Using \surrogate Surrogate Data" to Calibrate the Actual Rate of False Positives in Tests for Nonlinearity in Time Series

A distinction can be drawn between two approaches for using sur-rogate data to test for nonlinearity in a time series. The rst is a \typical realizations" model which can be implemented in terms of a direct autore-gressive (AR) t to the data, and the second is a \constrained realizations" model which is implemented using a Fourier transform (FT). Earlier comparisons of these two methods found that that the FT surrogate data test was the more powerful of the two for the same nominal false-positive rate, but that the actual false-positive rate was much lower for the AR test. In this paper, further comparisons are made of the FT and AR tests. A kind of \double bootstrap" approach (that involves taking surrogate data of surrogate data) is suggested for calibrating hypothesis tests so that their actual rate of false positives matches a speciied nominal false-positive rate. Numerical experiments are performed using a noise-corrupted high-dimensional strange attractor to generate the data, and a global nonlinear predictor as a discriminating statistic. These experiments indicate that the FT algorithm still provides a more powerful test than the AR algorithm, when compared at the same actual size, though the diierence is considerably less than when compared at the same nominal size. Further experiments comparing in-sample with out-of-sample prediction show that in-sample prediction error, despite its aws as an indicator of actual predictability, can provide a more powerful discriminating statistic when used for hypothesis testing. 1 A bridge between the natural and the statistical sciences Over thirty years ago, in a dozen dense pages in the Journal of the Atmospheric Sciences, Edward N. Lorenz described an extensive set of numerical experiments which explained and illustrated the notion of sensitive dependence on initial conditions from deterministic dynamical equations Lorenz 1963]. That same year, in a

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