基于幂律尾指数研究不同尺度系统对降水的影响

利用中国气象局国家气候中心740站点1960—2000年日降水观测资料，统计分析表明，各气候特征区30 mm以上日降水存在幂律尾分布特征，从中国的整体情况来看，幂律尾指数的均值超过了3.0，其对应的降水过程不存在平稳性成分，因此长期暴雨预报成为一个艰巨的任务.借助滤波方法进一步研究发现：日降水幂律尾分布特征是大气中各尺度系统相互作用的结果，其中一周尺度系统对30 mm以上日降水幂律尾指数影响最大. 关键词： 幂律尾指数 传递熵 暴雨     

Interacting gaps model,dynamics of order book,and stock-market fluctuations

Inspired by order-book models of financial fluctuations, we investigate the Interacting gaps model, which is the schematic one-dimensional system mimicking the order-book dynamics. We find by simulations the power-law tail in return distribution, power-law decay of volatility autocorrelation with exponent 0.5 and Hurst exponent close to 1/2. Surprisingly, when we make a mean-field approximation, i.e. replace the one-dimensional system by effectively infinite-dimensional one, we obtain analytically the return exponent 5/2, in perfect accord with one-dimensional simulations.     

Scaling in the distribution of intertrade durations of Chinese stocks

The distribution of intertrade durations, defined as the waiting times between two consecutive transactions, is investigated based upon the limit order book data of 23 liquid Chinese stocks listed on the Shenzhen Stock Exchange in the whole year 2003. A scaling pattern is observed in the distributions of intertrade durations, where the empirical density functions of the normalized intertrade durations of all 23 stocks collapse onto a single curve. The scaling pattern is also observed in the intertrade duration distributions for filled and partially filled trades and in the conditional distributions. The ensemble distributions for all stocks are modeled by the Weibull and the Tsallis q-exponential distributions. Maximum likelihood estimation shows that the Weibull distribution outperforms the q-exponential for not-too-large intertrade durations which account for more than 98.5% of the data. Alternatively, nonlinear least-squares estimation selects the q-exponential as a better model, in which the optimization is conducted on the distance between empirical and theoretical values of the logarithmic probability densities. The distribution of intertrade durations is Weibull followed by a power-law tail with an asymptotic tail exponent close to 3.     

Problems with fitting to the power-law distribution

This short communication uses a simple experiment to show that fitting to a power law distribution by using graphical methods based on linear fit on the log-log scale is biased and inaccurate. It shows that using maximum likelihood estimation (MLE) is far more robust. Finally, it presents a new table for performing the Kolmogorov-Smirnov test for goodness-of-fit tailored to power-law distributions in which the power-law exponent is estimated using MLE. The techniques presented here will advance the application of complex network theory by allowing reliable estimation of power-law models from data and further allowing quantitative assessment of goodness-of-fit of proposed power-law models to empirical data.Received: 18 June 2004, Published online: 12 October 2004PACS: 02.50.Ng Distribution theory and Monte Carlo studies - 05.10.Ln Monte Carlo methods - 89.75.-k Complex systems     

Generalized randomly amplified linear system driven by gaussian noises: extreme heavy tail and algebraic correlation decay in plasma turbulence

The extreme heavy tail and the power-law decay of the turbulent flux correlation observed in hot magnetically confined plasmas are modeled by a system of coupled Langevin equations describing a continuous time linear randomly amplified stochastic process where the amplification factor is driven by a superposition of colored noises which, in a suitable limit, generate a fractional Brownian motion. An exact analytical formula for the power-law tail exponent beta is derived. The extremely small value of the heavy tail exponent and the power-law distribution of laminar times also found experimentally are obtained, in a robust manner, for a wide range of input values, as a consequence of the (asymptotic) self-similarity property of the noise spectrum. As a by-product, a new representation of the persistent fractional Brownian motion is obtained.     

