Skip to main content
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
/content/aip/journal/chaos/17/1/10.1063/1.2430294
1.
1.H. D. I. Abarbanel, Analysis of Observed Chaotic Data (Springer, New York, 1996).
2.
2.H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University. Press, Cambridge, 1997).
3.
3.N. H. Packard, J. P. Crutchfield, J. D. Farmer et al., Phys. Rev. Lett. 45, 712 (1980).
http://dx.doi.org/10.1103/PhysRevLett.45.712
4.
4.F. Takens, in Dynamical Systems and Turbulence, Warwick 1980, edited by D. Rand and L.-S. Young (Springer, Berlin, 1981), p. 366.
5.
5.T. Sauer, J. A. Yorke, and M. Casdagli, J. Stat. Phys. 64, 579 (1991).
http://dx.doi.org/10.1007/BF01048307
6.
6.A. M. Fraser and H. L. Swinney, Phys. Rev. A 33, 1134 (1986).
http://dx.doi.org/10.1103/PhysRevA.33.1134
7.
7.G. Kember and A. C. Fowler, Physica D 58, 127 (1993);
http://dx.doi.org/10.1016/0167-2789(92)90104-U
7.M. T. Rosenstein, J. J. Collins, and C. J. De Luca, Physica D 73, 82 (1994).
http://dx.doi.org/10.1016/0167-2789(94)90226-7
8.
8.L. Cao, A. Mees, and J. K. Judd, Physica D 121, 75 (1998).
http://dx.doi.org/10.1016/S0167-2789(98)00151-1
9.
9.K. Judd and A. Mees, Physica D 120, 273 (1998).
http://dx.doi.org/10.1016/S0167-2789(98)00089-X
10.
10.D. Kugiumtzis, Physica D 95, 13 (1996).
http://dx.doi.org/10.1016/0167-2789(96)00054-1
11.
11.P. Grassberger, T. Schreiber, and C. Schaffrath, Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, 521 (1991).
http://dx.doi.org/10.1142/S0218127491000403
12.
12.W. Leibert, K. Pawlezik, and H. G. Schuster, Europhys. Lett. 14, 521 (1991).
13.
13.A. Fraser, IEEE Trans. Inf. Theory 35, 245 (1989).
http://dx.doi.org/10.1109/18.32121
14.
14.S. P. Garcia and J. S. Almeida, Phys. Rev. E 71, 037204 (2005).
http://dx.doi.org/10.1103/PhysRevE.71.037204
15.
15.M. Kennel and H. D. I. Abarbanel, Phys. Rev. E 66, 026209 (2002).
http://dx.doi.org/10.1103/PhysRevE.66.026209
16.
16.M. B. Kennel, R. Brown, and H. D. I. Abarbanel, Phys. Rev. A 45, 3403 (1992).
http://dx.doi.org/10.1103/PhysRevA.45.3403
17.
17.Lingyue Cao, Physica D 110, 43 (1997).
http://dx.doi.org/10.1016/S0167-2789(97)00118-8
18.
18.M. Casdagli, S. Eubank, J. D. Farmer et al., Physica D 51, 52 (1991);
http://dx.doi.org/10.1016/0167-2789(91)90222-U
18.J. F. Gibson, J. D. Farmer, M. Casdagli et al., Physica D 57, 1 (1992).
http://dx.doi.org/10.1016/0167-2789(92)90085-2
19.
19.S. Boccaletti, L. M. Pecora, and A. Pelaez, Phys. Rev. E 65, 066219 (2001).
http://dx.doi.org/10.1103/PhysRevE.63.066219
20.
20.D. Broomhead and G. P. King, Physica D 20, 217 (1986).
http://dx.doi.org/10.1016/0167-2789(86)90031-X
21.
21.A. P. M. Tsui, A. J. Jones, and A. G. de Oliveira, Neural Comput. 10, 318 (2002);
http://dx.doi.org/10.1007/s005210200004
21.D. Evans and A. J. Jones, Proc. - R. Soc. Edinburgh, Sect. A: Math. 458, 2759 (2002).
22.
22.H. Kantz and E. Olbrich, Chaos 7, 423 (1997).
http://dx.doi.org/10.1063/1.166215
