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Practical Guide to Principal Component Methods in R Multivariate Analysis Alboukadel KASSAMBARA

Practical Guide to Principal Component Methods in R

Preface

0.1 What you will learn Large data sets containing multiple samples and variables are collected everyday by researchers in various fields, such as in Bio-medical, marketing, and geospatial fields. Discovering knowledge from these data requires specific techniques for analyzing data sets containing multiple variables. Multivariate analysis (MVA) refers to a set of techniques used for analyzing a data set containing more than one variable. Among these techniques, there are: Cluster analysis for identifying groups of observations with similar profile according to a specific criteria. Principal component methods, which consist of summarizing and visualizing the most important information contained in a multivariate data set.

Previously, we published a book entitled "Practical Guide To Cluster Analysis in R" (https://goo.gl/DmJ5y5). The aim of the current book is to provide a solid practical guidance to principal component methods in R. Additionally, we developed an R package named factoextra to create, easily, a ggplot2-based elegant plots of the results of principal component method. Factoextra official online documentation: http://www.sthda.com/english/rpkgs/factoextra

One of the difficulties inherent in multivariate analysis is the problem of visualizing data that has many variables. In R, there are many functions and packages for displaying a graph of the relationship between two variables (http://www.sthda.com/english/wiki/data-visualization). There are also commands for displaying different three-dimensional views. But when there are more than three variables, it is more difficult to visualize their relationships. Fortunately, in data sets with many variables, some variables are often correlated. This can be explained by the fact that, more than one variable might be measuring the same driving principle governing the behavior of the system. Correlation indicates that there is redundancy in the data. When this happens, you can simplify

the problem by replacing a group of correlated variables with a single new variable. Principal component analysis is a rigorous statistical method used for achieving this simplification. The method creates a new set of variables, called principal components. Each principal component is a linear combination of the original variables. All the principal components are orthogonal to each other, so there is no redundant information. The type of principal component methods to use depends on variable types contained in the data set. This practical guide will describe the following methods: 1. Principal Component Analysis (PCA), which is one of the most popular multivariate analysis method. The goal of PCA is to summarize the information contained in a continuous (i.e, quantitative) multivariate data by reducing the dimensionality of the data without loosing important information. 2. Correspondence Analysis (CA), which is an extension of the principal component analysis for analyzing a large contingency table formed by two qualitative variables (or categorical data). 3. Multiple Correspondence Analysis (MCA), which is an adaptation of CA to a data table containing more than two categorical variables. 4. Factor Analysis of Mixed Data (FAMD), dedicated to analyze a data set containing both quantitative and qualitative variables. 5. Multiple Factor Analysis (MFA), dedicated to analyze data sets, in which variables are organized into groups (qualitative and/or quantitative variables). Additionally, we'll discuss the HCPC (Hierarchical Clustering on Principal Component) method. It applies agglomerative hierarchical clustering on the results of principal component methods (PCA, CA, MCA, FAMD, MFA). It allows us, for example, to perform clustering analysis on any type of data (quantitative, qualitative or mixed data). Figure 1 illustrates the type of analysis to be performed depending on the type of

variables contained in the data set.

Principal component methods

0.2 Key features of this book Although there are several good books on principal component methods and related topics, we felt that many of them are either too theoretical or too advanced. Our goal was to write a practical guide to multivariate analysis, visualization and interpretation, focusing on principal component methods. The book presents the basic principles of the different methods and provide many examples in R. This book offers solid guidance in data mining for students and researchers. Key features Covers principal component methods and implementation in R Short, self-contained chapters with tested examples that allow for flexibility in designing a course and for easy reference At the end of each chapter, we present R lab sections in which we systematically work through applications of the various methods discussed in that chapter. Additionally, we provide links to other resources and to our hand-curated list of videos on principal component methods for further learning.

