Exact Probability Distribution Research Articles

An accelerated Benders decomposition method for distributionally robust sustainable medical waste location and transportation problem

This study addresses the sustainable medical waste location and transportation (SMWLT) problem from the viewpoint of social risk, environmental impact, and economic performance, where model uncertainty includes risk and transportation costs. In practice, it is usually hard to obtain the exact probability distribution of uncertain parameters. To address this challenge, this study first constructs an ambiguity set to model the partial distribution information of uncertain parameters. Based on the constructed ambiguity set, this study develops a new multi-objective distributionally robust chance-constrained (DRCC) model for the SMWLT problem. Subsequently, this study adopts the robust counterpart (RC) approximation method to reformulate the proposed DRCC model as a computationally tractable mixed-integer linear programming (MILP) model. Furthermore, an accelerated Benders decomposition (BD) enhanced by valid inequalities is designed to solve the resulting MILP model, which significantly improves the solution efficiency compared with the classical BD algorithm and CPLEX solver. Finally, a practical case in Chongqing, China, is addressed to illustrate the effectiveness of our DRCC model and the accelerated BD solution method.

Anomalous scaling of heterogeneous elastic lines: A picture from sample-to-sample fluctuations.

We study a discrete model of an heterogeneous elastic line with internal disorder, submitted to thermal fluctuations. The monomers are connected through random springs with independent and identically distributed elastic constants drawn from p(k)∼k^{μ-1} for k→0. When μ>1, the scaling of the standard Edwards-Wilkinson model is recovered. When μ<1, the elastic line exhibits an anomalous scaling of the type observed in many growth models and experiments. Here we derive and use the exact probability distribution of the line shape at equilibrium, as well as the spectral properties of the matrix containing the random couplings, to fully characterize the sample to sample fluctuations. Our results lead to scaling predictions that partially disagree with previous works, but that are corroborated by numerical simulations. We also provide an interpretation of the anomalous scaling in terms of the abrupt jumps in the line's shape that dominate the average value of the observable.

Wind farm layout optimization under uncertainty

Wind power is a major source of green energy production. However, the energy generation of wind power is highly affected by uncertainty. Here, we consider the problem of designing the cable network that interconnects the turbines to the substation in wind farms, aiming to minimize both the infrastructure cost and the cost of the energy losses during the wind farm’s lifetime. Nonetheless, the energy losses depend on wind direction and speed, which are rarely known with certainty in real situations. Hence, the design of the network should consider these losses as uncertain parameters. We assume that the exact probability distribution of these parameters is unknown but belongs to an ambiguity set and propose a distributionally robust two-stage mixed integer model. The model is solved using a decomposition algorithm. Three enhancements are proposed given the computational difficulty in solving real problem instances. Computational results are reported based on real data.

Characteristic Function of the Tsallis q-Gaussian and Its Applications in Measurement and Metrology

The Tsallis q-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets and image processing. It belongs to the q-distribution family, which is characterized by a non-additive entropy. Due to their versatility and practicality, q-Gaussians are a natural choice for modeling input quantities in measurement models. This paper presents the characteristic function of a linear combination of independent q-Gaussian random variables and proposes a numerical method for its inversion. The proposed technique makes it possible to determine the exact probability distribution of the output quantity in linear measurement models, with the input quantities modeled as independent q-Gaussian random variables. It provides an alternative computational procedure to the Monte Carlo method for uncertainty analysis through the propagation of distributions.

Exact Probability Distribution for the ROC Area under Curve.

The Receiver Operating Characteristic (ROC) is a de facto standard for determining the accuracy of in vitro diagnostic (IVD) medical devices, and thus the exactness in its probability distribution is crucial toward accurate statistical inference. We show the exact probability distribution of the ROC AUC-value, hence exact critical values and p-values are readily obtained. Because the exact calculations are computationally intense, we demonstrate a method of geometric interpolation, which is exact in a special case but generally an approximation, vastly increasing computational speeds. The method is illustrated through open access data, demonstrating superiority of 26 composite biomarkers relative to a predicate device. Especially under correction for testing of multiple hypotheses, traditional asymptotic approximations are encumbered by considerable imprecision, adversely affecting IVD device development. The ability to obtain exact p-values will allow more efficient IVD device development.

