Nonparametric functional data analysis : theory and practice / Frâedâeric Ferraty, Philippe Vieu.

Format:

Book

Author/Creator:

Ferraty, Frâedâeric.

Contributor:

Vieu, Philippe.

Anne and Joseph Trachtman Memorial Book Fund.

Series:

Springer series in statistics

Language:

English

Subjects (All):

Nonparametric statistics.

Multivariate analysis.

Physical Description:

xx, 258 pages : illustrations ; 24 cm.

Place of Publication:

New York : Springer, [2006]

Summary:

Modern apparatuses allow us to collect samples of functional data, mainly curves but also images. On the other hand, nonparametric statistics produces useful tools for standard data exploration. This book links these two fields of modern statistics by explaining how functional data can be studied through parameterfree statistical ideas. This book starts from theoretical foundations including functional nonparametric modeling, description of the mathematical framework, construction of the statistical methods, and statements of their asymptotic behaviors. It proceeds to computational issues including R and S-PLUS routines. Several functional datasets in chemometrics, econometrics, and pattern recognition are used to emphasize the wide scope of nonparametric functional data analysis in applied sciences. The companion Web site includes R and S-PLUS routines, command lines for reproducing examples presented in the book, and the functional datasets.

Rather than set application against theory, this book is really an interface of these two features of statistics. A special effort has been made in writing this book to accommodate several levels of reading. The computational aspects are oriented toward practitioners whereas open problems emerging from this new field of statistics will attract Ph.D. students and academic researchers. Finally, this book is also accessible to graduate students starting in the area of functional statistics.

Contents:

