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Statistical Bases for Map Reconstructions and Comparisons

Jerry Platt

May 2005

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Preliminaries

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Outline• Motivation

– Do Different Maps “Differ”?

• Methods

– Singular-Value Decomposition

Multidimensional Scaling and PCA

Mantel Permutation TestProcrustean Fit and Permu. TestBidimensional Regression

• Working Example– Locational Attributes of Eight URSB Campuses

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Motivation• Comparing Maps Over Time

Accuracy of a 14th Century MapLeader Image Change in Great BritainWhere IS Wall Street, post-9/11?

• Comparing Maps Among Sub-samplesThings People Fear, M v. F Face-to-Face Comparisons

• Comparing Maps Across AttributesCompetitive Positioning of FirmsChinese Provinces & Human Dev. Indices

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Accuracy of a 14th Century Map

http://www.geog.ucsb.edu/~tobler/publications/pdf_docs/geog_analysis/Bi_Dim_Reg.pdf

6http://www.mori.com/pubinfo/rmw/two-triangulation-models.pdf

7http://igeographer.lib.indstate.edu/pohl.pdf

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http://www.analytictech.com/borgatti/papers/borgatti%2002%20-%20A%20statistical%20method%20for%20comparing.pdf

Things People Fear, F v. M

9http://www.multid.se/references/Chem%20Intell%20Lab%20Syst%2072,%20123%20(2004).pdf

Face-to-Face Comparisons

10http://www.gsoresearch.com/page2/map.htm

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MethodsEigen-Analysis and Singular-Value Decomposition

Multidimensional Scaling & Principal Comps.

Mantel Permutation Test

Procrustean Fit and Permutation Test

Bidimensional Regression

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Eigen-analysis

• C = an NxN variance-covariance matrix

• Find the N solutions to C = = the N Eigenvalues, with 1 ≥ 2 ≥ …

= the N associated Eigenvectors

• C = LDL’, where

L = matrix of s

D = diagonal matrix of s

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Singular Value Decomposition

• Every NxP matrix A has a SVD

• A = U D V’

• Columns of U = Eigenvectors of AA’

• Entries in Diagonal Matrix D = Singular Values

= SQRT of Eigenvalues of either AA’ or A’A

• Columns of V = Eigenvectors of A’A

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SVD

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Principal Component Analysis

• A is a column-centered data matrix

• A = U D V’

• V’ = Row-wise Principal Components

• D ~ Proportional to variance explained

• UD = Principal Component Scores

• DV’ = Principle Axes

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Multidimensional Scaling• A is a column-centered dissimilarity matrix

• B =

• B = U D V’

• B = XX’, where X = UD1/2

• Limit X to 2 Columns Coordinates to 2d MDS

'

1'

1

2

1 2 iiN

IAiiN

I

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A RandomPermutation

Test

Given DissimilarityMatrices A and B:

N! Permutations37! = 1.4*E+43 8! = 40,320

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Permutation Tests

PermuteList & rerun

ObservedTestStatisticTS = 25# CorrectOf 37 SB.

Is 25Significantly> 18.5?

Ho: TS = 18.5HA: TS > 18.5

P = .069P > .05Do NotReject Ho

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21http://www.entrenet.com/~groedmed/greekm/mythproc.html

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http://www.zoo.utoronto.ca/jackson/pro2.html

Centering &Scaling

MirrorReflection

Rotation &Dilation toMin ∑(є2)

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Procrustean Analysis

• Two NxP data configurations, X and Y• X’Y = U D V’• H = UV• OLS Min SSE = tr ∑(XH-Y)’(XH-Y)

= tr(XX’) + tr(YY’) -2tr(D)

= tr(XX’) + tr(YY’) – 2tr(VDV’)

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OLS Regression

• Y = X + • Y = Xb + e• X = UDV’• b = VrD-1Ur’Y, where r = first r columns (N>P)

• b = (X’X)-1X’Y

• b = VrVr’ • Estimated Y values = Ur Ur’Y

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Bidimensional Regression• (Y,X) = Coordinate pair in 2d Map 1

Y = 0 + 0X

• (A,B) = Coordinate pair in 2d Map 2

E[A] 1 1 -2 X 1

E[B] 1 2 1 Y 2

1 = Horizontal Translation

2 = Vertical Translation

= Scale Transformation = SQRT(12

+ 22)

= Angle Transformation = TAN-1(2 / 1 ) +1800

= + +

Iff 1 < 0

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Althoughr = 1,differ inlocation,scale, andangles ofrotationaroundorigin (0,0)

Horizontal& VerticalTranslation

Angle ofrotationaroundorigin (0,0)

Scaletransform,with < 1 ifcontration,& > 1 ifexpansion

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Working Example

• Eight URSB Campuses– RD, BK, TO, RC, SA, RV, SD, TA

• Data Sources– Locations– Housing Attributes– Tapestry Attributes

• Data Analyses

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Eight URSB Campuses

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87.5 miles

88.1 miles

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…

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EXAMPLE: Eight URSB Campuses

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SD

TA

RDRVRCBK

TOSA

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… and if DISTANCES available, but COORDINATES Unavailable?

• Treat Distance Matrix as Dissimilarity Matrix

• Apply Multidimensional Scaling

• Apply the two-dimension solution “as if” it represents latitude and longitude coordinates

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Distance Estimates Vary

… But Not “Significantly”

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MDS RepresentationInput = D; Output = 2d

D8x8

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SD

TA

RD

RVRC

BK

TO

SA

Errors“appear”

to bequitesmall

…BUT

is therea wayto test

if errorsare

“STATSIGNIF”

?

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Mantel Test

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Procrustean Test:MDS Map Recreation

CONCLUDE: Near-perfect Map Recreation

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Driving Distances

Do these differ “significantly” from linear distances?

STATISTICAL PRACTICAL

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DriveD = Driving DistancesEight URSB Locations

Multidimensional Scaling,with 2-dimension solution

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SD

TA

RD

RVRC

BK

TO

SA

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Bidimensional Regression:AB on YX

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PROTEST Comparison

BidimensionalRegression

ProcrusteanRotation

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Housing

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Tapestry (ESRI)

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Map Coordinates as Explanatory Variables in Linear Models

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Incremental Tests

So Map Coordinates seem sufficient as predictors

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Proxy Measures of lat-longin Linear Model

Translations& Transforms

Reduce 8

And ↑ R2

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Robust criterionwould help here:

Min (Med(є2))

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Bidimensional Regression

Is There a Linear RelationshipBetween Housing and Tapestry

Data?

r = 0.5449

MustStandardize

Data

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It’s Still a 3-d World

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