Time Series Surrogate Calculator

Why surrogates matter in time series physics

Surrogate data methods are widely used to test whether an observed pattern is genuinely nonlinear or can be explained by simpler constraints such as the amplitude distribution or the power spectrum. In experimental physics and complex systems, this supports hypothesis testing without assuming a specific parametric model. Surrogates are especially helpful when signals are noisy, short, or influenced by unknown processes.

Common null hypotheses and what they preserve

Different surrogate constructions correspond to different null hypotheses. A shuffle surrogate preserves the marginal distribution but removes temporal structure. Fourier-transform (phase randomized) surrogates preserve the magnitude of the spectrum, keeping linear autocorrelation structure consistent with a stationary linear process. AAFT and IAAFT add stronger control over the observed amplitude distribution.

Fourier magnitude preservation and spectral constraints

Phase randomization keeps |X(k)| fixed while drawing phases uniformly on [0, 2π). This conserves the power spectrum and therefore preserves second-order statistics closely tied to autocorrelation. In practice, the FFT implementation pads to the next power of two for speed, then truncates back to your original length after inversion.

AAFT: matching distribution with approximate spectrum control

AAFT first maps your samples to a Gaussian distribution using ranks, then applies phase randomization, and finally remaps values back to the original sorted amplitudes. This provides a practical compromise for nonlinear testing, although the final spectrum matching is approximate and can deviate for strongly non-Gaussian signals.

IAAFT: iterative refinement for stronger matching

IAAFT alternates two projections: enforcing the target spectral magnitudes and enforcing the original rank order of amplitudes. With more iterations, the surrogate typically converges toward tighter agreement in both constraints. For many datasets, 50–200 iterations gives good stability, while very long signals may benefit from more.

Interpreting output statistics for quick validation

The calculator reports mean, standard deviation, and lag-1 autocorrelation for the original and each surrogate. For shuffle surrogates, autocorrelation should drop toward zero. For FT/AAFT/IAAFT, autocorrelation should remain similar to the original if the signal is approximately stationary. Large deviations can indicate short length, heavy tails, or preprocessing choices.

Practical guidance on preprocessing and parameters

Removing the mean is common when analyzing oscillatory or centered fluctuations. Standardization (z-score) can help compare across experiments, but it changes the absolute amplitude scale. When using a seed, you can reproduce exact surrogate sets for publications. Increase surrogate count to improve Monte Carlo resolution in p-value estimates.

Limitations and responsible use

Surrogates assume your series is reasonably stationary over the analyzed window. Strong trends, abrupt regime shifts, or missing data can bias both the spectrum and rank mapping. When these occur, consider detrending or analyzing segments. Always match the surrogate method to the null hypothesis you truly want to test.

FAQs

1) Which surrogate method should I choose?

Use shuffle to remove time dependence, FT to preserve linear correlations, AAFT for approximate distribution matching, and IAAFT for stronger matching of both spectrum and amplitude ranks.

2) What does “phase randomized” preserve?

It preserves the Fourier magnitude spectrum, which largely preserves second-order structure like autocorrelation for stationary linear processes, while destroying phase-dependent nonlinear structure.

3) Why do IAAFT iterations matter?

More iterations usually improve agreement with both the original spectrum and the original amplitude distribution. Too few iterations may leave noticeable mismatches, especially for non-Gaussian signals.

4) How many surrogates are enough for testing?

For quick checks, 19–39 surrogates can help. For tighter p-values, use 99 or more if performance permits. More surrogates reduce Monte Carlo uncertainty.

5) My lag-1 autocorrelation changes a lot. Is that normal?

Large changes often indicate a short series, strong nonstationarity, or a heavy-tailed distribution. Try removing trends, using more samples, or choosing IAAFT with more iterations.

6) Does standardization change the surrogate meaning?

Standardization rescales values and can help comparisons, but it changes absolute amplitudes. It typically does not harm spectral matching, yet it can affect interpretation if physical units matter.

7) Can I export all surrogate values?

Yes. The CSV export includes the original series and every surrogate with index labels. The PDF option prints the on-screen report, including statistics and the preview table.