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# factoextra : Extract and Visualize the Results of Multivariate Data Analyses

[**factoextra**](https://rpkgs.datanovia.com/factoextra/index.html) is
an R package that makes it easy to *extract* and *visualize* the output
of exploratory **multivariate data analyses**, including:

1.  [Principal Component Analysis
    (PCA)](https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/112-pca-principal-component-analysis-essentials/),
    which is used to summarize the information contained in continuous
    (i.e., quantitative) multivariate data by reducing the
    dimensionality of the data without losing important information.

2.  [Correspondence Analysis
    (CA)](https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/113-ca-correspondence-analysis-in-r-essentials/),
    which is an extension of principal component analysis suited to
    analyzing a large contingency table formed by two *qualitative
    variables* (or categorical data).

3.  [Multiple Correspondence Analysis
    (MCA)](https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/114-mca-multiple-correspondence-analysis-in-r-essentials/),
    which is an adaptation of CA to a data table containing more than
    two categorical variables.

4.  [Multiple Factor Analysis
    (MFA)](https://rpkgs.datanovia.com/factoextra/reference/fviz_mfa.html)
    dedicated to datasets where variables are organized into groups
    (qualitative and/or quantitative variables).

5.  [Hierarchical Multiple Factor Analysis
    (HMFA)](https://rpkgs.datanovia.com/factoextra/reference/fviz_hmfa.html):
    An extension of MFA in a situation where the data are organized into
    a hierarchical structure.

6.  [Factor Analysis of Mixed Data
    (FAMD)](https://rpkgs.datanovia.com/factoextra/reference/fviz_famd.html),
    a particular case of the MFA, dedicated to analyze a data set
    containing both quantitative and qualitative variables.

There are a number of R packages implementing principal component
methods. These packages include: *FactoMineR*, *ade4*, *stats*, *ca*,
*MASS* and *ExPosition*.

However, the result is presented differently according to the used
packages. To help in the interpretation and in the visualization of
multivariate analysis - such as cluster analysis and dimensionality
reduction analysis - we developed an easy-to-use R package named
[factoextra](https://rpkgs.datanovia.com/factoextra/index.html).

  - The R package **factoextra** has flexible and easy-to-use methods to
    extract quickly, in a human readable standard data format, the
    analysis results from the different packages mentioned above.  
  - It produces a **ggplot2**-based **elegant data visualization** with
    less typing.
  - It also contains many functions that facilitate clustering analysis
    visualization.

> We’ll use i) the FactoMineR package (Sebastien Le, et al., 2008) to
> compute PCA, (M)CA, FAMD, MFA and HCPC; ii) and the factoextra package
> for extracting and visualizing the results.

The figure below shows methods whose outputs can be visualized using the
factoextra package. The official online documentation is available at:
<https://rpkgs.datanovia.com/factoextra/index.html>.

![factoextra R package](tools/factoextra-r-package.png)

## Why using factoextra?

1.  The *factoextra* R package can handle the results of PCA, CA, MCA,
    MFA, FAMD and HMFA from several packages, for extracting and
    visualizing the most important information contained in your data.

2.  *After PCA, CA, MCA, MFA, FAMD and HMFA, the most important
    row/column elements* can be highlighted using :  

  - their cos2 values corresponding to their quality of representation
    on the factor map
  - their contributions to the definition of the principal dimensions.

If you want to do this, the factoextra package
provides a convenient solution.

3.  *PCA and (M)CA are used sometimes for prediction problems* : one can
    predict the coordinates of new supplementary variables (quantitative
    and qualitative) and supplementary individuals using the information
    provided by the previously performed PCA or (M)CA. This can be done
    easily using the
    [FactoMineR](https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/112-pca-principal-component-analysis-essentials/)
    package.

If you want to make predictions with PCA/MCA and
to visualize the position of the supplementary variables/individuals on
the factor map using ggplot2: then factoextra can help you. It’s quick,
write less and do more…

4.  *Several functions from different packages - FactoMineR, ade4,
    ExPosition, stats - are available in R for performing PCA, CA or
    MCA*. However, The components of the output vary from package to
    package.

No matter the package you decided to use,
factoextra can give you a human understandable output.

