On the geometry of generalized linear models : submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Massey University, Palmerston North, New Zealand
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Date
2003
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Massey University
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Abstract
The perspective afforded by Euclidean geometry led to the rapid development of linear models in the early stages of the twentieth century: Fisher saw the data as a point in finite-dimensional Euclidean space, the model as a subspace and least squares fitting as projection of the observation vector onto the model space. From the late 1960s to early 1970s, Fienberg revealed geometry underlying loglinear models for two-way tables, while Haberman discussed geometry for the log-transformed case. Generalized linear models, however, have largely eluded geometers until recently. In 1997 an extension of Fisher's view to generalized linear models was given by Kass and Vos, using the language of differential geometry. The aim of this work is to develop a simple, general geometric framework for generalized linear models, closely related to the thinking of Fienberg and Haberman. Whereas Kass and Vos developed a geometric view which leads to the usual scoring method, we develop geometry which leads to a new algorithm. A linearization of this new algorithm yields the scoring method. The geometry discussed by Kass and Vos is based on the log-likelihood function whereas the geometry developed here depends on sufficiency. In the geometry of generalized linear models, developed through chapters 1 to 3, an observation with n values is viewed as a vector in Euclidian space Rn. This Euclidian space Rn is partitioned into two orthogonal spaces, the sufficiency space S and the auxiliary space A, with respect to a new basis. We focus on two mean sets relating to generalized linear models, one for the untransformed model space and another for the link-transformed model space. There are two critical properties of the maximum likelihood estimate of the parameters of a generalized linear model with canonical link. The first property is that the coefficients of the basis of the sufficiency space, the sufficient statistics, are preserved in the untransformed model space in the fitting process. The second property is that the coefficients of the basis of the auxiliary space are zeroed in the link-transformed model space in the fitting process. Linear models and loglinear models serve as special cases of generalized linear models with identity and log link respectively. Based on the geometric framework discussed in the thesis, a new algorithm is constructed for fitting generalized linear models with canonical link in Chapter 4. This algorithm, which relies on sufficient statistics for the parameters in the model rather than the likelihood function, takes two projections alternately, orthogonal projection onto a sufficiency affine plane and non-orthogonal projection onto the transformed model space. In the process, we match the model space and sufficient statistics iteratively until convergence. Linearization of the new algorithm induces the scoring method. In Chapter 5 we pay special attention to a subset of loglinear models, graphical loglinear models, those which are the intersection of a finite set of conditional independence statements. The model space of one conditional independence statement is described through the notions of "corresponding point convex hull" and "set convex hull". The fitting of one conditional independence statement is considered geometrically using a direct fitting method and the familiar iterative proportional fitting method.
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Geometry, Linear models