Multidimensional Probability Distribution Research Articles

Design Wave Height Estimation under the Influence of Typhoon Frequency, Distance, and Intensity

The extreme sea conditions caused by typhoons pose a threat to the design safety of marine and coastal engineering structures. In the past, design wave height calculation models that only considered the frequency of typhoons ignored the influence of other hazard factors of typhoons, resulting in lower design standards. In this paper, typhoon frequency, intensity, and distance are selected, and dimensional influences of different factors are eliminated through standardization processing. Based on the correlation between different hazard factors, we have obtained a multi-dimensional discrete joint probability distribution of typhoon hazard factors and constructed a new design wave height that considers the comprehensive effects of typhoon frequency, intensity, and distance. Our results show that the design wave height values of the 50-year, 100-year, and 200-year events are 12.59%, 8.10%, and 3.14% higher than the Gumbel distribution, which is more in line with the distribution of the wave height of waves under severe typhoon conditions. The new model can fully reflect the impact of typhoons on wave height, which can provide a reference for the design safety of marine engineering in the South China Sea.

Assessment of multi-dimensional joint probability distribution for uncertain mechanical strength parameters under small sample test data

The multi-dimensional joint probability distribution of uncertain mechanical strength parameters is crucial for reliability analysis in geotechnical engineering. However, due to the limited number of test data samples, the analysis faces significant challenges. This paper presents a modeling method for joint probability distribution based on multidimensional Gaussian copula, which is suitable for small sample conditions. Firstly, the copula theory is introduced, and a construction method for multidimensional Gaussian copula is proposed. Secondly, measurement data for 192 sets of coal seam roof strength parameters were collected from 24 coal mines in China. For the four-dimensional Gaussian copula, analyses of goodness of fit and simulation error were performed using the Pearson method, Kendall method, and Spearman method, respectively. Finally, the impact of different copula functions on the fitting capability of the original data is discussed. The results indicate that the established multi-dimensional Gaussian copula joint distribution model possesses an advantage in characterizing the uncertainty of relevant parameters. This model also offers an effective approach for studying the uncertainty of coal mine strength parameters. Different construction methods for multidimensional Gaussian copulas yield varying simulation outcomes. The Pearson method exhibits the most favorable fitting effect for constructing correlation parameters. The two rank correlation methods have the advantages of broad applicability, excellent fitting effect, and minimal variation in simulation error. For the bivariate copula fitting ability of uncertain parameters, the Gaussian Copula exhibits the best fitting ability under both positive and negative correlation scenarios. When the correlation is weak, Plackett copula and Frank copula can also serve as alternative candidate copula functions. These findings offer a significant theoretical foundation for conducting reliability analysis of a coal seam roof under limited test data conditions.

Computing the decomposable entropy of belief-function graphical models

In 2018, Jiroušek and Shenoy proposed a definition of entropy for Dempster-Shafer (D-S) belief functions called decomposable entropy (d-entropy). This paper provides an algorithm for computing the d-entropy of directed graphical D-S belief function models. We illustrate the algorithm using Almond's Captain's Problem example. For belief function undirected graphical models, assuming that the set of belief functions in the model is non-informative, the belief functions are distinct. We illustrate this using Haenni-Lehmann's Communication Network problem. As the joint belief function for this model is quasi-consonant, it follows from a property of d-entropy that the d-entropy of this model is zero, and no algorithm is required. For a class of undirected graphical models, we provide an algorithm for computing the d-entropy of such models. Finally, the d-entropy coincides with Shannon's entropy for the probability mass function of a single random variable and for a large multi-dimensional probability distribution expressed as a directed acyclic graph model called a Bayesian network. We illustrate this using Lauritzen-Spiegelhalter's Chest Clinic example represented as a belief-function directed graphical model.

