Functional and Operational Statistics
17
Functional and Operatorial Statistics
Transcript of Functional and Operational Statistics
Functional and Operatorial Statistics
● Frédéric Ferraty
StatisticsFunctional and Operatorial
Sophie Dabo-Niang
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Dr. Frédéric FerratyInstitut de Mathématiques de
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ISBN: 978-3-7908-2061-4 e-ISBN: 978-3-7908-2062-1
Laboratoire GREMARS-EQUIPPE
Preface
An increasing number of problems and methods involve in�nite-dimensionalaspects. This is due to the progress of technologies which allow us to storemore and more information while modern instruments are able to collectdata much more e�ectively due to their increasingly sophisticated design.This evolution directly concerns the statisticians who have to propose newmethodologies while taking into account such high-dimensional data (e.g. con-tinuous processes, functional data, etc.). The numerous applications (micro-arrays, paleo-ecological data, radar waveforms, spectrometric curves, speechrecognition, continuous time series, 3-D images, etc.) in various �elds (biol-ogy, econometrics, environmetrics, the food industry, medical sciences, paperindustry, speech recognition, etc.) make researching this statistical topic veryworthwhile. New challenges emerge both from theoretical and practical pointof views. This First International Workshop on Functional and Operator-ial Statistics (IWFOS) aims to emphasize this fascinating �eld of researchand this volume gathers the contributions presented in this conference. It isworth noting that this volume mixes applied works (with original datasetsand/or computational issues) as well as fundamental theoretical ones (withdeep mathematical developments). Therefore, this book should cover a largeaudience, like academic researchers (theoreticians and/or practitioners), grad-uate/PhD students and should appeal to anyone working in statistics withindustrial companies, research institutes or software developers.
This Workshop covers a wide scope of statistical aspects. Numerous worksdeal with classi�cation (see for instance chapters 6, 7, 17, 21 or 41), functionalPCA-based methods (see for instance chapters 2, 16, 30 or 37 ), mathemat-ical toolbox (see for instance chapters 11, 13, 24 or 29), regression (see forinstance chapters 3, 4, 5, 8, 10, 18, 19, 23, 26, 27, 33 or 34), spatial statistics(see for instance chapters 9, 22, 35, 36 or 42), time series (12, 28, 39, 40 or44). Other topics are also present as subsampling (see chapter 38) as well astransversal/explorative methodologies (see for instance chapters 14, 20, 31 or43). In addition, interesting works focus on original/motivating applications(see for instance chapters 15, 25 or 32). This splitting into topics (classi�ca-
vi Preface
tion, functional PCA-based methods, ...) is introduced just for giving an ideaon the contents but most of the time, one can assign a same work to severalsubjects. It is worth noting that numerous contributions deal with statisticalmethodologies for functional data which is certainly the main common de-nominator of IWFOS (see chapters 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17,18, 19, 20, 21, 22, 23, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 43, 44).
The scienti�c success of this event is obviously linked with the wide varietyof participants coming from about 20 countries covering all the continents.One would like to thank them gratefully and specially the invited speakersand all the contributors for the high quality of their submitted works.
Of course the hard core is the STAPH members (see chapter 1), whichmanaged and coordinated this Workshop both from a scienti�c and orga-nizational point of view. But this international conference would not existwithout the help of many people. In particular, K. Benhenni (France), B.Cadre (France), H. Cardot (France), A. Cuevas (Spain), A. Dahmani (Alge-ria), A. Goia (Italia), W. Gonzalez-Manteiga (Spain), W. Härdle (Germany),A. Kneip (Germany), Ali Laksaci (Algeria), A. Mas (France), E. Ould-saïd(France), M. Rachdi (France), E. Salinelli (Italia) and I. Van Keilegom (Bel-gium) have greatly contributed to the high quality of IWFOS'2008 and aregratefully thanked.
The �nal thanks go to Marie-Laure Ausset which took charge of sec-retary tasks as well as the institutions which have supported this Work-shop via grants or administrative supports (CNRS, Conseil Général de laHaute-Garonne, Conseil Régional Midi-Pyrénées, Laboratoire de Statistiqueet Probabilités, Institut de Mathématiques de Toulouse, Université PaulSabatier, Laboratoire Gremars-Equippe of university Charles De Gaulle,Équipe de Probabilités-Statistique of University Montpellier 2, laboratoireLMPA (Littoral) and University del Piemonte Orientale (Italy)).
