Identification of Fractal Properties in Geomagnetic Data of Southeast Asian Region during Various Solar Activity Levels

The fractal properties of geomagnetic northward component data (H-component) in the equatorial region during various phases of solar activity over Southeast Asia were investigated and then quantified using the parameter of the Hurst exponent (H). This study began with the identification of existence of spectral peaks and scaling properties in international quiet day H-component data which were measured during three levels of solar activity: low, intermediate, and high. Then, various cases of quiet and disturbed days during different solar activity levels were analyzed using the method that performed the best in the preceding part. In all the years analyzed, multifractal scaling and spectral peaks exist, signifying that the data have fractal properties and that there are external factors driving the fluctuations of geomagnetic activity other than solar activity. The analysis of various cases of quiet and disturbed days generally showed that quiet days had anti-persistence tendencies (H < 0.5) while disturbed days had persistence tendencies (H > 0.5)—generally a higher level of Hurst exponent compared to quiet days. As for long-term quiet day H-component data, it had a Hurst exponent value that was near H ≃ 0.50, while the long-term disturbed day H-component data showed higher values than that of the quiet day.

On the statistical properties of sea-ice lead fraction and heat fluxes in the Arctic

We explore several statistical properties of the observed and simulated Arctic sea-ice lead fraction, as well as the statistics of simulated Arctic ocean–atmosphere heat fluxes. First we show that the observed lead fraction in the Central Arctic has a monofractal spatial scaling, which we relate to the multifractal spatial scaling present in sea-ice deformation rates. We then show that the relevant statistics of the observed lead fraction in the Central Arctic are well represented by our model, neXtSIM. Given that the heat flux through leads may be up to 2 orders of magnitude larger than that through unbroken ice, we then explore the statistical properties (probability distribution function – PDF – and spatial scaling) of the heat fluxes simulated by neXtSIM. We demonstrate that the modelled heat fluxes present a multifractal scaling in the Central Arctic, where heat fluxes through leads dominate the high-flux tail of the PDF. This multifractal character relates to the multi- and monofractal character of deformation rates and the lead fraction. In the wider Arctic, the high-flux tail of the PDF is dominated by an exponential decay, which we attribute to the presence of coastal polynyas. Finally, we show that the scaling of the simulated lead fraction and heat fluxes depends weakly on the model resolution and discuss the role sub-grid-scale parameterisations of the ice heterogeneity may have in improving this result.

Multifractal scaling analyses of urban street network structure: The cases of twelve megacities in China

Traffic networks have been proved to be fractal systems. However, previous studies mainly focused on monofractal networks, while complex systems are of multifractal structure. This paper is devoted to exploring the general regularities of multifractal scaling processes in the street network of 12 Chinese cities. The city clustering algorithm is employed to identify urban boundaries for defining comparable study areas; box-counting method and the direct determination method are utilized to extract spatial data; the least squares calculation is employed to estimate the global and local multifractal parameters. The results showed multifractal structure of urban street networks. The global multifractal dimension spectrums are inverse S-shaped curves, while the local singularity spectrums are asymmetric unimodal curves. If the moment order q approaches negative infinity, the generalized correlation dimension will seriously exceed the embedding space dimension 2, and the local fractal dimension curve displays an abnormal decrease for most cities. The scaling relation of local fractal dimension gradually breaks if the q value is too high, but the different levels of the network always keep the scaling reflecting singularity exponent. The main conclusions are as follows. First, urban street networks follow multifractal scaling law, and scaling precedes local fractal structure. Second, the patterns of traffic networks take on characteristics of spatial concentration, but they also show the implied trend of spatial deconcentration. Third, the development space of central area and network intensive areas is limited, while the fringe zone and network sparse areas show the phenomenon of disordered evolution. This work may be revealing for understanding and further research on complex spatial networks by using multifractal theory.

Scaling Laws in the Stellar Mass Distribution and the Transition to Homogeneity

We present a new statistical analysis of the large-scale stellar mass distribution in the Sloan Digital Sky Survey (data release 7). A set of volume-limited samples shows that the stellar mass of galaxies is concentrated in a range of galaxy luminosities that is very different from the range selected by the usual analysis of galaxy positions. Nevertheless, the two-point correlation function is a power-law with the usual exponent γ = 1.71 − 1.82 , which varies with luminosity. The mass concentration property allows us to make a meaningful analysis of the angular distribution of the full flux-limited sample. With this analysis, after suppressing the shot noise, we extend further the scaling range and thus obtain γ = 1.83 and a clustering length r 0 = 5.8 − 7.0 h − 1 Mpc . Fractional statistical moments of the coarse-grained stellar mass density exhibit multifractal scaling. Our results support a multifractal model with a transition to homogeneity at about 10 h − 1 Mpc .

A Hypothesis Test Method for Detecting Multifractal Scaling, Applied to Bitcoin Prices

Multifractal processes reproduce some of the stylised features observed in financial time series, namely heavy tails found in asset returns distributions, and long-memory found in volatility. Multifractal scaling cannot be assumed, it should be established; however, this is not a straightforward task, particularly in the presence of heavy tails. We develop an empirical hypothesis test to identify whether a time series is likely to exhibit multifractal scaling in the presence of heavy tails. The test is constructed by comparing estimated scaling functions of financial time series to simulated scaling functions of both an iid Student t-distributed process and a Brownian Motion in Multifractal Time (BMMT), a multifractal processes constructed in Mandelbrot et al. (1997). Concavity measures of the respective scaling functions are estimated, and it is observed that the concavity measures form different distributions which allow us to construct a hypothesis test. We apply this method to test for multifractal scaling across several financial time series including Bitcoin. We observe that multifractal scaling cannot be ruled out for Bitcoin or the Nasdaq Composite Index, both technology driven assets.

On the statistical properties of sea ice lead fraction and heat fluxes in the Arctic

In this paper we explore several statistical properties of the observed and simulated Arctic sea-ice lead-fraction, as well as the statistics of simulated Arctic ocean-atmosphere heat fluxes. We first show that the probability density function (PDF) and the monofractal spatial scaling of the observed lead fraction in the Central Arctic are both well represented by our model, neXtSIM. Given that the heat flux through leads may be up to two orders of magnitude larger than that through unbroken ice we then explore the statistical properties (PDF and spatial scaling) of the heat fluxes simulated by neXtSIM. We demonstrate that the modelled heat fluxes present a multifractal scaling in the Central Arctic, where heat fluxes through leads dominate the high-flux tail of the PDF. In the Central Arctic, the high-flux tail of the PDF is dominated by an exponential decay, which we attribute to the presence of coastal polynyas. Finally, we show that the scaling of simulated lead fraction and heat fluxes depend weakly on the model resolution and discuss the role sub-grid scale parameterisations of the ice heterogeneity may have in improving this result.