Statistical bases for map reconstructions and comparisons
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Statistical Bases for Map Reconstructions and Comparisons. Jerry Platt May 2005. Preliminaries. Motivation Do Different Maps “Differ”? Methods Singular-Value Decomposition Multidimensional Scaling and PCA Mantel Permutation Test Procrustean Fit and Permu. Test

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Statistical Bases for Map Reconstructions and Comparisons

Jerry Platt

May 2005


Preliminaries


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

Outline


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

Motivation


Accuracy of a 14th Century Map

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


Things People Fear, F v. M

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

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


Face-to-Face Comparisons

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

Methods


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

Eigen-analysis


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

Singular Value Decomposition


SVD


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

Principal Component Analysis


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

Multidimensional Scaling


Given Dissimilarity

Matrices A and B:

A Random

Permutation

Test

N! Permutations

37! = 1.4*E+43

8! = 40,320


Permutation Tests

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’)

Procrustean Analysis


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

OLS 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

Bidimensional Regression

+

=

+

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


Working Example

  • 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



EXAMPLE: Eight URSB Campuses


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

… and if DISTANCES available, but COORDINATES Unavailable?


Distance Estimates Vary

… But Not “Significantly”


MDS RepresentationInput = D; Output = 2d

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


Mantel Test


Procrustean Test:MDS Map Recreation

CONCLUDE: Near-perfect Map Recreation


Driving Distances

Do these differ “significantly” from linear distances?

PRACTICAL

STATISTICAL


DriveD = Driving DistancesEight URSB Locations

Multidimensional Scaling,

with 2-dimension solution


RD

RV

RC

TA

SA

BK

SD

TO


Bidimensional Regression:AB on YX


PROTEST Comparison

Bidimensional

Regression

Procrustean

Rotation


Housing


Tapestry (ESRI)


Map Coordinates as Explanatory Variables in Linear Models


Incremental Tests

So Map Coordinates seem sufficient as predictors


Proxy Measures of lat-longin Linear Model

Translations

& Transforms

Reduce 8

And ↑ R2


Robust criterion

would help here:

Min (Med(є2))


Is There a Linear RelationshipBetween Housing and Tapestry Data?

Bidimensional Regression

r = 0.5449

Must

Standardize

Data


It’s Still a 3-d World


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