Modeling the dynamics of the frequent users of electronic commerce in Spain using optimization techniques for inverse problems with uncertainty. (English) Zbl 07093344

Burgos, Clara; Cortés, Juan-Carlos; Lombana, Iván-Camilo; Martínez-Rodríguez, David; Villanueva, Rafael-J.

J. Optim. Theory Appl. 182, No. 2, 785-796 (2019).

Summary: In this paper, we retrieve data about the frequent users of electronic commerce during the period 2011-2016 from the Spanish National Institute of Statistics. These data, coming from surveys, have intrinsic uncertainty that we describe using appropriate random variables. Then, we propose a stochastic model to study the dynamics of frequent users of electronic commerce. The goal of this paper is to solve the inverse problem that consists of determining the model parameters as suitable parametric random variables, in such a way the model output be capable of capturing the data uncertainty, at the time instants where sample data are available, via adequate probability density functions. To achieve the aforementioned goal, we propose a computational procedure that involves building a nonlinear objective function, based on statistical moment measures, to be minimized using a variation of the particle swarm optimization algorithm.

MSC:

65C20	Probabilistic models, generic numerical methods in probability and statistics
65C60	Computational problems in statistics (MSC2010)
65K10	Numerical optimization and variational techniques

Keywords:

inverse problem; uncertainty quantification; random optimization computational methods; nonlinear stochastic model; probability density function

References:

[1]	Dorini, FA; Cecconello, MS; Dorini, MB, On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density, Commun. Nonlinear Sci. Numer. Simul., 33, 160-173, (2016)
[2]	Hussein, A.; Selim, MM, Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique, Appl. Math. Comput., 218, 7193-7203, (2012) · Zbl 1246.65014
[3]	Dorini, F.; Cunha, M., Statistical moments of the random linear transport equation, J. Comput. Phys., 227, 8541-8550, (2008) · Zbl 1211.65005
[4]	Xu, Z.; Tipireddy, R.; Lin, G., Analytical approximation and numerical studies of one-dimensional elliptic equation with random coefficients, Appl. Math. Model., 40, 5542-5559, (2016)
[5]	Mourad, K.; Debbouche, A., Complete controllability of nonlocal fractional stochastic differential evolution equations with Poisson jumps in Hilbert spaces, Int. J. Adv. Appl. Math. Mech., 3, 41-48, (2015) · Zbl 1359.93052
[6]	Casabán, MC; Cortés, JC; Navarro-Quiles, A.; Romero, JV; Roselló, MD; Villanueva, RJ, A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique, Commun. Nonlinear Sci. Numer. Simul., 32, 199-210, (2016)
[7]	Casabán, MC; Cortés, JC; Navarro-Quiles, A.; Romero, JV; Roselló, MD; Villanueva, RJ, Computing probabilistic solutions of the Bernoulli random differential equation, J. Comput. Appl. Math., 309, 396-407, (2017) · Zbl 1468.65006
[8]	Cortés, JC; Santonja, FJ; Tarazona, AC; Villanueva, RJ; Villanueva-Oller, J., A probabilistic estimation and prediction technique for dynamic continuous social science models: the evolution of the attitude of the basque country population towards ETA as a case study, Appl. Math. Comput., 264, 13-20, (2015) · Zbl 1410.91393
[9]	Spanish INE: Encuesta sobre equipamiento y uso de tecnologías de información y comunicación en los hogares (Survey on equipment and use of the information technologies and communication in the household. http://www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica_C&cid=1254736176741&menu=resultados&idp=1254735576692. Accessed 20 June 2018
[10]	Spanish INE: Indicadores demográficos básicos (Basic demographic indicators). http://www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica_C&cid=1254736177003&menu=resultados&idp=1254735573002. Accessed 20 June 2018
[11]	Christakis, N.A., Fowler, J.H.: Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives. Little, Brown and Company, Boston (2009)
[12]	Brauer, F., Castillo-Chávez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, New York (2001). https://doi.org/10.1007/978-1-4757-3516-1 · Zbl 0967.92015
[13]	Norman, L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions. Wiley, London (1994) · Zbl 0811.62001
[14]	Khemka, N.; Jacob, C., Exploratory toolkit for evolutionary and swarm-based optimization, Math. J., 11, 376-391, (2010)
[15]	Acedo, L.; Burgos, C.; Hidalgo, JI; Sánchez-Alonso, V.; Villanueva, RJ; Villanueva-Oller, J., Calibrating a large network model describing the transmission dynamics of the human papillomavirus using a particle swarm optimization algorithm in a distributed computing environment, Int. J. High Perf. Comput. Appl., (2017)