| Type: | Package |
| Title: | Simultaneous Clustering and (or) Dimensionality Reduction |
| Version: | 0.1 |
| Date: | 2024-04-19 |
| Description: | Methods for simultaneous clustering and dimensionality reduction such as: Double k-means, Reduced k-means, Factorial k-means, Clustering with Disjoint PCA but also methods for exclusively dimensionality reduction: Disjoint PCA, Disjoint FA. The statistical methods implemented refer to the following articles: de Soete G., Carroll J. (1994) "K-means clustering in a low-dimensional Euclidean space" <doi:10.1007/978-3-642-51175-2_24> ; Vichi M. (2001) "Double k-means Clustering for Simultaneous Classification of Objects and Variables" <doi:10.1007/978-3-642-59471-7_6> ; Vichi M., Kiers H.A.L. (2001) "Factorial k-means analysis for two-way data" <doi:10.1016/S0167-9473(00)00064-5> ; Vichi M., Saporta G. (2009) "Clustering and disjoint principal component analysis" <doi:10.1016/j.csda.2008.05.028> ; Vichi M. (2017) "Disjoint factor analysis with cross-loadings" <doi:10.1007/s11634-016-0263-9>. |
| License: | GPL (≥ 3) |
| Encoding: | UTF-8 |
| Imports: | Rcpp, RcppArmadillo, fpc, cluster, factoextra, pheatmap |
| LinkingTo: | Rcpp, RcppArmadillo |
| RoxygenNote: | 7.3.1 |
| NeedsCompilation: | yes |
| Packaged: | 2024-04-19 15:48:27 UTC; Ionel |
| Author: | Ionel Prunila [aut, cre], Maurizio Vichi [aut] |
| Maintainer: | Ionel Prunila <ionel.prunila@uniroma1.it> |
| Repository: | CRAN |
| Date/Publication: | 2024-04-22 18:22:47 UTC |
Cronbach Alpha
Description
Computes the Cronbach Alpha index on a units x variables data matrix. It measures the internal reliability, i.e., the propensity of J variables of a data matrix (n units x J variables) to be concordantly correlated with a single factor (composite indicator).
Usage
CronbachAlpha(X)
Arguments
X |
Units x variables numeric data matrix. |
Value
as |
Cronbach's Alpha |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Cronbach L. J. (1951) "Coefficient alpha and the internal structure of tests" <doi:10.1007/BF02310555>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# standardizing the data
iris <- scale(iris)
# compute Cronbach's Alpha
as <- CronbachAlpha(iris)
pseudoF (pF or Calinski-Harabsz) index for choosing k in partitioning models
Description
Calculates and plots the CH index for k = 2, ..., maxK. The function provides an interval wide (2tol*pF) so that the choice of K is less conservative. Instead of just choosing the maximum pF, if it exists, picks the value such that its upper bound is larger than max pF.
Usage
apseudoF(data, maxK, tol, model, Q)
Arguments
data |
Units x variables numeric data matrix. |
maxK |
Maximum number of clusters for the units to be tested. |
tol |
Approximation value. It is half of the length of theinterval put for each pF. 0 <= tol < 1. Its default value is 0.05. |
model |
Partitioning Models to run for each value of k. (1 = doublekm; 2 = redkm; 3 = factkm; 4 = dpcakm) |
Q |
Number of principal components w.r.t. variables selected for the maxK -1 partitions to be tested. |
Value
bestK |
best value of K (scalar). |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Calinski T., Harabasz J. (1974) "A dendrite method for cluster analysis" <doi:10.1080/03610927408827101>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
apF <- apseudoF(iris, maxK=10, tol = 0.05, model = 3, Q = 2)
Ward-dendrogeam of centroids of partitioning models
Description
Plots the Ward-dendrogram of the centroids of a partitioning model. The plot is useful as a diagnosis tool for the choice o the number of clusters.
