Package 'varclust'

Simulates subspace clustering data

Description

Generates data for simulation with a low-rank subspace structure: variables are clustered and each cluster has a low-rank representation. Factors than span subspaces are not shared between clusters.

Usage

data.simulation(
  n = 100,
  SNR = 1,
  K = 10,
  numb.vars = 30,
  max.dim = 2,
  min.dim = 1,
  equal.dims = TRUE
)

Arguments

n	An integer, number of individuals.
SNR	A numeric, signal to noise ratio measured as variance of the variable, element of a subspace, to the variance of noise.
K	An integer, number of subspaces.
numb.vars	An integer, number of variables in each subspace.
max.dim	An integer, if equal.dims is TRUE then max.dim is dimension of each subspace. If equal.dims is FALSE then subspaces dimensions are drawn from uniform distribution on [min.dim,max.dim].
min.dim	An integer, minimal dimension of subspace .
equal.dims	A boolean, if TRUE (value set by default) all clusters are of the same dimension.

Value

A list consisting of:

X	matrix, generated data
signals	matrix, data without noise
dims	vector, dimensions of subspaces
factors	matrix, columns of which span subspaces
s	vector, true partiton of variables

Examples

sim.data <- data.simulation()
sim.data2 <- data.simulation(
  n = 30, SNR = 2, K = 5, numb.vars = 20,
  max.dim = 3, equal.dims = FALSE
)

---

Simulates subspace clustering data with shared factors

Description

Generating data for simulation with a low-rank subspace structure: variables are clustered and each cluster has a low-rank representation. Factors that span subspaces are shared between clusters.

Usage

data.simulation.factors(
  n = 100,
  SNR = 1,
  K = 10,
  numb.vars = 30,
  numb.factors = 10,
  min.dim = 1,
  max.dim = 2,
  equal.dims = TRUE,
  separation.parameter = 0.1
)

Arguments

n	An integer, number of individuals.
SNR	A numeric, signal to noise ratio measured as variance of the variable, element of a subspace, to the variance of noise.
K	An integer, number of subspaces.
numb.vars	An integer, number of variables in each subspace.
numb.factors	An integer, number of factors from which subspaces basis will be drawn.
min.dim	An integer, minimal dimension of subspace .
max.dim	An integer, if equal.dims is TRUE then max.dim is dimension of each subspace. If equal.dims is FALSE then subspaces dimensions are drawn from uniform distribution on [min.dim,max.dim].
equal.dims	A boolean, if TRUE (value set by default) all clusters are of the same dimension.
separation.parameter	a numeric, coefficients of variables in each subspace basis are drawn from range [separation.parameter,1]

Value

A list consisting of:

X	matrix, generated data
signals	matrix, data without noise
factors	matrix, columns of which span subspaces
indices	list of vectors, indices of factors that span subspaces
dims	vector, dimensions of subspaces
s	vector, true partiton of variables

Examples

sim.data <- data.simulation.factors()
sim.data2 <- data.simulation.factors(
  n = 30, SNR = 2, K = 5, numb.vars = 20,
  numb.factors = 10, max.dim = 3, equal.dims = FALSE, separation.parameter = 0.2
)

---

Computes integration and acontamination of the clustering

Description

Integartion and acontamination are measures of the quality of a clustering with a reference to a true partition. Let $X = (x_1, \ldots x_p)$ be the data set, $A$ be a partition into clusters $A_1, \ldots A_n$ (true partition) and $B$ be a partition into clusters $B_1, \ldots, B_m$. Then for cluster $A_j$ integration is eqaul to:

$Int(A_j) = \frac{max_{k = 1, \ldots, m} \# \{ i \in \{ 1, \ldots p \}: x_i \in A_j \wedge x_i \in B_k \} }{\# A_j}$

The $B_k$ for which the value is maximized is called the integrating cluster of $A_j$. Then the integration for the whole clustering equals is $Int(A,B) = \frac{1}{n} \sum_{j=1}^n Int(A_j)$ .The acontamination is defined by:

$Acont(A_j) = \frac{ \# \{ i \in \{ 1, \ldots p \}: x_i \in A_j \wedge x_i \in B_k \} }{\# B_k}$

where $B_k$ is the integrating cluster for $A_j$. Then the acontamination for the whole dataset is $Acont(A,B) = \frac{1}{n} \sum_{j=1}^n Acont(A_j)$

Usage

integration(group, true_group)

Arguments

group	A vector, first partition.
true_group	A vector, second (reference) partition.

