Cluster Analysis and Multidimensional Scaling

Abstract

Discriminant analysis is used to evaluate group separation and to develop rules for assigning observations to groups. Cluster analysis is concerned with group identification. The goal of cluster analysis is to partition a set of observations into a distinct number of unknown groups or clusters in such a manner that all observations within a group are similar, while observations in different groups are not similar. If data are represented as an n x p matrix Y = [y _ij ] where

$$ \mathop Y\limits_{n \times p} = \left[ \begin{gathered} y'_1 \\ y'_i \\ \vdots \\ y'_n \\ \end{gathered} \right] $$

the goal of cluster analysis is to develop a classification scheme that will partition the rows of Y into k distinct groups (clusters). The rows of the matrix usually represent items or objects. To uncover the groupings in the data, a measure of nearness, also called a proximity measure needs to be defined. Two natural measures of nearness are the degree of distance or “dissimilarity” and the degree of association or “similarity” between groups. The choice of the proximity measure depends on the subject matter, scale of measurement (nominal, ordinal, interval, ratio), and type of variables (continuous, categorical) being analyzed. In many applications of cluster analysis, one begins with a proximity matrix rather than a data matrix. Given the proximity matrix of order (n x n) say, the entries may represent dissimilarities [d _rs ] or similarities [s _rs ] between the r^th and s^th objects. Cluster analysis is a tool for classifying objects into groups and is not concerned with the geometric representation of the objects in a low-dimensional space. To explore the dimensionality of the space, one may use multidimensional scaling.