Package 'clusterSim'

Descriptive statistics calculated separately for each cluster and variable

Description

Descriptive statistics calculated separately for each cluster and variable: arithmetic mean and standard deviation, median and median absolute deviation, mode

Usage

cluster.Description(x, cl, sdType="sample",precission=4,modeAggregationChar=";")

Arguments

x	matrix or dataset
cl	a vector of integers indicating the cluster to which each object is allocated
sdType	type of standard deviation: for "sample" (n-1) or for "population" (n)
precission	Number of digits on the right side of decimal mark sign
modeAggregationChar	Character used for aggregation of mode values (if more than one value of mode appear in variable)

Value

Three-dimensional array:

First dimension contains cluster number

Second dimension contains original coordinate (variable) number from matrix or data set

Third dimension contains number from 1 to 5:

1 - arithmetic mean

2 - standard deviation

3 - median

4 - median absolute deviation (mad)

5 - mode (value of the variable which has the largest observed frequency. This formula is applicable for nominal and ordinal data only).

For example:

desc<-cluster.Description(x,cl)

desc[2,4,2] - standard deviation of fourth coordinate of second cluster

desc[3,1,5] - mode of first coordinate (variable) of third cluster

desc[1,,] - all statistics for all dimensions (variables) of first cluster

desc[,,3] - medians of all dimensions (variables) for each cluster

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

See Also

cluster.Sim, mean, sd, median, mad

Examples

library(clusterSim)
data(data_ratio)
cl <- pam(data_ratio,5)
desc <- cluster.Description(data_ratio,cl$cluster)
print(desc)

---

Random cluster generation with known structure of clusters

Description

Random cluster generation with known structure of clusters (optionally with noisy variables and outliers)

Usage

cluster.Gen(numObjects=50, means=NULL, cov=NULL, fixedCov=TRUE,
                   model=1, dataType="m",numCategories=NULL, 
                   numNoisyVar=0, numOutliers=0, rangeOutliers=
                   c(1,10), inputType="csv2", inputHeader=TRUE, 
                   inputRowNames=TRUE, outputCsv="", outputCsv2="", 
                   outputColNames=TRUE, outputRowNames=TRUE)

Arguments

numObjects	number of objects in each cluster - positive integer value or vector with the same size as nrow(means), e.g. numObjects=c(50,20)
means	matrix of cluster means (e.g. means=matrix(c(0,8,0,8),2,2)). If means = NULL matrix should be read from means_<modelNumber>.csv file
cov	covariance matrix (the same for each cluster, e.g. cov=matrix(c(1, 0, 0, 1), 2, 2)). If cov=NULL matrix should be read from cov_<modelNumber>.csv file. Note: you cannot use this argument for generation of clusters with different covariance matrices. Those kind of generation should be done by setting fixedCov to FALSE and using appropriate model
model	model number, model=1 - no cluster structure. Observations are simulated from uniform distribution over the unit hypercube in number of dimensions (variables) given in numNoisyVar argument; model=2 - means and covariances are taken from arguments means and cov (see Example 1); model=3,4,...,20 - see file $R_HOME\library\clusterSim\pdf\clusterGen_details.pdf; model=21,22,... - if fixedCov=TRUE means should be read from means_<modelNumber>.csv and covariance matrix for all clusters should be read from cov_<modelNumber>.csv and if fixedCov=FALSE means should be read from means_<modelNumber>.csv and covariance matrices should be read separately for each cluster from cov_<modelNumber>_<clusterNumber>.csv
fixedCov	if fixedCov=TRUE covariance matrix for all clusters is the same and if fixedCov=FALSE each cluster is generated from different covariance matrix - see model
dataType	"m" - metric (ratio, interval), "o" - ordinal, "s" - symbolic interval
numCategories	number of categories (for ordinal data only). Positive integer value or vector with the same size as ncol(means) plus number of noisy variables.
numNoisyVar	number of noisy variables. For model=1 it means number of variables
numOutliers	number of outliers (for metric and symbolic interval data only). If a positive integer - number of outliers, if value from <0,1> - percentage of outliers in whole data set
rangeOutliers	range for outliers (for metric and symbolic interval data only). The default range is [1, 10].The outliers are generated independently for each variable for the whole data set from uniform distribution. The generated values are randomly added to maximum of j-th variable or subtracted from minimum of j-th variable
inputType	"csv" - a dot as decimal point or "csv2" - a comma as decimal point in means_<modelNumber>.csv and cov_<modelNumber>.csv files
inputHeader	inputHeader=TRUE indicates that input files (means_<modelNumber>.csv; cov_<modelNumber...>.csv) contain header row
inputRowNames	inputRowNames=TRUE indicates that input files (means_<modelNumber>.csv; cov_<modelNumber...>.csv) contain first column with row names or with number of objects (positive integer values)
outputCsv	optional, name of csv file with generated data (first column contains id, second - number of cluster and others - data)
outputCsv2	optional, name of csv (a comma as decimal point and a semicolon as field separator) file with generated data (first column contains id, second - number of cluster and others - data)
outputColNames	outputColNames=TRUE indicates that output file (given by outputCsv and outputCsv2 parameters) contains first row with column names
outputRowNames	outputRowNames=TRUE indicates that output file (given by outputCsv and outputCsv2 parameters) contains a vector of row names

Details

See file $R_HOME\library\clusterSim\pdf\clusterGen_details.pdf for further details

Value

clusters	cluster number for each object, for model=1 each object belongs to its own cluster thus this variable contains objects numbers
data	generated data: for metric and ordinal data - matrix with objects in rows and variables in columns; for symbolic interval data three-dimensional structure: first dimension represents object number, second - variable number and third dimension contains lower- and upper-bounds of intervals

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Billard, L., Diday, E. (2006), Symbolic data analysis. Conceptual statistics and data mining, Wiley, Chichester. ISBN: 978-0-470-09016-9.

Qiu, W., Joe, H. (2006), Generation of random clusters with specified degree of separation, "Journal of Classification", vol. 23, 315-334. Available at: doi:10.1007/s00357-006-0018-y.

Steinley, D., Henson, R. (2005), OCLUS: an analytic method for generating clusters with known overlap, "Journal of Classification", vol. 22, 221-250. Available at: doi:10.1007/s00357-005-0015-6.

Walesiak, M., Dudek, A. (2008), Identification of noisy variables for nonmetric and symbolic data in cluster analysis, In: C. Preisach, H. Burkhardt, L. Schmidt-Thieme, R. Decker (Eds.), Data analysis, machine learning and applications, Springer-Verlag, Berlin, Heidelberg, 85-92. Available at: doi:10.1007/978-3-540-78246-9_11.

Walesiak, M. (2016), Uogólniona miara odległości GDM w statystycznej analizie wielowymiarowej z wykorzystaniem programu R. Wydanie 2 poprawione i rozszerzone [The Generalized Distance Measure GDM in multivariate statistical analysis with R], Wydawnictwo Uniwersytetu Ekonomicznego, Wroclaw.

Examples

# Example 1
library(clusterSim)
means <- matrix(c(0,7,0,7),2,2)
cov <- matrix(c(1,0,0,1),2,2)
grnd <- cluster.Gen(numObjects=60,means=means,cov=cov,model=2,
numOutliers=8)
colornames <- c("red","blue","green")
grnd$clusters[grnd$clusters==0]<-length(colornames)
plot(grnd$data,col=colornames[grnd$clusters],ask=TRUE)

# Example 2
library(clusterSim)
grnd <- cluster.Gen(50,model=4,dataType="m",numNoisyVar=2)
data <- as.matrix(grnd$data)
colornames <- c("red","blue","green")
plot(grnd$data,col=colornames[grnd$clusters],ask=TRUE)

# Example 3
library(clusterSim)
grnd<-cluster.Gen(50,model=4,dataType="o",numCategories=7, numNoisyVar=2)
plotCategorial(grnd$data,,grnd$clusters,ask=TRUE)

# Example 4 (1 nonnoisy variable and 2 noisy variables, 3 clusters)
library(clusterSim)
grnd <- cluster.Gen(c(40,60,20), model=2, means=c(2,14,25),
cov=c(1.5,1.5,1.5),numNoisyVar=2)
colornames <- c("red","blue","green")
plot(grnd$data,col=colornames[grnd$clusters],ask=TRUE)

# Example 5
library(clusterSim)
grnd <- cluster.Gen(c(20,35,20,25),model=14,dataType="m",numNoisyVar=1,
fixedCov=FALSE, numOutliers=0.1)
# or 
#grnd <- cluster.Gen(c(20,35,20,25),model=14,dataType="m",numNoisyVar=1,
#fixedCov=FALSE, numOutliers=0.1, outputCsv2="data14.csv")
data <- as.matrix(grnd$data)
colornames <- c("red","blue","green","brown","black")
grnd$clusters[grnd$clusters==0]<-length(colornames)
plot(grnd$data,col=colornames[grnd$clusters],ask=TRUE)

# Example 6 (this example needs files means_24.csv) 
# and cov_24.csv to be placed in working directory
# library(clusterSim)
# grnd<-cluster.Gen(c(50,80,20),model=24,dataType="m",numNoisyVar=1, 
# numOutliers=10, rangeOutliers=c(1,5))
# print(grnd)
# data <- as.data.frame(grnd$data)
# colornames<-c("red","blue","green","brown")
# grnd$clusters[grnd$clusters==0]<-length(colornames)
# plot(data,col=colornames[grnd$clusters],ask=TRUE)

# Example 7 (this example needs files means_25.csv and cov_25_1.csv) 
# cov_25_2.csv, cov_25_3.csv, cov_25_4.csv, cov_25_5.csv
# to be placed in working directory
# library(clusterSim)
# grnd<-cluster.Gen(c(40,30,20,35,45),model=25,numNoisyVar=3,fixedCov=F)
# data <- as.data.frame(grnd$data)
# colornames<-c("red","blue","green","magenta","brown")
# plot(data,col=colornames[grnd$clusters],ask=TRUE)

---

Determination of optimal clustering procedure for a data set

Description

Determination of optimal clustering procedure for a data set by varying all combinations of normalization formulas, distance measures, and clustering methods

Usage

cluster.Sim (x,p,minClusterNo,maxClusterNo,icq="S",
	outputCsv="",outputCsv2="",normalizations=NULL,
	distances=NULL,methods=NULL)

Arguments

x	matrix or dataset
p	path of simulation: 1 - ratio data, 2 - interval or mixed (ratio & interval) data, 3 - ordinal data, 4 - nominal data, 5 - binary data, 6 - ratio data without normalization, 7 - interval or mixed (ratio & interval) data without normalization, 8 - ratio data with k-means, 9 - interval or mixed (ratio & interval) data with k-means
minClusterNo	minimal number of clusters, between 2 and no. of objects - 1 (for G3 or C: no. of objects - 2)
maxClusterNo	maximal number of clusters, between 2 and no. of objects - 1 (for G3 or C: no. of objects - 2; for KL: no. of objects - 3), greater or equal minClusterNo
icq	Internal cluster quality index, "S" - Silhouette,"G1" - Calinski & Harabasz index, "G2" - Baker & Hubert index ,"G3" - G3 index,"C" - C index, "KL" - Krzanowski & Lai index
outputCsv	optional, name of csv file with results
outputCsv2	optional, name of csv (comma as decimal point sign) file with results
normalizations	optional, vector of normalization formulas that should be used in procedure
distances	optional, vector of distance measures that should be used in procedure
methods	optional, vector of classification methods that should be used in procedure

Details

Parameter normalizations for each path may be the subset of the following values

path 1: "n6" to "n11" (if measurement scale of variables is ratio and transformed measurement scale of variables is ratio) or "n1" to "n5" (if measurement scale of variables is ratio and transformed measurement scale of variables is interval)

path 2: "n1" to "n5"

path 3 to 7 : "n0"

path 8: "n1" to "n11"

path 9: "n1" to "n5"

Parameter distances for each path may be the subset of the following values

path 1: "d1" to "d7" (if measurement scale of variables is ratio and transformed measurement scale of variables is ratio) or "d1" to "d5" (if measurement scale of variables is ratio and transformed measurement scale of variables is interval)

path 2: "d1" to "d5"

path 3: "d8"

path 4: "d9"

path 5: "b1" to "b10"

path 6: "d1" to "d7"

path 7: "d1" to "d5"

path 8 and 9: N.A.

