Random Variable Transformation Research Articles

Folding Domain Functions (FDF): A Random Variable Transformation technique for the non-invertible case, with applications to RDEs

The Random Variable Transformation (RVT) method is a fundamental tool for determining the probability distribution function associated with a Random Variable (RV) Y=g(X), where X is a RV and g is a suitable transformation. In the usual applications of this method, one has to evaluate the derivative of h≡g−1. This can be a straightforward procedure when g is invertible, while difficulties may arise when g is non-invertible. The RVT method has received a great deal of attention in the recent years, because of its crucial relevance in many applications. In the present work we introduce a new approach which allows to determine the probability density function μY of the RV Y=g(X), when g is non-invertible due to its non-bijective nature. The main interest of our approach is that it can be easily implemented, from the numerical point of view, but mostly because of its low computational cost, which makes it very competitive. As a proof of concept, we apply our method to some numerical examples related to random differential equations, as well as discrete mappings, all of them of interest in the domain of applied Physics.

A random mathematical model to describe the antibiotic resistance depending on the antibiotic consumption: the Acinetobacter baumannii colistin-resistant case in Valencia, Spain

ABSTRACT The increase in antibiotic resistance in recent years, mainly due to the non-rational use of antibiotics, is one of the most important global public health threats. In this paper, we propose a mathematical dynamic random model describing the antibiotic resistance evolution of a bacteria and where antibiotic consumption is included is the main driving force in the resistance increase. The random model is solved using the Random Variable Transformation technique and is applied to study the case of Acinetobacter baumannii bacterium resistant to the antibiotic colistin in Valencia, Spain. Using the Multi-Objective Particle Swarm Optimization algorithm, the model has been calibrated with the A. baumannii colistin-resistance and colistin consumption data series. With the optimal model, four possible 7-year future scenarios with different antibiotic consumption trends have been simulated. The model results show how reducing antibiotic consumption does not easily stop the increase in resistance.

Statistical analysis of randomized pseudo-first/second order kinetic models. Application to study the adsorption on cadmium ions onto tree fern

Adsorption kinetics are commonly modeled using pseudo-first order (PFO) and pseudo-second order (PSO) rate laws. Both models are formulated via differential equations whose coefficients are commonly treated as deterministic quantities, which are calculated from experimental data using regression techniques. In this paper, we propose a full randomization of both models by assuming that all the parameters of the PFO and PSO models are random variables with arbitrary densities. Then, we probabilistically solve both models by determining semi-explicit expressions of the first probability density functions of the corresponding solution stochastic processes, as well as the densities of the time until a fixed adsorbed amount of reactant has been reached for each model too. The analysis is conducted under very general assumptions and is based on extensive application of the so-called Random Variable Transformation (RVT) technique. Finally, we apply all the aforementioned theoretical findings to model the adsorption of cadmium ions onto tree fern, using real data. We compare the results obtained by the randomized PFO and PSO models, using a Bayesian and a randomized version of the least mean square method to assign reasonable densities to model parameters. After comparing the results, the PSO model is selected, and, we then show how the RVT technique can be applied to obtain further key information, such as the second probability density function, of the solution and the covariance function.

Estimating the Magnitude of Completeness of Earthquake Catalogs Using a Simple Random Variable Transformation

Abstract The estimation of the magnitude of completeness for a seismic catalog is the starting point for most of the seismological statistical analyses. It is important, in particular, for the estimation of the Gutenberg–Richter (GR) law parameters. This law corresponds to an exponential distribution of the magnitudes. The recent studies have shown that the proper method to estimate the completeness is to check the exponentiality of the data. Here, we try to improve this method using a simple random variable transformation, which makes the method more robust in the case of catalogs with magnitudes coming from exponential distributions with different parameters, that is, catalogs with different b-values of the GR law.

Unit Distributions: A General Framework, Some Special Cases, and the Regression Unit-Dagum Models

In this work, we propose a general framework for models with support in the unit interval, which is obtained using the technique of random variable transformations. For this class, the general expressions of distribution and density functions are given, together with the principal characteristics, such as quantiles, moments, and hazard and reverse hazard functions. It is possible to verify that different proposals already present in the literature can be seen as particular cases of this general structure by choosing a suitable transformation. Moreover, we focus on the class of unit-Dagum distributions and, by specifying two different kinds of transformations, we propose the type I and type II unit-Dagum distributions. For these two models, we first consider the possibility of expressing the distribution in terms of indicators of interest, and then, through the regression approach, relate the indicators and covariates. Finally, some applications using data on the unit interval are reported.