A 3-queue polling system with join the shortest-serve the longest policy

In 1987, J.W. Cohen analyzed the so-called Serve the Longest Queue (SLQ) queueing system, where a single server attends two non-symmetric M/G/1-type queues, exercising a non-preemptive priority switching policy. Cohen further analyzed in 1998 a non-symmetric 2-queue Markovian system, where newly arriving customers follow the Join the Shortest Queue (JSQ) discipline. The current paper generalizes and extends Cohen’s works by studying a combined JSQ–SLQ model, and by broadening the scope of analysis to a non-symmetric 3-queue system, where arriving customers follow the JSQ strategy and a single server exercises the preemptive priority SLQ discipline. The system states’ multi-dimensional probability distribution function is derived while applying a non-conventional representation of the underlying process’s state-space. The analysis combines both Probability Generating Functions and Matrix Geometric methodologies. It is shown that the joint JSQ–SLQ operating policy achieves extremely well the goal of balancing between queue sizes. This is emphasized when calculating the Gini Index associated with the differences between mean queue sizes: the value of the coefficient is close to zero. Extensive numerical results are presented.

Motion Planning Under Uncertainty with Complex Agents and Environments via Hybrid Search

As autonomous systems and robots are applied to more real world situations, they must reason about uncertainty when planning actions. Mission success oftentimes cannot be guaranteed and the planner must reason about the probability of failure. Unfortunately, computing a trajectory that satisfies mission goals while constraining the probability of failure is difficult because of the need to reason about complex, multidimensional probability distributions. Recent methods have seen success using chance-constrained, model-based planning. However, the majority of these methods can only handle simple environment and agent models. We argue that there are two main drawbacks of current approaches to goal-directed motion planning under uncertainty. First, current methods suffer from an inability to deal with expressive environment models such as 3D non-convex obstacles. Second, most planners rely on considerable simplifications when computing trajectory risk including approximating the agent’s dynamics, geometry, and uncertainty. In this article, we apply hybrid search to the risk-bound, goal-directed planning problem. The hybrid search consists of a region planner and a trajectory planner. The region planner makes discrete choices by reasoning about geometric regions that the autonomous agent should visit in order to accomplish its mission. In formulating the region planner, we propose landmark regions that help produce obstacle-free paths. The region planner passes paths through the environment to a trajectory planner; the task of the trajectory planner is to optimize trajectories that respect the agent’s dynamics and the user’s desired risk of mission failure. We discuss three approaches to modeling trajectory risk: a CDF-based approach, a sampling-based collocation method, and an algorithm named Shooting Method Monte Carlo. These models allow computation of trajectory risk with more complex environments, agent dynamics, geometries, and models of uncertainty than past approaches. A variety of 2D and 3D test cases are presented including a linear case, a Dubins car model, and an underwater autonomous vehicle. The method is shown to outperform other methods in terms of speed and utility of the solution. Additionally, the models of trajectory risk are shown to better approximate risk in simulation.

On-orbit placement optimization of proximity defense shield for space station

Space station is under the increasing threaten caused by large amount of space debris on Earth orbit. Besides traditional active and passive defense measures, it is urgent to study active close-in defense strategy for protecting space station from space debris. In this paper, an active close-in defense mode is proposed for a space station based on a flexible and deployable proximity defense shield, while a hybrid procedure is formulated to optimize the on-orbit placement of the shield. First, an active close-in defense scheme for a space station is proposed. Second, the proximity defense process is modeled including orbit propagation and impact momentum transfer. Third, the orbital spatio-temporal evolution of the space debris and the proximity defense shield during the defense process are transformed into a multi-dimensional space probability distribution using the Poincaré section and the Gaussian mixture model. Then, based on probability optimization, the on-orbit placement of the proximity defense shield is optimized for protecting a space station from tiny space debris. Finally, comparison simulations demonstrate the effectiveness and feasibility of the proposed technique.

