Title:
Nonparametric functional data analysis : theory and practice
Personal Author:
Series:
Springer series in statistics
Publication Information:
New York, NY : Springer, 2006
ISBN:
9780387303697
Added Author:
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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010122177 | QA278.8 F47 2006 | Open Access Book | Book | Searching... |
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Summary
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 parameter-free statistical ideas. At the same time it shows how functional data can be studied through parameter-free statistical ideas, and offers an original presentation of new nonparametric statistical methods for functional data analysis.
Table of Contents
Preface | p. VII |
List of Abbreviations and Symbols | p. XVII |
List of Figures | p. XIX |
Part I Statistical Background for Nonparametric Statistics and Functional Data | |
1 Introduction to Functional Nonparametric Statistics | p. 5 |
1.1 What is a Functional Variable? | p. 5 |
1.2 What are Functional Datasets? | p. 6 |
1.3 What are Nonparametric Statistics for Functional Data | p. 7 |
1.4 Some Notation | p. 9 |
1.5 Scope of the Book | p. 10 |
2 Some Functional Datasets and Associated Statistical Problematics | p. 11 |
2.1 Functional Chemometric Data | p. 11 |
2.1.1 Description of Spectrometric Data | p. 12 |
2.1.2 First Study and Statistical Problems | p. 13 |
2.2 Speech Recognition Data | p. 15 |
2.2.1 What are Speech Recognition Data? | p. 15 |
2.2.2 First Study and Problematics | p. 15 |
2.3 Electricity Consumption Data | p. 17 |
2.3.1 The Data | p. 17 |
2.3.2 The Forecasting Problematic | p. 18 |
3 What is a Well-Adapted Space for Functional Data? | p. 21 |
3.1 Closeness Notions | p. 21 |
3.2 Semi-Metrics as Explanatory Tool | p. 22 |
3.3 What about the Curse of Dimensionality? | p. 25 |
3.4 Semi-Metrics in Practice | p. 28 |
3.4.1 Functional PCA: a Tool to Build Semi-Metrics | p. 28 |
3.4.2 PLS: a New Way to Build Semi-Metrics | p. 30 |
3.4.3 Semi-metrics Based on Derivatives | p. 32 |
3.5 R and S+ Implementations | p. 33 |
3.6 What About Unbalanced Functional Data? | p. 33 |
3.7 Semi-Metric Space: a Well-Adapted Framework | p. 35 |
4 Local Weighting of Functional Variables | p. 37 |
4.1 Why Use Kernel Methods for Functional Data? | p. 37 |
4.1.1 Real Case | p. 38 |
4.1.2 Multivariate Case | p. 39 |
4.1.3 Functional Case | p. 41 |
4.2 Local Weighting and Small Ball Probabilities | p. 42 |
4.3 A Few Basic Theoretical Advances | p. 43 |
Part II Nonparametric Prediction from Functional Data | |
5 Functional Nonparametric Prediction Methodologies | p. 49 |
5.1 Introduction | p. 49 |
5.2 Various Approaches to the Prediction Problem | p. 50 |
5.3 Functional Nonparametric Modelling for Prediction | p. 52 |
5.4 Kernel Estimators | p. 55 |
6 Some Selected Asymptotics | p. 61 |
6.1 Introduction | p. 61 |
6.2 Almost Complete Convergence | p. 62 |
6.2.1 Regression Estimation | p. 62 |
6.2.2 Conditional Median Estimation | p. 66 |
6.2.3 Conditional Mode Estimation | p. 70 |
6.2.4 Conditional Quantile Estimation | p. 76 |
6.2.5 Complements | p. 76 |
6.3 Rates of Convergence | p. 79 |
6.3.1 Regression Estimation | p. 79 |
6.3.2 Conditional Median Estimation | p. 80 |
6.3.3 Conditional Mode Estimation | p. 87 |
6.3.4 Conditional Quantile Estimation | p. 90 |
6.3.5 Complements | p. 92 |
6.4 Discussion, Bibliography and Open Problems | p. 93 |
6.4.1 Bibliography | p. 93 |
6.4.2 Going Back to Finite Dimensional Setting | p. 94 |
6.4.3 Some Tracks for the Future | p. 95 |
7 Computational Issues | p. 99 |
7.1 Computing Estimators | p. 99 |
7.1.1 Prediction via Regression | p. 100 |
7.1.2 Prediction via Functional Conditional Quantiles | p. 103 |
7.1.3 Prediction via Functional Conditional Mode | p. 104 |
7.2 Predicting Fat Content From Spectrometric Curves | p. 105 |
7.2.1 Chemometric Data and the Aim of the Problem | p. 105 |
7.2.2 Functional Prediction in Action | p. 106 |
7.3 Conclusion | p. 107 |
Part III Nonparametric Classification of Functional Data | |
8 Functional Nonparametric Supervised Classification | p. 113 |
8.1 Introduction and Problematic | p. 113 |
8.2 Method | p. 114 |
8.3 Computational Issues | p. 116 |
