Heavy-tailed Research Articles

Forecasting geomagnetic activity: Neural networks, moving windows and state transition models

Geomagnetic activity shows high degrees of nonlinear variability. The probability distribution has heavy tails, and there are intermittent outliers. This has led to increased interest in forecasting using neural networks and nonlinear regressions, which include time-varying coefficient techniques. Because geomagnetic storms pose the greatest threat to satellites and power grids, there is a particular interest in predicting outlying events. The model proposed here combines two techniques. Neural networks and regressions are trained over moving windows of observations, so that the weights or coefficients adjust to new data. Second, logistic regression is used to predict the periods of high activity, and the cumulative distribution function is used as a causal input in time series and machine learning models. The data set is the Aa index, corrected for secular drift. Forecasting experiments are run over horizons of 1–4 days. The other models include time-varying parameter regressions and a recurrent neural network with fixed weights. The model combining the neural net and logistic regression achieves the most accurate forecast, although the regression is a close second. The ability to predict outliers depends on the width of the moving window. With wider windows, the overall error is lower, but the forecasted values fall into a narrower range, missing the outliers. With narrower windows, the model predicts the outliers better but is vulnerable to calling them at the wrong times, so the average error is higher. Further, while the model achieves more accurate predictions at 1 day, at longer horizons the accuracy deteriorates quite rapidly.

Deep-water ambient sound on near-bottom receivers in the New England Seamount Chain

Ambient sound was recorded for about two months by three synchronized, single-hydrophone receivers that were moored at depths of 2573, 2994, and 4443 m on seamount flanks. Hydrophones were located within 5 m from the seafloor. The data reveal the power spectra and intermittency of the ambient sound intensity in a 13-octave frequency band from 0.5 to 4000 Hz. Statistical distribution of pressure amplitude exhibits much heavier tails than the expected Rayleigh distribution throughout the frequency band of observations. Spatial variability of the observed ambient sound is controlled by the seafloor properties, bathymetric shadowing, and nonuniform distribution of the noise sources on the sea surface due to the Gulf Stream and its meanders. Interferometry of the ambient sound recorded by the receivers with the horizontal separations of 7.0 km shows strong variations of the acoustic travel time between the receivers along a surface-reflected path due to the evolution of the Gulf Stream position. The magnitude of the variations of the passively measured acoustic travel time is consistent with the available contact measurements of the sound speed profiles. Environmental inferences derived from the ambient, shipping, and flow noise data will be discussed. [Work supported by the ONR TFO DRI.]

Fraud Power Laws

ABSTRACTUsing misstatement data, we find that the distribution of detected fraud features a heavy tail. We propose a theoretical mechanism that explains such a relatively high frequency of extreme frauds. In our dynamic model, a manager manipulates earnings for personal gain. A monitor of uncertain quality can detect fraud and punish the manager. As the monitor fails to detect fraud, the manager's posterior belief about the monitor's effectiveness decreases. Over time, the manager's learning leads to a slippery slope, in which the size of frauds grows steeply, and to a power law for detected fraud. Empirical analyses corroborate the slippery slope and the learning channel. As a policy implication, we establish that a higher detection intensity can increase fraud by enabling the manager to identify an ineffective monitor more quickly. Further, nondetection of frauds below a materiality threshold, paired with a sufficiently steep punishment scheme, can prevent large frauds.

On a novel skewed generalized t distribution: Properties, estimations, and its applications

With the progress of information technology, large amounts of asymmetric, leptokurtic, and heavy-tailed data are arising in various fields, such as finance, engineering, genetics, and medicine. It is very challenging to model those kinds of data, especially for extremely skewed data, accompanied by very high kurtosis or heavy tails. In this article, we propose a class of novel skewed generalized t distribution (SkeGTD) as a scale mixture of skewed generalized normal. The proposed SkeGTD has excellent adaptiveness to various data, because of its capability of allowing for a large range of skewness and kurtosis and its compatibility of the separated location, scale, skewness, and shape parameters. We investigate some important properties of this family of distributions. The maximum likelihood estimation, L-moments estimation, and two-step estimation for the SkeGTD are explored. To illustrate the usefulness of the proposed methodology, we present simulation studies and analyze two real datasets.

