多因子Poisson回归模型的D-最优设计

发布时间：2018-07-01 19:40

本文选题：Poisson回归 + D-最优　；参考：《上海师范大学》2016年硕士论文

【摘要】：当今社会信息高速发展,大量信息以数据的形式存在,信息的多样性使得数据形式也具有多样性。在处理离散型数据时,传统的线性模型具有很大的局限性,此时就需要借助广义线性模型。Poisson回归模型是广义线性模型的一个重要的分支,主要用来研究计数型数据。近年来,Poisson回归模型的最优设计问题逐渐引起大家的关注。现有的研究中多是基于模型中只含定量因子的情形,对含有定性因子的Poisson回归模型的研究较少。且多因子设计问题较为复杂,单因子设计问题则更为简单,其研究也比较成熟。所以本文旨在将复杂的多因子问题转化为简单的单因子问题,主要就以下两方面进行了研究：对含有多个定量因子的Poisson回归可加模型的D-最优设计问题,首先将回归函数作典则变换以消除对参数的依赖性;然后转化成求异方差线性可加模型的D-最优设计问题,接下来就此问题进行研究；对其异方差子模型,定义了一种新的最优准则,通过计算方向导数得到其等价条件并据此进行算法构造；最后对回归函数作中心化变换并借助一般等价性定理,证明了其D-最优设计是其异方差子模型与其同方差子模型的最优设计的乘积设计,从而使问题得到解决。对含有定性因子的多因子Poisson回归可加模型的D-最优设计问题,则按照惯例首先引入哑变量,然后将回归函数作典则变换,并将设计问题转化为该含定性因子的部分异方差线性可加模型的设计问题；最后利用第一部分的结论,解决此异方差模型的设计问题。两方面的研究都表明：多因子Poisson回归模型的设计问题可以转化为单因子设计问题来解决,从而使设计问题得到简化。此外,本文就这两类问题都给出了例题进行演示。
[Abstract]:Nowadays, with the rapid development of social information, a great deal of information exists in the form of data. When dealing with discrete data, the traditional linear model has great limitations. In this case, the generalized linear model .Poisson regression model is an important branch of the generalized linear model, which is mainly used to study the counting-type data. In recent years, the optimal design of Poisson regression model has attracted more and more attention. Most of the existing studies are based on the case where there are only quantitative factors in the model, but the Poisson regression model with qualitative factors is less studied. The problem of multi-factor design is more complex, the problem of single-factor design is more simple, and its research is more mature. Therefore, this paper aims to transform the complex multi-factor problem into a simple single-factor problem. This paper mainly studies the following two aspects: the D- optimal design problem for Poisson regression additive model with multiple quantitative factors. Firstly, the regression function is canonical transformed to eliminate the dependence on parameters, and then the D- optimal design problem of the linear additive model of heteroscedasticity is transformed into a D- optimal design problem, and the heteroscedasticity submodel is studied. In this paper, a new optimal criterion is defined, and the equivalent condition is obtained by calculating the directional derivative and the algorithm is constructed. Finally, the central transformation of the regression function is made and the general equivalence theorem is used. It is proved that the D- optimal design is the product design of the optimal design of its heteroscedasticity submodel and its isomorphic submodel, so that the problem can be solved. For the D- optimal design problem of multivariate Poisson regression additive model with qualitative factors, the dummy variable is first introduced according to the convention, and then the regression function is canonical transformed. The design problem is transformed into the design problem of the partial heteroscedasticity linear additive model with qualitative factors, and the design problem of the heteroscedasticity model is solved by using the conclusions in the first part. Both studies show that the design problem of multi-factor Poisson regression model can be transformed into a single-factor design problem and the design problem can be simplified. In addition, examples of both kinds of problems are given in this paper.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：F224

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