Tito Gregorio

Nicolas Sutton-Charani

Véronique Sals-Vettorel

Maria Trouche

Alexandre Mairin

Nicolas Verzelen

Brigitte Charnomordic

Christophe Abraham

Meïli Baragatti

Patrice Loisel

Martine Marco

Anne Tireau

Gabrielle Weinrott

Paul-Marie Grollemund

Arnaud Charleroy

Hazaël Jones

Céline Casenave

Besides

Eladio Ocaña

Félix Escalante

Keywords: Functional regression models, Functional data, Convolution Model, Concurrent Model, Historical Model. vi Ridge regression, functional data, concurrent model, varying coefficient model

Functional data analysis (FDA) is a statistical branch that is increasingly being used in many applied scientific fields such as biological experimentation, finance, physics, etc. A reason for this is the use of new data collection technologies that increase the number of observations during a time interval.

Functional datasets are realization samples of some random functions which are measurable functions defined on some probability space with values in an infinite dimensional functional space.

There are many questions that FDA studies, among which functional linear regression is one of the most studied, both in applications and in methodological development.

The objective of this thesis is the study of functional linear regression models when both the covariate X and the response Y are random functions and both of them are time-dependent. In particular we want to address the question of how the history of a random function X influences the current value of another random function Y at any given time t.

In order to do this we are mainly interested in three models: the functional concurrent model (FCCM), the functional convolution model (FCVM) and the historical functional linear model. In particular for the FCVM and FCCM we have proposed estimators which are consistent, robust and which are faster to compute compared to others already proposed in the literature.

Our estimation method in the FCCM extends the Ridge Regression method developed in the classical linear case to the functional data framework. We prove the probability convergence of this estimator, obtain a rate of convergence and develop an optimal selection procedure of the regularization parameter.

The FCVM allows to study the influence of the history of X on Y in a simple way through the convolution. In this case we use the continuous Fourier transform operator to define an estimator of the functional coefficient. This operator transforms the convolution model into a FCCM associated in the frequency domain. The consistency and rate of convergence of the estimator are derived from the FCCM.

The FCVM can be generalized to the historical functional linear model, which is itself a particular case of the fully functional linear model. Thanks to this we have used the Karhunen-Loève estimator of the historical kernel. The related question about the estimation of the covariance operator of the noise in the fully functional linear model is also treated.

Finally we use all the aforementioned models to study the interaction between Vapour Pressure Deficit (VPD) and Leaf Elongation Rate (LER) curves. This kind of data is obtained with high-throughput plant phenotyping platform and is well suited to be studied with FDA methods.

Résumé

L'Analyse des Données Fonctionnelles (ADF) est une branche de la statistique qui est de plus en plus utilisée dans de nombreux domaines scientifiques appliqués tels que l'expérimentation biologique, la finance, la physique, etc. Une raison à cela est l'utilisation des nouvelles technologies de collecte de données qui augmentent le nombre d'observations dans un intervalle de temps.

Les jeux de données fonctionnelles sont des échantillons de réalisations de fonctions aléatoires qui sont des fonctions mesurables définies sur un espace de probabilité à valeurs dans un espace fonctionnel de dimension infinie.

Parmi les nombreuses questions étudiées par l'ADF, la régression linéaire fonctionnelle est l'une des plus étudiées, aussi bien dans les applications que dans le développement méthodologique.

L'objectif de cette thèse est l'étude de modèles de régression linéaire fonctionnels lorsque la covariable X et la réponse Y sont des fonctions aléatoires et les deux dépendent du temps. En particulier, nous abordons la question de l'influence de l'histoire d'une fonction aléatoire X sur la valeur actuelle d'une autre fonction aléatoire Y à un instant donné t.

Pour ce faire, nous sommes surtout intéressés par trois modèles: le modèle fonctionnel de concurrence (Functional Concurrent Model: FCCM), le modèle fonctionnel de convolution (Functional Convolution Model: FCVM) et le modèle linéaire fonctionnel historique. En particulier pour le FCVM et FCCM nous avons proposé des estimateurs qui sont consistants, robustes et plus rapides à calculer par rapport à d'autres estimateurs déjà proposés dans la littérature.

Notre méthode d'estimation dans le FCCM étend la méthode de régression Ridge développée dans le cas linéaire classique au cadre de données fonctionnelles. Nous avons montré la convergence en probabilité de cet estimateur, obtenu une vitesse de convergence et développé une méthode de choix optimal du paramètre de régularisation.

Le FCVM permet d'étudier l'influence de l'histoire de X sur Y d'une manière simple par la convolution. Dans ce cas, nous utilisons la transformée de Fourier continue pour définir un estimateur du coefficient fonctionnel. Cet opérateur transforme le modèle de convolution en un FCCM associé dans le domaine des fréquences. La consistance et la vitesse de convergence de l'estimateur sont obtenues à partir du FCCM.

Le FCVM peut être généralisé au modèle linéaire fonctionnel historique, qui est lui-même un cas particulier du modèle linéaire entièrement fonctionnel. Grâce à cela, nous avons utilisé l'estimateur de Karhunen-Loève du noyau historique. La question connexe de l'estimation de l'opérateur de covariance du bruit dans le modèle linéaire entièrement fonctionnel est également traitée.

Finalement nous utilisons tous les modèles mentionnés ci-dessus pour étudier l'interaction entre le déficit de pression de vapeur (Vapour Pressure Deficit: VPD) et vitesse d'élongation foliaire (Leaf Elongation Rate: LER) courbes. Ce type de données est obtenu avec phénotypage végétal haut débit. L'étude est bien adaptée aux méthodes de l'ADF.

Mots-clefs : Données fonctionnelles, Régression linéaire fonctionnelle, Modèle de convolution, Modèle de concurrence, Modèle historique.

List of Figures

Log des intensités spectrales issu des données de spectrométrie de masse.

Les lignes noires sont les spectres tracés de 20 patients atteint de cancer du pancréas (solide) et de 20 patients contrôle (en pointillés), avec les spectres moyens pour le groupe avec cancer (rouge) et pour le groupe contrôle (bleu).

3.7

The real part of the function F -1 (Ψ n ) (the imaginary part is equal to constant zero) for setting 1 to 3. In green 50 examples of F -1 (Ψ n ) computed for samples of size n = 70. In red the cross-sectional mean in each case. . . . . . . . . . . . . . . 3.8 The plots of the function Φ for setting 1 to 3. . . . . . . . . . . . . . . . . . . 3.9 Estimators of θ for each setting. The cross-sectional mean of the FFDE estima- tor before and after removing the edge effect are the curves in green and in red respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Boxplots of MADE and WASE criteria before (FFDE) and after removing the edge effect (FFDE.no.ed) respectively. . . . . . . . . . . . . . . . . . . . . . . . .

Résumé Etendu

En biologie, les infrastructures expérimentales (ex. plates-formes de phénotypage, bioprocédés) disposent de nouveaux moyens techniques qui génèrent de grandes quantités de données, de qualité hétérogène, acquises à différentes échelles et dans le temps, de nature et de type variés. Valoriser et exploiter ces masses de données est un défi important pour produire de nouvelles connaissances, voir par exemple [START_REF] Ullah | Applications of functional data analysis: A systematic review[END_REF] ou [START_REF] Wang | Functional data analysis[END_REF]. Il faut donc développer des méthodes et fournir des outils dédiés à un traitement systématique de ces données, avec une prise en compte adaptée de leur dimension temporelle. En statistique, les analyses multivariées ont montré leur limite, voir [START_REF] Bickel | Some theory for fisher's linear discriminant function,'naive bayes', and some alternatives when there are many more variables than observations[END_REF], [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF], Hsing and Eubank (2015, p. 1), [START_REF] Ramsay | Functional data analysis[END_REF], Ch 1)... Les raisons sont par exemple : i) le nombre de pas d'observation p est plus grand que le nombre de réalisations n ( cf. Hsing and Eubank (2015, p. 2)); ii) les réalisations ne sont pas observées sur les mêmes grilles de temps (cf. [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF]); iii) il y a de fortes corrélations temporelles au sein des réalisations (cf. Ferraty and Vieu (2006, p. 7)) et iv) la régularité et les dérivées des fonctions aléatoires observées jouent un rôle important dans l'étude des données (cf. [START_REF] Mas | Functional linear regression with derivatives[END_REF]).

Une façon plus adaptée de traiter ce type de données est de les considérer comme des "réalisations de processus stochastiques à temps continu" (Hsing and Eubank (2015, Ch 1), Bosq (2000, Ch 1)). Cela permet d'introduire les notions de données fonctionnelles et fonctions aléatoires. Les données fonctionnelles sont des échantillons de réalisations de fonctions aléatoires. Une fonction aléatoire représente une évolution, discrète ou à temps continu, d'une variable aléatoire. D'un point de vue mathématique, les fonctions aléatoires sont des fonctions mesurables définies sur un espace de probabilité avec des valeurs dans un espace de dimension infini (Ferraty and Vieu (2006, Ch 1)).

Dans beaucoup d'applications, les fonctions observées sont univariées, dépendant du temps. Elles peuvent dépendre d'un paramètre de type différent, comme par exemple une longueur d'onde dans le cas de jeux de données de spectrométrie. Pour ces fonctions univariées, on parle aussi de "données courbe" (voir [START_REF] Gasser | Searching for structure in curve samples[END_REF]). Les fonctions peuvent être aussi multivariées, dépendant du temps, d'une longueur d'onde, de l'espace ou autres, voir [START_REF] Morris | Functional regression[END_REF], Wang et al. (2016, p. 1)).

Le modèle de régression linéaire fonctionnelle est l'un des sujets les plus traités en analyse de données fonctionnelles, que ce soit dans les applications ou pour les développements méthodologiques (Morris (2015, p. 3)). C'est un bon moyen pour étudier la relation entre fonctions aléatoires dans des domaines variés, voir par exemple [START_REF] Ullah | Applications of functional data analysis: A systematic review[END_REF] et [START_REF] Wang | Functional data analysis[END_REF].

L'objectif de la thèse est d'étudier les modèles de régression linéaire pour données fonctionnelles quand le régresseur X (la covariable, l'entrée) et la réponse Y (la sortie) sont tous deux des fonctions aléatoires dépendant du temps (on parle aussi de données "courbes" ou de données "longitudinales"). En particulier nous souhaitons répondre à la question de comment les valeurs passées du régresseur X influencent la valeur courante de la réponse Y à chaque pas de temps t. Dans ce sens, nous proposerons des méthodes d'estimation dans les modèles qui ont de bonnes propriétés (consistance, robustesse) et qui sont rapides à calculer par rapport à d'autres méthodes déjà développées dans la littérature. Dans ce cadre, nous nous intéressons principalement à deux modèles : le modèle linéaire fonctionnel historique et le modèle de convolution fonctionnel (FCVM) que nous introduisons à présent.

Le modèle linéaire fonctionnel historique, introduit par [START_REF] Malfait | The historical functional linear model[END_REF], est de la forme

Y (t) = t 0 K hist (s,t)X(s) ds + ε(t), (1)
où s,t ≥ 0, K hist (s,t) est la fonction coefficient de régression historique et ε est un bruit aléatoire fonctionnel avec E[ε] = 0. Ce modèle est un cas particulier du "modèle de régression complètement fonctionnel" (voir par exemple Horváth and Kokoszka (2012, p. 130) et Ramsay and Silverman (2005, Ch 16)), où X et Y sont reliés par un opérateur noyau plus général. Ce dernier s'écrit comme suit :

Y (t) = I K (s,t)X(s) ds + ε(t), (2)
où s,t ∈ R, I ⊆ R et K (•, •) est le noyau intégrable. Cependant l'interprétation et l'estimation sont facilitées quand le noyau K a la forme de K hist car cette fonction est définie sur le domaine triangulaire plus simple où s < t.

Un moyen encore plus simple d'étudier l'influence du passé de X sur la valeur courante de Y se fait au travers du modèle de convolution fonctionnel (FCVM) défini ci-dessous :

Y (t) = t 0 θ (s)X(t -s) ds + ε(t), (3)
où t ≥ 0, θ est le coefficient fonctionnel inconnu à estimer. Dans ce modèle, θ est une fonction qui dépend de s uniquement et pas du pas de temps courant t comme le fait K hist . Toutes ces fonctions sont considérées comme nulles pour t < 0, ce qui peut s'interpréter comme le fait que 0 est le point de départ des mesures. Au delà de ces deux modèles, historique et de convolution, nous avons aussi étudié le modèle concurrent fonctionnel (FCCM), défini par Ramsay and Silverman (2005, Ch 14) comme suit :

Y (t) = β (t) X(t) + ε(t), (4)
où t ∈ R et β est le coefficient fonctionnel inconnu à estimer. De même que le modèle de régression complètement fonctionnel déjà mentionné plus haut, c'est un modèle majeur de régression qui traite le cas des variables réponses fonctionnelles, voir Wang et al. (2016, p. 272)). Son importance est soulignée dans Ramsay and Silverman (2005, p. 220). Notre intérêt pour le FCCM vient du fait que le FCVM et le FCCM sont reliés grâce à la Transformée de Fourier Continue ; chaque fonction θ , associée à un modèle FCVM et définie sur le domaine temporel, a une transformée de Fourier β dans le domaine des fréquences qui est l'élément fonctionnel inconnu d'un modèle FCCM. Cela établit, sous certaines conditions, l'équivalence entre ces deux modèles (voir la section 1.3). Le modèle linéaire fonctionnel historique est aussi relié au modèle FCCM quand par exemple le noyau historique est exprimé comme le produit de deux fonctions univariées de la façon suivante : K (s,t) = β (t)γ(s) [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF], [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF]).

Dans la suite de ce résumé nous introduirons le cadre théorique, les notations principales et les définitions utilisés tout au long de la thèse. Nous donnerons tout d'abord des aspects généraux sur l'analyse des données fonctionnelles. Nous ferons un compte rendu de la littérature sur les modèles de régression pour données fonctionnelles à réponses fonctionnelles. Nous développerons ensuite notre procédure d'estimation de la fonction θ dans le FCVM, ainsi que son implémentation numérique. Nous décrirons brièvement nos résultats, aussi bien théoriques que pratiques (obtenus en simulation et sur données réelles).

Analyse de données fonctionnelles

Quelques exemples de jeux de données Voici deux exemples pour donner une idée intuitive de ce que l'on appelle des données "courbes" (voir Wang et al. (2016, p. 258)), type de données faciles à visualiser et que nous étudierons tout au long de ce document. Données de spectrométrie de masse Cet exemple vient de Koomen et al. (2005) et est commenté dans [START_REF] Morris | Functional regression[END_REF]. Ce jeu de données contient les spectres de masse d'une étude sur le cancer du pancréas menée à l'université du Texas. Du sérum a été prélevé du sang de 139 patients atteints de cancer et de 117 personnes en bonne santé. Alors, des spectres protéomiques X i (t) ont été obtenus avec un instrument de spectrométrie de masse pour chaque individu i = 1, • • • , 256. La grille des observations discrètes (masse moléculaire par unité de charge m/z) pour chaque courbe est de taille T = 12096. Un sous ensemble de ces données est montré à la figure 1.

Fonctions aléatoires

Les fonctions aléatoires sont une extension naturelle des variables aléatoires à valeurs réelles et prennent leurs valeurs dans des espaces fonctionnels, plutôt que dans R. Plus généralement Ferraty and Vieu (2006, p. 6) définissent une fonction aléatoire comme une fonction mesurable d'un espace de probabilité (Ω, A , P) dans un espace fonctionnel de dimension infini E. Il est courant de considérer E avec sa σ -algèbre de Borel généré par ses ensembles ouverts. Cet espace fonctionnel peut être un espace de Hilbert (voir Horváth and Kokoszka (2012, Ch 2)), un espace de Banach (voir Ledoux and Talagrand (1991, Ch 2)),un espace d'applications (voir Bosq (2000, p. 16)), etc.

Dans cette thèse nous nous intéressons au cas où les fonctions de E sont univariées, par exemple fonction du temps ou d'une fréquence. Comme nous l'avons signalé plus haut, on parle de données courbe ou de données longitudinales pour les données observées à partir de ces fonctions. C'est par exemple le cas quand E := L 2 (I), l'ensemble des fonctions Lebesgue de carré intégrable définies sur un intervalle I ⊂ R. Les méthodes qui traitent ce type de données sont étudiées dans le livre de [START_REF] Ramsay | Functional data analysis[END_REF] classiquement cité ou dans le livre plus récent de [START_REF] Hsing | Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators[END_REF].

Dans ce qui suit nous résumons les principaux éléments concernant les opérateurs espérance et covariance dans les espaces de Hilbert et de Banach (voir [START_REF] Hsing | Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators[END_REF] pour plus de détails).

Opérateur Espérance : Soit B un espace de Banach séparable et B * son espace dual. Une fonction aléatoire à valeurs dans B X :

(Ω, A , P) → B est dite faiblement intégrable si et seulement si i) la fonction composée f * (X) est intégrable pour tout f * ∈ B * et ii) il existe un élément de B, noté E[X], tel que pour tout f * ∈ B * E[ f * (X)] = f * (E[X]). L'élément E[X]
:= [E∥X∥ p B ] 1/p .
Opérateur Covariance : Pour une fonction aléatoire X ∈ L 2 B (P), avec E[X] = 0, nous définissons son opérateur de covariance comme l'opérateur linéaire borné suivant :

C X : B * → B f * → E[ f * (X) X]. Dans le cas où E[X] ̸ = 0, nous définissons C X := C X-E[X]
. Il est possible de généraliser cette définition pour définir l'opérateur de covariance croisée entre deux fonctions aléatoires. Pour ce faire, considérons

X ∈ L 2 B 1 (P) et Y ∈ L 2 B 2 (P), où B 1 et B 2 sont des espaces de Banach séparables tels que E[X] = 0 et E[Y ] = 0.
Les opérateurs de covariance croisée de X et Y sont les opérateurs linéaires bornés suivants:

C X,Y : B * 1 → B 2 f * → E[ f * (X)Y ] et C Y,X : B * 2 → B 1 g * → E[g * (X)Y ].
Dans le cas particulier d'un espace de Hilbert séparable H, doté du produit scalaire ⟨•, •⟩, la définition de l'espérance est similaire à celle définie pour les espaces de Banach.

En outre, la définition de l'opérateur de covariance sera plus simple, grâce au théorème de représentation de Riez pour l'espace dual H * . De cette façon, soit X une fonction aléatoire à valeurs dans

H telle que E[∥X∥ 2 H ] < ∞ et E[X] = 0. Alors, l'opérateur de covariance de X est le suivant C X : H → H x → E[⟨x, X⟩ X].
Il est connu que cet opérateur est symétrique, positif, nucléaire (voir Bosq (2000, p. 34) pour plus de détails). De même, l'opérateur de covariance croisée est défini comme suit :

C X,Y (x) = E[⟨X, x⟩Y ],
pour tout x ∈ H. Cet opérateur est aussi nucléaire (Bosq (2000, p. 34)).

Décomposition spectrale des opérateurs : Soit X une fonction aléatoire à valeurs dans

H telle que E[∥X∥ 2 H ] < ∞ et E[X] = 0.
La décomposition spectrale de X existe et est la suivante :

C X (x) = ∞ ∑ j=1 λ j ⟨x, v j ⟩v j
où les (v j ) j≥1 sont les fonctions propres de C X et forment une base orthonormale de H, les (λ j ) j≥1 sont les valeurs propres et satisfont :

∞ ∑ j=1 |λ j | = E[∥X∥ 2 H ] < ∞, puisque λ j = ⟨C X (v j ), v j ⟩ = E(⟨X, v j ⟩ 2 ).
Cette décomposition spectrale sert à projeter les fonctions aléatoires dans un sous-espace de dimension finie engendré par les K premières fonctions propres. Elle intervient dans de nombreuses méthodes bien connues d'analyses pour données fonctionnelles, comme l'analyse en composantes principales fonctionnelle (voir, (Ramsay and Silverman, 2005, Ch 8)) ou les méthodes d'estimation pour le modèle de régression linéaire fonctionnel de type Karhunen-Loeve (voir [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF]).

Analyses de données fonctionnelles (ADF) et analyses statistiques multivariées

Dans ce paragraphe, nous abordons la question de l'inadéquation de certaines méthodes multivariées pour traiter des données fonctionnelles. Comme nous l'avons mentionné plus tôt, Wang et al. (2016, p. 1) définit l'ADF comme " l'analyse et la théorie des données qui sont sous la forme de fonctions, d'images et de formes, ou d'objets plus généraux ". Quant à elle, l'analyse multivariée porte sur la compréhension et l'analyse de l'interaction de plusieurs variables statistiques qui sont en général mesurées simultanément (Johnson and Wichern, 2007, p. 1). Hsing and Eubank (2015, p. 1) Il existe deux approches principales pour répondre à ces questions. D'abord l'approche paramétrique fonctionnelle nécessite que Ψ appartienne à un sous-ensemble particulier S de l'ensemble des opérateurs linéaires continus C sur L 2 (I), qui peut être indexé par un nombre fini de certains opérateurs linéaires continus fixés sur L 2 (I) (Ferraty and Vieu (2006, p. 8)). Un exemple est celui de la régression linéaire fonctionnelle avec sortie fonctionnelle définie dans (2), où l'opérateur linéaire continu est uniquement paramétré avec un seul opérateur noyau intégral.

La seconde approche est l'approche non-paramétrique fonctionnelle. L'ensemble S n'a pas à être indexé par un ensemble fini d'opérateurs linéaires continus Ferraty and Vieu (2006, p. 8). Les méthodes non-paramétriques fonctionnelles tiennent compte de la régularité des éléments de S , par exemple les modèles additifs fonctionnels étudiés par [START_REF] Müller | Functional additive models[END_REF], les espaces de Hilbert à noyau reproduisant (voir [START_REF] Lian | Nonlinear functional models for functional responses in reproducing kernel hilbert spaces[END_REF], [START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF]) ou les méthodes à noyau [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF].

Tout au long de cette thèse, nous nous intéressons à l'approche paramétrique fonctionnelle, plus précisément au modèle de régression linéaire fonctionnelle avec réponse fonctionnelle. Dans ce qui suit, nous donnons une étude plus détaillée de ce modèle.

Deux modèles majeurs

Dans un article récent, [START_REF] Wang | Functional data analysis[END_REF] propose de diviser la classe des modèles de régression linéaire fonctionelle à sortie fonctionnelle en deux grandes catégories : le modèle concurrent fonctionnel (FCCM) et le modèle de régression entièrement fonctionnel (voir Horváth and Kokoszka (2012, p. 130)). Nous discutons brièvement des deux.

Modèle concurrent fonctionnel : Son équation est la suivante :

Y (t) = β 0 (t) + β 1 (t) X(t) + ε(t), (5)
où t ∈ R, β 0 (t) et β 1 (t) sont les coefficients fonctionnels du modèle à estimer. Des modèles étroitement liés ont déjà été discutés par plusieurs auteurs. Par exemple dans [START_REF] West | Dynamic generalized linear models and bayesian forecasting[END_REF], les auteurs définissent un modèle similaire appelé « modèle linéaire dynamique généralisé » et ils étudient le modèle d'un point de vue bayésien. [START_REF] Hastie | Varying-coefficient models[END_REF] proposent le 'Varying Coefficient Model'. Ce modèle a la forme

η = β 0 (R 0 ) + X 1 β 1 (R 1 ) + • • • X p β p (R p ), (6)
où η est un paramètre qui détermine la distribution de la variable aléatoire

Y , X 1 , • • • , X p et R 1 , • • • , R p sont des prédicteurs, β 1 , • • • , β p sont
Y = β 0 (R 0 ) + X 1 β 1 (R 1 ) + • • • X p β p (R p ) + ε, où E[ε] = 0 et var[ε] = σ 2 .
Y (t) = β 0 (t) + X 1 (t)β 1 (t) + • • • X p (t)β p (t) + ε.
Beaucoup d'auteurs ont étudié l'estimation des fonctions inconnues β i . Par exemple [START_REF] Wu | Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data[END_REF] proposent une approche basée sur les noyaux quand il y a k covariables fonctionnelles. [START_REF] Dreesman | Non-stationary conditional models for spatial data based on varying coefficients[END_REF]Tutz (2001) et Cai et al. (2000) proposent une estimation de type maximum de vraisemblance locale. [START_REF] Zhang | Variable bandwidth selection in varying-coefficient models[END_REF], [START_REF] Fan | Adaptive varying-coefficient linear models[END_REF] et [START_REF] Zhang | Local polynomial fitting in semivarying coefficient model[END_REF] proposent des approches par lissage polynomial local. [START_REF] Fan | Two-step estimation of functional linear models with applications to longitudinal data[END_REF] proposent une approche en deux étapes : réaliser tout d'abord des régressions de type moindres carrés ordinaires de manière ponctuelle puis faire un lissage de ces estimateurs grossiers. [START_REF] Huang | Polynomial spline estimation and inference for varying coefficient models with longitudinal data[END_REF] proposent d'estimer les fonctions β i par des B-splines et des méthodes de moindres carrés. Une étude plus poussée du "Varying Coefficient Model" multivarié est présentée dans [START_REF] Zhu | Spatially varying coefficient model for neuroimaging data with jump discontinuities[END_REF].

Nous voulons souligner que, bien que le "varying coefficient model" soit lié au FCCM, il a été conçu à l'origine pour le cas où t est une variable aléatoire et a été traité avec des méthodes d'analyse multivariée. Ainsi, les premiers travaux sur l'estimation ne tiennent pas compte de la nature fonctionnelle des données, comme l'a remarqué Ramsay and Silverman (2005, p. 259). En revanche, [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF] proposent une méthode d'estimation qui est plus proche de l'approche pour données fonctionnelles.

Modèle de régression entièrement fonctionnel

Nous rappelons la forme de ce modèle :

Y (t) = I K (s,t)X(s) ds + ε(t).
Ce modèle a été popularisé par le travail fondateur de [START_REF] Ramsay | Some tools for functional data analysis[END_REF]. L'estimation du noyau K est un problème inverse et nécessite une certaine régularisation pour que l'inversion soit possible. Ramsay and Dalzell (1991, p. 552) ont proposé une méthode des moindres carrés pénalisés pour le faire. [START_REF] James | Generalized linear models with functional predictors[END_REF] a proposé une méthode pénalisée avec des splines. [START_REF] Cuevas | Linear functional regression: the case of fixed design and functional response[END_REF] étudient cette régularisation pour un design fixe. Dans [START_REF] He | Extending correlation and regression from multivariate to functional data[END_REF], les auteurs définissent l' «équation normale fonctionnelle» qui généralise les équations normales multivariées et ils prouvent qu'il existe une solution unique sous certaines conditions. Ils utilisent la décomposition de Karhunen-Loève pour estimer K . [START_REF] Chiou | Functional response models[END_REF] donnent un bon résumé de cette approche. Cette idée a été revisitée dans Yao et al. (2005a) où les auteurs ont traité des données longitudinales parcimonieuses avec des temps d'observation irréguliers et aléatoires. De plus, un estimateur fondé sur des ondelettes est proposé dans [START_REF] Aguilera | Estimation of functional regression models for functional responses by wavelet approximation[END_REF], un estimateur basé sur des splines est proposé dans [START_REF] Antoch | Electricity consumption prediction with functional linear regression using spline estimators[END_REF], et un estimateur de type Karhunen-Loève dans [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF].

Dès que X et Y sont dépendants du temps, dans l'équation (2), on voit que des valeurs futures de X sont utilisées pour expliquer des valeurs passées de Y . Ce constat est contreintuitif. Pour cette raison, [START_REF] Malfait | The historical functional linear model[END_REF] s'intéressent au cas particulier où Y (t), la valeur de Y au temps t, dépend du passé {X(s) : 0 ≤ s ≤ t}. Cela conduit au modèle linéaire fonctionnel historique. Ce modèle particulier est discuté dans la sous-section qui suit.

Modèle de régression linéaire fonctionnelle historique HFLM

Ce modèle, proposé par [START_REF] Malfait | The historical functional linear model[END_REF], a été défini à l'équation (1) que nous rappelons :

Y (t) = t 0 K hist (s,t)X(s) ds + ε(t),
pour tout t > 0. Nous allons maintenant résumer l'approche de [START_REF] Malfait | The historical functional linear model[END_REF] pour estimer K hist . Les auteurs proposent d'utiliser une base d'éléments finis φ k (s,t). Cette base est faite de fonctions linéaires par morceaux définies sur une grille fixe et adaptée de points. Les auteurs utilisent une approximation K hist (s,t) ≈ ∑ K k=1 b k φ k (s,t) pour transformer le problème comme suit :

Y i (t) = K ∑ k=1 b k ψ ik (s,t) + ε i (t), où ψ ik (s,t) := t 0 X i (s) φ k (s,t) ds. Soient y(t) et e(t) les vecteurs de longueur N contenant les valeurs Y i (t) et ε i (t), re- spectivement. Soit aussi Ψ(t) la matrice N × K contenant les valeurs ψ ik (t) . On note b le vecteur des coefficients (b 1 , • • • , b K ) ′ . Alors la forme matricielle de l'équation (1) est pour tout t ∈ [0, ∞[: y(t) = Ψ(t) b + e(t).
Cela mène aux équations normales T 0 Ψ(t) ′ Ψ(t)dt b = T 0 Φ(t) ′ y(t)dt. Finalement les auteurs approchent et résolvent cette équation avec un modèle linéaire multivarié.

Avec une approche similaire, [START_REF] Harezlak | Penalized solutions to functional regression problems[END_REF] ont considéré la même représentation de la fonction K hist (s,t) à travers la base de fonctions φ k (s,t). Dans ce cas, les auteurs ont exploré deux techniques de régularisation différentes qui utilisent une troncature de la base, des pénalités de rugosité, et des pénalités de parcimonie. La première pénalise les valeurs absolues des différences de fonction de base de coefficients (approche LASSO) et la seconde pénalise les carrés de ces différences (méthodologie des splines pénalisées). Enfin, les auteurs ont évalué la qualité de l'estimation avec une extension du critère d'information d'Akaike.

Dans [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF], les auteurs s'intéressent à un type particulier de modèle historique. Ils considèrent les courbes X i et Y i comme des données longitudinales parcimonieuses, et la fonction K hist (s,t) décomposable comme suit :

K hist (s,t) = ∑ K k=1 b k (t)φ k (s)
, où les b k (t) sont les coefficients fonctionnels à estimer et les φ k (s) sont des fonctions de base prédéterminées, par exemple des B-splines.

Dans ce cas, le modèle (1) devient un " varying coefficient model" ou un un modèle fonctionnel concurrent (FCCM) après intégration. La procédure d'estimation utilise la décomposition en fonctions de base de Y et X. Ces décompositions sont nécessaires pour résoudre une équation normale particulière qui utilise les auto-covariances de X(s) et Y (t) et la covariance croisée de (X(s),Y (t)).

Un modèle similaire est étudié dans [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF]. Les auteurs considèrent la modification suivante de (1) [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF], ce cas particulier est fortement lié au FCCM. Il est clair que, après avoir estimé la fonction γ, le problème devient celui de l'estimation dans un FCCM. Dans [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF], une nouvelle méthode pour estimer les fonctions β 0 et β 1 est développée, qui tient compte de la nature fonctionnelle des données. Toutefois, l'idée essentielle pour estimer les fonctions inconnues (γ, β 0 et β 1 ) est d'utiliser une idée équivalente à l' «équation normale fonctionnelle» : relier l'auto-covariance de X avec la covariance croisée de X et Y , puis d'inverser l'auto-covariance de X pour obtenir la fonction inconnue.

Y (t) = β 0 (t) + ∆ 0 β 1 (t)γ(s)X(s) ds + ε(t), pour t ∈ [∆, T ] avec ∆ > 0 et un choix adapté de T > 0. De même que pour
Finalement il est possible de considérer une structure encore plus spécifique de la fonction K hist (s,t) qui ne dépend pas de la valeur courante t, mais uniquement de s. L'intégrale prend la forme d'une convolution. Pour ce faire, nous supposons qu'il existe une fonction θ telle que K hist (s,t) = θ (ts) et nous appliquons un changement de variables pour obtenir le modèle de convolution fonctionnel ((3), objet du paragraphe suivant.

Modèle de convolution fonctionnel (FCVM)

L'un des principaux objectifs de cette thèse est d'étudier l'influence du passé de la covariable fonctionnelle X sur la valeur actuelle de la réponse Y (t). Une façon de modéliser cette relation de dépendance est à travers le FCVM. Dans ce paragraphe, nous présentons l'origine de ce modèle et dans la section suivante, nous discutons des méthodes d'estimation de la fonction coefficient θ et la place du FCVM entre autres modèles dans la littérature.

Rappelons la forme du modèle FCVM (3), dérivé de Malfait and Ramsay (2003),

Y (t) = t 0 θ (s) X(t -s)ds + ε(t),
pour tout t ≥ 0. Nous nous intéressons à l'estimation de la fonction θ à partir d'un échantillon i.i.d.

