3
$\begingroup$

I am running simulations from a paper on graphical clustering based on latent positions. Essentially, the first step is to do Multidimensional Scaling on the Adjacency matrix, after which the authors select the 'best dimension' by doing an elbow fit. When they truncate the MD Scaled data to only use a few dimension, they also normalize each row before performing clustering. Is there any reason why you would want to normalize a low dimensional MDS output before clustering. I am attaching the R code that does it.

Xhat <- embed_adjacency_matrix(g1,dmax)$X #MDS on data
eval <- sqrt(colSums(Xhat^2))
(dhat <- getElbows(eval,3,plot=F)) #Elbow fit
sXhat <- Xhat[,1:dhat] / sqrt(rowSums(Xhat[,1:dhat]^2))#Normalizing?
$\endgroup$
1
$\begingroup$

One does this normalization because when one knows how characterized/located is each of our observations on each dimension, i.e. when one has Xhat[,1:dhat] in hand, one must normalize these dhat-dimensional characterizations so as to get a comparative point of view across observations. Let's do a numerical example.

Say that Xhat[,1:dhat] is $6$-dimensional (dhat<-6), (illustrated above via its two first rows)

obs1    2.508   1.080   0.072   0.772   0.061   0.094
obs2    9.821   2.508   0.660   0.715   0.883   0.266
...     ...     ...     ...     ...     ...     ...

One sees that along the first component, obs1 is at a ($1$-dimensional) distance of 2.508 from the origin, i.e. from not being categorized along this axis. The first dimension is the most influential for obs1. But would you say that for obs2 the second dimension is as influential as the dimension $1$ for obs1? To really address that sort of consideration, you must first compute sqrt(rowSums(Xhat[,1:dhat]^2)), i.e. the $6$-dimensional euclidean distance (from the origin for each individual), which returns

obs1    2.8
obs2    10.2
...     ... 

and then, get sXhat, i.e. a comparative view of how individual-characterizing each dimension is by normalizing each one-dimensional distance by its $6$d counterpart,

obs1    0.883   0.38    0.025   0.272   0.021   0.033
obs2    0.961   0.245   0.065   0.07    0.086   0.026

One cannot say that for obs2 the second dimension is as influential as the dimension $1$ for obs1. And this conclusion is permitted only by the normalization. Now one can cluster our observations without worrying about scale effects, just-controlled.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.