The temporal dynamics of climate processes are spread across different timescales and, as such, the study of these processes at only one selected timescale might not reveal the complete mechanisms and interactions within and between the (sub-)processes. To capture the non-linear interactions between climatic events, the method of event synchronization has found increasing attention recently. The main drawback with the present estimation of event synchronization is its restriction to analysing the time series at one reference timescale only. The study of event synchronization at multiple scales would be of great interest to comprehend the dynamics of the investigated climate processes. In this paper, the wavelet-based multi-scale event synchronization (MSES) method is proposed by combining the wavelet transform and event synchronization. Wavelets are used extensively to comprehend multi-scale processes and the dynamics of processes across various timescales. The proposed method allows the study of spatio-temporal patterns across different timescales. The method is tested on synthetic and real-world time series in order to check its replicability and applicability. The results indicate that MSES is able to capture relationships that exist between processes at different timescales.

Synchronization is a widespread phenomenon that can be observed in numerous climate-related processes, such as synchronized climate changes in the northern and southern polar regions (Rial, 2012), see-saw relationships between monsoon systems (Eroglu et al., 2016), or coherent fluctuations in flood activity across regions (Schmocker-Fackel and Naef, 2010) and among El Niño and the Indian summer monsoon (Maraun and Kurths, 2005; Mokhov et al., 2011). Synchronous occurrences of climate-related events can be of great societal relevance. The occurrence of strong precipitation or extreme runoff, for instance, at many locations within a short time period may overtax the disaster management capabilities.

Various methods for studying synchronization are available, based on recurrences (Marwan et al., 2007; Donner et al., 2010; Arnhold et al., 1999; Le Van Quyen et al., 1999; Quiroga et al., 2000, 2002; Schiff et al., 1996), phase differences (Schiff et al., 1996; Rosenblum et al., 1997), or the quasi-simultaneous appearance of events (Tass et al., 1998; Stolbova et al., 2014; Malik et al., 2012; Rheinwalt et al., 2016). For the latter, the method of event synchronization (ES) has received popularity owing to its simplicity, in particular within the fields of brain (Pfurtscheller and Silva 1999; Krause et al., 1996) and cardiovascular research (O'Connor et al., 2013), non-linear chaotic systems (Callahan et al., 1990), and climate sciences (Tass et al., 1998; Stolbova et al., 2014; Malik et al., 2012; Rheinwalt et al., 2016). ES has also been used to understand driver–response relationships, i.e. which process leads and possibly triggers another based on its asymmetric property. It has been shown that, for event-like data, ES delivers more robust results compared to classical measures such as correlation or coherence functions which are limited by the assumption of linearity (Liang et al., 2016).

Particularly in climate sciences, ES has been successfully applied to capture driver–response relationships, time delays between spatially distributed processes, strength of synchronization, and moisture source and rainfall propagation trajectories, and to determine typical spatio-temporal patterns in monsoon systems (Stolbova et al., 2014; Malik et al., 2012; Rheinwalt et al., 2016). Furthermore, extensions of the ES approach have been suggested to increase its robustness with respect to boundary effects (Stolbova et al., 2014; Malik et al., 2012) and number of events (Rheinwalt et al., 2016).

Even though ES has been successfully used, it is still limited by measuring the strength of the non-linear relationship at only one given temporal scale, i.e. it does not consider relationships at and between different temporal scales. However, climate-related processes typically show variability at a range of scales. Synchronization and interaction can occur at different temporal scales, as localized features, and can even change with time (Rathinasamy et al., 2014; Herlau et al., 2012; Steinhaeuser et al., 2012; Tsui et al., 2015). Features at a certain timescale might be hidden while examining the process at a different scale. Also, some of the natural processes are complex due to the presence of scale-emergent phenomena triggered by non-linear dynamical generating processes and long-range spatial and long-memory temporal relationships (Barrat et al., 2008). In addition, single-scale measures, such as correlation and ES, are valid and meaningful only for stationary systems. For non-stationary systems, they may underestimate or overestimate the strength of the relationship (Rathinasamy et al., 2014).

The wavelet transform can potentially convert a non-stationary time series into stationary components (Rathinasamy et al., 2014), and this can help in analysing non-stationary time series using the proposed method.

