Random Variable Transformation Research Articles

Approximating the Probability Density Function of a Transformation of Random Variables

We propose a general formula for the probability density function of transformations of continuous or discrete random variables. Approximations and estimations are derived. Particular cases are treated when transformations are sum or products of random variables. The formula has a simple form when probability density functions are expressed with respect to a reference measure which belongs to the class of natural exponential families with quadratic variance functions. Some numerical results are provided to illustrate the method.

Information geometry connecting Wasserstein distance and Kullback\u2013Leibler divergence via the entropy-relaxed transportation problem

Two geometrical structures have been extensively studied for a manifold of probability distributions. One is based on the Fisher information metric, which is invariant under reversible transformations of random variables, while the other is based on the Wasserstein distance of optimal transportation, which reflects the structure of the distance between underlying random variables. Here, we propose a new information-geometrical theory that provides a unified framework connecting the Wasserstein distance and Kullback–Leibler (KL) divergence. We primarily considered a discrete case consisting of n elements and studied the geometry of the probability simplex S_{n-1}, which is the set of all probability distributions over n elements. The Wasserstein distance was introduced in S_{n-1} by the optimal transportation of commodities from distribution {varvec{p}} to distribution {varvec{q}}, where {varvec{p}}, {varvec{q}} in S_{n-1}. We relaxed the optimal transportation by using entropy, which was introduced by Cuturi. The optimal solution was called the entropy-relaxed stochastic transportation plan. The entropy-relaxed optimal cost C({varvec{p}}, {varvec{q}}) was computationally much less demanding than the original Wasserstein distance but does not define a distance because it is not minimized at {varvec{p}}={varvec{q}}. To define a proper divergence while retaining the computational advantage, we first introduced a divergence function in the manifold S_{n-1} times S_{n-1} composed of all optimal transportation plans. We fully explored the information geometry of the manifold of the optimal transportation plans and subsequently constructed a new one-parameter family of divergences in S_{n-1} that are related to both the Wasserstein distance and the KL-divergence.

Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function

This paper deals with the approximate computation of the first probability density function of the solution stochastic process to second-order linear differential equations with random analytic coefficients about ordinary points under very general hypotheses. Our approach is based on considering approximations of the solution stochastic process via truncated power series solution obtained from the random regular power series method together with the so-called Random Variable Transformation technique. The validity of the proposed method is shown through several illustrative examples.

Some results about randomized binary Markov chains: theory, computing and applications

ABSTRACTThis paper is addressed to give a generalization of the classical Markov methodology allowing the treatment of the entries of the transition matrix and initial condition as random variables instead of deterministic values lying in the interval . This permits the computation of the first probability density function (1-PDF) of the solution stochastic process taking advantage of the so-called Random Variable Transformation technique. From the 1-PDF relevant probabilistic information about the evolution of Markov models can be calculated including all one-dimensional statistical moments. We are also interested in determining the computation of distribution of some important quantities related to randomized Markov chains (steady state, hitting times, etc.). All theoretical results are established under general assumptions and they are illustrated by modelling the spread of a technology using real data.

Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty

This paper is devoted to construct approximations of the probability density function of the non-autonomous first-order homogeneous linear random differential equation, where the initial condition and the diffusion coefficient are assumed to be a random variable and a stochastic process, respectively. We combine Random Variable Transformation technique and Karhunen–Loève expansion to construct reliable approximations under general conditions. Several numerical examples illustrate our theoretical findings.

A Gaussian Mixture Model Representation of Endmember Variability in Hyperspectral Unmixing

Hyperspectral unmixing while considering endmember variability is usually performed by the normal compositional model, where the endmembers for each pixel are assumed to be sampled from unimodal Gaussian distributions. However, in real applications, the distribution of a material is often not Gaussian. In this paper, we use Gaussian mixture models (GMM) to represent endmember variability. We show, given the GMM starting premise, that the distribution of the mixed pixel (under the linear mixing model) is also a GMM (and this is shown from two perspectives). The first perspective originates from random variable transformations and gives a conditional density function of the pixels given the abundances and GMM parameters. With proper smoothness and sparsity prior constraints on the abundances, the conditional density function leads to a standard maximum a posteriori (MAP ) problem which can be solved using generalized expectation maximization. The second perspective originates from marginalizing over the endmembers in the GMM, which provides us with a foundation to solve for the endmembers at each pixel. Hence, compared to the other distribution based methods, our model can not only estimate the abundances and distribution parameters, but also the distinct endmember set for each pixel. We tested the proposed GMM on several synthetic and real datasets, and showed its potential by comparing it to current popular methods.

