Multivariate permutation entropy, a Cartesian graph product approach

Entropy metrics are nonlinear measures to quantify the complexity of time series. Among them, permutation entropy is a commonly used metric due to its robustness and fast computation. Multivariate entropy metrics techniques are needed to analyse data consisting of more than one time series. To this end, we present a multivariate permutation entropy,
$\text{MPE}_{\mathrm{G}}$
, using a graph-based approach. Given a multivariate signal, the algorithm to compute
$\text{MPE}_{\mathrm{G}}$
involves two main steps: 1) we construct an underlying graph
$G$
as the Cartesian product of two graphs
$G_{1}$
and
$G_{2}$
, where
$G_{1}$
preserves temporal information of each times series and
$G_2$
models the relations between different channels; and 2) we consider the multivariate signal as samples defined on the regular graph
$G$
and apply the recently introduced permutation entropy for graphs. Our graph-based approach gives the flexibility to consider diverse types of cross channel relationships and signals, and it overcomes with the limitations of current multivariate permutation entropy.