## Exceptional Events as Evidence for Determinism (1994)

Venue: | Physica D |

Citations: | 37 - 3 self |

### BibTeX

@ARTICLE{Kaplan94exceptionalevents,

author = {Daniel Kaplan and Physica D},

title = {Exceptional Events as Evidence for Determinism},

journal = {Physica D},

year = {1994},

volume = {73},

pages = {38--48}

}

### Years of Citing Articles

### OpenURL

### Abstract

By focusing attention on close returns of a trajectory to itself, the existence of deterministic dynamics underlying a time series can be detected even in very short data sets. This provides a practical means of detecting determinism in moderate-dimensional (e.g. ß 7) noisy systems, or low-dimensional systems with large Lyapunov exponents such as computer random number generators. A large variety of methods have been developed to reconstruct dynamics from measured time series, and to characterize dynamics in terms of predictability or dynamical invariants such as the correlation dimension or spectrum of Lyapunov exponents. These characterizations are often applied to experimental or field time series in order to decide whether the data are consistent with a low-dimensional deterministic mechanism, or a stochastic (or extremely high-dimensional, hence, effectively stochastic) mechanism. Limits on the power of these methods arise from finite and often small lengths of data sets and measu...

### Citations

252 |
Measuring the strangeness of strange attractors
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(Show Context)
Citation Context ...ation is infinite dimensional, however the attractors are finite-dimensional. Specifically, for a = 0:2, b = 0:1, c = 10 and \Delta = 100, the attractor is estimated to be approximately 7-dimensional =-=[10]-=-, suggesting that an adequate representation of the dynamics can be achieved with a measurement ~z 2 IR m where m ? 7. (The equations were integrated using a fourth-order fixed-stepsize (dt = 0:1) Run... |

157 |
Nonlinear prediction of chaotic time series
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Citation Context ...the time series. The Mackey-Glass system (Eq. 15) at the parameters \Delta, a, b, and c used here has been studied previously in terms of correlation dimensions and nonlinear predictability. Casdagli =-=[11] used 9 \T-=-heta 10 4 data points (with sampling time T = 6) to construct an ad hoc model that makes "acceptable" predictions. This is consistent with the idea that methods that rely on typical events r... |

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Approximate Entropy as a Measure of System Complexity
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Citation Context ...erlying the time series. One can easily construct a time series that consists of a few repetitions of a segment of fixed pattern, surrounded by fundamentally stochastic episodes (e.g, the mix process =-=[19]-=-). At the smallest r, the points lying in the fixed pattern segment will dominate, and will indicate deterministic structure. Ignoring the interesting philosophical question of whether such a time ser... |

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Citation Context ... thumb is to selectssuch that \Psi( )s1=e. Ifsis too small, i.e., \Psi( )s1, then it can be more difficult to distinguish between deterministic and stochastic dynamics. In addition, following Theiler =-=[8]-=-, in calculating ffl(r) it is helpful to ignore those pairs j; k where j and k are close, e.g. where \Psi(jj \Gamma kj) ? 1=e. The analysis of time series from high-dimensional or continuous-time syst... |

69 |
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(Show Context)
Citation Context ...he time series provides evidence for some local form of P p\Gamma1 k=0 ff k x i\Gammak in Eq. 5 which diminishes the role ofsi compared to some other plausible stochastic model. For example, Casdagli =-=[14] and others have sug-=-gested that a better posed problem than "detecting determinism" is "detecting dynamical nonlinearities," and that one should seek to establish the length scale over which the linea... |

28 |
Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems
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(Show Context)
Citation Context ...ce the "scaling region" in r over which the dimension D = d ln C(r)=d ln r is calculated may include only very few pairs of points. In fact, the theoretical lower bound proposed by Eckmann a=-=nd Ruelle [12]-=- on the amount of data needed to estimate a valid dimension of D from a time series is founded on the smallest interpoint distance to be found in the data set. For a system of dimension Ds7, the Eckma... |

17 |
Method to distinguish possible chaos from colored noise and to determine embedding parameters
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Citation Context ... error bars on ffl(r). (A cumulative statistic, E(r), will be introduced below that eliminates the need to set \Deltar.) The method described here is similar to methods proposed by Kennel and Isabell =-=[2]-=- and Wayland et al. [3], who consider typical rather than exceptional events, and therefore do not categorize ffl j;k by ffi j;k or construct the average ffl(r). For points z j , z k that are very clo... |

9 |
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Citation Context ...sion Ds7, the Eckmann-Ruelle bound is Ns3000. This bound, however, includes an assumption about an appropriate size of a scaling region, which might not be appropriate in practice. Havstad and Ehlers =-=[13]-=- found that they could make meaningful estimates of the dimension of the Mackey-Glass data using much smaller N . The smallest set they consider has N = 50, but they present results only for averages ... |

