Intercluster distances and intracluster diameters  Internal Measures
Description
Two functions which find most popular intercluster distances and intracluster diameters.
Usage
1 2  cls.scatt.data(data, clust, dist="euclidean")
cls.scatt.diss.mx(diss.mx, clust)

Arguments
data 

diss.mx 
square, symmetric 
clust 
integer 
dist 
chosen metric: "euclidean" (default value), "manhattan", "correlation"
(variable enable only in 
Details
Six intercluster distances and three intracluster diameters can be used to
calculate such validity indices as Dunn and DaviesBouldin like.
Let d(x,y)
be a distance function between two objects comming from our data set.
Intracluster diameters
The complete diameter represents the distance between two the most remote objects belonging to the same cluster.
diam1(C) = max{ d(x,y): x,y belongs to cluster C }
The average diameter distance defines the average distance between all of the samples belonging to the same cluster.
diam2(C) = 1/C(C1) * sum{ forall x,y belongs to cluster C and x != y } d(x,y)
The centroid diameter distance reflects the double average distance between all of the samples and the cluster's center (v(C)  cluster center).
diam3(C) = 1/C * sum{ forall x belonging to cluster C} d(x,v(C))
Intercluster distances
The single linkage distance defines the closest distance between two samples belonging to two different clusters.
dist1(Ci,Cj) = min{ d(x,y): x belongs to Ci and y to Cj cluster }
The complete linkage distance represents the distance between the most remote samples belonging to two different clusters.
dist2(Ci,Cj) = max{ d(x,y): x belongs to Ci and y to Cj cluster }
The average linkage distance defines the average distance between all of the samples belonging to two different clusters.
dist3(Ci,Cj) = 1/(Ci*Cj) * sum{ forall x belongs Ci and y to Cj } d(x,y)
The centroid linkage distance reflects the distance between the centres of two clusters (v(i), v(j)  clusters' centers).
dist4(Ci,Cj) = d(v(i), V(j))
The average of centroids linkage represents the distance between the centre of a cluster and all of samples belonging to a different cluster.
dist5(Ci,Cj) = 1/(Ci+Cj) * ( sum{ forall x belongs Ci } d(x,v(j)) + sum{ forall y belongs Cj } d(y,v(i)) )
Hausdorff metrics are based on the discovery of a maximal distance from samples of one cluster to the nearest sample of another cluster.
dist6(Ci,Cj) = max{ distH(Ci,Cj), distH(Cj,Ci) }
where: distH(A,B) = max{ min{ d(x,y): y belongs to B}: x belongs to A }
Value
cls.scatt.data
returns an object of class "list"
.
Intracluster diameters:
intracls.complete
,
intracls.average
,
intracls.centroid
,
are stored in vectors and intercluster distances:
intercls.single
,
intercls.complete
,
intercls.average
,
intercls.centroid
,
intercls.ave_to_cent
,
intercls.hausdorff
in symmetric matrices.
Vectors' lengths and both dimensions of each matrix are equal to number of clusters.
Additionally in result list cluster.center
matrix (rows correspond to clusters centers)
and cluster.size
vector is given (information about size of each cluster).
cls.scatt.diss.mx
returns an object of class "list"
.
Intracluster diameters:
intracls.complete
,
intracls.average
,
are stored in vectors and intercluster distances:
intercls.single
,
intercls.complete
,
intercls.average
,
intercls.hausdorff
in symmetric matrices.
Vectors' lengths and both dimensions of each matrix are equal to number of clusters.
Additionally in result list cluster.size
vector is given (information about size of each cluster).
Author(s)
Lukasz Nieweglowski
References
J. Handl, J. Knowles and D. B. Kell Computational cluster validation in postgenomic data analysis, http://bioinformatics.oxfordjournals.org/cgi/reprint/21/15/3201?ijkey=VbTHU29vqzwkGs2&keytype=ref
N. Bolshakova, F. Azuajeb Cluster validation techniques for genome expression data, http://citeseer.ist.psu.edu/552250.html
See Also
Result used in: clv.Dunn
, clv.Davies.Bouldin
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17  # load and prepare data
library(clv)
data(iris)
iris.data < iris[,1:4]
# cluster data
pam.mod < pam(iris.data,5) # create five clusters
v.pred < as.integer(pam.mod$clustering) # get cluster ids associated to given data objects
# compute intercluster distances and intracluster diameters
cls.scatt1 < cls.scatt.data(iris.data, v.pred)
cls.scatt2 < cls.scatt.data(iris.data, v.pred, dist="manhattan")
cls.scatt3 < cls.scatt.data(iris.data, v.pred, dist="correlation")
# the same using dissimilarity matrix
iris.diss.mx < as.matrix(daisy(iris.data))
cls.scatt4 < cls.scatt.diss.mx(iris.diss.mx, v.pred)
