Narrow Results

This work explores the use of the Higuchi fractal dimension (HFD) to characterize the complexity of the Standard and Poor’s (S&P) Index for the period from 1928 to 2023. It is found that the fractal dimension is not constant but exhibits large time fluctuations. In line with adaptive market hypothesis notions, such a feature can be seen as the response of the stock market to a complex and changing environment formed by a diversity of participants and exogenous shocks. The concept of fractal dimension was extended to consider scale dependence and multifractality. It is shown that the fractality dimension approaches an integer value when the time scale increases, which reflects smoother price fluctuation profiles. It was also shown that the multifractal HFD exhibits large fluctuations for scales of weeks, months, and quarters, which can be linked to the seasonal periods of the operation of the stock market. The impact of salient events was also assessed. It was found that the 1987 and 2008 market crashes had the highest effect on the multifractal HFD, suggesting that these events involved multiple factors. Overall, the results in the present work showed that the fractal dimension tools and notions provide a useful and complementary framework for characterizing the behavior of financial indices.

By taking Bitcoin, Ethereum, and Ripple as research objects, this paper applies the multifractal detrended partial cross-correlation analysis (MF-DPXA) to study the intrinsic cross-correlation between cryptocurrencies. Combining MF-DPXA and time-delay DCCA methods, we develop the removing factors time-delayed detrended cross-correlation analysis (R-TD-DCCA) to study the risk transmission direction between cryptocurrencies after removing the influence of common factors. The results show that after removing the influence of cryptocurrencies, the persistence of the cross-correlation between cryptocurrencies is enhanced, and the multifractal degrees of the cross-correlation between Bitcoin and Ethereum and between Ethereum and Ripple are increased, but the multifractal degree of the cross-correlation between Bitcoin and Ripple is weakened. However, after removing the influence of the S&P 500 index, the multifractal degree of the cross-correlation between cryptocurrencies has weakened. The Hurst exponent of local dynamic cross-correlation between cryptocurrencies is almost always greater than 0.5. With the increase in time delay, the risk of Bitcoin is mainly transmitted to Ethereum and Ripple, and the risk of Ripple is mainly transmitted to Ethereum. When removing the impact of the S&P 500 index, the short-term risk of Bitcoin is mainly transmitted to Ethereum and Ripple. The findings of this study have several implications for re-understanding the intrinsic interdependence structure and portfolios.

We study percolation as a critical phenomenon on a random multifractal support. The scaling exponent β related to the mass of the infinite cluster and the fractal dimension of the percolating cluster d_f are quantities that have the same value as the ones from the standard two-dimensional regular lattice percolation. The scaling exponent ν related to the correlation length is sensitive to the local anisotropy and assumes a value different from standard percolation. We compare our results with those obtained from the percolation on a deterministic multifractal support. The analysis of ν indicates that the deterministic multifractal is more anisotropic than the random multifractal. We also analyze connections with correlated percolation problems and discuss some possible applications.

Evidence of discrete scale invariance (DSI) in daytime healthy heart rate variability (HRV) is presented based on the log-periodic power law scaling of the heart beat interval increment. Our analysis suggests multiple DSI groups and a dynamic cascading process. A cascade model is presented to simulate such a property.

Financial markets have been known to exhibit plentiful nonlinear complex volatility behaviors. In order to reproduce the volatility dynamics of financial price changes, the agent-based financial model is established by stochastic finite-range exclusion process. The exclusion process is a kind of statistical physics system, which is considered as modeling particle Markov motion with conserved number of particles. To measure the volatility of financial return series, a novel statistic called maximum monotonic volatility rate is put forward to measure the speed of monotonic volatility of returns. Meanwhile, average monotonic volatility duration of returns is also investigated, which can reflect the average volatility level. For verifying the rationality of the model, matching energy analysis that can detect chaos and complexity in nonlinear time series is applied to study the new statistics. Further, empirical mode decomposition and multifractal are employed to study the behaviors of monotonic volatility duration. The model has similar complexity behaviors with real markets in terms of monotonic volatility with matching energy analysis, and the proposed financial model and real markets both show multifractal and anti-correlation for average monotonic volatility series by MFDFA method. The results display that the model is feasible with respect to above volatility analyses.

