Recurrence plot: Difference between revisions

Revision as of 06:46, 15 February 2019

Revision as of 13:50, 11 May 2019

Line 35:

Cross recurrence plots consider the phase space trajectories of two different systems in the same phase space (Marwan & Kurths 2002):

:<math>\mathbf{CR}(i,j) = \Theta(\varepsilon - \| \vec{x}(i) - \vec{y}(j)\|), \quad \vec{x}(i),\, \vec{y}(i) \in \Bbbmathbb{R}^m, \quad i=1, \dots, N_x, \ j=1, \dots, N_y.</math>

The dimension of both systems must be the same, but the number of considered states (i.e. data length) can be different. Cross recurrence plots compare the occurrences of ''similar states'' of two systems. They can be used in order to analyse the similarity of the dynamical evolution between two different systems, to look for similar matching patterns in two systems, or to study the time-relationship of two similar systems, whose time-scale differ (Marwan & Kurths 2005).

Line 41:

Joint recurrence plots are the [[Matrix product#Hadamard product|Hadamard product]] of the recurrence plots of the considered sub-systems (Romano et al. 2004), e.g. for two systems <math>\vec{x}</math> and <math>\vec{y}</math> the joint recurrence plot is

:<math>\mathbf{JR}(i,j) = \Theta(\varepsilon_x - \| \vec{x}(i) - \vec{x}(j)\|) \cdot \Theta(\varepsilon_y - \| \vec{y}(i) - \vec{y}(j)\|), \quad \vec{x}(i) \in \Bbbmathbb{R}^m, \quad \vec{y}(i) \in \Bbbmathbb{R}^n,\quad i,j=1, \dots, N_{x,y}.</math>

In contrast to cross recurrence plots, joint recurrence plots compare the simultaneous occurrence of ''recurrences'' in two (or more) systems. Moreover, the dimension of the considered phase spaces can be different, but the number of the considered states has to be the same for all the sub-systems. Joint recurrence plots can be used in order to detect [[phase synchronization|phase synchronisation]].