NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-599-2017Multi-scale event synchronization analysis for unravelling climate
processes: a wavelet-based approachAgarwalAnkitaagarwal@uni-potsdam.deMarwanNorberthttps://orcid.org/0000-0003-1437-7039RathinasamyMaheswaranMerzBrunohttps://orcid.org/0000-0002-5992-1440KurthsJürgenUniversity of Potsdam, Institute of Earth and Environmental Science,
Karl-Liebknecht-Strasse 24–25, 14476 Potsdam, GermanyPotsdam Institute for Climate Impact Research, P.O. Box 60 12 03,
14412 Potsdam, GermanyGFZ German Research Centre for Geosciences, Section 5.4: Hydrology,
Telegrafenberg, Potsdam, GermanyCivil engineering department, MVGR college of Engineering, Vizianagaram, IndiaInstitute of Applied Physics of the Russian Academy of Sciences, 46
Ulyanova St., Nizhny Novgorod 603950, RussiaAnkit Agarwal (aagarwal@uni-potsdam.de)13October20172445996113May201712June20171September20179September2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/599/2017/npg-24-599-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/599/2017/npg-24-599-2017.pdf
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.
Introduction
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.
Methods
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.
Discrete wavelet transform
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 ψ(t) is a localized function which satisfies
certain admissibility conditions. The wavelet transform Ta,bx of a continuous function x(t) can be defined as a simple
convolution between x(t) and dilated and translated versions of the mother
wavelet ψ(t):
Ta,bx=∫-∞∞xtψa,btdt,
where a and b refer to the scale and location variables (real numbers)
and ψa,b is defined as
ψa,bt=1aψt-ba.
Depending on the way we sample parameters a and b, we get either a
continuous wavelet transform (CWT) or a discrete wavelet transform (DWT). A
natural way to sample a and b is to use a logarithmic discretization of
the scale and link this in turn to the size of steps taken between b
locations. This kind of discretization of the wavelet has the form
ψλ,qt=1a0λψt-qboaoλaoλ
where the integers λ and q control the wavelet dilation and
translation, respectively; ao is a specified fixed dilation step
parameter and bo>0 is the location parameter. The general choices of the
discrete wavelet parameters ao and bo are 2 and 1, respectively.
This is known as dyadic grid arrangement.
Schematic showing the decomposition tree for signal Xt
using DWT.
Using the dyadic grid wavelet, the DWT can be written as
Tλ,q=∫-∞∞xt1aoλψ(t-qboaoλaoλ)dt.Substitutinga0=2andbo=1,wegetTλ,q=∫-∞∞xt2-λ/2ψ2-λt-qdt
where Tλ,q are the discrete wavelet transform values given on
scale-location grid indexes λ and q. For the DWT, the values
Tλ,q are known as wavelet coefficients or detail coefficients.
The decomposition of the dyadic discrete wavelet is also associated with the
scaling function ϕλ,qt (Eq. 5) which represents
the smoothing of the signal and has the same form as the wavelet, given by
(Addison, 2005)
ϕλ,qt=2-λ2ϕ(2-λt-q).
The scaling function is orthonormal to the translation of itself, but not to
the dilation of itself. ϕλ,qt can be
convolved with the signal to produce approximation coefficients at a given scale as follows:
Aλ,q=∫-∞∞xtϕλ,qtdt.
The approximation coefficients at a specific scale λ are known as a
discrete approximation of the signal at that scale. As proven in Mallat
(1989), the wavelet function and the scaling function form multi-resolution
bases resulting in a pyramidal algorithm. The decomposition methodology is
schematically shown in Fig. 1.
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 λ=1, 2, 3, …, the approximation
coefficients Aλ correspond to the “coarse-grained” original
signal after removal of the details at scales λ, λ-1,…,1. In practical terms, considering a daily climatic time series at λ=0, the time series represents the original observations. At λ=1,
A1 represents the features beyond the 2-day scale (wavelet scale) which
is obtained by extracting T1 (2-day features) from the original time
series. Similarly, at λ=3, A3 represents the climatic variable
beyond the 8-day scale and is obtained after removing T1, T2, and
T3 (2-, 4-, and 8-day features) from the original signal. In essence,
A1, A2, A3,… represent the original signal at different
timescales. The schematic plot explaining the procedure and relationship
between signal, approximate component, and detailed component has been shown
in Fig. 2.
