Introdução à teoria dos modelos polinomiais fracionários

Abstract

We frequently deal with situations where we wish to model one variable’s behaviour as a function of other variables of interest. The set of statistical, mathematical and probabilistic techniques used in such situations is called regression models. For several decades, the linear models have been the most commonly used class of regression models. However, in many situations, these are not sufficient to model our variable of interest’s behaviour. One of the reasons of that is the fact that very often data do not satisfy the linear model assumptions. Therefore, several more flexible model classes have been proposed. Among them, we can cite polynomial models, which allow us to model the regression function through polynomials of the explanatory variables. These can also work as a good approximation in cases where this function is nonlinear. On the other hand, such models may not fit properly data with certain characteristics; for instance, this model class tends not to fit properly regression functions that have an asymptote and in some cases, is particularly vulnerable to influential points. Alternatives to polynomial models include regression models based on smoothing functions, such as lowess, which are non-parametric, and FPMs (fractional polynomial models). In particular, FPMs commonly outperform the usual polynomial models, mostly due to its flexibility, considering a fully parametric modeling. Such models have been used along with more complex model classes, such as GLMs (generalised linear models). In this work, we introduce the fundamentals of FPMs with computational support of the R software. We also present an application of FPMs in real data, aiming to illustrate the application of this model class.