Surrogates with random Fourier Phases

TL;DR

This paper investigates how existing surrogate data generation algorithms may inadvertently introduce phase correlations and proposes new methods that ensure true phase randomness at each iteration, improving the reliability of nonlinear data analysis.

Contribution

The paper identifies phase correlation issues in AAFT and IAAFT algorithms and introduces two new iterative methods that maintain phase randomness throughout.

Findings

Existing algorithms can produce surrogate data with phase correlations.

New methods successfully control phase randomization at every step.

Surrogates generated by new methods are truly linear and suitable for analysis.

Abstract

The method of surrogates is widely used in the field of nonlinear data analysis for testing for weak nonlinearities. The two most commonly used algorithms for generating surrogates are the amplitude adjusted Fourier transform (AAFT) and the iterated amplitude adjusted Fourier transfom (IAAFT) algorithm. Both the AAFT and IAAFT algorithm conserve the amplitude distribution in real space and reproduce the power spectrum (PS) of the original data set very accurately. The basic assumption in both algorithms is that higher-order correlations can be wiped out using a Fourier phase randomization procedure. In both cases, however, the randomness of the Fourier phases is only imposed before the (first) Fourier back tranformation. Until now, it has not been studied how the subsequent remapping and iteration steps may affect the randomness of the phases. Using the Lorenz system as an example, we show that both algorithms may create surrogate realizations containing Fourier phase correlations. We present two new iterative surrogate data generating methods being able to control the randomization of Fourier phases at every iteration step. The resulting surrogate realizations which are truly linear by construction display all properties needed for surrogate data.