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 Aλ,q of
the signal xt at scale λ as xλ.
Event synchronization
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 xt and yt at time tlx and
tmy, where l=1,2,3,4,…Sx, m=1,2,3,4,…Sy,
and Sx and Sy are the total number of events, respectively. In
our study, we derive events from a more-or-less continuous time series by
selecting all time steps with values above a threshold (α=95th
percentile). These events in xt and yt are
considered synchronized when they occur within a time lag ±τlmxy which is defined as follows:
τlmxy=mintl+1x-tlx,tlx-tl-1x,tm+1y-tmy,tmy-tm-1y}/2.
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 Cx|y an event occurs in x(t) after it appears in y(t)and vice versa (Cy|x):
Cx|y=∑l=1Sx∑m=1SyJxy
and
Jxy=1if0<tlx-tmy<τlmxy12iftlx=tmy0else.C(y|x) is calculated analogously but with exchanged x and y. From
these quantities we obtain the symmetric measure:
Qxy=Cx|y+Cy|x(Sx-2)(Sy-2).Qxy is a measure of the strength of event synchronization between
signals x(t) and y(t). It is normalized to 0≤Qxy≤1, with
Qxy=1 for perfect synchronization (coincidence of extreme events)
between signals x(t) and yt.
Multi-scale event synchronization (MSES) stepwise methodology.
(a) Signal 1 and its decomposed component along with corresponding
event series after applying the (95th percentile) threshold.
(b) Same for signal 2. (c) Event synchronization values
corresponding to each scale.
Recalling Eq. (6), the scale-wise approximation at different scales 0, 1, 2,
…, λ for any given time series xt is
given by xλ=Aλ,q where xλ
represents the approximation coefficients of signal xt at scale λ. Now, to determine the synchronization between
any two time series xt and y(t) at multiple scales,
the event synchronization is estimated between the scaled versions of
xt and y(t) for different λ resulting in
multi-scale event synchronization (MSES). The normalized strength of MSES
between signals x(t) and yt at scale λ is
then defined as
Qxλ,yλ=Cxλ|yλ+Cyλ|xλSxλ-2Syλ-2.Qxλyλ=1 for perfect synchronization, and
Qxλ,yλ=0 suggests the absence of any
synchronization at scale λ between x(t) and yt.
Figure 3 shows the stepwise methodology of multi-scale event synchronization.
Significance test for MSES
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 = 2λ.
Data and study design to test MSES
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.
Testing MSES with synthetic data
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.Case I.
A single synthetic stationary time series (S) is generated and
contaminated with two random white noise time series. Two sub-cases with
different noise–signal ratios are investigated (Table 1). This case allows
understanding of how the synchronization between two series is affected by
the presence of noise or high-frequency features. For climate variables such
situations can emerge when two signals originate from the same parent source
or mechanism (e.g. identical large-scale climatic mode, identical storm
tracks) but get covered by high-frequency fluctuations arising from local
features.
Case II(a).
Here we generate two stationary signals consisting of partly
shared long-term oscillations and autoregressive (AR1) noise St (see
Table 1). The long-term oscillations y1, y2, y3, and y4 have periods
of 16, 32, 64, and 128 units, respectively (Fig. 4, Panel I). The purpose of
case II(a) is to test the ability of MSES to identify synchronization in
processes which originate from different parent sources or different
mechanisms (e.g. two different climatic process, different storm tracks) but
have some common features (y1 and y4) at coarser scales.
Case II(b)
presents two signals (Fig. 4, Panel II) with no common features across all
scales. Feature y2 in signal S1 and feature y4 in signal S2 represent a
long-term oscillation of period 32 and 128 units, respectively. The idea is
to investigate the possibility of overprediction of synchronization if we
analyse at one scale only.
Case III.
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 (z1 to z5), whereas the square root of the location
term results in a gradual change in frequency. Two combinations are generated
of which case III(a) investigates the ability of MSES to deal with
non-stationarity signals. Case III(b) examines the capability of MSES to
capture processes emerging at lower scales (in this case at scales 5 and 6)
in the presence of short-lived transient features. For both combinations, the
signal is contaminated with white noise.
