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For two-component mixtures of shifted distributions a general formula for finding the value of the shift parameter ã0,s, at which the cumulant coefficients γs ( p ã) of any order s ≥ 3 are equal to zero, is obtained. An algorithm for mathematical and computer modeling of two component mixtures of shifted distributions with zero cumulant coefficients is formulated. General formulas for two-component mixtures of shifted gamma-distributions with zero cumulant coefficients of any order are obtained and examples of mixtures with zero skewness and kurtosis coefficients are given. General formulas of two-component mixtures of shifted Student’s distributions with zero cumulant coefficients of any order are obtained and examples of mixtures with zero kurtosis coefficient and coefficient γ6 ( p, ã) are given. The research results provide the practical possibility of using two-component mixtures of shifted distributions for mathematical and computer modeling of non-Gaussian random variables with zero cumulant coefficients of any order. [ABSTRACT FROM AUTHOR]

The dependence of the extremes and zeros of the excess kurtosis γ4(p, ã ) on the weight coefficient P is researched. Formulas for finding the extrema points, the values of the minimums and maximums of the excess kurtosis are obtained. Conditions on the shift parameter ã under which the extrema points belong to the interval [0;1] are determined. Formulas for finding the zeros of the excess kurtosis are obtained and conditions on shift parameter under which the roots of the equation γ4(p, ã ) are real and belong to the interval [0;1] are determined. Examples of calculating extremes and zeros of the excess kurtosis of two-component mixtures of shifted non-Gaussian distributions are considered. The results of the research justify the possibility of practical application of two-component mixtures of shifted distributions for mathematical and computer modeling of an infinite number of non-Gaussian random variables with negative, positive and zero excess kurtosis. [ABSTRACT FROM AUTHOR]

Extensive high-speed railway (HSR) network resembled the intricate vascular system of the human body, crisscrossing mainlands. Seismic events, known for their unpredictability, pose a significant threat to both trains and bridges, given the HSR's extended operational duration. Therefore, ensuring the running safety of train-bridge coupled (TBC) system, primarily composed of simply supported beam bridges, is paramount. Traditional methods like the Monte Carlo method fall short in analyzing this intricate system efficiently. Instead, efficient algorithm like the new point estimate method combined with moment expansion approximation (NPEM-MEA) is applied to study random responses of numerical simulation TBC systems. Validation of the NPEM-MEA's feasibility is conducted using the Monte Carlo method. Comparative analysis confirms the accuracy and efficiency of the method, with a recommended truncation order of four to six for the NPEM-MEA. Additionally, the influences of seismic magnitude and epicentral distance are discussed based on the random dynamic responses in the TBC system. This methodology not only facilitates seismic safety assessments for TBC systems but also contributes to standard-setting for these systems under earthquake conditions. [ABSTRACT FROM AUTHOR]

This paper presents a methodology for computing Value at Risk for financial assets that does not follow a normal distribution of return. A back-testing approach have been applied in order to select the best theoretical non-Gaussian distributions that can explain the behavior of the empirical data. In this study, Cauchy, Laplace, Logistic and Beta distributions have been considered. As benchmark, historical distribution, and Extreme Value Theory (EVT) method have been used. The experiment suggests differences in estimation of over 5 times between one method and another. [ABSTRACT FROM AUTHOR]

The estimation of higher-order co-moments of asset returns play an important role in higher-order moment portfolio selection. We improve the estimation of higher-order co-moments by using non-Gaussian components in the observed factor models and construct a portfolio selection method, labelled as Non-Gaussian Component (NGC) portfolio. We assume the non-normality of asset returns is driven by the independent non-Gaussian components in the observed factors. Through identifying and extracting those non-Gaussian components, the parameters in the portfolio objective function have been significantly decreased. We show that the non-Gaussian components can be estimated consistently by the independent component analysis and higher-order cumulant tests. Simulation studies confirm the good finite sample properties of our estimation procedure and further the performance of the NGC portfolio. Empirical results show that the NGC portfolio outperforms the benchmark portfolios, and only a few non-Gaussian components are needed to optimize the objective function. [ABSTRACT FROM AUTHOR]

