Multivariate data

  In Ref. [26], the flexibility of the approach was illustrated by a simultaneous recording of the breath rate and the instantaneous heart rate of a human subject during sleep. The interesting question was, how much of the structure in the heart rate data can be explained by linear dependence on the breath rate. In order to answer this question, surrogates were made that had the same autocorrelation structure but also the same cross-correlation with respect to the fixed input signal, the breath rate. While the linear cross-correlation with the breath rate explained the coherent structure of the heart rate, other features, in particular its asymmetry under time reversal, remained unexplained. Possible explanations include artefacts due to the peculiar way of deriving heart rate from inter-beat intervals, nonlinear coupling to the breath activity, nonlinearity in the cardiac system, and others.

Within the general framework, multivariate data can be treated very much the same way as scalar time series. In the above example, we chose to use one of the channels as a reference signal which was not randomised. The rationale behind this was that we were not looking for nonlinear structure in the breath rate itself and thus we didn't want to destroy any such structure in the surrogates. In other cases, we can decide either to keep or to destroy cross-correlations between channels. The former can be achieved by applying the same permutations to all channels. Due to the limited experience we have so far and the multitude of possible cases, multivariate problems have not been included in the TISEAN implementation yet.