clv (version 0.3-2.1)

dot.product: Cosine similarity measure - External Measure, Cluster Stability

Description

Similarity index based on dot product is the measure which estimates how those two different partitionings, that comming from one dataset, are different from each other.

Usage

dot.product(clust1, clust2)

Arguments

clust1

integer vector with information about cluster id the object is assigned to. If vector is not integer type, it will be coerced with warning.

clust2

integer vector with information about cluster id the object is assigned to. If vector is not integer type, it will be coerced with warning.

Value

dot.product returns a cosine similarity measure of two partitionings. NaN is returned when in any partitioning each cluster contains only one object.

Details

Two input vectors keep information about two different partitionings of the same subset comming from one data set. For each partitioning (let say P and P') its matrix representation is created. Let P[i,j] and P'[i,j] each defines as:

P[i,j] = 1 when object i and j belongs to the same cluster and i != j P[i,j] = 0 in other case

Two matrices are needed to compute dot product using formula:

= sum(forall i and j) P[i,j]*P'[i,j]

This dot product satisfy Cauchy-Schwartz inequality <= *. As result we get cosine similarity measure: /sqrt(*)

References

A. Ben-Hur and I. Guyon Detecting stable clusters using principal component analysis, http://citeseer.ist.psu.edu/528061.html

T. Lange, V. Roth, M. L. Braun and J. M. Buhmann Stability-Based Validation of Clustering Solutions, ml-pub.inf.ethz.ch/publications/papers/2004/lange.neco_stab.03.pdf

See Also

Other external measures: std.ext, similarity.index

Examples

# dot.product function(and also similarity.index) is used to compute 
# cluster stability, additional stability functions will be 
# defined - as its arguments some additional functions (wrappers) 
# will be needed

# define wrappers
pam.wrapp <-function(data)
{
	return( as.integer(data$clustering) )
}

identity <- function(data) { return( as.integer(data) ) }

agnes.average <- function(data, clust.num)
{
	return( cutree( agnes(data,method="average"), clust.num ) )
}

# define cluster stability function - cls.stabb

# cls.stabb arguments description:
# data - data to be clustered
# clust.num - number of clusters to which data will be clustered
# sample.num - number of pairs of data subsets to be clustered,
#              each clustered pair will be given as argument for 
#              dot.product and similarity.index functions 
# ratio - value comming from (0,1) section: 
#		  0 - means sample emtpy subset,
#		  1 - means chose all "data" objects
# method - cluster method (see wrapper functions)
# wrapp - function which extract information about cluster id assigned 
#         to each clustered object 

# as a result mean of dot.product (and similarity.index) results,
# computed for subsampled pairs of subsets is given
cls.stabb <- function( data, clust.num, sample.num , ratio, method, wrapp  )
{
	dot.pr  = 0
	sim.ind = 0
	obj.num = dim(data)[1]

	for( j in 1:sample.num )
	{
		smp1 = sort( sample( 1:obj.num, ratio*obj.num ) )
		smp2 = sort( sample( 1:obj.num, ratio*obj.num ) )

		d1 = data[smp1,]
		cls1 = wrapp( method(d1,clust.num) )

		d2 = data[smp2,]
		cls2 = wrapp( method(d2,clust.num) )

		clsm1 = t(rbind(smp1,cls1))
		clsm2 = t(rbind(smp2,cls2))

		m = cls.set.section(clsm1, clsm2)
		cls1 = as.integer(m[,2])
		cls2 = as.integer(m[,3])
		cnf.mx = confusion.matrix(cls1,cls2)
		std.ms = std.ext(cls1,cls2)
		
		# external measures - compare partitioning
		dt = dot.product(cls1,cls2)
		si = similarity.index(cnf.mx)

		if( !is.nan(dt) ) dot.pr = dot.pr + dt/sample.num 
		sim.ind = sim.ind + si/sample.num 
	}
	return( c(dot.pr, sim.ind) )
}

# load and prepare data
library(clv)
data(iris)
iris.data <- iris[,1:4]

# fix arguments for cls.stabb function
iter = c(2,3,4,5,6,7,9,12,15)
smp.num = 5
sub.smp.ratio = 0.8

# cluster stability for PAM
print("PAM method:")
for( i in iter )
{
	result = cls.stabb(iris.data, clust.num=i, sample.num=smp.num,
           ratio=sub.smp.ratio, method=pam, wrapp=pam.wrapp)
	print(result)
}

# cluster stability for Agnes (average-link)
print("Agnes (single) method:")
for( i in iter )
{
	result = cls.stabb(iris.data, clust.num=i, sample.num=smp.num,
            ratio=sub.smp.ratio, method=agnes.average, wrapp=identity)
	print(result)
}