clustering

Clustering manage the problem of arrange data in a series of $k$ cluster such as the intra-cluster similarity is maximized.

So given a set of $N$ object with $D$ features the result of clustering is a clustering scheme, a function that maps each object in a $[1….k]$ clusters or to noise

$$ \forall x_{1},x_{2} \in X clust(x_{1}) = clust(x_{2})\space if \space x_{1}\space is \space similar \space to \space x_{2} $$

$$ \forall x_{1},x_{2} \in X clust(x_{1}) \neq clust(x_{2})\space if \space x_{1}\space is \space not \space similar \space to \space x_{2} $$

centroid

A point which is the center of gravity of the cluster, for each cluster $k$ and dimension $d$, the $d$ coordinate of the centroid is

$$ centroid_{d}^{k}= \frac{1}{x_{i}: clust(x_{i})=k}\sum_{x_{i}: clust(x_{i})=k}{x_{id}} $$