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.
CaseMathematical expressionOther detailsReferences and figuresI(a)Sinusoidal stationary signal S1=S+Strongnoise1S=sin((2πt)/50)+cos((2πt)/60)Noise1signal∼2.8Rathinasamy etal. (2014)I(b)Sinusoidal stationary signal S2=S+Weaknoise2Noise2signal∼5Rathinasamy etal. (2014)II(a)Stationary signal (S1 and S2) S1=St1+y1+y2+y4; S2=St2+y1+y3+y4Two AR1 processes St=∅St-1+ϵtεt=uncorrelatedrandomnoise Parameter {∅1=0.60;∅2=.70}Yan and Gao (2007),Hu and Si (2016) Fig. 3: Panel III(b)Stationary dataset (S1 and S2) S1=y2+St1S2=y4+St2y1=sin2πt16;y2=sin2πt32; y3=sin2πt64; y4=sin2πt128; where t=1,2,3,…40177.Yan and Gao (2007),Hu and Si (2016) Fig. 3: Panel IIIII(a)Non-stationary dataset S1=z1+z2+z3+z4+z5S2=S1+randomnoise(uncorrelated)noisesignal∼2.781; where t=1,2,3,…40177.Z1=cos500πt10000.5, Z2=cos250πt10000.5, Z3=cos125πt10000.5, Z4=cos62.5πt10000.5, Z5=cos31.25πt10000.5.Yan and Gao (2007),Hu and Si (2016) Fig. 3: Panel IIIIII(b)Non-stationary dataset S1=z4+z5S2=S1+randomnoise(uncorrelated)noisesignal∼21.5664;where t=1,2,3,…40177.Z4=cos62.5πt10000.5, Z5=cos31.25πt10000.5.Yan and Gao (2007),Hu and Si (2016) Fig. 3: Panel IVTesting MSES with real-world data
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).
Case IV.
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.
Results
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 ∼ 1) at all scales, the ES
value at the observational scale (λ=0) is moderate (∼ 0.7),
leading to the interpretation that both signals are only weakly synchronized.
In contrast, the proposed MSES approach is able to capture the underlying
features (which were hidden in the original signal) at higher scales (λ≥1) by approaching ES values of 1, indicating the actual
synchronization between these signals. At the scale λ=0 the ES
measure is lower because of the heavy noise covering the underlying
information. Considering higher scales, the effect of noise is removed
through wavelet decomposition, allowing for a more reliable identification of
the actual underlying synchronization between the signals. Interestingly, the
slight decrease in the ES values at a high scale (λ≥7) (Fig. 6)
might indicate that the essential feature that is responsible for the
synchronization at that scale gets removed in the form of a detail component
(Fig. 2). If features are present at a particular scale λ and when
we go up to the next scale (λ+1), those features get removed in the
form of the details and essentially the synchronization is lost at the scale
λ+1.
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 (λ>3) and the MSES values remain unaltered because of the same
signal structure (Fig. 6). These findings confirm that the MSES approach is
able to capture the synchronization in the presence of noise.
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.
(a, b) MSES and significance level (1 %) values at
different scales for cases II(a) and II(b). The value at scale 0 is equal to
the single-scale ES analysis.
(a, b) MSES values and significance level (1 %) at
different scales for cases III(a) and III(b). The value at scale 0 is equal
to the single-scale ES analysis.
(a, b) MSES and significance level (1 %) values at
various scales for stations 1 and 2 and stations 1 and 3, respectively;
(c, d, e) WPS of precipitation of stations 1 (c), 2
(d), and 3 (e) (station ID: 20009, 20208, 25005),
respectively; (f, g, h) global wavelet spectrum of the same
stations. In (c)–(h) the y-axis represents the
corresponding Fourier period =2λ.
