Aplicação da regressão Dirichlet na modelagem do desempenho dos campeões dos principais campeonatos de futebol

Abstract

Compositional data describe parts of a whole, usually summing to one in the case of proportions, or one hundred in the case of percentages. Traditional multivariate analysis methods are not suitable for handling this type of data. This thesis addresses methods for analyzing compositional data, covering topics such as the appropriate sample space, known as the Simplex, compositional operations, log-ratio transformations, and data visualization. In particular, it focuses on cases where compositions are the predictor variables in a regression analysis. The Dirichlet regression model and the estimation process through the Maximum Likelihood Method are discussed. To evaluate the effectiveness of this regression model, simulations were performed using the Monte Carlo method for parameter estimation, revealing a tendency for the estimates to converge toward the parameters as the sample size increased. Subsequently, applications were made to the Arctic Lake dataset, Brazilian football champions, and national football champions in 2022. In these applications, visualization techniques, descriptive statistics, and regression appropriate for compositional data were employed. For the latter two datasets, the composition of wins, draws, and losses was chosen as the response variable, and a Dirichlet regression model was fitted, allowing for insights into the behavior of this composition among national football champions.