Mod: 11/ 2009What is MDS?

Prof APM Coxon, U Cardiff1

What is What is Multidimensional Multidimensional Scaling [MDS] ?Scaling [MDS] ? • Anthony P.M. Coxon

– Emeritus Professor of Sociological Research Methods, University of Wales

– Honorary Professor, Cardiff University– Honorary Professorial Research Fellow, University of

Edinburgh– Co-founder & Co- Director of MDS software packages,

• MDSX [OS] (freeware)and • NewMDSX for Windows (not-for-profit)• Website: www.newmdsx.com • Course materials: http://apmc.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 )

U Winchester 12/09What is MDS?

Prof APM Coxon, Cardiff U 2

ORIGINS / DEVELOPMENT OF MDSORIGINS / DEVELOPMENT OF MDS

• MDS (aka “Smallest Space Analysis”)– Has origins in Psychometrics in 1920-’60s:

• Scale construction and dimensionality reduction• Underwent major burst of development in 1960s due

to “non-metric revolution”(Coombs) and computing developments allowing iterative estimation

– Originally designed for analysis of LTM of dis/similarities data , taking a range of measures (not just PM correlations):

• “anything which, by an act of faith, can be considered a similarity” (Shepard)

– Extended rapidly to deal with wide range of other types of data

• Rectangular matrices ; triads, pair-comparisons, free-sorting• “stacks” of matrices (3-way scaling – INDSCAL)

3

CONSTRUCTING A MAP …– Given a map, it’s easy to calculate the distances

between the points … – MDS operates the other way round:

• Given the data [ interpreted as quasi “distances” ] it attempts to find the configuration [location of points] which generated the distances

» This is “Classic MDS”: developed in 1930s – but imperfect, not robust, & works only if data are ratio.

• Whereas more recent MDS can work when only the ordinal information exists: “Non-metric” = ordinal MDS (Coombs / Kruskal “non-metric revolution” )

• What?? You can create an accurate map from only the rank –order of the distances???

Yes! And it works!!

The RANK of distances can recover the Map…though not the coastline

NEWMDSX (RUNSCRIPT + SYNTAX)RUN NAME Rank of Scottish distances, COMMENT 1 = smallest; 120 = max; dissimilarity data F3.3, p48 The User’s Guide to MDSN OF STIMULI 16 PARAMETERS DATA TYPE(1)LABELS BERWICKEDINBURGHGLASGOWSTRANRAERAYRPERTHDUNDEEABERDEENSTIRLINGOBANFORT_WMINVERNESSKYLE_LOCHALSHBRAEMARULLAPOOLTHURSOREAD MATRIX1753 1192 68 3670 30 4 1134 7 19 83 4527 8 29 93 58 1= Perth-Dundee63 56 83 115 103 35 2443 4 2 58 21 3 14 66 99 57 26 72 36 43 63 98 28 96 60 39 89 62 41 49 79 33 6100 75 75 112 97 45 49 48 60 52 22 111 89 78 107 89 72 81 94 70 26 9 23 67 36 51 106 80 15 10 19 30 65 30 15 54114 105 101 119 109 85 87 86 88 68 40 16 17 55117 113 115 120 119 103 102 77 110 107 95 47 62 74 42COMPUTE = Stranraer - Thurso

U. Winchester12/09

What is MDS?Prof APM Coxon, Cardiff U

4

Uni Winchester 12/09

What is MDS?Prof APM Coxon, Cardiff Uni

5

WHAT IS MULTIDIMENSIONAL SCALING?

A student’s 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 provides …• a useful and easily-assimilable graphic visualisation of

all sorts of data– Tukey: “A picture is worth a thousand words”

• In a user-chosen (small) # of dimensions• providing a graphical representation of the structure

underlying a complex data set• And measure how well / badly the solution distances

match the data dissimilarities (Stress)

U Winchester, 12/2009

What is MDS?Prof APM Coxon, Cardiff U

6

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

» Basically, type of input data, defined by their “Way” and “Mode” [e.g. 2W1M]. (Cf observations vs data)

– (FUNCTION) data are then optimally re-scaled (according to permissible trans-formations for the data) in terms of …

» Choice of level of measurement [e.g. ordinal ]

– (MODEL) the assumptions of the model chosen to represent the data

» Usually (Euclidean) Distance model

Uni Winchester 12/09 What is MDS?Prof APM Coxon, Cardiff Uni

7

VARIANTS OF MDS due to type of data

MDS can be used with a wide variety of DATAe.g.: SORTS OF DATA

– direct data (pair comparisons, ratings, rankings, triads, counts)

– derived data (profiles, co-occurrence matrices, textual data, aggregated data)

– measures of association etc derived from simpler data, and

– tables of data.• TYPES of DATA • Described by WAY (2W=matrix; 3W=stack of matrices