Toulouse, France Sophie Dabo-NiangMay 2008 Frédéric Ferraty
Contents
1 Introduction to IWFOS'2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Alain Boudou, Frédéric Ferraty, Yves Romain, Pascal Sarda,Philippe Vieu and Sylvie Viguier-Pla1.1 Historical and scienti�c setting . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The STAPH group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The �rst IWFOS'2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Solving Multicollinearity in Functional MultinomialLogit Models for Nominal and Ordinal Responses . . . . . . . . 7Ana Aguilera and Manuel Escabias2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Functional multinomial response model . . . . . . . . . . . . . . . . . . 8
2.2.1 Nominal responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Ordinal responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Model estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Principal components approach . . . . . . . . . . . . . . . . . . . . . . . . . 12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Estimation of Functional Regression Models forFunctional Responses by Wavelet Approximation . . . . . . . . . 15Ana Aguilera, Francisco Ocaña and Mariano Valderrama3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Functional linear model for a functional response . . . . . . . . . . 163.3 Model estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Wavelet approximation of sample curves . . . . . . . . . . . . . . . . . 19References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Functional Linear Regression with Functional Response:Application to Prediction of Electricity Consumption . . . . . 23Jaromír Antoch, Lubo² Prchal, Maria Rosaria De Rosa andPascal Sarda
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4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Estimation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Computational aspects and simulations . . . . . . . . . . . . . . . . . . 264.4 Prediction of electricity consumption . . . . . . . . . . . . . . . . . . . . 27References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Asymptotic Normality of Robust NonparametricEstimator for Functional Dependent Data . . . . . . . . . . . . . . . . 31Mohammed Attouch, Ali Laksaci and Elias Ould-Saïd5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Model and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Hypothesis and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.4 Conditional con�dence curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 34References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Measuring Dissimilarity Between Curves by Means ofTheir Granulometric Size Distributions . . . . . . . . . . . . . . . . . . . 35Guillermo Ayala, Martin Gaston, Teresa Leon and Fermín Mallor6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3 Methods and experimental results . . . . . . . . . . . . . . . . . . . . . . . 37References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7 Supervised Classi�cation for Functional Data: ATheoretical Remark and Some Numerical Comparisons . . . 43Amparo Baíllo and Antonio Cuevas7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.2 Consistency of the nearest neighbour rule . . . . . . . . . . . . . . . . 457.3 Comparison of several classi�cation techniques . . . . . . . . . . . . 46References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8 Local Linear Regression for Functional Predictor andScalar Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Amparo Baíllo and Aurea Grané8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478.2 Local linear smoothing for functional data . . . . . . . . . . . . . . . . 498.3 Performance of the estimator m̂LL . . . . . . . . . . . . . . . . . . . . . . . 50References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
9 Spatio-temporal Functional Regression on PaleoecologicalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Avner Bar-Hen, Liliane Bel and Rachid Cheddadi9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549.3 Functional regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
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10 Local Linear Functional Regression Based on WeightedDistance-based Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Eva Boj, Pedro Delicado and Josep Fortiana10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.2 Weighted distance-based regression (WDBR) . . . . . . . . . . . . . 5910.3 Local linear distance-based regression . . . . . . . . . . . . . . . . . . . . 6010.4 A real data example: Spectrometric Data . . . . . . . . . . . . . . . . . 61References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
11 Singular Value Decomposition of Large Random Matrices(for Two-Way Classi�cation of Microarrays) . . . . . . . . . . . . . . 65Marianna Bolla, Katalin Friedl and András Krámli11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6511.2 Singular values of a noisy matrix . . . . . . . . . . . . . . . . . . . . . . . . 6611.3 Classi�cation via singular vector pairs . . . . . . . . . . . . . . . . . . . 6711.4 Perturbation results for correspondence matrices . . . . . . . . . . 6811.5 Recognizing the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
12 On Tensorial Products of Hilbertian Linear Processes . . . . 71Denis Bosq12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.2 The real case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.3 The hilbertian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
13 Recent Results on Random and Spectral Measures withSome Applications in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Alain Boudou, Emmanuel Cabral and Yves Romain13.1 Introduction and de�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.2 Convolution product of spectral measures . . . . . . . . . . . . . . . . 7913.3 Tensor and convolution products of random measures . . . . . . 81References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
