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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Outline

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


Motivation

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 14 th century map
Accuracy of a 14th Century Map

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

pdf_docs/geog_analysis/Bi_Dim_Reg.pdf




Things people fear f v m
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
Face-to-Face Comparisons

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



Methods

Eigen-Analysis and Singular-Value Decomposition

Multidimensional Scaling & Principal Comps.

Mantel Permutation Test

Procrustean Fit and Permutation Test

Bidimensional Regression

Methods


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

Eigen-analysis


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

Singular Value Decomposition



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

Principal Component Analysis


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

Multidimensional Scaling


Given Dissimilarity

Matrices A and B:

A Random

Permutation

Test

N! Permutations

37! = 1.4*E+43

8! = 40,320


Permutation tests
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



Centering &

Scaling

Rotation &

Dilation to

Min ∑(є2)

Mirror

Reflection

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


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’)

Procrustean Analysis


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

OLS Regression


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

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
Working Example

  • Eight URSB Campuses

    • RD, BK, TO, RC, SA, RV, SD, TA

  • Data Sources

    • Locations

    • Housing Attributes

    • Tapestry Attributes

  • Data Analyses



87.5 miles

88.1 miles



Example eight ursb campuses
EXAMPLE: Eight URSB Campuses


BK

RC

RD

RV

TO

SA

TA

SD


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

… and if DISTANCES available, but COORDINATES Unavailable?


Distance estimates vary
Distance Estimates Vary

… But Not “Significantly”


Mds representation input d output 2d
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



Procrustean test mds map recreation
Procrustean Test:MDS Map Recreation

CONCLUDE: Near-perfect Map Recreation


Driving distances
Driving Distances

Do these differ “significantly” from linear distances?

PRACTICAL

STATISTICAL


Drived driving distances eight ursb locations
DriveD = Driving DistancesEight URSB Locations

Multidimensional Scaling,

with 2-dimension solution


RD

RV

RC

TA

SA

BK

SD

TO



Protest comparison
PROTEST Comparison

Bidimensional

Regression

Procrustean

Rotation





Incremental tests
Incremental Tests

So Map Coordinates seem sufficient as predictors


Proxy measures of lat long in linear model
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 relationship between housing and tapestry data
Is There a Linear RelationshipBetween Housing and Tapestry Data?

Bidimensional Regression

r = 0.5449

Must

Standardize

Data



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