Multidimensional scaling (MDS)

Install required packages

vegan

wants <- c("vegan")
has   <- wants %in% rownames(installed.packages())
if(any(!has)) install.packages(wants[!has])

Metric MDS

Given a distance matrix

cities <- c("Augsburg", "Berlin", "Dresden", "Hamburg", "Hannover",
            "Karlsruhe", "Kiel", "Muenchen", "Rostock", "Stuttgart")
N      <- length(cities)
dstMat <- matrix(numeric(N^2), nrow=N)
##             B   DD   HH    H   KA   KI    M  HRO    S
cityDst <- c(596, 467, 743, 599, 226, 838,  65, 782, 160, ## AUG
                  194, 288, 286, 673, 353, 585, 231, 633, ## B
                       477, 367, 550, 542, 465, 420, 510, ## DD
                            157, 623,  96, 775, 187, 665, ## HH
                                 480, 247, 632, 330, 512, ## H
                                      723, 298, 805,  80, ## KA
                                           872, 206, 752, ## KI
                                                777, 220, ## M
                                                     824) ## HRO

dstMat[upper.tri(dstMat)] <- rev(cityDst)
dstMat <- t(dstMat[ , N:1])[ , N:1]
dstMat[lower.tri(dstMat)] <- t(dstMat)[lower.tri(dstMat)]
dimnames(dstMat) <- list(city=cities, city=cities)

(mds <- cmdscale(dstMat, k=2))

                [,1]       [,2]
Augsburg   399.60802  -70.51443
Berlin    -200.20462 -183.39465
Dresden    -18.47337 -213.01950
Hamburg   -316.39991  130.72245
Hannover  -161.38209  120.21761
Karlsruhe  333.38724  212.12523
Kiel      -409.08703  147.62226
Muenchen   408.55752 -144.46446
Rostock   -401.19605 -126.20651
Stuttgart  365.19030  126.91200

Given object-wise ratings

set.seed(123)
P   <- 3
obj <- matrix(sample(-20:20, N*P, replace=TRUE), ncol=P)
dst <- dist(obj, diag=TRUE, upper=TRUE)
cmdscale(dst, k=2)
# not shown

Plot

xLims <- range(mds[ , 1]) + c(0, 250)
plot(mds, xlim=xLims, xlab="North-South", ylab="East-West", pch=16,
     main="City locations according to MDS")
text(mds[ , 1]+50, mds[ , 2], adj=0, labels=cities)

Non-metric MDS

library(vegan)
(nmMDS <- monoMDS(dstMat, k=2))

Call:
monoMDS(dist = dstMat, k = 2) 

Non-metric Multidimensional Scaling

10 points, dissimilarity 'unknown'

Dimensions: 2 
Stress:     0.06803376 
Stress type 1, weak ties
Scores scaled to unit root mean square, rotated to principal components
Stopped after 55 iterations: Stress nearly unchanged (ratio > sratmax)

scores(nmMDS)

                 MDS1        MDS2
Augsburg  -1.20797386 -0.01719197
Berlin     0.59341232 -0.41182940
Dresden    0.09538295 -0.73450992
Hamburg    0.85903495  0.30137581
Hannover   0.49243793  0.26880268
Karlsruhe -1.00749610 -0.18900784
Kiel       1.16050702  0.01464720
Muenchen  -1.17092083  0.34936915
Rostock    1.17197472  0.23940430
Stuttgart -0.98635909  0.17894000
attr(,"pc")
[1] TRUE

Detach (automatically) loaded packages (if possible)

try(detach(package:vegan))
try(detach(package:permute))

Get the article source from GitHub

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