## Installing FactoMineR

The FactoMineR package can be installed and loaded as follow:

``` r
# Install
install.packages("FactoMineR")

# Load
library("FactoMineR")
```

## Installing and loading factoextra

  - factoextra can be installed from
    [CRAN](https://cran.r-project.org/package=factoextra) as follow:



``` r
install.packages("factoextra")
```

  - Or, install the latest version from
    [Github](https://github.com/kassambara/factoextra)



``` r
if(!require(remotes)) install.packages("remotes")
remotes::install_github("kassambara/factoextra")
```

The current maintenance baseline targets:

  - `R >= 4.1.0`

  - `ggplot2 >= 3.5.2`

  - `ggpubr >= 0.6.3` (CRAN)

  - `FactoMineR >= 2.13`

  - Load factoextra as follow :



``` r
library("factoextra")
```

## Recent maintenance highlights

The current development version includes:

  - helper-level `k = 1` handling for clustering diagnostics such as
    `fviz_nbclust(..., method = "wss")` and hierarchical `eclust()`
    auto-selection
  - silhouette diagnostics now omit the undefined `k = 1` point and keep
    the optimal-k guide aligned with the displayed cluster count
  - stricter validation for scaled clustering data and non-finite
    distance objects
  - clearer MCA `quanti.sup`, `axes`, `ncp`, and `parallel.iter`
    validation paths, including integer-like numeric support for
    `fviz_eig()`
  - refreshed examples and manuals for the updated clustering and
    validation workflows



``` r
data(iris)
iris.scaled <- scale(iris[, -5])
res.pca <- prcomp(iris[, -5], scale = TRUE)

# WSS now computes the k = 1 baseline internally
fviz_nbclust(iris.scaled, hcut, method = "wss", hc_method = "complete")

# Parallel analysis validation is explicit and reproducible
fviz_eig(res.pca, choice = "eigenvalue", parallel = TRUE,
         parallel.iter = 10, parallel.seed = 123)

# FactoMineR MCA quantitative supplementary variables are supported directly
library(FactoMineR)
data(poison)
res.mca <- MCA(poison, quanti.sup = 1:2, graph = FALSE)
get_mca(res.mca, "quanti.sup")

# If installation leaves stale 00LOCK-* directories in your library,
# remove those directories manually before reinstalling.
```

## Main functions in the factoextra package

See the online documentation
(<https://rpkgs.datanovia.com/factoextra/index.html>) for a complete
list.

### Visualizing dimension reduction analysis outputs

| Functions                         | Description                                                                                                                               |
| --------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
| *fviz\_eig (or fviz\_eigenvalue)* | Extract and visualize the eigenvalues/variances of dimensions.                                                                            |
| *fviz\_pca*                       | Graph of individuals/variables from the output of *Principal Component Analysis* (PCA).                                                   |
| *fviz\_ca*                        | Graph of column/row variables from the output of *Correspondence Analysis* (CA).                                                          |
| *fviz\_mca*                       | Graph of individuals/variables from the output of *Multiple Correspondence Analysis* (MCA).                                               |
| *fviz\_mfa*                       | Graph of individuals/variables from the output of *Multiple Factor Analysis* (MFA), including supplementary qualitative categories.       |
| *fviz\_famd*                      | Graph of individuals/variables from the output of *Factor Analysis of Mixed Data* (FAMD), including supplementary qualitative categories. |
| *fviz\_hmfa*                      | Graph of individuals/variables from the output of *Hierarchical Multiple Factor Analysis* (HMFA).                                         |
| *fviz\_ellipses*                  | Draw confidence ellipses around the categories.                                                                                           |
| *fviz\_cos2*                      | Visualize the quality of representation of the row/column variable from the results of PCA, CA, MCA functions.                            |
| *fviz\_contrib*                   | Visualize the contributions of row/column elements from the results of PCA, CA, MCA functions.                                            |

### Extracting data from dimension reduction analysis outputs

| Functions          | Description                                                                                                                                                  |
| ------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| *get\_eigenvalue*  | Extract and visualize the eigenvalues/variances of dimensions.                                                                                               |
| *get\_pca*         | Extract all the results (coordinates, squared cosine, contributions) for the active individuals/variables from *Principal Component Analysis* (PCA) outputs. |
| *get\_ca*          | Extract all the results (coordinates, squared cosine, contributions) for the active column/row variables from *Correspondence Analysis* outputs.             |
| *get\_mca*         | Extract results from *Multiple Correspondence Analysis* outputs.                                                                                             |
| *get\_mfa*         | Extract results from *Multiple Factor Analysis* outputs, including supplementary qualitative categories.                                                     |
| *get\_famd*        | Extract results from *Factor Analysis of Mixed Data* outputs, including supplementary qualitative categories.                                                |
| *get\_hmfa*        | Extract results from *Hierarchical Multiple Factor Analysis* outputs.                                                                                        |
| *facto\_summarize* | Subset and summarize the output of factor analyses.                                                                                                          |

