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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... 

The level crossing and inverse statistics analysis of DAX and oil price time series are given. We determine the average frequency of positive-slope crossings, να+, where Tα=1να+ is the average waiting time for observing the level α again. We estimate the probability P(K,α), which provides us the probability of observing K times of the level α with positive slope, in time scale Tα. For analyzed time series, we found that maximum K is about ≈6. We show that by using the level crossing analysis one can estimate how the DAX and oil time series will develop. We carry out the same analysis for the increments of DAX and oil price log-returns (which is known as inverse statistics), and provide the... 

This review addresses a central question in the field of complex systems: given a fluctuating (in time or space), sequentially measured set of experimental data, how should one analyze the data, assess their underlying trends, and discover the characteristics of the fluctuations that generate the experimental traces? In recent years, significant progress has been made in addressing this question for a class of stochastic processes that can be modeled by Langevin equations, including additive as well as multiplicative fluctuations or noise. Important results have emerged from the analysis of temporal data for such diverse fields as neuroscience, cardiology, finance, economy, surface science,... 

A master equation for the Kardar-Parisi-Zhang (KPZ) equation in 2+1 dimensions is developed. In the fully nonlinear regime we determine the finite time scale of the singularity formation in terms of the characteristics of forcing. The exact probability density function of the one point height field is obtained correspondingly  

Large-scale ab initio molecular-dynamics simulations have been carried out to compute, at human-body temperature, the vibrational modes and lifetimes of pure and hydrated dipalmitoylphosphatidylcholine (DPPC) lipids. The projected atomic vibrations calculated from the spectral energy density are used to compute the vibrational modes and the lifetimes. All the normal modes of the pure and hydrated DPPC and their frequencies are identified. The computed lifetimes incorporate the full anharmonicity of the atomic interactions. The vibrational modes of the water molecules close to the head group of DPPC are active (possess large projected spectrum amplitudes) in the frequency range 0.5-55 THz,... 

In this work, we search for signatures of gravitational millilensing in gamma-ray bursts (GRBs) in which the source-lens-observer geometry produces two images that manifest in the GRB light curve as superimposed peaks with identical temporal variability (or echoes), separated by the time delay between the two images. According to the sensitivity of our detection method, we consider millilensing events due to point-mass lenses in the range of 105 - 107 M o˙ at lens redshift about half that of the GRB, with a time delay on the order of 10 s. Current GRB observatories are capable of resolving and constraining this lensing scenario if the above conditions are met. We investigated the Fermi/GBM... 

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...