R/generate_data.R
piecewise_fractional_brownian_trajectory.RdThis function generates a realization of a piecewise fractional Brownian motion with random noise. A piecewise fractional Brownian motion is defined by a non constant Hurst parameter along the sampling points. We observe the process at regularly spaced time \(t_i = \frac{i}{M_n}\), where \(i = 0, \dots, M_n\). We define a segmentation \(\tau = (\tau_k)_{k=0, \dots, K+1}\), with \(0 = \tau_0 < \tau_1 < \dots < \tau_{K} < \tau_{K+1} = 1\). So, on the interval \([\tau_k, \tau_{k+1}]\), for \(k = 0, \dots, K\), the process is a fractional Brownian motion with Hurst parameter \(H_k\).
piecewise_fractional_brownian_trajectory(M, H, sigma, pdf = NULL)
| M | An integer, expected number of points in the trajectory. The number of points follows a Poisson distribution with mean \(M\). |
|---|---|
| H | A vector of numeric, Hurst coefficients. \(0 < H_k < 1\) |
| sigma | A vector of numeric, standard deviation of the noise to add to the piecewise fractional Brownian motion. Should have the length of H. It adds heteroscedastic noise to the data. |
A function for the generation of the sampling points. |
A tibble containing the following elements:
...1: The sampling points
...2 The true trajectory
...3 The trajectory contaminated by noise with standard deviation \(\sigma\)