Constructing reliable approximations of the random fractional Hermite equation: solution, moments and density

We extend the study of the random Hermite second-order ordinary differential equation to the fractional setting. We first construct a random generalized power series that solves the equation in the mean square sense under mild hypotheses on the random inputs (coefficients and initial conditions). From this representation of the solution, which is a parametric stochastic process, reliable approximations of the mean and the variance are explicitly given. Then, we take advantage of the random variable transformation technique to go further and construct convergent approximations of the first probability density function of the solution. Finally, several numerically simulations are carried out to illustrate the broad applicability of our theoretical findings.

Forward uncertainty quantification in random differential equation systems with delta‐impulsive terms: Theoretical study and applications

This contribution aims at studying a general class of random differential equations with Dirac‐delta impulse terms at a finite number of time instants. Our approach directly addresses calculating the so‐called first probability density function, from which all the relevant statistical information about the solution, a stochastic process, can be extracted. We combine the Liouville partial differential equation and the random variable transformation method to conduct our study. Finally, all our theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, for which numerical simulations are carried out.

A numerical modeling and its computational implementing simulation for generating distributions of the complicated random variable transformations with applications

This paper presents a mathematical modeling of new numerical techniques which are used to approximate the probability distribution of the univariate random variable transformations, to overcome the restrictions that prevent applying common analytical methods in the most cases. The proposed study is implemented by deducing mathematical formulas that make it easy to determine the cumulative distribution function numerically for any transformation of a random variable at a point from its range. The effectiveness of employing this procedure depends on finding numerically the values of inverse transformation images of that point and since it is difficult to deal analytically with the transformation function, this research presents an alternative method that depends on using Taylor's expansion to find a best polynomial function that converges strongly to the transformation function on its domain which facilitates generating the inverse images. Mathematical algorithms are designed to illustrate the computational implementation process of all former procedures. A stochastic model related to the heat equation is considered to obtain the distribution of its probabilistic solution which has a complicated nature impedes applying the usual analytical methods, therefore applying this current study has become effective in such cases. The results have been validated using Mathematica software.

Probabilistic analysis of a cantilever beam subjected to random loads via probability density functions

This paper addresses the probabilistic analysis of the deflection of a cantilever beam by means of a randomization of the classical governing fourth-order differential equation with null boundary conditions. The probabilistic study is based on the calculation of the first probability density function of the solution, which is a stochastic process, as well as the density function of further quantities of interest associated with this engineering problem such as the maximum slope and deflection at the free end of the cantilever beam, that are treated as random variables. In addition, the probability density function of the bending moment and the shear force will also be computed. The study takes extensive advantage of the so called Random Variable Transformation method, also known as Probability Transformation Method, that allows us to fully unify the probabilistic analysis in three relevant cases commonly studied in the deterministic setting. All the theoretical findings are illustrated via detailed numerical examples corresponding to each one of the three scenarios.

Probabilistic analysis of a general class of nonlinear random differential equations with state-dependent impulsive terms via probability density functions

In this contribution, we rigorously construct a pathwise solution to a general scalar random differential equation with state-dependent Dirac-delta impulse terms at a finite number of time instants. Furthermore, we obtain the first probability density function of the solution by combining two main results, firstly, the Liouville–Gibbs equation between the impulse instants, and secondly, the Random Variable Transformation technique at the impulse times. Finally, all theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, carrying on computational simulations.

Некоторые вопросы вероятности на конечных квазигруппах

We consider transformations of independent random variables by quasigroup operations. We explore the role of non-uniformity measures in the proof of convergence to uniformity and the estimation of convergence rate. We suggest a non-uniformity measure that allows an elementary proof of exponential convergence rate. The results on convergence rate are juxtaposed with known results on transformations of random variables in finited groups.

