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1 Colored Noise and MC-SSA

1∕f β power-law noise is known to be highly relevant in several astrophysical systems [3–5]. It includes the well-known white noise (β = 0), pink noise (β = 1) and red or Brownian noise (β = 2). Such kind of noises are very often not tied to instrumental disturbance but rather they are property of the observed sources that emit radiation varying in a stochastic manner. In the context of our work, it is sufficient to qualitatively characterize 1∕f β to generate a parsimonious hypothesis test in MC-SSA (see below). Subsequently, the dynamical origin of noise fluctuation or the extra-noise variance – according to acceptance or rejection of our null hypothesis – has to be determined by theoretical analysis.

Singular Spectrum Analysis (SSA) is an effective, data-adaptive and non-parametric method for the decomposition of a time series into a well-defined set of independent and interpretable components that include a non-linear trend, anharmonic, amplitude-modulated oscillations, and noise [6]. In its later developments, the Monte Carlo approach to signal-to-noise separation introduced by [1] has become known as Monte Carlo SSA (MC-SSA) [2]. The distinction between what is noise and what is signal is made through the measure of the resemblance of a given noise surrogate to the original data via eigendecomposition of the time-lagged covariance matrix. For this study the noise surrogates set is generated by using an AR(1) process. In practice, the AR(1) coefficients are estimated from the time-series under consideration by using a maximum-likelihood criterion. Subsequently, these AR(1) noise surrogates are corrupted by Poisson noise to mimic the effect of a photon counting detector.

2 Simulated 1∕f β Series Test

The procedure has been tested on a large sample of artificial time series of arbitrary colored noises. As a test of validity, we show several discrete colored noise vectors of length N = 5,000, with a power-law distribution of slope β ranging from 1 to 2 in steps of two. To obtain sequences of colored noises we use the Matlab library CNOISE.Footnote 1 A threshold of signal identification is incorporated by subtracting a mean value \(\sim \) 35 % of the maximum peak-flux, C max . In this way, we take into account episodes of emission during which the emission count rate drops to the background level. Finally, the time series are corrupted by Poisson noise to simulate the effect of the shot noise based on Poisson photon statistics. The resulting test series, are shown in the supplementary video.Footnote 2 The signal to noise ratio (SNR) of the artificial series test increases as the simulations run.

3 Conclusion and Future Work

We present a modified data-adaptive spectral method – Monte Carlo Singular Spectrum Analysis (MC-SSA) – in which a parsimonious (dynamic and instrumental) noise-model H 0 is adopted for astrophysical applications, i.e. H 0 = {AR(1) + Poisson noise}. AR(1) noise takes into account the long-term variability in the power-law slope −β of a light-curve’s spectral density, and the Poisson noise considers the short-term variability imposed by the quantum nature of light. Our simulations show a remarkable effectiveness of the model for colored noises with β values between 1.5 and 2 and with C max   >  1,000 counts; otherwise one should re-bin the light-curve to improve the signal-to-noise ratio of the source.

All analyses were performed by using SSA-MTM Toolkit freeware that has been developed by the Theoretical Climate Dynamics Group at UCLA.Footnote 3 To obtain presented results we relied on an advanced option in SSA-MTM Toolkit for MC-SSA that allows to read in ensemble of surrogates created off-line and corresponding to an arbitrary noise null-hypothesis H 0. In this regard, MC-SSA allows to iteratively refine the working hypothesis and adapt the null-hypothesis H 0 for various theoretical scenarios. We plan to extend further SSA-MTM Toolkit for astrophysical applications by directly including the algorithm for parsimonious H 0 in the freeware. Current and future work will focus on SSA as a promising technique for analysing complex spatio-temporal structures to detect possible periodicities and other statistical regularities due to various astrophysical processes.