Continuous Random Variables Research Articles

Some results on maximum of expected sums of sequential minima

Let be independent and identically distributed continuous nonnegative random variables and mn be the sample minimum. In the present paper, we propose an approach, which maximizes the value and makes it greater than . We further apply this approach to the standard uniform and exponential distributions and for finding the corresponding values solve proper algebraic equations by numerical methods.

Topology Estimation Using Graphical Models in Multi-Phase Power Distribution Grids

Power distribution grids are structurally operated radially, such that energized lines form a collection of trees with a substation at the root of each tree. The operational topology may change from time to time; however, tracking these changes, even though important for the distribution grid operation and control, is hindered by limited real-time monitoring. This paper develops a learning framework to reconstruct the radial operational structure of the distribution grid from synchronized voltage measurements. To detect operational lines, our learning algorithm uses conditional independence tests for continuous random variables that is applicable to a wide class of probability distributions, and in particular Gaussian, for injections. We validate the algorithm through extensive experiments on ac three-phase IEEE distribution grid test cases.

Estimation of Probability Distribution and Its Application in Bayesian Classification and Maximum Likelihood Regression

Nonparametric estimation of cumulative distribution function and probability density function of continuous random variables is a basic and central problem in probability theory and statistics. Although many methods such as kernel density estimation have been presented, it is still quite a challenging problem to be addressed to researchers. In this paper, we proposed a new method of spline regression, in which the spline function could consist of totally different types of functions for each segment with the result of Monte Carlo simulation. Based on the new spline regression, a new method to estimate the distribution and density function was provided, which showed significant advantages over the existing methods in the numerical experiments. Finally, the density function estimation of high dimensional random variables was discussed. It has shown the potential to apply the method in classification and regression models.

A distribuição Half-Normal generalizada discreta: uma distribuição alternativa na análise de dados de contagem

Geralmente, dados que são obtidos por processos de contagem, estritamente discretos ou discretizados (provenientes de truncamentos e/ou arredondamentos), são analisados, sem exaustão, pelas distribuições Geométrica, Logarítmica, Poisson e Binomial negativa. Nos últimos anos um grande número de distribuições discretas vêm sendo propostas na literatura a partir da discretização de variáveis aleatórias contínuas. Muitos dos métodos de discretização preservam uma ou mais características da versão contínua, sendo que a proposta de Nakagawa e Osaki (1975) é a mais utilizada. Neste artigo, a partir dessa metodologia, a qual faz uso da função de sobrevivência, é proposta a versão discreta da distribuição Half-Normal generalizada contínua, introduzida na literatura por Cooray e Ananda (2008). Discute-se algumas de suas propriedades e via simulações Monte Carlo avalia-se o viés e a acurácia das estimativas obtidas pelo método da máxima verossimilhança e pelo método dos momentos. Alguns conjuntos de dados discretos da literatura são considerados para ilustrar a aplicabilidade da distribuição proposta.

Further results and applications on continuous random variables

Further results and applications on continuous random variables

Learning moment closure in reaction-diffusion systems with spatial dynamic Boltzmann distributions.

Many physical systems are described by probability distributions that evolve in both time and space. Modeling these systems is often challenging due to their large state space and analytically intractable or computationally expensive dynamics. To address these problems, we study a machine-learning approach to model reduction based on the Boltzmann machine. Given the form of the reduced model Boltzmann distribution, we introduce an autonomous differential equation system for the interactions appearing in the energy function. The reduced model can treat systems in continuous space (described by continuous random variables), for which we formulate a variational learning problem using the adjoint method to determine the right-hand sides of the differential equations. This approach can be used to enforce a reduced physical model by a suitable parametrization of the differential equations. The parametrization we employ uses the basis functions from finite-element methods, which can be used to model any physical system. One application domain for such physics-informed learning algorithms is to modeling reaction-diffusion systems. We study a lattice version of the Rössler chaotic oscillator, which illustrates the accuracy of the moment closure approximation made by the method and its dimensionality reduction power.

The Nontransitivity Problem for Three Continuous Random Variables

The nontransitivity problem of the stochastic precedence relation for three independent random variables with distributions from a given class of continuous distributions is studied. Originally, this issue was formulated in one problem of strength theory. In recent time, nontransitivity has become a popular topic of research for the so-called nontransitive dice. Some criteria are first developed and then applied for proving that nontransitivity may not hold for many classical continuous distributions (uniform, exponential, Gaussian, logistic, Laplace, Cauchy, Simpson, one-parameter Weibull and others). The case of all distributions with a polynomial density on the unit interval is considered separately. Some promising directions of further investigations on the subject are outlined.

Measures of ageing tendency

AbstractA family of generalized ageing intensity functions of univariate absolutely continuous lifetime random variables is introduced and studied. They allow the analysis and measurement of the ageing tendency from various points of view. Some of these generalized ageing intensities characterize families of distributions dependent on a single parameter, while others determine distributions uniquely. In particular, it is shown that the elasticity functions of various transformations of distributions that appear in lifetime analysis and reliability theory uniquely characterize the parent distribution. Moreover, the recognition of the shape of a properly chosen generalized ageing intensity estimate admits a simple identification of the data lifetime distribution.