Case II(a) presents a system where synchronization between two signals exists
at a common long-term frequency (y1 and y4). This is particularly
relevant in studying the rainfall processes of two different regions, which
are governed by different local climatic processes but similar long-term
oscillations such as ENSO cycles. The MSES values (λ=0 to 7) are
smaller than the confidence level, except for scales 4 and 7 (Fig. 7a). The
synchronization emerging at scale 4 (λ=4) and scale 7 (λ=7)
corresponds to features present at those scales shown in the wavelet power
spectrum (Fig. 5, Panel I). The thick contour in the WPS indicates the
presence of significant features (at the 5 % significance level)
corresponding to y1, y2, y3, and y4 (Table 1). In the same figure,
the dashed curve represents the cone of influence (COI) of the wavelet
analysis. Outside of this region edge effects become more influential. Any
peak falling outside the COI has presumably been reduced in magnitude due to
zero padding necessary to deal with the finite length of the time series. To
test the statistical significance of WPS, a background Fourier spectrum is
chosen (Addison, 2005; Agarwal et al., 2016a, b).
For case II(b), we would expect that the ES value should be zero or
nonsignificant at scale λ=0. However, we find that the
synchronization between S1 and S2 at scale λ=0 is significant
(Fig. 7b), although there is no common feature by construction (Fig. 3,
Panel II).
Interestingly, the MSES does not find significant synchronization at any
scale (λ>0). Moreover, the MSES values become zero after scale 4
because signals S1 and S2 have no common feature beyond these scales.
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 2≤λ≤6. ES values at
lower scales λ≤1 are below the significance level, revealing
that the two signals are not synchronized (Fig. 8a). The ES for the signal
components of the larger timescales reveals significant synchronization up to
scale 6, which is expected because of the common features (scale 2 to
scale 6) in S1 and S2. After scale λ=6, the MSES value drops below
the significance level as the features responsible for synchronization are
removed in the form of the details component during decomposition. Results
from this case show the wavelet's ability in capturing the underlying
multiple non-stationarities that are common in both the time series which
otherwise go unnoticed using ES at the observation scale.
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 λ=1, 5, and 7 (Fig. 9a) by tracking the features present
in the WPS (Fig. 9c, d, and e). The significant ES value at the observational
scale (λ=0) might be due to the integrated effect of features
present at coarser scales (λ=1, 5, and 7). In order to emphasize the
features present in the data, we use the global wavelet spectrum (Fig. 9f, g,
and h) which is defined as the time average of the WPS (Agarwal et al.,
2016a, b; Mallat, 1989).
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 λ=7 (Fig. 9b). However, evaluating the ES in the
traditional way (i.e. λ=0) leads to the conclusion that both
stations are not significantly synchronized. Here, MSES plays a critical role
in identifying synchronization at specific temporal scales. Hence, MSES
provides further insights into the process, such as low-frequency features
that are present and the dominating scales causing the significant
synchronization at scale 0.
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.
Discussion
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.
Conclusions
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 https://opendata.dwd.de/. Further, preprocessing of the
data was done by Potsdam Institute for Climate Impact Research (Conradt et
al., 2012; Oesterle, 2001).
The authors declare that they have no conflict of
interest.
Acknowledgements
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
(http://www.uni-potsdam.de/natriskchange) and RSF support (support by
the Russian Science Foundation (grant no. 16-12-10198)). The third author
acknowledges the research funding from the Inspire Faculty Award, Department
of Science and Technology, India, for carrying out the research. Also, we
gratefully acknowledge the provision of precipitation data by the German
Weather Service. Edited by: Stéphane
Vannitsem Reviewed by: two anonymous referees
References
Addison, P. S.: Wavelet transforms and the ECG: a review, Physiol. Meas., 26,
R155, 2005.
Agarwal, A., Maheswaran, R., Sehgal, V., Khosa, R., Sivakumar, B., and
Bernhofer, C.: Hydrologic regionalization using wavelet-based multiscale
entropy method, J. Hydrol., 538, 22–32, 2016a.Agarwal, A., Maheswaran, R., Kurths, J., and Khosa, R.: Wavelet Spectrum and
Self-Organizing Maps-Based Approach for Hydrologic Regionalization – a Case
Study in the Western United States, Water Resour. Manag., 30, 4399–4413,
10.1007/s11269-016-1428-1, 2016b.
Arnhold, J., Grassberger, P., Lehnertz, K., and Elger, C. E.: A robust
method for detecting interdependences: application to intracranially
recorded EEG, Physica D, 134, 419–430, 1999.
