1 / 9

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

ornice
Download Presentation

Multidimensional Scaling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related