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

Abstract

A technique termed gradual multifractal reconstruction (GMR) is formulated. A continuum is deined from a signal that preserves the pointwise H lder 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, C. J. 2017. Phys. Rev. E 95, 032123] that fully randomises the wavelet phases. This is extended to the GMR formulation by selective phase randomisation depending on whether or not the wavelet coeicient amplitudes exceeds a threshold criterion. An econophysics application of the technique is presented. The relation between the normalised log-returns and their H lder exponents for the daily returns of eight inancial indices are compared. One particularly noticeable result is the change for the two american indices (NASDAQ 100 and S&P 500) from a non-signiicant to a strongly signiicant (as determined using GMR) cross-correlation between the returns and their H lder exponents from before the 2008 crash to afterwards. This is also relected in the skewness of the phase diference distributions, which exhibit a geographical structure, with asian markets not exhibiting signiicant skewness in contrast to those from elsewhere globally.

A general class of multifractional processes and stock price informativeness

“…However Asian markets don't exhibit significant cross-correlation to those from elsewhere globally. Our setting and goal in this section is different from [28]. We apply the functional form Φ(t, X(t)) to model equity prices and examine the PHE of individual stock series using the proposed LGQV method, from three markets.…”

“…Recently, Keylock [28] has formulated the gradual multifractal reconstruction approach and applied it to stock market returns spanning the 2008 crisis. By comparing the relation between the normalized log-returns and their Hölder exponent for the daily returns of eight financial indexes, Keylock observes that the change for NASDAQ 100 and S&P 500 from a non-significant to a strongly significant cross-correlation between the returns and their Hölder exponents from before the 2008 crash to afterwards.…”

Evaluating Landscape Complexity and the Contribution of Non‐Locality to Geomorphometry

“…An algorithm for this class of problems was developed by Keylock (2017) and was termed the iterated amplitude‐adjusted wavelet transform (IAAWT) method. It was integrated into a gradual reconstruction framework by Keylock (2018) and is described below.…”

“…Gradual multifractal reconstruction (GMR) (Keylock, 2018) generates synthetic data based on the IAAWT algorithm between limits of η = 0 (corresponding to realizations using the original IAAWT algorithm) and η = 1 (the original data set). We first define an energy measure that needs to account for the decimated nature of the dual‐tree complex transform by weighting the coefficients by a factor 2 j (i.e., we adopt an L1 norm): …”

Hypothesis Testing for Nonlinear Phenomena in the Geosciences Using Synthetic, Surrogate Data

“…In this case, it is the wavelet phase that is constrained rather than the Fourier phase in the randomization. This technique, termed gradual multifractal resolution was introduced by Keylock () in a time series context and is used in a spatial sense for the first time in Figure . Here φ is the control parameter used to index gradual multifractal resolution in a directly analogous fashion to the manner than ρ indexes the GWR technique.…”