Parameter methods for each path may be the subset of the following values

path 1 to 7 : "m1" to "m8"

path 8: "m9"

path 9: "m9"

See file ../doc/clusterSim_details.pdf for further details

Value

result	optimal value of icq for all classifications
normalization	normalization used to obtain optimal value of icq
distance	distance measure used to obtain optimal value of icq
method	clustering method used to obtain optimal value of icq
classes	number of clusters for optimal value of icq
time	time of all calculations for path

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Everitt, B.S., Landau, E., Leese, M. (2001), Cluster analysis, Arnold, London. ISBN 9780340761199.

Gatnar, E., Walesiak, M. (Eds.) (2004), Metody statystycznej analizy wielowymiarowej w badaniach marketingowych [Multivariate statistical analysis methods in marketing research], Wydawnictwo AE, Wroclaw, p. 338.

Gordon, A.D. (1999), Classification, Chapman & Hall/CRC, London. ISBN 9781584880134.

Milligan, G.W., Cooper, M.C. (1985), An examination of procedures of determining the number of cluster in a data set, "Psychometrika", vol. 50, no. 2, 159-179. Available at: doi:10.1007/BF02294245.

Milligan, G.W., Cooper, M.C. (1988), A study of standardization of variables in cluster analysis, "Journal of Classification", vol. 5, 181-204. Available at: doi:10.1007/BF01897163.

Walesiak, M., Dudek, A. (2006), Symulacyjna optymalizacja wyboru procedury klasyfikacyjnej dla danego typu danych - oprogramowanie komputerowe i wyniki badan, Prace Naukowe AE we Wroclawiu, 1126, 120-129.

Walesiak, M., Dudek, A. (2007), Symulacyjna optymalizacja wyboru procedury klasyfikacyjnej dla danego typu danych - charakterystyka problemu, Zeszyty Naukowe Uniwersytetu Szczecinskiego nr 450, 635-646.

See Also

data.Normalization, dist.GDM, dist.BC, dist.SM, index.G1, index.G2,

index.G3, index.C, index.S, index.KL, hclust, dist,

Examples

#library(clusterSim)
#data(data_ratio)
#cluster.Sim(data_ratio, 1, 2, 3, "G1", outputCsv="results1")
#data(data_interval)
#cluster.Sim(data_interval, 2, 2, 4, "G1", outputCsv="results2")
#data(data_ordinal)
#cluster.Sim(data_ordinal, 3, 2, 4,"G2", outputCsv2="results3")
#data(data_nominal)
#cluster.Sim(data_nominal, p=4, 2, 4, icq="G3", outputCsv="results4", methods=c("m2","m3","m5"))
#data(data_binary)
#cluster.Sim(as.matrix(data_binary), p=5, 2, 4, icq="S", 
#outputCsv="results5", distances=c("b1","b3","b6"))
#data(data_ratio)
#cluster.Sim(data_ratio, 1, 2, 4,"G1", outputCsv="results6",normalizations=c("n1","n3"),
#distances=c("d2","d5"),methods=c("m5","m3","m1"))

---

Calculate agreement indices between two partitions

Description

Calculate agreement indices between two partitions

Usage

comparing.Partitions(cl1,cl2,type="nowak")

Arguments

cl1	A vector of integers (or letters) indicating the cluster to which each object is allocated for first clustering
cl2	A vector of integers (or letters) indicating the cluster to which each object is allocated for second clustering
type	"rand" - for Rand index, "crand" - for adjusted Rand index or "nowak" for Nowak index

Details

See file $R_HOME\library\clusterSim\pdf\comparingPartitions_details.pdf for further details.

Rand and adjusted Rand indices uses classAgreement function from e1071 library.

Value

Returns value of index.

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Hubert, L., Arabie, P. (1985), Comparing partitions, "Journal of Classification", no. 1, 193-218. Available at: doi:10.1007/BF01908075.

Nowak, E. (1985), Wskaznik podobienstwa wynikow podzialow, "Przeglad Statystyczny" ["Statistical Review"], no. 1, 41-48.

Rand, W.M. (1971), Objective criteria for the evaluation of clustering methods, "Journal of the American Statistical Association", no. 336, 846-850. Available at: doi:10.1080/01621459.1971.10482356.

See Also

replication.Mod

Examples

# Example 1
library(clusterSim)
dataSet<-cluster.Gen(model=5)
cl1<-dataSet$clusters
cl2<-kmeans(dataSet$data,2)$cluster
print(comparing.Partitions(cl1,cl2,type="rand"))

# Example 2
library(clusterSim)
data(data_patternGDM1)
z<-data.Normalization(data_patternGDM1,type="n1")
d<-dist.GDM(z,method="GDM1")
cl1<-pam(d,3,diss=TRUE)$clustering
cl2<-pam(d,4,diss=TRUE)$clustering
print(comparing.Partitions(cl1,cl2,type="crand"))

# Example 3
library(clusterSim)
data(data_patternGDM1)
z<-data.Normalization(data_patternGDM1,type="n9")
d<-dist.GDM(z,method="GDM1")
cl1<-pam(d,3,diss=TRUE)$clustering
hc<-hclust(d, method="complete")
cl2<-cutree(hc,k=3)
print(comparing.Partitions(cl1,cl2,type="nowak"))

---

Binary data

Description

Binary variables for eight people

Format

data.frame: 8 objects, 10 variables

Source

Kaufman, L., Rousseeuw, P.J. (1990), Finding groups in data: an introduction to cluster analysis, Wiley, New York, p. 24.

Examples

#library(clusterSim)
#data(data_binary)
#cluster.Sim(as.matrix(data_binary), p=5, 2, 6, icq="S", 
#outputCsv="results5", distances=c("b1","b3","b6"))

---

Interval data

Description

Artificially generated interval data

Format

data.frame: 75 objects, 5 variables, 5-class structure

Source

Artificially generated data

Examples

#library(clusterSim)
#data(data_interval)
#cluster.Sim(data_interval, 2, 2, 3, "G1", outputHtml="results2")

---

Mixed data

Description

Artificial mixed data

Format

data.frame: 25 objects, 4 variables (1, 3 - interval variables, 2 - ordinal variable, 4, nominal variable

Source

Artificial data

Examples

library(clusterSim)
data(data_mixed)
r3 <- HINoV.Mod(data_mixed, type=c("m","n","m","n"), s=2, 3, distance="d1",
     method="complete", Index="cRAND")
print(r3$stopri)
plot(r3$stopri[,2], xlab="Variable number", ylab="topri", xaxt="n")
axis(1,at=c(1:max(r3$stopri[,1])),labels=r3$stopri[,1])

---

Nominal data

Description

Artificial nominal data

Format

data.frame: 26 objects, 12 variables

Source

Artificial data

Examples

#library(clusterSim)
#data(data_nominal)
#cluster.Sim(data_nominal, p=4, 2, 5, icq="G3",
#outputHtml="results4", methods=c("m2","m3","m5"))

---

Ordinal data

Description

Artificial ordinal data

Format

data.frame: 26 objects, 12 variables

Source

Artificial data

Examples

#library(clusterSim)
#data(data_ordinal)
#cluster.Sim(data_ordinal, 3, 3, 12,"S", 
#outputCsv2="results3")

---

Metric data with 17 objects and 10 variables (8 stimulant variables, 2 destimulant variables)

Description

Metric data with 17 objects and 10 variables (8 stimulant variables, 2 destimulant variables)

Data on the Polish voivodships, owing to the conditions of the population living in cities in 2007. The analysis includes the following variables:

x1 - dwellings in per cent fitted with water-line system,

x2 - dwellings in per cent fitted with lavatory,

x3 - dwellings in per cent fitted with bathroom,

x4 - dwellings in per cent fitted with gas-line system,

x5 - dwellings in per cent fitted with central heating,

x6 - average number of rooms per dwelling,

x7 - average number of persons per dwelling,

x8 - average number of persons per room,

x9 - usable floor space in square meter per dwelling,

x10 - usable floor space in square meter per person.

Types of performance variables:

x1 - x6, x9, x10 - stimulants,

x7, x8 - destimulants.

Format

data.frame: 17 objects, 10 variables

Source

Voivodships Statistical Yearbook, Poland 2008.

Examples

# Example 1
library(clusterSim)
data(data_patternGDM1)
res<-pattern.GDM1(data_patternGDM1,
performanceVariable=c("s","s","s","s","s","s","d","d","s","s"),
scaleType="r",nomOptValues=NULL,weightsType<-"equal",weights=NULL,
normalization<-"n4",patternType<-"lower",patternCoordinates<-"manual",
patternManual<-c(0,0,0,0,0,"min","max","max","min","min"),
nominalTransfMethod <-NULL)
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p),xlim=c(max(gdm_p),min(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)),xaxt="n",
xlab="Order of objects from the best to the worst", 
ylab="GDM distances from pattern object",lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

# Example 2
library(clusterSim)
data(data_patternGDM1)
res<-pattern.GDM1(data_patternGDM1,
performanceVariable=c("s","s","s","s","s","s","d","d","s","s"),
scaleType="r",nomOptValues=NULL,weightsType<-"equal",weights=NULL,
normalization<-"n2",patternType<-"upper",
patternCoordinates<-"dataBounds",patternManual<-NULL,
nominalTransfMethod <-NULL)
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p),xlim=c(min(gdm_p),max(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)),xaxt="n",
xlab="Order of objects from the best to the worst", 
ylab="GDM distances from pattern object", lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

---

Ordinal data with 27 objects and 6 variables (3 stimulant variables, 2 destimulant variables and 1 nominant variable)

Description

Ordinal data with 27 objects and 6 variables (3 stimulant variables, 2 destimulant variables and 1 nominant variable)

Residential housing properties were described by the following variables:

x1 - Location of environmental land, which is linked to a dwelling (1 - poor, 2 - inadequate, 3 - satisfactory, 4 - good, 5 - very good),

x2 - Standard utility of a dwelling (1 - bad, 2 - low, 3 - average, 4 - high),

x3 - Living conditions occurring on the land, which is linked to a dwelling (1 - bad, 2 - average, 3 - good),

x4 - Location of land, which is related to dwelling in the area of the city (1 - central, 2 - downtown, 3 - intermediate, 4 - peripheral),

x5 - Type of condominium (1 - low, 2 - large),

x6 - Area of land, which is related to dwelling (1 - below the contour of the building, 2 - outline of the building, 3 - the outline of the building with the environment acceptable, such as parking, playground, 4 - the outline of the building with the environment too much).

Types of performance variables:

x1, x2, x3 - stimulants,

x4, x5 - destimulants,

x6 - nominant (the nominal category: 3).

