Figure 3 shows the autocorrelation of CAAFT surrogates
compared with the original eight time series used
in this study. The MATLAB software associated with
(Kugiumtzis 2000) is used to generate the CAAFT surrogates.
It can be seen that they reproduce the original
autocorrelation well in most cases although those in the
lower right panel are biased downwards.
We next compare the ratio of linear vs. nonlinear insample
forecast error for the original and surrogate time
series. If there is nonlinearity present, we would expect
the nonlinear method to show an improvement over the
linear method. The linear method used is persistence, and
the nonlinear method is a zero-order nearest-neighbor
method which is described in Additional file 1: Section
II and implemented in the TISEAN function lzo-run
(Hegger et al. 1999). Figure 4 shows the result displayed
as a histogram, with the error ratio for the original series
shown as a dark line. None of the original time series
appears to benefit from nonlinear forecasting. The histogram
on the bottom-right of the figure shows that the
time series is better predicted by the linear method. This
is a result of the lower average correlation among the
surrogates for this time series, shown in Figure 3.
Finally, we compare the original and surrogate time
series using a time reversal asymmetry statistic. Asymmetry
of the time series when reversed in time can be a
signature of nonlinearity (Theiler et al. 1992). A measure
of time reversibility is the ratio of the mean cubed to the
mean squared differences,
Q = E[(yi+1 − yi)3]
E[(yi+1 − yi)2]
Figure 5 shows the Q statistic values for the surrogates
shown as a histogram and the statistic for the original
series shown as a dark line. There is no general evidence
of time asymmetry in the patients’ time series, again with
the exception of the series shown on the bottom right.