Package 'FCPS'

Title: Fundamental Clustering Problems Suite
Description: Over sixty clustering algorithms are provided in this package with consistent input and output, which enables the user to try out algorithms swiftly. Additionally, 26 statistical approaches for the estimation of the number of clusters as well as the mirrored density plot (MD-plot) of clusterability are implemented. The packages is published in Thrun, M.C., Stier Q.: "Fundamental Clustering Algorithms Suite" (2021), SoftwareX, <DOI:10.1016/j.softx.2020.100642>. Moreover, the fundamental clustering problems suite (FCPS) offers a variety of clustering challenges any algorithm should handle when facing real world data, see Thrun, M.C., Ultsch A.: "Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems" (2020), Data in Brief, <DOI:10.1016/j.dib.2020.105501>.
Authors: Michael Thrun [aut, cre, cph] , Peter Nahrgang [ctr, ctb], Felix Pape [ctr, ctb], Vasyl Pihur [ctb], Guy Brock [ctb], Susmita Datta [ctb], Somnath Datta [ctb], Luis Winckelmann [com], Alfred Ultsch [dtc, ctb], Quirin Stier [ctb, rev]
Maintainer: Michael Thrun <[email protected]>
License: GPL-3
Version: 1.3.4
Built: 2024-10-13 07:51:35 UTC
Source: CRAN

Help Index

Fundamental Clustering Problems Suite

Description

Over sixty clustering algorithms are provided in this package with consistent input and output, which enables the user to try out algorithms swiftly. Additionally, 26 statistical approaches for the estimation of the number of clusters as well as the mirrored density plot (MD-plot) of clusterability are implemented. The packages is published in Thrun, M.C., Stier Q.: "Fundamental Clustering Algorithms Suite" (2021), SoftwareX, <DOI:10.1016/j.softx.2020.100642>. Moreover, the fundamental clustering problems suite (FCPS) offers a variety of clustering challenges any algorithm should handle when facing real world data, see Thrun, M.C., Ultsch A.: "Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems" (2020), Data in Brief, <DOI:10.1016/j.dib.2020.105501>.

The package consists of many algorithms and fundamental datasets for clustering published in [Thrun/Stier, 2021]. Originally, the 'Fundamental Clustering Problems Suite' (FCPS) offered a variety of clustering problems any algorithm shall be able to handle when facing real world data. Nine of the here presented artificial datasets were priorly named FCPS with a fixed sample size in Ultsch, A.: "Clustering with SOM: U*C", In Workshop on Self-Organizing Maps, 2005. FCPS often served in the paper as an elementary benchmark for clustering algorithms. The FCPS package extends datasets, enables variable sample sizes for these datasets, and provides a standardized and easy access to many clustering algorithms.

https://www.deepbionics.org/

Details

FCPS datasets consists of data sets with known a priori classification to be reproduced by the algorithms. All data sets are intentionally created to be simple and might be visualized in two or three dimensions. Each data sets represents a certain problem that is solved by known clustering algorithms with varying success. This is done in order to reveal benefits and shortcomings of algorithms in question. Standard clustering methods, e.g. single-linkage, ward and k-means, are not able to solve all FCPS problems satisfactorily. "Lsun3D and each of the nine artificial data sets of "Fundamental Clustering Problems Suite" (FCPS) were defined separately for a specific clustering problem as cited (in [Thrun/Ultsch, 2020]). The original sample size defined in the respective first publication mentioning the data was used in [Thrun/Ultsch, 2020], but using the R function "ClusterChallenge" (...) any sample size can be drawn for all artificial data sets. [Thrun/Ultsch, 2020]

Index: This package was not yet installed at build time.

Author(s)

Michael Thrun [aut, cre, cph] (<https://orcid.org/0000-0001-9542-5543>), Peter Nahrgang [ctr, ctb], Felix Pape [ctr, ctb], Vasyl Pihur [ctb], Guy Brock [ctb], Susmita Datta [ctb], Somnath Datta [ctb], Luis Winckelmann [com], Alfred Ultsch [dtc, ctb], Quirin Stier [ctb, rev]

Maintainer: Michael Thrun <[email protected]>

References

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

[Thrun/Stier, 2021] Thrun, M. C., & Stier, Q.: Fundamental Clustering Algorithms Suite SoftwareX, Vol. 13(C), in press, pp. 100642. doi:10.1016/j.softx.2020.100642, 2021.

[Ultsch, 2005] Ultsch, A.: Clustering with SOM: U*C, In Proc. Workshop on Self-Organizing Maps, pp. 75-82, Paris, France, 2005.

(Adaptive) Density Peak Clustering algorithm using automatic parameter selection

Description

The algorithm was introduced in [Rodriguez/Laio, 2014] and here implemented by [Wang/Xu, 2017]. The algorithm is adaptive in the sense that only ClusterNo has to be set instead of the paramters of [Rodriguez/Laio, 2014] implemented in ADPclustering.

Usage

ADPclustering(Data,ClusterNo=NULL,PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

Optional, either: A number k which defines k different Clusters to be build by the algorithm, or a range of ClusterNo to let the algorithm choose from.
PlotIt

default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

The ADP algorithm decides the k number of clusters. This is contrary to the other version of the algorithm from another package which can be called with DensityPeakClustering.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Rodriguez/Laio, 2014] Rodriguez, A., & Laio, A.: Clustering by fast search and find of density peaks, Science, Vol. 344(6191), pp. 1492-1496. 2014.

[Wang/Xu, 2017] Wang, X.-F., & Xu, Y.: Fast clustering using adaptive density peak detection, Statistical methods in medical research, Vol. 26(6), pp. 2800-2811. 2017.

See Also

DensityPeakClustering

adpclust

Examples

data('Hepta')
out=ADPclustering(Hepta$Data,PlotIt=FALSE)

AGNES clustering

Description

Agglomerative hierarchical clustering (AGNES)of [Rousseeuw/Kaufman, 1990, pp. 199-252]

Usage

AgglomerativeNestingClustering(DataOrDistances, ClusterNo,

PlotIt = FALSE, Standardization = TRUE, ...)

Arguments

DataOrDistances

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases or d-dimensional data points. Every case has d attributes, variables or features. Alternatively, symmetric [1:n,1:n] distance matrix
ClusterNo

A number k which defines k different clusters to be built by the algorithm. if ClusterNo=0, the dendrogram is generated instead of a clustering to estimate the numbers of clusters.

PlotIt

Default: FALSE if ClusterNo!=0, If TRUE or ClusterNo=0 plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Standardization

DataOrDistances is standardized before calculating the dissimilarities. Measurements are standardized for each variable (column), by subtracting the variable's mean value and dividing by the variable's mean absolute deviation. If DataOrDistances is already a distance matrix, then this argument will be ignored.

...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Dendrogram

Dendrogram of hierarchical clustering algorithm
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Rousseeuw/Kaufman, 1990] Rousseeuw, P. J., & Kaufman, L.: Finding groups in data, Belgium, John Wiley & Sons Inc., ISBN: 0471735787, doi 10.1002/9780470316801, Online ISBN: 9780470316801, 1990.

[Struyf et al., 1996] Struyf,A., Hubert, M. and Rousseeuw, Peter J.: Clustering in an Object-Oriented Environment, Journal of Statistical Software, Vol. 1, doi: 10.18637/jss.v001.i04, 1996.

[Struyf et al., 1997] Struyf, A., Hubert, M. and Rousseeuw, P.J.: Integrating Robust Clustering Techniques in S-PLUS, Computational Statistics and Data Analysis, Vol. 26, pp. 17–37, 1997.

See Also

agnes

Examples

data('Hepta')
CA=AgglomerativeNestingClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)
## Not run: 
ClusterDendrogram(CA$Dendrogram,7,main='AGNES clustering')

print(CA$Object)
plot(CA$Object)

## End(Not run)

Affinity Propagation Clustering

Description

Affinity propagation clustering published by [Frey/Dueck, 2007] and implemented by [Bodenhofer et al., 2011].

Usage

APclustering(DataOrDistances,

InputPreference=NA,ExemplarPreferences=NA,

DistanceMethod="euclidean",

Seed=7568,PlotIt=FALSE,Data,...)

Arguments

DataOrDistances

[1:n,1:d] with: if d=n and symmetric then distance matrix assumed, otherwise: [1:n,1:d] matrix of dataset to be clustered. It consists of n cases or d-dimensional data points. Every case has d attributes, variables or features. In the latter case the Euclidean distances will be calculated.

InputPreference

Default parameter set, see apcluster

ExemplarPreferences

Default parameter set, see apcluster

DistanceMethod

DistanceMethod as in dist for similarities.

Seed

Set as integervalue to have reproducible results, see apcluster

PlotIt

Default: FALSE, If TRUE and dataset of [1:n,1:d] dimensions then a plot of the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls will be generated.
Data

[1:n,1:d] data matrix in the case that DataOrDistances is missing and partial matching does not work.
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

Distancematrix D is converted to similarity matrix S with S=-(D^2).

If data matrix is used, then euclidean similarities are calculated by similarities and a specifed distance method.

The AP algorithm decides the k number of clusters.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Frey/Dueck, 2007] Frey, B. J., & Dueck, D.: Clustering by passing messages between data points, Science, Vol. 315(5814), pp. 972-976, <doi:10.1126/science.1136800>, 2007.

[Bodenhofer et al., 2011] Bodenhofer, U., Kothmeier, A., & Hochreiter, S.: APCluster: an R package for affinity propagation clustering, Bioinformatics, Vol. 27(17), pp, 2463-2464, 2011.

Further details in http://www.bioinf.jku.at/software/apcluster/

See Also

apcluster

Examples

data('Hepta')
res=APclustering(Hepta$Data, PlotIt = FALSE)

Atom introduced in [Ultsch, 2004].

Description

Two nested spheres with different variances that are not linear not separable. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("Atom")

Details

Size 800, Dimensions 3, stored in Atom$Data

Classes 2, stored in Atom$Cls

References

[Ultsch, 2004] Ultsch, A.: Strategies for an artificial life system to cluster high dimensional data, Abstracting and Synthesizing the Principles of Living Systems, GWAL-6, U. Brggemann, H. Schaub, and F. Detje, Eds, pp. 128-137. 2004.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(Atom)
str(Atom)

Automatic Projection-Based Clustering

Description

Projection-based clustering AutomaticProjectionBasedClustering projects the data (nonlinear) into two dimensions and tries only to preserve relevant neighborhoods prior to clustering. The cluster analysis itself includes the high-dimensional distances in the clustering process. Performs non-interactive projection-based clustering based on non-linear projection methods [Thrun/Ultsch, 2017], [Thrun/Ultsch, 2020a].

Usage

AutomaticProjectionBasedClustering(DataOrDistances,ClusterNo,Type="NerV",

StructureType = TRUE,PlotIt=FALSE,PlotTree=FALSE,PlotMap=FALSE,...)

Arguments

DataOrDistances

Either nonsymmetric [1:n,1:d] numerical matrix of a dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.

or

symmetric [1:n,1:n] distance matrix, e.g. as.matrix(dist(Data,method))

ClusterNo

A number k which defines k different clusters to be built by the algorithm.
Type

Type of Projection method, either

NerV [Venna et al., 2010]

Pswarm [Thrun/Ultsch, 2020b]

MDS [Torgerson, 1952]

Uwot [McInnes et al., 2018]

CCA [Demartines/Herault, 1995]

Sammon [Sammon, 1969]

t-SNE [Van der Maaten/Hinton, 2008]

StructureType

Either compact (TRUE) or connected (FALSE), see discussion in [Thrun, 2018]

PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
PlotTree

TRUE: Plots the dendrogram, FALSE: no plot

PlotMap

Plots the topographic map [Thrun et al., 2016].

...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

The first idea of using non-PCA projections for clustering was published by [Bock, 1987] as a definition. However, to the knowledge of the author, it was not applied to any data. The coexistence of projection and clustering was introduced in [Thrun/Ultsch, 2017].

Projection-based clustering is based on a nonlinear projection of high-dimensional data into a two-dimensional space [Thrun/Ultsch, 2020b]. Typical projection-methods like t-distributed stochastic neighbor embedding (t-SNE) [Van der Maaten/Hinton, 2008], or neighbor retrieval visualizer (NerV) [Venna et al., 2010] are used project data explicitly into two dimensions disregarding the subspaces of higher dimension than two and preserving only relevant neighborhoods in high-dimensional data. In the next step, the Delaunay graph [Delaunay, 1934] between the projected points is calculated, and each vertex between two projected points is weighted with the high-dimensional distance between the corresponding high-dimensional data points. Thereafter the shortest path between every pair of points is computed using the Dijkstra algorithm [Dijkstra, 1959]. The shortest paths are then used in the clustering process, which involves two choices depending on the structure type in the high-dimensional data [Thrun/Ultsch, 2020b]. This Boolean choice can be decided by looking at the topographic map of high-dimensional structures [Thrun/Ultsch, 2020a]. In a benchmarking of 34 comparable clustering methods, projection-based clustering was the only algorithm that always was able to find the high-dimensional distance or density-based structure of the dataset [Thrun/Ultsch, 2020b].

It should be noted that it is preferable to use a visualization for the Generalized U-Matrix like the topographic map plotTopographicMap of [Thrun et al., 2016] to evaluate the choice of the boolean parameter StructureType and the clustering, improve it or set the number of clusters appropriately. A comparison with 32 clustering algorithms showed that PBC is always able to find the correct cluster structure while the best of the 32 clustering algorithms varies depending on the dataset [Thrun/Ultsch, 2020].

The first systematic comparison to other DR clustering methods like Projection-Pursuit Methods ProjectionPursuitClustering, supspace clustering methods SubspaceClustering, and CA-based clustering methods can be found in [Thrun/Ultsch, 2020a]. For PCA-based clustering methods please see TandemClustering.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. . Points which cannot be assigned to a cluster will be reported with 0.

Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Bock, 1987] Bock, H.: On the interface between cluster analysis, principal component analysis, and multidimensional scaling, Multivariate statistical modeling and data analysis, (pp. 17-34), Springer, 1987.

[Thrun/Ultsch, 2017] Thrun, M. C., & Ultsch, A.: Projection based Clustering, Proc. International Federation of Classification Societies (IFCS), pp. 250-251, Tokai University, Japanese Classification Society (JCS), Tokyo, Japan August 7-10, 2017.

[Thrun/Ultsch, 2020a] Thrun, M. C., & Ultsch, A.: Using Projection based Clustering to Find Distance and Density based Clusters in High-Dimensional Data, Journal of Classification, in press, doi 10.1007/s00357-020-09373-2, 2020.

[Thrun et al., 2016] Thrun, M. C., Lerch, F., Loetsch, J., & Ultsch, A.: Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG), Vol. 24, pp. 7-16, Plzen, http://wscg.zcu.cz/wscg2016/short/A43-full.pdf, 2016.

[McInnes et al., 2018] McInnes, L., Healy, J., & Melville, J.: Umap: Uniform manifold approximation and projection for dimension reduction, arXiv preprint arXiv:1802.03426, 2018.

[Demartines/Herault, 1995] Demartines, P., & Herault, J.: CCA:" Curvilinear component analysis", Proc. 15 Colloque sur le traitement du signal et des images, Vol. 199, GRETSI, Groupe d Etudes du Traitement du Signal et des Images, France 18-21 September, 1995.

[Sammon, 1969] Sammon, J. W.: A nonlinear mapping for data structure analysis, IEEE Transactions on computers, Vol. 18(5), pp. 401-409. doi doi:10.1109/t-c.1969.222678, 1969.

[Thrun/Ultsch, 2020b] Thrun, M. C., & Ultsch, A.: Swarm Intelligence for Self-Organized Clustering, Journal of Artificial Intelligence, Vol. in press, pp. doi 10.1016/j.artint.2020.103237, 2020.

[Torgerson, 1952] Torgerson, W. S.: Multidimensional scaling: I. Theory and method, Psychometrika, Vol. 17(4), pp. 401-419. 1952.

[Venna et al., 2010] Venna, J., Peltonen, J., Nybo, K., Aidos, H., & Kaski, S.: Information retrieval perspective to nonlinear dimensionality reduction for data visualization, The Journal of Machine Learning Research, Vol. 11, pp. 451-490. 2010.

[Van der Maaten/Hinton, 2008] Van der Maaten, L., & Hinton, G.: Visualizing Data using t-SNE, Journal of Machine Learning Research, Vol. 9(11), pp. 2579-2605. 2008.

Examples

data('Hepta')
out=AutomaticProjectionBasedClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

Chainlink introduced in [Ultsch et al., 1994; Ultsch, 1995].

Description

Two chains of rings. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("Chainlink")

Details

Size 1000, Dimensions 3, stored in Chainlink$Data

Classes 2, stored in Chainlink$Cls

References

[Ultsch et al., 1994] Ultsch, A., Guimaraes, G., Korus, D., & Li, H.: Knowledge extraction from artificial neural networks and applications, Parallele Datenverarbeitung mit dem Transputer, (pp. 148-162), Springer, 1994.

[Ultsch, 1995] Ultsch, A.: Self organizing neural networks perform different from statistical k-means clustering, Proc. Society for Information and Classification (GFKL), Vol. 1995, Basel 8th-10th March 1995.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(Chainlink)
str(Chainlink)

Clusterability MDplot

Description

Clusterability mirrored-density plot. Clusterability aims to quantify the degree of cluster structures [Adolfsson et al., 2019]. A dataset has a high probabilty to possess cluster structures, if the first component of the PCA projection is multimodal [Adolfsson et al., 2019]. As the dip test is less exact than the MDplot [Thrun et al., 2020] , pvalues above 0.05 can be given for MDplots which are clearly multimodal.

An alternative investigation of clusterability can be performed by inspecting the topographic map of the Generalized U-Matrix for a specfic projection method using the ProjectionBasesdClustering and GeneralizedUmatrix packages on CRAN, see [Thrun/Ultsch, 2021] for details.

Usage

ClusterabilityMDplot(DataOrDistance,Method,

na.rm=FALSE,PlotIt=TRUE,...)

Arguments

DataOrDistance

Either a dataset[1:n,1:d] of n cases and d features or a symmetric distance matrix [1:d,1:d] or multiple data sets or distances in a list

Method

"none" performs no dimension reduction.

"pca" uses the scores from the first principal component.

