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What is MDS? Prof APM Coxon, U Edinburgh

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MDS (aka Smallest Space Analysis ) ... Utilities: Comparison of configurations (by Procrustes rotation) WOMBATS (Measures & Data-manipulation) ... – PowerPoint PPT presentation

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Title: What is MDS? Prof APM Coxon, U Edinburgh


1
What is Multidimensional Scaling MDS ?
  • Tony Macmillan Coxon
  • Honorary Professorial Research Fellow, University
    of Edinburgh
  • Emeritus Professor of Sociological Research
    Methods, University of Wales
  • Co-founder Co- Director of MDS software
    packages,
  • MDSX OS (freeware)and
  • NewMDSX for Windows (not-for-profit)
  • Website www.newmdsx.com
  • see my entry on multidimensional scaling in
  • Lewis-Beck, M.S. et al, eds (2004) The Sage
    Encyclopaedia of Social Science Research Methods.
    London Sage Publications )

2
What is Multidimensional Scaling MDS ?
  • A quick answer
  • the most flexible, useful, user-friendly
  • but grossly under-used, under-taught
  • family of Research Methods in Social Science!

3
WHAT IS MULTIDIMENSIONAL SCALING?
  • MDS (aka Smallest Space Analysis)
  • Has origins in Psychometrics in 1920-60s
  • Analysis of similarity data as distances
  • Scale construction and dimensionality reduction
  • Underwent major burst of development in 1960s due
    to non-metric revolution(Coombs) and computing
    developments

4
WHAT IS MULTIDIMENSIONAL SCALING?
  • Given a map, its easy to calculate the
    distances between the points
  • MDS operates the other way round
  • Given the distances data find the
    configuration which generated them
  • .and can do so when all but the ordinal
    information has been jettisoned
  • (the non-metric revolution)
  • Even when there are missing data and in the
    presence of considerable noise/error
  • (MDS is robust)

5
WHAT IS MULTIDIMENSIONAL SCALING?
  • MDS is a family of models differentiated by
  • (DATA) the empirical inter-relationships between
    a set of objects/variables which are given in a
    set of dis/similarity data
  • (FUNCTION) are then re-scaled (according to
    permissible transformations for the data level of
    measurement), and in terms of
  • (MODEL) the assumptions of the model chosen to
    represent the data

6
WHAT IS MULTIDIMENSIONAL SCALING?
  • To produce a SOLUTION consisting of
  • a CONFIGURATION, which is a
  • pattern of points representing the objects
  • located in a space of a small number of
    dimensions (hence SSA)
  • where the distances (or angular separation)
    between the points represents the
    dis/similarities between the data-points
  • as perfectly as possible (the imperfection/badness
    of fit is measured by Stress No stress is
    perfection)

7
WHAT IS MULTIDIMENSIONAL SCALING?
  • MDS thus provides.
  • a useful and easily-assimilable graphic
    visualisation
  • Tukey A picture is worth a thousand words
  • and/or a graphical representation of the
    structure underlying a complex data set

8
VARIANTS OF MDS MODELS
  • MDS can be used with a wide variety of DATA
  • direct data (pair comparisons, ratings, rankings)
  • derived data (profiles, co-occurrence matrices,
    textual data, aggregated data)
  • measures of association etc derived from simpler
    data, and
  • tables of data.

9
VARIANTS OF MDS MODELS
  • MDS can also be used with a wide variety of
    Transformations (levels of measurement)
  • monotonic (ordinal),
  • linear/metric (interval),
  • but also
  • Splines (SPSS PROXSCAL)
  • log-interval (MRSCAL),
  • Power (MULTISCALE)
  • smoothness (local preservation of distance)
  • Models
  • chiefly distance (Euclidean, but also
    City-block),
  • Factor/vector, Composition (additive)

10
MDS PROGRAMS
  • Usually either General Purpose (e.g. SPSS
    PROXSCAL)
  • or specific to Data-shape, Trans Model (e.g.
    NewMDSX)
  • BASIC 2W1M SCALING
  • Non-metric (ordinal), Metric (linear, log-int)
  • Principal Components
  • Clustering (Hierarchical Non-hierarchical)
  • 2W2M (Rectangular) SCALING
  • Multidimensional Preference, Sorting, Unfolding
  • Property-fitting, Profile Analysis (continuity)
  • Triadic MD Scaling
  • 3W2M (and higher)
  • Individual Differences (INDSCAL), (Tucker)
    Points-of-View
  • Procrustean IndDiffs (Lingoes PINDIS)
  • Canonical Decomposition ( 3-7 Mode)
  • Utilities
  • Comparison of configurations (by Procrustes
    rotation)
  • WOMBATS (Measures Data-manipulation)

11
SITES SOFTWARE
  • SITES
  • NEWMDSX AND DOCUMENTATION
  • http//www.newmdsx.com
  • INTERACTIVE PERMAP (Heady) http//www.ucs.louisian
    a.edu/rbh8900/
  • THREE-WAY SCALING (Kroonenberg)
  • http//three-mode.leidenuniv.nl/encyclopedia/encyc
    lopedia.htm
  • FORREST YOUNGS VISTA (Visual Statistics)
  • http//forrest.psych.unc.edu/research/index.html

12
WHAT IS MDS? and now for an example
  • Perception of road accident causes
  • by Ward Vanlaar (Traffic Injury Research
    Foundation, Ottawa) APMC
  • An application of INDSCAL c
  • Other examples available at www.tonycoxon.com
  • Molinero Predicting Bank Failure
  • Kendall Maps from Marriages
  • Coxon Subjective Categorization of Drugs

13
WHAT IS MULTIDIMENSIONAL SCALING? APPENDIX
  • Multidimensional Scaling
  • Is often subsumed with Cluster analysis
    Seriation/sequencing as Combinatorial Data
    Analysis
  • (Arabie http//www.ec-securehost.com/SIAM/DT06.ht
    ml)
  • Forms basis of Graphical Data Visualization
  • http//lib.stat.cmu.edu/general/XGobi/
  • and Data Mining
  • http//www.the-data-mine.com/
  • As well as computer-assisted text-analysis
  • http//www.tlab.it/en/ (T-LAB)
  • http//www.apb.cwc.net/ (HAMLET)

14
WHAT IS MULTIDIMENSIONAL SCALING?APPENDIX
Related and Special-case Models
  • Metric Scalar Products SVDModels
  • PRINCIPAL COMPONENTS ANALYSIS
  • FACTOR ANALYSIS ( communalities)
  • Discrete Clustering
  • HIERARCHICAL CLUSTERING
  • PARTITION ADDITIVE CLUSTERING
  • Tables
  • SIMPLE (2W2M) AND MULTIPLE (3W)
    CORRESPONDENCE ANALYSIS
  • Other M/ANOVA SIMPLE COMPOSITION

15
SOME POSSIBLE WEAKNESSES in MDS There ARE
any??!
  • Relative ignorance of the sampling/inferential
    properties of stress
  • But, simulation (Spence), MLE estimation
  • Prone-ness to local minima solutions
  • but less so, and multiple starts interactive
    programs like PERMAP allow thousands of runs to
    check
  • A few forms of data/models are prone to
    degeneracies
  • especially MD Unfolding, but see new PREFSCAL
    in SPSS14)
  • difficulty in representing the asymmetry of
    causal models
  • though external analysis is very akin to
    dependent-independent modelling,
  • there are convergences with GLM in hybrid models
    such as CLASCAL (INDSCAL with parameterization of
    latent classes)
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