Analyzing the statistical features of fluctuation is remarkably significant for financial risk identification and measurement. In this study, the asymmetric detrended fluctuation analysis (A-DFA) method was applied to evaluate asymmetric multifractal scaling behaviors in the Shanghai and New York gold markets. Our findings showed that the multifractal features of the Chinese and international gold spot markets were asymmetric. The gold return series persisted longer in an increasing trend than in a decreasing trend. Moreover, the asymmetric degree of multifractals in the Chinese and international gold markets decreased with the increase in fluctuation range. In addition, the empirical analysis using sliding window technology indicated that multifractal asymmetry in the Chinese and international gold markets was characterized by its time-varying feature. However, the Shanghai and international gold markets basically shared a similar asymmetric degree evolution pattern. The American subprime mortgage crisis (2008) and the European debt crisis (2010) enhanced the asymmetric degree of the multifractal features of the Chinese and international gold markets. Furthermore, we also make statistical tests for the results of multifractatity and asymmetry, and discuss the origin of them. Finally, results of the empirical analysis using the threshold autoregressive conditional heteroskedasticity (TARCH) and exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models exhibited that good news had a more significant effect on the cyclical fluctuation of the gold market than bad news. Moreover, good news exerted a more significant effect on the Chinese gold market than on the international gold market.

On the basis of the lattice Boltzmann method for the Navier–Stokes equation, we have done a numerical experiment of a forced turbulence in real space and time. Our new findings are summarized into two points. Firstly, in the analysis of the mean-field behavior of the velocity field using the exit-time statistics, we have verified Kolmogorov's scaling and Taylor's hypothesis at the same time. Secondly, in the analysis of the intermittent velocity fluctuations using a non-equilibrium probability distribution function and the wavelet denoising, we have clarified that the coherent vortices sustain the power-law velocity correlation in the non-equilibrium state.

Based on the daily price data of Shanghai and London gold spot markets, we applied detrended cross-correlation analysis (DCCA) and detrended moving average cross-correlation analysis (DMCA) methods to quantify power-law cross-correlation between domestic and international gold markets. Results show that the cross-correlations between the Chinese domestic and international gold spot markets are multifractal. Furthermore, forward DMCA and backward DMCA seems to outperform DCCA and centered DMCA for short-range gold series, which confirms the comparison results of short-range artificial data in L. Y. He and S. P. Chen [Physica A 390 (2011) 3806–3814]. Finally, we analyzed the local multifractal characteristics of the cross-correlation between Chinese domestic and international gold markets. We show that multifractal characteristics of the cross-correlation between the Chinese domestic and international gold markets are time-varying and that multifractal characteristics were strengthened by the financial crisis in 2007–2008.

In applied engineering, there are tremendous optimization problems which are multiobjective problems. Meanwhile, a number of them require large amount of time to evaluate their expensive cost function during optimization procedures. This kind of problems can be either financially expensive due to significant computational resources being required or time expensive due to numerous computational complexity. Aiming to this kind of problems, this paper proposed a multilevel surrogate model-based evolutionary algorithm. The proposed method employs DACE modeling method at the beginning to obtain a global trend in the decision domain. When more and more samples are involved and the sample distribution presents a trend or a manifold, the SVR model is utilized as a second-level surrogate model to achieve a better local search. The model transition is determined by the multifractal analysis on the solution set. Experimental results on ZDT and DTLZ standard test cases demonstrate that the time for EGO modeling can be reduced, and the accuracy can be better balanced by comparing to existing SVR and EGO methods.

We establish a financial price process by continuum percolation system, in which we attribute price fluctuations to the investors’ attitudes towards the financial market, and consider the clusters in continuum percolation as the investors share the same investment opinion. We investigate the cross-correlations in two return time series, and analyze the multifractal behaviors in this relationship. Further, we study the corresponding behaviors for the real stock indexes of SSE and HSI as well as the liquid stocks pair of SPD and PAB by comparison. To quantify the multifractality in cross-correlation relationship, we employ multifractal detrended cross-correlation analysis method to perform an empirical research for the simulation data and the real markets data.

