Random Variable Transformation Research Articles

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.

Degenerate Gaussian factors for probabilistic inference

In this paper, we propose a parametrised factor that enables inference on Gaussian networks where linear dependencies exist among the random variables. Our factor representation is effectively a generalisation of traditional Gaussian parametrisations where the positive-definite constraint of the covariance matrix has been relaxed. For this purpose, we derive various statistical operations and results (such as marginalisation, multiplication and affine transformations of random variables) that extend the capabilities of Gaussian factors to these degenerate settings. By using this principled factor definition, degeneracies can be accommodated accurately and automatically at little additional computational cost. As illustration, we apply our methodology to a representative example involving recursive state estimation of cooperative mobile robots.

Uncertainty quantification for hybrid random logistic models with harvesting via density functions

The so-called logistic model with harvesting, p′(t)=rp(t)(1−p(t)K)−c(t)p(t), p(t0)=p0, is a classical ecological model that has been extensively studied and applied in the deterministic setting. It has also been studied, to some extent, in the stochastic framework using the Itô Calculus by formulating a Stochastic Differential Equation whose uncertainty is driven by the Gaussian white noise. In this paper, we present a new approach, based on the so-called theory of Random Differential Equations, that permits treating all model parameters as a random vector with an arbitrary join probability distribution (so, not just Gaussian). We take extensive advantage of the Random Variable Transformation method to probabilistically solve the full randomized version of the above logistic model with harvesting. It is done by exactly computing the first probability density function of the solution assuming that all model parameters are continuous random variables with an arbitrary join probability density function. The probabilistic solution is obtained in three relevant scenarios where the harvesting or influence function is mathematically described by discontinuous parametric stochastic processes having a biological meaning. The probabilistic analysis also includes the computation of the probability density function of the nontrivial equilibrium state, as well as the probability that stability is reached. All these results are new and extend their deterministic counterpart under very general assumptions. The theoretical findings are illustrated via two numerical examples. Finally, we show a detailed example where results are applied to describe the dynamics of stock of fishes over time using real data.

SNNKIQ IMPROVING DIESEL ENGINE DIAGNOSTICS RELIABILITY AT UNSTEADY LOAD

The issues of rational use of energy in the operation of machine-tractor units by reducing downtime, including hidden ones, are very relevant. An increase in the energy efficiency of machine-tractor units can be achieved by diagnosing diesel engines according to functional parameters that are used as the power plant of unit. The difference in engine loading parameters during diagnosis and operation can be taken into account by correction factors for the tolerance of diagnostic parameters, which are determined for each brand of engine. (Research purpose) The research purpose is developing the methodology for determining correction coefficients for the tolerances of diagnostic parameters, taking into account the distribution of input effects. (Materials and methods) The values of the functional parameters of tractors in real operation significantly differ from the indicators in bench conditions. The probabilistic, functional transformation of random variables and computational methods were applied. The article proposes calculating the mathematical expectations of the functional parameters of the engines using the functional transformation of random variables and probabilistic characteristics of the input effect to simplify the process. (Results and discussion) Without taking into account the correction coefficients, the error in determining the deviations of functional parameters can be significant. Deviations of functional parameters from the values of the typical characteristic are maximal for the arcsin function and minimal for the Gauss law. The methodology for determining the correction coefficients for the tolerances of functional parameters was substantiated using the example of the D-240 engine. (Conclusions) The article presents quantitative characteristics of correction coefficients for the transition from operational tolerance to diagnostic. For most technological operations, diagnosing diesel engines with an unsteady load will reduce the error in monitoring the permissible values of functional parameters by 10-24 percent.

Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions

<p style='text-indent:20px;'>We study a full randomization of the complete linear differential equation subject to an infinite train of Dirac's delta functions applied at different time instants. The initial condition and coefficients of the differential equation are assumed to be absolutely continuous random variables, while the external or forcing term is a stochastic process. We first approximate the forcing term using the Karhunen-Loève expansion, and then we take advantage of the Random Variable Transformation method to construct a formal approximation of the first probability density function (1-p.d.f.) of the solution. By imposing mild conditions on the model parameters, we prove the convergence of the aforementioned approximation to the exact 1-p.d.f. of the solution. All the theoretical findings are illustrated by means of two examples, where different types of probability distributions are assumed to model parameters.</p>

Development of a Probability Model for the Transmission Coefficient of Low Crested Breakwaters (LCB)

