Random Variable Transformation Research Articles

An approximate probabilistic solution of a random SIR-type epidemiological model using RVT technique

This paper adapts a multi-dimensional Random Variable Transformation technique to derive a comprehensive stochastic description of the Susceptible-Infected-Recovered epidemiological model. An approximate solution of a system of nonlinear differential equations, that characterizes this model, is deterministically obtained and, from this approximation, we derive the first probability density functions for the solution processes of susceptible, infected and recovered percentages. These probability density functions are used to find the approximate mean and variance functions, as well as the confidence intervals. Taking a general situation, the infection contact rate, the recovery rate and the initial conditions are taken to be random variables with arbitrary distributions. To test the validity of the theoretical findings associated to the proposed random epidemiological model, some numerical results are tabulated and graphically presented through an illustrative example.

A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique

A novel technique, named (KLE-RVT), is constructed to find a full probabilistic solution of a finite Milne problem of radiative transfer in spatially stochastic atmosphere. This technique is a combination between the random variable transformation (RVT) technique and the Karhunen–Loève expansion (KLE) of the input stochastic process (the total cross section of the medium) to find the probability density function of the solution stochastic process. The RVT technique is applicable only if the probability density function of the input random variable (process) is known in a closed form. To overcome this obstacle, KLE is applied to represent the spatially continuous random cross section, defined only by its mean and covariance function, in terms of a finite number of uncorrelated random variables with known probability density functions. By this technique, the probability density function of the solution process is evaluated dealing with the input process itself instead the integral transformation of it. This solution is general and valid for any input second order stochastic process. Numerical results of our findings are presented to realize this work.

Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay

We randomize the following class of linear differential equations with delay, xτ′(t)=axτ(t)+bxτ(t−τ),t > 0, and initial condition, xτ(t)=g(t),−τ≤t≤0, by assuming that coefficients a and b are random variables and the initial condition g(t) is a stochastic process. We consider two cases, depending on the functional form of the stochastic process g(t), and then we solve, from a probabilistic point of view, both random initial value problems by determining explicit expressions to the first probability density function, f(x, t; τ), of the corresponding solution stochastic processes. Afterwards, we establish sufficient conditions on the involved random input parameters in order to guarantee that f(x, t; τ) converges, as τ→0+, to the first probability density function, say f(x, t), of the corresponding associated random linear problem without delay (τ=0). The paper concludes with several numerical experiments illustrating our theoretical findings.

Reliability-based design for geotechnical engineering: An inverse FORM approach for practice

In order to advance the reliability analysis of practical engineering problems, a simplified iterative algorithm for forward and/or inverse first-order reliability method (FORM) is presented in this paper. The approach extends the well-known HLRF algorithms that are conventionally formulated in the uncorrelated standard normal space (u-space) into the correlated nonnormal space (x-space). Because the simplified algorithms are self-contained in x-space, the tedious mathematical transformations of random variables between statistical spaces as well as the inversion of Jacobian matrix for gradient vectors are not required, which are desirable for practitioners to implement FORM within standalone numerical codes. The convergence of the simplified iterative algorithms can be improved by properly adjusting the step length during the iterative computations. Verifications of the proposed algorithm are illustrated through a highly nonlinear problem for both forward and/or inverse reliability analysis. Based on the proposed inverse FORM algorithm, the geotechnical reliability-based designs (RBD) of a strip footing and an earth slope are performed and some insights gained.

Some Features of the Direct and Inverse Transformation of Random Variables

Always actual tasks of obtaining and processing experimental results in complex systems. Random obstacles (errors), measurement errors, imperfections and limitations of mathematical models and data processing algorithms can change the appearance of the distribution and lead to incorrect use of algorithms, for example, as is the case with Kalman filtering in control systems. Complex methods for the identification of distribution laws require the study of quantum systems, natural phenomena, environmental, biological, etc. processes, which are characterized by the presence of singularities and multimodality of distributions. Therefore, it is often not recommended to apply separate distribution laws to simulate probabilistic experimental data distributions, but a generalized distribution as a single statistical system, which known distributions include as individual partial cases. Thus, the generalized gamma distribution includes Rayleigh, Maxwell, Weibull, Levy, Hi-Square distributions, which are widely used in applied problems associated with statistical methods of physical processes research, remote sensing, in the theory of reliability, for describing the dispersion composition of particles fragmentation and calculation of the efficiency of phase separation in gas-liquid streams.

(CMMSE2018 paper) Solving the random Pielou logistic equation with the random variable transformation technique: Theory and applications

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.

Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models

In this paper, we deal with computational uncertainty quantification for stochastic models with one random input parameter. The goal of the paper is twofold: First, to approximate the set of probability density functions of the solution stochastic process, and second, to show the capability of our theoretical findings to deal with some important epidemiological models. The approximations are constructed in terms of a polynomial evaluated at the random input parameter, by means of generalized polynomial chaos expansions and the stochastic Galerkin projection technique. The probability density function of the aforementioned univariate polynomial is computed via the random variable transformation method, by taking into account the domains where the polynomial is strictly monotone. The algebraic/exponential convergence of the Galerkin projections gives rapid convergence of these density functions. The examples are based on fundamental epidemiological models formulated via linear and nonlinear differential and difference equations, where one of the input parameters is assumed to be a random variable.

Polynomial Transformations of Random Variables on Finite Sets

Polynomial Transformations of Random Variables on Finite Sets

Gradient Density Estimation in Arbitrary Finite Dimensions Using the Method of Stationary Phase

We prove that the density function of the gradient of a sufficiently smooth function , obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of as the free parameter . The frequencies act as gradient histogram bins. The result is shown using the stationary phase approximation and standard integration techniques and requires proper ordering of limits. We highlight a relationship with the well-known characteristic function approach to density estimation, and detail why our result is distinct from this method. Our framework for computing the joint density of gradients is extremely fast and straightforward to implement requiring a single Fourier transform operation without explicitly computing the gradients.

A probabilistic analysis of a Beverton-Holt-type discrete model: Theoretical and computing analysis

In this paper a randomized version of the Beverton-Holt type discrete model is proposed. Its solution stochastic process and the random steady state are determined. Its first probability density function and second probability density function are obtained by means of the random variable transformation method, providing a full probabilistic description of the solution. Finally, several numerical examples are shown.

Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique

This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic process. The main objective is to obtain the first probability density function, f1(p, t), of the solution stochastic process, P(t, ω). To achieve this goal, first the diffusion coefficient is represented via a truncation of order N of the Karhunen–Loève expansion, and second, the Random Variable Transformation technique is applied. In this manner, approximations, say f1N(p,t), of f1(p, t) are constructed. Afterwards, we rigorously prove that f1N(p,t)⟶f1(p,t) as N → ∞ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.

A Flexible Polynomial Expansion Method for Response Analysis with Random Parameters

The generalized Polynomial Chaos Expansion Method (gPCEM), which is a random uncertainty analysis method by employing the orthogonal polynomial bases from the Askey scheme to represent the random space, has been widely used in engineering applications due to its good performance in both computational efficiency and accuracy. But in gPCEM, a nonlinear transformation of random variables should always be used to adapt the generalized Polynomial Chaos theory for the analysis of random problems with complicated probability distributions, which may introduce nonlinearity in the procedure of random uncertainty propagation as well as leading to approximation errors on the probability distribution function (PDF) of random variables. This paper aims to develop a flexible polynomial expansion method for response analysis of the finite element system with bounded random variables following arbitrary probability distributions. Based on the large family of Jacobi polynomials, an Improved Jacobi Chaos Expansion Method (IJCEM) is proposed. In IJCEM, the response of random system is approximated by the Jacobi expansion with the Jacobi polynomial basis whose weight function is the closest to the probability density distribution (PDF) of the random variable. Subsequently, the moments of the response can be efficiently calculated though the Jacobi expansion. As the IJCEM avoids the necessity that the PDF should be represented in terms of the weight function of polynomial basis by using the variant transformation, neither the nonlinearity nor the errors on random models will be introduced in IJCEM. Numerical examples on two random problems show that compared with gPCEM, the IJCEM can achieve better efficiency and accuracy for random problems with complex probability distributions.

Directional splitting of Gaussian density in non‐linear random variable transformation

Transformation of a random variable is a common need in a design of many algorithms in signal processing, automatic control, and fault detection. Typically, the design is tied to an assumption on a probability density function of the random variable, often in the form of the Gaussian distribution. The assumption may be, however, difficult to be met in algorithms involving non-linear transformation of the random variable. This paper focuses on techniques capable to ensure validity of the Gaussian assumption of the non-linearly transformed Gaussian variable by approximating the to-be-transformed random variable distribution by a Gaussian mixture (GM) distribution. The stress is laid on an analysis and selection of design parameters of the approximate GM distribution to minimise the error imposed by the non-linear transformation such as the location and number of the GM terms. A special attention is devoted to the definition of the novel GM splitting directions based on the measures of non-Gaussianity. The proposed splitting directions are analysed and illustrated in numerical simulations.

Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

This paper deals with the randomized heat equation defined on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.

