Computing probabilistic solutions of the Bernoulli random differential equation

[EN] The random variable transformation technique is a powerful method to determine the probabilistic solution for random differential equations represented by the first probability density function of the solution stochastic process. In this paper, that technique is applied to construct a closed form expression of the solution for the Bernoulli random differential equation. In order to account for the general scenario, all the input parameters (coefficients and initial condition) are assumed to be absolutely continuous random variables with an arbitrary joint probability density function. The analysis is split into two cases for which an illustrative example is provided. Finally, a fish weight growth model is considered to illustrate the usefulness of the theoretical results previously established using real data.This work has been partially supported by the Ministerio de Economía y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. Contratos Predoctorales UPV 2014- Subprograma 1.Casabán, M.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics. 309:396-407. https://doi.org/10.1016/j.cam.2016.02.034S39640730

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

"This is the peer reviewed version of the following article: Cortés, J-C, Navarro-Quiles, A, Romero, J-V, Roselló, M-D. A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis. Comp and Math Methods. 2019; 1:e1013, which has been published in final form at https://doi.org/10.1002/cmm4.1013. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] In this paper a randomized version of the Beverton-Holt type discrete model is proposed. Its solution stochastic process and the random steady state are determined. Its first probability density function and second probability density function are obtained by means of the random variable transformation method, providing a full probabilistic description of the solution. Finally, several numerical examples are shown.This work has been partially supported by the Ministerio de Economía, Industria y Competitividad under grant MTM2017-89664-P. The authors express their deepest thanks and respect to the editors and reviewers for their valuable comments.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2019). A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis. Computational and Mathematical Methods. 1(1):1-12. https://doi.org/10.1002/cmm4.1013S11211Kwasnicki, W. (2013). Logistic growth of the global economy and competitiveness of nations. Technological Forecasting and Social Change, 80(1), 50-76. doi:10.1016/j.techfore.2012.07.007De la Sen, M. (2008). The generalized Beverton–Holt equation and the control of populations. Applied Mathematical Modelling, 32(11), 2312-2328. doi:10.1016/j.apm.2007.09.007Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty. Journal of Computational and Applied Mathematics, 337, 190-208. doi:10.1016/j.cam.2018.01.015Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics, 309, 396-407. doi:10.1016/j.cam.2016.02.034Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications. Journal of Computational and Applied Mathematics, 324, 225-240. doi:10.1016/j.cam.2017.04.040Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems. Applied Mathematics Letters, 68, 150-156. doi:10.1016/j.aml.2016.12.01

Feynman-Kac representation of fully nonlinear PDEs and applications

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance

Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

This paper presents a stochastic model predictive control approach for nonlinear systems subject to time-invariant probabilistic uncertainties in model parameters and initial conditions. The stochastic optimal control problem entails a cost function in terms of expected values and higher moments of the states, and chance constraints that ensure probabilistic constraint satisfaction. The generalized polynomial chaos framework is used to propagate the time-invariant stochastic uncertainties through the nonlinear system dynamics, and to efficiently sample from the probability densities of the states to approximate the satisfaction probability of the chance constraints. To increase computational efficiency by avoiding excessive sampling, a statistical analysis is proposed to systematically determine a-priori the least conservative constraint tightening required at a given sample size to guarantee a desired feasibility probability of the sample-approximated chance constraint optimization problem. In addition, a method is presented for sample-based approximation of the analytic gradients of the chance constraints, which increases the optimization efficiency significantly. The proposed stochastic nonlinear model predictive control approach is applicable to a broad class of nonlinear systems with the sufficient condition that each term is analytic with respect to the states, and separable with respect to the inputs, states and parameters. The closed-loop performance of the proposed approach is evaluated using the Williams-Otto reactor with seven states, and ten uncertain parameters and initial conditions. The results demonstrate the efficiency of the approach for real-time stochastic model predictive control and its capability to systematically account for probabilistic uncertainties in contrast to a nonlinear model predictive control approaches.Comment: Submitted to Journal of Process Contro

T-Crowd: Effective Crowdsourcing for Tabular Data

Crowdsourcing employs human workers to solve computer-hard problems, such as data cleaning, entity resolution, and sentiment analysis. When crowdsourcing tabular data, e.g., the attribute values of an entity set, a worker's answers on the different attributes (e.g., the nationality and age of a celebrity star) are often treated independently. This assumption is not always true and can lead to suboptimal crowdsourcing performance. In this paper, we present the T-Crowd system, which takes into consideration the intricate relationships among tasks, in order to converge faster to their true values. Particularly, T-Crowd integrates each worker's answers on different attributes to effectively learn his/her trustworthiness and the true data values. The attribute relationship information is also used to guide task allocation to workers. Finally, T-Crowd seamlessly supports categorical and continuous attributes, which are the two main datatypes found in typical databases. Our extensive experiments on real and synthetic datasets show that T-Crowd outperforms state-of-the-art methods in terms of truth inference and reducing the cost of crowdsourcing