Plotting MDS (multidimensional scaling)

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Plotting MDS (multidimensional scaling)

Daniel Malter
Hi,

I just encountered what I thought was strange behavior in MDS. However, it turned out that the mistake was mine. The lesson learned from my mistake is that one should plot on a square pane when plotting results of an MDS. Not doing so can be very misleading. Follow the example of an equilateral triangle below to see what I mean. I hope this helps others to avoid this kind of headache.

Let's say I have an equilateral triangle. Then, the three Euclidean distances between points A, B, and C  are all equal. That is, dist(AB)=dist(AC)=dist(BC). Let the points A, B, and C have (x,y)-coordinates  (0,0), (2,0), and (1,sqrt(3)). Then, MDS should reproduce an equilateral triangle, which it does if there are only three points.

require(MASS)
x=c(0,2,1,0,0,sqrt(3))
dim(x)=c(3,2)
d1=dist(x)
fit1<-isoMDS(d1)
plot(fit1$points, xlab="Coordinate 1", ylab="Coordinate 2",
  main="Metric MDS",type="n")
text(fit1$points, labels = c('A','B','C'), cex=1)

So far so good, until I add more points. Now assume, I add a fourth point D at {0,2*sqrt(3)}. This produces the rectangular triangle ABD with hypothenuse BD that encompasses the smaller triangle ABC such that C lies in the middle between B and D. Then, MDS should reproduce the rectangular triangle ABD and the equilateral triangle ABC within it. However, even though distance matrix d2 below still indicates that ABC is an equilateral triangle, the plot of the MDS does not confirm this.

x=c(0,2,1,0,0,0,sqrt(3),2*sqrt(3))
dim(x)=c(4,2)
d2=dist(x)
fit2<-isoMDS(d2)
plot(fit2$points, xlab="Coordinate 1", ylab="Coordinate 2",
  main="Metric MDS",type="n")
text(fit2$points, labels = c('A','B','C','D'), cex=1)

The reason for this is that the dimension of the plot is automatically scaled to fit the points. This distorts the visual impression of the distances, angular relationships, and relative locations. If you plot on a square pane, however, peace and order are restored in the galaxy.

plot(fit2$points, xlab="Coordinate 1", ylab="Coordinate 2",
  main="Metric MDS",type="n",xlim=c(-3,3),ylim=c(-3,3))
text(fit2$points, labels = c('A','B','C','D'), cex=1)

Best,
Daniel

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Re: Plotting MDS (multidimensional scaling)

Mario Valle
Also try the asp parameter in plot.

plot(fit2$points, xlab="Coordinate 1", ylab="Coordinate 2",main="Metric
MDS",type='n',asp=1)
text(fit2$points, labels = c('A','B','C','D'), cex=1)

Hope it helps
                                 mario

On 02-Apr-11 22:07, Daniel Malter wrote:

> Hi,
>
> I just encountered what I thought was strange behavior in MDS. However, it
> turned out that the mistake was mine. The lesson learned from my mistake is
> that one should plot on a square pane when plotting results of an MDS. Not
> doing so can be very misleading. Follow the example of an equilateral
> triangle below to see what I mean. I hope this helps others to avoid this
> kind of headache.
>
> Let's say I have an equilateral triangle. Then, the three Euclidean
> distances between points A, B, and C  are all equal. That is,
> dist(AB)=dist(AC)=dist(BC). Let the points A, B, and C have
> (x,y)-coordinates  (0,0), (2,0), and (1,sqrt(3)). Then, MDS should reproduce
> an equilateral triangle, which it does if there are only three points.
>
> require(MASS)
> x=c(0,2,1,0,0,sqrt(3))
> dim(x)=c(3,2)
> d1=dist(x)
> fit1<-isoMDS(d1)
> plot(fit1$points, xlab="Coordinate 1", ylab="Coordinate 2",
>    main="Metric MDS",type="n")
> text(fit1$points, labels = c('A','B','C'), cex=1)
>
> So far so good, until I add more points. Now assume, I add a fourth point D
> at {0,2*sqrt(3)}. This produces the rectangular triangle ABD with
> hypothenuse BD that encompasses the smaller triangle ABC such that C lies in
> the middle between B and D. Then, MDS should reproduce the rectangular
> triangle ABD and the equilateral triangle ABC within it. However, even
> though distance matrix d2 below still indicates that ABC is an equilateral
> triangle, the plot of the MDS does not confirm this.
>
> x=c(0,2,1,0,0,0,sqrt(3),2*sqrt(3))
> dim(x)=c(4,2)
> d2=dist(x)
> fit2<-isoMDS(d2)
> plot(fit2$points, xlab="Coordinate 1", ylab="Coordinate 2",
>    main="Metric MDS",type="n")
> text(fit2$points, labels = c('A','B','C','D'), cex=1)
>
> The reason for this is that the dimension of the plot is automatically
> scaled to fit the points. This distorts the visual impression of the
> distances, angular relationships, and relative locations. If you plot on a
> square pane, however, peace and order are restored in the galaxy.
>
> plot(fit2$points, xlab="Coordinate 1", ylab="Coordinate 2",
>    main="Metric MDS",type="n",xlim=c(-3,3),ylim=c(-3,3))
> text(fit2$points, labels = c('A','B','C','D'), cex=1)
>
> Best,
> Daniel
>
>
>
> --
> View this message in context: http://r.789695.n4.nabble.com/Plotting-MDS-multidimensional-scaling-tp3422670p3422670.html
> Sent from the R help mailing list archive at Nabble.com.
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

