We propose an extension to multivariate time series of the phase-randomized Fourier-transform algorithm for generating surrogate data. Such surrogate data sets must mimic not only the autocorrelations of each of the variables in the original data set, they must mimic the cross-correlations between all the variables as well. The method is applied both to a simulated example (the three components of the Lorenz equations) and to data from a multichannel electroencephalogram.

We compare two theoretically distinct approaches to generating articial (or \surrogate") data for testing hypotheses about a given data set. The rst and more straightforward approach is to t a single \best" model to the original data, and then to generate surrogate data sets that are \typical realizations" of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a speci ed set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) ts to the data. The comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal. A statistic is \pivotal" if its distribution is the same for all processes consistent with the null hypothesis. The typical-realization method requires that the discriminating statistic satisfy this property. The constrained-realization approach, on the other hand, does not share this requirement, and can provide an accurate and powerful test without having to sacri ce exibility in the choice of discriminating statistic.

Extensions to various information theoretic quantities used for nonlinear time series analysis are discussed, as well as their relationship to the generalized correlation integral. It is shown that calculating redundancies from the correlation integral can be more accurate and more efficient than direct box counting methods. It is also demonstrated that many commonly used nonlinear statistics have information theory based analogues. Furthermore, the relationship between the correlation integral and information theoretic statistics allows us to define ``local'' versions of many information theory based statistics; including a local version of the Kolmogorov-Sinai entropy, which gives an estimate of the local predictability.Comment: 19 pages uuencoded compressed postscript file (using uufiles script

Recent theories of the effects of ethanol on the brain have focused on its direct actions on neuronal membrane proteins. However, neuromolecular mechanisms whereby ethanol produces its CNS effects in low doses typically used by social drinkers (e.g., 2-3 drinks, 10-25 mM, 0.05-0.125 gm/dl) remain less well understood. We propose the hypothesis that ethanol may act by introducing a level of randomness or "noise" in brain electrical activity. We investigated the hypothesis by applying a battery of tests originally developed for nonlinear time series analysis and chaos theory to EEG data collected from 32 men who had participated in an ethanol/placebo challenge protocol. Because nonlinearity is a prerequisite for chaos and because we can detect nonlinearity more reliably than chaos, we concentrated on a series of measures that quantitated different aspects of nonlinearity. For each of these measures the method of surrogate data was used to assess the significance of evidence for nonlinear structure. Significant nonlinear structure was found in the EEG as evidenced by the measures of time asymmetry, determinism, and redundancy. In addition, the evidence for nonlinear structure in the placebo condition was found to be significantly greater than that for ethanol. Nonlinear measures, but not spectral measures, were found to correlate with a subject's overall feeling of intoxication. These findings are consistent with the notion that ethanol may act by introducing a level of randomness in neuronal processing as assessed by EEG nonlinear structure.