Measurement, Scaling, and Dimensional Analysis Summer 2017 METRIC MDS IN R
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1 Measurement, Scaling, and Dimensional Analysis Summer 2017 Bill Jacoby METRIC MDS IN R This handout shows the contents of an R session that carries out a metric multidimensional scaling analysis of the driving distances between ten U.S. cities. The analysis is first performed by hand. In other words, R functions are used to double-center the original matrix of distances. The latter is then factored to obtain the coordinates of the points representing the cities. The R function, cmdscale, is also shown. (First few lines of session omitted to save space) ##### ##### ### Metric multidimensional scaling of ### driving distances between US cities ##### ##### library(lattice) ### ### Read matrix of driving distances between ### US cities (in 1000-mile units) from file, ### "Cities, with names.txt", and convert ### from data frame to matrix. ### cities <- as.matrix( + read.table(file.choose(), header = T) ### ### Assign row names ### rownames(cities) <- colnames(cities) cities Atlanta Chicago Denver Houston LA Miami NYC SF Seattle DC Atlanta Chicago Denver Houston LA Miami NYC SF Seattle DC ### ### Double-center matrix of distances ### in order to obtain scalar products. ### ### First, square all entries in matrix ### dist.squared <- cities^2
2 Page 2 ### ### Create matrices of row and column means ### mean.vector <- apply(dist.squared, 1, FUN = mean) row.means <- mean.vector %*% + matrix(data = 1, nrow = 1, ncol = ncol(cities)) col.means <- matrix(data = 1, nrow = nrow(cities), ncol = 1) %*% + t(mean.vector) ### ### Create matrix of grand mean ### grand.mean <- matrix(data = mean(dist.squared), + nrow = nrow(cities), ncol = ncol(cities)) ### ### Create double-centered matrix ### double.centered < * (dist.squared - row.means - + col.means + grand.mean) ### ### Perform SVD (or eigendecomposition) of ### double-centered distance matrix ### svd.dc.dists <- svd(double.centered) ### ### Create two-dimensional coordinates ### by multiplying first two singular vectors ### and square roots of first two singular values. ### coords <- svd.dc.dists$u[,1:2] %*% + diag((svd.dc.dists$d[1:2])^.5) ### ### Round results and provide row labels ### coords <- round(coords, digits = 3) rownames(coords) <- rownames(cities) coords [,1] [,2] Atlanta Chicago Denver Houston LA Miami NYC SF Seattle DC
3 Page 3 ### ### Create scree plot to assess ### dimensionality of MDS solution ### xyplot(svd.dc.dists$d ~ 1:nrow(cities), + panel.xyplot(x, y, col = "black", + pch = 18, type = "b") + xlab = "Singular vector (or eigenvector)", + ylab = "Singular value (or eigenvalue)" ### ### Plot MDS configuration ### xyplot(coords[,2] ~ coords[,1], + panel.xyplot(x, y, col = "black") + panel.text(x, y-.02, labels = rownames(coords), cex =.75) + xlab = "First coordinate axis", + ylab = "Second coordinate axis" ### ### Reflect axes to bring configuration into "normal" ### geographic coordinates. Note importance of making ### scale units and range identical across axes! ### coords <- coords * -1 xyplot(coords[,2] ~ coords[,1], + panel.xyplot(x, y, col = "black") + panel.text(x, y-.05, labels = rownames(coords), cex =.75) + xlab = "First coordinate axis", + ylab = "Second coordinate axis", + xlim = c(-1.75, 1.25), + ylim = c(-1.5, 1.5)
4 Page 4 ### ### Alternative (much easier!) method: ### Use "cmdscale" function. ### cmdscale(cities) [,1] [,2] Atlanta Chicago Denver Houston LA Miami NYC SF Seattle DC Figure 1: Scree plot for metric multidimensional scaling analysis of driving distances between U.S. cities Singular value (or eigenvalue) Singular vector (or eigenvector)
5 Page 5 Figure 2: Plot of two-dimensional point configuration obtained from metric MDS of driving distances between U.S. cities. Note differing scale ranges on axes. 0.6 Miami Houston 0.4 LA Second coordinate axis Atlanta Denver SF -0.4 DC Chicago -0.6 NYC Seattle First coordinate axis Figure 3: Plot of two-dimensional point configuration obtained from metric MDS of driving distances between U.S. cities. Both axes have been reflected, and scale ranges are identical across axes. 1.0 Second coordinate axis Seattle SF LA Denver Chicago Atlanta DC NYC Houston Miami First coordinate axis
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