Physical approach to complex systems

Kwapień, Jarosław; Dróżdż, S.

doi:10.1016/j.physrep.2012.01.007

Cited by 408 publications

(377 citation statements)

References 299 publications

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“…Standard correlation measures like the Pearson correlation coefficient and the cross-correlation function require stationary data in order to provide reliable results, which is a requirement that is hard to fulfill in many real-world situations (the financial and physiological data are the negative examples here [1][2][3][4][5][6][7][8][9][10]). (By stationarity we mean stability of the probability distribution functions of the data over time; from this perspective nonstationarity can be produced both by the long-range autocorrelations and by the pdf's heavy tails that make any signal length effectively insufficient.)…”

Section: Introductionmentioning

confidence: 99%

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

2015

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The detrended cross-correlation coefficient ρDCCA has recently been proposed to quantify the strength of cross-correlations on different temporal scales in bivariate, non-stationary time series. It is based on the detrended cross-correlation and detrended fluctuation analyses (DCCA and DFA, respectively) and can be viewed as an analogue of the Pearson coefficient in the case of the fluctuation analysis. The coefficient ρDCCA works well in many practical situations but by construction its applicability is limited to detection of whether two signals are generally cross-correlated, without possibility to obtain information on the amplitude of fluctuations that are responsible for those cross-correlations. In order to introduce some related flexibility, here we propose an extension of ρDCCA that exploits the multifractal versions of DFA and DCCA: MFDFA and MFCCA, respectively. The resulting new coefficient ρq not only is able to quantify the strength of correlations, but also it allows one to identify the range of detrended fluctuation amplitudes that are correlated in two signals under study. We show how the coefficient ρq works in practical situations by applying it to stochastic time series representing processes with long memory: autoregressive and multiplicative ones. Such processes are often used to model signals recorded from complex systems and complex physical phenomena like turbulence, so we are convinced that this new measure can successfully be applied in time series analysis. In particular, we present an example of such application to highly complex empirical data from financial markets. The present formulation can straightforwardly be extended to multivariate data in terms of the q-dependent counterpart of the correlation matrices and then to the network representation.

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Section: Introductionmentioning

confidence: 99%

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

2015

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“…Under simplicity of the acceptable model we imply the proper hypothesis ('best fit' model) containing a minimal number of the fitting parameters (FP) that describe the behavior of the system considered quantitatively. The different approaches that exist nowadays for the description of these systems are collected in a recent review [7].…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

Detection of quasi-periodic processes in complex systems: how do we quantitatively describe their properties?

2013

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It has been shown that in reality at least two general scenarios of data structuring are possible: (a) a self-similar (SS) scenario when the measured data form an SS structure and (b) a quasi-periodic (QP) scenario when the repeated (strongly correlated) data form random sequences that are almost periodic with respect to each other. In the second case it becomes possible to describe their behavior and express a part of their randomness quantitatively in terms of the deterministic amplitude-frequency response belonging to the generalized Prony spectrum. This possibility allows us to re-examine the conventional concept of measurements and opens a new way for the description of a wide set of different data. In particular, it concerns different complex systems when the 'best-fit' model pretending to be the description of the data measured is absent but the barest necessity of description of these data in terms of the reduced number of quantitative parameters exists. The possibilities of the proposed approach and detection algorithm of the QP processes were demonstrated on actual data: spectroscopic data recorded for pure water and acoustic data for a test hole. The suggested methodology allows revising the accepted classification of different incommensurable and self-affine spatial structures and finding accurate interpretation of the generalized Prony spectroscopy that includes the Fourier spectroscopy as a partial case.

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“…The study of spectral fluctuations within the framework of Random Matrix Theory (RMT) is a standard tool in the statistical study of quantum chaos in the excitation spectra of quantum systems [1][2][3][4]. Recently, the approach has found new applications in many fields, such as in the study of eigenspectra of adjacency matrices of networks [5][6][7], and eigenspectra of empirical correlation matrices in finance [8][9][10], the climate [11], electro-and magnetoencephalography [12][13][14], and in complex systems [15]. The interest of the approach lies in the fact that the level density fluctuations ρ(E) = ρ(E) − ρ(E) around the smooth global density ρ(E) are universal and indicate the underlying symmetry class of the system [2,16].…”

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Random-matrix spectra as a time series

2013

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Spectra of ordered eigenvalues of finite Random Matrices are interpreted as a time series. Dataadaptive techniques from signal analysis are applied to decompose the spectrum in clearly differentiated trend and fluctuation modes, avoiding possible artifacts introduced by standard unfolding techniques. The fluctuation modes are scale invariant and follow different power laws for Poisson and Gaussian ensembles, which already during the unfolding allows to distinguish the two cases.PACS numbers: 05.45. Tp,05.45.Mt,02.50.Sk The study of spectral fluctuations within the framework of Random Matrix Theory (RMT) is a standard tool in the statistical study of quantum chaos in the excitation spectra of quantum systems [1][2][3][4]. Recently, the approach has found new applications in many fields, such as in the study of eigenspectra of adjacency matrices of networks [5][6][7], and eigenspectra of empirical correlation matrices in finance [8][9][10], the climate [11], electro-and magnetoencephalography [12][13][14], and in complex systems [15]. The interest of the approach lies in the fact that the level density fluctuations ρ(E) = ρ(E) − ρ(E) around the smooth global density ρ(E) are universal and indicate the underlying symmetry class of the system [2,16]. On the other hand, the global level density ρ(E) is system dependent, and an unfolding procedure needs to be performed, to separate the global and the fluctuating parts [1]. The unfolding is straightforward if an analytical formula is known to describe the global level density ρ(E) for the system under study, such as e.g. the gaussian and semicircle distributions for Possion and GOE matrix ensembles from RMT [2], or the MarchenkoPastur distribution for the Laguerre ensemble of random Wishart correlation matrices [17]. However, such analytical formulae are formally only adequate in the asymptotic limit for spectra with an infinite number of levels. Often, an analytical form for ρ(E) is unknown, as is the case for adjacency matrices [5]. In practical cases, having finite, albeit large matrices, the usual approach is then to project the sequence of ordered eigenvalues into unfolded values E(n) → N [E(n)], using a smooth (often polynomial) approximation N (E) to the accumulated density , 5, 20]. After unfolding, the short-range and long-range correlations can be quantified using standard fluctuations measures such as the Nearest-Neighbour Spacing (NNS) distribution, number variance Σ 2 and ∆ 3 . In a recent approach, the unfolded fluctuations of the accumulated level density function N (E) = N (E) − N (E) (also called δ n function) *

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Physical approach to complex systems

Cited by 408 publications

References 299 publications

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

Detection of quasi-periodic processes in complex systems: how do we quantitatively describe their properties?

Random-matrix spectra as a time series

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