Uncovering meaningful regularities in complex oscillatory signals is a challenging problem with applications across a wide range of disciplines. Here we present a novel approach, based on the Hilbert transform (HT). We show that temporal periodicity can be uncovered by averaging the signal in a moving window of appropriated length, $tau$, before applying the HT. By analyzing the variation of the mean rotation period, $overline{T}$, of the Hilbert phase as a function of $tau$, we discover well-defined plateaus. In many geographical regions the plateau corresponds to the expected one-year solar cycle; however, in regions where SAT dynamics is highly irregular, the plateaus reveal non-trivial periodicities, which can be interpreted in terms of climatic phenomena such as El Ni~no. In these regions, we also find that Fourier analysis is unable to detect the periodicity that emerges when $tau$ increases and gradually washes out SAT variability. The values of $overline{T}$ obtained for different $tau$s are then given to a standard machine learning algorithm. The results demonstrate that these features are informative and constitute a new approach for SAT time series classification. To support these results, we analyse synthetic time series generated with a simple model and confirm that our method extracts information that is fully consistent with our knowledge of the model that generates the data. Remarkably, the variation of $overline{T}$ with $tau$ in the synthetic data is similar to that observed in real SAT data. This suggests that our model contains the basic mechanisms underlying the unveiled periodicities. Our results demonstrate that Hilbert analysis combined with temporal averaging is a powerful new tool for discovering hidden temporal regularity in complex oscillatory signals.
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