Reliability analysis of laminated composite plates based on progressive failure method and universal gray system approach

The mechanical responses of composite structures are dependent on the parameter uncertainties, such as fiber orientations, ply thicknesses, and applied loads. In view of these problems, a universal gray system approach is applied to accurately reflect the impact of parameter uncertainties on the failure load and the initial damage of composite laminates. Deterministic failure criteria of composite laminates is modified based on nonlinear behavior and maximum stress criteria. The user subroutines USDFLD in Abaqus are used to modify the linear elastic material behavior. The maximum range of failure indexes are obtained and the lower and upper bounds of the uncertain parameters are drawn based on progressive failure method and universal gray system approach. The proposed method is then demonstrated by considering an open-hole composite laminate subjected to different types of load conditions. The results indicate that the framework can find the ranges of exact failure probability distributions in an efficient and accurate manner.

First-contact-breaking distributions in strained disordered crystals.

We derive exact probability distributions for the strain (ε) at which the first stress drop event occurs in uniformly strained disordered crystals, with quenched disorder introduced through polydispersity in particle sizes. We characterize these first stress drop events numerically as well as theoretically and identify them with the first-contact-breaking event in the system. Our theoretical results are corroborated with numerical simulations of quasistatic volumetric strain applied to disordered near-crystalline configurations of athermal soft particles. We develop a general technique to determine the distribution of strains at which the first stress drop events occur, through an exact mapping between the cumulative distribution of first-contact-breaking events and the volume of a convex polytope whose dimension is determined by the number of defects N_{d} in the system. An exact numerical computation of this polytope volume for systems with small numbers of defects displays a remarkable match with the distribution of strains generated through direct numerical simulations. Finally, we derive the distribution of strains at which the first stress drop occurs, assuming that individual contact-breaking events are uncorrelated, which accurately reproduces distributions obtained from direct numerical simulations.

Distributionally Robust Optimization Model for a Minimum Cost Consensus with Asymmetric Adjustment Costs Based on the Wasserstein Metric

When solving the problem of the minimum cost consensus with asymmetric adjustment costs, decision makers need to face various uncertain situations (such as individual opinions and unit adjustment costs for opinion modifications in the up and down directions). However, in the existing methods for dealing with this problem, robust optimization will lead to overly conservative results, and stochastic programming needs to know the exact probability distribution. In order to overcome these shortcomings, it is essential to develop a novelty consensus model. Thus, we propose three new minimum-cost consensus models with a distributionally robust method. Uncertain parameters (individual opinions, unit adjustment costs for opinion modifications in the up and down directions, the degree of tolerance, and the range of thresholds) were investigated by modeling the three new models, respectively. In the distributionally robust method, the construction of an ambiguous set is very important. Based on the historical data information, we chose the Wasserstein ambiguous set with the Wasserstein distance in this study. Then, three new models were transformed into a second-order cone programming problem to simplify the calculations. Further, a case from the EU Trade and Animal Welfare (TAW) program policy consultation was used to verify the practicability of the proposed models. Through comparison and sensitivity analysis, the numerical results showed that the three new models fit the complex decision environment better.

A new statistical test for distinguishing 2-partitions of a finite set

This paper considers a family of so-called 2-partitions of some finite set. Each of them divides the set under study into two disjoint parts. Under the assumption that two such partitions are chosen randomly, the exact probability distribution of the special cluster metric on this family is found. On this basis, a new statistical test for checking the significance of differences between 2-partitions is proposed. In addition, the distribution of the values of this metric is found for the case when both partitions are of the ledge type in ordering the set of objects in ascending order of values of some numerical indicator. This means that one of the parts of each partition, which in some sense is the main one, is a segment. The boundaries of such a segment are called normative. By comparing various estimates of the normative boundaries based on sample data, it is introduced the concept of indicative certainty of the numerical indicator. It can be regarded as the degree of confidence in this indicator as a basis for decision whether an object belongs to the main set of the ledge partition. Some application of the results to medical data processing is considered.

