The global morphological analysis of a time-delay embedding of the scalar time series

We have developed a new method for analyzing the long scalar experimental data. By the time delay embedding, the oscillatory nature of the data was transformed into a sequence of loops of the system phase space trajectory. The proposed method reduces the classification of all possible loop types and rules of their succession. Its true merit shows when dealing with adaptive complex systems, where dynamical models are oversimplistic or nonexistent, and it is virtually impossible to separate the systematic and stochastic parts of the data. Thus, the proposed method will be presented by analyzing the experimental data obtained from the Holter electrocardiogram of an arbitrarily chosen 79-year-old male patient and a 24-year-old healthy female individual. In the case of the patient, the system's trajectory consists of only 8 of the most simple loop types. Out of 64 possible loop transitions, only 49 play a significant role in the observed system dynamics. Further, we constructed a stochastic finite state machine capable of reproducing observed system trajectory statistically. We also investigated metamorphoses of the phase space distribution. The transformations of the distribution from clustered to branched were found to be manifestations of the identified Fold-Hopf bifurcations. The Holter electrocardiogram of a 24-year-old healthy individual showed no Fold-Hopf bifurcations, and the corresponding finite state machine has the same number of states but with radically different distributions of the transition probabilities. Our method shows more sensitivity in detecting pathological states than standard heart rate variability assessments. Its topological nature makes it very robust to the disturbing effect of noise. Although, additional research on a larger experimental sample is required before the usefulness of the proposed morphological approach can be rightfully assessed. Results presented for just the two subjects clearly show the great potential of the dynamist approach aided with constructing the finite state machine.