On signal-noise decomposition of time-series using the continuous wavelet transform: application to sunspot index

Abstract

We show that the continuous wavelet transform can provide a unique decomposition of a time‐series into ‘signal‐like’ and ‘noise‐like’ components. From the overall wavelet spectrum, two mutually independent skeleton spectra can be extracted, allowing the separate detection and monitoring in even non‐stationary time‐series of the evolution of (i) both stable but also transient, evolving periodicities, such as the output of low‐dimensional dynamical systems, and (ii) scale‐invariant structures, such as discontinuities, self‐similar structures or noise. The idea of the method is to keep from the overall wavelet expansion of the time‐series only the wavelet components of locally maximal amplitude at any given time or scale, thus obtaining the instantly maximal and scale maximal wavelet skeleton spectrum, respectively. The scale maximal spectrum was previously proposed for studying possible multifractal scaling properties of time‐series. The instantly maximal spectrum proposed here exhibits clearer spectral peaks and reduced noise, as compared to the overall wavelet spectrum. An indicative application to the monthly‐averaged sunspot index reveals, apart from the well‐known 11‐yr periodicity, three of its harmonics, the 2‐yr periodicity (quasi‐biennial oscillation, QBO) and several more (some of which have been detected previously in various solar, Earth–solar connection and climate indices), here proposed as just harmonics of the QBO, in all supporting the double‐cycle solar magnetic dynamo model. The scale maximal spectrum reveals the presence of 1/f fluctuations with time‐scales up to 1 yr in the sunspot number, indicating that the solar magnetic configurations involved in the transient solar activity phenomena with those characteristic time‐scales are in a self‐organized critical state, as previously proposed for the solar flare occurrence.

Long-term solar activity explored with wavelet methods

“…Skeleton spectrum (Polygiannakis et al, 2003) can be derived from scalograms. The scale maximal wavelet skeleton spectrum keeps only those wavelet components which are locally of maximum amplitude at any given time scale.…”

“…Skeleton can be used to discover transients, the start and end of processes and how scales change with time (Polygiannakis et al, 2003). The scale maximal wavelet skeleton spectrum keeps only those wavelet components that are locally of maximum amplitude at any given time scale.…”

A history of solar activity over millennia

“…However, it has been shown (e.g., Rozelot 1994;Weiss and Tobias 2000;Charbonneau 2001;Mininni et al 2002) that such an oversimplified approach depends on the chosen reference time interval and does not adequately describe the long-term evolution of solar activity. A multi-harmonic representation is based on an assumption of the stationarity of the benchmark series, but this assumption is broadly invalid for solar activity (e.g., Kremliovsky 1994;Sello 2000;Polygiannakis et al 2003). Moreover, a multi-harmonic representation cannot, for an apparent reason, be extrapolated to a timescale larger than that covered by the benchmark series.…”

Massively parallel non-stationary EEG data processing on GPGPU platforms with Morlet continuous wavelet transform

“…Fligge et al [9] used MCWT to objectively determine the length of sunspot cycle and carry out error analysis on long-term solar activities, e.g., sunspot number, sunspot area. MCWT has also been employed to extract two complementary wavelet skeleton spectra to discriminate the components of periodicities and of hierarchies of discontinuities from several largesize time series represent solar activity records [27]. Piöft [26] has used MCWT to process four long Czech mean monthly temperature series from 1775 to 2001, and then the temperature variability in the Czech Republic has been examined.…”