Doppler Spectrum Analysis of a Roadside Scatterer Model for Vehicle-to-Vehicle Channels: An Indirect Method

In vehicle-to-vehicle (V2V) channels, such roadside scatterers (RSSs) as houses, buildings, trees, and many more, play a crucial role in the determination of the Doppler power spectral density (DPSD) characteristics. However, the relevant research results are scarce due to the lack of computationally tractable analytic DPSD solutions. To fill this gap, we investigate an indirect method for the DPSD analysis of a generic two-dimensional (2D) RSS model for V2V channels. The indirect method, based on the Hoeher's theorem, employs successive transformations of random variables to obtain the DPSD. Compared to the conventional methods, leading to impractical multiple integral solutions, our method yields a single integral-form, more useful for analytic studies, model validation/parameter estimation, and fading simulator design. Using the new DPSD solution, the impact of different RSS layouts on the DPSD characteristics is further investigated, and several new insights are provided. The joint probability density function (PDF) of angle-of-departure and angle-of-arrival (AoA) and the joint Doppler-AoA PDF are newly presented in closed-forms and analyzed with respect to the DPSD shape. Comparisons with the DPSDs measured in highway and \FB{urban canyon} environments demonstrate not only the validity of the generic 2D RSS model, but also the significant contribution of RSSs to V2V channels.

Gaussian Process Quadrature Moment Transform

Computation of moments of transformed random variables is a problem appearing in many engineering applications. The current methods for moment transformation are mostly based on the classical quadrature rules, which cannot account for the approximation errors. Our aim is to design a method for moment transformation of Gaussian random variables, which accounts for the error in the numerically computed mean. We employ an instance of Bayesian quadrature, called Gaussian process quadrature (GPQ), which allows us to treat the integral itself as a random variable, where the integral variance informs us about the incurred integration error. Experiments on the coordinate transformation and nonlinear filtering examples show that the proposed GPQ moment transform performs better than the classical transforms.

Almost first-degree stochastic dominance for transformations and its application in insurance strategy

AbstractAlmost stochastic dominance is a relaxation of stochastic dominance, which allows small violations of stochastic dominance rules to avoid situations where most decision makers prefer one alternative to another but stochastic dominance cannot rank them. The authors first discuss the relations between almost first-degree stochastic dominance (AFSD) and the second-degree stochastic dominance (SSD), and demonstrate that the AFSD criterion is helpful to narrow down the SSD efficient set. Since the existing AFSD criterion is not convenient to rank transformations of random variables due to its relying heavily on cumulative distribution functions, the authors propose the AFSD criterion for transformations of random variables by means of transformation functions and the probability function of the original random variable. Moreover, they employ this method to analyze the transformations resulting from insurance and option strategy.

A spectral decomposition approach for the mechanical statistical characterization of distributed fiber-reinforced tissues

We discuss a spectral decomposition formulation for the mechanical statistical characterization of the anisotropic strain energy density of soft hyperelastic materials embedded with distributed fibers. We consider a generalized angular probability density function (PDF) of the reinforcement built upon the local eigenvalue and eigenvector system of the Cauchy–Green deformation tensor. We focus our analysis on material models dependent on the fourth pseudo-invariant of the deformation, I4, and on exponential forms of the fiber strain energy function. Within such a spectral reference system, we derive the closed-form expression of the PDF for I4 by generalizing the multi-value random variable transformation procedure recently developed in Gizzi et al. 2016. Our formulation bypasses the cumbersome extension–contraction switch, commonly adopted for shutting down the contribution of contracted fibers in models based on generalized structure tensors. Accordingly, we identify analytically the support of the fibers in pure extension for significant loading conditions. We can readily compute any statistics of the fourth pseudo-invariant and we can derive the direct definition of the average second Piola–Kirchhoff stress tensor according to the second order approximation.

A full probabilistic solution of the random linear fractional differential equation via the random variable transformation technique

This paper provides a full probabilistic solution of the randomized fractional linear nonhomogeneous differential equation with a random initial condition via the computation of the first probability density function of the solution stochastic process. To account for most generality in our analysis, we assume that uncertainty appears in all input parameters (diffusion coefficient, source term, and initial condition) and that a wide range of probabilistic distributions can be assigned to these parameters. Throughout our study, we will consider that the fractional order of Caputo derivative lies in ]0,1], that corresponds to the main standard case. To conduct our analysis, we take advantage of the random variable transformation technique to construct approximations of the first probability density function of the solution process from a suitable infinite series representation. We then prove these approximations do converge to the exact density assuming mild conditions on random input parameters. Our theoretical findings are illustrated through 2 numerical examples.

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.