Abstract
It is popular to study a time-dependent nonlinear system by encoding outcomes of measurements into sequences of symbols following certain symbolization schemes. Mostly, symbolizations by threshold crossings or variants of it are applied, but also, the relatively new symbolic approach, which goes back to innovative works of Bandt and Pompe—ordinal symbolic dynamics—plays an increasing role. In this paper, we discuss both approaches novelly in one breath with respect to the theoretical determination of the Kolmogorov-Sinai entropy (KS entropy). For this purpose, we propose and investigate a unifying approach to formalize symbolizations. By doing so, we can emphasize the main advantage of the ordinal approach if no symbolization scheme can be found that characterizes KS entropy directly: the ordinal approach, as well as generalizations of it provide, under very natural conditions, a direct route to KS entropy by default.
Highlights
Using symbolizations to study observed data plays an important role in today’s time series analysis
In order to estimate the KS entropy, a data analyst is always faced with the problem of choosing an adequate symbolization scheme
We show, by proposing a unifying approach to formalize symbolizations, that under relatively week assumptions, the search for a generating partition can be skipped if one chooses a symbolization scheme that regards a dependency between two measured values
Summary
Using symbolizations to study observed data plays an important role in today’s time series analysis (see for instance the review papers of Daw et al [1], Zanin et al [2], Amigó et al [3], and the examples in biology, medicine, artificial intelligence and data mining, just to mention a few, given therein). The relatively new ordinal approach could benefit from results achieved in “classical” symbolic dynamics, for instance to estimate a good symbolization scheme (see our ending remarks of the paper in Section 5 and for instance Steuer et al [7], Letellier [8] and, published most recently, Li and Ray [9], as well as the references given therein). Such topics exceed the scope of this paper
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