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Volumn 34, Issue 3, 1986, Pages 2427-2432

Spurious dimension from correlation algorithms applied to limited time-series data

Author keywords

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Indexed keywords

EID: 0001413326 PISSN: 10502947 EISSN: None Source Type: Journal
DOI: 10.1103/PhysRevA.34.2427 Document Type: Article

Times cited : (911)

References (16)

1
- 0000241853
- Deterministic Nonperiodic Flow
- (1963) Journal of the Atmospheric Sciences , vol.20 , pp. 130
- Lorenz, E.N.¹

2
- 35949018382
- For a comprehensive review of these and related methods, see
- (1985) Rev. Mod. Phys. , vol.57 , pp. 617
- Eckman, J.-P.¹ Ruelle, D.²

3
- 40749093037
- (1983) Physica , vol.9-500 vec , pp. 189
- Grassberger, P.¹ Procaccia, I.²

4
- 84927304036
- F. Takens, Dynamical Systems and Turbulence, Warwick, 1980, in Vol. 898 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981).

5
- 84927304035
- If the limit exists, then l'Hospital's rule assures us that the limit is the correct value; furthermore, the approach as r -> 0 is much quicker. There are attractors however for which the limit does not exist. One example is given by G. M. Zaslavskii, Phys. Lett. 69 A vec, 145 (1978). (This example is also quoted in Refs. 3 and 4.)

6
- 84927304034
- F. Takens, Dynamical Systems and Bifurcations, Groningen, 1984, Vol. 1125 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1985).

8
- 0000134542
- The Brownian Movement and Stochastic Equations
- (1942) The Annals of Mathematics , vol.43 , pp. 351
- Doob, J.¹

9
- 34548696055
- An excellent method for choosing T in the case of a strange attractor is given by
- (1986) Phys. Rev. A , vol.33 , pp. 1134
- Fraser, A.¹ Swinney, H.²

10
- 84927304032
- We note that although the method is not directly applicable to stochastic data, a value of T on the order of tau (or even τ /m, see Ref. 10) is probably most appropriate. The algebra is more complicated in this case, however, and no more enlightening. Note that in the numerical experiment, we take T= τ /2 to demonstrate that the assumption T >> tau is not essential.

11
- 84927304031
- Numerical experiments with other metrics, namely the Euclidean (L2) and the ``taxicab'' (L1) metric, yielded similar results.

12
- 84927304030
- P. Atten, J. G. Caputo, B. Malraison, and Y. Gagne, J. Mec. Theor. Appl., Numero special, 133, 1984. Attention is directed to Fig. 14 of this paper, where an effect similar to the one we describe is discussed.

14
- 84927304029
- Initial computations were performed on the Hypercube Mark II, a 32-node array of (Intel 8086) parallel processors. We are indebted to Dr. Geoffrey Fox and the Caltech Concurrent Computation Project for the use of this machine.

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