Non-stationarity

  It is quite common in bio-medical time series (and elsewhere) that otherwise harmless looking data once in a while are interrupted by a singular event, for example a spike. It is now debatable whether such spikes can be generated by a linear process by nonlinear rescaling. We do not want to enter such a discussion here but merely state that a time series that covers only one or a few such events is not suitable for the statistical study of the spike generation process. The best working assumption is that the spike comes in by some external process, thus rendering the time series non-stationary. In any case, the null hypotheses we are usually testing against are not likely to generate such singular events autonomously. Thus, typically, a series with a single spike will be found to violate the null hypothesis, but, arguably, the cause is non-stationarity rather than non-linearity. Let us discuss as a simple example the same AR(2) process considered previously, this time without any rescaling. Only at a single instant, n=1900, the system is kicked by a large impulse instead of the Gaussian variate . This impulse leads to the formation of a rather large spike. Such a sequence is shown in Fig. 23. Note that due to the correlations in the process, the spike covers more than a single measurement.

When we generate surrogate data, the first observation we make is that it takes the algorithm more than 400 iterations in order to converge to a reasonable tradeoff between the correct spectrum and the required distribution of points. Nevertheless, the accuracy is quite good -- the spectrum is correct within 0.1% of the rms amplitude. Visual inspection of the lower panel of Fig. 23 shows that the spectral content -- and the assumed values -- during the single spike are represented in the surrogates by a large number of shorter spikes. The surrogates cannot know of an external kick. The visual result can be confirmed by a statistical test with several surrogates, equally well (99% significance) by a time asymmetry statistic or a nonlinear prediction error.

If non-stationarity is known to be present, it is necessary to include it in the null hypothesis explicitly. This is in general very difficult but can be undertaken in some well behaved cases. In Sec. 6.1 we discussed the simplest situation of a slow drift in the calibration of the data. It has been shown empirically [52] that a slow drift in system parameters is not as harmful as expected [53]. It is possible to generate surrogates for sliding windows and restrict the discriminating statistics to exclude the points at the window boundaries. It is quite obvious that special care has to be taken in such an analysis.