Loading in 3 Seconds
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
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
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
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