The Fundamentals of Heavy Tails

Nair, Jayakrishnan; Wierman, Adam; Zwart, Bert

doi:10.1017/9781009053730

Cited by 29 publications

(32 citation statements)

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“…For the sum of dependent random variables, the tail heaviness of the aggregated random variable depends on the heaviness of the marginal tails and the dependence of the tails (Albrecher et al, 2006;Kortschak & Albrecher, 2009). For the additive aggregation of many random variables, the aggregation via the mean may remove heavy tails, as a consequence of a variation on the classical Central Limit Theorem (Billingsley, 1995;Nair et al, 2017). However, heavy tails are preserved in the limiting distribution of mean-aggregated processes, if the condition of finite variance of the components is not met, resulting in α-stable distribution (Table 1, for details, see Nair et al [2017]).…”

Section: Arithmetic Combination Of Random Variables (S1)mentioning

confidence: 99%

Understanding Heavy Tails of Flood Peak Distributions

Merz

Basso

Fischer

et al. 2022

Water Resources Research

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Floods often come as a surprise. Examples of extreme floods that have occurred unexpectedly and have led to disastrous socio-economic consequences abound in the literature (Merz et al., 2015). Figure 1 shows one example time series with such a surprising flood. The 2002 flood peak of the River Kamp, Austria, was about three times larger than the highest flood in the 100-year observational period before and has indeed caused enormous damage triggering desperate emergency measures in the region (Blöschl et al., 2006). From a statistical perspective, the occurrence of such an event is very unlikely if the extreme value behavior conforms to an asymptotically exponential (light-tailed) distribution. However, if the underlying probability distribution has a heavy tail, its occurrence is less unlikely. A heavy upper tail implies that the extreme values are more likely to occur than would be predicted by distributions with exponential asymptotic behavior, such as Exponential, Gamma, and Gumbel distributions (El Adlouni et al., 2008). Because human intuition tends to expect light tail behavior, processes that show heavy tail behavior often lead to surprise (Taleb, 2007).Heavy-tailed behavior of flood peak distributions is of the highest relevance for flood design and risk management. Neglecting heavy tail behavior, if it exists, results in underestimating the probability of occurrence of extremes. This underestimation may result in biased flood management measures, such as underestimated dike

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Section: Arithmetic Combination Of Random Variables (S1)mentioning

confidence: 99%

Understanding Heavy Tails of Flood Peak Distributions

Merz

Basso

Fischer

et al. 2022

Water Resources Research

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“…These observations are consistent with the corresponding PDFs shown in figure 4. Namely, because BG noise is a mixture of two Gaussian distributions, BG noise has rapidly decaying 'light' tails, whereas SαS noise has slowly decaying 'heavy' tails with a higher probability of extreme values [71].…”

Section: Impulsive Noisementioning

confidence: 99%

“…Consequently, for the impulsive noise types, we also investigated replacing the feature min-max scaling of the target data with a quantile transformation [78, section 7.4.1] applied independently to each channel to make the data approximately follow a standard normal distribution. The motivations for this transformation were twofold: (1) it ensured that the distribution of each channel was unimodal with 'light' tails [71], and (2) it effectively limited the impact of outliers.…”

Section: Quantile Data Transformation For Impulsive Noisementioning

confidence: 99%

Data-driven modeling of noise time series with convolutional generative adversarial networks ^∗

Wunderlich

Sklar

2023

Mach. Learn.: Sci. Technol.

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Random noise arising from physical processes is an inherent characteristic of measurements and a limiting factor for most signal processing and data analysis tasks. Given the recent interest in generative adversarial networks (GANs) for data-driven modeling, it is important to determine to what extent GANs can faithfully reproduce noise in target data sets. In this paper, we present an empirical investigation that aims to shed light on this issue for time series. Namely, we assess two general-purpose GANs for time series that are based on the popular deep convolutional GAN (DCGAN) architecture, a direct time-series model and an image-based model that uses a short-time Fourier transform (STFT) data representation. The GAN models are trained and quantitatively evaluated using distributions of simulated noise time series with known ground-truth parameters. Target time series distributions include a broad range of noise types commonly encountered in physical measurements, electronics, and communication systems: band-limited thermal noise, power law noise, shot noise, and impulsive noise. We ﬁnd that GANs are capable of learning many noise types, although they predictably struggle when the GAN architecture is not well suited to some aspects of the noise, e.g., impulsive time-series with extreme outliers. Our ﬁndings provide insights into the capabilities and potential limitations of current approaches to time-series GANs and highlight areas for further research. In addition, our battery of tests provides a useful benchmark to aid the development of deep generative models for time series.

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“…), making it sensitive to non-linear relationships in data [ 21 ]. It is well-known that Gaussian models can fail when the underlying generative dynamic is strongly non-normal, such as in heavy-tailed distributions (for example, the distribution of wealth in the United States) [ 22 , 23 ], temporally extended human dynamics, such as epidemic spreading [ 24 ] and these nonlinearities can lead to erroneous conclusions and incorrect models [ 25 ]. Finally, for researchers who do wish to leverage the power of linear models, or work with continuous data, closed-form Gaussian estimators of all major information-theoretic relationships exist: for example, the Pearson correlation is a function of the mutual information between Gaussian variables [ 21 ], the partial correlation maps to the conditional mutual information [ 26 ], and the Granger Causality is a special case of the more general transfer entropy [ 27 ].…”

Section: Introductionmentioning

confidence: 99%

Untangling Synergistic Effects of Intersecting Social Identities with Partial Information Decomposition

Varley

Kaminski

2022

Entropy

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The theory of intersectionality proposes that an individual’s experience of society has aspects that are irreducible to the sum of one’s various identities considered individually, but are “greater than the sum of their parts”. In recent years, this framework has become a frequent topic of discussion both in social sciences and among popular movements for social justice. In this work, we show that the effects of intersectional identities can be statistically observed in empirical data using information theory, particularly the partial information decomposition framework. We show that, when considering the predictive relationship between various identity categories such as race and sex, on outcomes such as income, health and wellness, robust statistical synergies appear. These synergies show that there are joint-effects of identities on outcomes that are irreducible to any identity considered individually and only appear when specific categories are considered together (for example, there is a large, synergistic effect of race and sex considered jointly on income irreducible to either race or sex). Furthermore, these synergies are robust over time, remaining largely constant year-to-year. We then show using synthetic data that the most widely used method of assessing intersectionalities in data (linear regression with multiplicative interaction coefficients) fails to disambiguate between truly synergistic, greater-than-the-sum-of-their-parts interactions, and redundant interactions. We explore the significance of these two distinct types of interactions in the context of making inferences about intersectional relationships in data and the importance of being able to reliably differentiate the two. Finally, we conclude that information theory, as a model-free framework sensitive to nonlinearities and synergies in data, is a natural method by which to explore the space of higher-order social dynamics.

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The Fundamentals of Heavy Tails

Cited by 29 publications

References 0 publications

Understanding Heavy Tails of Flood Peak Distributions

Understanding Heavy Tails of Flood Peak Distributions

Data-driven modeling of noise time series with convolutional generative adversarial networks ^∗

Untangling Synergistic Effects of Intersecting Social Identities with Partial Information Decomposition

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The Fundamentals of Heavy Tails

Cited by 29 publications

References 0 publications

Understanding Heavy Tails of Flood Peak Distributions

Understanding Heavy Tails of Flood Peak Distributions

Data-driven modeling of noise time series with convolutional generative adversarial networks ∗

Untangling Synergistic Effects of Intersecting Social Identities with Partial Information Decomposition

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Data-driven modeling of noise time series with convolutional generative adversarial networks ^∗