"distance" computes pairwise distances (using distance_metric as the metric).

na.rm

Statistical testing will not work with missing values, if TRUE values are imputed with averages
PlotIt

TRUE: print plot, otherwise do not plot directly, instead use Handle for further adjustment
...

Further arguments for functionMDplot4multiplevectors of package DataVisualizations like "main", and "Ordering"

Details

Use the method of [Adolfsson et al., 2019] specified as pca plus dip-test (PCA dip) per default without scaling or standardization of data because this step should never be done automatically. In [Thrun, 2020] the standardization and scaling did not improve the results.

If list is named, than the names of the list will be used and the MDplots will be re-ordered according to multimodality in the plot, otherwise only the pvalues of [Adolfsson et al., 2019] will be the names and the ordering of the MDplots is the same as the list.

Beware, as shown below, this test fails for almost touching clusters of Tetra and is difficult to intepret on WingNut but with overlayed with a roubustly estimated unimodal Gaussian distribution it can be interpreted as multimodal). However, it does not fail for chaining data contrary to the claim in [Adolfsson et al., 2019].

Based on [Thrun, 2020], the author of this function disagrees with [Adolfsson et al., 2019] as to the preference which clusterablity method should be used because the approach "distance" is not preferable for density-based cluster structures.

Value

List of

Handle

GGobject, plotter handle of ggplot2
Pvalue

One or more p-values of dip test depending on DataOrDistance

Note

"none" seems to call dip.test in clusterabilitytest with high-dimensional data. In that case dip.test just vectorizes the matrix of the data which does not make any sense. Since this could be a bug, the "none" option should not be used.

Imputation does not work for distance matrices. Imputation is still experimental. It is adviced to impute missing values before using this function

Author(s)

Michael Thrun

References

[Adolfsson et al., 2019] Adolfsson, A., Ackerman, M., & Brownstein, N. C.: To cluster, or not to cluster: An analysis of clusterability methods, Pattern Recognition, Vol. 88, pp. 13-26. 2019.

[Thrun et al., 2020] Thrun, M. C., Gehlert, T. & Ultsch, A.: Analyzing the Fine Structure of Distributions, PLoS ONE, Vol. 15(10), pp. 1-66, DOI doi:10.1371/journal.pone.0238835, 2020.

[Thrun/Ultsch, 2021] Thrun, M. C., and Ultsch, A.: Swarm Intelligence for Self-Organized Clustering, Artificial Intelligence, Vol. 290, pp. 103237, doi:10.1016/j.artint.2020.103237, 2021.

[Thrun, 2020] Thrun, M. C.: Improving the Sensitivity of Statistical Testing for Clusterability with Mirrored-Density Plot, in Archambault, D., Nabney, I. & Peltonen, J. (eds.), Machine Learning Methods in Visualisation for Big Data, The Eurographics Association, https://diglib.eg.org:443/handle/10.2312/mlvis20201102, Norrkoping, Sweden, May, 2020.

See Also

MDplot

Examples

##one dataset
data(Hepta)

ClusterabilityMDplot(Hepta$Data)

##multiple datasets
data(Atom)
data(Chainlink)
data(Lsun3D)
data(GolfBall)
data(EngyTime)
data(Target)
data(Tetra)
data(WingNut)
data(TwoDiamonds)

DataV = list(
  Atom = Atom$Data,
  Chainlink = Chainlink$Data,
  Hepta = Hepta$Data,
  Lsun3D = Lsun3D$Data,
  GolfBall = GolfBall$Data,
  EngyTime = EngyTime$Data,
  Target = Target$Data,
  Tetra = Tetra$Data,
  WingNut = WingNut$Data,
  TwoDiamonds = TwoDiamonds$Data
)

ClusterabilityMDplot(DataV)

Applies a function over grouped data

Description

Applies a given function to each dimension d of data separately for each cluster

Usage

ClusterApply(DataOrDistances,FUN,Cls,Simple=FALSE,...)

Arguments

DataOrDistances

[1:n,1:d] with: if d=n and symmetric then distance matrix assumed, otherwise:

[1:n,1:d] matrix of defining the dataset that consists of n cases or d-dimensional data points. Every case has d attributes, variables or features.

FUN

Function to be applied to each cluster of data and each column of data
Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Simple

Boolean, if TRUE, simplifies output
...

Additional parameters to be passed on to FUN

Details

Applies a given function to each feature of each cluster of data using the clustering stored in Cls which is the cluster identifiers for all rows in data. If missing, all data are in first cluster, The main output is FUNPerCluster[i] which is the result of FUN for the data points in cluster of UniqueClusters[i] named with the function's name used.

In case of a distance matrix an automatic classical multidimensional scaling transformation of distances to data is computed. Number of dimensions is selected by the minimal stress w.r.t. the possible output dimensions of cmdscale.

If FUN has not function name, then ResultPerCluster is given back.

Value

if(Simple==FALSE) List with

UniqueClusters

The unique clusters in Cls
FUNPerCluster

a matrix of [1:k,1:d] of d features and k clusters, the list element is named by the function FUN used

if(Simple==TRUE)

a matrix of [1:k,1:d] of d features and k clusters

Author(s)

Felix Pape, Michael Thrun

Examples

##one dataset
data(Hepta)
Data=Hepta$Data
Cls=Hepta$Cls
#mean per cluster
ClusterApply(Data,mean,Cls)

#Simplified
ClusterApply(Data,mean,Cls,Simple=TRUE)

# Mean per cluster of MDS transformation
# Beware, this is not the same!

ClusterApply(as.matrix(dist(Data)),mean,Cls)


## Not run: 
Iris=datasets::iris
Distances=as.matrix(Iris[,1:4])
SomeFactors=Iris$Species
V=ClusterCreateClassification(SomeFactors)
Cls=V$Cls
V$ClusterNames
ClusterApply(Distances,mean,Cls)

## End(Not run)
#special case of identity
## Not run: 
suppressPackageStartupMessages(library('prabclus',quietly = TRUE))
data(tetragonula)
#Generated Specific Distance Matrix
ta <- alleleconvert(strmatrix=as.matrix(tetragonula[1:236,]))
tai <- alleleinit(allelematrix=ta,distance="none")
Distance=alleledist((unbuild.charmatrix(tai$charmatrix,236,13)),236,13)

MDStrans=ClusterApply(Distance,identity)$identityPerCluster

## End(Not run)

Adjusted Rand index

Description

Adjusted Rand index for two clusterings that should be compared to each other. This index has expected value zero for independant clusterings and maximum value 1 (for identical clusterings).

Usage

ClusterARI(Cls1, Cls2,Fast=TRUE)

Arguments

Cls1

1:n numerical vector of numbers defining the classification as the main output of the first clustering or trial for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

Cls2

1:n numerical vector of numbers defining the classification as the main output of the second clustering algorithm trial for the n cases of data. It has p unique numbers representing the arbitrary labels of the clustering.

Fast

TRUE:uses mclust package which maybe does not integrate all published insights about ARI FALSE: uses partitionComparison package

Details

"The expected value of the Rand Index of two random partitions does not take a constant value (e.g. zero). Thus, Hubert and Arabie proposed an adjustment [Hubert & Arabie] which assumes a generalized hypergeometric distribution as null hypothesis: the two clusterings are drawn randomly with a fixed number of clusters and a fixed number of elements in each cluster (the number of clusters in the two clusterings need not be the same). Then the adjusted Rand Index is the (normalized) difference of the Rand Index and its expected value under the null hypothesis. The significance of this measure has to be put into question because of the strong assumptions it makes on the distribution. Meila [Meila, 2003] notes, that some pairs of clusterings may result in negative index values" [Wagner and Wagner, 2007].

Value

value of adjusted rand index

Note

the equation of adjusted random index ignores the labels themselve and measures only the agreement. Hence, one can compare clusterin solutions for k!=p unique numbers that represent the labels, see second example

Author(s)

Michael Thrun

References

[Rand, 1971] Rand, W. M.: Objective criteria for the evaluation of clustering methods, Journal of the American Statistical Association, Vol. 66(336), pp. 846-850, 1971.

[Hubert & Arabie] Hubert, L. and Arabie, P.: Comparing partitions, Journal of Classification. Vol. 2 (1), pp. 193-218. doi:10.1007/BF01908075, 1985.

[Ball/Geyer-Schulz, 2018] Ball, F., & Geyer-Schulz, A.: Invariant Graph Partition Comparison Measures, Symmetry, Vol. 10(10), pp. 1-27, 2018.

[Meila, 2003] Meila, Marina: Comparing Clusterings. COLT 2003.

[Wagner and Wagner, 2007] Wagner, Silke; Wagner, Dorothea. Comparing clusterings: an overview. Karlsruhe: Universitaet Karlsruhe, Fakultaet für Informatik, 2007.

See Also

adjustedRandIndex

Examples

data(Hepta)
#compare to baseline
Cls2=kmeansClustering(Hepta$Data,7,Type = "Steinley")$Cls
ClusterARI(Hepta$Cls,Cls2)
#compare different solutions
Cls3=kmeansClustering(Hepta$Data,5)$Cls
ClusterARI(Cls3,Cls2)

Generates a Fundamental Clustering Challenge based on specific artificial datasets.

Description

Lsun3D and FCPS datasets were introduced in various publications for a specific fixed size. This function generalizes them for any sample size.

Usage

ClusterChallenge(Name,SampleSize,

PlotIt=FALSE,PointSize=1,Plotter3D="rgl",...)

Arguments

Name

string, either 'Atom', 'Chainlink, 'EngyTime', 'GolfBall', 'Hepta', 'Lsun3D', 'Target' 'Tetra' 'TwoDiamonds' 'WingNut

SampleSize

Size of Sample higher than 300, preferable above 500

PlotIt

TRUE: Plots the challenge with ClusterPlotMDS

PointSize

If PlotIt=TRUE: see ClusterPlotMDS

Plotter3D

If PlotIt=TRUE: see ClusterPlotMDS

...

If PlotIt=TRUE: further arguments for ClusterPlotMDS

Details

A detailed description of the datasets can be found in [Thrun/Ultsch 2020]. Sampling works by combining Pareto Density Estimation with rejection sampling.

Value

LIST, with

Name

[1:SampleSize,1:d] data matrix
Cls

[1:SampleSize] numerical vector of classification

Author(s)

Michael Thrun

References

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. in press, pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

See Also

ClusterPlotMDS

Examples

## Not run: 
ClusterChallenge("Chainlink",2000,PlotIt=TRUE)

## End(Not run)

ClusterCount

Description

Calulates statistics for clustering in each group of the data points

Usage

ClusterCount(Cls,Ordered=TRUE,NonFinite=9999)

Arguments

Cls

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.
Ordered

Optional, boolean, if TRUE: the ouput is ordered increasingly by cluster labels in UniqueClusters.

NonFinite

Optional, If non finite values are given in the numerical vector, they are set to the scalar value defined here

Details

The ordering of the output is defined by the first occurence of every cluster label in Cls in the setting of Ordered=FALSE.

The function can be overloaded with non-numerical vectors. In this case, a cast via as.character() is applied to Cls, a warning is stated, and the statistics are still computed.

Value

UniqueClusters

[1:k] numerical vector of the k unique clusters in Cls
CountPerCluster

Named vector [1:k] with the number of data points in the corresponding unique clusters. Names are the UniqueClusters
NumberOfClusters

The number of clusters k
ClusterPercentages

[1:k] numerical vector of the percentages of datapoints belonging to a cluster for each cluster

Author(s)

Michael Thrun

Examples

data('Hepta')
Cls=Hepta$Cls
ClusterCount(Cls)

Create Classification for Cluster.. functions

Description

Creates a Cls from arbitrary list of objects

Usage

ClusterCreateClassification(Objects,Decreasing)

Arguments

Objects

Listed objects, for example factor

Decreasing

Boolean that can be missing. If given, sorts ClusterNames with either decreasing or increasing.

Details

ClusterNames can be sorted before the classification stored Cls is created. See example.

Value

LIST, with

Cls

[1:n] numerical vector with n numbers defining the labels of the classification. It has 1 to k unique numbers representing the arbitrary labels of the classification.
ClusterNames

ClusterNames defined which names belongs to which unique number

Author(s)

Michael Thrun

Examples

## Not run: 
  Iris=datasets::iris
  SomeFactors=Iris$Species
  V=ClusterCreateClassification(SomeFactors)
  Cls=V$Cls
  V$ClusterNames
  table(Cls,SomeFactors)
  
  #Increasing alphabetical order
  V=ClusterCreateClassification(SomeFactors,Decreasing=FALSE)
  Cls=V$Cls
  V$ClusterNames
  table(Cls,SomeFactors)
  
## End(Not run)

Davies Bouldin Index

Description

Internal (i.e. without prior classification) cluster quality measure called Davies Bouldin index for a given clustering published in [Davies/Bouldin, 1979].

Usage

ClusterDaviesBouldinIndex(Cls, Data,...)

Arguments

Cls

[1:n] numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

Data

matrix, [1:d,1:n] dataset of d variables and n cases

...

Further arguments passed on to the index.DB function of clusterSim

Details

Wrapper for index.DB. Davies Bouldin index is defined in [Davies/Bouldin, 1979]. Best clustering scheme essentially minimizes the Davies-Bouldin index because it is defined as the function of the ratio of the within cluster scatter, to the between cluster separation.[Davies/Bouldin, 1979].

Value

List of

DaviesBouldinIndex

scalar,Davies Bouldin index

Object

further information stored in index.DB

Author(s)

Michael Thrun

References

[Davies/Bouldin, 1979] Davies, D. L., & Bouldin, D. W.: A cluster separation measure, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 1(2), pp. 224-227. doi 10.1109/TPAMI.1979.4766909, 1979.

Examples

data("Hepta")
Cls=kmeansClustering(Hepta$Data,ClusterNo = 7,Type="Hartigan")$Cls
ClusterDaviesBouldinIndex(Cls,Hepta$Data)[1]


data("Hepta")
ClsWellSeperated=kmeansClustering(Hepta$Data,ClusterNo = 7,Type="Steinley")$Cls
ClusterDaviesBouldinIndex(ClsWellSeperated,Hepta$Data)[1]

Cluster Dendrogram

Description

Presents a dendrogram of a given tree using a colorsequence for the branches defined from the highest cluster size to the lowest cluster size.

Usage

ClusterDendrogram(TreeOrDendrogram, ClusterNo, 

Colorsequence,main='Name of Algorithm')

Arguments

TreeOrDendrogram

Either object of hcclust defining the tree, third list element of hierarchical cluster algorithms of this package

or

Object of class dendrogram, second list element of hierarchical cluster algorithms.

ClusterNo

k number of clusters for cutree.

Colorsequence

[1:k] character vector of colors, per default the colorsquence defined in the DataVisualizations is used

main

Title of plot

Details

Reqires the package dendextend to work correctly.

Value

In mode invisible:

[1:n] numerical vector defining the clustering of k clusters; this classification is the main output of the algorithm.

Author(s)

Michael Thrun

See Also

cutree, hclust

Examples

data(Lsun3D)
listofh=HierarchicalClustering(Lsun3D$Data,0,'SingleL')
Tree=listofh$Object
#given colors are per default:
#"magenta" "yellow"  "black"   "red"
ClusterDendrogram(Tree, 4,main='Single Linkage Clustering')

listofh=HierarchicalClustering(Lsun3D$Data,4)
ClusterCount(listofh$Cls)
#c1 is magenta, c2 is red, c3 is yellow, c4 is black
#because the order of the cluster sizes is
#c1,c3,c4,c2

ClusterDistances

Description

Computes intra-cluster distances which are the distance in-between each cluster.

Usage

ClusterDistances(FullDistanceMatrix, Cls,

Names, PlotIt = FALSE)

ClusterIntraDistances(FullDistanceMatrix, Cls,

Names, PlotIt = FALSE)

Arguments

FullDistanceMatrix

[1:n,1:n] symmetric distance matrix

Cls

[1:n] numerical vector of k classes

Names

Optional [1:k] character vector naming k classes

PlotIt

Optional, Plots if TRUE

Details

Cluster distances are given back as a matrix, one column per cluster and the vector of the full distance matrix without the diagonal elements and the upper half of the symmetric matrix. Details and definitons can be found in [Thrun, 2021].

Value

Matrix [1:m,1:(k+1)] of k clusters, each columns consists of the distances in a cluster, filled up with NaN at the end to be of the same length as the vector of the upper triangle of the complete distance matrix.

Author(s)

Michael Thrun

References

[Thrun, 2021] Thrun, M. C.: The Exploitation of Distance Distributions for Clustering, International Journal of Computational Intelligence and Applications, Vol. 20(3), pp. 2150016, DOI: doi:10.1142/S1469026821500164, 2021.

See Also

MDplot

ClusterInterDistances

Examples

data(Hepta)
Distance=as.matrix(dist(Hepta$Data))

interdists=ClusterDistances(Distance,Hepta$Cls)

Dunn Index

Description

Internal (i.e. without prior classification) cluster quality measure called Dunn index for a given clustering published in [Dunn, 1974].

Usage

ClusterDunnIndex(Cls,DataOrDistances,

DistanceMethod="euclidean",Silent=TRUE,Force=FALSE,...)

Arguments

Cls

[1:n] numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

DataOrDistances

matrix, DataOrDistance[1:n,1:n] symmetric matrix of dissimilarities, if variable unsymmetric DataOrDistance[1:d,1:n] is assumed as a dataset and the euclidean distances are calculated of d variables and n cases

DistanceMethod

Optional, one of 39 distance methods of parDist of package parallelDist, if Data matrix is chosen above

Silent

TRUE: Warnings are shown

Force

TRUE: force computing in case of numerical instability

...

Further arguments passed on to the parDist function, e.g. user_defined distance functions

Details

Dunn index is defined as Dunn=min(InterDist)/max(IntraDist). Well seperated clusters have usually a dunn index above 1, for details please see [Dunn, 1974].

Value

List of

Dunn

scalar, Dunn Index

IntraDist

[1:k] numerical vector of minimal intra cluster distances per given cluster

InterDist

[1:k] numerical vector of minimal inter cluster distances per given cluster

Author(s)

Michael Thrun

References

[Dunn, 1974] Dunn, J. C.: Well_separated clusters and optimal fuzzy partitions, Journal of cybernetics, Vol. 4(1), pp. 95-104. 1974.

