Yannick Pencolé
email: ypencole@laas.fr

Audine Subias
email: subias@laas.fr

Diagnosis of supervision patterns on bounded labeled Petri nets by Model Checking

This paper investigates the problem of pattern diagnosis of systems modeled as bounded labeled Petri nets that extends the diagnosis problem on single fault events to more complex behaviours. An effective method to solve the diagnosis problem is proposed. It relies on a matching relation between the system and the pattern that turns the pattern diagnosis problem into a model-checking problem.

Introduction

Fault diagnosis of Discrete Event Systems (DES) is a generic problem that has been widely studied [START_REF] Zaytoon | Overview of fault diagnosis methods for discrete event systems[END_REF]. The original definition has been introduced in [START_REF] Sampath | Diagnosability of discrete-event systems[END_REF] in the automaton formalism. The fault diagnosis problem was based on a system model and on an explicit set of fault events. From this early approach several contributions were proposed exploring different modeling formalisms among them, communicating automata ( [START_REF] Rozé | Diagnosing discrete-event systems : extending the diagnoser approach to deal with telecommunication networks[END_REF], [START_REF] Pencolé | A formal framework for the decentralised diagnosis of large scale discrete event systems and its application to telecommunication networks[END_REF], [START_REF] Lamperti | Flexible diagnosis of discrete-event systems by similaritybased reasoning techniques[END_REF]) or Petri nets ( [START_REF] Benveniste | Diagnosis of asynchronous discrete-event systems: a net unfolding approach[END_REF], [START_REF] Genc | Distributed diagnosis of place-bordered Petri nets[END_REF], [START_REF] Lefebvre | Diagnosis of DES with Petri net models[END_REF], [START_REF] Basile | On K-diagnosability of Petri nets via integer linear programming[END_REF], [START_REF] Paola Cabasino | Diagnosability of discrete-event systems using labeled Petri nets[END_REF]) leading to various diagnosis methods. This original problem has also been extended to consider more complex faulty behaviours than simple fault events. In [START_REF] Jiang | Failure diagnosis of discrete-event systems with linear-time temporal logic specifications[END_REF] the authors consider as faulty, behaviours that violate a given formal specification such as reaching a dead locking state, and propose to use linear-time temporal logic (LTL) to express these complex behaviours. In [START_REF] Thierry Jéron | Supervision patterns in discrete event systems diagnosis[END_REF] supervision patterns are introduced by means of labelled transition systems to define complex behaviours one may be interested in to diagnose. Supervision patterns as occurrence of one fault, occurrence of multiple faults, ordered occurrence of events faults but also multiple occurrence of a fault are considered by the authors. In [START_REF] Bozzano | Formal design of asynchronous fault detection and identification components using temporal epistemic logic[END_REF], such patterns are called diagnosis conditions and are formally represented with LTL properties with past operators. The advantage of such approaches is their generalizing power to tackle a large spectrum of diagnosis problems. Moreover, the use of patterns emphasizes on the separation between the behavioral model of the system and the objectives of the diagnosis tasks defined as patterns [START_REF] Zanella | Diagnosis of discrete-event systems by separation of concerns, knowledge compilation, and reuse[END_REF]. Another advantage is the possibility of pattern reusability to address others diagnosis problems on others systems.

In this article we revisit the diagnosis of patterns in the framework of Petri nets. We propose a concise way to model patterns that stresses pattern advantages. The diagnosis problem is posed as a pattern matching problem, so that the proposed approach does not derive a diagnoser-like machine like in [START_REF] Thierry Jéron | Supervision patterns in discrete event systems diagnosis[END_REF] but a machine that captures the matching between the system and a pattern. The construction of this machine is based on a specific product of Petri nets called the system-pattern product. The diagnosis problem of a supervision pattern for a given sequence of observations is then solved by determining the intersection between the system's evolutions that produce the input sequence and those that match the pattern. What we want to emphasize in this paper is how this task can be totally and efficiently entrusted to a model checker fed by Petri nets. An interesting point is that the proposed approach can be fully coupled with a pattern diagnosability analysis of systems [START_REF] Gougam | Diagnosability analysis of patterns on bounded labeled prioritized Petri nets[END_REF].

