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In this post we will talk about 7 different ways to perform a metric multidimensional scaling in R.

### Multidimensional Scaling

Multidimensional Scaling (MDS), is a set of multivariate data analysis methods that are used to analyze similarities or dissimilarities in data. One of the nice features of MDS is that it allows us to represent the (dis)similarities among pairs of objects as distances between points in a low-dimensional space. Put another way, MDS allows us to visualize the (dis)similarities in a low-dimensional space for exploration and inspection purposes.

The general approach behind MDS consists of calculating a (dis)similarity matrix among pairs of objects (i.e. observations, individuals, samples, etc), and then apply one of the several MDS “models” to obtain the low-dimensional representation. The MDS model to be applied depends on the type of data, and consequently, the type of (dis)similarity measurement that the analyst decides to use.

### Metric Multidimensional Scaling

Depending on the chosen measurement and the obtained (dis)similarity matrix, MDS can be divided in two main approaches: metric and nonmetric. If the analyzed matrix is based on a metric distance, we talk about metric MDS, otherwise we talk about nonmetric MDS.

Metric multidimensional scaling, also known as Principal Coordinate Analysis or Classical Scaling, transforms a distance matrix into a set of coordinates such that the (Euclidean) distances derived from these coordinates approximate as well as possible the original distances (do not confuse Principal Coordinate Analysis with Principal Component Analysis). In other words, the advantage of working with metric MDS, is that the relationships among objects can, in most cases, be fully represented in an Euclidean space.

### Metric Multidimensional Scaling in R

**R** has a number of ways to perform metric MDS. The following list shows you 7 different functions to perform metric MDS (with their corresponding packages in parentheses):

**cmdscale**(stats by R Development Core Team)**smacofSym**(smacof by Jan de Leeuw and Patrick Mair)**wcmdscale**(vegan by Jari Oksanen*et al*)**pco**(ecodist by Sarah Goslee and Dean Urban)**pco**(labdsv by David W. Roberts)**pcoa**(ape by Emmanuel Paradis*et al*)**dudi.pco**(ade4 by Daniel Chessel*et al*)

You should know that all the previous functions require a distance matrix as the main argument to work with. If you don’t have your data in (dis)similairty matrix format, you can calculate the distance matrix with the function **dist**. This is the “work-horse” function in R for calculating distances (euclidean, manhattan,binary, canberra and maximum). In addition, some of the packages mentioned above, provide their own functions for calculating other types of distances.

#### Installing packages

Except for **cmdscale**, the rest of the functions don’t come with the default distribution of R; this means that you have to install their corresponding packages:

# install packages install.packages(c("vegan", "ecodist", "labdsv", "ape", "ade4", "smacof")) # load packages library(vegan) library(ecodist) library(labdsv) library(ape) library(ade4) library(smacof)

#### Data eurodist

We will use the dataset **eurodist** that gives the road distances (in km) between 21 cities in Europe. Notice that **eurodist** is already an object of class **“dist”** (matrix distance). You can inspect the first 5 elements like so:

# convert eurodist to matrix euromat = as.matrix(eurodist) # inspect first five elements euromat[1:5,1:5]

Athens Barcelona Brussels Calais Cherbourg Athens 0 3313 2963 3175 3339 Barcelona 3313 0 1318 1326 1294 Brussels 2963 1318 0 204 583 Calais 3175 1326 204 0 460 Cherbourg 3339 1294 583 460 0

The goal is to apply metric MDS to get a visual representation of the distances between European cities.

#### 1) MDS with cmdscale

The most popular function to perform a classical scaling is **cmdscale** (which comes with the default distribution of R). Its general usage has the following form:

cmdscale(d, k = 2, eig = FALSE, add = FALSE, x.ret = FALSE)

# 1) MDS 'cmdscale' mds1 = cmdscale(eurodist, k=2) # plot plot(mds1[,1], mds1[,2], type = "n", xlab = "", ylab = "", axes = FALSE, main = "cmdscale (stats)") text(mds1[,1], mds1[,2], labels(eurodist), cex=0.9)

As you can see, the obtained graphic allows us to represent the distances between cities in a two-dimensional space. However, the representation is not identical to a geographical map of Europe: Athens is in the north while Stockholm is in the south. This “anomaly” reflects the fact that the representation is not unique; if we wanted to get a more accurate geographical representation, we would need to invert the vertical axis.

