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## The concept of surrogate data

The degree of nonlinearity can be measured in several ways. But how much nonlinear predictability, say, is necessary to exclude more trivial explanations? All quantifiers of nonlinearity show fluctuations but the distributions, or error bars if you wish, are not available analytically. It is therefore necessary to use Monte Carlo techniques to assess the significance of results. One important method in this context is the method of surrogate data [82]. A null hypothesis is formulated, for example that the data has been created by a stationary Gaussian linear process, and then it is attempted to reject this hypothesis by comparing results for the data to appropriate realizations of the null hypothesis. Since the null assumption is not a simple one but leaves room for free parameters, the Monte Carlo sample has to take these into account. One approach is to construct constrained realizations of the null hypothesis. The idea is that the free parameters left by the null are reflected by specific properties of the data. For example the unknown coefficients of an autoregressive process are reflected in the autocorrelation function. Constrained realizations are obtained by randomizing the data subject to the constraint that an appropriate set of parameters remains fixed. For example, random data with a given periodogram can be made by assuming random phases and taking the inverse Fourier transform of the given periodogram. Random data with the same distribution as a given data set can be generated by permuting the data randomly without replacement. Asking for a given spectrum and a given distribution at the same time poses already a much more difficult question.

Next: Iterative Fourier transform method Up: Testing for nonlinearity Previous: Testing for nonlinearity