Random Variable Transformation Research Articles

A computational probabilistic procedure to quantify the time of breast cancer recurrence after chemotherapy administration

We propose a random linear differential equation with jumps to model the dynamics of breast tumor growth using real patients’ data. The model considers the effect of chemotherapy administration and tumor resection. Also, the inherent randomness of the data and the model are taken into account.The model is probabilistically solved by combining two main methods belonging to Uncertainty Quantification: the principle of maximum entropy (PME) and the random variable transformation technique (RVT). The PME is applied to assign proper probability distributions to model parameters. The RVT technique is used to determine the probability density function (PDF) of the solution of the random differential equation, which is a stochastic process.We apply computational optimization techniques to determine the PDF of model parameters so that the PDF of the solution matches the ones assigned to real patients’ data.Once random tumor dynamics has been modeled, we simulate, after surgery, different adjuvant chemotherapy strategies with the aim of delaying tumor recurrence as long as possible. Results are in concordance with medical literature, and tumor relapse is delayed as more cycles of chemotherapy are administered.Although the proposed model has a common general structure, it is shown how it can be customized to patients in order to construct projections about the tumor’s growth. To better illustrate the applicability of the proposed approach, we carefully show how our model can be tailored to two real patients treated at the Hospital Clínico de Valencia (Spain). The obtained results have been overseen by doctors from this hospital.

Reliability inference for dual constant-stress accelerated life test with exponential distribution and progressively Type-II censoring

Accelerated life test provides a feasible and effective way to rapidly derive lifetime information by exposing products to higher-than-normal operating conditions. However, most of the previous research on accelerated life test has focused on the application of a single stress factor and a traditional censoring scheme. This article considers the reliability inference for a dual constant-stress accelerated life test model with exponential distribution and progressively Type-II censoring. Point estimates for model parameters are provided using maximum likelihood estimation and the weighted least squares method based on random variable transformation. In addition, we construct asymptotic confidence intervals, approximate confidence intervals, and bootstrap confidence intervals for the parameters of interest. Finally, extensive simulation studies and an illustrative example are presented to investigate the performance of the proposed methods.

Методи обґрунтування запасів запасних частин у системі управління транспортним процесом

The article formulates directions and analyzes the possibilities of substantiating and choosing the most optimal variant of applying mathematical models for forecasting the number and range of spare parts. The provided recommendations allow to take into account the multi-brand fleets of motor transport enterprises and trends for extrapolation in the short term, identified by the results of observations in real operating conditions. The main reasons for the occurrence of prolonged vehicle downtime due to technical malfunctions are considered. Traditional methods of planning spare parts inventories based on the use of reliability theory and results of mileage forecasting, standard time to failure; inventory management theory and statistical methods for forecasting random processes; methods of operations research; economic and mathematical methods; stochastic methods of transformation of random variables are analyzed. The article formalizes the grouping of factors that affect the size of stocks and the range of spare parts in the operation of rolling stock of enterprises, depending on the state of the system of organization of work of services for maintenance and repair of vehicles; structure of the fleet and technical characteristics of vehicles; level of development of the production base; availability, interest and qualification of personnel; operating conditions of automotive equipment. It is proposed to improve the logistics of spare parts inventory management and to formulate measures to improve the performance of carriers based on an assessment of the total costs incurred by the enterprise to maintain the reliability of rolling stock in operation and taking into account the amount of lost profit due to vehicle downtime in the technical service units. At the same time, it is recommended to use the mathematical apparatus of fuzzy set theory for decision-making, which allows formalizing the parameters of vaguely defined expectations of consumers of motor transport services and reducing the subjectivity of decision-making when solving problems of optimizing spare parts stocks in the transport process management system based on the analysis of the structure of consumer properties and service quality indicators.

A note on the application of the RVT method to general classes of single-species population models formulated by random differential equations

The Random Variable Transformation (RVT) technique has been applied in recent years to analyze a wide variety of dynamic models formulated via random differential equations. The applicability of this technique has usually been focused on problems where an explicit solution of the underlying deterministic problem is available. This fact limits the usefulness of the RVT method. This note aims to point out that the RVT technique can be successfully applied without this requirement by showing a wider range of potential applications including very general classes of single-species models.

Probabilistic analysis of a class of compartmental models formulated by random differential equations

This paper deals with the probabilistic analysis of a class of compartmental models formulated via a system of linear differential equations with time‐dependent non‐homogeneous terms. For the sake of generality, we assume that initial conditions and rates between compartments are random variables with arbitrary distributions while the source terms defining the flows entering the compartments are stochastic processes. We then take extensive advantage of the so‐called random variable transformation (RVT) technique to determine the first probability distribution of the solution of such a randomized model under very general hypotheses. In the simplest but relevant case of time‐independent source terms, we also obtain the probability distribution of the equilibrium, which is a random vector. Furthermore, we first particularize the aforementioned results for different types of integrable source terms that permit obtaining an explicit solution of the compartmental model: random constants, a train of Dirac delta impulses with random intensities, and a Wiener process. Secondly, we show an alternative approach based on the Liouville–Gibbs equation, which is useful when dealing with source terms that do not admit a closed‐form primitive. All the previous theoretical results are first illustrated through several numerical examples and simulations where a wide range of different probability distributions are assumed for the model parameters. The paper concludes by applying a compartmental model that describes the dynamics of oral drug administration through multiple chronologically spaced doses using synthetic data generated according to pharmacological references.

A mathematical model with uncertainty quantification for allelopathy with applications to real-world data

We revisit a deterministic model for studying the dynamics of allelopathy. The model is formulated in terms of a non-homogeneous linear system of differential equations whose forcing or source term is a piecewise constant function (square wave). To account for the inherent uncertainties present in this natural phenomenon, we reformulate the model as a system of random differential equations where all model parameters and the initial condition are assumed to be random variables, while the forcing term is a stochastic process. Taking extensive advantage of the so-called Random Variable Transformation (RVT) method, we obtain the solution of the randomized model by providing explicit expressions of the first probability density function of the solution under very general assumptions on the model data. We also determine the joint probability density function of the non-trivial equilibrium point, which is a random vector. If the source term is a time-dependent stochastic process, the RVT method might not be applicable since no explicit solution of the model is available. We then show an alternative approach to overcome this drawback by applying the Liouville–Gibbs partial differential equation. All the theoretical findings are illustrated through several examples, including the application of the randomized model to real-world data on alkaloid contents from leaching thornapple seed.

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.