Random Quantities Research Articles

A Quotient Space Formulation for Generative Statistical Analysis of Graphical Data

Complex analyses involving multiple, dependent random quantities often lead to graphical models—a set of nodes denoting variables of interest, and corresponding edges denoting statistical interactions between nodes. To develop statistical analyses for graphical data, especially towards generative modeling, one needs mathematical representations and metrics for matching and comparing graphs, and subsequent tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis, statistical testing, and modeling. We demonstrate the efficacy of this framework using datasets taken from several problem areas, including letters, biochemical structures, and social networks.

A Novel Stochastic Approach for Static Damage Identification of Beam Structures Using Homotopy Analysis Algorithm.

This paper proposes a new damage identification approach for beam structures with stochastic parameters based on uncertain static measurement data. This new approach considers not only the static measurement errors, but also the modelling error of the initial beam structures as random quantities, and can also address static damage identification problems with relatively large uncertainties. First, the stochastic damage identification equations with respect to the damage indexes were established. On this basis, a new homotopy analysis algorithm was used to solve the stochastic damage identification equations. During the process of solution, a static condensation technique and a L1 regularization method were employed to address the limited measurement data and ill-posed problems, respectively. Furthermore, the definition of damage probability index is presented to evaluate the possibility of existing damages. The results of two numerical examples show that the accuracy and efficiency of the proposed damage identification approach are good. In comparison to the first-order perturbation method, the proposed method can ensure better accuracy in damage identification with relatively large measurement errors and modelling error. Finally, according to the static tests of a simply supported concrete beam, the proposed method successfully identified the damage of the beam.

Dynamic Type Matching

Problem definition: We consider an intermediary’s problem of dynamically matching demand and supply of heterogeneous types in a periodic-review fashion. Specifically, there are two disjoint sets of demand and supply types, and a reward for each possible matching of a demand type and a supply type. In each period, demand and supply of various types arrive in random quantities. The platform decides on the optimal matching policy to maximize the expected total discounted rewards, given that unmatched demand and supply may incur waiting or holding costs, and will be fully or partially carried over to the next period. Academic/practical relevance: The problem is crucial to many intermediaries who manage matchings centrally in a sharing economy. Methodology: We formulate the problem as a dynamic program. We explore the structural properties of the optimal policy and propose heuristic policies. Results: We provide sufficient conditions on matching rewards such that the optimal matching policy follows a priority hierarchy among possible matching pairs. We show that those conditions are satisfied by vertically and unidirectionally horizontally differentiated types, for which quality and distance determine priority, respectively. Managerial implications: The priority property simplifies the matching decision within a period, and the trade-off reduces to a choice between matching in the current period and that in the future. Then the optimal matching policy has a match-down-to structure when considering a specific pair of demand and supply types in the priority hierarchy.

Structured Total Least Squares Method Based on Variable Projection

Objectives To solve the problem that the coefficient matrix and the observation vector have structured characteristics for many practical applications in the field of surveying and mapping, that is, the coefficient matrix and the observation vector contain fixed quantities (or even fixed columns) and random quantities, and the random quantities at different positions are linearly related. Methods Starting from the errors-in-variables (EIV) function model, the augmented matrix composed of the coefficient matrix and the observation vector is expressed as an affine function form, and the function model is reconstructed by the variable projection method. Then, a structured total least squares (STLS) estimation algorithm is derived by the Lagrange method. Results The example results show that the proposed algorithm is consistent with the existing weighted or structured total least squares estimation algorithms that can solve structured problem in the coefficient matrix and the observation vector. Compared with the weighted total least squares (WTLS) estimation algorithms, the proposed algorithm only needs to establish a positive definite weight matrix of independent random variables and reduces the number of estimates. Compared with other STLS estimation algorithms, the proposed algorithm takes into account the overall structure of the coefficient matrix and the observation vector. Conclusions This shows the effectiveness of the proposed algorithm. The STLS estimation and the WTLS estimation ensure the statistical optimality from the perspective of the function model and the stochastic model, respectively.