Part I Statistical Background for Nonparametric Statistics and Functional Data

1 Introduction to Functional Nonparametric Statistics 5

1.1 What is a Functional Variable? 5

1.2 What are Functional Datasets? 6

1.3 What are Nonparametric Statistics for Functional Data 7

1.4 Some Notation 9

2 Some Functional Datasets and Associated Statistical Problematics 11

2.1 Functional Chemometric Data 11

2.1.1 Description of Spectrometric Data 12

2.1.2 First Study and Statistical Problems 13

2.2 Speech Recognition Data 15

2.2.1 What are Speech Recognition Data? 15

2.2.2 First Study and Problematics 15

2.3 Electricity Consumption Data 17

2.3.1 The Data 17

2.3.2 The Forecasting Problematic 18

3 What is a Well-Adapted Space for Functional Data? 21

3.1 Closeness Notions 21

3.2 Semi-Metrics as Explanatory Tool 22

3.3 What about the Curse of Dimensionality? 25

3.4 Semi-Metrics in Practice 28

3.4.1 Functional PCA: a Tool to Build Semi-Metrics 28

3.4.2 PLS: a New Way to Build Semi-Metrics 30

3.4.3 Semi-metrics Based on Derivatives 32

3.5 R and S+ Implementations 33

3.6 What About Unbalanced Functional Data? 33

3.7 Semi-Metric Space: a Well-Adapted Framework 35

4 Local Weighting of Functional Variables 37

4.1 Why Use Kernel Methods for Functional Data? 37

4.1.1 Real Case 38

4.1.2 Multivariate Case 39

4.1.3 Functional Case 41

4.2 Local Weighting and Small Ball Probabilities 42

4.3 A Few Basic Theoretical Advances 43

Part II Nonparametric Prediction from Functional Data

5 Functional Nonparametric Prediction Methodologies 49

5.2 Various Approaches to the Prediction Problem 50

5.3 Functional Nonparametric Modelling for Prediction 52

5.4 Kernel Estimators 55

6 Some Selected Asymptotics 61

6.2 Almost Complete Convergence 62

6.2.1 Regression Estimation 62

6.2.2 Conditional Median Estimation 66

6.2.3 Conditional Mode Estimation 70

6.2.4 Conditional Quantile Estimation 76

6.2.5 Complements 76

6.3 Rates of Convergence 79

6.3.1 Regression Estimation 79

6.3.2 Conditional Median Estimation 80

6.3.3 Conditional Mode Estimation 87

6.3.4 Conditional Quantile Estimation 90

6.3.5 Complements 92

6.4 Discussion, Bibliography and Open Problems 93

6.4.2 Going Back to Finite Dimensional Setting 94

6.4.3 Some Tracks for the Future 95

7 Computational Issues 99

7.1 Computing Estimators 99

7.1.1 Prediction via Regression 100

7.1.2 Prediction via Functional Conditional Quantiles 103

7.1.3 Prediction via Functional Conditional Mode 104

7.2 Predicting Fat Content From Spectrometric Curves 105

7.2.1 Chemometric Data and the Aim of the Problem 105

7.2.2 Functional Prediction in Action 106

Part III Nonparametric Classification of Functional Data

8 Functional Nonparametric Supervised Classification 113

8.3 Computational Issues 116

8.3.1 kNN Estimator 116

8.3.2 Automatic Selection of the kNN Parameter 117

8.3.3 Implementation: R/S+ Routines 118

8.4 Functional Nonparametric Discrimination in Action 119

8.4.1 Speech Recognition Problem 119

8.4.2 Chemometric Data 122

8.5 Asymptotic Advances 122

8.6 Additional Bibliography and Comments 123

9 Functional Nonparametric Unsupervised Classification 125

9.2 Centrality Notions for Functional Variables 127

9.2.1 Mean 127

9.2.2 Median 129

9.2.3 Mode 130

9.3 Measuring Heterogeneity 131

9.4 A General Descending Hierarchical Method 131

9.4.1 How to Build a Partitioning Heterogeneity Index? 132

9.4.2 How to Build a Partition? 132

9.4.3 Classification Algorithm 134

9.4.4 Implementation: R/S+ Routines 135

9.5 Nonparametric Unsupervised Classification in Action 135

9.6 Theoretical Advances on the Functional Mode 137

9.6.1 Hypotheses on the Distribution 138

9.7 The Kernel Functional Mode Estimator 140

9.7.1 Construction of the Estimates 140

9.7.2 Density Pseudo-Estimator: a.co. Convergence 141

9.7.3 Mode Estimator: a.co. Convergence 144

9.7.4 Comments and Bibliography 145

Part IV Nonparametric Methods for Dependent Functional Data

10 Mixing, Nonparametric and Functional Statistics 153

10.1 Mixing: a Short Introduction 153

10.2 The Finite-Dimensional Setting: a Short Overview 154

10.3 Mixing in Functional Context 155

10.4 Mixing and Nonparametric Functional Statistics 156

11 Some Selected Asymptotics 159

11.2 Prediction with Kernel Regression Estimator 160

11.2.2 Complete Convergence Properties 161

11.2.3 An Application to the Geometrically Mixing Case 163

11.2.4 An Application to the Arithmetically Mixing Case 166

11.3 Prediction with Functional Conditional Quantiles 167

11.3.2 Complete Convergence Properties 168

11.3.3 Application to the Geometrically Mixing Case 171

11.3.4 Application to the Arithmetically Mixing Case 175

11.4 Prediction with Conditional Mode 177

11.4.2 Complete Convergence Properties 178

11.4.3 Application to the Geometrically Mixing Case 183

11.4.4 Application to the Arithmetically Mixing Case 184

11.5 Complements on Conditional Distribution Estimation 185

11.5.1 Convergence Results 185

11.5.2 Rates of Convergence 187

11.6 Nonparametric Discrimination of Dependent Curves 189

11.6.2 Complete Convergence Properties 190

11.7.2 Back to Finite Dimensional Setting 192

11.7.3 Some Open Problems 193

12 Application to Continuous Time Processes Prediction 195

12.1 Time Series and Nonparametric Statistics 195

12.2 Functional Approach to Time Series Prediction 197

12.3 Computational Issues 198

12.4 Forecasting Electricity Consumption 198

12.4.1 Presentation of the Study 198

12.4.2 The Forecasted Electrical Consumption 200

13 Small Ball Probabilities and Semi-metrics 205

13.2 The Role of Small Ball Probabilities 206

13.3 Some Special Infinite Dimensional Processes 207

13.3.1 Fractal-type Processes 207

13.3.2 Exponential-type Processes 209

13.3.3 Links with Semi-metric Choice 212

13.4 Back to the One-dimensional Setting 214

13.5 Back to the Multi- (but Finite) -Dimensional Setting 219

13.6 The Semi-metric: a Crucial Parameter 223

14 Some Perspectives 225

Appendix: Some Probabilistic Tools 227

A.1 Almost Complete Convergence 228

A.2 Exponential Inequalities for Independent r.r.v. 233

A.3 Inequalities for Mixing r.r.v. 235.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

0387303693

OCLC:

70261207