--
Ing. Mario Valle
Data Analysis and Visualization Group            | http://www.cscs.ch/~mvalle
Swiss National Supercomputing Centre (CSCS)      | Tel:  +41 (91) 610.82.60
v. Cantonale Galleria 2, 6928 Manno, Switzerland | Fax:  +41 (91) 610.82.82

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Re: Plotting MDS (multidimensional scaling)

Jari Oksanen
In reply to this post by Daniel Malter
Daniel Malter <daniel <at> umd.edu> writes:

>
> Let's say I have an equilateral triangle. Then, the three Euclidean
> distances between points A, B, and C  are all equal. That is,
> dist(AB)=dist(AC)=dist(BC). Let the points A, B, and C have
> (x,y)-coordinates  (0,0), (2,0), and (1,sqrt(3)). Then, MDS should reproduce
> an equilateral triangle, which it does if there are only three points.
>
> require(MASS)
> x=c(0,2,1,0,0,sqrt(3))
> dim(x)=c(3,2)
> d1=dist(x)
> fit1<-isoMDS(d1)
> plot(fit1$points, xlab="Coordinate 1", ylab="Coordinate 2",
>   main="Metric MDS",type="n")
> text(fit1$points, labels = c('A','B','C'), cex=1)
>
> So far so good, until I add more points. Now assume, I add a fourth point D
> at {0,2*sqrt(3)}. This produces the rectangular triangle ABD with
> hypothenuse BD that encompasses the smaller triangle ABC such that C lies in
> the middle between B and D. Then, MDS should reproduce the rectangular
> triangle ABD and the equilateral triangle ABC within it. However, even
> though distance matrix d2 below still indicates that ABC is an equilateral
> triangle, the plot of the MDS does not confirm this.
>
> x=c(0,2,1,0,0,0,sqrt(3),2*sqrt(3))
> dim(x)=c(4,2)
> d2=dist(x)
> fit2<-isoMDS(d2)
> plot(fit2$points, xlab="Coordinate 1", ylab="Coordinate 2",
>   main="Metric MDS",type="n")
> text(fit2$points, labels = c('A','B','C','D'), cex=1)
>
Daniel,

Mario Valle already told you about asp=1 in plot() to force equal aspect
ratio, and MASS also has eqscplot() function for plots with geometrically
equal scaling. However, your example above hints that there is
something else you should take care of: You label your plot as
"Metric MDS", but isoMDS does not do metric MDS. The title in its
documentation reads "Kruskal's Non-metric Multidimensional Scaling".
In this case you happened to have metric MDS, because isoMDS uses
metric scaling as its default starting configuration, and in this case
that starting configuration is a perfect fit (stress = 0), and isoMDS()
makes no iterations to change the starting configuration. If you want
to work with metric MDS, use cmdscale() which does metric MDS.

Cheers, jari Oksanen

> The reason for this is that the dimension of the plot is automatically
> scaled to fit the points. This distorts the visual impression of the
> distances, angular relationships, and relative locations. If you plot on a
> square pane, however, peace and order are restored in the galaxy.
>
> plot(fit2$points, xlab="Coordinate 1", ylab="Coordinate 2",
>   main="Metric MDS",type="n",xlim=c(-3,3),ylim=c(-3,3))
> text(fit2$points, labels = c('A','B','C','D'), cex=1)
>
> Best,
> Daniel
>
> --
> View this message in context: http://r.789695.n4.nabble.com/Plotting-MDS-multidimensional-
scaling-tp3422670p3422670.html
> Sent from the R help mailing list archive at Nabble.com.
>
>

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Re: Plotting MDS (multidimensional scaling)

Daniel Malter
The header "metric mds" was actually a leftover because I initially used cmdscale and did not bother changing it for the example.

Thanks,
Daniel