×

On the relationship between the theory of cointegration and the theory of phase synchronization. (English) Zbl 1403.62222

Summary: The theory of cointegration has been a leading theory in econometrics with powerful applications to macroeconomics during the last decades. On the other hand, the theory of phase synchronization for weakly coupled complex oscillators has been one of the leading theories in physics for many years with many applications to different areas of science. For example, in neuroscience phase synchronization is regarded as essential for functional coupling of different brain regions. In an abstract sense, both theories describe the dynamic fluctuation around some equilibrium. In this paper, we point out that there exists a very close connection between both theories. Apart from phase jumps, a stochastic version of the Kuramoto equations can be approximated by a cointegrated system of difference equations. As one consequence, the rich theory on statistical inference for cointegrated systems can immediately be applied for statistical inference on phase synchronization based on empirical data. This includes tests for phase synchronization, tests for unidirectional coupling and the identification of the equilibrium from data including phase shifts. We study two examples on a unidirectionally coupled Rössler-Lorenz system and on electrochemical oscillators. The methods from cointegration may also be used to investigate phase synchronization in complex networks. Conversely, there are many interesting results on phase synchronization which may inspire new research on cointegration.

MSC:

62P20 Applications of statistics to economics
62G05 Nonparametric estimation
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics

Software:

Systemfit
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Allefeld, C. and Kurths, J. (2004a). An approach to multivariate phase synchronization analysis and its application to event-related potentials. Internat. J. Bifur. Chaos Appl. Sci. Engrg.14 417–426. · Zbl 1099.37517 · doi:10.1142/S0218127404009521
[2] Allefeld, C. and Kurths, J. (2004b). Testing for phase synchronization. Internat. J. Bifur. Chaos Appl. Sci. Engrg.14 405–416. · Zbl 1099.37516 · doi:10.1142/S021812740400951X
[3] Banerjee, A., Galbraith, J., Dolado, J. and Hendry, D. (1993). Co-Integration, Error Correction, and the Econometric Analysis of Non-Stationary Data. Oxford Univ. Press, Oxford. · Zbl 0937.62650
[4] Baptista, M., Silva, T., Sartorelli, J., Caldas, I. and Rosa, E. Jr. (2003). Phase synchronization in the perturbed Chua circuit. Phys. Rev. E67 056212.
[5] Blasius, B., Huppert, A. and Stone, L. (1999). Complex dynamics and phase synchronization in spatially extended ecological systems. Nature399 354–359.
[6] Boccaletti, S., Pecora, L. and Pelaez, A. (2001). Unifying framework for synchronization of coupled dynamical systems. Phys. Rev. E63 066219.
[7] Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L. and Zhou, C. S. (2002). The synchronization of chaotic systems. Phys. Rep.366 1–101. · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0
[8] Brillinger, D. R. (2001). Time Series: Data Analysis and Theory. Classics in Applied Mathematics36. SIAM, Philadelphia, PA. Reprint of the 1981 edition. · Zbl 0983.62056
[9] Colgin, L. and Moser, E. (2010). Gamma oscillations in the hippocampus. Physiology25 319–329.
[10] Dahlhaus, R., Kurths, R., Maas, P. and Timmer, J., eds. (2008). Mathematical Methods in Time Series Analysis and Digital Image Processing. Springer, Berlin and Heidelberg.
[11] Dahlhaus, R., Dumont, T., Le Corff, S. and Neddermeyer, J. C. (2017). Statistical inference for oscillation processes. Statistics51 61–83. · Zbl 1369.62222 · doi:10.1080/02331888.2016.1266985
[12] David, O., Cosmelli, D., Lachaux, J.-P., Baillet, S., Garnero, L. and Martinerie, J. (2003). A theoretical and experimental introduction to the non-invasive study of large-scale neural phase synchronization in human beings. Internat. J. Comput. Cog.1 53–77.
[13] DeShazer, D. J., Breban, R., Ott, E. and Roy, R. (2001). Detecting phase synchronization in a chaotic laser array. Phys. Rev. Lett.87 044101. · Zbl 1060.78527 · doi:10.1142/S0218127404011302
[14] Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc.74 427–431. · Zbl 0413.62075
[15] Dickey, D. A. and Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica49 1057–1072. · Zbl 0471.62090 · doi:10.2307/1912517
[16] Elson, R., Selverston, A., Huerta, R., Rulkov, N., Rabinovich, M. and Abarbanel, H. (1998). Synchronous behavior of two coupled biological neurons. Phys. Rev. Lett.81 5692–5695.