Distributionally robust optimization of non-fossil fuels processing network under uncertainty

Under the ever more prominent climate issues concerning greenhouse gas emission reduction, governments have sped up the pace of “Decarbonization while meeting demand”, and traditional energy companies have also begun to advance strategy layout and transformation, of which gasoline substitute, diesel substitute, hydrogen and ammonia are the highly potential candidates in the future. This paper investigates the design of non-fossil fuels processing network with renewable feedstocks under various demand uncertainties in the framework of two-stage distributionally robust optimization (TSDRO) model. To handle the difficulties of estimating the exact probability distribution of the considered uncertainties, 1-norm Wasserstein metric is applied to construct the ambiguity set, and the resulting tractability issue is recast in terms of sufficiently expensive recourse and strong duality theorem. Compared with two-stage stochastic programming and two-stage robust optimization, optimal design decisions of the Wasserstein TSDRO of base case and net-zero emission case exhibit the balance between the optimality and robustness, which lay a foundation to ensure an orderly transition towards sustainable and resilient network systems under various demand uncertainties, as well as a pathway to alleviate the abandon or redundancy power phenomena existing in renewable power through employing non-fossil fuels as the storage medium.

Implementation of the characteristic functions approach to measurement uncertainty evaluation

Probability distributions suitable for modelling measurements and determining their uncertainties are usually based on a standard approximation approach as described in GUM, i.e. the GUM uncertainty framework (GUF), using the law of uncertainty propagation (also known as the delta method) or a more accurate method based on the law of probability propagation calculated using the Monte Carlo method (MCM). As an alternative to GUF and MCM, we present a characteristic function approach (CFA), which is suitable for determining measurement uncertainties by using the exact probability distribution of a measured quantity in linear measurement models by inverting the associated characteristic function (CF), which is defined as a Fourier transform of the probability density function (PDF). In this paper, we present the current state of the MATLAB implementation of the characteristic function approach (the toolbox CharFunTool) and illustrate the use and applicability of the CFA for determining the distribution and uncertainty evaluation with a simple example. The proposed approach is compared with GUM, MCM and the kurtosis uncertainty method (KUM).

A distributed dynamic load identification method based on the hierarchical-clustering-oriented radial basis function framework using acceleration signals under convex-fuzzy hybrid uncertainties

Load identification is a hotly studied topic due to the widespread recognition of its importance in structural design and health monitoring. This paper explores an effective identification method for the distributed dynamic load (DDL) varying in both time progress and space dimensions using limited acceleration responses. As for the reconstruction of spatial distribution, the radial basis function (RBF) interpolation strategy, whose hyper-parameters are determined by a hierarchical clustering algorithm, is applied to approximate the DDL and then transform the continuous function into finite dimensions. In the time domain, based on the inverse Newmark iteration, the RBF coefficients at each discrete instant are obtained by the least square solution of the modal forces. Considering the multi-source uncertainties lacking exact probability distributions, a multi-dimensional interval model is developed to quantify convex parameters and fuzzy parameters uniformly. Further, a Chebyshev-interval surrogate model with different orders is constructed to obtain the fuzzy-interval boundaries of DDLs. Eventually, three examples are discussed to demonstrate the feasibility of the developed DDL identification approach considering hybrid uncertainties. The results suggest its promising applications in different structures and loading conditions.

Extreme value statistics and arcsine laws for heterogeneous diffusion processes.

