Multi-span transition networks: a new unified framework for analyzing time series

Abstract

The paper seeks to overcome the limitations inherent in traditional transition network methods, which primarily concentrate on transition frequencies between adjacent symbols, neglecting broader transition relationships. We present a novel approach called “multi-span transition network.” This method excels at capturing dynamic information within time series by incorporating transitions across higher time-scale patterns. We also propose a conditional entropy measure to assess the complexity of time-series data derived from the multi-span transition network. With expanding dimensionality, the multi-span transition network adeptly discriminates between various types of time series and unveils concealed information. The conditional entropy of the multi-span transition network exhibits a robust correlation with the maximum Lyapunov exponent of the system. The conditional entropy of a multi-span network can distinguish the time series of different states and determine chaos degradation. Employing the multi-span transition network for the classification of epileptic EEG data resulted in a substantial enhancement in accuracy compared to conventional transition network methods. The method is a more general form of the traditional transition network and is more generalizable.

Appendix A

The CEMTN algorithm is capable of providing a certain level of discrimination for time series in the phase space that undergo affine transformations. Taking the Rössler system as an example, we solve its equations using a fourth-order Runge–Kutta method with a chosen step size of 0.01 and perform a rotation of the resulting three-dimensional coordinates, rotating the line around the z-axis by 45° and then around the y-axis by 30°. Subsequently, we extract the transformed x-variable coordinates to obtain the chaotic map and corresponding time-series data. The shape of the phase space is schematically illustrated in Fig. 

Fig. 9

Two phase space of Rössler systems undergoing rotational transformations

9.

Following a rotational transformation of the phase space, trajectories within the phase space are projected onto the x-axis, resulting in two distinct time series, as illustrated in Fig. 

Fig. 10

The time series produced by the projection of the phase space on the x-axis where the rotational transformation occurs

10. Despite their dissimilarity, both time series correspond to the same underlying phase space.

We select embedding dimension $d=10$, delay time $\tau =1$, and spanning dimension $m=20$. The CEMTN entropy values for these two time series, varying with spanning dimension, are depicted in Fig. 

Fig. 11

CEMTN algorithm values for different spanning dimensions at delay time $\tau$ = 1

11. If we adjust some parameters, such as changing the delay time $\tau$ from 1 to 30 while keeping the other parameters constant, the CEMTN entropy values for these two time series, varying with spanning dimension, are shown in Fig. 

Fig. 12

CEMTN algorithm values for different spanning dimensions at delay time $\tau$ = 30

12.

We observe that when the delay time $\tau$=1 during phase space reconstruction, the two time series undergoing affine transformations are almost indistinguishable. However, when selecting delay time $\tau$=30, the two time series undergoing affine transformations gradually separate as the spanning dimension increases.

For other choices of delay time $\tau$, the two curves also exhibit varying degrees of separation, while altering the embedding dimension $d$ hardly distinguishes the two curves. This phenomenon occurs because the different choices of delay time lead to a certain difference in the reconstructed phase space, and this difference is precisely detected by the CEMTN entropy values at different spanning dimensions.