The Petri net formalism used in the approach and the system and patterns modeling are presented in Section 2. Section 3 states the diagnosis problem into a pattern matching problem. Section 4 focuses on the construction of the Petri machine that capture the matching. The implementation of the diagnosis problem as a model checking problem is given Section 5. We present experimental results in Section 6. Finally, conclusions and some perspectives are given Section 7.

Modeling framework

Patterns aim at modeling complex faulty behaviours or any normal behaviour of interest. This section presents the modeling framework of patterns and systems.

To have a complete characterization of the diagnosis problem, we use Labeled Petri nets with transition priorities ( [START_REF] Berthomieu | Bridging the gap between timed automata and bounded time Petri nets[END_REF]). This formalism extends the classical Labeled Petri net. The use of priorities is a convenient way to manage transition conflicts in a model and then to enhance the expressiveness of the model. In particular, transition priorities will be used here to overcome the problem of modeling the interactions between the system and a pattern. This section first presents the formalism, called L-type Labeled Prioritized Petri Nets, that is used throughout this paper. Then the modeling of the system and the pattern are presented.

L-type Labeled Prioritized Petri Nets

Definition 1 (LLPPN). An L-type Labeled Prioritized Petri Net (LLPPN for short) is a 7-uple N = hP, T, A, , `, ⌃, Qi where:

• P is a finite set of nodes called places;

• T is a finite set of nodes called transitions and P \T = ?;

• A ✓ (P ⇥ T ) [ (T ⇥ P ) is a binary relation transitive, not reflexive and not symetrical [START_REF] Berthomieu | Model-checking bounded prioriterized time Petri nets[END_REF].

• ⌃ is a finite set of transition labels (events);

• `: T ! ⌃ is the transition labeling function;

• Q is the set of final markings.

The state of an LLPPN is defined as a marking. A marking M is a function M : P ! N which maps any place of the net to a number of tokens contained in this place.

A marked LLPPN is a couple hN, M 0 i also denoted as the 8-tuple hP, T, A, , `, ⌃, Q, M 0 i. The preset pre(n) of a node n is the set of nodes pre(n) = {n 0 2 P [ T :

(n 0 , n) 2 A} and the postset post(n) of a node n is the set of nodes post(n) = {n 0 2 P [ T : (n, n 0 ) 2 A}. By construction the preset and postset of a transition (respectively a place) are two sets of places (respectively two sets of transitions). An LLPPN is a classical Petri Net formalism with some additional constraints or pieces of information.

Static priorities between transitions are introduced (relation ) to manage transition conflicts. Priorities add expressiveness to the Petri net and modify the model's semantics. Indeed a transition cannot be fired if a transition with higher priority is firable at the same time: a net transition t is firable from a given marking M if and only if

(8p 2 pre(t), M(p) > 0) ^(8t 0 2 T, t 0 t =) 9p 0 2 pre(t 0 ), M(p 0 ) = 0)
. This firing rule ensures that any firable transition t is firable in the usual sense (when = ?) but it avoids the choice of firing transitions with a lower priority that are also firable in the usual sense.

Firing a transition t consumes a token from any place in the preset and adds a token in any place of the postset. From a marking M , the fire of a transition t leads to the new marking M 0 such that

M 0 = M \ pre(t) [ post(t), which is denoted M t ! M 0 . A marking M is reachable in a marked LLPPN if it exists a sequence of firable transitions M 0 t1 ! M 1 t2 ! . . . t k ! M , also denoted M 0 r ! M
where r is the sequence t 1 t 2 . . . t k , the sequence r is called a run of the net. For a given LLPPN N and a given marking M , the set of reachable markings from M is denoted: R(N, M ).

An LLPPN also contains a transition labeling function that maps any transition to a label of a set ⌃, this is the way to associate physical events with transitions. Last but not least, an LLPPN has a set of final markings Q. The set Q is the way to define the language that is associated with an LLPPN (L-type language of [START_REF] James L Peterson | Petri nets[END_REF]).

Let r = t 1 . . . t k 2 T ⇤ and let `(r) denote the sequence `(t 1 ) . . . `(t k ) 2 ⌃ ⇤ , the language generated by a given LLPPN is defined as follows:

Definition 2 (Language of an LLPPN). The language generated by an LLPPN N = hP, T, A, , `, ⌃, Q, M 0 i is:

L(N ) = {`(r) : r 2 T ⇤ ^M0 r ! M ^M 2 Q}.