[17] Engel, A., Fries, P. and Singer, W. (2001). Dynamic predictions: Oscillations and synchrony in top-down processing. Nat. Rev., Neurosci.2 704–716.
[18] Engle, R. F. and Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica55 251–276. · Zbl 0613.62140 · doi:10.2307/1913236
[19] Engle, R. and White, H. (1999). Cointegration, Causality and Forecasting. Oxford Univ. Press, Oxford.
[20] Fell, J. and Axmacher, N. (2011). The role of phase synchronization in memory processes. Nat. Rev., Neurosci.12 105–118.
[21] Fuller, W. A. (1996). Introduction to Statistical Time Series, 2nd ed. Wiley, New York. · Zbl 0851.62057
[22] Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification. J. Econometrics16 121–130.
[23] Greene, W. (2008). Econometric Analysis, 6th ed. Pearson Prentice Hall, New Jersey.
[24] Grossmann, A., Kronland-Martinet, R. and Morlet, J. (1989). Reading and understanding continuous wavelet transforms. In Wavelets, Time-Frequency Methods and Phase Space. Inverse Probl. Theoret. Imaging (J. Combes, ed.) 2–20. Springer, Berlin. · Zbl 0850.42006
[25] Guan, S., Lai, C.-H. and Wei, G. W. (2005). Phase synchronization between two essentially different chaotic systems. Phys. Rev. E (3) 72 016205.
[26] Hamilton, J. D. (1994). Time Series Analysis. Princeton Univ. Press, Princeton, NJ. · Zbl 0831.62061
[27] Hannan, E. J. (1973). The estimation of frequency. J. Appl. Probab.10 510–519. · Zbl 0271.62122 · doi:10.2307/3212772
[28] Henningsen, A. and Hamann, J. D. (2007). systemfit: A package for estimating systems of simultaneous equations in r. J. Stat. Softw.23 1–40.
[29] Horvath, M. T. K. and Watson, M. W. (1995). Testing for cointegration when some of the cointegrating vectors are prespecified. Econometric Theory11 984–1014. Trending multiple time series (New Haven, CT, 1993).
[30] Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica59 1551–1580. · Zbl 0755.62087 · doi:10.2307/2938278
[31] Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford Univ. Press, New York. · Zbl 0928.62069
[32] Juselius, K. (2006). The Cointegrated VAR Model: Methodology and Applications. Advanced Texts in Econometrics. Oxford Univ. Press, Oxford. · Zbl 1258.91006
[33] Kammerdiner, A. R. and Pardalos, P. M. (2010). Analysis of multichannel EEG recordings based on generalized phase synchronization and cointegrated VAR. In Computational Neuroscience. Springer Optim. Appl.38 317–339. Springer, New York.
[34] Kammerdiner, A., Boyko, N., Ye, N., He, J. and Pardalos, P. (2010). Integration of signals in complex biophysical system. In Dynamics of Information Systems (M. Hirsch, P. Pardalos and R. Murphey, eds.) 197–211. Springer, New York. · Zbl 1402.92091
[35] Kessler, M. and Rahbek, A. (2001). Asymptotic likelihood based inference for co-integrated homogenous Gaussian diffusions. Scand. J. Stat.28 455–470. · Zbl 0981.62069 · doi:10.1111/1467-9469.00248
[36] Kiss, I. Z. and Hudson, J. L. (2001). Phase synchronization and suppression of chaos through intermittency in forcing of an electrochemical oscillator. Phys. Rev. E64 046215. DOI:10.1103/PhysRevE.64.046215.
[37] Kiss, I. and Hudson, J. (2002). Phase synchronization of nonidentical chaotic electrochemical oscillators. Phys. Chem. Chem. Phys.4 2638–2647.
[38] Kiss, I., Lv, Q. and Hudson, J. (2005). Synchronization of non-phase-coherent chaotic electrochemical oscillations. Phys. Rev. E71 035201.
[39] Kocarev, L. and Parlitz, U. (1996). Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett.76 1816–1819.
[40] Kremers, J. J. M., Ericsson, N. R. and Dolado, J. J. (1992). The power of cointegration tests. Oxf. Bull. Econ. Stat.54 325–48.
[41] Kuramoto, Y. (1975). Self-entrainment of a population of coupled non-linear oscillators. In Lecture Notes in Phys.39 420–422. Springer, Berlin. · Zbl 0335.34021
[42] Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence. Springer Series in Synergetics19. Springer, Berlin. · Zbl 0558.76051
[43] Kwiatkowski, D., Phillips, P., Schmidt, P. and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? J. Econometrics54 159–178. · Zbl 0871.62100 · doi:10.1016/0304-4076(92)90104-Y
[44] Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer, Berlin. · Zbl 1072.62075
[45] Maraun, D. and Kurths, J. (2005). Epochs of phase coherence between El Niño/Southern Oscillation and Indian monsoon. Geophys. Res. Lett.32. DOI:10.1029/2005GL023225.
[46] Mormann, F., Lehnertz, K., David, P. and Elger C, E. (2000). Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients. Phys. D144 358–369. · Zbl 0962.92020 · doi:10.1016/S0167-2789(00)00087-7
[47] Mosconi, R. and Olivetti, F. (2005). Bivariate generalizations of the ACD models. Presented at the Journal of Applied Econometrics Annual Conference, Venezia.
