Data Analysis

Gérard Govaert

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848210981
• eBook ISBN: 9781118617861
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

The first part of this book is devoted to methods seeking relevant dimensions of data. The variables thus obtained provide a synthetic description which often results in a graphical representation of the data. After a general presentation of the discriminating analysis, the second part is devoted to clustering methods which constitute another method, often complementary to the methods described in the first part, to synthesize and to analyze the data. The book concludes by examining the links existing between data mining and data analysis.

Chapter 1

Principal Component Analysis: Application to Statistical Process Control ¹

1.1. Introduction

Principal component analysis (PCA) is an exploratory statistical method for graphical description of the information present in large datasets. In most applications, PCA consists of studying p variables measured on n individuals. When n and p are large, the aim is to synthesize the huge quantity of information into an easy and understandable form.

Unidimensional or bidimensional studies can be performed on variables using graphical tools (histograms, box plots) or numerical summaries (mean, variance, correlation). However, these simple preliminary studies in a multidimensional context are insufficient since they do not take into account the eventual relationships between variables, which is often the most important point.

Principal component analysis is often considered as the basic method of factor analysis, which aims to find linear combinations of the p variables called components used to visualize the observations in a simple way. Because it transforms a large number of correlated variables into a few uncorrelated principal components, PCA is a dimension reduction method. However, PCA can also be used as a multivariate outlier detection method, especially by studying the last principal components. This property is useful in multidimensional quality control.

1.2. Data table and related subspaces

1.2.1.Data and their characteristics

Data are generally represented in a rectangular table with n rows for the individuals and p columns corresponding to the variables. Choosing individuals and variables to analyze is a crucial phase which has an important influence on PCA results. This choice has to take into account the aim of the study; in particular, the variables have to describe the phenomenon being analyzed.

Usually PCA deals with numerical variables. However, ordinal variables such as ranks can also be processed by PCA. Later in this chapter, we present the concept of supplementary variables which afterwards integrates nominal variables.

1.2.1.1. Data table

Let X be the (n,p)matrix of observations:

where

is the value of individual i for variable j (denoted x^j ) which is identified with a vector of n components (

, . . . ,

)′. In a similar way, an individual i is identified to a vector x_i of p components with x_i = (

, . . . ,

)′.

Table 1.1 is an example of such a data matrix. Computations have been carried out using SPAD 5 software, version 5 ¹, kindly provided by J.-P. Gauchi.

The data file contains 57 brands of mineral water described by 11 variables defined in Table 1.2. The data come from the bottle labels. Numerical variables are homogenous; they are all active variables (see section 1.4.3). A variable of a different kind such as price would be considered as a supplementary variable. On the other hand, qualitative variables such as country, type and whether still or sparkling (PG) are necessarily supplementary variables.

Table 1.1. Data table

1.2.1.2. Summaries

1.2.1.2.1. Centroid

Let

be the vector of arithmetic means of each of the p variables, defining the centroid:

Table 1.2. Variable description

Name	Complete water name as labeled on the bottle
Country	Identified by the official car registration letters; sometimes it is necessary to add a letter, for example Crete: GRC (Greece Crete)
Type	M for mineral water, S for spring water
PG	P for still water, G for sparkling water
CA	Calcium ions (mg/litre)
MG	Magnesium ions (mg/litre)
NA	Sodium ions (mg/litre)
K	Potassium ions (mg/litre)
SUL	Sulfate ions (mg/litre)
NO3	Nitrate ions (mg/litre)
HCO3	Carbonate ions (mg/litre)
CL	Chloride ions (mg/litre)

where

.

If the data are collected following a random sampling, the n individuals all have the same importance in the computations of the sample characteristics. The same weight p_i = 1/n is therefore allocated to each observation.

However, it can be useful for some applications to use weight p_i varying from one individual to another as grouped data or a reweighted sample. These weights, which are positive numbers summing to 1, can be viewed as frequencies and are stored in a diagonal matrix of size n:

We then have the matrix expression

= X′D_p1_n where 1_n represents the vector of

ⁿ with all its components equal to 1. The centered data matrix associated with X is then Y with

and Y = X − 1_n

′ = (I_n − 1_n1_n′ D_p)X, where I_n is the unity matrix of dimension n.

1.2.1.2.2. Covariance matrix and correlation matrix

Let

and

, the variance of variab...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Chapter 1: Principal Component Analysis: Application to Statistical Process Control
• Chapter 2: Correspondence Analysis: Extensions and Applications to the Statistical Analysis of Sensory Data
• Chapter 3: Exploratory Projection Pursuit
• Chapter 4: The Analysis of Proximity Data
• Chapter 5: Statistical Modeling of Functional Data
• Chapter 6: Discriminant Analysis
• Chapter 7: Cluster Analysis
• Chapter 8: Clustering and the Mixture Model
• Chapter 9: Spatial Data Clustering
• List of Authors
• Index