Influence of α-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis

Non-negative matrix factorization (NMF) has been used in various applications, including local damage detection in rotating machines. Recent studies highlight the limitations of diagnostic techniques in the presence of non-Gaussian noise. The authors examine the impact of non-Gaussianity levels on the extraction of the signal of interest (SOI). The simple additive model of the signal is proposed: SOI and non-Gaussian noise. As a model of the random component, i.e., noise, a heavy-tailed α-stable distribution with two important parameters (σ and α) was proposed. If SOI is masked by noise (controlled by σ), the influence of non-Gaussianity level (controlled by α) is more critical. We performed an empirical analysis of how these parameters affect SOI extraction effectiveness using NMF. Finally, we applied two NMF algorithms to several (both vibration and acoustic) signals from a machine with faulty bearings at different levels of non-Gaussian disturbances and the obtained results align with the simulations. The main conclusion of this study is that NMF is a very powerful tool for analyzing non-Gaussian data and can provide satisfactory results in a wide range of a non-Gaussian noise levels.

What controls the tail behaviour of flood series: rainfall or runoff generation?

Abstract. Many observed time series of precipitation and streamflow show heavy-tail behaviour. For heavy-tailed distributions, the occurrence of extreme events has a higher probability than for distributions with an exponentially receding tail. If we neglect heavy-tail behaviour we might underestimate the magnitude of rarely observed, high-impact events. Robust estimation of upper-tail behaviour is often hindered by the limited length of observational records. Using long time series and a better understanding of the relevant process controls can help with achieving more robust tail estimations. Here, a simulation-based approach is used to analyse the effect of precipitation and runoff generation characteristics on the upper tail of flood peak distributions. Long, synthetic precipitation time series with different tail behaviour are produced by a stochastic weather generator. These are used to force a conceptual rainfall–runoff model. In addition, catchment characteristics linked to a threshold process in the runoff generation are varied between model runs. We characterize the upper-tail behaviour of the simulated precipitation and discharge time series with the shape parameter of the generalized extreme value (GEV) distribution. Our analysis shows that runoff generation can strongly modulate the tail behaviour of flood peak distributions. In particular, threshold processes in the runoff generation lead to heavier tails. Beyond a certain return period, the influence of catchment processes decreases and the tail of the rainfall distribution asymptotically governs the tail of the flood peak distribution. Beyond which return period this is the case depends on the catchment storage in relation to the mean annual rainfall amount.

Human emergency behaviour and psychological stress characteristic mining based on large-scale emergencies

AbstractHuman emergency behaviour and psychological stress response in emergencies are important scientific issues in basic emergency management research. The analysis of the dynamic characteristics of large-scale human behaviour based on electronic footprint data provides a new method for quantitative research on this problem. Previous studies usually assumed that human behaviors were randomly distributed in time, but few studies have studied the psychological stress response of human groups under the influence of emergencies and carried out prediction methods through social media data. Based on the data from five emergencies and daily events in the Qzone, this paper explores the statistical characteristics of human communication behaviors such as time, space and social interaction. The research results reveal the psychological evolution of human groups when they encounter public security emergencies by analysing the causes of individual psychological stress responses in the group. We find that the time interval between people’s posting behaviour and interactive comment behaviour in mobile QQ space before and after an emergency can be approximately described by a power-law distribution. The time interval distribution of Posting and reply is an obvious heavy-tailed distribution. These behavioural characteristics are consistent with people’s psychological stress characteristics. Individual psychological stress responses gradually evolve into social-psychological responses with changes in behavioural characteristics. The greater the social-psychological stress response is, the more panic the public will be, which will cause the outbreak of group irrational behaviour. The research results are theoretically helpful in understanding the impact of emergencies on human communication behaviour patterns and reveal the psychological stress process of mass panic in large-scale emergencies.

