Surrogate Data Generation By Gradual Wavelet Reconstruction (GWR): A General Method with Applications to Simulation, Hypothesis Testing and Uncertainty Analysis

Abstract:

This presentation will introduce the Gradual Wavelet Reconstruction (GWR) method and highlight the diversity of potential applications of the technique in hydrology, geophysics and beyond.

The starting point for the method is the Iterated Amplitude Adjusted Fourier Transform (IAAFT) method introduced nearly twenty years ago by Schreiber and Schmitz (Physical Review Letters, 1996). Given a chosen significance level, α, and a γ = {1,2}-tailed statistical test, if (γ/α) -1 surrogate series have been generated with IAAFT, if the value for a metric of nonlinearity for the original data lies outside the range for the surrogates then a significant difference is deemed to exist (the data are assumed non-linear).

GWR generalises this idea, by postulating a continuum from ρ = 0 (phase randomised data) to ρ = 1 (the original data). Thus, given rejection of the null hypothesis using IAAFT surrogates, the question of how nonlinear the data are may be answered for the first time by determining the critical value for ρ. This then opens up other research possibilities including:

(1) A method for generating synthetic data with an appropriate degree of nonlinearity;

(2) Novel approaches to confidence limits for extreme value problems based on the surrogates in (1); and,

(3) The testing of the sensitivity of different metrics for nonlinearity.

GWR surrogates are produced in the wavelet domain rather than the Fourier one. The parameter ρ is the total energy of the time series that is fixed in place and not randomised. That is, given a wavelet coefficient, w_j,k at scale, j, and position, k, the total wavelet energy is the summation of w²_j,k over all scales and positions, Σw²_j,k. If all the w²_j,k are placed in descending rank order, GWR fixes in place n wavelet coefficients such that the total energy of these coefficients is ρ×Σw²_j,k. The other coefficients are randomised such that the fidelity of the wavelet filtering operation is preserved.

Because it is a completely generic method, applications are possible in any subject where the generation of synthetic data with appropriate nonlinear properties is desirable. Because the derivative skewness of the typical discharge time series is a nonlinear feature, applications in hydrology are potentially wide-ranging. Various examples of using GWR in hydrology will be presented.