Park, S.-H.; Choi, J.-G., and Cho, Y.-J., 2021. Development of a probability model for the transmission coefficient of Low Crested Breakwaters (LCB). In: Lee, J.L.; Suh, K.-S.; Lee, B.; Shin, S., and Lee, J. (eds.), Crisis and Integrated Management for Coastal and Marine Safety. Journal of Coastal Research, Special Issue No. 114, pp. 484–488. Coconut Creek (Florida), ISSN 0749-0208. Low Crested Breakwaters (LCB) have been the most preferred structural type among many countermeasures against beach erosion. The extent of shore protection by LCB depends on transmittance coefficients, which are random due to the variability inherent in the marine environment. However, a probability model for transmittance coefficients is hard to find in the literature, indispensable for the reliability-based optimal design of LCB. In this rationale, first, a probability model of transmission coefficients is empirically developed. In doing so, transmittance characteristics of LCB were examined using the situ-wave data at the up-wave and down-wave sides of LCB at Bong-Po and Sok-Cho from 2019.8.6 to 2019.8.21 in order to identify the pertinent random variables affecting LCB transmittance. It turns out that sea wave conditions significantly affect the transmittance coefficients, and the roughness of sea wave conditions can be quantified in terms of wave height and its associated wave slope. Wave height and its associated wave slope are highly correlated in the mild sea, but they behave independently as wave heights get more enhanced. These complicated interrelations between wave height and wave slope can be described using the joint distribution of wave amplitude and its associated period by Longuet-Higgins (1983). In this rationale, we analytically derived a probability model for transmittance coefficients of LCB from the joint distribution of wave amplitude and its associated period by Longuet-Higgins (1983) and d'Angremond, van der Meer, and de Jong (1996) model using the standard technique of transformation of random variables (Papoulis, 1984). Numerical simulation shows that as sea wave conditions are getting harsh, a non-negligible probability mass shifts toward the low transmittance coefficients due to long waves appearing in the random wave field by sub-harmonic resonance wave-wave interaction, which is well-known after the studies of Hasselmann(1967) and Phillips(1980) on the development of wind waves. It was also shown that the probability mass shift toward the low transmission coefficients is accompanied by the probability mass shift toward higher transmission coefficients, which also comply with the studies of Hasselmann (1967) and Phillips (1980).

On three-valued random variable transformations by bivariate functions

On three-valued random variable transformations by bivariate functions

Full probabilistic analysis of random first‐order linear differential equations with Dirac delta impulses appearing in control

We study, from a probabilistic standpoint, first‐order impulsive linear differential equations, where all its parameters (initial condition and coefficients) are absolutely continuous random variables with a joint probability density function. We assume an infinite train of Dirac delta impulse applications at given time instants to control the model output. We take extensive advantage of the random variable transformation method to determine, first, an explicit expression for the probability density of the solution stochastic process, and secondly, of the random sequences for the maxima and minima. From these sequences, we determine the probability of stability of the solution stochastic process. This analysis is extended in the case that the application times are evenly spaced, via a period T, which is assumed to be a random variable. All the theoretical results are illustrated by means of several numerical examples, where we also perform a sensitivity analysis, via Sobol indexes, to highlight those model parameters that most explain the variability of the model response.

A new statistical inference method for multi-stress accelerated life testing based on random variable transformation

The limited sample size in multi-stress accelerated life testing makes the conventional large-sample-based inference methods inefficient. To overcome this problem, this paper develops a new statistical inference method based on random variable transformation for multi-stress accelerated life testing with Weibull distribution and progressive Type-II censoring. Firstly, a χ2 statistic is constructed using the random variable transformation method, and the exact point estimates of model parameters are derived based on the χ2 statistic. Subsequently, based on the χ2 statistic, the exact confidence interval of shape parameter is provided, while the generalized confidence intervals of the accelerated model parameters are calculated by constructing the new multivariate generalized pivotal quantities. Furthermore, the generalized confidence intervals of some commonly used reliability indexes are provided for better reliability management and decision making. Finally, a simulation study and a real case study are conducted to illustrate the implementation and effectiveness of the proposed method.

On Transformations of Three-Valued Random Variables by Bivariate Functions

On Transformations of Three-Valued Random Variables by Bivariate Functions

OPTIMIZATION OF PREVENTIVE THRESHOLD FOR CONDITIONBASED MAINTENANCE OF RADIO ELECTRONIC EQUIPMENT

Context. Operation costs throughout the life cycle of radio electronic equipment are very significant, which value far exceeds the initial cost of the equipment. Therefore, the up-to-date scientific and technical problem is to minimize operation costs. One of the ways to solve this problem is the introduction of statistical data processing technologies in the operation systems of radio electronic equipment.
 Objective. The goal of the paper is to improve the efficiency of thecondition-based maintenance with the determining parameters monitoring, which is widely used in civil aviation.
 Method. The solution of this problem is based on finding the functional dependence of the efficiency indicator in the form of specific operation costs on the basic parameters of radio electronic equipment and its operation system. To determine this dependence, the probability-event model is used,as well as methods of probability theory and mathematical statistics, in particular methods of statistical classification of sample sets and functional transformations of random variables. To determine the optimal level of the preventive threshold by the criterion of minimizing operation costs, the method of statistical simulation of Monte-Carlo is used.
 Results. Maintenance strategy with the determining parameters monitoring based on additional statistical data processing and technology of the optimal preventive threshold calculation are improved.
 Conclusions. The obtained results can be used during the development and modernization of operation systems of radio electronic equipment in terms of application of statistical data processing procedures. A comparative analysis of the two maintenance strategies showed that the use of additional statistical data processing might reduce specific operation costs. The proposed technology for determining the optimal preventive threshold can be extended to use during the operation of complex technical systems, in particular for those whose technical condition is associated with the values of the determining parameters.

