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Gradual multifractal reconstruction of time-series: Formulation of the method and an application to the coupling between stock market indices and their Hoelder exponents

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journal contribution
posted on 04.06.2018, 08:43 by Chris Keylock
A technique termed gradual multifractal reconstruction (GMR) is formulated. A continuum is defined from a signal that preserves the pointwise Hoelder exponent (multifractal) structure of a signal but randomises the locations of the original data values with respect to this (φ=0), to the original signal itself(φ=1). We demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dual-tree complex wavelet transform. That is, the φ=0 end of the continuum is realised using the recently proposed iterated, amplitude adjusted wavelet transform algorithm (Keylock, 2017) that fully randomises the wavelet phases. This is extended to the GMR formulation by selective phase randomisation depending on whether or not the wavelet coefficient amplitudes exceeds a threshold criterion. An econophysics application of the technique is presented. The relation between the normalised log-returns and their Hoelder exponents for the daily returns of eight financial indices are compared. One particularly noticeable result is the change for the two American indices (NASDAQ 100 and S & P 500) from a non-significant to a strongly significant (as determined using GMR) cross-correlation between the returns and their Hoelder exponents from before the 2008 crash to afterwards. This is also reflected in the skewness of the phase difference distributions, which exhibit a geographical structure, with Asian markets not exhibiting significant skewness in contrast to those from elsewhere globally.

Funding

This research was supported by NERC grant NE/F00415X/1, EPSRC Grant EP/K007688/1, and Royal Academy of Engineering/Leverhulme Senior Research Fellowship LTSRF1516-12-89.

History

School

  • Architecture, Building and Civil Engineering

Published in

Physica D: Nonlinear Phenomena

Volume

368

Pages

1 - 9

Citation

KEYLOCK, C.J., 2017. Gradual multifractal reconstruction of time-series: Formulation of the method and an application to the coupling between stock market indices and their Hoelder exponents. Physica D: Nonlinear Phenomena, 368, pp.1-9.

Publisher

© Elsevier

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2017

Notes

This paper was accepted for publication in the journal Physica D: Nonlinear Phenomena and the definitive published version is available at https://doi.org/10.1016/j.physd.2017.11.011

ISSN

0167-2789

Language

en

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