摘要
判别分析的理念是在样本类别信息已知的情况下，建立判别分析模型来判别新观察的所属类别。判别分析同时也是数据预处理环节中的降维手段之一，它是一种有监督的分类。通过提取出最有利于分类的特征，以此空间做分类，这即是判别分析的主要任务。另外，本文考虑把在向量基础上的特征提取视作一维方法，把在矩阵基础上的特征提取视作二维方法。一维方法例如线性判别分析（LDA），其方法的目的就是确认出如何让费舍尔准则函数取出极值，并以取到此极值的向量即为最佳的投影方向。这样会改变样本在此最佳投影方向上的投影，使其具有最大的类间散布的同时也具有最小的类内散布。二维方法例如二维线性判别分析（2DLDA），是LDA处理矩阵型数据上的拓展，2DLDA最突出的优点是无需考虑如何把高维矩阵型数据转变成向量，故可以达到降低计算量这一目的。同时，一部分通过引入正则化过程的判别分析算法也被提出与优化，其考虑带来的额外信息进行新的估计。
然而，无论是已有的单边二维判别分析还是正则化判别分析，都分别存在着较高维度的类间和类内散布矩阵、特征维数较大等缺陷，为了进一步在特征提取上更加高效准确，故通过拓展正则化的判别分析到单边二维线性判别分析，得到了正则化的单边二维线性判别分析（R2DLDA）。
主要研究方法为在单边二维线性判别分析的基础上，加入正则化过程，正则化过程通过交叉验证的方法确定最优正则化参数 ，后寻最佳投影矩阵，进而判断分类效果。
后续本文为了验证R2DLDA的判别分类性能，考虑通过数据加以验证，即
选取了五个真实数据集，设计实验，并分别在这些数据集上加以实验。通过实验及得出的相应降维结果与分类错误率，同2DLDA、2DPCA进行对比，证实了R2DLDA其在分类准确率及降维的效果上是更优秀的。
关键词：判别分析；正则化；降维；分类
Abstract
The idea of discriminant analysis is to use some known research methods to determine the group to which a new observation sample belongs after dividing the target research object into several groups under certain number of samples. Discriminant analysis is also one of the dimensionality reduction methods in data preprocessing. It is a supervised classification. The main task of discriminant analysis is to extract the features that are most conducive to classification and use this space for classification. In addition, this paper considers the feature extraction based on the vector as a one-dimensional method and the feature extraction based on the matrix as a two-dimensional method. The one-dimensional method, such as LDA, aims to determine how to get the extreme value of the Fisher criterion function, and the vector that takes this extreme value is the best projection direction. This will
change the projection of the sample in this optimal projection direction, so that it has the largest inter-class dispersion and the smallest intra-class dispersion. Two-dimensional methods such as 2DLDA are an extension of LDA's processing of matrix data. The most prominent advantage of 2DLDA is that it does not need to consider how to convert high-dimensional matrix data into vectors, so it can achieve the goal of reducing our calculation amount. At the same time, part of the discriminant analysis algorithm introduced by the regularization process has also been continuously developed and improved, and it takes new information into consideration to make new estimates.
However, both the existing unilateral two-dimensional discriminant analysis and regularized discriminant analysis have defects such as higher-dimensional inter-class and intra-class dispersion matrices and larger feature dimensions. In order to further improve feature extraction, Efficient and accurate, so by extending the regularized linear discriminant analysis to the unilateral two-dimensional linear discriminant analysis, a regularized unilateral two-dimensional linear discriminant analysis (R2DLDA) is obtained.
The main research method is to add a regularization process based on unilateral two-dimensional linear discriminant analysis. The regularization process determines
the optimal regularization parameter by cross-validation, and then finds the best projection matrix to judge the classification effect.
In the following article, in order to verify the discriminative classification performance of R2DLDA, consider verifying with data, that is, five real data sets are selected, experiments are designed, and experiments are performed on these data sets. Through experiments and the corresponding dimensionality reduction results and classification error rates, compared with 2DLDA and 2DPCA, it is confirmed that R2DLDA is superior in classification accuracy and dimensionality reduction effect.
Key Words: Discriminant analysis；Regularization；Dimension reduction；Classification
目录
摘要............................................................................................................. I ABSTRACT ............................................................................................... I 第一章引言.. (1)
第一节选题背景 (1)
第二节问题提出 (2)
第三节研究目的和意义 (2)
第四节文献综述 (2)
一、传统判别分析 (3)
二、相关研究发展 (4)
第五节研究方法 (6)
第六节论文结构安排 (6)
第二章前期相关工作 (8)
第一节线性判别分析（LDA） (8)
一、基本思想 (8)
二、算法原理 (9)
三、主要步骤 (9)
第二节单边二维线性判别分析（2DLDA） (10)
一、基本思想 (10)
二、算法原理 (10)
三、主要步骤 (11)
第三节正则化判别分析（RDA） (11)
一、基本思想 (11)
二、算法原理 (12)
三、主要步骤 (12)
第四节主成分分析（PCA） (13)
一、基本思想 (13)
二、算法原理 (13)
三、主要步骤 (14)
第五节单边二维主成分分析（2DPCA） (15)
一、基本思想 (15)
二、算法原理 (15)
三、主要步骤 (15)
第六节本章小结 (16)
第三章正则化及其参数选择 (17)
第一节正则化及其作用 (17)
第二节常用方法 (18)
一、L1&L2正则化 (18)
二、作用及区别 (19)
第三节正则化参数的选择 (20)
一、交叉验证基本思想 (20)
二、方法介绍 (21)
第四节正则化在R2DLDA中的应用 (23)
第五节本章小结 (24)
第四章正则化的单边二维线性判别分析 (25)
第一节基本思想 (25)
第二节算法原理 (26)
第三节主要步骤 (27)
第四节本章小结 (28)
第五章实验 (29)
第一节数据集介绍 (29)
一、wafer数据集 (29)
二、AUSLAN数据集 (29)
三、Japanese V owels数据集 (30)
四、BCI数据集 (30)
五、ECG数据集 (30)
第二节在真实数据上的实验 (31)
>正则化最小二乘问题

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