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Statistical Bases for Map Reconstructions and Comparisons

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Statistical Bases for Map Reconstructions and Comparisons

Jerry Platt

May 2005

Motivation

Do Different Maps “Differ”?

Methods

Singular-Value Decomposition

Multidimensional Scaling and PCA

Mantel Permutation Test

Procrustean Fit and Permu. Test

Bidimensional Regression

Working Example

Locational Attributes of Eight URSB Campuses

Comparing Maps Over Time

Accuracy of a 14th Century Map

Leader Image Change in Great Britain

Where IS Wall Street, post-9/11?

Comparing Maps Among Sub-samples

Things People Fear, M v. F

Face-to-Face Comparisons

Comparing Maps Across Attributes

Competitive Positioning of Firms

Chinese Provinces & Human Dev. Indices

http://www.geog.ucsb.edu/~tobler/publications/

pdf_docs/geog_analysis/Bi_Dim_Reg.pdf

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

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

http://www.analytictech.com/borgatti/papers/borgatti

%2002%20-%20A%20statistical%20method%20for%20comparing.pdf

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

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

Eigen-Analysis and Singular-Value Decomposition

Multidimensional Scaling & Principal Comps.

Mantel Permutation Test

Procrustean Fit and Permutation Test

Bidimensional Regression

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

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

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

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

Given Dissimilarity

Matrices A and B:

A Random

Permutation

Test

N! Permutations

37! = 1.4*E+43

8! = 40,320

Observed

Test

Statistic

TS = 25

# Correct

Of 37 SB.

Is 25

Significantly

> 18.5?

Ho: TS = 18.5

HA: TS > 18.5

P = .069

P > .05

Do Not

Reject Ho

Permute

List & rerun

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

Centering &

Scaling

Rotation &

Dilation to

Min ∑(є2)

Mirror

Reflection

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

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’)

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

(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

Angle of

rotation

around

origin (0,0)

Horizontal

& Vertical

Translation

Although

r = 1,

differ in

location,

scale, and

angles of

rotation

around

origin (0,0)

Scale

transform,

with

< 1 if

contration,

& > 1 if

expansion

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

- Data Sources
- Locations
- Housing Attributes
- Tapestry Attributes

- Data Analyses

Eight URSB Campuses

87.5 miles

88.1 miles

…

BK

RC

RD

RV

TO

SA

TA

SD

Treat Distance Matrix as Dissimilarity Matrix

Apply Multidimensional Scaling

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

… But Not “Significantly”

D

8x8

Errors

“appear”

to be

quite

small

…

BUT

is there

a way

to test

if errors

are

“STAT

SIGNIF”

?

RD

RV

RC

TA

BK

SD

SA

TO

CONCLUDE: Near-perfect Map Recreation

Do these differ “significantly” from linear distances?

PRACTICAL

STATISTICAL

Multidimensional Scaling,

with 2-dimension solution

RD

RV

RC

TA

SA

BK

SD

TO

Bidimensional

Regression

Procrustean

Rotation

So Map Coordinates seem sufficient as predictors

Translations

& Transforms

Reduce 8

And ↑ R2

Robust criterion

would help here:

Min (Med(є2))

Bidimensional Regression

r = 0.5449

Must

Standardize

Data