Usage
centree(drclust_out)
Arguments
drclust_out |
Output of either doublekm, redkm, factkm or dpcakm. |
Value
centroids-dkm |
Centroids x centroids distance matrix. |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Ward J. H. (1963) "Hierarchical Grouping to Optimize an Objective Function" <doi:10.1080/01621459.1963.10500845>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
dc_out <- dpcakm(iris, 20, 3)
d <- centree(dc_out)
classification variable
Description
Recodes the binary and row-stochastic membership matrix U into the classification variable (similar to the "cluster" output returned by kmeans()).
Usage
cluster(U)
Arguments
U |
Binary and row-stochastic matrix. |
Value
cl |
vector of length n indicating, for each element, the index of the cluster to which it has been assigned. |
Author(s)
Ionel Prunila, Maurizio Vichi
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# standardizing the data
iris <- scale(iris)
# double k-means with 3 unit-clusters and 2 components for the variables
p1 <- redkm(iris, K = 3, Q = 2)
cl <- cluster(p1$U)
Disjoint Factor Analysis
Description
Performs disjoint factor analysis, i.e., a Factor Analysis with a simple structure. In fact, each factor is defined by a disjoint subset of variables, resulting thus, in a simplified, easier to interpret loading matrix A and factors. Estimation is carried out via Maximum Likelihood.
Usage
disfa(X, Q, Rndstart, verbose, maxiter, tol, constr, prep, print)
Arguments
X |
Units x variables numeric data matrix. |
Q |
Number of factors. |
Rndstart |
Number of runs to be performed (Defaults is 20). |
verbose |
Outputs basic summary statistics for each run (1 = enabled; 0 = disabled, default option). |
maxiter |
Maximum number of iterations allowed (if convergence is not yet reached. Default is 100). |
tol |
Tolerance threshold (maximum difference between the values of the objective function of two consecutive iterations such that convergence is assumed. Default is 1e-6). |
constr |
is a vector of length J = nr. of variables, pre-specifying to which cluster some of the variables must be assigned. Each component of the vector can assume integer values from 1 o Q (See example for more details), or 0 if no constraint on the variable is imposed (i.e., it will be assigned based on the plain algorithm). |
prep |
Pre-processing of the data. 1 performs the z-score transform (default choice); 2 performs the min-max transform; 0 leaves the data un-pre-processed. |
print |
Prints summary statistics of the performed method (1 = enabled; 0 = disabled, default option). |
Value
returns a list of estimates and some descriptive quantities of the final results.
V |
Variables x factors membership matrix (binary and row-stochastic). Each row is a dummy variable indicating to which cluster each variable has been assigned. |
A |
Variables x components loading matrix. |
Psi |
Specific variance of each observed variable, not accounted for by the common factors (matrix). |
discrepancy |
Value of the objective function, to be minimized. Difference between the observed and estimated covariance matrices (scalar). |
RMSEA |
Adjusted Root Mean Squared Error (scalar). |
AIC |
Aikake Information Criterion (scalar). |
BIC |
Bayesian Information Criterion (scalar). |
GFI |
Goodness of Fit Index (scalar). |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Vichi M. (2017) "Disjoint factor analysis with cross-loadings" <doi:10.1007/s11634-016-0263-9>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# No constraint on variables
out <- disfa(iris, Q = 2)
# Constraint: the first two variables must contribute to the same factor.
outc <- disfa(iris, Q = 2, constr = c(1,1,0,0))
Disjoint Principal Components Analysis
Description
Performs disjoint PCA, that is, a simplified version of PCA. Computes each one of the Q principal components from a different subset of the J variables (resulting thus, in a simplified, easier to interpret loading matrix A).