Research on Optimization Algorithm for Resource Allocation of Heterogeneous Car Networking Engineering Cloud System Based on Big Data

A big data-based heterogeneous Internet of Vehicles engineering cloud system resource allocation optimization algorithm is proposed for the sake of meeting the needs of Internet of Vehicles applications and improving the rationality and efficiency of cloud system resource allocation. Based on taking the minimum cloud system delay as the resource allocation target, a multislot cloud system delay optimization model and its indicative function are constructed, the probability distribution function is derived according to the obtained multidimensional probability distribution function set, and the available channels of the vehicle in different time periods are determined. In this way, the matching degree between the vehicle and the channel is solved, the delay optimization model is turned into a convex optimization problem with independent variables, and the resource allocation algorithm for different task offload destinations is optimized. Meanwhile, by building a heterogeneous vehicle network simulation system, the performance of the algorithm is evaluated from the perspectives of resource rental cost, weighted resource utilization, and bit loss rate. As can be learned from the simulation results, the proposed algorithm can effectively reduce the cost of resource rental, and at the same time, the advantages of resource utilization and bit loss rate are relatively significant, so it has certain effectiveness and practicability.

Classification of Unconstrained Respiratory States Utilising Multidimensional Probability Distribution Based on Respiratory Frequency Information at Each Time Step

Classification of Unconstrained Respiratory States Utilising Multidimensional Probability Distribution Based on Respiratory Frequency Information at Each Time Step

Modeling and Reconstruction of Data with Applications as Matrix of Pixels

Probabilistic Features Combination method (PFC), which is proposed by the author, is the approach of multidimensional data modeling, extrapolation and interpolation using the set of high-dimensional feature vectors. This method is a hybridization of numerical methods and probabilistic methods. Identification of faces or fingerprints need modeling and each model of the pattern is built by a choice of multi-dimensional probability distribution function and feature combination. PFC modeling via nodes combination and parameter γ as N-dimensional probability distribution function enables data interpolation for feature vectors. Multi-dimensional data is modeled and interpolated via nodes combination and different functions as probability distribution functions for each feature treated as random variable.

Analysis of the Transformation of Random Signals and Noise Using Poly-Gaussian Models

Analysis performed transformation of random signals and noise in linear and nonlinear systems based on the use of poly-Gaussian models and multidimensional PDF of the output paths of information-measuring and radio systems. The classification of elements of these systems, as well as expressions describing the input action and output response of the system are given. It is shown that the analysis of information-measuring and systems can be carried out using poly-Gaussian models. The analysis is carried out with a series connection of a linear system and a nonlinear element, a series connection of a nonlinear element and a linear system, as well as with a parallel connection of the named links. The output response in all cases will be a mixture of a poly-Gaussian distribution with a number of components. An example of the analysis of signal transmission through an intermediate frequency amplifier and a linear detector against a background of non-Gaussian noise is given. The resulting probability density distribution of the sum of the signal and non-Gaussian noise at the output of the detector will be poly-Rice. The multidimensional probability distribution density of the output processes of the nonlinear signal envelope detector is also obtained. The results of modeling the found distribution densities are presented. It is shown that the use of the poly-Gaussian representation of signals and noise, as well as the impulse response of the system, makes it possible to effectively analyze inertial systems in the time domain.

Coordinate-wise transformation of probability distributions to achieve a Stein-type identity

It is shown that for any given multi-dimensional probability distribution with regularity conditions, there exists a unique coordinate-wise transformation such that the transformed distribution satisfies a Stein-type identity. A sufficient condition for the existence is referred to as copositivity of distributions. The proof is based on an energy minimization problem over a totally geodesic subset of the Wasserstein space. The result is considered as an alternative to Sklar’s theorem regarding copulas, and is also interpreted as a generalization of a diagonal scaling theorem. The Stein-type identity is applied to a rating problem of multivariate data. A numerical procedure for piece-wise uniform densities is provided. Some open problems are also discussed.

