Fractional Brownian motion realizations.

## Project description

Exact methods for simulating fractional Brownian motion (fBm) or fractional Gaussian noise (fGn) in python.

*Approximate*simulation of multifractional Brownian motion (mBm) or multifractional Gaussian noise (mGn).

## Installation

The fbm package is available on PyPI and can be installed via pip:

`pip install fbm`

## fractional Brownian motion

Fractional Brownian motion can be generated via either Hosking’s method, the Cholesky method, or the Davies-Harte method. All three methods are theoretically exact in generating a discretely sampled fBm/fGn.

Usage:

```
from fbm import FBM
f = FBM(n=1024, hurst=0.75, length=1, method='daviesharte')
# or
f = FBM(1024, 0.75)
# Generate a fBm realization
fbm_sample = f.fbm()
# Generate a fGn realization
fgn_sample = f.fgn()
# Get the times associated with the fBm
t_values = f.times()
```

where n is the number of equispaced increments desired for a fBm with Hurst parameter hurst on the interval [0, length]. Method can be either 'hosking', 'cholesky', or 'daviesharte'. The fbm() method returns a length n+1 array of discrete values for the fBm (includes 0). The fgn() method returns a length n array of fBm increments, or fGn. The times() method returns a length n+1 array of times corresponding to the fBm realizations.

The n and hurst parameters are required. The length parameter defaults to 1 and method defaults to 'daviesharte'.

For simulating multiple realizations use the FBM class provided as above. Some intermediate values are cached for repeated simulation.

For one-off samples of fBm or fGn there are separate functions available:

```
from fbm import fbm, fgn, times
# Generate a fBm realization
fbm_sample = fbm(n=1024, hurst=0.75, length=1, method='daviesharte')
# Generate a fGn realization
fgn_sample = fgn(n=1024, hurst=0.75, length=1, method='daviesharte')
# Get the times associated with the fBm
t_values = times(n=1024, length=1)
```

For fastest performance use the Davies and Harte method. Note that the Davies and Harte method can fail if the Hurst parameter hurst is close to 1 and there are a small amount of increments n. If this occurs, a warning is printed to the console and it will fallback to using Hosking’s method to generate the realization. See page 412 of the following paper for a more detailed explanation:

Wood, Andrew TA, and Grace Chan. “Simulation of stationary Gaussian processes in [0, 1] d.” Journal of computational and graphical statistics 3, no. 4 (1994): 409-432.

**Hosking’s method:**

Hosking, Jonathan RM. “Modeling persistence in hydrological time series using fractional differencing.” Water resources research 20, no. 12 (1984): 1898-1908.

**Cholesky method:**

Asmussen, Søren. Stochastic simulation with a view towards stochastic processes. University of Aarhus. Centre for Mathematical Physics and Stochastics (MaPhySto)[MPS], 1998.

**Davies Harte method:**

Davies, Robert B., and D. S. Harte. “Tests for Hurst effect.” Biometrika 74, no. 1 (1987): 95-101.

## multifractional Brownian motion

This package supports *approximate* generation of multifractional
Brownian motion. The current method uses the Riemann–Liouville fractional
integral representation of mBm.

Usage:

```
import math
from fbm import MBM
# Example Hurst function with respect to time.
def h(t):
return 0.25 * math.sin(20*t) + 0.5
m = MBM(n=1024, hurst=h, length=1, method='riemannliouville')
# or
m = MBM(1024, h)
# Generate a mBm realization
mbm_sample = m.mbm()
# Generate a mGn realization
mgn_sample = m.mgn()
# Get the times associated with the mBm
t_values = m.times()
```

The hurst argument here should be a callable that accepts one argument and returns a float in (0, 1).

For one-off samples of mBm or mGn there are separate functions available:

```
from fbm import mbm, mgn, times
# Define a hurst function
def h(t):
return 0.75 - 0.5 * t
# Generate a mbm realization
mbm_sample = mbm(n=1024, hurst=h, length=1, method='riemannliouville')
# Generate a fGn realization
mgn_sample = mgn(n=1024, hurst=h, length=1, method='riemannliouville')
# Get the times associated with the mBm
t_values = times(n=1024, length=1)
```

**Riemann-Liouville representation method:**

*Approximate* method originally proposed for fBm in

Rambaldi, Sandro, and Ombretta Pinazza. “An accurate fractional Brownian motion generator.” Physica A: Statistical Mechanics and its Applications 208, no. 1 (1994): 21-30.

Adapted to approximate mBm in

Muniandy, S. V., and S. C. Lim. “Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type.” Physical Review E 63, no. 4 (2001): 046104.

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