Therefore, the multi-scale analysis of climatic processes holds the promise of better understanding the system dynamics that may be missed when analysing processes at one timescale only (Perra et al., 2012; Miritello et al., 2013). According to this background, we propose a novel method, multi-scale event synchronization (MSES), which integrates ES and the wavelet approach in order to analyse synchronization between event time series at multiple temporal scales. To test the effectiveness of the proposed methodology, we apply it to several synthetic and real-world test cases.

The paper is organized as follows: Sect. 2 describes the proposed methodology and Sect. 3 introduces selected case studies. The results are discussed in Sect. 4. Conclusions are summarized in Sect. 5.

Here we describe the methodology for the proposed MSES approach. In this we combine two already well-established approaches (DWT and ES) to analyse synchronization at multiple temporal scales. The following sub-sections briefly introduce wavelets and ES and subsequently provide the mathematical framework for estimating MSES.

Wavelet analysis has become an important method in spectral analysis due to
its multi-resolution and localization capability in both time and frequency
domains. A wavelet transform converts a function (or signal) into another
form which makes certain features of the signal more amenable to study
(Addison, 2005). A wavelet

Depending on the way we sample parameters

Schematic showing the decomposition tree for signal

Using the dyadic grid wavelet, the DWT can be written as

The decomposition of the dyadic discrete wavelet is also associated with the
scaling function

The scaling function is orthonormal to the translation of itself, but not to
the dilation of itself.

The approximation coefficients at a specific scale

In this study, to calculate the synchronization at multiple scales, we only consider the approximation coefficients (not detail coefficients) at that particular scale because the aim is to separate the effects of time-localized features and high-frequency components from the signal.

For different

Scheme of multi-scale decomposition of signals using discrete wavelet transformation (DWT). The relationship between signal, approximate component, and detailed component is shown.

For simplicity we denote the approximation coefficient

To quantify the synchronous occurrence of events in different time series, we use the event synchronization (ES) method proposed by Quiroga et al. (2002). ES can be used for any time series in which we can define events, such as single-neuron recordings, eptiform spikes in EEGs, heart beats, stock market crashes, or abrupt weather events, such as heavy rainfall events. However, ES is not limited to this definition of events. It could also be applied to time series which are pure event time series (e.g. heart beats). In principle, when dealing with signals of different characters, the events could be defined differently in each time series, since their common cause might manifest itself differently in each (Quiroga et al., 2002). ES has advantages over other time-delayed correlation techniques (e.g. Pearson lag correlation), as it allows us to study interrelations between series of non-Gaussian data or data with heavy tails, or to use a dynamical (non-constant) time delay (Tass et al., 1998; Stolbova et al., 2014). The latter refers to a time delay that is dynamically adjusted according to the two time series being compared, which allows for better adaptation to the region of interest. Furthermore, ES has been specifically designed to calculate non-linear linkages between time series. Various modifications of ES have been proposed, such as solving the problems of boundary effects and bias due to an infinite number of events (Stolbova et al., 2014; Malik et al., 2012; Rheinwalt et al., 2016).

The modified algorithm proposed by Stolbova et al. (2014), Malik et
al. (2012), and Rheinwalt et al. (2016) works as follows: an event occurs in
signals

This definition of the time lag helps to separate independent events, as it
is the minimum time between two succeeding events. Then we count the number
of times

Multi-scale event synchronization (MSES) stepwise methodology.

Recalling Eq. (6), the scale-wise approximation at different scales 0, 1, 2,
…,

Figure 3 shows the stepwise methodology of multi-scale event synchronization.

To evaluate the statistical significance of ES values, a surrogate test will be used (Rheinwalt et al., 2016). We randomly reshuffle each time series 100 times (an arbitrary number). Reshuffling is done without replacement because estimating the expected number of simultaneous events in independent time series is equivalent to the combinatorial problem of sampling without replacement (Rheinwalt et al., 2016). Then, for each pair of time series, we calculate the MSES values for the different scales. At each scale, the empirical test distribution of the 100 MSES values for the reshuffled time series is compared to the MSES values of the original time series. Using a 1 % significance level, we assume that synchronization cannot be explained by chance, if the MSES value at a certain scale of the original time series is larger than the 99th percentile of the test distribution.

