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In this dissertation, a new non-global regression model - the partial linear threshold regression model (PLTRM) - is proposed. Various issues related to the PLTRM are discussed. In the first main section of the dissertation (Chapter 2), we define what is meant by the term “non-global regression model”, and we provide a brief review of the current literature associated with such models. In particular, we focus on their advantages and disadvantages in terms of their statistical properties. Because there are some weaknesses in the existing non-global regression models, we propose the PLTRM. The PLTRM combines non-parametric modelling with the traditional threshold regression models (TRMs), and hence can be thought of as an extension of the later models. We verify the performance of the PLTRM through a series of Monte Carlo simulation experiments. These experiments use a simulated data set that exhibits partial linear and partial nonlinear characteristics, and the PLTRM out-performs several competing parametric and non-parametric models in terms of the Mean Squared Error (MSE) of the within-sample fit. In the second main section of this dissertation (Chapter 3), we propose a method of estimation for the PLTRM. This requires estimating the parameters of the parametric part of the model; estimating the threshold; and fitting the non-parametric component of the model. An “unbalanced penalized least squares” approach is used. This involves using restricted penalized regression spline and smoothing spline techniques for the non-parametric component of the model; the least squares method for the linear parametric part of the model; together with a search procedure to estimate the threshold value. This estimation procedure is discussed for three mutually exclusive situations, which are classified according to the way in which the two components of the PLTRM “join” at the threshold. Bootstrap sampling distributions of the estimators are provided using the parametric bootstrap technique. The various estimators appear to have good sampling properties in most of the situations that are considered. Inference issues such as hypothesis testing and confidence interval construction for the PLTRM are also investigated. In the third main section of the dissertation (Chapter 4), we illustrate the usefulness of the PLTRM, and the application of the proposed estimation methods, by modelling various real-world data sets. These examples demonstrate both the good statistical performance, and the great application potential, of the PLTRM. / Graduate

[pt] Este trabalho propõe um novo modelo linear por partes para a extração de regras de conhecimento de banco de dados. O modelo é uma heurística baseada em análise de árvore de regressão, como introduzido por Friedman (1979) e discutido em detalhe por Breiman (1984). A motivação desta pesquisa é trazer uma nova abordagem combinando técnicas estatísticas de modelagem e um algoritmo de busca por quebras eficiente. A decisão de quebra usada no algoritmo de busca leva em consideração informações do ajuste de equações lineares e foi implementado tendo por inspiração o trabalho de Tsay (1989). Neste, ele sugere um procedimento para construção um modelo para a análise de séries temporais chamado TAR (threshold autoregressive model), introduzido por Tong (1978) e discutido em detalhes por Tong e Lim (1980) e Tong (1983). O modelo TAR é um modelo linear por partes cuja idéia central é alterar os parâmetros do modelo linear autoregressivo de acordo com o valor de uma variável observada, chamada de variável limiar. No trabalho de Tsay, a Identificação do número e localização do potencial limiar era baseada na analise de gráficos. A idéia foi então criar um novo algoritmo todo automatizado. Este processo é um algoritmo que preserva o método de regressão por mínimos quadrados recursivo (MQR) usado no trabalho de Tsay. Esta talvez seja uma das grandes vantagens da metodologia introduzida neste trabalho, visto que Cooper (1998) em seu trabalho de análise de múltiplos regimes afirma não ser possível testar cada quebra. Da combinação da árvore de decisão com a técnica de regressão (MQR), o modelo se tornou o TS-TARX (Tree Structured - Threshold AutoRegression with eXternal variables). O procedimento consiste numa busca em árvore binária calculando a estatística F para a seleção das variáveis e o critério de informação BIC para a seleção dos modelos. Ao final, o algoritmo gera como resposta uma árvore de decisão (por meio de regras) e as equações de regressão estimadas para cada regime da partição. A principal característica deste tipo de resposta é sua fácil interpretação. O trabalho conclui com algumas aplicações em bases de dados padrões encontradas na literatura e outras que auxiliarão o entendimento do processo implementado. / [en] This research work proposes a new piecewise linear model to extract knowledge rules from databases. The model is an heuristic based on analysis of regression trees, introduced by Friedman (1979) and discussed in detail by Breiman (1984). The motivation of this research is to come up with a new approach combining both statistical modeling techniques and an efficient split search algorithm. The split decision used in the split search algorithm counts on information from adjusted linear equation and was implemented inspired by the work of Tsay (1989). In his work, he suggests a model-building procedure for a nonlinear time series model called by TAR (threshold autoregressive model), first proposed by Tong (1978) and discussed in detail by Tong and Lim (1980) and Tong (1983). The TAR model is a piecewise linear model which main idea is to set the coefficients of a linear autoregressive process in accordance with a value of observed variable, called by threshold variable. Tsay`s identification of the number and location of the potential thresholds was based on supplementary graphic devices. The idea is to get the whole process automatic on a new model-building process. This process is an algorithm that preserves the method of regression by recursive least squares (RLS) used in Tsay`s work. This regression method allowed the test of all possibilities of data split. Perhaps that is the main advantage of the methodology introduced in this work, seeing that Cooper, S. (1998) said about the impossibility of testing each break.Thus, combining decision tree methodology with a regression technique (RLS), the model became the TS-TARX (Tree Structured - Threshold AutoRegression with eXternal variables). It searches on a binary tree calculating F statistics for variable selection and the information criteria BIC for model selection. In the end, the algorithm produces as result a decision tree and a regression equation adjusted to each regime of the partition defined by the decision tree. Its major advantage is easy interpretation.This research work concludes with some applications in benchmark databases from literature and others that helps the understanding of the algorithm process.