Impact of scaling range on the effectiveness of detrending methods

We make the comparative study of scaling range properties for detrended fluctuation analysis (DFA), detrended moving average analysis (DMA) and recently proposed new technique called modified detrended moving average analysis (MDMA). Basic properties of scaling ranges for these techniques are reviewed. The efficiency and exactness of all three methods towards proper determination of scaling exponent $H$ is discussed, particularly for short series of uncorrelated or persistent data. \end{abstract}Comment: 16 pages, 9 figure

On the scaling ranges of detrended fluctuation analysis for long-memory correlated short series of data

We examine the scaling regime for the detrended fluctuation analysis (DFA) - the most popular method used to detect the presence of long memory in data and the fractal structure of time series. First, the scaling range for DFA is studied for uncorrelated data as a function of length $L$ of time series and regression line coefficient $R^2$ at various confidence levels. Next, an analysis of artificial short series with long memory is performed. In both cases the scaling range $\lambda$ is found to change linearly -- both with $L$ and $R^2$. We show how this dependence can be generalized to a simple unified model describing the relation $\lambda=\lambda(L, R^2, H)$ where $H$ ($1/2\leq H \leq 1$) stands for the Hurst exponent of long range autocorrelated data. Our findings should be useful in all applications of DFA technique, particularly for instantaneous (local) DFA where enormous number of short time series has to be examined at once, without possibility for preliminary check of the scaling range of each series separately.Comment: 27 pages, 17 figure

Multifractal dynamics of stock markets

We present a comparative analysis of multifractal properties of financial time series built on stock indices from developing (WIG) and developed (S&P500) financial markets. It is shown how the multifractal image of the market is altered with the change of the length of time series and with the economic situation on the market. We emphasize that the proper adjustment of scaling range for multiscaling power laws is essential to obtain the multifractal image of time series. We analyze in this paper multifractal properties of real financial time series using H\"older $f(\alpha)$ representation and multifractal-DFA method. It is also investigated how multifractal properties of stocks change with variety of "surgeries" done on the initial real financial time series. This way we reveal main phenomena on the market influencing its multifractal dynamics. In particular, we focus on examining how multifractal picture of real time series changes when one cuts off extreme events like crashes or rupture points, and how fluctuations around the main trend in time series influence the multifractal behavior of financial series in the long-time horizon for both developed and developing markets

Identification of cross and autocorrelations in time series within an approach based on Wigner eigenspectrum of random matrices

We present an original and novel method based on random matrix approach that enables to distinguish the respective role of temporal autocorrelations inside given time series and cross correlations between various time series. The proposed algorithm is based on properties of Wigner eigenspectrum of random matrices instead of commonly used Wishart eigenspectrum methodology. The proposed approach is then qualitatively and quantitatively applied to financial data in stocks building WIG (Warsaw Stock Exchange Index).Comment: 12 pages, 6 figure

On the scaling range of power-laws originated from fluctuation analysis

We extend our previous study of scaling range properties done for detrended fluctuation analysis (DFA) \cite{former_paper} to other techniques of fluctuation analysis (FA). The new technique called Modified Detrended Moving Average Analysis (MDMA) is introduced and its scaling range properties are examined and compared with those of detrended moving average analysis (DMA) and DFA. It is shown that contrary to DFA, DMA and MDMA techniques exhibit power law dependence of the scaling range with respect to the length of the searched signal and with respect to the accuracy $R^2$ of the fit to the considered scaling law imposed by DMA or MDMA schemes. This power law dependence is satisfied for both uncorrelated and autocorrelated data. We find also a simple generalization of this power law relation for series with different level of autocorrelations measured in terms of the Hurst exponent. Basic relations between scaling ranges for different techniques are also discussed. Our findings should be particularly useful for local FA in e.g., econophysics, finances or physiology, where the huge number of short time series has to be examined at once and wherever the preliminary check of the scaling range regime for each of the series separately is neither effective nor possible.Comment: 20 pages, 10 figure

Numerical Simulation of the Perrin - like Experiments

A simple model of random Brownian walk of a spherical mesoscopic particle in viscous liquids is proposed. The model can be both solved analytically and simulated numerically. The analytic solution gives the known Eistein-Smoluchowski diffusion law $= Dt$ where the diffusion constant $D$ is expressed by the mass and geometry of a particle, the viscosity of a liquid and the average effective time between consecutive collisions of the tracked particle with liquid molecules. The latter allows to make a simulation of the Perrin experiment and verify in detailed study the influence of the statistics on the expected theoretical results. To avoid the problem of small statistics causing departures from the diffusion law we introduce in the second part of the paper the idea of so called Artificially Increased Statistics (AIS) and prove that within this method of experimental data analysis one can confirm the diffusion law and get a good prediction for the diffusion constant even if trajectories of just few particles immersed in a liquid are considered.Comment: LaTeX, 10 pages, 14 figures (included). Three figures adde

