Mean, cluster size and center - cluster utilities

Description

Mean, center of each cluster, number of objects in each cluster - informations retrieved from partitioned data using cls.attrib.

Usage

cls.attrib(data, clust)

Arguments

Value

As a result function returns object of list type which contains three objects with information about:
mean - numeric vector which represents mean of given data,
cluster.center - numeric matrix where columns correspond to variables and rows to observations,
cluster.size - integer vector with information about size of each cluster.

Author(s)

Lukasz Nieweglowski

See Also

Result of function is mostly used to compute following indicies: clv.Dis, wcls.matrix, bcls.matrix.

Examples

# create "data" matrix
mx <- matrix(0,4,2)
mx[2,1] = mx[3,2] = mx[4,1] = mx[4,2] = 1
# give information about cluster assignment
clust = as.integer(c(1,1,2,2))
cls.attrib(mx,clust)

Intercluster distances and intracluster diameters - Internal Measures

Description

Two functions which find most popular intercluster distances and intracluster diameters.

Usage

cls.scatt.data(data, clust, dist="euclidean")
cls.scatt.diss.mx(diss.mx, clust)

Arguments

Details

Six intercluster distances and three intracluster diameters can be used to calculate such validity indices as Dunn and Davies-Bouldin 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|(|C|-1) * 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 post-genomic 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

# 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)

Section of two subsets - External Measure utilities

Description

Function finds section of two different subsets comming from the same data set.

Usage

cls.set.section(clust1, clust2)

Arguments

Details

Let A and B be two different subsamples of the same data set. Each subset is partitioned into P(A) and P(B) cluster sets. Information about object and cluster id's for pairs (A,P(A)) and (B,P(B)) are stored in matrices clust1 and clust2. Function creates matrix which represents section of A and B.

Value

cls.set.section returns a n x 3 integer matrix. First column gives information about object number in dataset in increasing order. Second column store information about cluster id the object is assigned to. Information is taken from clust1 vector The same is for the third column but cluster id is taken from vector clust2.

Author(s)

Lukasz Nieweglowski

See Also

Function preapres data for further computation. Result mostly is used in: std.ext, dot.product, confusion.matrix

Examples

# create two different subsamples 
mx1 <- matrix(as.integer( c(1,2,3,4,5,6,1,1,2,2,3,3) ), 6, 2 )
mx2 <- matrix(as.integer( c(1,2,4,5,6,7,1,1,2,2,3,3) ), 6, 2 )
# find section
m = cls.set.section(mx1,mx2)

Cluster Stability - Similarity Index and Pattern-wise Stability Approaches

Description

cls.stab.sim.ind and cls.stab.opt.assign reports validation measures for clustering results. Both functions return lists of cluster stability results computed according to similarity index and pattern-wise stability approaches.

Usage

cls.stab.sim.ind( data, cl.num, rep.num, subset.ratio, clust.method,
                   method.type, sim.ind.type, fast, ... )
cls.stab.opt.assign( data, cl.num, rep.num, subset.ratio, clust.method,
                      method.type, fast, ... )

Arguments

Details

Both functions realize cluster stability approaches described in Detecting stable clusters using principal component analysis (see references).

The cls.stab.sim.ind function realizes algorithm given in chapter 3.1 where only cosine similarity index (see dot.product) is introduced as a similarity index between two different partitionings. This function realize this cluster stability approach also for other similarity indices such us similarity.index, clv.Rand and clv.Jaccard. The important thing is that similarity index (if chosen) produced by this function is not exactly the same as index produced by similarity.index function. The value of the similarity.index is a number which depends on number of clusters. Eg. if two "n-clusters" partitionings are compared the value always will be a number which belong to the [1/n, 1] section. That means the results produced by this similarity index are not comparable for different number of clusters. That's why each result is scaled thanks to the linear function f:[1/n, 1] -> [0, 1] where "n" is a number of clusters. The results' layout is described in Value section.

The cls.stab.opt.assign function realizes algorithm given in chapter 3.2 where pattern-wise agreement and pattern-wise stability was introduced. Function returns the lowest pattern-wise stability value for given number of clusters. The results' layout is described in Value section.

It often happens that clustering algorithms can't produce amount of clusters that user wants. In this situation only the warning is produced and cluster stability is computed for partitionings with unequal number of clusters.

