Multiscale structure of time series revealed by the monotony spectrum

Observation of complex systems produces time series with specific dynamics at different time scales. The majority of the existing numerical methods for multiscale analysis first decompose the time series into several simpler components and the multiscale structure is given by the properties of their components. We present a numerical method which describes the multiscale structure of arbitrary time series without decomposing them. It is based on the monotony spectrum defined as the variation of the mean amplitude of the monotonic segments with respect to the mean local time scale during successive averagings of the time series, the local time scales being the durations of the monotonic segments. The maxima of the monotony spectrum indicate the time scales which dominate the variations of the time series. We show that the monotony spectrum can correctly analyze a diversity of artificial time series and can discriminate the existence of deterministic variations at large time scales from the random fluctuations. As an application we analyze the multifractal structure of some hydrological time series.

Figure 1

(a) A noisy time series and some of its averaged forms. For the readability of the graph the plots are displaced vertically by the quantities ${\delta }_i\in \{2,4,6,7\}$. (b) MS of the time series in panel (a) ($\circ$ markers) and the noneffective MMVs ($\times$ markers). The local extrema ($•$ markers) correspond to the smoothed time series plotted in panel (a).

Figure 2

MSs of the trend of the time series in Fig. 1 for three different samplings characterized by their time steps. The continuous line with point markers is the MS in Fig. 1.

Figure 3

Frequencies estimated by the MS for the sinusoidal (a) and nonsinusoidal (b) components, as a function of $\varphi$. The continuous lines mark the principal frequencies and the dashed line one of the secondary frequencies of the nonsinusoidal component.

Figure 4

(a) Global MS of 100 Gaussian white noises. (b) Relative frequencies of the number of the monotonic segments of a Gaussian white noise with $N=20000$ with respect to their lengths for the first four effective averagings.

Figure 5

Autocorrelation function induced on an i.i.d. time series by the averaging (1) with $K_i=1$ for several values of $i$.

Figure 6

(a) The average MSs of statistical ensembles with 1000 time series of AR(1) type with different values of $\varphi <1$ and 1000 Brownian motions ($\varphi =1$). (b) The relative standard deviations of ${\langle A^{*}\rangle }_k$.

Figure 7

The average MSs of statistical ensembles with 1000 time series obtained by superposing a Gaussian white noise on a deterministic trend with different amplitudes.

Figure 8

The average MSs of the signal in Fig. 3 for $\varphi =0$ on which Gaussian white noises with different amplitudes are superposed. The continuous lines mark the semiperiods of the largest terms in Eq. (3), while the dashed lines mark those of the small terms.

Figure 9

The average MSs of statistical ensembles with 1000 discrete fBms with different values $H$ and of two Brownian motions ($H=0.5$) on which a linear trend (asterisk markers) and a sinusoid ($x$ markers) are superposed.

Figure 10

Three average MSs of statistical ensembles of multifractal time series.