References:

Introduction

VarSelLCM permits a full model selection (detection of the relevant features for clustering and selection of the number of clusters) in model-based clustering, according to classical information criteria (BIC, MICL or AIC).

Data to analyzed can be composed of continuous, integer and/or categorical features. Moreover, missing values are managed, without any pre-processing, by the model used to cluster with the assumption that values are missing completely at random. Thus, VarSelLCM can also be used for data imputation via mixture models.

An R-Shiny application is implemented to easily interpret the clustering results

Clustering for cytological diagnosis

This section performs the whole analysis of the Golub data set. The first column indicates the type of leukemia. Clustering is made on the other (continuous) features.

library(VarSelLCM)
# please install the package multtest to get the data
# source("https://bioconductor.org/biocLite.R")
# biocLite("multtest")
data(golub, package = "multtest") 

# one row = one individual
golub <- t(golub)

Clustering is done with and without variable selection. Here ICL and MICL criteria are used because the number of observations is less than the number of features (thus, BIC is not relevant).

# Please indicate the number of cores you wan to use for parallelisation
nb.CPU <- 4
# clustering without variable selection (about than 10/20 sec on 4 CPU)
res.noselec <- VarSelCluster(golub, 2, crit.varsel = "ICL", 
                             vbleSelec = FALSE, 
                             nbcores = nb.CPU)

# clustering with variable selection (about than 10/20 sec on 4 CPU)
res.selec <- VarSelCluster(golub, 2, crit.varsel = "MICL",
                           vbleSelec = TRUE, 
                           nbcores = nb.CPU)

Summary of the results

summary(res.noselec)
Data set:
   Number of individuals: 38 
   Number of continuous variables: 3051 

Model:
   Number of components: 2 
   Model selection has been performed according to the ICL  criterion 
Information Criteria:
   loglike: -71047.52 
   AIC:     -83252.52 
   BIC:     -93245.89 
   ICL:     -114089.7 
summary(res.selec)
Data set:
   Number of individuals: 38 
   Number of continuous variables: 3051 

Model:
   Number of components: 2 
   Model selection has been performed according to the MICL  criterion 
   Variable selection has been performed, 553  ( 18.13 % ) of the variables are relevant for clustering 
   
Information Criteria:
   loglike: -77235.87 
   AIC:     -84444.87 
   BIC:     -90347.55 
   ICL:     -103858.8 
   MICL:    -103858.8 
   Best values has been found  5 times

Comparison of the estimated partitions with the Adjusted Rand Index

# Comparison of the Adjusted Rand Index (variable selection increases the partition accuracy)
ARI(res.noselec@partitions@zMAP, golub.cl)
[1] 0.6992238
ARI(res.selec@partitions@zMAP, golub.cl)
[1] 0.7927409

Mixed-type data analysis

This section performs the whole analysis of the Heart data set. Warning continuous features must be stored in numeric, integer features must be stored in integer and categorical features must be stored in factor.

library(VarSelLCM)
# Data loading
data("heart")
head(heart)
  Age Sex ChestPainType RestBloodPressure SerumCholestoral
1  70   1             4               130              322
2  67   0             3               115              564
3  57   1             2               124              261
4  64   1             4               128              263
5  74   0             2               120              269
6  65   1             4               120              177
  FastingBloodSugar ResElectrocardiographic MaxHeartRate ExerciseInduced
1                 0                       2          109               0
2                 0                       2          160               0
3                 0                       0          141               0
4                 0                       0          105               1
5                 0                       2          121               1
6                 0                       0          140               0
  Slope MajorVessels Thal Class
1     2            3    3     2
2     2            0    7     1
3     1            0    7     2
4     2            1    7     1
5     1            1    3     1
6     1            0    7     1

Clustering is performed with variable selection. Model selection is done with BIC because n>>d. The number of components is between 1 and 4. Do not hesitate to use parallelisation (here only two cores are used).

# Add a missing value artificially (just to show that it works!)
heart[1,1] <- NA
# Clustering with variable selection and a number of cluster betwee 1 and 4
# Model selection is BIC (to use MICL, the option must be specified)
out <- VarSelCluster(heart[,-13], 1:4, nbcores = 2)

Now, all the results can be analyzed by the Shiny application…

# Start the shiny application
 VarSelShiny(out)

… but this analysis can also be done on R.

To get a summary of the selected model.

# Summary of the best model
summary(out)
Data set:
   Number of individuals: 270 
   Number of continuous variables: 3 
   Number of count variables: 1 
   Percentile of missing values for the integer variables: 0.37 
   Number of categorical variables: 8 

Model:
   Number of components: 2 
   Model selection has been performed according to the BIC  criterion 
   Variable selection has been performed, 8  ( 66.67 % ) of the variables are relevant for clustering 
   
Information Criteria:
   loglike: -6403.136 
   AIC:     -6441.136 
   BIC:     -6509.506 
   ICL:     -6638.116 

Model interpretation should focus on the most discriminative variables. These variables can be found with the following plot.

# Discriminative power of the variables (here, the most discriminative variable is MaxHeartRate)
plot(out, type="bar")

Interpretation of the most discriminative variable is based on its distribution per cluster.

# Boxplot for continuous (or interger) variable
plot(out, y="MaxHeartRate", type="boxplot")

We can check that the distribution used to cluster is relevant.

# Empirical and theoretical distributions (to check that clustering is pertinent)
plot(out, y="MaxHeartRate", type="cdf")

Interpretation of a categorical variable is based on its distribution per cluster.

# Summary of categorical variable
plot(out, y="Sex")

Interpretation of the cluster overlaps by using the probabilities of missclassification.

# Summary of the probabilities of missclassification
plot(out, type="probs-class")

Missing values can be imputed.

# Imputation by posterior mean for the first observation
not.imputed <- heart[1,-13]
imputed <- VarSelImputation(out)[1,]
rbind(not.imputed, imputed)
       Age Sex ChestPainType RestBloodPressure SerumCholestoral
1       NA   1             4               130              322
2 58.11332   1             4               130              322
  FastingBloodSugar ResElectrocardiographic MaxHeartRate ExerciseInduced
1                 0                       2          109               0
2                 0                       2          109               0
  Slope MajorVessels Thal
1     2            3    3
2     2            3    3