Multidimensional Scaling
摘要
This chapter deals with a slightly different approach, but still associated with PCA. The dataset is not a set of variables and features, or a contingency table, but a pairwise distance array between some items. It is assumed implicitly that the distance array contains some information about the structure of the set of items. To reveal this structure, the objective is to recover a point cloud in a Euclidean space with a one-to-one correspondence between items and points, such that the Euclidean distance between points mimics as much as possible the distances as given by the distance array. Then, it is possible to derive a low-dimensional approximation of the point cloud with PCA, which offers the possibility to study its structure in a low-dimensional space, i.e., in a much easier way than in a high-dimensional space. Not all distance arrays can lead to such a procedure: In some cases, there exists no point cloud in a Euclidean space which can exactly mimic the distances as given. It is shown that, in such a case, a quadratic embedding provides a way to associate with the distance array a point cloud in a quadratic space endowed with a pseudo-Euclidean structure. How to reveal a shape in a pseudo-Euclidean space seems to be an open question, much less mature than in a Euclidean space.