Examples

data("Hepta")
Cls=kmeansClustering(Hepta$Data,ClusterNo = 7,Type="Hartigan")$Cls
ClusterDunnIndex(Cls,Hepta$Data)


data("Hepta")
ClsWellSeperated=kmeansClustering(Hepta$Data,ClusterNo = 7,Type="Steinley")$Cls
ClusterDunnIndex(ClsWellSeperated,Hepta$Data)

ClusterEqualWeighting

Description

Weights clusters equally

Usage

ClusterEqualWeighting(Cls, Data, MinClusterSize)

Arguments

Cls

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

Data

Optional, [1:n,1:d] matrix of dataset consisting of n cases of d-dimensional data points. Every case has d attributes, variables or features.

MinClusterSize

Optional, scalar defining the number of cases m that each cluster should have

Details

Balance clusters such that their sizes are the same by subsampling the larger cluster. If MinClusterSize is missing the number of cases per cluster is set to the smallest cluster size. For clusters sizes smaller than MinClusterSize, sampling with replacement is turned on, i.e. up sampling. For clusters sizes equal to MinClusterSize, no sampling is performed.

Value

List of

BalancedCls

Vector of Cls such that all clusters have the same sizes spezified by MinClusterSize
BalancedInd

index such that BalancedCls = Cls[BalancedInd]
BalancedData

NULL if missing, otherwise, Data[BalancedInd,]

Author(s)

Alfred Ultsch (matlab), reimplemented by Michael Thrun

Examples

data(Hepta)
ClusterEqualWeighting(Hepta$Cls,Hepta$Data,5)

ClusterAccuracy

Description

ClusterAccuracy

Usage

ClusterAccuracy(PriorCls,CurrentCls,K=9)

Arguments

PriorCls

Ground truth,[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the clustering.

CurrentCls

Main output of the clustering, [1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the clustering.

K

Maximal number of classes for computation.

Details

Here, accuracy is defined as the normalized sum over all true positive labeled data points of a clustering algorithm. The best of all permutation of labels with the highest accuracy is selected in every trial because algorithms arbitrarily define the labels [Thrun et al., 2018]. Beware that in contrast to ClusterMCC, the labels can be arbitrary. However, accuracy is a only a valid quality measure if the clusters are balanced (of) nearly equal size). Ohterwise please use ClusterMCC.

In contrast to the F-measure, "Accuracy tends to be naturally unbiased, because it can be expressed in terms of a binomial distribution: A success in the underlying Bernoulli trial would be defined as sampling an example for which a classifier under consideration makes the right prediction. By definition, the success probability is identical to the accuracy of the classifier. The i.i.d. assumption implies that each example of the test set is sampled independently, so the expected fraction of correctly classified samples is identical to the probability of seeing a success above. Averaging over multiple folds is identical to increasing the number of repetitions of the Binomial trial. This does not affect the posterior distribution of accuracy if the test sets are of equal size, or if we weight each estimate by the size of each test set." [Forman/Scholz, 2010]

Value

Single scalar of Accuracy between zero and one

Author(s)

Michael Thrun

References

[Thrun et al., 2018] Michael C. Thrun, Felix Pape, Alfred Ultsch: Benchmarking Cluster Analysis Methods in the Case of Distance and Density-based Structures Defined by a Prior Classification Using PDE-Optimized Violin Plots, ECDA, Potsdam, 2018

[Forman/Scholz, 2010] Forman, G., and Scholz, M.: Apples-to-apples in cross-validation studies: pitfalls in classifier performance measurement, ACM SIGKDD Explorations Newsletter, Vol. 12(1), pp. 49-57. 2010.

See Also

ClusterMCC

Examples

#Influence of random sets/ random starts on k-means

data('Hepta')
Cls=kmeansClustering(Hepta$Data,7,Type = "Hartigan",nstart=1)
table(Cls$Cls,Hepta$Cls)
ClusterAccuracy(Hepta$Cls,Cls$Cls)



data('Hepta')
Cls=kmeansClustering(Hepta$Data,7,Type = "Hartigan",nstart=100)
table(Cls$Cls,Hepta$Cls)
ClusterAccuracy(Hepta$Cls,Cls$Cls)

Computes Inter-Cluster Distances

Description

Computes inter-cluster distances which are the distance between each cluster and all other clusters

Usage

ClusterInterDistances(FullDistanceMatrix, Cls,

Names,PlotIt=FALSE)

Arguments

FullDistanceMatrix

[1:n,1:n] symmetric distance matrix

Cls

[1:n] numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

Names

Optional [1:k] character vector naming k classes

PlotIt

Optional, Plots if TRUE

Details

Cluster distances are given back as a matrix, one column per cluster and the vector of the full distance matrix without the diagonal elements and the upper half of the symmetric matrix. Details and definitons can be found in [Thrun, 2021].

Value

Matrix [1:m,1:(k+1)] of k clusters, each columns consists of the distances between a cluster and all other clusters, filled up with NaN at the end to be of the same lenght as the vector of the upper triangle of the complete distance matrix.

Author(s)

Michael Thrun

References

[Thrun, 2021] Thrun, M. C.: The Exploitation of Distance Distributions for Clustering, International Journal of Computational Intelligence and Applications, Vol. 20(3), pp. 2150016, DOI: doi:10.1142/S1469026821500164, 2021.

See Also

MDplot

ClusterDistances

Examples

data(Hepta)
Distance=as.matrix(dist(Hepta$Data))

interdists=ClusterInterDistances(Distance,Hepta$Cls)

Matthews Correlation Coefficient (MCC)

Description

Matthews correlation coefficient eneralized to the multiclass case (a.k.a. R_K statistic).

Usage

ClusterMCC(PriorCls, CurrentCls,Force=TRUE)

Arguments

PriorCls

Ground truth,[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the labels of the clustering.

CurrentCls

Main output of the clustering, [1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the labels of the clustering.

Force

Boolean, if is TRUE: forces code even if one or more than one of the k numbers given in PriorCls is missing in CurrentCls or vice versa. In this case, one label per missing number is added ad the end of the vectors.

Details

Contrary to accuracy, the MCC is balanced measure which can be used even if the classes are of very different sizes. When there are more than two labels the MCC will no longer range between -1 and +1. Instead the minimum value will be between -1 and 0 depending on the true distribution. The maximum value is always +1. Beware that in contrast to ClusterAccuracy, the labels cannot be arbitrary. Instead each label of PriorCls and CurrentCls has to be mapped to the same cluster of data points. Typically this has to be ensured manually.

Value

Single scalar of MCC in a range described in details.

Note

If No. of Clusters is not equivalent, internally the number is allgined with zero datapoints belonging to the missing clusters.

Author(s)

Michael Thrun

References

Matthews, B. W.: Comparison of the predicted and observed secondary structure of T4 phage lysozyme, Biochimica et Biophysica Acta (BBA), Protein Structure, Vol. 405(2), pp. 442-451, 1975.

Boughorbel, S.B: Optimal classifier for imbalanced data using Matthews Correlation Coefficient metric, PLOS ONE, Vol. 12(6), pp. e0177678, 2017.

Chicco, D.; Toetsch, N. and Jurman, G.: The Matthews correlation coefficient (MCC) is more reliable than balanced accuracy, bookmaker informedness, and markedness in two_class confusion matrix evaluation. BioData Mining. Vol. 14., 2021.

See Also

ClusterAccuracy

Examples

#Beware that algorithm arbitrary defines the labels
data(Hepta)
V=kmeansClustering(Hepta$Data,Type = "Hartigan",7)
table(V$Cls,Hepta$Cls)
#result is only valid if the above issue is resolved manually
ClusterMCC(Hepta$Cls,V$Cls)

Estimates Number of Clusters using up to 26 Indicators

Description

Calculation of up to 26 indicators and the recommendations based on them for the number of clusters in data sets. For a given dataset and clusterings for this dataset, key indicators mentioned in details are calculated and based on this a recommendation regarding the number of clusters is given for each indicator.

An alternative estimation of the cluster number can be done by counting the valleys of the topographic map of the generalized U-Matrix for a specfic projection method using the ProjectionBasesdClustering and GeneralizedUmatrix packages on CRAN, see [Thrun/Ultsch, 2021] for details.

Usage

ClusterNoEstimation(DataOrDistances, ClsMatrix = NULL, MaxClusterNo, 

ClusterIndex = "all", Method = NULL, MinClusterNo = 2,

Silent = TRUE,PlotIt=FALSE,SelectByABC=TRUE,Colorsequence,...)

Arguments

DataOrDistances

Either [1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.

or

Symmetric [1:n,1:n] distance matrix

ClsMatrix

[1:n,1:(MaxClusterNo)] matrix of clusterings each columns is defined as:

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

(see also details (2) and (3)), must be specified if method = NULL

MaxClusterNo

Highest number of clusters to be checked

Method

Cluster procedure, with which the clusterings are created (see details (4) for possible methods), must be specified if ClsMatrix = NULL

Optional:

ClusterIndex

String or vector of strings with the indicators to be calculated (see details (1)), default = "all

MinClusterNo

Lowest number of clusters to be checked, default = 2

Silent

If TRUE status messages are output, default = FALSE

PlotIt

If TRUE plots fanplot with proposed cluster numbers

SelectByABC

If PlotIt=TRUE, TRUE: Plots group A of ABCanalysis of the most important ones (highest overlap in indicators), FALSE: plots all indicators

Colorsequence

Optional, character vector of sufficient length of colors for the fan plot.If the sequence is too long the first part of the sequence is used.

...

Optional, further arguents used if clustering methods if Method is set.

Details

Each column of ClsMatrix has to have at least two unqiue clusters defined. Otherwise the function will stop.

(1)

The following 26 indicators can be calculated: "ball", "beale", "calinski", "ccc", "cindex", "db", "duda", "dunn", "frey", "friedman", "hartigan", "kl", "marriot", "mcclain", "pseudot2", "ptbiserial", "ratkowsky", "rubin", "scott", "sdbw", "sdindex", "silhouette", "ssi", "tracew", "trcovw", "xuindex".

These can be specified individually or as a vector via the parameter index. If you enter 'all', all key figures are calculated.

(2)

The indicators kl, duda, pseudot2, beale, frey and mcclain require a clustering for MaxClusterNo+1 clusters. If these key figures are to be calculated, this clustering must be specified in cls.

(3)

The indicator kl requires a clustering for MinClusterNo-1 clusters. If this key figure is to be calculated, this clustering must also be specified in cls. For the case MinClusterNo = 2 no clustering for 1 has to be given.

(4)

The following methods can be used to create clusterings:

"kmeans," "DBSclustering","DivisiveAnalysisClustering","FannyClustering", "ModelBasedClustering","SpectralClustering" or all methods found in HierarchicalClustering.

(5)

The indicators duda, pseudot2, beale and frey are only intended for use in hierarchical cluster procedures.

If a distances matrix is given, then ProjectionBasedClustering is required to be accessible.

Value

Indicators

A table of the calculated indicators except Duda, Pseudot2 and Beale

ClusterNo

The recommended number of clusters for each calculated indicator

ClsMatrix

[1:n,MinClusterNo:(MaxClusterNo)] Output of the clusterings used for the calculation

HierarchicalIndicators

Either NULL or the values for the indicators Duda, Pseudot2 and Beale in case of hierarchical cluster procedures, if calculated

Note

Code of "calinski", "cindex", "db", "hartigan", "ratkowsky", "scott", "marriot", "ball", "trcovw", "tracew", "friedman", "rubin", "ssi" of package cclust ist adapted for the purpose of this function.

Colorsequence works if DataVisualizations 1.1.13 is installed (currently only on github available).

Author(s)

Peter Nahrgang, revised by Michael Thrun (2021)

References

Charrad, Malika, et al. "Package 'NbClust', J. Stat. Soft Vol. 61, pp. 1-36, 2014.

Dimtriadou, E. "cclust: Convex Clustering Methods and Clustering Indexes." R package version 0.6-16, URL https://CRAN.R-project.org/package=cclust, 2009.

[Thrun/Ultsch, 2021] Thrun, M. C., and Ultsch, A.: Swarm Intelligence for Self-Organized Clustering, Artificial Intelligence, Vol. 290, pp. 103237, doi:10.1016/j.artint.2020.103237, 2021.

Examples

# Reading the iris dataset from the standard R-Package datasets
data <- as.matrix(iris[,1:4])
MaxClusterNo = 7
# Creating the clusterings for the data set
#(here with method complete) for the number of clusters 2 to 8
hc <- hclust(dist(data), method = "complete")
clsm <- matrix(data = 0, nrow = dim(data)[1],

ncol = MaxClusterNo)
for (i in 2:(MaxClusterNo+1)) {
  clsm[,i-1] <- cutree(hc,i)
}

# Calculation of all indicators and recommendations for the number of clusters
indicatorsList=ClusterNoEstimation(Data = data, 

ClsMatrix = clsm, MaxClusterNo = MaxClusterNo)

# Alternatively, the same calculation as above can be executed with the following call
ClusterNoEstimation(Data = data, MaxClusterNo = 7, Method = "CompleteL")
# In this variant, the function clusterumbers also takes over the clustering

Cluster Normalize

Description

Values in Cls are consistently recoded to positive consecutive integers

Usage

ClusterNormalize(Cls)

Arguments

Cls

[1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

Details

For recoding depending on cluster size please see ClusterRenameDescendingSize.

Value

The renamed classification. A vector of clusters recoded to positive consecutive integers.

Author(s)

.

See Also

ClusterRenameDescendingSize

Examples

data('Lsun3D')
Cls=Lsun3D$Cls
#not desceending cluster numbers
Cls[Cls==1]=543
Cls[Cls==4]=1

# Now ordered consecutively
ClusterNormalize(Cls)

Plot Clustering using Dimensionality Reduction by MDS

Description

This function uses a projection method to perform dimensionality reduction (DR) on order to visualize the data as 3D data points colored by a clustering.

Usage

ClusterPlotMDS(DataOrDistances, Cls, main = "Clustering",

DistanceMethod = "euclidean", OutputDimension = 3,

PointSize=1,Plotter3D="rgl",Colorsequence, ...)

Arguments

DataOrDistances

Either nonsymmetric [1:n,1:d] datamatrix of n cases and d features or symmetric [1:n,1:n] distance matrix

Cls

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

main

String, title of plot

DistanceMethod

Method to compute distances, default "euclidean"

OutputDimension

Either two or three depending on user choice

PointSize

Scalar defining the size of points

Plotter3D

In case of 3 dimensions, choose either "plotly" or "rgl",

Colorsequence

[1:k] character vector of colors, per default the colorsquence defined in the DataVisualizations is used

...

Please see Plot3D in DataVisualizations

Details

If dataset has more than 3 dimesions, mds is performed as defined in the smacof [De Leeuw/Mair, 2011]. If smacof package is not installed, classical metric MDS (see Def. in [Thrun, 2018]) is performed. In both cases, the first OutputDimension are visualized. Points are colored by the labels (Cls).

In the special case that the dataset has not more than 3 dimensions, all dimensions are visualized and no DR is performed.

Value

The rgl or plotly plot handler depending on Plotter3D

Note

If DataVisualizations is not installed a 2D plot using native plot function is shown.

If MASS is not installed, classicial metric MDS is used, see [Thrun, 2018] for definition.

Author(s)

Michael Thrun

References

[De Leeuw/Mair, 2011] De Leeuw, J., & Mair, P.: Multidimensional scaling using majorization: SMACOF in R, Journal of statistical Software, Vol. 31(3), pp. 1-30. 2011.

[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, ISBN: 978-3-658-20539-3, Heidelberg, 2018.

See Also

Plot3D

Examples

data(Hepta)
ClusterPlotMDS(Hepta$Data,Hepta$Cls)

data(Leukemia)
ClusterPlotMDS(Leukemia$DistanceMatrix,Leukemia$Cls)

Redfines Clustering

Description

Redfines some or all Clusters of Clustering such that the names of the numerical vectors are defined by

Usage

ClusterRedefine(Cls, NewLabels,OldLabels)

Arguments

Cls

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

NewLabels

[1:p], p<=k labels (identifiers) of clusters to be changed with

OldLabels

Optional, [1:p], p<=k labels(identifiers) of clusters to be changed, default [1:k] unique cluster Ids of Cls

Details

The same ordering of NewLabels and OldLabels is assumend, i.e., the mapping is defined by OldLabels[i] -> NewLabels[i] with i in [1:p]. NewLabels can also be a vector for strings, for example for plotting.

Value

Cls[1:n] numerical vector named after the row names of data

Author(s)

Michael Thrun

Examples

data('Lsun3D')
Cls=Lsun3D$Cls
Data=Lsun3D$Data#
#prior
ClsNew=unique(Cls)+10
#Redfined Clustering
NewCls=ClusterRedefine(Cls,ClsNew)

table(Cls,NewCls)

#require(DataVisualizations)
n=length(unique(Cls))
NewCls=ClusterRedefine(Cls,LETTERS[1:n])
#DataVisualizations package required
if(requireNamespace("DataVisualizations"))
  DataVisualizations::Classplot(Data[,1],Data[,2],
  Cls,Names=NewCls,Plotter="ggplot",Size =1.5)

Renames Clustering

Description

Renames Clustering such that the names of the numerical vectors are the row names of DataOrDistances

Usage

ClusterRename(Cls, DataOrDistances)

Arguments

Cls

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

DataOrDistances

Either nonsymmetric [1:n,1:d] datamatrix of n cases and d features or symmetric [1:n,1:n] distance matrix

Details

If DataOrDistances is missing or if inconsistent length, nothing is done.

Value

Cls[1:n] numerical vector named after the row names of data

Author(s)

Michael Thrun

Examples

data('Hepta')
Cls=Hepta$Cls
Data=Hepta$Data#
#prior
Cls
#Named Clustering
ClusterRename(Cls,Data)

Cluster Rename Descending Size

Description

Renames the clusters of a classification in descending order.

Usage

ClusterRenameDescendingSize(Cls,

ProvideClusterNames=FALSE)

Arguments

Cls

[1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.
ProvideClusterNames

TRUE: Provides in seperate output new and old k numbers, FALSE: simple output

Details

Beware: output changes in this function depending on ProvideClusterNames in order to be congruent to prior code in a large varierity of other packages.