Format

data.frame: 27 objects, 6 variables

Source

data from real estate market

Examples

# Example 1
library(clusterSim)
data(data_patternGDM2)
res<-pattern.GDM2(data_patternGDM2,
performanceVariable=c("s","s","s","d","d","n"),
nomOptValues=c(NA,NA,NA,NA,NA,3), weightsType<-"equal", weights=NULL,
patternType="lower", patternCoordinates="manual",
patternManual=c("min","min",0,5,"max","max"),
nominalTransfMethod="symmetrical")
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p),xlim=c(max(gdm_p),min(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)), 
xaxt="n",xlab="Order of objects from the best to the worst",
ylab="GDM distances from pattern object",lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

# Example 2
library(clusterSim)
data(data_patternGDM2)
res<-pattern.GDM2(data_patternGDM2,
performanceVariable=c("s","s","s","d","d","n"),
nomOptValues=c(NA,NA,NA,NA,NA,3), weightsType<-"equal", weights=NULL,
patternType="upper", patternCoordinates="dataBounds",
patternManual=NULL, nominalTransfMethod="database")
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p), xlim=c(min(gdm_p),max(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)), 
xaxt="n",xlab="Order of objects from the best to the worst",
ylab="GDM distances from pattern object", lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

---

Ratio data

Description

Artificially generated ratio data

Format

data.frame: 75 objects, 5 variables, 5-class structure

Source

Artificially generated data

Examples

#library(clusterSim)
#data(data_ratio)
#c <- pam(data_ratio,10)
#index.G1(data_ratio, c$clustering)

---

Symbolic interval data

Description

Artificially generated symbolic interval data

Format

3-dimensional array: 125 objects, 6 variables, third dimension represents begining and end of interval, 5-class structure

Source

Artificially generated data

Examples

library(clusterSim)
data(data_symbolic)
r<- HINoV.Symbolic(data_symbolic, u=5)
print(r$stopri)
plot(r$stopri[,2], xlab="Variable number", ylab="topri",
xaxt="n", type="b")
axis(1,at=c(1:max(r$stopri[,1])),labels=r$stopri[,1])

---

The evaluation of Polish voivodships tourism attractiveness level

Description

The empirical study uses the statistical data referring to the attractiveness level of 18 objects (16 Polish NUTS-2 regions - voivodships, pattern and anti-pattern object).

Two-stage data collection was performed. Firstly, data on tourist attractiveness were collected for 380 counties using 9 classic metric variables (measured on a ratio scale):

x1 - beds in hotels per 1000 inhabitants of a county,

x2 - number of nights spent daily by resident tourists per 1000 inhabitants of a county,

x3 - number of nights spent daily by foreign tourists per 1000 inhabitants of a county,

x4 - dust pollution emission in tons per 10 km2 of a county area,

x5 - gas pollution emission in tons per 1 km2 of a county area,

x6 - number of criminal offences, crimes against life and health and property crimes per 1000 inhabitants of a county,

x7 - forest cover of the county in

x8 - participants of mass events per 1000 inhabitants of a county,

x9 - number of tourist economy entities (sections: I, N79) registered in the system REGON per 1000 inhabitants of a county.

The three variables (x4, x5 and x6) are destimulants. Other variables are stimulants.

In the second step, the data were aggregated to the level of the voivodships (NUTS-2), giving the symbolic interval-valued data. The lower bound of the interval for each symbolic interval-valued variable in the voivodship was obtained by calculating the first quartile on the basis of data from counties. The upper bound of the interval was obtained by calculating the third quartile.

Format

Tree-dimansional array: 18 objects (16 Polish NUTS-2 regions - voivodships, pattern and anti-pattern object), 9 symbolic interval-valued variables with lower and upper values of interval in third dimension. The coordinates of an pattern object cover the most preferred preference variable values. The coordinates of an anti-pattern object cover the least preferred preference variable values.

Source

The statistical data were collected in 2016 and come from the Local Data Bank of the Central Statistical Office of Poland.

Examples

library(clusterSim)
data(data_symbolic_interval_polish_voivodships)
print(data_symbolic_interval_polish_voivodships)

---

Types of variable (column) and object (row) normalization formulas

Description

Types of variable (column) and object (row) normalization formulas

Usage

data.Normalization (x,type="n0",normalization="column",...)

Arguments

x	vector, matrix or dataset
type	type of normalization:

n0 - without normalization

n1 - standardization ((x-mean)/sd)

n2 - positional standardization ((x-median)/mad)

n3 - unitization ((x-mean)/range)

n3a - positional unitization ((x-median)/range)

n4 - unitization with zero minimum ((x-min)/range)

n5 - normalization in range <-1,1> ((x-mean)/max(abs(x-mean)))

n5a - positional normalization in range <-1,1> ((x-median)/max(abs(x-median)))

n6 - quotient transformation (x/sd)

n6a - positional quotient transformation (x/mad)

n7 - quotient transformation (x/range)

n8 - quotient transformation (x/max)

n9 - quotient transformation (x/mean)

n9a - positional quotient transformation (x/median)

n10 - quotient transformation (x/sum)

n11 - quotient transformation (x/sqrt(SSQ))

n12 - normalization ((x-mean)/sqrt(sum((x-mean)^2)))

n12a - positional normalization ((x-median)/sqrt(sum((x-median)^2)))

n13 - normalization with zero being the central point ((x-midrange)/(range/2))

normalization	"column" - normalization by variable, "row" - normalization by object
...	arguments passed to sum, mean, min sd, mad and other aggregation functions. In particular: na.rm - a logical value indicating whether NA values should be stripped before the computation

Details

See file ../doc/dataNormalization_details.pdf for further details

Thanks Wolfgang Lederer (<[email protected]>) for reporting n4/vector error

Value

Normalized data The numeric shifts and scalings used (if any) are returned as attributes "normalized:shift" and "normalized:scale"

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Anderberg, M.R. (1973), Cluster analysis for applications, Academic Press, New York, San Francisco, London. ISBN 9780120576500.

Gatnar, E., Walesiak, M. (Eds.) (2004), Metody statystycznej analizy wielowymiarowej w badaniach marketingowych [Multivariate statistical analysis methods in marketing research], Wydawnictwo AE, Wroclaw, 35-38.

Jajuga, K., Walesiak, M. (2000), Standardisation of data set under different measurement scales, In: R. Decker, W. Gaul (Eds.), Classification and information processing at the turn of the millennium, Springer-Verlag, Berlin, Heidelberg, 105-112. Available at: doi:10.1007/978-3-642-57280-7_11.

Milligan, G.W., Cooper, M.C. (1988), A study of standardization of variables in cluster analysis, "Journal of Classification", vol. 5, 181-204. Available at: doi:10.1007/BF01897163.

Mlodak, A. (2006), Analiza taksonomiczna w statystyce regionalnej, Difin, Warszawa. ISBN 83-7251-605-7.

Walesiak, M. (2014), Przeglad formul normalizacji wartosci zmiennych oraz ich wlasnosci w statystycznej analizie wielowymiarowej [Data normalization in multivariate data analysis. An overview and properties], "Przeglad Statystyczny" ("Statistical Review"), vol. 61, no. 4, 363-372. Available at: doi:10.5604/01.3001.0016.1740.

See Also

cluster.Sim

Examples

library(clusterSim)
data(data_ratio)
z1 <- data.Normalization(data_ratio,type="n1",normalization="column",na.rm=FALSE)
z2 <- data.Normalization(data_ratio,type="n10",normalization="row",na.rm=FALSE)

---

Calculates Bray-Curtis distance measure for ratio data

Description

Calculates Bray-Curtis distance measure for ratio data

Usage

dist.BC (x)

Arguments

x	matrix or dataset

Details

See file $R_HOME\library\clusterSim\pdf\distBC_details.pdf for further details

Value

object with calculated distance

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Cormack, R.M. (1971), A review of classification (with discussion), "Journal of the Royal Statistical Society", ser. A, part 3, 321-367.

Gatnar, E., Walesiak, M. (Eds.) (2004), Metody statystycznej analizy wielowymiarowej w badaniach marketingowych [Multivariate statistical analysis methods in marketing research], Wydawnictwo AE, Wroclaw, p. 41.

See Also

dist.GDM, dist.SM, dist

Examples

library(clusterSim)
sampleData <- cbind(c(2,3,5),c(4,5,6),c(5,3,4))
d <- dist.BC(sampleData)

---

Calculates Generalized Distance Measure

Description

Calculates Generalized Distance Measure for variables measured on metric scale (ratio & interval) or ordinal scale

Usage

dist.GDM(x, method="GDM1", weightsType="equal", weights=NULL)
GDM(x, method="GDM1", weightsType="equal", weights=NULL)
GDM1(x, weightsType="equal", weights=NULL)
GDM2(x, weightsType="equal", weights=NULL)

Arguments

x	matrix or data set
method	GDM1 or GDM2

"GDM1" - metric scale (ratio & interval)

"GDM2" - ordinal scale

weightsType

equal or different1 or different2

"equal" - equal weights

"different1" - vector of different weights should satisfy conditions: each weight takes value from interval [0; 1] and sum of weights equals one

"different2" - vector of different weights should satisfy conditions: each weight takes value from interval [0; m] and sum of weights equals m (m - the number of variables)

weights

vector of weights

Details

See file $R_HOME\library\clusterSim\pdf\distGDM_details.pdf for further details

Value

object with calculated distance

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Jajuga, K., Walesiak, M., Bak, A. (2003), On the general distance measure, In: M. Schwaiger, O. Opitz (Eds.), Exploratory data analysis in empirical research, Springer-Verlag, Berlin, Heidelberg, 104-109. Available at: doi:10.1007/978-3-642-55721-7_12.

Walesiak, M. (1999), Distance Measure for Ordinal Data, "Argumenta Oeconomica", No. 2 (8), 167-173.

Walesiak, M. (2006), Uogolniona miara odleglosci w statystycznej analizie wielowymiarowej [The Generalized Distance Measure in multivariate statistical analysis], Wydawnictwo AE, Wroclaw.

See Also

dist.BC, dist.SM, dist

Examples

#Example 1
library(clusterSim)
data(data_ratio)
d1 <- GDM(data_ratio, method="GDM1")
data(data_ordinal)
d2 <- GDM(data_ordinal, method="GDM2")
d3 <- GDM1(data_ratio)
d4 <- GDM2(data_ordinal)

#Example 2
library(clusterSim)
data(data_ratio)
d1w <- GDM(data_ratio, method="GDM1", weightsType="different1",
weights=c(0.4,0.1,0.3,0.15,0.05))
data(data_ordinal)
d2w <- GDM(data_ordinal, method="GDM2", weightsType="different2",
weights=c(1,3,0.5,1.5,1.8,0.2,0.4,0.6,0.2,0.4,0.9,1.5))

---

Calculates Sokal-Michener distance measure for nominal variables

Description

Calculates Sokal-Michener distance measure for nominal variables

Usage

dist.SM(x)

Arguments

x	matrix or data set

Details

See file $R_HOME\library\clusterSim\pdf\distSM_details.pdf for further details

Value

object with calculated distance

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Gatnar, E., Walesiak, M. (Eds.) (2004), Metody statystycznej analizy wielowymiarowej w badaniach marketingowych [Multivariate statistical analysis methods in marketing research], Wydawnictwo AE, Wroclaw, p. 43.

Kaufman, L., Rousseeuw, P.J. (1990), Finding groups in data: an introduction to cluster analysis, Wiley, New York, p. 28. ISBN: 978-0-471-73578-6.