Sufficient conditions of stochastic dominance for general transformations and its application in option strategy

Abstract A counterexample is presented to show that the sufficient condition for one transformation dominating another by the second degree stochastic dominance, proposed by Theorem 5 of Levy (Stochastic dominance and expected utility: Survey and analysis, 1992), does not hold. Then, by restricting the monotone property of the dominating transformation, a revised exact sufficient condition for one transformation dominating another is given. Next, the stochastic dominance criteria, proposed by Meyer (Stochastic dominance and transformations of random variables, 1989) and developed by Levy (1992), are extended to the most general transformations. Moreover, such criteria are further generalized to transformations on discrete random variables. Finally, the authors employ this method to analyze the transformations resulting from holding a stock with the corresponding call option.

Transformation of circular random variables based on circular distribution functions

In this paper, we propose a new transformation of circular random variables based on circular distribution functions, which we shall call inverse distribution function (id f ) transformation. We show that M?bius transformation is a special case of our id f transformation. Very general results are provided for the properties of the proposed family of id f transformations, including their trigonometric moments, maximum entropy, random variate generation, finite mixture and modality properties. In particular, we shall focus our attention on a subfamily of the general family when id f transformation is based on the cardioid circular distribution function. Modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for the resulting distribution by applying the id f transformation to a random variable following a von Mises distribution. In fact, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of gun crimes in Pittsburgh, Pennsylvania.

On the ratio distribution of energy windowing algorithms for radiation portal monitors

This paper introduces a windowing algorithm for radiation portal monitors based on a transformation of random variables to calculate the ratio between the radiation intensity in two energy windows. Procedures to evaluate critical limits and establish detection limits are described. Simulations and experimental data illustrate a good agreement between derived and observed ratio distributions. The algorithm was compared with commonly-used algorithms. The results illustrate proper evaluation of critical and detection limits even in the case of shadow shielding by vehicles entering a radiation portal monitor.

Solution of the finite Milne problem in stochastic media with RVT Technique

This paper presents the solution to the Milne problem in the steady state with isotropic scattering phase function. The properties of the medium are considered as stochastic ones with Gaussian or exponential distributions and hence the problem treated as a stochastic integro-differential equation. To get an explicit form for the radiant energy density, the linear extrapolation distance, reflectivity and transmissivity in the deterministic case the problem is solved using the Pomraning-Eddington method. The obtained solution is found to be dependent on the optical space variable and thickness of the medium which are considered as random variables. The random variable transformation (RVT) technique is used to find the first probability density function (1-PDF) of the solution process. Then the stochastic linear extrapolation distance, reflectivity and transmissivity are calculated. For illustration, numerical results with conclusions are provided.

Gathered Village Location Optimization for Chinese Sustainable Urbanization Using an Integrated MODM Approach under Bi-Uncertain Environment

Urbanization has become a main challenge all over developing countries in the 21st Century. However, decision making should take into account the different national situations with their complex factors to achieve sustainable development. As standards of living have risen in urban areas, local/neighbor urbanization has become a coming trend in China. With this in mind, the paper focuses on the optimization of nearby gathered village locations in Population Migration (PM) with consideration of both qualitative and quantitative criteria. Therefore, an integrated multiple objective decision making approach (MODM) under a bi-uncertain environment is proposed to solve this problem, which is based on the comprehensive Economy-Society-Ecology-Resource-Religion (ESERR) urbanization concept. The first step is to establish a bi-uncertain multiple objective programming model orienting the problem. Secondly, the model process is composed of fuzzy random variable transformation and the expected value model based on a new fuzzy measure, which is given accordingly to obtain the equivalent model. Thirdly, in order to describe the model efficiently, the Multi-Objective Adaptive Global Local Neighbor Particle Swarm Optimization (MOAGLNPSO) with three-dimensional Pareto optimal judgment criteria is designed. Finally, a case study is tested to validate the effectiveness and to illustrate the advantages of the whole approach. This novel approach can help optimize sustainable urbanization strategies and ensure their realistic application.

