基于非参数方法的HIV模型统计诊断问题的研究

发布时间：2018-06-02 06:44

本文选题：非参数回归模型 + 核估计　；参考：《西安电子科技大学》2015年博士论文

【摘要】：非参数方法因其良好的灵活性在经济、医药、生物等领域有着广泛的应用.近年来,随着生物学的迅猛发展以及生物数据的大量采集,生物统计的重要性日益显著,其中对艾滋病的研究已经引起了科学家的极大关注.艾滋病(AIDS)威胁着全人类的生命安全,目前对艾滋病病毒(HIV)的动态研究已成为AIDS研究领域的热门问题.描述HIV动态变化的HIV模型是一组非线性常微分方程,其中包括HIV病毒死亡率,CD4+T细胞的新生率等多个重要未知参数,准确估计这些参数可以给AIDS致病机理以及药效评估提供重要依据.然而相对于参数估计方法,一般的非参数估计方法,如核估计和局部线性估计的收敛速度较慢且对窗宽的选择敏感；另一方面,由于实际测量数据的波动性,HIV模型的参数估计常常是不稳健的.本文针对非参数估计方法和HIV模型统计诊断的相关问题进行研究,主要研究成果如下：1.将复合方法的思想应用于非参数估计法中,提出复合非参数估计法：首先选择不同的窗宽作相应的非参数估计,然后通过一个参数回归方法将备选的估计量重新组合.当回归函数具有2k阶连续导数时,所得到的新估计量的最优均方收敛速度达到O(n-4k/(4k+1)),即当回归函数足够光滑时,新估计量的收敛速度接近参数估计的收敛速度.最优窗宽的阶数为O(n-1/(4k+1)),并且即使采用的窗宽不是最优的,但在条件hj=O(n-α),且1/(5k)α1/5成立时,新估计量仍具有比普通非参数估计更小的均方误差,这说明新估计量对窗宽的选择稳健.由此从收敛速度和窗宽选择两方面改进了通常的非参数估计.模拟研究证实了复合方法的有效性.2.对HIV模型的参数估计进行了实证研究.针对观测数据含有测量误差的情形,利用两步估计法和广义光滑方法对HIV模型进行了参数估计,随机模拟实验证实了两种方法良好的估计效果.通过比较两种方法的表现得出：两步估计法的计算更加简单,而广义光滑方法的精度更高.3.针对HIV模型的两步估计方法,研究了基于均值漂移模型的统计诊断问题.给出了漂移参数的计算公式,构造了Score检验统计量,并给出其大样本分布.随机模拟实验和实例研究结果表明：(1)所构造的检验统计量的极限分布是经验分布的一个合理近似；(2)所构造的统计量可有效地检测中度以上的偏移；(3)相对于内点,边界点对HIV模型的参数估计具有更大的影响.基于以上结论,在对HIV模型进行参数估计时,我们应对边界点加以格外的关注.4.针对HIV模型的广义光滑估计方法,分别讨论了基于数据删除模型的统计诊断和基于似然距离的局部影响分析方法,并且分别给出了数据删除模型和扰动模型下的似然距离近似计算公式.通过模拟数据和临床数据分析发现：两种方法均能有效地检测出模型中的异常点.
[Abstract]:Non-parametric method has been widely used in economy, medicine, biology and so on because of its good flexibility. In recent years, with the rapid development of biology and a large number of biological data collection, the importance of biological statistics has become increasingly significant, among which the study of AIDS has attracted great attention of scientists. HIV / AIDS (HIV / AIDS) is a threat to the safety of human life. The dynamic study of HIV / AIDS (HIV) has become a hot issue in the field of AIDS research. The HIV model for describing the dynamic changes of HIV is a set of nonlinear ordinary differential equations, which include several important unknown parameters, such as the HIV virus mortality rate of CD4 T cells, and so on. Accurate estimation of these parameters can provide an important basis for the pathogenesis of AIDS and the evaluation of its efficacy. However, compared with parameter estimation methods, general nonparametric estimation methods, such as kernel estimation and local linear estimation, converge slowly and are sensitive to the choice of window width; on the other hand, Parameter estimation of HIV models is often not robust because of the volatility of measured data. In this paper, the nonparametric estimation method and the statistical diagnosis of HIV model are studied. The main research results are as follows: 1. The idea of compound method is applied to the nonparametric estimation method, and a compound nonparametric estimation method is proposed: firstly, different window widths are selected for the corresponding nonparametric estimation, and then the alternative estimators are recombined by a parametric regression method. When the regression function has a continuous derivative of order 2k, the optimal mean square convergence rate of the new estimator reaches O(n-4k/(4k 1, that is, when the regression function is smooth enough, the convergence rate of the new estimator is close to that of the parameter estimation. The order of the optimal window width is O(n-1/(4k 1 / 1, and even if the window width is not optimal, the new estimator still has a smaller mean square error than the ordinary nonparametric estimator when the condition hjn- 伪, and 1 / 5) 伪 / 1 / 5, is established, which shows that the new estimator is robust to the window width selection. Thus, the general nonparametric estimation is improved in terms of convergence rate and window width selection. The simulation results show that the composite method is effective. 2. The parameter estimation of HIV model is studied empirically. Two step estimation method and generalized smoothing method are used to estimate the parameters of HIV model for the case where the observed data contain measurement errors. The results of stochastic simulation show that the two methods have good estimation effect. By comparing the performance of the two methods, it is found that the two-step estimation method is simpler, while the generalized smoothing method has higher accuracy. For the two-step estimation of HIV model, the statistical diagnosis problem based on mean shift model is studied. The calculation formula of drift parameter is given, the Score test statistic is constructed, and its large sample distribution is given. The results of random simulation experiments and case studies show that the limit distribution of the test statistics constructed by: 1) is a reasonable approximation of the empirical distribution. The boundary point has more influence on the parameter estimation of HIV model. Based on the above conclusions, we should pay special attention to the boundary point when we estimate the parameters of HIV model. For the generalized smooth estimation method of HIV model, the statistical diagnosis based on data deletion model and the local impact analysis method based on likelihood distance are discussed respectively. The approximate formulas of the likelihood distance under the data deletion model and the perturbation model are given respectively. Through the analysis of simulation data and clinical data, it is found that both methods can effectively detect abnormal points in the model.
【学位授予单位】：西安电子科技大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O212.1

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