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This book focuses on a central question in the field of complex systems: Given a fluctuating (in time or space), uni- or multi-variant sequentially measured set of experimental data (even noisy data), how should one analyse non-parametrically the data, assess underlying trends, uncover characteristics of the fluctuations (including diffusion and jump contributions), and construct a stochastic evolution equation?
Here, the term "non-parametrically" exemplifies that all the functions and parameters of the constructed stochastic evolution equation can be determined directly from the measured data.
The book provides an overview of methods that have been developed for the analysis of... 

Stochastic differential equations (SDE) play an important role in a range of application areas, including biology, physics, chemistry, epidemiology, mechanics, microelectronics, economics, and finance [1]. However, most SDEs, especially nonlinear SDEs, do not have analytical solutions, so that one must resort to numerical approximation schemes in order to simulate trajectories of the solutions to the given equation. The simplest effective computational method for approximation of ordinary differential equations is the Euler’s method. The Euler–Maruyama method is the analogue of the Euler’s method for ordinary differential equations for numerical simulation of the SDEs [2]. Another numerical... 

In this chapter we study stochastic processes in the presence of jump discontinuity, and discuss the meaning of non-vanishing higher-order Kramers–Moyal coefficients. We describe in details the stochastic properties of Poisson jump processes. We derive the statistical moments of the Poisson process and the Kramers–Moyal coefficients for pure Poisson jump events. Growing evidence shows that continuous stochastic modeling (white noise-driven Langevin equation) of time series of complex systems should account for the presence of discontinuous jump components [1–6]. Such time series have some distinct important characteristics, such as heavy tails and occasionally sudden large jumps.... 

In this chapter we reconstruct stochastic dynamical equations with known drift and diffusion coefficients, as well as known properties of jumps, jump amplitude and jump rate from synthetic time series, sampled with time interval τ. The examples have Langevin (white noise- and Lévy noise-driven) and jump-diffusion dynamical equations. We also study the estimation of the Kramers–Moyal coefficients for “phase” dynamics that enable us to investigate the phenomenon of synchronisation in systems with interaction. © 2019, Springer Nature Switzerland AG  

Data sampled at discrete times appear as successions of discontinuous jump events, even if the underlying trajectory is continuous. In this chapter we study finite sampling τ expansion of the Kramers-Moyal conditional moments for the Langevin and jump-diffusion dynamics. Using the expansion for the Langevin dynamics, we introduce a criterion to validate the method numerically, namely, the Pawula theorem, to judge whether the fourth-order KM moment tends to zero. The criterion is a relation between the fourth- and second-order KM conditional moments for small time lag τ [1]. © 2019, Springer Nature Switzerland AG  

The method outlined in the Chaps. 15 – 21 has been used for revealing nonlinear deterministic and stochastic behaviors in a variety of problems, ranging from physics, to neuroscience, biology and medicine. In most cases, alternative procedures with strong emphasis on deterministic features have been only partly successful, due to their inappropriate treatment of the dynamical fluctuations [1]. In this chapter, we provide a list of the investigated phenomena using the introduced reconstruction method. In the “outlook” possible research directions for future are discussed. © 2019, Springer Nature Switzerland AG  

In Chaps. 16 – 21 we address a central question in the field of complex systems: Given a fluctuating (in time or space), sequentially uni- or multi-variant measured set of experimental data (even noisy data), how should one analyse the data non-parametrically, assess their underlying trends, discover the characteristics of the fluctuations, including diffusion and jump parts, and construct stochastic evolution equation for the data? © 2019, Springer Nature Switzerland AG  

In this chapter we show the equivalence between the Langevin approach and the Fokker–Planck equation, and derive the equation for the statistical moments of the process whose dynamics is described by the Langevin equation. © 2019, Springer Nature Switzerland AG  

In this chapter we introduce the Langevin equation and Wiener process. All the statistical properties of Wiener process will be presented and discussed. © 2019, Springer Nature Switzerland AG  

Complex systems are composed of a large number of subsystems that may interact with each other. The typically nonlinear and multiscale interactions often lead to large-scale behaviors, which are not easily predicted from the knowledge of only the behavior of individual subsystems. © 2019, Springer Nature Switzerland AG  

As an example of the analysis and reconstruction of nonlinear stochastic time series, we consider an important medical problem, namely, epileptic brain dynamics. © 2019, Springer Nature Switzerland AG  

Most real world time series have transient behaviours and are non-stationary. They exhibit different type of non-stationarities, such as trends, cycles, random-walking, and generally exhibit strong intermittency. Therefore local stochastic characteristics of time series, such as the drift and diffusion coefficients, as well as the jump rate and jump amplitude, will provide very important information for understanding and quantifying “real time” variability of time series. For diffusive processes the systems have a longer memory and a higher correlation time scale and, therefore, one expects the stochastic features of dynamics to change slowly. In contrast, a rapid change of dynamics with... 

In this chapter we study stochastic properties of spatially- and temporally-disordered structures, such as turbulence and rough surfaces, or temporal fluctuations of given time series, in scale. Experimental observables include the field increments, such as the difference in the velocity field between two points separated by a distance r, or difference of time series in a time lag r. Therefore, the lag r can be either spatial distance or a time interval. The change of the increments’ fluctuations as a function of the scale r can then be viewed as a stochastic process in a length or time scale and can, quite often, after pioneering work by Friedrich & Peinke, be mapped onto the mathematical... 

Jumps are discontinuous variations in time series and with large amplitude can be considered as an extreme event. We expect the higher the jump activity to cause higher uncertainty in the stochastic behaviour of measured time series. Therefore, building statistical evidence to detect real jump seems of primary importance. In addition jump events can participate in the observed non-Gaussian feature of the increments’ (ramp up and ramp down) statistics of many time series [1]. This is the reason that most of the jump detection techniques are based on threshold values for differential of time series. There is not, however, a robust method for detection and characterisation of such discontinuous... 

In this chapter we define notions of stochastic continuity and differentiability and describe Lindeberg’s condition for continuity of stochastic Markovian trajectories. We also show that the Fokker–Planck equation describes a continuous stochastic process. Finally, we derive the stationary solutions of the Fokker–Planck equation and define potential function of dynamics. © 2019, Springer Nature Switzerland AG  

In this chapter, we introduce Lévy noise-driven Langevin and fractional Fokker–Planck equations, and derive short-time propagator of the Lévy-driven processes. Then, we provide the details of the limit theorems for the Wiener and Lévy processes. Finally, non-parametric reconstruction of the Lévy-driven Langevin dynamics from time series is described. © 2019, Springer Nature Switzerland AG  

In this chapter we introduce jump-diffusion processes and provide a theoretical framework that justifies the nonparametric (data-based) extraction of the parameters and functions controlling the arrival of a jump and the distribution of the jump size from the estimated conditional Kramers–Moyal moments. The method and the results are applicable to both stationary and nonstationary time series in the presence of discontinuous jump components; see Chap. 17. © 2019, Springer Nature Switzerland AG  

In this chapter we provide a generalization of jump-diffusion precesses (12.1) in two dimensions by considering a class of coupled systems that are described by a bivariate state vector x(t) contained in a two-dimensional state space {x}. The evolution of the state vector x(t) is assumed to be governed by a deterministic part to which diffusion parts and jump contributions are added. Generalization of the results to higher dimensions is straightforward. We note that for N multivariate time series, by assuming the presence of two-body type of interactions between time series, the analysis will reduce to analysing N(N − 1)/2 pairwise bivariate time series. © 2019, Springer Nature Switzerland...