A Topological Approach of Principal Component Analysis

摘要

Large datasets are increasingly widespread in many disciplines. The exponential growth of data requires the development of more data analysis methods in order to process information more efficiently. In order to better visualize the data, many methods such as Principal Component Analysis (PCA) and MultiDimensional Scaling (MDS) allow to extract a low-dimensional structure from high-dimensional data set. The proposed approach, called Topological Principal Component Analysis (TPCA), is a multidimensional descriptive method witch studies a homogeneous set of continuous variables defined on the same set of individuals. It is a topological method of data analysis that consists of comparing and classifying proximity measures from among some of the most widely used proximity measures for continuous data. Proximity measures play an important role in many areas of data analysis, the results strongly depend on the proximity measure chosen. So, among the many existing measures, which one is most useful? Are they all equivalent? How to identify the one that is most appropriate to analyze the correlation structure of a set of quantitative variables. TPCA proposes an appropriate adjacency matrix associated to an unknown proximity measure according to the data under consideration, then analyzes and visualizes, with graphic representations, the relationship structure of the variables relating to, the well known PCA problem. Its uses the concept of neighborhood graphs and compares a set of proximity measures for continuous data which can be more-or-less equivalent a topological equivalence criterion between two proximity measures is defined and statistically tested according to the topological correlation between the variables considered. An example on real data illustrates the proposed approach.