The dependence of the cumulant coefficients of two-component mixtures of shifted nonGaussian distributions on the weight coefficient is analyzed and conditions are determined under which the cumulant coefficients of any orders are equal to zero. The dependence of the cumulant coefficients of two-component mixtures on the shear parameter is investigated and the parameter values are determined at which the cumulant coefficients of any orders have extrema and zeros. The dependence of the skewness and excess kurtosis of a two-component mixture of shifted Gumbel distributions of type 1 on the weight coefficient and the shear parameter is investigated and their values are obtained at which the skewness and excess kurtosis of the mixture are equal to zero. The features of computer modeling of random variables, the probability density of which is a two-component mixture of shifted distributions, are considered. [ABSTRACT FROM AUTHOR]

• Pairwise coupling of Nataf-based stochastic models through disaggregation. • Generation of multi-scale timeseries with any distribution and correlation structure. • Applicable for multivariate disaggregation/simulation problems. The generation of hydrometeorological time series that exhibit a given probabilistic and stochastic behavior across multiple temporal levels, traditionally expressed in terms of specific statistical characteristics of the observed data, is a crucial task for risk-based water resources studies, and simultaneously a puzzle for the community of stochastics. The main challenge stems from the fact that the reproduction of a specific behavior at a certain temporal level does not imply the reproduction of the desirable behavior at any other level of aggregation. In this respect, we first introduce a pairwise coupling of Nataf-based stochastic models within a disaggregation scheme, and next we propose their puzzle-type configuration to provide a generic stochastic simulation framework for multivariate processes exhibiting any distribution and any correlation structure. Within case studies we demonstrate two characteristic configurations, i.e., a three-level one, operating at daily, monthly and annual basis, and a two-level one to disaggregate daily to hourly data. The first configuration is applied to generate correlated daily rainfall and runoff data at the river basin of Achelous, Western Greece, which preserves the stochastic behavior of the two processes at the three temporal levels. The second configuration disaggregates daily rainfall, obtained from a meteorological station at Germany, to hourly. The two studies reveal the ability of the proposed framework to represent the peculiar behavior of hydrometeorological processes at multiple temporal resolutions, as well as its flexibility on formulating generic simulation schemes. [ABSTRACT FROM AUTHOR]

Geostatistical analysis has been introduced over half a century ago to allow quantifying seemingly random spatial variations in earth quantities such as rock mineral content or permeability. The traditional approach has been to view such quantities as multivariate Gaussian random functions characterized by one or a few well-defined spatial correlation scales. There is, however, mounting evidence that many spatially varying quantities exhibit non-Gaussian behavior over a multiplicity of scales. The purpose of this minireview is not to paint a broad picture of the subject and its treatment in the literature. Instead, we focus on very recent advances in the recognition and analysis of this ubiquitous phenomenon, which transcends hydrology and the Earth sciences, brought about largely by our own work. In particular, we use porosity data from a deep borehole to illustrate typical aspects of such scalable non-Gaussian behavior, describe a very recent theoretical model that (for the first time) captures all these behavioral aspects in a comprehensive manner, show how this allows generating random realizations of the quantity conditional on sampled values, point toward ways of incorporating scalable non-Gaussian behavior in hydrologic analysis, highlight the significance of doing so, and list open questions requiring further research. [ABSTRACT FROM AUTHOR]

This article focuses on investigation of statistical approaches to the task of control performance assessment. Different statistical measures with Gaussian and non-Gaussian probabilistic distributions are taken into consideration. Analysis starts with the observations for simulated proportional–integral–derivative control error histograms followed by its statistical investigation using selected probabilistic distribution functions. Simulation experiments are followed by the analysis of control data originating from real industrial loops. Shadowing effect of long-tail control error histograms is identified, as it may significantly disable proper loop quality assessment. Results show that non-Gaussian approach with Cauchy or α-stable distributions seems to be reasonable assessment alternative in case of disturbances existing in industrial processes. [ABSTRACT FROM AUTHOR]

