Likelihood Function through the Delta Approximation in Mixed SDE Models

Abstract

Stochastic differential equations (SDE) appropriately describe a variety of phenomena occurring in random environments, such as the growth dynamics of individual animals. Using appropriate weight transformations and a variant of the Ornstein–Uhlenbeck model, one obtains a general model for the evolution of cattle weight. The model parameters are α, the average transformed weight at maturity, β, a growth parameter, and σ, a measure of environmental fluctuations intensity. We briefly review our previous work on estimation and prediction issues for this model and some generalizations, considering fixed parameters. In order to incorporate individual characteristics of the animals, we now consider that the parameters α and β are Gaussian random variables varying from animal to animal, which results in SDE mixed models. We estimate parameters by maximum likelihood, but, since a closed-form expression for the likelihood function is usually not possible, we approximate it using our proposed delta approximation method. Using simulated data, we estimate the model parameters and compare them with existing methodologies, showing that the proposed method is a good alternative. It also overcomes the existing methodologies requirement of having all animals weighed at the same ages; thus, we apply it to real data, where such a requirement fails.

Soft Labeling and Expectation-Maximization Algorithm to the Learning Problem of Uncertain Labeling in Vehicular Networks

“…The existence of these sub-populations is introduced into the model via a latent variable Y with value in a discrete set 𝑦 = {𝑐 1 , c 2 , … , c 𝐾 }. More concretely, mixture models assume a data generation model of the following form: The concept of likelihood has important links with the notions of possibility and more generally of likelihood, as many authors have highlighted [71][72] [73]. Moreover, selecting the parameter vector of the mixture model having the maximum likelihood knowing the observations is a natural decision strategy within the framework of belief function theory [74].…”

Stochastic differential equations mixed model for individual growth with the inclusion of genetic characteristics

“…We have applied the maximum likelihood estimation method, [9,14,15], to estimate the parameter vector p = (α, β, σ). From (3), using the fact that Y i (t) is a Markov process, we know that, given Y i,0 = y i,0 (assumed known), the Y i joint probability density function for individual i takes the form…”

“…We have just described the general SDE model (1) for the complete growth curve of the animals where the model's parameters α, β, and σ are assumed common to all individuals. In [14], we consider a generalization to SDE mixed models to take into account that different animals, due to their individual characteristics, may have different values of the parameters. We have denoted by b the d-dimensional vector of parameters that vary randomly among animals and assumed that the distribution of b among animals has probability density function (p.d.f.)…”

“…For the general situation, with no such restrictions, it is possible to explicitly compute the integral in the likelihood function, resulting in a final closed-form expression for this function. This is shown in [14,15], where the exact log-likelihood function for all animals is given by…”

“…In the last four decades, Stochastic Differential Equations (SDE) have been studied and applied to individual growth modeling by several authors using SDE non-mixed models [9,10,11,12,13,19,20,22,23] where, for instance, authors have modeled the growth dynamic of animals, trees or glucose, and where the parameters of the SDE have fixed, although unknown, values that need to be estimated from data. When the model parameters are considered random, SDE mixed models are used to model individual growth, with applications in various fields, such as animal growth and pharmacokinetics, among others [5,6,7,14,17,18,21,25].…”