Barrat, A., Barthelemy, M., and Vespignani, A.: Dynamical processes on
complex networks, Cambridge university press, 2008.
Callahan, D., Kennedy, K., and Subhlok, J.: Analysis of event
synchronization in a parallel programming tool, ACM SIGPLAN Notices, 21–30,
1990.Conradt, T., Koch, H., Hattermann, F. F., and Wechsung, F.: Precipitation or
evapotranspiration? Bayesian analysis of potential error sources in the
simulation of sub-basin discharges in the Czech Elbe River basin, Reg.
Environ. Change, 12, 649–661, 10.1007/s10113-012-0280-y, 2012.Donner, R. V., Zou, Y., Donges, J. F., Marwan, N., and Kurths, J.:
Recurrence networks – a novel paradigm for nonlinear time series analysis,
New J. Phys., 12, 033025, 10.1088/1367-2630/12/3/033025, 2010.Eroglu, D., McRobie, F. H., Ozken, I., Stemler, T., Wyrwoll, K.-H.,
Breitenbach, S. F., Marwan, N., and Kurths, J.: See-saw relationship of the
Holocene East Asian-Australian summer monsoon, Nat. Commun., 7, 12929, 10.1038/ncomms12929,
2016.
Herlau, T., Mørup, M., Schmidt, M. N., and Hansen, L. K.: Modelling dense
relational data, Machine Learning for Signal Processing (MLSP), IEEE
International Workshop, 1–6, 2012,Hu, W. and Si, B. C.: Technical note: Multiple wavelet coherence for
untangling scale-specific and localized multivariate relationships in
geosciences, Hydrol. Earth Syst. Sci., 20, 3183–3191,
10.5194/hess-20-3183-2016, 2016.
Krause, C. M., Lang, A. H., Laine, M., Kuusisto, M., and Pörn, B.:
Event-related. EEG desynchronization and synchronization during an auditory
memory task, Electroen. Clin. Neuro., 98, 319–326, 1996.
Le Van Quyen, M., Martinerie, J., Adam, C., and Varela, F. J.: Nonlinear
analyses of interictal EEG map the brain interdependences in human focal
epilepsy, Physica D, 127, 250–266, 1999.
Liang, Z., Ren, Y., Yan, J., Li, D., Voss, L. J., Sleigh, J. W., and Li, X.:
A comparison of different synchronization measures in electroencephalogram
during propofol anesthesia, J. Clin. Monitor. Comp., 30, 451–466, 2016.
Malik, N., Bookhagen, B., Marwan, N., and Kurths, J.: Analysis of spatial and
temporal extreme monsoonal rainfall over South Asia using complex networks,
Clim. Dynam., 39, 971–987, 2012.
Mallat, S. G.: A theory for multiresolution signal decomposition: the wavelet
representation, IEEE T. Pattern Anal., 11, 674–693, 1989.Maraun, D. and Kurths, J.: Epochs of phase coherence between El Nino/Southern
Oscillation and Indian monsoon, Geophys. Res. Lett., 32, 10.1029/2005GL023225, 2005.
Marwan, N., Romano, M. C., Thiel, M., and Kurths, J.: Recurrence plots for
the analysis of complex systems, Phys. Rep., 438, 237–329, 2007.
Miritello, G., Moro, E., Lara, R., Martínez-López, R., Belchamber,
J., Roberts, S. G., and Dunbar, R. I.: Time as a limited resource:
Communication strategy in mobile phone networks, Soc. Networks, 35, 89–95,
2013.Mokhov, I. I., Smirnov, D. A., Nakonechny, P. I., Kozlenko, S. S., Seleznev,
E. P., and Kurths, J.: Alternating mutual influence of El-Niño/Southern
Oscillation and Indian monsoon, Geophys. Res. Lett., 38, 47–56, 10.1134/S0001433812010082, 2011.O'Connor, J. M., Pretorius, P. H., Johnson, K., and King, M. A.: A method to
synchronize signals from multiple patient monitoring devices through a single
input channel for inclusion in list-mode acquisitions, Med. Phys.,
40, 122502, 10.1118/1.4828844, 2013.Oesterle, H.: Reconstruction of daily global radiation for past years for use
in agricultural models, Phys. Chem. Earth Part B, 26, 253–256,
10.1016/S1464-1909(00)00248-3, 2001.
