Abstract
In view of the current availability and variety of measured data, there is an increasing demand for powerful signal processing tools that can cope successfully with the associated problems that often arise when data are being analysed. In practice many of the data-generating systems are not only time-variable, but also influenced by neighbouring systems and subject to random fluctuations (noise) from their environments. To encompass problems of this kind, we present a tutorial about the dynamical Bayesian inference of time-evolving coupled systems in the presence of noise. It includes the necessary theoretical description and the algorithms for its implementation. For general programming purposes, a pseudocode description is also given. Examples based on coupled phase and limit-cycle oscillators illustrate the salient features of phase dynamics inference. State domain inference is illustrated with an example of coupled chaotic oscillators. The applicability of the latter example to secure communications based on the modulation of coupling functions is outlined. MatLab codes for implementation of the method, as well as for the explicit examples, accompany the tutorial.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Stankovski, A. Duggento, P.V.E. McClintock, A. Stefanovska, Phys. Rev. Lett. 109, 024101 (2012)
D.A. Smirnov, B.P. Bezruchko, Phys. Rev. E 79(4), 046204 (2009)
K.J. Friston, NeuroImage 16(2), 513 (2002)
M. Sanjeev Arulampalam, S. Maskell, N. Gordon, T. Clapp, Signal Processing, IEEE Trans. 50(2), 174 (2002)
H.U. Voss, J. Timmer, J. Kurths, Int. J. Bifurcat. Chaos 14(06), 1905 (2004)
I.Z. Kiss, Y. Zhai, J.L. Hudson, Phys. Rev. Lett. 94, 248301 (2005)
R.E. Kalman, J. Fluid. Eng. 82(1), 35 (1960)
S.J. Julier, J.K. Uhlmann, New extension of the Kalman filter to nonlinear systems, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 3068 (1997) p. 182
T. Bayes, Phil. Trans. 53, 370 (1763)
A. Duggento, T. Stankovski, P.V.E. McClintock, A. Stefanovska, Phys. Rev. E 86, 061126 (2012)
T. Stankovski, P.V.E. McClintock, A. Stefanovska, Phys. Rev. X 4, 011026 (2014)
V.N. Smelyanskiy, D.G. Luchinsky, A. Stefanovska, P.V.E. McClintock, Phys. Rev. Lett. 94(9), 098101 (2005)
U. von Toussaint, Rev. Mod. Phys. 83(3), 943 (2011)
V.N. Smelyanskiy, D.G. Luchinsky, D.A. Timuchin, A. Bandrivskyy, Phys. Rev. E 72(2), 026202 (2005)
W.D. Penny, V. Litvak, L. Fuentemilla, E. Duzel, K. Friston, J. Neurosci. Meth. 183(1), 19 (2009)
H. Haken, Rev. Mod. Phys. 47, 67 (1975)
P.E. Kloeden, M. Rasmussen, Nonautonomous Dynamical Systems (AMS Mathematical Surveys and Monographs, New York, 2011)
Y.F. Suprunenko, P.T. Clemson, A. Stefanovska, Phys. Rev. Lett. 111(2), 024101 (2013)
P.T. Clemson, Y.F. Suprunenko, T. Stankovski, A. Stefanovska, Phys. Rev. E 89, 032904 (2014)
J. Daunizeau, K.J. Friston, S.J. Kiebel, Physica D 238(21), 2089 (2009)
D.G. Luchinsky, V.N. Smelyanskiy, A. Duggento, P.V.E. McClintock, Phys. Rev. E 77(6), 061105 (2008)
A. Duggento, D.G. Luchinsky, V.N. Smelyanskiy, I. Khovanov, P.V.E. McClintock, Phys. Rev. E 77(6), 061106 (2008)
A. Stefanovska, H. Haken, P.V.E. McClintock, M. Hožič, F. Bajrović, S. Ribarič, Phys. Rev. Lett. 85(22), 4831 (2000)
Y. Shiogai, A. Stefanovska, P.V.E. McClintock, Phys. Rep. 488, 51 (2010)
D. Rudrauf, A. Douiri, C. Kovach, J.P. Lachaux, D. Cosmelli, M. Chavez, C. Adam, B. Renault, J. Martinerie, M.L. Van Quyen, Neuroimage 31(1), 209 (2006)
V. Jirsa, V. Müller, Frontiers Comput. Neurosci. 7, 78 (2013)
M.G. Rosenblum, A.S. Pikovsky, Phys. Rev. E. 64(4), 045202 (2001)
M. Paluš, A. Stefanovska, Phys. Rev. E 67, 055201(R) (2003)
A. Porta, L. Faes, V. Bari, A. Marchi, T. Bassani, G. Nollo, N.M. Perseguini, J. Milan, V. Minatel, A. Borghi-Silva et al., PloS one 9(2), e89463 (2014)
