8 - Multidimensional Scaling

Summary

In mathematics you don't understand things. You just get used to them (John von Neumann, 1903–1957; in Gary Zukav (1979), The Dancing Wu Li Masters).

Introduction

Suppose that we have n objects and that for each pair of objects a numeric quantity or a ranking describes the relationship between objects. The objects could be geographic locations, with a distance describing the relationship between locations. Other examples are different types of food or drink, with judges comparing items pairwise and providing a score for each pair. Multidimensional Scaling combines such pairwise information into a whole picture of the data and leads to a visual representation of the relationships.

Visual and geometric aspects have been essential parts of Multidimensional Scaling. For geographic locations, they lead to a map (see Figure 8.1). From comparisons and rankings of foods, drinks, perfumes or laptops, one typically reconstructs low-dimensional representations of the data and displays these representations graphically in order to gain insight into the relationships between the different objects of interest. In addition to these graphical representations, in a ranking of wines, for example, we might want to know which features result in wines that will sell well; the type of grape, the alcohol content and the region might be of interest.