See Also

dist.GDM, dist.BC, dist

Examples

library(clusterSim)
data(data_nominal)
d <- dist.SM(data_nominal)

---

Calculates distance between interval-valued symbolic data

Description

Calculates distance between interval-valued symbolic data for four distance types

Usage

dist.Symbolic(data,type="U_2",gamma=0.5,power=2)

Arguments

data	symbolic data
type	type of distance used for symbolic interval-valued data U_2 - Ichino and Yaguchi distance M - distance between points given by means of intervals (for interval-values variables), H - Hausdorff distance, S - sum of distances between all corresponding vertices of hyperrectangles given by symbolic objects with interval-valued variables
gamma	parameter for calculating Ichino and Yaguchi distance
power	parameter for calculating distance: Ichino and Yaguchi distance, Hausdorff distance

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Billard, L., Diday, E. (2006), Symbolic data analysis. Conceptual statistics and data mining, Wiley, Chichester. ISBN: 978-0-470-09016-9.

Ichino, M., Yaguchi, H. (1994), Generalized Minkowski metrics for mixed feature type data analysis, "IEEE Transactions on Systems Man and Cybernetics", Vol. 24, Issue 4, 698-708. Http://dx.doi.org/10.1109/21.286391.

See Also

symbolicDA::dist.SDA

Examples

library(clusterSim)
dataSymbolic<-cluster.Gen(numObjects=10,model=5,dataType="s")$data
print(dist.Symbolic(dataSymbolic))

---

Modification of Carmone, Kara & Maxwell Heuristic Identification of Noisy Variables (HINoV) method

Description

Modification of Heuristic Identification of Noisy Variables (HINoV) method

Usage

HINoV.Mod (x, type="metric", s = 2, u, distance=NULL, 
	method = "kmeans", Index ="cRAND")

Arguments

x	data matrix
type	"metric" (default) - all variables are metric (ratio, interval), "nonmetric" - all variables are nonmetric (ordinal, nominal) or vector containing for each variable value "m"(metric) or "n"(nonmetric) for mixed variables (metric and nonmetric), e.g. type=c("m", "n", "n", "m")
s	for metric data only: 1 - ratio data, 2 - interval or mixed (ratio & interval) data
u	number of clusters (for metric data only)
distance	NULL for kmeans method (based on data matrix) and nonmetric data for ratio data: "d1" - Manhattan, "d2" - Euclidean, "d3" - Chebychev (max), "d4" - squared Euclidean, "d5" - GDM1, "d6" - Canberra, "d7" - Bray-Curtis for interval or mixed (ratio & interval) data: "d1", "d2", "d3", "d4", "d5"
method	NULL for nonmetric data clustering method: "kmeans" (default) , "single", "ward.D", "ward.D2", "complete", "average", "mcquitty", "median", "centroid", "pam"
Index	"cRAND" - corrected Rand index (default); "RAND" - Rand index

Details

See file ../doc/HINoVMod_details.pdf for further details

Value

parim	m x m symmetric matrix (m - number of variables). Matrix contains pairwise corrected Rand (Rand) indices for partitions formed by the j-th variable with partitions formed by the l-th variable
topri	sum of rows of parim
stopri	ranked values of topri in decreasing order

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Carmone, F.J., Kara, A., Maxwell, S. (1999), HINoV: a new method to improve market segment definition by identifying noisy variables, "Journal of Marketing Research", November, vol. 36, 501-509.

Hubert, L.J., Arabie, P. (1985), Comparing partitions, "Journal of Classification", no. 1, 193-218. Available at: doi:10.1007/BF01908075.

Rand, W.M. (1971), Objective criteria for the evaluation of clustering methods, "Journal of the American Statistical Association", no. 336, 846-850. Available at: doi:10.1080/01621459.1971.10482356.

Walesiak, M. (2005), Variable selection for cluster analysis - approaches, problems, methods, Plenary Session of the Committee on Statistics and Econometrics of the Polish Academy of Sciences, 15 March, Wroclaw.

Walesiak, M., Dudek, A. (2008), Identification of noisy variables for nonmetric and symbolic data in cluster analysis, In: C. Preisach, H. Burkhardt, L. Schmidt-Thieme, R. Decker (Eds.), Data analysis, machine learning and applications, Springer-Verlag, Berlin, Heidelberg, 85-92. Available at: doi:10.1007/978-3-540-78246-9_11

See Also

hclust, kmeans, dist, dist.GDM, dist.BC, dist.SM, cluster.Sim

Examples

# for metric data
library(clusterSim)
data(data_ratio)
r1<- HINoV.Mod(data_ratio, type="metric", s=1, 4, method="kmeans",
     Index="cRAND")
print(r1$stopri)
plot(r1$stopri[,2],xlab="Variable number", ylab="topri",
xaxt="n", type="b")
axis(1,at=c(1:max(r1$stopri[,1])),labels=r1$stopri[,1])

# for nonmetric data
library(clusterSim)
data(data_nominal)
r2<- HINoV.Mod (data_nominal, type="nonmetric", Index = "cRAND")
print(r2$stopri)
plot(r2$stopri[,2], xlab="Variable number", ylab="topri",
xaxt="n", type="b")
axis(1,at=c(1:max(r2$stopri[,1])),labels=r2$stopri[,1])

# for mixed data
library(clusterSim)
data(data_mixed)
r3<- HINoV.Mod(data_mixed, type=c("m","n","m","n"), s=2, 3, distance="d1",
     method="complete", Index="cRAND")
print(r3$stopri)
plot(r3$stopri[,2], xlab="Variable number", ylab="topri",
xaxt="n", type="b")
axis(1,at=c(1:max(r3$stopri[,1])),labels=r3$stopri[,1])

---

Modification of Carmone, Kara & Maxwell Heuristic Identification of Noisy Variables (HINoV) method for symbolic interval data

Description

Modification of Heuristic Identification of Noisy Variables (HINoV) method for symbolic interval data

Usage

HINoV.Symbolic(x, u=NULL, distance="H", method = "pam", 
	Index = "cRAND")

Arguments

x	symbolic interval data: a 3-dimensional table, first dimension represents object number, second dimension - variable number, and third dimension contains lower- and upper-bounds of intervals
u	number of clusters
distance	"M" - minimal distance between all vertices of hyper-cubes defined by symbolic interval variables; "H" - Hausdorff distance; "S" - sum of squares of distance between all vertices of hyper-cubes defined by symbolic interval variables
method	clustering method: "single", "ward.D", "ward.D2", "complete", "average", "mcquitty", "median", "centroid", "pam" (default)
Index	"cRAND" - corrected Rand index (default); "RAND" - Rand index

Details

See file ../doc/HINoVSymbolic_details.pdf for further details

Value

parim	m x m symmetric matrix (m - number of variables). Matrix contains pairwise corrected Rand (Rand) indices for partitions formed by the j-th variable with partitions formed by the l-th variable
topri	sum of rows of parim
stopri	ranked values of topri in decreasing order

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Carmone, F.J., Kara, A., Maxwell, S. (1999), HINoV: a new method to improve market segment definition by identifying noisy variables, "Journal of Marketing Research", November, vol. 36, 501-509.

Hubert, L.J., Arabie, P. (1985), Comparing partitions, "Journal of Classification", no. 1, 193-218. Available at: doi:10.1007/BF01908075.

Rand, W.M. (1971), Objective criteria for the evaluation of clustering methods, "Journal of the American Statistical Association", no. 336, 846-850. Available at: doi:10.1080/01621459.1971.10482356.

Walesiak, M., Dudek, A. (2008), Identification of noisy variables for nonmetric and symbolic data in cluster analysis, In: C. Preisach, H. Burkhardt, L. Schmidt-Thieme, R. Decker (Eds.), Data analysis, machine learning and applications, Springer-Verlag, Berlin, Heidelberg, 85-92. Available at: doi:10.1007/978-3-540-78246-9_11.

See Also

hclust, kmeans, cluster.Sim

Examples

library(clusterSim)
data(data_symbolic)
r<- HINoV.Symbolic(data_symbolic, u=5)
print(r$stopri)
plot(r$stopri[,2], xlab="Variable number", ylab="topri",
xaxt="n", type="b")
axis(1,at=c(1:max(r$stopri[,1])),labels=r$stopri[,1])

#symbolic data from .csv file
#library(clusterSim)
#dsym<-as.matrix(read.csv2(file="csv/symbolic.csv"))
#dim(dsym)<-c(dim(dsym)[1],dim(dsym)[2]%/%2,2)          
#r<- HINoV.Symbolic(dsym, u=5)
#print(r$stopri)
#plot(r$stopri[,2], xlab="Variable number", ylab="topri",
#xaxt="n", type="b")
#axis(1,at=c(1:max(r$stopri[,1])),labels=r$stopri[,1])

---

Calculates Hubert & Levin C index - internal cluster quality index

Description

Calculates Hubert & Levin C index - internal cluster quality index

Usage

index.C(d,cl)

Arguments

d	'dist' object
cl	A vector of integers indicating the cluster to which each object is allocated

Details

See file $R_HOME\library\clusterSim\pdf\indexC_details.pdf for further details

Thanks to Özge Sahin from Technical University of Munich for for pointing the difference between index.G3 and index.C.

Value

calculated C index

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Hubert, L.J., Levin, J.R. (1976), A General Statistical Framework for Assessing Categorical Clustering in Free Recall, Psychological Bulletin, Vol. 83, No. 6, 1072-1080.

See Also

index.G1, index.G2, index.G3, index.S, index.H, index.KL, index.Gap, index.DB

Examples

# Example 1
library(clusterSim)
data(data_ratio)
d <- dist.GDM(data_ratio)
c <- pam(d, 5, diss = TRUE)
icq <- index.C(d,c$clustering)
print(icq)

# Example 2
library(clusterSim)
data(data_ordinal)
d <- dist.GDM(data_ordinal, method="GDM2")
# nc - number_of_clusters
min_nc=2
max_nc=6
res <- array(0,c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
hc <- hclust(d, method="complete")
cl2 <- cutree(hc, k=nc)
res[nc-min_nc+1,2] <- C <- index.C(d,cl2)
clusters <- rbind(clusters,cl2)
}
print(paste("min C for",(min_nc:max_nc)[which.min(res[,2])],"clusters=",min(res[,2])))
print("clustering for min C-index")
print(clusters[which.min(res[,2]),])
#write.table(res,file="C_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res, type="p", pch=0, xlab="Number of clusters", ylab="C", xaxt="n")
axis(1, c(min_nc:max_nc))

---

Calculates Davies-Bouldin's index

Description

Calculates Davies-Bouldin's cluster separation measure

Usage

index.DB(x, cl, d=NULL, centrotypes="centroids", p=2, q=2)

Arguments

x	data
cl	vector of integers indicating the cluster to which each object is allocated
d	optional distance matrix, used for calculations if centrotypes="medoids"
centrotypes	"centroids" or "medoids"
p	the power of the Minkowski distance between centroids or medoids of clusters: p=1 - Manhattan distance; p=2 - Euclidean distance
q	the power of dispersion measure of a cluster: q=1 - the average distance of objects in the r-th cluster to the centroid or medoid of the r-th cluster; q=2 - the standard deviation of the distance of objects in the r-th cluster to the centroid or medoid of the r-th cluster

Details

See file ../doc/indexDB_details.pdf for further details

Thanks to prof. Christian Hennig [email protected] for finding and fixing the "immutable p" error

Value

DB	Davies-Bouldin's index
r	vector of maximal R values for each cluster
R	R matrix $(S_r+S_s)/d_{rs}$
d	matrix of distances between centroids or medoids of clusters
S	vector of dispersion measures for each cluster
centers	coordinates of centroids or medoids for all clusters

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Davies, D.L., Bouldin, D.W. (1979), A cluster separation measure, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 1, no. 2, 224-227. Available at: doi:10.1109/TPAMI.1979.4766909.