#### 2) MDS with wcmdscale

The package ‘vegan’ provides the function **wcmdscale** (Weighted Classical Multidimensional Scaling). Its general usage has the following form:

wcmdscale(d, k, eig = FALSE, add = FALSE, x.ret = FALSE, w)

If we specify the vector of the weights **w** as a vector of ones, **wcmdscale** will give ordinary multidimensional scaling.

# 2) MDS 'wcmdscale' mds2 = wcmdscale(eurodist, k=2, w=rep(1,21)) # plot plot(mds2[,1], mds2[,2], type = "n", xlab = "", ylab = "", axes = FALSE, main = "wcmdscale (vegan)") text(mds2[,1], mds2[,2], labels(eurodist), cex = 0.9)

#### 3) MDS with pco (package ecodist)

The package ‘ecodist’ provides the function **pco** (Principal Coordinates Analysis). Its general usage has the following form:

pco(x, negvals = "zero", dround = 0)

# 3) MDS 'pco' mds3 = pco(eurodist) # plot plot(mds3$vectors[,1], mds3$vectors[,2], type = "n", xlab = "", ylab = "", axes = FALSE, main = "pco (ecodist)") text(mds3$vectors[,1], mds3$vectors[,2], labels(eurodist), cex = 0.9)

#### 4) MDS with pco (package labdsv)

The package ‘labdsv’ also provides a function **pco** (Principal Coordinates Analysis). Its general usage has the following form:

pco(dis, k=2)

# 4) MDS 'pco' mds4 = pco(eurodist, k=2) # plot plot(mds4$points[,1], mds4$points[,2], type = "n", xlab = "", ylab = "", axes = FALSE, main = "pco (labdsv)") text(mds4$points[,1], mds4$points[,2], labels(eurodist), cex = 0.9)

#### 5) MDS with pcoa

The package ‘ape’ provides the function **pcoa** (Principal Coordinates Analysis). Its general usage has the following form:

pcoa(D, correction="none", rn=NULL)

# 5) MDS 'pcoa' mds5 = pcoa(eurodist) # plot plot(mds5$vectors[,1], mds5$vectors[,2], type = "n", xlab = "", ylab = "", axes = FALSE, main = "pcoa (ape)") text(mds5$vectors[,1], mds5$vectors[,2], labels(eurodist), cex = 0.9)

#### 6) MDS with dudi.pco

The package ‘ade4′ provides the function **dudi.pco** (Principal Coordinates Analysis). Its general usage has the following form:

dudi.pco(d, row.w = "uniform", scannf = TRUE, nf = 2, full = FALSE, tol = 1e-07)

# 6) MDS 'dudi.pco' mds6 = dudi.pco(eurodist, scannf=FALSE, nf=2) # plot plot(mds6$li[,1], mds6$li[,2], type = "n", xlab = "", ylab = "", axes = FALSE, main = "dudi.pco (ade4)") text(mds6$li[,1], mds6$li[,2], labels(eurodist), cex = 0.9)

#### 7) MDS with smacofSym

The package ‘smacof’ provides the function **smacofSym** (Multidimensional scaling (stress minimization: SMACOF) on symmetric dissimilarity matrix.). This function uses a majorization approach to get the solution (more info in this vignette). Its general usage has the following form:

smacofSym(delta, ndim = 2, weightmat = NULL, init = NULL, metric = TRUE, ties = "primary", verbose = FALSE, relax = FALSE, modulus = 1, itmax = 1000, eps = 1e-06)

# 7) MDS 'smacofSym' mds7 = smacofSym(eurodist, ndim=2) # plot plot(mds7$conf[,1], mds7$conf[,2], type = "n", xlab = "", ylab = "", axes = FALSE, main = "smacofSym (smacof)") text(mds7$conf[,1], mds7$conf[,2], labels(eurodist), cex = 0.9)

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