Efficient implementation of the Gaussian kernel algorithm in estimating invariants and noise level from noisy time series data

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 61, Issue 4, 2000, Pages 3750-3756

  Yu, Dejin ^a   Small, Michael ^a   Harrison, Robert G ^a   Diks, Cees ^b  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF AMSTERDAM   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

GAUSSIAN DISTRIBUTION; TIME SERIES;

CORRELATION DIMENSIONS; CORRELATION INTEGRAL; EFFICIENT IMPLEMENTATION; GAUSSIAN KERNELS; NOISE LEVELS; NOISY DATA; REDUCING COMPUTING TIME; TIME-SERIES DATA;

ALGORITHMS;

EID: 0001238862     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.61.3750     Document Type: Article

References (30)

1. Measures of Complexity and Chaos, edited by N. B. Abraham, A. M. Albano, A. Passamante, and P. E. Rapp (Plenum, New York, 1989).
2. P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983)
3. P. GrassbergerI. ProcacciaPhysica D 9, 189 (1983).
4. Dimensions and Entropies in Chaotic Systems, edited by G. Mayer-Kress (Springer-Verlag, Berlin, 1986).
5. J. Theiler, J. Opt. Soc. Am. A 7, 1055 (1990).
6. T. Schreiber and H. Kantz, in Predictability of Complex Dynamical Systems, edited by Y. A. Kravtsov and J. B. Kadtke (Springer, New York, 1996).
7. E. Ott, E. D. Yorke, and J. A. Yorke, Physica D 16, 62 (1985).
8. M. Möller, W. Lange, F. Mitschke, N. B. Abraham, and U. Hübner, Phys. Lett. A 138, 176 (1989).
9. R. L. Smith, J. R. Stat. Soc., Ser. B Methodol. 54, 329 (1992).
10. G. G. Szpiro, Physica D 65, 289 (1993).
11. T. Schreiber, Phys. Rev. E 48, R13 (1993).
12. J. C. Schouten, F. Takens, and C. M. van den Bleek, Phys. Rev. E 50, 1851 (1994).
13. C. Diks, Phys. Rev. E 53, R4263 (1996).
14. H. Oltmans and P. J. T. Verheijen, Phys. Rev. E 56, 1160 (1997).
15. D. Kugiumtzis, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 1283 (1997).
16. J. Argyris, I. Anderadis, G. Pavlos, and M. Athanasiou, Chaos, Solitons and Fractals 9, 343 (1998).
17. T. Schreiber, Phys. Rev. E 56, 274 (1997).
18. J. G. Caputo and P. Atten, Phys. Rev. A 35, 1311 (1987).
19. F. Takens, in Dynamical Systems and Turbulence, Warwick, 1980, edited by D. A. Rand and L. S. Young (Springer-Verlag, New York, 1981), Vol. 898, pp. 366–381.
20. J. M. Ghez and S. Vaienti, Nonlinearity 5, 777 (1992)
21. J. M. Ghez, E. Orlandini, M. C. Tesi, and S. Vaienti, Physica D 63, 282 (1993).
22. M. Frank, H.-R. Blank, J. Heindl, M. Kaltenhäuser, H. Köchner, W. Kreische, N. Müller, S. Poscher, R. Sporer, and T. Wagner, Physica D 65, 359 (1993).
23. M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, Physica D 51, 52 (1991).
24. W. H. Press, S. A. Teukolski, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge University Press, Cambridge, 1992).
25. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Physica D 16, 285 (1985).
26. T. Schreiber, Phys. Rep. 308, 1 (1999).
27. A large error in estimating D, K, and σ may result. This is because Eq. (17) is derived under the condition that (Formula presented) and (Formula presented) This is not the case when (Formula presented) is numerically computed.
28. J. Theiler, Phys. Rev. A 36, 4456 (1987).
29. Dejin Yu, M. Small, R. G. Harrison, C. Robertson, G. Clegg, M. Holzer, and F. Sterz, Phys. Lett. A 265, 68 (2000).
30. J. Theiler, Phys. Rev. A 34, 2427 (1986).