Probability Integral Method in the Influence of Ground Settlement on a Certain Construction Land

There is a potential threat of geological hazards in the construction land adjacent to a mined-out area on the west side. To accurately determine the degree of influence of the mined-out area in this area, this article has collected and sorted out five reports on a gypsum mine since 2010, combined with various parameters in the relevant reports, and used the integral probability method. Quantitatively calculate the surface displacement value, surface tilt value, absolute value of horizontal deformation and the maximum settlement value in the goaf area. The actual calculation results show that the construction land here is not affected by the goal of the gypsum mine.

A Unifying Framework for Probabilistic Validation Metrics

Probabilistic modeling methods are increasingly being employed in engineering applications. These approaches make inferences about the distribution for output quantities of interest. A challenge in applying probabilistic computer models (simulators) is validating output distributions against samples from observational data. An ideal validation metric is one that intuitively provides information on key differences between the simulator output and observational distributions, such as statistical distances/divergences. Within the literature, only a small set of statistical distances/divergences have been utilized for this task; often selected based on user experience and without reference to the wider variety available. As a result, this paper offers a unifying framework of statistical distances/divergences, categorizing those implemented within the literature, providing a greater understanding of their benefits, and offering new potential measures as validation metrics. In this paper, two families of measures for quantifying differences between distributions, that encompass the existing statistical distances/divergences within the literature, are analyzed: f-divergence and integral probability metrics (IPMs). Specific measures from these families are highlighted, providing an assessment of current and new validation metrics, with a discussion of their merits in determining simulator adequacy, offering validation metrics with greater sensitivity in quantifying differences across the range of probability mass.

A review of conjectured laws of total mass of Bacry–Muzy GMC measures on the interval and circle and their applications

Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry–Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic potentials on the unit interval and circle, respectively. The construction and properties of these distributions are reviewed from three perspectives: Analytic based on several representations of the Mellin transform, asymptotic based on low intermittency expansions, and probabilistic based on the theory of Barnes beta probability distributions. In particular, positive and negative integer moments, infinite factorizations and involution invariance of the Mellin transform, analytic and probabilistic proofs of infinite divisibility of the logarithm, factorizations into products of Barnes beta distributions, and Stieltjes moment problems of these distributions are presented in detail. Applications are given in the form of conjectured mod-Gaussian limit theorems, laws of derivative martingales, distribution of extrema of [Formula: see text] noises, and calculations of inverse participation ratios in the Fyodorov–Bouchaud model.

Plain Interpretation of Freak Waves Phenomenon

An extremely large (‘freak’) wave is a typical though quite a rare phenomenon observed in the sea. Special theories (for example, the modulational instability theory) were developed to explain the mechanics and appearance of freak waves as a result of nonlinear wave-wave interactions. This paper demonstrates that freak wave appearance can be also explained by superposition of linear modes with a realistic spectrum. The integral probability of trough-to-crest waves is calculated by two methods: the first one is based on the results of a numerical simulation of wave field evolution, performed with one-dimensional and two-dimensional nonlinear models. The second method is based on the calculation of the same probability over ensembles of wave fields, constructed as a superposition of linear waves with random phases and a spectrum similar to that used in nonlinear simulations. It is shown that the integral probabilities for nonlinear and linear cases are of the same order of values. One-dimensional model was used for performing thousands of exact short-term simulations of evolution of two superposed wave trains with different steepness and wavenumbers to investigate the effect of wave crests merging. The nonlinear sharpening of merging crests is demonstrated. It is suggested that such effect may be responsible for appearance of typical sharp crests of surface waves, as well as for the wave breaking.

An Evaluation of Validation Metrics for Probabilistic Model Outputs

Probabilistic modelling methods are increasingly being employed in engineering applications. These approaches make inferences about the distribution, or summary statistical moments, for output quantities. A challenge in applying probabilistic models is validating output distributions. An ideal validation metric is one that intuitively provides information on key divergences between the output and validation distributions. Furthermore, it should be interpretable across different problems in order to informatively select the appropriate statistical method. In this paper, two families of measures for quantifying differences between distributions are compared: f-divergence and integral probability metrics (IPMs). Discussions and evaluation of these measures as validation metrics are performed with comments on ease of computation, interpretability and quantity of information provided.

Linear and nonlinear statistics of extreme waves

The probability of extremely high waves is calculated by two methods. The first method is based on the direct numerical simulation of two-dimensional wave field using a three-dimensional nonlinear model. The second method consists in calculation of the probability of wave heights over ensemble of fields representing a superposition of linear waves with random phases and a spectrum similar to that obtained in the nonlinear model. It is shown that the integral probability of extreme waves are very close to each other in both cases. This implies that the role of nonlinearity in the generation of extreme waves is probably not so important as it was assumed in most papers considering this phenomenon.