Regression Methods Based on Nearest Neighbors with Adaptive Distance Metrics Applied to a Polymerization Process

Empirical models based on sampled data can be useful for complex chemical engineering processes such as the free radical polymerization of methyl methacrylate achieved in a batch bulk process. In this case, the goal is to predict the monomer conversion, the numerical average molecular weight and the gravimetrical average molecular weight. This process is characterized by non-linear gel and glass effects caused by the sharp increase in the viscosity as the reaction progresses. To increase accuracy, one needs more samples in the areas with higher variation and this is achieved with adaptive sampling. An extensive comparative study is performed between three regression algorithms for this chemical process. The first two are based on the concept of a large margin, typical of support vector machines, but used for regression, in conjunction with an instance-based method. The learning of problem-specific distance metrics can be performed by means of either an evolutionary algorithm or an approximate differential approach. Having a set of prototypes with different distance metrics is especially useful when a large number of instances should be handled. Another original regression method is based on the idea of denoising autoencoders, i.e., the prototype weights and positions are set in such a way as to minimize the mean square error on a slightly corrupted version of the training set, where the instances inputs are slightly changed with a small random quantity. Several combinations of parameters and ways of splitting the data into training and testing sets are used in order to assess the performance of the algorithms in different scenarios.

Tensor Computations of Stochastic Dynamic Fields

This article attempts to delve a kind of quantities—tensor for a stochastic dynamic domain, making its principles of computations such as gradient, divergence and interaction. The notation “stochastic dynamic field (SDF)”, proposed by the author, refers to such a dynamic field that it appears or annihilates, moves, acts or interacts at random. The study aims to consider not only randomness of vector fields in a continuous dynamic system, but also discontinuity with time variable t, i.e. more randomness for t, in a discontinuous dynamic system. To do so, representation forms of tensors areproperly transformed, which apply multiple-points tensor, and introduce the quantity of compound tensor like an expression of quantum state of combination of uncertainties. The work bears out some new results: (1) discontinuous mutation, even emerge or disappear, of a point in a tensor field, analyzed using a tensor with matrix structure, would not defer tensor computations of that field, while wouldn’t be computed commonly using accustomed representation of tensor with only a single-point structure. (2) Random quantity relations between phrases, no matter smooth nearby, non-connected or not, all can be computed. (3) Some classical physical laws analyzing field properties of space-relevant factors such as Euler equation and Gauss theorem are available in new forms with a random nature.

Quantitative Representation of Aleatoric Uncertainties in Network-Like Topological Structural Systems

Abstract The complex topological characteristics of network-like structural systems, such as lattice structures, cellular metamaterials, and mass transport networks, pose a great challenge for uncertainty qualification (UQ). Various UQ approaches have been developed to quantify parametric uncertainties or high dimensional random quantities distributed in a simply connected space (e.g., line section, rectangular area, etc.), but it is still challenging to consider the topological characteristics of the spatial domain for uncertainty representation and quantification. To resolve this issue, a network distance-based Gaussian random process uncertainty representation approach is proposed. By representing the topological input space as a node-edge network, the network distance is employed to replace the Euclidean distance in characterizing the spatial correlations. Furthermore, a conditional simulation-based sampling approach is proposed for generating realizations from the uncertainty representation model. Network node values are modeled by a multivariate Gaussian distribution, and the network edge values are simulated conditionally on the node values and the known network edge values. The effectiveness of the proposed approach is demonstrated on two engineering case studies: thermal conduction analysis of 3D lattice structures with stochastic properties and characterization of the distortion patterns of additively manufactured cellular structures.