System model

We propose in this paper to model a system over the set of events ⌃ as an LLPPN ⇥ = hP, T, A, , `, ⌃, Q, M 0 i. As introduced in [START_REF] Sampath | Diagnosability of discrete-event systems[END_REF], systems generate prefix-closed regular languages: for any run r of the system, any of its prefix r 0 : r = r 0 r 00 is indeed a run. It follows that such a system can be represented by a bounded LLPPN (regular language) and with a set of final markings corresponding to the set of reachable markings Q = R(⇥, M 0 ) (prefix-closed language). The set of events ⌃ generated by the system is partitioned into two subsets: ⌃ o is the set of observable events and ⌃ u the set of unobservable ones.

Definition 3 (System model). The model of a system is an Bold events c and d are the observable events of the system whereas the events {a, b, e, f } = ⌃ u are the unobservable events. The dashed line represents the priority between the transitions of event b and a. It ensures that the transition labeled with b is always fired before the one labeled with a in a marking M such that M (p 0 ) 1 ^M (p 1 ) 1 (as the initial marking of the system).

LLPPN ⇥ = hP, T, A, , `, ⌃ o [ ⌃ u , Q, M 0 i such that: • Q is the set of reachable markings R(⇥, M 0 ); • the Petri net is bounded (9n 2 N + : 8M 2 R(⇥, M 0 ), 8p 2 P, M (p)  n).

Pattern model

The notion of pattern requires the definition of a general and flexible framework allowing to design a large spectrum of diagnosis problems including simple faulty event, multiple faults, fault repetitions....and more generally any behaviour of interest (faulty or not). The use of fault pattern helps to clearly separate the system and the diagnosis objectives (see [START_REF] Zanella | Diagnosis of discrete-event systems by separation of concerns, knowledge compilation, and reuse[END_REF]). The reusability is then reinforced to tackle new problems. For complex patterns this separation principle improves the applicability. Indeed, a modification of the system model in order to merge with the pattern is a tricky task that might even be impracticable in real cases. This would suppose to search in the system model for all the traces where the pattern has occurred and to modify them accordingly to the pattern model. Moreover, the system model would have to be modified each time a new pattern has to be analyzed.

Definition 4 (Pattern

). A pattern ⌦ is an LLPPN ⌦ = hP, T, A, , `, ⌃, Q, M 0 i such that 1. M 0 6 2 Q i.e
. the language of the pattern does not contain "; 2. from any reachable marking M of R(⌦, M 0 ) there exists a sequence of transitions r 2 T ⇤ such that M r ! M 0 and M 0 2 Q; 3. from any reachable marking M of R(⌦, M 0 ) there is no event e 2 ⌃ that labels more than one firable transition;

Q ⌦a!e = {M : M (pp 3 ) = 1} and Q ⌦ a!b = {M : M (pp 0 3 ) = 1}. 4. 8M 2 Q, 8M 0 : (9t 2 T, M t ! M 0 ) ) M 0 2 Q;
5. = ?, (no priority). Firstly, Figure 2 presents two similar patterns that will be used as running examples. The set of final markings of ⌦ a!e only contains one reachable marking, the one with pp 3 = 1. The language associated to ⌦ a!e is L(⌦ a!e ) = {ae} that is the sequence of event a followed by e.

Q ⌦1 = {M : M (p 2 ) = k}.
Figure 3 shows a pattern ⌦ 1 representing k occurrences of the event e (k 2 N + ). Figure 4 presents the LLPPN model of a pattern ⌦ 2 representing the occurrence of n events {e 1 , e 2 , . . . , e n } in no fixed order. The language L(⌦ 1 ) generated by pattern ⌦ 1 is L(⌦ 1 ) = {e k } that is the sequence of exactly k events e. The language L(⌦ 2 ) gathers any possible interleavings of the events {e 1 , . . . , e n }, the cardinality of the language is |L(⌦ In this way, a pattern can easily be modeled directly disregarding any notion of system language contrarily to [START_REF] Thierry Jéron | Supervision patterns in discrete event systems diagnosis[END_REF] or to ([19]). This concise way to model pattern reinforces the separation between the system and the diagnosis objectives. The way to handle the complex interactions between a system and a pattern is then not part of the modeling phase. It is managed by a specific product operator that is defined in Section 6.