Gaussian Mixture Reduction for Time-Constrained Approximate Inference in Hybrid Bayesian Networks

Hybrid Bayesian Networks (HBNs), which contain both discrete and continuous variables, arise naturally in many application areas (e.g., image understanding, data fusion, medical diagnosis, fraud detection). This paper concerns inference in an important subclass of HBNs, the conditional Gaussian (CG) networks, in which all continuous random variables have Gaussian distributions and all children of continuous random variables must be continuous. Inference in CG networks can be NP-hard even for special-case structures, such as poly-trees, where inference in discrete Bayesian networks can be performed in polynomial time. Therefore, approximate inference is required. In approximate inference, it is often necessary to trade off accuracy against solution time. This paper presents an extension to the Hybrid Message Passing inference algorithm for general CG networks and an algorithm for optimizing its accuracy given a bound on computation time. The extended algorithm uses Gaussian mixture reduction to prevent an exponential increase in the number of Gaussian mixture components. The trade-off algorithm performs pre-processing to find optimal run-time settings for the extended algorithm. Experimental results for four CG networks compare performance of the extended algorithm with existing algorithms and show the optimal settings for these CG networks.

A compromise solution method for the multiobjective minimum risk problem

We develop an approach which enables the decision maker to search for a compromise solution to a multiobjective stochastic linear programming (MOSLP) problem where the objective functions depend on parameters which are continuous random variables with normal multivariate distributions. The minimum-risk criterion is used to transform the MOSLP problem into its corresponding deterministic equivalent which in turn is reduced to a Chebyshev problem. An algorithm based on the combined use of the bisection method and the probabilities of achieving goals is developed to obtain the optimal or epsilon optimal solution of this specific problem. An illustrated example is included in this paper to clarify the developed theory.

Alternative expectation formulas for real-valued random vectors

When the elements of a random vector take any real values, formulas of product moments are obtained for continuous and discrete random variables using distribution/survival functions. The random product can be that of strictly increasing functions of random variables. For continuous cases, the derivation based on iterated integrals is employed. It is shown that Hoeffding’s covariance lemma is algebraically equal to a special case of this result. For discrete cases, the elements of a random vector can be non-integers and/or unequally spaced. A discrete version of Hoeffding’s covariance lemma is derived for real-valued random variables.

Novel entropy estimators of a continuous random variable

This paper presents some novel entropy estimators of a continuous random variable using simple random sampling (SRS), ranked set sampling (RSS), and double RSS (DRSS) schemes. The theoretical results of the proposed entropy estimators are derived. The proposed entropy estimators are compared in terms of the bias and the root mean squared errors, theoretically and numerically, with the Vasicek O. [A test for normality based on sample entropy, J. R. Stat. Soc. B 38:54–59, 1976.] entropy estimators using SRS, RSS, and DRSS schemes. It turns out that the new novel entropy estimators are substantially better than the existing Vasicek’s entropy estimators.

Multiellipsoidal extended target tracking with known extent using sequential Monte Carlo framework

In this paper, we consider a variant of the extended target tracking (ETT) problem, namely the multiel- lipsoidal ETT problem. In multiellipsoidal ETT, target extent is represented by multiple ellipses, which correspond to the origin of the measurements on the target surface. The problem involves estimating the target’s kinematic state and solving the association problem between the measurements and the ellipses. We cast the problem in a sequential Monte Carlo (SMC) framework and investigate different marginalization strategies to find an efficient particle filter. Under the known extent assumption, we define association variables to find the correct association between the measurements and the ellipses; hence, the posterior involves both discrete and continuous random variables. By expressing the measurement likelihood as a mixture of Gaussians we derive and employ a marginalized particle filter for the independent association variables without sampling the discrete states. We compare the performance of the method with its alternatives and illustrate the gain in nonstandard marginalization.

