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Statistical Bases for Map Reconstructions and Comparisons

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

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  1. Statistical Bases for Map Reconstructions and Comparisons Jerry Platt May 2005

  2. Preliminaries

  3. 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

  4. 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

  5. Accuracy of a 14th Century Map http://www.geog.ucsb.edu/~tobler/publications/ pdf_docs/geog_analysis/Bi_Dim_Reg.pdf

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

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

  8. Things People Fear, F v. M http://www.analytictech.com/borgatti/papers/borgatti %2002%20-%20A%20statistical%20method%20for%20comparing.pdf

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

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

  11. Eigen-Analysis and Singular-Value Decomposition Multidimensional Scaling & Principal Comps. Mantel Permutation Test Procrustean Fit and Permutation Test Bidimensional Regression Methods

  12. 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

  13. 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

  14. SVD

  15. 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

  16. 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

  17. Given Dissimilarity Matrices A and B: A Random Permutation Test N! Permutations 37! = 1.4*E+43 8! = 40,320

  18. 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

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

  20. Centering & Scaling Rotation & Dilation to Min ∑(є2) Mirror Reflection http://www.zoo.utoronto.ca/jackson/pro2.html

  21. 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

  22. 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

  23. (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

  24. 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

  25. Working Example • Eight URSB Campuses • RD, BK, TO, RC, SA, RV, SD, TA • Data Sources • Locations • Housing Attributes • Tapestry Attributes • Data Analyses

  26. Eight URSB Campuses

  27. 87.5 miles 88.1 miles

  28. …

  29. EXAMPLE: Eight URSB Campuses

  30. BK RC RD RV TO SA TA SD

  31. 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?

  32. Distance Estimates Vary … But Not “Significantly”

  33. MDS RepresentationInput = D; Output = 2d D 8x8

  34. 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

  35. Mantel Test

  36. Procrustean Test:MDS Map Recreation CONCLUDE: Near-perfect Map Recreation

  37. Driving Distances Do these differ “significantly” from linear distances? PRACTICAL STATISTICAL

  38. DriveD = Driving DistancesEight URSB Locations Multidimensional Scaling, with 2-dimension solution

  39. RD RV RC TA SA BK SD TO

  40. Bidimensional Regression:AB on YX

  41. PROTEST Comparison Bidimensional Regression Procrustean Rotation

  42. Housing

  43. Tapestry (ESRI)

  44. Map Coordinates as Explanatory Variables in Linear Models

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