Patterns › Dimension reduction

MDS (classical)

Classical (metric) multi-dimensional scaling embeds rows into a low-dimensional space whose pairwise Euclidean distances best preserve the original metric distances.

What is MDS (classical)?

Given an n × n distance matrix D, classical MDS finds the k-dimensional Euclidean configuration whose distances best match D in the least-squares sense. For Euclidean input distances, MDS coordinates are essentially PCA scores on the original features — so for numeric feature data, PCA is the more direct route.

For non-Euclidean distances (Manhattan, correlation, custom), MDS is the right tool — it works on any symmetric distance matrix without needing the underlying coordinates.

When should I use MDS (classical)?

  • Visualising any pairwise distance matrix in 2D.
  • Non-Euclidean similarity / distance representations.
  • Switch to non-metric MDS when only the rank order of distances is reliable.

What data does it need?

≥ 2 numeric feature columns + distance metric + n components.

What does it report?

Coordinates + stress / eigenvalue summary.

How do I interpret the result?

Cumulative variance explained by the first k components measures how well the embedding represents the original distances; > 80% with k = 2 ⇒ the 2D plot captures the essential structure.

See also

References

  • Torgerson (1958). Theory and Methods of Scaling.