[48] Osipov, G. V., Kurths, J. and Zhou, C. (2007). Synchronization in Oscillatory Networks. Springer Series in Synergetics. Springer, Berlin. · Zbl 1137.37018
[49] Paluš, M. and Vejmelka, M. (2007). Directionality of coupling from bivariate time series: How to avoid false causalities and missed connections. Phys. Rev. E (3) 75 056211.
[50] Paluš, M., Komárek, V., Hrnčíř, Z. and Štěrbová, K. (2001). Synchronization as adjustment of information rates: Detection from bivariate time series. Phys. Rev. E63 046211. DOI:10.1103/PhysRevE.63.046211.
[51] Palut, Y. and Zanone, P.-G. (2005). A dynamical analysis of tennis: Concepts and data. J. Sports Sci.23 1021–1032.
[52] Paraschakis, K. and Dahlhaus, R. (2012). Frequency and phase estimation in time series with quasi periodic components. J. Time Series Anal.33 13–31. · Zbl 1300.62081 · doi:10.1111/j.1467-9892.2011.00736.x
[53] Pecora, L. M. and Carroll, T. L. (1990). Synchronization in chaotic systems. Phys. Rev. Lett.64 821–824. · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[54] Pfaff, B. (2008). Analysis of Integrated and Cointegrated Time Series with R, 2nd ed. Use R! Springer, New York. · Zbl 1165.62068
[55] Phillips, P. C. B. (1991). Error correction and long-run equilibrium in continuous time. Econometrica59 967–980. · Zbl 0725.62101 · doi:10.2307/2938169
[56] Phillips, P. C. B. and Ouliaris, S. (1990). Asymptotic properties of residual based tests for cointegration. Econometrica58 165–193. · Zbl 0733.62100 · doi:10.2307/2938339
[57] Pikovsky, A., Rosenblum, M. and Kurths, J. (2001a). Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge Nonlinear Science Series12. Cambridge Univ. Press, Cambridge, MA. · Zbl 0993.37002
[58] Pikovsky, A. S., Rosenblum, M. G., Osipov, G. V. and Kurths, J. (1997). Phase synchronization of chaotic oscillators by external driving. Phys. D104 219–238. · Zbl 0898.70015 · doi:10.1016/S0167-2789(96)00301-6
[59] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in C, 2nd ed. Cambridge Univ. Press, Cambridge. · Zbl 0778.65003
[60] Pujol-Pere, A., Calvo, O., Matias, M. and Kurths, J. (2003). Experimental study of imperfect phase synchronization in the forced Lorenz system. Chaos13 319–326.
[61] Quian Quiroga, R., Kreuz, T. and Grassberger, P. (2000). Learning driver-response relationships from synchronization patterns. Phys. Rev. E61 5142–5148.
[62] Quian Quiroga, R., Kreuz, T. and Grassberger, P. (2002). Performance of different synchronization measures in real data: A case study on electroencephalographic signals. Phys. Rev. E65 041903. DOI:10.1103/PhysRevE.65.041903.
[63] Rosenblum, M., Pikovsky, A. and Kurths, J. (1996). Phase synchronization of chaotic oscillators. Phys. Rev. Lett.76. 1804–1807.
[64] Saikkonen, P. and Lütkepohl, H. (2000). Testing for the cointegrating rank of a VAR process with an intercept. Econometric Theory16 373–406. · Zbl 1054.62585
[65] Schelter, B., Winterhalder, M., Timmer, J. and Peifer, M. (2007). Testing for phase synchronization. Phys. Lett. A366 382–390.
[66] Stefanovska, A. (2002). Cardiorespiratory interactions. Nonlinear Phenom. Complex Syst.5 462–469.
[67] Stefanovska, A., Haken, H., McClintock, P. V. E., Hožič, M., Bajrović, F. and Ribarič, S. (2000). Reversible transitions between synchronization states of the cardiorespiratory system. Phys. Rev. Lett.85 4831–4834. DOI:10.1103/PhysRevLett.85.4831.
[68] Strogatz, S. H. (2000). From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Phys. D143 1–20. Bifurcations, patterns and symmetry. · Zbl 0983.34022 · doi:10.1016/S0167-2789(00)00094-4
[69] Tass, P., Rosenblum, M. G., Weule, J., Kurths, J., Pikovsky, A., Volkmann, J., Schnitzler, A. and Freund, H. J. (1998). Detection of n:m phase locking from noisy data: Application to magnetoencephalography. Phys. Rev. Lett.81 3291–3294.
[70] Van Leeuwen, P., Geue, D., Thiel, D., Cysarz, D., Lange, S., Romano, M., Wessel, N., Kurths, J. and Grönemeyer, D. (2009). Influence of paced maternal breathing on fetal-maternal heart rate coordination. In Proceedings of the National Academy of Sciences of the United States of America (PNAS) 106 13661–13666.
[71] Varela, F., Lachaux, J.-P., Rodriguez, E. and Martinerie, J. (2001). The brainweb: Phase synchronization and large-scale integration. Nat. Rev. Neurosci.2 229–239.
[72] Winfree, A. T. (1967). Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol.16 15–42.
[73] Womelsdorf, T., Schoffelen, J.-M., Oostenveld, R., Singer, W., Desimone, R., Engel, A. K. and Fries, P. (2007). Modulation of neuronal interactions through neuronal synchronization. Science316 1609–1612.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.