On the probability distribution of stock returns in the Mike-Farmer model

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not only the well-known power-law tails, together with several other important stylized facts. There are three key ingredients in the Mike-Farmer (MF) model: the long memory of order signs characterized by the Hurst index H_s, the distribution of relative order prices x in reference to the same best price described by a Student distribution (or Tsallis’ q-Gaussian), and the dynamics of order cancelation. They showed that different values of the Hurst index H_s and the freedom degree α_x of the Student distribution can always produce power-law tails in the return distribution f_r(r) with different tail exponent α_r. In this paper, we study the origin of the power-law tails of the return distribution f_r(r) in the MF model, based on extensive simulations with different combinations of the left part L(x) for x < 0 and the right part R(x) for x > 0 of f_x(x). We find that power-law tails appear only when L(x) has a power-law tail, no matter R(x) has a power-law tail or not. In addition, we find that the distributions of returns in the MF model at different timescales can be well modeled by the Student distributions, whose tail exponents are close to the well-known cubic law and increase with the timescale.     

The distribution of first-passage times and durations in FOREX and future markets

Possible distributions are discussed for intertrade durations and first-passage processes in financial markets. The view-point of renewal theory is assumed. In order to represent market data with relatively long durations, two types of distributions are used, namely a distribution derived from the Mittag-Leffler survival function and the Weibull distribution. For the Mittag-Leffler type distribution, the average waiting time (residual life time) is strongly dependent on the choice of a cut-off parameter t_max, whereas the results based on the Weibull distribution do not depend on such a cut-off. Therefore, a Weibull distribution is more convenient than a Mittag-Leffler type if one wishes to evaluate relevant statistics such as average waiting time in financial markets with long durations. On the other hand, we find that the Gini index is rather independent of the cut-off parameter. Based on the above considerations, we propose a good candidate for describing the distribution of first-passage time in a market: The Weibull distribution with a power-law tail. This distribution compensates the gap between theoretical and empirical results more efficiently than a simple Weibull distribution. It should be stressed that a Weibull distribution with a power-law tail is more flexible than the Mittag-Leffler distribution, which itself can be approximated by a Weibull distribution and a power-law. Indeed, the key point is that in the former case there is freedom of choice for the exponent of the power-law attached to the Weibull distribution, which can exceed 1 in order to reproduce decays faster than possible with a Mittag-Leffler distribution. We also give a useful formula to determine an optimal crossover point minimizing the difference between the empirical average waiting time and the one predicted from renewal theory. Moreover, we discuss the limitation of our distributions by applying our distribution to the analysis of the BTP future and calculating the average waiting time. We find that our distribution is applicable as long as durations follow a Weibull law for short times and do not have too heavy a tail.     

Experimental evidence for the interplay between individual wealth and transaction network

We conduct a market experiment with human agents in order to explore the structure of transaction networks and to study the dynamics of wealth accumulation. The experiment is carried out on our platform for 97 days with 2,095 effective participants and 16,936 times of transactions. From these data, the hybrid distribution (log-normal bulk and power-law tail) in the wealth is observed and we demonstrate that the transaction networks in our market are always scale-free and disassortative even for those with the size of the order of few hundred. We further discover that the individual wealth is correlated with its degree by a power-law function which allows us to relate the exponent of the transaction network degree distribution to the Pareto index in wealth distribution.     

The power-law tail exponent of income distributions

In this paper we tackle the problem of estimating the power-law tail exponent of income distributions by using the Hill's estimator. A subsample semi-parametric bootstrap procedure minimizing the mean squared error is used to choose the power-law cutoff value optimally. This technique is applied to personal income data for Australia and Italy.     

