Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem

Burnecki, Krzysztof; Wyłomańska, Agnieszka; Chechkin, Aleksei V.

doi:10.1371/journal.pone.0145604

Cited by 34 publications

(20 citation statements)

References 65 publications

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“…To distinguish between Gaussian and non-Gaussian distributions we also advocate the application of the discrimination algorithm based on examining the rate of convergence to the Gaussian law by means of the central limit theorem [81]. The idea of the algorithm is to analyse the convergence of the estimated index α of stability for sequential bootstrapped samples from the analysed data.…”

Section: Non-gaussianitymentioning

confidence: 99%

Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

2019

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Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as 'superstatistics' or 'diffusing diffusivity'. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models. We start from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.

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Section: Non-gaussianitymentioning

confidence: 99%

Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

2019

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“…The whiskers are lines extending from each end of the box to show the extent of the rest of the data. Points are drawn as outliers if they are larger than Q3+1.5(Q3−Q1) or smaller than Q1−1.5(Q3−Q1), where Q1 and Q3 are lower and upper quartiles, respectively [61].…”

Section: Fbm Time-changed By Gamma Processmentioning

confidence: 99%

Fractional Brownian motion time-changed by gamma and inverse gamma process

Kumar

Wyłomańska

Połoczański

et al. 2017

Physica A: Statistical Mechanics and its Applications

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Abstract. Many real time-series exhibit behavior adequate to long range dependent data. Additionally very often these time-series have constant time periods and also have characteristics similar to Gaussian processes although they are not Gaussian. Therefore there is need to consider new classes of systems to model these kind of empirical behavior. Motivated by this fact in this paper we analyze two processes which exhibit long range dependence property and have additional interesting characteristics which may be observed in real phenomena. Both of them are constructed as the superposition of fractional Brownian motion (FBM) and other process. In the first case the internal process, which plays role of the time, is the gamma process while in the second case the internal process is its inverse. We present in detail their main properties paying main attention to the long range dependence property. Moreover, we show how to simulate these processes and estimate their parameters. We propose to use a novel method based on rescaled modified cumulative distribution function for estimation of parameters of the second considered process. This method is very useful in description of rounded data, like waiting times of subordinated processes delayed by inverse subordinators. By using the Monte Carlo method we show the effectiveness of proposed estimation procedures.Keywors: subordination, gamma process, inverse gamma process, simulation, estimation

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“…In the first step we specify the distribution category expressed in the language of the domain of attraction of the distribution corresponding to analysed time series, while in the second step we try to fit the specific distribution that in best way describes the behavior of time series. In the preliminary analysis (step one) we use the graphical test that was introduced in order to recognize if analysed dataset belongs to domain of attraction of Gaussian or strictly stable distribution [26]. Here we can not answer the question which distribution is the best one for analysed data, we only specify category of distribution (fat or heavy tailed).…”

Section: Methodsmentioning

confidence: 99%

Discrimination of particulate matter emission sources using stochastic methods

Szczurek

Maciejewska

Wyłomańska

et al. 2016

Physica A: Statistical Mechanics and its Applications

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Particulate matter (PM) is one of the criteria pollutants which has been determined as harmful to public health and the environment. For this reason the ability to recognize its emission sources is very important. There are a number of measurement methods which allow to characterize PM in terms of concentration, particles size distribution, and chemical composition. All these information are useful to establish a link between the dust found in the air, its emission sources and influence on human as well as the environment. However, the methods are typically quite sophisticated and not applicable outside laboratories. In this work, we considered PM emission source discrimination method which is based on continuous measurements of PM concentration with a relatively cheap instrument and stochastic analysis of the obtained data. The stochastic analysis is focused on the temporal variation of PM concentration and it involves two steps: 1) recognition of the category of distribution for the data i.e. stable or the domain of attraction of stable distribution and 2) finding best matching distribution out of Gaussian, stable and normal-inverse Gaussian (NIG). We examined six PM emission sources. They were associated with material processing in industrial environment, namely machining and welding aluminum, forged carbon steel and plastic with various tools. As shown by the obtained results, PM emission sources may be distinguished based on statistical distribution of PM concentration variations. Major factor responsible for the differences detectable with our method, was the type of material processing and the tool applied. In case different materials were processed by the same tool the distinction of emission sources was difficult. For successful discrimination it was crucial to consider size-segregated mass fraction concentrations. In our opinion the presented approach is very promising. It deserves further study and development.

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Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem

Cited by 34 publications

References 65 publications

Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

Fractional Brownian motion time-changed by gamma and inverse gamma process

Discrimination of particulate matter emission sources using stochastic methods

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