Type:	Package
Title:	Tandem Clustering with Invariant Coordinate Selection
Version:	0.1.0
Date:	2023-09-20
Description:	Implementation of tandem clustering with invariant coordinate selection with different scatter matrices and several choices for the selection of components as described in Alfons, A., Archimbaud, A., Nordhausen, K.and Ruiz-Gazen, A. (2022) <doi:10.48550/arXiv.2212.06108>.
License:	GPL (≥ 3)
Encoding:	UTF-8
Depends:	ICS (≥ 1.4-0), ggplot2
Imports:	cluster, fpc, GGally, heplots, mclust, moments, mvtnorm, otrimle, RcppRoll, rrcov, scales, tclust
LinkingTo:	Rcpp, RcppArmadillo
Suggests:	testthat (≥ 3.0.0)
URL:	https://github.com/AuroreAA/ICSClust
BugReports:	https://github.com/AuroreAA/ICSClust/issues
Author:	Aurore Archimbaud [aut, cre], Andreas Alfons [aut], Klaus Nordhausen [aut], Anne Ruiz-Gazen [aut]
Maintainer:	Aurore Archimbaud <aurore.archimbaud@live.fr>
RoxygenNote:	7.2.3
Config/testthat/edition:	3
NeedsCompilation:	yes
Packaged:	2023-09-20 16:53:56 UTC; auror
Repository:	CRAN
Date/Publication:	2023-09-21 13:20:02 UTC

Tandem Clustering with Invariant Coordinate Selection

Description

Implementation of tandem clustering with invariant coordinate selection with different scatter matrices and several choices for the selection of components as described in Alfons, A., Archimbaud, A., Nordhausen, K.and Ruiz-Gazen, A. (2022) <arXiv:2212.06108>.

Details

The DESCRIPTION file:

Index of help topics:

ICSClust                Tandem clustering with ICS
ICSClust-package        Tandem Clustering with Invariant Coordinate
                        Selection
ICS_lcov                Local Shape Scatter Estimates for ICS
ICS_mcd                 MCD location and Scatter Estimates for ICS
ICS_mlc                 Cauchy location and Scatter Estimates for ICS
ICS_tcov                Pairwise one-step M-estimate of scatter for ICS
ICS_ucov                Simple robust estimates of scatter for ICS
component_plot          Scatterplot Matrix with densities on the
                        diagonal
discriminatory_crit     Selection of ICS components based on
                        discriminatory power
kmeans_clust            _k_-means clustering
mclust_clust            Model-Based Clustering
med_crit                Selection of Invariant components using the med
                        criterion
mixture_sim             Simulation of a mixture of Gaussian
                        distributions
normal_crit             Selection of Non-normal Invariant Components
                        Using Marginal Normality Tests
pam_clust               Partitioning Around Medoids clustering
plot.ICSClust           Scatterplot Matrix with densities on the
                        diagonal
print.ICSClust_summary
                        Print of an 'ICSClust_summary' object
rimle_clust             Robust Improper Maximum Likelihood Clustering
runif_outside_range     Uniform distribution outside a given range
select_plot             Plot of the Generalized Kurtosis Values of the
                        ICS Transformation
summary.ICSClust        Summary of an 'ICSClust' object
tcov                    Pairwise one-step M-estimate of scatter
tkmeans_clust           Trimmed k-means clustering
ucov                    Simple robust estimates of scatter
var_crit                Selection of Invariant components using the var
                        criterion

Author(s)

Aurore Archimbaud [aut, cre] (<https://orcid.org/0000-0002-6511-9091>), Andreas Alfons [aut] (<https://orcid.org/0000-0002-2513-3788>), Klaus Nordhausen [aut] (<https://orcid.org/0000-0002-3758-8501>), Anne Ruiz-Gazen [aut] (<https://orcid.org/0000-0001-8970-8061>)

Maintainer: Aurore Archimbaud <aurore.archimbaud@live.fr>

References

Alfons, A., Archimbaud, A., Nordhausen, K., & Ruiz-Gazen, A. (2022). Tandem clustering with invariant coordinate selection. arXiv preprint arXiv:2212.06108.

