Recurrence Plot
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In descriptive
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
and
chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
returns to the previous state at i, i.e., when the phase space trajectory visits roughly the same area in the phase space as at time j. In other words, it is a plot of :\vec(i)\approx \vec(j), showing i on a horizontal axis and j on a vertical axis, where \vec is the state of the system (or its phase space trajectory).


Background

Natural processes can have a distinct recurrent behaviour, e.g. periodicities (as
seasonal A season is a division of the year based on changes in weather, ecology, and the number of daylight hours in a given region. On Earth, seasons are the result of the axial parallelism of Earth's tilted orbit around the Sun. In temperate and po ...
or
Milankovich cycle Milankovitch cycles describe the collective effects of changes in the Earth's movements on its climate over thousands of years. The term was coined and named after Serbian geophysicist and astronomer Milutin Milanković. In the 1920s, he hypot ...
s), but also irregular cyclicities (as
El Niño El Niño (; ; ) is the warm phase of the El Niño–Southern Oscillation (ENSO) and is associated with a band of warm ocean water that develops in the central and east-central equatorial Pacific (approximately between the International Date L ...
Southern Oscillation, heart beat intervals). Moreover, the recurrence of states, in the meaning that states are again arbitrarily close after some time of
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
, is a fundamental property of deterministic
dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
and is typical for
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
or
chaotic system Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have ...
s (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).


Detailed description

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. However, making a recurrence plot enables us to investigate certain aspects of the ''m''-dimensional phase space trajectory through a two-dimensional representation. At a recurrence the trajectory returns to a location in phase space it has visited before up to a small error \varepsilon (i.e., the system returns to a state that it has before). The recurrence plot represents the collection of pairs of times such recurrences, i.e., the set of (i,j) with \vec(i) \approx \vec(j), with i and j discrete points of time and \vec(i the state of the system at time i (location of the trajectory at time i). Mathematically, this can be expressed by the binary recurrence matrix :R(i,j) = \begin 1 &\text \quad \, \vec(i) - \vec(j)\, \le \varepsilon \\ 0 & \text, \end where \, \cdot \, is a norm and \varepsilon the recurrence threshold. The recurrence plot visualises \mathbf with coloured (mostly black) dot at coordinates (i,j) if R(i,j)=1, with time at the x- and y-axes. If only a
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
is available, the phase space can be reconstructed by using a time delay embedding (see
Takens' theorem In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the prope ...
): :\vec(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1)), where u(i) is the time series, m the embedding dimension and \tau the time delay. Phase space reconstruction is not essential part of the recurrence plot (although often stated in literature), because it is based on phase space trajectories which could be derived from the system's variables directly (e.g., from the three variables of the
Lorenz system The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lo ...
). The visual appearance of a recurrence plot gives hints about the dynamics of the system. Caused by characteristic behaviour of the phase space trajectory, a recurrence plot contains typical small-scale structures, as single dots, diagonal lines and vertical/horizontal lines (or a mixture of the latter, which combines to extended clusters). The large-scale structure, also called ''texture'', can be visually characterised by ''homogenous'', ''periodic'', ''drift'' or ''disrupted''. For example, the plot can show if the trajectory is strictly periodic with period T, then all such pairs of times will be separated by a multiple of T and visible as diagonal lines. The small-scale structures in RPs are used by the
recurrence quantification analysis Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space tr ...
(Zbilut & Webber 1992; Marwan et al. 2002). This quantification allows us to describe the RPs in a quantitative way and to study transitions or nonlinear parameters of the system. In contrast to the heuristic approach of the recurrence quantification analysis, which depends on the choice of the embedding parameters, some dynamical invariants as
correlation dimension In chaos theory, the correlation dimension (denoted by ''ν'') is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For example, if we have a set of random points on t ...
,
K2 entropy K, or k, is the eleventh letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''kay'' (pronounced ), plural ''kays''. The letter K u ...
or
mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such ...
, which are independent on the embedding, can also be derived from recurrence plots. The base for these dynamical invariants are the recurrence rate and the distribution of the lengths of the diagonal lines. Close returns plots are similar to recurrence plots. The difference is that the relative time between recurrences is used for the y-axis (instead of absolute time). The main advantage of recurrence plots is that they provide useful information even for short and non-stationary data, where other methods fail.


Extensions

Multivariate extensions of recurrence plots were developed as cross recurrence plots and joint recurrence plots. Cross recurrence plots consider the phase space trajectories of two different systems in the same phase space (Marwan & Kurths 2002): :\mathbf(i,j) = \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i),\, \vec(i) \in \mathbb^m, \quad i=1, \dots, N_x, \ j=1, \dots, N_y. 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). Joint recurrence plots are the Hadamard product of the recurrence plots of the considered sub-systems (Romano et al. 2004), e.g. for two systems \vec and \vec the joint recurrence plot is :\mathbf(i,j) = \Theta(\varepsilon_x - \, \vec(i) - \vec(j)\, ) \cdot \Theta(\varepsilon_y - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m, \quad \vec(i) \in \mathbb^n,\quad i,j=1, \dots, N_. 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 synchronisation {{no footnotes, date=June 2017 Phase synchronization is the process by which two or more cyclic signals tend to oscillate with a repeating sequence of relative phase angles. Phase synchronisation is usually applied to two waveforms of the same fr ...
.


Example


See also

*
Poincaré plot A Poincaré plot, named after Henri Poincaré, is a type of recurrence plot used to quantify self-similarity in processes, usually periodic functions. It is also known as a return map. Poincaré plots can be used to distinguish chaos from random ...
*
Recurrence period density entropy Recurrence period density entropy (RPDE) is a method, in the fields of dynamical systems, stochastic processes, and time series analysis, for determining the periodicity, or repetitiveness of a signal. Overview Recurrence period density entropy i ...
, an information-theoretic method for summarising the recurrence properties of both deterministic and stochastic dynamical systems. *
Recurrence quantification analysis Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space tr ...
, a heuristic approach to quantify recurrence plots. *
Self-similarity matrix In data analysis, the self-similarity matrix is a graphical representation of similarity measure, similar sequences in a data series. Similarity can be explained by different measures, like spatial distance (distance matrix), correlation, or compar ...
*
Dot plot (bioinformatics) In bioinformatics a dot plot is a graphical method for comparing two biological sequences and identifying regions of close similarity after sequence alignment. It is a type of recurrence plot. History One way to visualize the similarity between ...


References

* * * {{cite journal , author=N. Marwan , title=A historical review of recurrence plots , journal=European Physical Journal ST , volume=164 , issue=1 , year=2008 , pages=3–12 , url= https://zenodo.org/record/996840 , doi=10.1140/epjst/e2008-00829-1 , bibcode = 2008EPJST.164....3M , arxiv=1709.09971


External links


Recurrence Plot
Plots (graphics) Signal processing Dynamical systems Visualization (graphics) Chaos theory Scaling symmetries