### Clustering analysis and visualization

| Functions                                | Description                                                 |
| ---------------------------------------- | ----------------------------------------------------------- |
| *dist*(fviz\_dist, get\_dist)            | Enhanced Distance Matrix Computation and Visualization.     |
| *get\_clust\_tendency*                   | Assessing Clustering Tendency.                              |
| *fviz\_nbclust*(fviz\_gap\_stat)         | Determining and Visualizing the Optimal Number of Clusters. |
| *fviz\_dend*                             | Enhanced Visualization of Dendrogram                        |
| *fviz\_cluster*                          | Visualize Clustering Results                                |
| *fviz\_mclust*                           | Visualize Model-based Clustering Results                    |
| *fviz\_silhouette*                       | Visualize Silhouette Information from Clustering.           |
| *hcut*                                   | Computes Hierarchical Clustering and Cut the Tree           |
| *hkmeans* (hkmeans\_tree, print.hkmeans) | Hierarchical k-means clustering.                            |
| *eclust*                                 | Visual enhancement of clustering analysis                   |

## Dimension reduction and factoextra

As depicted in the figure below, the type of analysis to be performed
depends on the data set formats and structures.

![dimension reduction and
factoextra](tools/multivariate-analysis-factoextra.png)

In this section we start by illustrating classical methods - such as
PCA, CA and MCA - for analyzing a data set containing continuous
variables, contingency table and qualitative variables, respectively.

We continue by discussing advanced methods - such as FAMD, MFA and HMFA
- for analyzing a data set containing a mix of variables (qualitatives &
quantitatives) organized or not into groups.

Finally, we show how to perform hierarchical clustering on principal
components (HCPC), which useful for performing clustering with a data
set containing only qualitative variables or with a mixed data of
qualitative and quantitative variables.

### Principal component analysis

  - Data: *decathlon2* \[in *factoextra* package\]
  - PCA function: *FactoMineR::PCA*()
  - Visualization *factoextra::fviz\_pca*()

Read more about computing and interpreting
principal component analysis at: [**Principal Component Analysis**
(PCA)](https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/112-pca-principal-component-analysis-essentials/).

1.  **Loading data**

``` r
library("factoextra")
#> Loading required package: ggplot2
#> Welcome to factoextra!
#> Want to learn more? See two factoextra-related books at https://www.datanovia.com/en/product/practical-guide-to-principal-component-methods-in-r/
data("decathlon2")
df <- decathlon2[1:23, 1:10]
```

2.  **Principal component analysis**



``` r
library("FactoMineR")
res.pca <- PCA(df,  graph = FALSE)
```

3.  **Extract and visualize eigenvalues/variances**:



``` r
# Extract eigenvalues/variances
get_eig(res.pca)
#>        eigenvalue variance.percent cumulative.variance.percent
#> Dim.1   4.1242133        41.242133                    41.24213
#> Dim.2   1.8385309        18.385309                    59.62744
#> Dim.3   1.2391403        12.391403                    72.01885
#> Dim.4   0.8194402         8.194402                    80.21325
#> Dim.5   0.7015528         7.015528                    87.22878
#> Dim.6   0.4228828         4.228828                    91.45760
#> Dim.7   0.3025817         3.025817                    94.48342
#> Dim.8   0.2744700         2.744700                    97.22812
#> Dim.9   0.1552169         1.552169                    98.78029
#> Dim.10  0.1219710         1.219710                   100.00000
# Visualize eigenvalues/variances
fviz_screeplot(res.pca, addlabels = TRUE, ylim = c(0, 50))
```

![](tools/README-pca-eigenvalue-1.png)

