Detection of higher order covariate relations embedded in multi-dimensional data

本研究的目的是开发新的方法，适用于检测单调和局部单调高阶协变量关系嵌入多维符号数据。我们得到了以下三个主要结果。(1)嵌入在多维符号数据中的局部单调链结构的检测：我们开发了一种能够检测高阶协变量关系的方法。该方法不需要函数识别过程，就能检测出多维符号数据表中的高阶多项式结构、正弦结构等。(2)广义相关系数：通过将众所周知的相关系数应用于与每个数据样本相关联的局部区域，并通过聚合局部相关性，我们有一个广义相关系数，它能够评估两个特征变量之间的高阶协变量关系。(3)单调结构的嵌套特性及其在符号数据分析中的应用：我们经常使用直方图表示来减少给定的庞大数据表。利用累积分布函数的单调性，将分位数方法推广到直方图型数据表的主成分分析中。分位数方法也可以处理符号数据分析中的其他研究问题。

This research aims to develop new methods applicable to detect monotone and locally monotone higher order covariate relations embedded in multidimensional symbolic data. We obtained the following three major results.(1) Detection of locally monotonic chain structures embedded in multidimensional symbolic data : We developed a method that is able to detect higher order covariate relations. By this method we can detect higher order polynomial structures, sinusoidal structures, and others in multidimensional symbolic data table without functional identification process.(2) A generalized correlation coefficient : By applying a well known correlation coefficient to local regions associated with each data sample and by aggregating the local correlations, we have a generalized correlation coefficient that is able to evaluate higher order covariate relations between two feature variables.(3) The characterization of monotone structures by the nesting property and its application to symbolic data analysis : We frequently use histogram representations in order to reduce given huge data tables. By the virtue of monotone property of the cumulative distribution functions, we developed the quantile method to the principal component analysis for histogram valued data tables. The quantile method may also be able to treat other research problems in symbolic data analysis.