8.3.1 kNN Estimator | p. 116 |
8.3.2 Automatic Selection of the kNN Parameter | p. 117 |
8.3.3 Implementation: R/S+ Routines | p. 118 |
8.4 Functional Nonparametric Discrimination in Action | p. 119 |
8.4.1 Speech Recognition Problem | p. 119 |
8.4.2 Chemometric Data | p. 122 |
8.5 Asymptotic Advances | p. 122 |
8.6 Additional Bibliography and Comments | p. 123 |
9 Functional Nonparametric Unsupervised Classification | p. 125 |
9.1 Introduction and Problematic | p. 125 |
9.2 Centrality Notions for Functional Variables | p. 127 |
9.2.1 Mean | p. 127 |
9.2.2 Median | p. 129 |
9.2.3 Mode | p. 130 |
9.3 Measuring Heterogeneity | p. 131 |
9.4 A General Descending Hierarchical Method | p. 131 |
9.4.1 How to Build a Partitioning Heterogeneity Index? | p. 132 |
9.4.2 How to Build a Partition? | p. 132 |
9.4.3 Classification Algorithm | p. 134 |
9.4.4 Implementation: R/S+ Routines | p. 135 |
9.5 Nonparametric Unsupervised Classification in Action | p. 135 |
9.6 Theoretical Advances on the Functional Mode | p. 137 |
9.6.1 Hypotheses on the Distribution | p. 138 |
9.7 The Kernel Functional Mode Estimator | p. 140 |
9.7.1 Construction of the Estimates | p. 140 |
9.7.2 Density Pseudo-Estimator: a.co. Convergence | p. 141 |
9.7.3 Mode Estimator: a.co. Convergence | p. 144 |
9.7.4 Comments and Bibliography | p. 145 |
9.8 Conclusions | p. 146 |
Part IV Nonparametric Methods for Dependent Functional Data | |
10 Mixing, Nonparametric and Functional Statistics | p. 153 |
10.1 Mixing: a Short Introduction | p. 153 |
10.2 The Finite-Dimensional Setting: a Short Overview | p. 154 |
10.3 Mixing in Functional Context | p. 155 |
10.4 Mixing and Nonparametric Functional Statistics | p. 156 |
11 Some Selected Asymptotics | p. 159 |
11.1 Introduction | p. 159 |
11.2 Prediction with Kernel Regression Estimator | p. 160 |
11.2.1 Introduction and Notation | p. 160 |
11.2.2 Complete Convergence Properties | p. 161 |
11.2.3 An Application to the Geometrically Mixing Case | p. 163 |
11.2.4 An Application to the Arithmetically Mixing Case | p. 166 |
11.3 Prediction with Functional Conditional Quantiles | p. 167 |
11.3.1 Introduction and Notation | p. 167 |
11.3.2 Complete Convergence Properties | p. 168 |
11.3.3 Application to the Geometrically Mixing Case | p. 171 |
11.3.4 Application to the Arithmetically Mixing Case | p. 175 |
11.4 Prediction with Conditional Mode | p. 177 |
11.4.1 Introduction and Notation | p. 177 |
11.4.2 Complete Convergence Properties | p. 178 |
11.4.3 Application to the Geometrically Mixing Case | p. 183 |
11.4.4 Application to the Arithmetically Mixing Case | p. 184 |
11.5 Complements on Conditional Distribution Estimation | p. 185 |
11.5.1 Convergence Results | p. 185 |
11.5.2 Rates of Convergence | p. 187 |
11.6 Nonparametric Discrimination of Dependent Curves | p. 189 |
11.6.1 Introduction and Notation | p. 189 |
11.6.2 Complete Convergence Properties | p. 190 |
11.7 Discussion | p. 192 |
11.7.1 Bibliography | p. 192 |
11.7.2 Back to Finite Dimensional Setting | p. 192 |
11.7.3 Some Open Problems | p. 193 |
12 Application to Continuous Time Processes Prediction | p. 195 |
12.1 Time Series and Nonparametric Statistics | p. 195 |
12.2 Functional Approach to Time Series Prediction | p. 197 |
12.3 Computational Issues | p. 198 |
12.4 Forecasting Electricity Consumption | p. 198 |
12.4.1 Presentation of the Study | p. 198 |
12.4.2 The Forecasted Electrical Consumption | p. 200 |
12.4.3 Conclusions | p. 201 |
Part V Conclusions | |
13 Small Ball Probabilities and Semi-metrics | p. 205 |
13.1 Introduction | p. 205 |
13.2 The Role of Small Ball Probabilities | p. 206 |
13.3 Some Special Infinite Dimensional Processes | p. 207 |
13.3.1 Fractal-type Processes | p. 207 |
13.3.2 Exponential-type Processes | p. 209 |
13.3.3 Links with Semi-metric Choice | p. 212 |
13.4 Back to the One-dimensional Setting | p. 214 |
13.5 Back to the Multi- (but Finite) -Dimensional Setting | p. 219 |
13.6 The Semi-metric: a Crucial Parameter | p. 223 |
14 Some Perspectives | p. 225 |
Appendix: Some Probabilistic Tools | p. 227 |
A.1 Almost Complete Convergence | p. 228 |
A.2 Exponential Inequalities for Independent r.r.v. | p. 233 |
A.3 Inequalities for Mixing r.r.v. | p. 235 |
References | p. 239 |
Index | p. 255 |