# Using original centroid as cluster identifier after applying PCA

Take a look at my original data. (masked with purely random alphabetic here) :

    a b c d e
f g h i j
A = k l m n o
p q r s t
u v w x y


I'm running kMeans (3 cluster) on the data, resulting final centroid like this :

    aa aa aa aa aa
B = bb bb bb bb bb
cc cc cc cc cc


Now, before run kMean again, I applied PCA on data and took only first three principal component :

    xa yb zc
xd ye zf
C = xg yh zi
xj yk zl
xm yn zo


After that, I ran kMeans for 3 centroid and, of course, resulting 3 centroid :

    xaa yaa zcc
D = xbb ybb zbb
xcc ycc zcc


The cluster result with PCA are exactly same with the first test (without PCA). My question : After finishing kMeans with PCA, can I say that this cluster (say cluster 1) has centroid aa aa aa aa aa, rather than saying that cluster 1 has centroid xaa yaa zcc?

• Since both cluster analyses in your specific case gave the same results - the same distribution of cases among clusters - then yes you may of course say it, it's obvious. Apr 22, 2016 at 7:01

$$M\cdot \mu = M\cdot (\frac{1}{N}\sum_i \vec{x}_i) = \frac{1}{N}\sum_i M\cdot\vec{x}_i$$