Tandem clustering with ICS

Description

Sequential clustering approach: (i) dimension reduction through the Invariant Coordinate Selection method using the ICS function and (ii) clustering of the transformed data.

Usage

ICSClust(
  X,
  nb_select = NULL,
  nb_clusters = NULL,
  ICS_args = list(),
  criterion = c("med_crit", "normal_crit", "var_crit", "discriminatory_crit"),
  ICS_crit_args = list(),
  method = c("kmeans_clust", "tkmeans_clust", "pam_clust", "mclust_clust",
    "rmclust_clust", "rimle_clust"),
  clustering_args = list(),
  clusters = NULL
)

Arguments

Details

Tandem clustering with ICS is a sequential method:

• ICS is performed.

• only a subset of the first and/or the last few components are selected based on a criterion.

• the clustering method is performed only on the subspace of the selected components.

• wrapper for several different clustering methods are provided. Users can however also write wrappers for other clustering methods.

Value

An object of class "ICSClust" with the following components:

• ICS_out: An object of class "ICS". See ICS

• select: a vector of the names of the selected invariant coordinates.

• clusters: a vector of the new partition of the data, i.e a vector of integers (from 1:k) indicating the cluster to which each observation is allocated. 0 indicates outlying observations.

summary() and plot() methods are available.

Author(s)

Aurore Archimbaud

References

Alfons, A., Archimbaud, A., Nordhausen, K., & Ruiz-Gazen, A. (2022). Tandem clustering with invariant coordinate selection. arXiv preprint arXiv:2212.06108..

See Also

med_crit(), normal_crit(), var_crit(), ICS, discriminatory_crit(), kmeans_clust(), tkmeans_clust(), pam_clust(), rimle_clust(), mclust_clust() summary() and plot() methods

Examples

X <- iris[,1:4]

# indicating the number of components to retain for the dimension reduction
# step as well as the number of clusters searched for.
out <- ICSClust(X, nb_select = 2, nb_clusters = 3)
summary(out)
plot(out)

# changing the scatter pair to consider in ICS
out <- ICSClust(X, nb_select = 1, nb_clusters = 3,
ICS_args = list(S1 = ICS_mcd_raw, S2 = ICS_cov,S1_args = list(alpha = 0.5)))
summary(out)
plot(out)
 
# changing the criterion for choosing the invariant coordinates
out <- ICSClust(X, nb_clusters = 3, criterion = "normal_crit",
ICS_crit_args = list(level = 0.1, test = "anscombe.test", max_select = NULL))
summary(out)
plot(out)

# changing the clustering method
out <- ICSClust(X, nb_clusters = 3, method  = "tkmeans_clust", 
clustering_args = list(alpha = 0.1))
summary(out)
plot(out)

Local Shape Scatter Estimates for ICS

Description

It is a wrapper for the local shape estimator of scatter as computed by fpc::localshape().

Usage

ICS_lcov(x, mscatter = "cov", proportion = 0.1, ...)

Arguments

Value

An object of class "ICS_scatter" with the following components:

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

fpc::localshape()

MCD location and Scatter Estimates for ICS

Description

It is a wrapper for the (reweighted) MCD estimators of location and scatter as computed by rrcov::CovMcd().

Usage

ICS_mcd_raw(x, location = FALSE, nsamp = "deterministic", alpha = 0.5, ...)

ICS_mcd_rwt(x, location = FALSE, nsamp = "deterministic", alpha = 0.5, ...)

Arguments

Details

• ICS_mcd_raw(): computes the raw MCD estimates.

• ICS_mcd_rwt(): computes the reweighted MCD estimates.

Value

An object of class "ICS_scatter" with the following components:

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

rrcov::CovMcd()

Cauchy location and Scatter Estimates for ICS

Description

It is a wrapper for the Cauchy estimator of location and scatter for a multivariate t-distribution, as computed by ICS::tM().

Usage

ICS_mlc(x, location = FALSE, ...)