Statistical Analysis of a General Adsorption Kinetic Model with Randomness in Its Formulation. An Application to Real Data

The general adsorption kinetic model, also called pseudo- order (PNO) equation, is revisited using random differential equations. We provide a full probabilistic solution of the model, which is a stochastic process, by computing its first probability density function under very general hypotheses on its parameters, that are treated as absolutely continuous random variables with an arbitrary joint probability density function. The analysis is based on the so called Random Variable Transformation technique. From the first probability density function, we compute relevant information of the PNO model, such that, the mean, the variance and confidence interval. We also provide explicit expressions for the probability density functions of other significant quantities as the time required to reach a specific level of absorbed substance or the rate coefficient of the chemical reaction. All the theoretical findings are illustrated by means of real data. The application includes a thorough discussion about two important uncertainty quantification inverse methods, namely, the Random Least Mean Square and the Bayesian technique, to assign appropriate probability density functions to all the PNO model parameters so that the solution captures data uncertainties.

Compressibility Measures for Affinely Singular Random Vectors

The notion of compressibility of a random measure is a rather general concept which find applications in many contexts from data compression, to signal quantization, and parameter estimation. While compressibility for discrete and continuous measures is generally well understood, the case of discrete-continuous measures is quite subtle. In this paper, we focus on a class of multi-dimensional random measures that have singularities on affine lower-dimensional subsets. We refer to this class of random variables as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">affinely singular</i> . Affinely singular random vectors naturally arises when considering linear transformation of component-wise independent discrete-continuous random variables. To measure the compressibility of such distributions, we introduce the new notion of dimensional-rate bias (DRB) which is closely related to the entropy and differential entropy in discrete and continuous cases, respectively. Similar to entropy and differential entropy, DRB is useful in evaluating the mutual information between distributions of the aforementioned type. Besides the DRB, we also evaluate the the RID of these distributions. We further provide an upper-bound for the RID of multi-dimensional random measures that are obtained by Lipschitz functions of component-wise independent discrete-continuous random variables (X). The upper-bound is shown to be achievable when the Lipschitz function is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A \mathrm {X}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> satisfies <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm{ SPARK}}({A_{m\times n}}) = m+1$ </tex-math></inline-formula> (e.g., Vandermonde matrices). When considering discrete-domain moving-average processes with non-Gaussian excitation noise, the above results allow us to evaluate the block-average RID and DRB, as well as to determine a relationship between these parameters and other existing compressibility measures.

On the generalized logistic random differential equation: Theoretical analysis and numerical simulations with real-world data

Based on the previous literature about the random logistic and Gompertz models, the aim of this paper is to extend the investigations to the generalized logistic differential equation in the random setting. First, this is done by rigorously constructing its solution in two different ways, namely, the sample-path approach and the mean-square calculus. Secondly, the probability density function at each time instant is derived in two ways: by applying the random variable transformation technique and by solving the associated Liouville’s partial differential equation. It is also proved that both the stochastic solution and its density function converge, under specific conditions, to the corresponding solution and density function of the logistic and Gompertz models, respectively. The investigation finishes showing some examples, where a number of computational techniques are combined to construct reliable approximations of the probability density of the stochastic solution. In particular, we show, step-by-step, how our findings can be applied to a real-world problem.

Real-time remaining useful life prediction based on adaptive kernel window width density

Remaining useful life (RUL) prediction plays an important role in improving the availability and productivity of systems. To improve the accuracy of real-time RUL prediction during system operation, we propose a modeling method for real-time RUL prediction based on adaptive kernel window width density. First, a non-parametric kernel density estimation (KDE) real-time RUL prediction model is proposed, and a window width model with adaptive kernel window width density is established by introducing a local density factor in the window width selection. The local density of sample points is calculated by the k-nearest neighbor distance, and the KDE is performed by adaptively selecting the window width value according to the local density of sample points in the region of non-uniform distribution of sample points. As the monitoring data changes in real time, the kernel density estimates of known samples are used to recursively update the kernel density estimates of new samples. Moreover, the logarithmic transformation of random variables and space mapping are used in the establishment of the RUL prediction model. A model of logarithmic kernel diffeomorphism transformation is established to solve the boundary shift problem of kernel estimation in the prediction to improve the prediction accuracy. Finally, the validity of the method is verified through case studies, and the accuracy of the model is judged using evaluation quasi-measures.