(X i ,Y i ) i∈{1,••• ,n} des fonctions aléatoires X et Y .
Le FCVM est une extension fonctionnelle des modèles à retards échelonnés en séries temporelles (Greene (2003, Ch 19)). Le modèle de régression dynamique de forme générale

y t = α + ∞ ∑ i=0 β i x t-i + ε t .
pour tout t ≥ 0, en est un exemple. Si nous supposons que x i = 0 pour tout i < 0 (c'est à dire qu'il y a un point de départ), alors la somme est finie. C'est une discrétisation du FCVM. Les applications de ce modèle sont commentées dans Greene (2003, Ch 19).

La question de l'estimation de θ est centrale dans l'étude du FCVM. C'est le sujet de la section suivante. Nous détaillerons plus précisément la connexion entre le FCVM et le FCCM obtenue avec l'utilisation de la transformée de Fourier continue.

Estimation de θ dans le FCVM

Le FCVM a quatre caractéristiques principales : i) la covariable (entrée) et la réponse (sortie) sont des fonctions aléatoires, ii) la convolution est non périodique (i.e. nous ne considérons pas les fonctions périodiques), iii) la taille de l'échantillon est n > 1, en outre, nous sommes intéressés par le comportement asymptotique (n → ∞) et enfin iv) le bruit est fonctionnel. A notre connaissance, il y a peu de documents qui étudient un modèle avec de telles caractéristiques. Cependant, il existe de nombreux modèles qui sont proches de FCVM. Dans ce qui suit, nous explorons certains d'entre eux. [START_REF] Asencio | Functional convolution models[END_REF] étudient un problème lié, dans lequel ils considèrent plus de fonctions covariables (prédicteurs). L'estimation de θ est faite en projetant les fonctions dans une base spline de dimension finie et en utilisant une approche de moindres carrés ordinaires pénalisés pour estimer les coefficients dans cette base. Une autre approche consiste à utiliser le modèle linéaire fonctionnel historique [START_REF] Malfait | The historical functional linear model[END_REF]) pour estimer θ , en tenant compte de la forme particulière que la fonction noyau K hist doit avoir dans ce cas particulier. De la même manière, nous pouvons utiliser l'approche de Harezlak et al. (2007), ou même dans un cas plus restreint les approches de [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF] et [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF]. Le FCVM peut aussi être considéré comme un cas particulier du modèle proposé par [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF] Şentürk andMüller (2010, p. 1259) proposent une méthode pour estimer θ quand le FCVM a la forme restreinte suivante :

, quand K = 1, et φ K = θ dans K hist (s,t) = ∑ K k=1 b k (t)φ k (s).
Y (t) = ∆ 0 θ (s) X(t -s)ds + ε(t),
où ∆ > 0 est une valeur fixe. L'estimation de θ dans ce cas est faite en utilisant la décomposition de Karhunen-Loève de l'opérateur de covariance de la fonction aléatoire Z t (s) := X(t -s), où t est fixé et s ∈ [0, ∆]. Les auteurs expriment θ dans la base de fonctions propres de cet opérateur, puis estiment les coefficients avec une procédure de moindres carrés ordinaires. Cela produit un estimateur pour chaque pas de temps t. Ils considèrent une grille de temps d'observation, puis ils prennent la moyenne de tous ces estimateurs. Cette approche est similaire à celle de [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF].

A notre connaissance, seuls les articles mentionnées ci-dessus ont abordé l'étude de l'estimation de θ en tenant compte des quatre caractéristiques du FCVM mentionnées plus tôt. L'approche que nous développons dans le chapitre 3 est une nouvelle façon de répondre à cette question. Nous n'utilisons pas de projection dans une base de fonctions de dimension finie. De plus, nous étudions les propriétés asymptotiques de l'estimateur, ce qui n'est fait dans aucune des approches précédentes.

Estimateur par déconvolution de Fourier fonctionnelle (FFDE)

Nous définissons l'estimateur par déconvolution de Fourier fonctionnelle en trois étapes. i) D'abord, nous utilisons la transformée de Fourier continue (F ) pour transformer la convolution dans le domaine temporel en une multiplication dans le domaine des fréquences, voir (4). ii) Une fois dans le domaine des fréquences, nous estimons β avec l'estimateur de régression Ridge fonctionnelle (FRRE) défini dans [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] (voir Chapitre 2), qui est une extension de la méthode de régularisation Ridge [START_REF] Hoerl | Application of ridge analysis to regression problems[END_REF]) introduite pour traiter des problèmes mal posés dans la régression linéaire classique. iii) La dernière étape consiste à utiliser la transformée de Fourier continue inverse pour estimer θ . Cette définition est formalisée mathématiquement comme suit.

Soit

(X i ,Y i ) i=1,••• ,n un échantillon i.i.d issu du FCVM (3).
Etape i) Nous utilisons la transformée de Fourier continue (F ) définie par

F ( f )(ξ ) = +∞ t=-∞ f (t)e -2πitξ dt, où ξ ∈ R et f ∈ L 2 .
Cet opérateur est utilisé pour transformer le FCVM (3) défini dans le domaine temporel en un modèle équivalent dans le domaine des fréquences :

Y (ξ ) = β (ξ ) X (ξ ) + ε(ξ ), (7)
où ξ ∈ R, β := F (θ ) est le coefficient fonctionnel à estimer. X := F (X) et Y := F (Y ) sont les transformées de Fourier de X et Y . Enfin, ε := F (ε) est un bruit additif fonctionnel. Le modèle (7) dans le domaine des fréquences est un modèle de convolution fonctionnel FCCM de type (4). Clairement l'estimation de β impliquera l'estimation de θ grâce à la transformée de Fourier inverse F -1 .

Etape ii) L'estimateur de régression Ridge fonctionnelle (FRRE) de β dans le FCCM ( 4) ou (7) est défini comme suit :

βn := 1 n ∑ n i=1 Y i X * i 1 n ∑ n i=1 |X i | 2 + λ n n , (8)
où l'exposant * indique le conjugué complexe et λ n est un paramètre de régularisation positif.

Nous avons choisi un estimateur Ridge de β dans (8), car ainsi il est naturel d'utiliser la transformée de Fourier inverse (F -1 ) pour estimer θ . Les propriétés de consistance en norme L 2 de l'estimateur de β sont aussi conservées pour l'estimateur de θ . Nous bénéficions de plus de l'efficacité de calcul de l'algorithme de la transformée de Fourier rapide.

Comme nous l'avons vu auparavant, l'idée de transformer le modèle linéaire fonctionnel historique en un FCCM a déjà été proposée par [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF] et d'une manière différente par [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF]. Dans ces deux articles, les auteurs ont utilisé des structures spéciales pour la fonction noyau K hist . Ces structures leur permettent de transformer le modèle historique en FCCM. Dans notre cas, nous utilisons une approche différente. Nous n'imposons pas une structure particulière à la fonction du noyau. Nous transformons le modèle FCVM dans le domaine temporel en son équivalent dans le domaine fréquentiel. En conséquence, cela ouvre la possibilité d'utiliser également d'autres méthodes d'estimation de β dans le FCCM afin d'estimer θ dans le FCVM.

Etape iii) L'estimateur par déconvolution de Fourier fonctionnelle (FFDE) de θ dans (3) est défini par θn := F -1 ( βn ).

Notons que l'estimateur θn (FFDE) est à valeurs dans R et appartient à L 2 (R, R) (voir Chapitre 3. Une autre hypothèse importante est que le FFDE se décompose comme suit :

θn = θ - λ n n F -1 F (θ ) 1 n ∑ n i=1 |F (X i )| 2 + λ n n + F -1 1 n ∑ n j=1 F (ε j )F (X j ) 1 n ∑ n i=1 |F (X i )| 2 + λ n n . (10)
L'étude de cette décomposition nous permettra de démontrer la consistance de cet estimateur. Notez l'importance de l'équivalence entre le FCVM et le FCCM, en raison de l'utilisation de deux représentations équivalentes de la même information (dans le domaine temporel et le domaine fréquentiel) obtenue grâce à la transformée de Fourier continue. L'estimateur par déconvolution de Fourier fonctionnelle (FFDE) de θ dans le FCVM est étudié dans le chapitre 3. Nous avons choisi de proposer un tel estimateur pour tirer parti de l'équivalence des modèles en temps et en fréquence, et des propriétés mathématiques de la transformée de Fourier continue. Les avantages de cet estimateur sont à la fois théoriques et pratiques : théoriques, car nous développons une approche construite avec des fonctions aléatoires et des espaces fonctionnels, et pratiques parce que pour implémenter cette méthode, nous utilisons la transformée de Fourier discrète et l'algorithme de transformée de Fourier rapide (FFT) qui améliore la vitesse de calcul des estimateurs de façon significative par rapport à d'autres estimateurs possibles. Nous décrivons dans la suite d'autres estimateurs possibles adaptés de la littérature.

Les méthodes de déconvolution dans la littérature

Considérons à présent d'autres modèles indirectement liés au FCVM. De cela, nous serons en mesure d'adapter certaines techniques pour estimer la fonction θ .

Nous commençons avec le modèle de déconvolution multicanal (voir par exemple De Canditiis and [START_REF] De Canditiis | Simultaneous wavelet deconvolution in periodic setting[END_REF], [START_REF] Pensky | On convergence rates equivalency and sampling strategies in functional deconvolution models[END_REF] et [START_REF] Kulik | Multichannel deconvolution with long range dependence: Upper bounds on the lp-risk[END_REF]). Ce modèle est considéré par les méthodes de traitement du signal. De même que pour le FCVM, l'entrée et la sortie sont fonctionnelles (signaux, données courbes), il y a beaucoup de réalisations (n > 1, multicanaux) et le bruit est fonctionnel. Mais la différence avec le FCVM est que les auteurs étudient le cas périodique (les signaux sont périodiques, ainsi que que la convolution). En outre, les auteurs ne traitent pas du comportement asymptotique des estimateurs.

Le problème de déconvolution multicanal est une façon de généraliser le problème de la déconvolution en traitement du signal (voir, par exemple [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF], [START_REF] Brown | Introduction to Random Signals and Applied Kalman Filtering with MATLAB Exercises[END_REF], [START_REF] Gonzalez | Digital Image Processing Using MATLAB[END_REF]). Les auteurs utilisent la convolution (périodique ou non) pour modéliser comment une fonction réponse h transforme un signal g (inconnu) à travers l'équation suivante

f (t) = D h(s)g(t -s)ds + ε(t),
où D est le domaine d'intégration ([0, T ] dans la cas périodique pour un T fixé, et [0,t] ou R dans le cas non périodique), f est le signal observé et ε le bruit. Il y a plusieurs méthodes pour estimer g étant données les fonctions h et f , par exemple la méthode de déconvolution paramétrique de Wiener (Gonzalez and Eddins (2009, Ch 5)).

Si on interprète f comme Y , h comme X et g comme θ , alors on peut appliquer ces méthodes pour estimer θ . Nous remarquons que, bien que ce problème d'estimation est relié au FCVM, il ne traite que du cas n = 1. Il n'y a pas d'étude du comportement asymptotique des estimateurs proposés.

De façon similaire, les méthodes de déconvolution en statistique non paramétrique (voir [START_REF] Meister | Deconvolution Problems in Nonparametric Statistics[END_REF], [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF]) traitent du cas n = 1 et ne considèrent pas les bruits fonctionnels. Le but ici est d'estimer la densité de probabilité d'une variable aléatoire réelle X à partir de l'observation d'une autre variable aléatoire réelle Y telle que Y = X + Z, la densité de probabilité de Z étant connue. Pour résoudre ce problème, les auteurs utilisent le fait que la densité de probabilité de la somme de deux variables aléatoires est la convolution de leurs densités respectives. Il pourrait être possible d'adapter ces techniques pour estimer θ dans le FCVM, mais nous pensons que l'estimation serait pire que celle des méthodes de traitement du signal, parce que dans le premier cas le bruit fonctionnel est pas considéré.

En outre, grâce à une approximation numérique de la convolution comme un opérateur matriciel, l'estimation dans le FCVM devient un problème linéaire inverse pour chaque couple (X i ,Y i ). Dans ce cas, pour chaque i ∈ {1, • • • , n}, nous pouvons estimer θ avec des techniques comme la régularisation de Tikhonov, la méthode de décomposition en valeurs singulières, ou des méthodes basées sur des ondelettes (voir par exemple [START_REF] Tikhonov | Solutions of ill-posed problems[END_REF], [START_REF] O'sullivan | A statistical perspective on ill-posed inverse problems[END_REF], [START_REF] Donoho | Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition[END_REF], [START_REF] Abramovich | Wavelet decomposition approaches to statistical inverse problems[END_REF]). Notez encore une fois que ces méthodes ne traitent que le cas n = 1. Les propriétés asymptotiques ne sont pas étudiées.

Enfin, une autre méthode apparentée est la déconvolution de Laplace introduite par [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging[END_REF]. Cette méthode traite également du cas n = 1. Les auteurs considèrent à la fois la convolution non périodique, comme dans le FCVM, et un bruit fonctionnel.

Dans le chapitre 3 nous avons adapté la déconvolution paramétrique de Wiener, la méthode de décomposition en valeurs singulières, la régularisation de Tikhonov et la déconvolution de Laplace pour estimer θ dans le FCVM.

Contribution de la thèse

Dans cette thèse, nous cherchons à savoir comment le passé du régresseur fonctionnel X influe sur la valeur actuelle de la fonction de réponse Y dans les modèles de régression linéaires.

La thèse, écrite en anglais, est divisée en six chapitres. Le chapitre 1 est une introduction générale, plus exhaustive que ce résumé étendu en français. Les contributions principales de cette thèse sont détaillées dans les chapitres 2 à 4, où nous étudions respectivement le modèle fonctionnel concurrent (chapitre 2), le modèle fonctionnel de convolution (chapitre 3) et le modèle entièrement fonctionnel (chapitre 4). Une illustration sur un jeu de données réelles est faite au chapitre 5. Enfin nous présentons au chapitre 6 les conclusions et les perspectives de cette thèse.

Voici un court résumé de chacun de ces chapitres.

Chapitre 1

Nous y donnons les principales idées sur les modèles de régression linéaire fonctionnelle avec réponse fonctionnelle, ainsi que les définitions et l'arrière-plan théorique utilisé dans les chapitres suivants. Les étapes de l'implémentation numérique des estimateurs sont également détaillées.

Chapitre 2

Dans ce chapitre, nous proposons une approche fonctionnelle pour estimer la fonction inconnue dans le modèle fonctionnel concurrent (FCCM). Cette approche est une généralisation aux données fonctionnelles de la méthode de régression Ridge classique. L'estimateur que nous construisons est ainsi nommé l'estimateur de régression Ridge fonctionnelle (FRRE). L'importance du modèle FCCM a été mise en évidence dans certains articles et livres, parce que c'est un modèle général auquel tous les modèles linéaires fonctionnels peuvent être réduits (voir par exemple [START_REF] Ramsay | Functional data analysis[END_REF], [START_REF] Morris | Functional regression[END_REF], [START_REF] Wang | Functional data analysis[END_REF]).

Nous avons prouvé la consistance du FRRE pour la norme L 2 , et obtenu sa vitesse de convergence sur l'ensemble des réels, et non pas seulement sur les compacts. Nous avons également fourni une procédure de sélection du paramètre optimal de régularisation λ n par validation croisée prédictive et par validation croisée généralisée. Les simulations ont montré de bonnes propriétés du FRRE, même sous un très faible rapport signal-bruit. Compte tenu de sa définition simple, le FRRE est plus rapide à calculer que d'autres estimateurs pour le modèle FCCM trouvés dans la littérature, comme celui proposé par [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF].

La définition de cet estimateur le rend apte à être utilisé dans une étape de la procédure d'estimation dans le modèle de convolution fonctionnel, sujet au coeur du Chapitre 3.

Le chapitre 2 est un article que nous avons soumis à Electronic Journal of Statistics.

Chapitre 3

Dans ce chapitre, nous étudions l'estimateur par déconvolution de Fourier fonctionnelle (FFDE) du coefficient fonctionnel dans le modèle (FCVM). Pour ce faire, nous avons mis au point une nouvelle approche qui utilise la dualité des domaines temporel et fréquentiel à travers la transformée de Fourier continue. Grâce à cette dualité nous associons les modèles FCCM et FCVM et nous pouvons utiliser l'estimateur de régression Ridge fonctionnelle dans le domaine fréquentiel pour définir le FFDE. Cela nous a permis de démontrer la consistance du FFDE pour la norme L 2 et d'obtenir une vitesse de convergence sur l'ensemble des réels. Nous avons également fourni une procédure de sélection du paramètre optimal de régularisation λ n par validation croisée prédictive avec exclusion.

Nous avons défini d'autres estimateurs pour le FCVM, que nous avons adaptés de différentes méthodes trouvées dans la littérature sur le " problème de déconvolution ". Nous avons ainsi comparé les performances du FFDE avec ces estimateurs. Les simulations ont montré la robustesse, la précision et le temps de calcul rapide du FFDE par rapport aux autres. Le calcul du FFDE est rapide car nous utilisons la transformée de Fourier discrète dans la mise en oeuvre numérique. Ceci est une propriété très utile du FFDE.

Ce chapitre est un article bientôt prêt à être soumis.

Chapitre 4

Dans ce chapitre, nous proposons deux estimateurs de l'opérateur de covariance du bruit (Γ ε ) dans la régression linéaire fonctionnelle lorsque la réponse et la covariable sont fonctionnelles, voir le modèle complètement fonctionnel (2). Nous avons étudié les propriétés asymptotiques de ces estimateurs et leur comportement en simulations. Plus particulièrement, nous avons estimé la trace de l'opérateur de covariance du bruit (σ 2 ε = tr(Γ ε )). L'estimation de σ 2 ε rendra possible la construction de tests d'hypothèses dans le cadre du modèle complètement fonctionnel. De plus, σ 2 ε est impliqué dans la majoration de l'erreur quadratique de prédiction, qui sert à déterminer la vitesse de convergence [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF]). Ainsi avoir un estimateur de σ 2 ε renseignera sur la qualité de prédiction dans le modèle complètement fonctionnel.

Ce chapitre est un article que nous avons publié dans Statistics and Probability Letters (Volume 113, June 2016, Pages 7-15).

Chapitre 5

Ce chapitre est une illustration de l'implémentation des résultats présentés au chapitre 3. Nous avons utilisé le modèle FCVM (3) et le modèle linéaire fonctionnel historique pour étudier comment la demande évaporative (VPD) influence la vitesse d'élongation foliaire (LER) de plants de maïs. Les données sont des données réelles obtenues dans des platesformes de phénotypage haut-débit de plantes, lors de deux expériences menées en 2014, T72A et T73A. Pour les deux expériences, le modèle FCVM est trop simple pour apporter de la connaissance sur l'interaction entre VPD et LER. En revanche, le modèle fonctionnel historique est plus utile pour comprendre cette interaction, car c'est un modèle plus riche.

Pour estimer le noyau historique K hist , nous avons proposé deux estimateurs : l'estimateur de Karhunen-Loève restreint et l'estimateur fonctionnel de Tikhonov. Des deux estimateurs, celui de Tikhonov montre des résultats plus cohérents pour les deux expériences. Thanks to new data collection technologies it is possible to increase the number of observations during a time interval to measure how some quantitative variables dynamically evolve for each of many experimentation subjects (longitudinal data, curve data, time series, etc.). This can be found for instance in biological experimentation, finance, physics, etc. Valorize and exploit these masses of data is a current challenge which will be helpful for these scientific fields (see, e.g., [START_REF] Ullah | Applications of functional data analysis: A systematic review[END_REF] and [START_REF] Wang | Functional data analysis[END_REF]).

When analyzing these data some classical methods from multivariate statistical analysis have been shown to be unsuitable (see, e.g., [START_REF] Bickel | Some theory for fisher's linear discriminant function,'naive bayes', and some alternatives when there are many more variables than observations[END_REF], [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF], Hsing and Eubank (2015, p. 1), Ramsay and Silverman (2005, Ch 1), etc). In this sense, some reasons for this inadequacy are for instance: i) the number of observation times p is bigger than the number of realizations n (see, e.g., Hsing and Eubank (2015, p. 2)); ii) the grid of observation times could slightly differ from one realization to another (see, e.g., [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF]); iii) high correlations between close observation times (see, e.g., Ferraty and Vieu (2006, p. 7)) and iv) the fact that the smoothness and derivatives of the random functions play a major role to study the data (see, e.g., [START_REF] Mas | Functional linear regression with derivatives[END_REF].

A more suitable way to deal with these kind of data is to consider them as "realizations from continuous time stochastic processes" (Hsing and Eubank (2015, Ch 1), Bosq (2000, Ch 1)). This leads to the introduction of more appropriate definitions such as functional data (datasets) and random functions. Functional datasets are realization samples of some random functions. A random function is a random phenomena which has functions as realizations. From a mathematical viewpoint random functions are "measurable functions defined on some probability space with values in an infinite dimensional functional space" (Ferraty and Vieu (2006, Ch 1)). Thorough expositions of these concepts can be found in Section 1.1.

In many applications these functions are time-dependent but they could depend on another unidimensional variable, for instance frequency in the case of spectrometric datasets. In any case whenever these functions are univariate they are also called curve data (see, e.g., [START_REF] Gasser | Searching for structure in curve samples[END_REF]). More generally these functions can be multivariate, for instance when they depend on frequency, time, space or other variables [START_REF] Morris | Functional regression[END_REF], Wang et al. (2016, p. 1)). Examples of this more complex kind of datasets are brain and neuroimaging data.

There are many questions that FDA studies, among which functional linear regression is one of the most studied, both in applications and in methodological development (Morris (2015, p. 3)). This is due to the fact that regression models are a good way to study the interrelationship of random functions in diverse fields (see, e.g., [START_REF] Ullah | Applications of functional data analysis: A systematic review[END_REF] and [START_REF] Wang | Functional data analysis[END_REF]).

The objective of this thesis is the study of functional linear regression models when both the regressor X (covariate, input) and the response Y (output) are random functions and time-dependent (i.e. curve data, longitudinal data). In particular we want to address the question of how the history of a random function X (regressor) influences the current value of another random function Y (response) at any given time t. Along with this objective, we want to propose estimation methods which have good properties (consistency, robustness) and which are faster to compute compared to others already proposed in the literature. In order to do this we are mainly interested in two models: the historical functional linear model and the functional convolution model (FCVM) which are introduced in what follows.

The historical functional linear model, introduced by [START_REF] Malfait | The historical functional linear model[END_REF], has the form

Y (t) = t 0 K hist (s,t)X(s) ds + ε(t), (1.1)
where s,t ≥ 0, K hist (s,t) is the history regression coefficient function and ε is some functional random noise with E[ε] = 0. This model is a special case of the fully functional regression model (see, e.g., Horváth and Kokoszka (2012, p. 130) and Ramsay and Silverman (2005, Ch 16)), where X and Y are related through a more general kernel operator. This latter can be written as follows

Y (t) = I K (s,t)X(s) ds + ε(t), (1.2)
where s,t ∈ I ⊆ R and K (•, •) is the integrable kernel. However the interpretation and the estimation are more easily done when the kernel K has the simpler form of K hist because this function is defined over a simpler domain, namely the triangular domain where s < t.

An even simplified way to study the influence of the history of X over the current value of Y is through the functional convolution model (FCVM) which is defined next

Y (t) = t 0 θ (s)X(t -s) ds + ε(t), (1.3)
where t ≥ 0 and θ is the functional coefficient to be estimated. In this model θ is a function which only depends on s and not on the current time t as K hist does. Note that all these functions are considered to be equal to zero for all t < 0, which is interpreted as the fact that zero is the starting point of the measurements. Besides these two models (the historical and the FCVM) we have also studied the functional concurrent model (FCCM) defined in Ramsay and Silverman (2005, Ch 14) as follows

Y (t) = β (t) X(t) + ε(t), (1.4)
where t ∈ R and β is the functional coefficient to be estimated. This is one of the two major kinds of functional regression models which deal with functional responses (the other one is the fully functional regression model defined in equation (1.2), see Wang et al. (2016, p. 272)) and its importance was already remarked in Ramsay and Silverman (2005, p. 220).

Our interest in the FCCM arises from the fact that the FCVM and the FCCM are related in a deeper way through the Continuous Fourier Transform, which associates to each convolution estimation problem of θ in the time domain the estimation of β in the frequency domain. This establishes, under some particular conditions, the equivalence between both models (see Section 1.3). Furthermore the historical functional linear model is also related to the FCCM when for instance the historical kernel can be expressed as the product of two univariate functions in the following way, K hist (s,t) = β (t)γ(s) [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF], [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF]).

In the following pages of this chapter we will introduce the main notations, definitions and the theoretical framework which will be used throughout this thesis. Section 1.1 is about general aspects about FDA. In Section 1.2 we review the literature about functional regression models with functional response. In particular we study the major categories of these models, the historical functional linear regression model and the functional convolution model. Then we focus our attention on the estimation of θ in the FCVM in Section 1.3. The numerical implementation of the functional Fourier deconvolution estimator is addressed in Section 1.4. Finally in the Section 1.5 we discuss the contributions of this thesis in theory and practice.

Functional Data Analysis

Examples of Functional Data Sets

Let us start with three examples of functional datasets. Through these examples we want to give an intuitive idea of what functional data (dataset) refer to. Afterwards we will discuss more theoretical aspects of functional data and random functions. Note that all these examples show a particular kind of functional data referred to as curve data (Wang et al. (2016, p. 258)). We have chosen examples with curve data because they are easier to visualize and because this is the kind of functional data we are concerned with in this thesis.

Mass Spectrometric Data Set : The first example comes from Koomen et al. (2005) and is commented in [START_REF] Morris | Functional regression[END_REF]. The dataset contains mass spectra from a study about pancreatic cancer performed at the University of Texas. To obtain these data blood serum was taken from 139 pancreatic cancer patients and 117 healthy controls. Then with a mass spectrometry instrument it was obtained the proteomic spectra X i (t) for each individual i = 1, • • • , 256. The grid of discrete observations (molecular mass per unit charge m/z) for each curve has size T = 12096. A subset of this dataset is shown in Figure 1.1. Oil data set : The second example comes from Ramsay et al. (2009, Ch 1). This is a case in which two random functions arise as input/output pairs and there is a dependency relationship. This dataset has been collected from an oil refinery in Texas. It is shown in Figure 1.2. It represents two variables measured over time in a grid of 193 points. The first variable is the amount of petroleum product at tray level 47 in a distillation column in an oil refinery and the second is the flow of a vapor into the tray (reflux flow). In this case it is known that the amount of petroleum product reacts to the change in the flow of a vapor into the tray. Some functional linear regression models are useful to characterize this dependency.

Here both the regressor (input, predictor) and the response (output) are functions.

VPD and LER data set : Finally the last example of functional dataset is about highthroughput plant phenotyping data which was obtained in the project PHENOME. In Figure 1.3 we show 13 pairs of Vapor Pressure Deficit (VPD) and the Leaf Elongation Rate (LER) curves. All these curves have been measured 96 times during one day (one observation every 15 minutes). Again here we have two random functions that arise as input/output pairs with a dependency relationship. It is known that the VPD influences the LER. In these three examples, each observed curve is a realization of some underlying random function. Thus a functional dataset is a set of many realizations of some random functions. For instance, let X and Y be two random functions. Then the sample (X i ,Y i ) i=1,••• ,n , where each couple (X i ,Y i ) represents two functions (curves) observed over the interval domain I ⊆ R, is called a functional dataset. In practice these curves have been observed in a finite grid of points. A more general and rigorous theoretical definition of a random function is given in the following subsection 1.1.2.

Random Functions

We start with the definition of random functions. They are a natural extension of real valued random variables, which take values in functional spaces and not in R. Generally speaking Ferraty and Vieu (2006, p. 6) define a random function as a measurable function from a probability space (Ω, A , P) into a infinite dimensional functional space E. It is common to use E with its Borel σ -algebra generated by its open sets. This functional space may be a Hilbert space (see e.g. Horváth and Kokoszka (2012, Ch 2)), a Banach space (see e.g. Ledoux and Talagrand (1991, Ch 2)), a space of mappings (see e.g. Bosq (2000, p. 16)), etc.

In this thesis we are interested in the case where the functions of E have only one variable (univariate), for instance time or frequency. As stated before, this sort of data are referred to as curve data or sometimes longitudinal data. For example this is the case when E := L 2 (I), the set of Lebesgue square integrable functions defined over an interval I ⊂ R. The study of the methods that deal with this kind of data is the main subject of the classical monograph [START_REF] Ramsay | Functional data analysis[END_REF] or of the recent one [START_REF] Hsing | Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators[END_REF].

In what follows we summarize important facts about the definition of the expectation and the covariance operator in Banach spaces and Hilbert spaces (see [START_REF] Hsing | Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators[END_REF] for more details).

Expectation : Let B be a separable Banach space and B * its dual space. A B-valued random function X : (Ω, A , P) → B is said to be weakly integrable if and only if i) the composed function f * (X) is integrable for all f * ∈ B * and ii) there exists an element of B,

denoted E[X], such that for all f * ∈ B * E[ f * (X)] = f * (E[X]).
The element E[X] is referred to as the weak integral of X. Additionally a B-valued random function X will be defined as strongly integrable

(or integrable) if E[∥X∥ B ] < ∞, where ∥ • ∥ B is the norm of B. Whenever X is integrable (strongly) it is common to denote E[X]
with XdP equivalently.

Let us consider the following equivalence relation among B-random functions X and Y , X ∼ Y if and only if X = Y almost surely (a.s.). Using the corresponding equivalence classes we define the space L 1 B (P) of equivalence classes of integrable B-random functions. If we define the norm ∥X∥ L 1

B := E∥X∥ B , L 1 B (P) becomes a Banach space. Analogously for p ∈]1, ∞[, we define the spaces L p B (P) of the classes of B-random functions X such that E[∥X∥ p B ] < ∞.
Similarly to L 1 B (P) this space turns to be a Banach space if we define the norm

∥X∥ L p B := [E∥X∥ p B ] 1/p .
Covariance Operator: For a random function X ∈ L 2 B (P), with E[X] = 0, we define its covariance operator as the following bounded linear operator,

C X : B * → B f * → E[ f * (X) X].
In the case where

E[X] ̸ = 0, we define C X := C X-E[X]
. It is possible to generalize this definition to define the cross-covariance operator between two random functions. In order to do this let us consider X ∈ L 2 B 1 (P) and Y ∈ L 2 B 2 (P), where B 1 and B 2 are separable Banach spaces such that E[X] = 0 and E[Y ] = 0. The cross-covariance operators of X and Y are the following bounded linear operators:

C X,Y : B * 1 → B 2 f * → E[ f * (X)Y ] and C Y,X : B * 2 → B 1 g * → E[g * (X)Y ].
In the particular case of a separable Hilbert Space H, with inner product ⟨•, •⟩, the definition of the expectation is similar to the one defined for Banach spaces. Moreover the definition of the covariance operator will be simpler, because of the Riez representation theorem of the dual space H * . In this way let X be a H-random function such that E[∥X∥ 2H ] < ∞ and E[X] = 0. Then the covariance operator of X is the following

C X : H → H x → E[⟨x, X⟩ X].
It is known that this operator is symmetric, positive and nuclear (see Bosq (2000, p. 34) for more details). Similarly the cross-covariance operator is defined as follows

C X,Y (x) = E[⟨X, x⟩Y ],
for every x ∈ H. This operator is also nuclear (Bosq (2000, p. 34)).

Spectral Decomposition of the Operators : A quite useful fact about the covariance operator C X , when X is a H-random function such that E[∥X∥ 2 H ] < ∞ and E[X] = 0, is that there exists the following spectral decomposition

C X (x) = ∞ ∑ j=1 λ j ⟨x, v j ⟩v j ,
where the set (v j ) j≥1 is an orthonormal basis of H called the eigenfunctions of C X , and (λ j ) j≥1 are the eigenvalues of C X which satisfy

∞ ∑ j=1 |λ j | = E[∥X∥ 2 H ] < ∞, since λ j = ⟨C X (v j ), v j ⟩ = E(⟨X, v j ⟩ 2 ).
This spectral decomposition is used to project the random functions into a finite dimensional subspace generated by the first K eigenfunctions in many well-known methods of Functional Analysis such as the Functional Principal Components Analysis (see, e.g., (Ramsay and Silverman, 2005, Ch 8)) and Karhunen-Loève (see, e.g., [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF]) estimation methods for the functional linear regression model.

Sequence of Random Functions in Hilbert Spaces

In a similar way as for real valued random variables the Strong Law of Large Numbers (SLLN) apply for independent and identically distributed (i.i.d) sequences of H-random functions (see [START_REF] Bosq | Linear Processes in Function Spaces: Theory and Applications[END_REF]).

Theorem

1. Let (X i ) i≥1 be a sequence of i.i.d. H-random functions with expectation E[X] ∈ H. Then ∑ n i=1 X i n a.s. --→ E[X],
when n → ∞.

This theorem is also true for i.i.d. B-random functions with B a separable Banach space. A generalization of the Central Limit Theorem (CLT) for Hilbert spaces was also obtained by Varadhan (1962).

Theorem 2. Let (X i ) i≥1 be a sequence of i.i.d. H-random functions, where H is separable. If E[X] = m ∈ H, E∥X∥ 2 H < ∞ and its covariance operator is C X . Then ∑ n i=1 X i -m √ n d - → N,
when n → ∞, where N ∼ N (0,C X ) is a Gaussian process with mean zero and covariance operator C X .