The cluster stability will not be calculated for all cluster numbers that are bigger than the subset size. For example if data contains about 20 objects and the subset.ratio equals 0.5 then the highest cluster number to calculate is 10. In that case all elements above 10 will be removed from cl.num vector.

Value

cls.stab.sim.ind returns a list of lists of matrices. Each matrix consists of the set of external similarity indices (which one similarity index see below) where number of columns is equal to cl.num vector length and row number is equal to rep.num value what means that each column contain a set of similarity indices computed for fixed number of clusters. The order of the matricides depends on three input arguments: clust.method, method.type, and sim.ind.type. Combination of clust.method and method.type give a names for elements listed in the first list. Each element of this list is also a list type where each element name correspond to one of similarity index type chosen thanks to sim.ind.type argument. The order of the names exactly match to the order given in those arguments description. It is easy to understand after considering the following example.
Let say we are running cls.stab.sim.ind with default arguments then the results will be given in the following order: $agnes.single$dot.pr, $agnes.single$sim.ind, $agnes.average$dot.pr, $agnes.average$sim.ind, $pam$dot.pr, $pam$sim.ind.

cls.stab.opt.assign returns a list of vectors. Each vector consists of the set of cluster stability indices described in Detecting stable clusters using principal component analysis (see references). Vector length is equal to cl.num vector length what means that each position in vector is assigned to proper clusters' number given in cl.num argument. The order of the vectors depends on two input arguments: clust.method, method.type. The order of the names exactly match to the order given in arguments description. It is easy to understand after considering the following example.
Let say we are running cls.stab.opt.assign with c("pam", "kmeans", "hclust", "agnes") as clust.method and c("ward","average") as method.type then the results will be given in the following order: $hclust.average, $hclust.ward, $agnes.average, $agnes.ward, $kmeans, $pam.

Author(s)

Lukasz Nieweglowski

References

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

C. D. Giurcaneanu, I. Tabus, I. Shmulevich, W. Zhang Stability-Based Cluster Analysis Applied To Microarray Data, http://citeseerx.ist.psu.edu/.

T. Lange, V. Roth, M. L. Braun and J. M. Buhmann Stability-Based Validation of Clustering Solutions, doi:10.1162/089976604773717621

See Also

Advanced cluster stability functions: cls.stab.sim.ind.usr, cls.stab.opt.assign.usr.

Functions that compare two different partitionings: clv.Rand, dot.product, similarity.index.

Examples

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

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

res1 = cls.stab.sim.ind( iris.data, iter, rep.num=smp.num, subset.ratio=0.7,
                         sim.ind.type=c("rand","dot.pr","sim.ind") )
res2 = cls.stab.opt.assign( iris.data, iter, clust.method=c("hclust","kmeans"),
                             method.type=c("single","average") )

print(res1)
boxplot(res1$agnes.average$sim.ind)
plot(res2$hclust.single)

Cluster Stability - Similarity Index and Pattern-wise Stability Approaches with User Defined Cluster Algorithms

Description

cls.stab.sim.ind.usr and cls.stab.opt.assign.usr reports validation measures for clustering results. Both functions return lists of cluster stability results computed for user defined cluster algorithms according to similarity index and pattern-wise stability approaches.

Usage

cls.stab.sim.ind.usr( data, cl.num, clust.alg, sim.ind.type, rep.num, subset.ratio )
cls.stab.opt.assign.usr( data, cl.num, clust.alg, rep.num, subset.ratio )
cls.alg( clust.method, clust.wrap, fast )

Arguments

Details

Both functions realize cluster stability approaches described in Detecting stable clusters using principal component analysis chapters 3.1 and 3.2 (see references).

The cls.stab.sim.ind.usr as well as cls.stab.opt.assign.usr do the same thing as cls.stab.sim.ind and cls.stab.opt.assign functions. Main difference is that using this functions user is able to define and apply its own cluster algorithm to measure its cluster stability. For that reason clust.alg argument is introduced. This argument may represent partitioning algorithm (by passing it directly as a function) or hierarchical algorithm (by passing an object of "cls.alg" type produced by cls.alg function).

If a partitioning algorithm is going to be used the decalration of this function that represents this algorithm should always look like this: function(data, clust.num) { ... return(integer.vector)} . As an output function should always return integer vector that represents single clustering result on data.