Accessibility of SPDEs driven by pure jump noise and its applications

In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump (possibly degenerate) noise, including stochastic 2D Navier-Stokes equations, stochastic Burgers equations, stochastic singular p p -Laplace equations, and stochastic fast diffusion equations. As a further application, we establish the ergodicity of stochastic singular p p -Laplace equations and stochastic fast diffusion equations driven by additive pure jump noise, and we remark that the driving noises could be Compound Poisson processes or Lévy processes with heavy tails.

Non-Gaussian anomalous diffusion of optical vortices.

Anomalous diffusion of different particlelike entities, the deviation from typical Brownian motion, is ubiquitous in complex physical and biological systems. While optical vortices move randomly in evolving speckle fields, optical vortices have only been observed to exhibit pure Brownian motion in random speckle fields. Here we present direct experimental evidence of the anomalous diffusion of optical vortices in temporally varying speckle patterns from multiple-scattering viscoelastic media. Moreover, we observe two characteristic features, i.e., the self-similarity and the antipersistent correlation of the optical vortex motion, indicating that the mechanism of the observed subdiffusion of optical vortices can only be attributed to fractional Brownian motion (FBM). We further demonstrate that the vortex displacements exhibit a non-Gaussian heavy-tailed distribution. Additionally, we modulate the extent of subdiffusion, such as diffusive scaling exponents, and the non-Gaussianity of optical vortices by altering the viscoelasticity of samples. The discovery of the complex FBM but non-Gaussian subdiffusion of optical vortices may not only offer insight into certain fundamental physics, including the anomalous diffusion of vortices in fluids and the decoupling between Brownianity and Gaussianity, but also suggest a strong potential for utilizing optical vortices as tracers in microrheology instead of the introduced exogenous probe particles in particle tracking microrheology.

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.

Contaminated Kent mixture model for clustering non-spherical directional data with heavy tails or scatter

To cluster asymmetrically distributed data on a sphere, a Kent mixture model is commonly used. However, the performance of such a model can be severely affected by the presence of heavy tails or outliers. A novel contaminated Kent mixture model is proposed to alleviate this issue. As demonstrated via a series of simulation studies and applications to real-life data sets, the developed model shows superior performance over existing alternatives for non-spherical heavy-tailed data.

Fractional-Differential Models of the Time Series Evolution of Socio-Dynamic Processes with Possible Self-Organization and Memory

This article describes the solution of two problems. First, based on the fractional diffusion equation, a boundary problem with arbitrary values of derivative indicators was formulated and solved, describing more general cases than existing solutions. Secondly, from the consideration of the probability schemes of transitions between states of the process, which can be observed in complex systems, a fractional-differential equation of the telegraph type with multiples is obtained (in time: β, 2β, 3β, … and state: α, 2α, 3α, …) using orders of fractional derivatives and its analytical solution for one particular boundary problem is considered. In solving edge problems, the Fourier method was used. This makes it possible to represent the solution in the form of a nested time series (one in time t, the second in state x), each of which is a function of the Mittag-Leffler type. The eigenvalues of the Mittag-Leffler function for describing states can be found using boundary conditions and the Fourier coefficient based on the initial condition and orthogonality conditions of the eigenfunctions. An analysis of the characteristics of time series of changes in the emotional color of users’ comments on published news in online mass media and the electoral campaigns of the US presidential elections showed that for the mathematical expectation of amplitudes of deviations of series levels from the size of the amplitude calculation interval (“sliding window”), a root dependence of fractional degree was observed; for dispersion, a power law with a fractional index greater than 1.5 was observed; and the behavior of the excess showed the presence of so-called “heavy tails”. The obtained results indicate that time series have unsteady non-locality, both in time and state. This provides the rationale for using differential equations with partial fractional derivatives to describe time series dynamics.