Value

An array containing values of integration and acontamination.

References

M. Sołtys. Metody analizy skupień. Master’s thesis, Wrocław University of Technology, 2010

Examples

sim.data <- data.simulation(n = 20, SNR = 1, K = 2, numb.vars = 50, max.dim = 2)
true_segmentation <- rep(1:2, each = 50)
mlcc.fit <- mlcc.reps(sim.data$X, numb.clusters = 2, max.dim = 2, numb.cores = 1)
integration(mlcc.fit$segmentation, true_segmentation)

---

Computes misclassification rate

Description

Missclasification is a commonly used performance measure in subspace clustering. It allows to compare two partitions with the same number of clusters.

Usage

misclassification(group, true_group, M, K)

Arguments

group	A vector, first partition.
true_group	A vector, second (reference) partition.
M	An integer, maximal number of elements in one class.
K	An integer, number of classes.

Details

As getting exact value of misclassification requires checking all permutations and is therefore intrackable even for modest number of clusters, a heuristic approach is proposed. It is assumed that there are K classes of maximum M elements. Additional requirement is that classes labels are from range [1, K].

Value

Misclassification rate.

References

R. Vidal. Subspace clustering. Signal Processing Magazine, IEEE, 28(2):52-68,2011

Examples

sim.data <- data.simulation(n = 100, SNR = 1, K = 5, numb.vars = 30, max.dim = 2)
mlcc.fit <- mlcc.reps(sim.data$X, numb.clusters = 5, numb.runs = 20, max.dim = 2, numb.cores = 1)
misclassification(mlcc.fit$segmentation, sim.data$s, 30, 5)


# one can use this function not only for clusters
partition1 <- sample(10, 300, replace = TRUE)
partition2 <- sample(10, 300, replace = TRUE)
misclassification(partition1, partition1, max(table(partition1)), 10)
misclassification(partition1, partition2, max(table(partition2)), 10)

---

Multiple Latent Components Clustering - Subspace clustering with automatic estimation of number of clusters and their dimension

Description

This function is an implementation of Multiple Latent Components Clustering (MLCC) algorithm which clusteres quantitative variables into a number, chosen using mBIC, of groups. For each considered number of clusters in numb.clusters mlcc.reps function is called. It invokes K-means based algorithm (mlcc.kmeans) finding local minimum of mBIC, which is run a given number of times (numb.runs) with different initializations. The best partition is choosen with mBIC (see mlcc.reps function).

Usage

mlcc.bic(
  X,
  numb.clusters = 1:10,
  numb.runs = 30,
  stop.criterion = 1,
  max.iter = 30,
  max.dim = 4,
  scale = TRUE,
  numb.cores = NULL,
  greedy = TRUE,
  estimate.dimensions = TRUE,
  verbose = FALSE,
  flat.prior = FALSE,
  show.warnings = FALSE
)

Arguments

X	A data frame or a matrix with only continuous variables.
numb.clusters	A vector, numbers of clusters to be checked.
numb.runs	An integer, number of runs (initializations) of mlcc.kmeans.
stop.criterion	An integer, if an iteration of mlcc.kmeans algorithm makes less changes in partitions than stop.criterion, mlcc.kmeans stops.
max.iter	An integer, maximum number of iterations of the loop in mlcc.kmeans algorithm.
max.dim	An integer, if estimate.dimensions is FALSE then max.dim is dimension of each subspace. If estimate.dimensions is TRUE then subspaces dimensions are estimated from the range [1, max.dim].
scale	A boolean, if TRUE (value set by default) then variables in dataset are scaled to zero mean and unit variance.
numb.cores	An integer, number of cores to be used, by default all cores are used.
greedy	A boolean, if TRUE (value set by default) the clusters are estimated in a greedy way - first local minimum of mBIC is chosen.
estimate.dimensions	A boolean, if TRUE (value set by default) subspaces dimensions are estimated.
verbose	A boolean, if TRUE plot with mBIC values for different numbers of clusters is produced and values of mBIC, computed for every number of clusters and subspaces dimensions, are printed (value set by default is FALSE).
flat.prior	A boolean, if TRUE then, instead of an informative prior that takes into account number of models for a given number of clusters, flat prior is used.
show.warnings	A boolean, if set to TRUE all warnings are displayed, default value is FALSE.