If a hierarchical algorithm is going to be used user has to use helper cls.alg function that produces an object of "cls.alg" type. This object encapsulates a pair of methods that are used in hierarchical version (which is faster if the fast argument is not FALSE) of cluster stability approach. These methods are:
1. clust.method - which builds hierarchical structure that might be cut. The declaration of this function should always look like this one: function(data) { ... return(hierarchical.struct) } ,
2. clust.wrap - which cuts this hierarchical structure to clust.num clusters. This function definition should always look like this one: function(clust.res, clust.num) { ... return(integer.vector)} . As an output function should always return integer vector that represents single clustering result on clust.res.

cls.alg function has also third argument that indicates if fast computation should be taken (when TRUE) or if these two methods should be converted to one partitioning algorithm and to be run as a normal partitioning algorithm.

Well defined cluster functions "f" should always follow this rules (size(data) means number of object to be partitioned, res - integer vector with cluster ids):
1. when data is empty or cl.num is less than 2 or more than size(data) then f(data, cl.num) returns error. 2. if f(data, cl.num) -> res then length(res) == size(data),
3. if f(data, cl.num) -> res then for all "elem" in "res" the folowing condition is true: 0 < elem <= cl.num.

It often happens that clustering algorithms can't produce amount of clusters that user wants. In this situation only the warning is produced and cluster stability is computed for partitionings with unequal number of clusters.

The cluster stability will not be calculated for all cluster numbers that are bigger than the subset size. For example if data contains about 20 objects and the subset.ratio equals 0.5 then the highest cluster number to calculate is 10. In that case all elements above 10 will be removed from cl.num vector.

Value

cls.stab.sim.ind.usr returns a lists of matrices. Each matrix consists of the set of external similarity indices (which one similarity index see below) where number of columns is equal to cl.num vector length and row number is equal to rep.num value what means that each column contain a set of similarity indices computed for fixed number of clusters. The order of the matrices depends on sim.ind.type argument. Each element of this list correspond to one of similarity index type chosen thanks to sim.ind.type argument. The order of the names exactly match to the order given in those arguments description.

cls.stab.opt.assign.usr returns a vector. The vector consists of the set of cluster stability indices described in Detecting stable clusters using principal component analysis chapter 3.2 (see references). Vector length is equal to cl.num vector length what means that each position in vector is assigned to proper clusters' number given in cl.num argument.

Author(s)

Lukasz Nieweglowski

References

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

C. D. Giurcaneanu, I. Tabus, I. Shmulevich, W. Zhang Stability-Based Cluster Analysis Applied To Microarray Data, http://citeseerx.ist.psu.edu/.

T. Lange, V. Roth, M. L. Braun and J. M. Buhmann Stability-Based Validation of Clustering Solutions, doi:10.1162/089976604773717621

See Also

Other cluster stability methods: cls.stab.sim.ind, cls.stab.opt.assign.

Functions that compare two different partitionings: clv.Rand, dot.product,similarity.index.

Examples

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

# example of wrapper for partitioning algorithm 
pam.clust <- function(data, clust.num) pam(data, clust.num, cluster.only=TRUE)

# example of wrapper for hierarchical algorithm
cutree.wrap <- function(clust.res, clust.num)  cutree(clust.res, clust.num)
agnes.single <- function(data) agnes(data, method="single") 

# converting hierarchical algorithm to partitioning one
agnes.part1 <- function(data, clust.num) cutree.wrap( agnes.single(data), clust.num )
# the same using "cls.alg"
agnes.part2 <- cls.alg(agnes.single, cutree.wrap, fast=FALSE)

# fix arguments for cls.stab.* function
iter = c(2,4,5,7,9,12,15)

res1 = cls.stab.sim.ind.usr( iris.data, iter, pam.clust, 
    sim.ind.type=c("rand","dot.pr","sim.ind"), rep.num=5, subset.ratio=0.7 )
res2 = cls.stab.opt.assign.usr( iris.data, iter, clust.alg=cls.alg(agnes.single, cutree.wrap) )

res3 = cls.stab.sim.ind.usr( iris.data, iter, agnes.part1,
     sim.ind.type=c("rand","dot.pr","sim.ind"), rep.num=5, subset.ratio=0.7 )
res4 = cls.stab.opt.assign.usr( iris.data, iter, clust.alg=agnes.part2 )

print(res1)
boxplot(res1$sim.ind)
plot(res2)

Davies-Bouldin Index - Internal Measure

Description

Function computes Dunn index - internal measure for given data and its partitioning.