A Robust TCPHD Filter for Multi-Sensor Multitarget Tracking Based on a Gaussian–Student’s t-Mixture Model

To realize multitarget trajectory tracking under non-Gaussian heavy-tailed noise, we propose a Gaussian–Student t-mixture distribution-based trajectory cardinality probability hypothesis density filter (GSTM-TCPHD). We introduce the multi-sensor GSTM-TCPHD (MS-GSTM-TCPHD) filter to enhance tracking performance. Conventional cardinality probability hypothesis density (CPHD) filters typically assume Gaussian noise and struggle to accurately establish target trajectories when faced with heavy-tailed non-Gaussian distributions. Heavy-tailed noise leads to significant estimation errors and filter dispersion. Moreover, the exact trajectory of the target is crucial for tracking and prediction. Our proposed GSTM-TCPHD filter utilizes the GSTM distribution to model heavy-tailed noise, reducing modeling errors and generating a set of potential target trajectories. Since single sensors have a limited field of view and limited measurement information, we extend the filter to a multi-sensor scenario. To tackle the issue of data explosion from multiple sensors, we employed a greedy approximation method to assess measurements and introduced the MS-GSTM-TCPHD filter. The simulation results demonstrate that our proposed filter outperforms the CPHD/TCPHD filter and Student’s t-based TCPHD filter in terms of accurately estimating the trajectories of multiple targets during tracking while also achieving improved accuracy and shorter processing time.

A flexible extension of asymmetric power-t distribution

In this article, a new generalized modified slash-power-t (GMSLPT) distribution, as an extension of the power t (PT) distribution, is constructed by using independent PT and one parameter gamma random variables. The resulting distribution is an asymmetric model whose kurtosis is greater than other slash distributions or power distributions to fit skewed and heavy-tailed data. The probability density function and identifiability of GMSLPT distribution are obtained. The hazard rate function, main properties, moments, skewness, and kurtosis coefficients are studied in detail. The parameters in the model are estimated by the maximum likelihood method. A simulation study is given to illustrate the good behavior of MLEs. The advantages of the GMSLPT distribution are confirmed by two actual data examples to model asymmetric data with heavy tail and excess kurtosis.

Randomly Stopped Minimum, Maximum, Minimum of Sums and Maximum of Sums with Generalized Subexponential Distributions

Randomly Stopped Minimum, Maximum, Minimum of Sums and Maximum of Sums with Generalized Subexponential Distributions

Floating-point histograms for exploratory analysis of large scale real-world data sets

Histograms are among the most popular methods used in exploratory analysis to summarize univariate distributions. In particular, irregular histograms are good non-parametric density estimators that require very few parameters: the number of bins with their lengths and frequencies. Although many approaches have been proposed in the literature to infer these parameters, most existing histogram methods are difficult to exploit for exploratory analysis in the case of real-world data sets, with scalability issues, truncated data, outliers or heavy-tailed distributions. In this paper, we focus on the G-Enum histogram method, which exploits the Minimum Description Length (MDL) principle to build histograms without any user parameter. We then propose to extend this method by exploiting a new modeling space based on floating-point representation, with the objective of building histograms resistant to outliers or heavy-tailed distributions. We also suggest several heuristics and a methodology suitable for the exploratory analysis of large scale real-world data sets, whose underlying patterns are difficult to recover for digitization reasons. Extensive experiments show the benefits of the approach, evaluated with a dual objective: the accuracy of density estimation in the case of outliers or heavy-tailed distributions, and the effectiveness of the approach for exploratory data analysis.

Ground-Based Mobile Measurements to Track Urban Methane Emissions from Natural Gas in 12 Cities across Eight Countries.

To mitigate methane emission from urban natural gas distribution systems, it is crucial to understand local leak rates and occurrence rates. To explore urban methane emissions in cities outside the U.S., where significant emissions were found previously, mobile measurements were performed in 12 cities across eight countries. The surveyed cities range from medium size, like Groningen, NL, to large size, like Toronto, CA, and London, UK. Furthermore, this survey spanned across European regions from Barcelona, ES, to Bucharest, RO. The joint analysis of all data allows us to focus on general emission behavior for cities with different infrastructure and environmental conditions. We find that all cities have a spectrum of small, medium, and large methane sources in their domain. The emission rates found follow a heavy-tailed distribution, and the top 10% of emitters account for 60-80% of total emissions, which implies that strategic repair planning could help reduce emissions quickly. Furthermore, we compare our findings with inventory estimates for urban natural gas-related methane emissions from this sector in Europe. While cities with larger reported emissions were found to generally also have larger observed emissions, we find clear discrepancies between observation-based and inventory-based emission estimates for our 12 cities.