Heterogeneous diffusion with a spatially changing diffusion coefficient arises in many experimental systems such as protein dynamics in the cell cytoplasm, mobility of cajal bodies, and confined hard-sphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient D(x) varies in a power-law way, i.e., D(x)∼|x|^{-α} with the exponent α>-1. This model is known to exhibit anomalous scaling of the mean-squared displacement (MSD) of the form ∼t^{2/2+α} and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all α, the exact probability distributions of the maximum spatial displacement M(t) and arg-maximum t_{m}(t) (i.e., the time at which this maximum is reached) till duration t. In the second part of our paper, we analyze the statistical properties of the residence time t_{r}(t) and the last-passage time t_{ℓ}(t) and compute their distributions exactly for all values of α. Our study unravels that the heterogeneous version (α≠0) displays many rich and contrasting features compared to that of the standard Brownian motion (BM). For example, while for BM (α=0), the distributions of t_{m}(t),t_{r}(t), and t_{ℓ}(t) are all identical (á la "arcsine laws" due to Lévy), they turn out to be significantly different for nonzero α. Another interesting property of t_{r}(t) is the existence of a critical α (which we denote by α_{c}=-0.3182) such that the distribution exhibits a local maximum at t_{r}=t/2 for α<α_{c} whereas it has minima at t_{r}=t/2 for α≥α_{c}. The underlying reasoning for this difference hints at the very contrasting natures of the process for α≥α_{c} and α<α_{c} which we thoroughly examine in our paper. All our analytical results are backed by extensive numerical simulations.

A practical, effective calculation of gamma difference distributions with open data science tools

At present, there is still no officially accepted and extensively verified implementation of computing the gamma difference distribution allowing unequal shape parameters. We explore four computational ways of the gamma difference distribution with different shape parameters resulting from time series kriging, a forecasting approach based on the best linear unbiased prediction and linear mixed models. The results of our numerical study, with emphasis on using open data science tools, demonstrate that our computational algorithm implemented in high-performance Python(with Numba) is exponentially fast, highly accurate and very reliable. It combines numerical inversion of the characteristic function and the trapezoidal rule with the double exponential oscillatory transformation (DE quadrature). At the double 53-bit precision, our open tool outperformed the speed of the analytical computation based on Tricomi's function in CAS software (commercial Mathematica, open SageMath) by 1.5–2 orders. At the default precision of scientific numerical computational tools, it exceeded open SciPy, NumPy and commercial MATLAB 5–10 times. The potential future application of our tool for a mixture of characteristic functions could open new possibilities for fast data analysis based on exact probability distributions in areas like multidimensional statistics, measurement uncertainty analysis in metrology, as well as in financial mathematics, and risk analysis.

The containment profile of hyper-recursive trees

AbstractWe investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned with counts of the number of vertices of a particular containment level.For the local containment profile, we obtain the exact mean, variance, and probability distribution in terms of standard combinatorial quantities such as generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an asymptotically normal distribution, intermediate vertices have a Poisson distribution, and late vertices have a degenerate distribution.As for the global containment profile, we establish an asymptotically normal distribution for the number of vertices at the smallest containment level as well as their covariances with the number of vertices at the second smallest containment level and the variances of these numbers. The orders in the variance–covariance matrix establish concentration laws.

Ultralarge-scale approximations and galaxy clustering: Debiasing constraints on cosmological parameters

ABSTRACT Upcoming galaxy surveys will allow us to probe the growth of the cosmic large-scale structure with improved sensitivity compared to current missions, and will also map larger areas of the sky. This means that in addition to the increased precision in observations, future surveys will also access the ultralarge-scale regime, where commonly neglected effects such as lensing, redshift-space distortions, and relativistic corrections become important for calculating correlation functions of galaxy positions. At the same time, several approximations usually made in these calculations such as the Limber approximation break down at those scales. The need to abandon these approximations and simplifying assumptions at large scales creates severe issues for parameter estimation methods. On the one hand, exact calculations of theoretical angular power spectra become computationally expensive, and the need to perform them thousands of times to reconstruct posterior probability distributions for cosmological parameters makes the approach unfeasible. On the other hand, neglecting relativistic effects and relying on approximations may significantly bias the estimates of cosmological parameters. In this work, we quantify this bias and investigate how an incomplete modelling of various effects on ultralarge scales could lead to false detections of new physics beyond the standard ΛCDM model. Furthermore, we propose a simple debiasing method that allows us to recover true cosmologies without running the full parameter estimation pipeline with exact theoretical calculations. This method can therefore provide a fast way of obtaining accurate values of cosmological parameters and estimates of exact posterior probability distributions from ultralarge-scale observations.

