47 have also shown that the standard deviation of the estimated decay rate of missing ordinal patterns decreases with an increasing length of the patterns. More precisely, missing ordinal patterns are more persistent in time series with higher correlation structures. Results show that for a fixed pattern length, the decay rate of missing ordinal patterns in stochastic processes depends not only on the series length but also on their correlation structures. We are speaking of fractional Brownian motion (fBm), fractional Gaussian noise (fGn), and k-noises with k ≥ 0. 47 to the analysis of missing ordinal patterns in stochastic processes with different degrees of correlation. This methodology was extended by Carpi et al. Amigó 12,44 proposed a test that uses missing ordinal patterns to distinguish determinism (chaos) from pure randomness in finite time series contaminated with observational white noise (uncorrelated noise). II D, the existence of “missing ordinal patterns” could be either related to stochastic processes (correlated or uncorrelated) or to deterministic noisy processes, which is the case for observational time series. In our works, we follow the lexicographic order described by Lehmer 36 in the generation of the Bandt–Pompe PDF. Thus, a different ordering of the summands in (6) would lead to a different value of FIM, hence its local nature. The summands can be regarded as a kind of “distance” between two contiguous probabilities. The local sensitivity of FIM for discrete PDFs is reflected in the fact that the specific “ j-ordering” of the discrete values p j must be seriously taken into account in evaluating the sum in Eq. One can state that the general FIM behavior of the present discrete version is opposite to that of the Shannon entropy, except for periodic motions. On the other hand, when the system under study is in a very disordered state, that is, when all the p i’s oscillate around the same value, we obtain H ≈ 1 while F ≈ 0. If our system is in a very ordered state, which occurs when almost all the probabilities p i are zero, we have a normalized Shannon entropy H ≈ 0, and a normalized FIM F ≈ 1. Tools of the ordinal methodology based on probabilities As an additional asset in practical applications, ordinal patterns (and derived quantities for that matter) can be computed in real time since knowledge of the data range is not required. Another advantage in the case of random processes is that the time series need not be stationary for the empirical probabilities to converge to the true probabilities of the patterns (with probability 1) in the limit of arbitrarily long time series it suffices that the increments of the random process are stationary, which includes non-stationary processes such as the fractional Brownian motion. Although an ordinal representation loses details of the amplitude of the original time series, it is still suitable for the analysis of experimental data, since it avoids amplitude threshold dependencies that mar other methods based on range partitions, for example. The transformation proposed by Bandt and Pompe is robust to the presence of observational and dynamical noise, as well as invariant under nonlinear monotonous transformations. Regarding the selection of the parameters D and τ and the subtleties involved, see, e.g. Therefore, ordinal representations have two parameters: the length of the ordinal patterns D (sometimes called the embedding dimension) and the delay time τ. The transformation of a real-valued time series into a sequence of ordinal patterns (a discrete-valued time series) is called an ordinal representation. Therefore, the Bandt–Pompe symbolization procedure maps blocks of D data to the set of D ! possible ordinal patterns of length D (the “alphabet”), and it is able to capture their temporal structure since ordinal patterns are related to the temporal correlation of the physical phenomena being considered. These patterns (also called permutations or rank vectors) are obtained by means of the “ ≤” relationship between D successive entries of the series if the delay time τ = 1 or τ-spaced data samples for τ > 1. This methodology is based on the transformation of a time series into a sequence of symbols called ordinal patterns of length D. The use of quantifiers based on Information Theory, which incorporate in their evaluation the “time causality,” are a viable alternative, and is just the methodology proposed by Bandt and Pompe in their cornerstone contribution of 2002, 2 usually known as ordinal methodology.
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