In general the CLT does not hold in Banach spaces. But under stronger conditions it is possible to obtain it. In particular Ledoux and Talagrand (1991, Ch 9 and 10) show that the geometry of the space (type and cotype structure) is intrinsically linked to the CLT property. The authors give the characterization of the CLT in separable Banach spaces through the small ball criterion (Ledoux and Talagrand (1991, Section 10.3)).

FDA and Multivariate Statistical Analysis

In this subsection we address the question of the inadequacy of some methods of the Multivariate (Statistical) Analysis to deal with functional data. As we mentioned earlier Wang et al. (2016, p. 1) define FDA as "the analysis and theory of data that are in the form of functions, images and shapes, or more general objects", whereas Multivariate Analysis is concerned in understanding and analyzing the interaction of many statistical outcome variables which in general are measured simultaneously (Johnson and Wichern, 2007, p. 1).

Hsing and Eubank (2015, p. 1) and Ramsay and Silverman (2005, Ch 1) among others have shown that some classical methods from Multivariate Analysis (MVA) are unsuitable to deal with functional data. There are four main reasons for this inadequacy:

1. One requirement to apply MVA methods to functional data is that the grid of observations must be fixed and the same for all the realizations. This is not required by FDA and thus it could be applied to more cases, for instance when the observation times are random and independent among the realizations (see e.g. Şentürk andMüller (2010, p. 1257), Yao et al. (2005b, p. 578)).

2. The number of observation times p is in general bigger than the number of realizations n (Hsing and Eubank (2015, p. 2)). The study of statistical inference under this condition is not possible with classical methods from multivariate analysis. Highdimensional statistics (Bühlmann and van de Geer (2011, Ch 1)) and FDA are suitable for this type of data. One of the reasons why p >> n is problematic for multivariate analysis is the fact that it makes the covariance operators to be non-invertible (illconditioning). This in turn makes difficult to solve linear systems in regression models which are widely used in MVA.

3. High correlations of the measured variables when observed in close observation times (see e.g. Yao et al. (2005b), Ferraty and Vieu (2006, p. 7)) is an ill-conditioned problem and then is not suitable for MVA methods because it makes difficult to solve linear systems.

4. Finally in the case where the smoothness and derivatives of the random functions play a major role to study the data (see e.g. [START_REF] Mas | Functional linear regression with derivatives[END_REF], Ramsay and Silverman (2005, Ch 17)), it is necessary to consider the functional nature of the data, and this cannot be accomplished with MVA approach.

Functional Linear Regression Models with Functional

Response

The aim of this section is to present a succinct review of some models that are used to study the dependency relationship between two random functions. Here we consider random functions defined in the Hilbert space L 2 (I), i.e. the space of Lebesgue square integrable functions defined on the interval I ⊆ R. Let X and Y be two random functions and consider X as the regressor (predictor, input, explanatory) and Y as the response (output, dependent). A natural model that relates X and Y is

Y = Ψ(X) + ε,
where Ψ is a functional operator and ε is a noise random variable.

In this model Ψ summarizes the way how X acts upon Y . Thus estimating Ψ is a key element in order to understand this relationship. The estimation question is stated as follows: how to define an estimator from a sample of n i.i.d. realizations of X and Y , (X i ,Y i ) i=1,••• ,n , to estimate Ψ and how is the asymptotic behavior of this estimator (consistency and rate of convergence).

There are two main approaches to deal with this problem. First the functional parametric approach requires Ψ to belong to a particular subset S of the continuous linear operators C on L 2 (I), which can be indexed by a finite number of some fixed continuous linear operators on L 2 (I) (Ferraty and Vieu (2006, p. 8)). An example of this case is the functional linear regression model with functional response defined with equation (1.2), where the continuous linear operator is parametrized solely with one kernel integral operator.

The second approach is the non-parametric functional. In this approach the set S is not required to be indexed by a finite set of continuous linear operators (Ferraty and Vieu (2006, p. 8)). The only constraint is the regularity of the elements of S . Some ways to accomplish this are through the functional additive models studied by [START_REF] Müller | Functional additive models[END_REF], the Reproducing kernel Hilbert spaces (see e.g. [START_REF] Lian | Nonlinear functional models for functional responses in reproducing kernel hilbert spaces[END_REF], [START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF]) and kernel methods [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF]).

Throughout this thesis we are interested in the functional parametric approach, more specifically in the functional linear regression model with Functional Response. In what follows we give a more detailed study of this model.

Two Major Models

The goal of this section is to review the literature about the Functional Linear Regression Models with Functional Response, that is when both the covariate (input) and the response (output) are functional. In a recent article, [START_REF] Wang | Functional data analysis[END_REF] propose to divide this class of models into two major categories. The functional concurrent model (FCCM) and the fully functional regression model (see Horváth and Kokoszka (2012, p. 130)). We briefly discuss both of them.

Functional Concurrent Model :

The first one is referred to as the functional concurrent model and its equation was given in (1.4), let us recall it

Y (t) = β (t) X(t) + ε(t),
where t ∈ R. Here β 0 (t) and β 1 (t) are the functional coefficients of the model to be estimated. Some related models have already been discussed by several authors. For instance [START_REF] West | Dynamic generalized linear models and bayesian forecasting[END_REF] defined a similar model called 'dynamic generalized linear model' and they study the model from a Bayesian point of view. [START_REF] Hastie | Varying-coefficient models[END_REF] proposed the 'varying-coefficients model': In this model Y is supposed to be a random variable whose distribution depends on a parameter η, of the form

η = β 0 (R 0 ) + X 1 β 1 (R 1 ) + • • • X p β p (R p ), (1.5) where X 1 , • • • , X p and R 1 , • • • , R
Y = β 0 (R 0 ) + X 1 β 1 (R 1 ) + • • • X p β p (R p ) + ε,
where E[ε] = 0 and var[ε] = σ 2 . In the case where all the covariates R 1 , • • • , R p are the time variable and when the covariates X 1 , • • • , X p are time-dependent the model can take the form of the functional concurrent model, that is

Y (t) = β 0 (t) + X 1 (t)β 1 (t) + • • • X p (t)β p (t) + ε.
Afterwards many people studied this model trying to estimate the unknown smooth regression functions β i . For instance [START_REF] Wu | Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data[END_REF] proposes a kernel based method when there are k functional covariates. [START_REF] Dreesman | Non-stationary conditional models for spatial data based on varying coefficients[END_REF] and [START_REF] Cai | Efficient estimation and inferences for varying-coefficient models[END_REF] propose a local maximum likelihood type of estimation. [START_REF] Zhang | Variable bandwidth selection in varying-coefficient models[END_REF]; [START_REF] Fan | Adaptive varying-coefficient linear models[END_REF]; and [START_REF] Zhang | Local polynomial fitting in semivarying coefficient model[END_REF] propose local polynomial smoothing methods. [START_REF] Fan | Two-step estimation of functional linear models with applications to longitudinal data[END_REF] propose a two-step approach, first performing pointwise Ordinary Least Squares regressions and then smoothing these raw estimators. [START_REF] Huang | Polynomial spline estimation and inference for varying coefficient models with longitudinal data[END_REF] propose the use of B-splines to represent the functions β i and to use the least square method to perform the estimation. A good review about these methods can be found in [START_REF] Fan | Statistical methods with varying coefficient models[END_REF]. A further development of the multivariate varying-coefficient model is presented in [START_REF] Zhu | Spatially varying coefficient model for neuroimaging data with jump discontinuities[END_REF].

We want to highlight that although the varying coefficient model is related to the FCCM it was originally designed for the case where t is a random variable and was treated with multivariate analysis methods. Thus the first works about estimation did not take into account the functional nature of the data, as noticed by Ramsay and Silverman (2005, p. 259). In contrast, [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF] propose an estimation method which is closer to the functional data approach.

Fully Functional Regression Model : This model has been defined in the equation (1.2) and is recalled here :

Y (t) = I K (s,t)X(s) ds + ε(t).
This model have been first studied by [START_REF] Ramsay | Some tools for functional data analysis[END_REF]. Estimation of the kernel K is an inverse problem and requires some sort of regularization to achieve inversion. Ramsay and Dalzell (1991, p. 552) proposed a penalized least square method to do this. [START_REF] James | Generalized linear models with functional predictors[END_REF] proposed penalized polynomial splines. [START_REF] Cuevas | Linear functional regression: the case of fixed design and functional response[END_REF] studied this regularization when the design is fixed. [START_REF] He | Extending correlation and regression from multivariate to functional data[END_REF] defined the 'Functional Normal Equation' which generalizes the multivariate normal equations and they proved that there exists a unique solution under certain conditions. They used the Karhunen-Loève decomposition to estimate K . [START_REF] Chiou | Functional response models[END_REF] gave a good summary of this approach. This idea has been revisited in Yao et al. (2005a) where they dealt with sparse longitudinal data and when the observation times are irregular and random. Additionally [START_REF] Aguilera | Estimation of functional regression models for functional responses by wavelet approximation[END_REF] proposed a wavelet based estimator. [START_REF] Antoch | Electricity consumption prediction with functional linear regression using spline estimators[END_REF] studied spline estimator. Finally [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF] have studied a Karhunen-Loève type estimator.

Whenever Y and X are time-dependent, in the equation (1.2) future values of X are used to explain past values of Y . This fact is counterintuitive and should not be used. For this reason [START_REF] Malfait | The historical functional linear model[END_REF] are concerned with the particular case where Y (t), the value of Y at time t, depends on the history {X(s) : 0 ≤ s ≤ t}. This leads to the historical functional linear model defined in equation (1.1). This particular type of model will be discussed in the following subsection.

Historical Functional Linear Regression Model

As mentioned earlier the historical functional linear model has been proposed in [START_REF] Malfait | The historical functional linear model[END_REF]. It was defined in equation (1.1), namely

Y (t) = t 0 K hist (s,t)X(s) ds + ε(t),
for all t ∈ [0, T ] and T ∈ R. Now we want to summarize the approach of [START_REF] Malfait | The historical functional linear model[END_REF] to estimate K hist . The authors propose to use a finite element basis φ k (s,t). This basis is made of piecewise linear functions defined over a fixed and suitable grid of points. Then the authors use the approximation K hist (s,t) ≈ ∑ K k=1 b k φ k (s,t) to transform the problem as follows

Y i (t) = K ∑ k=1 b k ψ ik (s,t) + ε i (t),
where ψ ik (s,t) := t 0 X i (s) φ k (s,t) ds. Let y(t) and e(t) be the vectors of length N containing the values Y i (t) and ε i (t), respectively. Let also Ψ(t) be the N × K matrix containing values of ψ ik (t) and let the coefficient

vector b be (b 1 , • • • , b K ) ′ .
Then we have the matrix expression of (1.1) for each t ∈ [0, T ],

y(t) = Ψ(t) b + e(t).

This leads to the normal equations

T 0 Ψ(t) ′ Ψ(t)dt b = T 0 Φ(t) ′ y(t)dt.
Finally they approach and solve this equation with a multivariate linear model with penalization.

Following a similar approach, [START_REF] Harezlak | Penalized solutions to functional regression problems[END_REF] considered the same representation of the function K hist (s,t) through the basis functions φ k (s,t). But in this case the authors explored two different regularization techniques which use basis truncation, roughness penalties, and sparsity penalties. The first one penalizes the absolute values of the basis function coefficient differences (LASSO approach) and the second one penalizes the squares of these differences (penalized spline methodology). Finally they assessed the estimation quality with an extension of the Akaike Information Criterion. [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF] are interested in a particular type of the historical functional linear model. They consider the curves X i and Y i as being sparse longitudinal data, and the function K hist (s,t) to be decomposable as follows,

K hist (s,t) = ∑ K k=1 d k (t)φ k (s)
, where d k (t) are the functional coefficients to be estimated and φ k (s) are predetermined basis functions, for instance a B-spline basis.

In this case the model (1.1) becomes a varying-coefficients model or functional concurrent model (FCCM) after integration. The estimation procedure uses the functional principal components decompositions for both Y and X. These decompositions are needed to solve the normal equation which uses the auto-covariances of X(s) and Y (t) and the cross-covariance of (X(s),Y (t)). [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF] are interested in a similar model. The authors consider the following modification of (1.1)

Y (t) = β 0 (t) + β 1 (t) ∆ 0 γ(s)X(s) ds + ε(t),
for t ∈ [∆, T ] with ∆ > 0 and a suitable T > 0. Similarly to [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF] this particular case is strongly related to the FCCM. It is clear that after estimating the function γ, the problem becomes a FCCM. [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF] develop a new method to estimate the functions β 0 and β 1 , taking into account the functional nature of the data. However the essential idea to estimate the unknown functions here (γ, β 0 and β 1 ) is to use an equivalent idea to the 'functional normal equation', that is to relate the auto-covariance of X with the cross-covariance of X and Y , and then to invert the auto-covariance of X to obtain the unknown function.

Finally it is possible to consider a more specific structure for the function K hist (s,t) which does not depend on the current value t but only on s and the integral takes the form of a convolution. In order to do this first we suppose there is a function θ such that K hist (s,t) = θ (ts) and then we apply a change of variables to obtain the functional convolution model (1.3). The study of this model is the goal of the following subsection.

Functional Convolution Model (FCVM)

One of the main goals of this thesis was to study the influence of the history of the functional covariate X on the current value of the functional response Y (t). One way to model this dependency relationship is through the FCVM. In this subsection we present the origin of this model and in the following section we discuss many estimation methods of the functional coefficient θ and the place of the FCVM among other models in the literature.

Let us recall the equation (1.3) which defines the FCVM, that is

Y (t) = t 0 θ (s) X(t -s)ds + ε(t),
for all t ≥ 0. This model was derived from [START_REF] Malfait | The historical functional linear model[END_REF]. We are interested in the estimation of θ for a given i.i.d. sample

(X i ,Y i ) i∈{1,••• ,n} of the random functions X and Y .
The FCVM is a functional extension of distributed lag models in time series (Greene (2003, Ch 19)). One of these models is the dynamic regression model, which has the following general form

y t = α + ∞ ∑ i=0 β i x t-i + ε t .
for all t ≥ 0. In this model if we suppose that x i = 0 for all i < 0 (i.e. there is a starting point) then the sum becomes finite and this is clearly the discretization of the FCVM. Applications of this model are commented in Greene (2003, Ch 19).

The question of the estimation of θ is central in our study of the FCVM. We address this question in the next section. Besides we comment on a deeper connection between the FCVM and the FCCM obtained through the use of the Continuous Fourier Transform.

Estimation of θ in the FCVM

There are four main characteristics of the FCVM : i) the covariate (input) and the response (output) are random functions, ii) the convolution is non-periodic (i.e. we do not consider periodic functions), iii) the sample size is n > 1, moreover we are interested in the asymptotic behavior (n → ∞) and finally iv) the noise is functional. To the best of our knowledge there are few papers studying this model satisfying these four characteristics. However there are many models which are close to FCVM. In what follows we explore some of them. [START_REF] Asencio | Functional convolution models[END_REF] study a related problem, in which they consider more covariate functions (predictors). The estimation of θ is done by projecting the functions into finitedimensional spline basis and using a penalized ordinary least square approach to estimate the coefficients in this basis. Another approach is to use the Historical Functional Linear Model [START_REF] Malfait | The historical functional linear model[END_REF]) to estimate θ , by taking into account the special shape that the kernel function K hist shall have in this particular case. In the same way we can use the approach of [START_REF] Harezlak | Penalized solutions to functional regression problems[END_REF], or even in a more restricted case the approaches of [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF] and [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF]. Note that the FCVM could be seen as a particular case of the model proposed by [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF], namely when K = 1, and Şentürk andMüller (2010, p. 1259) proposed a method to estimate θ when the FCVM has the following more restricted form

φ K = θ in K hist (s,t) = ∑ K k=1 b k (t)φ k (s).
Y (t) = ∆ 0 θ (s) X(t -s)ds + ε(t),
where ∆ > 0 is a fixed value. The estimation of θ in this case is made by using the Karhunen-Loève decomposition of the covariance operator of the random function Z t (s) := X(ts), where t is fixed and s ∈ [0, ∆]. They express θ in the basis of eigenfunctions of this operator and then they estimate the coefficients with an ordinary least squares procedure. This produces one estimator for each time t. They consider a grid of observation times and then they take the mean of all these estimators. This approach is similar to that of [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF].

To the best of our knowledge only these papers have addressed the study of the estimation of θ under the four characteristics of the FCVM mentioned earlier. The approach we propose in Chapter 3 is a new way to answer this question. We do not use projection into finite dimensional basis nor kernel estimation with finite elements methods. Moreover we study the asymptotic properties of the estimator, which has not been done either in the approaches previously mentioned.

Functional Fourier Deconvolution Estimator (FFDE)

We propose the Functional Fourier Deconvolution Estimator (FFDE) which is defined in three steps. i) First we use the Continuous Fourier Transform (F ) to transform the convolution in the time domain into a multiplication in the frequency domain (see (3.2)). ii) Once in the frequency domain, we estimate β with the Functional Ridge Regression Estimator (FRRE) defined in [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] (see Ch 2), which is an extension of the Ridge Regularization method [START_REF] Hoerl | Application of ridge analysis to regression problems[END_REF]) that deals with ill-posed problems in the classical linear regression. iii) The last step consists in using the Inverse Continuous Fourier Transform to estimate θ . This definition is formalized mathematically as follows.

Let (X i ,Y i ) i=1,••• ,n be an i.i.d sample following the FCVM (1.3).

Step i) We use the Continuous Fourier Transform (F ) defined as follows

F ( f )(ξ ) = +∞ t=-∞ f (t)e -2πitξ dt,
where ξ ∈ R and f ∈ L 2 . This operator is used to transform the FCVM (1.3) which is defined in the time domain into its equivalent in the frequency domain. Thus equation (1.3) becomes

Y (ξ ) = β (ξ ) X (ξ ) + ε(ξ ), (1.6)
where ξ ∈ R, β := F (θ ) is the functional coefficient to be estimated. X := F (X) and Y := F (Y ) are Fourier transforms of X and Y . Lastly ε := F (ε) is an additive functional noise.

The equivalent problem (eq. (1.6)) in the frequency domain is a particular case of the FCCM (eq. (1.4)). Clearly the estimation of β implies the estimation of θ through F -1 .

Step ii) The functional Ridge regression estimator (FRRE) of β in the FCCM (1.4) or in (1.6) is defined as follows

βn := 1 n ∑ n i=1 Y i X * i 1 n ∑ n i=1 |X i | 2 + λ n n , (1.7)
where the exponent * stands for the complex conjugate and λ n is a positive regularization parameter.

We have defined the functional Ridge regression estimator of β , see (1.7), because with this estimator it is natural to use the inverse Fourier transform (F -1 ) to estimate θ and to prove the consistency property under the L 2 -norm. Besides this we wanted to use the computation efficiency of the Fast Fourier Transform Algorithm.

As we saw before the idea of transforming the historical functional linear model into a FCCM was already proposed by [START_REF] Kim | Recent history functional linear models for sparse longitudinal data[END_REF] and in a different way by [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF]. In both papers the authors used special structures for the kernel function K hist . These structures allow them to transform the historical model into the FCCM. In our case we use a different approach. We do not impose a particular structure to the kernel function, but we transform the whole model FCVM in the time domain into its equivalent in the frequency domain. As a consequence, it opens the possibility to also use other estimation methods of β in the FCCM (see Subsection 1.2.1) in order to estimate θ in the FCVM.

Step iii) The FFDE of θ in (1.3) is defined by θn := F -1 ( βn ).

(1.8)

Note that the estimator θn (FFDE) is real valued and belongs to L 2 (R, R) (see Chapter 3). Another important property is that the FFDE can be decomposed as follows

θn = θ - λ n n F -1 F (θ ) 1 n ∑ n i=1 |F (X i )| 2 + λ n n + F -1 1 n ∑ n j=1 F (ε j )F (X j ) 1 n ∑ n i=1 |F (X i )| 2 + λ n n
.

(1.9)

The study of this decomposition will allow us to prove the consistency of this estimator. Note the importance of the equivalence between the FCVM and the FCCM, because of the use of two equivalent representations of the same information (time domain and frequency domain) obtained thanks to the Continuous Fourier Transform.

The functional Fourier deconvolution estimator (FFDE) of θ in the FCVM is further studied in Chapter 3. The aim of proposing such an estimator was to take advantage of the equivalence between the time and frequency domains as well as of the mathematical properties of the Continuous Fourier Transform. The advantages of this estimator are both theoretical and practical: theoretical because we develop an approach which uses primarily the fact that we work with random functions and functional spaces, and practical because for implementing this method we use the Fast Fourier Transform (FFT) algorithm which increases the computation speed of the estimators in a significant way over other possible estimators. We describe in the following subsection other possible estimators adapted from the literature.

Deconvolution Methods in the Literature

Next let us consider other models which are indirectly related to the FCVM. From this we will be able to adapt some techniques to estimate θ in the FCVM.

We start with the multichannel deconvolution problem (see e.g. De Canditiis and Pensky (2006), [START_REF] Pensky | On convergence rates equivalency and sampling strategies in functional deconvolution models[END_REF] and [START_REF] Kulik | Multichannel deconvolution with long range dependence: Upper bounds on the lp-risk[END_REF]). This problem belongs to Signal Processing methods. Similarly to the FCVM, here the input and output are functionals (i.e. signals, curve data), there are many realizations (n > 1, multichannel) and the noise is functional. But the difference with the FCVM is that they study the periodic case (the signals are periodical and so is the convolution). Besides the authors do not deal with the asymptotic behavior of the estimator.

The multichannel deconvolution problem is one way to generalize the deconvolution problem in Signal Processing (see e.g [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF], [START_REF] Brown | Introduction to Random Signals and Applied Kalman Filtering with MATLAB Exercises[END_REF], [START_REF] Gonzalez | Digital Image Processing Using MATLAB[END_REF]). In this problem they use the convolution (periodic or not) to model how an impulse response function h transforms an original signal g (unknown)

through the following equation

f (t) = D h(s)g(t -s)ds + ε(t),
where D is the domain of integration ([0, T ] in the periodic case for a fixed period T , and [0,t] or R in the non-periodic one), f is the observed signal and ε the noise. There are several methods to estimate g given the functions h and f , for instance the Parametric Wiener Deconvolution (Gonzalez and Eddins (2009, Ch 5)).

It is clear that if we take one couple (X i ,Y i ) and we interpret f as Y , h as X and g as θ we can apply these methods to estimate θ (apply the deconvolution to Y to obtain θ in (1.3) ). At this point we notice that although this problem is related to the FCVM it only deals with the case n = 1, and so there is no study of the asymptotic behavior of the estimator.

In a similar way the Deconvolution Problem in Non-parametric statistics (see e.g. [START_REF] Meister | Deconvolution Problems in Nonparametric Statistics[END_REF], [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF]) deals with the case n = 1 and does not consider a functional noise. The goal here is to estimate the probability density function (pdf) of a real random variable X from the observation of another real random variable Y such that Y = X + Z, the pdf of Z being known. To solve this problem they use the fact that the pdf of the sum of two random variables is the convolution of their respective pdf. It might be possible to adapt these techniques to estimate θ in the FCVM, but we think that the estimation would be worse than the one with signal processing methods, because in the former case the functional noise is not considered. Also, through a numerical approximation of the convolution as a matrix operator, the FCVM becomes a Linear Inverse Problem for each couple (X i ,Y i ). In this case for each i ∈ {1, • • • , n}, we can estimate θ with some of the techniques to solve the linear inverse problem, such as the Tikhonov regularization, the singular value decomposition method, or wavelet based methods (see, e.g., [START_REF] Tikhonov | Solutions of ill-posed problems[END_REF], O'Sullivan (1986), [START_REF] Donoho | Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition[END_REF], [START_REF] Abramovich | Wavelet decomposition approaches to statistical inverse problems[END_REF]). Note again that these methods only deal with the case n = 1. They do not study the asymptotic case.

Finally another related method is the Laplace deconvolution introduced by [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging[END_REF]. This method also deals with the case n = 1. But the authors consider both the non-periodic convolution, as in the FCVM, and a functional noise.

In Chapter 3 we have adapted the parametric Wiener deconvolution, the singular value decomposition method, the Tikhonov regularization and the Laplace deconvolution to estimate θ in the FCVM.

The Section 1.4 deals with the numerical implementation of the FFDE.

Numerical Implementation of the Functional Fourier Deconvolution Estimator

In this section we discuss how we estimate θ in the FCVM in practice. In particular we describe the necessity to rethink the FCVM in a finite discrete way, and to use the Discrete Fourier Transform as the discrete equivalent of the Continuous Fourier Transform in this new context. We start by describing the discretization of the convolution. To do this properly we start with some definitions.

Throughout this section we use ∆ as the discretization step between two observation times (for instance ∆ = 0.01). The observation times are defined for every j ∈ Z as t j := j * ∆ and thus they define the grid G ∆ over R. We use a fix grid in this section. With this grid we transform each function f : R → C to a vector f d ∈ C Z infinite dimensional, with elements

f d j := f (t j ) ∈ C.
In what follows the superscript d will denote this discretization.

In this section all the functions will have compact support. Otherwise we should compute convolution of infinite vectors which cannot be done in practice. For simplicity we consider all the functions defined over a compact interval [0, T ] with T large enough. Thus we will consider

f d = f d 0 , • • • , f d q-1 ∈ C q , where q -1 = max{ j ∈ N |t j ∈ [0, T ]}.
Let RM (rectangular method) be the operator which associates to an integral over R, its numerical approximation by the rectangular method over the grid of points we have already defined. So for a given integral

J = R f (s)ds = T 0 f (s)ds, RM(J) := ∆ ∑ q-1 j=0 f (t j ) = ∆ ∑ q-1 j=0 f d j .
Understanding how to compute numerically the convolution of two functions is a key element to implement the estimator developed for the FCVM.

We start our discussion by describing the discretization of the convolution of two functions with support included on [0, T ],

f * g(t) := +∞ -∞ f (s)g(t -s)ds = T 0 f (s)g(t -s)ds.
Approximating this convolution with the Rectangular Method we obtain for every j ∈ N,

RM( f * g)(t j ) = q-1 ∑ l=0 f (t l ) g(t j-l ) ∆ = ∆ q-1 ∑ l=0 f d l g d j-l .
(1.10)

The last sum in equation (3.21) is the convolution between vectors. Thus we can rewrite this equation as follows

RM( f * g)(t j ) = ∆ ( f d * g d ) j . for j ∈ [0, • • • , 2p -2] and where ( f d * g d ) j := ∑ q-1 l=0 f d l g d j-l . Besides note that for j / ∈ [0, • • • , 2p -2]
we have RM( f * g)(t j ) = 0 since f and g have compact support.

Additionally we can compute the vector ((

f d * g d ) 0 , • • • , ( f d * g d ) 2q-2 ) using matrices as follows ( f d * g d )(0), • • • , ( f d * g d )(2q -2) T = MC G ( f d 0 , • • • , f d q-1 ) T , (1.11)
where MC G is the matrix associated to the convolution discretized over the grid G, defined as follows

MC G :=                         g d 0 0 0 0 • • • 0 g d 1 g d 0 0 0 • • • 0 g d 2 g d 1 g d 0 0 • • • 0 . . . . . . . . . . . . • • • . . . g d q-2 • • • • • • g d 1 g d 0 0 g d q-1 g d q-2 • • • • • • • • • g d 0 0 g d q-1 g d q-2 • • • g d 2 g d 1 0 0 g d q-1 • • • • • • g d 2 . . . . . . . . . . . . • • • . . . 0 0 • • • 0 g d q-1 g d q-2 0 0 0 • • • 0 g d q-1                         ∈ R (2q-1)×q .
Remark : From this we note that the convolution could have a larger support. This arises because an important property of the convolution is that supp( f * g) ⊂ supp( f ) + supp(g) (Brezis (2010, p. 106)). Thus in our case supp( f * g) ⊂ [0, 2 T ]. However afterwards we will take T large enough to contain even the convolution. In this way, every time we will consider the convolution of two functions f and g we suppose supp( f ) + supp(g) ⊂ [0, T ]. In this case the number of discretization points q will be defined as before, namely q -1 = max{ j ∈ N |t j ∈ [0, T ]} but now for all j ≥ q, ( f d * g d ) j = 0. Besides the matrix representation of the convolution through MC G will still be correct.

In the following subsection we explore the parallel between the continuous convolution of two functions and the convolution of two vectors with respect to the whole model FCVM.

The Discretization of the FCVM and the FFDE

We have defined the functional Fourier deconvolution estimator of θ in the FCVM using the continuous Fourier transform and its inverse (equations (1.7) and (1.8)). Given that both operators are integral operators, we need to use some kind of numerical approach to compute them. The goal of this subsection is to show that the proper way for doing this is by using a discrete model which behaves like the FCVM. This model will be based on the convolution of finite dimensional vectors. It will be studied through the discrete Fourier transform and its inverse instead of their continuous counterparts.

First let us show that it is not practical to compute the Functional Fourier Deconvolution estimator by direct approximation of the Continuous Fourier Transform and its inverse. This is not possible because these two operators are integrals defined over the whole R. To see why this is a problem let us consider a function f ∈ L 2 with compact support. Then although it is possible to use the Rectangular Method to compute F ( f )(ξ ) for every value ξ , we cannot ensure that F ( f ) has compact support ( (Kammler, 2008, p. 130)). This implies that we need to know the values of F ( f ) for all the infinite values of the grid G ∆ to approximate the F -1 , which is impossible in practice. Note that even if F ( f ) has a compact support we cannot know how large it is and in this case we will need to compute F ( f ) over too many points of the grid which again makes the approximation unpractical.

Instead of using the direct approximation of the Continuous Fourier Transform and its inverse, another approach is to propose a finite discretized version of the FCVM, which reflects the main characteristics of the FCVM. In order to achieve this, note two important things: i) the convolution of two functions can be approached by as the convolution of two vectors and ii) the convolution of two vectors is transformed into a multiplication with the Discrete Fourier Transform ( (Kammler, 2008, p. 102), Oppenheim and Schafer (2011, p. 60)).

Here we use the definition of the Discrete Fourier Transform found in Kammler (2008, p. 291) or in Bloomfield (2004, p. 41), defined for vectors of C q as follows

F d : C q → C q f := ( f 0 , • • • , f q-1 ) → (F d ( f )(0), • • • , F d ( f )(q -1)) ,
where for every l = 0, • • • , q -1,

F d ( f )(l) := 1 q q-1 ∑ r=0 f r ω rl ∈ C.
(1.12)

with ω := e -2πi/q . If we define the matrix

Ω q :=         1 1 1 • • • 1 1 (ω 1 ) 1 (ω 1 ) 2 • • • (ω 1 ) (q-1) 1 (ω 2 ) 1 (ω 2 ) 2 • • • (ω 2 ) (q-1) . . . . . . . . . . . . . . . 1 (ω (q-1) ) 1 (ω (q-1) ) 2 • • • (ω (q-1) ) (q-1)         (1.13)
we can write

F d ( f ) = 1 q Ω k f ∈ C q . (1.14)
Furthermore from this definition we can deduce

F -1 d = Ω * q , (1.15)
where Ω * q is the conjugate transpose of Ω q . Remark: We can see that the definition of F d depends on the number q, which is the length of the vector. In this way when we apply F d to a vector of size p we need to redefine the matrix Ω p by using ω := e -2πi/p . Finite Discrete version of the FCVM Let us take T large enough such that [0, T ] contains supp(X) + supp(θ ). Thus the supports of θ , X and Y are also contained in [0, T ] (Brezis (2010, p. 106)). Let us define q -1 = max{ j ∈ N |t j ∈ [0, T ]}. Now take the discretization of the each function X i and Y i of the sample (X i ,Y i ) i=1,••• ,n over the grid [t 0 , • • • ,t q-1 ], so all these functions will become vectors in R q ⊂ C q , that is X d i ,Y d i ∈ C q for every i = 1, • • • , n. Given that the matrix Ω q has the property of transforming finite convolutions into multiplications, we can use the three steps method as the one used to define the estimator θn for the continuous case, namely i) transform the problem with the matrix Ω q from the time-domain to the frequency one, ii) use the ridge estimator in this domain, and iii) finally come back with the inverse of Ω q .

The comparison between the continuous and the discrete cases is done next. Note that in the discrete case the multiplication and the division is done the element by element between vectors of same length. Furthermore, * d is discrete convolution, ∆ is the step of discretization and we use P q : R 2q-1 → R q , the projection into the first q components, to have vectors of the same length.

CONTINUOUS

Data and conditions: θ ∈ L 2 ([0, T ]).