Arguments

Value

An object of class "ICS_scatter" with the following components:

Author(s)

Andreas Alfons and Aurore Archimbaud

See Also

ICS::tM(), ICS::ICS_tM()

Pairwise one-step M-estimate of scatter for ICS

Description

Wrapper function for the pairwise one-step M-estimator of scatter with weights based on pairwise Mahalanobis distances, as computed by tcov(). Note that this estimator is based on pairwise differences and therefore no location estimate is returned.

Usage

ICS_tcov(x, beta = 2)

Arguments

Value

An object of class "ICS_scatter" with the following components:

Author(s)

Andreas Alfons

See Also

ICS()

tcov(), ucov(), ICS_ucov()

Simple robust estimates of scatter for ICS

Description

Wrapper functions for the one-step M-estimator of scatter with weights based on Mahalanobis distances as computed by scov(), or the simple related estimator that is based on a transformation as computed by ucov().

Usage

ICS_scov(x, location = TRUE, beta = 0.2)

ICS_ucov(x, location = TRUE, beta = 0.2)

Arguments

Value

An object of class "ICS_scatter" with the following components:

Author(s)

Andreas Alfons

See Also

ICS()

tcov(), ICS_tcov(), ucov()

Scatterplot Matrix with densities on the diagonal

Description

Produces a gg-scatterplot matrix of the variables of a given dataframe or an invariant coordinate system obtained via an ICS transformation with densities on the diagonal for each cluster.

Usage

component_plot(
  object,
  select = TRUE,
  clusters = NULL,
  text_size_factor = 8/6.5,
  colors = NULL
)

Arguments

Value

An object of class "ggmatrix" (see GGally::ggpairs()).

Author(s)

Andreas Alfons and Aurore Archimbaud

Examples

X <- iris[,1:4]
component_plot(X)
out <- ICS(X)
component_plot(out, select = c(1,4))

Selection of ICS components based on discriminatory power

Description

Identifies invariant coordinates associated to the highest discriminatory power (by default "eta2").

Usage

discriminatory_crit(object, ...)

## S3 method for class 'ICS'
discriminatory_crit(
  object,
  clusters,
  method = "eta2",
  nb_select = NULL,
  select_only = FALSE,
  ...
)

## Default S3 method:
discriminatory_crit(
  object,
  clusters,
  method = "eta2",
  nb_select = NULL,
  select_only = FALSE,
  gen_kurtosis = NULL,
  ...
)

Arguments

Details

The discriminatory power \eta^{2} = 1 - \Lambda, where \Lambda denotes Wilks' lambda, is evaluated for each combination of the first and/or last combinations of nb_select components. The combination achieving the highest discriminatory power is selected.

More specifically, we compute

\eta^{2} = 1 - \frac{\det(E)}{\det(T)},

where E is the within-group sum of squares and cross-products matrix and T is the total sum of squares and cross-products matrix.

Value

If select_only is TRUE a vector of the names of the invariant components or variables to select. If FALSE an object of class "ICS_crit" is returned with the following objects:

• crit: the name of the criterion "discriminatory".

• method: the name of the discriminatory power.

• nb_select: the number of components to select.

• select: the names of the invariant components or variables to select.

• power_combinations: the discriminatory values for each of the considered combinations of nb_select components.

• gen_kurtosis: the vector of generalized kurtosis values in case of ICS object.

Author(s)

Aurore Archimbaud and Anne Ruiz-Gazen

References

Alfons, A., Archimbaud, A., Nordhausen, K., & Ruiz-Gazen, A. (2022). Tandem clustering with invariant coordinate selection. arXiv preprint arXiv:2212.06108..

See Also

normal_crit(), med_crit(), var_crit().

Examples

X <- iris[,-5]
out <- ICS(X)
discriminatory_crit(out, clusters = iris[,5], select_only = FALSE)

k-means clustering

Description

Wrapper for performing k-means clustering from stats::kmeans().

Usage

kmeans_clust(X, k, clusters_only = FALSE, iter.max = 100, nstart = 20, ...)

Arguments

Value

If clusters_only is TRUE a vector of the new partition of the data is returned, i.e a vector of integers (from 1:k) indicating the cluster to which each observation is allocated.

