GHBMP: Simulation of Gaussian Haar-based multifractional processes
In Rmfrac: Simulation and Statistical Analysis of Multifractional Processes

GHBMP

R Documentation

Simulation of Gaussian Haar-based multifractional processes

Description

This function simulates a realisation of a Gaussian Haar-based multifractional process at any time point or time sequence on the interval [0,1].

Usage

GHBMP(t, H, J = 15, num.cores = availableCores(omit = 1))

Arguments

`t`	Time point or time sequence on the interval `[0,1]`.
`H`	Hurst function which depends on `t` `(H(t))`. See Examples for usage.
`J`	Positive integer. `J` is recommended to be greater than `\log_2(length(t))`. For large `J` values could be rather time consuming. Default is set to 15.
`num.cores`	Number of cores to set up the clusters for parallel computing.

Details

The following formula defined in Ayache, A., Olenko, A. & Samarakoon, N. (2025) was used in simulating Gaussian Haar-based multifractional process.

X(t) := \sum_{j=0}^{+\infty} \sum_{k=0}^{2^{j}-1}\left(\int_{0}^{1} (t-s)_{+}^{H_{j}(k/{2^j})-{1}/{2}} h_{j,k}(s)ds \right)\varepsilon_{j,k},

where

\int_{0}^{1} (t-s)_{+}^{H_{j,k}-\frac{1}{2}} h_{j,k} (s) ds = 2^{-j H_{j,k}} h^{[H_{j,k}]} (2^jt-k)

with h^{[\lambda]} (x) = \int_{\mathbb{R}} (x-s)_{+}^{\lambda-\frac{1}{2}} h(s) ds. h is the Haar mother wavelet, j and k are positive integers, t is time, H is the Hurst function and \varepsilon_{j,k} is a sequence of independent \mathcal{N}(0,1) Gaussian random variables. For simulations, the truncated version of this formula with first summation up to J is used.

Value

A data frame of class "mp" where the first column is time moments t and second column is simulated values of X(t).

Note

See Examples for the usage of constant, time-varying, piecewise or step Hurst functions.

References

Ayache, A., Olenko, A. and Samarakoon, N. (2025). On Construction, Properties and Simulation of Haar-Based Multifractional Processes. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.48550/arXiv.2503.07286")}. (submitted).

Examples


#Constant Hurst function
t <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.4 + 0*t)}
GHBMP(t, H)

#Linear Hurst function
t <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.2 + 0.45*t)}
GHBMP(t, H)

#Oscillating Hurst function
t <- seq(0, 1, by = (1/2)^10)
H <- function(t) {return(0.5 - 0.4 * sin(6 * 3.14 * t))}
GHBMP(t, H)

#Piecewise Hurst function
t <- seq(0, 1, by = (1/2)^10)
H <- function(x) {
ifelse(x >= 0 & x <= 0.8, 0.375 * x + 0.2,
      ifelse(x > 0.8 & x <= 1,-1.5 * x + 1.7, NA))
}
GHBMP(t, H)

Rmfrac documentation built on Sept. 10, 2025, 10:31 a.m.