For many stochastic phenomena, we observe statistical distributions that have fat-tails and high-peaks compared to the Gaussian distribution. In this paper, we will explain how observable statistical distributions in the macroscopic world could be related to the randomness in the subatomic world. We show that fat-tailed (leptokurtic) phenomena in our everyday macroscopic world are ultimately rooted in Gaussian – or very close to Gaussian-distributed subatomic particle randomness, but they are not, in a strict sense, Gaussian distributions. By running a truly random experiment over a three and a half-year period, we observed a type of random behavior in trillions of photons. Combining our results with simple logic, we find that fat-tailed and high-peaked statistical distributions are exactly what we would expect to observe if the subatomic world is quantized and not continuously divisible. We extend our analysis to the fact that one typically observes fat-tails and high-peaks relative to the Gaussian distribution in stocks and commodity prices and many aspects of the natural world; these instances are all observable and documentable macro phenomena that strongly suggest that the ultimate building blocks of nature are discrete (e.g. they appear in quanta). [ABSTRACT FROM AUTHOR]

Uncertainty-aware design and management of urban water systems lies on the generation of synthetic series that should precisely reproduce the distributional and dependence properties of residential water demand process (i.e., significant deviation from Gaussianity, intermittent behaviour, high spatial and temporal variability and a variety of dependence structures) at various temporal and spatial scales of operational interest. This is of high importance since these properties govern the dynamics of the overall system, while prominent simulation methods, such as pulse-based schemes, address partially this issue by preserving part of the marginal behaviour of the process (e.g., low-order statistics) or neglecting the significant aspect of temporal dependence. In this work, we present a single stochastic modelling strategy, applicable at any fine time scale to explicitly preserve both the distributional and dependence properties of the process. The strategy builds upon the Nataf’s joint distribution model and particularly on the quantile mapping of an auxiliary Gaussian process, generated by a suitable linear stochastic model, to establish processes with the target marginal distribution and correlation structure. The three real-world case studies examined, reveal the efficiency (suitability) of the simulation strategy in terms of reproducing the variety of marginal and dependence properties encountered in water demand records from 1-min up to 1-h.

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Distributions of data taken from crystal structure databases, display unusual statistical properties like non-Gaussian shape, polymodality, and heavy tails. These features, typical for statistics of numerical data in social systems, appear in bigger sets (from hundreds to many thousand points) of database-originated parameters. The non-classic statistics of physical data collected through many years reflects a strong impact of social factors (financial support, exchange of information, competition, etc.) on a research activity. The values of some definite structural parameter, determined (and deposited to a database) by different authors, generally are neither independent nor random; their handling by common statistical tools may result in incorrect conclusions and wrong predictions. These statements are illustrated by several sets of reliable structural data whose statistics looks 'bad,' or inconclusive, in contemporary structural chemistry. [ABSTRACT FROM AUTHOR]

The impact of the logconductivity ( Y = ln K ) distribution f Y on transport at the MADE site is analyzed. Our principal interest is in non-Gaussian f Y characterized by heavier tails than the Gaussian. Both the logconductivity moments and f Y itself are inferred, taking advantage of the detailed measurements of Bohling et al. (2012). The resulting logconductivity distribution displays heavier tails than the Gaussian, although the departure from Gaussianity is not significant. The effect of the logconductivity distribution on the breakthrough curve (BTC) is studied through an analytical, physically based model. It is found that the non-Gaussianity of the MADE logconductivity distribution does not strongly affect the BTC. Counterintuitively, assuming heavier tailed distributions for Y , with same variance, leads to BTCs which are more symmetrical than those for the Gaussian f Y , with less pronounced preferential flow. Results indicate that the impact of strongly non-Gaussian, heavy tailed distributions on solute transport in heterogeneous porous formations can be significant, especially in the presence of high heterogeneity, resulting in reduced preferential flow and retarded peak arrivals. [ABSTRACT FROM AUTHOR]

