## Detecting Nonlinearity in Experimental Data (1997)

Venue: | International Journal of Bifurcation and Chaos Submitted |

Citations: | 5 - 5 self |

### BibTeX

@ARTICLE{Small97detectingnonlinearity,

author = {Michael Small and Kevin Judd},

title = {Detecting Nonlinearity in Experimental Data},

journal = {International Journal of Bifurcation and Chaos Submitted},

year = {1997},

volume = {8},

pages = {1231--1244}

}

### OpenURL

### Abstract

The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data are clearly distinct from the different classes of linear systems we are unable to distinguish between our data and surrogates generated by nonlinear models. Hence we conclude that human respiration is likely to be a nonlinear system with more than 2 degrees of freedom with a limit cycle that is driven by high dimensional dynamics or noise.

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Citation Context ...ods [Rapp 1994] for choosing an appropriate value of the lags; the first zero of the autocorrelation function [Albano et al. 1988; Albano et al. 1991] and the first minimum of the mutual information [=-=Abarbanel et al. 1993-=-; Fraser and Swinney 1986; Martinerie et al. 1992]. The rationale of both of them, however, is to choose the lag so that the coordinate components of v t are reasonably uncorrelated while still being ... |

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Citation Context ..., except that it tests temporal correlation between cycles, not data points. In other studies we have examined the correlation between cycles directly, by reducing each cycle to a single measurement [=-=Small et al. 1996-=-; Small et al. a]. It is then possible not only to test algorithm 0 type hypotheses but also algorithm 1 and 2. However, reducing each cycle to a single measurement can result in substantial loss of i... |

3 |
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Citation Context ...test statistics are equally good. Furthermore, not all hypotheses are as straightforward, or interesting, as they may appear. It is possible that one of the surrogate generating algorithms is flawed [=-=Schreiber and Schmitz 1996-=-] and the choice of test statistic and surrogate generation algorithm should be made very carefully [Theiler and Prichard 1996]. 1 Because nonlinear measures are of particular interest they are often ... |

2 |
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Citation Context ...iations from this regular periodic behavior. Because we wish to observe such fine detail we did not filter signals. Filtering methods, such as linear filters and singular-value decomposition methods [=-=Pilgram et al. 1995-=-], can remove some features that we wish to observe. Furthermore, filtering has been shown in some cases to lead to erroneous identification of chaos [Mees et al. 1987; Palus and Dvorak 1992]. 4.2 Com... |

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Citation Context ...50Hz using a 12 bit analogue to digital convertor. From measurements of electroencephalogram, electromyogram and electrooculogram, sleep stage was determined using standard polysomnographic criteria [=-=Anders et al. 1971-=-]. During quiet sleep (stages 3--4) breathing often appears relatively regular. The possibly chaotic features of most interest are the small variations from this regular periodic behavior. Because we ... |

1 |
Preprint submitted to International Journal of Bifurcation and Chaos 24
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(Show Context)
Citation Context ...s; the existing methods, our nonlinear modeling approach, and suitability of test statistics. We then introduce correlation dimension, we discuss some pitfalls of the Grassberger-Procaccia algorithm [=-=Grassberger and Procaccia 1983-=-a; Grassberger and Procaccia 1983b] and the benefits of Judd's method [Judd 1992; Judd 1994], and consider the appropriateness of correlation dimension as a test statistic. Finally we present some num... |

1 |
Preprint submitted to International Journal of Bifurcation and Chaos 26
- Theiler
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(Show Context)
Citation Context ...straightforward. Thieler's original work on surrogate methods [Theiler et al. 1992], suggested a "hierarchy" of hypotheses that should be tested with a "battery" of test statistics=-=. More recent work [Theiler 1995-=-; Theiler and Rapp 1996] has demonstrated that not all test statistics are equally good. Furthermore, not all hypotheses are as straightforward, or interesting, as they may appear. It is possible that... |