Simulation of two-dimensional distribution laws of random correlated quantities of natural-climatic factors in context of probabilistic assessment of reliability of hydraulic structures of cascades of hydroschemes

When performing calculations to assess reliability of hydraulic structures of cascades of hydroschemes on the basis of probabilistic methods, the necessity to simulate random natural-climatic phenomena producing loads and effects on hydraulic structures arises. In particular, statistical series of random quantities of such important natural-climatic phenomena are considered: annual lowest average monthly temperatures, annual maximal amplitudes of average monthly temperatures. Each of the enumerated natural-climatic phenomena is characterized by presence of close correlation connections between random quantities when passing from one hydroscheme of the cascade to another. The necessity to consider correlation connections requires construction (simulation) of joint distribution law of random quantities system. The purpose of the work is simulation of joint distribution law of system of random variables that do not satisfy the normal distributions, taking into account correlation connections between random variables when passing from one hydroscheme of the cascade to another. Methods of the theory of correlation and methods of mathematical statistics with the use of software package MathCad were used in the course of the investigation. Simulation of joint law of distribution of system of random variables that do not satisfy normal distributions, taking into account correlation connections between random variables when passing from one hydroscheme of the cascade to another, and also assessment of accuracy of results, that were performed, have shown advantages of this approach from the viewpoint of accuracy of results obtained by different procedures. The results can be used in probabilistic calculations of reliability of hydraulic structures and cascades of hydroschemes.

Features of functional dependence of random phenomena and values in social being in conditions of its unstability (the environmental position)

The analysis of the phenomenon of “random” and the principle of the relationship of random phenomena in social reality in the conditions of its instability has been conducted. On this basis, the key task was the conceptualization of the random phenomena in the scales of typology of fundamental and social interactions. It has been confirmed that the concept of “random” in terms of instability is more effectively described through ontological, phenomenological, transcendental and functional approaches and in the context of environmental position. A probability principle was applied when describing the randomness of abstract values. It has been proved that even minor aberrations at one or another point of space at different systemic levels of the material world can profoundly change the metric properties of systems, cause their instability. The results of the research confirmed that the unstable social system does not return to the state of equilibrium from which it has came out for different reasons, but continuously it moves away from it or makes unacceptably large fluctuations around it. And functional dependence between random phenomena and quantities under conditions of social instability is possible as stability of a different kind.

Simulation of Life Game Based on Cellular Automata

Vs2010 is used as the development environment, so as to realize the visual programming of Game of Life, and explore the life evolution process of cell group in different sizes and states. According to cell forms such as circulation and disappearance, it reflects the complex changes of Game of Life. Setting different initial states through the code and observing the final generated graphics, you can see that the complex and simple initial states can achieve the same result. It is also concluded that a suitable initial state can reach the final state in fewer steps, which can greatly simplify the evolution process. The entire system is completely closed and has certain limitations. Meanwhile, the evolution process of symmetrical initial state is also symmetrically distributed. I introduce random quantities into the system to make the simulation results closer to the actual situation. By setting a random initial state to make the chaotic and disorderly situation simple, the concept of “determinism and randomness” can be better expressed. In the process of change, some local structures remain fixed, and some local structures present periodic cycles. These structures interact in complex ways to understand the concept of “whole and part”. The game of life enlightens us: the simplest logical rules can produce complex and interesting activities, and a complex system may be iterated by simple rules.

Dynamics of Data-driven Ambiguity Sets for Hyperbolic Conservation Laws with Uncertain Inputs

Ambiguity sets of probability distributions are used to hedge against uncertainty about the true probabilities of uncertain inputs and random quantities of interest (QoIs). When available, these am...

