Variable Transformation Technique 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.

A High-Order Numerical Method Based on a Spatial Compact Exponential Scheme for Solving the Time-Fractional Black–Scholes Model

This paper investigates a high-order numerical method based on a spatial compact exponential scheme for solving the time-fractional Black–Scholes model. Firstly, the original time-fractional Black–Scholes model is converted into an equivalent time-fractional advection–diffusion reaction model by means of a variable transformation technique. Secondly, a novel high-order numerical method is constructed with (2−α) accuracy in time and fourth-order accuracy in space based on a spatial compact exponential scheme, where α is the fractional derivative. The uniqueness of solvability of the derived numerical method is rigorously discussed. Thirdly, the unconditional stability and convergence of the derived numerical method are rigorously analyzed using the Fourier analysis technique. Finally, numerical examples are presented to test the effectiveness of the derived numerical method. The proposed numerical method is also applied to solve the time-fractional Black–Scholes model, whose exact analytical solution is unknown; numerical results are illustrated graphically.

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.

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.

Finite‐time stabilization for a class of more general p‐normal nonlinear systems with input and output nonlinearities

AbstractThis study investigates the global finite‐time state feedback stabilization for a class of more general p‐normal nonlinear systems subjects to input time‐delay and uncertain output function under rational powers. A novel control input compensation signal comprising a finite integral of previous control values is presented, and an equivalent system with control input dependent‐free time delay is obtained by introducing a variable transformation technique that includes the compensation signal. A suitable controller is designed in line with the homogeneous domination method to ensure the globally finite‐time stability of the closed‐loop system. Finally, the effectiveness of the proposed control scheme is verified through a numerical simulation and an induction heater circuit system.

Towards neutrosophic Circumstances goal programming approach for solving multi-objective linear fractional programming problems

Indeterminacy is common in practical decision-making circumstances. So the mathematical models of decision making represent the situations in a better way if the parameters are considered as neutrosophic. This article studies a general framework of multi-objective neutrosophic linear fractional programming problem (MONLFPP) and proposes unique approach. The issue's parameters are thought of as triangular neutrosophic numbers. The problem is converted into an equal crisp multi-objective linear programming problem (MOLPP) with the help of variable transformation technique and a ranking function. Fuzzy goal programming is used to solve the MOLPP that has been obtained. Finally, the usefulness of the proposed technique is established using two mathematical models.

Research on the Selection of the most Popular Product Categories in TikTok based on Linear Regression

In the e-commerce industry, live streaming has become an increasingly popular way to promote and sell products. With the rise of social media platforms like Facebook, Instagram, and TikTok, more and more businesses are using live streaming to engage with their customers and boost sales. This study investigates the relationship between sales and the number of live viewers to find out the most popular partition for live sales of TikTok’s e-commerce by employing simple linear regression, deseasonalization, and variable transformation techniques to optimize the models. This study improves the accuracy and interpretability of regression models in the e-commerce industry, specifically focusing on the relationship between real-time viewership and Gross Merchandise Volume (GMV). The findings indicate that removing seasonality and applying log-to-log transformation provide more reliable and accurate models, ultimately helping businesses better understand and optimize their sales strategies.

Needed MIS Competencies to the Job Market: Students’ Perspective

The purpose of this paper is to investigate and identify the sets of competencies that are commensurate with employability requirements. The study follows a quantitative, deductive, and positivist approach. A sample of 62 Lebanese students majoring in management information technology and management information systems (MIT/MIS) was selected conveniently based on the participants' willingness to participate. Students belonged to three Lebanese universities. The research capitalizes on a survey questionnaire divided into three sections. The first section includes knowledge questions, the second section constitutes five parts that address different sets of competencies, and the third section constitutes demographics. Questions and statements followed different styles. Collected data were analyzed using the IBM SPSS version 26.0 package. Data analysis used descriptive, factor, and linear regression analyses. The validity and reliability of the questionnaire were very favorable. Results identified seven factors, each factor constituting 3 to 10 elements. These were condensed using a variable transformation technique and obtained weighted sums modeled using regression analysis. As a result, a competency model was assessed based on the students' current competencies against the necessary employability competencies. The research outcomes serve as a boost to the MIT/MIS competencies theoretical foundations in the context of Lebanon. Moreover, results, though not generalized, serve as an eye-opener for future research. Several recommendations posit the implications to policymakers in higher education and employers in the ICT sectors and other technology-based companies.