Value

ProvideClusterNames==FALSE:

RenamedCls

The renamed classification. A vector of clusters, were the largest cluster is C1 and so forth

ProvideClusterNames==TRUE: List V with

RenamedCls

The renamed classification. A vector of clusters, were the largest cluster is C1 and so forth
ClusterName

[1:k,1:2] matrix of k new numbers and prior numbers

Author(s)

Michael Thrun, Alfred Ultsch

See Also

ClusterNormalize

Examples

data('Lsun3D')
Cls=Lsun3D$Cls
#not desceending cluster numbers
Cls[Cls==1]=543
Cls[Cls==4]=1

# Now ordered per cluster size and descending
ClusterRenameDescendingSize(Cls)

Shannon Information

Description

Shannon Information [Shannon, 1948] for each column in ClsMatrix.

Usage

ClusterShannonInfo(ClsMatrix)

Arguments

ClsMatrix

[1:n,1:C] matrix of C clusterings each columns is defined as:

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

Details

Info[1:d] = sum(-p * log(p)/MaxInfo) for all unique cases with probability p in ClsMatrix[,c] for a column with k clusters MaxInfo = -(1/k)*log(1/k)

Value

Info

[1:max.nc,1:C] matrix of Shannin informaton as defined in details, each column represents one Cls of ClsMatrix,each row yields the information of one cluster up the ClusterNo k, if k<max.nc (highest number of clusters) then NaN are filled.

ClusterNo

Number of Clusters k found for each Cls respectively

MaxInfo

max per column of Info

MinInfo

min per column of Info

MedianInfo

median per column of Info

MeanInfo

mean per column of Info

Note

reeimplemented from Alfred's Ultsch Matlab version but not verified yet.

Author(s)

Michael Thrun

References

[Shannon, 1948] Shannon, C. E.: A Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27(3), pp. 379-423. doi doi:10.1002/j.1538-7305.1948.tb01338.x, 1948.

Examples

# Reading the iris dataset from the standard R-Package datasets
data <- as.matrix(iris[,1:4])
max.nc = 7
# Creating the clusterings for the data set
#(here with method complete) for the number of classes 2 to 8
hc <- hclust(dist(data), method = "complete")
clsm <- matrix(data = 0, nrow = dim(data)[1],

ncol = max.nc)
for (i in 2:(max.nc+1)) {
  clsm[,i-1] <- cutree(hc,i)
}

ClusterShannonInfo(clsm)

Cluster Up Sampling using SMOTE for minority cluster

Description

Wrapper for one specific internal function of L. Torgo who implemented there the relevant part of the SMOTE algorithm [Chawla et al., 2002].

Usage

ClusterUpsamplingMinority(Cls, Data, MinorityCluster,

Percentage = 200, knn = 5, PlotIt = FALSE)

Arguments

Cls

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.

Data

[1:n,1:d] datamatrix of n cases and d features

MinorityCluster

scalar defining the number of the cluster to be upsampeled

Percentage

pecentage above 100 of who many samples should be taken

knn

k nearest neighbors of SMOTE algorithm
PlotIt

TRUE: plots the result using ClusterPlotMDS

Details

the number of items m is defined by the scalar Percentage and the up sampling is combined with the Data and the Cls to DataExt and ClsExt such that the sample is placed thereafter.

Value

List with

ClsExt

1:(n+m) numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.
DataExt

[1:(n+m),1:d] datamatrix of n cases and d features

.

Author(s)

L. Torgo

References

[Chawla et al., 2002] Chawla, N. V., Bowyer, K. W., Hall, L. O., & Kegelmeyer, W. P.: SMOTE: synthetic minority over-sampling technique, Journal of artificial intelligence research, Vol. 16, pp. 321-357. 2002.

Examples

data(Lsun3D)
Data=Lsun3D$Data
Cls=Lsun3D$Cls
table(Cls)

V=ClusterUpsamplingMinority(Cls,Data,4,1000)
table(V$ClsExt)

Cross-Entropy Clustering

Description

Cross-entropy clustering published by [Tabor/Spurek, 2014] and implemented by [Spurek et al., 2017].

Usage

CrossEntropyClustering(Data, ClusterNo,PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

Contrary to most of the other implemented algorithms in this package, the results on the easiest clustering challenge of Hepta are unstable for cross-entropy clustering in the sense that the clustering is not always correct. Reproducibilty experiments should be performed (see [Tabor/Spurek, 2014]).

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Spurek et al., 2017] Spurek, P., Kamieniecki, K., Tabor, J., Misztal, K., & Śmieja, M.: R package cec, Neurocomputing, Vol. 237, pp. 410-413. 2017.

[Tabor/Spurek, 2014] Tabor, J., & Spurek, P.: Cross-entropy clustering, Pattern Recognition, Vol. 47(9), pp. 3046-3059. 2014.

Examples

data('Hepta')
out=CrossEntropyClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

DBSCAN

Description

Density-Based Spatial Clustering of Applications with Noise of [Ester et al., 1996].

Usage

DBSCAN(Data,Radius,minPts,Rcpp=TRUE,

PlotIt=FALSE,UpperLimitRadius,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
Radius

Eps [Ester et al., 1996, p. 227] neighborhood in the R-ball graph/unit disk graph), size of the epsilon neighborhood. If NULL, automatic estimation is performed using insights of [Ultsch, 2005].

minPts

Number of minimum points in the eps region (for core points). In principle minimum number of points in the unit disk, if the unit disk is within the cluster (core) [Ester et al., 1996, p. 228]. If NULL, 2.5 percent of points is selected.

Rcpp

If TRUE: fast Rcpp implementation of mlpack is used. FALSE uses dbscan package.

PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
UpperLimitRadius

Limit for radius search, experimental
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Value

List of

Cls

[1:n] numerical vector defining the clustering; this classification is the main output of the algorithm. Points which cannot be assigned to a cluster will be reported as members of the noise cluster with 0.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Ester et al., 1996] Ester, M., Kriegel, H.-P., Sander, J., & Xu, X.: A density-based algorithm for discovering clusters in large spatial databases with noise, Proc. Kdd, Vol. 96, pp. 226-231, 1996.

[Ultsch, 2005] Ultsch, A.: Pareto density estimation: A density estimation for knowledge discovery, In Baier, D. & Werrnecke, K. D. (Eds.), Innovations in classification, data science, and information systems, (Vol. 27, pp. 91-100), Berlin, Germany, Springer, 2005.

Examples

data('Hepta')

out=DBSCAN(Hepta$Data,Radius=NULL,minPts=NULL,PlotIt=FALSE)

## Not run: 
#search for right parameter setting by grid search
data("WingNut")
Data = WingNut$Data
DBSGrid <- expand.grid(
  Radius = seq(from = 0.01, to = 0.3, by = 0.02),
  minPTs = seq(from = 1, to = 50, by = 2)
)
BestAcc = c()
for (i in seq_len(nrow(DBSGrid))) {
  parameters <- DBSGrid[i,]
  Cls9 = DBSCAN(
    Data,
    minPts = parameters$minPTs,
    
    Radius = parameters$Radius,
    PlotIt = F,
    
    UpperLimitRadius = parameters$Radius
  )$Cls
  if (length(unique(Cls9)) < 5)
    BestAcc[i] = ClusterAccuracy(WingNut$Cls,
                                    
                                    Cls9) * 100
  else
    BestAcc[i] = 50
}
max(BestAcc)
which.max(BestAcc)
parameters <- DBSGrid[13,]

Cls9 = DBSCAN(
  Data,
  minPts = parameters$minPTs,
  Radius = parameters$Radius,
  UpperLimitRadius = parameters$Radius, 
  PlotIt = TRUE
)$Cls

## End(Not run)

Databionic Swarm (DBS) Clustering and Visualization

Description

Swarm-based clustering by exploting self-organization, emergence, swarm intelligence and game theory published in [Thrun/Ultsch, 2021].

Usage

DatabionicSwarmClustering(DataOrDistances, ClusterNo = 0,

StructureType = TRUE, DistancesMethod = NULL,

PlotTree = FALSE, PlotMap = FALSE,PlotIt=FALSE, 

Parallel = FALSE)

Arguments

DataOrDistances

Either nonsymmetric [1:n,1:d] numerical matrix of a dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.

or

symmetric [1:n,1:n] distance matrix, e.g. as.matrix(dist(Data,method))

ClusterNo

Number of Clusters, if zero a the topographic map is ploted. Number of valleys equals number of clusters.

StructureType

Either TRUE or FALSE, has to be tested against the visualization. If colored points of clusters a divided by mountain ranges, parameter is incorrect.

DistancesMethod

Optional, if data matrix given, annon Euclidean distance can be selected

PlotTree

Optional, if TRUE: dendrogram is plotted.

PlotMap

Optional, if TRUE: topographic map is plotted if GeneralizedUmatrix is installed. See details.

PlotIt

Default: FALSE, If TRUE and dataset of [1:n,1:d] dimensions then a plot of the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls will be generated.
Parallel

FALSE: default implementatiomn, TRUE faster Cpp parallel implementation, for this the subsequent packages have to be installed from github, as they are not available on CRAN yet.

Details

This function does not enable the user first to project the data and then to test the Boolean parameter defining the type of structure contrary to the DatabionicSwarm which is an inappropriate approach in case of exploratory data analysis.

Instead, this function is implemented for the purpose of automatic benchmarking because in such a case nobody will investigate many trials with one visualization per trial.

If one would like to perform a clustering exploratively (in the sense that a prior clustering is not given for evaluation purposes), then please use the DatabionicSwarm package directly and read the vignette there. Databionic swarm is like k-means a stochastic algorithm meaning that the clustering and visualization may change between trials.

If PlotMap==TRUE and ClusterNo=0 a topview of the topographic map is shown, in which the points are not labeled, i.e. colored by the same color. If PlotMap==TRUE and ClusterNo>0, then the points are colored by their cluster labels. If you would like to look an 3D topogrpahic map that can be interactively rotated or use 3D printing of the high-dimensional structures [Thrun et al., 2016], please see plotTopographicMap for further details.

Value

List of

Cls

1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.
Object

List of further output of DBS

Note

Current implementation is not efficient enough to cluster more than N=4000 cases as in that case it takes longer than a day for a result.

Author(s)

Michael Thrun

References

[Thrun/Ultsch, 2021] Thrun, M. C., and Ultsch, A.: Swarm Intelligence for Self-Organized Clustering, Artificial Intelligence, Vol. 290, pp. 103237, doi:10.1016/j.artint.2020.103237, 2021.

[Thrun/Ultsch, 2021] Thrun, M. C., & Ultsch, A.: Swarm Intelligence for Self-Organized Clustering (Extended Abstract), in Bessiere, C. (Ed.), 29th International Joint Conference on Artificial Intelligence (IJCAI), Vol. IJCAI-20, pp. 5125–5129, doi:10.24963/ijcai.2020/720, Yokohama, Japan, Jan., 2021.

[Thrun et al., 2016] Thrun, M. C., Lerch, F., Lötsch, J., & Ultsch, A. : Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG), Vol. 24, Plzen, 2016.

See Also

Pswarm, DBSclustering,GeneratePswarmVisualization

Examples

# Generate random but small non-structured data set
data = cbind(
  sample(1:100, 300, replace = TRUE),
  sample(1:100, 300, replace = TRUE),
  sample(1:100, 300, replace = TRUE)
)
# Make sure there are no structures
# (sample size is small and still could generate structures randomly)
if(requireNamespace('DataVisualizations',quietly = TRUE)){
Data = DataVisualizations::RobustNormalization(data, Centered = TRUE)
#DataVisualizations::Plot3D(Data)

# No structres are visible
# Topographic map looks like "egg carton"
# with every point in its own valley
ClsV = DatabionicSwarmClustering(Data, 0, PlotMap = TRUE)
}else{
# only for testing purposes of CRAN!
# in case CRAN tests with no suggest packages available
# please use alpways some kind of standardization!
ClsV = DatabionicSwarmClustering(data, 0, PlotMap = TRUE)
}


# Distance based cluster structures
# 7 valleys are visible, thus ClusterNo=7

data(Hepta)
#DataVisualizations::Plot3D(Hepta$Data)

ClsV = DatabionicSwarmClustering(Hepta$Data, 0, PlotMap = TRUE)


#entagled, complex, and non-linear seperable structures 
## Not run: 
#takes too long for CRAN tests
data(Chainlink)
#DataVisualizations::Plot3D(Chainlink$Data)

# 2 valleys are visible, thus ClusterNo=2
ClsV = DatabionicSwarmClustering(Chainlink$Data, 0, PlotMap = TRUE)

# Experiment with parameter StructureType only
# reveals that clustering is appropriate
# if StructureType=FALSE
ClsV2 = DatabionicSwarmClustering(Chainlink$Data,
                                2,
                                StructureType = FALSE,
                                PlotMap = TRUE)

# Here clusters (colored points)
# are not seperated by valleys
ClsV = DatabionicSwarmClustering(Chainlink$Data,
                                2,
                                StructureType = TRUE,
                                PlotMap = TRUE)

## End(Not run)

Density Peak Clustering algorithm using the Decision Graph

Description

Density peaks clustering of [Rodriguez/Laio, 2014] is here implemented by [Pedersen et al., 2017] with estimation of [Wang et al, 2015] meaning its non adaptive in the sense of ADPclustering.

Usage

DensityPeakClustering(DataOrDistances, Rho,Delta,Dc,Knn=7, 

DistanceMethod = "euclidean", PlotIt = FALSE, Data, ...)

Arguments

DataOrDistances

Either [1:n,1:n] symmetric distance matrix or [1:n,1:d] non symmetric data matrix of n cases and d variables

Rho

Local density of a point, see [Rodriguez/Laio, 2014] for explanation

Delta

Minimum distance between a point and any other point, see [Rodriguez/Laio, 2014] for explanation

Dc

Optional, cutoff distance, will either be estimated by [Pedersen et al., 2017] or [Wang et al, 2015] (see example below)

Knn

Optional k nearest neighbors

DistanceMethod

Optional distance method of data, default is euclid, see parDist for details

PlotIt

Optional TRUE: Plots 2d or 3d result with clustering

Data

[1:n,1:d] data matrix in the case that DataOrDistances is missing and partial matching does not work.
...

Optional, further arguments for densityClust

Details

The densityClust algorithm does not decide the k number of clusters, this has to be done by the parameter setting. This is contrary to the other version of the algorithm from another package which can be called with ADPclustering.

The plot shows the density peaks (Cluster centers). Set Rho and Delta as boundaries below the number of relevant cluster centers for your problem. (see example below).

Value

If Rho and Delta are set:

list of

Cls

[1:n numerical vector of numbers defining the classification as the main output of the clustering algorithm for the n cases of data. It has k unique numbers representing the arbitrary labels of the clustering.
Object

output of [Pedersen et al., 2017] algorithm

If Rho and Delta are missing:

p

object of plot_ly for the decision graph is returned

Author(s)

Michael Thrun

References

[Wang et al., 2015] Wang, S., Wang, D., Li, C., & Li, Y.: Comment on" Clustering by fast search and find of density peaks", arXiv preprint arXiv:1501.04267, 2015.

[Pedersen et al., 2017] Thomas Lin Pedersen, Sean Hughes and Xiaojie Qiu: densityClust: Clustering by Fast Search and Find of Density Peaks. R package version 0.3. https://CRAN.R-project.org/package=densityClust, 2017.

[Rodriguez/Laio, 2014] Rodriguez, A., & Laio, A.: Clustering by fast search and find of density peaks, Science, Vol. 344(6191), pp. 1492-1496. 2014.

See Also

ADPclustering

densityClust

Examples

data(Hepta)
H=EntropyOfDataField(Hepta$Data,  seq(from=0,to=1.5,by=0.05),PlotIt=FALSE)
Sigmamin=names(H)[which.min(H)]
Dc=3/sqrt(2)*as.numeric(names(H)[which.min(H)])
# Look at the plot and estimate rho and delta

DensityPeakClustering(Hepta$Data, Knn = 7,Dc=Dc)
Cls=DensityPeakClustering(Hepta$Data,Dc=Dc,Rho = 0.028,

Delta = 22,Knn = 7,PlotIt = TRUE)$Cls

Large DivisiveAnalysisClustering Clustering

Description

Divisive Analysis Clustering (diana) of [Rousseeuw/Kaufman, 1990, pp. 253-279]

Usage

DivisiveAnalysisClustering(DataOrDistances, ClusterNo,

PlotIt=FALSE,Standardization=TRUE,PlotTree=FALSE,Data,...)

Arguments

DataOrDistances

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features. Alternatively, symmetric [1:n,1:n] distance matrix
ClusterNo

A number k which defines k different clusters to be build by the algorithm. if ClusterNo=0 and PlotTree=TRUE, the dendrogram is generated instead of a clustering to estimate the numbers of clusters.

PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Standardization

DataOrDistances Is standardized before calculating the dissimilarities. Measurements are standardized for each variable (column), by subtracting the variable's mean value and dividing by the variable's mean absolute deviation.If DataOrDistances Is already a distance matrix, then this argument will be ignored.

PlotTree

TRUE: Plots the dendrogram, FALSE: no plot

Data

[1:n,1:d] data matrix in the case that DataOrDistances is missing and partial matching does not work.
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Dendrogram

Dendrogram of hierarchical clustering algorithm
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Rousseeuw/Kaufman, 1990] Rousseeuw, P. J., & Kaufman, L.: Finding groups in data, Belgium, John Wiley & Sons Inc., ISBN: 0471735787, doi: 10.1002/9780470316801, Online ISBN: 9780470316801, 1990.

Examples

data('Hepta')
CA=DivisiveAnalysisClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

print(CA$Object)
plot(CA$Object)
ClusterDendrogram(CA$Dendrogram,7,main='DIANA')

EngyTime introduced in [Baggenstoss, 2002].

Description

Gaussian mixture. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("EngyTime")

Details

Size 4096, Dimensions 2, stored in EngyTime$Data

Classes 2, stored in EngyTime$Cls

References

[Baggenstoss, 2002] Baggenstoss, P. M.: Statistical modeling using gaussian mixtures and hmms with matlab, Naval Undersea Warfare Center, Newport RI, 2002.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(EngyTime)
str(EngyTime)

Entropy Of a Data Field [Wang et al., 2011].

