How to find out which system leads the other by using CRPs

[Norbert]

In order to find out which system drives the other, from the CRP you could compute a RQA measure defined diagonal-wise (i.e. not for the entire RP but only for the kth diagonal parallel to the LOI), e.g. RR(k). The measure RR(k) or RR(-k) means the recurrence density in the diagonal above or below the LOI (with distance k from the LOI). The variable k hat the meaning of a time lag.

Then calculate a symmetry

Q(k) = ( RR(k) + RR(-k) ) / 2

and an asymmetry measure

q(k) = ( RR(k) - RR(-k) ) / 2.

The graph of both measures over the time lag k shows whether both systems are coupled/synchronised and which system leads the other. These measures can also be calculated from the other RQA measures.


This plot shows these proposed symmetry and asymmetry measures derived from RR(k) for two coupled Roessler oscillators. The maxima of the (A) symmetry measure at lag 0.3 reveals the lag-coupling, and (B) the negative value of the asymmetry measure athe the same lag reveals that the second system leads the first one.

I got this idea by reading
Quian Quiroga et al, Event synchronization: A simple and fast method to measure synchronicity and time delay patterns, PRE 66, 041904, 2002

Perhaps someone find it useful.

Regards,
Norbert

[geo]

Just what my doctor ordened

Thanks very much.

Georges

[sultornsanee]

I am not clear about your answer here. Also an image is already gone. Could you explain to me again? ... Please

[Norbert]

You can find these information in our Physics Reports paper.
N. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329, 2007

Best
Norbert