Value

An object of class mlcc.fit consisting of

segmentation	a vector containing the partition of the variables
BIC	numeric, value of mBIC
subspacesDimensions	a list containing dimensions of the subspaces
nClusters	an integer, estimated number of clusters
factors	a list of matrices, basis for each subspace
all.fit	a list of segmentation, mBIC, subspaces dimension for all numbers of clusters considered for an estimated subspace dimensions
all.fit.dims	a list of lists of segmentation, mBIC, subspaces dimension for all numbers of clusters and subspaces dimensions considered

Examples

sim.data <- data.simulation(n = 50, SNR = 1, K = 3, numb.vars = 50, max.dim = 3)
mlcc.res <- mlcc.bic(sim.data$X, numb.clusters = 1:5, numb.runs = 20,
numb.cores = 1, verbose = TRUE)
show.clusters(sim.data$X, mlcc.res$segmentation)

---

Multiple Latent Components Clustering - kmeans algorithm

Description

Performs k-means based subspace clustering. Center of each cluster is some number of principal components. This number can be fixed or estimated by PESEL. Similarity measure between variable and a cluster is calculated using BIC.

Usage

mlcc.kmeans(
  X,
  number.clusters = 2,
  stop.criterion = 1,
  max.iter = 30,
  max.subspace.dim = 4,
  initial.segmentation = NULL,
  estimate.dimensions = TRUE,
  show.warnings = FALSE
)

Arguments

X	A matrix with only continuous variables.
number.clusters	An integer, number of clusters to be used.
stop.criterion	An integer indicating how many changes in partitions triggers stopping the algorithm.
max.iter	An integer, maximum number of iterations of k-means loop.
max.subspace.dim	An integer, maximum dimension of subspaces.
initial.segmentation	A vector, initial segmentation of variables to clusters.
estimate.dimensions	A boolean, if TRUE (value set by default) subspaces dimensions are estimated.
show.warnings	A boolean, if set to TRUE all warnings are displayed, default value is FALSE.

Value

A list consisting of:

segmentation	a vector containing the partition of the variables
pcas	a list of matrices, basis vectors for each cluster (subspace)

References

Bayesian dimensionality reduction with PCA using penalized semi-integrated likelihood, Piotr Sobczyk, Malgorzata Bogdan, Julie Josse

Examples

sim.data <- data.simulation(n = 50, SNR = 1, K = 5, numb.vars = 50, max.dim = 3)
mlcc.res <- mlcc.kmeans(sim.data$X, number.clusters = 5, max.iter = 20, max.subspace.dim = 3)
show.clusters(sim.data$X, mlcc.res$segmentation)

---

Multiple Latent Components Clustering - Subspace clustering assuming that the number of clusters is known

Description

For a fixed number of cluster function returns the best partition and basis for each subspace.