See Also

index.G1, index.G2, index.G3, index.C, index.S, index.H, index.Gap, index.KL

Examples

# Example 1
library(clusterSim)
data(data_ratio)
cl1 <- pam(data_ratio, 4)
d<-dist(data_ratio)
print(index.DB(data_ratio, cl1$clustering,d, centrotypes="medoids"))

# Example 2
library(clusterSim)
data(data_ratio)
cl2 <- pam(data_ratio, 5)
print(index.DB(data_ratio, cl2$clustering, centrotypes="centroids"))

# Example 3
library(clusterSim)
data(data_ratio)
md <- dist(data_ratio, method="euclidean")
# nc - number_of_clusters
min_nc=2
max_nc=8
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
hc <- hclust(md, method="complete")
cl2 <- cutree(hc, k=nc)
res[nc-min_nc+1, 2] <- DB <- index.DB(data_ratio, cl2, centrotypes="centroids")$DB
clusters <- rbind(clusters, cl2)
}
print(paste("min DB for",(min_nc:max_nc)[which.min(res[,2])],"clusters=",min(res[,2])))
print("clustering for min DB")
print(clusters[which.min(res[,2]),])
#write.table(res,file="DB_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res, type="p", pch=0, xlab="Number of clusters", ylab="DB", xaxt="n")
axis(1, c(min_nc:max_nc))

# Example 4
library(clusterSim)
data(data_ordinal)
md <- dist.GDM(data_ordinal, method="GDM2")
# nc - number_of_clusters
min_nc=2
max_nc=6
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
hc <- hclust(md, method="complete")
cl2 <- cutree(hc, k=nc)
res[nc-min_nc+1,2] <- DB <- index.DB(data_ordinal,cl2,d=md,centrotypes="medoids")$DB
clusters <- rbind(clusters, cl2)
}
print(paste("min DB for",(min_nc:max_nc)[which.min(res[,2])],"clusters=",min(res[,2])))
print("clustering for min DB")
print(clusters[which.min(res[,2]),])
#write.table(res,file="DB_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res, type="p", pch=0, xlab="Number of clusters", ylab="DB", xaxt="n")
axis(1, c(min_nc:max_nc))

---

Calculates Calinski-Harabasz pseudo F-statistic

Description

Calculates Calinski-Harabasz pseudo F-statistic

Usage

index.G1 (x,cl,d=NULL,centrotypes="centroids")

Arguments

x	data
cl	A vector of integers indicating the cluster to which each object is allocated
d	optional distance matrix, used for calculations if centrotypes="medoids"
centrotypes	"centroids" or "medoids"

Details

See file ../doc/indexG1_details.pdf for further details.

thank to Nejc Ilc from University of Ljubljana for fixing error for one-element clusters.

Value

Calinski-Harabasz pseudo F-statistic

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Calinski, T., Harabasz, J. (1974), A dendrite method for cluster analysis, "Communications in Statistics", vol. 3, 1-27. Available at: doi:10.1080/03610927408827101.

Everitt, B.S., Landau, E., Leese, M. (2001), Cluster analysis, Arnold, London, p. 103. ISBN 9780340761199.

Gatnar, E., Walesiak, M. (Eds.) (2004), Metody statystycznej analizy wielowymiarowej w badaniach marketingowych [Multivariate statistical analysis methods in marketing research], Wydawnictwo AE, Wroclaw, p. 338.

Gordon, A.D. (1999), Classification, Chapman & Hall/CRC, London, p. 62. ISBN 9781584880134.

Milligan, G.W., Cooper, M.C. (1985), An examination of procedures of determining the number of cluster in a data set, "Psychometrika", vol. 50, no. 2, 159-179. Available at: doi:10.1007/BF02294245.

See Also

index.G2,index.G3,index.S, index.C, index.H,index.KL,index.Gap, index.DB

Examples

# Example 1
library(clusterSim)
data(data_ratio)
c<- pam(data_ratio,10)
index.G1(data_ratio,c$clustering)

# Example 2
library(clusterSim)
data(data_ratio)
md <- dist(data_ratio, method="euclidean")
# nc - number_of_clusters
min_nc=2
max_nc=20
res <- array(0,c(max_nc-min_nc+1,2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
cl2 <- pam(md, nc, diss=TRUE)
res[nc-min_nc+1,2] <- G1 <- index.G1(data_ratio,cl2$cluster,centrotypes="centroids")
clusters <- rbind(clusters, cl2$cluster)
}
print(paste("max G1 for",(min_nc:max_nc)[which.max(res[,2])],"clusters=",max(res[,2])))
print("clustering for max G1")
print(clusters[which.max(res[,2]),])
#write.table(res,file="G1_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res, type="p", pch=0, xlab="Number of clusters", ylab="G1", xaxt="n")
axis(1, c(min_nc:max_nc))

---

Calculates G2 internal cluster quality index

Description

Calculates G2 internal cluster quality index - Baker & Hubert adaptation of Goodman & Kruskal's Gamma statistic

Usage

index.G2(d,cl)

Arguments

d	'dist' object
cl	A vector of integers indicating the cluster to which each object is allocated

Details

See file $R_HOME\library\clusterSim\pdf\indexG2_details.pdf for further details

Value

calculated G2 index

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Everitt, B.S., Landau, E., Leese, M. (2001), Cluster analysis, Arnold, London, p. 104. ISBN 9780340761199.

Gatnar, E., Walesiak, M. (Eds.) (2004), Metody statystycznej analizy wielowymiarowej w badaniach marketingowych [Multivariate statistical analysis methods in marketing research], Wydawnictwo AE, Wroclaw, p. 339.

Gordon, A.D. (1999), Classification, Chapman & Hall/CRC, London, p. 62. ISBN 9781584880134.

Hubert, L. (1974), Approximate evaluation technique for the single-link and complete-link hierarchical clustering procedures, "Journal of the American Statistical Association", vol. 69, no. 347, 698-704. Available at: doi:10.1080/01621459.1974.10480191.

Milligan, G.W., Cooper, M.C. (1985), An examination of procedures of determining the number of cluster in a data set, "Psychometrika", vol. 50, no. 2, 159-179. Available at: doi:10.1007/BF02294245.

See Also

index.G1, index.G3, index.S, index.H, index.KL, index.Gap, index.C, index.DB

Examples

# Example 1
library(clusterSim)
data(data_ratio)
d <- dist.GDM(data_ratio)
c <- pam(d, 5, diss = TRUE)
icq <- index.G2(d,c$clustering)
#print(icq)

# Example 2
library(clusterSim)
data(data_ordinal)
d <- dist.GDM(data_ordinal, method="GDM2")
# nc - number_of_clusters
min_nc=2
max_nc=6
res <- array(0,c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
  cl2 <- pam(d, nc, diss=TRUE)
  res[nc-min_nc+1,2] <- G2 <- index.G2(d,cl2$cluster)
  clusters <- rbind(clusters,cl2$cluster)
}
print(paste("max G2 for",(min_nc:max_nc)[which.max(res[,2])],"clusters=",max(res[,2])))
print("clustering for max G2")
print(clusters[which.max(res[,2]),])
plot(res, type="p", pch=0, xlab="Number of clusters", ylab="G2", xaxt="n")
axis(1, c(min_nc:max_nc))

---

Calculates G3 internal cluster quality index

Description

Calculates G3 internal cluster quality index

Usage

index.G3(d,cl)

Arguments

d	'dist' object
cl	A vector of integers indicating the cluster to which each object is allocated

Details

See file $R_HOME\library\clusterSim\pdf\indexG3_details.pdf for further details

Value

calculated G3 index

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Gordon, A.D. (1999), Classification, Chapman & Hall/CRC, London, p. 62. ISBN 9781584880134.

See Also

index.G1, index.G2, index.S, index.C, index.H, index.KL, index.Gap, index.DB

Examples

# Example 1
library(clusterSim)
data(data_ratio)
d <- dist.GDM(data_ratio)
c <- pam(d, 5, diss = TRUE)
icq <- index.G3(d,c$clustering)
print(icq)

# Example 2
library(clusterSim)
data(data_ordinal)
d <- dist.GDM(data_ordinal, method="GDM2")
# nc - number_of_clusters
min_nc=2
max_nc=6
res <- array(0,c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
hc <- hclust(d, method="complete")
cl2 <- cutree(hc, k=nc)
res[nc-min_nc+1,2] <- G3 <- index.G3(d,cl2)
clusters <- rbind(clusters,cl2)
}
print(paste("min G3 for",(min_nc:max_nc)[which.min(res[,2])],"clusters=",min(res[,2])))
print("clustering for min G3")
print(clusters[which.min(res[,2]),])
#write.table(res,file="G3_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res, type="p", pch=0, xlab="Number of clusters", ylab="G3", xaxt="n")
axis(1, c(min_nc:max_nc))

---

Calculates Tibshirani, Walther and Hastie gap index

Description

Calculates Tibshirani, Walther and Hastie gap index

Usage

index.Gap (x, clall, reference.distribution="unif", B=10, 
	method="pam",d=NULL,centrotypes="centroids")

Arguments

x	data
clall	Two vectors of integers indicating the cluster to which each object is allocated in partition of n objects into u, and u+1 clusters
reference.distribution	"unif" - generate each reference variable uniformly over the range of the observed values for that variable or "pc" - generate the reference variables from a uniform distribution over a box aligned with the principal components of the data. In detail, if $X=\{x_{ij}\}$ is our n x m data matrix, assume that the columns have mean 0 and compute the singular value decomposition $X=UDV^T$. We transform via $X'=XV$ and then draw uniform features Z' over the ranges of the columns of X' , as in method a) above. Finally we back-transform via $Z=Z'V^T$ to give reference data Z
B	the number of simulations used to compute the gap statistic
method	the cluster analysis method to be used. This should be one of: "ward.D", "ward.D2", "single", "complete", "average", "mcquitty", "median", "centroid", "pam", "k-means","diana"
d	optional distance matrix, used for calculations if centrotypes="medoids"
centrotypes	"centroids" or "medoids"

Details

See file ../doc/indexGap_details.pdf for further details

Thanks to dr Michael P. Fay from National Institute of Allergy and Infectious Diseases for finding "one column error".

Value

Gap	Tibshirani, Walther and Hastie gap index for u clusters
diffu	necessary value for choosing correct number of clusters via gap statistic Gap(u)-[Gap(u+1)-s(u+1)]

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Tibshirani, R., Walther, G., Hastie, T. (2001), Estimating the number of clusters in a data set via the gap statistic, "Journal of the Royal Statistical Society", ser. B, vol. 63, part 2, 411-423. Available at: doi:10.1111/1467-9868.00293.

