Motivated by the recent demonstration of its use as a tool for the detection
and characterization of phase-shape correlations in multivariate time series,
we show that eigenvalue decomposition can also be applied to a matrix of
indices of bivariate phase synchronization strength. The resulting method is
able to identify clusters of synchronized oscillators, and to quantify their
strength as well as the degree of involvement of an oscillator in a cluster.
Since for the case of a single cluster the method gives similar results as our
previous approach, it can be seen as a generalized Synchronization Cluster
Analysis, extending its field of application to more complex situations. The
performance of the method is tested by applying it to simulation data.Comment: Submitted Oct 2005, accepted Jan 2006, "published" Oct 2007, actually
  available Jan 200

Anderson T. W.

CARSTEN ALLEFELD

JÜRGEN KURTHS

MARKUS MÜLLER

English

arXiv

Crossref

EIGENVALUE DECOMPOSITION AS A GENERALIZED SYNCHRONIZATION CLUSTER ANALYSIS

We report results on the synchronization of two organ pipes positioned side by side. Special attention is put on the synchronization of the higher harmonics. As possible explanation, classical theory provides the amplitude death as explanation for the reduction to almost silence of two coupled organ pipes. With our measurements we exclude this scenario. The higher harmonics show a behavior in perfect coincidence with synchronization theory. In addition we investigate the dependence on the coupling of two pipes by varying their distance. In the context of synchronization in networks, a new synchronization effect is observed for extended systems with two distributed, slightly different delays

Allefeld, C.

Mueller, M.

Kurths, J.