Uncertainty Quantification of Microstructural Properties due to Experimental Variations

Electron backscatter diffraction scans are an important experimental input for microstructure generation and homogenization. Multiple electron backscatter diffraction scans can be used to sample the uncertainty in orientation distribution function: both point to point within a specimen as well as across multiple specimens that originate from the same manufacturing process. However, microstructure analysis methods typically employ only the mean values of the orientation distribution function to predict properties, and the stochastic information is lost. In this work, analytical methods are developed to account for the uncertainty in the electron backscatter diffraction data during property analysis. To this end, a linear smoothing scheme is developed in the Rodrigues fundamental region to compute the orientation distribution function from the electron backscatter diffraction data. The joint multivariate probability distributions of the orientation distribution function are then modeled using a Gaussian assumption. The uncertainty in engineering properties that are obtained by homogenization are also computed. It is shown that the uncertainty in nonlinear properties can be analytically obtained using direct transformation of random variables in the homogenization approach.

Computing the two first probability density functions of the random Cauchy-Euler differential equation: Study about regular-singular points

Abstract In this paper the randomized Cauchy-Euler differential equation is studied. With this aim, from a statistical point of view, both the first and second probability density functions of the solution stochastic process are computed. Then, the main statistical functions, namely, the mean, the variance and the covariance functions are determined as well. The study includes the computation of the first and second probability density functions of the regular-singular infinite point via an adequate mapping transforming the problem about the origin. The study is strongly based upon the Random Variable Transformation technique along with some results that have been recently published by some of authors to the random homogeneous linear second-order differential equation. Finally, an illustrative example is shown.

Pedagogy of Random Variable Transformations: A Matlab Workbook

A course in probability and random variables is a key component of the core curriculum of undergraduate electrical engineering programs. Students often find the course a little bit difficult. Educational software such as Matlab, Mathematica and Maple are increasingly being used to enhance the understanding of complex topics including probability and random variables. A number of demos in Matlab were created to illustrate the transformation of random variables, a topic, students usually find challenging. Suggestions of the students were sought on improving these problem specific demos. Taking these into account, a Matlab workbook has been created. With simple textural input, the Matlab workbook generates the density functions using two approaches along with the range of the output. In addition, it verifies the validity of the pdf and all these analytical steps are displayed as figures. Students were very enthused about the final version of the workbook. It is expected that the workbook will make the study of the transformation of random variables simple, straightforward and easy. The p-coded version of the Matlab workbook (examples, instruction on its use, etc.) will be accessible to the readers at the journal’s website following the publication of the manuscript.

Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications

Classical Markov models are defined through a stochastic transition matrix, i.e., a matrix whose columns (or rows) are deterministic values representing transition probabilities. However, in practice these quantities could often not be known in a deterministic manner, therefore, it is more realistic to consider them as random variables. Following this approach, this paper is aimed to give a technical generalization of classical Markov methodology in order to improve modelling of stroke disease when dealing with real data. With this goal, we randomize the entries of the transition matrix of a Markov chain with three states (susceptible, reliant and deceased) that has been previously proposed to model the stroke disease. This randomization of the classical Markov model permits the computation of the first probability density function of the solution stochastic process taking advantage of the so-called Random Variable Transformation technique. Afterwards, punctual and probabilistic predictions are computed from the first probability density function. In addition, the probability density functions of the time instants until a certain proportion of the total population remains susceptible, reliant and deceased are also computed. The study is completed showing the usefulness of our computational approach to determine, from a probabilistic point of view, key quantities in medical decision making, such as the cost-effectiveness ratio. • A randomization of a Markov model for study the stroke is proposed. • Random Variable Transformation method gives a probabilistic solution of the model. • First probability density of the solution to randomized Markov model is determined. • Time until a certain proportion of susceptibles remains in the population is given. • Key quantities in medical decision making are determined.