2 )| = n!. p 11 p 12 • • • p 1n t 1 e 1 t 2 e 2 • • • t n e n p 21 p 22 • • • p 2n

Diagnosis problem

Intuitively speaking, pattern diagnosis is the problem of determining the set of patterns that have effectively occurred and that could explain a sequence of observations. The formal definition of the diagnosis problem then depends on the notion of pattern occurrence that is defined below.

A pattern ⌦ has occurred in a run r of the system if it is possible to extract from the event sequence ⇢ = `(r) 2 L(⇥) an ordered set of events that is a word of the language L(⌦) generated by the pattern, in this case we say that the sequence ⇢ matches the pattern ⌦ and is denoted ⇢ c ⌦ (we also say for convenience that the run r matches the pattern

⌦ if `(r) c ⌦).
Definition 5 (Matching of a pattern). A sequence ⇢ 2 ⌃ ⇤ matches a pattern ⌦ (⇢ c ⌦) if there exists at least a sequence of L(⌦) that is matched by ⇢ (⇢ c ).

Definition 5 relies on the definition of sequence matching.

Definition 6 (Sequence matching). A sequence ⇢ 2 ⌃ ⇤ matches another sequence 2 ⌃ ⇤ , denoted ⇢ c , if: • is empty ( = "); or • = s. 1 , s 2 ⌃, 1 2 ⌃ ⇤
and there exist two sequences ⇢ 0 , ⇢ 1 2 ⌃ ⇤ such that:

1. ⇢ = ⇢ 0 s⇢ 1 ; 2. ⇢ 1 c 1 .
A sequence ⇢ matches a sequence if ⇢ contains any event of the sequence and these events occur in the same order as in . Obviously ⇢ can contain several collections of events that gather the events of in the same order; in this case occurs several times in ⇢. The pattern ⇢ matches if occurs at least once. Figure 5 illustrates how the sequence ⇢ = bcaabcac matches the sequence = acc. Definition 6 is recursively applied three times till 3 = ". In this example, occurs twice in ⇢. Back to the example of Figure 1, the system ⇥ 1 matches both the patterns ⌦ a!e and ⌦ a!b of Figure 2. The sequence bade is a possible run of ⇥ 1 and bade c ae. ⇥ 1 also matches ⌦ a!b as the sequence badeebd is a possible run of ⇥ 1 and badeebd c ab.

Based on the notion of pattern, the classical fault diagnosis problem on DESs can be generalized as proposed in [START_REF] Thierry Jéron | Supervision patterns in discrete event systems diagnosis[END_REF]. Let P ⌃1!⌃2 : ⌃ ⇤ 1 ! ⌃ ⇤ 2 denote the projection function of any sequence of ⌃ ⇤ 1 to a sequence of ⌃ ⇤ 2 defined by: 8 < :

P ⌃1!⌃2 (") = ", P ⌃1!⌃2 (⇢.s) = P ⌃1!⌃2 (⇢) if ⇢ 2 ⌃ ⇤ 1 , s 6 2 ⌃ 2 , P ⌃1!⌃2 (⇢.s) = P ⌃1!⌃2 (⇢).s if ⇢ 2 ⌃ ⇤ 1 , s 2 ⌃ 2 .
For a given pattern ⌦, the diagnosis problem consists in defining an ⌦-diagnoser function that takes as input a sequence of observable events and that produces one of the three symbols {⌦ certain, ⌦ safe, ⌦ ambiguous} ( [START_REF] Pencolé | Towards low-cost fault diagnosis in large component-based systems[END_REF]).

Definition 7 (⌦-diagnoser).

Let ⇥ be the model of a system based on the set of events Considering now a set of patterns ⌦ 1 , . . . , ⌦ n , the diagnoser function of a system can be defined as

