increments)==length(lfsm_full$levy_increments)