For i = 1, • • • , n, X i ,Y i , ε i ∈ L 2 ([0, T ]), Y i = θ * X i + ε i .
Estimation steps:

1. For i = 1, • • • , n, F (Y i ) = F (θ )F (X i ) + F (ε i ). 2. F (θ ) n := ∑ n i=1 F (Y i )F (X i ) ∑ n i=1 |F (X i )| 2 + λ n 3. θn := F -1 ( F (θ ) n )

DISCRETE

Data and conditions:

θ d ∈ R q . For i = 1, • • • , n, X d i ,Y d i , ε d i ∈ R q , Y d i = ∆ P q (θ d * d X d i ) + ε d i .
Estimation steps:

1. For i = 1, • • • , n, Ω q (Y d i ) = ∆Ω d q (θ d )•Ω q (X d i )+Ω q (ε d i ). 2. Ωq (θ d ) n := 1 ∆ ∑ n i=1 Ω q (Y d i )Ω q (X d i ) ∑ n i=1 |Ω q (X d i )| 2 + ⃗ λ n , where ⃗ λ n := (λ n , • • • , λ n ) ∈ R q . 3. θ d n := Ω -1 q ( Ωq (θ d ) n ).
From this comparison we can define the numerical estimator of θ over the grid [t 0 , • • • ,t q-1 ] as follows

θ d n := 1 ∆ Ω -1 q ∑ n i=1 Ω q Y d i • Ω q X d i ∑ n i=1 |Ω q X d i | 2 + ⃗ λ n
.

(1.16)

Compact Supports and Grid of Observations

From now on we will compute θn numerically with equation (3.27). The important question we want to address here is how large the grid of observation points should be to properly estimate θ ? In this regard understanding the relationship between the supports of X and θ and the one of their convolution (Y ) is an essential element to answer this question. We know that (Brezis (2010, p. 106)),

supp(Y ) = supp(θ * X) ⊂ supp(X) + supp(θ ).
Then as mentioned before whenever our grid of observations contains the interval [0, T ] and [0, T ] contains supp(X) + supp(θ ) we will be able to estimate θ over its whole compact support.

The problem arises from the fact that we do not know θ and as a consequence neither supp(θ ) nor supp(X) + supp(θ ). Then how big T should be in order to estimate θ correctly?

There are several cases to consider. First let us suppose that the grid of observations covers [0, T 1 ] and supp(X), supp(Y ) ⊂ [0, T 1 ] then we can choose T > T 1 big enough and estimate θ over [0, T ]. To see this more clearly let us say that the grid of observations over [0, T 1 ] is t 0 , • • • ,t q 1 and over [0, T ] is t 0 , • • • ,t q , with q > q 1 . Given that we have only observed the curves over [0, T 1 ] we only know the vectors

(X d i ,Y d i ) i=1,••• ,n ⊂ R q 1 .
Then the only thing we need to do before applying equation (3.27) properly is to redefine the vectors X d i and Y d i by adding zeros such that they will belong to R q , for instance

X d i := (X d i , 0, • • • , 0) ∈ R q .
This procedure is known as zero padding the signal (Gonzalez and Eddins (2009, p. 111)).

In this case equation (3.27) is well defined and we will compute θ over [0, T ]. Note also that supp(θ ) could be bigger than [0, T ] but the estimation of θ over [0, T ] is still correct.

Secondly we have the case where the grid of observations covers [0, T 1 ] and we know supp(X) ⊂ [0, T 1 ] and supp(Y ) \ [0, T 1 ] ̸ = / 0. Under these hypotheses we cannot add more zeros to the vectors Y d i because if we did it would imply that Y has zero values outside [0, T 1 ] which contradicts supp(Y ) \ [0, T 1 ] ̸ = / 0. Thus we cannot apply the property of Ω q to transform the convolution into a multiplication correctly. This is one restriction to the correct application of the FCVM.

Finally if the grid of observations covers [0,

T 1 ], supp(X) \ [0, T 1 ] ̸ = / 0 and supp(Y ) \ [0, T 1 ] ̸ = /
0 we have the same phenomenon, that is we cannot add more zeros to the vectors X d i and Y d i to belong to R q . Thus it is not possible to transform the convolution into a multiplication because q 1 is not big enough. Note that Ω q 1 is quite different from Ω q (see definition 3.24) and the property of transforming the convolution into a multiplication of two vectors only holds when Ω q is applied to the entire convolution of both vectors, that is q is big enough to contain the convolution.

In any case in order to estimate θ with the functional Fourier deconvolution estimator, the grid of observations should cover supp(X) and supp(Y ). This is an important restriction of this estimator.

FFT Algorithm and fast computing : One of the main advantages of the Functional Fourier Deconvolution estimator is that it is calculated very fast. This is due to the fact that it uses the Fast Fourier Transform to compute the Discrete Fourier Transform. It is known that this algorithm computes the Discrete Fourier Transform of an n-dimensional signal in O(n log(n)) time. The publication of the Cooley-Tukey Fast Fourier transform (FFT) algorithm in 1965 [START_REF] Cooley | An algorithm for the machine calculation of complex fourier series[END_REF]) revolutionized the area of digital signal processing because it reduced the order of complexity of the Fourier transform and of the convolution from n 2 to n log(n), where n is the problem size. Then over the last years new algorithms have improved the performance of the Cooley-Tukey algorithm under some conditions (split-radix FFT, Winograd FFT, etc). Among the recent improvements we highlight the Nearly Optimal Sparse Fourier Transform [START_REF] Hassanieh | Nearly optimal sparse fourier transform[END_REF]).

Contribution of this thesis

In this thesis, we want to know how the history of the functional regressor X influences the current value of the functional response Y in functional linear regression models.

This thesis is divided in 6 chapters. We present in Chapter 1 a general introduction of the theoretical background used in the following chapters. The theoretical and practical contributions of this thesis are from Chapter 2 to Chapter 4. In these chapters we studied the functional concurrent model (Chapter 2), the functional convolution model (Chapter 3) and the fully functional model (Chapter 4). An illustration on real datasets is given in Chapter 5. Finally we present in Chapter 6 the conclusions and perspectives of this thesis.

A more detailed review of the contributions is given below.

Chapter 2

In this chapter we propose a functional approach to estimate the unknown function in the Functional Concurrent Model (FCCM). This method is a generalization of the classic Ridge regression method to the functional data framework. For this reason we named this new estimator the Functional Ridge Regression Estimator (FRRE).

We proved the consistency of the FREE for the L 2 -norm, and obtained a rate of convergence over the whole real line, and not only on compact sets. We also provided a selection procedure of the optimal regularization parameter λ n through the Leave-One-Out Predictive Cross-Validation and the General Cross-Validation. The whole estimation procedure has been experienced on simulation trials, which showed good properties of the FRRE under very low Signal-to-Noise ratio. Thanks to its simpler definition, the FRRE is faster to compute than other estimators in the FCCM, such as the one proposed in [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF].

Finally the definition of the FRRE is suitable to be used as a step of the estimation procedure in the Functional Convolution Model, which is the focus of the Chapter 3.

This chapter is an article we have submitted to the Electronic Journal of Statistics.

Chapter 3

In this chapter we propose the Functional Fourier Deconvolution Estimator (FFDE) of the functional coefficient in the Functional Convolution Model (FCVM). To do this we implemented a new approach which uses the duality between the time domain and frequency domain spaces through the continuous Fourier transform.

Thanks to this duality we associate the FCCM to the FCVM and we can use the Functional Ridge Regression Estimator in the frequency domain to define the FFDE. This fact allowed us to prove the consistency of the FFDE for the L 2 -norm, and obtained a rate of convergence over the whole real line. We also provided a selection procedure of the optimal regularization parameter λ n through the Leave-One-Out Predictive Cross-Validation.

We have defined other estimators for the FCVM, which we adapted from different methods found in the literature about the "deconvolution problem". Then we compared the performance of the FFDE with these alternative estimators. The simulations have shown the robustness, the accuracy and the fast computation time of the FFDE compared to the others. The reason why the FFDE is calculated very fast is that we use the Discrete Fourier Transform for its numerical implementation. This is a very useful property of the FFDE.

This chapter is an article will be submitted to the Electronic Journal of Statistics.

Chapter 4

In this chapter we have proposed two estimators of the covariance operator of the noise (Γ ε ) in functional linear regression when both the response and the covariate are functional (see the fully functional model (1.2)). We studied the asymptotic properties of these estimators and their behavior on simulations. More particularly we have estimated the trace of the covariance operator of the noise (σ 2 ε = tr(Γ ε )). The estimation of σ 2 ε would make possible the construction of hypothesis testing in connection with fully functional model. Furthermore σ 2 ε is involved in the square prediction error bound that participates to determine the convergence rate [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF]). Thus an estimator of σ 2 ε will provide details on the prediction quality in the fully functional model.

This chapter is an article published in Statistics and Probability Letters (Volume 113, June 2016, Pages 7-15)

Chapter 5

This chapter is an illustration of the implementation of the results presented in Chapter 3. We have used the FCVM (1.3) and the historical functional linear model (1.1) to study how the Vapour Pressure Deficit (VPD) influences Leaf Elongation Rate (LER) curves obtained on high-throughput plant phenotyping platforms, from experiments carried out in 2014, named here as T72A and T73A.

In both experiments the FCVM is too simple to bring light about the interaction of the VPD and LER. On contrast the historical functional model is more helpful to understand this interaction because it is more complex.

To estimate the historical kernel K hist we have proposed two estimators: first the Karhunen-Loève estimator satisfying the historical restriction and secondly a Tikhonovtype functional estimator. Among these two estimators the latter shows more consistent results across both experiments.

Introduction

Functional Data Analysis (FDA) proposes very good tools to handle data that are functions of some covariate (e.g. time, when dealing with longitudinal data), see [START_REF] Hsing | Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators[END_REF] or [START_REF] Horváth | Inference for Functional Data with Applications[END_REF]. These tools allow for better modelling of complex relationships than classical multivariate data analysis, as noticed by Ramsay and Silverman (2005, Ch. 1), Yao et al. (2005a,b), among others.

There are several models in FDA for studying the relationship between two variables. In particular in this paper we are interested in the Functional Concurrent Model (FCM) which is defined as follows

Y (t) = β (t) X(t) + ε(t), (2.1)
where t ∈ R, β is the unknown function to be estimated, X,Y are random functions and ε is a noise random function. As stated by Ramsay and Silverman (2005, p. 220), all functional linear models can be reduced to this form. This model is also related to the functional varying coefficient model (VCM) and has been studied for example by [START_REF] Wu | Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data[END_REF] or more recently by [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF].

Another practical advantage of model (3.2) is that it allows to simplify the study of the following convolution model

W (s) = +∞ -∞ θ (u) Z(s -u)du + η(s) (2.2) through the Fourier transform F with Y = F (W ), β = F (θ ), X = F (Z) and ε = F (η).
As far as we know, despite the abundant literature related to FCM or functional VCM, there is hardly any paper providing a strictly functional approach (i.e. with random functions defined inside normed functional spaces and studying the convergence with their own norms). As noticed by Ramsay and Silverman (2005, p. 259), all these methods come from a multivariate data analysis approach rather than from a functional one. For some applications, for example when the observations are highly auto-correlated, taking this functional nature into account may be decisive. If not, multivariate approaches may cause a loss of information because, as noticed by Şentürk andMüller (2010, p. 1257), they "do not take full advantage of the functional nature of the underlying data". In practice this loss of information may reduce the accuracy of estimation and prediction. To circumvent this problem, [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF] propose a three-step functional approach based on smoothing and least square estimation. However, the convergence results obtained on compact sets do not allow to study specific models like (2.2), for which convergence on the whole real line is required.

The objective of the present paper is to propose an estimator of the function β in the FCM (3.2) for which such asymptotic results hold. Our estimation approach is based on the Ridge Regression method developed in the classical linear case, see [START_REF] Hoerl | Application of ridge analysis to regression problems[END_REF]. We extended it to the functional data framework of model (3.2). First we establish the consistency of the estimator and get a rate of convergence. Then we propose a method for selecting the regularization parameter. Finally we present some simulation trials which show the accuracy of the estimator in fitting the unknown function β , despite a very low signal-to-noise ratio (SNR).

All the proofs are postponed to Section 2.7.

Model and Estimator

To remain as general as possible, all considered functions are complex valued functions. Before studying the FCM let us define some useful notations. We define L 2 (R, C) = L 2 the set of square integrable complex valued functions, with the

L 2 - norm ∥ f ∥ L 2 := R | f (x)| 2 dx
1/2 , and given a subset

K ⊂ R, ∥ f ∥ L 2 (K) := K | f (x)| 2 dx 1/2 , where | • |
denotes the complex modulus.

The theoretical results given in the next sections are proved on the whole real line. For this reason, we need to restrict the study to the set of functions that vanish at infinity. Let C 0 (R, C) = C 0 be the space of complex valued continuous functions, which satisfies : for all ζ > 0 there exists a R > 0 such that for all

|t| > R, | f (t)| < ζ . We use the supremum norm ∥ f ∥ C 0 := sup x∈R | f (x)|. In particular for a subset K ⊂ R, ∥ f ∥ C 0 (K) := sup x∈K | f (x)|.
Finally throughout this paper the support of a continuous function f : R → C is the set supp( f

) := {t ∈ R : | f (t)| ̸ = 0}.
This set is open because f is continuous. Besides we define the boundary of a set S, as ∂ (S) := S \ int(S), where S is the closure of S and int(S) is its interior.

General Hypotheses of the FCM

The space C 0 is too large. For instance, its geometry does not allow for the application of the Central Limit Theorem (CLT) under the general hypothesis of the existence of the covariance operator, that is Ledoux and Talagrand (1991, Ch 10)). To circumvent this difficulty, we consider functions that belong to the space C 0 ∩ L 2 . Here are general hypotheses that will be used all along the paper:

E(∥X∥ 2 C 0 ) < ∞ (see
(HA1 FCM ) X, ε are independent C 0 ∩ L 2 valued random functions, such that E(ε) = E(X) = 0, (HA2 FCM ) β ∈ C 0 ∩ L 2 , (HA3 FCM ) E(∥ε∥ 2 C 0 ), E(∥X∥ 2 C 0 ), E(∥ε∥ 2 L 2
) and E(∥X∥ 2 L 2 ) are all finite.

Functional Ridge Regression Estimator (FRRE)

The definition of the estimator of β is inspired by the estimator introduced by Hoerl (1962) used in the Ridge Regularization method that deal with ill-posed problems in the classical linear regression.

Let

(X i ,Y i ) i=1,••• ,n be an i.i.d sample of FCM (3.
2) and a regularization parameter λ n > 0. We define the estimator of β as follows

βn := 1 n ∑ n i=1 Y i X * i 1 n ∑ n i=1 |X i | 2 + λ n n , (2.3)
where the exponent * stands for the complex conjugate. In the classical linear regression case, Hoerl and Kennard (1970, p. 62) proved that there is always a regularization parameter for which the ridge estimator is better than the Ordinary Linear Squares (OLS) estimator. [START_REF] Huh | Asymptotic aspects of ordinary ridge regression[END_REF] made a study of some asymptotic properties of the ridge estimator in this context.

Asymptotic Properties of the FRRE

From the definition (3.3), it is easy to show that the FRRE βn decomposes as follows:

βn = β + λ n n β 1 n ∑ n i=1 |X i | 2 + λ n n + 1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n . (2.4)
The main results of this paper are the probability convergence of the FRRE and the rate of convergence

∥ βn -β ∥ L 2 = O P max λ n n , √ n λ n ,
under very large conditions.

Consistency of the Estimator

Theorem 3. Let us consider the FCM with the general hypotheses (HA1 FCM ), (HA2 FCM ) and (HA3 FCM ). Let (X i ,Y i ) i≥1 be i.i.d. realizations. We suppose moreover that

(A1) supp(|β |) ⊆ supp(E[|X|]), (A2) (λ n ) n≥1 ⊂ R + is such that λ n n → 0 and √ n λ n → 0 as n → +∞.
Then

lim n→+∞ ∥ βn -β ∥ L 2 = 0 in probability. (2.5)
Let us make some comments about the hypotheses.

Remark. Hypothesis (A2) is classic in the context of ridge regression. Hypothesis (A1) specifies that it is not possible to estimate β outside the support of the modulus of X. From model (3.2), it is clear that β cannot be estimated in the intervals where the function X is zero, as proved in the following proposition:

Proposition 4. Let (X i ,Y i ) i=1,••• ,n be an i.i.d. sample of FCM in C 0 ∩ L 2 which satisfies hypothesis (A2) and (nA1) There exists t 0 ∈ supp(|β |) and δ > 0 such that E[∥X∥ C 0 ([t 0 -δ ,t 0 +δ ]) ] = 0.
Then there exists a constant C > 0 such that almost surely

∥ βn -β ∥ L 2 ≥ C. (2.6)
Proof. For all the independent realizations of X, we have E[∥X n ∥ C 0 ([t 0 -δ ,t 0 +δ ]) ] = 0. Then for all n ∈ N, the function X n restricted to the interval [t 0δ ,t 0 + δ ] is equal to zero almost surely. Thus over this interval βn = 0 (a.s.). If we define

C := ∥β ∥ L 2 ([t 0 -δ ,t 0 +δ ]) we obtain ∥ βn -β ∥ L 2 ≥ ∥ βn -β ∥ L 2 ([t 0 -δ ,t 0 +δ ]) = C (a.s.).
Hypothesis (nA1) is stronger than the negation of (A1). It provides that there exists some t 0 in supp(|β |), such that X is zero almost surely in a neighborhood of t 0 .

The geometry of L 2 helps a lot in the proof of Theorem 3. By paying attention to the geometry of L p spaces, it is also possible to generalize this result for those spaces. Theorem 5. Let us consider the FCM with the general hypotheses (HA1 FCM ), (HA2 FCM ) and (HA3 FCM ). We assume additionally that (A1) holds, together with :

Rate of Convergence

(A3) E[∥X∥ 4 L 2 ] < ∞. (A4) |β | E[|X| 2 ] 1 supp(β )\∂ (supp(E[|X|])) L 2 < +∞.
(A5) There exist positive real numbers α > 0, M 0 , M 1 , M 2 > 0 such that (a) For every p ∈ C β ,∂ X , there exists an open neighborhood J p ⊂ supp(|β |) such that

E[|X| 2 (t)] ≥ |t -p| α ,
for every t ∈ J p and 1

E[|X| 2 ] L 2 (J p \{p}) ≤ M 0 , (b) ∑ p∈C β ,∂ X ∥β ∥ 2 C 0 (J p ) < M 1 , (c) |β | E[|X| 2 ] 1 supp(|β |)\J < M 2 , where J = p∈C β ,∂ X J p .
(A6) For n ≥ 1,

λ n := n 1-1 4α+2 ,
where α > 0 comes from the hypothesis (A5).

Then

∥ βn -β ∥ L 2 = O P n -γ , (2.7)
where γ := min 1 2(2α+1) , 1 2 -1 2(2α+1) .

The following corollary specifies the rate of convergence for α = 1/2.

Corollary 6. Under the hypotheses of Theorem 5, n -γ = max λ n n ,

√ n λ n and in particular if α = 1/2 ∥ βn -β ∥ L 2 = O P 1 n 1/4 .
Remark. Hypothesis (A3) is classic and allows to apply the CLT on the denominator of βn . Hypothesis (A4) is needed because the second term in (3.5), namely Finally hypothesis (A6) replaces (A2) in Theorem 3, as the rate of convergence strongly depends on the behavior of β E[|X| 2 ] around the points of C β ,∂ X , which depends on α. We can see that (A6) always implies (A2).

β 1 n ∑ n i=1 |X i | 2 + λn
It is possible to get the same convergence results as that of Theorem 5 under assumptions easier to verify, in particular when C β ,∂ X = / 0, which is a stronger assumption than hypothesis (A4bis) in Corollary 20.

Corollary 7. Under hypotheses (A1), (A2) and (A3) and if additionally we assume

(A4bis) |β | E[|X| 2 ] 1 supp(|β |) ∈ L 2 ∩ L ∞ , then ∥ βn -β ∥ L 2 = O P max λ n n , √ n λ n . (2.8)
Hypothesis (A4bis) is a reformulation of (A4) and part (c) of (A5). It is required to control the second term of (3.5) and the decreasing rate of β with respect to E[|X| 2 ] around infinity (tails control).

Finally, Theorem 21 deals with the convergence rate on compact subsets of the support of

E[|X| 2 ].
Theorem 8. Under hypotheses (A1), ( A2) and (A3), for every compact subset K ⊂ supp(E[|X| 2 ]),

we have ∥ βn -β ∥ L 2 (K) = O P max λ n n , √ n λ n .
(2.9)

Moreover if the support of β is compact, we deduce the following corollary.

Corollary 9. Under the hypotheses (A1), (A2) and (A3), if supp(β ) is compact and is a subset of

supp(E[|X|]), then ∥ βn -β ∥ L 2 = O P max λ n n , √ n λ n .

Selection of the Regularization Parameter

Predictive Cross-Validation (PCV) and Generalized Cross-Validation (GCV)

This section is devoted to developing a selection procedure of the regularization parameter λ n for a given sample

(X i ,Y i ) i∈{1,••• ,n} .
To solve this problem we chose the Predictive Cross-Validation (PCV) criterion. Its definition, see for instance Febrero-Bande and Oviedo de la Fuente (2012, p. 17) or Hall and Hosseini-Nasab (2006, p. 117), is the following

PCV (λ n ) := 1 n n ∑ i=1 ∥Y i - β (-i) n X i ∥ 2 L 2 ,
where

β (-i) n is computed with the sample (X j ,Y j ) j∈{1,••• ,i-1,i+1,••• ,n} .
The selection method consists in choosing the value λ n which minimizes the function PCV (•).

In this subsection we give results that allow for computing faster the PCV by processing only one regression, instead of n. These results use similar ideas as in Green and Silverman (1994, pp. 31-33) about the smoothing parameter selection for smoothing splines.

Proposition 10. We have

PCV (λ n ) = 1 n n ∑ i=1 Y i -βn X i 1 -A i,i 2 L 2 , ( 2

.10)

where A i,i ∈ L 2 is defined as follows A i,i :

= |X i | 2 /(∑ n j=1 |X j | 2 + λ n ).
This last proposition allows to write the PCV without excluding the ith observation. We then introduce the following Generalized Cross-Validation (GCV), computationally faster than the PCV:

GCV (λ n ) := 1 n n ∑ i=1 Y i -βn X i 1 -A 2 L 2 , where A ∈ L 2 is A := ( 1 n ∑ n i=1 |X i | 2 )/(∑ n j=1 |X j | 2 + λ n ).
Remark: From the definition of A, we have that, for every t ∈ R, 0 ≤ A(t) ≤ 1/n, then 1 ≤ 1 1-A(t) ≤ n n-1 , which yields that the GCV criterion is bounded as follows:

1 n n ∑ i=1 Y i -βn X i 2 L 2 ≤ GCV (λ n ) ≤ 1 n -1 n ∑ i=1 Y i -βn X i 2 L 2 .
This last inequality gives thus quickly an idea of the GCV values.

Regularization function Parameter

As we are working with functional data, another possibility is to use a time-dependent function Λ n (t) in the estimator defined in (3.3), instead of a constant number λ n . We shall optimize, for each time t, the choice of Λ n (t). To that aim, we have to compute the PCV for each time t ∈ R,

PCV (Λ n (t)) := 1 n n ∑ i=1 |Y i (t) - β (-i) n (t) X i (t)| 2 , where β (-i) n (t) is computed with the sample (X j (t),Y j (t)) j∈{1,••• ,n}\{i} .
As above, we obtain a simpler formula for PCV (Λ n (t)) (see next proposition bellow), which yields a faster computation.

Proposition 11. We have

PCV (Λ n (t)) = 1 n n ∑ i=1 Y i (t) -βn (t) X i (t) 1 -A i,i (t) 2 , ( 2

.11)

where A i,i (t) :=

|X i (t)| 2 ∑ n j=1 |X j (t)| 2 +λ n (t) .
This criterion is discussed in the next section about simulation studies. Its performance is evaluated and compared to that of GCV (λ n ).

Simulation study

The simulation study follows model (3.2) with an intercept term:

Y i (t) = β 0 (t) + β 1 (t)X i (t) + ε i (t), ∀i = 1, . . . , n, ∀t ∈ [0, T ].
(2.12)

We evaluate our estimation procedure in the case of a low Signal-to-Noise-Ratio (SNR). Both approaches using λ n and Λ n (t) are compared. We first give the estimation procedure adapted to model ( 2.12) and we introduce three criteria to measure the estimation error when estimating β 0 and β 1 .

Estimation procedure and evaluation criteria

We compute the mean curve X := 1 n ∑ n i=1 X i and Ȳ := 1 n ∑ n i=1 Y i . Thus we use the FRRE to compute the estimators β1 and β0 , as follows

β1 := ∑ n i=1 (Y i -Ȳ ) (X i -X) * ∑ n i=1 |(X i -X)| 2 + λ n , (2.13) β0 := Ȳ -β1 X.
We use 500 Monte Carlo runs to evaluate the mean absolute deviation error (MADE), the weighted average squared error (WASE) and the unweighted average squared error (UASE), defined in the same way as in Şentürk andMüller (2010, p. 1261),

MADE := 1 2T T 0 |β 0 (t) -β0 (t)|dt range(β 0 ) + T 0 |β 1 (t) -β1 (t)|dt range(β 1 ) , WASE := 1 2T T 0 |β 0 (t) -β0 (t)| 2 dt range 2 (β 0 ) + T 0 |β 1 (t) -β1 (t)| 2 dt range 2 (β 1 ) , UASE := 1 2T T 0 |β 0 (t) -β0 (t)| 2 dt + T 0 |β 1 (t) -β1 (t)| 2 dt ,
where range(β r ) is the range of the function β r .

Setting

The data were simulated on the interval [0, T ] (T = 24), discretized over p = 100 equispaced observation times. More precisely for j ∈ [1, 100] ∩ N, t j := j * 24/101. The simulated input curves X i , for i = 1, . . . , n with n = 150, were generated with mean function µ X (t) = t + sin(t) and covariance function constructed from the 10 first eigenfunctions of the Wiener Process with its correspondent eigenvalues. Namely for 0 ≤ t ≤ 24 we have X i (t) = µ X (t) + ∑ 10 j=1 ρ j ξ i j φ j (t), where for j ≥ 1, φ j (t) = √ 2 sin(( j -1/2)πt), ρ j = 1/(( j -1/2)π) 2 and the ξ i j were generated from N(0, 1). The functions β 0 and β 1 are respectively:

β 0 (t) = 2 18 2 (t -6) 2 + 1 and β 1 (t) =      -2 16 (t -6) 2 + 2 if t ∈ [2, 10], - 2
16 (t -18) 2 + 2 if t ∈ [14, 22], 0 otherwise.
The noise ε i was defined as follows:

ε i (t) = c ε ∑ 20 j=11 ρ j ξ i j φ j (t)
, where c ε is a constant such that the signal-to-noise ratio (SNR) is equal to 2, where SNR := (tr(Cov(X)))/(tr(Cov(ε)), with Cov(X) :=

E (< X, . > X-< E(X), . > E(X)), Cov(ε) := E (< ε, . > ε)
and tr is the trace of an operator.

The general hypotheses (HA1 FCM ) -(HA3 FCM ) are satisfied. The regularization parameter λ n is optimized over the interval [0, 10].

Results

The simulation results are presented in Figures 2.1 and 2.2, and Table 2.1. The performance of the estimators with both regularization parameter λ and regularization curve Λ are illustrated. We can see that, even in conditions of high noise (SNR = 2), the estimation is really good. This shows the robustness of the FRRE. Moreover, the FRRE β (2) 0 and β (2) 1 computed with an optimal regularization curve Λ 150 give in average better estimations of the functions β 0 and β 1 . In this simulation setting, the mean of the optimal regularization curve Λ n is almost constant (equal to 0.015 where β 1 ̸ = 0) with constant value close to the mean optimal regularization parameter λ n (0.0156).

Conclusions

In this paper we have generalized the Ridge Regression method to estimate the unknown function of the FCM with the FRRE. We proved its consistency for the L 2 -norm, and obtained its rate of convergence over the whole real line, and not only on compact sets. This strong result opens new perspectives for studying models related to the FCM, like the convolution model (2.2).

We also provided a selection procedure of the optimal regularization parameter λ n through PCV. The simulations showed good properties of the FRRE under very low SNR.

This work shows some similarities with [START_REF] Şentürk | Functional varying coefficient models for longitudinal data[END_REF], where the model studied is close to the FCM (3.2) with i) an intercept term, ii) data X and Y observed up to additive noise and iii) a sparse random design. The estimation of the unknown function is done after three preliminary steps, first a smoothing step, then the computation of the raw covariances, and finally the smoothing of them. This three-step procedure requires the choice of several smoothing parameters, while the FRRE only requires the choice of one. In addition, the computation of the raw covariances is done over a square domain, whereas the FRRE directly calculates them over its diagonal. For these reasons, the FRRE is simpler to compute.

Main Proofs

Before proving Theorem 3, let us first introduce a useful technical lemma. Here we will denote

ϕ := E[|X| 2 ] ∈ C 0 .
Lemma 12. Under hypotheses (A1) and (A2) of Theorem 3, if there exists a sequence of functions

( f n ) n≥1 ⊂ C 0 such that ∥ f n -ϕ∥ C 0 → 0, then there exists 1. a sequence (C j ) j≥1 of subsets of R such that m lim sup j→+∞ C j = m ∩ J≥1 [∪ J j=1 C j ] = 0,
where m is the Lebesgue measure, 2. a strictly increasing sequence of natural numbers (N j ) j≥1 ⊂ N and a sequence of real numbers

(d n ) n≥1 ⊂ R, with lim n→+∞ d n = 0, such that for every j ≥ 1 and n ∈ {N j , • • • , N j+1 }, λ n n β f n + λ n n C 0 (R\C j ) ≤ d n .
(2.14)

Proof of Lemma 12. To start the proof, we notice that supp(ϕ) = supp(E[|X|]), hence supp(|β |) ⊆ supp(ϕ) by hypothesis (A1).

We define the sequence α r := λ r r which is decreasing to 0, and the sets K ϕ r := ϕ -1 ([α r , +∞[) and K β q := |β | -1 ([1/q, +∞[) for r, q ∈ N + . All these sets are compacts and cover the supports of ϕ and β respectively, that is

∪ ∞ r=1 ↑ K ϕ r = supp(ϕ) and ∪ ∞ q=1 ↑ K β q = supp(β ).
Without loss of generality, we can suppose that there exists some

Q 1 ∈ N such that K β Q 1 ̸ = / 0
(otherwise β ≡ 0). Then we redefine for all q ∈ N, K

β q := K β Q 1 +q .
Let us take a sequence δ s decreasing to 0 and define for all s ∈ N,

D s := B δ s (∂ supp(ϕ)) = ∪ a∈∂ supp(ϕ) B δ s (a) and C s := K β s ∩ D s , with B δ s (a) :=]a -δ s , a + δ s [. Clearly K β 1 \C 1 ⊂ int(supp(ϕ)) = supp(ϕ) = ∪ ∞ r=1 K ϕ r .
since the supports of continuous functions are open. Thus, from the definition of K ϕ r and the fact that α r goes to zero, there exists r 1 ∈ N such that for all r ≥ r 1 , K

β 1 \C 1 ⊂ K ϕ r .
Moreover, from (A2) there exists r1 > r 1 such that, for all r ≥ r1 , max

r≥r 1 λ r r ≤ λ r 1 r 1 .
Considering K β 1 \C 1 , from the definition of K ϕ r 1 and the uniform convergence of ( f n ) n≥1 towards ϕ, we deduce that there exists N 1 > r1 such that for all n ≥ N 1 and t ∈ K

ϕ r 1 , 3 4 α r 1 ≤ f n (t) + λ n n .
Thus for all n such that n ≥ N 1 ,

| λ n n | β f n + λ n n C 0 (K ϕ r 1 ) ≤ | λ n n | 4 3α r 1 ∥β ∥ C 0 (R) ≤ max s≥r 1 [| λ s s |] 4 3α r 1 ∥β ∥ C 0 (R) .
In particular we can deduce, for all n ≥ N 1 > r 1 ,

| λ n n | β f n + λ n n C 0 (K β 1 \C 1 ) ≤ | λ r 1 r 1 | 4 3α r 1 ∥β ∥ C 0 (R) ≤ λ r 1 r 1 4 3 ∥β ∥ C 0 (R) .
because of the definition of α r 1 .

