Modelos de Crescimento de Bovinos Mertolengos em Ambiente Aleatório

Apresentamos modelos de crescimento individual em ambiente aleatório para descrever a evolução do peso de bovinos mertolengos da estirpe rosilho. Tendo como objectivo obter modelos que incluam o efeito das variações aleatórias do ambiente na evolução do peso, recorremos a equações diferenciais estocásticas. Os modelos usados para o crescimento individual de animais em termos do tamanho X(t) no instante t têm geralmente a forma dY(t)/dt = b(A-Y(t)), onde se fez a mudança de variável Y(t)=g(X(t)) com g estritamente crescente. Aqui A=g(a), onde a representa o tamanho assintótico do animal, e b é o coeficiente de crescimento que regula a velocidade de aproximação a A. No caso de haver flutuações aleatórias do ambiente, considerámos o modelo dY(t) = b(A-Y(t))dt + dW(t), onde mede a intensidade das flutuações e W(t) é um processo de Wiener padrão. Aplicámos o modelo e estudámos os problemas de estimação e de previsão para uma trajectória (um animal). Foi também estudada a extensão a várias trajectórias (vários animais) .Considerámos o caso do modelo de Bertalanffy-Richards (g(x)=xc com c>0) e do modelo de Gompertz (g(x)=ln x). Foram também utilizados métodos bootstrap para estudar o problema de estimação

Modelos Multifásicos de Crescimento de Animais em Ambiente Aleatório

Em trabalhos anteriores, estudámos um modelo geral de crescimento individual de animais em ambiente aleatório da forma dY(t) = b(g(a)-Y(t))dt+sdW(t), com Y(t)=g(X(t)), sendo g estritamente crescente (g(x)=x^c e g(x)=ln x são casos particulares típicos), X(t) o peso do animal no instante t, W(t) um processo de Wiener padrão, b o coeficiente de crescimento e a o peso assintótico. Este modelo, que usa uma equação diferencial estocástica, já não sofre do problema dos modelos clássicos de regressão em que um atraso de crescimento num determinado momento não se repercute nos pesos futuros. Aplicámos este modelo monofásico (uma única forma funcional descreve a dinâmica média para toda a curva de crescimento) e estudámos os problemas de estimação. Na literatura têm sido propostos modelos determinísticos multifásicos para o crescimento de bovinos, já que a curva de crescimento observada sugere a existência de pelo menos duas fases. Aqui estudamos a generalização do modelo estocástico referido ao caso multifásico, em que admitimos que o coeficiente de crescimento b tem valores diferentes para diferentes fases da vida do animal. Por simplicidade, consideramos duas fases com coeficientes de crescimento b1 e b2. Aplicamos a dados de bovinos

Random differential operational calculus: Theory and applications

A product rule and a chain rule for mean square derivatives are obtained using fourth order properties. Applications to the mean square solution of random differential equations are shown

Random differential operational calculus: theory and applications

In this article, we obtain a product rule and a chain rule for mean square derivatives. An application of the chain rule to the mean square solution of random differential equations is shown. However, to achieve such mean square differentiation rules, fourth order properties were needed and, therefore, we first studied a mean fourth order differential and integral calculus. Results are applied to solve random linear variable coefficient differential problems

Stochastic Differential Equations General Models of Individual Growth in Uncertain Environments and Application to Profit Optimization in Livestock Production

Individual animal growth in a randomly varying environment is modeled using stochastic differential equation models. These models are generalizations of the classical deterministic growth models used in regression methods, but incorporate a random dynamical term describing the effects of environmental and other random fluctuations on the growth process. We describe parameter estimation and prediction methods, illustrating with data on cow growth of the Mertolengo breed raised in Alentejo (Portugal) under natural conditions; see models, their properties and statistical techniques in the references below. We first show that these models outperform the traditional regression models in predictive power for they take into account the dynamical nature of the growth process and its interaction with environmental fluctuations. We then apply the models to profit optimization in livestock production, taking into account production costs and sales revenues. Assuming the animal is to be sold when it reaches some prescribed age, we determine the optimal age at which an animal should be sold in order to maximize profit. Another possibility is to sell the animal when it reaches a prescribed size. The first passage time distribution through a prescribed size is studied and used to determine the optimal size at which the animal should be sold. We then determine which policy (selling at a fixed age or selling at a fixed size) is preferable in terms of profit and compare results with the deterministic case