Linear multifractional stable motion for modeling of fluid-filled regions in retinal optical coherence tomography images

Fluid appears in several retinal disorders can be visualized by optical coherence tomography (OCT) images. Since manual survey of OCT images is challenging, automatic fluid detection is desirable. This paper develops a generalized multifractal framework to model the local scale-invariant property of retinal OCT signals and accordingly proposes a simple methodology for automatic localization of fluid-filled regions. The proposed framework comprises both local self-similarity and heavy tailed distribution of OCT signals by generalization of fractional Levy stable motion (fLsm) to the Riemann-Liouville multifractional Levy stable motion (RL-mLsm). In order to measure the multifractality appears in RL-mLsm, the scaling properties of Levy flights and stochastic differential equations (SDEs) are used to develop a fractional momentbased algorithm namely, multi-fractional Levy fluctuation analysis (MLFA) algorithm. Here, we apply MLFA to model OCT images. Modeling results indicate the significant difference between the multifractal bandwidth in non-fluid and fluid regions. In addition, by utilizing various surrogate series of the signals, we discover the source of multifractality of OCT signals. As an application of the proposed multifractal modeling, we propose a new method, based on simple clustering of estimated local scaling exponent of modeled OCT signals, to localize fluid regions in OCT images. The localization method is validated, and the experimental results reveal the superiority of proposed method. Overall, the proposed multifractal framework has full advantages on excellent improvement of fluid localization, which promises the proposed model is suitable for effective description of self-similar nature of normal and abnormal OCT images independent of capturing device.

Renyi entropy based design of heavy tailed distribution for return of financial assets

It is well-known that returns of financial assets exhibit heavy tail property and there has been no distribution which can reliably capture this characteristic so far. To contribute to the solution of this problem, we derive a new heavy tail distribution using the maximum entropy principle for Renyi entropy under the absolute moment constraints. Our newly derived distribution with two shape parameters forms a family of distributions. They are smooth, scaleable, symmetric and may be heavy tailed if their shape parameter attains the appropriate value. As a result, parameters of this distribution can be estimated by maximum likelihood estimation technique. The ability of the derived distribution to model the heavy tail property of financial assets is verified on a range of financial instruments. The results we obtained show that it can be a better option for modeling the returns of financial assets compared to other well-known heavy tailed distributions.

Under the weight of heavy tails: A power law perspective on the emergence of outliers in entrepreneurship

A fundamental discovery in entrepreneurship is that firm outcomes do not follow a symmetrical Gaussian curve. Instead, most are heavily right-skewed distributions in which a few extreme outliers (e.g., rock star firms like Airbnb, Tesla, and Uber) account for a disproportionate amount of the output. Although past research usually described outcome distributions as shaped following the power law, our study asks the following question: What other less extreme distributions of generalizable firm outcomes exist in entrepreneurship? Our investigation leverages four representative datasets from the U.S., Europe, and Australia, comprising 32 samples with about 22,000 ventures. We implemented a precise data-analytic approach that compares each sample (i.e., empirical distribution) against multiple theoretical distribution shapes to identify the best fit. Results showed that, across nearly all samples, the pure power law was not the dominant distribution. Instead, the annual revenue distribution is shaped as a power law with an exponential cutoff, and the number of employees distribution is shaped lognormally. Combined, these suggest the existence of top-down limitations on the highest performing firms. Accordingly, we offer an agenda for future research focused on (a) identifying and releasing systemic constraints, (b) examining and falsifying the underlying generative mechanisms that cause the emergence of heavy-tailed distributions and the outliers therein, and (c) conducting multi-level, mixed-method studies to investigate how micro-level interactions aggregate into macro-level heavy-tailed distributions. Our paper makes significant contributions to the power law perspective and future efforts to explain and predict the emergence of rock star firms in entrepreneurship.