Similarly

K β 2 \C 2 ⊂ int(supp(ϕ)),
and there exists r 2 > r 1 such that for all r ≥ r 2 , K

β 2 \C δ 2 ⊂ K ϕ r . From (A2) there exists r2 > r 2 such that max r≥r 2 λ r r ≤ λ r 2 r 2 .
Again, given the definition of K ϕ r 2 and the uniform convergence of ( f n ) n g eq1 towards ϕ, we deduce that there exists N 2 > r2 such that for all n ≥ N 2 and t ∈ K

ϕ r 2 , 3 4 α r 2 ≤ f n (t) + λ n n .
This yields that, for all n such that n ≥ N 2 > r 2 ,

λ n n β f n + λ n n C 0 (K β 2 \C 2 ) ≤ λ r 2 r 2 4 3 ∥β ∥ C 0 (R) .
We continue this way to build three strictly increasing sequences r j ↑ ∞, r j ↑ ∞ and N j ↑ ∞ such that for all j ∈ N,

1. N j > r j > r j , 2. ∀r ≥ r j , K β j \C j ⊂ K ϕ r , 3. max r≥r j [ λ r r ] ≤ λ r j r j , 4. ∀n ≥ N j , | λ n n | β f n + λn n C 0 (K β j \C j ) ≤ λ r j r j 4 3 ∥β ∥ C 0 (R) .
Let n be an integer greater than N 1 . Then there exists an integer j such that n belongs to the set {N j , N j + 1, • • • , N j+1 -1}. The following sequence (d n ) is then defined as follows:

d n := max 4 3 λ r j r j ∥β ∥ C 0 (R) , 1 j . (2.15)
It is easy to see that this sequence goes to zero and from (2.15) we conclude that for all n ∈ {N j ,

N j + 1, • • • , N j+1 -1}, | λ n n | β f n + λ n n C 0 (R\C j ) ≤ d n , (2.16) because of R \C δ j = [K β j \C δ j ] ∩ [(K β j ) c \C δ j ] and the definition of K β j (outside K β j , β is bounded by 1/ j).
Proof of Theorem 3. From the decomposition (3.5), we obtain

∥ βn -β ∥ L 2 ≤ λ n n β 1 n ∑ n i=1 |X i | 2 + λ n n L 2 + 1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n L 2 .
Let us start by showing that

1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n L 2 = O P √ n λ n .
(2.17)

First we have

E[∥ε X * ∥ 2 L 2 ] ≤ E[∥ε∥ 2 C 0 ] E[∥X∥ 2 L 2 ] < +∞,
because of (HA1 FCM ) and (HA3 FCM ). Now due to the moment monotonicity E[∥ε X∥ L 2 ] < +∞, ε X is strongly integrable with the L 2 -norm, so there exists a function E[ε X] ∈ L 2 which is the zero function because of (HA1 FCM ). We conclude that

E[ε X] = 0 and E[∥ε X∥ 2 L 2 ] < +∞,
which, from the CLT in L 2 (see Theorem 2.7 in Bosq (2000, p. 51) and Ledoux and Talagrand (1991, p. 276) for the rate of convergence), yields to

1 n n ∑ i=1 ε i X * i L 2 = O P 1 √ n .
Finally (3.17) is obtained from the fact that

1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |W i | 2 + λ n n L 2 ≤ n λ n 1 n n ∑ i=1 ε i X * i L 2 = O P √ n λ n .

As

√ n

λ n → 0 by (A3), we obtain the probability convergence of this part.

To conclude the proof, it is enough to show that

λ n n β 1 n ∑ n i=1 |X i | 2 + λ n n L 2 a.s.
--→ 0.

(2.18)

To that purpose, we use the fact that

1 n n ∑ i=1 |X i | 2 -E[|X| 2 ] C 0 a.s.
--→ 0, which can be obtained by applying the Strong Law of Large Numbers (SLLN) (see Bosq (2000, p. 47)) to the random function |X| 2 . Notice here that

E[|X| 2 ] ∈ C 0 . Now for S := {ω ∈ Ω : ∥ 1 n ∑ n i |X(ω)| 2 -ϕ∥ C 0 → 0}, P(S ) = 1.
Let us take an arbitrary and fixed value ω ∈ S . Then for n ≥ 1 we define the sequence of functions

f n := 1 n ∑ n i=1 |X i (ω)| 2 .
Clearly this sequence belongs to C 0 and ∥ f n -ϕ∥ C 0 → 0. Thus we can use Lemma 12 which implies that there exists a sequence of subsets of R, (C j ) j≥1 , a strictly increasing sequence of natural numbers (N j ) j≥1 ⊂ N and a sequence of real numbers (d n ) n≥1 ⊂ R converging to zero, such that inequality (2.14) holds.

At this point we define for n ≥ N 1 , R n := 1 d n → ∞ and the intervals Īn :=

[-R n , +R n ]. For n ∈ {N j , N j + 1, • • • , N j+1 -1}
, by the triangular inequality and inequality (2.14),

| λ n n | β f n + λn n L 2 (R) ≤ | λ n n | β f n + λn n L 2 ( Īn ∩C j ) + | λ n n | β f n + λn n L 2 ( Īn ∩C c j ) + + ∥β ∥ L 2 ( Īc n ) ≤ ∥β ∥ L 2 (C j ) + | λ n n | β f n + λn n C 0 (R\C j ) √ 2 R n + ∥β ∥ L 2 ( Īc n ) .
In this way we obtain for every n ∈ {N j ,

N j + 1, • • • , N j+1 -1}, | λ n n | β f n + λ n n L 2 (R) ≤ ∥β ∥ L 2 (C j ) + d n 2 d n + ∥β ∥ L 2 ( Īc n ) .
Thus

L := lim n→∞ | λ n n | β f n + λ n n L 2 (R) ≤ lim j→∞ ∥β • 1 C j ∥ L 2 (R) .
Finally the sequence of functions |β • 1 C j | is bounded by β and is pointwise convergent to zero almost everywhere because {t ∈ R :

β • 1 C j (t) → 0} c ⊂ ∩ ∞ l=1 ∪ s≥l C s ⊂ ∩ ∞ l=1 ∪ s≥l D s ⊂ ∩ ∞ l=1 D l ⊂ ∂ supp(ϕ)
which is countable then with measure zero.

By the dominated convergence theorem, lim j→∞ ∥β • 1 C j ∥ L 2 = 0. Thus L = 0 and so (3.18) is proved because ω is an arbitrary element of S and P(S ) = 1.

Proof of Theorem 5. We use (3.5) and the triangle inequality to obtain

∥ βn -β ∥ L 2 ≤ λ n n β 1 n ∑ n i=1 |X i | 2 + λn n L 2 (supp(|β |)) + 1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λn n L 2 . The proof of 1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n L 1 = O P √ n λ n
is the same as in Theorem 3. Hence, to finish the proof of Theorem 5, we have to show that

β 1 n ∑ n i=1 |X i | 2 + λ n n 2 L 2 (supp(|β |)\J) + β 1 n ∑ n i=1 |X i | 2 + λ n n 2 L 2 (J) = O P (1), (2.19) which will lead to ∥ βn -β ∥ L 2 = λ n n O P (1) + O P √ n λ n = O P n -γ .
The proof of (2.19) is based on the two following lemmas.

Lemma 13. Under the assumptions of Theorem 5, we have

β 1 n ∑ n i=1 |X i | 2 + λ n n 2 L 2 (supp(|β |)\J)
= O P (1).

Proof of Lemma 13. Throughout the proof, we use the following notations to simplify the writing.

For all n ≥ 1, λn :

= λ n n , S n := ∑ n i=1 |X i | 2 , Sn := S n n , A n := |β |/( Sn + λn ). The support of function ϕ := E[|X| 2 ] is supp(ϕ) = supp(E[|X|]), so that C β ,∂ X = supp(|β |) \ ∂ (supp(ϕ)). Finally, the set C := supp(|β |) \ J satisfies C ⊂ supp(ϕ).
Let us define for j ≥ 1, r j := ∥ϕ∥ C 0 /2 j , r 0 := ∥ϕ∥ C 0 + 1, the compact sets K 0 := / 0, K j := ϕ -1 ([r j , ∞[), and D j := K j \ K j-1 . So we have ∪ j≥1 ↑ K j = supp(ϕ) and we can cover

C = ∪ j≥1 (C ∩ D j ).
We obtain

∥A n ∥ 2 L 2 (C) = ∑ j≥1 ∥A n 1 Sn ∈[0,r j /2] ∥ 2 L 2 (C∩D j ) + ∑ j≥1 ∥A n 1 Sn >r j /2 ∥ 2 L 2 (C∩D j ) ≤ 1 λ 2 n ∑ j≥1 ∥β ∥ 2 C 0 (C∩D j ) m( Sn ∈ [0, r j /2] ∩C ∩ D j ) + + ∑ j≥1 2 2 r 2 j r 2 j-1 r 2 j-1 ∥β ∥ 2 L 2 (C∩D j ) .
Now for each j ≥ 1, r j-1 r j ≤ r 0 r 1 and in the set

C ∩ D j , β r j-1 < β ϕ ≤ β r j . Then ∥β ∥ C 0 ≤ M 2 r j-1 because of part (c) of (A5). Thus ∥A n ∥ 2 L 2 (C) ≤ 1 λ 2 n M 2 2 ( r 0 r 1 ) 2 ∑ j≥1 r 2 j-1 m( Sn ∈ [0, r j /2] ∩C ∩ D j )+ + 4( r 0 r 1 ) 2 ∑ j≥1 ∥ β ϕ ∥ 2 L 2 (C∩D j ) . Moreover ∑ j≥1 r 2 j 4 m( Sn ∈ [0, r j /2] ∩C ∩ D j ) ≤ ∑ j≥1 ∥(ϕ -Sn ) 1 Sn ∈[0,r j /2] ∥ 2 L 2 (C∩D j ) ≤ ∥ϕ -Sn ∥ 2 L 2 (C) .

Now we can bound

A n ∥A n ∥ 2 L 2 (C) ≤ 1 λ 2 n M 2 2 ( r 0 r 1 ) 2 × 4∥ϕ -Sn ∥ 2 L 2 (C) + 4( r 0 r 1 ) 2 ∥ β ϕ ∥ 2 L 2 (C) = 4M 2 2 ( r 0 r 1 ) 2 O P (( √ n λ n ) 2 ) + 4( r 0 r 1 ) 2 ∥ β ϕ ∥ 2 L 2 (C) = O P (1).
Lemma 14. Under the assumptions of Theorem 5, we have

β 1 n ∑ n i=1 |X i | 2 + λ n n 2 L 2 (J)
= O P (1).

Proof of Lemma 14. Now we suppose that card(C β ,∂ X ) = +∞ (the case where this set is finite is similar). We denote its elements as p v , with v ≥ 1. So J is the union of disjoints intervals J = ∪ v≥1 J v , where J v := J p v , and part (b) of (A5) can be written as

∑ v≥1 ∥β ∥ 2 C 0 (J v ) < M 1 . For n ≥ 1, let us define ξ n := λ 2α n .
Clearly from (A6), ξ n ↓ 0. We define for l ≥ 1, the compact sets

K ξ 0 := / 0, K ξ l := ϕ -1 ([ξ l , ∞[), and
D ξ l := K ξ l \ K ξ l-1 . So we have ∪ ↑ K ξ l = supp(ϕ) and we can cover J v \ {p v } = ∪ j≥1 (J v ∩ D ξ j ) for each fixed v ≥ 1. Moreover in D ξ l , 1 ξ l-1 < 1 ϕ ≤ 1 ξ l .
Let us take a fixed v ≥ 1. Given the fact that ξ l is strictly decreasing to zero, by hypothesis (A6), there exists a unique number N v ∈ N such that

ξ N v < max t∈∂ (J v ) |t -p v | α ≤ ξ N v -1 . Then for every n ≥ N v , ∥A n ∥ 2 L 2 (J v ) = ∑ n l=N v ∥A n ∥ 2 L 2 (J v ∩D ξ l ) + ∥A n ∥ 2 L 2 (J v \K ξ n ) = ∑ n l=N v ∥A n 1 Sn ∈[0,ξ n /2] ∥ 2 L 2 (J v ∩D ξ l ) + + ∑ n l=N v ∥A n 1 Sn ≥ξ n /2 ∥ 2 L 2 (J v ∩D ξ l ) + ∥A n ∥ 2 L 2 (J v \K ξ n ) ≤ ∥β ∥ 2 C 0 (J v ) 1 λ 2 n ∑ n l=N v m( Sn ∈ [0, ξ n /2] ∩ J v ∩ D ξ l ) + ∥β ∥ 2 C 0 (J v ) 4 ξ 2 n ∑ n l=N v m(J v ∩ D ξ l ) + 1 λ 2 n m(J v \ K ξ n ) .
Using the inequality

ξ 2 n 4 n ∑ l=N v m( Sn ∈ [0, ξ n /2] ∩ J v ∩ D ξ l ) ≤ ∥ϕ -Sn ∥ L 2 (J v ) ,
we obtain

∥A n ∥ 2 L 2 (J v ) ≤ ∥β ∥ 2 C 0 (J v ) 1 λ 2 n 4 ρ 2 n ∥ϕ -Sn ∥ L 2 (J v ) + 4 ∑ n l=N v ξ 2 l-1 ξ 2 l m(J v ∩D ξ l ) ξ 2 l-1 + + 1 λ 2 n m(J v \ K ξ n ) .
Because of (A6), there exists

M 3 > 0 such that for l ≥ 1, | λ l-1 λ l | ≤ M 3 . Thus for n ≥ N v , ∥A n ∥ 2 L 2 (J v ) ≤ ∥β ∥ 2 C 0 (J v ) 4 λ 2+4α n ∥ϕ -Sn ∥ L 2 (J v ) + + 4M 2 3 ∥ 1 ϕ ∥ 2 L 2 (J v ∩K ξ n ) + 1 λ 2 n m(J v \ K ξ n ) . Moreover, if t ∈ J v \ K ξ n , 0 ≤ ϕ(t) < ξ n hence |t -p v | α ≤ ϕ(t) < ξ n and in particular J v \ K ξ n ⊂ [p v -ξ 1/α n , p v + ξ 1/α n ]. In this way we can prove that for n ≥ N v , m(J v \ K ξ n ) ≤ 2ξ 1/α n ≤ 2 λ 2 n . We obtain from this that for every n ∈ {1, • • • , N v -1}, ∥A n ∥ 2 L 2 (J v ) ≤ 1 λ 2 n ∥β ∥ 2 L 2 (J v ) ,
and for n ≥ N v ,

∥A n ∥ 2 L 2 (J v ) ≤ 4∥β ∥ 2 C 0 (J v ) n∥ Sn -ϕ∥ 2 L 2 (J v ) + M 2 3 ∥ 1 ϕ ∥ 2 L 2 (J v ) + 1/2 .
To finish the proof of this lemma, we bound the sequence

∥A n ∥ 2 L 2 (J) = ∑ v≥1 ∥A n ∥ 2 L 2 (J v ) .
In order to do this we define for each n ≥ 1, the set

C n := {n ≥ 1 : n ∈ [1, • • • , N v -1]}. We obtain ∥A n ∥ 2 L 2 (J) ≤ 1 λ 2 n ∥β ∥ 2 L 2 (∪ v∈Cn J v ) + + 4 ∑ v≥1 ∥β ∥ 2 C 0 (J v ) n∥ Sn -ϕ∥ 2 L 2 (J) + M 2 3 M 2 0 + 1/2 ≤ 1 λ 2 n ∥β ∥ 2 L 2 (∪ v∈Cn J v ) + 4M 1 O P (1) + M 2 3 M 2 0 + 1/2 . For each n ≥ 1, v ∈ C n then n < N v , hence ξ n ≥ max t∈∂ J v (t -p v ) α , from what we deduce that m(J v ) ≤ 2 ξ 1/α n . We obtain for n ≥ 1 ∥β ∥ 2 L 2 (∪ v∈Cn J v ) ≤ 2ξ 1/α n ∑ v∈C n ∥β ∥ 2 C 0 (J v ) ≤ 2ξ 1/α n ∑ v≥1 ∥β ∥ 2 C 0 (J v ) = 2ξ 1/α n [M 1 /4] ,
and thus for n ≥ 1,

∥A n ∥ 2 L 2 (J) ≤ 1 λ 2 n 2ξ 1/α n M 1 4 + 4M 1 O P (1) + M 2 3 M 2 0 + 1/2 ≤ M 1 2 + M 4 O P (1) + 4M 1 M 2 3 M 2 0 + 1/2 = O P (1).
Proof of Theorem 20. It is a particular case of Theorem 5. First, (A4bis) implies that, for all t ∈ supp(β ), |β (t)|/ϕ(t) is finite. Thus supp(β ) ⊂ supp(ϕ) and supp(β ) ∩ ∂ (supp(ϕ)) = / 0. Because of this, parts (a) and (b) of hypothesis (A5) are true by default.

Moreover, (A4bis) implies (A4), and if we have J := / 0, supp(β ) ∩ ∂ (supp(ϕ)) = / 0 implies part (c) of (A5). Finally, equation (2.19) in the proof of Theorem 5 is replaced by

β 1 n ∑ n i=1 |X i | 2 + λ n n 2 L 2 (supp(|β |)) = O P (1),
which is proved with the same technique.

Proof of Theorem 21. We start with the decomposition

∥ βn -β ∥ L 2 (K) = λ n n β 1 n ∑ n i=1 |X i | 2 + λ n n L 2 (K) + 1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n L 2 (K)
.

The proof of

1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λn n L 2 (K) = O P ( √ n
λ n ) is the same as in Theorem 3. We finish the proof of the theorem by showing .20) Given that K ⊂ supp(ϕ), there exists a positive number

β 1 n ∑ n i=1 |X i | 2 + λ n n L 2 (K) = O P (1). ( 2
s 1 > 0 such that K ⊂ K ϕ s 1 , where K ϕ s 1 := ϕ -1 ([s 1 , ∞[) is a compact in R.
We define s := s 1 /2. We have for every n ∈ N,

β Sn + λn L 2 (K) ≤ β Sn + λn 1 Sn ∈[0,s] L 2 (K) + β Sn + λn 1 Sn ∈[s,∞[ L 2 (K)
.

Clearly, the first part above is bounded by

β Sn + λn 1 Sn ∈[s,∞[ L 2 (K) ≤ 1 s ∥β ∥ L 2 (K) = O P (1).
To bound the other part we have

β Sn + λn 1 Sn ∈[0,s] L 2 (K) ≤ 1 λn β 1 Sn ∈[0,s] L 2 (K) ≤ ∥β ∥ C 0 λn m(K ∩ Sn ∈ [0, s]).
Moreover, thanks to hypothesis (A3), we have

∥ Sn -ϕ∥ L 2 (K) = O P ( 1 √ n ).
This inequality, together with the fact that | Sn -ϕ| > s whenever Sn ∈ [0, s], allows us to obtain

∥ Sn -ϕ∥ L 2 (K) ≥ ∥( Sn -ϕ)1 Sn ∈[0,s] ∥ L 2 (K) ≥ K |s| 2 1 Sn ∈[0,s] dm ≥ |s| m(K ∩ Sn ∈ [0, s]).
In this way, m(K

∩ Sn ∈ [0, s]) = O P ( 1 √ n
) and as a consequence

β Sn + λn 1 Sn ∈[0,s] L 2 (K) ≤ ∥β ∥ C 0 λn O P ( 1 √ n ) = O P ( √ n λ n ),
which finishes to prove (3.19).

Proof of Proposition 22. We only need to prove that for every Our approach is a new way to answer this question by using the equivalent representation of the model in the frequency domain as we explain later.

i ∈ {1, • • • , n}, Y i - β (-i) n X i = Y i -βn X i 1 -A i,i . ( 2
Ỹj := Y j if j ̸ = i, β (-i) n X j otherwise. Because β (-i) n = ∑ n l̸ =i Y l X l ∑ n l̸ =i |X l | 2 +λ n by definition, we have ∑ n l=1 Ỹl X l S n +λ n = ∑ n l̸ =i Y l X l S n +λ n + β (-i) n |X i | 2 S n +λ n = β (-i) n ∑ n l̸ =i |X l | 2 +λ n S n +λ n + |X i | 2 S n +λ n = β (-i) n . Then βn X i - β (-i) n X i = ∑ n l=1 Y l X l -∑ n l=1 Ỹl X l S n +λ n X i = Y i -β (-i) n X i S n +λ n |X i | 2 , from what we obtain 1 - Y i -βn X i Y i - β (-i) n X i = βn X i - β (-i) n X i Y i - β (-i) n X i = |X i | 2 S n + λ n = A i,i ,
In the literature the FCVM is related to the multichannel deconvolution problem (see e.g. De Canditiis and [START_REF] De Canditiis | Simultaneous wavelet deconvolution in periodic setting[END_REF], [START_REF] Pensky | On convergence rates equivalency and sampling strategies in functional deconvolution models[END_REF] and [START_REF] Kulik | Multichannel deconvolution with long range dependence: Upper bounds on the lp-risk[END_REF]) which treats the periodic case with n > 1 (multichannel). On the other hand all the following methods deal with the case n = 1 : The Deconvolution Problem in Non-parametric statistics (see e.g. [START_REF] Meister | Deconvolution Problems in Nonparametric Statistics[END_REF], [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF], [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging[END_REF]), in Signal Processing (see e.g. [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF], [START_REF] Brown | Introduction to Random Signals and Applied Kalman Filtering with MATLAB Exercises[END_REF]) or in Linear Inverse Problems (see e.g. [START_REF] Abramovich | Wavelet decomposition approaches to statistical inverse problems[END_REF]) and the Laplace Deconvolution (see e.g. [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging[END_REF]). In this paper we are interested to study the estimation in the non periodic case with n > 1 and the asymptotic behavior when n goes to infinity.

We develop another approach to estimate θ . The main idea is to use the Continuous Fourier Transform (CFT) to transform the problem into its equivalent in the frequency domain. This yields to a particular case of the Functional Concurrent Model (FCCM) (Ramsay and Silverman (2005, Ch 14)). The equation of the associated FCCM is

Y (ξ ) = β (ξ ) X (ξ ) + ε(ξ ), (3.2)
where ξ ∈ R, β is an unknown function to be estimated, X := F (X), Y := F (Y ) and ε := F (ε) are Fourier transforms of X and Y , and ε respectively. Once in the frequency domain we use an estimator of β for the FCCM, and then come back to the time domain through the Inverse Continuous Fourier Transform (ICFT). We called the estimator defined in this way the Functional Fourier Deconvolution Estimator (FFDE).

The estimation of β have already been discussed by several authors in the context of some related models to the FCCM. For instance [START_REF] Hastie | Varying-coefficient models[END_REF] has proposed a generalization of FCCM called 'varying coefficient model'. Until now many methods have been developed to estimate the unknown smooth regression function β , for instance by local maximum likelihood estimation (see e.g. [START_REF] Dreesman | Non-stationary conditional models for spatial data based on varying coefficients[END_REF]), or by local polynomial smoothing (see e.g. [START_REF] Fan | Adaptive varying-coefficient linear models[END_REF]). These methods use techniques from multivariate data analysis, as noticed by Ramsay and Silverman (2005, p. 259). In our case we do not use these methods because we want to estimate β = F (θ ) over the whole frequency domain (i.e. for all ξ ∈ R). To do this we use the Functional Ridge Regression Estimator of β defined in [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF].

In this paper we prove the consistency of the FFDE and obtain a rate of convergence. We present some simulations to compare the performance of this estimator with others we have adapted from different methods found in the literature. These simulations show the robustness, the accuracy and the fast computation time of our estimator compared to the others.

All the proofs are postponed to Appendix 3.B, where we use the main theorems of Manrique et al. (2016) (Appendix 3.A).

Model and Estimator

Let us first define some useful notations. We define L 1 (R, C) the space of integrable complex valued functions, with the

L 1 -norm ∥ f ∥ L 1 := [ R | f (x)|dx] where | • | denotes the complex modulus. L 1 (R, R)
is the subspace of integrable real valued functions. In a similar way L 2 (R, C) is the space of square integrable complex valued functions, with the

L 2 -norm ∥ f ∥ L 2 := R | f (x)| 2 dx 1/2 ,
and L 2 (R, R) is the subspace of square integrable real valued functions. Given a subset

K ⊂ R, ∥ f ∥ L 2 (K) := K | f (x)| 2 dx 1/2 .
Let C 0 (R, C) = C 0 be the space of complex valued continuous functions f that vanish at infinity, that is, for all ζ > 0 there exists a R > 0 such that for all

|t| > R, | f (t)| < ζ . We use the supremum norm ∥ f ∥ C 0 := sup x∈R | f (x)|. For a subset K ⊂ R, this norm is ∥ f ∥ C 0 (K) := sup x∈K | f (x)|.
The Continuous Fourier Transform (CFT) is denoted by F and its inverse (ICFT) by F -1 . Lastly throughout this paper the support of a continuous function f : R → C is the set supp( f

) := {t ∈ R : | f (t)| ̸ = 0}.
This set is open because f is continuous. Besides we define the boundary of a set S, as

∂ (S) := S \ int(S)
, where S is the closure of S and int(S) is its interior.

General Hypotheses of the FCVM

We present the general hypotheses that will be used along the paper.

(

HA1 FCV M ) X, ε are independent L 1 (R, R) ∩ L 2 (R, R)
valued random functions such that E(ε) = 0 and for every t < 0, we have

ε(t) = X(t) = 0. (HA2 FCV M ) θ ∈ L 2 (R, R) and for every t < 0, θ (t) = 0. (HA3 FCV M ) The expectations E(∥ε∥ 2 L 1 ), E(∥X∥ 2 L 1 ), E(∥ε∥ 2 L 2
) and E(∥X∥ 2 L 2 ) are all finite. Under the assumption that for every t < 0, θ (t) = X(t) = 0, obtained from (HA1 FCV M ) and (HA2 FCV M ), it is possible to write the integral in model (3.1) over the whole real line, that is Y (t) = +∞ -∞ θ (s)X(ts)ds + ε(t). It allows to use the CFT to transform the convolution into a multiplication.

Functional Fourier Deconvolution Estimator (FFDE)

To define the FFDE we proceed in three steps. i) First we use the CFT to transform the convolution in the time domain into a simple multiplication in the frequency domain (see (3.2)). ii) Once in the frequency domain, we estimate β with the Functional Ridge Regression Estimator (FRRE) defined in [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF], which is an extension of the Ridge Regularization method [START_REF] Hoerl | Application of ridge analysis to regression problems[END_REF]) that deals with ill-posed problems in the classical linear regression. iii) The last step consists in using the ICFT to estimate θ . Let (X i ,Y i ) i=1,••• ,n be an i.i.d sample of FCVM (3.1) and λ n be a positive regularization parameter. The FRRE of β in the FCCM (3.2) is defined as follows

βn := 1 n ∑ n i=1 Y i X * i 1 n ∑ n i=1 |X i | 2 + λ n n , (3.3)
where the exponent * stands for the complex conjugate,

Y i = F (Y i ) and X i = F (X i ).
Finally the FFDE of θ in (3.1) is defined by θn := F -1 ( βn ).

(3.4)

Asymptotic Properties of the FFDE

The main result of this paper is the probability convergence of the FFDE with a rate of convergence

∥ θn -θ ∥ L 2 = O P max λ n n , √ n λ n ,
under large conditions. The CFT is an isometry in the L 2 -space. Thus the study of the asymptotic behavior of ∥ θnθ ∥ L 2 is equivalent to that of ∥ βnβ ∥ L 2 . We use this important fact to show the consistency of the FFDE. In addition, it is useful to notice that βnβ can be decomposed as follows

βn -β = - λ n n β 1 n ∑ n i=1 |X i | 2 + λ n n + 1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n .
(3.5) Equation (3.5) shows the relationship between the frequencies of θ and X (β and X respectively) through the ratio β /|X | 2 . This ratio plays a central role in the asymptotic behavior of the FFDE. The consistency is obtained in Theorem 15 under assumption (A1) related to this relationship.

Consistency of the Estimator

Theorem 15. Let us consider the FCVM with the general hypotheses (HA1 FCV M ), (HA2 FCV M ) and (HA3 FCV M ). Let (X i ,Y i ) i≥1 be i.i.d. realizations of (X,Y ) according to this model. Additionally we suppose that

(A1) supp(|F (θ )|) ⊆ supp(E[|F (X)|]), (A2) (λ n ) n≥1 ⊂ R + is such that λ n n → 0 and √ n λ n → 0 as n → +∞.
Then

lim n→+∞ ∥ θn -θ ∥ L 2 = 0 in probability. (3.6)
Remark. Hypothesis (A2) is classic when studying asymptotic behavior. Hypothesis (A1) specifies that it is not possible to estimate F (θ ) outside the support of E[|F (X)|]. In the intervals where

E[|X |] = 0, model (3.
2) reduces to Y = ε and no estimation of β is possible.

A practical interpretation of Hypothesis (A1) is that the FFDE will only converge toward θ if all the non zero frequencies of θ belong to the set of the non-zero frequencies of E[X], that is, there are enough non-zero frequencies of E[X] to "save" the information about the non-zero frequencies of θ . This allows to reconstruct θ through the ICFT.

Rate of Convergence

To obtain the rate of convergence we need to assume a stronger hypothesis about the relationship between the frequencies of θ and X. Hypothesis (A4) in Theorem 16 is one way to do this.

Theorem 16. Let us consider the FCVM with the general hypotheses (HA1 FCV M ), (HA2 FCV M ) and (HA3 FCV M ). We assume additionally that:

(A3) E[∥X∥ 4 L 2 ] < ∞, (A4) |F (θ )| E[|F (X)| 2 ] 1 supp(|F (θ )|) ∈ L 2 ∩ L ∞ , then ∥ θn -θ ∥ L 2 = O P max λ n n , √ n λ n . (3.7)
Remark. Hypothesis (A3) is classic and allows to apply the CLT on the denominator of the decomposition of βnβ in (3.5). Hypothesis (A4) implies (A1) and is required to control the first term of (3.5) and the decreasing rate of F (θ ) with respect to E[|F (X)| 2 ] when frequencies go to infinity (tails control). More precisely (A4) implies the strict inclusion, supp(F (θ )) ⊂ supp(E[|F (X)|]). This condition might be too strong when for instance supp(E[|F (X)|]) = R\S, where S is a non-dense and countable set and supp(F (θ )) ∩ S ̸ = / 0. In Theorem 23, given in Appendix 3.C, we weaken Hypothesis (A4) to also obtain the rate of convergence in this case. Nevertheless the hypotheses which replace (A4), namely (A4bis) and (A5) are more difficult to interpret.

The ratio |F (θ )| E[|F (X)| 2 ]
in Hypothesis (A4), is a way to measure the regularity of F (θ ) with respect to that of E[|F (X)| 2 ]. The fact that this ratio belongs to L 2 ∩ L ∞ can be interpreted as the fact that θ is required to be more regular than E[X]. For example it implies that F (θ ) decreases to 0 at infinity faster than E[|F (X)| 2 ]. This kind of regularity assumption is commonly used in Functional Linear Regression problems, where the unknown function of the regression model is more regular than the covariate X (see e.g. [START_REF] Cardot | Functional linear model[END_REF] and [START_REF] Cardot | Spline estimators for the functional linear model[END_REF]).

Finally Theorem 17 deals with the convergence rate on compact subsets of the support of

E[|F (X)| 2 ]
without using the Hypothesis (A4). This theorem is useful when the support of F (θ ) is compact.

Theorem 17. Under hypotheses (A1) and (A3), for every compact subset K

⊂ supp(E[|F (X)|]), we have ∥ θn -θ ∥ L 2 (K) = O P max λ n n , √ n λ n . (3.8) In particular if supp(F (θ )) is compact and is a subset of supp(E[|F (X)|]), then ∥ θn -θ ∥ L 2 = O P max λ n n , √ n λ n .

Selection of the Regularization Parameter

In this section we introduce a selection procedure of the regularization parameter λ n for a given sample

(X i ,Y i ) i∈{1,••• ,n} .
We chose the Leave-one-out Predictive Cross-Validation (LOOPCV) criterion. Its definition can be found in Febrero-Bande and Oviedo de la Fuente (2012, p. 17) or Hall and Hosseini-Nasab (2006, p. 117) and is the following

LOOPCV (λ n ) := 1 n n ∑ i=1 ∥Y i -Conv( θ (-i) n , X i )∥ 2 L 2 , where θ (-i) n is computed with the sample (X j ,Y j ) j∈{1,••• ,i-1,i+1,••• ,n} and for t ≥ 0, Conv( θ (-i) n , X i )(t) := t 0 θ (-i) n
(s) X i (ts) ds denotes the convolution.

The selection method consists in choosing the value λ n which minimizes the function LOOPCV . Proposition 18 shows a way to compute the LOOPCV with only one regression instead of n while working directly in the frequency domain. The proof is based on the fact that the CFT is an isometry and on similar ideas as in Green and Silverman (1994, pp. 31-33) about the smoothing parameter selection for smoothing splines.

Proposition 18. We have

LOOPCV (λ n ) = 1 n n ∑ i=1 Y i -βn X i 1 -A i,i 2 L 2 , (3.9) where A i,i ∈ L 2 is defined as follows A i,i := |X | 2 i /(∑ n j=1 |X j | 2 + λ n )
and βn , X i and Y i are defined as in (3.3).

Simulation study

The simulation study follows model (3.1) when the supports of all the involved functions are included in the interval [0, T ], for some fixed value T > 0. We want to illustrate the performance of the FFDE To calibrate the parameters of both estimators we do not use the LOOPCV but the 10-fold Predictive Cross Validation (see Seni and Elder (2010, Ch 3)) to avoid redundancy in calculations due to the use of the mean before inverting M X in the first step of these two methods.

Laplace estimator (Lap): We use the adapted version of the Laplace estimator introduced by [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging[END_REF], denoted here as θLap . We start by calculating the mean of all realizations to eliminate the noise as much as possible since this estimator is designed to solve the problem when n = 1 (one couple of X and Y ). Thus we obtain for j

= 1, • • • , p -1, Ȳ (t j ) = t i 0 θ (s) X(t j -s)ds + ε(t j ) (3.12)
where t 0 , • • • ,t p-1 ∈ R are the observation times.