0.3 How this book is organized This book is divided into 4 parts and 6 chapters. Part I provides a quick introduction to R (chapter 2) and presents required R packages for the analysis and visualization (chapter 3). In Part II, we describe classical multivariate analysis methods: Principal Component Analysis - PCA (chapter 4) Correspondence Analysis - CA (chapter 5) Multiple Correspondence Analysis - MCA (chapter 6) In part III, we continue by discussing advanced methods for analyzing a data set containing a mix of variables (qualitative & quantitative) organized or not into groups: Factor Analysis of Mixed Data - FAMD (chapter 7) and, Multiple Factor Analysis - MFA (chapter 8). Finally, we show in Part IV, how to perform hierarchical clustering on principal components (HCPC) (chapter 9), which is useful for performing clustering with a data set containing only qualitative variables or with a mixed data of qualitative and quantitative variables. Some examples of plots generated in this book are shown hereafter. You'll learn how to create, customize and interpret these plots. 1. Eigenvalues/variances of principal components. Proportion of information retained by each principal component.

2. PCA - Graph of variables: Control variable colors using their contributions to the principal components.

Highlight the most contributing variables to each principal dimension:

3. PCA - Graph of individuals: Control automatically the color of individuals using the cos2 (the quality of the individuals on the factor map)

Change the point size according to the cos2 of the corresponding individuals:

4. PCA - Biplot of individuals and variables

5. Correspondence analysis. Association between categorical variables.

6. FAMD - Analyzing mixed data

7. Clustering on principal components

0.4 Book website The website for this book is located at : http://www.sthda.com/english/. It contains number of resources.

0.5 Executing the R codes from the PDF For a single line R code, you can just copy the code from the PDF to the R console. For a multiple-line R codes, an error is generated, sometimes, when you copy and paste directly the R code from the PDF to the R console. If this happens, a solution is to: Paste firstly the code in your R code editor or in your text editor Copy the code from your text/code editor to the R console

0.6 Acknowledgment I sincerely thank all developers for their efforts behind the packages that factoextra depends on, namely, ggplot2 (Hadley Wickham, Springer-Verlag New York, 2009), FactoMineR (Sebastien Le et al., Journal of Statistical Software, 2008), dendextend (Tal Galili, Bioinformatics, 2015), cluster (Martin Maechler et al., 2016) and more.

0.7 Colophon This book was built with: R 3.3.2 factoextra 1.0.5 FactoMineR 1.36 ggpubr 0.1.5 dplyr 0.7.2 bookdown 0.4.3

1 About the author Alboukadel Kassambara is a PhD in Bioinformatics and Cancer Biology. He works since many years on genomic data analysis and visualization (read more: http://www.alboukadel.com/). He has work experiences in statistical and computational methods to identify prognostic and predictive biomarker signatures through integrative analysis of large-scale genomic and clinical data sets. He created a bioinformatics web-tool named GenomicScape (www.genomicscape.com) which is an easy-to-use web tool for gene expression data analysis and visualization. He developed also a training website on data science, named STHDA (Statistical Tools for High-throughput Data Analysis, www.sthda.com/english), which contains many tutorials on data analysis and visualization using R software and packages. He is the author of many popular R packages for: multivariate data analysis (factoextra, http://www.sthda.com/english/rpkgs/factoextra), survival analysis (survminer, http://www.sthda.com/english/rpkgs/survminer/), correlation analysis (ggcorrplot, http://www.sthda.com/english/wiki/ggcorrplot-visualization-of-acorrelation-matrix-using-ggplot2), creating publication ready plots in R (ggpubr, http://www.sthda.com/english/rpkgs/ggpubr). Recently, he published three books on data analysis and visualization: 1. Practical Guide to Cluster Analysis in R (https://goo.gl/DmJ5y5) 2. Guide to Create Beautiful Graphics in R (https://goo.gl/vJ0OYb). 3. Complete Guide to 3D Plots in R (https://goo.gl/v5gwl0).

2 Introduction to R R is a free and powerful statistical software for analyzing and visualizing data. If you want to learn easily the essential of R programming, visit our series of tutorials available on STHDA: http://www.sthda.com/english/wiki/r-basicsquick-and-easy. In this chapter, we provide a very brief introduction to R, for installing R/RStudio as well as importing your data into R for computing principal component methods.

2.1 Installing R and RStudio R and RStudio can be installed on Windows, MAC OSX and Linux platforms. RStudio is an integrated development environment for R that makes using R easier. It includes a console, code editor and tools for plotting. 1. R can be downloaded and installed from the Comprehensive R Archive Network (CRAN) webpage (http://cran.r-project.org/) 2. After installing R software, install also the RStudio software available at: http://www.rstudio.com/products/RStudio/. 3. Launch RStudio and start use R inside R studio.

