Characterizing the structure of nonlinear systems using gradual wavelet reconstruction

Abstract

Abstract. In this paper, classical surrogate data methods for testing hypotheses concerning nonlinearity in time-series data are extended using a wavelet-based scheme. This gives a method for systematically exploring the properties of a signal relative to some metric or set of metrics. A signal continuum is defined from a linear variant of the original signal (same histogram and approximately the same Fourier spectrum) to the exact replication of the original signal. Surrogate data are generated along this continuum with the wavelet transform fixing in place an increasing proportion of the properties of the original signal. Eventually, chaotic or nonlinear behaviour will be preserved in the surrogates. The technique permits various research questions to be answered and examples covered in the paper include identifying a threshold level at which signals or models for those signals may be considered similar on some metric, analysing the complexity of the Lorenz attractor, characterising the differential sensitivity of metrics to the presence of multifractality for a turbulence time-series, and determining the amplitude of variability of the Hölder exponents in a multifractional Brownian motion that is detectable by a calculation method. Thus, a wide class of analyses of relevance to geophysics can be undertaken within this framework.

Intermittency and multifractional Brownian character of geomagnetic time series

“…The evaluation of local Hurst exponent is a nontrivial target, and for this reason different approaches have been proposed in the past years (e.g., Peng et al, 1994;Peltier and Lévy-Vehel, 1995;Muniandy and Lim, 2001;Chen et al, 2002;Alessio et al, 2002;Carbone et al, 2004;Keylock, 2010). One of the most accurate, fast and simple methods for nonstandard Gaussian multifractional Brownian motions is the detrending moving average (DMA) technique introduced by Alessio et al (2002) and Carbone et al (2004).…”

Gradual wavelet reconstruction of the velocity increments for turbulent wakes

“…Other advantages of this approach are given in the text by Percival and Walden 48 and the original GWR paper. 23 The Appendix briefly reviews the continuous, discrete, and MODWT wavelet methods.…”

“…For ϑ ≥ ϑ thresh , the surrogates and data are similar and the former may be considered legitimate realisations of the latter with respect to the metric of nonlinearity adopted. Thus, ϑ thresh can be used to determine the degree of nonlinearity in two datasets, to measure the sensitivity of metrics of nonlinearity applied to turbulence time series, 23 or as the value for which to generate synthetic data that preserve the nonlinearity of a particular process. 46 Alternatively, synthetic velocity time series may be generated as a function of ϑ and used to form the inlet boundary conditions of eddy-resolving numerical simulations, highlighting any dependencies on the prescribed inlet condition.…”

“…45 The IAAFT method is more suitable for formal hypothesis testing than a simple phase randomization because not only the Fourier amplitudes but also the values of the original data series are identical to those in the synthetic data. Gradual wavelet reconstruction 23 was introduced to study a rejection of the null hypothesis based on IAAFT surrogates (finding nonlinearity). By constraining the degree of phase randomization, surrogate series closer in nature to the original data are generated.…”

“…As a consequence, turbulence decay for these flows differed from that classically assumed in the Richardson-Kolmogorov view of turbulence. 1, 22 The aim of this paper is to study the conditional velocity increment distribution functions and thus, intermittency of jet and fractal-generated turbulence using a controlled randomization technique (termed Gradual Wavelet Reconstruction (GWR) 23 ). Hence, in the strongly randomized synthetic datasets, we try to destroy the properties of the turbulence.…”