Wavelet power spectra (WPS) of the
test signals (Table 1). Panel I: original signal S1 (left) and S2 (right),
respectively, for case II(a); Panel II: original signal S1 (left) and S2
(right), respectively, for case II(b); Panel III: original signal S1 for
case III(a); Panel IV: original signal S1 for case III(b). In all the panels,
the y-axis represents the corresponding Fourier period

The proposed method is tested using synthetic and real-world data. The aim of these tests is to understand whether MSES is advantageous, compared to ES, in understanding the system interaction and the scale-emerging natural processes.

Following the approach of Rathinasamy et al. (2014), Yan and Gao (2007), and
Hu and Si (2016), we test MSES using a set of case studies including
stationary and non-stationary synthetic data. The details of the case studies
and the wavelet power spectra are given in Table 1 and Fig. 4, respectively.

A single synthetic stationary time series (

Here we generate two stationary signals consisting of partly
shared long-term oscillations and autoregressive (AR1) noise

presents two signals (Fig. 4, Panel II) with no common features across all
scales. Feature

Here, MSES is tested using non-stationary signals (Fig. 4, Panel III and IV) generated as
proposed by Yan and Gao (2007) and Hu and Si (2016). The signal encompasses
five cosine waves (

The time series of case III have features that are often found in climatic and geophysical data, where high-frequency, small-scale processes are superimposed on low-frequency, coarse-scale processes (Hu and Si, 2016). Such structures are widespread in time series of seismic signals, turbulence, air temperature, precipitation, hydrologic fluxes, or the El Niño–Southern Oscillation. They can also be found in spatial data, e.g. in ocean waves, seafloor bathymetry, or land surface topography (Hu and Si, 2016).

Details of synthetic test cases.

To test MSES with real-world data, we use precipitation data from stations in Germany (Fig. 5): 110 years of daily data, from 1 January 1901 to 31 December 2010, are available from various stations operated by the German Weather Service. Data processing and quality control were performed according to Österle et al. (2006).

We use daily rainfall data from the three stations: Kahl/Main, Freigericht-Somborn, and Hechingen (station ID: 20009, 20208, and 25005). Considering Kahl/Main (station 1) as the reference station, the distance to the other two stations, Freigericht-Somborn (station 2) and Hechingen (station 3), are 14.88 and 185.62 km, respectively (Fig. 5). Rainfall is a point process with large spatial and temporal discontinuities ranging from very weak to strong events within small temporal and spatial scales (Malik et al., 2012). This case explores the ability of MSES, in comparison to ES, to improve the understanding of synchronization given such time series features.

Geographical locations of rainfall stations considered in case study IV.

To evaluate the synchronization between two signals, which can be expressed in terms of events, at multiple scales, we decompose the given time series up to a maximum scale beyond which there is no significant number event. The number of events at a scale is a function of the nature of the time series and also the length of the time series under consideration. In most cases it was found that the number of events was significantly reduced after seven or eight levels of decomposition. We use the Haar wavelet as this is one of the simplest but most basic mother wavelets. There are several other mother wavelets which could be used for wavelet decomposition; however, it has been demonstrated that the choice of the mother wavelets does not affect the results to a great extent for rainfall (Rathinasamy et al., 2014).

In case I(a) the noise–signal ratio is quite high in the range of 2.7–3
(Table 1), such that the effect of the noise is felt up to scale 7 (Fig. 6).
Although both signals stem from the same parent source and hence ideally they
should possess perfect synchronization (ES

While repeating the same analysis but with a lower noise–signal ratio (i.e.
case I(b)), we find that the effect of noise is almost completely removed
after

The significance test (Sect. 2.4) underlines the high level of synchronization as indicated by the quite high ES values (Fig. 6). Based on this example we find that the MSES analysis captures the synchronization at multiple scales.

MSES values for case I(a) and case I(b), including significance test values for the significance level of 1 %. The value at scale 0 is equal to the single-scale ES analysis.

Case II(a) presents a system where synchronization between two signals exists
at a common long-term frequency (

For case II(b), we would expect that the ES value should be zero or
nonsignificant at scale

Interestingly, the MSES does not find significant synchronization at any
scale

As seen clearly, the ES at only one scale overpredicts the actual synchronicity between the two series. This behaviour may be due to the integrated effect of all scales, and hence some spurious synchronization (although rather small but still significant) is indicated.