City Research Online

              City, University of London Institutional RepositoryCitation: Allefeld, C. ORCID: 0000-0002-1037-2735, Mueller, M. and Kurths, J. (2007). Eigenvalue decomposition as a generalized synchronization cluster analysis. International Journal of Bifurcation and Chaos, 17(10), pp. 3483-3491. doi: 10.1142/S0218127407019251 This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link:  http://openaccess.city.ac.uk/id/eprint/22839/Link to published version: http://dx.doi.org/10.1142/S0218127407019251Copyright and reuse: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to.City Research Online:            http://openaccess.city.ac.uk/            publications@city.ac.ukCity Research OnlinearXiv:0706.3375v3  [physics.data-an]  3 Sep 2008Electronic version of an article published as:International Journal of Bifurcation and Chaos, 17(10), 2007, 3493–3497. doi:10.1142/S0218127407019251c©World Scientific Publishing Company http://www.worldscinet.com/ijbc/EIGENVALUE DECOMPOSITION AS AGENERALIZED SYNCHRONIZATION CLUSTER ANALYSISCARSTEN ALLEFELDNonlinear Dynamics Group and Research Unit “Conflicting Rules”University of Potsdam, P.O. Box 601553, 14415 Potsdam, Germanyallefeld@ling.uni-potsdam.dehttp://www.agnld.uni-potsdam.de/∼allefeldMARKUS M ¨ULLERFacultad de Ciencias, Universidad Auto´noma del Estado de Morelos62210 Cuernavaca, Morelos, MexicoJ ¨URGEN KURTHSNonlinear Dynamics Group and Research Unit “Conflicting Rules”, University of PotsdamReceived October 18, 2005Revised January 13, 2006Motivated by the recent demonstration of its use as a tool for the detection and characterization ofphase-shape correlations in multivariate time series, we show that eigenvalue decomposition can alsobe applied to a matrix of indices of bivariate phase synchronization strength. The resulting methodis able to identify clusters of synchronized oscillators, and to quantify their strength as well as thedegree of involvement of an oscillator in a cluster. Since for the case of a single cluster the methodgives similar results as our previous approach, it can be seen as a generalized Synchronization ClusterAnalysis, extending its field of application to more complex situations. The performance of the methodis tested by applying it to simulation data.Keywords: eigenvalue decomposition; synchronization matrix; phase synchronization; cluster analysis.1. IntroductionThe eigenvalue decomposition of the equal-time correlation matrix of a set of signals isone of the standard tools of multivariate data analysis (cf. Anderson [2003]). Recently,Mu¨ller et al. [2005] demonstrated the usefulness of the eigenvalue decomposition of thecorrelation matrix specifically as a tool for the detection of phase-shape correlations inmultivariate data sets. They showed that changes in the degree of synchronization in all or asubset of signals are reflected in coordinated changes in the highest and lowest eigenvalues,and that information about the channels involved and the type of their interaction can beobtained from the corresponding eigenvectors.While the correlation of time series may indicate the synchronization of the oscillators12 C. Allefeld, M. Mu¨ller, & J. Kurthsthey are obtained from, the physical concept of synchronization refers specifically to theadjustment of the rhythms of oscillators, i.e. to the relative dynamics of their phases ratherthan their amplitudes [Pikovsky et al. (2001)]. Moreover, there is a regime in the dynamicsof coupled chaotic oscillators in which the phase difference is bounded while the ampli-tudes remain uncorrelated [Rosenblum et al. (1996)], called phase synchronization. In thispaper, we show that in order to focus the analysis on synchronization relations, it is possibleto replace the matrix of correlation coefficients with a matrix of indices of bivariate phasesynchronization strength. Combined with an additional step of sorting signals into groups,eigenvalue decomposition can operate as a Synchronization Cluster Analysis, generalizingthe previous approach of Allefeld and Kurths [2004].2. Eigenvalue Decomposition of the Synchronization MatrixThe correlation matrix C of a set of data channels xi, i = 1 . . .N, consists of the correlationcoefficients Ci j ∈ [−1; 1] between channels. Its eigenvalues λk and eigenvectors ~vk aredefined by the equationC ~vk = λk ~vk, (1)which in general has N different solutions, k = 1 . . .N. In the following we assume that theeigenvectors are normalized, |~vk | = 1, and the solutions have been sorted according to theeigenvalues, λ1 ≤ λ2 ≤ . . . ≤ λN .The eigenvectors and -values of C are real-valued, and the eigenvalues are non-negative.Because a matrix becomes the diagonal matrix of its eigenvalues by being transformed intothe basis of its eigenvectors and the trace of a matrix is invariant under such a transform, forevery correlation matrix holds ∑ λk = tr(C) = N. In the uncorrelated case this is triviallyfulfilled by λk = 1 for all k. With a deviation from this, each increase of an eigenvalueabove 1 has to be compensated for by at least one other eigenvalue becoming smaller than1, such that this value gives a natural distinction between “large” and “small” eigenvalues.The quantification of phase synchronization is based on the instantaneous phase φi ofeach oscillator i = 1 . . .N. How these phases are determined in the special case is notimportant here; if the given data are time series, the standard approach is the Hilbert trans-form for narrowband data, or the Morlet wavelet transform for broadband signals (for adiscussion, see Pikovsky et al. [1997] or Allefeld [2004], Sec. 3.2). The statistical strengthof phase synchronization of two oscillators i and j can then be defined as the “peakedness”of the distribution of the phase difference φ j − φi; here we use the measureRi j =∣∣∣∣∣∣∣1nn∑l=1exp(i (φ jl − φil))∣∣∣∣∣∣∣ , (2)where l = 1 . . .n enumerates the realizations in the given sample. For the continuum fromno to perfect phase synchronization, this measure takes on values from 0 to 1. R can beseen as the modulus of the complex correlation coefficient of signals xi = exp(i φi), and itsdecomposition shares the properties given above for C.Since R is a nonlinear measure, in this case the eigenvalue decomposition can no longerbe interpreted with regard to a linear transform of data channels into source channels. ButEigenvalue decomposition as a generalized Synchronization Cluster Analysis 32 4 6 8 10 1224681012matrixosc #osc #R0 0.5 1#12:   3.28osc ##11:   2.69osc #2 4 6 8 1012#10:   2.12osc ##9:   0.72osc #eigenvectors#8:   0.60osc #2 4 6 8 1012#7:   0.55osc ##6:   0.48osc ##5:   0.45osc #2 4 6 8 1012#4:   0.39osc ##3:   0.35osc ##2:   0.27osc #2 4 6 8 1012#1:   0.10osc #Fig. 1. Left: Synchronization matrix consisting of three clusters of oscillators. Right: Its eigenvectors and -values.Eigenvectors corresponding to eigenvalues > 1 describe the cluster structure.2 4 6 8 10 1224681012matrixosc #osc #R0 0.5 1#12:   4.80osc ##11:   2.40osc #2 4 6 8 1012#10:   1.75osc ##9:   0.48osc #eigenvectors#8:   0.48osc #2 4 6 8 1012#7:   0.48osc ##6:   0.39osc ##5:   0.36osc #2 4 6 8 1012#4:   0.35osc ##3:   0.24osc ##2:   0.21osc #2 4 6 8 1012#1:   0.06osc #Fig. 2. Synchronization matrix consisting of three clusters with additional inter-cluster synchronization. The clus-ter structure appears in the eigenvectors for λk > 1, but only in different superpositions.still the result of the decomposition can be used as a means to analyze the structure ofsynchronization relations. We will demonstrate this with two basic, artificially constructedexamples of synchronization matrices.Figure 1 shows the synchronization matrix for a system consisting of three clustersof synchronized oscillators (with no synchronization between clusters and a different de-gree of involvement of each oscillator in its cluster) along with the result of the eigenvaluedecomposition. There are three eigenvalues larger than one, and the corresponding eigen-vectors describe clearly the extent of the clusters as well as the degree of involvement ofthe individual oscillators. There is also a correspondence between the size and strength ofinternal synchronization of the clusters and the three eigenvalues. In contrast, the remainingeigenvectors seem not to contribute to the description of the synchronization clusters. The4 C. Allefeld, M. Mu¨ller, & J. Kurths1 2 3 4 5 6 7 8 9 10 11 1200.20.40.60.81original participation indicesosc #2 4 6 8 10 1224681012trimmed matrixR0 0.5 1 1 2 3 4 5 6 7 8 9 10 11 1200.20.40.60.81clusters3.902.92 2.12osc #Fig. 3. Synchronization Cluster Analysis. Left: Participation indices corresponding to the λk > 1 for the synchro-nization matrix of Fig. 2. Oscillators are attributed to that cluster for which its participation index is maximal:blue, green, or red. Center: Synchronization matrix with removed inter-cluster synchronization. Right: Result ofthe eigenvalue decomposition of the trimmed matrix; shown are the participation indices and cluster strengths.interpretation of the eigenvalue decomposition of R therefore has these aspects: 1) Synchro-nization clusters are identified by eigenvalues λk > 1. The eigenvalues themselves quantifythe strength of the clusters. 2) For each cluster, the corresponding eigenvector describesits internal structure. Because∑i v2ik = 1, the index v2ik quantifies the relative involvementof channel (oscillator) i in cluster k. 3) Combining both, the “absolute” involvement ofchannel i in cluster k can be quantified by the participation index λkv2ik.Figure 2 gives the result for a matrix consisting of the same three clusters, but withadditional inter-cluster synchronization. Because of the coupling between them, the threeclusters no longer appear in separate components of its eigenvalue decomposition. Thereare still three λk > 1, but the eigenvectors consist of superpositions of the clusters. To ac-count for this, the oscillators belonging to the three clusters have to be identified explicitely.This can be done by means of the participation indices; the procedure is illustrated inFig. 3. Each oscillator is attributed to that cluster for which its participation is maximal.In this way, the three clusters consisting of oscillators #1–5 (blue), 6–9 (green), and 10–12(red) are correctly identified. In a second step, all of the indices for inter-cluster synchro-nizations are set to zero, and the eigenvalue decomposition is repeated on the trimmedmatrix. For the result of this decomposition, the interpretation given above is valid again.3. Generalized Synchronization Cluster AnalysisThe procedure described in the last section is an approach to synchronization cluster anal-ysis. The method identifies clusters of synchronized oscillators and quantifies the strengthof the clusters as well as the degree of involvement of each oscillator in its cluster. In aprevious paper, Allefeld and Kurths [2004] introduced another approach that was limitedto a single cluster of synchronization; it assumed that all of the oscillators belong to thesame cluster, and focused on the quantification of the degree of oscillator participation.The algorithm derived from the observation that under relatively general conditions thesynchronization indices within a cluster can be written as the product of factors RiC,Ri j = RiC R jC for i , j (Rii = 1), (3)Eigenvalue decomposition as a generalized Synchronization Cluster Analysis 50 0.2 0.4 0.6 0.8 100.20.40.60.81squared to−cluster synchronization strengthsparticipation indicesFig. 