Usage
dispca(X, Q, Rndstart, verbose, maxiter, tol, prep, print, constr)
Arguments
X |
Units x variables numeric data matrix. |
Q |
Number of factors. |
Rndstart |
Number of runs to be performed (Defaults is 20). |
verbose |
Outputs basic summary statistics for each run (1 = enabled; 0 = disabled, default option). |
maxiter |
Maximum number of iterations allowed (if convergence is not yet reached. Default is 100). |
tol |
Tolerance threshold (maximum difference between the values of the objective function of two consecutive iterations such that convergence is assumed). Default is 1e-6. |
prep |
Pre-processing of the data. 1 performs the z-score transform (default choice); 2 performs the min-max transform; 0 leaves the data un-pre-processed. |
print |
Prints summary statistics of the results (1 = enabled; 0 = disabled, default option). |
constr |
is a vector of length J = nr. of variables, pre-specifying to which cluster some of the variables must be assigned. Each component of the vector can assume integer values from 1 o Q (See example for more details), or 0 if no constraint on the variable is imposed (i.e., it will be assigned based on the plain algorithm). |
Value
returns a list of estimates and some descriptive quantities of the final results.
V |
Variables x factors membership matrix (binary and row-stochastic). Each row is a dummy variable indicating to which cluster it has been assigned. |
A |
Variables x components loading matrix. |
betweenss |
Amount of deviance captured by the model (scalar). |
totss |
total amount of deviance (scalar). |
size |
Number of variables assigned to each column-cluster (vector). |
loop |
The index of the (best) run from which the results have been chosen. |
it |
the number of iterations performed during the (best) run. |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Vichi M., Saporta G. (2009) "Clustering and disjoint principal component analysis" <doi:10.1016/j.csda.2008.05.028>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# No constraint on variables
out <- dispca(iris, Q = 2)
# Constraint: the first two variables must contribute to the same factor.
outc <- dispca(iris, Q = 2, constr = c(1,1,0,0))
Double k-means Clustering
Description
Performs simultaneous k-means partitioning on units and variables (rows and columns of the data matrix).
Usage
doublekm(Xs, K, Q, Rndstart, verbose, maxiter, tol, prep, print)
Arguments
Xs |
Units x variables numeric data matrix. |
K |
Number of clusters for the units. |
Q |
Number of clusters for the variables. |
Rndstart |
Number of runs to be performed (Defaults is 20). |
verbose |
Outputs basic summary statistics for each run (1 = enabled; 0 = disabled, default option). |
maxiter |
Maximum number of iterations allowed (if convergence is not yet reached. Default is 100). |
tol |
Tolerance threshold. It is the maximum difference between the values of the objective function of two consecutive iterations such that convergence is assumed (default is 1e-6). |
prep |
Pre-processing of the data. 1 performs the z-score transform (default choice); 2 performs the min-max transform; 0 leaves the data un-pre-processed. |
print |
Prints summary statistics of the results (1 = enabled; 0 = disabled, default option). |
Value
returns a list of estimates and some descriptive quantities of the final results.
U |
Units x clusters membership matrix (binary and row-stochastic). Each row is a dummy variable indicating to which unit-cluster each unit has been assigned. |
V |
Variables x clusters membership matrix (binary and row-stochastic). Each row is a dummy variable indicating to which variable-cluster each variable has been assigned. |
centers |
K x Q matrix of centers containing the row means expressed in terms of column means. |
totss |
The total sum of squares (scalar). |
withinss |
Vector of within-row-cluster sum of squares, one component per cluster. |
columnwise_withinss |
Vector of within-column-cluster sum of squares, one component per cluster. |
betweenss |
Amount of deviance captured by the model (scalar). |
K-size |
Number of units assigned to each row-cluster (vector). |
Q-size |
Number of variables assigned to each column-cluster (vector). |
pseudoF |
Calinski-Harabasz index of the resulting (row-) partition (scalar). |
loop |
The index of the (best) run from which the results have been chosen. |
it |
the number of iterations performed during the (best) run. |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Vichi M. (2001) "Double k-means Clustering for Simultaneous Classification of Objects and Variables" <doi:10.1007/978-3-642-59471-7_6>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# double k-means with 3 unit-clusters and 2 variable-clusters
out <- doublekm(iris, K = 3, Q = 2)
Clustering with Disjoint Principal Components Analysis
Description
Performs simultaneously k-means partitioning on units and disjoint PCA on the variables, computing each principal component from a different subset of variables. The result is a simplified, easier to interpret loading matrix A, the principal components and the clustering. The reduced subspace is identified by the centroids.