Reliability assessment of power supply systems with powerful nonlinear loads

Purpose. To substantiate the expediency of applying distribution laws to model the reliability of power supply systems with powerful nonlinear loads. Methodology. Application of exponential and normal distribution laws, and Weibull's law. Findings. In a comprehensive assessment of the problem of electromagnetic compatibility and the development of multicriteria parameters of energy efficiency of power supply systems of industrial enterprises, it is most appropriate to use the reliability of the elements that are mathematically described by the Weibull distribution. At the same time, the results of modeling of typical electrical modes and variations of substitution schemes of enterprises showed the sensitivity of reliability indicators to changes in power system capacity. Originality. New theoretical foundations for research in modern power supply systems with increasing nonlinear load capacity have developed the scientific basis for analysis and synthesis of energy processes in multiphase electrical systems and propose methods of correction taking into account the complex action of sources of electromagnetic interference. And also to offer methods of estimation of their influence on reliability and efficiency of work of the basic electrotechnological equipment. Practical value. The possibility of applying the obtained results to assess the reliability of power supply systems with powerful nonlinear loads is substantiated. The necessary accuracy and reliability of the decision at the levels of random events, quantities and processes is provided. The mathematical apparatus for the analysis of the main indicators of reliability in the presence of non-sinusoidal voltage is given. Conclusions are made on the adequacy of the calculation results at different stages of electricity distribution. It is shown that the simultaneous optimization of reliability and voltage quality are components of a complex scientific problem of ensuring energy efficiency of electrical networks in the implementation of decentralized models of the energy system.

Information-Theoretic Limits for the Matrix Tensor Product

This article studies a high-dimensional inference problem involving the matrix tensor product of random matrices. This problem generalizes a number of contemporary data science problems including the spiked matrix models used in sparse principal component analysis and covariance estimation and the stochastic block model used in network analysis. The main results are single-letter formulas (i.e., analytical expressions that can be approximated numerically) for the mutual information and the minimum mean-squared error (MMSE) in the Bayes optimal setting where the distributions of all random quantities are known. We provide non-asymptotic bounds and show that our formulas describe exactly the leading order terms in the mutual information and MMSE in the high-dimensional regime where the number of rows n and number of columns d scale with d = O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> ) for some α <; 1/20. On the technical side, this article introduces some new techniques for the analysis of high-dimensional matrix-valued signals. Specific contributions include a novel extension of the adaptive interpolation method that uses order-preserving positive semidefinite interpolation paths, and a variance inequality between the overlap and the free energy that is based on continuous-time I-MMSE relations.

Non-Parametric Probability Distributions Embedded Inside of a Linear Space Provided with a Quadratic Metric

There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributions.

On the use of the maximum entropy method for reliability evaluation involving stochastic process modeling

From a probabilistic perspective, a stochastic process (or random field) is completely specified by its joint probability distribution, which can be further decomposed into marginal distributions and a multivariate copula function. This paper proposes a non-parametric approach based on the maximum entropy theory to build the two parts of the distribution of stochastic processes. The specification of a stochastic process is interpreted as moment constraints and classified into two categories. The first category describes the uncertainty of the random quantity at an arbitrary point. The second category defines the dependence between the random quantities at any two points. The marginal distribution and copula function are then developed by maximizing the entropy under two classes of constraints, which are formulated as two optimization problems. The proposed method is applied to two reliability problems: a beam reliability evaluation considering time-dependent corrosion, and a tunnel reliability evaluation considering soil spatial variability. The first example illustrates the method when only partial information (marginal distribution and autocorrelation function) is available, while the second example shows the development of a random field model based on data. The comparison of the non-parametric approach with typical parametric models indicates the flexibility of the proposed method in capturing the variation of random quantities in time or space.

Parametric models analysed with linear maps

Parametric entities appear in many contexts, be it in optimisation, control, modelling of random quantities, or uncertainty quantification. These are all fields where reduced order models (ROMs) have a place to alleviate the computational burden. Assuming that the parametric entity takes values in a linear space, we show how is is associated to a linear map or operator. This provides a general point of view on how to consider and analyse different representations of such entities. Analysis of the associated linear map in turn connects such representations with reproducing kernel Hilbert spaces and affine-/linear-representations in terms of tensor products. A generalised correlation operator is defined through the associated linear map, and its spectral analysis helps to shed light on the approximation properties of ROMs. This point of view thus unifies many such representations under a functional analytic roof, leading to a deeper understanding and making them available for appropriate analysis.