…)• And MODE (# sets of distinct objects – eg variables,

subjects)– E.G. 2W1M; 2W2M; 3W2M … 7W4M

Uni Winchester 12/09

What is MDS?Prof APM Coxon, Cardiff U

8

VARIANTS OF MDS MODELS due to TRANSFORMATIONS

MDS can also be used with a wide variety of: Transformations (“levels of measurement”)

• monotonic (ordinal), • linear/metric (interval),

… but also

– Splines (SPSS PROXSCAL)• local preservation of distance

– log-interval (MRSCAL),– Power (MULTISCALE)– “smoothness”

VARIANTS OF MDS due to type of MODEL• DISTANCE “Minkowski-r”

– Usually Euclidean (r=2)• Less often “City Block”,

r=1– Sometimes

“Dominance”, r=∞≈ 32• SCALAR PRODUCTS/Factor• scalar product : a ・ b = |a| |b| cos θ

– E.g. Covariance, PM Correlation– As used in PCA, FA, MDPREF

• COMPOSITION– Most usually, Additive (cf ANOVA),

as in Impression Formation:– X(i.j) = a(i) + b(j) + …

• nb Ordinal.non-metric ANOVA– But also, difference, product,

mixed

Uni Winchester 12/09

What is MDS?Prof APM Coxon, Cardiff Uni

9

HOW DOES MDS WORK?• Iteratively! • START: Produce Init. “Guestimate” Configuration

• (a) FIT – Calculate distances (d)– Compare with data (δ) [via Ordinal regression]– Calculate overall badness -of-fit measure

» Stress (d- δ) … well, almost! Actually more complex

» Perfect/Acceptable? EXIT• (b) IMPROVE: For each point,

– find direction of improvement (don’t ask: calculus! Derivatives!)

– How far to move? Step-size (call it ‘heuristic’ ; “parachute & mist”)

• (c) MOVE configuration/points• BACK TO (a)

Uni Winchester 12/09

What is MDS?Prof APM Coxon, Cardiff Uni

10

Uni Winchester 12/2009What is MDS?

Prof APM Coxon, Cardiff Uni 11

MDS PROGRAMS: 1. Usually either “General Purpose” Package (SPSS )

– Basic Model for 2W1M data: PROXSCAL and 3W2M INDSCAL

– Also contains CORRESP, HICLUS and (in >SPSS13 ) PREFSCAL (2W2M)2. or “Library” : set of programs, each specific to

Data-shape, Trans & Model (e.g. NewMDSX for Windows); includes

– BASIC 2W1M SCALING: • Non-metric (ordinal) MINISSA , Metric (MRSCAL) linear, • Clustering (Hierarchical & Non-hierarchical)

– 2W2M (“Rectangular”) SCALING:• Multidimensional … Preference, Triads, Unfolding, Sorting

– 3W2M (and higher) SCALING:• Individual Differences (INDSCAL), (Tucker) Points-of-View • Procrustean IndDiffs (Lingoes’ PINDIS)

• Or “ Interactive “ Package (PERMAP via NewMDSX) • primarily for basic model • Visually animated• Superb diagnostic procedures

Uni Winchester, 12/09

What is MDS?Prof APM Coxon, Cardiff Uni

12

SITES & SOFTWARE:SITES

– NEWMDSX AND DOCUMENTATION: http://www.newmdsx.com

– INTERACTIVE PERMAP (Heady)

» (presently obtained via NewMDSX)

– THREE-WAY SCALING (Kroonenberg)– http://www.leidenuniv.nl/fsw/three-mode/content.htm

– FORREST YOUNG’S VISTA (Visual Statistics)

http://forrest.psych.unc.edu/research/index.html

UniWinchester 12/09

What is MDS?Prof APM Coxon, Cardiff Uni

13

WHAT IS MDS?

… and now for an example!

APPENDICES

1. Interpretation: Headlines2. MVA & MDS

Professor APM Coxon 14

MDS: Interpretation: Headlines For Euclidean Distance MDS: "What information is

stable/significant?“ Beware Local minima [PERMAP] Remember: You may translate, reflect, (rigidly) rotate

the configuration: do so! [e.g. NewMDSX Graphics; PERMAP]

“CLEARING UP” Configuration: [PERMAP] Map Evaluation & Diagnostics; Points and Links;

selective removal and hints of structure via Waern’s Graphic links.

BASIC STRUCTURES: Regional: what points are close to each other and distant

from others? CLUSTERING [(HI)CLUS, SPSS] Linear: directions in space where some property is

increasing: External properties [PRO-FIT NewMDSX], If you must ... dimensions -- remember changing the

origin or dimensional orientation has no effect on relative distance. Most MDS rotated at end to PCA ... Unlike FA, dimensions may/ may not have importance.

SIMPLE STRUCTURES dimensions, yes -- but also other simple structures

(“horseshoes”, radex/circumplex). Professor APM Coxon 15

MDS & other “Dimensional” Multivariate Analysis models

Uni Winchester 12/09

What is MDS?Prof APM Coxon, Cardiff Uni

16

Research Methods Festival 2006

What is MDS?Prof APM Coxon, U Edinburgh

17

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)