Recurrence plot

In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment ${\displaystyle j}$ in time, the times at which the state of a dynamical system returns to the previous state at ${\displaystyle i}$, i.e., when the phase space trajectory visits roughly the same area in the phase space as at time ${\displaystyle j}$. In other words, it is a plot of

showing ${\displaystyle i}$ on a horizontal axis and ${\displaystyle j}$ on a vertical axis, where ${\displaystyle {\vec {x}}}$ is the state of the system (or its phase space trajectory).

Natural processes can have a distinct recurrent behaviour, e.g. periodicities (as seasonal or Milankovich cycles), but also irregular cyclicities (as El Niño Southern Oscillation, heart beat intervals). Moreover, the recurrence of states, in the meaning that states are again arbitrarily close after some time of divergence, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems (cf. Poincaré recurrence theorem). The recurrence of states in nature has been known for a long time and has also been discussed in early work (e.g. Henri Poincaré 1890).

One way to visualize the recurring nature of states by their trajectory through a phase space is the recurrence plot, introduced by Eckmann et al. (1987). Often, the phase space does not have a low enough dimension (two or three) to be pictured, since higher-dimensional phase spaces can only be visualized by projection into the two or three-dimensional sub-spaces. One frequently used tool to study the behaviour of such phase space trajectories is then the Poincaré map. Another tool is the recurrence plot, which enables us to investigate many aspects of the m-dimensional phase space trajectory through a two-dimensional representation.

At a recurrence the trajectory returns to a location (state) in phase space it has visited before up to a small error ${\displaystyle \varepsilon }$ . The recurrence plot represents the collection of pairs of times of such recurrences, i.e., the set of ${\displaystyle (i,j)}$ with ${\displaystyle {\vec {x}}(i)\approx {\vec {x}}(j)}$, with ${\displaystyle i}$ and ${\displaystyle j}$ discrete points of time and ${\displaystyle {\vec {x}}(i)}$ the state of the system at time ${\displaystyle i}$ (location of the trajectory at time ${\displaystyle i}$). Mathematically, this is expressed by the binary recurrence matrix

where ${\displaystyle \|\cdot \|}$ is a norm and ${\displaystyle \varepsilon }$ the recurrence threshold. An alternative, more formal expression is using the Heaviside step function ${\displaystyle R(i,j)=\Theta (\varepsilon -D_{i,j})}$ with ${\displaystyle D_{i,j}=\|{\vec {x}}(i)-{\vec {x}}(j)\|}$ the norm of distance vector between ${\displaystyle {\vec {x}}(i)}$ and ${\displaystyle {\vec {x}}(j)}$. Alternative recurrence definitions consider different distances ${\displaystyle D_{i,j}}$, e.g., angular distance, fuzzy distance, or edit distance.

The recurrence plot visualises ${\displaystyle \mathbf {R} }$ with coloured (mostly black) dot at coordinates ${\displaystyle (i,j)}$ if ${\displaystyle R(i,j)=1}$, with time at the ${\displaystyle x}$- and ${\displaystyle y}$-axes.

If only a univariate time series ${\displaystyle u(t)}$ is available, the phase space can be reconstructed, e.g., by using a time delay embedding (see Takens' theorem):