Correlations Between Record Events in Sequences of Random Variables with a Linear Trend

Wergen, Gregor; Franke, Jasper; Krug, Joachim

doi:10.1007/s10955-011-0307-7

Cited by 21 publications

(54 citation statements)

References 34 publications

(99 reference statements)

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“…processes, P n|m is simply equal to P n . For a symmetric random walk, records are not independent, and the conditional probability of two successive records is given by P n+1|n = 1/2 for large n [37]. This shows that the record events in symmetric random walks effectively attract each other, i.e., there is an increased probability for the occurrence of a record if another record has occurred in the previous time step [37].…”

Section: General Concepts In Records Statisticsmentioning

confidence: 99%

“…To include more complicated systems, many research studies have been performed. For example, the record statistics of independent random variables from non-identical distributions [35], broadening distributions [35,36], and time-dependent distributions (linear drift) [37,38] are also well understood. However, many time series in real situations are correlated.…”

Section: Introductionmentioning

confidence: 99%

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Records in fractal stochastic processes

Aliakbari

Manshour

Salehi

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The records statistics in stationary and non-stationary fractal time series is studied extensively. By calculating various concepts in record dynamics, we find some interesting results. In stationary fractional Gaussian noises, we observe a universal behavior for the whole range of Hurst exponents. However, for non-stationary fractional Brownian motions the record dynamics is crucially dependent on the memory, which plays the role of a non-stationarity index, here. Indeed, the deviation from the results of the stationary case increases by increasing the Hurst exponent in fractional Brownian motions. We demonstrate that the memory governs the dynamics of the records as long as it causes non-stationarity in fractal stochastic processes, otherwise, it has no impact on the records statistics.

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Section: General Concepts In Records Statisticsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Records in fractal stochastic processes

Aliakbari

Manshour

Salehi

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…In [21] both cases were found depending on the distribution F (η). Specifically, an expansion to first order in c yields l n,n−1 (c) = 1 + cJ(n) + O(c 2 ) with J(n) ≈ − 1 2 n 4 (I(n) − I(n − 1)) − n 3 I(n) where…”

mentioning

confidence: 91%

“…RVs with distribution F (η) and density f (η), correlations between record events were quantified by considering the ratio [21] l n,n−1 (c) = p n,n−1 (c)…”

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confidence: 99%

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Correlations of Record Events as a Test for Heavy-Tailed Distributions

2012

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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.Determining the probability distribution underlying a given data set or at least its behavior for large argument is of pivotal importance for predicting the behavior of the system: If the data is drawn from a distribution with heavy tails, one needs to prepare for large events. Of particular relevance is the case when the probability density displays a power law decay, as this implies a drastic enhancement of the probability of extreme events. This is one of the reasons for the persistent interest in the observation and modeling of power law distributions, which have been associated with critical, scale-invariant behavior [1, 2] in diverse contexts ranging from complex networks [3] to paleontology [4], foraging behavior of animals [5], citation distributions [6] and many more [7].However, when trying to infer the tail behavior of the underlying distribution from a finite data set, one faces the problem that the number of entries of large absolute value is very small. This implies that even though binning the entries by magnitude and plotting them would yield an approximate representation of the probability density, this process becomes inconclusive in particular in the tail of the probability density. Furthermore, in small data sets, extreme outliers can strongly affect the results of methods like maximum likelihood estimators such that leaving out even one of these extreme and possibly spurious data points renders the outcome of the test insignificant. A case in point is the problem of estimating the distribution of fitness effects of beneficial mutations in evolution experiments, which are expected on theoretical ground to conform to one of the universality classes of extreme value theory (EVT) [8]. Because beneficial mutations are rare, the corresponding data sets are typically limited to a few dozen values, and the determination of the tail behavior can be very challenging [9,10].In this Letter we present a method for detecting heavy tails in empirical data that works reliably for small data sets (on the order of a few dozen entries) and is robust with respect to removal of extreme entries. The test is based on the statistics of records of subsamples of the data set. Similar to conventional record-based statistical tests [11][12][13], and in contrast to the bulk of methods available in this field [7], our approach is non-parametric and, hence, does not require any hypothesis about the underlying distribution. Rather tha...

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Record occurrence and record values in daily and monthly temperatures

2013

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We analyze the occurrence and the values of record-breaking temperatures in daily and monthly temperature observations. Our aim is to better understand and quantify the statistics of temperature records in the context of global warming. Similar to earlier work we employ a simple mathematical model of independent and identically distributed random variables with a linearly growing expectation value. This model proved to be useful in predicting the increase (decrease) in upper (lower) temperature records in a warming climate. Using both station and re-analysis data from Europe and the United States we further investigate the statistics of temperature records and the validity of this model. The most important new contribution in this article is an analysis of the statistics of record values for our simple model and European reanalysis data. We estimate how much the mean values and the distributions of record temperatures are affected by the large scale warming trend. In this context we consider both the values of records that occur at a certain time and the values of records that have a certain record number in the series of record events. We compare the observational data both to simple analytical computations and numerical simulations. We find that it is more difficult to describe the values of record breaking temperatures within the framework of our linear drift model. Observations from the summer months fit well into the model with Gaussian random variables under the observed linear warming, in the sense that record breaking temperatures are more extreme in the summer. In winter however a significant asymmetry of the daily temperature distribution hides the effect of the slow warming trends. Therefore very extreme cold records are still possible in winter. This effect is even more pronounced if one considers only data from subpolar regions.Comment: 16 pages, 20 figures, revised version, published in Climate Dynamic

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Correlations Between Record Events in Sequences of Random Variables with a Linear Trend

Cited by 21 publications

References 34 publications

Records in fractal stochastic processes

Records in fractal stochastic processes

Correlations of Record Events as a Test for Heavy-Tailed Distributions

Record occurrence and record values in daily and monthly temperatures

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