Regular Series

Vol. 36 (2005), No. 8, pp. 2403 – 2599

Statistical Properties of Old and New Techniques in Detrended Analysis of Time Series


Recently introduced Detrended Moving Average (DMA) method is examined and compared with Detrended Fluctuation Analysis (DFA) technique for artificial stochastic Brownian time series of various length \(L \sim 10^3 \div 10^{5}\). Our analysis reveals some statistical properties of the Hurst exponent values measured with the use of DFA and DMA methods. Good agreement between DFA and DMA techniques is found for long time series \(L\sim 10^{5}\), however for shorter series two methods are clearly distinguishable. No clear systematic relation previously postulated in literature between DFA and DMA results is found. However, it is shown that on the average, DMA method gives overestimation of the Hurst exponent compared with DFA technique.

Is Discretization of the Stochastic Continuous-Time Processes a Reason for the Non-Linear Long-Term Autocorrelations Observed in High-Frequency Financial Time-Series?


By using regular time-steps we define discrete-time random walks and flights on subordinate (directed) Continuous-Time Hierarchical (or Weierstrass-Mandelbrot) Walks and Flights, respectively. The obtained results can be considered as a kind of warning that indicates some persistent, non-linear, long-term autocorrelations (artifacts) accompanying the recording of empirical high-frequency financial (and probably other types of) time-series by regular time-steps, indeed.

Statistical Properties of Stock Market Eigensignals


By using the correlation matrix approach, we decompose the evolution of a set of the 100 largest American companies into the components (portfolios) defined by the eigenvectors of the correlation matrix. Among the results, we show that a number of the non-random components exceeds the previous estimates based on much shorter time series of daily returns. This indicates that for short signals the bulk of random eigenvalues defined by Random Matrix Theory can comprise also a significant amount of information. We also show that the components corresponding to a few largest eigenvalues and describing the most collective part of the market evolution reveal strong nonlinear correlation structure in contrast to the other components. All the components are multifractal. Moreover, by using a modified definition of the correlation matrix, we are able to decompose the daily pattern of the German DAX30 index into components which can characterize the recurrent events occurring at precise moments of a trading day.

all authors

W. Bachnik, P. Chomiuk, Sz. Fałtynowicz, M. Gawin, W. Gorajek, J. Kedzierski, K. Kosk, A. Kucharczyk, P. Leszczyński, R. Podsiadło, D. Makowiec

2004 on Warsaw Stock Exchange via Zipf Analysis, Scatter and Lag Plots


This paper presents the last year on Warsaw Stock Exchange (WSE) and world stock exchanges by graphical analysis: Scatter Plot, Zipf Analysis and Lag Plot of selected Polish (WIG, WIG20, WIG-BANKI, TECHWIG) and foreign (NIKKEI, DOW JONES Industrial Average) indices, and also selected companies listed on WSE. Zipf analysis proves that although, generally, holding securities was the best way to earn money in the last year, however, Zipf based strategy also could be profitable. Scatter Plots show no similarities between Polish and foreign indices, however, behaviour of Polish ones is similar. The volatility of indices and most companies was highest on Monday and lowest on Friday. Distribution of returns in continuous trading is neither Gaussian nor uniform.

Investigating Multifractality of Stock Market Fluctuations Using Wavelet and Detrending Fluctuation Methods


We apply the Multifractal Detrended Fluctuation Analysis and the Wavelet Transform Modulus Maxima to investigate multifractal properties of stock price fluctuations. By applying both methods to the same data sets coming from the German and the American stock markets and based on our earlier knowledge of how these methods detect multifractality while employed to well-known mathematical models, we compare the results given by both methods and infer which one can be preferable in the case of the financial data. We argue that the Multifractal Detrended Fluctuation Analysis acts better for a global detection of multifractal behavior, while the Wavelet Transform Modulus Maxima method is the optimal tool for the local characterization of the scaling properties of signals.

Detecting Subtle Effects of Persistence in the Stock Market Dynamics


The conventional formal tool to detect effects of the financial persistence is in terms of the Hurst exponent. A typical corresponding result is that its value comes out close to \(0.5\), as characteristic for geometric Brownian motion, with at most small departures from this value in either direction depending on the market and on the time scales involved. We study the high frequency price changes on the American and on the German stock markets. For both corresponding indices, the Dow Jones and the DAX respectively, the Hurst exponent analysis results in values close to \(0.5\). However, by decomposing the market dynamics into pairs of steps such that an elementary move up (down) is followed by another move up (down) and explicitly counting the resulting conditional probabilities we find values typically close to \(60\%\). This effect of persistence is particularly visible on the short time scales ranging from 1 up to 3 minutes, decreasing gradually to \(50\%\) and even significantly below this value on the larger time scales. We also detect some asymmetry in persistence related to the moves up and down, respectively. This indicates a subtle nature of the financial persistence whose characteristics escape detection within the conventional Hurst exponent formalism.