4\.**Extract and visualize results for variables**:

``` r
# Extract the results for variables
var <- get_pca_var(res.pca)
var
#> Principal Component Analysis Results for variables
#>  ===================================================
#>   Name       Description                                    
#> 1 "$coord"   "Coordinates for the variables"                
#> 2 "$cor"     "Correlations between variables and dimensions"
#> 3 "$cos2"    "Cos2 for the variables"                       
#> 4 "$contrib" "contributions of the variables"
# Coordinates of variables
head(var$coord)
#>                   Dim.1       Dim.2      Dim.3       Dim.4      Dim.5
#> X100m        -0.8506257 -0.17939806  0.3015564  0.03357320 -0.1944440
#> Long.jump     0.7941806  0.28085695 -0.1905465 -0.11538956  0.2331567
#> Shot.put      0.7339127  0.08540412  0.5175978  0.12846837 -0.2488129
#> High.jump     0.6100840 -0.46521415  0.3300852  0.14455012  0.4027002
#> X400m        -0.7016034  0.29017826  0.2835329  0.43082552  0.1039085
#> X110m.hurdle -0.7641252 -0.02474081  0.4488873 -0.01689589  0.2242200
# Contribution of variables
head(var$contrib)
#>                  Dim.1      Dim.2     Dim.3       Dim.4     Dim.5
#> X100m        17.544293  1.7505098  7.338659  0.13755240  5.389252
#> Long.jump    15.293168  4.2904162  2.930094  1.62485936  7.748815
#> Shot.put     13.060137  0.3967224 21.620432  2.01407269  8.824401
#> High.jump     9.024811 11.7715838  8.792888  2.54987951 23.115504
#> X400m        11.935544  4.5799296  6.487636 22.65090599  1.539012
#> X110m.hurdle 14.157544  0.0332933 16.261261  0.03483735  7.166193
# Graph of variables: default plot
fviz_pca_var(res.pca, col.var = "black")
```

![](tools/README-pca-variables-1.png)

It’s possible to control variable colors using their contributions
(“contrib”) to the principal axes:

``` r
# Control variable colors using their contributions
fviz_pca_var(res.pca, col.var="contrib",
             gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
             repel = TRUE # Avoid text overlapping
             )
```

![](tools/README-pca-variable-colors-by-contributions-1.png)

5.  **Variable contributions to the principal axes**:



``` r
# Contributions of variables to PC1
fviz_contrib(res.pca, choice = "var", axes = 1, top = 10)

# Contributions of variables to PC2
fviz_contrib(res.pca, choice = "var", axes = 2, top = 10)
```

![](tools/README-pca-variable-contributions-1.png)![](tools/README-pca-variable-contributions-2.png)

6.  **Extract and visualize results for individuals**:



``` r
# Extract the results for individuals
ind <- get_pca_ind(res.pca)
ind
#> Principal Component Analysis Results for individuals
#>  ===================================================
#>   Name       Description                       
#> 1 "$coord"   "Coordinates for the individuals" 
#> 2 "$cos2"    "Cos2 for the individuals"        
#> 3 "$contrib" "contributions of the individuals"
# Coordinates of individuals
head(ind$coord)
#>                Dim.1      Dim.2      Dim.3       Dim.4       Dim.5
#> SEBRLE     0.1955047  1.5890567  0.6424912  0.08389652  1.16829387
#> CLAY       0.8078795  2.4748137 -1.3873827  1.29838232 -0.82498206
#> BERNARD   -1.3591340  1.6480950  0.2005584 -1.96409420  0.08419345
#> YURKOV    -0.8889532 -0.4426067  2.5295843  0.71290837  0.40782264
#> ZSIVOCZKY -0.1081216 -2.0688377 -1.3342591 -0.10152796 -0.20145217
#> McMULLEN   0.1212195 -1.0139102 -0.8625170  1.34164291  1.62151286
# Graph of individuals
# 1. Use repel = TRUE to avoid overplotting
# 2. Control automatically the color of individuals using the cos2
    # cos2 = the quality of the individuals on the factor map
    # Use points only
# 3. Use gradient color
fviz_pca_ind(res.pca, col.ind = "cos2", 
             gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
             repel = TRUE # Avoid text overlapping (slow if many points)
             )
```

![](tools/README-principal-component-analysis-data-mining-1.png)

``` r
# Biplot of individuals and variables
fviz_pca_biplot(res.pca, repel = TRUE)
```

![](tools/README-principal-component-analysis-data-mining-2.png)

7.  **Color individuals by groups**:



``` r
# Compute PCA on the iris data set
# The variable Species (index = 5) is removed
# before PCA analysis
iris.pca <- PCA(iris[,-5], graph = FALSE)

# Visualize
# Use habillage to specify groups for coloring
fviz_pca_ind(iris.pca,
             label = "none", # hide individual labels
             habillage = iris$Species, # color by groups
             palette = c("#00AFBB", "#E7B800", "#FC4E07"),
             addEllipses = TRUE # Concentration ellipses
             )
```