We consider random vectors drawn from a multivariate normal distribution and compute the sample statistics in the presence of stochastic correlations. For this purpose, we construct an ensemble of random correlation matrices and average the normal distribution over this ensemble. The resulting distribution contains a modified Bessel function of the second kind whose behavior differs significantly from the multivariate normal distribution, in the central part as well as in the tails. This result is then applied to asset returns. We compare with empirical return distributions using daily data from the NASDAQ Composite Index in the period from 1992 to 2012. The comparison reveals good agreement, the average portfolio return distribution describes the data well especially in the central part of the distribution. This in turn confirms our ansatz to model the nonstationarity by an ensemble average. [ABSTRACT FROM AUTHOR]

The Ensemble Kalman Filter (EnKF) is a Monte Carlo-based technique for assisted history matching and real-time updating of reservoir models. However, it often fails to detect precise locations of distinct facies boundaries and their proportions, as the facies distributions are non-Gaussian, while geologic data for reservoir modeling is usually insufficient. In this paper, a new re-sampling step is introduced to the conventional EnKF formulation; after certain number of assimilation steps, the updated ensemble is used to generate a new ensemble with a novel probability weighted re-sampling scheme. The new ensemble samples from a probability density function that is conditional to both the geological information and the early production data. After the re-sampling step, the forecast model is applied to the new ensemble from the beginning up to the last update step (without any intermediate Kalman updates). Full EnKF is again applied on the ensemble members to assimilate the remaining production history. Combination of EnKF and regenerating new members using the re-sampling method demonstrates reasonable improvement and reduction of uncertainty in history matching of reservoir models with multiple facies. The histogram and the experimental variogram of the updated ensemble members are more consistent with the static geologic information. Moreover, the technique helps maintaining ensemble variance which is essential for uncertainty estimation in the posterior probability distribution of facies proportions. [ABSTRACT FROM AUTHOR]

The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism, non-standard limit distributions are obtained, and related to a form of ergodicity breaking in the underlying non-homogeneous Hidden Markov Chain. The link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions. [ABSTRACT FROM AUTHOR]

In this paper, we perform molecular dynamics (MD) simulations to study the random packing of spheres with different particle size distributions. In particular, we deal with non-Gaussian distributions by means of the Lévy distributions. The initial positions as well as the radii of five thousand non-overlapping particles are assigned inside a confining rectangular box. After that, the system is allowed to settle under gravity towards the bottom of the box. Both the translational and rotational movements of each particle are considered in the simulations. In order to deal with interacting particles, we take into account both the contact and long-range cohesive forces. The normal viscoelastic force is calculated according to the nonlinear Hertz model, whereas the tangential force is calculated through an accurate nonlinear-spring model. Assuming a molecular approach, we account for the long-range cohesive forces using a Lennard-Jones (LJ)-like potential. The packing processes are studied assuming different long-range interaction strengths. • Packing dynamics is sensitive to both the distribution type and long-range forces. • The packing density gradually decays as the ε values increase. • The mean coordination number gradually decays as the ε values increase. • The Lévy type II distribution yields the highest packing densities. [ABSTRACT FROM AUTHOR]

Uncertainty-aware design and management of urban water systems lies on the generation of synthetic series that should precisely reproduce the distributional and dependence properties of residential water demand process (i.e., significant deviation from Gaussianity, intermittent behaviour, high spatial and temporal variability and a variety of dependence structures) at various temporal and spatial scales of operational interest. This is of high importance since these properties govern the dynamics of the overall system, while prominent simulation methods, such as pulse-based schemes, address partially this issue by preserving part of the marginal behaviour of the process (e.g., low-order statistics) or neglecting the significant aspect of temporal dependence. In this work, we present a single stochastic modelling strategy, applicable at any fine time scale to explicitly preserve both the distributional and dependence properties of the process. The strategy builds upon the Nataf's joint distribution model and particularly on the quantile mapping of an auxiliary Gaussian process, generated by a suitable linear stochastic model, to establish processes with the target marginal distribution and correlation structure. The three real-world case studies examined, reveal the efficiency (suitability) of the simulation strategy in terms of reproducing the variety of marginal and dependence properties encountered in water demand records from 1-min up to 1-h. [ABSTRACT FROM AUTHOR]