Description

Calculates the Potential Entropy Of a Data Field for a given ranges of impact factors sigma

Usage

EntropyOfDataField(Data, 

sigmarange = c(0.01, 0.1, 0.5, 1, 2, 5, 8, 10, 100)

, PlotIt = FALSE)

Arguments

Data

[1:n,1:d] data matrix

sigmarange

Numeric vector [1:s] of relevant sigmas

PlotIt

FALSE: disable plot, TRUE: Plot with upper boundary of H after [Wang et al., 2011].

Details

In theory there should be a curve with a clear minimum of Entropy [Wang et al.,2011]. Then the choice for the impact factor sigma is the minimum of the entropy to define the correct data field. It follows, that the influence radius is 3/sqrt(2)*sigma (3B rule of gaussian distribution) for clustering algorithms like density peak clustering [Wang et al.,2011].

Value

[1:s] named vector of the Entropy of data field. The names are the impact factor sigma.

Author(s)

Michael Thrun

References

[Wang et al., 2015] Wang, S., Wang, D., Li, C., & Li, Y.: Comment on" Clustering by fast search and find of density peaks", arXiv preprint arXiv:1501.04267, 2015.

[Wang et al., 2011] Wang, S., Gan, W., Li, D., & Li, D.: Data field for hierarchical clustering, International Journal of Data Warehousing and Mining (IJDWM), Vol. 7(4), pp. 43-63. 2011.

Examples

data(Hepta)
H=EntropyOfDataField(Hepta$Data,PlotIt=FALSE)
Sigmamin=names(H)[which.min(H)]
Dc=3/sqrt(2)*as.numeric(names(H)[which.min(H)])

Estimate Radius By Distance

Description

Published in [Thrun et al, 2016] for the case of automatically estimating the radius of the P-matrix. Can also be used to estimate the radius parameter for distance based clustering algorithms.

Usage

EstimateRadiusByDistance(DistanceMatrix)

Arguments

DistanceMatrix

[1:n,1:n] symmetric distance Matrix of n cases

Details

For density-based clustering algorithms like DBSCAN it is not always usefull.

Value

Numerical scalar defining the radius

Note

Symmetric matrix is assumed.

Author(s)

Michael Thrun

References

[Thrun et al., 2016] Thrun, M. C., Lerch, F., Loetsch, J., & Ultsch, A.: Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG), Vol. 24, pp. 7-16, Plzen, http://wscg.zcu.cz/wscg2016/short/A43-full.pdf, 2016.

See Also

GeneratePmatrix

Examples

data('Hepta')
DistanceMatrix=as.matrix(dist(Hepta$Data))
Radius=EstimateRadiusByDistance(DistanceMatrix)

Fuzzy Analysis Clustering [Rousseeuw/Kaufman, 1990, p. 253-279]

Description

...

Usage

FannyClustering(DataOrDistances,ClusterNo,

PlotIt=FALSE,Standardization=TRUE,...)

Arguments

DataOrDistances

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases or d-dimensional data points. Every case has d attributes, variables or features. Alternatively, symmetric [1:n,1:n] distance matrix
ClusterNo

A number k which defines k different clusters to be build by the algorithm.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Standardization

DataOrDistances is standardized before calculating the dissimilarities. Measurements are standardized for each variable (column), by subtracting the variable's mean value and dividing by the variable's mean absolute deviation. If DataOrDistances is already a distance matrix, then this argument will be ignored.

...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

...

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Points which cannot be assigned to a cluster will be reported with 0.

Object

Object defined by clustering algorithm as the second output of this algorithm

Author(s)

Michael Thrun

References

[Rousseeuw/Kaufman, 1990] Rousseeuw, P. J., & Kaufman, L.: Finding groups in data, Belgium, John Wiley & Sons Inc., ISBN: 0471735787, doi: 10.1002/9780470316801, Online ISBN: 9780470316801, 1990.

Examples

data('Hepta')
out=FannyClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

Gap Statistic

Description

Gap Statistic

Usage

GapStatistic(Data, ClusterNoMax, ClusterFun, ...)

Arguments

Data

[1:n,1:d] data matrix

ClusterNoMax

max no of clusters to beinvestigated

ClusterFun

which clustering algorithm to investigate

...

further arguments passed on

Details

does not work on hepta, see example

Value

tobedocumented

Note

Wrapper only

Author(s)

Michael Thrun

References

Tibshirani, R., Walther, G. and Hastie, T: Estimating the number of data clusters via the Gap statistic, Journal of the Royal Statistical Society B, Vol. 63, pp. 411-423, 2003.

Examples

data(Hepta)
#GapStatistic(Hepta$Data,10,ClusterFun = kmeans)

Genie Clustering by Gini Index

Description

Outlier Resistant Hierarchical Clustering Algorithm of [Gagolewski/Bartoszuk, 2016].

Usage

GenieClustering(DataOrDistances, ClusterNo = 0,
DistanceMethod="euclidean", ColorTreshold = 0,...)

Arguments

DataOrDistances

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features. Alternatively, symmetric [1:n,1:n] distance matrix
ClusterNo

A number k which defines k different clusters to be build by the algorithm.
DistanceMethod

See parDist, for example 'euclidean','mahalanobis','manhatten' (cityblock),'fJaccard','binary', 'canberra', 'maximum'. Any unambiguous substring can be given.
ColorTreshold

Draws cutline w.r.t. dendogram y-axis (height), height of line as scalar should be given
...

furter argument to genie like:

thresholdGini Single numeric value in [0,1], threshold for the Gini index, 1 gives the standard single linkage algorithm

Details

Wrapper for Genie algorithm.

Value

List of

Cls

If, ClusterNo>0: [1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Otherwise for ClusterNo=0: NULL
Dendrogram

Dendrogram of hierarchical clustering algorithm
Object

Ultrametric tree of hierarchical clustering algorithm

Author(s)

Michael Thrun

References

[Gagolewski/Bartoszuk, 2016] Gagolewski M., Bartoszuk M., Cena A., Genie: A new, fast, and outlier-resistant hierarchical clustering algorithm, Information Sciences, Vol. 363, pp. 8-23, 2016.

See Also

HierarchicalClustering

Examples

data('Hepta')
Clust=GenieClustering(Hepta$Data,ClusterNo=7)

GolfBall introduced in [Ultsch, 2005]

Description

No clusters at all. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("GolfBall")

Details

Size 4002, Dimensions 3, stored in GolfBall$Data

Classes 1, stored in GolfBall$Cls

References

[Ultsch, 2005] Ultsch, A.: Clustering wih SOM: U* C, Proc. Proceedings of the 5th Workshop on Self-Organizing Maps, Vol. 2, pp. 75-82, 2005.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(GolfBall)
str(GolfBall)

On-line Update (Hard Competitive learning) method

Description

Hard Competitive learning clustering published by [Ripley, 2007].

Usage

HCLclustering(Data, ClusterNo,PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be build by the algorithm.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Dimitriadou, 2002] Dimitriadou, E.: cclust-convex clustering methods and clustering indexes. R package, 2002,

[Ripley, 2007] Ripley, B. D.: Pattern recognition and neural networks, Cambridge university press, ISBN: 0521717701, 2007.

Examples

data('Hepta')
out=HCLclustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

HDD clustering is a model-based clustering method of [Bouveyron et al., 2007].

Description

HDD clustering is based on the Gaussian Mixture Model and on the idea that the data lives in subspaces with a lower dimension than the dimension of the original space. It uses the EM algorithm to estimate the parameters of the model [Berge et al., 2012].

Usage

HDDClustering(Data, ClusterNo, PlotIt=F,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

Optional, Numeric indicating either the number of cluster or a vector of 1:k to indicate the maximal expected number of clusters.
PlotIt

(optional) Boolean. Default = FALSE = No plotting performed.
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used, see hddc for details.

Details

HDD clustering maximises the BIC criterion for a range of possible number of cluster up to ClusterNo. Per default the most general model is used, alternetively the parameter model="ALL" can be used to evaluate all possible models with BIC [Berge et al., 2012]. If specific properties of Data are known priorly please see hddc for specific model selection.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Quirin Stier

References

[Berge et al., 2012] L. Berge, C. Bouveyron and S. Girard, HDclassif: an R Package for Model-Based Clustering and Discriminant Analysis of High-Dimensional Data, Journal of Statistical Software, vol. 42 (6), pp. 1-29, 2012.

[Bouveyron et al., 2007] Bouveyron, C. Girard, S. and Schmid, C: High-Dimensional Data Clustering, Computational Statistics and Data Analysis, vol. 52 (1), pp. 502-519, 2007.

Examples

# Hepta
data("Hepta")
Data = Hepta$Data
#Non-default parameter model
#can be set to evaulate all possible models
V = HDDClustering(Data=Data,ClusterNo=7,model="ALL")
Cls = V$Cls

ClusterAccuracy(Hepta$Cls, Cls)

## Not run: 
library(HDclassif)
data(Crabs)
Data = Crabs[,-1]
V = HDDClustering(Data=Data,ClusterNo=4,com_dim=1)

## End(Not run)

Hepta introduced in [Ultsch, 2003]

Description

Clearly defined clusters, different variances. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("Hepta")

Details

Size 212, Dimensions 3, stored in Hepta$Data

Classes 7, stored in Hepta$Cls

References

[Ultsch, 2003] Ultsch, A.: Maps for the visualization of high-dimensional data spaces, Proc. Workshop on Self organizing Maps (WSOM), pp. 225-230, Kyushu, Japan, 2003.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(Hepta)
str(Hepta)

Internal function of Hierarchical Clusterering of Data

Description

Please use HierarchicalClustering. Hierarchical cluster analysis on a set of dissimilarities and methods for analyzing it. Uses stats package function 'hclust'.

Usage

HierarchicalClusterData(Data,ClusterNo=0,

Type="ward.D2",DistanceMethod="euclidean",

ColorTreshold=0,Fast=FALSE,Cls=NULL,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be build by the algorithm.
Type

Methode der Clusterung: "ward.D", "ward.D2", "single", "complete", "average", "mcquitty", "median" or "centroid".
DistanceMethod

see parDist, for example 'euclidean','mahalanobis','manhatten' (cityblock),'fJaccard','binary', 'canberra', 'maximum'. Any unambiguous substring can be given.
ColorTreshold

Draws cutline w.r.t. dendrogram y-axis (height), height of line as scalar should be given
Fast

If TRUE and fastcluster installed, then a faster implementation of the methods above can be used
Cls

[1:n] classification vector for coloring of dendrogram in plot
...

In case of plotting further argument for plot, see as.dendrogram

Value

List of

Cls

If, ClusterNo>0: [1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Otherwise for ClusterNo=0: NULL
Dendrogram

Dendrogram of hierarchical clustering algorithm
Object

Ultrametric tree of hierarchical clustering algorithm

Author(s)

Michael Thrun

See Also

HierarchicalClusterData

HierarchicalClusterDists

HierarchicalClustering

Examples

data('Hepta')
#out=HierarchicalClusterData(Hepta$Data,ClusterNo=7)

Internal Function of Hierarchical Clustering with Distances

Description

Please use HierarchicalClustering. Cluster analysis on a set of dissimilarities and methods for analyzing it. Uses stats package function 'hclust'.

Usage

HierarchicalClusterDists(pDist,ClusterNo=0,Type="ward.D2",

ColorTreshold=0,Fast=FALSE,...)

Arguments

pDist

Distances as either matrix [1:n,1:n] or dist object
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
Type

Method of cluster analysis: "ward.D", "ward.D2", "single", "complete", "average", "mcquitty", "median" or "centroid".
ColorTreshold

Draws cutline w.r.t. dendogram y-axis (height), height of line as scalar should be given
Fast

If TRUE and fastcluster installed, then a faster implementation of the methods above can be used
...

In case of plotting further argument for plot, see as.dendrogram

Value

List of

Cls

If, ClusterNo>0: [1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Otherwise for ClusterNo=0: NULL
Dendrogram

Dendrogram of hierarchical clustering algorithm
Object

Ultrametric tree of hierarchical clustering algorithm

Author(s)

Michael Thrun

See Also

HierarchicalClusterData

HierarchicalClusterDists

HierarchicalClustering

Examples

data('Hepta')
#out=HierarchicalClusterDists(as.matrix(dist(Hepta$Data)),ClusterNo=7)

Hierarchical Clustering

Description

Wrapper for various agglomerative hierarchical clustering algorithms.

Usage

HierarchicalClustering(DataOrDistances,ClusterNo,Type='SingleL',Fast=TRUE,Data,...)

Arguments

DataOrDistances

Either nonsymmetric [1:n,1:d] numerical matrix of a dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.

or

symmetric [1:n,1:n] distance matrix, e.g. as.matrix(dist(Data,method))

ClusterNo

A number k which defines k different clusters to be built by the algorithm.
Type

Method of cluster analysis: "Ward", "SingleL", "CompleteL", "AverageL" (UPGMA), "WPGMA" (mcquitty), "MedianL" (WPGMC), "CentroidL" (UPGMC), "Minimax", "MinEnergy", "Gini","HDBSCAN", or "Sparse"
Fast

If TRUE and fastcluster installed, then a faster implementation of the methods above can be used except for "Minimax", "MinEnergy", "Gini" or "HDBSCAN"
Data

[1:n,1:d] data matrix in the case that DataOrDistances is missing and partial matching does not work.
...

Further arguments passed on to either HierarchicalClusterData, HierarchicalClusterDists, MinimalEnergyClustering or GenieClustering (for "Gini"), HierarchicalDBSCAN (for HDBSCAN) or SparseClustering (for Sparse).

Details

Please see HierarchicalClusterData and HierarchicalClusterDists or the other functions listed above.

It should be noted that in case of "HDBSCAN" the number of clusters is manually selected by cutree to have the same convention as the other algorithms. Usually, "HDBSCAN" selects the number of clusters automatically.

Value

List of

Cls

If, ClusterNo>0: [1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Otherwise for ClusterNo=0: NULL
Dendrogram

Dendrogram of hierarchical clustering algorithm
Object

Ultrametric tree of hierarchical clustering algorithm

Author(s)

Michael Thrun

See Also

HierarchicalClusterData

HierarchicalClusterDists,

MinimalEnergyClustering.

Examples

data('Hepta')
out=HierarchicalClustering(Hepta$Data,ClusterNo=7)

Hierarchical DBSCAN

Description

Hierarchical DBSCAN clustering [Campello et al., 2015].

Usage

HierarchicalDBSCAN(DataOrDistances,minPts=4,

PlotTree=FALSE,PlotIt=FALSE,...)

Arguments

DataOrDistances

Either a [1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.

or a [1:n,1:n] symmetric distance matrix.

minPts

Classic smoothing factor in density estimates [Campello et al., 2015, p.9]

PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
PlotTree

Default: FALSE, If TRUE plots the dendrogram. If minPts is missing, PlotTree is set to TRUE.
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

"Computes the hierarchical cluster tree representing density estimates along with the stability-based flat cluster extraction proposed by Campello et al. (2013). HDBSCAN essentially computes the hierarchy of all DBSCAN* clusterings, and then uses a stability-based extraction method to find optimal cuts in the hierarchy, thus producing a flat solution."[Hahsler et al., 2019]

It is claimed by the inventors that the minPts parameter is noncritical [Campello et al., 2015, p.35]. minPts is reported to be set to 4 on all experiments [Campello et al., 2015, p.35].

Value

List of

Cls

[1:n] numerical vector defining the clustering; this classification is the main output of the algorithm. Points which cannot be assigned to a cluster will be reported as members of the noise cluster with 0.
Dendrogram

Dendrogram of hierarchical clustering algorithm
Tree

Ultrametric tree of hierarchical clustering algorithm
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Campello et al., 2015] Campello, R. J., Moulavi, D., Zimek, A., & Sander, J.: Hierarchical density estimates for data clustering, visualization, and outlier detection, ACM Transactions on Knowledge Discovery from Data (TKDD), Vol. 10(1), pp. 1-51. 2015.

[Hahsler et al., 2019] Hahsler M, Piekenbrock M, Doran D: dbscan: Fast Density-Based Clustering with R. Journal of Statistical Software, 91(1), pp. 1-30. doi: 10.18637/jss.v091.i01, 2019

Examples

data('Hepta')

out=HierarchicalDBSCAN(Hepta$Data,PlotIt=FALSE)


data('Leukemia')
set.seed(1234)
CA=HierarchicalDBSCAN(Leukemia$DistanceMatrix)
#ClusterCount(CA$Cls)
#ClusterDendrogram(CA$Dendrogram,5,main='H-DBscan')

K-Means Clustering

Description

Perform k-means clustering on a data matrix.

Usage

kmeansClustering(DataOrDistances, ClusterNo,

 Type = 'LBG',RandomNo=5000, CategoricalData,
 
 PlotIt=FALSE, Verbose = FALSE,... )

Arguments

DataOrDistances

Either nonsymmetric [1:n,1:d] datamatrix of n cases and d numerical features or

symmetric [1:n,1:n] distance matrix
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
Type

Choice of Kmeans algorithm, currently either " Hartigan" [Hartigan/Wong, 1979], "LBG" [Linde et al., 1980], "Sparse" sparse k-means proposed in [Witten/Tibshirani, 2010], "Steinley" best method of [Steinley/Brusco, 2007] proposed in Steinley 2003, "Lloyd" [Lloyd, 1982], "Forgy"[Forgy, 1965], MacQueen [MacQueen, 1967], kcentroids [Leisch, 2006], "kprototypes" [Szepannek, 2018], "Pelleg-moore" [Pelleg & Moores,2000], "Elkan" [Elkan, 2003], "kmeans++"" [Arthur & Vassilvitskii], Hamerly"[Hamerly, 2010] ,Dualtree" or Dualtree-covertree [Curtin, 2017]"

RandomNo

Only for " Steinley" or in case of distance matrix, number of random initializations with searching for minimal SSE, see [Steinley/Brusco, 2007]
CategoricalData

Only for " kprototypes", [1:n,1:m] matrix of categorical features]
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Verbose

Print details, if true
...

Further arguments like iter.max, nstart, for kcentroids please see kcca function of the flexclust package, or KMeansSparseCluster

Details

Uses either stats package function 'kmeans', cclust package implemention, flexclust package implemention or own code. In case of a distance matrix, RandomNo should be significantly lower than 5000, otherwise a long computation time is to be expected.

Value

List V of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.

