## Nonparametric Functional Data Analysis: Theory and PracticeModern 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. 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. |

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### Contents

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922 Median | 129 |

923 Mode | 130 |

93 Measuring Heterogeneity | 131 |

941 How to Build a Partitioning Heterogeneity Index? | 132 |

943 Classiﬁcation Algorithm | 134 |

RS+ Routines | 135 |

96 Theoretical Advances on the Functional Mode | 137 |

961 Hypotheses on the Distribution | 138 |

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What is a WellAdapted Space for Functional Data? | 21 |

32 SemiMetrics as Explanatory Tool | 22 |

34 SemiMetrics in Practice | 28 |

a New Way to Build SemiMetrics | 30 |

343 Semimetrics Based on Derivatives | 32 |

35 R and S+ Implementations | 33 |

a WellAdapted Framework | 35 |

Local Weighting of Functional Variables | 36 |

411 Real Case | 38 |

412 Multivariate Case | 39 |

413 Functional Case | 41 |

42 Local Weighting and Small Ball Probabilities | 42 |

43 A Few Basic Theoretical Advances | 43 |

Nonparametric Prediction from Functional Data | 45 |

Functional Nonparametric Prediction Methodologies | 48 |

52 Various Approaches to the Prediction Problem | 50 |

53 Functional Nonparametric Modelling for Prediction | 52 |

54 Kernel Estimators | 55 |

Some Selected Asymptotics | 61 |

62 Almost Complete Convergence | 62 |

622 Conditional Median Estimation | 66 |

623 Conditional Mode Estimation | 70 |

624 Conditional Quantile Estimation | 76 |

63 Rates of Convergence | 79 |

632 Conditional Median Estimation | 80 |

633 Conditional Mode Estimation | 87 |

634 Conditional Quantile Estimation | 90 |

635 Complements on Conditional Distribution Estimation | 92 |

64 Discussion Bibliography and Open Problems | 93 |

642 Going Back to Finite Dimensional Setting | 94 |

643 Some Tracks for the Future | 95 |

Computational Issues | 99 |

711 Prediction via Regression | 100 |

712 Prediction via Functional Conditional Quantiles | 103 |

713 Prediction via Functional Conditional Mode | 104 |

72 Predicting Fat Content From Spectrometric Curves | 105 |

722 Functional Prediction in Action | 106 |

73 Conclusion | 107 |

Nonparametric Classiﬁcation of Functional Data | 110 |

Functional Nonparametric Supervised Classiﬁcation | 113 |

82 Method | 114 |

83 Computational Issues | 116 |

832 Automatic Selection of the kNN Parameter | 117 |

RS+ Routines | 118 |

84 Functional Nonparametric Discrimination in Action | 119 |

842 Chemometric Data | 122 |

86 Additional Bibliography and Comments | 123 |

Functional Nonparametric Unsupervised Classiﬁcation | 125 |

92 Centrality Notions for Functional Variables | 127 |

97 The Kernel Functional Mode Estimator | 140 |

aco Convergence | 141 |

aco Convergence | 144 |

974 Comments and Bibliography | 145 |

98 Conclusions | 146 |

Nonparametric Methods for Dependent Functional Data | 150 |

Mixing Nonparametric and Functional Statistics | 153 |

a Short Overview | 154 |

Some General Considerations | 155 |

104 Mixing and Nonparametric Functional Statistics | 156 |

Some Selected Asymptotics | 158 |

112 Prediction with Kernel Regression Estimator | 160 |

1122 Complete Convergence Properties | 161 |

1123 An Application to the Geometrically Mixing Case | 163 |

1124 An Application to the Arithmetically Mixing Case | 166 |

113 Prediction with Functional Conditional Quantiles | 167 |

1132 Complete Convergence Properties | 168 |

1133 Application to the Geometrically Mixing Case | 171 |

1134 Application to the Arithmetically Mixing Case | 175 |

114 Prediction with Conditional Mode | 177 |

1142 Complete Convergence Properties | 178 |

1143 Application to the Geometrically Mixing Case | 183 |

1144 Application to the Arithmetically Mixing Case | 184 |

115 Complements on Conditional Distribution Estimation | 185 |

1152 Rates of Convergence | 187 |

116 Nonparametric Discrimination of Dependent Curves | 189 |

1162 Complete Convergence Properties | 190 |

117 Discussion | 192 |

1173 Some Open Problems | 193 |

Application to Continuous Time Processes Prediction | 195 |

122 Functional Approach to Time Series Prediction | 197 |

123 Computational Issues | 198 |

1242 The Forecasted Electrical Consumption | 200 |

1243 Conclusions | 201 |

Conclusions | 202 |

Small Ball Probabilities and Semimetrics | 203 |

132 The Role of Small Ball Probabilities | 206 |

133 Some Special Inﬁnite Dimensional Processes | 207 |

1332 Exponentialtype Processes | 209 |

1333 Links with Semimetric Choice | 212 |

134 Back to the Onedimensional Setting | 214 |

135 Back to the Multi but Finite Dimensional Setting | 219 |

a Crucial Parameter | 223 |

Some Perspectives | 224 |

Some Probabilistic Tools | 225 |

A1 Almost Complete Convergence | 228 |

A2 Exponential Inequalities for Independent rrv | 233 |

A3 Inequalities for Mixing rrv | 235 |

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### Common terms and phrases

a.co applied Assume ball probability function bandwidth c.d.f. estimate Chapter Chemometric complete convergence concerning conditional c.d.f. conditional density conditional expectation conditional median conditional mode conditional quantile conditions of Theorem Corollary covariance curse of dimensionality deﬁned Deﬁnition denote density estimate dependent diﬀerent directly discrimination discussed eﬀect estimate can reach ﬁeld ﬁnd ﬁnite dimensional setting ﬁrst ﬁx ﬁxed func functional conditional functional data functional dataset functional kernel functional nonparametric functional variable inﬁnite dimensional instance kernel conditional kernel estimator kernel function kernel regression Lebesgue measure Lemma literature logn methodology metric mode estimate multivariate nonparametric functional nonparametric models nonparametric regression nonparametric statistics notation Oa.co posterior probability prediction problem Proposition quantile estimate rate of convergence real number resp results presented sample satisﬁes scalar response Section semi-metric sequence small ball probability smoothing parameter speciﬁc spectrometric curves Spectrometric Data Speech Recognition tional ϕχ