Stretched exponential distributions in Nature and Economy: ``Fat tails'' with characteristic scales

To account quantitatively for many reported ``natural'' fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok temperature variations, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also briefly discuss its potential for the distribution of earthquake sizes and fault displacements and earth temperature variations over the last 400 000 years. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions.Comment: 17 pages, 16 sets of figures, in press in European Physical Journal

Citation Statistics from 110 Years of Physical Review

Publicly available data reveal long-term systematic features about citation statistics and how papers are referenced. The data also tell fascinating citation histories of individual articles.Comment: This is esssentially identical to the article that appeared in the June 2005 issue of Physics Toda

Finding Scientific Gems with Google

We apply the Google PageRank algorithm to assess the relative importance of all publications in the Physical Review family of journals from 1893--2003. While the Google number and the number of citations for each publication are positively correlated, outliers from this linear relation identify some exceptional papers or "gems" that are universally familiar to physicists.Comment: 6 pages, 4 figures, 2 tables, 2-column revtex4 forma

Evidence for Power-law tail of the Wealth Distribution in India

The higher-end tail of the wealth distribution in India is studied using recently published lists of the wealth of richest Indians between the years 2002-4. The resulting rank distribution seems to imply a power-law tail for the wealth distribution, with a Pareto exponent between 0.81 and 0.92 (depending on the year under analysis). This provides a comparison with previous studies of wealth distribution, which have all been confined to Western advanced capitalist economies. We conclude with a discussion on the appropriateness of multiplicative stochastic process as a model for asset accumulation, the relation between the wealth and income distributions (we estimate the Pareto exponent for the latter to be around 1.5 for India), as well as possible sources of error in measuring the Pareto exponent for wealth.Comment: 8 pages, 3 figure

Tagging Scientific Publications using Wikipedia and Natural Language Processing Tools. Comparison on the ArXiv Dataset

In this work, we compare two simple methods of tagging scientific publications with labels reflecting their content. As a first source of labels Wikipedia is employed, second label set is constructed from the noun phrases occurring in the analyzed corpus. We examine the statistical properties and the effectiveness of both approaches on the dataset consisting of abstracts from 0.7 million of scientific documents deposited in the ArXiv preprint collection. We believe that obtained tags can be later on applied as useful document features in various machine learning tasks (document similarity, clustering, topic modelling, etc.)

Statistical Asynchronous Regression: Determining the Relationship Between two Quantities that are not Measured Simultaneously

We introduce the Statistical Asynchronous Regression (SAR) method: a technique for determining a relationship between two time varying quantities without simultaneous measurements of both quantities. We require that there is a time invariant, monotonic function Y = u(X) relating the two quantities, Y and X. In order to determine u(X), we only need to know the statistical distributions of X and Y. We show that u(X) is the change of variables that converts the distribution of X into the distribution of Y, while conserving probability. We describe an algorithm for implementing this method and apply it to several example distributions. We also demonstrate how the method can separate spatial and temporal variations from a time series of energetic electron flux measurements made by a spacecraft in geosynchronous orbit. We expect this method will be useful to the general problem of spacecraft instrument calibration. We also suggest some applications of the SAR method outside of space physics.Comment: 27 pages, 10 figures, stronger motivations and rewriting to make the paper more accessible to a general audience. in press in J. Geophys. Res. (Space Physics

k-Generalized Statistics in Personal Income Distribution

Starting from the generalized exponential function $\exp_{\kappa}(x)=(\sqrt{1+\kappa^{2}x^{2}}+\kappa x)^{1/\kappa}$, with $\exp_{0}(x)=\exp(x)$, proposed in Ref. [G. Kaniadakis, Physica A \textbf{296}, 405 (2001)], the survival function $P_{>}(x)=\exp_{\kappa}(-\beta x^{\alpha})$, where $x\in\mathbf{R}^{+}$, $\alpha,\beta>0$, and $\kappa\in[0,1)$, is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function $P_{>}^{0}(x)=\exp(-\beta x^{\alpha})$\textemdash to which reduces as $\kappa$ approaches zero\textemdash behaving in very different way in the $x\to0$ and $x\to\infty$ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law $P_{>}(x)\sim(2\beta\kappa)^{-1/\kappa}x^{-\alpha/\kappa}$. This makes the $\kappa$-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range.Comment: Latex2e v1.6; 14 pages with 12 figures; for inclusion in the APFA5 Proceeding

