Table 3 RQA measures

Recurrence Rate (RR) or %recurrence

\(RR = \frac{1}{{N^{2} }}\mathop \sum \limits_{i,j = 1}^{N} R_{i,j}\)

\(R_{i,j} = \left\{ {\begin{array}{*{20}c} {1, \left( {i,j} \right) recurrent} \\ {0, otherwise} \\ \end{array} } \right.\)

High Recurrence Rate values indicate recurrent states that could be due to a deterministic behavior while chaotic states are associated with low values of the Recurrence Rate

Determinism (DET)

\(DET = \frac{{\mathop \sum \nolimits_{{l = l_{min} }}^{N} l \cdot Pd\left( l \right)}}{{\mathop \sum \nolimits_{i,j}^{N} R_{i,j} }}\)

where Pd(l) is the histogram of the lengths l of the diagonal lines

The presence of diagonal lines indicates the existence of a deterministic structure. DET reveals also information relevant to the duration of a stable interaction

Average Length (L)

\(L = \frac{{\mathop \sum \nolimits_{{l = l_{min} }}^{N} l \cdot Pd\left( l \right)}}{{\mathop \sum \nolimits_{{l = l_{min} }}^{N} Pd\left( l \right)}}\)

This measure refers to the diagonal line length. Small L values reveal processes with stochastic or chaotic behavior while big values indicate a deterministic process. The L value is related to the DET value, as both count the number of recurrent points on the diagonal structures

Laminarity (LAM)

\(LAM = \frac{{\mathop \sum \nolimits_{{l = l_{min} }}^{N} l \cdot Pv\left( l \right)}}{{\mathop \sum \nolimits_{l = 1}^{N} lPv\left( l \right)}}\)

where Pv(l) is the histogram of the lengths of the vertical lines included in the recurrence plot

Laminarity is a measure of the appearance of laminar states indicative of intermittency. The lower the LAM value, the more stable the system is

Trapping Time (TT)

\(TT = \frac{{\mathop \sum \nolimits_{{l = l_{min} }}^{N} lPv\left( l \right)}}{{\mathop \sum \nolimits_{{l = l_{min} }}^{N} Pv\left( l \right)}}\)

Wh where Pv(l) is the histogram of the lengths of the vertical lines

The Trapping Time measure indicates the slowing variation of the values of the time series through time. High TT values indicate a non-fluctuating, slow changing series