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Multiple Factor Analysis by Example Using R

Name: Multiple Factor Analysis by Example Using R
Author: Jérôme Pagès
ISBN: 9781498786690

by Jérôme Pagès

November 2014

Intermediate to advanced

272 pages

6h 57m

English

Chapman and Hall/CRC

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Preliminaries
Preface
Chapter 1 Principal Component Analysis
Vocabulary: Factor Analysis or Factorial Analysis?1.1 Data, Notations1.2 Why Analyse a Table with PCA?1.3 Clouds of Individuals and VariablesCloud of Individuals NICloud of Variables NK1.4 Centring and Reducing1.5 Fitting Clouds NI and NK1.5.1 General Principles and Formalising CriteriaFitting NI in ℝKFitting NK in ℝI1.5.2 Interpreting Criteria1.5.3 SolutionIn the Individuals’ SpaceIn the Variables’ Space1.5.4 Relationships Between the Analyses of the Two Clouds1.5.5 Representing the Variables1.5.6 Number of Axes1.5.7 Vocabulary: Axes and Factors1.6 Interpretation Aids1.6.1 Percentage of Inertia Associated with an Axis1.6.2 Contribution of One Point to the Inertia of an AxisCase of an IndividualCase of a Variable1.6.3 Quality of Representation of a Point by an Axis1.7 First Example: 909 Baccalaureate Candidates1.7.1 Projected Inertia (Eigenvalues)1.7.2 Interpreting the Axes1.7.3 Methodological RemarksInterpretability and Percentage of InertiaTwo Readings of the Correlation CircleValidating the InterpretationPCA and Synthetic Visualisation1.8 Supplementary ElementsRemark1.9 Qualitative Variables in PCAPrincipleBaccalaureate Grades Example1.10 Second Example: Six Orange Juices1.11 PCA in FactoMineRDrop-Down Menu in R CommanderMain Menu (See Figure 1.11)Graphical Options (See Figure 1.12)Examples of CommandsRemarkScript for Analysing the Orange Juice DataFigure 1.1Figure 1.2Figure 1.3Figure 1.4Figure 1.5Figure 1.6Figure 1.7Figure 1.8Figure 1.9Figure 1.10Figure 1.11Figure 1.12Table 1.1Table 1.2Table 1.3Table 1.4Table 1.5Table 1.6
Chapter 2 Multiple Correspondence Analysis
2.1 Data2.2 Complete Disjunctive Table2.3 Questioning2.4 Clouds of Individuals and Variables2.4.1 Cloud of IndividualsDistance Between an Individual i and the Centre of Gravity of NITotal Inertia of NI (With Respect to GI)Distance Between Two Individuals i and lRemark2.4.2 Cloud of Categories2.4.3 Qualitative Variables2.5 Fitting Clouds NI and NK2.5.1 Cloud of Individuals2.5.2 Cloud of Categories2.5.3 Relationships Between the Two Analyses2.6 Representing Individuals, Categories and VariablesRepresenting Individuals and CategoriesRepresenting the VariablesNumber of Axes2.7 Interpretation AidsPercentage of Inertia Associated with an AxisContributionsSupplementary Elements2.8 Example: Five Educational Tools Evaluated by 25 Students2.8.1 Data2.8.2 Analyses and RepresentationsRepresenting IndividualsRepresenting Categories2.8.3 MCA/PCA Comparison for Ordinal Variables2.9 MCA in FactoMineRDrop-Down Menu in R CommanderFactoMineR offers three types of graph for MCA (see Figure 2.7).Examples of CommandsFigure 2.1Figure 2.2Figure 2.3Figure 2.4Figure 2.5Figure 2.6Figure 2.7Table 2.1Table 2.2
Chapter 3 Factorial Analysis of Mixed Data
3.1 Data, Notations3.2 Representing Variables3.3 Representing Individuals3.4 Transition RelationsRelationships from ℝK Toward ℝ1Relationship from ℝI Toward ℝKRemark3.5 Implementation3.6 Example: Biometry of Six Individuals3.7 FAMD in FactoMineRDrop-Down Menu in R CommanderMain Menu (see Figure 3.5)Graphical Options (see Figure 3.6)Examples of CommandsFigure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 3.6Table 3.1Table 3.2Table 3.3
Chapter 4 Weighting Groups of Variables