Usage

clv.Davies.Bouldin( index.list, intracls, intercls)

Arguments

Details

Davies-Bouldin index is given by equation:

DB = (1/|C|) sum{forall i in 1:|C|} max[ i != j ] { (diam(Ci) + diam(Cj))/dist(Ci,Cj) }

Value

As output user gets the matrix of Davies-Bouldin indices. Matrix dimension depends on how many diam and dist measures are chosen by the user, normally dim(D)=c(length(intercls),length(intracls)). Each pair: (inter-cluster dist, intra-cluster diam) have its own position in result matrix.

Author(s)

Lukasz Nieweglowski

References

M. Halkidi, Y. Batistakis, M. Vazirgiannis Clustering Validity Checking Methods : Part II, http://citeseer.ist.psu.edu/537304.html

See Also

Functions which produce index.list input argument: cls.scatt.data, cls.scatt.diss.mx. Related functions: clv.Dunn.

Examples

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

# cluster data
agnes.mod <- agnes(iris.data) # create cluster tree 
v.pred <- as.integer(cutree(agnes.mod,5)) # "cut" the tree 

intraclust = c("complete","average","centroid")
interclust = c("single", "complete", "average","centroid", "aveToCent", "hausdorff")

# compute Davies-Bouldin indicies (also Dunn indicies)
# 1. optimal solution:

# compute intercluster distances and intracluster diameters
cls.scatt <- cls.scatt.data(iris.data, v.pred, dist="manhattan")

# once computed valuse use in both functions
dunn1 <- clv.Dunn(cls.scatt, intraclust, interclust)
davies1 <- clv.Davies.Bouldin(cls.scatt, intraclust, interclust)

# 2. functional solution:

# define new Dunn and Davies.Bouldin functions
Dunn <- function(data,clust) 
  clv.Dunn( cls.scatt.data(data,clust),
     intracls = c("complete","average","centroid"), 
     intercls = c("single", "complete", "average","centroid", "aveToCent", "hausdorff")
  )
Davies.Bouldin <- function(data,clust) 
  clv.Davies.Bouldin( cls.scatt.data(data,clust),
    intracls = c("complete","average","centroid"),
    intercls = c("single", "complete", "average","centroid", "aveToCent", "hausdorff")
  )

# compute indicies
dunn2 <- Dunn(iris.data, v.pred)
davies2 <- Davies.Bouldin(iris.data, v.pred)

Inter-cluster density - Internal Measure

Description

Function computes inter-cluster density.

Usage

clv.DensBw(data, clust, scatt.obj, dist="euclidean")

Arguments

Details

The definition of inter-cluster density is given by equation:

Dens_bw = 1/(|C|*(|C|-1)) * sum{forall i in 1:|C|} sum{forall j in 1:|C| and j != i} density(u(i,j))/max{density(v(i)), density(v(j))}

where:

Let define function f(x,u):

Function f is used in definition of density(u):

density(u) = sum{forall i in 1:n(i,j)} f(xi,u)

where n(i,j) is the number of objects which belongs to clusters i and j and xi is such object.

This value is used by clv.SDbw.

Value

As result Dens_bw value is returned.

Author(s)

Lukasz Nieweglowski

See Also

clv.SD and clv.SDbw

Examples

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

# cluster data
agnes.mod <- agnes(iris.data) # create cluster tree 
v.pred <- as.integer(cutree(agnes.mod,5)) # "cut" the tree 

# compute Dens_bw index
scatt <- clv.Scatt(iris.data, v.pred)
dens.bw <- clv.DensBw(iris.data, v.pred, scatt)

Total separation between clusters - Internal Measure

Description

Function computes total separation between clusters.

Usage

clv.Dis(cluster.center)

Arguments

Details

The definition of total separation between clusters is given by equation:

Dis = (Dmax/Dmin) * sum{forall i in 1:|C|} 1 /( sum{forall j in 1:|C|} ||vi - vj|| )

where:

This value is a part of clv.SD and clv.SDbw.

Value

As result Dis value is returned.

Author(s)

Lukasz Nieweglowski

References

M. Haldiki, Y. Batistakis, M. Vazirgiannis On Clustering Validation Techniques, http://citeseer.ist.psu.edu/513619.html

See Also

clv.SD and clv.SDbw

Examples

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

# cluster data
agnes.mod <- agnes(iris.data) # create cluster tree 
v.pred <- as.integer(cutree(agnes.mod,5)) # "cut" the tree 

# compute Dis index
scatt <- clv.Scatt(iris.data, v.pred)
dis <- clv.Dis(scatt$cluster.center)

Dunn Index - Internal Measure

Description

Function computes Dunn index - internal measure for given data and its partitioning.

