  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.