Rstudio interface

2.2 Installing and loading R packages An R package is an extension of R containing data sets and specific R functions to solve specific questions. For example, in this book, you'll learn how to compute and visualize principal component methods using FactoMineR and factoextra R packages. There are thousands other R packages available for download and installation from CRAN, Bioconductor (biology related R packages) and GitHub repositories. 1. How to install packages from CRAN? Use the function install.packages(): install.packages("FactoMineR") install.packages("factoextra")

2. How to install packages from GitHub? You should first install devtools if you don't have it already installed on your computer: For example, the following R code installs the latest developmental version of factoextra R package developed by A. Kassambara (https://github.com/kassambara/facoextra) for multivariate data analysis and elegant visualization. install.packages("devtools") devtools::install_github("kassambara/factoextra")

Note that, GitHub contains the latest developmental version of R packages. 3. After installation, you must first load the package for using the functions in the package. The function library() is used for this task. library("FactoMineR") library("factoextra")

Now, we can use R functions, such as PCA() [in the FactoMineR package] for performing principal component analysis.

2.3 Getting help with functions in R If you want to learn more about a given function, say PCA(), type this in R console: ?PCA

2.4 Importing your data into R 1. Prepare your file as follow: Use the first row as column names. Generally, columns represent variables Use the first column as row names. Generally rows represent observations or individuals. Each row/column name should be unique, so remove duplicated names. Avoid names with blank spaces. Good column names: Long_jump or Long.jump. Bad column name: Long jump. Avoid names with special symbols: ?, $, *, +, #, (, ), -, /, }, {, |, >, < etc. Only underscore can be used. Avoid beginning variable names with a number. Use letter instead. Good column names: sport_100m or x100m. Bad column name: 100m R is case sensitive. This means that Name is different from Name or NAME. Avoid blank rows in your data. Delete any comments in your file. Replace missing values by NA (for not available) If you have a column containing date, use the four digit format. Good format: 01/01/2016. Bad format: 01/01/16 2. The final file should look like this:

General data format for importation into R 3. Save your file We recommend to save your file into .txt (tab-delimited text file) or .csv (comma separated value file) format. 4. Get your data into R: Use the R code below. You will be asked to choose a file: # .txt file: Read tab separated values my_data

View more...
Practical Guide to Principal Component Methods in R

Preface

0.1 What you will learn Large data sets containing multiple samples and variables are collected everyday by researchers in various fields, such as in Bio-medical, marketing, and geospatial fields. Discovering knowledge from these data requires specific techniques for analyzing data sets containing multiple variables. Multivariate analysis (MVA) refers to a set of techniques used for analyzing a data set containing more than one variable. Among these techniques, there are: Cluster analysis for identifying groups of observations with similar profile according to a specific criteria. Principal component methods, which consist of summarizing and visualizing the most important information contained in a multivariate data set.

Previously, we published a book entitled "Practical Guide To Cluster Analysis in R" (https://goo.gl/DmJ5y5). The aim of the current book is to provide a solid practical guidance to principal component methods in R. Additionally, we developed an R package named factoextra to create, easily, a ggplot2-based elegant plots of the results of principal component method. Factoextra official online documentation: http://www.sthda.com/english/rpkgs/factoextra

One of the difficulties inherent in multivariate analysis is the problem of visualizing data that has many variables. In R, there are many functions and packages for displaying a graph of the relationship between two variables (http://www.sthda.com/english/wiki/data-visualization). There are also commands for displaying different three-dimensional views. But when there are more than three variables, it is more difficult to visualize their relationships. Fortunately, in data sets with many variables, some variables are often correlated. This can be explained by the fact that, more than one variable might be measuring the same driving principle governing the behavior of the system. Correlation indicates that there is redundancy in the data. When this happens, you can simplify

