Statistical Bases for Map Reconstructions and Comparisons

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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 Test
• Procrustean Fit and Permu. Test
• Bidimensional Regression
• Working Example
• Locational Attributes of Eight URSB Campuses

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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

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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
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http//www.mori.com/pubinfo/rmw/two-triangulation-
models.pdf
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http//igeographer.lib.indstate.edu/pohl.pdf
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Things People Fear, F v. M
http//www.analytictech.com/borgatti/papers/borgat
ti 200220-20A20statistical20method20for20co
mparing.pdf
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Face-to-Face Comparisons
http//www.multid.se/references/Chem20Intell20La
b20Syst2072,2012320(2004).pdf
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http//www.gsoresearch.com/page2/map.htm
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Methods

• Eigen-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 AA
• Columns of V Eigenvectors of AA

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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

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Given Dissimilarity Matrices A and B
A Random Permutation Test
N! Permutations 37! 1.4E43 8! 40,320
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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
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http//www.entrenet.com/groedmed/greekm/mythproc.
html
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Centering Scaling
Rotation Dilation to Min ?(?2)
Mirror Reflection
http//www.zoo.utoronto.ca/jackson/pro2.html
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Procrustean Analysis

• Two NxP data configurations, X and Y
• XY 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-1UrY, where r first r columns (NP)
• b (XX)-1XY
• b VrVr ?
• Estimated Y values Ur UrY

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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
• EA ?1 ?1 -?2 X ?1
• EB ?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 26
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 expansion
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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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EXAMPLE Eight URSB Campuses
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BK
RC
RD
RV
TO
SA
TA
SD
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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
D 8x8
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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
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Mantel Test
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Procrustean TestMDS Map Recreation
CONCLUDE Near-perfect Map Recreation
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Driving Distances
Do these differ significantly from linear
distances?
PRACTICAL
STATISTICAL
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DriveD Driving DistancesEight URSB Locations
Multidimensional Scaling, with 2-dimension
solution
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RD
RV
RC
TA
SA
BK
SD
TO
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Bidimensional RegressionAB on YX
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PROTEST Comparison
Bidimensional Regression
Procrustean Rotation
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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 criterion would help here Min (Med(?2))
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Is There a Linear RelationshipBetween Housing
and Tapestry Data?
Bidimensional Regression
r 0.5449
Must Standardize Data
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Its Still a 3-d World
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