Usage
dpcakm(X, K, Q, Rndstart, verbose, maxiter, tol, constr, print, prep)
Arguments
X |
Units x variables numeric data matrix. |
K |
Number of clusters for the units. |
Q |
Number of principal components. |
Rndstart |
Number of runs to be performed (Defaults is 20). |
verbose |
Outputs basic summary statistics for each run (1 = enabled; 0 = disabled, default option). |
maxiter |
Maximum number of iterations allowed (if convergence is not yet reached. Default is 100). |
tol |
Tolerance threshold (maximum difference between the values of the objective function of two consecutive iterations such that convergence is assumed. Default is 1e-6). |
constr |
is a vector of length J = nr. of variables, pre-specifying to which cluster some of the variables must be assigned. Each component of the vector can assume integer values from 1 o Q = nr. of variable-cluster / principal components (See examples for more details), or 0 if no constraint on the variable is imposed (i.e., it will be assigned based on the plain algorithm). |
print |
Prints summary statistics of the results (1 = enabled; 0 = disabled, default option). |
prep |
Pre-processing of the data. 1 performs the z-score transform (default choice); 2 performs the min-max transform; 0 leaves the data un-pre-processed. |
Value
returns a list of estimates and some descriptive quantities of the final results.
V |
Variables x factors membership matrix (binary and row-stochastic). Each row is a dummy variable indicating to which cluster each variable has been assigned. |
U |
Units x clusters membership matrix (binary and row-stochastic). Each row is a dummy variable indicating to which cluster each unit has been assigned. |
A |
Variables x components loading matrix. |
centers |
K x Q matrix of centers containing the row means expressed in the reduced space of Q principal components. |
totss |
The total sum of squares (scalar). |
withinss |
Vector of within-cluster sum of squares, one component per cluster. |
betweenss |
Amount of deviance captured by the model (scalar). |
K-size |
Number of units assigned to each row-cluster (vector). |
Q-size |
Number of variables assigned to each column-cluster (vector). |
pseudoF |
Calinski-Harabasz index of the resulting partition (scalar). |
loop |
The index of the (best) run from which the results have been chosen. |
it |
the number of iterations performed during the (best) run. |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Vichi M., Saporta G. (2009) "Clustering and disjoint principal component analysis" <doi:10.1016/j.csda.2008.05.028>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# No constraint on variables
out <- dpcakm(iris, K = 3, Q = 2, Rndstart = 5)
# Constraint: the first two variables must contribute to the same factor.
outc <- dpcakm(iris, K = 3, Q = 2, Rndstart = 5,constr = c(1,1,0,0))
double pseudoF (Calinski-Harabsz) index
Description
A pseudoF version for double partitioning, for the choice of the number of clusters of the units and variables (rows and columns of the data matrix). It is a diagnostic tool for inspecting simultaneously the optimal number of unit-clusters and variable-clusters.
Usage
dpseudoF(data, maxK, maxQ)
Arguments
data |
Units x variables numeric data matrix. |
maxK |
Maximum number of clusters for the units to be tested. |
maxQ |
Maximum number of clusters for the variables to be tested. |
Value
dpseudoF |
matrix containing the pF value for each pair of K and Q within the specified range |
Author(s)
Ionel Prunila, Maurizio Vichi
References
R. Rocci, M. Vichi (2008)" Two-mode multi-partitioning" <doi:10.1016/j.csda.2007.06.025>
T. Calinski & J. Harabasz (1974). A dendrite method for cluster analysis. Communications in Statistics, 3:1, 1-27
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
dpeudoF <- dpseudoF(iris, maxK=10, maxQ = 3)
Factorial k-means
Description
Performs simultaneously k-means partitioning on units and principal component analysis on the variables. Identifies the best partition in a Least-Squares sense in the best reduced space of the data. Both the data and the centroids are used to identify the best Least-Squares reduced subspace, where also their distances is measured.
