A STUDY OF SYMBOLIC DATA ANALYSIS BASED ON NEIGHBORHOOD GRAPHS

符号数据分析（Symbolic Data Analysis，SDA）是广义数据表的一个新的研究领域，在广义数据表中，每一个样本不仅由定量数据描述，而且由定性数据描述。我们报告的研究结果：（1）在分类问题的特征选择方法;（2）一种新的相关性分析的方法。（1）分类问题的特征选择当我们只有有限数量的样本时，通过添加用于描述样本的新特征可能无法提高分类性能。这意味着我们必须在类间的可扩展性和类描述的通用性之间取得平衡。为此，我们引入了两个新的邻域图：广义有序互邻域图和广义有序类间互邻域图。通过使用这些新的邻域图，我们得到了一个简单的特征选择算法，它达到了上述的平衡。（2）广义相关系数Pearson相关系数可以用来检测特征变量之间的因果关系。然而，这个众所周知的工具是不适用于一般的非线性因果关系。如果两个特征变量遵循函数结构，则关于特征变量的样本分布具有几何上薄的结构。从这个角度来看，我们开发了两个功能的卡尔霍恩相关系数。为了处理三维或高维特征空间中的几何厚度问题，我们引入了一种称为“广义有序相对邻域图”的邻域图。作为一个基本的结果，我们发现，我们可以评估几何厚度的许多分布，如绳在高维特征空间。

The purpose of this research is to develop some methods for Symbolic Data Analysis (SDA).SDA is a new research field for generalized data table in which each sample is described by not only quantitative data but also qualitative data. We report research results : (1)a feature selection method in classification problems ; and (2)an approach to new correlation analysis.(1)Feature selection for classification problemsWhen we have only a finite number of samples, the classification performance may not be improved by the addition of new features used to describe samples. This means that we have to strike a balance between the interclass distinguish-ability and the generality of class descriptions. For this purpose, we introduced two new neighborhood graphs called "the generality ordered mutual neighborhood graph" and "the generality ordered interclass mutual neighborhood graph". By using these new neighborhood graphs, we obtained a simple feature selection algorithm which strikes the balance described in the above.(2)Generalized correlation coefficientPearson's correlation coefficient is useful to detect causality between feature variables. However, this well known tool is not applicable to general nonlinear causal relations. If two feature variables follow to a functional structure, the sample distribution with respect to the feature variables has a geometrically thin structure. From this viewpoint, we developed the Calhoun correlation coefficient for two features. We introduced a neighborhood graph called "generality ordered relative neighborhood graph" in order to treat geometrical thickness in three or more high dimensional feature spaces. As a basic result, we found that we can evaluate the geometrical thickness for many distributions like ropes in high dimensional feature spaces.