Generalized boosted additive models

The doctoral thesis is focused on non-parametric nonlinear regression and additive modeling. Regression analysis is a central method of statistical data analysis. Linear regression concerns the conditional distribution of a dependent variable, Y , as a function of one or more predictors, or independent variables. The main characteristics of this model are its parametric form and the hypothesis that the underlying relationship between the outcome and the predictors is linear. For this reason this method is often inappropriate to model this relationship when it is characterized by complex nonlinear patterns and it can fail to capture important features of the data. In such cases, nonparametric regression, which allows to determine the functional form between the dependent variable, Y , and the explicative variables by the data themselves, is more suitable. Hence, nonparametric methods become increasingly popular and apply to many area of research and practical problems. These methods show a great exibility compared to parametric ones, but they also present an important drawback known as curse of dimensionality, which involves that the precision of the estimates obtained via these methods is in inverse proportion to the number of explicative variables that are included in the model. To overcome this problem Generalized Additive Models (GAM) were introduced. GAMs are based on the assumption that the conditional value of the outcome variable can be expressed as the sum of a certain number of univariate nonlinear functions, one for each predictor that is included in the model. One major concern to the use of the GAM is, therefore, when concurvity is present in the data. Concurvity can be de_ned as the presence of nonlinear dependencies among transformations of the explanatory variables considered in the model. One of the most common case of concurvity directly follows from the presence of collinearity among the untransformed predictors. In the context of generalized additive models the presence of concurvity leads to biased estimates of the model parameters and of their standard errors. For such reasons we explore an alternative class of models, CATREG, based on the Regression with Transformation approach, applying the optimal scaling methodology as presented in the Gi_ system. When we use this class of models in the presence of collinearity among untransformed predictors, applying nonlinear transformations through optimal scaling implies that interdependence among these predictor decreases. Moreover in the framework of nonlinear regression with optimal scaling, we follow the approach proposed by Meulman (2003) of considering models in which, applying the basic idea of a forward stagewise boosting procedure, we introduce in the model nonlinear prediction components in a sequential way with the aim of improving the predictive power of the model itself. We call this approach the Generalized Boosted Additive Model (GBAM).