稳健加权主成分及因子聚类方法的研究

【作者】 刘玲；

【导师】 刘惠篮；

【作者基本信息】 贵州大学 ， 数学， 2022， 硕士

【摘要】 聚类分析方法是一种十分经典有效的分类方法,但当数据中出现异常值或者存在高度相关性时,传统聚类方法的结果会失真.而主成分分析和因子分析是能够处理数据中高度相关性的方法,因此常将两种方法和聚类分析结合起来使用;但传统主成分聚类、因子聚类方法忽略了不同主成分、不同因子对聚类的重要性差异.而加权主成分距离聚类分析方法既可以处理数据之间的相关性,同时又考虑了不同主成分对聚类的差异.稳健主成分、稳健因子分析可以有效抵抗离群值,又可以处理数据中的高相关性.因此,本文将稳健主成分、稳健因子分析与加权主成分距离聚类分析相结合,既能够处理数据中的高度相关性,又能有效抵御异常值的影响,且又能考虑到不同主成分、不同因子对聚类的重要性差异.本文的研究内容主要包括以下两个方面:一、基于稳健主成分聚类分析方法和加权主成分距离聚类分析方法,提出了一种可行的稳健加权主成分聚类方法.该方法集稳健主成分聚类与加权主成分距离聚类方法的优点于一身,既可以减少异常值的影响,又考虑了不同主成分对聚类的影响.并在不同异常值比例下,证明了该方法的稳健性,并通过数值模拟和实例分析,进一步说明了所提方法的表现效果.二、基于稳健因子分析和加权主成分距离聚类分析方法,提出了稳健加权因子聚类方法,该方法以FAST-MCD方法为稳健估计方法,又考虑了不同因子本身的差异.并通过数值模拟,在不同的异常值比例下,证明所提方法的有效性.更多还原

【Abstract】 Cluster analysis method is a very classic and effective classification method,but when there are outliers in the data or there is a high degree of correlation,the results of the traditional clustering method will be distorted.And principal components analysis and factor analysis are the methods that can handle the high correlation of the data,so the two methods are often combined with clustering analysis.However,traditional principal components clustering and factor clustering methods ignore the importance of different principal components and factors for clustering.The weighted principal components distance clustering analysis method can not only deal with the correlation of the data,but also consider the differences between different principal components on the cluster.Robust principal components and robust factor analysis can effectively resist outliers,and can deal with high correlation of the data.Therefore,this paper combines robust principal components,robust factor analysis and weighted principal components distance clustering analysis,which can not only deal with the high correlation of the data,and can effectively resist the influence of outliers,and can take into account different principal components,differences in the importance of different factors for clustering.The research content of this paper mainly includes the following two aspects:1.Based on the robust principal components clustering analysis method and the weighted principal components distance clustering analysis method,a feasible robust weighted principal components clustering method is proposed.The method combines the advantages of robust principal components clustering and weighted principal components distance clustering method,which can reduce the influence of outliers and consider the influence of different principal components for clustering.And at different outlier ratios,the robustness of the method is proved,and the performance effect of the proposed method is further proved by numerical simulation and case analysis.2.Based on the robust factor analysis and weighted principal components distance clustering method,a robust weighted factor clustering method is proposed,which takes the FAST-MCD method as the robust estimation method,and considers the differences between different factors themselves.And through numerical simulation,the effectiveness of the proposed method is proved under different outlier ratios.更多还原