dorsal/arxiv
View SchemaRecurrence plot statistics and the effect of embedding
| Authors | T. K. March, S. C. Chapman, R. O. Dendy |
|---|---|
| Categories | |
| ArXiv ID | physics/0502042 |
| URL | https://arxiv.org/abs/physics/0502042 |
| DOI | 10.1016/j.physd.2004.11.002 |
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 (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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"abstract": "Recurrence plots provide a graphical representation of the recurrent patterns\nin a timeseries, the quantification of which is a relatively new field. Here we\nderive analytical expressions which relate the values of key statistics,\nnotably determinism and entropy of line length distribution, to the correlation\nsum as a function of embedding dimension. These expressions are obtained by\nderiving the transformation which generates an embedded recurrence plot from an\nunembedded plot. A single unembedded recurrence plot thus provides the\nstatistics of all possible embedded recurrence plots. If the correlation sum\nscales exponentially with embedding dimension, we show that these statistics\nare determined entirely by the exponent of the exponential. This explains the\nresults of Iwanski and Bradley (Chaos 8 [1998] 861-871) who found that certain\nrecurrence plot statistics are apparently invariant to embedding dimension for\ncertain low-dimensional systems. We also examine the relationship between the\nmutual information content of two timeseries and the common recurrent structure\nseen in their recurrence plots. This allows time-localized contributions to\nmutual information to be visualized. This technique is demonstrated using\ngeomagnetic index data; we show that the AU and AL geomagnetic indices share\nhalf their information, and find the timescale on which mutual features appear.",
"arxiv_id": "physics/0502042",
"authors": [
"T. K. March",
"S. C. Chapman",
"R. O. Dendy"
],
"categories": [
"physics.data-an"
],
"doi": "10.1016/j.physd.2004.11.002",
"title": "Recurrence plot statistics and the effect of embedding",
"url": "https://arxiv.org/abs/physics/0502042"
},
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