Surrogates with random Fourier Phases

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.

💡 Research Summary

The paper revisits the surrogate‑data methodology that underpins most weak‑nonlinearity tests in time‑series analysis. The two standard algorithms—Amplitude Adjusted Fourier Transform (AAFT) and its iterated version (IAAFT)—are designed to preserve the original series’ amplitude distribution while reproducing its power spectrum. Both rely on a single randomisation of Fourier phases before the first inverse transform, assuming that this step eliminates all higher‑order correlations. The authors demonstrate that this assumption is flawed: the subsequent remapping of amplitudes (and, for IAAFT, the iterative refinement of the spectrum) can re‑introduce phase correlations, thereby contaminating the surrogate with residual nonlinear structure.

To expose the problem, the authors generate surrogates from a chaotic Lorenz‑system time series. By computing phase‑phase correlation functions, phase histograms, and complex‑phase autocorrelations, they show that both AAFT and IAAFT produce surrogates whose Fourier phases are not truly independent. Consequently, standard nonlinear statistics (mutual information, nonlinear prediction error, correlation dimension) applied to these surrogates can yield inflated values, leading to false‑positive detections of nonlinearity.

In response, the paper proposes two new iterative surrogate‑generation schemes that enforce phase randomness at every iteration. The first, Phase‑Controlled AAFT (PC‑AAFT), inserts a full phase‑randomisation step after each inverse Fourier transform before the amplitude‑remapping stage. The second, Iterative Phase‑Controlled IAAFT (IP‑IAAFT), augments the classic IAAFT loop with the same phase‑randomisation at each iteration, while still iteratively matching both the power spectrum and the empirical amplitude distribution. Convergence criteria are defined by the mean‑square error of the spectrum and a Kolmogorov‑Smirnov statistic for the amplitude distribution.

Extensive numerical tests reveal that both PC‑AAFT and IP‑IAAFT drive the phase‑correlation coefficients to values indistinguishable from zero, while preserving the target spectrum and distribution to within machine precision. When the newly generated surrogates are subjected to the same nonlinear diagnostics, the statistics match those of a purely linear Gaussian process, confirming that the surrogates are truly linear by construction. In contrast, the traditional AAFT/IAAFT surrogates display significant deviations, illustrating the risk of spurious nonlinearity detection.

The authors conclude that rigorous control of Fourier‑phase randomness is essential for reliable surrogate‑based hypothesis testing. Their phase‑controlled algorithms close a long‑standing gap in the methodology, offering a robust tool for fields ranging from climate science to biomedical signal analysis where accurate discrimination between linear stochasticity and genuine nonlinear dynamics is critical.