The ordinal Kolmogorov-Sinai entropy: A generalized approximation

We introduce the multi-dimensional ordinal arrays complexity as a generalized approximation of the ordinal Komogorov-Sinai entropy. The ordinal arrays entropy (OAE) is defined as the Shannon entropy of a series of m-ordinal patterns encoded symbols, while the ordinal arrays complexity (OAC) is defined as the differential of the OAE with respect to m. We theoretically establish that the OAC provides a better estimate of the complexity measure for short length time series. Simulations were carried out using discrete maps, and confirm the efficiency of the OAC as complexity measure from a small data set even in a noisy environment.