In [START_REF] Comte | Laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging[END_REF] this equation is interpreted as a discrete noisy version of the linear Volterra equation of the first kind, where the goal is to estimate θ . More precisely the authors use a model where ε

(t i ) are i.i.d sub-Gaussian random variables such that E[ε(t i )] = 0 and E[|ε(t i )| 2 ] = σ 2 .
To estimate θ , the authors use the Laguerre functions, defined for k ∈ N, t ≥ 0 and some fixed a > 0 as follows

φ k (t) := √ 2ae -at k ∑ j=0 (-1) j k j t j j! .
First they use these functions as an orthonormal basis of L 2 (R + , R) to transform equation (3.12) into an infinite system of linear equations with coefficients obtained from the expansion in the Laguerre basis. They chose the Laguerre functions because the convolution of a couple of these functions is easy to obtain, and satisfies that for k, l ≥ 0,

t 0 φ k (s)φ l (t -s)ds = (2a) -1/2 [φ k+l (t) -φ k+l+1 (t)].
Thanks to this fact the latter system is simplified and becomes an infinite lower triangular system of linear equations. Next they solve the finite subsystem of the first M linear equations to compute the estimators of the first M coefficients of θ on the Laguerre basis. The numerical computation of their estimator is done with the R package LaplaceDeconv. In order to avoid numerical instability, due to the computation of Laguerre functions in R, we resize the curves from [0, T ] to the interval [0, 10] (stretching the curves) but keeping the SNR equal to 10. In this way to estimate θ we use the initial curves X i and Y i stretched to [0, 10] together with the noise with standard deviation equal to σ /n. After computing the Laplace estimator with this data we multiply this one by 10/T to resize it. Notice that the true value of σ is necessary to compute θlap both theoretically (the authors use it to penalize the estimator during the calibration of parameters) and numerically.

Remark : In practice after computing all the estimators defined in this section and the FFDE we have used the spline smoothing method to smooth all of them. This step improves their estimation performance.

Settings

We compared these estimation procedures in three different simulation settings. The goal is to compare how well the FFDE estimates θ with respect to the performance of the others. In the first setting the X variable is such that E[X] = 0 which is a situation where the estimation is more difficult, in particular for SVD and Tik, because they need to invert the associated matrix M X (see definition of the SVD estimator). The second setting uses E[X] ̸ = 0 and here the inversion of M X is numerically more stable. In this setting the shape of θ has some periodicity, thus one goal is to asses how well the methods can estimate this periodicity and another is to experience FFDE under favorable conditions for SVD and Tik. The last setting uses θ and X which are well represented with the Laguerre functions. This is a favorable condition for the Laplace estimator (Lap). We want to see how the others perform under this condition.

Let us detail each setting. For settings 1 and 2 the data were simulated on the interval [0, 1] (T = 1), discretized over p = 100 equispaced observation times, t j := j/100, with j = 0, • • • , 99. Whereas for Setting 3 the interval is [0, 8] (T = 8), with p = 100 equispaced observation times t j := 8 j/100, for j = 0, • • • , 99.

In the Table 3.1 we describe the curves X i and the functions θ for each setting. In that table BB i stands for the Brownian Bridge on the interval [0, 0.5] with the process pinned at the origin at both t = 0 and t = 0.5, for every i = 1, • • • , n. On the other hand for settings 1 and 2 we use 1 [0,0.5] , the indicator function of the interval [0, 0.5], because we want the support of Y to be [0, 1] given that supp(Y ) = supp(X) + supp(θ ). In contrast to those settings, in setting 3 the supp(Y ) is bigger than [0, 8], however the estimation with FFDE is still possible due to the fact that the values of Y (t) for t > 8 are relatively small compared to the values for t ∈ [0, 8]. Note that in general the condition supp(X) + supp(θ ) = supp(Y ) ⊆ [0, T ] is necessary to compute numerically the FFDE. Indeed, in this case the CFT can properly transform the convolution between X and θ into a multiplication in the frequency domain.

For all these settings the noise ε is the White Gaussian Noise defined with a standard deviation σ (σ is constant and for every t

∈ [0, T ], σ 2 = E[|ε(t)| 2 ]
) chosen for each setting such that the Signal-to-Noise-Ratio (SNR) is equal to 10 (interpreted as 10% of noise). Here the SNR is defined as

SNR := E[∥θ * X∥ 2 L 2 ]/σ 2 .
Note also that for each setting we have numerically verified that the general hypotheses (HA1 FCV M ) -(HA3 FCV M ) are satisfied.

We evaluate our estimation procedure for sample of sizes n = 70 and n = 400. Additionally we use the two following criteria to measure the estimation error. Now we discuss the estimation performance for each setting separately because they have been chosen to assess various properties of the FFDE under different situations. More specifically we can see in Table 3.3 and in the box plots in Figure 3.2 that FFDE and ParWD are the best estimators, whereas SVD is the worst of all of them. When the sample size increases to n = 400, FFDE is the one which has improved the most.

In this setting FFDE and ParWD handle well the case where E[X] = 0, because they use the Fast Fourier Algorithm (FFT) to directly deconvolve the convolution of X and θ , whereas SVD and Tik perform badly because they cannot properly invert the matrix M X used in their definitions. Besides note that Lap does not improve the estimation because we apply it to the mean equation (3.12), which is almost the same when n = 70 and n = 400, this fact will also be true for Settings 2 and 3. Finally although SVD and Tik use the mean equation (3.12), they slightly improve due to the strong dependency of the inversion of M X on the noise. In this setting Lap performs better than the others because both X and θ are functions well represented with the Laguerre functions. However all the other estimators show a great improvement when n = 400. This shows that SVD and Tik give good estimations as long as E[X] ̸ = 0. Finally FFDE is almost as good as SVD.

Setting 2 :

A further discussion about FFDE

In each of the three settings we have seen that the FFDE performed well with very fast computation time and convergence towards θ as the sample size increases. It gives a good estimation in these three settings, even in the disadvantageous case where E[X] = 0 and thus the noise plays a major role.

We note an edge effect for small sample sizes that decreases as n goes to infinity. This effect comes from the second component of the decomposition of θn derived from (3.5), namely

F -1 (Ψ n ) := - λ n n F -1 β 1 n ∑ n i=1 |X i | 2 + λ n n .
In Figure 3.7 we see the F -1 (Ψ n ) components for each of the three settings. One of the reasons of this shape is that Φ := E[|X | 2 ] (denominator) is highly concentrated on the borders. This is shown in Figure 3.8, where for each setting we approximate Φ by the empirical mean with n = 7000. Note that all these functions are positive over the whole interval despite what might be assumed from Figure 3.8. From these reasons the difference θnθ will have higher values close to the borders (edge effect) since θnθ ≈ F -1 (Ψ n ). Note that when n increases we have λ n /n → 0 and thus Ψ n → 0, so the edge effect will decrease. This fact is observed in the simulation studies when we increase the sample size to n = 400.

Whenever E[|X | 2 ] has higher values close to the borders the edge effect should be expected. In that case we propose the practical solution of using the estimation over an appropriate interval before the borders to extrapolate the estimation in the borders. The results of this method are shown in Figure 3.9. We took 10% before the borders (last 5% in each side) to extrapolate over them, we did this for each one of the 100 realizations. The cross-sectional mean of the FFDE estimator before and after removing the edge effect are in green and in red respectively. Fig. 3.9 Estimators of θ for each setting. The cross-sectional mean of the FFDE estimator before and after removing the edge effect are the curves in green and in red respectively.

The boxplots of the MADE and WASE criteria before (FFDE) and after removing the edge effect (FFDE.no.ed) are shown in Figure 3.10. We see there that a major improvement in the estimation is done in the setting 1, whereas in settings 2 and 3 this improvement is small and WASE is changing the most.

Conclusions

In this paper we have defined the FFDE for the FCVM. We proved its consistency for the L 2 -norm and obtained a rate of convergence. We also provided a selection procedure of the regularization parameter λ n through the LOOPCV criterion. The simulations showed the robustness of the FFDE despite some irregularities on the borders (edge effect). This effect can be reduced by using the estimation over an appropriate interval before these borders.

Compared to other estimation methods adapted from the literature, FFDE is almost as good as the best estimator in all the three settings and always with the fastest computation time. Proposition 22 (Proposition 4.1). We have

PCV (λ n ) = 1 n n ∑ i=1 Y i -βn X i 1 -A i,i 2 L 2 , (3.16)
where A i,i ∈ L 2 is defined as follows A i,i :

= |X i | 2 /(∑ n j=1 |X j | 2 + λ n ).

3.B Proofs

Throughout these proofs we use the notation of the associated functional concurrent model (3.2).

Proof of Theorem 15. We use a modified version of Theorem 3.1 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] to prove Theorem 15 in this paper. In order to do this let us recall the three general hypotheses used to prove Theorem 3.1 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] rewritten with the notations of (3.2).

(HA1 FCM ) X , ε are independent

C 0 ∩ L 2 valued random functions, such that E(ε) = E(X ) = 0, (HA2 FCM ) β ∈ C 0 ∩ L 2 , (HA3 FCM ) E(∥ε∥ 2 C 0 ), E(∥X ∥ 2 C 0 ), E(∥ε∥ 2 L 2 ) and E(∥X ∥ 2 L 2
) are all finite. Given that we are interested in a more general version of Theorem 3.1, we will change (HA1 FCM ) for (HA1bis FCM ) defined as follows (HA1bis FCM ) X , ε are independent C 0 ∩ L 2 valued random functions, such that E(ε) = 0.

Our goal is to prove that (HA1bis FCM ), (HA2 FCM ) and (HA3 FCM ) are implied by the general hypotheses of the FCVM (see subsection 3.2.1), and then to prove a generalization of Theorem 3.1 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] with (HA1bis FCM ) instead of (HA1 FCM ).

First we show that the hypotheses (HA1bis FCM ) and (HA2 FCM ) are satisfied. Given that θ ∈ L 1 , then β ∈ C 0 (R, C) (see Pinsky (2002, Ch. 2)). Moreover since F is an isometry in L 2 we obtain β ∈ L 2 (R, C). Thus hypothesis (HA2 FCM ) holds. In a similar way we prove that X , ε ∈ C 0 (R, C) ∩ L 2 (R, C). The linearity of F implies E[F (ε)] = 0 so (HA1bis FCM ) holds too.

We use the contraction property of F , namely Pinsky (2002, Ch. 2)) and again the fact that F is an isometry to prove that (HA3 FCM ) holds.

∥F ( f )∥ C 0 ≤ ∥ f ∥ L 1 (see
Next we outline the proof of a generalization of Theorem 3.1 (see Theorem 19 in Appendix 3.A), in which we use (HA1bis FCM ) instead of (HA1 FCM ). First we need to prove Let us prove 3.17. We have

1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n L 2 = O P √ n λ n , (3.17
E[∥ε X * ∥ 2 L 2 ] ≤ E[∥ε∥ 2 C 0 ] E[∥X ∥ 2 L 2 ] < ∞,
because of (HA1bis FCM ) and (HA3 FCM ). Now due to the moment monotonicity E[∥ε X * ∥ L 2 ] < ∞, ε X * is strongly integrable with the L 2 -norm, so there exists the expectation E[ε X * ] ∈ L 2 which is the zero function because

E[ε] = 0. We conclude that E[ε X * ] = 0 and E[∥ε X * ∥ 2 L 2 ]
< ∞ which, from the CLT in L 2 (see Theorem 2.7 in Bosq (2000, p. 51) and Ledoux and Talagrand (1991, p. 276) for the rate of convergence), yields to

1 n n ∑ i=1 ε i X * i L 2 = O P 1 √ n .
Finally (3.17) is obtained from the fact that

1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n L 2 ≤ n λ n 1 n n ∑ i=1 ε i X * i L 2 = O P √ n λ n .
Notice that hypotheses (A1) and (A2) of Theorem 3.1 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] (Theorem 19 in Appendix 3.A) are implied by hypotheses (A1) and (A2) of Theorem 15, and the normed functions in (3.17) converge in probability to zero.

Finally with the same argument as in the proof of Theorem 3.1 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] (Theorem 19 in Appendix 3.A) it is possible to prove

λ n n β 1 n ∑ n i=1 |X i | 2 + λ n n L 2
a.s.

--→ 0, (3.18)

and thus the triangular inequality applied to the decomposition (3.5) implies that ∥ βnβ ∥ L 2 goes to zero in probability.

Proof of Theorem 16. This is a direct consequence of Corollary 3.7 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] because hypotheses (A3) and (A4bis) of this corollary are consequences of (A3) and (A4) in Theorem 16.

Proof of Theorem 17. We start with the triangle inequality applied to (3.5) but restricted to the compact subset K,

∥ βn -β ∥ L 2 (K) ≤ λ n n β 1 n ∑ n i=1 |X i | 2 + λ n n L 2 (K) + 1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λ n n L 2 (K)
.

The proof of

1 n ∑ n i=1 ε i X * i 1 n ∑ n i=1 |X i | 2 + λn n L 2 (K) = O P ( √ n
λ n ) is the same as in Theorem 15. To finish the proof of this theorem we prove

β 1 n ∑ n i=1 |X i | 2 + λ n n L 2 (K) = O P (1), (3.19)
which is done with the same method used in Theorem 3.8 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF].

Proof of Proposition 18. This is a direct consequence of Proposition 4.1 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF].

3.C Generalization of Theorem 16

Theorem 23. For the FCVM which satisfies the general hypotheses (HA1 FCV M ), (HA2 FCV M ) and (HA3 FCV M ), hypotheses (A1) in Theorem 15 and (A3) in Theorem 16, we additionally assume (A4bis) (3.20) where γ := min 1 2(2α+1) , 1 2 -1 2(2α+1) .

|F (θ )| E[|F (X)| 2 ] 1 supp(F (θ ))\∂ (supp(E[|F (X)|])) L 2 < ∞, (
E[|F (X)| 2 ] L 2 (J p \{p}) ≤ M 0 , (b) ∑ p∈C θ ,∂ X ∥β ∥ 2 C 0 (J p ) < M 1 , (c) |F (θ )| E[|F (X)| 2 ] 1 supp(|F (θ )|)\J < M 2 , where J = p∈C θ ,∂ X J p , (A6) For n ≥ 1, λ n := n 1-1 4α+2 . Then ∥ θn -θ ∥ L 2 = O P n -γ ,
Proof. As in the proof of Theorem 15, it is easy to show that X , ε, β and Y satisfy all the hypotheses of Theorem 3.4 of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF], then ∥ βnβ ∥ L 2 = O P (n -γ ). The isometry property of the CFT ends the proof.

3.D Numerical Implementation of the FFDE

In this appendix we discuss how we estimate θ in the FCVM in practice. In particular we describe the necessity to rethink the FCVM in a finite discrete way, and to use the Discrete Fourier Transform as the discrete equivalent of the Continuous Fourier Transform in this new context. We start by describing the discretization of the convolution. To do this properly we start with some definitions. Throughout this appendix we use ∆ as the discretization step between two observation times (for instance ∆ = 0.01). The observation times are defined for every j ∈ Z as t j := j * ∆ and thus they define the grid G ∆ over R. We use a fix grid in this appendix. With this grid we transform each function f : R → C to a vector f d ∈ C Z infinite dimensional, with elements

f d j := f (t j ) ∈ C.
In what follows the superscript d will denote this discretization.

Besides here all the functions will have compact support. Otherwise we should compute the convolution of infinite vectors which cannot be done in practice. For simplicity we consider all the functions defined over a compact interval [0, T ] with T large enough. Thus we will consider

f d = ( f d 0 , • • • , f d q-1 ) ∈ C q , where q -1 = max{ j ∈ N |t j ∈ [0, T ]}.
Let RM (rectangular method) be the operator which associates to an integral over R, its numerical approximation by the rectangular method over the grid of points we have already defined. Thus for a given integral J = R f (s)ds = T 0 f (s)ds we associate RM(J) := ∆ ∑ q-1 j=0 f (t j ) = ∆ ∑ q-1 j=0 f d j . Understanding how to compute numerically the convolution of two functions is a key element to implement the estimator developed for the FCVM.

We start our discussion by describing the discretization of the convolution of two functions with support included on [0, T ],

f * g(t) := +∞ -∞ f (s)g(t -s)ds = T 0 f (s)g(t -s)ds.
Approximating this convolution with the rectangular method we obtain for every j ∈ N,

RM( f * g)(t j ) = q-1 ∑ l=0 f (t l ) g(t j-l ) ∆ = ∆ q-1 ∑ l=0 f d l g d j-l . (3.21)
The last sum in equation (3.21) is the convolution between vectors. Thus we can rewrite this equation as follows

RM( f * g)(t j ) = ∆ ( f d * g d ) j . for j ∈ [0, • • • , 2p -2] and where ( f d * g d ) j := ∑ q-1 l=0 f d l g d j-l . Besides note that for j / ∈ [0, • • • , 2p -2]
we have RM( f * g)(t j ) = 0 since f and g have compact support.

Additionally we can compute the vector ((

f d * g d ) 0 , • • • , ( f d * g d ) 2q-2
) using matrices as follows (3.22) where MC G is the matrix associated to the convolution discretized over the grid G, defined as follows

( f d * g d )(0), • • • , ( f d * g d )(2q -2) T = MC G ( f d 0 , • • • , f d q-1 ) T ,
MC G :=                         g d 0 0 0 0 • • • 0 g d 1 g d 0 0 0 • • • 0 g d 2 g d 1 g d 0 0 • • • 0 . . . . . . . . . . . . • • • . . . g d q-2 • • • • • • g d 1 g d 0 0 g d q-1 g d q-2 • • • • • • • • • g d 0 0 g d q-1 g d q-2 • • • g d 2 g d 1 0 0 g d q-1 • • • • • • g d 2 . . . . . . . . . . . . • • • . . . 0 0 • • • 0 g d q-1 g d q-2 0 0 0 • • • 0 g d q-1                         ∈ R (2q-1)×q .
Remark : From this fact we note that the convolution could have a larger support. This arises because an important property of the convolution is that supp( f * g) ⊂ supp( f ) + supp(g) (Brezis (2010, p. 106)). Thus in our case supp( f * g) ⊂ [0, 2 T ]. However afterwards we will take T large enough to contain even the convolution. In this way, every time we will consider the convolution of two functions f and g we suppose supp( f ) + supp(g) ⊂ [0, T ]. In this case the number of discretization points q will be defined as before, namely q -1 = max{ j ∈ N |t j ∈ [0, T ]} but now for all j ≥ q, ( f d * g d ) j = 0. Besides the matrix representation of the convolution through MC G will still be correct.

In the following subsection we explore the parallel between the continuous convolution of two functions and the convolution of two vectors with respect to the whole model FCVM.

3.D.1 The Discretization of the FCVM and the FFDE

We have defined the functional Fourier deconvolution estimator of θ in the FCVM using the continuous Fourier transform and its inverse (equations (3.3) and (3.4)). Given that both operators are integral operators, we need to use some kind of numerical approach to compute them. The goal of this subsection is to show that the proper way for doing this is by using a discrete model which behaves like the FCVM. This model will be based on the convolution of finite dimensional vectors. It will be studied through the discrete Fourier transform and its inverse instead of their continuous counterparts.

First let us show that it is not practical to compute the functional Fourier deconvolution estimator by direct approximation of the continuous Fourier transform and its inverse. This is not possible because these two operators are integrals defined over the whole R. To see why this is a problem let us consider a function f ∈ L 2 with compact support. Then although it is possible to use the Rectangular Method to compute F ( f )(ξ ) for every value ξ , we cannot ensure that F ( f ) has compact support ( (Kammler, 2008, p. 130)). This implies that we need to know the values of F ( f ) for all the infinite values of the grid G ∆ to approximate the F -1 , which is impossible in practice. Note that even if F ( f ) has a compact support we cannot know how large it is and in this case we will need to compute F ( f ) over too many points of the grid which again makes the approximation unpractical.

Instead of using the direct approximation of the continuous Fourier transform and its inverse, another approach is to propose a finite discretized version of the FCVM, which reflects the main characteristics of the FCVM. In order to achieve this, note two important things: i) the convolution of two functions can be approached by as the convolution of two vectors and ii) the convolution of two vectors is transformed into a multiplication with the discrete Fourier transform ( (Kammler, 2008, p. 102), Oppenheim and Schafer (2011, p. 60)).

Here we use the definition of the discrete Fourier transform found in Kammler (2008, p. 291) or in Bloomfield (2004, p. 41), defined for vectors of C q as follows

F d : C q → C q f := ( f 0 , • • • , f q-1 ) → (F d ( f )(0), • • • , F d ( f )(q -1)) ,
where for every l = 0, • • • , q -1,

F d ( f )(l) := 1 q q-1 ∑ r=0 f r ω rl ∈ C. (3.23)
with ω := e -2πi/q . If we define the matrix

Ω q :=          1 1 1 • • • 1 1 (ω 1 ) 1 (ω 1 ) 2 • • • (ω 1 ) (q-1) 1 (ω 2 ) 1 (ω 2 ) 2 • • • (ω 2 ) (q-1) . . . . . . . . . . . . . . . 1 (ω (q-1) ) 1 (ω (q-1) ) 2 • • • (ω (q-1) ) (q-1)          (3.24)
we can write

F d ( f ) = 1 q Ω k f ∈ C q . (3.25)
Furthermore from this definition we can deduce

F -1 d = Ω * q , (3.26)
where Ω * q is the conjugate transpose of Ω q .

Remark: We can see that the definition of F d depends on the number q, which is the length of the vector. In this way when we apply F d to a vector of size p we need to redefine the matrix Ω p by using ω := e -2πi/p .

Finite Discrete version of the FCVM Let us take T large enough such that [0, T ] contains supp(X) + supp(θ ). Thus the supports of θ , X and Y are also contained in [0, T ] (Brezis (2010, p.

106)). Let us define q -1 = max{ j ∈ N |t j ∈ [0, T ]}. Now take the discretization of each function X i and Y i of the sample

(X i ,Y i ) i=1,••• ,n over the grid [t 0 , • • • ,t q-1 ]
, so all these functions will become

vectors in R q ⊂ C q , that is X d i ,Y d i ∈ C q for every i = 1, • • • , n.
Given that the matrix Ω q has the property of transforming finite convolutions into multiplications, we can use the three steps method as the one used to define the estimator θn for the continuous case, namely i) transform the problem with the matrix Ω q from the time-domain to the frequency one, ii) use the ridge estimator in this domain, and iii) finally come back with the inverse of Ω q .

The comparison between the continuous and the discrete cases is done next. Note that in the discrete case the multiplication and the division is done the element by element between vectors of same length. Furthermore, * d is discrete convolution, ∆ is the step of discretization and we use P q : R 2q-1 → R q , the projection into the first q components, to have vectors of the same length.

CONTINUOUS

Data and conditions

: θ ∈ L 2 ([0, T ]). For i = 1, • • • , n, X i ,Y i , ε i ∈ L 2 ([0, T ]), Y i = θ * X i + ε i .
Estimation steps:

1. For i = 1, • • • , n, F (Y i ) = F (θ )F (X i ) + F (ε i ). 2. F (θ ) n := ∑ n i=1 F (Y i )F (X i ) ∑ n i=1 |F (X i )| 2 + λ n 3. θn := F -1 ( F (θ ) n )

DISCRETE

Data and conditions:

θ d ∈ R q . For i = 1, • • • , n, X d i ,Y d i , ε d i ∈ R q , Y d i = ∆ P q (θ d * d X d i ) + ε d i .
Estimation steps:

1. For i = 1, • • • , n, Ω q (Y d i ) = ∆Ω d q (θ d )•Ω q (X d i )+Ω q (ε d i ). 2. Ωq (θ d ) n := 1 ∆ ∑ n i=1 Ω q (Y d i )Ω q (X d i ) ∑ n i=1 |Ω q (X d i )| 2 + ⃗ λ n , where ⃗ λ n := (λ n , • • • , λ n ) ∈ R q . 3. θ d n := Ω -1 q ( Ωq (θ d ) n ).
the convolution into a multiplication of two vectors only holds when Ω q is applied to the entire convolution of both vectors, that is q is big enough to contain the convolution.

In any case in order to estimate θ with the functional Fourier deconvolution estimator, the grid of observations should cover supp(X) and supp(Y ). This is an important restriction of this estimator.

FFT Algorithm and fast computing : One of the main advantages of the functional Fourier deconvolution estimator is that it is calculated very fast. This is due to the fact that it uses the Fast Fourier Transform (FFT) to compute the discrete Fourier transform. It is known that this algorithm computes the discrete Fourier transform of an n-dimensional signal in O(n log(n)) time. The publication of the Cooley-Tukey FFT algorithm in 1965 (Cooley and[START_REF] Cooley | An algorithm for the machine calculation of complex fourier series[END_REF]) revolutionized the area of digital signal processing because it reduced the order of complexity of the Fourier transform and of the convolution from n 2 to n log(n), where n is the problem size. Then over the last years new algorithms have improved the performance of the Cooley-Tukey algorithm under some conditions (split-radix FFT, Winograd FFT, etc). Among the recent improvements we highlight the Nearly Optimal Sparse Fourier Transform [START_REF] Hassanieh | Nearly optimal sparse fourier transform[END_REF]).

For any integer k, we define Π k := ∑ k j=1 v j ⊗v j the projection operator on the sub-space ⟨v 1 , • • • , v k ⟩. By projecting Γ on this sub-space, we get :

Γ| ⟨v 1 ,••• ,v k ⟩ := ΓΠ k = k ∑ j=1 λ j v j ⊗ v j .

Construction of the estimator of S

We start from the moment equation

∆ = S Γ. (4.3) On the sub-space ⟨v 1 , • • • , v k ⟩, the operator Γ is invertible, more precisely (ΓΠ k ) -1 = ∑ k j=1 λ -1 j v j ⊗ v j .
As a consequence, with equation ( 4.3) and the fact that Π k ΓΠ k = ΓΠ k we get, on the sub-space

⟨v 1 , • • • , v k ⟩, ∆Π k = (S Π k ) (ΓΠ k ). We deduce that S Π k = ∆Π k (ΓΠ k ) -1 . Now, taking k = k n , denoting Πk n := ∑ k n j=1 v j ⊗ v j and the generalized inverse Γ+ k n := ( Γn Πk n ) -1
, we are able to define the estimator of S. We have

Γn = ∞ ∑ j=1 λ j v j ⊗ v j = n ∑ j=1 λ j v j ⊗ v j , with eigenvalues λ1 ≥ • • • ≥ λn ≥ 0 = λn+1 = λn+2 = • • • ∈ R 1 and orthonormal eigenfunctions v1 , v2 , • • • ∈ H. By taking λk n > 0, with k n < n, we define the operator Γk n = ∑ k n j=1 λ j v j ⊗ v j and we get Γ+ k n = ∑ k n j=1 ( λ j ) -1 v j ⊗ v j .
Hence we define the estimator of S as follows

Ŝk n = ∆n Γ+ k n . (4.4)
Finally, the associated kernel of Ŝk n , estimating S , is

Ŝk n (t, s) = 1 n n ∑ i=1 k n ∑ j=1 1 λ j 1 0 X i (r) v j (r)dr Y i (t) v j (s) . (4.5)
4.3 Estimation of Γ ε and its trace

The plug-in estimator

The plug-in estimator of Γ ε is given by Γε

,n := 1 n -k n n ∑ i=1 (Y i -Ŝk n X i ) ⊗ (Y i -Ŝk n X i ) = 1 n -k n n ∑ i=1 εi ⊗ εi . (4.6)
This estimator is biased, for a fixed n, as stated in the next theorem:

Theorem 24. Let (X i ,Y i ) i=1,...,n be a sample of i.i.d. observations following model (4.1). Let k n < n be an integer. We have

E[ Γε,n ] = Γ ε + n n -k n S E n ∑ i=k n +1 λi vi ⊗ vi S ′ . (4.7)
The proof of Theorem 24 is postponed in Section 4.5.1. As Γk n = ∑ n i=1 λi vi ⊗ vi and Π(k n +1):n := ∑ n i=k n +1 vi ⊗ vi , we deduce the following result:

Corollary 25. We have

E[ Γε,n ] = Γ ε + n n -k n S E Π(k n +1):n Γn S ′ , (4.8)
where Π(k n +1):n is the projection on the sub-space ⟨ vk n +1 , • • • , vn ⟩.

Under some additional assumptions, we prove that the plug-in estimator (4.6) of Γ ε is asymptotically unbiased. Let us consider the following assumptions:

(A.1) The operator S is a nuclear operator, in other words ∥S∥ N < +∞.

(A.

2) The variable X satisfies E ∥X∥ 4 < +∞. (4.10)

Other estimation of Γ ε

Without loss of generality, we assume in this section that n is a multiple of 3. In formula (4.8), the bias of the plug-in estimator is related to S E Π(k n +1):n Γn S ′ . Another way of estimating Γ ε is thus to subtract an estimator of the bias to the plug-in estimator Γε,n . To achieve this, we split the n-sample into three sub-samples with size m = n/3 to keep good theoretical properties thanks to the independence of the sub-samples. As a consequence, we define

Bn := Ŝ[2] 2k m Π[1] (k m +1):m Γ[1] m Ŝ[3] 2k m ′ , (4.11)
where the quantities with superscripts [1], [2] and [3] are respectively estimated with the first, second and third part of the sample. We use 2k m eigenvalues (where k m ≤ n/2) in the estimation of S with the second and third sub-sample in order to avoid orthogonality between Ŝ

[2] 2k m , Ŝ[3] 2k m and Π[1] (k m +1):m Γ[1] m .
We are now in a position to define another estimator of Γ ε :

Γε,n := Γ[1] ε,m - m m -k m
Bn .

(4.12)

The following result is established.

Theorem 28. Under the assumptions of Theorem 26, we have

lim n→+∞ E Γε,n -Γ ε N = 0. ( 4.13)
The above result can also be written using the trace.

Comments on both estimators

Subtracting an estimator of the bias to the plug-in estimator Γε,n does not provide an unbiased estimator of Γ ε,n . The situation is completely different to that of multivariate multiple regression models, see [START_REF] Johnson | Applied Multivariate Statistical Analysis[END_REF], where an unbiased estimator of the noise covariance is easily produced. Both estimators Γε,n and Γε,n are consistent. We can see from the proofs of Theorems 26 and 28 that

E Γε,n -Γ ε N ≤ n n-k n ∥S∥ N ∥S ′ ∥ ∞ E λk n +1 , and E Γε,n -Γ ε N ≤ 2 n n-3k m ∥S∥ N ∥S ′ ∥ ∞ E λk m +1
. Number 2 in the estimation bound of Γε,n is due to the use of the triangle inequality. In this way, we cannot prove that subtracting the bias may improve the estimation of Γ ε , nor of its trace. We will study the behavior of both estimators by simulations in the next section.

Cross validation and Generalized cross validation

Whatever the estimator, we have to choose a dimension k n of principal components in order to compute the estimator. We chose to select it with cross validation and generalized cross validation. First, we define the usual cross validation criterion (in the framework of functional response) Simulation 1 The operator S is S = Π 20 := ∑ 20 j=1 v j ⊗ v j , where v j (t) := √ 2 sin(( j -1/2)π t) are the the eigenfunctions of the covariance operator X.

Simulation 2

The operator S is the integral operator defined by S X = 1 0 S (t, s)X(s)ds, where the kernel of S is S (t, s) = t 2 + s 2 .

Three estimators

We consider three different estimators of the trace of the covariance operator of ε: (i) the plug-in estimator given in (4.6), (ii) the corrected estimator given in (4.11) and (4.12), and (iii) the estimator Γε,n := Γε,nn n-k n Ŝn,2k n ( Π(k n +1):n Γn )( Ŝn,2k n ) ′ . The third estimator uses the whole sample when trying to remove the bias term, so it is not possible to obtain an immediate consistency result for this estimator because we do not have anymore the independence between the terms Ŝn,2k n and Π(k n +1):n Γn , but we can see its practical behavior.

Results

We present in table 4.1 (simulation 1) and table 4.2 (simulation 2) the mean values of the trace obtained for the three estimators on N = 100 simulations, as well as the CV and GCV criteria. The criteria have a convex form, that allows to choose a value for k. ). All values are given up to a factor of 10 -3 (the standard deviation is given in brackets up to a factor of 10 -4 ).

In simulation 1, the true value of k is known (k = 20) and the values chosen by CV and GCV are k = 22 or k = 24. For these values of k, the best estimator is tr( Γε,n ) for n = 300 and n = 1 500. The overestimation of tr( Γε,n ) seems to be well corrected by tr( Γε,n ), even if the usefulness of this bias removal cannot be theoretically proved. On this simulation, the estimator tr( Γε,n ) does not behave better than the others, especially for small sample sizes. ). All values are given up to a factor of 10 -3 (the standard deviation is given in brackets up to a factor of 10 -4 ).

n k CV (k) GCV (k) tr( Γε,n ) tr( Γε,n ) tr( Γε,n ) n=300 2
In simulation 2, the true value of k is unknown and the value chosen by CV and GCV is k = 4 (for n = 300) or k = 6 (for n = 1 500). The estimator tr( Γε,n ) is slightly better than the two others for n = 300. For n = 1500, tr( Γε,n ) is slightly better.

On both simulations, tr( Γε,n ) and tr( Γε,n ) show a good estimation accuracy and are quite equivalent. From a practical point of view, tr( Γε,n ) may be preferred as it is easy to implement. The bias removal of tr( Γε,n ) will give a more precise estimation.

