Recurrence plot statistics and the effect of embedding |
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| Affiliation: | 1. Space and Astrophysics Group, Department of Physics, Warwick University, Coventry CV4 7AL, UK;2. UKAEA Culham Division, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, UK;1. Potsdam Institute for Climate Impact Research, Telegraphenberg A 31, 14473 Potsdam, Germany;2. Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany;3. Department of Physics, Humboldt University Berlin, 12489 Berlin, Germany;4. Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom;5. Department of Epileptology, University of Bonn, Sigmund-Freud-Straße 25, 53105 Bonn, Germany;6. Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14–16, 53115 Bonn, Germany;7. Interdisciplinary Center for Complex Systems, University of Bonn, Brühler Straße 7, 53175 Bonn, Germany;1. Department of Industrial and Management Systems Engineering, University of South Florida, Tampa, FL 33620, USA;2. Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, PA 16802, USA;1. Department of Civil Engineering, University of Birmingham, Edgbaston, Birmingham, United Kingdom;2. Hong Kong Observatory, Kowloon, Hong Kong;3. Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong;4. Joint Research Center for Engineering Structure Disaster Prevention and Control, Guangzhou University, China;5. Chongqing University, Key Laboratory of New Technology for Construction of Cities in Mountain Area, Ministry of Education, School of Civil Engineering, Chongqing, China;1. Department of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia;2. ARC Industrial Transformation Training Centre (Transforming Maintenance through Data Science), Curtin University, Bentley, WA 6102, Australia |
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| Abstract: | ![]()
Recurrence plots provide a graphical representation of the recurrent patterns in a timeseries, the quantification of which is a relatively new field. Here we derive analytical expressions which relate the values of key statistics, notably determinism and entropy of line length distribution, to the correlation sum as a function of embedding dimension. These expressions are obtained by deriving the transformation which generates an embedded recurrence plot from an unembedded plot. A single unembedded recurrence plot thus provides the statistics of all possible embedded recurrence plots. If the correlation sum scales exponentially with embedding dimension, we show that these statistics are determined entirely by the exponent of the exponential. This explains the results of Iwanski and Bradley [J.S. Iwanski, E. Bradley, Recurrence plots of experimental data: to embed or not to embed? Chaos 8 (1998) 861–871] who found that certain recurrence plot statistics are apparently invariant to embedding dimension for certain low-dimensional systems. We also examine the relationship between the mutual information content of two timeseries and the common recurrent structure seen in their recurrence plots. This allows time-localized contributions to mutual information to be visualized. This technique is demonstrated using geomagnetic index data; we show that the AU and AL geomagnetic indices share half their information, and find the timescale on which mutual features appear. |
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