The authors introduce a complexity measure for nonlinear time series data that bases on the reccurence plot (RP) and the Shannon information entropy of its microstates. This complexity measure is easy and efficient to compute and approximates the maximum Lyapunov exponent of the data. It can also be used to discriminate between deterministic, chaotic, and stochastic data.

For $x_i\in\R^d$, $i=1,\ldots,K$, the RP is a matrix $\tens{R}=(R_{ij})_{ij}$ given by

\begin{equation} \nonumber R_{ij} = \begin{cases} 1,\, & |x_i-x_j| \lt \varepsilon,\newline 0,\, & |x_i-x_j| \geq \varepsilon, \end{cases} \end{equation} where $\varepsilon>0$ is called vicinity threshold. Diagonal structures of $1$s parallel to the main diagonal display recurrence patters and are signs for determinism. iThe idea here is now to fix a small natural number $N$, typically $N\in\{1,2,3,4\}$, and look at ($N\times N)$-submatrices of $\tens{R}$. A fixed number $\bar{N}$ of such structures is selected randomly. The total number of possible microstates is $N^\ast=2^{(N^2)}$ and with $P_i=n_i/\bar{N}$, where $n_i$ is the number of occurences of the specific microstate $i$, we get the entropy

\begin{equation} \nonumber S(N^\ast) = -\suml_{i=1}^{N^\ast} P_i\,\log P_i. \end{equation}

Although $N^\ast$ grows quickly as a function of $N$, usually just a small number of microstates are effectively populated. So, the effective set of microstates needed to compute adequately the entropy can be populated by just $\bar{N}$ random samples obtained from the recurrence matrix, and a fast convergence is expected. In general, we found that usually $\bar{N} \ll N^\ast$ for $N > 3$ such that $\bar{N} \sim 10,000$ is enough. This makes the method extremely fast even for moderate values of microstate sizes $N$. This observation also points out that a microstate size $N = 4$ is sufficient for many dynamical and stochastic systems.

The maximum entropy occurs when all microstates are equally likely, i.e. $P_i=1/N^\ast$, and is given by

\begin{equation} \nonumber S(N^\ast) = N^2\,\log2 . \end{equation}

The closer $S(N)$ is to $S(N^\ast)$, the more stochastic are the data.