Convex algebras of probability distributions induced by finite associative rings

Abstract We consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables.

Dealing with variability in ecological modelling: An analysis of a random non‐autonomous logistic population model

This paper presents a methodology to deal with the randomness associated to ecological modelling. Data variability makes it necessary to analyse the impact of random perturbations on the fitted model parameters. We conduct such analysis for the logistic growth model with a certain sigmoid functional form of the carrying capacity, which was proposed in the literature for the study of parasite growth during infection. We show how the probability distributions of the parameters are set via the maximum entropy principle. Then the random variable transformation method allows for computing the density function of the population.

A full probabilistic analysis of a randomized kinetic model for reaction–deactivation of hydrogen peroxide decomposition with applications to real data

The classical kinetic equation has been broadly used to describe reaction and deactivation processes in chemistry. The mathematical formulation of this deterministic nonlinear differential equation depends on reaction and deactivation rate constants. In practice, these rates must be calculated via laboratory experiments, hence involving measurement errors. Therefore, it is more realistic to treat these rates as random variables rather than deterministic constants. This leads to the randomization of the kinetic equation, and hence its solution becomes a stochastic process. In this paper we address the probabilistic analysis of a randomized kinetic model to describe reaction and deactivation by catalase of hydrogen peroxide decomposition at a given initial concentration. In the first part of the paper, we determine closed-form expressions for the probability density functions of important quantities of the aforementioned chemical process (the fractional conversion of hydrogen peroxide, the time until a fixed quantity of this fractional conversion is reached and the activity of the catalase). These expressions are obtained by taking extensive advantage of the so called Random Variable Transformation technique. In the second part, we apply the theoretical results obtained in the first part together with the principle of maximum entropy to model the hydrogen peroxide decomposition and aspergillus niger catalase deactivation using real data excerpted from the recent literature. Our results show full agreement with previous reported analysis but having as additional benefit that they provide a more complete description of both model inputs and outputs since we take into account the intrinsic uncertainties involved in modelling process.

A full probabilistic solution of a stochastic red blood cells model using RVT technique

Mathematical modeling is one of the most interesting ways to express the problem that describe the dynamics of the number of red blood cells count in the bloodstream of the human body. This problem has been deterministically solved based on continuous or discrete differential models. However, the stochastic models of this problem are rarely available and inadequate. This paper is organized to solve the random homogeneous linear second-order difference equation that describes a stochastic discrete red blood cells model. A complete probabilistic solution of this problem is conducted via applying the random variable transformation technique. This is achieved by deriving the first probability density function of the solution processes. The probabilistic behavior of the steady-state case (when time tends to infinity) is also studied. Moreover, the fundamental statistical measures, related to the stochastic solutions, such as the mean, the variance and the confidence intervals, are obtained. For the sake of clarity, numerical results (for pre-assigned distributions to the model parameters and initial conditions) with conclusions are presented.

On the Random Non-Autonomous Logistic Equation with Time-Dependent Coefficients

In this paper, we deal with the non-autonomous logistic growth model with time-dependent intrinsic growth rate and carrying capacity. Accounting for errors in recorded data, randomness is incorporated into the equation by assuming that the input parameters are random variables. The uncertainty of the model output is quantified by approximations of the first probability density function via the random variable transformation method. A numerical example illustrates the results.

Solving fully randomized first-order linear control systems: Application to study the dynamics of a damped oscillator with parametric noise under stochastic control

This paper is devoted to study random linear control systems where the initial condition, the final target, and the elements of matrices defining the coefficients are random variables, while the control is a stochastic process. The so-called Random Variable Transformation technique is adapted to obtain closed-form expressions of the probability density functions of the solution and of the control. The theoretical findings are applied to study the dynamics of a damped oscillator subject to parametric noise.

On Necessary Conditions of Finite-Valued Random Variable Algebraic Approximation

We consider transformations of random variables on finite sets by algebraic operations. A system of operations is said to be approximation complete if any random variable may be approximated with arbitrary precision by applying the given operations to mutually independent identically distributed random variables whose distributions have no zero components. We establish some necessary conditions for a function system to be approximation complete and construct examples of approximation incomplete systems.