14 Parameter Cascading for High Dimensional Models . . . . . . . 85David Campbell, Jiguo Cao, Giles Hooker and James Ramsay14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8514.2 Inner optimization: nuisance parameters . . . . . . . . . . . . . . . . . . 8614.3 Middle optimization: structural parameters . . . . . . . . . . . . . . . 8614.4 Outer optimization: complexity parameters . . . . . . . . . . . . . . . 8714.5 Parameter cascading precedents . . . . . . . . . . . . . . . . . . . . . . . . . 8714.6 Parameter cascading advantages . . . . . . . . . . . . . . . . . . . . . . . . 88References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
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15 Advances in Human Protein Interactome Inference . . . . . . . 89Enrico Capobianco and Elisabetta Marras15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.3 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
16 Functional Principal Components Analysis with SurveyData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Hervé Cardot, Mohamed Chaouch, Camelia Goga and CatherineLabruère16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.2 FPCA and sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
16.2.1 FPCA in a �nite population setting . . . . . . . . . . . . . . 9716.2.2 The Horvitz-Thompson estimator . . . . . . . . . . . . . . . . 98
16.3 Linearization by in�uence function . . . . . . . . . . . . . . . . . . . . . . 9816.3.1 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 9916.3.2 Variance approximation and estimation . . . . . . . . . . . 100
16.4 A Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
17 Functional Clustering of Longitudinal Data . . . . . . . . . . . . . . . 103Jeng-Min Chiou and Pai-Ling Li17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10317.2 The methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10417.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
18 Robust Nonparametric Estimation for Functional Data . . . 109Christophe Crambes, Laurent Delsol and Ali Laksaci18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10918.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11018.3 Asymptotic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
18.3.1 Convergence in probability and asymptotic normality11118.3.2 A uniform integrability result . . . . . . . . . . . . . . . . . . . . 11118.3.3 Moments convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 112
18.4 Application to time series prediction . . . . . . . . . . . . . . . . . . . . . 113References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
19 Estimation of the Functional Linear Regression withSmoothing Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Christophe Crambes, Alois Kneip and Pascal Sarda19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11719.2 Construction of the estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 11819.3 Convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
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20 A Random Functional Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Juan Cuesta-Albertos and Alicia Nieto-Reyes20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12120.2 Functional depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12320.3 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12420.4 Analysis of a real data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
21 Parametric Families of Probability Distributions forFunctional Data Using Quasi-Arithmetic Means withArchimedean Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Etienne Cuvelier and Monique Noirhomme-Fraiture21.1 QAMML distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12721.2 Gateaux density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12921.3 GQAMML distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13021.4 CQAMML distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13121.5 Supervised classi�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13121.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
22 Point-wise Kriging for Spatial Prediction of FunctionalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Pedro Delicado, Ramón Giraldo and Jorge Mateu22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13522.2 Point-wise kriging for functional Data . . . . . . . . . . . . . . . . . . . . 13622.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
23 Nonparametric Regression on Functional Variable andStructural Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Laurent Delsol23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14323.2 Nonparametric estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14523.3 Structural tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14623.4 Bootstrap procedures and simulations . . . . . . . . . . . . . . . . . . . . 147References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
24 Vector Integration and Stochastic Integration in BanachSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Nicolae Dinculeanu24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15124.2 Vector integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
24.2.1 The classical integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 15224.2.2 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 15224.2.3 Integration with respect to a vector-measure with
�nite variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
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24.2.4 Integration with respect to a vector-measure with�nite semivariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
24.3 The stochastic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
25 Multivariate Functional Data Discrimination Using ICA:Analysis of Hippocampal Di�erences in Alzheimer'sDisease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Irene Epifanio and Noelia Ventura25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15725.2 Brain MR scans processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15825.3 Methodology: ICA and linear discriminant analysis . . . . . . . . 15925.4 Results of the hippocampus study . . . . . . . . . . . . . . . . . . . . . . . 160References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