From Randomness to Periodicity — the Effect of Polarization in the Minority Game Strategy Space


Properties and behavior of modified minority game are analyzed. It appears that results of the game depend strongly on the way how we draw strategies for players. The probability that given strategy will be chosen is determined by the polarization parameter \(P\). This parameter differs between strategies that go along or against the existing market trend. Strong polarization of the space leads to the periodic dynamics and, finally, for negative \(P\) values to the domination of the single strategy in the system. When \(P\) is changed the variability of the process decreases, showing kind of the phase transition region. The dependence of the variability on the polarization parameter can be understood on the basis of the crowd–anticrowd theory.

Correlations Between the Most Developed (G7) Countries. A Moving Average Window Size Optimisation


Different distance matrices are defined and applied to look for correlations between the gross domestic product of G7 countries. Results are illustrated through displays obtained from various graph methods. Significant similarities between results are obtained. A procedure for choosing the best distance function is proposed taking into account the size of the window in which correlations are averaged.

Optimum Finite Impulse Response (FIR) Low-Pass Filtering of Market Data


The goal of this contribution is to compare 9 cases of FIR (Finite Impulse Response) type filters, by using approximation theory based norms for the following output parameters: delay and correlation between input and output, and “smoothness” of the output derivative. It was found that the most commonly used rectangular shape of impulse response is in general not an optimum solution. Indications concerning the optimum shape of impulse response subject to the assumed criteria are shortly presented and the triangular shape of impulse response is recommended.

Applications of Chebyshev Minimax Deconvolution Filtering to the Estimation of Detrended Data


Signal derivative at the output of Chebyshev deconvolution filter with respect to the odd pair of impulses is an estimator, very close to the de-trended input signal. Computations using a significantly large database of market quotations show that average “closeness” in terms of normalized in \(L_2\) covariance coefficient is above 96 percent.

Log-Periodic Oscillations in Degree Distributions of Hierarchical Scale-Free Networks


Hierarchical models of scale free networks are introduced where numbers of nodes in clusters of a given hierarchy are stochastic variables. Our models show periodic oscillations of degree distribution \(P(k)\) in the log–log scale. Periods and amplitudes of such oscillations depend on network parameters. Numerical simulations are in a good agreement to analytical calculations.

Information Theory Point of View on Stochastic Networks


Stochastic networks represent very important subject of research because they have been found in almost all branches of modern science, including also sociology and economy. We provide a information theory point of view, mostly based on its nonextensive version, on their most characteristic properties illustrating it with some examples.

Matrix Representation of Evolving Networks


We present the distance matrix evolution for different types of networks: exponential, scale-free and classical random ones. Statistical properties of these matrices are discussed as well as topological features of the networks. Numerical data on the degree and distance distributions are compared with theoretical predictions.

Sznajd Model and Its Applications


In 2000 we proposed a sociophysics model of opinion formation, which was based on trade union maxim “United we Stand, Divided we Fall” (USDF) and latter due to Dietrich Stauffer became known as the Sznajd model (SM). The main difference between SM compared to voter or Ising-type models is that information flows outward. In this paper we review the modifications and applications of SM that have been proposed in the literature.

The Heider Balance and Social Distance


The Heider balance is a state of a group of people with established mutual relations between them. These relations, friendly or hostile, can be measured in the Bogardus scale of the social distance. In previous works on the Heider balance, these relations have been described with integers \(0\) and \(\pm 1\). Recently we have proposed real numbers instead. Also, differential equations have been used to simulate the time evolution of the relations, which were allowed to vary within a given range. In this work, we investigate an influence of this allowed range on the system dynamics. As a result, we have found that a narrowing of the range of relations leads to a large delay in achieving the Heider balance. Another point is that a slight shift of the initial distribution of the social distance towards friendship can lead to a total elimination of hostility.

Measuring the Social Relations: Social Distance in Social Structure — a Study of Prison Community


Social relations and their influence on various phenomena are one of the key issues not only in sociology. The crucial problem, however, is how to measure the social relations and their implications in society. We try to adapt a physical perspective to the “typical” sociological analysis and to measure the qualitative nature of human community adapting the category of social distance. This category is used to explore the properties of social relations in the structure and the communication system of prison community. The issues that are discussed: the specific properties of social relations as the constitutive factors for different type of group structure and type of communication. How the elementary social networks (short-range group structures) form the dynamics of prison community? What is the role of the numerical force of the group for prison community? Is there the interplay between the microstructures and macrostructures? The work is based on our research carried out in 17 prisons in Poland in 2003, 2004 and 2005. There were about 2000 prisoners in the sample.

Value at Risk in the Presence of the Power Laws


The aim of this paper is to determine the Value at Risk (VaR) of the portfolio consisting of several long positions in risky assets. We consider the case when the tail parts of distributions of logarithmic returns of these assets follow the power law of the same degree and the lower tail of associated copula \(C\) follows the power law of degree 1. We provide the asymptotic formula for Value at Risk and determine the optimal portfolio. We show that the part of the capital invested in the \(i\)-th asset should be equal to the conditional probability that the drop of the value of the \(i\)-th asset will be smaller than the others under the condition that the value of the all assets will be smaller than \(c\) times their initial value (\(c\ll 1\)).

Effective Portfolios — Econometrics and Statistics in Search of Profitable Investments


Methods of constructing effective portfolios based on mathematical statistics are presented and compared with methods based on econometrics, multi-criteria optimization and on-line investments.


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