4. Relation between the result of the single-cluster analysis (horizontal scale: estimate of R2iC) and the parti-cipation indices given by eigenvalue decomposition (vertical scale: λN v2iN ) in 20 simulation runs.which can be interpreted as the synchronization strength between oscillator i and the clusteritself, its “to-cluster synchronization strength”.We investigated the relationship between the results of the two methods by applyingthem to simulation data that conforms to the presupposition of a single factorizable syn-chronization cluster. For N = 20 oscillators, the RiC were drawn randomly from the uniformdistribution over [0; 1]; then the Ri j were calculated based on 100 samples of the phasedifference from a wrapped normal distribution (cf. Allefeld and Kurths [2004]). Figure 4shows the result for the relation between the output of the earlier method and the participa-tion indices for the strongest cluster. Though the mathematical background of both methodsis clearly different, the plot shows an almost functional dependency, which can be roughlydescribed by R2iC = λNv2iN . The region of zero participation indices for R2iC below about 0.2comes from the attribution of weakly synchronized oscillators to another cluster by the newmethod; in this range the observed value can no longer be reliably distinguished from nosynchronization for the given sample size. Since there is a clear relationship between theresults of both methods in the case where the assumptions of the earlier method hold, thecluster analysis based on the eigenvalue decomposition of the synchronization matrix canbe seen as a generalization of our previous approach to synchronization cluster analysis.4. Application to Simulated Phase SynchronizationTo check the performance of the new method, we applied it to data obtained from the nu-merical simulation of a system that is known to exhibit clusters of phase synchronization,previously investigated by Osipov et al. [1997] (see Eq. 1 & Sec. IV A). The system con-sists of a chain of 20 phase-coherent Ro¨ssler oscillators with a diffusive coupling in they-component of strength ǫ = 0.007. The natural frequencies of the oscillators increase lin-early along the chain, in the range ω = 1 . . .1.004. Phases were defined as the angle inthe x/y-plane. To also test for the influence of small sample size and noise, we addition-ally applied the algorithm to the synchronization matrix obtained from data reduced to 100approximately independent samples with 50% white noise added to the time series data.The result is shown in Fig. 5. The analysis identifies two clusters of synchronized os-cillators (#1–16 & 17–20). There is an almost constant high participation of oscillators6 C. Allefeld, M. Mu¨ller, & J. Kurths1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2000.20.40.60.8114.58 3.63osc #full, clean dataset1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012.20 3.23osc #few samples, noisyFig. 5. Analysis results (participation indices and cluster strengths) for a chain of nonidentical Ro¨ssler oscillators.Left: Based on original simulation data. Right: Data reduced to 100 samples, plus 50% measurement noise.in their cluster, resulting in cluster strengths close to the number of involved oscillators.Decreased participation occurs near the border between the clusters, where the couplingalong the chain pulls oscillators away from the common dynamics of their group. The re-sult for reduced sample size and measurement noise is very similar to the original. Thoughparticipation indices and cluster strengths are slightly smaller due to the noise, the sametwo clusters are clearly identified. This result indicates that the cluster analysis based oneigenvalue decomposition is relatively robust against small sample size and noise.Conclusion and AcknowledgementsWe introduced a new approach to synchronization cluster analysis based on eigenvaluedecomposition. The new method can be seen as a generalization of our previous approach,extending its field of application to situations including multiple clusters. The algorithmwas tested on simulation data and shown to be robust against small sample size and noise.Future work will be on the use of the method to investigate synchronization patterns in EEGdata. A Matlab implementation of the algorithm can be obtained from the correspondingauthor.—This work has been supported by Deutsche Forschungsgemeinschaft.ReferencesAllefeld, C. (2004). Phase Synchronization Analysis of Event-Related Brain Potentials in LanguageProcessing, Ph.D. thesis, University of Potsdam.Allefeld, C. and Kurths, J. (2004). An approach to multivariate phase synchronization analysis andits application to event-related potentials. Int. J. Bifurcat. Chaos, 14 (2): 417–426.Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, Wiley.Mu¨ller, M., Baier, G., Galka, A., Stephani, U. and Muhle, H. (2005). Detection and characterizationof changes of the correlation structure in multivariate time series. Phys. Rev. E, 71: 046116.Osipov, G. V., Pikovsky, A. S., Rosenblum, M. G. and Kurths, J. (1997). Phase synchronization effectsin a lattice of nonidentical Ro¨ssler oscillators. Phys. Rev. E, 55 (3): 2353–2361.Pikovsky, A., Rosenblum, M. and Kurths, J. (2001). Synchronization: A Universal Concept in Non-linear Sciences, Cambridge University Press.Pikovsky, A. S., Rosenblum, M. G., Osipov, G. V. and Kurths, J. (1997). Phase synchronization ofchaotic oscillators by external driving. Physica D, 104: 219–238.Rosenblum, M. G., Pikovsky, A. S. and Kurths, J. (1996). Phase synchronization of chaotic oscilla-tors. Phys. Rev. Lett., 76: 1804–1807.

Eigenvalue decomposition as a generalized synchronization cluster analysis

Eigenvalue Decomposition as a Generalized Synchronization Cluster
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Eigenvalue Decomposition as a Generalized Synchronization Cluster Analysis

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