# Why normalize data after doing Multidimensional scaling?

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?  ## 1 Answer 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. • That makes sense! But I am thinking that won't it make our clustering worse if we suppose had$dhat=2 $and$X[1] = 9.5, 9$and$X[2] = 1, 1.1$. Because normalizing will reduce these and put them very close to each other and our clustering algorithm would cluster$X[1]$and$X[2]\$ together while they are actually in 2 different clusters. – daniel Nov 22 '17 at 18:28
• @daniel nothing stops you from removing the normalization line and see if the so-obtained clusters make sense. – keepAlive Nov 22 '17 at 18:48
• @dianiel, however, I think that the scale effect you are thinking of, should be considered as a new dimension, kept by getElbows if significantly categorizing. – keepAlive Nov 22 '17 at 18:51
• – keepAlive Nov 22 '17 at 19:37