![](tools/README-individuals-factor-map-color-by-groups-1.png)

### Correspondence analysis

  - Data: *housetasks* \[in factoextra\]
  - CA function *FactoMineR::CA*()
  - Visualize with *factoextra::fviz\_ca*()

Read more about computing and interpreting
correspondence analysis at: [**Correspondence Analysis**
(CA)](https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/113-ca-correspondence-analysis-in-r-essentials/).

  - **Compute CA**:

``` r
 # Loading data
data("housetasks")

 # Computing CA
library("FactoMineR")
res.ca <- CA(housetasks, graph = FALSE)
```

  - **Extract results for row/column variables**:



``` r
# Result for row variables
get_ca_row(res.ca)
  
# Result for column variables
get_ca_col(res.ca)
```

  - **Biplot of rows and columns**



``` r
fviz_ca_biplot(res.ca, repel = TRUE)
```

![](tools/README-correspondence-analysis-biplot-1.png)

To visualize only row points or column points, type this:

``` r
# Graph of row points
fviz_ca_row(res.ca, repel = TRUE)

# Graph of column points
fviz_ca_col(res.ca)

# Visualize row contributions on axes 1
fviz_contrib(res.ca, choice ="row", axes = 1)
     
# Visualize column contributions on axes 1
fviz_contrib(res.ca, choice ="col", axes = 1)
```

### Multiple correspondence analysis

  - Data: **poison** \[in factoextra\]
  - MCA function **FactoMineR::MCA**()
  - Visualization **factoextra::fviz\_mca**()

Read more about computing and interpreting
multiple correspondence analysis at: [**Multiple Correspondence
Analysis**
(MCA)](https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/114-mca-multiple-correspondence-analysis-in-r-essentials/).

1.  **Computing MCA**:

``` r
library(FactoMineR)
data(poison)
res.mca <- MCA(poison, quanti.sup = 1:2,
              quali.sup = 3:4, graph=FALSE)
```

2.  **Extract results for variables and individuals**:



``` r
# Extract the results for variable categories
get_mca_var(res.mca)

# Extract the results for individuals
get_mca_ind(res.mca)
```

3.  **Contribution of variables and individuals to the principal axes**:



``` r
# Visualize variable category contributions on axes 1
fviz_contrib(res.mca, choice ="var", axes = 1)

# Visualize individual contributions on axes 1
# select the top 20
fviz_contrib(res.mca, choice ="ind", axes = 1, top = 20)
```

4.  **Graph of individuals**



``` r
# Color by groups
# Add concentration ellipses
# Use repel = TRUE to avoid overplotting
grp <- as.factor(poison[, "Vomiting"])
fviz_mca_ind(res.mca,  habillage = grp,
             addEllipses = TRUE, repel = TRUE)
```

![](tools/README-mca-graph-of-individuals-1.png)

5.  **Graph of variable categories**:



``` r
fviz_mca_var(res.mca, repel = TRUE)
```

![](tools/README-mca-graph-variables-1.png)

6.  **Biplot of individuals and variables**:



``` r
fviz_mca_biplot(res.mca, repel = TRUE)
```

![](tools/README-mca-biplot-1.png)

### Advanced methods

The factoextra R package has also functions that support the
visualization of advanced methods such:

  - Factor Analysis of Mixed Data (FAMD): : [FAMD
    Examples](https://rpkgs.datanovia.com/factoextra/reference/fviz_famd.html)
  - Multiple Factor Analysis (MFA): [MFA
    Examples](https://rpkgs.datanovia.com/factoextra/reference/fviz_mfa.html)
  - Hierarchical Multiple Factor Analysis (HMFA): [HMFA
    Examples](https://rpkgs.datanovia.com/factoextra/reference/fviz_hmfa.html)
  - [Hierachical Clustering on Principal Components
    (HCPC)](https://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/117-hcpc-hierarchical-clustering-on-principal-components-essentials/)

## Cluster analysis and factoextra

To learn more about cluster analysis, you can refer to the book
available at: [Practical Guide to Cluster Analysis in
R](https://www.datanovia.com/en/product/practical-guide-to-cluster-analysis-in-r/)

The main parts of the book include:

  - *distance measures*,
  - *partitioning clustering*,
  - *hierarchical clustering*,
  - *cluster validation methods*, as well as,
  - *advanced clustering methods* such as fuzzy clustering,
    density-based clustering and model-based clustering.