Usage

clv.Dunn( index.list, intracls, intercls)

Arguments

Details

Dunn index:

D = [ min{ k,l - numbers of clusters } dist(Ck, Cl) ]/[ max{ m - cluster number } diam(Cm) ]

Value

As output user gets matrix of Dunn indices. Matrix dimension depends on how many diam and dist measures are chosen by the user, normally dim(D)=c(length(intercls),length(intracls)). Each pair: (inter-cluster dist, intra-cluster diam) have its own position in result matrix.

Author(s)

Lukasz Nieweglowski

References

M. Halkidi, Y. Batistakis, M. Vazirgiannis Clustering Validity Checking Methods : Part II, http://citeseer.ist.psu.edu/537304.html

See Also

Functions which produce index.list input argument: cls.scatt.data, cls.scatt.diss.mx. Related functions: clv.Davies.Bouldin.

Examples

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

# cluster data
agnes.mod <- agnes(iris.data) # create cluster tree 
v.pred <- as.integer(cutree(agnes.mod,5)) # "cut" the tree 

intraclust = c("complete","average","centroid")
interclust = c("single", "complete", "average","centroid", "aveToCent", "hausdorff")

# compute Dunn indicies (also Davies-Bouldin indicies)
# 1. optimal solution:

# compute intercluster distances and intracluster diameters
cls.scatt <- cls.scatt.data(iris.data, v.pred, dist="manhattan")

# once computed valuse use in both functions
dunn1 <- clv.Dunn(cls.scatt, intraclust, interclust)
davies1 <- clv.Davies.Bouldin(cls.scatt, intraclust, interclust)

# 2. functional solution:

# define new Dunn and Davies.Bouldin functions
Dunn <- function(data,clust) 
  clv.Dunn( cls.scatt.data(data,clust),
     intracls = c("complete","average","centroid"), 
     intercls = c("single", "complete", "average","centroid", "aveToCent", "hausdorff")
  )
Davies.Bouldin <- function(data,clust) 
  clv.Davies.Bouldin( cls.scatt.data(data,clust),
    intracls = c("complete","average","centroid"),
    intercls = c("single", "complete", "average","centroid", "aveToCent", "hausdorff")
  )

# compute indicies
dunn2 <- Dunn(iris.data, v.pred)
davies2 <- Davies.Bouldin(iris.data, v.pred)

SD, SDbw - Internal Measures

Description

Function computes SD and \textrm{S\_Dbw} validity indices.

Usage

clv.SD(scatt, dis, alfa)
clv.SDbw(scatt, dens)

Arguments

Details

SD validity index is defined by equation:

SD = scatt*alfa + dis

where scatt means average scattering for clusters defined in clv.Scatt. \textrm{S\_Dbw} validity index is defined by equation:

\textrm{S\_Dbw} = scatt + dens

where dens is defined in clv.DensBw.

Value

As result of clv.SD function SD validity index is returned. As result of clv.SDbw function \textrm{S\_Dbw} validity index is returned.

Author(s)

Lukasz Nieweglowski

References

M. Haldiki, Y. Batistakis, M. Vazirgiannis On Clustering Validation Techniques, http://citeseer.ist.psu.edu/513619.html

See Also

clv.Scatt, clv.Dis and clv.DensBw

Examples

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

# cluster data
agnes.mod <- agnes(iris.data) # create cluster tree 
v.pred <- as.integer(cutree(agnes.mod,5)) # "cut" the tree 

# prepare proper input data for SD and S_Dbw indicies
scatt <- clv.Scatt(iris.data, v.pred)
dis <- clv.Dis(scatt$cluster.center)
dens.bw <- clv.DensBw(iris.data, v.pred, scatt)

# compute  SD and S_Dbw indicies
SD <- clv.SD(scatt$Scatt, dis, alfa=5) # alfa is equal to number of clusters 
SDbw <- clv.SDbw(scatt$Scatt, dens.bw)

Average scattering for clusters - Internal Measure

Description

Function computes average scattering for clusters.

