  The complex behavior of many systems in nature requires the application of
robust methodologies capable of identifying changes in their dynamics. In the
case of time series (which are sensed values of a system during a time
interval), several methods have been proposed to evaluate their irregularity.
However, for some types of dynamics such as stochastic and chaotic, new
approaches are required that can provide a better characterization of them. In
this paper we present the simplicial complex approximate entropy (SCAE), which
is based on the conditional probability of the occurrence of elements of a
simplicial complex. Our results show that this entropy measure provides a wide
range of values with details not easily identifiable with standard methods. In
particular, we show that our method is able to quantify the irregularity in
simulated random sequences and those from low-dimensional chaotic dynamics.
Furthermore, it is possible to consistently differentiate cardiac interbeat
sequences from healthy subjects and from patients with heart failure, as well
as to identify changes between dynamical states of coupled chaotic maps. Our
results highlight the importance of the structures revealed by the simplicial
complexes, which holds promise for applications of this approach in various
contexts.
