Fractals

According to current literature, the Mike-Farmer (MF) model¹ is constructed empirically based on the continuous double auction mechanism in an order-driven market, which can successfully capture the diffusive behavior of stock prices at the transaction level. In our paper, we revisit the statistical properties of the generated series of prices based on the MF model to clarify whether it can reproduce the stylized facts in real world markets. However, the Detrended Fluctuation Analysis (DFA) scaling exponent of volatility H_v ≈ 0.6, which may be slightly lower than that in real markets; while a modified version of the MF model proposed by Gu and Zhou² can improve the DFA scaling exponent of volatility H_v ≈ 0.75, which is closer to the empirical findings. Finally, we test the existence of another commonly found two stylized facts in the real world: the volatility clustering, and leverage effect.

This paper presents the basic scheme and the log-normal, log-gamma and log-negative-inverted-gamma scenarios to establish the Rényi function for infinite products of homogeneous isotropic random fields on Rⁿ; in particular for random fields on the sphere in R³. The motivation of this paper is the test of (non-)Gaussianity on the Cosmic Microwave Background Radiation (CMBR) in Cosmology. In the presentation, we need to employ spherical harmonics for some concrete computations.

Public transport networks (PTNs) are often researched without reference to their geographical embedding. The question arises if there is any underlying structure or principle characterizing the observed behavior of geographically embedded transport routes. Here, we focus on the scaling properties of PTNs in Space L through fractal analysis and consider the effect of the real bus routes, which reflects the human movement between stations indirectly. We find that the PTN in Space L is the better basal one to mimic the human migration in city. Furthermore, we also research the scaling property of the correlation between stations and the distribution of node's weight, which shows the heterogeneous property of human activity between different stations. Our present work provides some new perspective and tools to realize the human migration on spatial networks.

This paper explores the three different methods to explicitly recognize the healthy and epileptic EEG signals: Modified, Improved, and Advanced forms of Generalized Fractal Dimensions (GFD). The newly proposed scheme is based on GFD and the discrete wavelet transform (DWT) for analyzing the EEG signals. First EEG signals are decomposed into approximation and detail coefficients using DWT and then GFD values of the original EEGs, approximation and detail coefficients are computed. Significant differences are observed among the GFD values of the healthy and epileptic EEGs allowing us to classify seizures with high accuracy. It is shown that the classification rate is very less accurate without DWT as a preprocessing step. The proposed idea is illustrated through the graphical and statistical tools. The EEG data is further tested for linearity by using normal probability plot and we proved that epileptic EEG had significant nonlinearity whereas healthy EEG distributed normally and similar to Gaussian linear process. Therefore, we conclude that the GFD and the wavelet decomposition through DWT are the strong indicators of the state of illness of epileptic patients.

The FARIMA models, which have long-range-dependence (LRD), are widely used in many areas. Through the derivation of a precise characterization of the spectrum and variance time function, we show that this family is very atypical among LRD processes, being extremely close to the fractional Gaussian noise in a precise sense which results in ultra-fast convergence to fGn under rescaling. Furthermore, we show that this closeness property is not robust to additive noise. We argue that the use of FARIMA, and more generally fractionally differenced time series, should be reassessed in some contexts, in particular when convergence rate under rescaling is important and noise is expected.

Tortuosity (τ) of two-dimensional fractal model of porous media is investigated to study their relationship with porosity. Square full-walk technique is applied to obtain τ in a two-dimensional fractal model of porous substance constructed by Randomized Sierspinski Carpets. The numerical result is in good agreement with previous results and empirical relation between tortuosity and porosity given by τ ~ p(1 - ϕ) + 1 that was found by other using Lattice Gas Automata method for solving flow equation on two-dimensional porous substance constructed by randomly placed rectangles of equal size and with unrestricted overlap. Average tortuosity of the flow path decreases linearly as fractal dimension of pore increases at each fractal iteration. Both fractal dimension and iteration give almost the same linearly tortuosity–porosity relation. The type of random algorithm for constructing Randomized Sierspinski Carpets has no significant influence on the tortuosity–porosity relation.

In this paper, we focus on the "minimal payment" which minimizes the number of coins left after the payment. Two kinds of multifractal properties of the minimal payment are studied. The first one is a frequency distribution of change amounts, and the second one is a visiting probability on the delay plot of successive change amounts. When the face values of coins are power of two (1, 2, 4, 8, …), we find that these two distributions are related to well-known multifractal models, and derive analytical expressions of multifractal spectra.