Usage
factkm(X, K, Q, Rndstart, verbose, maxiter, tol, rot, prep, print)
Arguments
X |
Units x variables numeric data matrix. |
K |
Number of clusters for the units. |
Q |
Number of principal components w.r.t. variables. |
Rndstart |
Number of runs to be performed (Defaults is 20). |
verbose |
Outputs basic summary statistics for each run (1 = enabled; 0 = disabled, default option). |
maxiter |
Maximum number of iterations allowed (if convergence is not yet reached. Default is 100). |
tol |
Tolerance threshold (maximum difference in the values of the objective function of two consecutive iterations such that convergence is assumed. Default is 1e-6). |
rot |
performs varimax rotation of axes obtained via PCA. (=1 enabled; =0 disabled, default option) |
prep |
Pre-processing of the data. 1 performs the z-score transform (default choice); 2 performs the min-max transform; 0 leaves the data un-pre-processed. |
print |
Prints summary statistics of the results (1 = enabled; 0 = disabled, default option). |
Value
returns a list of estimates and some descriptive quantities of the final results.
U |
Units x clusters membership matrix (binary and row-stochastic). Each row is a dummy variable indicating to which cluster each unit has been assigned. |
A |
Variables x components loading matrix (orthonormal). |
centers |
K x Q matrix of centers containing the row means expressed in the reduced space of Q principal components. |
totss |
The total sum of squares. |
withinss |
Vector of within-cluster sum of squares, one component per cluster. |
betweenss |
amount of deviance captured by the model. |
size |
Number of units assigned to each cluster. |
pseudoF |
Calinski-Harabasz index of the resulting partition. |
loop |
The index of the (best) run from which the results have been chosen. |
it |
the number of iterations performed during the (best) run. |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Vichi M., Kiers H.A.L. (2001) "Factorial k-means analysis for two-way data" <doi:10.1016/S0167-9473(00)00064-5>
Kaiser H.F. (1958) "The varimax criterion for analytic rotation in factor analysis" <doi:10.1007/BF02289233>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# factorial k-means with 3 unit-clusters and 2 components for the variables
out <- factkm(iris, K = 3, Q = 2, Rndstart = 15, verbose = 0, maxiter = 100, tol = 1e-7, rot = 1)
Heatmap of a partition in a reduced subspace
Description
Plots the heatmap of a partition on a reduced subspace obtained via either: doublekm, redkm, factkm or dpcakm.
Usage
heatm(data, drclust_out)
Arguments
data |
Units x variables data matrix. |
drclust_out |
Out of either doublekm, redkm, factkm or dpcakm. |
Value
No return value, called for side effects
Author(s)
Ionel Prunila, Maurizio Vichi
References
Kolde R. (2019) "pheatmap: Pretty Heatmaps" <https://cran.r-project.org/web/packages/pheatmap/index.html>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# standardizing the data
iris <- scale(iris)
# applying a clustering algorithm
drclust_out <- dpcakm(iris, 20, 3)
# obtain a heatmap based on the output of the clustering algorithm and the data
h <- heatm(iris, drclust_out)
Selecting the number of principal components to be extracted from a dataset
Description
Selects the optimal number of principal components to be extracted from a dataset based on Kaiser's criterion
Usage
kaiserCrit(data)
Arguments
data |
Units x variables data matrix. |
Value
bestQ |
Number of components to be extracted (scalar). |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Kaiser H. F. (1960) "The Application of Electronic Computers to Factor Analysis" <doi:10.1177/001316446002000>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- scale(as.matrix(iris[,-5]))
# Apply the Kaiser rule
h <- kaiserCrit(iris)
Adjusted Rand Index
Description
Performs the Adjusted Rand Index on a confusion matrix (row-by-column product of two partition-matrices). ARI is a measure of the similarity between two data clusterings.