23.
23.E. Olbrich and H. Kantz, Phys. Lett. A 232, 63 (1997).
http://dx.doi.org/10.1016/S0375-9601(97)00351-4
24.
24.Wendell Fleming, Functions of Several Variables (Springer-Verlag, New York, 1977).
25.
25.L. Pecora, T. Carroll, and J. Heagy, Phys. Rev. E 52, 3420 (1995).
http://dx.doi.org/10.1103/PhysRevE.52.3420
26.
26.Louis M. Pecora, Thomas L. Carroll, and James F. Heagy, in Nonlinear Dynamics and Time Series: Building a Bridge Between the Natural and Statistical Sciences, Fields Institute Communications, edited by C. D. Cutler and D. T. Kaplan (American Mathematical Society, Providence, 1996), Vol. 11, p. 49.
27.
27.L. Moniz, L. Pecora, J. Nichols et al., Struct. Health Monit. 3, 199 (2003).
28.
28.David Middleton, An Introduction to Statistical Communication Theory (IEEE, Piscataway, 1996).
29.
29.E. T. Jaynes, Probability Theory, The Logic of Science (Cambridge University Press, Cambridge, 2003).
30.
30.J. Theiler, Phys. Rev. A 34, 2427 (1986).
http://dx.doi.org/10.1103/PhysRevA.34.2427
31.
31.E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
http://dx.doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
32.
32.G. Strang, Introduction to Applied Mathematics (Wellesley-Cambridge, Wellesley, 1986), p. 611.
33.
33.W. H. Press, B. P. Flannery, S. A. Teukolsky et al., Numerical Recipes (Cambridge University Press, New York, 1990).
34.
34.R. C. Elson, A. I. Selverston, R. Huerta et al., Phys. Rev. Lett. 81, 5692 (1998).
http://dx.doi.org/10.1103/PhysRevLett.81.5692
35.
35.J. L. Hindmarsh and R. M. Rose, Proc. R. Soc. London, Ser. B 221, 87 (1984).
36.
36.Martin Falcke, Ramón Huerta, Mikhail I. Rabinovich et al., Biol. Cybern. 82, 517 (2000).
http://dx.doi.org/10.1007/s004220050604
37.
37.L. Aguirre and C. Letellier, J. Phys. A 38, 6311 (2005).
http://dx.doi.org/10.1088/0305-4470/38/28/004
http://aip.metastore.ingenta.com/content/aip/journal/chaos/17/1/10.1063/1.2430294
Loading
/content/aip/journal/chaos/17/1/10.1063/1.2430294
Loading

Data & Media loading...

Loading

Article metrics loading...

/content/aip/journal/chaos/17/1/10.1063/1.2430294
2007-02-13
2016-09-28

Abstract

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several longstanding, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction.

Loading

Full text loading...

/deliver/fulltext/aip/journal/chaos/17/1/1.2430294.html;jsessionid=WV7nJld_OPG4TtlKUJDskLIM.x-aip-live-06?itemId=/content/aip/journal/chaos/17/1/10.1063/1.2430294&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/chaos
true
true

Access Key

  • FFree Content
  • OAOpen Access Content
  • SSubscribed Content
  • TFree Trial Content
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
/content/realmedia?fmt=ahah&adPositionList=
&advertTargetUrl=//oascentral.aip.org/RealMedia/ads/&sitePageValue=chaos.aip.org/17/1/10.1063/1.2430294&pageURL=http://scitation.aip.org/content/aip/journal/chaos/17/1/10.1063/1.2430294'
Right1,Right2,Right3,