Erratum

Abstract

Real-world systems possess deterministic trajectories, phase singularities and noise. Dynamic trajectories have been studied in temporal and frequency domains, but these are linear approaches. Basic to the field of nonlinear dynamics is the representation of trajectories in phase space. A variety of nonlinear tools such as the Lyapunov exponent, Kolmogorov–Sinai entropy, correlation dimension, etc. have successfully characterized trajectories in phase space, provided the systems studied were stationary in time. Ubiquitous in nature, however, are systems that are nonlinear and nonstationary, existing in noisy environments all of which are assumption breaking to otherwise powerful linear tools. What has been unfolding over the last quarter of a century, however, is the timely discovery and practical demonstration that the recurrences of system trajectories in phase space can provide important clues to the system designs from which they derive. In this chapter we will introduce the basics of recurrence plots (RP) and their quantification analysis (RQA). We will begin by summarizing the concept of phase space reconstructions. Then we will provide the mathematical underpinnings of recurrence plots followed by the details of recurrence quantifications. Finally, we will discuss computational approaches that have been implemented to make recurrence strategies feasible and N. Marwan (!) Potsdam Institute for Climate Impact Research, Postdam, Germany e-mail: [email protected] C.L. Webber, Jr. Loyola University Chicago, Chicago, IL, USA e-mail: [email protected] C.L. Webber, Jr. and N. Marwan (eds.), Recurrence Quantification Analysis, Understanding Complex Systems, DOI 10.1007/978-3-319-07155-8__1, © Springer International Publishing Switzerland 2015 3 4 N. Marwan and C.L. Webber, Jr. useful. As computers become faster and computer languages advance, younger generations of researchers will be stimulated and encouraged to capture nonlinear recurrence patterns and quantification in even better formats. This particular branch of nonlinear dynamics remains wide open for the definition of new recurrence variables and new applications untouched to date. 1.1 Phase Space Trajectories Systems in nature or engineering typically exist in either quasi-stationary states or in non-stationary states as they move or transition between states. These complicated processes derive mostly from complex systems (nonlinear, many coupled variables, polluted by noise, etc.) and defy meaningful analysis. Still, approximate investigations of these processes remain an important focus among numerous scientific disciplines (e.g. meteorology). To the extent that systems are deterministic (ruledriven) there still remains the hope and challenge of describing dynamical system changes to such a degree rendering it possible to predict future states of the system (e.g. make forecasts). Practically, the usual aim is to find mathematical models which can be adapted to the real processes (mimicry) and then used for solving given problems. The measuring of a state (which leads to observations of the state but not to the state itself) and subsequent data analysis are the first steps toward the understanding of a process. Well known and approved methods for data analysis are those based on linear concepts as estimations of moments, correlations, power spectra, or principal components analyses etc. In the last two decades this zoo of analytical methods has been enriched with methods of the theory of nonlinear dynamics. Some of these new methods are rooted in the topological analysis of the phase space of the underlying dynamics or on an appropriate reconstruction of it [1, 2] The state of a system can be described by its d state variables x1.t/; x2.t/; : : : ; xd .t/; (1.1) for example the two state variables temperature and pressure in a thermodynamic system. The d state variables at time t form a vector x.t/ in a d -dimensional space which is called phase space. This vector moves in time and in the direction that is specified by its velocity vector P x.t/ D @tx.t/ D F.x/: (1.2) The temporary succession of the phase space vectors forms a trajectory (phase space trajectory, orbit). The velocity field F.x/ is tangent to this trajectory. For autonomous systems the trajectory must not cross itself. The time evolution of the trajectory explains the dynamics of the system, i.e., the attractor of the system. If F.x/ is known, the state at a given time can be determined by integrating the equation system [Eq. (1.2)]. However, a graphical visualization of the trajectory enables the determination of a state without integrating the equations. The shape 1 Mathematical and Computational Foundations of Recurrence Quantifications 5 of the trajectory gives hints about the system; periodic or chaotic systems have characteristic phase space portraits. The observation of a real process usually does not yield all possible state variables. Either not all state variables are known or not all of them can be measured. Most often only one observation u.t/ is available. Since measurements result in discrete time series, the observations will be written in the following as ui , where t D i !t and !t is the sampling rate of the measurement. [Henceforth, variables with a subscribed index are in this work time discrete (e.g. xi , Ri;j ), whereas a braced t denotes continuous variables (e.g. x.t/, R.t1; t2/).] Couplings between the system’s components imply that each single component contains essential information about the dynamics of the whole system. Therefore, an equivalent phase space trajectory, which preserves the topological structures of the original phase space trajectory, can be reconstructed by using only one observation or time series, respectively [2, 3]. A method frequently used for reconstructing such a trajectory O x.t/ is the time delay method: O xi D .ui ; uiC ; : : : ; uiC.m!1/ / , where m is the embedding dimension and is the time delay (index based; the real time delay is !t). The preservation of the topological structures of the original trajectory is guaranteed ifm ! 2dC1, where d is the dimension of the attractor [2]. Both embedding parameters, the dimensionm and the delay , have to be chosen appropriately. Different approaches are applicable for the determination of the smallest sufficient embedding dimension [1,4]. Here we focus on an approach which uses the number of false nearest neighbours. There are various methods that use false nearest neighbours in order to determine the embedding dimension. The basic idea is that by decreasing the dimension an increasing amount of phase space points will be projected into the neighbourhood of any phase space point, even if they are not real neighbours. Such points are called false nearest neighbours (FNNs). The simplest method uses the amount of these FNNs as a function of the embedding dimension in order to find the minimal embedding dimension [1]. Such a dimension has to be taken where the FNNs vanish. Other methods use the ratios of the distances between the same neighbouring points for different dimensions [4, 5]. Random errors and low measurement precision can lead to a linear dependence between the subsequent vectors xi . Hence, the delay has to be chosen in such a way that such dependences vanishes. One possible means of determining the delay is by using the autocovariance function C. / D hui ui! i (using the assumption huii D 0). A delay may be appropriate when the autocovariance approaches zero. This minimizes the linear correlation between the components but does not have to mean they are independent. However, the converse is true: if two variables are independent they will be uncorrelated. Therefore, another well established possibility for determining the delay is the mutual information [6]