Exploratory Multivariate Analysis by Example Using R

Francois Husson, Sebastien Le, Jérôme Pagès

• Publisher: Chapman and Hall/CRC
• Year: 2017
• Topic: Mathematics
• eBook ISBN:
  9781315301853
• Subtopic: Computer Science General
• Index: Mathematics

Full of real-world case studies and practical advice, Exploratory Multivariate Analysis by Example Using R, Second Edition focuses on four fundamental methods of multivariate exploratory data analysis that are most suitable for applications. It covers principal component analysis (PCA) when variables are quantitative, correspondence analysis (CA) a

1

Principal Component Analysis (PCA)

1.1 Data — Notation — Examples

Principal component analysis (PCA) applies to data tables where rows are considered as individuals and columns as quantitative variables. Let x_ik be the value taken by individual i for variable k, where i varies from 1 to I and k from 1 to K.

Let ${\overline{x}}_k$ denote the mean of variable k calculated over all individual instances of I:

${\overline{x}}_k=\frac{1}{I}{\displaystyle \sum _{i=1}^I{x}_{ik}},$

and s_k the standard deviation of the sample of variable k (uncorrected):

$s_k=\sqrt{\frac{1}{I}{\displaystyle \sum _{i=1}^I{\left(x_{ik}-{\overline{x}}_k\right)}^2}}.$

Data subjected to a PCA can be very diverse in nature; some examples are listed in Table 1.1.

This first chapter will be illustrated using the “orange juice” dataset chosen for its simplicity since it comprises only six statistical individuals or observations. The six orange juices were evaluated by a panel of experts according to seven sensory variables (odour intensity, odour typicality, pulp content, intensity of taste, acidity, bitterness, sweetness). The panel’s evaluations are summarised in Table 1.2.

1.2 Objectives

The data table can be considered either as a set of rows (individuals) or as a set of columns (variables), thus raising a number of questions relating to these different types of objects.

TABLE 1.1
Some Examples of Datasets

Field	Individuals	Variables	xik
Ecology	Rivers	Concentration of pollutants	Concentration of pollutant k in river i
Economics	Years	Economic indicators	Indicator value k for year i
Genetics	Patients	Genes	Expression of gene k for patient i
Marketing	Brands	Measures of satisfaction	Value of measure k for brand i
Pedology	Soils	Granulometric composition	Content of component k in soil i
Biology	Animals	Measurements	Measure k for animal i
Sociology	Social classes	Time by activity	Time spent on activity k by individuals from social class i

TABLE 1.2
The Orange Juice Data

1.2.1 Studying Individuals

Figure 1.1 illustrates the types of questions posed during the study of individuals. This diagram represents three different situations where 40 individuals are described in terms of two variables: j and k. In graph A, we can clearly identify two distinct classes of individuals. Graph B illustrates a dimension of variability which opposes extreme individuals, much like graph A, but which also contains less extreme individuals. The cloud of individuals is therefore long and thin. Graph C depicts a more uniform cloud (i.e., with no specific structure).

Interpreting the data depicted in these examples is relatively straightforward as they are two dimensional. However, when individuals are described by a large number of variables, we require a tool to explore the space in which these individuals evolve. Studying individuals means identifying the similarities between individuals from the point of view of all the variables. In other words, to provide a typology of the individuals: which are the most similar individuals (and the most dissimilar)? Are there groups of individuals which are homogeneous in terms of their similarities? In addition, we should look for common dimensions of variability which oppose extreme and intermediate individuals.

FIGURE 1.1
Representation of 40 individuals described by two variables: j and k.

In the example, two orange juices are considered similar if they were evaluated in the same way according to all the sensory descriptors. In such cases, the two orange juices have the same main dimensions of variability and are thus said to have the same sensory “profile.” More generally, we want to know whether or not there are groups of orange juices with similar profiles, that is, sensory dimensions which might oppose extreme juices with more intermediate juices.

1.2.2 Studying Variables

Following the approach taken to study the individuals, might it also be possible to interpret the data from the variables? PCA focuses on the linear relationships between variables. More complex links also exist, such as quadratic relationships, logarithmics, exponential functions, and so forth, but they are not studied in PCA. This may seem restrictive, but in practice many relationships can be considered linear, at least for an initial approximation.

Let us consider the example of the four variables (j, k, l, and m) in Figure 1.2. The clouds of points constructed by working from pairs of variables show that variables j and k (graph A) as well as variables l and m (graph F) are strongly correlated (positively for j and k and negatively for l and m). However, the other graphs do not show any signs of relationships between variables. The study of these variables also suggests that the four variables are split into two groups of two variables, (j, k) and (l, m), and that, within one group, the variables are strongly correlated, whereas between groups, the variables are uncorrelated. In exactly the same way as for constructing groups of individuals, creating groups of variables may be useful with a view to synthesis. As for the individuals, we identify a continuum with groups of both very unusual variables and intermediate variables, which are to some extent linked to both groups. In the example, each group can be represented by one single variable as the variables within each group are very strongly correlated. We refer to these variables as synthetic variables.

FIGURE 1.2
Representation of the relationships between four variables: j, k, l, and m, taken two-by-two.

When confronted with a very small number of variables, it is possible to draw conclusions from the clouds of points, or from the correlation matrix which groups together all of the linear correlation coefficients r(j, k) between the pairs of variables. However, when working with a great...

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Principal Component Analysis (PCA)
• 2 Correspondence Analysis (CA)
• 3 Multiple Correspondence Analysis (MCA)
• 4 Clustering
• 5 Visualisation
• Appendix
• Bibliography of Software Packages
• Bibliography
• Index