A Distributionally Robust Chance-Constrained Unit Commitment with N-1 Security and Renewable Generation

With the increasing penetration of renewable energy generation, one of the major challenges is the problem of how to express the stochastic process of wind power and photovoltaic output as the exact probability density and distribution, in order to improve the security and accuracy of unit commitment results, a distributed robust security-constrained optimization model based on moment uncertainty is proposed, in which the uncertainty of wind and photovoltaic power is captured by two uncertain sets of first- and second-order moments, respectively. The two sets contain the probability distribution of the forecast error of the wind and photovoltaic power, and in the model, the energy storage is considered. In order to solve the model effectively, firstly, based on the traditional chance-constrained second-order cone transformation, according to the first- and second-order moments polyhedron expression of the distribution set, a cutting plane method is proposed to solve the distributed robust chance constraints. Secondly, the modified IEEE-RTS 24 bus system is selected to establish a simulation example, an improved generalized Benders decomposition algorithm is developed to solve the model to optimality. The results show that the unit commitment results with different emphasis on economy and security can be obtained by setting different conservative coefficients and confidence levels and, then, provide a reasonable decision-making basis for dispatching operation.

Scenario-robust pre-disaster planning for multiple relief items.

The increasing vulnerability of the population from frequent disasters requires quick and effective responses to provide the required relief through effective humanitarian supply chain distribution networks. We develop scenario-robust optimization models for stocking multiple disaster relief items at strategic facility locations for disaster response. Our models improve the robustness of solutions by easing the difficult, and usually impossible, task of providing exact probability distributions for uncertain parameters in a stochastic programming model. Our models allow decision makers to specify uncertainty parameters (i.e., point and probability estimates) based on their degrees of knowledge, using distribution-free uncertainty sets in the form of ranges. The applicability of our generalized approach is illustrated via a case study of hurricane preparedness in the Southeastern United States. In addition, we conduct simulation studies to show the effectiveness of our approach when conditions deviate from the model assumptions.

Engineering analysis with probability boxes: A review on computational methods

The consideration of imprecise probability in engineering analysis to account for missing, vague or incomplete data in the description of model uncertainties is a fast-growing field of research. Probability-boxes (p-boxes) are of particular interest in an engineering context, since they offer a mathematically straightforward description of imprecise probabilities, as well as allow for an intuitive visualisation. In essence, p-boxes are defined via lower and upper bounds on the cumulative distribution function of a random variable whose exact probability distribution is unknown. However, the propagation of p-boxes on model inputs towards bounds on probabilistic measures describing the uncertainty on the model responses is numerically still very demanding, and hence is subject of intensive research. In order to provide an overview on the available methods, this paper gives a state-of-the art review for the modelling and propagation of p-boxes with a special focus on structural reliability analysis.

Reliability Analysis of Mechanical Systems Based on the First Four Moments of Input Parameters

Abstract The performance of a mechanical or structural system can be improved through a proper selection of its design parameters such as the geometric dimensions, external actions (loads), and material characteristics. The computation of the reliability of a system, in general, requires a knowledge of the probability distributions of the parameters of the system. It is known that for most practical systems, the exact probability distributions of the parameters are not known. However, the first few moments of the parameters of the system may be readily available in many cases from experimental data. The determination of the reliability and the sensitivity of reliability to variations or fluctuations in the parameters of the system starts with the establishment of a suitable limit state equation. This work presents an approximate reliability analysis for mechanical and structural systems using the fourth-order moment function for approximating the first four moments of the limit state function. By combining the fourth-order moment function with the probabilistic perturbation method, numerical methods are developed for finding the reliability and sensitivity of reliability of the system. An automobile brake and a power screw are considered for demonstrating the methodology and effectiveness of the proposed computational approach. The results of the automobile brake are compared with those given by the Monte Carlo method.