Abstract
Recurrence structures in univariate time series are challenging to detect. We propose a combination of symbolic and recurrence analysis in order to identify recurrence domains in the signal. This method allows to obtain a symbolic representation of the data. Recurrence analysis produces valid results for multidimensional data, however, in the case of univariate time series one should perform phase space reconstruction first. In this chapter, we propose a new method of phase space reconstruction based on the signal’s time-frequency representation and compare it to the delay embedding method. We argue that the proposed method outperforms the delay embedding reconstruction in the case of oscillatory signals. We also propose to use recurrence complexity as a quantitative feature of a signal. We evaluate our method on synthetic data and show its application to experimental EEG signals.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Addison, P.S.: The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance. Institute of Physics Publishing, Bristol, UK, Philadelphia (2002)
Allefeld, C., Atmanspacher, H., Wackermann, J.: Mental states as macrostates emerging from EEG dynamics. Chaos 19, 015102 (2009)
beim Graben, P., Hutt, A.: Detecting recurrence domains of dynamical systems by symbolic dynamics. Phys. Rev. Lett. 110(15), 154101 (2013)
beim Graben, P., Hutt, A.: Detecting event-related recurrences by symbolic analysis: applications to human language processing. Philos. Trans. A Math. Phys. Eng. Sci. 373(2034) (2015)
beim Graben, P., Sellers, K.K., Fröhlich, F., Hutt, A.: Optimal estimation of recurrence structures from time series. EPL 114(3), 38003 (2016)
Donner, R., Hinrichs, U., Scholz-Reiter, B.: Symbolic recurrence plots: A new quantitative framework for performance analysis of manufacturing networks. Eur. Phys. J. Spec. Top. 164(1), 85–104 (2008)
Eckmann, J.P., Kamphorst, S.O., Ruelle, D.: Recurrence Plots of Dynamical Systems. Europhys. Lett. EPL 4(9), 973–977 (1987)
Faure, P., Lesne, A.: Recurrence plots for symbolic sequences. Int. J. Bifurc. Chaos 20(06), 1731–1749 (2010)
Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33(2), 1134–1140 (1986)
Freeman, W.J.: Evidence from human scalp EEG of global chaotic itinerancy. Chaos 13(3), 1069 (2003)
Friedrich, R., Uhl, C.: Spatio-temporal analysis of human electroencephalograms: Petit-Mal epilepsy. Phys. D 98, 171–182 (1996)
Hu, J., Gao, J., Principe, J.C.: Analysis of biomedical signals by the Lempel-Ziv Complexity: the effect of finite data size. IEEE Trans. Biomed. Eng. 53(12), 2606–2609 (2006)
Hutt, A., Riedel, H.: Analysis and modeling of quasi-stationary multivariate time series and their application to middle latency auditory evoked potentials. Phys. D 177, 203–232 (2003)
Kennel, M.B., Abarbanel, H.D.I.: False neighbors and false strands: A reliable minimum embedding dimension algorithm. Phys. Rev. E 66(2), 026209 (2002)
Kennel, M.B., Brown, R., Abarbanel, H.D.I.: Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45(6), 3403–3411 (1992)
Kugiumtzis, D., Christophersen, N.D.: State space reconstruction: method of delays vs singular spectrum approach. Res. Rep. Httpurn Nb NoURN NBN No-35645 (1997)
Larralde, H., Leyvraz, F.: Metastability for Markov processes with detailed balance. Phys. Rev. Lett. 94(16), 160201 (2005)
Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Inf. Theory 222(1), 75–81 (1976)
Liebert, W., Schuster, H.G.: Proper choice of the time delay for the analysis of chaotic time series. Phys. Lett. A 142(2), 107–111 (1989)
Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)
Marwan, N., Kurths, J.: Line structures in recurrence plots. Phys. Lett. A 336, 349–357 (2005)
Marwan, N., Romano, M.C., Thiel, M., Kurths, J.: Recurrence plots for the analysis of complex systems. Phys. Rep. 438(5–6), 237–329 (2007)
Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S.: Geometry from a time series. Phys. Rev. Lett. 45(9), 712 (1980)
Poincaré, H.: Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13(1), 3–270 (1890)
Sleigh, J.W., Leslie, K., Voss, L.: The effect of skin incision on the electroencephalogram during general anesthesia maintained with propofol or desflurane. J. Clin. Mon. Comput. 24, 307–318 (2010)
Takens, F.: Detecting Strange Attractors in Turbulence. Springer (1981)
Tošić, T., Sellers, K.K., Fröhlich, F., Fedotenkova, M., beim Graben, P., Hutt, A.: Statistical frequency-dependent analysis of trial-to-trial variability in single time series by recurrence plots. Fronti. Syst. Neurosci. 9(184) (2016)
Yildiz, I.B., Kiebel, S.J.: A hierarchical neuronal model for generation and online recognition of birdsongs. PLoS Comput. Biol. 7, e1002303 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Fedotenkova, M., Graben, P.b., Sleigh, J.W., Hutt, A. (2017). Time-Frequency Representations as Phase Space Reconstruction in Symbolic Recurrence Structure Analysis. In: Rojas, I., Pomares, H., Valenzuela, O. (eds) Advances in Time Series Analysis and Forecasting. ITISE 2016. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-55789-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-55789-2_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-55788-5
Online ISBN: 978-3-319-55789-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)