See Also

index.G1, index.G2, index.G3, index.C, index.S, index.H, index.KL, index.DB

Examples

# Example 1
library(clusterSim)
data(data_ratio)
cl1<-pam(data_ratio,4)
cl2<-pam(data_ratio,5)
clall<-cbind(cl1$clustering,cl2$clustering)
g<-index.Gap(data_ratio, clall, reference.distribution="unif", B=10,
   method="pam")
print(g)

# Example 2
library(clusterSim)
means <- matrix(c(0,2,4,0,3,6), 3, 2)
cov <- matrix(c(1,-0.9,-0.9,1), 2, 2)
x <- cluster.Gen(numObjects=40, means=means, cov=cov, model=2)
x <- x$data
md <- dist(x, method="euclidean")^2
# nc - number_of_clusters
min_nc=1
max_nc=5
min <- 0
clopt <- NULL
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
found <- FALSE
for (nc in min_nc:max_nc){
  cl1 <- pam(md, nc, diss=TRUE)
  cl2 <- pam(md, nc+1, diss=TRUE)
  clall <- cbind(cl1$clustering, cl2$clustering)
  gap <- index.Gap(x,clall,B=20,method="pam",centrotypes="centroids")
  res[nc-min_nc+1, 2] <- diffu <- gap$diffu
  if ((res[nc-min_nc+1, 2] >=0) && (!found)){
    nc1 <- nc
    min <- diffu
    clopt <- cl1$cluster
    found <- TRUE
  }
}
if (found){
print(paste("Minimal nc where diffu>=0 is",nc1,"for diffu=",round(min,4)),quote=FALSE)
}else{
print("I have not found clustering with diffu>=0", quote=FALSE)
}
plot(res,type="p",pch=0,xlab="Number of clusters",ylab="diffu",xaxt="n")
abline(h=0, untf=FALSE)
axis(1, c(min_nc:max_nc))

# Example 3
library(clusterSim)
means <- matrix(c(0,2,4,0,3,6), 3, 2)
cov <- matrix(c(1,-0.9,-0.9,1), 2, 2)
x <- cluster.Gen(numObjects=40, means=means, cov=cov, model=2)
x <- x$data
md <- dist(x, method="euclidean")^2
# nc - number_of_clusters
min_nc=1
max_nc=5
min <- 0
clopt <- NULL
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
found <- FALSE
for (nc in min_nc:max_nc){
  cl1 <- pam(md, nc, diss=TRUE)
  cl2 <- pam(md, nc+1, diss=TRUE)
  clall <- cbind(cl1$clustering, cl2$clustering)
  gap <- index.Gap(x,clall,B=20,method="pam",d=md,centrotypes="medoids")
  res[nc-min_nc+1, 2] <- diffu <- gap$diffu
  if ((res[nc-min_nc+1, 2] >=0) && (!found)){
    nc1 <- nc
    min <- diffu
    clopt <- cl1$cluster
    found <- TRUE
  }
}
if (found){
print(paste("Minimal nc where diffu>=0 is",nc1,"for diffu=",round(min,4)),quote=FALSE)
}else{
print("I have not found clustering with diffu>=0",quote=FALSE)
}
plot(res, type="p", pch=0, xlab="Number of clusters", ylab="diffu", xaxt="n")
abline(h=0, untf=FALSE)
axis(1, c(min_nc:max_nc))

---

Calculates Hartigan index

Description

Calculates Hartigan index

Usage

index.H (x,clall,d=NULL,centrotypes="centroids")

Arguments

x	data
clall	Two vectors of integers indicating the cluster to which each object is allocated in partition of n objects into u and u+1 clusters
d	optional distance matrix, used for calculations if centrotypes="medoids"
centrotypes	"centroids" or "medoids"

Details

See file $R_HOME\library\clusterSim\pdf\indexH_details.pdf for further details

Value

Hartigan index

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Hartigan, J. (1975), Clustering algorithms, Wiley, New York. ISBN 047135645X.

Milligan, G.W., Cooper, M.C. (1985), An examination of procedures of determining the number of cluster in a data set, "Psychometrika", vol. 50, no. 2, 159-179. Available at: doi:10.1007/BF02294245.

Tibshirani, R., Walther, G., Hastie, T. (2001), Estimating the number of clusters in a data set via the gap statistic, "Journal of the Royal Statistical Society", ser. B, vol. 63, part 2, 411-423. Available at: doi:10.1111/1467-9868.00293.

See Also

index.G1, index.G2, index.G3, index.C, index.S, index.KL, index.Gap, index.DB

Examples

# Example 1
library(clusterSim)
data(data_ratio)
cl1<-pam(data_ratio,4)
cl2<-pam(data_ratio,5)
clall<-cbind(cl1$clustering,cl2$clustering)
index.H(data_ratio,clall)

# Example 2
library(clusterSim)
data(data_ratio)
md <- dist(data_ratio, method="euclidean")
# nc - number_of_clusters
min_nc=1
max_nc=20
min <- 0
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
found <- FALSE
clusters <- NULL
for (nc in min_nc:max_nc)
{
	print(nc)
	hc <- hclust(md, method="complete")
	cl1 <- cutree(hc, k=nc)
	cl2 <- cutree(hc, k=nc+1)
	clall <- cbind(cl1,cl2)
	res[nc-min_nc+1,2] <- H <- index.H(data_ratio,clall,centrotypes="centroids")
	if ((res[nc-min_nc+1, 2]<10) && (!found)){
       nc1 <- nc
       min <- H
       clopt <- cl1
		   found <- TRUE
	}
}
if (found)
{
	print(paste("minimal nc for H<=10 equals",nc1,"for H=",min))
	print("clustering for minimal nc where H<=10")
	print(clopt)
}else
{
	print("Clustering not found with H<=10")
}
#write.table(res,file="H_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res,type="p",pch=0,xlab="Number of clusters",ylab="H",xaxt="n")
abline(h=10, untf=FALSE)
axis(1, c(min_nc:max_nc))

# Example 3
library(clusterSim)
data(data_ratio)
md <- dist(data_ratio, method="manhattan")
# nc - number_of_clusters
min_nc=1
max_nc=20
min <- 0
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
found <- FALSE
clusters <- NULL
for (nc in min_nc:max_nc)
{
	print(nc)
	hc <- hclust(md, method="complete")
	cl1 <- cutree(hc, k=nc)
	cl2 <- cutree(hc, k=nc+1)
	clall <- cbind(cl1,cl2)
	res[nc-min_nc+1,2] <- H <- index.H(data_ratio,clall,d=md,centrotypes="medoids")
	if ((res[nc-min_nc+1, 2]<10) && (!found)){
       nc1 <- nc
       min <- H
       clopt <- cl1
		   found <- TRUE
	}
}
if (found)
{
	print(paste("minimal nc for H<=10 equals",nc1,"for H=",min))
	print("clustering for minimal nc where H<=10")
	print(clopt)
}else
{
	print("Clustering not found with H<=10")
}
#write.table(res,file="H_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res,type="p",pch=0,xlab="Number of clusters",ylab="H",xaxt="n")
abline(h=10, untf=FALSE)
axis(1, c(min_nc:max_nc))

---

Calculates Krzanowski-Lai index

Description

Calculates Krzanowski-Lai index

Usage

index.KL (x,clall,d=NULL,centrotypes="centroids")

Arguments

x	data
clall	Three vectors of integers indicating the cluster to which each object is allocated in partition of n objects into u-1, u, and u+1 clusters
d	optional distance matrix, used for calculations if centrotypes="medoids"
centrotypes	"centroids" or "medoids"

Details

See file ../doc/indexKL_details.pdf for further details

Value

Krzanowski-Lai index

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Krzanowski, W.J., Lai, Y.T. (1988), A criterion for determining the number of groups in a data set using sum of squares clustering, "Biometrics", 44, 23-34.

Milligan, G.W., Cooper, M.C. (1985), An examination of procedures of determining the number of cluster in a data set, "Psychometrika", vol. 50, no. 2, 159-179. Available at: doi:10.1007/BF02294245.

Tibshirani, R., Walther, G., Hastie, T. (2001), Estimating the number of clusters in a data set via the gap statistic, "Journal of the Royal Statistical Society", ser. B, vol. 63, part 2, 411-423. Available at: doi:10.1111/1467-9868.00293.

See Also

index.G1, index.G2, index.G3, index.C, index.S, index.H, index.Gap, index.DB

Examples

# Example 1
library(clusterSim)
data(data_ratio)
cl1<-pam(data_ratio,4)
cl2<-pam(data_ratio,5)
cl3<-pam(data_ratio,6)
clall<-cbind(cl1$clustering,cl2$clustering,cl3$clustering)
index.KL(data_ratio,clall)

# Example 2
library(clusterSim)
data(data_ratio)
md <- dist(data_ratio, method="manhattan")
# nc - number_of_clusters
min_nc=2
max_nc=15
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
  if(nc-1==1){
    clustering1<-rep(1,nrow(data_ratio))
  }
  else{
    clustering1 <- pam(md, nc-1, diss=TRUE)$clustering
  }
  clustering2 <- pam(md, nc, diss=TRUE)$clustering
  clustering3 <- pam(md, nc+1, diss=TRUE)$clustering
  clall<- cbind(clustering1, clustering2, clustering3)
  res[nc-min_nc+1,2] <- KL <- index.KL(data_ratio,clall,centrotypes="centroids")
  clusters <- rbind(clusters, clustering2)
} 
print(paste("max KL for",(min_nc:max_nc)[which.max(res[,2])],"clusters=",max(res[,2])))
print("clustering for max KL")
print(clusters[which.max(res[,2]),])
#write.table(res,file="KL_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res,type="p",pch=0,xlab="Number of clusters",ylab="KL",xaxt="n")
axis(1, c(min_nc:max_nc))

# Example 3
library(clusterSim)
data(data_ratio)
md <- dist(data_ratio, method="manhattan")
# nc - number_of_clusters
min_nc=2
max_nc=15
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
  if(nc-1==1){
    clustering1<-rep(1,nrow(data_ratio))
  }
  else{
    clustering1 <- pam(md, nc-1, diss=TRUE)$clustering
  }
  clustering2 <- pam(md, nc, diss=TRUE)$clustering
  clustering3 <- pam(md, nc+1, diss=TRUE)$clustering
  clall<- cbind(clustering1, clustering2, clustering3)
  res[nc-min_nc+1,2] <- KL <- index.KL(data_ratio,clall,d=md,centrotypes="medoids")
  clusters <- rbind(clusters, clustering2)
} 
print(paste("max KL for",(min_nc:max_nc)[which.max(res[,2])],"clusters=",max(res[,2])))
print("clustering for max KL")
print(clusters[which.max(res[,2]),])
#write.table(res,file="KL_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res,type="p",pch=0,xlab="Number of clusters",ylab="KL",xaxt="n")
axis(1, c(min_nc:max_nc))

---

Calculates Rousseeuw's Silhouette internal cluster quality index

Description

Calculates Rousseeuw's Silhouette internal cluster quality index

Usage

index.S(d,cl,singleObject=0)

Arguments

d	'dist' object
cl	A vector of integers indicating the cluster to which each object is allocated
singleObject	0 - s(i)=0 or 1 - s(i)=1. When cluster contains a single object, it is unclear how a(i) of Silhouette index should be defined (see Kaufman & Rousseeuw (1990), p. 85).

Details

See file $R_HOME\library\clusterSim\pdf\indexS_details.pdf for further details

Value

calculated Silhouette index

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Gatnar, E., Walesiak, M. (Eds.) (2004), Metody statystycznej analizy wielowymiarowej w badaniach marketingowych [Multivariate statistical analysis methods in marketing research], Wydawnictwo AE, Wroclaw, 342-343, erratum.

Kaufman, L., Rousseeuw, P.J. (1990), Finding groups in data: an introduction to cluster analysis, Wiley, New York, pp. 83-88. ISBN: 978-0-471-73578-6.

