what is multidimensional scaling (uw)

8/8/2019 What is Multidimensional Scaling (UW)

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Mod: 11/ 2009

What is MDS?

Prof APM Coxon, U Cardiff 1

What isWhat is

MultidimensionalMultidimensional

Scaling [MDS] ?Scaling [MDS] ? Anthony P.M. Coxon

Emeritus Professor of Sociological Research Methods, Universityof 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 inLewis-Beck, M.S. et al, eds (2004) The Sage Encyclopaedia ofSocial Science Research Methods. London Sage Publications )
http://www.newmdsx.com/http://apmc.newmdsx.com/http://apmc.newmdsx.com/http://www.newmdsx.com/


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U Winchester 12/09 What 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 tonon-metric revolution(Coombs) and computingdevelopments 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)


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3

CONSTRUCTING A MAP

Given a map, its easy to calculate the distances between

the points MDS operates the other way round:

Given the data [ interpreted as quasi distances ] itattempts to find the configuration [location ofpoints] which generated the distances

This is Classic MDS: developed in 1930s butimperfect, not robust, & works only if data are ratio.

Whereas more recent MDS can work when only theordinal information exists: Non-metric = ordinalMDS (Coombs / Kruskal non-metric revolution)

What?? You can create an accurate map from onlythe rankorder of the distances???

Yes! And it works!!


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The RANK of distances can recover the Mapthough not the coastline

NEWMDSX(RUNSCRIPT + SYNTAX)

RUN NAME Rank of Scottish distances,

COMMENT 1 = smallest; 120 = max; dissimilarity data

F3.3, p48 The Users Guide to MDS

N OF STIMULI 16

PARAMETERS DATA TYPE(1)

LABELS BERWICK

EDINBURGH

GLASGOW

STRANRAER

AYR

PERTH

DUNDEE

ABERDEEN

STIRLING

OBANFORT_WM

INVERNESS

KYLE_LOCHALSH

BRAEMAR

ULLAPOOL

THURSO

READ MATRIX

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53 11

92 68 36

70 30 4 11

34 7 19 83 45

27 8 29 93 58 1= Perth-Dundee63 56 83 115 103 35 24

43 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 6

100 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 54

114 105 101 119 109 85 87 86 88 68 40 16 17 55

117 113 115 120 119 103 102 77 110 107 95 47 62 74 42

COMPUTE

= Stranraer - Thurso

U. Winchester12/09

What is MDS?Prof APM Coxon, Cardiff U

4


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Uni Winchester 12/09

What is MDS?

Prof APM Coxon, Cardiff Uni 5

WHAT IS MULTIDIMENSIONALSCALING?

A students definition: If you are interested in how certain objects relate to each

other and if you would like to present these relationshipsin 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 matchthe data dissimilarities (Stress)


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U Winchester, 12/2009

What is MDS?

Prof APM Coxon, Cardiff U 6

MDS is a family of modelsdifferentiated by

(DATA) the empirical inter-relationships between

a set of objects/variables which are given in aset of dis/similarity data

Basically, type of input data, defined by their Way andMode [e.g. 2W1M]. (Cf observations vs data)

(FUNCTION) data are then optimally re-scaled

(according to permissible trans-formations for thedata) in terms of

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

(MODEL) the assumptions of the model chosen torepresent the data

Usually (Euclidean) Distance model


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

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data

MDS can be used with a wide variety ofDATA

e.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


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What is MDS?

Prof APM Coxon, Cardiff U 8

VARIANTS OF MDS MODELS due toTRANSFORMATIONS

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


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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

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


What is MDS?



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HOW DOES MDS WORK? Iteratively!

START: Produce Init. Guestimate Configuration (a) FIT

Calculate distances (d)

Compare with data () [via Ordinal regression]

Calculate overall badness -of-fit measure

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

Perfect/Acceptable? EXIT

(b) IMPROVE: For each point, find direction of improvement (dont ask: calculus! Derivatives!)

How far to move? Step-size (call it heuristic ; parachute & mist)

(c) MOVE configuration/points

BACK TO (a)


What is MDS?



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What is MDS?


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


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Uni Winchester, 12/09

What is MDS?


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 YOUNGS VISTA (Visual Statistics)http://forrest.psych.unc.edu/research/index.html


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UniWinchester 12/09

What is MDS?


WHAT IS MDS?

and now for anexample!


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APPENDICES

1. Interpretation: Headlines

2. MVA & MDS

Professor APM Coxon 14


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MDS: Interpretation: Headlines

For Euclidean Distance MDS: "What information isstable/significant?

Beware Local minima [PERMAP]

Remember: You may translate, reflect, (rigidly) rotatethe 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 WaernsGraphic 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 isincreasing: External properties [PRO-FIT NewMDSX],

If you must ... dimensions -- remember changing theorigin or dimensional orientation has no effect onrelative 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


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MDS & other DimensionalMultivariate Analysis models


What is MDS?



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Research MethodsFestival 2006

What is MDS?

Prof APM Coxon, U Edinburgh 17

SOME POSSIBLE WEAKNESSES inMDS

There ARE any??! Relative ignorance of the sampling/inferential properties ofstress 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)

what is multidimensional scaling (uw)

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