26 In�uence in the Functional Linear Model with ScalarResponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Manuel Febrero, Pedro Galeano andWenceslao González-Manteiga26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16526.2 The functional linear model with scalar response . . . . . . . . . . 16726.3 In�uence measures for the functional linear model . . . . . . . . . 169References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
27 Is it Always Optimal to Impose Constraints onNonparametric Functional Estimators? Some Evidenceon the Smoothing Parameter Choice . . . . . . . . . . . . . . . . . . . . . . 173Jean-Pierre Florens and Anne Vanhems27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17327.2 General constrained solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17527.3 Regularized estimated solutions . . . . . . . . . . . . . . . . . . . . . . . . . 17627.4 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
28 Dynamic Semiparametric Factor Models in PricingKernels Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Enzo Giacomini and Wolfgang Härdle28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18128.2 Pricing kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18228.3 Pricing kernels estimation with DSFM . . . . . . . . . . . . . . . . . . . 18328.4 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
29 The Operator Trigonometry in Statistics . . . . . . . . . . . . . . . . . 189Karl Gustafson29.1 The origins of the operator trigonometry . . . . . . . . . . . . . . . . . 18929.2 The essentials of the operator trigonometry . . . . . . . . . . . . . . . 19029.3 The operator trigonometry in statistics . . . . . . . . . . . . . . . . . . . 191
Contents xiii
29.4 Operator trigonometry in general . . . . . . . . . . . . . . . . . . . . . . . . 19229.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
30 Selecting and Ordering Components in Functional-DataLinear Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Peter Hall30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19530.2 Linear prediction in a general setting . . . . . . . . . . . . . . . . . . . . 19630.3 Arguments for and against the principal component basis . . 19730.4 Theoretical argument in support of the ordering (3) of the
principal component basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19830.5 Other approaches to basis choice . . . . . . . . . . . . . . . . . . . . . . . . 199References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
31 Bagplots, Boxplots and Outlier Detection for FunctionalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Rob Hyndman and Han Lin Shang31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20131.2 Functional bagplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20331.3 Functional HDR boxplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20531.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
32 Marketing Applications of Functional Data Analysis . . . . . . 209Gareth James, Ashish Sood and Gerard Tellis32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20932.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21132.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
33 Nonparametric Estimation in Functional Linear Model . . . 215Jan Johannes33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21533.2 De�nition of the estimator of β . . . . . . . . . . . . . . . . . . . . . . . . . 21733.3 Risk bound when X is second order stationary . . . . . . . . . . . . 21933.4 Risk bound when X is not second order stationary . . . . . . . . 220References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
34 Presmoothing in Functional Linear Regression . . . . . . . . . . . . 223Adela Martínez-Calvo34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22334.2 Back to the multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . 22534.3 Presmoothing Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22634.4 Presmoothing X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22734.5 Comments about e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
xiv Contents
35 Probability Density Functions of the Empirical WaveletCoe�cients of Multidimensional Poisson Intensities . . . . . . . 231José Carlos Simon de Miranda35.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23135.2 Some basics and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23235.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23335.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
36 A Cokriging Method for Spatial Functional Data withApplications in Oceanology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Pascal Monestiez and David Nerini36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23736.2 Spatial linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23836.3 Cokriging on coe�cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23936.4 Dealing with real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
37 On the E�ect of Curve Alignment and Functional PCA . . . 243Juhyun Park37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
38 K-sample Subsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247Dimitris Politis and Joseph Romano38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24738.2 Subsampling hypothesis tests in K samples . . . . . . . . . . . . . . . 24938.3 Subsampling con�dence sets in K samples . . . . . . . . . . . . . . . . 25138.4 Random subsamples and the K�sample bootstrap . . . . . . . . . 252References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
39 Inference for Stationary Processes Using BandedCovariance Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Mohsen Pourahmadi and Wei Biao Wu39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25539.2 The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
39.2.1 A class of nonlinear processes . . . . . . . . . . . . . . . . . . . . 25739.2.2 Convergence of banded covariance estimators . . . . . . 25739.2.3 Band selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