See Also

index.G1, index.G2, index.G3, index.C, index.KL, index.H, index.Gap, index.DB

Examples

# Example 1
library(clusterSim)
data(data_ratio)
d <- dist.GDM(data_ratio)
c <- pam(d, 5, diss = TRUE)
icq <- index.S(d,c$clustering)
print(icq)

# Example 2
library(clusterSim)
data(data_ratio)
md <- dist(data_ratio, method="manhattan")
# nc - number_of_clusters
min_nc=2
max_nc=20
res <- array(0, c(max_nc-min_nc+1, 2))
res[,1] <- min_nc:max_nc
clusters <- NULL
for (nc in min_nc:max_nc)
{
cl2 <- pam(md, nc, diss=TRUE)
res[nc-min_nc+1, 2] <- S <- index.S(md,cl2$cluster)
clusters <- rbind(clusters, cl2$cluster)
}
print(paste("max S for",(min_nc:max_nc)[which.max(res[,2])],"clusters=",max(res[,2])))
print("clustering for max S")
print(clusters[which.max(res[,2]),])
#write.table(res,file="S_res.csv",sep=";",dec=",",row.names=TRUE,col.names=FALSE)
plot(res,type="p",pch=0,xlab="Number of clusters",ylab="S",xaxt="n")
axis(1, c(min_nc:max_nc))

---

Calculation of initial clusters centers for k-means like alghoritms

Description

Function calculates initial clusters centers for k-means like alghoritms with the following alghoritm (similar to SPSS QuickCluster function)

(a) if the distance between $x_k$ and its closest cluster center is greater than the distance between the two closest centers ($M_m$ and $M_n$ ), then $x_k$ replaces either $M_m$ or $M_n$, whichever is closer to $x_k$.

(b) If $x_k$ does not replace a cluster initial center in (a), a second test is made: If that distance $d_q$ greater than the distance between $M_q$ and its closest $M_i$, then $x_k$ replaces $M_q$.

where:

$M_i$ - initial center of i-th cluster

$x_k$ - vector of k-th observation

$d(...,...)$ - Euclidean distance

$d_{mn}$ = $min_{ij} d(M_i,M_j)$

$d_q = min_i d(x_k,M_i)$

Usage

initial.Centers(x, k)

Arguments

x	matrix or dataset
k	number of initial cluster centers

Value

Numbers of objects choosen as initial cluster centers

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Hartigan, J. (1975), Clustering algorithms, Wiley, New York. ISBN 0-471-35645-X.

See Also

cluster.Sim

Examples

#Example 1 (numbers of objects choosen as initial cluster centers)
library(clusterSim)
data(data_ratio)
ic <- initial.Centers(data_ratio, 10)
print(ic)

#Example 2 (application with kmeans algorithm)
library(clusterSim)
data(data_ratio)
kmeans(data_ratio,data_ratio[initial.Centers(data_ratio, 10),])

---

Types of normalization formulas for interval-valued symbolic variables

Description

Types of normalization formulas for interval-valued symbolic variables

Usage

interval_normalization(x,dataType="simple",type="n0",y=NULL,...)

Arguments

x	matrix dataset or symbolic table object
dataType	Type of symbolic data table passed to function, 'sda' - full symbolicDA format object; 'simple' - three dimensional array with lower and upper bound of intervals in third dimension; 'separate_tables' - lower bounds of intervals in x, upper bounds in y; 'rows' - lower and upper bound of intervals in neighbouring rows; 'columns' - lower and upper bound of intervals in neighbouring columns
type	type of normalization:

n0 - without normalization

n1 - standardization ((x-mean)/sd)

n2 - positional standardization ((x-median)/mad)

n3 - unitization ((x-mean)/range)

n3a - positional unitization ((x-median)/range)

n4 - unitization with zero minimum ((x-min)/range)

n5 - normalization in range <-1,1> ((x-mean)/max(abs(x-mean)))

n5a - positional normalization in range <-1,1> ((x-median)/max(abs(x-median)))

n6 - quotient transformation (x/sd)

n6a - positional quotient transformation (x/mad)

n7 - quotient transformation (x/range)

n8 - quotient transformation (x/max)

n9 - quotient transformation (x/mean)

n9a - positional quotient transformation (x/median)

n10 - quotient transformation (x/sum)

n11 - quotient transformation (x/sqrt(SSQ))

n12 - normalization ((x-mean)/sqrt(sum((x-mean)^2)))

n12a - positional normalization ((x-median)/sqrt(sum((x-median)^2)))

n13 - normalization with zero being the central point ((x-midrange)/(range/2))

y	matrix or dataset with upper bounds of intervals if argument dataType is uuqual to "separate_tables"
...	arguments passed to sum, mean, min sd, mad and other aggregation functions. In particular: na.rm - a logical value indicating whether NA values should be stripped before the computation

Value

Normalized data

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Jajuga, K., Walesiak, M. (2000), Standardisation of data set under different measurement scales, In: R. Decker, W. Gaul (Eds.), Classification and information processing at the turn of the millennium, Springer-Verlag, Berlin, Heidelberg, 105-112. Available at: doi:10.1007/978-3-642-57280-7_11.

Milligan, G.W., Cooper, M.C. (1988), A study of standardization of variables in cluster analysis, "Journal of Classification", vol. 5, 181-204. Available at: doi:10.1007/BF01897163.

Walesiak, M. (2014), Przeglad formul normalizacji wartosci zmiennych oraz ich wlasnosci w statystycznej analizie wielowymiarowej [Data normalization in multivariate data analysis. An overview and properties], "Przeglad Statystyczny" ("Statistical Review"), vol. 61, no. 4, 363-372. Available at: doi:10.5604/01.3001.0016.1740.

Walesiak, M., Dudek, A. (2017), Selecting the Optimal Multidimensional Scaling Procedure for Metric Data with R Environment, „STATISTICS IN TRANSITION new series”, September, Vol. 18, No. 3, pp. 521-540. Available at: doi:10.59170/stattrans-2017-027.

See Also

data.Normalization

Examples

library(clusterSim)
data(data_symbolic_interval_polish_voivodships)
n<-interval_normalization(data_symbolic_interval_polish_voivodships,dataType="simple",type="n2")
plotInterval(n$simple)

---

Reinforcing measurement scale for ordinal data

Description

Reinforcing measurement scale for ordinal data (ordinal to metric scale)

Usage

ordinalToMetric(data,scaleType="o",patternCoordinates)

Arguments

data	matrix or dataset
scaleType	"o" - variables measured on ordinal scale, "m" - variables measured on metric scale, "o/m" - vector with mixed variables - e.g. c("o","m","m","o","o","m")
patternCoordinates	vector containing pattern coordinates c(...) given by the reaseracher for data (for metric variables - NA, for ordinal variables - one of the categories for each ordinal variable (e.g. maximum category))

Details

See file ../doc/ordinalToMetric_details.pdf for further details

Value

pdata	raw (primary) data matrix
tdata	data matrix after transformation of ordinal variables into metric variables
cpattern	vector containing pattern coordinates

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, Wroclaw University of Economics, Poland

References

Jajuga, K., Walesiak, M., Bak, A. (2003), On the general distance measure, In: M. Schwaiger, O. Opitz (Eds.), Exploratory data analysis in empirical research, Springer-Verlag, Berlin, Heidelberg, 104-109. Available at: doi:10.1007/978-3-642-55721-7_12.

Walesiak, M. (1993), Statystyczna analiza wielowymiarowa w badaniach marketingowych [Multivariate statistical analysis in marketing research]. Wroclaw University of Economics, Research Papers no. 654.

Walesiak, M. (1999), Distance Measure for Ordinal Data, "Argumenta Oeconomica", No. 2 (8), 167-173.

Walesiak, M. (2011), Uogólniona miara odległości GDM w statystycznej analizie wielowymiarowej z wykorzystaniem programu R [The Generalized Distance Measure GDM in multivariate statistical analysis with R], Wydawnictwo Uniwersytetu Ekonomicznego, Wroclaw.

Walesiak, M. (2014), Wzmacnianie skali pomiaru dla danych porządkowych w statystycznej analizie wielowymiarowej [Reinforcing measurement scale for ordinal data in multivariate statistical analysis], Taksonomia 22, Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu no. 327, 60-68.

See Also

dist.GDM

Examples

# Example 1
library(clusterSim)
data(data_patternGDM2)
res1<-ordinalToMetric(data_patternGDM2,scaleType="o",patternCoordinates=c(5,4,3,1,1,3))
print(res1)

# Example 2
library(clusterSim)
data(data_patternGDM2)
res2<-ordinalToMetric(data_patternGDM2,scaleType="o",patternCoordinates=c(5,4,3,4,2,4))
print(res2)

---

An application of GDM1 distance for metric data to compute the distances of objects from the pattern object (upper or lower)

Description

An application of GDM1 distance for metric data to compute the distances of objects from the upper (ideal point co-ordinates) or lower (anti-ideal point co-ordinates) pattern object

Usage

pattern.GDM1(data, performanceVariable, scaleType="i",
nomOptValues=NULL, weightsType="equal", weights=NULL,
normalization="n0", patternType="upper",
patternCoordinates="dataBounds", patternManual=NULL,
nominalTransfMethod=NULL)

Arguments

data	matrix or dataset
performanceVariable	vector containing three types of performance variables: s for stimulants where higher value means better performance d for destimulants where low values indicate better performance n for nominants where the best value is implied. Object performance is positively assessed if the measure has implied value
scaleType	"i" - variables measured on interval scale, "r" - variables measured on ratio scale, "r/i" - vector with mixed variables
nomOptValues	vector containing optimal values for nominant variables and NA values for stimulants and destimulants. If performanceVariable do not contain nominant variables this nomOptValues may be set to NULL
weightsType	equal or different1 or different2 "equal" - equal weights "different1" - vector of different weights should satisfy conditions: each weight takes value from interval [0; 1] and sum of weights equals one "different2" - vector of different weights should satisfy conditions: each weight takes value from interval [0; m] and sum of weights equals m (m - the number of variables)
normalization	normalization formulas as in data.Normalization function
weights	vector of weights
patternType	"upper" - ideal point co-ordinates consists of the best variables' values "lower" - anti-ideal point co-ordinates consists of the worst variables' values
patternCoordinates	"dataBounds" - pattern should be calculated as following: "upper" pattern (maximum for stimulants, minimum for destimulants), "lower" pattern (minimum for stimulants, maximum for destimulants) "manual" - pattern should be given in patternManual variable
patternManual	Pattern co-ordinates contain: real numbers "min" - for minimal value of variable "max" - for maximal value of variable
nominalTransfMethod	method of transformation of nominant to stimulant variable: "q" - quotient transformation "d" - difference transformation

Details

See file ../doc/patternGDM1_details.pdf for further details

Value

pdata	raw (primary) data matrix
tdata	data matrix after transformation of nominant variables (with pattern in last row)
data	data matrix after normalization (with pattern in last row)
distances	GDM1 distances from pattern object
sortedDistances	sorted GDM1 distances from pattern object

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Jajuga, K., Walesiak, M., Bak, A. (2003), On the general distance measure, In: M. Schwaiger, O. Opitz (Eds.), Exploratory data analysis in empirical research, Springer-Verlag, Berlin, Heidelberg, 104-109. Available at: doi:10.1007/978-3-642-55721-7_12.

Walesiak, M. (1993), Statystyczna analiza wielowymiarowa w badaniach marketingowych [Multivariate statistical analysis in marketing research]. Wroclaw University of Economics, Research Papers no. 654.

Walesiak, M. (2006), Uogolniona miara odleglosci w statystycznej analizie wielowymiarowej [The Generalized Distance Measure in multivariate statistical analysis], Wydawnictwo AE, Wroclaw.

