利用广义特征值改进多元方差分析效率的探讨

发布时间：2018-04-20 14:06

本文选题：多元方差分析 + wilks检验统计量　；参考：《兰州财经大学》2017年硕士论文

【摘要】：多元方差分析是一种非常重要的多元统计分析方法,主要任务包括:检验各个因素对实验指标的影响是否显著、估计出各个因子不同水平的效应值、估计出各个因子水平之间的交互效应值、估计出协方差阵等等。其中首要任务是检验各个因素对实验指标的影响是否显著。为此,需要进行假设检验,较常使用的检验统计量有:wilks检验统计量、Hotelling迹检验统计量、Pillai-Bartlett准则检验统计量以及Roy最大特征值检验统计量。这些检验统计量经过适当变形,可以转化为服从F分布的检验统计量。这些检验统计量的导出过程计算量较大,将它们转化为服从F分布检验统计量的推导证明过程较难理解,需要用到高深的数理统计知识。为了克服上述问题,可以利用投影技术将高维数据降低为一维数据,能够证明投影后的数据仍然满足同方差且服从正态分布,可以直接基于线性变换后的数据构建F分布检验统计量进行多元方差分析,这种分析方法在很大程度上改进了多元方差分析的效率。基本思路为:该F检验统计量与一般的F统计量有所不同,其中投影方向事先并不知道,所以无法计算出检验统计量具体数值。投影技术可以使组间差异最大化,也可以使组间差异最小化,为了排除不同投影方向对检验结果的干扰,可以向使组间差异最小化方向投影,然后计算F检验统计量具体取值F_1,若该值落入拒绝域中,则向任何方向投影计算F检验统计量具体取值均落入拒绝域,故各组样本均值向量之间确实存在差异,依据小概率事件在一次实验中不发生原则,有充分理由拒绝原假设;如果F_1没有落入拒绝中,有可能向其他方向投影,计算的F检验统计量具体取值落入拒绝域,为了排除这种情况,可以计算出F检验统计量最大值F_2,若F统计量最大值仍然落入接受域,则向任何方向投影,计算F检验统计量具体取值均落入接受域,即各组样本的均值向量确实没有差异,故不拒绝原假设;如果F_1落入接受域,F_2落入拒绝域,则可以调整显著性水平,直到出现F_1落入拒绝域或者F_2落入接受域。
[Abstract]:Multivariate variance analysis is a very important method of multivariate statistical analysis. The main tasks include: to test whether the influence of each factor on the experimental index is significant, and to estimate the effect value of each factor at different levels. We estimate the interaction effect between each factor level, covariance matrix and so on. The primary task is to test whether the influence of each factor on the experimental indicators is significant. For this reason, we need to carry out hypothesis test. The commonly used test statistics are the: Wilks test statistic / Hotelling trace test statistic and the Pillai-Bartlett criterion test statistic and the Roy maximum eigenvalue test statistic. These test statistics can be transformed into test statistics from F distribution after proper deformation. It is difficult to understand the derivation process of these test statistics, which is difficult to understand when they are converted into the derivation of F distribution test statistics, so it is necessary to use advanced mathematical and statistical knowledge. In order to overcome the above problems, the projection technique can be used to reduce the high-dimensional data to one-dimensional data, and it can be proved that the projected data still satisfy the same square difference and follow the normal distribution. It is possible to construct F distribution test statistics directly based on linear transformation data for multivariate ANOVA. This method improves the efficiency of multivariate ANOVA to a great extent. The basic idea is that the F test statistic is different from the general F statistic, and the projection direction is not known in advance, so the concrete value of the test statistic can not be calculated. Projection technique can maximize or minimize the differences between groups. In order to eliminate interference of different projection directions to test results, it can be projected to minimize the differences between groups. If the value falls into the rejection domain, then the concrete value of F test statistic falls into the rejection domain, so there is a difference between the mean vectors of the samples in each group, and if the value of F test statistic falls into the rejection domain, the value of F test statistic can be calculated in any direction. According to the principle that a small probability event does not occur in an experiment, there is good reason to reject the original hypothesis. If FSP _ 1 does not fall into the rejection, it is possible to project in other direction, and the calculated F-test statistic falls into the rejection domain. In order to eliminate this situation, we can calculate the maximum value of F test statistic F2s. If the maximum value of F test statistic still falls into the acceptance domain, then if the maximum F statistic still falls into the acceptance domain, then projection in any direction to calculate the specific value of the F test statistic will fall into the acceptance domain. That is, there is no difference in the mean vector of each sample, so we do not reject the original hypothesis. If FS-1 falls into the receptive domain and FSt2 falls into the rejection domain, the significant level can be adjusted until FSP 1 falls into the rejection domain or FStus fall into the receptive domain.
【学位授予单位】：兰州财经大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.4

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