40 Automatic Local Spectral Envelope . . . . . . . . . . . . . . . . . . . . . . . 263Ori Rosen and David Sto�er40.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26340.2 Basic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
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41 Recent Advances in the Use of SVM for Functional DataClassi�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273Fabrice Rossi and Nathalie Villa41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27341.2 SVM classi�ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
41.2.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27441.2.2 Universal consistency of SVM . . . . . . . . . . . . . . . . . . . 275
41.3 Using SVM to classify functional data . . . . . . . . . . . . . . . . . . . 27641.3.1 Kernels for functional data . . . . . . . . . . . . . . . . . . . . . . 27641.3.2 Projection approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27641.3.3 Di�erentiation approach . . . . . . . . . . . . . . . . . . . . . . . . 277
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
42 Wavelet Thresholding Methods Applied to TestingSigni�cance Di�erences Between AutoregressiveHilbertian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281María Ruiz-Medina42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28142.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28242.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
42.3.1 Comparing two sequences of SFD in the ARHcontext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
43 Explorative Functional Data Analysis for 3D-geometriesof the Inner Carotid Artery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Laura Maria Sangalli, Piercesare Secchi and Simone Vantini43.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28943.2 E�cient estimation of 3D vessel centerlines and their
curvature functions by free knot regression splines . . . . . . . . . 29043.3 Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29243.4 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
44 Inference on Periodograms of In�nite DimensionalDiscrete Time Periodically Correlated Processes . . . . . . . . . . 297Zohreh Shishebor, Ahmad Reza Soltani and Ahmad Zamani44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29744.2 Preliminaries and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
List of Contributors
Ana Aguilera (chapters 2, 3)Universidad de Granada, Spain, e-mail: [email protected] Antoch (chapter 4)University of Prague, Tchequia, e-mail: [email protected] Attouch (chapters 5)Université Sidi Bel Abbes, Algérie, e-mail: [email protected] Baillo (chapters 7, 8)Universidad Autonoma Madrid, Spain, e-mail: [email protected] Bel (chapter 9)AgroParisTech, France, e-mail: [email protected] Bolla (chapter 11)Budapest University of Technology and Economics, Hungry, e-mail:[email protected] Bosq (chapter 12)Université Paris 6, France, e-mail: [email protected] Cabral (chapter 13)Université de Toulouse III, France, e-mail: [email protected] Capobianco (chapter 15)Technology Park of Sardinia, Italia, e-mail: [email protected] Chaouch (chapter 16)Université de Bourgogne, France, e-mail: mohamed.chaouch@u-bourgogneJeng-Min Chiou (chapter 17)Taiwan University, Taiwan, e-mail: [email protected] Crambes (chapters 18, 19)Université Montpellier II, France, e-mail: [email protected] Cuevas (chapter 7)Universidad Autonoma Madrid, Spain, e-mail: [email protected] Cuvelier (chapter 21)Université Namur, Belgique, e-mail: [email protected]
xviii List of Contributors
Pedro Delicado (chapters 10, 22)University Politecnica Catalunya, Spain, e-mail: [email protected] Delsol (chapters 18, 23)Université Toulouse III, France, e-mail: [email protected] Dinculeanu (chapter 24)University of Florida, USA, e-mail: [email protected] Epifanio (chapter 25)Universitat Jaume I Castello, Spain, e-mail: [email protected] Escabias (chapter 2)Universidad de Granada, Spain, e-mail: [email protected] Giacomini (chapter 28)Humboldt Universitat zu Berlin, Germany,e-mail: [email protected] Giraldo (chapter 22)Universidad Nacional de Colombia, Colombia,e-mail: [email protected] González-Manteiga (chapter 26)Universidad de Santiago de Compostela, Spain, e-mail: [email protected] Gustafson (chapter 29)University of Colorado, USA, e-mail: [email protected] Hall (chapter 30)Melbourne University, Australia, e-mail: [email protected] James (chapter 32)University Southern California, USA, e-mail: [email protected] Johannes (chapter 33)Universitat Heidelberg, Germany,e-mail: [email protected] Laksaci (chapters 5, 18)University Sidi Bel Abbes, Algeria, e-mail: [email protected] Leon (chapter 6)Universidad de Valencia, Spain, e-mail: [email protected] Martinez-Calvo (chapter 34)University de Santiago de Compostela, Spain, e-mail: [email protected]é Carlos Simon de Miranda (chapter 35)University of Sao Paulo, Brazil, e-mail: [email protected] Nerini (chapter 36)Centre d'Océanologie de Marseille, France,e-mail: [email protected] Nieto-Reyes (chapter 20)Universidad de Cantabria, Spain, e-mail: [email protected] Park (chapter 37)Lancaster University, United Kingdom,e-mail: [email protected] Politis (chapter 38)University of San Diego, USA, e-mail: [email protected]
List of Contributors xix
Moshen Pourahmadi (chapter 39)University Northern Illinois, USA, e-mail: [email protected] Ramsay (chapter 14)McGill University, Canada, e-mail: [email protected] Ruiz-Medina (chapter 42)Granada University, Spain, e-mail: [email protected] Secchi (chapter 43)Politecnico di Milano, Italia, e-mail: [email protected] Lin Shang (chapter 31)University of Clayton, Australia, e-mail: [email protected] Reza Soltani (chapter 44)Kuwait University, Kuwait, e-mail: [email protected] Sto�er (chapter 40)University of Pittsburgh, USA, e-mail: [email protected] Vanhems (chapter 27)Toulouse Business School, France, e-mail: [email protected] Villa (chapter 41)University of Toulouse, France,e-mail: [email protected]