Modelling conflicts with cluster dynamics in networks

We introduce cluster dynamical models of conflicts in which only the largest cluster can be involved in an action. This mimics the situations in which an attack is planned by a central body, and the largest attack force is used. We study the model in its annealed random graph version, on a fixed network, and on a network evolving through the actions. The sizes of actions are distributed with a power-law tail, however, the exponent is non-universal and depends on the frequency of actions and sparseness of the available connections between units. Allowing the network reconstruction over time in a self-organized manner, e.g., by adding the links based on previous liaisons between units, we find that the power-law exponent depends on the evolution time of the network. Its lower limit is given by the universal value 5/2, derived analytically for the case of random fragmentation processes. In the temporal patterns behind the size of actions we find long-range correlations in the time series of the number of clusters and the non-trivial distribution of time that a unit waits between two actions. In the case of an evolving network the distribution develops a power-law tail, indicating that through repeated actions, the system develops an internal structure with a hierarchy of units.     

Analysis of precipitation characteristics of South and North China based on the power-law tail exponents

Precipitation sequence is a typical nonlinear and chaotic observational series, and studies on precipitation forecasts are restricted to the use of traditional linear statistical methods, especially when analysing the regional characteristics of precipitation. In the context of 20 stations＇ daily precipitation series （from 1956 to 2000） in South China （SC） and North China （NC）, we divide each precipitation series into many self-stationary segments by using the heuristic segmentation algorithm （briefly BG algorithm）. For each station＇s precipitation series, we calculate the exponent of power-law tall （EPT） of the cumulative probability distribution of segments with a length larger than l for precipitation and temperature series. Our results show that the power-law decay of the cumulative probability distribution of stationary segments might be a common attribution for precipitation and other nonstationary time series; the EPT somewhat indicates the precipitation duration and its spatial distribution that might be different from area to area. The EPT in NC is larger than in SC; Meanwhile, EPT might be another effective way to study the abrupt changes in nonlinear and nonstationary time series.     

Runaway events dominate the heavy tail of citation distributions

Statistical distributions with heavy tails are ubiquitous in natural and social phenomena. Since the entries in heavy tail have unproportional significance, the knowledge of its exact shape is very important. Citations of scientific papers form one of the best-known heavy tail distributions. Even in this case there is a considerable debate whether citation distribution follows the log-normal or power-law fit. The goal of our study is to solve this debate by measuring citation distribution for a very large and homogeneous data. We measured citation distribution for 418, 438 Physics papers published in 1980–1989 and cited by 2008. While the log-normal fit deviates too strong from the data, the discrete power-law function with the exponent γ = 3.15 does better and fits 99.955% of the data. However, the extreme tail of the distribution deviates upward even from the power-law fit and exhibits a dramatic “runaway” behavior. The onset of the runaway regime is revealed macroscopically as the paper garners 1000-1500 citations, however the microscopic measurements of autocorrelation in citation rates are able to predict this behavior in advance.     

Changes of firm size distribution: The case of Korea

In this paper, the distribution and inequality of firm sizes is evaluated for the Korean firms listed on the stock markets. Using the amount of sales, total assets, capital, and the number of employees, respectively, as a proxy for firm sizes, we find that the upper tail of the Korean firm size distribution can be described by power-law distributions rather than lognormal distributions. Then, we estimate the Zipf parameters of the firm sizes and assess the changes in the magnitude of the exponents. The results show that the calculated Zipf exponents over time increased prior to the financial crisis, but decreased after the crisis. This pattern implies that the degree of inequality in Korean firm sizes had severely deepened prior to the crisis, but lessened after the crisis. Overall, the distribution of Korean firm sizes changes over time, and Zipf’s law is not universal but does hold as a special case.     