Thesaurus as a complex network

A thesaurus is one, out of many, possible representations of term (or word) connectivity. The terms of a thesaurus are seen as the nodes and their relationship as the links of a directed graph. The directionality of the links retains all the thesaurus information and allows the measurement of several quantities. This has lead to a new term classification according to the characteristics of the nodes, for example, nodes with no links in, no links out, etc. Using an electronic available thesaurus we have obtained the incoming and outgoing link distributions. While the incoming link distribution follows a stretched exponential function, the lower bound for the outgoing link distribution has the same envelope of the scientific paper citation distribution proposed by Albuquerque and Tsallis. However, a better fit is obtained by simpler function which is the solution of Ricatti's differential equation. We conjecture that this differential equation is the continuous limit of a stochastic growth model of the thesaurus network. We also propose a new manner to arrange a thesaurus using the ``inversion method''.Comment: Contribution to the Proceedings of `Trends and Perspectives in Extensive and Nonextensive Statistical Mechanics', in honour of Constantino Tsallis' 60th birthday (submitted Physica A

Fokker-Planck equation of distributions of financial returns and power laws

Our purpose is to relate the Fokker-Planck formalism proposed by [Friedrich et al., Phys. Rev. Lett. 84, 5224 (2000)] for the distribution of stock market returns to the empirically well-established power law distribution with an exponent in the range 3-5. We show how to use Friedrich et al.'s formalism to predict that the distribution of returns is indeed asymptotically a power law with an exponent mu that can be determined from the Kramers-Moyal coefficients determined by Friedrich et al. However, with their values determined for the U.S. dollar-German mark exchange rates, the exponent mu predicted from their theory is found around 12, in disagreement with the often-quoted value between 3 and 5. This could be explained by the fact that the large asymptotic value of 12 does not apply to real data that lie still far from the stationary state of the Fokker-Planck description. Another possibility is that power laws are inadequate. The mechanism for the power law is based on the presence of multiplicative noise across time-scales, which is different from the multiplicative noise at fixed time-scales implicit in the ARCH models developed in the Finance literature.Comment: 11 pages, in press in Physica

Stock Market Speculation: Spontaneous Symmetry Breaking of Economic Valuation

Firm foundation theory estimates a security's firm fundamental value based on four determinants: expected growth rate, expected dividend payout, the market interest rate and the degree of risk. In contrast, other views of decision-making in the stock market, using alternatives such as human psychology and behavior, bounded rationality, agent-based modeling and evolutionary game theory, expound that speculative and crowd behavior of investors may play a major role in shaping market prices. Here, we propose that the two views refer to two classes of companies connected through a ``phase transition''. Our theory is based on 1) the identification of the fundamental parity symmetry of prices ($p \to -p$), which results from the relative direction of payment flux compared to commodity flux and 2) the observation that a company's risk-adjusted growth rate discounted by the market interest rate behaves as a control parameter for the observable price. We find a critical value of this control parameter at which a spontaneous symmetry-breaking of prices occurs, leading to a spontaneous valuation in absence of earnings, similarly to the emergence of a spontaneous magnetization in Ising models in absence of a magnetic field. The low growth rate phase is described by the firm foundation theory while the large growth rate phase is the regime of speculation and crowd behavior. In practice, while large ``finite-time horizon'' effects round off the predicted singularities, our symmetry-breaking speculation theory accounts for the apparent over-pricing and the high volatility of fast growing companies on the stock markets.Comment: 23 pages, 10 figure