On the multifractal effects generated by monofractal signals

We study quantitatively the level of false multifractal signal one may encounter while analyzing multifractal phenomena in time series within multifractal detrended fluctuation analysis (MF-DFA). The investigated effect appears as a result of finite length of used data series and is additionally amplified by the long-term memory the data eventually may contain. We provide the detailed quantitative description of such apparent multifractal background signal as a threshold in spread of generalized Hurst exponent values $\Delta h$ or a threshold in the width of multifractal spectrum $\Delta \alpha$ below which multifractal properties of the system are only apparent, i.e. do not exist, despite $\Delta\alpha\neq0$ or $\Delta h\neq 0$. We find this effect quite important for shorter or persistent series and we argue it is linear with respect to autocorrelation exponent $\gamma$. Its strength decays according to power law with respect to the length of time series. The influence of basic linear and nonlinear transformations applied to initial data in finite time series with various level of long memory is also investigated. This provides additional set of semi-analytical results. The obtained formulas are significant in any interdisciplinary application of multifractality, including physics, financial data analysis or physiology, because they allow to separate the 'true' multifractal phenomena from the apparent (artificial) multifractal effects. They should be a helpful tool of the first choice to decide whether we do in particular case with the signal with real multiscaling properties or not.Comment: 36 pages, 41 figure

How much multifractality is included in monofractal signals?

We investigate the presence of residual multifractal background for monofractal signals which appears due to the finite length of the signals and (or) due to the long memory the signals reveal. This phenomenon is investigated numerically within the multifractal detrended fluctuation analysis (MF-DFA) for artificially generated time series. Next, the analytical formulas enabling to describe the multifractal content in such signals are provided. Final results are shown in the frequently used generalized Hurst exponent h(q) multifractal scenario and are presented as a function of time series length L and the autocorrelation exponent value {\gamma}. The multifractal spectrum ({\alpha}, f ({\alpha})) approach is also discussed. The obtained results may be significant in any practical application of multifractality, including financial data analysis, because the "true" multifractal effect should be clearly separated from the so called "multifractal noise". Examples from finance in this context are given. The provided formulas may help to decide whether we do deal with the signal of real multifractal origin or not.Comment: 21 pages, 14 figures, 2 tables, extended and corrected list of reference

Quantitative approach to multifractality induced by correlations and broad distribution of data

We analyze quantitatively the effect of spurious multifractality induced by the presence of fat-tailed symmetric and asymmetric probability distributions of fluctuations in time series. In the presented approach different kinds of symmetric and asymmetric broad probability distributions of synthetic data are examined starting from Levy regime up to those with finite variance. We use nonextensive Tsallis statistics to construct all considered data in order to have good analytical description of frequencies of fluctuations in the whole range of their magnitude and simultaneously the full control over exponent of power-law decay for tails of probability distribution. The semi-analytical compact formulas are then provided to express the level of spurious multifractality generated by the presence of fat tails in terms of Tsallis parameter $\tilde{q}$ and the scaling exponent $\beta$ of the asymptotic decay of cumulated probability density function (CDF).The results are presented in Hurst and H\"{o}lder languages - more often used in study of multifractal phenomena. According to the provided semi-analytical relations, it is argued how one can make a clear quantitative distinction for any real data between true multifractality caused by the presence of nonlinear correlations, spurious multifractality generated by fat-tailed shape of distributions - eventually with their asymmetry, and the correction due to linear autocorrelations in analyzed time series of finite length. In particular, the spurious multifractal effect of fat tails is found basic for proper quantitative estimation of all spurious multifractal effects. Examples from stock market data are presented to support these findings.Comment: 27 page

Second Loop Corrections from Superheavy Gauge Sector to Gauge Coupling Unification

We deal with extensions of the Standard Model (SM) adding horizontal interactions between particle generations. We calculate two loop corrections caused by the presence of coupling between hypothetical horizontal gauge bosons and matter field at high energy. It is shown that coupling of such extra bosons does not affect up to two loop level the positive features of unified and extended SM with horizontal symmetry discussed in former publications. Corrections from bosonic horizontal sector make about tenth part of those caused by fermionic sector. Although small they are however larger than accuracy of some electroweak measurements and therefore they might be important for future verification of various proposed horizontal models. 0