aleatory.processes.fBM

class aleatory.processes.fBM(hurst=0.5, T=1.0, rng=None)

Fractional Brownian motion

Notes

A fractional Brownian motion (fBM) is a continuous-time Gaussian process \(B_H(t)\) on \([0,T]\) that starts at zero, has expectation zero for all \(t \in [0,T]\) and has the following covariance function:

\[E\left[B_H(t) B_H(s) \right] = \frac{1}{2}(|t|^{2H}+ |s|^{2H}- |t-s|^{2H}),\]

where \(H\) is a real number in (0,1), called the Hurst or Hurst parameter.

Constructor, Methods, and Attributes

__init__(hurst=0.5, T=1.0, rng=None)

Parameters:

• hurst (float) – the Hurst parameter

• T (float) – the right hand endpoint of the time interval \([0,T]\) for the process

• rng (numpy.random.Generator) – a custom random number generator

Methods

__init__([hurst, T, rng])	parameter float hurst: the Hurst parameter
draw(n, N[, T, marginal, envelope, title, ...])	Simulates and plots paths/trajectories from the instanced stochastic process.
estimate_covariances([times])
estimate_expectations()
estimate_quantiles(q)
estimate_stds()
estimate_variances()
get_marginal(t)
marginal_expectation([times])
marginal_stds([times])
marginal_variance([times])
plot(n, N[, T, title, suptitle])	Simulates and plots paths/trajectories from the instanced stochastic process.
plot_covariance([times, title])
plot_kernel([times, colormap, matrix_shape, ...])
plot_kernel3d([times, title])
plot_mean_variance([times, title])
plot_paths_and_kernel(n, N[, T, title, ...])	Plots the paths of the process and the covariance kernel.
process_covariance([times])
process_expectation()
process_stds()
process_variance()
sample(n)
simulate(n, N[, T])	Simulate paths/trajectories from the instanced stochastic process.

Attributes

T	End time of the process.
rng