Otherwise a list is returned with the following components:

.

Author(s)

Aurore Archimbaud

See Also

stats::kmeans()

Examples

kmeans_clust(iris[,1:4], k = 3, clusters_only = TRUE)

Model-Based Clustering

Description

Wrapper for performing Model-Based Clustering from mclust::Mclust() allowing noise or not.

Usage

mclust_clust(X, k, clusters_only = FALSE, ...)

rmclust_clust(X, k, clusters_only = FALSE, ...)

Arguments

Details

• mclust_clust(): does not allow noise

• rmclust_clust(): allows noise

Value

If clusters_only is TRUE a vector of the new partition of the data is returned, i.e a vector of integers (from 1:k) indicating the cluster to which each observation is allocated. 0 indicates trimmed observations.

Otherwise a list is returned with the following components:

Author(s)

Aurore Archimbaud

See Also

mclust::Mclust()

Examples

mclust_clust(iris[,1:4], k = 3, clusters_only = TRUE)

Selection of Invariant components using the med criterion

Description

Identifies as interesting invariant coordinates whose generalized eigenvalues are the furthermost away from the median of all generalized eigenvalues.

Usage

med_crit(object, ...)

## S3 method for class 'ICS'
med_crit(object, nb_select = NULL, select_only = FALSE, ...)

## Default S3 method:
med_crit(object, nb_select = NULL, select_only = FALSE, ...)

Arguments

Details

If more than half of the components are "uninteresting" and have the same generalized eigenvalue then the median of all generalized eigenvalues corresponds to the uninteresting component generalized eigenvalue. The components of interest are the ones whose generalized eigenvalues differ the most from the median. The motivation of this criterion depends therefore on the assumption that at least half of the components have equal generalized eigenvalues.

Value

If select_only is TRUE a vector of the names of the invariant components or variables to select. If FALSE an object of class "ICS_crit" is returned with the following objects:

• crit: the name of the criterion "med".

• nb_select: the number of components to select.

• gen_kurtosis: the vector of generalized kurtosis values.

• med_gen_kurtosis: the median of the generalized kurtosis values.

• gen_kurtosis_diff_med: the absolute differences between the generalized kurtosis values and the median.

• select: the names of the invariant components or variables to select.

Author(s)

Andreas Alfons, Aurore Archimbaud and Klaus Nordhausen

References

Alfons, A., Archimbaud, A., Nordhausen, K., & Ruiz-Gazen, A. (2022). Tandem clustering with invariant coordinate selection. arXiv preprint arXiv:2212.06108..

See Also

normal_crit(), var_crit(), discriminatory_crit().

Examples

X <- iris[,-5]
out <- ICS(X)
med_crit(out, nb_select = 2, select_only = FALSE)

Simulation of a mixture of Gaussian distributions

Description

Simulation of a n \times p data frame according to a mixture of q Gaussian distributions with q < p, different location parameters \mu_1, \dots, \mu_q, and the identity matrix as the covariance matrix.

Usage

mixture_sim(pct_clusters = c(0.5, 0.5), n = 500, p = 10, delta = 10)

Arguments

Details

Let X be a p-variate real random vector distributed according to a mixture of q Gaussian distributions with q < p, different location parameters \mu_1, \dots, \mu_q, and the same positive definite covariance matrix I_p:

X \sim \sum_{h=1}^{q} \epsilon_h \, {\cal N}(\mu_h,I_p),

where \epsilon_{1}, \dots, \epsilon_{q} are mixture weights with \epsilon_1 + \cdots + \epsilon_q = 1, \mu_1 = 0_p, and \mu_{h+1} = \delta e_h with h = 1, \dots, q-1.

Value

A dataframe of n observations and p+1 variables with the first variable indicating the cluster assignment using a character string.

Author(s)

Aurore Archimbaud

References

Alfons, A., Archimbaud, A., Nordhausen, K., & Ruiz-Gazen, A. (2022). Tandem clustering with invariant coordinate selection. arXiv preprint arXiv:2212.06108..

