Clustering

This section performs the whole analysis of the Heart data set. Warning the univariate margin distribution are defined by class of the features: numeric columns imply Gaussian distributions, integer columns imply Poisson distribution while factor (or ordered) columns imply multinomial distribution

library(VarSelLCM)

Attaching package: 'VarSelLCM'

The following object is masked from 'package:stats':

    predict

# Data loading:
# x contains the observed variables
# z the known status (i.e. 1: absence and 2: presence of heart disease)
data(heart)
ztrue <- heart[,"Class"]
x <- heart[,-13]
# Add a missing value artificially (just to show that it works!)
x[1,1] <- NA

Clustering is performed with variable selection. Model selection is done with BIC because the number of observations is large (compared to the number of features). The number of components is between 1 and 3. Do not hesitate to use parallelization (here only two cores are used).

# Cluster analysis without variable selection
res_without <- VarSelCluster(x, gvals = 1:3, vbleSelec = FALSE, crit.varsel = "BIC")

# Cluster analysis with variable selection (with parallelisation)
res_with <- VarSelCluster(x, gvals = 1:3, nbcores = 2, crit.varsel = "BIC")

Comparison of the BIC for both models: variable selection permits to improve the BIC

BIC(res_without)

[1] -6516.216

BIC(res_with)

[1] -6509.506

Comparison of the partition accuracy. ARI is computed between the true partition (ztrue) and its estimators. ARI is an index between 0 (partitions are independent) and 1 (partitions are equals). Variable selection permits to improve the ARI. Note that ARI cannot be used for model selection in clustering, because there is no true partition.

ARI(ztrue, fitted(res_without))

[1] 0.2218655

ARI(ztrue, fitted(res_with))

[1] 0.2661321

To obtained the partition and the probabilities of classification

# Estimated partition
fitted(res_with)

  [1] 2 2 1 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2 1 2 1 1 1 1 1 2 1 2 2 2 2 1 2 2 2 2
 [38] 2 1 1 1 1 1 1 2 1 2 1 2 2 2 1 1 1 1 1 2 2 2 2 1 2 1 1 2 2 1 1 1 1 2 1 1 2
 [75] 2 2 2 1 1 1 2 2 2 1 2 1 1 2 1 2 1 1 2 2 1 2 2 2 2 1 1 2 1 2 2 2 2 2 2 1 2
[112] 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 1 2 2 2 1 2 1 1 1 2 1 2 2 1 2 2 1 2 1 2
[149] 1 1 1 1 1 2 1 1 2 1 2 2 2 2 1 2 1 2 1 1 2 2 2 2 2 2 1 2 2 1 2 1 1 2 1 2 1
[186] 1 2 2 1 2 1 2 1 1 1 1 2 1 2 2 2 2 2 2 2 1 1 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2
[223] 1 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 1 1 2 1 2 2 1 2 2 1 1 2 2 1 1 2 1 1 2 2 1
[260] 1 1 2 1 1 1 1 1 1 2 2

# Estimated probabilities of classification
head(fitted(res_with, type="probability"))

          class-1   class-2
[1,] 8.270815e-06 0.9999917
[2,] 3.667447e-01 0.6332553
[3,] 8.245960e-01 0.1754040
[4,] 4.451354e-08 1.0000000
[5,] 3.888408e-03 0.9961116
[6,] 4.526259e-02 0.9547374

To get a summary of the selected model.

# Summary of the best model
summary(res_with)

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 
   

Discriminative power of the variables (here, the most discriminative variable is MaxHeartRate). The greater this index, the more the variable distinguishes the clusters.

plot(res_with)

Distribution of the most discriminative variable per clusters

# Boxplot for the continuous variable MaxHeartRate
plot(x=res_with, y="MaxHeartRate")

Empirical and theoretical distributions of the most discriminative variable (to check that the distribution is well-fitted)

# Empirical and theoretical distributions (to check that the distribution is well-fitted)
plot(res_with, y="MaxHeartRate", type="cdf")

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.
ℹ The deprecated feature was likely used in the VarSelLCM package.
  Please report the issue to the authors.
This warning is displayed once per session.
Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
generated.

Distribution of a categorical variable per clusters

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

To have details about the selected model

# More detailed output
print(res_with)

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 

To print the parameters

# Print model parameter
coef(res_with)

An object of class "VSLCMparam"
Slot "pi":
  class-1   class-2 
0.4779414 0.5220586 

Slot "paramContinuous":
An object of class "VSLCMparamContinuous"
Slot "pi":
numeric(0)

Slot "mu":
                   class-1  class-2
RestBloodPressure 131.3444 131.3444
SerumCholestoral  249.6593 249.6593
MaxHeartRate      165.2579 135.4143

Slot "sd":
                   class-1  class-2
RestBloodPressure 17.82850 17.82850
SerumCholestoral  51.59043 51.59043
MaxHeartRate      13.14888 20.98067


Slot "paramInteger":
An object of class "VSLCMparamInteger"
Slot "pi":
numeric(0)

Slot "lambda":
     class-1  class-2
Age 50.32103 58.11379


Slot "paramCategorical":
An object of class "VSLCMparamCategorical"
Slot "pi":
numeric(0)

Slot "alpha":
$Sex
                0         1
class-1 0.4166157 0.5833843
class-2 0.2358055 0.7641945

$ChestPainType
                 1          2         3         4
class-1 0.05754103 0.28951585 0.4222930 0.2306501
class-2 0.08920998 0.03291573 0.1738526 0.7040217

$FastingBloodSugar
                0         1
class-1 0.8518519 0.1481481
class-2 0.8518519 0.1481481

$ResElectrocardiographic
                0           1         2
class-1 0.4851852 0.007407407 0.5074074
class-2 0.4851852 0.007407407 0.5074074

$ExerciseInduced
                0          1
class-1 0.9127861 0.08721392
class-2 0.4484403 0.55155974

$Slope
                1         2          3
class-1 0.7598751 0.1934119 0.04671297
class-2 0.2266138 0.6884521 0.08493415

$MajorVessels
                0         1          2           3
class-1 0.7915841 0.1402742 0.05988695 0.008254773
class-2 0.4104171 0.2830563 0.17928979 0.127236821

$Thal
                3            6         7
class-1 0.8302076 3.583226e-09 0.1697924
class-2 0.3183021 9.932190e-02 0.5823760

Probabilities of classification for new observations

# Probabilities of classification for new observations 
predict(res_with, newdata = x[1:3,])

          class-1   class-2
[1,] 8.644360e-06 0.9999914
[2,] 3.770629e-01 0.6229371
[3,] 8.308939e-01 0.1691061

The model can be used for imputation (of the clustered data or of a new observation)

# Imputation by posterior mean for the first observation
not.imputed <- x[1,]
imputed <- VarSelImputation(res_with, x[1,], method = "sampling")
rbind(not.imputed, imputed)

  Age Sex ChestPainType RestBloodPressure SerumCholestoral FastingBloodSugar
1  NA   1             4               130              322                 0
2  53   1             4               130              322                 0
  ResElectrocardiographic MaxHeartRate ExerciseInduced Slope MajorVessels Thal
1                       2          109               0     2            3    3
2                       2          109               0     2            3    3

Shiny application

All the results can be analyzed by the Shiny application…

# Start the shiny application
VarSelShiny(res_with)

… but this analysis can also be done on R.