When constructing a statistical model, nonlinearity detection has always been an interesting topic and a difficult problem. To balance precision of parametric modeling and robustness of nonparametric modeling, the semi-parametric modeling method has shown very good performance. The specific example, spline fitting, can very well estimate nonlinear patterns. However, as the number of spline bases goes up, the method can generate a large amount of parameters to estimate, especially for multiple dimensional case. It's been discussed in the literature to treat additional slopes of spline bases as random terms, then those slopes can be controlled with a single variance term. The semi-parametric model then becomes a linear mixed effect problem.
Data of large dimensions has become a serious computation burden, especially when it comes to nonlinearity. A good dimension reduction technique is needed to ease this situation. Methods like LASSO type penalties have very good performance in linear regression. Traditional LASSO add a restriction on slopes to the model. Parameters can be shrunk to 0. Here we extend that method to semi-parametric spline fitting, making it possible to reduce dimensions of nonlinearity. The problem of nonlinearity detection is then transformed to a model selection problem. The penalty is taken on variance terms which control nonlinearity in each dimension. As the limit value changes, variance terms can be shrunk to 0. When one variance term is reduced to 0, the nonlinear part of that dimension is removed from the model. AIC/BIC criteria are used to choose the final model. This method is very challenging since testing is almost impossible due to the boundary situation.
The method is further extended to generalized additive model. Quasi-likelihood is adopted to simplify the problem, making it similar to partially linear additive case. LASSO type penalties are again performed on variance components of each dimension, making dimension reduction possible for nonlinear terms. Conditional AIC/BIC is used to select the model.
The dissertation is consisted of five parts.
In Chapter 1, we have a thorough literature review. All previous works including semi-parametric modeling, penalized spline fitting, linear mixed effect modeling, variable selection methods, and generalized nonparametric modeling are all introduced here.
In Chapter 2, the model construction is explained in detail for single dimension case. It includes derivation of iteration procedures, computation technique discussion, simulation studies including power analysis, and discussions of other parameter estimation methods.
In Chapter 3, the model is extended to multiple dimensional case. In addition to model construction, derivation of iteration procedures, computation technique discussion and simulation studies, we have a real data example, using plasma beta-carotene data from a nutritional study. The result shows advantage of nonlinearity detection.
In Chapter 4, generalized additive modeling is considered. We especially focus on the two most commonly used distributions, Bernoulli distribution and Poisson distribution. Model is constructed using Quasi-likelihood. Two iteration methods are introduced here. Simulation studies are performed on both distributions of one dimensional and multiple dimensional case. We have a real data example using Pima Indian diabetes study dataset. The result also shows advantage of nonlinearity detection.
In Chapter 5, some possible future works are dicussed. The topics include more complicated covariance matrix structure of random terms, dimension reduction for both linearity and nonlinearity at the same time, bootstrap method with model selection taken into account, and higher degree p-spline setup.
|Commitee:||Barut, Emre, Lai, Yinglei, Li, Yuanzhang, Wang, Huixia|
|School:||The George Washington University|
|School Location:||United States -- District of Columbia|
|Source:||DAI-B 80/01(E), Dissertation Abstracts International|
|Keywords:||Generalized additive models, Mixed effect, Model selection, Nonlinearity detection, Semiparametric, Spline|
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