Examples

X <- mixture_sim()
summary(X)

Selection of Non-normal Invariant Components Using Marginal Normality Tests

Description

Identifies invariant coordinates that are non normal using univariate normality tests as in the comp.norm.test function from the ICSOutlier package, with the difference that both the first and last few components are investigated.

Usage

normal_crit(object, ...)

## S3 method for class 'ICS'
normal_crit(
  object,
  level = 0.05,
  test = c("agostino.test", "jarque.test", "anscombe.test", "bonett.test",
    "shapiro.test"),
  max_select = NULL,
  select_only = FALSE,
  ...
)

## Default S3 method:
normal_crit(
  object,
  level = 0.05,
  test = c("agostino.test", "jarque.test", "anscombe.test", "bonett.test",
    "shapiro.test"),
  max_select = NULL,
  select_only = FALSE,
  gen_kurtosis = NULL,
  ...
)

Arguments

Details

The procedure sequentially tests the first and the last components until finding no additional components as non-normal. The quantile levels are adjusted for multiple testing by taking the level as level/j for the jth component.

Value

If select_only is TRUE a vector of the names of the invariant components or variables to select. If FALSE an object of class "ICS_crit" is returned with the following objects:

• crit: the name of the criterion "normal".

• level: the level of the test.

• max_select: the maximal number of components to select.

• test: name of the normality test to be used.

• pvalues: the p-values of the tests.

• adjusted_levels: the adjusted levels.

• select: the names of the invariant components or variables to select.

• gen_kurtosis: the vector of generalized kurtosis values in case of ICS object.

Author(s)

Andreas Alfons, Aurore Archimbaud, Klaus Nordhausen and Anne Ruiz-Gazen

References

Alfons, A., Archimbaud, A., Nordhausen, K., & Ruiz-Gazen, A. (2022). Tandem clustering with invariant coordinate selection. arXiv preprint arXiv:2212.06108..

Archimbaud, A., Nordhausen, K., and Ruiz-Gazen, A. (2018). ICSOutlier: Unsupervised Outlier Detection for Low-Dimensional Contamination Structure, The RJournal, Vol. 10(1):234–250. doi:10.32614/RJ-2018-034

Archimbaud, A., Nordhausen, K., and Ruiz-Gazen, A. (2016). ICSOutlier: Outlier Detection Using Invariant Coordinate Selection. R package version 0.3-0

See Also

med_crit(), var_crit(), discriminatory_crit(), jarque.test(), anscombe.test(), bonett.test(), agostino.test(), stats::shapiro.test().

Examples

X <- iris[,-5]
out <- ICS(X)
normal_crit(out, level = 0.1, select_only = FALSE)

Partitioning Around Medoids clustering

Description

Wrapper for performing Partitioning Around Medoids clustering from cluster::pam().

Usage

pam_clust(X, k, clusters_only = FALSE, ...)

Arguments

Value

If clusters_only is TRUE a vector of the new partition of the data is returned, i.e a vector of integers (from 1:k) indicating the cluster to which each observation is allocated. 0 indicates trimmed observations.

Otherwise a list is returned with the following components:

.

Author(s)

Aurore Archimbaud

See Also

cluster::pam()

Examples

pam_clust(iris[,1:4], k = 3, clusters_only = TRUE)
 

Scatterplot Matrix with densities on the diagonal

Description

Wrapper for component_plot().

Usage

## S3 method for class 'ICSClust'
plot(x, ...)

Arguments

Value

An object of class "ggmatrix" (see GGally::ggpairs()).

Author(s)

Aurore Archimbaud

Print of an ICSClust_summary object

Description

Prints an ICSClust_summary object in an informative way.

Usage

## S3 method for class 'ICSClust_summary'
print(x, info = FALSE, digits = 4L, ...)

Arguments

Value

The supplied object of class "ICSClust_summary" is returned invisibly.

Author(s)

Aurore Archimbaud

Robust Improper Maximum Likelihood Clustering

Description

Wrapper for performing Robust Improper Maximum Likelihood Clustering clustering from otrimle::rimle().

Usage

rimle_clust(X, k, clusters_only = FALSE, ...)

