## Detecting Determinism in Time Series: The method of Surrogate Data (2003)

Venue: | IEEE TRANS. ON CIRCUITS AND SYSTEMS-I FUNDAMENTAL THEORY AND APPLICATIONS |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Small03detectingdeterminism,

author = {Michael Small and Chi K. Tse},

title = {Detecting Determinism in Time Series: The method of Surrogate Data},

journal = {IEEE TRANS. ON CIRCUITS AND SYSTEMS-I FUNDAMENTAL THEORY AND APPLICATIONS},

year = {2003},

volume = {50},

pages = {663--672}

}

### OpenURL

### Abstract

We review a relatively new statistical test that may be applied to determine whether an observed time series is inconsistent with a specific class of dynamical systems. These surrogate data methods may test an observed time series against the hypotheses of: i) independent and identically distributed noise; ii) linearly filtered noise; and iii) a monotonic nonlinear transformation of linearly filtered noise. A recently suggested fourth algorithm for testing the hypothesis of a periodic orbit with uncorrelated noise is also described. We propose several novel applications of these methods for various engineering problems, including: identifying a deterministic (message) signal in a noisy time series; and separating deterministic and stochastic components. When employed to separate deterministic and noise components, we show that the application of surrogate methods to the residuals of nonlinear models is equivalent to fitting that model subject to an information theoretic model selection criteria.

### Citations

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186 |
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(Show Context)
Citation Context ...ies has statistically significant deterministic component. Three standard linear techniques are widely applied in the physical and biological sciences (see for example, [1]) to test the hypotheses of =-=[2]-=-: i) independent and identically distributed (i.i.d.) noise1 ; ii) linearly filtered noise; and (iii) static monotonic nonlinear transformation of linearly filtered noise. Surrogate methods require on... |

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Citation Context ...e series (for model residual and surrogates) are equivalent. As an alternative, we employ a test statistic that is much more sensitive to temporal dependencies in the data: nonlinear prediction error =-=[18]-=-. Nonlinear prediction error is a measure of the amount of determinism in the time series. For a delay embedding such as (1) one predicts the future behavior of state by building a linear model based ... |

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Citation Context ... the amount of determinism in the time series. For a delay embedding such as (1) one predicts the future behavior of state by building a linear model based on the behavior of the spatial neighbors of =-=[19]-=-. Nonlinear prediction of one indicates no temporal determinism, less than one indicates nonlinear determinism. The second panel of Fig. 7 shows the results of this calculation for the model described... |

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Citation Context ...ty distributions. 1057-7122/03$17.00 © 2003 IEEE the hypothesis of interest. One then statistically compares the data and surrogates. Many extensions of this basic method have been suggested ([3] and =-=[4]-=-) but, in this current work, we are particularly interested in a method that may be applied to test for nonlinear nonperiodic determinism in apparently periodic time series [5], [6]. Surrogate methods... |

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Citation Context ...time series is , the scalar first components of . The above algorithm has three parameters: , , and . Selection of embedding parameters and is discussed at length in the literature (see, for example, =-=[16]-=- and the references therein) and we do not consider this problem here. The noise radius is selected according to the suggestions of [5]. We choose such that the expected number of sequences of length ... |

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Citation Context ...ere originally introduced as a “sanity test” for correlation dimension estimation [2], correlation dimension is a popular choice [1], [6]. A complete discussion of choice of statistic may be found in =-=[13]-=-, and Small and Judd [3] demonstrate that correlation dimension is indeed a good choice. We will paraphrase these results and say that the chosen statistic should be independent of the surrogate gener... |

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Citation Context ...]. Surrogate methods identify whether an observed time series contains determinism. By themselves, they cannot separate the noise and deterministic components. We employ nonlinear modeling techniques =-=[7]-=- that utilize the information theoretic minimum description length (MDL) [8] to fit data without over fitting. These methods have been successfully employed to identify and model the deterministic com... |

15 |
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Citation Context ...namical systems being tested (thus, nonlinear prediction is a good choice for the three linear hypotheses). The results in this paper use a correlation dimension estimation algorithm proposed by Judd =-=[14]-=-, [15] that has been shown to be a good choice of test statistic for many classes of dynamical systems of interest [3]. For the sake of brevity, we do not define correlation dimension here, other than... |

13 |
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Citation Context ...ne whether an observed time series has statistically significant deterministic component. Three standard linear techniques are widely applied in the physical and biological sciences (see for example, =-=[1]-=-) to test the hypotheses of [2]: i) independent and identically distributed (i.i.d.) noise1 ; ii) linearly filtered noise; and (iii) static monotonic nonlinear transformation of linearly filtered nois... |

13 |
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Citation Context ... separate certain classes of noise, such as i.i.d. or colored noise, from complex deterministic dynamics in a data set. A similar interpretation of surrogate data methods has been suggested by Takens =-=[12]-=-. Takens considered fitting some model to a time series and applying i.i.d. surrogate tests to the residuals. He showed that if the residuals are i.i.d., then the model offers a good fit to the data. ... |

12 |
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(Show Context)
Citation Context ...been suggested ([3] and [4]) but, in this current work, we are particularly interested in a method that may be applied to test for nonlinear nonperiodic determinism in apparently periodic time series =-=[5]-=-, [6]. Surrogate methods identify whether an observed time series contains determinism. By themselves, they cannot separate the noise and deterministic components. We employ nonlinear modeling techniq... |

11 |
Applying the method of surrogate data to cyclic time series. Physica D: Nonlinear Phenomena
- Small, Tse
- 2002
(Show Context)
Citation Context ...operiodic dynamics, rejection of this class of systems implies the existence of nonperiodic deterministic dynamics. However, for time series that do not exhibit pseudoperiodic dynamics, we have shown =-=[6]-=- that surrogates generated by this method are actually consistent with short-term deterministic dynamics. Rejection of this test therefore is evidence for long-term deterministic dynamics. The distinc... |

7 | Comparison of new nonlinear modelling techniques with applications to infant respiration
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(Show Context)
Citation Context ...cription length (MDL) [8] to fit data without over fitting. These methods have been successfully employed to identify and model the deterministic component of time series of many experimental systems =-=[9]-=-–[11]. We show that these methods may be employed to separate the noise and signal in an observed time series. Alternatively, we show that surrogate data methods applied to the model residuals may als... |

7 |
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Citation Context ...dels. Conversely, in [3] and [12], it has been shown that noisy iterated predictions of a model may be used as a form of hypothesis testing. For a scalar time series we perform a time delay embedding =-=[17]-=- according to (There is a minor notational change between this embedding and that in Section II-B. This distinction is cosmetic and largely a matter of convenience.) We then model the dynamics of the ... |

5 | Correlation dimension: A pivotal statistic for non-constrained realizations of composite hypotheses in surrogate data analysis
- Small, Judd
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(Show Context)
Citation Context ...robability distributions. 1057-7122/03$17.00 © 2003 IEEE the hypothesis of interest. One then statistically compares the data and surrogates. Many extensions of this basic method have been suggested (=-=[3]-=- and [4]) but, in this current work, we are particularly interested in a method that may be applied to test for nonlinear nonperiodic determinism in apparently periodic time series [5], [6]. Surrogate... |

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1 |
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(Show Context)
Citation Context ...tion length (MDL) [8] to fit data without over fitting. These methods have been successfully employed to identify and model the deterministic component of time series of many experimental systems [9]–=-=[11]-=-. We show that these methods may be employed to separate the noise and signal in an observed time series. Alternatively, we show that surrogate data methods applied to the model residuals may also be ... |