Bayesian reliability analysis for copula based step-stress partially accelerated dependent competing risks model

The reliability of high-reliable and long-lifespan products in the presence of competing risks are commonly derived based on a known acceleration model and a latent independent competing risks model. For the cases with unknown acceleration models and dependent competing risks, this paper proposes a copula based partially accelerated competing risks model using a tampered random variable transformation. The reliability function and dependence structure of the partially accelerated competing risks data are derived between the real lifetime and the accelerated lifetime of each failure cause. In consideration of the parametric constraints and complicated joint posterior distribution, Hamiltonian Monte Carlo method within MCMC procedures is utilized to obtain Bayesian estimation of model parameters and reliability characteristics by defining unconstrained parameters using a specific transformation function. A simulation study is conducted to investigate the estimation performance, showing that the transformation forms have little influence on the point estimation of model parameters, and the constrained parametric Bayesian estimation method is efficient to evaluate the reliability for the proposed model. Finally, a real data set from a light-emitting diode partially accelerated life test is presented for further illustration of Bayesian reliability analysis.

Pedagogy of transformation of a random variable, censoring, and truncation of data

AbstractThe author has been incorporating data analytics into the probability course for students pursuing baccalaureate programs in engineering through weekly assignments and end‐of‐the‐term projects. Data analytics makes the probability course much more relevant to the current trends in business, industry, and medicine that rely on data. The discussion related to the coronavirus pandemic about missing and incomplete data, the inability to pinpoint the exact start of the disease, and so forth offered an opportunity to expand activities in data analytics by examining “truncation” and “censoring” of data. In simple terms, truncation implies the exclusion of data (either not collected or ignored) in a certain range. Censoring implies redefining the data in a certain range while keeping the rest of the original data. A demo was created to explore the association of the transformation of random variables (mixed and conditional) to censoring and truncation. The demo utilized the theory of random variables, simulation, and computational techniques to examine and establish the link between the mixed transformation of variables and censoring. A similar link of conditional transformation of variables to truncation was also seen. The demo was used as the basis for a weekly assignment to the students. The analysis and results suggest that it is possible to expand the scope of the course in engineering probability to provide training in contemporary topics in data analytics.

Study of nonhomogeneous linear second‐order discrete dynamical systems with uncertainties: Solution and stability with applications

We study, from a probabilistic standpoint, a full randomization of nonhomogeneous second‐order linear difference equations assuming that its data (initial conditions, coefficients, and forcing term) are random variables. Our analysis consists of computing the so‐called first probability density function of the solution, which is a stochastic process, and then analyzing the stability of the solution assuming that all data have an arbitrary joint probability density function. To achieve these goals, we take extensive advantage of the so‐called random variable transformation technique. The theoretical results extend their deterministic counterpart, and then, they have many applications in real‐world problems where uncertainty plays a key role. Our findings are first illustrated by means of several numerical examples, where different simulations are carried out, and, second, by means of a model belonging to biomathematics.

Diagonal nonlinear transformations preserve structure in covariance and precision matrices

For a multivariate normal distribution, the sparsity of the covariance and precision matrices encodes complete information about independence and conditional independence properties. For general distributions, the covariance and precision matrices reveal correlations and so-called partial correlations between variables, but these do not, in general, have any correspondence with respect to independence properties. In this paper, we prove that, for a certain class of non-Gaussian distributions, these correspondences still hold, exactly for the covariance and approximately for the precision. The distributions—sometimes referred to as “nonparanormal”—are given by diagonal transformations of multivariate normal random variables. We provide several analytic and numerical examples illustrating these results.

Probabilistic solution of a homogeneous linear second-order differential equation with randomized complex coefficients

In this paper, an exact expression for the first probability density function of the solution stochastic process to a randomized homogeneous linear second-order complex differential equation is determined. To complete the probabilistic analysis, the first probability density functions of the real and complex contributions of the solution stochastic process are also calculated. To compute the densities, the random variable transformation method is applied under general hypothesis, all coefficients and initial conditions are absolutely continuous complex random variables. The capability of the theoretical results established is demonstrated by several numerical examples. Finally, we show the applicability of the method in engineering, by analysing the solution of a randomized simple harmonic oscillator, defined on the complex domain, and comparing our results with those obtained by using Monte Carlo simulations.