Quick

1
Quick Simple Introduction to Multidimensional
Scaling

• Professor Tony Coxon
• Hon. Professorial Research Fellow, University of
  Edinburgh ( apm.coxon_at_ed.ac.uk )
• see www.tonycoxon.com for information on me
• see www.newmdsx.com for information resource on
  MDS and NewMDSX programs/doc.
• See
• The Users Guide to MDS and
• Key Texts in MDS (readings), Heineman 1982
• Available as pdf at 15 from newmdsx

2
What is Multidimensional Scaling?

• A students definition
• If you are interested in how certain objects
  relate to each other and if you would like to
  present these relationships in the form of a map
  then MDS is the technique you need (Mr Gawels,
  KUB) A good start!
• MDS is a family of models structured by D-T-M
• (DATA) the empirical information on
  inter-relationships between a set of
  objects/variables are given in a set of
  dis/similarity data
• (TRANSFORMATION) which are then re-scaled (
  according to permissible transformations for the
  data / level of measurement) , in terms of
• (MODEL) the assumptions of the model chosen to
  represent the data

3
MDS Solution

•      to produce a SOLUTION, consisting of
• a CONFIGURATION, which is a
•         i.      pattern of points representing
  the objects
•         ii.     located in a space of a small
  number of dimensions
• (hence SSA Smallest-Space Analysis)
•        iii.      where the distances between the
  points represent the dis/similarities between the
  data-points
•        iv.      as perfectly as possible
• (the imperfection/badness of fit is measured by
  Stress)
• Low stress is desirable No stress is
  perfection

4
Distances Maps

• Given a map, its easy to calculate the
  (Euclidean) distances between the points
• MDS operates the other way round
• Given the distances data find the map
  configuration which generated them
• and MDS can do so when all but ordinal
  information has been jettisoned (fruit of the
  non-metric revolution)
• even when there are missing data and in the
  presence of considerable noise/error (MDS is
  robust).
• MDS thus provides at least
• exploratory a useful and easily-assimilable
  graphic visualization of a complex data set
  (Tukey A picture is worth a thousand words)

5
What is like MDS?

• Related and Special-case Models
• Metric Scalar Products Models
• PRINCIPAL COMPONENTS ANALYSIS
• FACTOR ANALYSIS ( communalities)
• Metric and Non-Metric Ultrametric Distance,
  Discrete models
• Hierarchical Clustering
• Partition Clustering (CONPAR)
• Additive Clustering ( 2 and 3-way)
• Metric Chi-squared Distance Model for 2W2M and 3W
  data / Tables
• Simple (2W2M) and Multiple (3W) Correspondence
  Analysis
• BECAUSE OF NON-METRIC (MONOTONE) REGRESSION, MDS
  ALSO OFFERS ORDINAL EQUIVALENTS OF
• ANOVA
• other simple composition models UNICON
• (All models with asterisk exist as programs
  within NewMDSX) 

6
How does MDS differ from other Multivariate
Methods?

• Compared to other multivariate methods, MDS
  models are usually
• distribution-free
• (though MLE models do exist Ramsays
  MULTISCALE)
• make conservative (non-metric) demands on the
  structure of the data,
• are relatively unaffected by non-systematic
  missing data,
• can be used with a very wide variety of types of
  data
• direct data (pair comparisons, ratings, rankings,
  triads, sortings)
• derived data (profiles, co-occurrence matrices,
  textual data, aggregated data)
• measures of association/correlation etc derived
  from simpler data, and
• tables of data.
• range of transformations
• monotonic (ordinal), linear/metric (interval),
  but also log-interval, power, smoothness even
  maximum variance non-dimensional scaling
  (Shepard)

7
How does MDS differ from other Multivariate
Methods (2)?

• Compared to other multivariate methods, MDS
  models are also offer
• range of models (chiefly distance (Euclidean,
  but also City-block), factor/vector
  (scalar-products), simple composition (additive).
  
• Also there are hierarchies of models
• Similarity models 2W1M METRIC 3W2M INDSCAL
  IDIOSCAL (honest!)
• Preference models Vector-distance-weighted
  distance-rotated, weighted (PREFMAP)
• Procrustes rotation for putting configurations
  into maximum conformity, and then increasingly
  complex transformations PINDIS
• the solutions are visually assimilable readily
  interpretable
• the structure is not limited to dimensional
  information also other simple structures
  (horseshoes, radex/circumplex, clusters,
  directions).

8
Weaknesses in MDS There ARE any??!

• Relative ignorance of the sampling properties of
  stress
• prone-ness to local minima solutions
• (but less so, and 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 SPSS)
• 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)

9
CHARACTERIZATION OF BASIC MDS TERMINOLOGY

• Structure of MDS specifiable in terms of D-T-M
• DATA (specifies input data shape and content)
• DATA MATRIX INPUT
• WAY dimensionality of array (2,3,4 ...)
• MODALITY No of distinct sets (to be represented)
  (1,2,3 )
• NB Modality lt or Way
• Common examples
• 2W1M basic models (LTM,UTM,FSM)
• 2W2M rectangular, joint (conditional )mapping
• 3W2M (stack of 2W1M) Individual differences
  Scaling

10
CHARACTERIZATION OF BASIC MDS (2)

• TRANSFORMATION (form or type of rescaling
  performed on data)
• Non-Metric /Ordinal ? M(d)
• Monotonic Increasing (sims) or Decreasing
  (dissims)
• Order/inequality
• Strong / Guttman ?(j,k) gt ?(l,m) -gt d(j,k) gt
  d(l,m)
• weak/Kruskal ?(j,k) gt ?(l,m) -gt d(j,k) ?
  d(l,m)
• Equality / ties
• Primary ?(j,k) ?(l,m) -gt d(j,k) OR?
  d(l,m)
• 2ndary ?(j,k) ?(l,m) -gt d(j,k) d(l,m)
• Metric / Linear
• Linear ? L(d)
• ? a b(d)

11
CHARACTERIZATION OF BASIC MDS (3)

• MODEL Euclidean Distance
• where x(i,a) is the co-ordinate of point i on
  dimension a in the solution configuration X of
  low dimension
• The basic model is Euclidean distance, but other
  Minkowski metrics are available, including
• City Block Model

12
(Badness of) FIT Stress
13
Types of Analysis

• INTERNAL
• If the analysis depends solely on the input data,
  it is termed internal, but
• EXTERNAL
• If the analysis uses additionally to the input
  data / solution information relating to the same
  points (but from another source), it is termed
  external.