Physical quantities, such as entropy, dimensions and multifractal characteristics of multiplicity distributions of charged particles produced in ${}^{197}$Au–AgBr collisions, are examined and the findings are compared with the predictions of Monte Carlo model Ultra-Relativistic Quantum Molecular Dynamics (URQMD) and Heavy Ion Jet INteraction Generator (HIJING) and also with the results reported earlier in hadron–hadron and nucleus–nucleus collisions at different energies. Based on their azimuth distribution, the charged particles produced within narrow-bins exhibit two kinds of substructures, namely, ring-like and jet-like substructures. Thus, on applying the suitable criteria, the two different types of events are identified and analyzed separately. It is observed that the maximum entropy production occurs around a narrow mid-rapidity region. The analyses of ring-like and jet-like events suggest that the entropy production is much larger in ring-like events as compared to that in jet-like events. Furthermore, Rényi’s order-q information entropy is used to estimate the multifractal specific heat and to construct the spectrum of scaling indices. The findings reveal that the value of multifractal specific heat is higher in ring-like events as compared to that in jet-like events. The studies of generalized dimension and multifractal spectrum indicate that the multifractality is rather, more pronounced in ring-like events as compared to jet-like events. Various features of the experimental data are noticed to be nicely reproduced by the URQMD model.

Natural disasters — earthquakes, hurricanes and other storms — cause substantial property damage and loss of life in many parts of the world. The relative infrequency and importance of extreme cases leads to a preferential use of simulation models over historical statistical/actuarial models in studying the impact of such catastrophes on insurance systems. Given the increasing awareness of the highly intermittent nature of geophysical phenomena, modelers need to revisit their assumptions not only of the geophysical fields, but also of the geographical distribution of insured property as well. This paper explores the distribution of insured property through the lens of multifractal theory.

Rainfall is a highly intermittent field over a wide range of time and space scales. We study a high resolution rainfall time series exhibiting large intensity fluctuations and localized events. We consider the return times of a given intensity, and show that the time series composed of these return times is itself also very intermittent, obeying to a hyperbolic probability density, entailing that the mean return time diverges. This is an unexpected property since mean return times are often introduced in meteorology, especially for the study of risk associated to extreme events. It suggests that the intermittency of first return times of extreme events should be taken into account when making statistical predictions.

The fractal component in the long-term heart rate variability (HRV) in health and in certain heart disease conditions were studied in the framework of dyadic random bounded cascade. The physiology of the fractal component was also proposed and tested in the simulation of HRV in autonomic nervous system blockade. Numerical results suggest the intrinsic mechanism behind HRV is of multiplicative nature and a "failure mechanism" due to the change of the fractal generating mechanism in certain heart diseases and in autonomic blockade.

The use of multifractals in heterogeneous chemistry has proven useful in the characterization of some complex phenomena. This paper proposes a multifractal analysis in order to study the distribution of adsorbed particles for a reaction of a monomer and a dimer that dissociates, both of which are adsorbed on a surface and react. For a range of a control parameter, there is a steady reactive window that changes when adsorbate-adsorbate lateral interaction is considered. τ(q) and f(α) for different values of control parameter and interactions are calculated, studying distribution of adsorbed particles in relation to multifractal properties.

Ballistic particles interacting with irregular surfaces are representative of many physical problems in the Knudsen diffusion regime. In this paper, the collisions of ballistic particles interacting with an irregular surface modeled by a quadratic Koch curve, are studied numerically. The q moments of the source spatial distribution of collision numbers μ(x) are characterized by a sequence of "collision exponent" τ(q). The measure μ(x) is found to be multifractal even when a random micro-roughness (or random re-emission) of the surface exists. The dimensions f(α), obtained by a Legendre transformation from τ(q), consist of two parabolas corresponding to a trinomial multifractal. This is demonstrated for a particular case by obtaining an exact f(α) for a multiplicative trinomial mass distribution. The trinomial nature of the multifractality is related to the type of surface macro-irregularity considered here and is independent of the micro-roughness of the surface which, however, influences the values of α_min and α_max. The information dimension D_I increases significantly with the micro-roughness of the surface. Interestingly, in contrast with this point of view, the surface seems to work uniformly. This corresponds to an absence of screening effects in Knudsen diffusion.