Walesiak, M. (2011), Uogólniona miara odległości GDM w statystycznej analizie wielowymiarowej z wykorzystaniem programu R [The Generalized Distance Measure GDM in multivariate statistical analysis with R], Wydawnictwo Uniwersytetu Ekonomicznego, Wroclaw.

Walesiak, M. (2016), Uogólniona miara odległości GDM w statystycznej analizie wielowymiarowej z wykorzystaniem programu R. Wydanie 2 poprawione i rozszerzone [The Generalized Distance Measure GDM in multivariate statistical analysis with R], Wydawnictwo Uniwersytetu Ekonomicznego, Wroclaw.

See Also

dist.GDM,data.Normalization

Examples

# Example 1
library(clusterSim)
data(data_patternGDM1)
res<-pattern.GDM1(data_patternGDM1,
performanceVariable=c("s","s","s","s","s","s","d","d","s","s"),
scaleType="r",nomOptValues=NULL,weightsType<-"equal",weights=NULL,
normalization<-"n4",patternType<-"lower",patternCoordinates<-"manual",
patternManual<-c("min","min","min","min","min","min","max","max","min","min"),
nominalTransfMethod <-NULL)
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p),xlim=c(max(gdm_p),min(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)),xaxt="n",
xlab="Order of objects from the best to the worst", 
ylab="GDM distances from pattern object", lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

# Example 2
library(clusterSim)
data(data_patternGDM1)
res<-pattern.GDM1(data_patternGDM1,
performanceVariable=c("s","s","s","s","s","s","d","d","s","s"),
scaleType="r",nomOptValues=NULL,weightsType<-"equal",weights=NULL,
normalization<-"n2",patternType<-"upper",
patternCoordinates<-"dataBounds",patternManual<-NULL,
nominalTransfMethod<-NULL)
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p),xlim=c(min(gdm_p),max(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)),xaxt="n",
xlab="Order of objects from the best to the worst", 
ylab="GDM distances from pattern object", lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

# Example 3
library(clusterSim)
data(data_patternGDM1)
res<-pattern.GDM1(data_patternGDM1,
performanceVariable=c("s","s","s","s","s","s","d","d","s","s"),
scaleType="r",nomOptValues=NULL,weightsType<-"different2",
weights=c(1.1,1.15,1.15,1.1,1.1,0.7,0.7,1.2,0.8,1.0),
normalization<-"n6",patternType<-"upper",patternCoordinates<-"manual",
patternManual<-c(100,100,100,100,100,"max","min","min","max","max"),
nominalTransfMethod <-NULL)
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p),xlim=c(min(gdm_p),max(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)),xaxt="n",
xlab="Order of objects from the best to the worst", 
ylab="GDM distances from pattern object", lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

---

An application of GDM2 distance for ordinal data to compute the distances of objects from the pattern object (upper or lower)

Description

An application of GDM2 distance for ordinal data to compute the distances of objects from the upper (ideal point co-ordinates) or lower (anti-ideal point co-ordinates) pattern object

Usage

pattern.GDM2(data, performanceVariable, nomOptValues=NULL,
weightsType="equal", weights=NULL, patternType="upper",
patternCoordinates="dataBounds", patternManual=NULL,
nominalTransfMethod=NULL)

Arguments

data	matrix or dataset
performanceVariable	vector containing three types of performance variables: s for stimulants where higher value means better performance d for destimulants where low values indicate better performance n for nominants where the best value is implied. Object performance is positively assessed if the measure has implied value
nomOptValues	vector containing optimal values for nominant variables and NA values for stimulants and destimulants. If performanceVariable do not contain nominant variables this nomOptValues may be set to NULL
weightsType	equal or different1 or different2 "equal" - equal weights "different1" - vector of different weights should satisfy conditions: each weight takes value from interval [0; 1] and sum of weights equals one "different2" - vector of different weights should satisfy conditions: each weight takes value from interval [0; m] and sum of weights equals m (m - the number of variables)
weights	vector of weights
patternType	"upper" - ideal point co-ordinates consists of the best variables' values "lower" - anti-ideal point co-ordinates consists of the worst variables' values
patternCoordinates	"dataBounds" - pattern should be calculated as following: "upper" pattern (maximum for stimulants, minimum for destimulants, nominal value for nominants), "lower" pattern (minimum for stimulants, maximum for destimulants) "manual" - pattern should be given in patternManual variable
patternManual	Pattern co-ordinates contain: real numbers "min" - for minimal value of variable "max" - for maximal value of variable "nom" - for nominal value of variable (for upper pattern only - given in nomOptValues vector)
nominalTransfMethod	method of transformation of nominant to destimulant variable for patternType="lower": "database" - for each nominant separately GDM2 distance is calculated between each nominant observation (with repetitions - all variable values are used in calculation) and nominal value. Next the variable observations are replaced by those distances "symmetrical" - for each nominant separately GDM2 distance is calculated between each nominant observation (without repetition - each observation is used once) and nominal value. Next the variable observations are replaced by those distances

Details

See file ../doc/patternGDM2_details.pdf for further details

Value

pdata	raw (primary) data matrix
data	data matrix after transformation of nominant variables (with pattern in last row)
distances	GDM2 distances from pattern object
sortedDistances	sorted GDM2 distances from pattern object

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

epartment of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

References

Jajuga, K., Walesiak, M., Bak, A. (2003), On the general distance measure, In: M. Schwaiger, O. Opitz (Eds.), Exploratory data analysis in empirical research, Springer-Verlag, Berlin, Heidelberg, 104-109. Available at: doi:10.1007/978-3-642-55721-7_12.

Walesiak, M. (1993), Statystyczna analiza wielowymiarowa w badaniach marketingowych [Multivariate statistical analysis in marketing research]. Wroclaw University of Economics, Research Papers no. 654.

Walesiak, M. (1999), Distance Measure for Ordinal Data, "Argumenta Oeconomica", No. 2 (8), 167-173.

Walesiak, M. (2006), Uogolniona miara odleglosci w statystycznej analizie wielowymiarowej [The Generalized Distance Measure in multivariate statistical analysis], Wydawnictwo AE, Wroclaw.

Walesiak, M. (2011), Uogólniona miara odległości GDM w statystycznej analizie wielowymiarowej z wykorzystaniem programu R [The Generalized Distance Measure GDM in multivariate statistical analysis with R], Wydawnictwo Uniwersytetu Ekonomicznego, Wroclaw.

Walesiak, M. (2016), Uogólniona miara odległości GDM w statystycznej analizie wielowymiarowej z wykorzystaniem programu R. Wydanie 2 poprawione i rozszerzone [The Generalized Distance Measure GDM in multivariate statistical analysis with R], Wydawnictwo Uniwersytetu Ekonomicznego, Wroclaw.

See Also

dist.GDM

Examples

# Example 1
library(clusterSim)
data(data_patternGDM2)
res<-pattern.GDM2(data_patternGDM2,
performanceVariable=c("s","s","s","d","d","n"),
nomOptValues=c(NA,NA,NA,NA,NA,3), weightsType<-"equal", weights=NULL,
patternType="lower", patternCoordinates="manual",
patternManual=c("min","min",0,5,"max","max"),
nominalTransfMethod="symmetrical")
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p),xlim=c(max(gdm_p),min(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)), 
xaxt="n",xlab="Order of objects from the best to the worst",
ylab="GDM distances from pattern object", lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

# Example 2
library(clusterSim)
data(data_patternGDM2)
res<-pattern.GDM2(data_patternGDM2,
performanceVariable=c("s","s","s","d","d","n"),
nomOptValues=c(NA,NA,NA,NA,NA,3), weightsType<-"equal", weights=NULL,
patternType="upper", patternCoordinates="dataBounds",
patternManual=NULL, nominalTransfMethod="database")
print(res)
gdm_p<-res$distances
plot(cbind(gdm_p,gdm_p), xlim=c(min(gdm_p),max(gdm_p)),
ylim=c(min(gdm_p),max(gdm_p)), 
xaxt="n",xlab="Order of objects from the best to the worst",
ylab="GDM distances from pattern object", lwd=1.6)
axis(1, at=gdm_p,labels=names(gdm_p), cex.axis=0.5)

---

Plot categorial data on a scatterplot matrix

Description

Plot categorial data on a scatterplot matrix (optionally with clusters)

Usage

plotCategorial(x, pairsofVar=NULL, cl=NULL, clColors=NULL,...)

Arguments

x	data matrix (rows correspond to observations and columns correspond to variables)
pairsofVar	pairs of variables - all variables (pairsofVar=NULL) or selected variables, e.g. pairsofVar=c(1,3,4)
cl	cluster membership vector
clColors	The colors of clusters. The colors are given arbitrary (clColors=TRUE) or by hand, e.g. clColors=c("red","blue","green"). The number of colors equals the number of clusters
...	Arguments to be passed to methods, such as graphical parameters (see par).

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

See Also

colors, pairs

Examples

# Example 1
library(clusterSim)
data(data_ordinal)
plotCategorial(data_ordinal, pairsofVar=c(1,3,4,9), cl=NULL, 
clColors = NULL)

# Example 2
library(clusterSim)
grnd <- cluster.Gen(50,model=5,dataType="o",numCategories=5)
plotCategorial(grnd$data, pairsofVar=NULL, cl=grnd$clusters, 
clColors=TRUE)

# Example 3
library(clusterSim)
grnd<-cluster.Gen(50,model=4,dataType="o",numCategories=7, numNoisyVar=2)
plotCategorial(grnd$data, pairsofVar=NULL, cl=grnd$clusters, 
clColors = c("red","blue","green"))

---

Plot symbolic interval-valued data on a scatterplot matrix

Description

Plot symbolic interval-valued data on a scatterplot matrix (optionally with clusters)

Usage

plotInterval(x, pairsofsVar=NULL, cl=NULL, clColors=NULL,...)

Arguments

x	symbolic interval-valued data
pairsofsVar	pairs of symbolic interval variables - all variables (pairsofsVar=NULL) or selected variables, e.g. pairsofsVar=c(1,3,4)
cl	cluster membership vector
clColors	The colors of clusters. The colors are given arbitrary (clColors=TRUE) or by hand, e.g. clColors=c("red","blue","green"). The number of colors equals the number of clusters
...	Arguments to be passed to methods, such as graphical parameters (see par).

Author(s)

Marek Walesiak [email protected], Andrzej Dudek [email protected]

Department of Econometrics and Computer Science, University of Economics, Wroclaw, Poland

See Also

plotCategorial, colors, pairs

Examples

# Example 1
library(clusterSim)
data(data_symbolic)
plotInterval(data_symbolic, pairsofsVar=c(1,3,4,6), cl=NULL,
clColors=NULL)

# Example 2
library(clusterSim)
grnd <- cluster.Gen(60, model=5, dataType="s", numNoisyVar=1, 
numOutliers=10, rangeOutliers=c(1,5))
grnd$clusters[grnd$clusters==0] <- max(grnd$clusters)+1			
# To colour outliers
plotInterval(grnd$data, pairsofsVar=NULL, cl=grnd$clusters,
clColors=TRUE)

# Example 3
library(clusterSim)
grnd <- cluster.Gen(50, model=4, dataType="s", numNoisyVar=2, 
numOutliers=10, rangeOutliers=c(1,4))
grnd$clusters[grnd$clusters==0] <- max(grnd$clusters)+1			
# To colour outliers
plotInterval(grnd$data, pairsofsVar=NULL, cl=grnd$clusters, 
clColors=c("red","blue","green","yellow"))

---