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## Rescaled Gaussian linear process

The two null hypotheses discussed so far (independent random numbers and Gaussian linear processes) are not what we want to test against in most realistic situations. In particular, the most obvious deviation from the Gaussian linear process is usually that the data do not follow a Gaussian single time probability distribution. This is quite obvious for data obtained by measuring intervals between events, e.g. heart beats since intervals are strictly positive. There is however a simple generalisation of the null hypothesis that explains deviations from the normal distribution by the action of an invertible, static measurement function:
We want to regard a time series from such a process as essentially linear since the only nonlinearity is contained in the -- in principle invertible -- measurement function .

Let us mention right away that the restriction that must be invertible is quite severe and often undesired. The reason why we have to impose it is that otherwise we couldn't give a complete specification of the process in terms of observables and constraints. The problem is further illustrated in Sec. 7.1 below.

The most common method to create surrogate data sets for this null hypothesis essentially attempts to invert by rescaling the time series to conform with a Gaussian distribution. The rescaled version is then phase randomised (conserving Gaussianity on average) and the result is rescaled to the empirical distribution of . The rescaling is done by simple rank ordering. Suppose we want to rescale the sequence so that the rescaled sequence takes on the same values as some reference sequence (e.g. draws from a Gaussian distribution). Let be sorted in ascending order and denote the ascending rank of , e.g. if is the 3rd smallest element of . Then the rescaled sequence is given by
The amplitude adjusted Fourier transform (AAFT) method has been originally proposed by Theiler et al. [6]. It results in a correct test when N is large, the correlation in the data is not too strong and is close to the identity. Otherwise, there is a certain bias towards a too flat spectrum, to be discussed in the following section.

Next: Flatness bias of AAFT Up: Fourier based surrogates Previous: Fourier based surrogates