Object

Object of the clustering algorithm used if existent, otherwise

SumDistsToCentroids: Vector of within-cluster sum of squares, one component per cluster

Centroids

the final cluster centers.

Note

The version using a distance matrix is still in the test phase and not yet verified.

Author(s)

Michael Thrun

References

[Hartigan/Wong, 1979] Hartigan, J. A., & Wong, M. A.: Algorithm AS 136: A k-means clustering algorithm, Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 28(1), pp. 100-108. 1979.

[Linde et al., 1980] Linde, Y., Buzo, A., & Gray, R.: An algorithm for vector quantizer design, IEEE Transactions on communications, Vol. 28(1), pp. 84-95. 1980.

[Steinley/Brusco, 2007] Steinley, D., & Brusco, M. J.: Initializing k-means batch clustering: A critical evaluation of several techniques, Journal of Classification, Vol. 24(1), pp. 99-121. 2007.

[Forgy, 1965] Forgy, E. W.: Cluster analysis of multivariate data: efficiency versus interpretability of classifications, Biometrics, Vol. 21, pp. 768-769. 1965.

[MacQueen, 1967] MacQueen, J.: Some methods for classification and analysis of multivariate observations, Proc. Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, Vol. 1, pp. 281-297, Oakland, CA, USA., 1967.

[Pelleg & Moores,2000] Pelleg, Dan, and Andrew W. Moore. X-means: Extending k-means with efficient estimation of the number of clusters, ICML. Vol. 1. 2000.

[Elkan, 2003] Elkan, Charles: Using the triangle inequality to acceler- ate k-means, In Tom Fawcett and Nina Mishra, editors, ICML, pages Vol.3, 147-153. AAAI Press, 2003.

[Lloyd, 1982] Lloyd, S.: Least squares quantization in PCM, IEEE transactions on information theory, Vol. 28(2), pp. 129-137. 1982.

[Leisch, 2006] Leisch, F.: A toolbox for k-centroids cluster analysis, Computational Statistics & Data Analysis, Vol. 51(2), pp. 526-544. 2006.

[Arthur & Vassilvitskii] Arthur, David, and Vassilvitskii, Sergei: K-means++ the advantages of careful seeding, Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms. 2007

[Witten/Tibshirani, 2010] Witten, D. and Tibshirani, R.: A Framework for Feature Selection in Clustering. Journal of the American Statistical Association, Vol. 105(490), pp. 713-726, 2010.

[Hamerly, 2010] Hamerly, Greg: Making k-means even faster, Proceedings of the 2010 SIAM international conference on data mining, Society for Industrial and Applied Mathematics, pp. 130-140, 2010.

[Szepannek, 2018] Szepannek, G.: clustMixType: User-Friendly Clustering of Mixed-Type Data in R, The R Journal, Vol. 10/2, pp. 200-208, doi:10.32614/RJ2018048, 2018.

[Curtin, 2017] Curtin, Ryan R: A dual-tree algorithm for fast k-means clustering with large k, Proceedings of the 2017 SIAM International Conference on Data Mining, Society for Industrial and Applied Mathematics, 2017.

Examples

data('Hepta')
out=kmeansClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)


data('Leukemia')
# As expected does not perform well
# For non-spherical cluster structures:
out=kmeansClustering(Leukemia$DistanceMatrix,ClusterNo=6,RandomNo =10,PlotIt=TRUE)




data('Hepta')
out=kmeansClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE,Type="Steinley")



data('Hepta')
out=kmeansClustering(Hepta$Data,ClusterNo = 7,
Type = "kprototypes",CategoricalData = as.matrix(Hepta$Cls))

k-means Clustering using a distance matrix

Description

Perform k-means clustering on a distance matrix

Usage

kmeansDist(Distance, ClusterNo=2,Centers=NULL,

RandomNo=1,maxIt = 2000, 

PlotIt=FALSE,verbose = F)

Arguments

Distance

Distance matrix. For n data points of the dimension n x n

ClusterNo

A number k which defines k different clusters to be built by the algorithm.
Centers

Default(NULL) a set of initial (distinct) cluster centres.
RandomNo

If>1: Number of random initializations with searching for minimal SSE is defined by this scalar
maxIt

Optional: Maximum number of iterations before the algorithm terminates.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
verbose

Optional: Algorithm always outputs current iteration.

Value

Cls[1:n]

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
centerids[1:k]

Indices of the centroids from which the cluster Cls was created

Note

Currently an experimental version

Author(s)

Felix Pape, Michael Thrun

Examples

data('Hepta')
#out=kmeansDist(as.matrix(dist(Hepta$Data)),ClusterNo=7,PlotIt=FALSE,RandomNo = 10)

## Not run: 
data('Leukemia')
#as expected does not perform well
#for non-spherical cluster structures:
#out=kmeansDist(Leukemia$DistanceMatrix,ClusterNo=6,PlotIt=TRUE,RandomNo=10)

## End(Not run)

Large Application Clustering

Description

Clustering Large Applications (clara) of [Rousseeuw/Kaufman, 1990, pp. 126-163]

Usage

LargeApplicationClustering(Data, ClusterNo,

PlotIt=FALSE,Standardization=TRUE,Samples=50,Random=TRUE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Standardization

Data is standardized before calculating the dissimilarities. Measurements are standardized for each variable (column), by subtracting the variable's mean value and dividing by the variable's mean absolute deviation.

Samples

Integer, say N, the number of samples to be drawn from the dataset. Default value set as recommended by documentation of clara
Random

Logical indicating if R's random number generator should be used instead of the primitive clara()-builtin one.
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

It is recommended to use set.seed if clustering output should be always the same instead of setting Random=FALSE in order to use the primitive clara()-builtin random number generator.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Rousseeuw/Kaufman, 1990] Rousseeuw, P. J., & Kaufman, L.: Finding groups in data, Belgium, John Wiley & Sons Inc., ISBN: 0471735787, doi 10.1002/9780470316801, Online ISBN: 9780470316801, 1990.

Examples

data('Hepta')
out=LargeApplicationClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

Leukemia distance matrix and classificiation used in [Thrun, 2018]

Description

Data is anonymized. Original dataset was published in [Haferlach et al., 2010]. Original dataset had around 12.000 dimensions. Detailed description of preprocessed dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("Leukemia")

Details

554x554 distance matrix. Cls defines the following clusters:

1= APL Outlier

2=APL

3=Healthy

4=AML

5=CLL

6=CLL Outlier

References

[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.

[Haferlach et al., 2010] Haferlach, T., Kohlmann, A., Wieczorek, L., Basso, G., Te Kronnie, G., Bene, M.-C., . . . Mills, K. I.: Clinical utility of microarray-based gene expression profiling in the diagnosis and subclassification of leukemia: report from the International Microarray Innovations in Leukemia Study Group, Journal of Clinical Oncology, Vol. 28(15), pp. 2529-2537. 2010.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(Leukemia)
str(Leukemia)
Cls=Leukemia$Cls
Distance=Leukemia$DistanceMatrix
isSymmetric(Distance)

Lsun3D inspired by FCPS introduced in [Thrun, 2018]

Description

Clearly defined clusters, different variances. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("Lsun3D")

Details

Size 404, Dimensions 3

Dataset defines discontinuites, where the clusters have different variances. Three main clusters, and four outliers (in cluster 4). For a more detailed description see [Thrun, 2018].

References

[Thrun, 2018] Thrun, M. C.: Projection Based Clustering through Self-Organization and Swarm Intelligence, doctoral dissertation 2017, Springer, Heidelberg, ISBN: 978-3-658-20539-3, doi:10.1007/978-3-658-20540-9, 2018.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(Lsun3D)
str(Lsun3D)
Cls=Lsun3D$Cls
Data=Lsun3D$Data

Markov Clustering

Description

Graph clustering algorithm introduced by [van Dongen, 2000].

Usage

MarkovClustering(DataOrDistances=NULL,Adjacency=NULL,

Radius=TRUE,DistanceMethod="euclidean",addLoops = TRUE,PlotIt=FALSE,...)

Arguments

DataOrDistances

NULL or: Either [1:n,1:n] symmetric distance matrix or [1:n,1:d] not symmetric data matrix of n cases and d variables
Adjacency

Used if Data is NULL, matrix [1:n,1:n] defining which points are adjacent to each other by the number 1; not adjacent: 0
Radius

Scalar, Radius for unit disk graph (r-ball graph) if adjacency matrix is missing. Automatic estimation can be done either with =TRUE [Ultsch, 2005] or FALSE [Thrun et al., 2016] if Data instead of Distances are given.
DistanceMethod

Optional distance method of data, default is euclid, see parDist for details

addLoops

Logical; if TRUE, self-loops with weight 1 are added to each vertex of x (see mcl of CRAN package MCL).

PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

DataOrDistances is used to compute the Adjecency matrix if this input is missing. Then a unit-disk (R-ball) graph is calculated.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Points which cannot be assigned to a cluster will be reported with 0.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[van Dongen, 2000] van Dongen, S.M. Graph Clustering by Flow Simulation. Ph.D. thesis, Universtiy of Utrecht. Utrecht University Repository: http://dspace.library.uu.nl/handle/1874/848, 2000

[Thrun et al., 2016] Thrun, M. C., Lerch, F., Loetsch, J., & Ultsch, A. : Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG), Vol. 24, Plzen, 2016.

[Ultsch, 2005] Ultsch, A.: Pareto density estimation: A density estimation for knowledge discovery, In Baier, D. & Werrnecke, K. D. (Eds.), Innovations in classification, data science, and information systems, (Vol. 27, pp. 91-100), Berlin, Germany, Springer, 2005.

Examples

data('Hepta')
out=MarkovClustering(Data=Hepta$Data,PlotIt=FALSE)

Mean Shift Clustering

Description

Mean Shift Clustering of [Cheng, 1995]

Usage

MeanShiftClustering(Data,

PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

the radius used for search can be specified with the "radius" parameter. The maximum number of iterations before algorithm termination is controlled with the "max_iterations" parameter.

If the distance between two centroids is less than the given radius, one will be removed. A radius of 0 or less means an estimate will be calculated and used for the radius. Default value "0" (numeric).

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Cheng, 1995] Cheng, Yizong: Mean Shift, Mode Seeking, and Clustering, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17 (8), pp. 790-799, doi:10.1109/34.400568, 1995.

Examples

data('Hepta')
out=MeanShiftClustering(Hepta$Data,PlotIt=FALSE,radius=1)

Minimal Energy Clustering

Description

Hierchical Clustering using the minimal energy approach of [Szekely/Rizzo, 2005].

Usage

MinimalEnergyClustering(DataOrDistances, ClusterNo = 0,
DistanceMethod="euclidean", ColorTreshold = 0,Data,...)

Arguments

DataOrDistances

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features. Alternatively, symmetric [1:n,1:n] distance matrix
ClusterNo

A number k which defines k different clusters to be build by the algorithm.
DistanceMethod

See parDist, for example 'euclidean','mahalanobis','manhatten' (cityblock),'fJaccard','binary', 'canberra', 'maximum'. Any unambiguous substring can be given.
ColorTreshold

Draws cutline w.r.t. dendogram y-axis (height), height of line as scalar should be given
Data

[1:n,1:d] data matrix in the case that DataOrDistances is missing and partial matching does not work.
...

In case of plotting further argument for plot, see as.dendrogram

Value

List of

Cls

If ClusterNo>0: [1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Otherwise ClusterNo=0: NULL
Dendrogram

Dendrogram of hierarchical clustering algorithm
Object

Ultrametric tree of hierarchical clustering algorithm

Author(s)

Michael Thrun

References

[Szekely/Rizzo, 2005] Szekely, G. J. and Rizzo, M. L.: Hierarchical Clustering via Joint Between-Within Distances: Extending Ward's Minimum Variance Method, Journal of Classification, 22(2) 151-183.http://dx.doi.org/10.1007/s00357-005-0012-9, 2005.

See Also

HierarchicalClustering

Examples

data('Hepta')
out=MinimalEnergyClustering(Hepta$Data,ClusterNo=7)

Minimax Linkage Hierarchical Clustering

Description

In the minimax linkage hierarchical clustering every cluster has an associated prototype element that represents that cluster [Bien/Tibshirani, 2011].

Usage

MinimaxLinkageClustering(DataOrDistances, ClusterNo = 0,
DistanceMethod="euclidean", ColorTreshold = 0,...)

Arguments

DataOrDistances

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases or d-dimensional data points. Every case has d attributes, variables or features. Alternatively, symmetric [1:n,1:n] distance matrix
ClusterNo

A number k which defines k different clusters to be build by the algorithm.
DistanceMethod

See parDist, for example 'euclidean','mahalanobis','manhatten' (cityblock),'fJaccard','binary', 'canberra', 'maximum'. Any unambiguous substring can be given.
ColorTreshold

Draws cutline w.r.t. dendogram y-axis (height), height of line as scalar should be given
...

In case of plotting further argument for plot, see as.dendrogram

Value

List of

Cls

If, ClusterNo>0: [1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Otherwise for ClusterNo=0: NULL
Dendrogram

Dendrogram of hierarchical clustering algorithm
Object

Ultrametric tree of hierarchical clustering algorithm

Author(s)

Michael Thrun

References

[Bien/Tibshirani, 2011] Bien, J., and Tibshirani, R.: Hierarchical Clustering with Prototypes via Minimax Linkage, The Journal of the American Statistical Association, Vol. 106(495), pp. 1075-1084, 2011.

See Also

HierarchicalClustering

Examples

data('Hepta')
out=MinimaxLinkageClustering(Hepta$Data,ClusterNo=7)

Model Based Clustering

Description

Calls Model based clustering of [Fraley/Raftery, 2006] which models a Mixture Of Gaussians (MoG).

Usage

ModelBasedClustering(Data,ClusterNo=2,PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

see [Thrun, 2017, p. 23] or [Fraley/Raftery, 2002] and [Fraley/Raftery, 2006].

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Note

MoGclustering used in [Thrun, 2017] was renamed to ModelBasedClustering in this package.

Author(s)

Michael Thrun

References

[Thrun, 2017] Thrun, M. C.:A System for Projection Based Clustering through Self-Organization and Swarm Intelligence, (Doctoral dissertation), Philipps-Universitaet Marburg, Marburg, 2017.

[Fraley/Raftery, 2002] Fraley, C., and Raftery, A. E.: Model-based clustering, discriminant analysis, and density estimation, Journal of the American Statistical Association, Vol. 97(458), pp. 611-631. 2002.

[Fraley/Raftery, 2006] Fraley, C., and Raftery, A. E.MCLUST version 3: an R package for normal mixture modeling and model-based clustering,DTIC Document, 2006.

See Also

MoGclustering

Examples

data('Hepta')
out=ModelBasedClustering(Hepta$Data,PlotIt=FALSE)

Model Based Clustering with Variable Selection

Description

Model-based clustering with variable selection and estimation of the number of clusters which is either based on [Marbac/Sedki, 2017],[Marbac et al., 2020], or on [Scrucca and Raftery, 2014].

Usage

ModelBasedVarSelClustering(Data,ClusterNo,Type,PlotIt=FALSE, ...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

Numeric which defines number of cluster to search for.
Type

String, either VarSelLCM [Marbac/Sedki, 2017],[Marbac et al., 2020], or clustvarsel [Scrucca and Raftery, 2014].
PlotIt

(optional) Boolean. Default = FALSE = No plotting performed.
...

Further arguments passed on to VarSelCluster or clustvarsel.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Quirin Stier, Michael Thrun

References

[Marbac/Sedki, 2017] Marbac, M. and Sedki, M.: Variable selection for model-based clustering using the integrated complete-data likelihood. Statistics and Computing, 27(4), pp. 1049-1063, 2017.

[Marbac et al., 2020] Marbac, M., Sedki, M., & Patin, T.: Variable selection for mixed data clustering: application in human population genomics, Journal of Classification, Vol. 37(1), pp. 124-142. 2020.

Examples

# Hepta
data("Hepta")
Data = Hepta$Data
V = ModelBasedVarSelClustering(Data, ClusterNo=7,Type="VarSelLCM")
Cls = V$Cls
ClusterAccuracy(Hepta$Cls, Cls, K = 7)

V = ModelBasedVarSelClustering(Data, ClusterNo=7,Type="clustvarsel")
Cls = V$Cls
ClusterAccuracy(Hepta$Cls, Cls, K = 7)

## Not run: 
# Hearts
heart=VarSelLCM::heart
ztrue <- heart[,"Class"]
Data <- heart[,-13]
V <- ModelBasedVarSelClustering(Data,2,Type="VarSelLCM")
Cls = V$Cls
ClusterAccuracy(ztrue, Cls, K = 2)

## End(Not run)

Mixture of Gaussians Clustering using EM

Description

MixtureOfGaussians (MoG) clustering based on Expectation Maximization (EM) of [Chen et al., 2012] or algorithms closely resembling EM of [Benaglia/Chauveau/Hunter, 2009].

Usage

MoGclustering(Data,ClusterNo=2,Type,PlotIt=FALSE,Silent=TRUE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
Type

string defining approach to select: initialization approach of "EM" or "kmeans" of [Chen et al., 2012], or other methods "mvnormalmixEM" [McLachlan/Peel, 2000], "npEM"[Benaglia et al., 2009] or its extension "mvnpEM" [Chauveau/Hoang, 2016].

PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Silent

(optional) Boolean: print output or not (Default = FALSE = no output)
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used, see package mixtools EMCluster or mixtools for details.

Details

Algorithms for clustering through EM or its close resembles.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Note

MoG used in [Thrun, 2017] was renamed to ModelBasedClustering in this package. Type="mvnormalmixEM" sometimes fails

Author(s)

Michael Thrun

References

[Chen et al., 2012] Chen, W., Maitra, R., & Melnykov, V.: EMCluster: EM Algorithm for Model-Based Clustering of Finite Mixture Gaussian Distribution, R Package, URL http://cran. r-project. org/package= EMCluster, 2012.

[Chauveau/Hoang, 2016] Chauveau, D., & Hoang, V. T. L.: Nonparametric mixture models with conditionally independent multivariate component densities, Computational Statistics & Data Analysis, Vol. 103, pp. 1-16. 2016.

[Benaglia et al., 2009] Benaglia, T., Chauveau, D., and Hunter, D. R.: An EM-like algorithm for semi-and nonparametric estimation in multivariate mixtures. Journal of Computational and Graphical Statistics, 18(2), pp. 505-526, 2009.

