Cover image for Nonparametric functional data analysis : theory and practice
Title:
Nonparametric functional data analysis : theory and practice
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Springer series in statistics
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New York, NY : Springer, 2006
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9780387303697
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30000010122177 QA278.8 F47 2006 Open Access Book
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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

Prefacep. VII
List of Abbreviations and Symbolsp. XVII
List of Figuresp. XIX
Part I Statistical Background for Nonparametric Statistics and Functional Data
1 Introduction to Functional Nonparametric Statisticsp. 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 Datap. 7
1.4 Some Notationp. 9
1.5 Scope of the Bookp. 10
2 Some Functional Datasets and Associated Statistical Problematicsp. 11
2.1 Functional Chemometric Datap. 11
2.1.1 Description of Spectrometric Datap. 12
2.1.2 First Study and Statistical Problemsp. 13
2.2 Speech Recognition Datap. 15
2.2.1 What are Speech Recognition Data?p. 15
2.2.2 First Study and Problematicsp. 15
2.3 Electricity Consumption Datap. 17
2.3.1 The Datap. 17
2.3.2 The Forecasting Problematicp. 18
3 What is a Well-Adapted Space for Functional Data?p. 21
3.1 Closeness Notionsp. 21
3.2 Semi-Metrics as Explanatory Toolp. 22
3.3 What about the Curse of Dimensionality?p. 25
3.4 Semi-Metrics in Practicep. 28
3.4.1 Functional PCA: a Tool to Build Semi-Metricsp. 28
3.4.2 PLS: a New Way to Build Semi-Metricsp. 30
3.4.3 Semi-metrics Based on Derivativesp. 32
3.5 R and S+ Implementationsp. 33
3.6 What About Unbalanced Functional Data?p. 33
3.7 Semi-Metric Space: a Well-Adapted Frameworkp. 35
4 Local Weighting of Functional Variablesp. 37
4.1 Why Use Kernel Methods for Functional Data?p. 37
4.1.1 Real Casep. 38
4.1.2 Multivariate Casep. 39
4.1.3 Functional Casep. 41
4.2 Local Weighting and Small Ball Probabilitiesp. 42
4.3 A Few Basic Theoretical Advancesp. 43
Part II Nonparametric Prediction from Functional Data
5 Functional Nonparametric Prediction Methodologiesp. 49
5.1 Introductionp. 49
5.2 Various Approaches to the Prediction Problemp. 50
5.3 Functional Nonparametric Modelling for Predictionp. 52
5.4 Kernel Estimatorsp. 55
6 Some Selected Asymptoticsp. 61
6.1 Introductionp. 61
6.2 Almost Complete Convergencep. 62
6.2.1 Regression Estimationp. 62
6.2.2 Conditional Median Estimationp. 66
6.2.3 Conditional Mode Estimationp. 70
6.2.4 Conditional Quantile Estimationp. 76
6.2.5 Complementsp. 76
6.3 Rates of Convergencep. 79
6.3.1 Regression Estimationp. 79
6.3.2 Conditional Median Estimationp. 80
6.3.3 Conditional Mode Estimationp. 87
6.3.4 Conditional Quantile Estimationp. 90
6.3.5 Complementsp. 92
6.4 Discussion, Bibliography and Open Problemsp. 93
6.4.1 Bibliographyp. 93
6.4.2 Going Back to Finite Dimensional Settingp. 94
6.4.3 Some Tracks for the Futurep. 95
7 Computational Issuesp. 99
7.1 Computing Estimatorsp. 99
7.1.1 Prediction via Regressionp. 100
7.1.2 Prediction via Functional Conditional Quantilesp. 103
7.1.3 Prediction via Functional Conditional Modep. 104
7.2 Predicting Fat Content From Spectrometric Curvesp. 105
7.2.1 Chemometric Data and the Aim of the Problemp. 105
7.2.2 Functional Prediction in Actionp. 106
7.3 Conclusionp. 107
Part III Nonparametric Classification of Functional Data
8 Functional Nonparametric Supervised Classificationp. 113
8.1 Introduction and Problematicp. 113
8.2 Methodp. 114
8.3 Computational Issuesp. 116
8.3.1 kNN Estimatorp. 116
8.3.2 Automatic Selection of the kNN Parameterp. 117
