A new surrogate data method for nonstationary time series

Hypothesis testing based on surrogate data has emerged as a popular way to test the null hypothesis that a signal is a realization of a linear stochastic process. Typically, this is done by generating surrogates which are made to conform to autocorrelation (power spectra) and amplitude distribution of the data (this is not necessary if data are Gaussian). Recently, a new algorithm was proposed, the null hypothesis addressed by this algorithm is that data are a realization of a non stationary linear stochastic process, surrogates generated by this algorithm preserve the autocorrelation and local mean and variance of data. Unfortunately, the assumption of Gaussian amplitude distribution is not always valid. Here we propose a new algorithm; the hypothesis addressed by our algorithm is that data are a realization of a nonlinear static transformation of a non stationary linear stochastic process. Surrogates generated by our algorithm preserve the autocorrelation, amplitude distribution and local mean and variance of data. We present some numerical examples where the previously proposed surrogate data methods fail, but our algorithm is able to discriminate between linear and nonlinear data, whether they are stationary or not. Using our algorithm we also confirm the presence of nonlinearity in the monthly global average temperature and in a small segment of a signal from a Micro Electrode Recording.

💡 Research Summary

Surrogate data methods have become a standard tool for testing whether a measured time series could be generated by a prescribed stochastic model. Classical approaches such as the Amplitude Adjusted Fourier Transform (AAFT) and its iterative refinement (IAAFT) generate surrogate series that share the original’s power spectrum (hence autocorrelation) and amplitude distribution. These techniques work well when the underlying process is stationary and, in the case of AAFT, when the data are Gaussian. However, many natural and biomedical signals are non‑stationary: their mean and variance drift over time, and their amplitude histograms are often far from Gaussian. In such cases, conventional surrogates either destroy the local statistical structure or implicitly assume a Gaussian amplitude distribution, leading to misleading hypothesis tests.

A recent surrogate algorithm attempted to address non‑stationarity by preserving the local mean and variance in addition to the power spectrum. While this method improves the handling of drifting statistics, it still relies on the assumption that the marginal amplitude distribution is Gaussian. Consequently, it fails when the data exhibit heavy tails, multimodality, or other non‑Gaussian features.

The present paper proposes a new surrogate‑generation scheme that targets a broader null hypothesis: the observed series is a nonlinear static transformation of a non‑stationary linear stochastic process. In other words, the data can be written as

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