[McLachlan/Peel, 2000] D. McLachlan, G. J. and Peel, D.: Finite Mixture Models, John Wiley and Sons, Inc, 2000.

See Also

ModelBasedClustering

Examples

data('Hepta')
Data = Hepta$Data
out=MoGclustering(Data,ClusterNo=7,Type="EM",PlotIt=FALSE)
V=out$Cls


V1 = MoGclustering(Data,ClusterNo=7,Type="mvnpEM")
Cls1 = V1$Cls

V2 = MoGclustering(Data,ClusterNo=7,Type="npEM")
Cls2 = V2$Cls


## Not run: 
#does not work always
  V3 = MoGclustering(Data,ClusterNo=7,Type="mvnormalmixEM")
  Cls3 = V3$Cls

## End(Not run)

MST-kNN clustering algorithm [Inostroza-Ponta, 2008].

Description

Performs the MST-kNN clustering algorithm which generate a clustering solution with automatic k determination using two proximity graphs: Minimal Spanning Tree (MST) and k-Nearest Neighbor (kNN) which are recursively intersected.

Usage

MSTclustering(DataOrDistances, DistanceMethod = "euclidean",PlotIt=FALSE, ...)

Arguments

DataOrDistances

Either [1:n,1:n] symmetric distance matrix or [1:n,1:d] not symmetric data matrix of n cases and d variables

DistanceMethod

Optional distance method of data, default is euclid, see parDist for details

PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Optional, further arguments for mst.knn

Details

Does not work on Hepta with euclidean distances.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Inostroza-Ponta, 2008] Inostroza-Ponta, M.: An integrated and scalable approach based on combinatorial optimization techniques for the analysis of microarray data, University of Newcastle, ISBN, 2008

See Also

mst.knn

Examples

data(Hepta)

MSTclustering(Hepta$Data)

Network Clustering

Description

Either leiden [Traag et al., 2019] or louvain [Blondel et al., 2008] clustering

Usage

NetworkClustering(DataOrDistances=NULL,Adjacency=NULL,

Type="louvain",Radius=FALSE,PlotIt=FALSE,...)

Arguments

DataOrDistances

NULL or: [1:n,1:d] matrix of dataset to be clustered. It consists of n cases or d-dimensional data points. Every case has d attributes, variables or features. Alternatively, symmetric [1:n,1:n] distance matrix
Adjacency

Used if DataOrDistances is NULL, matrix [1:n,1:n] defining which points are adjacent to each other by the number 1; not adjacent: 0
Type

Either "louvain" or "leiden"

Radius

Scalar, Radius for unit disk graph (r-ball graph) if adjacency matrix is missing. Automatic estimation can be done either with =TRUE [Ultsch, 2005] or FALSE [Thrun et al., 2016]
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

DataOrDistances is used to compute the Adjecency matrix if this input is missing. Then a unit-disk (R-ball) graph is calculated. Radius=TRUE only works if data matrix is given.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Points which cannot be assigned to a cluster will be reported with 0.
Object

Object defined by clustering algorithm as the other output of this algorithm

Note

leiden requires igraph package and an installed python version. automatic installation may not work. manual call in console has to be in this case conda install -c conda-forge leidenalg

Author(s)

Michael Thrun

References

[Blondel et al., 2008] Blondel, V. D., Guillaume, J.-L., Lambiotte, R., & Lefebvre, E.: Fast unfolding of communities in large networks, Journal of statistical mechanics: theory and experiment, Vol. 2008(10), pp. P10008. 2008.

[Traag et al., 2019] Traag, V. A., Waltman, L., & van Eck, N. J.: From Louvain to Leiden: guaranteeing well-connected communities, Scientific reports, Vol. 9(1), pp. 1-12. 2019.

Examples

data('Hepta')
#out=NetworkClustering(Hepta$Data,PlotIt=FALSE)

Neural gas algorithm for clustering

Description

Neural gas clustering published by [Martinetz et al., 1993]] and implemented by [Bodenhofer et al., 2011].

Usage

NeuralGasClustering(Data, ClusterNo,PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Dimitriadou, 2002] Dimitriadou, E.: cclust-convex clustering methods and clustering indexes. R package, 2002,

[Martinetz et al., 1993] Martinetz, T. M., Berkovich, S. G., & Schulten, K. J.: 'Neural-gas' network for vector quantization and its application to time-series prediction, IEEE Transactions on Neural Networks, Vol. 4(4), pp. 558-569. 1993.

Examples

data('Hepta')
out=NeuralGasClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

OPTICS Clustering

Description

OPTICS (Ordering points to identify the clustering structure) clustering algorithm [Ankerst et al.,1999].

Usage

OPTICSclustering(Data, MaxRadius,RadiusThreshold, minPts = 5, PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
MaxRadius

Upper limit neighborhood in the R-ball graph/unit disk graph), size of the epsilon neighborhood (eps) [Ester et al., 1996, p. 227]. If NULL, automatic estimation is done using insights of [Ultsch, 2005].

RadiusThreshold

Threshold to identify clusters (RadiusThreshold <= MaxRadius), if NULL 0.9*MaxRadius is set.

minPts

Number of minimum points in the eps region (for core points). In principle minimum number of points in the unit disk, if the unit disk is within the cluster (core) [Ester et al., 1996, p. 228]. If NULL, its 2.5 percent of points.

PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

...

Value

List of

Cls

[1:n] numerical vector defining the clustering; this classification is the main output of the algorithm. Points which cannot be assigned to a cluster will be reported as members of the noise cluster with 0.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Ankerst et al.,1999] Mihael Ankerst, Markus M. Breunig, Hans-Peter Kriegel, Joerg Sander: OPTICS: Ordering Points To Identify the Clustering Structure, ACM SIGMOD international conference on Management of data, ACM Press, pp. 49-60, 1999.

[Ester et al., 1996] Ester, M., Kriegel, H.-P., Sander, J., & Xu, X.: A density-based algorithm for discovering clusters in large spatial databases with noise, Proc. Kdd, Vol. 96, pp. 226-231, 1996.

[Ultsch, 2005] Ultsch, A.: Pareto density estimation: A density estimation for knowledge discovery, In Baier, D. & Werrnecke, K. D. (Eds.), Innovations in classification, data science, and information systems, (Vol. 27, pp. 91-100), Berlin, Germany, Springer, 2005.

See Also

optics

Examples

data('Hepta')
out=OPTICSclustering(Hepta$Data,MaxRadius=NULL,RadiusThreshold=NULL,minPts=NULL,PlotIt = FALSE)

Partitioning Around Medoids (PAM)

Description

Partitioning (clustering) of the data into k clusters around medoids, a more robust version of k-means [Rousseeuw/Kaufman, 1990, p. 68-125] .

Usage

PAMclustering(DataOrDistances,ClusterNo,

PlotIt=FALSE,Standardization=TRUE,Data,...)

Arguments

DataOrDistances

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features. Alternatively, symmetric [1:n,1:n] distance matrix
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Standardization

DataOrDistances is standardized before calculating the dissimilarities. Measurements are standardized for each variable (column), by subtracting the variable's mean value and dividing by the variable's mean absolute deviation.If DataOrDistances is already a distance matrix, then this argument will be ignored.

Data

[1:n,1:d] data matrix in the case that DataOrDistances is missing and partial matching does not work.
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

[Rousseeuw/Kaufman, 1990, chapter 2] or [Reynolds et al., 1992].

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Rousseeuw/Kaufman, 1990] Rousseeuw, P. J., & Kaufman, L.: Finding groups in data, Belgium, John Wiley & Sons Inc., ISBN: 0471735787, doi:10.1002/9780470316801, Online ISBN: 9780470316801, 1990.

[Reynolds et al., 1992] Reynolds, A., Richards, G.,de la Iglesia, B. and Rayward-Smith, V.: Clustering rules: A comparison of partitioning and hierarchical clustering algorithms, Journal of Mathematical Modelling and Algorithms 5, 475-504, DOI:10.1007/s10852-005-9022-1, 1992.

Examples

data('Hepta')
out=PAMclustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

Probability Density Distribution Clustering

Description

Clustering via non parametric density estimation

Usage

pdfClustering(Data, PlotIt = FALSE, ...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

Cluster analysis is performed by the density-based procedures described in Azzalini and Torelli (2007) and Menardi and Azzalini (2014), and summarized in Azzalini and Menardi (2014).

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

Azzalini, A., Menardi, G. (2014). Clustering via nonparametric density estimation: the R package pdfCluster. Journal of Statistical Software, 57(11), 1-26, URL http://www.jstatsoft.org/v57/i11/.

Azzalini A., Torelli N. (2007). Clustering via nonparametric density estimation. Statistics and Computing. 17, 71-80.

Menardi, G., Azzalini, A. (2014). An advancement in clustering via nonparametric density estimation. Statistics and Computing. DOI: 10.1007/s11222-013-9400-x.

Examples

data('Hepta')
out=pdfClustering(Hepta$Data,PlotIt=FALSE)

Penalized Regression-Based Clustering of [Wu et al., 2016].

Description

Clustering is performed through penalized regression with grouping pursuit

Usage

PenalizedRegressionBasedClustering(Data, FirstLambda, 

SecondLambda, Tau, PlotIt = FALSE, ...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.

FirstLambda

Set 1 for quadratic penalty based algorithm, 0.4 for revised ADMM.

SecondLambda

The magnitude of grouping penalty.

Tau

Tuning parameter: tau, related to grouping penalty.

PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls

...

Further arguments for PRclust, enables also usage of [Pan et al., 2013].

Details

Parameters are rather challenging to choose.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.

Object

Object defined by clustering algorithm as the other output of this algorithm

Note

Data matrix is internally transposed in order to fit the definition of the algorithm.

Author(s)

Michael Thrun

References

[Pan et al., 2013] Pan, W., Shen, X., & Liu, B.: Cluster analysis: unsupervised learning via supervised learning with a non-convex penalty, The Journal of Machine Learning Research, Vol. 14(1), pp. 1865-1889. 2013.

[Wu et al., 2016] Wu, C., Kwon, S., Shen, X., & Pan, W.: A new algorithm and theory for penalized regression-based clustering, The Journal of Machine Learning Research, Vol. 17(1), pp. 6479-6503. 2016.

Examples

data(Hepta)
Data=Hepta$Data
out=PenalizedRegressionBasedClustering(Data,0.4,1,2,PlotIt=FALSE)
table(out$Cls,Hepta$Cls)

Cluster Identification using Projection Pursuit as described in [Hofmeyr/Pavlidis, 2019].

Description

Summarizes recent projection pursuit methods for clustering based on [Hofmeyr/Pavlidis, 2015], [Hofmeyr, 2016] and [Pavlidis et al., 2016] .

Usage

ProjectionPursuitClustering(Data,ClusterNo,Type="MinimumDensity",

PlotIt=FALSE,PlotSolution=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
Type

Either MinimumDensity[Pavlidis et al., 2016]

MaximumClusterbility[Hofmeyr/Pavlidis, 2015]], or

NormalisedCut [Hofmeyr, 2016] or KernelPCA [Hofmeyr/Pavlidis, 2019].

PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
PlotSolution

Plots the partioning solution as a tree as described in

...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

The details of the options for projection pursuit and partioning of data are defined in [Hofmeyr/Pavlidis, 2019].

"KernelPCA" uses additionally the package kernlab and is implemented as given in the fifth example on page 21, section "extension" of [Hofmeyr/Pavlidis, 2019].

The first idea of using non-PCA projections for clustering was published by [Bock, 1987] as an definition. However, to the knowledge of the author it was not applied to any data. The first systematic comparison to Projection-Pursuit Methods ProjectionPursuitClustering and AutomaticProjectionBasedClustering can be found in [Thrun/Ultsch, 2018]. For PCA-based clustering methods please see TandemClustering

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Points which cannot be assigned to a cluster will be reported with 0.

Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Hofmeyr/Pavlidis, 2015] Hofmeyr, D., & Pavlidis, N.: Maximum clusterability divisive clustering, Proc. 2015 IEEE Symposium Series on Computational Intelligence, pp. 780-786, IEEE, 2015.

[Hofmeyr/Pavlidis, 2019] Hofmeyr, D., & Pavlidis, N.: PPCI: an R Package for Cluster Identification using Projection Pursuit, The R Journal, 2019.

[Hofmeyr, 2016] Hofmeyr, D. P.: Clustering by minimum cut hyperplanes, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 39(8), pp. 1547-1560. 2016.

[Pavlidis et al., 2016] Pavlidis, N. G., Hofmeyr, D. P., & Tasoulis, S. K.: Minimum density hyperplanes, The Journal of Machine Learning Research, Vol. 17(1), pp. 5414-5446. 2016.

[Thrun/Ultsch, 2018] Thrun, M. C., & Ultsch, A.: Using Projection based Clustering to Find Distance and Density based Clusters in High-Dimensional Data, Journal of Classification, Vol. in revision, 2018.

[Bock, 1987] Bock, H.: On the interface between cluster analysis, principal component analysis, and multidimensional scaling, Multivariate statistical modeling and data analysis, (pp. 17-34), Springer, 1987.

Examples

data('Hepta')
out=ProjectionPursuitClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

Stochastic QT Clustering

Description

Stochastic quality clustering of [Heyer et al., 1999] with an improved implementation by [Scharl/Leisch, 2006].

Usage

QTclustering(Data,Radius,PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
Radius

Maximum radius of clusters. If NULL, automatic estimation can be done with [Thrun et al., 2016] if not otherwise set.
PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Points which cannot be assigned to a cluster will be reported with 0.

Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Heyer et al., 1999] Heyer, L. J., Kruglyak, S., & Yooseph, S.: Exploring expression data: identification and analysis of coexpressed genes, Genome research, Vol. 9(11), pp. 1106-1115. 1999.

[Scharl/Leisch, 2006] Scharl, T., & Leisch, F.: The stochastic QT-clust algorithm: evaluation of stability and variance on time-course microarray data, in Rizzi , A. & Vichi, M. (eds.), Proc. Proceedings in Computational Statistics (Compstat), pp. 1015-1022, Physica Verlag, Heidelberg, Germany, 2006.

[Thrun et al., 2016] Thrun, M. C., Lerch, F., Loetsch, J., & Ultsch, A. : Visualization and 3D Printing of Multivariate Data of Biomarkers, in Skala, V. (Ed.), International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG), Vol. 24, Plzen, 2016.

[Ultsch, 2005] Ultsch, A.: Pareto density estimation: A density estimation for knowledge discovery, In Baier, D. & Werrnecke, K. D. (Eds.), Innovations in classification, data science, and information systems, (Vol. 27, pp. 91-100), Berlin, Germany, Springer, 2005.

Examples

data('Hepta')
out=QTclustering(Hepta$Data,Radius=NULL,PlotIt=FALSE)

Robust Trimmed Clustering

Description

Robust Trimmed Clustering invented by [Garcia-Escudero et al., 2008] and implemented by [Fritz et al., 2012].

Usage

RobustTrimmedClustering(Data, ClusterNo,

Alpha=0.05,PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Alpha

No trimming is done equals to alpha =0, otherwise proportion of datapoints to be trimmed, tclust uses 0.05 as default.

...

Further arguments to be set for the clustering algorithm, e.g. ,nstart (number of random initializations),iter.max (maximum number of concentration steps),restr and restr.fact described in details. If not set, default arguments are used.

Details

"This iterative algorithm initializes k clusters randomly and performs "concentration steps" in order to improve the current cluster assignment. The number of maximum concentration steps to be performed is given by iter.max. For approximately obtaining the global optimum, the system is initialized nstart times and concentration steps are performed until convergence or iter.max is reached. When processing more complex data sets higher values of nstart and iter.max have to be specified (obviously implying extra computation time). ... The larger restr.fact is chosen, the looser is the restriction on the scatter matrices, allowing for more heterogeneity among the clusters. On the contrary, small values of restr.fact close to 1 imply very equally scattered clusters. This idea of constraining cluster scatters to avoid spurious solutions goes back to Hathaway (1985), who proposed it in mixture fitting problems" [Fritz et al., 2012]. The type of constraint restr can be set to "eigen", "deter" or "sigma.". Please see tclust for further parameter description.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Garcia-Escudero et al., 2008] Garcia-Escudero, L. A., Gordaliza, A., Matran, C., & Mayo-Iscar, A.: A general trimming approach to robust cluster analysis, The annals of Statistics, Vol. 36(3), pp. 1324-1345. 2008.

[Fritz et al., 2012] Fritz, H., Garcia-Escudero, L. A., & Mayo-Iscar, A.: tclust: An R package for a trimming approach to cluster analysis, Journal of statistical Software, Vol. 47(12), pp. 1-26. 2012.

Examples

data('Hepta')
out=RobustTrimmedClustering(Hepta$Data,ClusterNo=7,Alpha=0,PlotIt=FALSE)

SNN clustering

Description

Shared Nearest Neighbor Clustering of [Ertoz et al., 2003].

Usage

SharedNearestNeighborClustering(Data,Knn,

Radius,minPts,PlotIt=FALSE,

UpperLimitRadius,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
Knn

Number of neighbors to consider to calculate the shared nearest neighbors.

Radius

Eps [Ester et al., 1996, p. 227] neighborhood in the R-ball graph/unit disk graph), size of the epsilon neighborhood. If NULL, automatic estimation is done using insights of [Ultsch, 2005].

minPts

Number of minimum points in the eps region (for core points). In principle minimum number of points in the unit disk, if the unit disk is within the cluster (core) [Ester et al., 1996, p. 228]. if NULL, its 2.5 percent of points.

PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
UpperLimitRadius

Limit for radius search, experimental
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

..

Value

List of

Cls

[1:n] numerical vector defining the clustering; this classification is the main output of the algorithm. Points which cannot be assigned to a cluster will be reported as members of the noise cluster with 0.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Ertoz et al., 2003] Levent Ertoz, Michael Steinbach, Vipin Kumar: Finding Clusters of Different Sizes, Shapes, and Densities in Noisy, High Dimensional Data, SIAM International Conference on Data Mining, 47-59, 2003.

See Also

sNNclust

Examples

data('Hepta')
out=SharedNearestNeighborClustering(
Hepta$Data, Knn=7,Radius=NULL,minPts=NULL,PlotIt = FALSE)

self-organizing maps based clustering implemented by [Wherens, Buydens, 2017].