Usage

clv.Scatt(data, clust, dist="euclidean")

Arguments

Details

Let scatter for set X assigned as sigma(X) be defined as vector of variances computed for particular dimensions. Average scattering for clusters is defined as:

Scatt = (1/|C|) * sum{forall i in 1:|C|} ||sigma(Ci)||/||sigma(X)||

where:

Standard deviation is defined as:

stdev = (1/|C|) * sqrt( sum{forall i in 1:|C|} ||sigma(Ci)|| )

Value

As result list with three values is returned.

Author(s)

Lukasz Nieweglowski

References

M. Haldiki, Y. Batistakis, M. Vazirgiannis On Clustering Validation Techniques, http://citeseer.ist.psu.edu/513619.html

See Also

clv.SD and clv.SDbw

Examples

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

# cluster data
agnes.mod <- agnes(iris.data) # create cluster tree 
v.pred <- as.integer(cutree(agnes.mod,5)) # "cut" the tree 

# compute Scatt index
scatt <- clv.Scatt(iris.data, v.pred)

Confusion Matrix - External Measures, Cluster Stability

Description

For two different partitioning function computes confusion matrix.

Usage

confusion.matrix(clust1, clust2)

Arguments

Details

Let P and P' be two different partitioning of the same data. Partitionings are represent as two vectors clust1, clust2. Both vectors should have the same length. Confusion matrix measures the size of intersection between clusters comming from P and P' according to equation:

M[i,j] = | intersection of P(i) and P'(j) |

where:

Value

cls.set.section returns a n x m integer matrix where n = |P| and m = |P'| defined above.

Author(s)

Lukasz Nieweglowski

See Also

Result used in similarity.index.

Examples

# create two different subsamples 
mx1 <- matrix(as.integer( c(1,2,3,4,5,6,1,1,2,2,3,3) ), 6, 2 )
mx2 <- matrix(as.integer( c(1,2,4,5,6,7,1,1,2,2,3,3) ), 6, 2 )
# find section
m = cls.set.section(mx1,mx2)
confusion.matrix(as.integer(m[,2]),as.integer(m[,3]))

Connectivity Index - Internal Measure

Description

Function evaluates connectivity index.

Usage

connectivity(data,clust,neighbour.num, dist="euclidean")
connectivity.diss.mx(diss.mx,clust,neighbour.num)

Arguments

Details

For given data and its partitioning connectivity index is computed. For choosen pattern neighbour.num nearest neighbours are found and sorted from closest to most further. Alghorithm checks if those neighbours are assigned to the same cluster. At the beggining connectivity value is equal 0 and increase with value:

Procedure is repeated for all patterns which comming from our data set. All values received for particular pattern are added and creates main connectivity index.

Value

connectivity returns a connectivity value.

Author(s)

Lukasz Nieweglowski

References

J. Handl, J. Knowles and D. B. Kell Sumplementary material to computational cluster validation in post-genomic data analysis, http://dbkgroup.org/handl/clustervalidation/supplementary.pdf

Examples

# 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 gived data objects

# compute connectivity index using data and its clusterization
conn1 <- connectivity(iris.data, v.pred, 10)
conn2 <- connectivity(iris.data, v.pred, 10, dist="manhattan")
conn3 <- connectivity(iris.data, v.pred, 10, dist="correlation")

# the same using dissimilarity matrix
iris.diss.mx <- as.matrix(daisy(iris.data))
conn4 <- connectivity.diss.mx(iris.diss.mx, v.pred, 10)

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

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:

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

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

Value

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

Author(s)

Lukasz Nieweglowski

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, doi:10.1162/089976604773717621

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)
}

Similarity index based on confusion matrix - External Measure, Cluster Stability

Description

Similarity index based on confusion matrix is the measure which estimates how those two different partitionings, that comming from one dataset, are different from each other. For given matrix returned by confusion.matrix function similarity index is found.

Usage

similarity.index(cnf.mx)

Arguments

Details

Let M is n x m (n <= m) confusion matrix for partitionings P and P'. Any one to one function sigma: {1,2,...,n} -> {1,2,... ,m}. is called assignment (or also association). Using set of assignment functions, A(P,P') index defined as:

A(P,P') = max{ sum( forall i in 1:length(sigma) ) M[i,sigma(i)]: sigma is an assignment }

is found. (Assignment which satisfy above equation is called optimal assignment). Using this value we can compute similarity index S(P.P') = (A(P,P') - 1)/(N - 1) where N is quantity of partitioned objects (here is equal to sum(M)).

Value

similarity.index returns value from section [0,1] which is a measure of similarity between two different partitionings. Value 1 means that we have two the same partitionings.