4.1 Objectives4.2 Introductory Numerical Example4.3 Weighting Variables in MFA4.4 Application to the Six Orange Juices4.5 Relationships with Separate Analyses4.6 Conclusion4.7 MFA in FactoMineR (First Results)Drop-Down Menu in R CommanderDefining the Groups (See Figure 4.9)Graphical Options (See Figure 4.10)Command ExamplesTable 4.5Simplified OutputFigure 4.1Figure 4.2Figure 4.3Figure 4.4Figure 4.5Figure 4.6Figure 4.7Figure 4.8Figure 4.9Figure 4.10Table 4.1Table 4.2Table 4.3Table 4.4Table 4.5Table 4.6Table 4.7
Chapter 5 Comparing Clouds of Partial Individuals
5.1 Objectives5.2 Method5.3 Application to the Six Orange Juices5.4 Interpretation AidsRemark5.5 Distortions in Superimposed Representations5.5.1 Example (Trapeziums Data)5.5.2 Geometric InterpretationRemark5.5.3 Algebra ApproachNotations (Reminders and Additions)Reconstitution of NjINumerical Example5.6 Superimposed Representation: Conclusion5.7 MFA Partial Clouds in FactoMineRThe Drop-Down MenuCommand LinesFigure 5.1Figure 5.2Figure 5.3Figure 5.4Figure 5.5Figure 5.6Figure 5.7Figure 5.8Figure 5.9Figure 5.10Table 5.1Table 5.2Table 5.3Table 5.4Table 5.5
Chapter 6 Factors Common to Different Groups of Variables
6.1 Objectives6.1.1 Measuring the Relationship between a Variable and a Group6.1.2 Factors Common to Several Groups of Variables6.1.3 Back to the Six Orange Juices6.1.4 Canonical Analysis6.2 Relationship Between a Variable and Groups of Variables6.3 Searching for Common Factors6.4 Searching for Canonical VariablesRemark6.5 Interpretation Aids6.5.1 Lg Relationship Measurement6.5.2 Canonical Correlation CoefficientsFigure 6.1Table 6.1
Chapter 7 Comparing Groups of Variables and Indscal Model
7.1 Cloud NJ of Groups of VariablesRemark7.2 Scalar Product and Relationship Between Groups of VariablesLg and RV7.3 Norm in the Groups’ Space7.4 Representation of Cloud NJ7.4.1 PrincipleRemark7.4.2 Criterion7.5 Interpretation Aids7.6 The Indscal Model7.6.1 Model7.6.2 Estimating Parameters and Properties7.6.3 Example of an Indscal model via MFA (cards)7.6.4 Ten Touraine White WinesData (Wines)ResultsComparing the Two Estimations: Quality of the FitNature of the FactorsConclusion7.7 MFA in FactoMineR (groups)Figure 7.1Figure 7.2Figure 7.3Figure 7.4Figure 7.5Figure 7.6Figure 7.7Figure 7.8Figure 7.9Table 7.1Table 7.2Table 7.3Table 7.4Table 7.5
Chapter 8 Qualitative and Mixed Data
8.1 Weighted MCA8.1.1 Cloud of Categories in Weighted MCA8.1.2 Transition Relations in Weighted MCA8.2 MFA of Qualitative Variables8.2.1 From the Perspective of Factorial AnalysisPrinciple of Weighting Groups of VariablesMFA of Qualitative Variables Is Based on a Weighted MCARemark8.2.2 From the Perspective of Multicanonical AnalysisLg Measurement for Qualitative VariablesRemarkLooking for General Variables8.2.3 Representing Partial IndividualsRemark8.2.4 Representing Partial CategoriesRemark8.2.5 Analysing in Space of Groups of Variables (ℝI2)Cloud of Groups of VariablesInterpreting the Scalar Product Between Two Groups8.3 Mixed Data8.3.1 Weighting the Variables8.3.2 PropertiesRepresenting the VariablesRepresenting Clouds of Partial IndividualsMulticanonical AnalysisAnalysis in ℝI28.4 Application (Biometry2)8.4.1 Separate AnalysesInertias (Table 8.4)Correlations Between the Factors of the Separate Analyses (Table 8.5)8.4.2 Inertias in the Overall Analysis8.4.3 Coordinates of the Factors of the Separate Analyses8.4.4 First FactorIndividuals and Variables (See Figure 8.2)Partial Individuals (See Figure 8.3)Transition RelationsPartial Categories (see Figure 8.3)Conclusion8.4.5 Second Factor8.4.6 Third Factor8.4.7 Representing Groups of Variables8.4.8 Conclusion8.5 MFA of Mixed Data in FactoMineRTable 8.5 (Correlations Between Partial Axes)Table 8.7 (Contributions of the Cells to Partial Individuals)Table 8.8 (Qualities of Representation in ℝI2)Table 8.9 (Lg an RV Relationship Indicators)Figure 8.1Figure 8.2Figure 8.3Figure 8.4Table 8.1Table 8.2Table 8.3Table 8.4Table 8.5Table 8.6Table 8.7Table 8.8Table 8.9