The book presents the basic principles of these tasks and provide many
examples in R. It offers solid guidance in data mining for students and
researchers.

### Partitioning clustering

![Partitioning cluster analysis](tools/partitioning-clustering.png)

``` r
# 1. Loading and preparing data
data("USArrests")
df <- scale(USArrests)

# 2. Compute k-means
set.seed(123)
km.res <- kmeans(scale(USArrests), 4, nstart = 25)

# 3. Visualize
library("factoextra")
fviz_cluster(km.res, data = df,
             palette = c("#00AFBB","#2E9FDF", "#E7B800", "#FC4E07"),
             ggtheme = theme_minimal(),
             main = "Partitioning Clustering Plot"
             )
```

![](tools/README-partitioning-clustering-1.png)

Read more:

1.  [Cluster analysis in R: All what you should
    know](https://www.datanovia.com/en/blog/types-of-clustering-methods-overview-and-quick-start-r-code/).

2.  [Partitioning cluster
    analysis](https://www.datanovia.com/en/courses/partitional-clustering-in-r-the-essentials/).

### Hierarchical clustering

``` r
library("factoextra")
# Compute hierarchical clustering and cut into 4 clusters
res <- hcut(USArrests, k = 4, stand = TRUE)

# Visualize
fviz_dend(res, rect = TRUE, cex = 0.5,
          k_colors = c("#00AFBB","#2E9FDF", "#E7B800", "#FC4E07"))
```

![](tools/README-hierarchical-clustering-1.png)

``` r
if (requireNamespace("igraph", quietly = TRUE)) {
  # The default compatibility alias "layout.auto" still works, and
  # modern igraph layout names are accepted as well.
  fviz_dend(res, type = "phylogenic", phylo_layout = "layout_nicely",
            show_labels = FALSE)
}
```

![](tools/README-hierarchical-clustering-phylogenic-1.png)

Read more:

1.  [Cluster analysis in R: All what you should
    know](https://www.datanovia.com/en/blog/types-of-clustering-methods-overview-and-quick-start-r-code/)

2.  [Hierarchical clustering
    essentials](https://www.datanovia.com/en/courses/hierarchical-clustering-in-r-the-essentials/)

### Determine the optimal number of clusters

``` r
# Optimal number of clusters for k-means
library("factoextra")
my_data <- scale(USArrests)
fviz_nbclust(my_data, kmeans, method = "gap_stat")
```

![](tools/README-determine-the-number-of-clusters-gap-statistics-1.png)

## Acknowledgment

Original package authors:

  - [Alboukadel Kassambara](https://github.com/kassambara)
  - [Fabian Mundt](https://github.com/inventionate)

Contributor:

  - Laszlo Erdey (Faculty of Economics and Business, University of
    Debrecen, Hungary)

We sincerely thank all developers for their efforts behind the packages
that **factoextra** depends on, namely,
[ggplot2](https://cran.r-project.org/package=ggplot2) (Hadley Wickham,
Springer-Verlag New York, 2009),
[FactoMineR](https://cran.r-project.org/package=FactoMineR) (Sebastien
Le et al., Journal of Statistical Software, 2008),
[dendextend](https://cran.r-project.org/package=dendextend) (Tal Galili,
Bioinformatics, 2015),
[cluster](https://cran.r-project.org/package=dendextend) (Martin
Maechler et al., 2016) and more …..

## References

  - H. Wickham (2009). ggplot2: Elegant Graphics for Data Analysis.
    Springer-Verlag New York.
  - Maechler, M., Rousseeuw, P., Struyf, A., Hubert, M., Hornik,
    K.(2016). cluster: Cluster Analysis Basics and Extensions. R package
    version 2.0.5.
  - Sebastien Le, Julie Josse, Francois Husson (2008). FactoMineR: An R
    Package for Multivariate Analysis. Journal of Statistical Software,
    25(1), 1-18. 10.18637/jss.v025.i01
  - Tal Galili (2015). dendextend: an R package for visualizing,
    adjusting, and comparing trees of hierarchical clustering.
    Bioinformatics. DOI: 10.1093/bioinformatics/btv428
