Identifying hidden common causes from bivariate time series: A method using recurrence plots

We propose a method for inferring the existence of hidden common causes from observations of bivariate time series. We detect related time series by excessive simultaneous recurrences in the corresponding recurrence plots. We also use a noncoverage property of a recurrence plot by the other to deny the existence of a directional coupling. We apply the proposed method to real wind data.

Figure 1

Elementary relations between two time series $x$ and $y$.

Figure 2

(Color online) Algorithm for identifying coupling types. Directional couplings are shown in single arrows and generalized synchronization is shown in double arrows. Elements which may not exist are shown in gray.

Figure 3

(Color online) Effects of (a) (b) embedding dimension, (c) coupling strength, (d) noise level, (e) length of time series, (f) recurrence rates during the independence test, and (g) recurrence rate during denying directional couplings. For (a), we used clean data. For (b), we added 10% observational noise. For each graph, the black dotted line corresponds to case (1), the blue dashed line corresponds to case (2a), the green dash-dotted line corresponds to case (3), and the red solid line corresponds to case (4). For (f) and (g), we used the 5-dimensional embedding space.

Figure 4

(Color online) Inferred network from coupled Rössler systems. We assumed that generalized synchronization does not exist.

Figure 5

(Color online) Comparisons of recurrence plots of (a) $x_1$ (blue ×) and $x_2$ (red $+$), (b) $x_2$( red $+$), and $x_3$ (green ○), and (c) $x_1$ (blue ×) and $x_4$ (black ▽). Parts of recurrence plots are shown to emphasize the details. Recurrence plots of $x_1$ and $x_2$ and those of $x_2$ and $x_3$ overlapped significantly ($p$-value $<0.001$), respectively, while the overlaps between recurrence plots of $x_1$ and $x_4$ are not significant ($p$-value $>0.05$).

Figure 6

Tests for directional couplings in coupled Rössler systems. (a) $x_1\to x_2$, (b) $x_2\to x_1$, (c) $x_2\to x_3$, and (d) $x_4\to x_5$. Recurrence plots for candidates of driving forces are shown in gray dots and those for candidates of forced systems are shown in black circles. Parts of recurrence plots are shown to emphasize the details.

Figure 7

(a) Observed wind data. The east and north winds in Sapporo are shown in gray and black, respectively. (b) The recurrence plots of east wind in Sapporo (gray dots) cover those of north wind in Sapporo (black circles). Part of recurrence plots is shown here to emphasize the details.