Multidimensional Scaling (MDS)
- Visual method that places similar objects close together and dissimilar objects farther apart in a low-dimensional plot (typically 2 or 3 dimensions).
- Starts from pairwise similarities (or dissimilarities), converts them to distances, and finds low-dimensional coordinates that best preserve those distances.
- Classical scaling minimizes “stress” (the difference between original distances and distances in the low-dimensional representation).
Definition
Section titled “Definition”Multidimensional scaling (MDS) is a technique used to visualize the underlying similarities and dissimilarities between a set of objects by representing the objects in a low-dimensional space (typically two or three dimensions) while preserving the distances between the objects as much as possible.
Explanation
Section titled “Explanation”MDS represents similarity relationships between objects as positions in a low-dimensional spatial representation: objects that are similar are placed close together, and objects that are dissimilar are placed further apart. The typical workflow described in the example is:
- Represent pairwise similarities in a similarity matrix.
- Compute a distance matrix from the similarity matrix (in the example, by taking the inverse of similarity values).
- Use a method such as classical scaling to find coordinates in a low-dimensional space that minimize the stress, i.e., the discrepancy between distances in the low-dimensional space and the distances in the distance matrix.
Classical scaling seeks coordinates that best preserve the original pairwise distances according to a chosen stress criterion.
Examples
Section titled “Examples”Example: four objects A, B, C, and D.
Similarity matrix:
| A | B | C | D | |
|---|---|---|---|---|
| A | 1.0 | 0.8 | 0.6 | 0.5 |
| B | 0.8 | 1.0 | 0.7 | 0.4 |
| C | 0.6 | 0.7 | 1.0 | 0.3 |
| D | 0.5 | 0.4 | 0.3 | 1.0 |
Distance matrix (computed as the inverse of the similarity matrix in this example):
| A | B | C | D | |
|---|---|---|---|---|
| A | 1.0 | 1.25 | 1.67 | 2.00 |
| B | 1.25 | 1.0 | 1.43 | 2.50 |
| C | 1.67 | 1.43 | 1.0 | 3.33 |
| D | 2.00 | 2.50 | 3.33 | 1.0 |
Two-dimensional representation found by classical scaling (rows X and Y give coordinate components in the example):
| A | B | C | D | |
|---|---|---|---|---|
| X | 1.0 | 0.8 | 0.6 | 0.5 |
| Y | 0.8 | 1.0 | 0.7 | 0.4 |
As shown, A and B appear close together in the plot, reflecting their higher similarity, while C and D are further apart, reflecting lower similarity.
Use cases
Section titled “Use cases”- Market research: represent customer preferences for products so similar preferences appear near each other in the MDS plot.
- Psychology: represent similarity between personality traits so similar traits appear close together.
- Biology: visualize similarity or dissimilarity relationships among biological entities.
Related terms
Section titled “Related terms”- Classical scaling
- Stress (stress function)
- Similarity matrix
- Distance matrix