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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We present a forecasting technique for chaotic data. After embedding a time series in a state space using delay coordinates, we "learn" the induced nonlinear mapping using a local approximation. This allows us to make short-term predictions of the future behavior of a time series, using information based only on past values. We present an error estimate for this technique, and demonstrate its eAectiveness by applying it to several examples, including data from the Mackey-Glass delay differential equation, Rayleigh-Benard convection, and Taylor-Couette flow.