Mixing Bandt-Pompe and Lempel-Ziv approaches : another way to analyze the complexity of continuous-state sequences
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Date
2014Author
Zozor, Steeve
Mateos, Diego M.
Lamberti, Pedro W.
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In this paper, we propose to mix the approach underlying Bandt-Pompe permutation entropy
with Lempel-Ziv complexity, to design what we call Lempel-Ziv permutation complexity. The principle
consists of two steps: (i) transformation of a continuous-state series that is intrinsically multivariate or
arises from embedding into a sequence of permutation vectors, where the components are the positions of
the components of the initial vector when re-arranged; (ii) performing the Lempel-Ziv complexity for this
series of ‘symbols’, as part of a discrete finite-size alphabet. On the one hand, the permutation entropy
of Bandt-Pompe aims at the study of the entropy of such a sequence; i.e., the entropy of patterns in a
sequence (e.g., local increases or decreases). On the other hand, the Lempel-Ziv complexity of a discrete-
state sequence aims at the study of the temporal organization of the symbols (i.e., the rate of compressibility
of the sequence). Thus, the Lempel-Ziv permutation complexity aims to take advantage of both of these
methods. The potential from such a combined approach – of a permutation procedure and a complexity
analysis – is evaluated through the illustration of some simulated data and some real data. In both cases,
we compare the individual approaches and the combined approach.