Arguments

Value

If clusters_only is TRUE a vector of the new partition of the data is returned, i.e a vector of integers (from 1:k) indicating the cluster to which each observation is allocated. 0 indicates trimmed observations.

Otherwise a list is returned with the following components:

Author(s)

Aurore Archimbaud

See Also

otrimle::rimle()

Examples

rimle_clust(iris[,1:4], k = 3, clusters_only = TRUE)

Uniform distribution outside a given range

Description

Draw from a multivariate uniform distribution outside a given range. Intuitively speaking, the observations are drawn from a multivariate uniform distribution on a hyperrectangle with a hole in the middle (in the shape of a smaller hyperrectangle). This is useful, e.g., for adding random noise to a data set such that the noise consists of large values that do not overlap the initial data.

Usage

runif_outside_range(n, min = 0, max = 1, mult = 2)

Arguments

Value

A matrix of generated points.

Author(s)

Andreas Alfons

References

#' Alfons, A., Archimbaud, A., Nordhausen, K., & Ruiz-Gazen, A. (2022). Tandem clustering with invariant coordinate selection. arXiv preprint arXiv:2212.06108.

Examples

## illustrations for argument 'mult'

# draw observations with argument 'mult = 2'
xy2 <- runif_outside_range(1000, min = rep(-1, 2), max = rep(1, 2), 
                           mult = 2)
# each side of the larger hyperrectangle is twice as long as 
# the corresponding side of the smaller rectanglar cut-out
df2 <- data.frame(x = xy2[, 1], y = xy2[, 2])
ggplot(data = df2, mapping = aes(x = x, y = y)) + 
  geom_point()

# draw observations with argument 'mult = 4'
xy4 <- runif_outside_range(1000, min = rep(-1, 2), max = rep(1, 2), 
                           mult = 4)
# each side of the larger hyperrectangle is four times as long 
# as the corresponding side of the smaller rectanglar cut-out
df4 <- data.frame(x = xy4[, 1], y = xy4[, 2])
ggplot(data = df4, mapping = aes(x = x, y = y)) + 
  geom_point()

Plot of the Generalized Kurtosis Values of the ICS Transformation

Description

Extracts the generalized kurtosis values of the components obtained via an ICS transformation and draws either a screeplot or a specific plot for a given criterion. If an object of class "ICS_crit" is given, then the selected components are shaded on the plot.

Usage

select_plot(object, ...)

## Default S3 method:
select_plot(
  object,
  select = NULL,
  scale = FALSE,
  screeplot = TRUE,
  type = c("dots", "lines"),
  width = 0.2,
  color = "grey",
  alpha = 0.3,
  size = 3,
  ...
)

## S3 method for class 'data.frame'
select_plot(
  object,
  type = c("dots", "lines"),
  width = 0.2,
  color = "grey",
  alpha = 0.3,
  ...
)

## S3 method for class 'ICS_crit'
select_plot(
  object,
  type = c("dots", "lines"),
  width = 0.2,
  color = "grey",
  alpha = 0.3,
  size = 3,
  screeplot = TRUE,
  ...
)

Arguments

Value

An object of class "ggplot" (see ggplot2::ggplot()).

Author(s)

Andreas Alfons and Aurore Archimbaud

Examples

X <- iris[,-5]
out <- ICS(X)

# on an ICS object
select_plot(out)
select_plot(out, type = "lines")

# on an ICS_crit object 
# median criterion
out_med <- med_crit(out, nb_select = 1, select_only = FALSE)
select_plot(out_med, type = "lines")
select_plot(out_med, screeplot = FALSE, type = "lines", 
color = "lightblue")

# discriminatory criterion
out_disc <- discriminatory_crit(out, clusters = iris[,5], 
 select_only = FALSE)
select_plot(out_disc)

Summary of an ICSClust object

Description

Summarizes an ICSClust object in an informative way.

