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Multidimensional Scaling

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. Contents. Motivations

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Multidimensional Scaling

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

3. Motivations MDS attempts to • Identify abstract variables which have generated the inter-object similarity measures • Reduce the dimension of the data in a non-linear fashion • 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 • Genuine distances (continuos data) • Simple matching coefficients, Jaccard coefficients (categorical data) • Scaled ranks (ordinal data) • Gower’s dissimilarity for mixed data:

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

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

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

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

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

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