Chapter 9 Multiple Factor Analysis and Procrustes Analysis
9.1 Procrustes Analysis9.1.1 Data, Notations9.1.2 Objectives9.1.3 Methods and VariationsDepending on the Number of CloudsRemarkDepending on the Number of DimensionsInfluence of the Dimensions on the Objectives9.2 Comparing MFA and GPA9.2.1 Representing NjIIn MFAIn GPA9.2.2 Mean CloudIn MFAIn GPA9.2.3 Objective, Criterion, AlgorithmIn MFAIn GPACriterionRemark9.2.4 Properties of the Representations of NjIIn GPAIn MFA9.2.5 A First Appraisal9.2.6 Harmonising the Inertia of NjI9.2.7 Relationships Between Homologous FactorsRemark9.2.8 Representing Individuals9.2.9 Interpretation AidsDecomposition by DimensionDecomposition by GroupDecomposition by IndividualSummary9.2.10 Representing the Variables9.3 Application (Data 23−1)9.3.1 Data 23−19.3.2 Results of the MFAProjected Inertias of the Mean CloudRelationship Measurements Between Factors and GroupsInertias of Individuals in the Mean CloudSuperimposed Representation9.3.3 Results of the GPAProjected Inertias of the Mean CloudRepresenting IndividualsIndicators of the Discrepancy to the Procrustes ModelConclusion9.4 Application to the Ten Touraine Wines9.5 Conclusion9.6 GPA in FactoMineRProcrustes MFA (PMFA; Figure 9.8)Figure 9.1Figure 9.2Figure 9.3Figure 9.4Figure 9.5Figure 9.6Figure 9.7Figure 9.8Table 9.1Table 9.2Table 9.3Table 9.4Table 9.5
Chapter 10 Hierarchical Multiple Factor Analysis
10.1 Data, Examples10.2 Hierarchy and Partitions10.3 Weighting the VariablesRemark10.4 Representing Partial Individuals10.4.1 Method10.4.2 Application to the Six Orange Juices10.5 Canonical Correlation Coefficients10.6 Representing the Nodes10.7 Application to Mixed Data: Sorted Napping®10.7.1 Data and Methodology10.7.2 Intermediary Analysis: MFA on a Sorted Nappe10.7.3 Decompositions of Inertia10.7.4 Representing Partial and Mean IndividualsRepresentation of Groups of Variables (see Figure 10.13)10.8 HMFA in FactoMineRRemarkFigure 10.1Figure 10.2Figure 10.3Figure 10.4Figure 10.5Figure 10.6Figure 10.7Figure 10.8Figure 10.9Figure 10.10Figure 10.11Figure 10.12Figure 10.13Figure 10.14Table 10.1Table 10.2Table 10.3Table 10.4Table 10.5Table 10.6Table 10.7
Chapter 11 Matrix Calculus and Euclidean Vector Space
11.1 Matrix CalculusDefinitionsTranspositionMatrix MultiplicationTrace of a Square MatrixMatrix and Function, Orthogonal Matrix, Diagonalisation11.2 Euclidean Vector Space11.2.1 Vector Space Endowed with the Usual DistanceTwo-Dimensional SpaceVector Space with n Dimensions11.2.2 Euclidean Space Endowed with a Diagonal Metric11.2.3 Visualising a Cloud in a Space Endowed with a Metric Different from the IdentityFigure 11.1Figure 11.2Table 11.1Table 11.2
Bibliography

Content preview from Multiple Factor Analysis by Example Using R

Chapter 9

Multiple Factor Analysis and Procrustes Analysis

Originally, the question which motivated Procrustes analysis (PA) was, ‘How can one of two homologous clouds of points situated within the same space be turned to make them coincide as closely as possible?’ Today, generalised Procrustes analysis (GPA) is applied to sets of more than two clouds.

Chapter 5 describes the solution proposed by multiple factor analysis (MFA) to obtain a superimposed representation of homologous clouds of points ( $N_{I}^{j}$ ). The two approaches must be compared.

As GPA is not well known outside the field of sensory analysis, we start by presenting some basic elements of this method.

9.1 Procrustes Analysis

9.1.1 Data, Notations

There are J clouds (denoted

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Multiple Factor Analysis by Example Using R

by Jérôme Pagès

Multiple Factor Analysis and Procrustes Analysis