An Approach to Error-Estimation in the Application of Dimension Algorithms

Holzfuss, Joachim; Mayer‐Kress, Gottfried

doi:10.1007/978-3-642-71001-8_15

Cited by 130 publications

(63 citation statements)

References 3 publications

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“…The delay time, τ, for the phase-space reconstruction is computed using the auto-correlation function method and is taken as the lag time at which the auto-correlation function first crosses the zero line (e.g. Holzfuss and Mayer-Kress, 1986). function against the lag time for the daily rainfall series.…”

Section: Analyses Results and Discussionmentioning

confidence: 99%

Rainfall dynamics at different temporal scales: A chaotic perspective

Sivakumar

2001

Hydrol. Earth Syst. Sci.

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This study of the behaviour of rainfall dynamics at different temporal scales identifies the type of approach most suitable for transformation of rainfall data from one scale to another. Rainfall data of four different temporal scales, i.e. daily, 2-day, 4-day and 8-day, observed over a period of about 25 years at the Leaf River basin, Mississippi, USA, are analysed. The correlation dimension method is employed to identify the behaviour of rainfall dynamics. The finite correlation dimensions obtained for the four rainfall series (4.82, 5.26, 6.42 and 8.87, respectively) indicate the possible existence of chaotic behaviour in the rainfall observed at the four scales. A possible implication of this might be that the rainfall processes at these scales are related through a chaotic (scale-invariant) behaviour. However, a comparison of the correlation dimension and coefficient of variation of each of the time series reveals an inverse relationship between the two (higher dimension for lower coefficient of variation and vice versa). The presence of a large number of zeros in the higher resolution time series (that could result in an underestimation of the dimension) and the possible presence of a higher level of noise in the lower resolution time series (that could result in an overestimation of the dimension) might account for such results. In view of these problems, it is concluded that the results must be verified using other chaos identification methods and the existence of chaos must be substantiated with additional evidence.

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Section: Analyses Results and Discussionmentioning

confidence: 99%

Rainfall dynamics at different temporal scales: A chaotic perspective

Sivakumar

2001

Hydrol. Earth Syst. Sci.

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“…Various methods/guidelines have been proposed for τ selection to have the best separation of neighboring trajectories, including autocorrelation function (e.g. Holzfuss and Mayer-Kress, 1986), mutual information (e.g. Fraser and Swinney, 1986), and correlation integral (Liebert and Schuster, 1989).…”

Section: Discussionmentioning

confidence: 99%

Hydrologic system complexity and nonlinear dynamic concepts for a catchment classification framework

Sivakumar

Singh

2012

Hydrol. Earth Syst. Sci.

134

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Abstract. The absence of a generic modeling framework in hydrology has long been recognized. With our current practice of developing more and more complex models for specific individual situations, there is an increasing emphasis and urgency on this issue. There have been some attempts to provide guidelines for a catchment classification framework, but research in this area is still in a state of infancy. To move forward on this classification framework, identification of an appropriate basis and development of a suitable methodology for its representation are vital. The present study argues that hydrologic system complexity is an appropriate basis for this classification framework and nonlinear dynamic concepts constitute a suitable methodology. The study employs a popular nonlinear dynamic method for identification of the level of complexity of streamflow and for its classification. The correlation dimension method, which has its base on data reconstruction and nearest neighbor concepts, is applied to monthly streamflow time series from a large network of 117 gaging stations across 11 states in the western United States (US). The dimensionality of the time series forms the basis for identification of system complexity and, accordingly, streamflows are classified into four major categories: low-dimensional, medium-dimensional, highdimensional, and unidentifiable. The dimension estimates show some "homogeneity" in flow complexity within certain regions of the western US, but there are also strong exceptions.

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“…One can guard against this by attempting to identify the various sources of error (both systematic and statistical), and then putting error bars on the estimate I ": (see, for example, Refs. [6][7][8][9][10][11][12]). But this can be problematic for nonlinear algorithms like dimension estimators: first, assignment of error bars requires some model of the underlying process, and that is exactly what is not known; further, even if the underlying process were known, the computation of an error bar may be analytically difficult if not intractable.…”

Section: Introductionmentioning

confidence: 99%

Testing for nonlinearity in time series: the method of surrogate data

Theiler

Eubank

Longtin

et al. 1992

Physica D: Nonlinear Phenomena

3,407

2,533

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Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or o_herwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any ager, cy thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.By acceplance of mta article. Thepubhsher rec_gn_-"eS that the U.S Government retains a nonexcluslve, royalty-free hcenlm1opubhll_ or reproduce tr_e0ubhSr_edform Of Ih,S conlr_buhOn,or tO allow omer$ to do so. for U S Government purposes T_e Los Alamos National Laboratory requests thal the 10ubllsheridefltlfy this art,cle as work performed under the ouspfceI of the U S Depo_menl at Energy ASTEB Lo8LosAa os a, ona a0o a,or ABSTRACTWe describe a statistical approach for identifying nonlinearity in time series; in particular, we want to avoid claims of chaos when simpler models (such as linearly correlated noise) can explain the data. The method requires a careful statement of the null hypothesis which characterizes a candidate linear process, the generation of an ensemble of "surrogate" data sets which are similar to the original time series but consistent with the null hypothesis, and the computation of a discriminating statistic for the original and for each of the surrogate data sets. The idea is to test the original time series against the null hypothesis by checking whether the discriminating statistic computed for the original time series differs significantly from the statistics computed for each of the surrogate sets. We present algorithms for generating surrogate data under various null hypotheses, and we show the results of numerical experiments on artificial data using correlation dimension, Lyapunov exponent, and forecasting error as discriminating statistics.Finally, we consider a number of experimental time series m including sunspots, electroencephalogram (EEG) signals, and fluid convection --and evaluate the statistical significance of the evidence for nonlinear structure in each case.

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An Approach to Error-Estimation in the Application of Dimension Algorithms

Cited by 130 publications

References 3 publications

Rainfall dynamics at different temporal scales: A chaotic perspective

Rainfall dynamics at different temporal scales: A chaotic perspective

Hydrologic system complexity and nonlinear dynamic concepts for a catchment classification framework

Testing for nonlinearity in time series: the method of surrogate data

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