Mixing Bandt-Pompe and Lempel-Ziv approaches: another way to analyze the complexity of continuous-states sequences

• Published in: The European physical journal. B, Condensed matter physics Vol. 87; no. 5
• Main Authors: Zozor, Steeve, Mateos, D., Lamberti, P. W.
• Format: Journal Article
• Language: English
• Published: Springer-Verlag 2014
• Subjects: Computer Science; Data Analysis, Statistics and Probability; Engineering Sciences; Physics; Signal and Image processing; Lempel-Ziv permutation complexity; continuous-state data analysis; permutation vectors; quantization
• ISSN: 1434-6028; 1434-6036
• DOI: 10.1140/epjb/e2014-41018-5

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.