Material intrinsic heterogeneity: statistical derivation

The value of a fully statistical sampling theory is that it is possible to quantify a measure of material intrinsic heterogeneity and, on this basis, provide the entire distribution of the analyte content of potential samples to be extracted from the lot. The analyte content of a sample of a given mass is a random quantity as samples of nominally equal masses taken from a lot in a given state of comminution will not have exactly the sample analyte content. The analyte content of a sample is correctly described as a random variable and to characterise a random variable completely it is necessary to know either the probability density function or distribution function for the random variable, or all of the moments of the random variable (mean, variance and all the higher moments). The following discussion derives the fundamental sampling variance from a purely mathematical statistics basis, relying on the assumption that the number of particles of any one type (size and analyte content) that fall into a sample taken in a mechanically correct manner (following the principle of equiprobable sampling) follows a Poisson distribution. In addition, the Poisson distributions of particle numbers are statistically independent. A more fully argued substantiation of this fundamental assumption, partial experimental evidence and standard statistical introduction to the definition and properties of the Poisson distribution, and reasons for its use, can be found at the end of this article. © Materials Sampling &amp; Consulting 2020

Mean lifetime of a false vacuum in terms of the Krylov-Fock nonescape probability

The Krylov-Fock expression of non-decay (or survival) probability, which allows to evaluate the deviations from the exponential decay law (nowadays well established experimentally), is more informative as it readily provides the distribution function for the lifetime as a random quantity. Guided by the well established formalism for describing nuclear alpha decay, we use this distribution function to figure out the mean value of lifetime and its fluctuation rate. This theoretical framework is of considerable interest inasmuch as it allows an experimental verification. Next, we apply the Krylov-Fock approach to the decay of a metastable state at a finite temperature in the framework of thermo-field dynamics. In contrast to the existing formalism, this approach shows the interference effect between the tunnelings from different metastable states as well as between the tunneling and the barrier hopping. This effect looks quite natural in the framework of consistent quantum mechanical description as a manifestation of the "double-slit experiment". In the end we discuss the field theory applications of the results obtained.

Measurement protocols, random-variable-valued measurements, and response process error: Estimation and inference when sample data are not deterministic.

Random-variable-valued measurements (RVVMs) are proposed as a new framework for treating measurement processes that generate non-deterministic sample data. They operate by assigning a probability measure to each observed sample instantiation of a global measurement process for some particular random quantity of interest, thus allowing for the explicit quantification of response process error. Common methodologies to date treat only measurement processes that generate fixed values for each sample unit, thus generating full (though possibly inaccurate) information on the random quantity of interest. However, many applied research situations in the non-experimental sciences naturally contain response process error, e.g. when psychologists assess patient agreement with various diagnostic survey items or when conservation biologists perform formal assessments to classify species-at-risk. Ignoring the sample-unit-level uncertainty of response process error in such measurement processes can greatly compromise the quality of resulting inferences. In this paper, a general theory of RVVMs is proposed to handle response process error, and several applications are considered.

A nonparametric approach for characterizing soil spatial variability based on cone penetration test data

Probabilistic site characterization usually requires a parametric model such as the Gaussian random field to begin. This paper proposes a nonparametric approach to characterizing soil spatial variability based on the maximum entropy method. The finite-dimensional distribution specifying a random field is decomposed into marginal distributions and vine-structured copulas. Then, the distribution is adaptively developed under the moment constraints, which are classified into two categories. The first category describes the uncertainty of the random quantity at an arbitrary point, while the second category defines the dependence between the random quantities at any two points. The marginal distribution and vine-structured copulas are developed under the two types of moment constraints, respectively, which are formulated as two optimization problems. The Akaike information criterion (AIC) is adopted for model selection, and the hypothesis test based on the probability integral transformation (PIT) is used to examine whether the spatial variability is modeled appropriately. Analysis results of a hypothetical case and several real soil profiles indicate that the nonparametric approach is feasible. The nonparametric models are comparable or superior to the parametric or Gaussian models, and no prior assumption is made on the distribution of the random field.