Ordinal symbolic analysis and its application to biomedical recordings

Abstract

Ordinal symbolic analysis opens an interesting and powerful perspective on time-series analysis. Here, we review this relatively new approach and highlight its relation to symbolic dynamics and representations. Our exposition reaches from the general ideas up to recent developments, with special emphasis on its applications to biomedical recordings. The latter will be illustrated with epilepsy data.

Keywords: ordinal patterns; permutation entropy; symbolic dynamics; time-series analysis.

Figure 1.

(a) Graph of the tent map. (b) Graphs of the identity map and of f,f² for f(x)=4x(1−x) determining the ordinal partition π^1,3 of [0,1], vertical dashed lines separate subintervals with ordinal patterns (0,1,2),(0,2,1),(2,0,1),(1,0,2) and (1,2,0).

Figure 2.

Complexity of the quadratic family: (a) the Feigenbaum diagram, (b) the ordinal version of the Feigenbaum diagram and (c) the corresponding Lyapunov exponents (black), permutation entropy h^1,10 (light grey) and the conditional entropies of ordinal patterns (dark grey).

Figure 3.

EEG example 1. (a) Original time series with epilepsy caused by inflammation (epileptic seizure is marked in grey), (b) corresponding time series of ordinal patterns (T=1, L=7) as numbers in [0, 1], (c) relative frequencies of the six ordinal 3-patterns (cf. table 1) in a sliding window of 2 s with partitioning in the vertical direction (ordinal pattern distribution, T=1, L=3) and (d) permutation entropy (T=1, L=3) for a sliding window of 2 s.

Figure 4.

EEG example 2. (a) Original time series with epilepsy caused by hippocampal sclerosis (epileptic seizure is marked in grey), (b) corresponding time series of ordinal patterns (T=1, L=7) as numbers in [0, 1], (c) relative frequencies of the six ordinal 3-patterns (cf. table 1) in a sliding window of 2 s with partitioning in the vertical direction (ordinal pattern distribution, T=1, L=3) and (d) permutation entropy (T=1, L=3) for a sliding window of 2 s.