the problem by replacing a group of correlated variables with a single new variable. Principal component analysis is a rigorous statistical method used for achieving this simplification. The method creates a new set of variables, called principal components. Each principal component is a linear combination of the original variables. All the principal components are orthogonal to each other, so there is no redundant information. The type of principal component methods to use depends on variable types contained in the data set. This practical guide will describe the following methods: 1. Principal Component Analysis (PCA), which is one of the most popular multivariate analysis method. The goal of PCA is to summarize the information contained in a continuous (i.e, quantitative) multivariate data by reducing the dimensionality of the data without loosing important information. 2. Correspondence Analysis (CA), which is an extension of the principal component analysis for analyzing a large contingency table formed by two qualitative variables (or categorical data). 3. Multiple Correspondence Analysis (MCA), which is an adaptation of CA to a data table containing more than two categorical variables. 4. Factor Analysis of Mixed Data (FAMD), dedicated to analyze a data set containing both quantitative and qualitative variables. 5. Multiple Factor Analysis (MFA), dedicated to analyze data sets, in which variables are organized into groups (qualitative and/or quantitative variables). Additionally, we'll discuss the HCPC (Hierarchical Clustering on Principal Component) method. It applies agglomerative hierarchical clustering on the results of principal component methods (PCA, CA, MCA, FAMD, MFA). It allows us, for example, to perform clustering analysis on any type of data (quantitative, qualitative or mixed data). Figure 1 illustrates the type of analysis to be performed depending on the type of

variables contained in the data set.

Principal component methods

0.2 Key features of this book Although there are several good books on principal component methods and related topics, we felt that many of them are either too theoretical or too advanced. Our goal was to write a practical guide to multivariate analysis, visualization and interpretation, focusing on principal component methods. The book presents the basic principles of the different methods and provide many examples in R. This book offers solid guidance in data mining for students and researchers. Key features Covers principal component methods and implementation in R Short, self-contained chapters with tested examples that allow for flexibility in designing a course and for easy reference At the end of each chapter, we present R lab sections in which we systematically work through applications of the various methods discussed in that chapter. Additionally, we provide links to other resources and to our hand-curated list of videos on principal component methods for further learning.

0.3 How this book is organized This book is divided into 4 parts and 6 chapters. Part I provides a quick introduction to R (chapter 2) and presents required R packages for the analysis and visualization (chapter 3). In Part II, we describe classical multivariate analysis methods: Principal Component Analysis - PCA (chapter 4) Correspondence Analysis - CA (chapter 5) Multiple Correspondence Analysis - MCA (chapter 6) In part III, we continue by discussing advanced methods for analyzing a data set containing a mix of variables (qualitative & quantitative) organized or not into groups: Factor Analysis of Mixed Data - FAMD (chapter 7) and, Multiple Factor Analysis - MFA (chapter 8). Finally, we show in Part IV, how to perform hierarchical clustering on principal components (HCPC) (chapter 9), which is useful for performing clustering with a data set containing only qualitative variables or with a mixed data of qualitative and quantitative variables. Some examples of plots generated in this book are shown hereafter. You'll learn how to create, customize and interpret these plots. 1. Eigenvalues/variances of principal components. Proportion of information retained by each principal component.

2. PCA - Graph of variables: Control variable colors using their contributions to the principal components.

Highlight the most contributing variables to each principal dimension:

3. PCA - Graph of individuals: Control automatically the color of individuals using the cos2 (the quality of the individuals on the factor map)

Change the point size according to the cos2 of the corresponding individuals:

4. PCA - Biplot of individuals and variables

5. Correspondence analysis. Association between categorical variables.

6. FAMD - Analyzing mixed data

7. Clustering on principal components

0.4 Book website The website for this book is located at : http://www.sthda.com/english/. It contains number of resources.

0.5 Executing the R codes from the PDF For a single line R code, you can just copy the code from the PDF to the R console. For a multiple-line R codes, an error is generated, sometimes, when you copy and paste directly the R code from the PDF to the R console. If this happens, a solution is to: Paste firstly the code in your R code editor or in your text editor Copy the code from your text/code editor to the R console

0.6 Acknowledgment I sincerely thank all developers for their efforts behind the packages that factoextra depends on, namely, ggplot2 (Hadley Wickham, Springer-Verlag New York, 2009), FactoMineR (Sebastien Le et al., Journal of Statistical Software, 2008), dendextend (Tal Galili, Bioinformatics, 2015), cluster (Martin Maechler et al., 2016) and more.