J. Jamšek, M. Paluš, A. Stefanovska, Phys. Rev. E 81(3), 036207 (2010)
P. Tass, M.G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, H.J. Freund, Phys. Rev. Lett. 81(15), 3291 (1998)
F. Mormann, K. Lehnertz, P. David, C.E. Elger, Physica D 144(3-4), 358 (2000)
N.F. Rulkov, M.M. Sushchik, L.S. Tsimring, H.D.I. Abarbanel, Phys. Rev. E 51(2), 980 (1995)
B. Kralemann, M. Frühwirth, A. Pikovsky, M. Rosenblum, T. Kenner, J. Schaefer, M. Moser, Nat. Commun. 4, 2418 (2013)
D. Iatsenko, A. Bernjak, T. Stankovski, Y. Shiogai, P.J. Owen-Lynch, P.B.M. Clarkson, P.V.E. McClintock, A. Stefanovska, Phil. Trans. R. Soc. Lond. A 371(1997), 20110622 (2013)
I. Manolopoulou, M.P. Matheu, M.D. Cahalan, M. West, T.B. Kepler, J. Amer. Stat. Assoc. 107(499), 855 (2012)
P. Lemey, A. Rambaut, A.J. Drummond, M.A. Suchard, PLOS Comp. Biol. 5(9), e1000520 (2009)
M. Staniek, K. Lehnertz, Phys. Rev. Lett. 100, 158101 (2008)
A. Bahraminasab, F. Ghasemi, A. Stefanovska, P.V.E. McClintock, H. Kantz, Phys. Rev. Lett. 100(8), 084101 (2008)
B. Kralemann, L. Cimponeriu, M. Rosenblum, A. Pikovsky, R. Mrowka, Phys. Rev. E 76(5), 055201 (2007)
I.T. Tokuda, S. Jain, I.Z. Kiss, J.L. Hudson, Phys. Rev. Lett. 99, 064101 (2007)
J. Miyazaki, S. Kinoshita, Phys. Rev. Lett. 96, 194101 (2006)
M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 76(11), 1804 (1996)
L. Kocarev, U. Parlitz, Phys. Rev. Lett. 76(11), 1816 (1996)
H.J. Park, K. Friston, Science 342(6158), 1238411 (2013)
U. Parlitz, L. Junge, W. Lauterborn, L. Kocarev, Phys. Rev. E 54, 2115 (1996)
T. Stankovski, P.V.E. McClintock, A. Stefanovska, Phys. Rev. E 89, 062909 (2014)
Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, Berlin, 1984)
B. Kralemann, L. Cimponeriu, M. Rosenblum, A. Pikovsky, R. Mrowka, Phys. Rev. E 77(6, Part 2), 066205 (2008)
A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization – A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001)
I. Daubechies, J. Lu, H. Wu, Appl. Comput. Harmon. Anal. 30(2), 243 (2011)
R.F. Galán, G.B. Ermentrout, N.N. Urban, Phys. Rev. Lett. 94, 158101 (2005)
T. Pereira, J. Eldering, M. Rasmussen, A. Veneziani, Nonlinearity 27(3), 501 (2014)
V.N. Smelyanskiy, D.A. Timucin, D.G. Luchinsky, A. Stefanovska, A. Bandrivskyy, P.V.E. McClintock, in IUTAM Symposium on Nonlinear Stochastic Dynamics, edited by N.S. Namachchivaya, Y.K. Lin (Kluwer, Amsterdam, 2003), p. 455
K.M. Cuomo, A.V. Oppenheim, Phys. Rev. Lett. 71, 65 (1993)
L. Kocarev, U. Parlitz, Phys. Rev. Lett. 74(25), 5028 (1995)
A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C.R. Mirasso, L. Pesquera, K.A. Shore, Nature 438(7066), 343 (2005)
G. Alvarez, S. Li, Intern. J. Bifurc. Chaos 16(8), 2129 (2006)
G. Pérez, H.A. Cerdeira, Phys. Rev. Lett. 74, 1970 (1995)
B. Kralemann, A. Pikovsky, M. Rosenblum, Chaos 21, 025104 (2011)
B. Kralemann, A. Pikovsky, M. Rosenblum, Phys. Rev. E 87, 052904 (2013)
K.J. Friston, Brain. Connect. 1(1), 13 (2011)
T. Schreiber, A. Schmitz, Physica D 142(3-4), 346 (2000)
L. Ljung, System identification (Springer, 1998)
K.P. Burnham, D.R. Anderson, Model selection and multimodel inference: a practical information-theoretic approach (Springer, 2002)
C.W. Granger, Econ: J. Econ. Soc., 424 (1969)
A.K. Seth, Network: Comput. Neural Syst. 16(1), 35 (2005)
T. Schreiber, Phys. Rev. Lett. 85(2), 461 (2000)
A.B. Barrett, L. Barnett, Front. Neuroinf. 7 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Stankovski, T., Duggento, A., McClintock, P.V.E. et al. A tutorial on time-evolving dynamical Bayesian inference. Eur. Phys. J. Spec. Top. 223, 2685–2703 (2014). https://doi.org/10.1140/epjst/e2014-02286-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2014-02286-7