Proofs

Proof of Theorem 24

We begin with preliminary lemmas.

Lemma 31. Ŝk n = S Πk n + 1 n ∑ n i=1 ε i ⊗ ( Γ+ k n X i ) .
Proof : From the definition of the estimator Ŝk n := ∆n Γ+ k n , we get

Ŝk n = 1 n n ∑ i=1 Y i ⊗ X i Γ+ k n = S 1 n n ∑ i=1 X i ⊗ X i + 1 n n ∑ i=1 ε i ⊗ X i Γ+ k n ,
and the result comes from the fact that Γn Γ+

k n = Πk n . □ Lemma 32. We have n ∑ i=1 n ∑ j=1 ⟨ Γ+ k n X j , X i ⟩ 2 H = n 2 k n and n ∑ i=1 ⟨ Γ+ k n X i , X i ⟩ H = n k n . Proof : We denote A the n × n matrix defined, for r, s ∈ 1, • • • , n, by A (r,s) := ⟨ Γ+ k n X r , X s ⟩ H = k n ∑ l=1 λ -1 l ⟨X r , vl ⟩ H ⟨X s , vl ⟩ H .
Let us remark that A = X Λ -1 X ′ , where X is introduced in Proposition 30 and Λ is the diagonal matrix

Λ := diag( λ1 , • • • , λk n ). We obtain n ∑ i, j=1 ⟨ Γ+ k n X j , X i ⟩ 2 H = tr(A ′ A) = tr XΛ -1 (nI n )X ′ = n tr (X ′ X)Λ -1 = n 2 k n .
The second part of the lemma can be obtained in a similar way. □ Now, coming back to the proof of Theorem 24, we can write

Γε,n = 1 n-k n ∑ n i=1 [(S -Ŝk n )(X i ) + ε i ] ⊗ [(S -Ŝk n )(X i ) + ε i ],
hence we have E( Γε,n ) = E[P I + P II + P III + P IV ] with

P I := 1 n-k n ∑ n i=1 (S -Ŝk n )(X i ) ⊗ (S -Ŝk n )(X i ), P II := 1 n-k n ∑ n i=1 (S -Ŝk n )(X i ) ⊗ ε i , P III := 1 n-k n ∑ n i=1 ε i ⊗ (S -Ŝk n )(X i ), P IV := 1 n-k n ∑ n i=1 ε i ⊗ ε i .
We start with P I . Using Lemma 31, we have, for i = 1, . . . , n,

(S -Ŝk n ) X i = S(I -Πk n )X i - 1 n n ∑ j=1 ⟨ Γ+ k n X j , X i ⟩ H ε j , (4.16)
and we can decompose P I = P

(1)

I + P (2)
I + P

(3)

I + P (4)
I , where

P (1) I = 1 n-k n ∑ n i=1 S(I -Πk n )X i ⊗ S(I -Πk n )X i , P (2) I = 1 n-k n ∑ n i=1 [-1 n ∑ n j=1 ⟨ Γ+ k n X j , X i ⟩ H ε j ] ⊗ S(I -Πk n )X i , P (3) I = 1 n-k n ∑ n i=1 S(I -Πk n )X i ⊗ [-1 n ∑ n j=1 ⟨ Γ+ k n X j , X i ⟩ H ε j ], P (4) I = 1 n-k n ∑ n i=1 [-1 n ∑ n j=1 ⟨ Γ+ k n X j , X i ⟩ H ε j ] ⊗ [-1 n ∑ n j=1 ⟨ Γ+ k n X j , X i ⟩ H ε j ].
First we have P

(1)

I = n n-k n S ∑ n i=k n +1 λi vi ⊗ vi S ′ .
From the independence between X and ε, we have

E[P (2)
I ] = E[P (3)
I ] = 0. Finally, we get

P (4) I = 1 n-k n ∑ n i=1 1 n 2 ∑ n j,l=1 ⟨ Γ+ k n X j , X i ⟩ H ⟨ Γ+ k n X l , X i ⟩ H ε j ⊗ ε l , hence E[P (4)
I ] = 1 n-k n ∑ n i=1 1 n 2 ∑ n j=1 E[⟨ Γ+ k n X j , X i ⟩ 2 H ] E(ε j ⊗ ε j ) = 1 n 2 (n-k n ) E ∑ n i=1 ∑ n j=1 ⟨ Γ+ k n X j , X i ⟩ 2 H Γ ε ,
and Lemma 32 gives E[P

I ] = k n n-k n Γ ε . So, we have shown that

E[P I ] = n n -k n S E n ∑ i=k n +1 λi vi ⊗ vi S ′ + k n n -k n Γ ε . (4.17)
Now, we decompose P II in the following way

P II = 1 n -k n n ∑ i=1 [S(I -Πk )(X i )] ⊗ ε i + 1 n -k n n ∑ i=1 - 1 n n ∑ j=1 ⟨Γ + k n X j , X i ⟩ H ε j ⊗ ε i .
By the independence between X and ε, the result of Lemma 32, and a similar computation for P III , we obtain

E[P II ] = E[P III ] = - k n n -k n Γ ε . (4.18)
Finally, coming back to the computation of E( Γε,n ), Theorem 24 is a direct consequence of (4.17) and (4.18). □

Proof of Theorem 26

The proof is based on the two following lemmas.

Lemma 33. Under the assumptions of Theorem 26, we have

lim n→+∞ E λk n = 0. (4.19) Proof : We have E λk n 2 ≤ 2λ 2 k n + 2E λk n -λ k n 2 .
From Lemma 2.2 and Theorem 2.5 in [START_REF] Horváth | Inference for Functional Data with Applications[END_REF] with assumption (A.2), we obtain

E λk n 2 ≤ 2λ 2 k n + 2 Γn -Γ 2 ∞ ≤ 2λ 2 k n + 2 n E ∥X∥ 4 ,

Proof of Theorem 28

We begin with the following lemmas.

Lemma 35. Under the assumptions of Theorem 26, we have

E Γε,n = Γ ε + m m -k m SE Π[1] (k m +1):m Γ[1] m S ′ - m m -k m SE Π[2] 2k m E Π[1] (k m +1):m Γ[1] m E Π[3] 2k m ′ S ′ . Proof : We first note that Ŝ2k m = ∆m Γ+ 2k m = S 1 m m ∑ i=1 X i ⊗ X i + 1 m m ∑ i=1 ε i ⊗ X i Γ + 2k m = S Π2k m + 1 m m ∑ i=1 ε i ⊗ Γ + 2k m X i .
As X and ε are independent, we get that E Ŝ2k m = SE Π2k m , which, combined with Corollary 25 and the fact that the three sub-samples are independent, ends the proof. □ Lemma 36. Under the assumptions of Theorem 26, we have

SE Π[2] 2k m E Π[1] (k m +1):m Γ[1] m E Π[3] 2k m ′ S ′ N ≤ ∥S∥ N S ′ ∞ E λk m +1 .
Proof : The proof is based on the same ideas as that used for proving Lemma 34. We remind that the infinite norm of projection operators are equal to one. □

The proof of Theorem 28 is now a simple combination of Lemmas 33, 34, 35 and 36 and using the triangle inequality.

Proof of Proposition 30

We consider the model Y i (t) = ∑ k n j=1 ⟨X i , v j ⟩ H α j (t) + η i (t), for i = 1, . . . , n and for all t. Here η

i (t) = ε i (t) + ∑ ∞ j=k n +1 ⟨X i , v j ⟩ H α j (t).
Writing this model in a matrix form, we have

Y(t) = Xα(t) + η,
where Y and η are the vectors with size n and respective general terms Y i and η i and α is the vector with size k n and general term α j . We can easily see that the associated mean square estimator is α

(t) = X ′ X -1 X ′ Y(t) = ⟨ Ŝk n (t, .), v1 ⟩ H , . . . , ⟨ Ŝk n (t, .), vk n ⟩ H ′ , (4.21)
where Ŝk n (t, s) is the estimator of S . Now, denoting Y ⋆ the vector with size n such that

Y ⋆ r = Y r for r ̸ = i, Y ⋆ i = Ŷ [-i] i
, X [-i] the matrix X without the i th row and α[-i] (t) the estimator of α(t) using the whole sample except the ith observation, we have, for any vector a = (a 1 , . . . , a k n ) ′ of functions of H and for any t

∥Y ⋆ (t) -Xa(t)∥ n ≥ Y [-i] (t) -X [-i] α[-i] (t) n-1 ≥ Y ⋆ (t) -X α[-i] (t) n . The fact that X ′ X -1 X ′ Y ⋆ (t) minimizes ∥Y ⋆ (t) -Xa(t)∥ n leads to α[-i] (t) = X ′ X -1 X ′ Y ⋆ (t), hence X α[-i] (t) = HY ⋆ (t).
The end of the proof comes from

Y i - Ŷ [-i] i = Y i -(HY ⋆ ) i = Y i - n ∑ r=1 r̸ =i H ir Y r -H ii Ŷ [-i] i = Y i -Ŷi + H ii Y i - Ŷ [-i] i .
Chapter 5

Modelling of High-throughput Plant Phenotyping with the FCVM The purpose of this chapter is to illustrate the implementation of the FCVM on a real dataset acquired in plant science experiments. The dataset consists in curves of Vapour Pressure Deficit (VPD) and Leaf Elongation Rate (LER) obtained on two high-throughput plant phenotyping platforms. The Vapour Pressure Deficit (VPD) is the difference (deficit) between the amount of moisture in the air and how much moisture the air can hold when it is saturated. In addition, the Leaf Elongation Rate (LER) is an important variable that characterize the growth of a plant.

The history of the VPD influences the LER curve. This can be modeled through the historical functional linear model (1.1) or the FCVM (1.3). The objective of this chapter is to understand better how the VPD influences the LER.

Datasets

Dataset T72A

In this dataset the VPD and LER of 18 plants were measured every 15 minutes from Day 159 to Day 168 of the year 2014 (June and July). This gives 96 observation times per day.

There were two platforms for this experiment: a growth chamber and a greenhouse. In the growth chamber the VPD is repeated, whereas in the greenhouse the VPD is not stable and changes all along the day and among days (sunny or cloudy days). The VPD curves depends on the environment and then they are the same for plants in the same platform in each day. This implies collinearity among these input curves.

For each day the first measurement of a plant could be 7:15am or 0:00am. This depends on whether the plant has been moved from the greenhouse to the growth chamber or vice versa at the previous day. For this reason there are missing values for some plants and some days. In total there is around 12% of missing data. Moreover some plants have not been studied during certain days due to the difference in development speed and phenological stages among plants.

We have extracted the curves which do not have zero values and have at most 5 NA's (missing observations). We used the R function approx to reconstruct these curves. We kept only the LER curves that have values to ensure that the plant were not stressed.

R Data-frames :

The dataset T72A contains other information (variables) than the VPD and LER measures. Besides as mentioned before there are missing data. For this reason we have extracted two datasets (R data-frames), each of which contain the name of the plants, the dates and either the VPD or LER curves respectively.

It is numerically more stable to apply the deconvolution methods to curves which starts with its support (non-zero part). That is why the VPD and LER curves start at 4am in the morning.

Each of these data-frames has 35 rows and 98 columns. The two first columns contain the name of the plant and the date. The remaining 96 columns represent the variable measured at the 96 observation times starting at 4am until 4am the next day. In Figure 5.1 we plot the VPD and LER curves of these two data-frames from the experiment T72A. The conditions of this experiment are similar to those of T72A. There were two experimental platforms: a growth chamber and a greenhouse. There are around 15% of missing data. There are collinearity among some of the VPD curves.

Again we have extracted the curves which do not have zero values and have at most have 5 NAs (missing observations). We used the R function approx to reconstruct these curves. But in contrast with T72A the LER curves do not have values higher than 3 in this experiment which implies that the plant were stressed.

R Data-frames : In the same way as T72A we have extracted two datasets (R data-frames).

Each of these datasets has 380 rows and 98 columns. The two first columns contain the name of the plant and the date. The remaining 96 columns represent the variable measured at the 96 observation times starting at 4:30am until 4:30am the next day. In Figure 5.2 we plot the VPD and LER curves of these two data-frames from the experiment T73A.

Functional Convolution Model

In this section we add a functional intercept µ to the model (1.3) to have a larger set of estimators of θ . Then the new FCVM has the form Y (t) = µ(t) + t 0 θ (s)X(ts) ds + ε(t).

(5.1)

Next we describe how to estimate µ and θ in this new situation.

The Estimators : From equation (5.1) it is easy to see that

E[Y ] = µ + θ * E[X],
where * is the convolution. So if we center the data X and Y we obtain (Lap). In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison.

Y -E[Y ] = θ * (X -E[X]) + ε. ( 5

Estimation with Experiment T73A

The results of the estimation of θ and µ are shown in Figure 5.5. In a similar way to the results with the experiment T72A we see that there are three subgroups among these estimators: first Fourier (FFDE) and Wiener (ParWD), secondly SVD and Tikhonov (Tik) and lastly Laplace. This is due to the use of the different methods to compute the estimators as we commented in the experiment T72A.

In contrast to the Fourier and Wiener estimators for the experiment T72A shown in Figure 5.3 we see here that these estimators have a more complex shape, whereas the SVD and Tikhonov are similar to the previous ones.

The optimized parameters of regression for Fourier and Wiener are λ n = 88.71029 and α = 0.03373 respectively (see subsection 3.5.1 Chapter 3). And for the SVD and Tikhonov these parameters are d = 2 and ρ = 10000 respectively.

The residuals for each estimators are shown in Figure 5.6. Again the prediction of Y i of these methods does not outperform the empirical mean estimator of E[Y ] (plot (a) in Figure 5.6). In particular the SVD and the Tikhonov methods give worse Estimation than Fourier and Wiener. Furthermore, Laplace cannot predict the Y i curves.

The results in both experiments show that the use of the FCVM does not improve the prediction over the empirical mean estimator of E[Y ]. This suggests that a more complex model better explains The estimators of the functional intercept (µ) are shown in Figure 5.8. Both are quite similar which is consistent with the similarity of the kernel estimators. Besides the residuals of each estimation method are shown in Figure 5.9. In that figure we see that the prediction when using this estimators improves over the FCVM (smaller residuals).

Estimation with Experiment T73A

The top view (level plot) of the Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ) are shown in Figure 5.10. Again both of them have a similar structure, in particular the diagonal shape for the sub-matrix of the first 60 rows and 60 columns.

We use the same methods to optimize the parameters of regression used for the experiment T72A. The optimal parameters now are k n = 16 for Karhunen-Loève and ρ = 0.5892068 for Tikhonov.

The estimators of the functional intercept (µ) are shown in Figure 5.11. We find again that both are similar. Additionally the residuals of each estimation method are shown in Figure 5.12. Again there is an slight improvement of the prediction of the Y i curves over the FCVM.

In both experiments we have improved the quality of prediction and thus the understanding of the interaction between VPD and LER. Nevertheless we need to deal more carefully with some features of the data. In particular the problem of the collinearity among the VPD curves should be addressed. The objective of the following section is to deal with this question and the necessary restriction on the estimators to follow the historical restriction: "the future does not influence the past".

s 1 over LER at each time t > s 1 remains almost the same (constant). Additionally note that the µ estimators are too wavy which makes harder the interpretation of the results. Finally the residuals are shown in Figure 5.15. We see there that the prediction of Y i improves greatly after 15 hours. This improvement is due to the non-collinearity of the VPD curves and the invertibility of the covariance matrix. To see this clearly note that the prediction starts to be 'perfect' precisely when the support of VPD ends.

Estimation with Experiment T73A

The Karhunen-Loève and Tikhonov estimators of K hist and their corresponding functional intercepts µ are shown in Figure 5.16. The optimal parameters in this case are d = 3 and ρ = 0.005880569 for Karhunen-Loève and Tikhonov respectively.

In this case both estimators of K hist differ a lot, the Karhunen-Loève estimator being close to zero compared to Tikhonov. Nevertheless this difference is due to a numerical instability in the computation of the generalized inverse Γ+ ρ of the covariance operator (see equation 5.5). In this way when ρ increase to ρ = 10 we obtain similar matrices and again with the same structure.

The Tikhonov estimator still contains parallel rows (s fixed) and is similar to the estimator for the experiment T72A. Additionally note that the µ estimators are less wavy than those for T72A.

Finally the residuals are shown in Figure 5.17. In this case, although the prediction of Y i improves over the mean empirical estimator, this improvement is not as important as for the experiment T72A.

Conclusions:

The historical functional model seems to predict better the LER curves than the FCVM. For this reason it could be more useful to understand how the VPD influences the LER. The estimators of the historical kernel K hist in both experiments have a similar structure. In particular we note the almost constant rows in each of them. This may suggests that the effect of the VPD on the LER remains almost constant over time. Finally in order to have a better assessment of this result, it would be interesting to compare it with functional non-parametric estimation methods.

Chapter 6

Conclusions and Perspectives

General Conclusions

This thesis has contributed to the study of how the history of the functional regressor X influences the current value of the functional response Y in functional linear regression models with functional response. In this regard, we have studied the theoretical and practical questions about the estimation for the following models:

1. The Functional Concurrent Model (FCCM), where only the instantaneous action is considered (Chapter 2).

2. The Functional Convolution Model (FCVM), where a fixed historical functional coefficient is used (Chapter 3).

3. The fully functional model, where we were interested in the estimation of the noise covariance operator (Chapter 4).

For the FCVM and the FCCM, the consistency and a rate of convergence were obtained, along with the numerical study of the robustness of the estimators. Additionally the shorter computation time of both estimators compared to others from the literature has also been shown.

Finally in Chapter 5 we apply these models and also the historical functional model to study how the Vapour Pressure Deficit (VPD) influences the Leaf Elongation Rate (LER) with a real dataset. This is a starting point for future research.

Perspectives

There are still many questions to be studied in future research. Here we outline some of them.

• The optimal rate of convergence of the functional Ridge regression estimator (3.3) and the functional Fourier deconvolution (3.4) are still unknown. One way to deal with this question is by considering estimators with other types of penalization like thresholding. This could give better theoretical properties but maybe with numerical instabilities.

• We can use the FCVM or the historical functional model in the context of a functional ANCOVA model where a qualitative is introduced a genotype factor for example.

In this way, for instance the FCVM (1.3) will generalize as follows. For t ∈ [0, ∞[, j ∈ {1, • • • , J} and k ∈ {1, • • • , n j } (replications)

Y jk (t) = µ j (t) + t 0 θ j (s) X jk (ts)ds + ε jk (t).

Potentially these functional ANCOVA models will be useful to differentiate and compare the VPD and LER interaction among different genotypes.

• The introduction of more functional covariates which have an instantaneous or historical influence over the response variable is an important generalization of the models studied in this thesis. For instance the following model: for i ∈ {1, • • • , n} and t ∈ [0, T ],

Y i (t) = µ(t) + β (t)X 1,i + t 0 K hist (t, s)X 2,i + ε i (t),

where X 1 and X 2 are two functional covariates which influence Y in a different way.

• The historical functional model applied to the VPD and LER interaction has shown that the estimator of the historical kernel (K hist ) has a structure that might be interpreted such that the influence of VPD at time s 1 over LER at each time t > s 1 remains almost the same (rows with almost constant values). This interpretation might be useful but it would be interesting to compare this result with functional non-parametric estimation methods [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF]) to better understand this structure.

Functional Linear Regression Models. Application to High-throughput Plant Phenotyping Functional Data.

Functional data analysis (FDA) is a statistical branch that is increasingly being used in many applied scientific fields such as biological experimentation, finance, physics, etc. A reason for this is the use of new data collection technologies that increase the number of observations during a time interval. Functional datasets are realization samples of some random functions which are measurable functions defined on some probability space with values in an infinite dimensional functional space. There are many questions that FDA studies, among which functional linear regression is one of the most studied, both in applications and in methodological development.

The objective of this thesis is the study of functional linear regression models when both the covariate X and the response Y are random functions and both of them are time-dependent. In particular we want to address the question of how the history of a random function X influences the current value of another random function Y at any given time t. In order to do this we are mainly interested in three models: the functional concurrent model (FCCM), the functional convolution model (FCVM) and the historical functional linear model. In particular for the FCVM and FCCM we have proposed estimators which are consistent, robust and which are faster to compute compared to others already proposed in the literature. Our estimation method in the FCCM extends the Ridge Regression method developed in the classical linear case to the functional data framework. We prove the probability convergence of this estimator, obtain a rate of convergence and develop an optimal selection procedure of the regularization parameter. The FCVM allows to study the influence of the history of X on Y in a simple way through the convolution. In this case we use the continuous Fourier transform operator to define an estimator of the functional coefficient. This operator transforms the convolution model into a FCCM associated in the frequency domain. The consistency and rate of convergence of the estimator are derived from the FCCM. The FCVM can be generalized to the historical functional linear model, which is itself a particular case of the fully functional linear model. Thanks to this we have used the Karhunen-Loève estimator of the historical kernel. The related question about the estimation of the covariance operator of the noise in the fully functional linear model is also treated. Finally we use all the aforementioned models to study the interaction between Vapour Pressure Deficit (VPD) and Leaf Elongation Rate (LER) curves. This kind of data is obtained with high-throughput plant phenotyping platform and is well suited to be studied with FDA methods.

Keywords : Functional regression models, Functional data, Convolution Model, Concurrent Model, Historical Model.

2

 2 Exemple de 13 couples de courbes de VPD et LER observées sur 96 pas de temps tout au long d'une journée. . . . . . . . . . . . . . . . . . . . . . . . 1.1 Log spectral intensities from the mass spectrometry data set. Black lines are plotted spectra from 20 pancreatic cancer (solid) and 20 control (dashed) patients, with mean spectra for pancreatic cancer (red) and control (blue). . 1.2 The top panel shows 193 measurements of the amount of petroleum product at tray level 47 in a distillation column of an oil refinery. The bottom panel shows the flow of a vapor into that tray during the experiment. . . . . . . . 1.3 Example of 13 pairs of VPD and LER curves observed 96 times during one day. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The true functions β 0 and β 1 (solid) compared to the cross-sectional mean curves of the FRRE β (1) 0 and β (1) 1 (red dashed) computed with the optimal regularization parameter λ 150 , and to the cross-sectional mean curves of the FRRE β ) computed with an optimal regularization curve Λ 150 . . . . . . . . . 2.2 Distribution of the evaluation criteria MADE, WASE and UASE in the cases of an optimal regularization parameter (left panel) and of an optimal regularization curve (right panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The true function θ (black) compared to the cross-sectional mean curves of the five estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Boxplots of the two criteria over N = 100 simulations with sample sizes n = 70 and n = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Function θ (black) and the cross-sectional mean curves of the five estimators. . . xv 3.4 Boxplots of the two criteria over N = 100 simulations with sample sizes n = 70 and n = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The function θ (black) and the cross-sectional mean curves of the five estimators. 3.6 Boxplots of the two criteria over N = 100 simulations with sample sizes n = 70 and n = 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. 1

 1 VPD and LER curves from the experiment T72A. . . . . . . . . . . . . . . 5.2 VPD and LER curves from the experiment T73A. . . . . . . . . . . . . . . 5.3 Estimation of θ and µ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Residuals of the estimators in the FCVM. (a) Residuals of the empirical mean estimator ( Y i -Ȳn ). (b) Residuals of the Fourier estimator (FFDE). (c) Residuals of Wiener (ParWD). (d) Residuals of SVD. (e) Residuals of Tikhonov (Tik). (f) Residuals of Laplace (Lap). In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Estimation of θ and µ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Residuals of the estimators in the FCVM. (a) Residuals of the empirical mean estimator ( Y i -Ȳn ). (b) Residuals of the Fourier estimator (FFDE). (c) Residuals of Wiener (ParWD). (d) Residuals of SVD. (e) Residuals of Tikhonov (Tik). (f) Residuals of Laplace (Lap). In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison.127 5.7 Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Estimators of µ when the Karhunen-Loève and Tikhonov estimators are use to estimate K hist in equation (5.4). . . . . . . . . . . . . . . . . . . . . . . xvi 5.9 Residuals of the estimators. Left, residuals of the empirical mean estimator ( Y i -Ȳn ). Center, residuals of the Karhunen-Loève estimator. Right, residuals of the Tikhonov functional estimator. In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison. . . . . 5.10 Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Estimators of µ when the Karhunen-Loève and Tikhonov estimators are used to estimate K hist in equation (5.4). . . . . . . . . . . . . . . . . . . . . . . 5.12 Residuals of the estimators. Left, residuals of the empirical mean estimator ( Y i -Ȳn ). Center, residuals of the Karhunen-Loève estimator. Right, residuals of the Tikhonov functional estimator.In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison. . . . . 5.13 VPD and LER curves from the experiments T72A and T73A which are not collinear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Top left and right: Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ) for the experiment T72A. These two estimators satisfy the historical restriction. Bottom left and right: Estimators of µ when the Karhunen-Loève and Tikhonov estimators are used to estimate K hist in equation (5.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Residuals of the estimators for the experiment T72A. Left, residuals of the empirical mean estimator ( Y i -Ȳn ). Center, residuals of the Karhunen-Loève estimator. Right, residuals of the Tikhonov functional estimator. In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 Top left and right: Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ) for the experiment T73A. These two estimators satisfy the historical restriction. Bottom left and right: Estimators of µ when the Karhunen-Loève and Tikhonov estimators are used to estimate K hist in equation (5.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Residuals of the estimators for the experiment T73A. Left, residuals of the empirical mean estimator ( Y i -Ȳn ). Center, residuals of the Karhunen-Loève estimator. Right, residuals of the Tikhonov functional estimator. In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables 2.1 Means and standard deviations of the evaluation criteria MADE, WASE and UASE in the cases of optimal regularization parameter and curve. . . . . . . . . . . . . 3.1 Curves X i and functions θ for each simulation setting. . . . . . . . . . . . . . . 3.2 Computation time (in seconds) of the estimators for a given sample and setting. . 3.3 Mean and standard deviation (sd) of the two criteria, computed from N = 100 simulations with sample sizes n = 70 and n = 400. . . . . . . . . . . . . . . . 3.4 Mean and standard deviation (sd) of the two criteria, computed from N = 100 simulations with sample sizes n = 70 and n = 400. . . . . . . . . . . . . . . . 3.5 Mean and standard deviation (sd) of the two criteria, computed from N = 100 simulations with sample size n = 70 and n = 400. . . . . . . . . . . . . . . . . 4.1 CV and GCV criteria for different values of k and mean values for the estimators of Tr(Γ ε ) (simulation 1 with n = 300 and n = 1500). All values are given up to a factor of 10 -3 (the standard deviation is given in brackets up to a factor of 10 -4 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 CV and GCV criteria for different values of k and mean values for the estimators of Tr(Γ ε ) (simulation 2 with n = 300 and n = 1500). All values are given up to a factor of 10 -3 (the standard deviation is given in brackets up to a factor of 10 -4 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 1

 1 Fig. 1 Log des intensités spectrales issu des données de spectrométrie de masse. Les lignes noires sont les spectres tracés de 20 patients atteint de cancer du pancréas (solide) et de 20 patients contrôle (en pointillés), avec les spectres moyens pour le groupe avec cancer (rouge) et pour le groupe contrôle (bleu).

Fig. 2

 2 Fig. 2 Exemple de 13 couples de courbes de VPD et LER observées sur 96 pas de temps tout au long d'une journée.

  des fonctions à estimer. Dans le cas le plus simple d'un modèle Gaussien, l'équation (6) prend la forme

  1.4.1 The Discretization of the FCVM and the FFDE . . . . . . . . . . . 1.4.2 Compact Supports and Grid of Observations . . . . . . . . . . . . 1.5 Contribution of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 1

 1 Fig. 1.1 Log spectral intensities from the mass spectrometry data set. Black lines are plotted spectra from 20 pancreatic cancer (solid) and 20 control (dashed) patients, with mean spectra for pancreatic cancer (red) and control (blue).

Fig. 1 . 2

 12 Fig. 1.2 The top panel shows 193 measurements of the amount of petroleum product at tray level 47 in a distillation column of an oil refinery. The bottom panel shows the flow of a vapor into that tray during the experiment.

Fig. 1

 1 Fig. 1.3 Example of 13 pairs of VPD and LER curves observed 96 times during one day.

  p are the predictors. Here β 1 , • • • , β p are the functions to be estimated.In the simplest case of the Gaussian model, η = E[Y ] and Y is normally distributed with mean η and equation (1.5) takes the form

  To obtain a rate of convergence, we need to control the shapes of the functions β and E[|X|] on the borders of the support of E[|X|]. Theorem 5 handles the general case where |β |/E[|X| 2 ] goes to infinity over the points of the set C β ,∂ X := supp(|β |) ∩ ∂ (supp(E[|X|])).

2 -

 2 bounded under this condition. Next (A5a) requires that around the points p ∈ supp(β ) ∩ ∂ (supp(E[|X|])) the function E[|X| 2 ] goes to zero slower than a polynomial of degree α, which implies that the second term in (3.5) behaves like β E[|X| 2 ] and determines the rate of convergence. Parts (b) and (c) of (A5) help us controlling the tails of β and |X| around infinity. They are useful only when card(C β ,∂ X ) = +∞. Note that the set C β ,∂ X is always countable (see the proof of Theorem 5).

Fig. 2 .FRRE

 2 Fig. 2.1 The true functions β 0 and β 1 (solid) compared to the cross-sectional mean curves of the FRRE β (1) 0 and β (1) 1 (red dashed) computed with the optimal regularization parameter λ 150 , and to the cross-sectional mean curves of the FRRE β (2) 0 and β (2) 1 (blue dotted) computed with an optimal regularization curve Λ 150 .

Fig. 2 . 2

 22 Fig. 2.2 Distribution of the evaluation criteria MADE, WASE and UASE in the cases of an optimal regularization parameter (left panel) and of an optimal regularization curve (right panel).

  We start the proof by considering the set C β ,∂ X . As supp(ϕ) is an open set in R, it is an union of open intervals. Because of this, ∂ (supp(ϕ)) is countable. Besides, by hypothesis (A5), for every p ∈ C β ,∂ X , there is an open neighborhood J p , in which (a) holds. Thus for all p ∈ C β ,∂ X , J p ∩ ∂ (supp(ϕ)) = {p}. These intervals J p are countable and pairwise disjoint.

  .21) Let us take an arbitrary i ∈ {1, • • • , n}. We define for each j ∈ {1, • • • , n},

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model and Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 General Hypotheses of the FCVM . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Functional Fourier Deconvolution Estimator (FFDE) . . . . . . . . . . . . . 3.3 Asymptotic Properties of the FFDE . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Consistency of the Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Selection of the Regularization Parameter . . . . . . . . . . . . . . . . . . . . . . . 3.5 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Competing techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 A further discussion about FFDE . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.A Main Theorems of Manrique et al. (2016) . . . . . . . . . . . . . . . . . . . . . . . 3.B Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.C Generalization of Theorem 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.D Numerical Implementation of the FFDE . . . . . . . . . . . . . . . . . . . . . . . . 3.D.1 The Discretization of the FCVM and the FFDE . . . . . . . . . . . . . . . . 3.D.2 Compact Supports and Grid of Observations . . . . . . . . . . . . . . . . . convolution over [0, ∞].

Setting 1 :

 1 First Figure 3.1 shows the true function θ and the cross-sectional mean curves of its five estimators computed from N = 100 simulations. The best estimators are FFDE and ParWD, both of them are close to each other. Note that FFDE have difficulty to estimate θ close to the borders. SVD and Tik are wavy, whereas Lap estimates poorly the quadratic part of θ over the subinterval [0, 0.3]. Finally all the estimators except Lap show an improvement when the sample size increases to n = 400, in particular FFDE improves considerably.

Fig. 3

 3 Fig. 3.1 The true function θ (black) compared to the cross-sectional mean curves of the five estimators.

  Figure 3.3 shows the true function θ and the cross-sectional mean curves of the five estimators. The best estimators are SVD and Tik, both of them behave similarly. FFDE gives a better (0.03702) 0.01120 (0.01170) Lap 0.18968 (0.13129) 0.15172 (0.28369) Table 3.3 Mean and standard deviation (sd) of the two criteria, computed from N = 100 simulations with sample sizes n = 70 and n = 400.

Fig. 3 . 2

 32 Fig. 3.2 Boxplots of the two criteria over N = 100 simulations with sample sizes n = 70 and n = 400.

Fig. 3 . 5

 35 Fig. 3.5 The function θ (black) and the cross-sectional mean curves of the five estimators.

Fig. 3 . 6

 36 Fig. 3.6 Boxplots of the two criteria over N = 100 simulations with sample sizes n = 70 and n = 400.

Fig. 3

 3 Fig. 3.7 The real part of the function F -1 (Ψ n ) (the imaginary part is equal to constant zero) for setting 1 to 3. In green 50 examples of F -1 (Ψ n ) computed for samples of size n = 70. In red the cross-sectional mean in each case.

Fig. 3

 3 Fig. 3.8 The plots of the function Φ for setting 1 to 3.

Fig. 3 .

 3 Fig. 3.10 Boxplots of MADE and WASE criteria before (FFDE) and after removing the edge effect (FFDE.no.ed) respectively.

  ) which helps us to bound the second term of ∥ βnβ ∥ L 2 in the decomposition (3.5).