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.

Electrical performance of the 3 × 6 × n cobweb cascade resistance network

SummaryIn this paper, a kind of multi‐stage cobweb resistance network consisting of n single‐stage cobwebs, namely, a 3 × 6 × n cobweb cascade resistance network (CCRN), was studied. To calculate the equivalent resistance of such a large‐scale complex network, we used a modified recursion‐transform (MRT) method. Firstly, the resistance network to be solved was simplified to a simple equivalent network. Thereafter, the recursive relation of the equivalent network was established according to the basic law of circuit. Then, the nonlinear recursive relation was transformed into the linear recursive relation by variable transformation technique. Finally, the equivalent resistance was gained by resolving the linear recursive relation. By this method, we obtained the exact analytical expression of the equivalent resistance of the 3 × 6 × n CCRN. The computation results show that the 3 × 6 × n CCRN's equivalent resistance is decided by the number of circuit stages n, and as n goes to infinity, these equivalent resistances all tend to a definite limit value.

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.

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.

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.

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.

An Application of Fully Intuitionistic Fuzzy Multi-objective Linear Fractional Programming Problem in E-education System

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

Optimal Decision Making for Fractional Multi-commodity Network Flow Problem in a Multi-choice Fuzzy Stochastic Hybrid Environment

In this paper, a fractional multi-commodity network flow problem with multi-choice parameters is studied under hybrid fuzzy-stochastic conditions. In this problem, coefficients of the objective function in both the nominator and denominator take the form of multi-choice parameters, with the alternative choices for the nominator and denominator of the fraction being represented by fuzzy-stochastic and fuzzy variables, respectively. The arc capacities are also considered as fuzzy-stochastic variables. The main goal of the present research is to provide the decision-maker with a model by the help of which he/she can manage unknown factors across a multi-commodity network. Given that this problem is herein investigated in a hybrid fuzzy-stochastic environment and includes multi-choice parameters, we use the probability-possibility approach, Lagrange interpolating polynomial, and the Charnes–Cooper variable transformation technique to convert the problem into a deterministic one. Finally, efficiency of the proposed model is evaluated by presenting a couple of numerical instances.

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.

A generalized Ross infiltration model and the application on hole irrigation based on the biconjugate gradient stabilized algorithm

Soil water dynamics are very important for understanding and managing water movement in agricultural irrigation; Therefore, it is of great significance to establish an efficient and accurate infiltration model to simulate soil water dynamics. In this work, based on the finite volume method discrete format and principal variable transformation technology, the Taylor first-order expansion is used to solve the Richards equation, the adaptive time step is controlled by the change of soil maximum saturation, the CMS and BiCGSTAB are used to solve the large sparse matrix generated by the implicit format, and the generalized Ross infiltration model is generated. The validity of the model was verified by mass conservation and field experiment, and the soil water dynamic change law of hole irrigation with two irrigation methods (HSI and HAI) was explored by the model. The result shows that the generalized Ross infiltration model based on CMS and BiCGSTAB can significantly improve the computational efficiency of the model; the main variable transformation technique of the generalized Ross infiltration model improves the mass conservation of the model. Balancing the model accuracy and calculation efficiency, a scientific and reasonable time step can be determined by controlling the change in maximum effective saturation (∆Semax), the generalized Ross infiltration model is found to accurately simulate soil-water dynamics under hole irrigation (R2 > 0.9). It is found that HSI and HAI in four-hole irrigation provides more possibilities for further realizing the movement of soil nutrients with water without considering the evaporation of irrigation holes or small evaporation. If the time and infiltration volume fitting curve determined with the indoor single-hole infiltration test is used to control the water volume, HSI requires longer infiltration time to reach the same irrigation volume.