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title: "VarSelLCM"
author: "Variable Selection for Model-Based Clustering of Mixed-Type Data Set with  Missing Values"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
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  %\VignetteIndexEntry{Vignette VarSelLCM}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
  
  
  
  **References:** 
  
  * Marbac, M. and Sedki, M. (2017), [Variable selection for model-based clustering using the integrated complete-data likelihood](https://link.springer.com/article/10.1007/s11222-016-9670-1), Statistics and Computing, Volume 27, Issue 4, pp 1049–1063.
  * Marbac, M., Patin, E. and Sedki, M. (2018), [Variable selection for mixed data clustering: Application in human population genomics](https://arxiv.org/abs/1703.02293), Journal of Classification, to appear.
  
# Introduction
  *[VarSelLCM](https://CRAN.R-project.org/package=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
  
  
# Mixed-type data analysis
## 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*
```{r, comment=""}
library(VarSelLCM)
# 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).

```{r, comment=""}
# 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
```{r, comment=""}
BIC(res_without)
BIC(res_with)
```

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.
```{r, comment=""}
ARI(ztrue, fitted(res_without))
ARI(ztrue, fitted(res_with))
```

To obtained the partition and the probabilities of classification
```{r, comment=""}
# Estimated partition
fitted(res_with)

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

To get a summary of the selected model.
```{r, comment=""}
# Summary of the best model
summary(res_with)
```

Discriminative power of the variables (here, the most discriminative variable is MaxHeartRate). The greater this index, the more the variable distinguishes the clusters.
```{r, comment=""}
plot(res_with)
```

Distribution of the most discriminative variable per clusters
```{r, comment=""}
# 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)
```{r, comment=""}
# Empirical and theoretical distributions (to check that the distribution is well-fitted)
plot(res_with, y="MaxHeartRate", type="cdf")
```

Distribution of a categorical variable per clusters
```{r, comment=""}
# Summary of categorical variable
plot(res_with, y="Sex")
```

To have details about the selected model
```{r, comment=""}
# More detailed output
print(res_with)
```

To print the parameters
```{r, comment=""}
# Print model parameter
coef(res_with)
```

Probabilities of classification for new observations 
```{r, comment=""}
# Probabilities of classification for new observations 
predict(res_with, newdata = x[1:3,])
```

The model can be used for imputation (of the clustered data or of a new observation)
```{r, comment=""}
# Imputation by posterior mean for the first observation
not.imputed <- x[1,]
imputed <- VarSelImputation(res_with, x[1,], method = "sampling")
rbind(not.imputed, imputed)
```

## Shiny application
All the results can be analyzed by the Shiny application...
```{r, eval=FALSE, comment="", include=TRUE}
# Start the shiny application
VarSelShiny(res_with)
```
... but this analysis can also be done on R.