10.2 Continous RQA measures

The RQA measures for continuous data are the same as discussed above, except their interpretation can be more explicitly defined in terms of system dynamics. Here, we briefly re-visit some of the most common RQA measures.

The probability of a cell in the matrix being a recurrent point is expressed by the Recurrence Rate, the measure \(RR\):

\[\begin{equation} RR = \frac{1}{N^2} \sum_{i,j=1}^N \mathbf{R}(i,j) \end{equation}\]

Time-adjacent recurrent points form diagonal lines \(\ell\), which are considered a recurrent trajectory through (reconstructed) phase space. If the distribution of diagonal line lengths is \(P(\ell)\) then the deterministic structure of the reconstructed attractor (\(DET\)) is quantified by the proportion of recurrent points that form a diagonal line:

\[\begin{equation} DET = \frac{ \sum_{\ell \geq \ell_{\min}}^{\ell_{\max}} \ell\ P(\ell)}{\sum_{i,j=1} \mathbf{R}(i,j)} \end{equation}\]

Based on the distribution of line lengths, measures can be calculated such as the maximum and mean diagonal line length (\(DL_{max}\), \(DL_{mean}\)). The entropy of the line length distribution, \(H(\ell)\), is based on the observed frequencies, see equation (eq:pl?):

\[\begin{equation} p(\ell) = \frac{P(\ell)}{\sum_{\ell \geq \ell_{\min}}^{\ell_{\max}} P(\ell)} H(\ell) = -\sum_{\ell \geq \ell_{\min}}^{\ell_{\max}} p(\ell) \ln p(\ell) \tag{10.2} \end{equation}\]

\(DET\) together with \(H(\ell)\) (\(ENT_{dl}\)) quantify different aspects of the (coupled) dynamics (Marwan & Kurths, 2002; Shockley, Butwill, Zbilut, & Webber Jr, 2002), a high determinism (\(DET\)) and a high entropy of the distribution \(P(\ell)\) would represent a meta-, or, multi-stable regime (Kelso, 2012), whereas a high determinism coinciding with low entropy of \(P(\ell)\) would indicate a relatively stable dynamical regime. Similar measures can be calculated based on the vertical (or horizontal) line structures \(P(\nu)\): Laminarity (\(LAM\)) is the proportion of recurrent trajectories that reflect recurrence of the same state, a laminar phase; Focusing on the vertical lines for a moment, Trapping time (\(TT\), or \(MEAN_{vl}\)) is the average duration of a laminar phase; \(MAX_{vl}\) the maximal vertical line length; and \(H(\nu)\) (\(ENT_{vl}\)), the entropy of the distribution of vertical lines.

The absolute values of these measures depend on the choice of the recurrence threshold \(\varepsilon\), but also on the length of the time series. For the purpose of comparing measures calculated from different recurrence plots it can make sense to fix the recurrence rate by choosing a different threshold value for each analysis. It is also possible to pick a fixed threshold and study how the recurrence rate varies between different plots. Several methods exist for the optimization of choosing \(\varepsilon\), some of which will yield different thresholds for different recurrence measures (e.g. the signal detection method, Schinkel, Dimigen, & Marwan, 2008). For these and other considerations when determining the recurrence analysis parameters see Marwan (2011) and the references therein. It is important to note that \(\varepsilon\) is a crucial analysis parameter for recurrence-based methods and its value and selection should always be reported and justified. In general, it is recommended to study the effects of choosing different sets of parameters when comparing the recurrence measures obtained from different (samples of) time series.

In Table 10.1, several common recurrence measures are provided for each of the phase spaces considered so far. The values for Determinism and laminarity are not surprising, after all, this is a completely deterministic system! The point here is to show that the dynamically invariant properties of the original attractor are retained for reconstructions based on \(Y_1\), \(Y_2\), \(Y_4\). The reconstruction based on \(Y_3\) was expected to be less accurate due to the lack of support (stationary level at zero for a large period of time). The recurrence threshold \(\varepsilon\) has to be extremely small compared to the other values in order to get \(RR = 0.045\). There are techniques that can be used to estimate how well a reconstructed phase space from, say, the first 75% of the time series predicts the remaining 25% (nonlinear prediction skill), or how well it predicts other observed dimensions (convergent cross-mapping) (Sugihara et al., 2012). Convergent cross-mapping can also be used to detect coupling direction and causality in dynamic or delayed interactions (Clark et al., 2015; Sugihara et al., 2012; see e.g., Ye, Deyle, Gilarranz, & Sugihara, 2015).

Table 10.1: Recurrence Measures Based on Different Phase Space Representations.
Original Y1 Y2 Y3 Y4
Recurrence Threshold \(\varepsilon\) 0.045 0.046 0.069 0.009 0.047
Recurrence Rate 0.050 0.040 0.040 0.040 0.040
DETerminism 0.998 0.997 0.998 0.997 0.996
Maximum DL 860.000 807.000 946.000 242.000 946.000
Entropy DL 4.500 3.900 3.900 3.800 4.200
LAMinarity 0.994 0.988 0.991 0.991 0.991
Maximum VL 131.000 90.000 100.000 168.000 108.000
Entropy VL 3.700 3.400 3.300 3.700 3.400