Solving Riccati time-dependent models with random quadratic coefficient

This paper deals with the construction of approximate solutions of a random logistic differential equation whose nonlinear coefficient is assumed to be an analytic stochastic process and the initial condition is a random variable. Applying p-mean stochastic calculus, the nonlinear equation is transformed into a random linear equation whose coefficients keep analyticity. Next, an approximate solution of the nonlinear problem is constructed in terms of a random power series solution of the associate linear problem. Approximations of the average and variance of the solution are provided. The proposed technique is illustrated through an example where comparisons with respect to Monte Carlo simulations are shown. © 2011 Elsevier Ltd. All rights reserved.This work has been partially supported by the Spanish M.C.Y.T. grants MTM2009-08587, DPI2010-20891-C02-01, Universitat Politecnica de Valencia grant PAID06-09-2588 and Mexican Conacyt.Cortés López, JC.; Jódar Sánchez, LA.; Company Rossi, R.; Villafuerte Altuzar, L. (2011). Solving Riccati time-dependent models with random quadratic coefficient. Applied Mathematics Letters. 24(12):2193-2196. https://doi.org/10.1016/j.aml.2011.06.024S21932196241

Multiphasic individual growth models in random environments

The evolution of the growth of an individual in a random environment can be described through stochastic differential equations of the form dY(t) = b(a-Y(t))dt+sdW(t), where Y(t)=h(X(t)), X(t) is the size of the individual at age t, h is a strictly increasing continuously differentiable function, a=h(A), where A is the average asymptotic size, and b represents the rate of approach to maturity. The parameter s measures the intensity of the effect of random fluctuations on growth and W(t) is the standard Wiener process. We have previously applied this monophasic model, in which there is only one functional form describing the average dynamics of the complete growth curve, and studied the estimation issues. Here, we present the generalization of the above stochastic model to the multiphasic case, in which we consider that the growth coefficient b assumes different values for different phases of the animal life. For simplicity, we consider two phases with growth coefficients b1 and b2. Results and methods are illustrated using bovine growth data

Modelling Animal Growth in Random Environments: An Application Using Nonparametric Estimation

We study a stochastic differential equation growth model to describe individual growth in random environments. In particular, in this paper, we discuss the estimation of the drift and the diffusion coefficients using nonparametric methods for the case of non-equidistant data for several trajectories. We illustrate the methodology by using bovine growth data. Our goal is to assess: a) if the parametric models (with specific functional forms for the drift and the diffusion coefficients) previously used by us to describe the evolution of bovine weight were adequate choices; b) whether some alternative specific parameterized functional forms of these coefficients might be suggested for further parametric analysis of this data

Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation

In this paper we study the use of the generalized polynomial chaos method to differential equations describing a model that depends on more than one random input. This random input can be in the form of parameters or of initial or boundary conditions. We investigate the effect of the choice of the probability density functions for the inputs on the output stochastic processes. The study is performed on the Airy¿s differential equation. This equation is a good test case since its solutions are highly oscillatory and errors can develop both in the amplitude and the phase. Several different situations are considered and, finally, conclusions are presented.This work has been partially supported by the Spanish M.C.Y.T. and FEDER Grants MTM2009-08587, DPI2010-20891-C02-01 as well as the Universitat Politecnica de Valencia Grants PAID-00-11 (Ref. 2751) and PAID-06-11 (Ref. 2070).Chen Charpentier, BM.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2013). Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Applied Mathematics and Computation. 219(9):4208-4218. https://doi.org/10.1016/j.amc.2012.11.007S42084218219

Solving linear and quadratic random matrix differential equations: A mean square approach

[EN] In this paper linear and Riccati random matrix differential equations are solved taking advantage of the so called L-p-random calculus. Uncertainty is assumed in coefficients and initial conditions. Existence of the solution in the L-p-random sense as well as its construction are addressed. Numerical examples illustrate the computation of the expectation and variance functions of the solution stochastic process. (C) 2016 Elsevier Inc. All rights reserved.This work has been partially supported by the Spanish Ministerio de Economia y Competitividad grant MTM2013-41765-P and by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance).Casabán Bartual, MC.; Cortés López, JC.; Jódar Sánchez, LA. (2016). Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling. 40(21-22):9362-9377. https://doi.org/10.1016/j.apm.2016.06.017S936293774021-2