Usage

## S3 method for class 'ICSClust'
summary(object, ...)

Arguments

Value

An object of class "ICSClust_summary" with the following components:

• ICS_out: ICS_out object

• nb_comp: number of selected components

• select: vector of names of selected components

• nb_clusters: number of clusters

• table_clusters: frequency table of clusters

Author(s)

Aurore Archimbaud

Pairwise one-step M-estimate of scatter

Description

Computes a pairwise one-step M-estimate of scatter with weights based on pairwise Mahalanobis distances. Note that it is based on pairwise differences and therefore does not require a location estimate.

Usage

tcov(x, beta = 2)

Arguments

Details

For a sample \boldsymbol{X}_{n} = (\mathbf{x}_{1}, \dots, \mathbf{x}_n)^{\top}, a positive and decreasing weight function w, and a tuning parameter \beta > 0, the pairwise one-step M-estimator of scatter is defined as

\mathrm{TCOV}_{\beta}(\boldsymbol{X}_{n}) = \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} w(\beta \, r^{2}(\mathbf{x}_{i}, \mathbf{x}_{j})) (\mathbf{x}_{i} - \mathbf{x}_{j}) (\mathbf{x}_{i} - \mathbf{x}_{j})^{\top}}{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} w(\beta \, r^{2}(\mathbf{x}_{i}, \mathbf{x}_{j}))},

where

r^{2}(\mathbf{x}_{i}, \mathbf{x}_{j}) = (\mathbf{x}_{i} - \mathbf{x}_{j})^{\top} \mathrm{COV}(\boldsymbol{X}_n)^{-1} (\mathbf{x}_{i} - \mathbf{x}_{j})

denotes the squared pairwise Mahalanobis distance between observations \mathbf{x}_{i} and \mathbf{x}_{j} based on the sample covariance matrix \mathrm{COV}(\boldsymbol{X}_n). Here, the weight function w(x) = \exp(-x/2) is used.

Value

A numeric matrix giving the pairwise one-step M-estimate of scatter.

Author(s)

Andreas Alfons and Aurore Archimbaud

References

Caussinus, H. and Ruiz-Gazen, A. (1993) Projection Pursuit and Generalized Principal Component Analysis. In Morgenthaler, S., Ronchetti, E., Stahel, W.A. (eds.) New Directions in Statistical Data Analysis and Robustness, 35-46. Monte Verita, Proceedings of the Centro Stefano Franciscini Ascona Series. Springer-Verlag.

Caussinus, H. and Ruiz-Gazen, A. (1995) Metrics for Finding Typical Structures by Means of Principal Component Analysis. In Data Science and its Applications, 177-192. Academic Press.

See Also

ICS_tcov(), ucov(), ICS_ucov()

Trimmed k-means clustering

Description

Wrapper for performing trimmed k-means clustering from tclust::tkmeans().

Usage

tkmeans_clust(X, k, clusters_only = FALSE, alpha = 0.05, ...)

Arguments

Value

If clusters_only is TRUE a vector of the new partition of the data is returned, i.e a vector of integers (from 1:k) indicating the cluster to which each observation is allocated. 0 indicates trimmed observations.

Otherwise a list is returned with the following components:

.

Author(s)

Aurore Archimbaud

See Also

tclust::tkmeans()

Examples

tkmeans_clust(iris[,1:4], k = 3, alpha = 0.1, clusters_only = TRUE)

Simple robust estimates of scatter

Description

Compute a one-step M-estimator of scatter with weights based on Mahalanobis distances, or a simple related estimator that is based on a transformation.

Usage

scov(x, beta = 0.2)

ucov(x, beta = 0.2)

Arguments

Details

For a sample \boldsymbol{X}_{n} = (\mathbf{x}_{1}, \dots, \mathbf{x}_n)^{\top}, a positive and decreasing weight function w, and a tuning parameter \beta > 0, the one-step M-estimator of scatter is defined as