Österle, H., Werner, P., and Gerstengarbe, F.: Qualitätsprüfung,
Ergänzung und Homogenisierung der täglichen Datenreihen in
Deutschland, 1951–2003: ein neuer Datensatz, 7. Deutsche Klimatagung,
Klimatrends: Vergangenheit und Zukunft, 9.–11. Oktober 2006, München,
2006.
Perra, N., Gonçalves, B., Pastor-Satorras, R., and Vespignani, A.:
Activity driven modeling of time varying networks, arXiv preprint
arXiv:1203.5351, 2012.
Pfurtscheller, G. and Da Silva, F. L.: Event-related EEG/MEG synchronization
and desynchronization: basic principles, Clin. Neurophysiol., 110,
1842–1857, 1999.Quiroga, R. Q., Arnhold, J., and Grassberger, P.: Learning driver-response
relationships from synchronization patterns, Phys. Rev. E, 61,
5142, 10.1103/PhysRevE.61.5142, 2000.Quiroga, R. Q., Kraskov, A., Kreuz, T., and Grassberger, P.: Performance of
different synchronization measures in real data: a case study on
electroencephalographic signals, Phys. Rev. E, 65, 041903, 10.1103/PhysRevE.65.041903, 2002.
Rathinasamy, M., Khosa, R., Adamowski, J., Partheepan, G., Anand, J., and
Narsimlu, B.: Wavelet-based multiscale performance analysis: An approach to
assess and improve hydrological models, Water Resour. Res., 50, 9721–9737,
2014.
Rheinwalt, A., Boers, N., Marwan, N., Kurths, J., Hoffmann, P., Gerstengarbe,
F.-W., and Werner, P.: Non-linear time series analysis of precipitation
events using regional climate networks for Germany, Clim. Dynam., 46,
1065–1074, 2016.
Rial, J. A.: Synchronization of polar climate variability over the last ice
age: in search of simple rules at the heart of climate's complexity, Am. J.
Sci., 312, 417–448, 2012.Rosenblum, M. G., Pikovsky, A. S., and Kurths, J.: From phase to lag
synchronization in coupled chaotic oscillators, Phys. Rev. Lett., 78,
4193–4196, 10.1103/PhysRevLett.78.4193, 1997.Schiff, S. J., So, P., Chang, T., Burke, R. E., and Sauer, T.: Detecting
dynamical interdependence and generalized synchrony through mutual prediction
in a neural ensemble, Phys. Rev. E, 54, 6708–6724, 10.1103/PhysRevE.54.6708, 1996.Schmocker-Fackel, P. and Naef, F.: Changes in flood frequencies in
Switzerland since 1500, Hydrol. Earth Syst. Sci., 14, 1581–1594,
10.5194/hess-14-1581-2010, 2010.
Steinhaeuser, K., Ganguly, A. R., and Chawla, N. V.: Multivariate and
multiscale dependence in the global climate system revealed through complex
networks, Clim. Dynam., 39, 889–895, 2012.
Stolbova, V., Martin, P., Bookhagen, B., Marwan, N., and Kurths, J.: Topology
and seasonal evolution of the network of extreme precipitation over the
Indian subcontinent and Sri Lanka, Nonlinear Proc. Geoph., 21, 901–917,
2014.Tass, P., Rosenblum, M. G., Weule, J., Kurths, J., Pikovsky, A., Volkmann,
J., Schnitzler, A., and Freund, H.-J.: Detection of n: m phase locking from
noisy data: application to magnetoencephalography, Phys. Rev. Lett., 81,
3291–3294, 10.1103/PhysRevLett.81.3291, 1998.Tsui, C. Y.: A Multiscale Analysis Method and its Application to Mesoscale
Rainfall System, Universität zu Köln, 2015.
Yan, R. and Gao, R. X.: A tour of the tour of the Hilbert-Huang transform: an
empirical tool for signal analysis, IEEE Instru. Meas. Mag., 10, 40–45,
2007.