  A5) There exist positive real numbers α > 0, M 0 , M 1 , M 2 > 0 such that (a) For every p ∈ C θ ,∂ X , with C θ ,∂ X := supp(|F (θ )|) ∩ ∂ (supp(E[|F (X)|])), there exists an open interval neighborhood J p ⊂ supp(|F (θ )|) such that first E[|F (X)| 2 (ξ )] ≥ |ξ -p| α ,for every ξ ∈ J p and secondly 1

  (A.3) We have almost surely λ1 > λ2 > . . . > λk n > 0. (A.4) We have λ 1 > λ 2 > . . . > 0. Our main result is then the following.Theorem 26. Under assumptions (A.1)-(A.4), if (k n ) n≥1 is a sequence such that lim n→+∞ k n = +∞ and lim n→+∞ k n /n = 0, we havelim n→+∞ E Γε,n -Γ ε N = 0. (4.9)The proof is postponed in Section 4.5.2. From the definition of the nuclear norm, we immediately get the following corollary: Corollary 27. Under the assumptions of Theorem 26, we have lim n→+∞ E tr Γε,n = tr (Γ ε ) .

Corollary 29 .

 29 Under the assumptions of Theorem 26, we have lim n→+∞ E tr Γε,n = tr (Γ ε ) . (4.14)

Fig. 5

 5 Fig. 5.1 VPD and LER curves from the experiment T72A.

  In this dataset the VPD and LER of 108 plants where measured every 15 minutes (96 observation times per day). But in this case there are three subsets of 36 plants which have been sown in different dates. The whole experiment took place between the Day 322 to the Day 350 of the year 2014 (November and December).

  123

Fig. 5

 5 Fig. 5.2 VPD and LER curves from the experiment T73A.

. 2 )

 2 Fig. 5.4 Residuals of the estimators in the FCVM. (a) Residuals of the empirical mean estimator ( Y i -Ȳn ). (b) Residuals of the Fourier estimator (FFDE). (c) Residuals of Wiener (ParWD). (d) Residuals of SVD. (e) Residuals of Tikhonov (Tik). (f) Residuals of Laplace (Lap). In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison.

Fig. 5

 5 Fig. 5.5 Estimation of θ and µ.

Fig. 5

 5 Fig. 5.6 Residuals of the estimators in the FCVM. (a) Residuals of the empirical mean estimator ( Y i -Ȳn ). (b) Residuals of the Fourier estimator (FFDE). (c) Residuals of Wiener (ParWD). (d) Residuals of SVD. (e) Residuals of Tikhonov (Tik). (f) Residuals of Laplace (Lap). In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison.

  predictive cross-validation with k = 5 for the Tikhonov estimator. The optimal parameters are k n = 5 for Karhunen-Loève and ρ = 0.001046277 for Tikhonov.

Fig. 5

 5 Fig. 5.7 Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ).

Fig. 5

 5 Fig. 5.8 Estimators of µ when the Karhunen-Loève and Tikhonov estimators are use to estimate K hist in equation (5.4).

Fig. 5

 5 Fig. 5.9 Residuals of the estimators. Left, residuals of the empirical mean estimator ( Y i -Ȳn ). Center, residuals of the Karhunen-Loève estimator. Right, residuals of the Tikhonov functional estimator. In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison.

Fig. 5 .

 5 Fig. 5.10 Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ).

Fig. 5 .

 5 Fig. 5.11 Estimators of µ when the Karhunen-Loève and Tikhonov estimators are used to estimate K hist in equation (5.4).

Fig. 5 .

 5 Fig. 5.12 Residuals of the estimators. Left, residuals of the empirical mean estimator ( Y i -Ȳn ). Center, residuals of the Karhunen-Loève estimator. Right, residuals of the Tikhonov functional estimator.In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison.

Fig. 5 .

 5 Fig. 5.14 Top left and right: Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ) for the experiment T72A. These two estimators satisfy the historical restriction. Bottom left and right: Estimators of µ when the Karhunen-Loève and Tikhonov estimators are used to estimate K hist in equation (5.4).

Fig. 5 .

 5 Fig. 5.15 Residuals of the estimators for the experiment T72A. Left, residuals of the empirical mean estimator ( Y i -Ȳn ). Center, residuals of the Karhunen-Loève estimator. Right, residuals of the Tikhonov functional estimator. In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison.

Fig. 5 .

 5 Fig. 5.16 Top left and right: Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ) for the experiment T73A. These two estimators satisfy the historical restriction. Bottom left and right: Estimators of µ when the Karhunen-Loève and Tikhonov estimators are used to estimate K hist in equation (5.4).

Fig. 5 .

 5 Fig. 5.17 Residuals of the estimators for the experiment T73A. Left, residuals of the empirical mean estimator ( Y i -Ȳn ). Center, residuals of the Karhunen-Loève estimator. Right, residuals of the Tikhonov functional estimator. In all the pictures we plot green lines (constant values -0.5 and 0.5 respectively) to help the comparison.

  est désigné sous le nom de intégrale faible de X. Une fonction aléatoire X à valeurs dans B sera définie comme fortement intégrable (ou intégrable) si E[∥X∥ B ] < ∞, où ∥ • ∥ B est la norme de B. Chaque fois que X est intégrable (fortement), il est courant de noter E[X] ou XdP de manière équivalente. Considérons la relation suivante d'équivalence entre fonctions aléatoires X et Y à valeurs dans B : X ∼ Y si et seulement si X = Y presque sûrement (a.s.). En utilisant les classes d'équivalence correspondantes, nous définissons l'espace L 1 B (P) des classes d'équivalence des fonctions intégrables aléatoires à valeurs dans B. Si l'on définit la norme ∥X∥ L 1

B := E∥X∥ B , alors L 1 B (P) est un espace de Banach. De manière analogue pour p ∈]1, ∞[, nous définissons les espaces L p B (P) des classes de fonctions aléatoires à valeurs dans B telles que E[∥X∥ p B ] < ∞. Similairement à L 1 B (P), cet espace devient un Banach si on définit la norme ∥X∥ L p B

  et Ramsay and Silverman (2005, Ch 1), parmi d'autres références, ont montré que certaines méthodes multivariées classiques sont inadaptées pour le traitement des données fonctionnelles. Il y a quatre raisons principales à cela : 1. Une exigence pour appliquer des méthodes multivariées aux données fonctionnelles est que la grille d'observations doit être fixe et la même pour toutes les réalisations. Ce n'est pas requis par l'ADF qui peut être appliquée à d'autres cas, par exemple lorsque les temps d'observation sont aléatoires et indépendants parmi les réalisations (voir par exemple Şentürk and Müller (2010, p. 1257), Yao et al. (2005b, p. 578)). Nous donnons dans cette section une courte bibliographie des modèles utilisés pour étudier la dépendance entre deux fonctions aléatoires. Nous considérons ici des fonctions aléatoires définies dans l'espace de Hilbert L 2 (I), c'est-à-dire l'espace des fonctions de carré intégrable au sens de Lebesgue sur l'intervalle I ⊆ R. Soient X et Y deux fonctions aléatoires, X la variable explicative, Y la variable réponse. Un modèle naturel qui lie X et Y est le suivant

2. Le nombre de pas de temps d'observation p est plus grand que le nombre de réalisations n (Hsing and Eubank (2015, p. 2)). L'inférence statistique sous cette condition est impossible avec les méthodes classiques de l'analyse multivariée. Les méthodes statistiques pour données à grande dimension (Bühlmann and van de Geer (2011, Ch 1)) et l'ADF sont adaptées à ce type de données. L'une des raisons pour lesquelles le cas (p >> n) est problématique pour l'analyse multivariée est que les opérateurs de covariance sont non-inversibles (mal-conditionnés). Cela rend difficile la résolution des systèmes linéaires dans les modèles de régression, qui sont largement utilisés en analyse multivariée. 3. Les corrélations élevées des mesures successives des variables, lorsqu'elles sont observées sur des pas de temps proches, impliquent un problème mal conditionné qui n'est pas adapté aux méthodes multivariées, car il rend difficile la résolution des systèmes linéaires (voir Yao et al. (2005b), Ferraty and Vieu (2006, p. 7)). 4. Enfin, dans le cas où la régularité et les dérivés des fonctions aléatoires jouent un rôle majeur pour étudier les données (voir par exemple Mas and Pumo (2009), Ramsay and Silverman (2005, Ch 17)), il est nécessaire de considérer la nature fonctionnelle des données, et cela ne peut pas être accompli avec l'approche multivariée. Modèles de régression linéaire fonctionnelle avec réponse fonctionnelle Y = Ψ(X) + ε, où Ψ est un opérateur fonctionnel et ε est un bruit. Dans ce modèle Ψ résume la façon dont X agit sur Y . Donc estimer Ψ est essentiel pour comprendre cette relation. Se posent alors les questions suivantes : comment définir un estimateur de Ψ à partir d'un échantillon (X i ,Y i ) i=1,••• ,n de n réalisations i.i.d. de X et Y , et quelles sont les propriétés de cet estimateur (consistance et vitesse de convergence).

Table 2 .

 2 1 Means and standard deviations of the evaluation criteria MADE, WASE and UASE in the cases of optimal regularization parameter and curve.

	λ	Constant λ 150	Curve Λ 150
	stats	mean	sd	mean	sd
	MADE 0.05421 0.01551 0.04719 0.01383
	WASE 0.01907 0.01788 0.01366 0.01454
	UASE 0.07235 0.06786 0.05185 0.05517

Table 4 .

 4 1 CV and GCV criteria for different values of k and mean values for the estimators of Tr(Γ ε ) (simulation 1 with n = 300 and n = 1500

Table 4 .

 4 2 CV and GCV criteria for different values of k and mean values for the estimators of Tr(Γ

ε ) (simulation 2 with n = 300 and n = 1500

  Lemma 34. Under the assumptions of Theorem 26, we haveSE Π(k n +1):n Γn S ′ N ≤ ∥S∥ N S ′ ∞ E λk n +1 . (4.20) Proof : Immediate properties of norms ∥.∥ ∞ and ∥.∥ N give SE Π(k n +1):n Γn S ′ N ≤ ∥S∥ N S ′ ∞ E Π(k n +1):n Γn ∞ ,which yields (4.20) as the norm ∥.∥ ∞ corresponds to the largest eigenvalue of the operator. □

	Theorem 26 is proved by combining Corollary 25 with Lemmas 33 and 34, and taking assumption
	(A.1) into account.

which concludes the proof of the lemma. □

  Functional Convolution Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.2.1 Estimation with Experiment T72A . . . . . . . . . . . . . . . . . . . . . . . 125 5.2.2 Estimation with Experiment T73A . . . . . . . . . . . . . . . . . . . . . . . 126 5.3 Historical Functional Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.3.1 Estimation with Experiment T72A . . . . . . . . . . . . . . . . . . . . . . . 128 5.3.2 Estimation with Experiment T73A . . . . . . . . . . . . . . . . . . . . . . . 129 5.4 Collinearity and Historical Restriction . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.4.1 Estimation with Experiment T72A . . . . . . . . . . . . . . . . . . . . . . . 133 5.4.2 Estimation with Experiment T73A . . . . . . . . . . . . . . . . . . . . . . . 135

Contents 5.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.1.1 Dataset T72A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.1.2 Dataset T73A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2

Acknowledgements

Acknowledgements: We are grateful to both anonymous referees for valuable comments that helped us to improve the paper.

I gratefully acknowledge the financial support of INRA and the Labex NUMEV (convention ANR-10-LABX-20) for funding my PhD thesis (under project 2013-1-007).

Chapter 2

Ridge Regression for the Functional Concurrent Model Contents which implies (2.21).

Proof of Proposition 11. It is similar to that of Proposition 22.

Abstract: The aim of this paper is to propose an estimator of the unknown function in the Functional Convolution Model (FCVM), which models the relationship between a functional covariate X(t) and a functional response Y (t) through the following equation Y (t) = t 0 θ (s)X(ts)ds + ε(t), where θ is the function to be estimated and ε is an additional functional noise. In this way we can study the influence of the history of X on Y (t). We use the continuous Fourier transform (F ) and its inverse (F -1 ) to define an estimator of θ . First we transform the problem into an equivalent one in the frequency domain, where we use the Functional Ridge Regression Estimator (FREE) to estimate F (θ ). Then we use F -1 to estimate θ in the time domain. We establish consistency properties of the proposed estimator. Afterwards we present some simulations to compare the performance of this estimator with others already developed in the literature. Keywords and phrases: Convolution model; Concurrent model; Functional data; Historical functional linear model.

Introduction

Functional Data Analysis (FDA) deals among other things with longitudinal data (e.g. random curves) for which the number of observation times (p) is much bigger than the sample size (n). In this situations, it has been showed that multivariate data analysis methods could fail (see [START_REF] Hsing | Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators[END_REF]) and it is more appropriate to use FDA methods that take into account the functional nature of the data. In particular to study the interaction of two such random curves we can consider some Functional linear regression models. In this paper we are interested in the Functional Convolution Model (FCVM)

where t ≥ 0, θ is the function to be estimated, X,Y are random functions and ε is a noise random function. All these functions are considered to be equal to zero for all t < 0. Integral in model (3.1) is called a convolution.

We can use this model to study the influence of the history of X on Y (t). It was derived from [START_REF] Malfait | The historical functional linear model[END_REF] by requiring that the function θ remains the same for each time t (i.e it only depends on s). In this paper we study how to estimate θ with a given i.i.d. sample

of the random functions X and Y . Some related models have been treated in the literature. On the one hand Asencio et al. ( 2014) studies a related problem, in which they consider more covariate functions. The estimation of θ is done by projecting the functions into finite-dimensional spline subspaces. On the other hand [START_REF] Malfait | The historical functional linear model[END_REF] consider the case where θ in the model (3.1) depends also on t (i.e. θ (s,t)), thus the integral becomes a kernel operator and they use a Finite Elements method to estimate it over an appropriate domain. Both papers are the most relevant among those in the literature which study the estimation of θ with an i.i.d. sample (X i ,Y i ) i∈{1,••• ,n} and where X and Y are related through the and compare it with that of other existing estimators which we adapted for the FCVM (3.1). We present these competing techniques in Subsection 3.5.1.

We have chosen three different simulation settings in Subsection 3.5.2 to do this comparison. Each one uses different functions θ and X in order to see the strengths and weaknesses of the FFDE respect to the others. In Subsection 3.5.3 we present the simulation results and describe the performance of the estimators. We finish this section with a further discussion about the behavior of the FFDE and its expected performance.

The numerical implementation of the FFDE is postponed to Appendix 3.D.1.

Competing techniques

To the best of our knowledge there are few estimation techniques which could be adapted to estimate θ in the context of the FCVM, that is when i) the input and output are random functions, ii) the convolution is non-periodic, iii) the sample size is n > 1 and iv) the noise is functional. In what follows we describe how we have adapted these techniques.

Parametric Wiener Deconvolution (ParWD): This method belongs to the family of signal processing methods (see Gonzalez and Eddins (2009, Ch 5)). For each realization (X i ,Y i ), X i is understood as the impulse response and Y i as the observed signal. Then we use the ParWD to estimate the 'unknown signal' θ for each couple

In this way we obtain n estimators of θ . Their mean will be the final estimator of θ in (3.1), that is θParWD := 1 n ∑ n i=1 θwie,i . Here we need to calibrate the parameter α, which is a constant number. We use the LOOPCV criteria to choose it. Note that the parameter α replaces the Noise-to-Signal power ratio E[|F (ε)| 2 ]/|F (θ )| 2 of the original Wiener deconvolution method (see Gonzalez and Eddins (2009, p. 241)).

Comparing definition (3.10) with the FFDE (3.4) we see that both are related; whereas for Wiener we start computing an estimator for each realization (X i ,Y i ) and then taking the mean, Fourier starts computing the mean in the frequency domain to estimate F (θ ) and then uses the IFCT to come back to the time domain. Due to this fact we will see that both estimators have a similar behavior.

Ill-posed linear inverse problems: The convolution can be understood as a special kind of matrix multiplication and consequently the deconvolution as a matrix inversion problem. We consider two well-known techniques to solve this matrix inversion problem, the Singular Value Decomposition (SVD) and the Tikhonov regularization (Tik) (see [START_REF] Tikhonov | Solutions of ill-posed problems[END_REF]).

Given that the solution is highly sensitive to the noise (because this matrix inversion is an Hadamard ill-posed problem, see Tikhonov and Arsenin (1977, Ch 1)), we want to get rid of this noise as much as possible before inverting the matrix. In this way we start by calculating the mean of all realizations and obtain for t ∈ [0, T ], (3.11) where X := 1 n ∑ n i=1 X i and Ȳ and ε are defined in a similar way. Now the noise ε(t) is close to zero. Note that this procedure contrasts with the ParWD method, where the mean is computed in a second step because the ParWD was devised to deconvolve one signal at a time taking already the noise into account.

Next we compute the numerical matrix approximation of this integral equation by using the rectangular method over a uniform grid of observation times t

′ and M X is the corresponding lower triangular matrix which approximates the convolution (3.11) on these time steps, namely

We consider the SVD of M X , that is M X = USV ′ where S is a diagonal matrix with the singular values of M X (which are the square roots of the eigenvalues of M ′ X M X ) and U and V are orthogonal matrices.

The Tikhonov estimator is defined as

where ρ is a regularization parameter.

The SVD estimator is defined as

where S k is a diagonal matrix with the same first non-zero k diagonal elements as S and zero elsewhere, and S + k is the pseudo-inverse of S k , which is obtained by replacing the non-zero elements of the diagonal of S k by their reciprocals and then transposing the resulting matrix. Here the dimension k is the regularization parameter.

Setting Curves X i

Function θ

Table 3.1 Curves X i and functions θ for each simulation setting.

Evaluation criteria: We use 100 Monte Carlo runs to evaluate for each simulated sample the mean absolute deviation error (MADE) and the weighted average squared error (WASE) as defined in Şentürk andMüller (2010, p. 1261),

where range(θ ) is the range of the function θ .

Simulation Results

All the computations have been implemented in R on a 2.9 GHz x 4 Intel Core i7-3520M processor, with a 4000KB cache size and 8GB total physical memory. Thanks to Proposition 18, it is possible to compute the FFDE with optimized parameter quickly. For the other estimators we have optimized numerically their respective parameters. The computation times are shown in Table 3 In this setting SVD and Tik perform better than the other ones, because this time the matrix M X is not close to zero and is easier to invert. However FFDE and ParWD are quite good, this shows that the use of FFT by FFDE and ParWD is stable in both cases whether E[X] = 0 or not. Furthermore when the sample size increases, FFDE is almost as good as SVD.

Setting 3: Figure 3.5 shows the function θ and the cross-sectional mean curves of the five estimators. In contrast to Settings 1 and 2, the best estimator is Lap, whereas the others perform quite similarly. Again FFDE has difficulties to estimate θ close to the borders. Finally all the estimators except Lap improve when the sample size increases to n = 400. Moreover all of them become better than Lap, and SVD gives the best estimation.

In Table 3.5 and in the boxplots of Figure 3.6 we can see that Lap outperforms the others when n = 70. The others give equivalent estimations. FFDE has a bigger variation for the WASE criteria.

In the case where the sample size is n = 400 we obtain an improvement in the estimation, SVD being the one improving the most.

Acknowledgments

The authors would like to thank the Labex NUMEV (convention ANR-10-LABX-20) for partly funding the PhD thesis of Tito Manrique (under project 2013-1-007). Appendix 3.A Main Theorems of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] The general hypotheses used in [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] and the results are rewritten with the notation of the associated concurrent model (3.2) to avoid confusion. The general hypotheses are:

) are all finite. The main results of [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF] used in this paper are presented next.

Theorem 19 (Theorem 3.1 in [START_REF] Manrique | Ridge regression for the functional concurrent model[END_REF]). Let us consider the FCM with the general hypotheses (HA1 FCM ), (HA2 FCM ) and (HA3 FCM ). Let (X i , Y i ) i≥1 be i.i.d. realizations. We suppose moreover that

Then

Corollary 20 (Corollary 3.7). Under hypotheses (A1), (A2) and if additionally we assume

Theorem 21 (Theorem 3.8). Under hypotheses (A1), (A2) and (A3), for every compact subset

From this comparison we can define the numerical estimator of θ over the grid [t 0 , • • • ,t q-1 ] as follows

(3.27)

3.D.2 Compact Supports and Grid of Observations

From now on we will compute θn numerically with equation (3.27). The important question we want to address here is how large the grid of observation points should be to properly estimate θ ?

In this regard understanding the relationship between the supports of X and θ and the one of their convolution (Y ) is an essential element to answer this question. We know that (Brezis (2010, p. 106)),

Then as mentioned before whenever our grid of observations contains the interval [0, T ] and [0, T ] contains supp(X) + supp(θ ) we will be able to estimate θ over its whole compact support.

The problem arises from the fact that we do not know θ and as a consequence neither supp(θ ) nor supp(X) + supp(θ ). Then how big T should be in order to estimate θ correctly?

There are several cases to consider. First let us suppose that the grid of observations covers [0, T 1 ] and supp(X), supp(Y ) ⊂ [0, T 1 ] then we can choose T > T 1 big enough and estimate θ over [0, T ]. To see this more clearly let us say that the grid of observations over [0, T 1 ] is t 0 , • • • ,t q 1 and over [0, T ] is t 0 , • • • ,t q , with q > q 1 . Given that we have only observed the curves over [0, T 1 ] we only know the vectors

Then the only thing we need to do before applying equation (3.27) properly is to redefine the vectors X d i and Y d i by adding zeros such that they will belong to R q , for instance

This procedure is known as zero padding the signal (Gonzalez and Eddins (2009, p. 111)). In this case equation (3.27) is well defined and we will compute θ over [0, T ]. Note also that supp(θ ) could be bigger than [0, T ] but the estimation of θ over [0, T ] is still correct.

Secondly we have the case where the grid of observations covers [0, T 1 ] and we know supp(X) ⊂ [0, T 1 ] and supp(Y ) \ [0, T 1 ] ̸ = / 0. Under these hypotheses we cannot add more zeros to the vectors Y d i because if we did it would imply that Y has zero values outside [0, T 1 ] which contradicts supp(Y ) \ [0, T 1 ] ̸ = / 0. Thus we cannot apply the property of Ω q to transform the convolution into a multiplication correctly. This is one restriction to the correct application of the FCVM.

Finally if the grid of observations covers [0, T 1 ], supp(X) \ [0, T 1 ] ̸ = / 0 and supp(Y ) \ [0, T 1 ] ̸ = / 0 we have the same phenomenon, that is we cannot add more zeros to the vectors X d i and Y d i to belong to R q . Thus it is not possible to transform the convolution into a multiplication because q 1 is not big enough. Note that Ω q 1 is quite different from Ω q (see definition 3.24) and the property of transforming Abstract : This work deals with the estimation of the noise in functional linear regression when both the response and the covariate are functional. Namely, we propose two estimators of the covariance operator of the noise. We give some asymptotic properties of these estimators, and we study their behavior on simulations. Keywords and phrases: functional linear regression, functional response, noise covariance operator

Introduction

We consider the following functional linear regression model where the functional output Y (.) is related to a random function X(.) through

Here S (., .) is an unknown integrable kernel: 1 0 1 0 |S (t, s)|dtds < ∞, to be estimated. ε is a noise random variable, independent of X. The functional variables X, Y and ε are random functions taking values on the interval I = [0, 1] of R. Considering this particular interval is equivalent to considering any other interval [a, b] in what follows. For the sake of clarity, we assume moreover that the random functions X and ε are centered. The case of non centered X and Y functions can be equivalently studied by adding an additive non random intercept function in model (4.1).

In all the sequel we consider a sample (X i ,Y i ) i=1,...,n of independent and identically distributed observations, following (4.1) and taking values in the same Hilbert space H = L 2 ([0, 1]), the space of all real valued square integrable functions defined on [0, 1]. The objective of this paper is to estimate the unknown noise covariance operator Γ ε of ε and its trace σ 2 ε := tr(Γ ε ) from these data sets. The estimation of the noise covariance operator Γ ε is well known in the context of multivariate multiple regression models, see for example Johnson and Wichern Johnson and Wichern (2007, section 7.7). The question is a little more tricky in the context of functional data. Answering it will then make possible the construction of hypothesis testing in connection with model (4.1).

Functional data analysis has given rise to many theoretical results applied in various domains (economics, biology, finance, etc...). The monograph by [START_REF] Ramsay | Functional data analysis[END_REF] is a major reference that gives an overview on the subject and highlights the drawbacks of considering a multivariate point of view. Novel asymptotic developments and illustrations on simulated and real data sets are also provided in [START_REF] Horváth | Inference for Functional Data with Applications[END_REF]. We follow here the approach of Crambes & Mas [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF] that studied the prediction in the model (4.1) revisited as:

where S : H → H is a general linear integral operator defined by S( f )(t) = 1 0 S (t, s) f (s)ds for any function f in H. The authors showed that the trace σ 2 ε is an important constant involved in the square prediction error bound that participate to determine the convergence rate. The estimation of σ 2 ε will thus provide details on the prediction quality in model (4.1).

In this context of functional linear regression, it is well known that the covariance operator of X cannot be inverted directly (see [START_REF] Cardot | Functional linear model[END_REF]), thus a regularization is needed. In [START_REF] Crambes | Asymptotics of prediction in functional linear regression with functional outputs[END_REF], it is based on the Karhunen-Loève expansion and the functional principal component analysis of the (X i ). This approach is also often used in functional linear models with scalar output, see for example [START_REF] Cardot | Functional linear model[END_REF].

The construction of the estimator Ŝ is introduced in Section 4.2. Section 4.3 is devoted to the estimation of Γ ε and its trace. Two types of estimators are given. Convergence properties are established and discussed. The proofs are postponed in Section 4.5. The results are illustrated on simulation trials in Section 4.4.

Estimation of S

Preliminaries

We denote respectively < ., . > H and ∥.∥ H the inner product and the corresponding norm in the Hilbert space H. We shall recall that < f , g > H = 1 0 f (t)g(t)dt, for all functions f and g in L 2 ([0, 1]). In contrast, < ., . > n and ∥.∥ n stand for the inner product and the Euclidean norm in R n . The tensor product is denoted ⊗ and defined by f ⊗ g =< g, . > H f for any functions f , g ∈ H.

We assume that X and ε have a second moment, that is:

The covariance operator of X is the linear operator defined on H as follows: Γ := E[X ⊗ X]. The cross covariance operator of X and Y is defined as

The empirical counterparts of these operators are: Γn := 1 n ∑ n i=1 X i ⊗ X i and ∆n := 1 n ∑ n i=1 Y i ⊗ X i . An objective of the paper is to study the trace σ 2 ε . We thus introduce the nuclear norm defined by ∥A∥ N = ∑ +∞ j=1 |µ j |, for any operator A such that ∑ +∞ j=1 |µ j | < +∞ where (µ j ) j≥1 is the sequence of the eigenvalues of A. We denote ∥.∥ ∞ the operator norm defined by ∥A∥ ∞ = sup ∥u∥=1 ∥Au∥.

Spectral decomposition of Γ

It is well known that Γ is a symmetric, positive trace-class operator, and thus diagonalizable in an orthonormal basis (see for instance [START_REF] Hsing | Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators[END_REF]). Let (λ j ) j≥1 be its non-increasing sequence of eigenvalues, and (v j ) j≥1 the corresponding eigenfunctions in H. Then Γ decomposes as follows:

is the predicted value of Y i using the whole sample except the ith observation, namely Ŷ

k n is the estimator of the operator S using the whole sample except the ith observation. Note that the criterion is based on the residuals.

The following property allows to introduce the generalized cross validation criterion.

Proposition 30. We denote X the matrix with size n × k n with general term ⟨X i , v j ⟩ H for i = 1, . . . , n and j = 1, . . . , k n , and

where Ŷi is the predicted value of Y i using the whole sample and H ii is the ith diagonal term of the matrix H.

This proposition allows to write the expression

without excluding the ith observation, and allows to get the generalized cross validation criterion, which is computationally faster than the cross validation criterion (see for example [START_REF] Wahba | Spline Models for Observational Data[END_REF]). The term H ii can be replaced by the mean tr(H)/n. Then, after noticing that tr(H) = tr(I k n ) = k n , where I k n is the identity matrix with size k n , we get

Simulations

Setting

The variable X is simulated as a standard Brownian motion on [0, 1], with its Karhunen Loève expansion, given by Ash & Gardner [START_REF] Ash | Topics in Stochastic Processes: By[END_REF]

where the v j (t) := √ 2 sin(( j -1/2)π t) and λ j = 1 ( j-0.5) 2 π 2 are the the eigenfunctions and the eigenvalues of the covariance operator of X. In practice, X(t) has been simulated using a truncated version with 1 000 eigenfunctions. The considered observation times are [ 1 1000 , 2 1000 , • • • , 1000 1000 ]. We simulate a sample with sizes n = 300 and n = 1 500.

We simulate the noise ε as a Standard Brownian motion multiplied by 0.1 (ratio noise-signal = 10%). Thus the trace of the covariance operator of ε will be tr(Γ ε ) = 0.005.

where Xn and Ȳn are the empirical estimators of the mean functions.

Estimation with Experiment T72A

The results of the estimation of θ and µ are shown in Figure 5.3. We can see three subgroups of estimators. First the Fourier (FFDE) and the Wiener (ParWD) approaches are similar, both of them are monotone decreasing functions. Secondly the SVD and Tikhonov (Tik) approaches are related to each other, and both of them differ of the first two estimators. Lastly the Laplace estimator is quite different from the other ones.

The difference among these subgroups is due to the use of the different methods to compute the estimators: Fourier and Wiener use the discrete Fourier transform, SVD and Tikhonov use the pseudo-inverse of the matrix associated to the convolution, and Laplace uses the Laguerre functions to project the convolution onto a finite dimensional subspace.

All the aforementioned estimators except Laplace use optimized parameters of regression. In the case of the Fourier and Wiener we use the Leave-one-out predictive cross-validation (LOOPCV). The optimal parameters for these are λ n = 0 and α = 0.04465 respectively (see subsection 3.5.1 in Chapter 3). For the SVD and Tikhonov we use the k-fold predictive cross-validation with k = 5 to obtain the optimal parameters d = 2 (dimension of inversion for the SVD) and ρ = 10000 respectively. In particular the SVD and the Tikhonov methods give worse predictions than Fourier and Wiener. Moreover, Laplace cannot predict the Y i curves.

how the VPD influences the LER. For this reason we use the historical functional linear model in the following section.

Historical Functional Linear Model

Estimators: Again we add a functional intercept µ to model (1.1) to have a larger set of estimators of the kernel K hist and to have a similar model to the FCVM with intercept (5.1). Then the new historical model has the form

(5.4)

We estimate µ in a similar way to equation (5.3), that is, we use the centered curves to estimate K hist and then we use the empirical means to estimate µ through μn (t) := Ȳn (t) -t 0 Khist (s,t) Xn (s) ds.

The estimation of K hist is done with two estimators: the Karhunen-Loève estimator (subsection 4.2.3 in Ch 4) and the Tikhonov functional estimator defined below.

Tikhonov Functional Estimator: This estimator is a variation of the Karhunen-Loève one.

To define it we use the same elements used in the definition of the Karhunen-Loève estimator (see subsection 4.2.3 in Ch 4). In particular we use the moment equation (4.3). But instead of taking the first k n dimensions to compute the generalized inverse Γ+ k n of the covariance operator, we use a positive number ρ > 0 which will be the Tikhonov (ridge) regularization parameter. With this value we define the Tikhonov generalized inverse as follows

and the Tikhonov Functional Estimator as Ŝρ = ∆n Γ+ ρ .

(5.5)

Estimation with Experiment T72A

The top view (level plot) of the Karhunen-Loève and Tikhonov functional estimators of the historical kernel (K hist ) are shown in Figure 5.7. We can see that they have both a similar structure, in particular the sub-matrix around the ordered pair (40, 40).

To optimize the parameters of regression for these estimators we have used the generalized cross-validation (see subsection 4.3.4 Chapter 4) for the Karhunen-Loève estimator and the k-fold

Collinearity and Historical Restriction

Collinearity: In both experiments (T72A and T73A), the VPD curves are repeated many times.

In order to avoid collinearity and identifiability issues we have extracted different VPD curves. After this we have 10 VPD and LER curves for the experiment T72A and 40 for T73A. These curves have been reconstructed with the R function approx (linear method) and then saved into the R data-frames. We show these curves in Figure 5.13. Historical Restriction: By this restriction we mean that "the future does not influence the past".

To implement this in the kernel estimation methods we must project the estimators onto the subspace where K hist in equation (5.4) satisfies K hist (s,t) = 0 for all s > t.

The results of the estimation are shown in the following two subsections.

Estimation with Experiment T72A

The Karhunen-Loève and Tikhonov estimators of K hist and their corresponding functional intercepts µ are shown in Figure 5.14. Both use the same calibration of parameters as the one used in section 5.3, namely the generalized cross-validation and the k-fold predictive cross-validation. The optimal parameters were: d = 24 and ρ = 4.947984e -05 for Karhunen-Loève and Tikhonov respectively. We can see that both estimators of K hist are similar. Each of these estimators have some rows with almost constant values (s fixed). This can be interpreted as that the influence of VPD at time