自对耦无序分布随机链Potts模型的临界普适性研究

以蒙特卡罗模拟方法对自对耦分布二维随机链q态Potts模型的短时临界行为进行了数值研究.利用初始非平衡演化阶段存在的普适幂指数和有限体积标度行为，数值模拟了在不同形式随机分布时q＝3和q＝8态Potts模型磁临界指数η和动力学临界指数z.计算结果发现η不依赖于自对偶无序分布的具体形式， 从而以数值方法给出了一个关于淬火掺杂自旋系统的临界普适行为的验证. 关键词： 随机链Potts模型 动力学蒙特卡罗模拟 临界普适性     

Most probable degree distribution at fixed structural entropy

The structural entropy is the entropy of the ensemble of uncorrelated networks with given degree sequence. Here we derive the most probable degree distribution emerging when we distribute stubs (or half-edges) randomly through the nodes of the network by keeping fixed the structural entropy. This degree distribution is found to decay as a Poisson distribution when the entropy is maximized and to have a power-law tail with an exponent γ → 2 when the entropy is minimized.      

Empirical shape function of limit-order books in the Chinese stock market

We have analyzed the statistical probabilities of limit-order book (LOB) shape through building the book using the ultra-high-frequency data from 23 liquid stocks traded on the Shenzhen Stock Exchange in 2003. We find that the averaged LOB shape has a maximum away from the same best price for both buy and sell sides of the LOB. The LOB shape function has nice exponential form in the right tail. The buy side of the LOB is found to be abnormally thicker for the price levels close to the same best although there are much more sell orders on the book. We also find that the LOB shape functions for both buy and sell sides have periodic peaks with a period of five. The 1-min averaged volumes at fixed tick level follow log-normal distributions except for the left tails which display power-law behaviors, exhibit abnormal intraday patterns with increasing trend, and possess long memory that cannot be explained by the intraday patterns. Academic implications of our empirical results are also briefly discussed.     

幂律指数在1与3之间的一类无标度网络

借助排队系统中顾客批量到达的概念，提出节点批量到达的Poisson网络模型.节点按照到达率为λ的Poisson过程批量到达系统.模型1，批量按照到达批次的幂律非线性增长，其幂律指数为θ(0≤θ<+∞).BA模型是在θ＝0时的特例.利用Poisson过程理论和连续化方法进行分析，发现这个网络稳态平均度分布是幂律分布，而且幂律指数在1和3之间.模型2，批量按照节点到达批次的对数非线性增长，得出当批量增长较缓慢时，稳态度分布幂律指数为3.因此，节点批量到达的Poisson网络模型不仅是BA模型的推广，也为许多幂律指数在1和2之间的现实网络提供了理论依据.     

Preferred numbers and the distributions of trade sizes and trading volumes in the Chinese stock market

The distributions of trade sizes and trading volumes are investigated based on the limit order book data of 22 liquid Chinese stocks listed on the Shenzhen Stock Exchange in the whole year 2003. We observe that the size distribution of trades for individualstocks exhibits jumps, which is caused by the number preference of traders when placing orders. We analyze the applicability of the “q-Gamma” function for fitting the distribution by the Cramér-von Mises criterion. The empirical PDFs of tradingvolumes at different timescales Δt ranging from 1 min to 240 min can be well modeled. The applicability of the q-Gamma functions for multiple trades is restricted to the transaction numbers Δn≤ 8. We find that all the PDFs have power-law tails for large volumes. Using careful estimation of the average tail exponents α of the distributions of trade sizes and trading volumes, we get α> 2, well outside the Lévy regime.     

Parameter estimation for power-law distributions by maximum likelihood methods

Distributions following a power-law are an ubiquitous phenomenon. Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods.     

On the growth of primary industry and population of China’s counties

The growth dynamics of complex organizations have attracted much interest of econophysicists and sociophysicists in recent years. However, most of the studies are done for developed countries. We investigate the growth dynamics of the primary industry and the population of 2079 counties in mainland China using the data from the China County Statistical Yearbooks from 2000 to 2006. We find that the annual growth rates are distributed according to Student’s t distribution with the tail exponent less than 2. We find power-law relationships between the sample standard deviation of the growth rates and the initial size. The scaling exponent is less than 0.5 for the primary industry and close to 0.5 for the population.