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

C J Keylock

doi:10.1029/2018ea000435

Abstract

Studying nonlinear and potentially chaotic phenomena in geophysics from measured signals is problematic when system noise interferes with the dynamic processes that one is trying to infer. In such circumstances, a framework for statistical hypothesis testing is necessary but the nonlinear nature of the phenomena studied makes the formulation of standard hypothesis tests, such as analysis of variance, problematic as they are based on underlying linear, Gaussian assumptions. One approach to this problem is the method of surrogate data, which is the technique explained in this paper. In particular, we focus on (i) hypothesis testing for nonlinearity by generating linearized surrogates as a null hypothesis, (ii) a variant of this that is perhaps more appropriate for image data where structural nonlinearities are common and should be retained in the surrogates, and (iii) gradual reconstruction where we systematically constrain the surrogates until there is no significant difference between data and surrogates and use this to understand geophysical processes. In addition to time series of sunspot activity, solutions to the Lorenz equations, and spatial maps of enstrophy in a turbulent channel flow, two examples are considered in detail. The first concerns gradual wavelet reconstruction testing of the significance of a specific vortical flow structure from turbulence time series acquired at a point. In the second, the degree of nonlinearity in the spatial profiles of river curvature is shown to be affected by the occurrence of meander cutoff processes but in a more complex fashion than previously envisaged.

Highlights

There is no double-blind, replicated trial for a tsunami, an earthquake, a tornado or an ionospheric perturbation (Simpson, 1963)
It follows that it is not sufficient to apply some measure of nonlinearity, such as those reviewed by Schreiber and Schmitz (1997), to a geophysical system and state that the value is x; rather, one should test to see if this value is statistically significantly different to what one might expect by chance
The key aspect to hypothesis testing with surrogate data is the formulation of the null hypothesis, which will imply some measure of the data, which needs to be calculated for both data and surrogates

Summary

Introduction

There is no double-blind, replicated trial for a tsunami, an earthquake, a tornado or an ionospheric perturbation (Simpson, 1963). Despite the excellent preservation of the Fourier amplitudes, and while the surrogates and original data consist of exactly the same values, it is clear that there are some differences in the shape of the time series. Because the Fourier transform is periodic, a harmonic is propagated into the surrogates if there is a major discrepancy in the first and last values of the original data set (implying a dramatic, near-instantaneous change from the Nth value to the first in a periodic domain) The standard deviation of Rmτ ax(γ1, γ2) for 39 surrogates including those shown was 2 × 10−16 with a mean of 0.91 as required

Two-Dimensional Data

Undertaking Hypothesis Testing With Surrogate Data

Measures Adopted

Example Applications

Findings

Summary and Conclusion