Description

Either the variant k-batch or k-online is possible in which every unit can be seen approximately as an cluster.

Usage

SOMclustering(Data,LC=c(1,2),ClusterNo=NULL,

Mode="online",PlotIt=FALSE,rlen=100,alpha = c(0.05, 0.01),...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
LC

Lines and Columns of a very small SOM, usually every unit is a cluster, will be ignored if ClusterNo is not NULL.
ClusterNo

Optional, A number k which defines k different clusters to be built by the algorithm. LC will then be set accordingly.
Mode

Either "batch" or "online"
PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
rlen

Please see supersom
alpha

Please see supersom
...

Further arguments to be set for the clustering algorithm in somgrid, if not set, default arguments are used.

Details

This clustering algorithm is based on very small maps and, hence, not emergent (c.f. [Thrun, 2018, p.37]). A 3x3 map means 9 units leading to 9 clusters.

Batch is a deterministic clustering approach whereas online is a stochastic clustering approach and research indicates that online should be preferred (c.f. [Thrun, 2018, p.37]).

Value

List of

Cls

[1:n] numerical vector defining the classification as the main output of the clustering algorithm
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Wherens, Buydens, 2017] R. Wehrens and L.M.C. Buydens, J. Stat. Softw. 21 (5), 2007; R. Wehrens and J. Kruisselbrink, submitted, 2017.

[Thrun, 2018] Thrun, M.C., Projection Based Clustering through Self-Organization and Swarm Intelligence. 2018, Heidelberg: Springer.

Examples

data('Hepta')
out=SOMclustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

SOTA Clustering

Description

Self-organizing Tree Algorithm (SOTA) introduced by [Herrero et al., 2001].

Usage

SOTAclustering(Data, ClusterNo,PlotIt=FALSE,UnrestGrowth,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
UnrestGrowth

TRUE: forces the ClusterNo option to uphold. FALSE: enables the algorithm to find its own number of clusters, in this cases ClusterNo should contain a high number because it is internally set as the number of iterations which is either reached or the max diversity criteria is satisfied priorly.

...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
sotaObject

Object defined by clustering algorithm as the other output of this algorithm

Note

*Luis Winckelman intergrated several function from clValid because it's ORPHANED.

Author(s)

Luis Winckelmann*, Vasyl Pihur, Guy Brock, Susmita Datta, Somnath Datta

References

[Herrero et al., 2001] Herrero, J., Valencia, A., & Dopazo, J.: A hierarchical unsupervised growing neural network for clustering gene expression patterns, Bioinformatics, Vol. 17(2), pp. 126-136. 2001.

Examples

#Does Work
data('Hepta')
out=SOTAclustering(Hepta$Data,ClusterNo=7)
table(Hepta$Cls,out$Cls)


#Does not work well
data('Lsun3D')
out=SOTAclustering(Lsun3D$Data,ClusterNo=100,PlotIt=FALSE,UnrestGrowth=FALSE)

Sparse Clustering

Description

Implements the sparse clustering methods of [Witten/Tibshirani, 2010].

Usage

SparseClustering(DataOrDistances, ClusterNo, Type="Hierarchical",

PlotIt=F,Silent=FALSE, NoPerms=10,Wbounds, ...)

Arguments

DataOrDistances

Either a [1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.

or a [1:n,1:n] symmetric distance matrix.

ClusterNo

Numeric indicating number to cluster to find in Tree/ Dendrogramm in case of Type="Hierachical" or numer of cluster to use in Type="kmeans"
Type

(optional) Char selecting methods Hierarchical or kmeans. Default: "Hierarchical"
PlotIt

(optional) Boolean. Default = FALSE = No plotting performed.
Silent

(optional) Boolean: print output or not (Default = FALSE = no output)
NoPerms

(optional), numeric scalar, Number of permutations.
Wbounds

(optional) numeric vector, range of tuning parameters to consider. This is the L1 bound on w, the feature weights [Witten/Tibshirani, 2010].
...

Further arguments passed on to sparcl HierarchicalSparseCluster or KMeansSparseCluster depending on Type.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm
Tree

Object Tree if Type="Hierachical" is used.

Note

Quality of clustering results varies between sparse hierarchical if data is given in comparison to the case that distances are given.

Author(s)

Quirin Stier, Michael Thrun

References

[Witten/Tibshirani, 2010] Witten, D. and Tibshirani, R.: A Framework for Feature Selection in Clustering. Journal of the American Statistical Association, Vol. 105(490), pp. 713-726, 2010.

Examples

# Hepta
data("Hepta")
Data = Hepta$Data
V1 = SparseClustering(Data, ClusterNo=7, Type="kmeans")
Cls1 = V1$Cls

V2 = SparseClustering(Data, ClusterNo=7, Type="Hierarchical")
Cls2 = V2$Cls

InputDistances = parallelDist::parDist(Data, method="euclidean")
DistanceMatrix = as.matrix(InputDistances)
V3 = SparseClustering(DistanceMatrix, ClusterNo=7, Type="Hierarchical")
Cls3 = V3$Cls

## Not run: 
set.seed(1)
Data = matrix(rnorm(100*50),ncol=50)
y    = c(rep(1,50),rep(2,50))
Data[y==1,1:25] = Data[y==1,1:25]+2

V1 = SparseClustering(Data, ClusterNo=2, Type="kmeans")
Cls1 = V1$Cls

## End(Not run)

Spectral Clustering

Description

Clusters the Data into "ClusterNo" different clusters using the Spectral Clustering method

Usage

SpectralClustering(Data, ClusterNo,PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
PlotIt

default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used. e.g.:

kernel : Kernelmethod, possible options: rbfdot Radial Basis kernel function "Gaussian" polydot Polynomial kernel function vanilladot Linear kernel function tanhdot Hyperbolic tangent kernel function laplacedot Laplacian kernel function besseldot Bessel kernel function anovadot ANOVA RBF kernel function splinedot Spline kernel stringdot String kernel

kpar : Kernelparameter: a character string or the list of hyper-parameters (kernel parameters). The default character string "automatic" uses a heuristic to determine a suitable value for the width parameter of the RBF kernel. "local" (local scaling) uses a more advanced heuristic and sets a width parameter for every point in the data set. A list can also be used containing the parameters to be used with the kernel function.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[Ng et al., 2002] Ng, A. Y., Jordan, M. I., & Weiss, Y.: On spectral clustering: Analysis and an algorithm, Advances in neural information processing systems, Vol. 2, pp. 849-856. 2002.

Examples

data('Hepta')
out=SpectralClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

Fast Adaptive Spectral Clustering [John et al, 2020]

Description

Spectrum is a self-tuning spectral clustering method for single or multi-view data. In this wrapper restricted to the standard use in other clustering algorithms.

Usage

Spectrum(Data, Type = 2, ClusterNo = NULL, 

PlotIt = FALSE, Silent = TRUE,PlotResults = FALSE, ...)

Arguments

Data

1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.

Type

Type=1: default eigengap method (Gaussian clusters)

Type=2: multimodality gap method (Gaussian/ non-Gaussian clusters)

Type=3: Allows to setClusterNo

ClusterNo

Optional, A number k which defines k different clusters to be built by the algorithm. For default ClusterNo=NULL please see details.

PlotIt

Default: FALSE, If TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
Silent

Silent progress of algorithm=TRUE

PlotResults

Plots result of spectrum with plot function

...

Method: numerical value: 1 = default eigengap method (Gaussian clusters), 2 = multimodality gap method (Gaussian/ non-Gaussian clusters), 3 = no automatic method (see fixk param)

Other parameters defined in Spectrum packages

Details

Spectrum is a partitioning algorithm and either uses the eigengap or multimodality gap heuristics to determine the number of clusters, please see Spectrum package for details

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[John et al, 2020] John, C. R., Watson, D., Barnes, M. R., Pitzalis, C., & Lewis, M. J.: Spectrum: Fast density-aware spectral clustering for single and multi-omic data. Bioinformatics, Vol. 36(4), pp. 1159-1166, 2020.

See Also

Spectrum

Examples

data('Hepta')
out=Spectrum(Hepta$Data,PlotIt=FALSE)

out=Spectrum(Hepta$Data,PlotIt=TRUE)

Pareto Density Estimation

Description

Density estimation for ggplot with a clear model behind it.

Format

The format is: Classes 'StatPDEdensity', 'Stat', 'ggproto' <ggproto object: Class StatPDEdensity, Stat> aesthetics: function compute_group: function compute_layer: function compute_panel: function default_aes: uneval extra_params: na.rm finish_layer: function non_missing_aes: parameters: function required_aes: x y retransform: TRUE setup_data: function setup_params: function super: <ggproto object: Class Stat>

Details

PDE was published in [Ultsch, 2005], short explanation in [Thrun, Ultsch 2018] and the PDE optimized violin plot was published in [Thrun et al., 2018].

References

[Ultsch,2005] Ultsch, A.: Pareto density estimation: A density estimation for knowledge discovery, in Baier, D.; Werrnecke, K. D., (Eds), Innovations in classification, data science, and information systems, Proc Gfkl 2003, pp 91-100, Springer, Berlin, 2005.

[Thrun, Ultsch 2018] Thrun, M. C., & Ultsch, A. : Effects of the payout system of income taxes to municipalities in Germany, in Papiez, M. & Smiech,, S. (eds.), Proc. 12th Professor Aleksander Zelias International Conference on Modelling and Forecasting of Socio-Economic Phenomena, pp. 533-542, Cracow: Foundation of the Cracow University of Economics, Cracow, Poland, 2018.

[Thrun et al, 2018] Thrun, M. C., Pape, F., & Ultsch, A. : Benchmarking Cluster Analysis Methods using PDE-Optimized Violin Plots, Proc. European Conference on Data Analysis (ECDA), accepted, Paderborn, Germany, 2018.

Algorithms for Subspace clustering

Description

Subspace (projected) clustering is a technique which finds clusters within different subspaces (a selection of one or more dimensions).

Usage

SubspaceClustering(Data,ClusterNo,DimSubspace,

Type='Orclus',PlotIt=FALSE,OrclusInitialClustersNo=ClusterNo+2,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases or d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the proclus or orclust algorithm.
DimSubspace

Numerical number defining the dimensionality in which clusters should be search in in the orclust algorithm, for proclus it is an optional parameter
Type

'Orclus', subspace clustering based on arbitrarily oriented projected cluster generation [Aggarwal and Yu, 2000]

'ProClus' ProClus algorithm for subspace clustering [Aggarwal/Wolf, 1999]

'Clique' ProClus algorithm finds subspaces of high-density clusters [Agrawal et al., 1999] and [Agrawal et al., 2005]

'SubClu' SubClu algorithm is a density-connected approach for subspace clustering [Kailing et al.,2004]

PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
OrclusInitialClustersNo

Only for Orclus algorithm: Initial number of clusters (that are computed in the entire data space) must be greater than k. The number of clusters is iteratively decreased by a factor until the final number of k clusters is reached.
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

For Subclue: "epsilon" and "minSupport", see DBSCAN

For Clique: "xi" (number of intervals for each dimension) and "tau" (Density Threshold), see DBSCAN

Details

Subspace clustering algorithms have the goal to finde one or more subspaces with the assumation that sufficient dimensionality reduction is dimensionality reduction without loss of information. Hence subspace clustering aums at finding a linear subspace sucht that the subspace contains as much predictive information as the input space. The subspace is usually higher than two but lower than the input space. In contrast, projection-based clustering AutomaticProjectionBasedClustering projects the data (nonlinear) into two dimensions and tries only to preerve relevant neighborhoods.

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering.
Object

Object defined by clustering algorithm as the other output of this algorithm

Note

JAVA_HOME has to be set for rJava to the ProClus algorithm (in windows set PATH env. variable to .../bin path of Java. The architecture of R and Java have to match. Java automatically downloads the Java version of the browser which may not be installed in the architecture in R. In such a case choose a Java version manually.

Author(s)

Michael Thrun

References

[Aggarwal/Wolf et al., 1999] Aggarwal, C. C., Wolf, J. L., Yu, P. S., Procopiuc, C., & Park, J. S.: Fast algorithms for projected clustering, Proc. ACM SIGMoD Record, Vol. 28, pp. 61-72, ACM, 1999.

[Aggarwal/Yu, 2000] Aggarwal, C. C., & Yu, P. S.: Finding generalized projected clusters in high dimensional spaces, (Vol. 29), ACM, ISBN: 1581132174, 2000.

[Agrawal et al., 1999]: Rakesh Agrawal, Johannes Gehrke, Dimitrios Gunopulos, and Prabhakar Raghavan: Automatic Subspace Clustering of High Dimensional Data for Data Mining Applications, In Proc. ACM SIGMOD, 1999.

[Agrawal et al., 2005] Agrawal, R., Gehrke, J., Gunopulos, D., & Raghavan, P.: Automatic subspace clustering of high dimensional data, Data Mining and Knowledge Discovery, Vol. 11(1), pp. 5-33. 2005.

[Kailing et al.,2004] Kailing, Karin, Hans-Peter Kriegel, and Peer Kroeger: Density-connected subspace clustering for high-dimensional data, Proceedings of the 2004 SIAM international conference on data mining. Society for Industrial and Applied Mathematics, 2004

Examples

data('Hepta')
out=SubspaceClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

Tandem Clustering

Description

Summarizes clustering methods that combine k-means and pca

Usage

TandemClustering(Data,ClusterNo,Type="Reduced",PlotIt=FALSE,...)

Arguments

Data

[1:n,1:d] matrix of dataset to be clustered. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
ClusterNo

A number k which defines k different clusters to be built by the algorithm.
Type

Reduced: Reduced k-means (RKM) [De Soete/Carroll, 1994].

Factorial: Factorial k-mean (FKM) [Vichi/Kiers, 2001]

KernelPCA: Kernel PCA with minimum normalised cut hyperplanes [Hofmeyr/Pavlidis, 2019]

PlotIt

Default: FALSE, if TRUE plots the first three dimensions of the dataset with colored three-dimensional data points defined by the clustering stored in Cls
...

Further arguments to be set for the clustering algorithm, if not set, default arguments are used.

Details

If the ClusterNo exceeds the number of dimensions, than the function is called recursively with ClusterNo=2. In each iteration the cluster with the highest number of overall points is clustered again, until the number of clusters is met.

"KernelPCA" uses addtionally the package kernlab and is implemented as given in the fifth example on page 18, section "extension" of [Hofmeyr/Pavlidis, 2019]

The first idea of using non-PCA projections for clustering was published by [Bock, 1987] as an definition. However, to the knowledge of the author it was not applied to any data. The first systematic comparison to Projection-Pursuit Methods ProjectionPursuitClustering and AutomaticProjectionBasedClustering can be found in [Thrun/Ultsch, 2018].

Value

List of

Cls

[1:n] numerical vector with n numbers defining the classification as the main output of the clustering algorithm. It has k unique numbers representing the arbitrary labels of the clustering. Points which cannot be assigned to a cluster will be reported with 0.

Object

Object defined by clustering algorithm as the other output of this algorithm

Author(s)

Michael Thrun

References

[De Soete/Carroll, 1994] De Soete, G., & Carroll, J. D.: K-means clustering in a low-dimensional Euclidean space, New approaches in classification and data analysis, (pp. 212-219), Springer, 1994.

[Hofmeyr/Pavlidis, 2019] Hofmeyr, D., & Pavlidis, N.: PPCI: an R Package for Cluster Identification using Projection Pursuit, The R Journal, 2019.

[Vichi/Kiers, 2001] Vichi, M., & Kiers, H. A.: Factorial k-means analysis for two-way data, Computational Statistics & Data Analysis, Vol. 37(1), pp. 49-64. 2001.

[Thrun/Ultsch, 2018] Thrun, M. C., & Ultsch, A.: Using Projection based Clustering to Find Distance and Density based Clusters in High-Dimensional Data, Journal of Classification, Vol. in revision, 2018.

[Bock, 1987] Bock, H.: On the interface between cluster analysis, principal component analysis, and multidimensional scaling, Multivariate statistical modeling and data analysis, (pp. 17-34), Springer, 1987.

Examples

data('Hepta')
out=TandemClustering(Hepta$Data,ClusterNo=7,PlotIt=FALSE)

Target introduced in [Ultsch, 2005].

Description

Detailed description of dataset and its clustering challenge of outliers is provided in [Thrun/Ultsch, 2020]

Usage

data("Target")

Details

Size 770, Dimensions 2, stored in Target$Data

Classes 6, stored in Target$Cls

References

[Ultsch, 2005] Ultsch, A.: U* C: Self-organized Clustering with Emergent Feature Maps, Proc. Lernen, Wissensentdeckung und Adaptivitaet (LWA/FGML), pp. 240-244, Saarbruecken, Germany, 2005.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(Target)
str(Target)

Tetra introduced in [Ultsch, 1993]

Description

Almost touching clusters. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("Tetra")

Details

Size 400, Dimensions 3, stored in Tetra$Data

Classes 4, stored in Tetra$Cls

References

[Ultsch, 1993] Ultsch, A.: Self-organizing neural networks for visualisation and classification, Information and classification, (pp. 307-313), Springer, 1993.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(Tetra)
str(Tetra)

TwoDiamonds introduced in [Ultsch, 2003a, 2003b]

Description

Cluster border defined by density. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("TwoDiamonds")

Details

Size 800, Dimensions 2, stored in TwoDiamonds$Data

Classes 2, stored in TwoDiamonds$Cls

References

[Ultsch, 2003a] Ultsch, A.Optimal density estimation in data containing clusters of unknown structure, technical report, Vol. 34,University of Marburg, Department of Mathematics and Computer Science, 2003.

[Ultsch, 2003b] Ultsch, A.: U*-matrix: a tool to visualize clusters in high dimensional data, Fachbereich Mathematik und Informatik, 2003.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(TwoDiamonds)
str(TwoDiamonds)

WingNut introduced in [Ultsch, 2005]

Description

Density vs. distance. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("WingNut")

Details

Size 1016, Dimensions 2, stored in WingNut$Data

Classes 2, stored in WingNut$Cls

References

[Ultsch, 2005] Ultsch, A.: Clustering wih SOM: U* C, Proc. Proceedings of the 5th Workshop on Self-Organizing Maps, Vol. 2, pp. 75-82, 2005.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(WingNut)
str(WingNut)