4 |
HDI Abarbanel, "Determining minimum embedding dimension using a geometrical construction," Phys. Rev. A
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Citation Context ... be a small-dimensional attractor, or the ability to predict the image of a point from the images of its neighbors [14], or persistence of neighboring points as the embedding dimension m is increased =-=[17]-=-, or the parallelism of secants to nearby passes of the trajectory [18]. The structure we consider here is extremely simple and arises directly from an assumption of piecewise continuity of a determin... |

3 |
Calculating the Dimension of Attractors from Small Data Sets, Physics Letters A 114
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Citation Context ...al spacing between nearest neighbor points on a D-dimensional attractor reconstructed from N data points is of order L=N 1=D , where L is the order of the largest interpoint distance on the attractor =-=[1]. This sca-=-ling comes from the following argument: the overall "bulk" of the attractor is L D . If the data points were uniformly distributed, each sitting at the center of a D-dimensional hypercube of... |

3 |
personal communication
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(Show Context)
Citation Context ...(r): E(r) j ffl j;k for j; k s.t. ffi j;k ! r: (7) E contains much the same information as ffl(r) but since it includes all the pairs where ffi j;k ! r, it can sometimes reduce the size of error bars =-=[5]-=-. Calculation of E avoids the need to set a value for \Deltar. 3 Probing continuity using exceptional events It is not necessary for the measured time series to describe the functional form of the map... |

1 |
A Passamante, "Recognizing determinism in a time series
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(Show Context)
Citation Context ...(A cumulative statistic, E(r), will be introduced below that eliminates the need to set \Deltar.) The method described here is similar to methods proposed by Kennel and Isabell [2] and Wayland et al. =-=[3]-=-, who consider typical rather than exceptional events, and therefore do not categorize ffl j;k by ffi j;k or construct the average ffl(r). For points z j , z k that are very close together, the separa... |

1 |
A Longtin, S Eubank and JD Farmer, "Using surrogate data to detect nonlinearity in time series
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(Show Context)
Citation Context ... in which case E can be calculated by a suitable averaging process. Given a stochastic model for a time series, one can calculate E either analytically for certain models or using Monte Carlo methods =-=[4]-=-. From the time series itself, one can calculate ffl(r) and also the correlation integral C(r). If lim r!0 ffl(r) ! E in a statistically significant way, then the 3 stochastic model has been ruled out... |

1 |
The "Minimal Standard" generator is ran0(). The authors of Numerical Recipes recommend generally the use of another generator, ran1(), in order to be able avoid the serial correlations that are detectable with the test in the output of ran0() wh
- Chap
(Show Context)
Citation Context ...en termed "pseudo-random." A common type is the linear congruential generator x i+1 = ax i + c (mod b) ; z i = x i =b (11) which is piecewise continuous. The "Minimal Standard" gen=-=erator suggested in [6]-=- has values a = 16807, c = 0, and b = 2 31 \Gamma 1. This map consists of 16807 piecewise linear segments. Knowing the structure of the map, and that E = 1=3 for uniformly distributed random numbers i... |

1 |
Other methods can also be used, such as measuring multiple signals from the dynamical system
- Sauer, Yorke, et al.
- 1991
(Show Context)
Citation Context ...ven if we make a series of scalar measurements (e.g., z i 2 IR, z i = h(x(t 0 + iT ))), we can still create a representation of the time series fz i g in IR m , by using the method of delay-embedding =-=[7]-=- ~z i j (z i ; z i\Gamma ; \Delta \Delta \Delta ; z i\Gamma(m\Gamma1) ); (13) wheresis the embedding lag. This same situation applies to discrete-time dynamical systems with p ? 1. Since we do not kno... |

1 |
H-F Graf, "Dimension analysis of El Ni~no/Southern Oscillation time series", Ann. Geophysicae
- Gober, Herzel
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(Show Context)
Citation Context ... (e.g., fluctuations in the numbers of blood cells), or are produced at a slow, fixed rate (e.g., the numbers of sunspots, or climatological data such as the El Ni~no/Southern Oscillation time series =-=[16]-=-) or because a changing environment or changing system necessitate the use of short data segments during which the assumption of environmental or system stationarity might be justified. The meaning of... |

1 |
Coarse-grained Embeddings of Time Series
- Kaplan, Glass
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(Show Context)
Citation Context ...of a point from the images of its neighbors [14], or persistence of neighboring points as the embedding dimension m is increased [17], or the parallelism of secants to nearby passes of the trajectory =-=[18]-=-. The structure we consider here is extremely simple and arises directly from an assumption of piecewise continuity of a deterministic dynamical map: nearby points tending to have nearby images. Even ... |