1. Multidimensional Scaling Vuokko Vuori 20.10.1999 Based on: Data Exploration Using Self-Organizing Maps, Samuel Kaski, Ph.D. Thesis, 1997 Multivariate Statistical Analysis, A Conceptual Introduction, Kachigan Pattern Recognition and Neural Networks, B. D. Ripley

2. Contents l Motivations l Dissimilarity matrix l Multidimensional scaling (MDS) l Sammon’s mapping l Self-Organizing maps Comparison between MDS, Sammon’s mapping, and SOM

3. Motivations MDS attempts to l Identify abstract variables which have generated the inter-object similarity measures l Reduce the dimension of the data in a non-linear fashion l Reproduce non-linear higher-dimen- sional structures on a lower-dimen- sional display

4. Dissimilarity Matrix In MDS, the dissimilarities between every pair of observations are given l Genuine distances (continuos data) l Simple matching coefficients, Jaccard coefficients (categorical data) l Scaled ranks (ordinal data) l Gower’s dissimilarity for mixed data:

5. Multidimensional Scaling Metric MDS: l Distances between data items are given, a configuration of points which gives rise to those distances is sought l Can be used for non-linear projection l Objective function which is minimized:

6. Nonmetric MDS: l Only the rank order of the distances is important l A monotonically increasing function that acts on the original distances is introduced: the rank order can be better preserved l Normalized objective function: l For given projection, is always chosen to minimize

7. Sammon’s Mapping l Closely related to metric MDS l Tries to preserve pairwise distances l Errors in distance preservation are normalized with the original distance l Objective function:

8. Self-Organizing Maps l Algorithm that performs clustering and non-linear projection onto lower dimen- sion at the same time l Finds and orders a set of reference vectors located on a discrete lattice l Learning rule: l Objective function: (Discrete data, fixed neighbourhood kernel)

9. Comparison Between MDS, Sammon’s Mapping and SOM l MDS tries to preserve the metric (ordering relations) of the original space, long distances dominate over the shorter ones l SOM tries to preserve the topology (local neighbourhood relations), items projected to nearby locations are similar l Sammon’s lies in the middle: it is like MDS but puts more emphasis on small distances