Usage
mrand(N)
Arguments
N |
Confusion matrix. |
Value
mri |
Adjusted Rand Index of a confusion matrix (scalar). |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Rand W. M. (1971) "Objective criteria for the evaluation of clustering methods" <doi:10.2307/2284239>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# standardizing the data
iris <- scale(iris)
# double k-means with 3 unit-clusters and 2 components for the variables
p1 <- redkm(iris, K = 3, Q = 2, Rndstart = 10)
p2 <- doublekm(iris, K=3, Q=2, Rndstart = 10)
mri <- mrand(t(p1$U)%*%p2$U)
k-means on a reduced subspace
Description
Performs simultaneously k-means partitioning on units and principal component analysis on the variables.
Usage
redkm(X, K, Q, Rndstart, verbose, maxiter, tol, rot, prep, print)
Arguments
X |
Units x variables numeric data matrix. |
K |
Number of clusters for the units. |
Q |
Number of principal components w.r.t. variables. |
Rndstart |
Number of runs to be performed (Defaults is 20). |
verbose |
Outputs basic summary statistics for each run (1 = enabled; 0 = disabled, default option). |
maxiter |
Maximum number of iterations allowed (if convergence is not yet reached. Default is 100). |
tol |
Tolerance threshold (maximum difference between the values of the objective function of two consecutive iterations such that convergence is assumed. Default is 1e-6). |
rot |
performs varimax rotation of axes obtained via PCA. (=1 enabled; =0 disabled, default option) |
prep |
Pre-processing of the data. 1 performs the z-score transform (default choice); 2 performs the min-max transform; 0 leaves the data un-pre-processed. |
print |
Tolerancestats summary statistics of the performed method (1 = enabled; 0 = disabled, default option). |
Value
returns a list of estimates and some descriptive quantities of the final results.
U |
Units x clusters membership matrix (binary and row-stochastic). Each row is a dummy variable indicating to which cluster each unit has been assigned. |
A |
Variables x components loading matrix (orthonormal). |
centers |
K x Q matrix of centers containing the row means expressed in the reduced space of Q principal components. |
totss |
The total sum of squares (scalar). |
withinss |
Vector of within-cluster sum of squares, one component per cluster. |
betweenss |
Amount of deviance captured by the model (scalar). |
size |
Number of units assigned to each cluster (vector). |
pseudoF |
Calinski-Harabasz index of the resulting partition (scalar). |
loop |
The index of the (best) run from which the results have been chosen. |
it |
the number of iterations performed during the (best) run. |
Author(s)
Ionel Prunila, Maurizio Vichi
References
de Soete G., Carroll J. (1994) "K-means clustering in a low-dimensional Euclidean space" <doi:10.1007/978-3-642-51175-2_24>
Kaiser H.F. (1958) "The varimax criterion for analytic rotation in factor analysis" <doi:10.1007/BF02289233>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
# reduced k-means with 3 unit-clusters and 2 components for the variables
out <- redkm(iris, K = 3, Q = 2, Rndstart = 15, verbose = 0, maxiter = 100, tol = 1e-7, rot = 1)
Silhouette
Description
Computes and plots the silhouette of a partition
Usage
silhouette(data, drclust_out)
Arguments
data |
Units x variables data matrix. |
drclust_out |
Out of either doublekm, redkm, factkm or dpcakm. |
Value
cl.silhouette |
Silhouette index for the given partition, for each object (matrix). |
fe.silhouette |
Factoextra silhouette graphical object |
Author(s)
Ionel Prunila, Maurizio Vichi
References
Rousseeuw P. J. (1987) "Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis" <doi:10.1016/0377-0427(87)90125-7>
Maechler M. et al. (2023) "cluster: Cluster Analysis Basics and Extensions" <https://CRAN.R-project.org/package=cluster>
Kassambara A. (2022) "factoextra: Extract and Visualize the Results of Multivariate Data Analyses" <https://cran.r-project.org/web/packages/factoextra/index.html>
Examples
# Iris data
# Loading the numeric variables of iris data
iris <- as.matrix(iris[,-5])
#standardizing the data
iris <- scale(iris)
#applying a clustering algorithm
drclust_out <- dpcakm(iris, 20, 3)
#silhouette based on the data and the output of the clustering algorithm
d <- silhouette(iris, drclust_out)