fractal-analysis 0.1.13

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Fractal Analysis

Fractal and multifractal methods, including

• fractional Brownian motion (FBM) tester
• multifractional Brownian motion (MBM) tester
• IR hurst exponents estimator of multifractional Brownian motion (MBM)
• QV hurst exponents estimator of multifractional Brownian motion (MBM)

FBM / MBM tester

Test if a series is FBM (MBM) given the hurst parameter (hurst exponents series). The implementation is based on the following papers:

Michał Balcerek, Krzysztof Burnecki. (2020)
Testing of fractional Brownian motion in a noisy environment.
Chaos, Solitons & Fractals, Volume 140, 110097.
https://doi.org/10.1016/j.chaos.2020.110097

Balcerek, Michał, and Krzysztof Burnecki. (2020)
Testing of Multifractional Brownian Motion. Entropy 22, no. 12: 1403.
https://doi.org/10.3390/e22121403

We added the following improvements to the FBM and/or MBM tester:

• option for automatically estimating sigma
  • based on Theorem 2.3 of the following paper:

    Ayache A., Peng Q. (2012)
    Stochastic Volatility and Multifractional Brownian Motion.
    In: Zili M., Filatova D. (eds) Stochastic Differential Equations and Processes. Springer Proceedings in Mathematics, vol 7. Springer, Berlin, Heidelberg.
    https://doi.org/10.1007/978-3-642-22368-6_6

  • a detailed introduction can be found in the section 5 of the following paper:

    todo: add paper name

• option for testing if the series itself is a FBM (MBM)
• option for testing if the increment of the series is the increment of a FBM (MBM)
• option for testing if the series is a FBM (MBM) with an add-on noise
• option for testing if the increment of the series is the increment of a FBM (MBM) with an add-on noise

IR / QV hurst estimator of MBM

Estimate the hurst parameter (hurst exponent series) of a MBM. The implementation is based on the following paper:

Bardet, Jean-Marc & Surgailis, Donatas, 2013.
Nonparametric estimation of the local Hurst function of multifractional Gaussian processes.
Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1004-1045.

Bardet, the author in the above paper, provides a Matlab code that can be found at:

http://samm.univ-paris1.fr/Sofwares-Logiciels
Software for estimating the Hurst function H of a Multifractional Brownian Motion: Quadratic Variation estimator and IR estimator

To install

To get started, simply do:

pip install fractal-analysis

or check out the code from out GitHub repository.

You can now use the series tester module in Python with:

from fractal_analysis import tester

or use the hurst estimator with

from fractal_analysis import estimator

Examples

FBM / MBM tester

Import:

from fractal_analysis.tester.series_tester import MBMSeriesTester, FBMSeriesTester
from fractal_analysis.tester.critical_surface import CriticalSurfaceFBM, CriticalSurfaceMFBM

To test if a series series is FBM, one needs to use CriticalSurfaceFBM with length of the series N, the significance level alpha (look at quantiles of order alpha/2 and 1 − alpha/2), and choose to test on the series itself or its increment series using is_increment_series (default is False, meaning to test on the series itself),

fbm_tester = FBMSeriesTester(critical_surface=CriticalSurfaceFBM(N=N, alpha=0.05, is_increment_series=False))

To test if the series is FBM with hurst parameter 0.3 and use auto estimated sigma square (set sig2=None):

is_fbm, sig2 = fbm_tester.test(h=0.3, x=series, sig2=None, add_on_sig2=0)

If the output contains, for example:

Bad auto sigma square calculated with error 6.239236333681868. Suggest to give sigma square and rerun.

The auto sigma square estimated is not accurate. One may want to manually choose a sigma square and rerun. For example:

is_fbm, sig2 = fbm_tester.test(h=0.3, x=series, sig2=1, add_on_sig2=0)

If one wants to test with an add-no noise, change the value of add_on_sig2.

To test if the series is MBM, one needs to use CriticalSurfaceMFBM with length of the series N and the significance level alpha (look at quantiles of order alpha/2 and 1 − alpha/2)

mbm_tester = MBMSeriesTester(critical_surface=CriticalSurfaceMFBM(N=N, alpha=0.05, is_increment_series=False))

To test if the series is MBM with a given holder exponent series h_mbm_series and use auto estimated sigma square:

is_mbm, sig2 = mbm_tester.test(h=h_mbm_series, x=series, sig2=None, add_on_sig2=0)

Be aware that MBMSeriesTester requires len(h_mbm_series)==len(series).

Use of cache

Use caching to speed up the testing process. If the series x for testing is unchanged and multiple h and/or sig2 are used, one may want to set is_cache_stat=True to allow cache variable stat. If h and sig2 are unchanged and multiple x are used, one may want to set is_cache_quantile=True to allow cache variable quantile. For example:

mbm_tester = MBMSeriesTester(critical_surface=CriticalSurfaceMFBM(N=N, alpha=0.05), is_cache_stat=True, is_cache_quantile=False)

IR / QV hurst estimator of MBM

Import:

from fractal_analysis.estimator.hurst_estimator import IrHurstEstimator, QvHurstEstimator
import numpy as np
import math

Generate a standard brownian motion

N = 100
series = np.random.randn(N) * 0.5 * math.sqrt(1 / N)
series = np.cumsum(series)

To estimate the hurst exponents series of the above series with alpha=0.2 using IR estimator,

estimator = IrHurstEstimator(mbm_series=series, alpha=0.2)
print(estimator.holder_exponents)

To estimate the hurst exponents series of the above series with alpha=0.2 using QV estimator,

estimator = QvHurstEstimator(mbm_series=series, alpha=0.2)
print(estimator.holder_exponents)

Here the value of alpha decides how many observations on the mbm_series is used to estimate a point of the holder exponent; small alpha means more observations are used for a single point and therefore the variance is small.