8.3.3 Implementation: R/S+ Routinesp. 118
8.4 Functional Nonparametric Discrimination in Actionp. 119
8.4.1 Speech Recognition Problemp. 119
8.4.2 Chemometric Datap. 122
8.5 Asymptotic Advancesp. 122
8.6 Additional Bibliography and Commentsp. 123
9 Functional Nonparametric Unsupervised Classificationp. 125
9.1 Introduction and Problematicp. 125
9.2 Centrality Notions for Functional Variablesp. 127
9.2.1 Meanp. 127
9.2.2 Medianp. 129
9.2.3 Modep. 130
9.3 Measuring Heterogeneityp. 131
9.4 A General Descending Hierarchical Methodp. 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 Algorithmp. 134
9.4.4 Implementation: R/S+ Routinesp. 135
9.5 Nonparametric Unsupervised Classification in Actionp. 135
9.6 Theoretical Advances on the Functional Modep. 137
9.6.1 Hypotheses on the Distributionp. 138
9.7 The Kernel Functional Mode Estimatorp. 140
9.7.1 Construction of the Estimatesp. 140
9.7.2 Density Pseudo-Estimator: a.co. Convergencep. 141
9.7.3 Mode Estimator: a.co. Convergencep. 144
9.7.4 Comments and Bibliographyp. 145
9.8 Conclusionsp. 146
Part IV Nonparametric Methods for Dependent Functional Data
10 Mixing, Nonparametric and Functional Statisticsp. 153
10.1 Mixing: a Short Introductionp. 153
10.2 The Finite-Dimensional Setting: a Short Overviewp. 154
10.3 Mixing in Functional Contextp. 155
10.4 Mixing and Nonparametric Functional Statisticsp. 156
11 Some Selected Asymptoticsp. 159
11.1 Introductionp. 159
11.2 Prediction with Kernel Regression Estimatorp. 160
11.2.1 Introduction and Notationp. 160
11.2.2 Complete Convergence Propertiesp. 161
11.2.3 An Application to the Geometrically Mixing Casep. 163
11.2.4 An Application to the Arithmetically Mixing Casep. 166
11.3 Prediction with Functional Conditional Quantilesp. 167
11.3.1 Introduction and Notationp. 167
11.3.2 Complete Convergence Propertiesp. 168
11.3.3 Application to the Geometrically Mixing Casep. 171
11.3.4 Application to the Arithmetically Mixing Casep. 175
11.4 Prediction with Conditional Modep. 177
11.4.1 Introduction and Notationp. 177
11.4.2 Complete Convergence Propertiesp. 178
11.4.3 Application to the Geometrically Mixing Casep. 183
11.4.4 Application to the Arithmetically Mixing Casep. 184
11.5 Complements on Conditional Distribution Estimationp. 185
11.5.1 Convergence Resultsp. 185
11.5.2 Rates of Convergencep. 187
11.6 Nonparametric Discrimination of Dependent Curvesp. 189
11.6.1 Introduction and Notationp. 189
11.6.2 Complete Convergence Propertiesp. 190
11.7 Discussionp. 192
11.7.1 Bibliographyp. 192
11.7.2 Back to Finite Dimensional Settingp. 192
11.7.3 Some Open Problemsp. 193
12 Application to Continuous Time Processes Predictionp. 195
12.1 Time Series and Nonparametric Statisticsp. 195
12.2 Functional Approach to Time Series Predictionp. 197
12.3 Computational Issuesp. 198
12.4 Forecasting Electricity Consumptionp. 198
12.4.1 Presentation of the Studyp. 198
12.4.2 The Forecasted Electrical Consumptionp. 200
12.4.3 Conclusionsp. 201
Part V Conclusions
13 Small Ball Probabilities and Semi-metricsp. 205
13.1 Introductionp. 205
13.2 The Role of Small Ball Probabilitiesp. 206
13.3 Some Special Infinite Dimensional Processesp. 207
13.3.1 Fractal-type Processesp. 207
13.3.2 Exponential-type Processesp. 209
13.3.3 Links with Semi-metric Choicep. 212
13.4 Back to the One-dimensional Settingp. 214
13.5 Back to the Multi- (but Finite) -Dimensional Settingp. 219
13.6 The Semi-metric: a Crucial Parameterp. 223
14 Some Perspectivesp. 225
Appendix: Some Probabilistic Toolsp. 227
A.1 Almost Complete Convergencep. 228
A.2 Exponential Inequalities for Independent r.r.v.p. 233
A.3 Inequalities for Mixing r.r.v.p. 235
Referencesp. 239
Indexp. 255