| Abstract: |
| Ordinal persistent homology (OPH) and its generalizations were proposed as a framework for investigating coupling complexity in multivariate time series. Total persistence (TP), as defined in OPH, is a measure of coupling complexity in the following sense: it is zero for completely synchronized time series, takes a specific small value with high probability for asynchronous time series satisfying a certain independence condition, and can be large somewhere between the two extreme cases. TP has been successfully applied to analyze coupling complexity in several high-dimensional nonlinear systems. In this presentation, we propose an alternative, simpler homological method to study coupling complexity in multivariate time series. In this method, we consider two sufficiently long time intervals separated in time and partition the set of time series using the ordinal patterns for each time interval. We construct a simplicial complex from the two partitions by assigning a simplex to each part of the partitions. We show that the first Betti number of the simplicial complex can serve as a coupling complexity measure better than TP: it takes zero not only in the synchronous case, but also in the asynchronous case with high probability. |
|