Record statistics and persistence for a random walk with a drift

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length \eta drawn from a continuous distribution f(\eta) which is symmetric around a constant drift c. We focus in particular on the case were f(\eta) is a symmetric stable law with a L\'evy index 0 < \mu \leq 2. The record statistics depends crucially on the persistence probability which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter \mu. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and \mu) of the jump distribution. We compute the asymptotic mean record number after n steps as well as its full distribution P(R,n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < \mu \leq 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.Comment: 51 pages, 22 figures. Published version (typos have been corrected

Records and sequences of records from random variables with a linear trend

We consider records and sequences of records drawn from discrete time series of the form $X_{n}=Y_{n}+cn$, where the $Y_{n}$ are independent and identically distributed random variables and $c$ is a constant drift. For very small and very large drift velocities, we investigate the asymptotic behavior of the probability $p_n(c)$ of a record occurring in the $n$th step and the probability $P_N(c)$ that all $N$ entries are records, i.e. that $X_1 < X_2 < ... < X_N$. Our work is motivated by the analysis of temperature time series in climatology, and by the study of mutational pathways in evolutionary biology.Comment: 21 pages, 7 figure

Correlations of record events as a test for heavy-tailed distributions

A record is an entry in a time series that is larger or smaller than all previous entries. If the time series consists of independent, identically distributed random variables with a superimposed linear trend, record events are positively (negatively) correlated when the tail of the distribution is heavier (lighter) than exponential. Here we use these correlations to detect heavy-tailed behavior in small sets of independent random variables. The method consists of converting random subsets of the data into time series with a tunable linear drift and computing the resulting record correlations.Comment: Revised version, to appear in Physical Review Letter

Evaluation of European Land Data Assimilation System (ELDAS) products using in site observations

Three land-surface models with land-data assimilation scheme (DA) were evaluated for one growing season using in situ observations obtained across Europe. To avoid drifts in the land-surface state in the models, soil moisture corrections are derived from errors in screen-level atmospheric quantities. With the in situ data it is assessed whether these land-surface schemes produce adequate results regarding the annual range of the soil water content, the monthly mean soil moisture content in the root zone and evaporative fraction (the ratio of evapotranspiration to energy available at the surface). DA considerably reduced bias in net precipitation, while slightly reducing RMSE as well. Evaporative fraction was improved in dry conditions but was hardly affected in moist conditions. The amplitude of soil moisture variations tended to be underestimated. The impact of improved land-surface properties like Leaf Area Index, water holding capacity and rooting depth may be as large as corrections of the DA systems. Because soil moisture memorizes errors in the hydrological cycle of the models, DA will remain necessary in forecast mode. Model improvements should be balanced against improvements of DA per se. Model bias appearing from persistent analysis increments arising from DA systems should be addressed by model improvement

Record statistics for biased random walks, with an application to financial data

We consider the occurrence of record-breaking events in random walks with asymmetric jump distributions. The statistics of records in symmetric random walks was previously analyzed by Majumdar and Ziff and is well understood. Unlike the case of symmetric jump distributions, in the asymmetric case the statistics of records depends on the choice of the jump distribution. We compute the record rate $P_n(c)$, defined as the probability for the $n$th value to be larger than all previous values, for a Gaussian jump distribution with standard deviation $\sigma$ that is shifted by a constant drift $c$. For small drift, in the sense of $c/\sigma \ll n^{-1/2}$, the correction to $P_n(c)$ grows proportional to arctan$(\sqrt{n})$ and saturates at the value $\frac{c}{\sqrt{2} \sigma}$. For large $n$ the record rate approaches a constant, which is approximately given by $1-(\sigma/\sqrt{2\pi}c)\textrm{exp}(-c^2/2\sigma^2)$ for $c/\sigma \gg 1$. These asymptotic results carry over to other continuous jump distributions with finite variance. As an application, we compare our analytical results to the record statistics of 366 daily stock prices from the Standard & Poors 500 index. The biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices, and after detrending the number of upper records is in good agreement with the symmetric random walk. However the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified.Comment: 16 pages, 7 figure

Gaming with eutrophication: Contribution to integrating water quantity and quality management at catchment level

