Springer - Applied Multivariate Statistical Analysis 2003.pdf

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Applied Multivariate

Statistical Analysis

Wolfgang Hardle

Leopold Simar

Version: 29th April 2003

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Contents

I

Descriptive Techniques

11

1

Comparison of Batches

13

1.1

Boxplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.2

Histograms

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

1.3

Kernel Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

1.4

Scatterplots

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

1.5

Cherno-Flury Faces

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

1.6

Andrews' Curves

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

1.7

Parallel Coordinates Plots

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

1.8

Boston Housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

1.9

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

II

Multivariate Random Variables

55

2

A Short Excursion into Matrix Algebra

57

2.1

Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.2

Spectral Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

2.3

Quadratic Forms

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

2.4

Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

2.5

Partitioned Matrices

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

2

Contents

2.6

Geometrical Aspects

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

2.7

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

3

Moving to Higher Dimensions

81

3.1

Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3.2

Correlation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

3.3

Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

3.4

Linear Model for Two Variables . . . . . . . . . . . . . . . . . . . . . . . . .

95

3.5

Simple Analysis of Variance

. . . . . . . . . . . . . . . . . . . . . . . . . . .

103

3.6

Multiple Linear Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108

3.7

Boston Housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

3.8

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

4

Multivariate Distributions

119

4.1

Distribution and Density Function . . . . . . . . . . . . . . . . . . . . . . . .

120

4.2

Moments and Characteristic Functions

. . . . . . . . . . . . . . . . . . . . .

125

4.3

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

4.4

The Multinormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .

137

4.5

Sampling Distributions and Limit Theorems

. . . . . . . . . . . . . . . . . .

142

4.6

Bootstrap

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

148

4.7

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152

5

Theory of the Multinormal

155

5.1

Elementary Properties of the Multinormal

. . . . . . . . . . . . . . . . . . .

155

5.2

The Wishart Distribution

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

162

5.3

Hotelling Distribution

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

5.4

Spherical and Elliptical Distributions . . . . . . . . . . . . . . . . . . . . . .

167

5.5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

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Contents

3

6

Theory of Estimation

173

6.1

The Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

6.2

The Cramer-Rao Lower Bound

. . . . . . . . . . . . . . . . . . . . . . . . .

178

6.3

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181

7

Hypothesis Testing

183

7.1

Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

184

7.2

Linear Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192

7.3

Boston Housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209

7.4

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212

III Multivariate Techniques

217

8

Decomposition of Data Matrices by Factors

219

8.1

The Geometric Point of View

. . . . . . . . . . . . . . . . . . . . . . . . . .

220

8.2

Fitting the p-dimensional Point Cloud

. . . . . . . . . . . . . . . . . . . . .

221

8.3

Fitting the n-dimensional Point Cloud

. . . . . . . . . . . . . . . . . . . . .

225

8.4

Relations between Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

8.5

Practical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

228

8.6

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

232

9

Principal Components Analysis

233

9.1

Standardized Linear Combinations

. . . . . . . . . . . . . . . . . . . . . . .

234

9.2

Principal Components in Practice . . . . . . . . . . . . . . . . . . . . . . . .

238

9.3

Interpretation of the PCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241

9.4

Asymptotic Properties of the PCs . . . . . . . . . . . . . . . . . . . . . . . .

246

9.5

Normalized Principal Components Analysis . . . . . . . . . . . . . . . . . . .

249

9.6

Principal Components as a Factorial Method . . . . . . . . . . . . . . . . . .

250

9.7

Common Principal Components . . . . . . . . . . . . . . . . . . . . . . . . .

256

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