Case III(a) is used as an analogue of dynamics and features of natural
processes (Table 1). Its WPS (Fig. 4, Panel III) shows non-stationary,
time-dependent features at higher scales

The similar case III(b) is used to investigate the behaviour of MSES in a scale-emerging process in a non-stationary regime (Table 1). As the wavelet spectrum of the signal reveals, only features at scales 5 and 6 are present (Fig. 4, Panel IV). The corresponding MSES values are significant only at those scales (Fig. 8b), revealing the synchronization at scales 5 and 6. This case illustrates that MSES reveals only the relevant timescales and does not mix them with the observation scale. In reality, there may be situations where the causative events act only at certain timescales and remain unconnected at other timescales. Under such situations MSES is useful for unravelling the relevant scale-emerging relationships.

After testing the efficacy of the proposed MSES approach by using some
prototypical situations, we apply the approach to real observed rainfall data
(case IV). We find significant ES values between station 1 and station 2 at
the scales

Applying ES in the traditional way, i.e. analysing only at scale 0, we find synchronization. However, only when we consider multiple scales are we able to find that the synchronization is the result of high- and low-frequency components present at scales 1, 5, and 7.

For station 1 and station 3 synchronization is significant at scale 7

The results for the real-world case study suggest that proximity of stations (station 1 and station 2) does not necessarily indicate synchronization at all scales. For stations 1 and 3, which are comparatively far from each other, we find insignificant synchronization at the observational scale. However, considering the scales separately, MSES detects significant synchronization at scale 7 as both stations might be sharing some common climatic cycle at this scale.

We have compared our novel MSES method with the traditional ES approach by systematically applying both methods to a range of prototypical situations. For test cases I and II we find that the ES value at the observation scale is influenced by noise, thereby reducing the ES values of two actually synchronized time series. When using MSES, the synchronization between the two time series can be much better detected even in the presence of strong noise. Another important aspect related to the analysis of these cases is that MSES has the ability to unravel synchronization between two stationary systems at timescales which are not obvious at the observation scale (scale-emerging processes). From these observations, it becomes clear that (i) event synchronization only at a single scale of reference is less robust, and (ii) the dependency measure of two given processes based on ES changes with the timescale depending on the features present in these processes.

Case study III illustrates that for a non-stationary system with synchronization changing over temporal scales, the single-scale ES is not robust. In contrast, MSES uncovers the underlying synchronization clearly. MSES is able to track the scale-emerging processes, scale of dominance in the process, and features present.

The real-world case study IV shows that the synchronization between climate time series can differ with temporal scales. The strength of synchronization as a function of temporal scale might result from different dynamics of the underlying processes. MSES has the ability to uncover the scale of dominance in the natural process.

Our series of test cases confirms the importance of applying a multi-scale view in order to investigate the relationship between processes that exist at different timescales. We suggest that investigating synchronization just at a single, i.e. observational, scale could give limited insight. The proposed extension offers the possibility of deciphering synchronization at different timescales, which is important in the case of climate systems where feedbacks and synchronization occur only at certain timescales and are absent at other scales.

We have proposed a novel method which combines wavelet transforms with event synchronization, thereby allowing us to investigate the synchronization between event time series at a range of temporal scales. Using a range of prototypical situations and a real-world case study, we have shown that the proposed methodology is superior compared to the traditional event synchronization method. MSES is able to provide more insight into the interaction between the analysed time series. Also, the effect of noise and local disturbance can be reduced to a greater extent and the underlying interrelationship becomes more prominent. This is attributed to the fact that wavelet decomposition provides a multi-resolution representation which helps to improve the estimation of synchronization. Another advantage of the proposed approach is its ability to deal with non-stationarity. Wavelets being made on local bases can pick up the non-stationary, transient features of a system, thereby improving the estimation of ES. Finally, it can be concluded that the proposed method is more robust and reliable than the traditional event synchronization in estimating the relationship between two processes.

The authors used Germany's precipitation data which is
maintained and provided by German Weather Service. The data is publicly
accessible at

The authors declare that they have no conflict of interest.

This research was funded by the Deutsche Forschungsgemeinschaft (DFG) (GRK
2043/1) within graduate research training group Natural risk in a changing
world (NatRiskChange) at the University of Potsdam
(