\mathrm{SCOV}_{\beta}(\boldsymbol{X}_{n}) = \frac{\sum_{i=1}^{n} w(\beta \, r^{2}(\mathbf{x}_{i})) (\mathbf{x}_{i} - \mathbf{\bar{x}}_{n}) (\mathbf{x}_{i} - \mathbf{\bar{x}}_{n})^{\top}}{\sum_{i=1}^{n} w(\beta \, r^{2}(\mathbf{x}_{i}))},

where

r^{2}(\mathbf{x}_{i}) = (\mathbf{x}_{i} - \mathbf{\bar{x}}_{n})^{\top} \mathrm{COV}(\boldsymbol{X}_n)^{-1} (\mathbf{x}_{i} - \mathbf{\bar{x}}_{n})

denotes the squared Mahalanobis distance of observation \mathbf{x}_{i} from the sample mean \mathbf{\bar{x}}_{n} based on the sample covariance matrix \mathrm{COV}(\boldsymbol{X}_n). Here, the weight function w(x) = \exp(-x/2) is used.

A simple robust estimator that is consistent under normality is obtained via the transformation

\mathrm{UCOV}_{\beta}(\boldsymbol{X}_{n}) = (\mathrm{SCOV}_{\beta}(\boldsymbol{X}_{n})^{-1} - \beta \, \mathrm{COV}(\boldsymbol{X}_{n})^{-1})^{-1}.

Value

A numeric matrix giving the estimate of the scatter matrix.

Author(s)

Andreas Alfons and Aurore Archimbaud

References

Caussinus, H. and Ruiz-Gazen, A. (1993) Projection Pursuit and Generalized Principal Component Analysis. In Morgenthaler, S., Ronchetti, E., Stahel, W.A. (eds.) New Directions in Statistical Data Analysis and Robustness, 35-46. Monte Verita, Proceedings of the Centro Stefano Franciscini Ascona Series. Springer-Verlag.

Caussinus, H. and Ruiz-Gazen, A. (1995) Metrics for Finding Typical Structures by Means of Principal Component Analysis. In Data Science and its Applications, 177-192. Academic Press.

Ruiz-Gazen, A. (1996) A Very Simple Robust Estimator of a Dispersion Matrix. Computational Statistics & Data Analysis, 21(2), 149-162. doi:10.1016/0167-9473(95)00009-7.

See Also

ICS_ucov(), tcov(), ICS_tcov()

Selection of Invariant components using the var criterion

Description

Identifies the interesting invariant coordinates based on the rolling variance criterion as used in the ICSboot function of the ICtest package. It computes rolling variances on the generalized eigenvalues obtained through ICS::ICS().

Usage

var_crit(object, ...)

## S3 method for class 'ICS'
var_crit(object, nb_select = NULL, select_only = FALSE, ...)

## Default S3 method:
var_crit(object, nb_select = NULL, select_only = FALSE, ...)

Arguments

Details

Assuming that the generalized eigenvalues of the uninformative components are all the same means that the variance of these generalized eigenvalues must be minimal. Therefore when nb_select components should be selected, the method identifies the p - nb_select neighboring generalized eigenvalues with minimal variance, where p is the total number of components. The number of interesting components should be at most p-2 as at least two uninteresting components are needed to compute a variance.

Value

If select_only is TRUE a vector of the names of the invariant components or variables to select. If FALSE an object of class "ICS_crit" is returned with the following objects:

• crit: the name of the criterion "var".

• nb_select: the number of components to select.

• gen_kurtosis: the vector of generalized kurtosis values.

• select: the names of the invariant components or variables to select.

• RollVarX: the rolling variances of order d-nb_select.

• Order: indexes of the ordered invariant components such that the ones associated to the smallest variances of the eigenvalues are at the end.

Author(s)

Andreas Alfons, Aurore Archimbaud and Klaus Nordhausen

References

Alfons, A., Archimbaud, A., Nordhausen, K., & Ruiz-Gazen, A. (2022). Tandem clustering with invariant coordinate selection. arXiv preprint arXiv:2212.06108..

Radojicic, U., & Nordhausen, K. (2019). Non-gaussian component analysis: Testing the dimension of the signal subspace. In Workshop on Analytical Methods in Statistics (pp. 101–123). Springer. doi:10.1007/978-3-030-48814-7_6.

See Also

normal_crit(), med_crit(), discriminatory_crit().

Examples

X <- iris[,-5]
out <- ICS(X)
var_crit(out, nb_select = 2, select_only = FALSE)