Eur. Phys. J. B (2014) 87: 107
https://doi.org/10.1140/epjb/e2014-41018-5

Regular Article

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

¹ Laboratoire Grenoblois d’Image, Parole, Signal et Automatique (GIPSA-Lab), CNRS et Université de Grenoble , 11 rue des Mathématiques, 38402 Saint Martin d’Hères, France
² Instituto de Física de La Plata (IFLP), CONICET and Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina
³ Facultad de Matemática, Astronomía y Física (FaMAF), CONICET and Universidad Nacional de Córdoba, Avenidad Medina Allende, Ciudad Universitaria , X5000 HUA Córdoba, Argentina

^a e-mail: steeve.zozor@gipsa-lab.grenoble-inp.fr

Received: 18 November 2013
Received in final form: 13 March 2014
Published online: 8 May 2014

Abstract

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.