The moment-based box counting method of multifractal analysis is widely used for estimating generalized dimensions, D_q, from two-dimensional grayscale images. An evaluation of the accuracy of this method is needed to establish confidence in the resulting estimates of D_q. We estimated D_q from q = -10 to +10 for 23 random geometrical multifractal fields with different grid sizes, and known analytical D_q versus q functions. The fields were transformed to give normalized grayscale values between zero and one. Comparison of the estimated and analytical functions indicated the moment-based box counting method overestimates D_q by as much as 6.9% when q ≪ 0. The root mean square error, RMSE, for the entire range of q values examined ranged from 7.81 × 10^-6 to 1.35 × 10^-1, with a geometric mean of 6.50 × 10^-3. The RMSE decreased with decreasing grid size and increasing heterogeneity. These trends appear to be largely due to the presence of zeros in the normalized grayscale fields. Variations in the slope of the log-transformed partition function, ln[χ(q,δ)], with box size resulted in the overestimation of D_q when q ≪ 0. An alternative procedure for estimating D_q was developed based on the numerical first derivatives of ln[χ(q,δ)]. Using this approach the maximum deviation in D_q values was only 1.2%, while the RMSE varied from 3.11 × 10^-6 to 2.72 × 10^-2, with a geometric mean of 2.57 × 10^-4. When analyzing normalized grayscale fields, moment-based estimates of D_q should be interpreted with care. An order of magnitude increase in the accuracy of D_q can be achieved for such fields if the numerical first derivatives of ln[χ(q,δ)] are used in the analysis instead of standard linear regression.

It seems evident that the understanding of the complex temporal evolution dynamic characteristics of air pollution indexes (APIs) can contribute to developing advanced techniques for air pollution forecasting. In this work, monofractal and multifractal methods have been successfully used to characterize the temporal fluctuations of APIs in Shanghai. It shows that APIs in Shanghai are characterized by scale invariance, long range dependence and multifractal scaling. Introducing a kind of variable window sizes method, we analyze the temporal evolution of the monofractal and multifractal behaviors of APIs. Some information about the temporal evolution dynamics of APIs is revealed. Our study suggests that the variable window sizes method can be help the analysis of air pollution time series temporal evolution dynamical mechanisms. This work could play an important role in the research of air pollution.

Multifractal box counting analysis has been widely applied to study the scaling characteristics of grayscale images. Since bit depth is an important property of such images it is desirable know the impact of varying bit depths on the estimation of the generalized dimensions (D_q). We generated random geometrical multifractal grayscale fields, which were then transformed from double precision to 16, 13, and 8 bit depths. Digitized grayscale images of soil thin sections at 13 bit depth were also selected for study and transformed to 8 bit depth. The moment based box counting method was applied to evaluate the bit depth effects on D_q. The partition functions for the multifractal fields became noticeably nonlinear on a log-log scale when q ≪ 0 as the bit depth decreased. This trend can be attributed to loss of grayscale details, changes in the local mass distribution, and the occurrence of zeros due to the bit depth transformation and data normalization processes. These effects were most pronounced for positively skewed multifractal fields, with a high proportion of extremely small mass fractions. As a result, the generalized dimensions estimated by linear regression were not always accurate, and an alternative method based on numerical derivatives was explored. The numerical method significantly improved the accuracy of the multifractal analyses; the maximum absolute difference between the analytical and numerically-derived estimates of D_q was only 9.62 × 10^-3. However, when applied to situations in which the box counting scale factor did not match the scale factor used to generate the multifractal field, the numerically-derived estimates of D_q were severely biased. In this case, the linear regression method is preferable even though some error may occur due to limited bit depths. All of the soil grayscale images exhibited multifractal scaling characteristics, although there was little effect of bit depth on the resulting D_q values. Because of random fluctuations in the partition functions, the linear regression method proved to be more robust than the numerical derivative method for estimating the generalized dimensions of natural grayscale images.

The water pH series from Dongting Lake Inlet and Outlet in China are analyzed by detrended fluctuation analysis (DFA), spectral analysis and multifractal methods. The results show that these pH series are characterized by long-term memory, 1/f noise and multifractal scaling, and these characteristics have obvious difference between the Lake Inlet and Outlet. The comparison results show that monofractal (DFA exponent) and multifractal (Δα, Δf, B) parameters can be quantitative dynamical indexes reflecting the capability of anti-acidification of Dongting Lake. Furthermore, we investigate the frequency-size distribution of pH series from Dongting Lake Inlet and Outlet. Our findings suggest that water pH is an example of a self-organized criticality (SOC) process. Based on concept of self-organized ctiticality, we analysis the cause that different scale-free power-law behavior between pH series from Dongting Lake Inlet and Outlet. This work can be helpful to improvement of modeling of lake water quality.