Definition of the Subject
Measurements of time signals of complex systems of the inanimate and the animate world like turbulent fluid motions, traffic flow, or human brain activity yield fluctuating time series. In recent years, methods have been devised which allow for a detailed analysis of such data. In particular, methods for parameter free estimations of the underlying stochastic equations have been proposed. The present entry gives an overview on the achievements obtained so far for analyzing stochastic data and describes results obtained for a variety of complex systems ranging from electrical nonlinear circuits, fluid turbulence, to traffic flow and financial market data. The systems will be divided into two classes, namely systems with complexity in time and systems with complexity in scale.
Introduction
The central theme of the present entry is exhibited in Fig. 1. Given a fluctuating, sequentially measured set of experimental data one can pose the question whether it is...
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Abbreviations
- Complexity in Time:
-
Complex structures may be characterized by nonregular time behavior of a describing variable \( q\in {\mathbf{R}}^d \). Thus the challenge is to understand or to model the time dependence of q(t), which may be achieved by a differential equation \( \frac{\mathrm{d}}{\mathrm{d}t}q(t)=\dots \) or by the discrete dynamics \( q\left(t+\tau \right)=f\left(q(t),\dots \right) \) fixing the evolution in the future. Of special interest are not only nonlinear equations leading to chaotic dynamics but also those which include general noise terms, too.
- Complexity in Space:
-
Complex structures may be characterized by their spatial disorder. The disorder on a selected scale l may be measured at the location x by some scale dependent quantities, q(l, x), like wavelets, increments, and so on. The challenge is to understand or to model the features of the disorder variable q(l, x) on different scales l. If the moments of q show power behavior \( \left\langle q{\left(l,\;x\right)}^n\right\rangle \propto {l}^{\zeta (n)} \) , the complex structures are called fractals. Well-known examples of spatial complex structures are turbulence or financial market data. In the first case, the complexity of velocity fluctuations over different distances l is investigated; in the second case the complexity of price changes over different time steps (time scale) is of interest.
- Fokker-Planck Equation:
-
The evolution of a variable x(t) from x′ at t′ to X at t, with \( {t}^{\prime }>t \), is described in a statistical manner by the conditional probability distribution \( p\left(\mathbf{x},t\Big|{\mathbf{x}}^{\mathbf{\prime}},\;{t}^{\prime}\right) \). The conditional probability is subject to a Fokker-Planck equation, also known as second Kolmogorov equation, if
$$ \begin{array}{l}\frac{\partial }{\partial t}p\left(\mathbf{x},t\Big|{\mathbf{x}}^{\mathbf{\prime}},\;{t}^{\prime}\right)=\\ {} -{\displaystyle \sum_{i=1}^d\frac{\partial }{\partial {x}_i}{D}_i^{(1)}\left(\mathbf{x},t\right)p\left(\mathbf{x},t\Big|{\mathbf{x}}^{\mathbf{\prime}},{t}^{\prime}\right)}\\ {} +\frac{1}{2}{\displaystyle \sum_{i,j=1}^d\frac{\partial^2}{\partial {x}_i\partial {x}_j}{D}_{ij}^{(2)}\left(\mathbf{x},t\right)p\left(\mathbf{x},t\Big|{\mathbf{x}}^{\mathbf{\prime}},{t}^{\prime}\right).}\end{array} $$holds. Here D(1) and D(2) are the drift vector and the diffusion matrix, respectively.
- Kramers–Moyal coefficients:
-
Knowing for a stochastic process the conditional probability distribution \( p\left(\mathbf{x}\left(\mathbf{t}\right), t\Big|{\mathbf{x}}^{\mathbf{\prime}},\ {t}^{\prime}\right), \) for all t and t′ the Kramers-Moyal coefficients can be estimated as nth order moments of the conditional probability distribution. In this way also the drift and diffusion coefficient of the Fokker-Planck equation can be obtained from the empirically measured conditional probability distributions.
- Langevin Equation:
-
The time evolution of a variable x(t) is described by Langevin equation if for x(t) it holds:
$$ \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{x}\left(\mathbf{t}\right)={\mathrm{D}}^{(1)}\left(\mathbf{x}, \mathbf{t}\right) \cdot \tau +\sqrt{{\mathrm{D}}^{(2)}\left(\mathbf{x}, \mathbf{t}\right)}\cdot \boldsymbol{\varGamma} \left({\mathbf{t}}_i\right). $$Using Itô’s interpretation, the deterministic part of the differential equation is equal to the drift term; the noise amplitude is equal to the square root of the diffusion term of a corresponding Fokker-Planck equation. Note, for vanishing noise a purely deterministic dynamics is included in this description.
- Stochastic Process in Scale:
-
For the description of complex system with spatial or scale disorder usually a measure of disorder on different scales q(l, x) is used. A stochastic process in scale is now a description of the l evolution of q(l, x) by means of stochastic equations. As a special case, the single event q(l, x) follows a Langevin equation, whereas the probability p(q(l)) follows a Fokker-Planck equation.
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Acknowledgments
The scientific results reported in this review have been worked out in close collaboration with many colleagues and students. We mention St. Barth, F. Böttcher, F. Ghasemi, I. Grabec, J. Gradisek, M. Haase, A. Kittel, D. Kleinhans, St. Lück, A. Nawroth, Chr. Renner, M. Siefert, and S. Siegert.
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Friedrich, R., Peinke, J., Tabar, M.R.R. (2015). Fluctuations, Importance of: Complexity in the View of Stochastic Processes. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_212-4
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