Probabilistic stability analysis of functionally graded graphene reinforced porous beams

This paper presents the first attempt to study the probabilistic stability characteristics of functionally graded (FG) graphene platelets (GPLs) reinforced beams by taking into account the multidimensional probability distributions, such as stochastic porosity and GPL distribution patterns as well as random material properties. For this purpose, a non-inclusive Chebyshev metamodel (CMM), which is implemented on deterministic analysis using discrete singular convolution (DSC) method with excellent computational efficiency and accuracy, is proposed and used to obtain both deterministic and probabilistic results including probability density functions (PDFs), cumulative density functions (CDFs), means and standard deviations of the critical buckling load. The present analysis is rigorously validated through direct comparisons against the results obtained by a direct quasi-Monte Carlo simulation (QMCS) method and those available in open literature. The influences of material properties, porosity distribution, GPL dispersion pattern and boundary condition on probabilistic buckling behaviour of the FG-GPL beam are comprehensively investigated. The global sensitivity analysis is also conducted. The results suggest that the critical buckling load of the FG-GPL beam is most sensitive to porosity distribution, followed by porosity coefficient and GPL weight fraction.

Approximation of Probabilistic Constraints in Stochastic Programming Problems with a Probability Measure Kernel

We consider a linear stochastic programming problem with a deterministic objective function and individual probabilistic constraints. Each probabilistic constraint is a lower bound on the probability function equal to the probability of the fulfillment of a certain linear inequality. We propose to first represent probabilistic constraints in the form of equivalent inequalities for the quantile functions. After that, each quantile function is approximated using the confidence method. The main analytic tool is based on polyhedral approximation of the p-kernel for the multidimensional probability distribution. For the case when probability functions are defined by linear inequalities, constraints on quantile functions are with arbitrary accuracy approximated by systems of deterministic linear inequalities. As a result, the original problem is approximated by a linear programming problem.

Rock burst prediction probability model based on case analysis

Most of grading results obtained by the traditional rock burst prediction model are qualitatively expressed, the corresponding relationship between the rock burst prediction grade and occurrence probability has not been studied, and it is difficult to obtain the quantitative occurrence possibility of rock burst. For the above problems, a novel tunnel rock burst prediction probability model is proposed based on actual rock burst cases analysis. Firstly, six quantitatively characteristic parameters, namely the maximum tangential stress, the uniaxial compressive strength, the uniaxial tensile strength, the stress coefficient, the rock brittleness coefficient and the elastic energy index were extracted from rock burst cases, and the probability distribution function and correlation of those parameters were determined. Secondly, the multi-dimensional joint probability distribution function of six characteristic parameters was constructed under the framework of Copula theory, and then the Least Squares Support Vector Machine (LSSVM) which was optimized by particle swarm optimization algorithm served as the intelligent response surface model to reflect the nonlinear mapping relationship between six parameters and the tunnel rock burst prediction level. Subsequently, the Copula-LSSVM rock burst prediction probability model was established, and the Weibull distribution function of rock burst prediction grade was obtained with the application of the Monte Carlo simulation method. Lastly, six rock burst cases of the Jinping II Hydropower Station are used to demonstrate the effectiveness of the proposed method, and analysis was done on the influence of parameter uncertainty and model uncertainty on the prediction results. It is evidenced that the prediction probability obtained can be used for rock burst quantitative risk assessment of hard rock tunnels.

Generation of Pairwise Potentials Using Multidimensional Data Mining.

The rapid development of molecular structural databases provides the chemistry community access to an enormous array of experimental data that can be used to build and validate computational models. Using radial distribution functions collected from experimentally available X-ray and NMR structures, a number of so-called statistical potentials have been developed over the years using the structural data mining strategy. These potentials have been developed within the context of the two-particle Kirkwood equation by extending its original use for isotropic monatomic systems to anisotropic biomolecular systems. However, the accuracy and the unclear physical meaning of statistical potentials have long formed the central arguments against such methods. In this work, we present a new approach to generate molecular energy functions using structural data mining. Instead of employing the Kirkwood equation and introducing the "reference state" approximation, we model the multidimensional probability distributions of the molecular system using graphical models and generate the target pairwise Boltzmann probabilities using the Bayesian field theory. Different from the current statistical potentials that mimic the "knowledge-based" PMF based on the 2-particle Kirkwood equation, the graphical-model-based structure-derived potential developed in this study focuses on the generation of lower-dimensional Boltzmann distributions of atoms through reduction of dimensionality. We have named this new scoring function GARF, and in this work we focus on the mathematical derivation of our novel approach followed by validation studies on its ability to predict protein-ligand interactions.

Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry

Convex optimization is a well-established research area with applications in almost all fields. Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a more general set of convex programs known as semi-definite programs and second-order cone programs. However, it has been established that these methods are excessively slow for high dimensions, i.e., they suffer from the curse of dimensionality. On the other hand, optimization algorithms on manifold have shown great ability in finding solutions to nonconvex problems in reasonable time. This paper is interested in solving a subset of convex optimization using a different approach. The main idea behind Riemannian optimization is to view the constrained optimization problem as an unconstrained one over a restricted search space. The paper introduces three manifolds to solve convex programs under particular box constraints. The manifolds, called the doubly stochastic, symmetric and the definite multinomial manifolds, generalize the simplex also known as the multinomial manifold. The proposed manifolds and algorithms are well-adapted to solving convex programs in which the variable of interest is a multidimensional probability distribution function. Theoretical analysis and simulation results testify the efficiency of the proposed method over state of the art methods. In particular, they reveal that the proposed framework outperforms conventional generic and specialized solvers, especially in high dimensions.

2D and 3D curve modeling — multidimensional data recovery

Proposed method, called Probabilistic Features Combination (PFC), is the method of multi-dimensional data modeling, extrapolation and interpolation using the set of high-dimensional feature vectors. This method is a hybridization of numerical methods and probabilistic methods. Identification of faces or fingerprints need modeling and each model of the pattern is built by a choice of multi-dimensional probability distribution function and feature combination. PFC modeling via nodes combination and parameter γ as N-dimensional probability distribution function enables data parameterization and interpolation for feature vectors. Multi-dimensional data is modeled and interpolated via nodes combination and different functions as probability distribution functions for each feature treated as random variable: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

Zbieżność stochastyczna ciągów wektorów losowych

The paper presents a multidimensional generalisation (known for one-dimensional random variables) of two theorems regarding stochastic convergence – that is, convergence by probability. The generalised theorems are Markov’s and Chinchyn’s weak laws of great numbers. Both lead to the theory that, with the appropriate assumptions, a sequence of arithmetic averages of the random vectors converges their expected values to the arithmetic average. The proof for this thesis uses „whole moments of the multidimensional probability distribution”, which the author has proposed elsewhere. Their basis is a definition of the power of a vector in a space with a scalar product.

Covariance approximation of nonlinear regression

A nonlinear regression model on the basis of the covariance approximation of a multidimensional probability distribution is constructed. The model is represented by an expansion in the basis functions in the form of partial derivatives of the logarithm of the joint factor probability distribution. The weight coefficients of the expansion are the covariances of the resulting and explanatory variables. On particular examples, the efficiency of the Bayesian approximation of the proposed regression model in which the factor distribution is described by a finite mixture of ellipsoidally symmetric densities is demonstrated.

Obtaining T1-T2 distribution functions from 1-dimensional T1 and T2 measurements: The pseudo 2-D relaxation model

We present the pseudo 2-D relaxation model (P2DRM), a method to estimate multidimensional probability distributions of material parameters from independent 1-D measurements. We illustrate its use on 1-D T1 and T2 relaxation measurements of saturated rock and evaluate it on both simulated and experimental T1-T2 correlation measurement data sets. Results were in excellent agreement with the actual, known 2-D distribution in the case of the simulated data set. In both the simulated and experimental case, the functional relationships between T1 and T2 were in good agreement with the T1-T2 correlation maps from the 2-D inverse Laplace transform of the full 2-D data sets. When a 1-D CPMG experiment is combined with a rapid T1 measurement, the P2DRM provides a double-shot method for obtaining a T1-T2 relationship, with significantly decreased experimental time in comparison to the full T1-T2 correlation measurement.