Author(s)

Lukasz Nieweglowski

References

C. D. Giurcaneanu, I. Tabus, I. Shmulevich, W. Zhang Stability-Based Cluster Analysis Applied To Microarray Data, http://citeseer.ist.psu.edu/577114.html.

T. Lange, V. Roth, M. L. Braun and J. M. Buhmann Stability-Based Validation of Clustering Solutions, doi:10.1162/089976604773717621

See Also

confusion.matrix as matrix representation of two partitionings. Other functions created to compare two different partitionings: std.ext, dot.product

Examples

# similarity.index function(and also dot.product) 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 similarity.index (and dot.product) 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)
}

Standard External Measures: Rand index, Jaccard coefficient etc.

Description

Group of functions which compute standard external measures such as: Rand statistic and Folkes and Mallows index, Jaccard coefficient etc.

Usage

std.ext(clust1, clust2)
clv.Rand(external.ind)
clv.Jaccard(external.ind)
clv.Folkes.Mallows(external.ind)
clv.Phi(external.ind)
clv.Russel.Rao(external.ind)

Arguments

Details

Two input vectors keep information about two different partitionings (let say P and P') of the same data set X. We refer to a pair of points (xi, xj) (we assume that i != j) from the data set using the following terms:

Those values are used to compute (M = SS + SD + DS +DD):

Value

Author(s)

Lukasz Nieweglowski

References

G. Saporta and G. Youness Comparing two partitions: Some Proposals and Experiments. http://cedric.cnam.fr/PUBLIS/RC405.pdf

See Also

Other measures created to compare two partitionings: dot.product, similarity.index

Examples

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

# cluster data
pam.mod <- pam(iris.data,3) # create three clusters
v.pred <- as.integer(pam.mod$clustering) # get cluster ids associated to given data objects
v.real <- as.integer(iris$Species) # get also real cluster ids

# compare true clustering with those given by the algorithm
# 1. optimal solution:

# use only once std.ext function
std <- std.ext(v.pred, v.real)
# to compute three indicies based on std.ext result
rand1 <- clv.Rand(std)
jaccard1 <- clv.Jaccard(std)
folk.mal1 <- clv.Folkes.Mallows(std)

# 2. functional solution:

# prepare set of functions which compare two clusterizations
Rand <- function(clust1,clust2) clv.Rand(std.ext(clust1,clust2))
Jaccard <- function(clust1,clust2) clv.Jaccard(std.ext(clust1,clust2))
Folkes.Mallows <- function(clust1,clust2) clv.Folkes.Mallows(std.ext(clust1,clust2))

# compute indicies
rand2 <- Rand(v.pred,v.real)
jaccard2 <- Jaccard(v.pred,v.real)
folk.mal2 <- Folkes.Mallows(v.pred,v.real)

Matrix Cluster Scatter Measures

Description

Functions compute two base matrix cluster scatter measures.

Usage

wcls.matrix(data,clust,cluster.center)
bcls.matrix(cluster.center,cluster.size,mean)

Arguments

Details

There are two base matrix scatter measures.

1. within-cluster scatter measure defined as:

W = sum(forall k in 1:cluster.num) W(k)

where W(k) = sum(forall x) (x - m(k))*(x - m(k))'

2. between-cluster scatter measure defined as:

B = sum(forall k in 1:cluster.num) |C(k)|*( m(k) - m )*( m(k) - m )'

Value

Author(s)

Lukasz Nieweglowski

References

T. Hastie, R. Tibshirani, G. Walther Estimating the number of data clusters via the Gap statistic, http://citeseer.ist.psu.edu/tibshirani00estimating.html

Examples

# 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 cluster sizes, center of each cluster 
# and mean from data objects
cls.attr <- cls.attrib(iris.data, v.pred)
center <- cls.attr$cluster.center
size <- cls.attr$cluster.size
iris.mean <- cls.attr$mean

# compute matrix scatter measures
W.matrix <- wcls.matrix(iris.data, v.pred, center)
B.matrix <- bcls.matrix(center, size, iris.mean)
T.matrix <- W.matrix + B.matrix

# example of indices based on W, B i T matrices
mx.scatt.crit1 = sum(diag(W.matrix))
mx.scatt.crit2 = sum(diag(B.matrix))/sum(diag(W.matrix))
mx.scatt.crit3 = det(W.matrix)/det(T.matrix)