0.7 Colophon This book was built with: R 3.3.2 factoextra 1.0.5 FactoMineR 1.36 ggpubr 0.1.5 dplyr 0.7.2 bookdown 0.4.3

1 About the author Alboukadel Kassambara is a PhD in Bioinformatics and Cancer Biology. He works since many years on genomic data analysis and visualization (read more: http://www.alboukadel.com/). He has work experiences in statistical and computational methods to identify prognostic and predictive biomarker signatures through integrative analysis of large-scale genomic and clinical data sets. He created a bioinformatics web-tool named GenomicScape (www.genomicscape.com) which is an easy-to-use web tool for gene expression data analysis and visualization. He developed also a training website on data science, named STHDA (Statistical Tools for High-throughput Data Analysis, www.sthda.com/english), which contains many tutorials on data analysis and visualization using R software and packages. He is the author of many popular R packages for: multivariate data analysis (factoextra, http://www.sthda.com/english/rpkgs/factoextra), survival analysis (survminer, http://www.sthda.com/english/rpkgs/survminer/), correlation analysis (ggcorrplot, http://www.sthda.com/english/wiki/ggcorrplot-visualization-of-acorrelation-matrix-using-ggplot2), creating publication ready plots in R (ggpubr, http://www.sthda.com/english/rpkgs/ggpubr). Recently, he published three books on data analysis and visualization: 1. Practical Guide to Cluster Analysis in R (https://goo.gl/DmJ5y5) 2. Guide to Create Beautiful Graphics in R (https://goo.gl/vJ0OYb). 3. Complete Guide to 3D Plots in R (https://goo.gl/v5gwl0).

2 Introduction to R R is a free and powerful statistical software for analyzing and visualizing data. If you want to learn easily the essential of R programming, visit our series of tutorials available on STHDA: http://www.sthda.com/english/wiki/r-basicsquick-and-easy. In this chapter, we provide a very brief introduction to R, for installing R/RStudio as well as importing your data into R for computing principal component methods.

2.1 Installing R and RStudio R and RStudio can be installed on Windows, MAC OSX and Linux platforms. RStudio is an integrated development environment for R that makes using R easier. It includes a console, code editor and tools for plotting. 1. R can be downloaded and installed from the Comprehensive R Archive Network (CRAN) webpage (http://cran.r-project.org/) 2. After installing R software, install also the RStudio software available at: http://www.rstudio.com/products/RStudio/. 3. Launch RStudio and start use R inside R studio.

Rstudio interface

2.2 Installing and loading R packages An R package is an extension of R containing data sets and specific R functions to solve specific questions. For example, in this book, you'll learn how to compute and visualize principal component methods using FactoMineR and factoextra R packages. There are thousands other R packages available for download and installation from CRAN, Bioconductor (biology related R packages) and GitHub repositories. 1. How to install packages from CRAN? Use the function install.packages(): install.packages("FactoMineR") install.packages("factoextra")

2. How to install packages from GitHub? You should first install devtools if you don't have it already installed on your computer: For example, the following R code installs the latest developmental version of factoextra R package developed by A. Kassambara (https://github.com/kassambara/facoextra) for multivariate data analysis and elegant visualization. install.packages("devtools") devtools::install_github("kassambara/factoextra")

Note that, GitHub contains the latest developmental version of R packages. 3. After installation, you must first load the package for using the functions in the package. The function library() is used for this task. library("FactoMineR") library("factoextra")

Now, we can use R functions, such as PCA() [in the FactoMineR package] for performing principal component analysis.

2.3 Getting help with functions in R If you want to learn more about a given function, say PCA(), type this in R console: ?PCA

2.4 Importing your data into R 1. Prepare your file as follow: Use the first row as column names. Generally, columns represent variables Use the first column as row names. Generally rows represent observations or individuals. Each row/column name should be unique, so remove duplicated names. Avoid names with blank spaces. Good column names: Long_jump or Long.jump. Bad column name: Long jump. Avoid names with special symbols: ?, $, *, +, #, (, ), -, /, }, {, |, >, < etc. Only underscore can be used. Avoid beginning variable names with a number. Use letter instead. Good column names: sport_100m or x100m. Bad column name: 100m R is case sensitive. This means that Name is different from Name or NAME. Avoid blank rows in your data. Delete any comments in your file. Replace missing values by NA (for not available) If you have a column containing date, use the four digit format. Good format: 01/01/2016. Bad format: 01/01/16 2. The final file should look like this:

General data format for importation into R 3. Save your file We recommend to save your file into .txt (tab-delimited text file) or .csv (comma separated value file) format. 4. Get your data into R: Use the R code below. You will be asked to choose a file: # .txt file: Read tab separated values my_data

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