The Metropolitan Region of Sao Paulo (MRSP) hosts 18 million inhabitants. A complex system of 23 interconnected reservoirs was built to ensure its water supply. Half of the potable water produced for MRSP's population (35 m3/s) is imported from a neighbour catchment, the other half is produced within the Alto Tietê catchment, where 99% of the population lives. Perimeters of land use restriction were defined to contain uncontrolled urbanization, as domestic effluents were causing increasing eutrophication of some of these reservoirs. In the 90's catchment committees and sub committees were created to promote discussion between stakeholders and develop catchment plans. The committees are very well structured "on paper". However, they are not very well organised and face a lack of experience. The objective of this work was to design tools that would strengthen their discussion capacities. The specific objective of the AguAloca process was to integrate the quality issue and its relation to catchment management as a whole in these discussions. The work was developed in the Alto Tietê Cabeceiras sub-catchment, one of the 5 sub catchments of the Alto-Tietê. It contains 5 interconnected dams, and presents competitive uses such as water supply, industry, effluent dilution and irrigated agriculture. A RPG was designed following a companion modelling approach (Etienne et al., 2003). It contains a friendly game-board, a set of individual and collective rules and a computerized biophysical model. The biophysical model is used to simulate water allocation and quality processes at catchment level. It articulates 3 modules. A simplified nutrient discharge model permits the estimation of land use nutrient exportation. An arc-node model simulates water flows and associated nutrient charges from one point of the hydrographical network to another. The Vollenweider model is used for simulating specific reservoir dynamics. The RPG allows players to make individual and collective decisions related to water allocation and the management of its quality. Impacts of these decisions are then simulated using the biophysical model. Specific indicators of the game are then updated and may influence player's behaviour (actions) in following rounds. To introduce discussions on the management of water quality at a catchment level, an issue that is rarely explicitly dealt with, four game sessions were implemented involving representatives of basin committees and water and sanitation engineers. During the game session, the participants took advantage of the water quality output of the biophysical model to test management alternatives such as rural sewage collection or effluent dilution. The biophysical model accelerated calculations of flows and eutrophication rates that were then returned to the game board with explicit indicators of quantity and quality. Players could easily test decisions impacting on qualitative water processes and visualize the simulation results directly on the game board that was representing a friendly, virtual and simplified catchment. The Agualoca game proved its ability to turn complex water processes understandable for a non totally initiated public. This experience contributed to a better understanding of multiple-use water management and also of joint management of water quality and quantity. (Résumé d'auteur

Record Statistics for Multiple Random Walks

We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance \sigma^2 of the jump distribution is finite and (II) when \sigma^2 is divergent as in the case of L\'evy flights with index 0 < \mu < 2. In both cases we find that the mean record number grows universally as \sim \alpha_N \sqrt{n} for large n, but with a very different behavior of the amplitude \alpha_N for N > 1 in the two cases. We find that for large N, \alpha_N \approx 2 \sqrt{\log N} independently of \sigma^2 in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, \alpha_N \approx 4/\sqrt{\pi}, independently of 0<\mu<2. For finite \sigma^2 we argue, and this is confirmed by our numerical simulations, that the full distribution of (R_{n,N}/\sqrt{n} - 2 \sqrt{\log N}) \sqrt{\log N} converges to a Gumbel law as n \to \infty and N \to \infty. In case II, our numerical simulations indicate that the distribution of R_{n,N}/\sqrt{n} converges, for n \to \infty and N \to \infty, to a universal nontrivial distribution, independently of \mu. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poors 500 index.Comment: 25 pages, 8 figure

Universality, limits and predictability of gold-medal performances at the Olympic Games

Inspired by the Games held in ancient Greece, modern Olympics represent the world's largest pageant of athletic skill and competitive spirit. Performances of athletes at the Olympic Games mirror, since 1896, human potentialities in sports, and thus provide an optimal source of information for studying the evolution of sport achievements and predicting the limits that athletes can reach. Unfortunately, the models introduced so far for the description of athlete performances at the Olympics are either sophisticated or unrealistic, and more importantly, do not provide a unified theory for sport performances. Here, we address this issue by showing that relative performance improvements of medal winners at the Olympics are normally distributed, implying that the evolution of performance values can be described in good approximation as an exponential approach to an a priori unknown limiting performance value. This law holds for all specialties in athletics-including running, jumping, and throwing-and swimming. We present a self-consistent method, based on normality hypothesis testing, able to predict limiting performance values in all specialties. We further quantify the most likely years in which athletes will breach challenging performance walls in running, jumping, throwing, and swimming events, as well as the probability that new world records will be established at the next edition of the Olympic Games.Comment: 8 pages, 3 figures, 1 table. Supporting information files and data are available at filrad.homelinux.or

Description of Atmospheric Conditions at the Pierre Auger Observatory using the Global Data Assimilation System (GDAS)

Atmospheric conditions at the site of a cosmic ray observatory must be known for reconstructing observed extensive air showers. The Global Data Assimilation System (GDAS) is a global atmospheric model predicated on meteorological measurements and numerical weather predictions. GDAS provides altitude-dependent profiles of the main state variables of the atmosphere like temperature, pressure, and humidity. The original data and their application to the air shower reconstruction of the Pierre Auger Observatory are described. By comparisons with radiosonde and weather station measurements obtained on-site in Malarg\"ue and averaged monthly models, the utility of the GDAS data is shown

Record-breaking temperatures reveal a warming climate

We present a mathematical analysis of records drawn from independent random variables with a drifting mean. To leading order the change in the record rate is proportional to the ratio of the drift velocity to the standard deviation of the underlying distribution. We apply the theory to time series of daily temperatures for given calendar days, obtained from historical climate recordings of European and American weather stations as well as re-analysis data. We conclude that the change in the mean temperature has increased the rate of record-breaking events in a moderate but significant way: for the European station data covering the time period 1976-2005, we find that about 5 of the 17 high temperature records observed on average in 2005 can be attributed to the warming climate. Copyright (C) EPLA, 201