Usage

mlcc.reps(
  X,
  numb.clusters = 2,
  numb.runs = 30,
  stop.criterion = 1,
  max.iter = 30,
  initial.segmentations = NULL,
  max.dim = 4,
  scale = TRUE,
  numb.cores = NULL,
  estimate.dimensions = TRUE,
  flat.prior = FALSE,
  show.warnings = FALSE
)

Arguments

X	A data frame or a matrix with only continuous variables.
numb.clusters	An integer, number of cluster.
numb.runs	An integer, number of runs of mlcc.kmeans algorithm with random initialization.
stop.criterion	An integer, if an iteration of mlcc.kmeans algorithm makes less changes in partitions than stop.criterion, mlcc.kmeans stops.
max.iter	max.iter An integer, maximum number of iterations of the loop in mlcc.kmeans algorithm.
initial.segmentations	A list of vectors, segmentations that user wants to be used as an initial segmentation in mlcc.kmeans algorithm.
max.dim	An integer, maximal dimension of subspaces.
scale	A boolean, if TRUE (value set by default) then variables in dataset are scaled to zero mean and unit variance.
numb.cores	An integer, number of cores to be used, by default all cores are used.
estimate.dimensions	A boolean, if TRUE (value set by default) subspaces dimensions are estimated.
flat.prior	A boolean, if TRUE then, instead of a prior that takes into account number of models for a given number of clusters, flat prior is used.
show.warnings	A boolean, if set to TRUE all warnings are displayed, default value is FALSE.

Details

In more detail, an algorithm mlcc.kmeans is run a numb.runs of times with random or custom initializations. The best partition is selected according to the BIC.

Value

A list consisting of

segmentation	a vector containing the partition of the variables
BIC	a numeric, value of the mBIC
basis	a list of matrices, the factors for each of the subspaces

Examples

sim.data <- data.simulation(n = 50, SNR = 1, K = 5, numb.vars = 50, max.dim = 3)
mlcc.res <- mlcc.reps(sim.data$X, numb.clusters = 5, numb.runs = 20, max.dim = 4, numb.cores = 1)
show.clusters(sim.data$X, mlcc.res$segmentation)

---

Print clusters obtained from MLCC

Description

Print clusters obtained from MLCC

Usage

show.clusters(data, segmentation)

Arguments

data	The original data set.
segmentation	A vector, segmentation of variables into clusters.

---

Variable Clustering with Multiple Latent Components Clustering algorithm

Description

Package varclust performs clustering of variables, according to a probabilistic model, which assumes that each cluster lies in a low dimensional subspace. Segmentation of variables, number of clusters and their dimensions are selected based on the appropriate implementation of the Bayesian Information Criterion.

Details

The best candidate models are identified by the specific implementation of K-means algorithm, in which cluster centers are represented by some number of orthogonal factors(principal components of the variables within a cluster) and similarity between a given variable and a cluster center depends on residuals from a linear model fit. Based on the Bayesian Information Criterion (BIC), sums of squares of residuals are appropriately scaled, which allows to avoid an over-excessive attraction by clusters with larger dimensions. To reduce the chance that the local minimum of modified BIC (mBIC) is obtained instead of the global one, for every fixed number of clusters in a given range K-means algorithm is run large number of times, with different random initializations of cluster centers.

The main function of package varclust is mlcc.bic which allows clustering variables in a data with unknown number of clusters. Variable partition is computed with k-means based algorithm. Number of clusters and their dimensions are estimated using mBIC and PESEL respectively. If the number of clusters is known one might use function mlcc.reps, which takes number of clusters as a parameter. For mlcc.reps one might specify as well some initial segmentation for k-means algorithm. This can be useful if user has some a priori knowledge about clustering.

We provide also two functions to simulate datasets with described structure. The function data.simulation generates the data so that the subspaces are indepentend and data.simulation.factors generates the data where some factores are shared between the subspaces.

We also provide function measures of quality of clustering. misclassification computes misclassification rate between two partitions. This performance measure is extensively used in image segmentation. The other measure is implemented as integration function.

Version: 0.9.5

Author(s)

Piotr Sobczyk, Stanislaw Wilczynski, Julie Josse, Malgorzata Bogdan

Maintainer: Piotr Sobczyk [email protected]

Examples

sim.data <- data.simulation(n = 50, SNR = 1, K = 3, numb.vars = 50, max.dim = 3)
mlcc.bic(sim.data$X, numb.clusters = 1:5, numb.runs = 20, numb.cores = 1, verbose = TRUE)
mlcc.reps(sim.data$X, numb.clusters = 3, numb.runs = 20, numb.cores = 1)

---