Transformation To Normality Based On Empirical Distribution Functions

The paper examines an effi cient alternative to the Box-Cox and Yeo-Johnson’s transformation to normality procedures which works under very general conditions. The method hinges on two fundamental results : the fact that the cumulative distribution function F(x) of a random variable X always has a U(0,1) distribution and the Box-Mueller transformation of uniform random variables to standard normal random variables. Given two observations x and y, we computed Fn(x) and Fn(y) , which for large n, are approximately uniform random variables. These values are then inputted into the Box-Mueller transformations. Bounds for the Kolmogorov-Smirnov statistic between the distribution of the transformed observations and the normal distribution are provided through numerical simulation and by appealing to the Dvoretzky-Kiefer-Wolfowitz inequality.

Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique

Generalized polynomial chaos (gPC) is a spectral technique in random space to represent random variables and stochastic processes in terms of orthogonal polynomials of the Askey scheme. One of its most fruitful applications consists of solving random differential equations. With gPC, stochastic solutions are expressed as orthogonal polynomials of the input random parameters. Different types of orthogonal polynomials can be chosen to achieve better convergence. This choice is dictated by the key correspondence between the weight function associated to orthogonal polynomials in the Askey scheme and the probability density functions of standard random variables. Otherwise, adaptive gPC constitutes a complementary spectral method to deal with arbitrary random variables in random differential equations. In its original formulation, adaptive gPC requires that both the unknowns and input random parameters enter polynomially in random differential equations. Regarding the inputs, if they appear as non-polynomial mappings of themselves, polynomial approximations are required and, as a consequence, loss of accuracy will be carried out in computations. In this paper an extended version of adaptive gPC is developed to circumvent these limitations of adaptive gPC by taking advantage of the random variable transformation method. A number of illustrative examples show the superiority of the extended adaptive gPC for solving nonlinear random differential equations. In addition, for the sake of completeness, in all examples randomness is tackled by nonlinear expressions.

FORM reliability analysis using a parallel evolutionary algorithm

This paper presents a parallel evolutionary algorithm to solve reliability problems with accuracy and repeatability of results. The last characteristic is usually overlooked; however, it is critical to the reliability of the calculation method itself. Note that evolutionary algorithms are stochastic processes and may not always generate identical results. The optimisation problem resulting from the first order reliability method is considered with an implicit state function that can include a call to a finite element analysis (FEA). A strategy to handle failures from the transformation of random variables or from the finite element call during the evolution process is explained in detail. Several benchmark tests are studied, including some involving bounded random variables that introduce strong non-linearities in the mapping to standard Gaussian space. In addition, the solutions of 2D and 3D frame problems using the finite element method illustrate the capabilities of the algorithm including the convenience of the algorithm in handling discrete limit state functions. Finally, the ability to obtain similar results after many runs is demonstrated.

A note on the logistic equation subject to uncertainties in parameters

This paper discusses the logistic equation subject to uncertainties in the intrinsic growth rate, \(\alpha \), in the initial population density, \(N_0\), and in the environmental carrying capacity, K. These parameters are treated as independent random variables. The random variable transformation method is applied to compute the first probability density function of the time–population density, N(t), and of its inflection point, \(t^{*}\). Results for the density functions of N(t), for a fixed \(t>0\), and \(t^{*}\) are also provided for \(\alpha \), \(N_0\) and K uniformly distributed. Finally, numerical experiments illustrate the proposed theoretical results.

Functional generalizations of Hoeffding’s covariance lemma and a formula for Kendall’s tau

This note extends Hoeffding’s lemma from the covariance between two random variables to that between transformations of random variables, and provides clarification of several existing functional generalizations. In the same spirit as these results, an explicit integral formula of Kendall’s tau for general, possibly discontinuous random variables is also determined.