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

Estimation Efficiency Under Privacy Constraints

We investigate the problem of estimating a random variable $Y\in \mathcal{Y}$ under a privacy constraint dictated by another random variable $X\in \mathcal{X}$, where estimation efficiency and privacy are assessed in terms of two different loss functions. In the discrete case, we use the Hamming loss function and express the corresponding utility-privacy tradeoff in terms of the privacy-constrained guessing probability $h(P_{XY}, \epsilon)$, the maximum probability $\mathsf{P}_\mathsf{c}(Y|Z)$ of correctly guessing $Y$ given an auxiliary random variable $Z\in \mathcal{Z}$, where the maximization is taken over all $P_{Z|Y}$ ensuring that $\mathsf{P}_\mathsf{c}(X|Z)\leq \epsilon$ for a given privacy threshold $\epsilon \geq 0$. We prove that $h(P_{XY}, \cdot)$ is concave and piecewise linear, which allows us to derive its expression in closed form for any $\epsilon$ when $X$ and $Y$ are binary. In the non-binary case, we derive $h(P_{XY}, \epsilon)$ in the high utility regime (i.e., for sufficiently large values of $\epsilon$) under the assumption that $Z$ takes values in $\mathcal{Y}$. We also analyze the privacy-constrained guessing probability for two binary vector scenarios. When $X$ and $Y$ are continuous random variables, we use the squared-error loss function and express the corresponding utility-privacy tradeoff in terms of $\mathsf{sENSR}(P_{XY}, \epsilon)$, which is the smallest normalized minimum mean squared-error (mmse) incurred in estimating $Y$ from its Gaussian perturbation $Z$, such that the mmse of $f(X)$ given $Z$ is within $\epsilon$ of the variance of $f(X)$ for any non-constant real-valued function $f$. We derive tight upper and lower bounds for $\mathsf{sENSR}$ when $Y$ is Gaussian. We also obtain a tight lower bound for $\mathsf{sENSR}(P_{XY}, \epsilon)$ for general absolutely continuous random variables when $\epsilon$ is sufficiently small.

A geometrization of quantum mutual information

It is well known that quantum mechanics admits a geometric formulation on the complex projective space as a Kähler manifold. In this paper, we consider the notion of mutual information among continuous random variables in relation to the geometric description of a composite quantum system introducing a new measure of total correlations that can be computed in terms of Gaussian integrals.

Error patterns in determining combined probability functions from continuous random variables

The purpose of this research is to analyze the patterns of error in determining the combined probability functions of continuous random variables. The type of this research is qualitative research. The data, which is obtained in this research, is then analyzed descriptively. Subjects selection are based on purposive sampling technique by taking 9 subjects with consideration/criteria for high, medium and low academic abilities. The method used to collect data in this research is a written test in the form of a description. The triangulation used in this research is time triangulation that collects data from 6th semester students in the academic year of 2015/2016, 2016/2017 and 2017/2018 who took mathematics statistics course. Conclusion indicates that the pattern of errors experienced by the subjects is concept errors, errors related to the concept of pre-requisite material and calculation errors.

Symbulate: Simulation in the Language of Probability

Simulation is an effective tool for analyzing probability models as well as for facilitating understanding of concepts in probability and statistics. Unfortunately, implementing a simulation from scratch often requires users to think about programming issues that are not relevant to the simulation itself. We have developed a Python package called Symbulate (https://github.com/dlsun/symbulate) which provides a user friendly framework for conducting simulations involving probability models. The syntax of Symbulate reflects the “language of probability” and makes it intuitive to specify, run, analyze, and visualize the results of a simulation. Moreover, Symbulate’s consistency with the mathematics of probability reinforces understanding of probabilistic concepts. Symbulate can be used in introductory through graduate courses, with a wide variety of probability concepts and problems, including: probability spaces; events; discrete and continuous random variables; joint, conditional, and marginal distributions; stochastic processes; discrete- and continuous-time Markov chains; Poisson processes; and Gaussian processes, including Brownian motion. In this work, we demonstrate Symbulate, discuss its main pedagogical features, present examples of Symbulate graphics, and share some of our experiences using Symbulate in courses.

Optimal design of inspection times for interval censoring

We treat optimal equidistant and optimal non-equidistant inspection times for interval censoring of exponential distributions. We provide in particular a new approach for determining the optimal non-equidistant inspection times. The resulting recursive formula is related to a formula for optimal spacing of quantiles for asymptotically best linear estimates based on order statistics and to a formula for optimal cutpoints by the discretisation of continuous random variables. Moreover, we show that by the censoring with the optimal non-equidistant inspection times as well as with optimal equidistant inspection times, there is no loss of information if the number of inspections is converging to infinity. Since optimal equidistant inspection times are easier to calculate and easier to handle in practice, we study the efficiency of optimal equidistant inspection times with respect to optimal non-equidistant inspection times. Moreover, since the optimal inspection times are only locally optimal, we also provide some results concerning maximin efficient designs.

Toward Breaking the Curse of Dimensionality: An FPTAS for Stochastic Dynamic Programs with Multidimensional Actions and Scalar States

We propose a Fully Polynomial-Time Approximation Scheme (FPTAS) for stochastic dynamic programs with multidimensional action, scalar state, convex costs and linear state transition function. The action spaces are polyhedral and described by parametric linear programs. This type of problems finds applications in the area of optimal planning under uncertainty, and can be thought of as the problem of optimally managing a single non-discrete resource over a finite time horizon. We show that under a value oracle model for the cost functions this result for one-dimensional state space is "best possible", because a similar dynamic programming model with two-dimensional state space does not admit a PTAS. The FPTAS relies on the solution of polynomial-sized linear programs to recursively compute an approximation of the value function at each stage. Our paper enlarges the class of dynamic programs that admit an FPTAS by showing, under suitable conditions, how to deal with multidimensional action spaces and with vectors of continuous random variables with bounded support. These results bring us one step closer to overcoming the curse of dimensionality of dynamic programming.