⌃ = ⌃ u [ ⌃ o , an ⌦-diagnoser is a function ⌦ : ⌃ ⇤ o ! {⌦ certain, ⌦ safe, ⌦ ambiguous} such that: • ⌦ ( ) = ⌦ certain if for any run ⇢ 2 L(⇥) that is consistent with (i.e. P ⌃!⌃o (⇢) = ), ⇢ c ⌦; ⇢ b c a a b c a c a c c ⇢ 0 s 0 ⇢ 1 s 0 1 ⇢ b c a a b c a c a c c ⇢ 10 s 1 ⇢ 2 s 1 2 ⇢ b c a a b c a c a c c ⇢ 20 s 2 s 2 3 = "
: ⌃ ⇤ o ! Q n i=1 {⌦ i certain, ⌦ i safe, ⌦ i ambiguous} such that ( ) = ( ⌦1 ( ), . . . , ⌦n ( )).
As an example, consider first the following observable sequence 1 = dd from the system ⇥ 1 (Figure 1) then ⌦a!e ( 1 ) = ⌦ a!e ambiguous and ⌦ a!b ( 1 ) = ⌦ a!b ambiguous. In both cases, it is possible to find in ⇥ 1 two runs which are consistent with 1 (their observable projection on ⌃ o is exactly 1 ), one run matches the pattern but not the other one. For instance consider the run badeebd of ⇥ 1 that matches both the pattern ⌦ a!e and the pattern ⌦ a!b and the run badf d that does not match any of the two patterns. Consider now the observable sequence 2 = cddc, then ⌦a!e ( 2 ) = ⌦ a!e safe and

⌦ a!b ( 2 ) = ⌦ a!b certain.
There is no run in ⇥ 1 consistent with 2 that contains an event e so none of them can match ⌦ a!e . On the other hand, any run consistent with 2 starts with ba and must contain one more b so any of them matches ⌦ a!b .

Matching determination

We introduce a specific product operator to combine the system ⇥ and the pattern ⌦ in order to retain in the system any run that effectively matches the pattern: this product is called the system-pattern product and is denoted: ⇥n⌦. Informally speaking, ⇥ n ⌦ consists of the union of the places of ⇥ and ⌦. The set of transitions is composed of the set of transitions of ⇥ and a set of synchronized and prioritized transitions replacing the transitions from ⌦. Definition 8 (system-pattern product).

Let ⇥ = hP ⇥ , T ⇥ , A ⇥ , ⇥ , `⇥, ⌃ ⇥ , Q ⇥ , M ⇥0 i, ⌦ = hP ⌦ , T ⌦ , A ⌦ , ?, `⌦, ⌃ ⌦ , Q ⌦ , M ⌦0 i
be respectively a system and a pattern such that P ⇥ \ P ⌦ = ?, T ⇥ \ T ⌦ = ? and ⌃ ⌦ ✓ ⌃ ⇥ , the system-pattern product ⇥ n ⌦ is the LLPPN hP, T, A, , `, ⌃, Q, M 0 i defined as follows.

•

P = P ⇥ [ P ⌦ . • T = T ⇥ [ T s such that -T s = S l2⌃⌦ {t ⇥ kt ⌦ : 9t ⇥ 2 T ⇥ ^9t ⌦ 2 T ⌦ ⇥(t ⇥ ) = `⌦(t ⌦ ) = l}
is the set of synchronized transitions, the transition denoted t ⇥ kt ⌦ resulting from the synchronization of t ⇥ 2 T ⇥ and t ⌦ 2 T ⌦ .

• A = A ⇥ [ A s such that A s = {(p, t) : p 2 P ⇥ , t = t ⇥ kt ⌦ , (p, t ⇥ ) 2 A ⇥ } [ {(p, t) : p 2 P ⌦ , t = t ⇥ kt ⌦ , (p, t ⌦ ) 2 A ⌦ } [ {(t, p) : p 2 P ⇥ , t = t ⇥ kt ⌦ , (t ⇥ , p) 2 A ⇥ } [ {(t, p) : p 2 P ⌦ , t = t ⇥ kt ⌦ , (t ⌦ , p) 2 A ⌦ }. • = ⇥ [ s , with: s = {(t s , t ⇥ ) : t s = t ⇥ kt ⌦ 2 T s }[ [ ts2Ts ({(t s , t) : (t ⇥ , t) 2 ⇥ } [ {(t, t s ) : (t, t ⇥ ) 2 ⇥ }) [ {(t 1 ⇥ k t 1 ⌦ , t 2 ⇥ k t 2 ⌦ ) : t 1 ⇥ k t 1 ⌦ 2 T s , t 2 ⇥ k t 2 ⌦ 2 T s , (t 1 ⇥ , t 2 ⇥ ) 2 ⇥ }. • 8t = t ⇥ ||t ⌦ 2 T s , `(t) = `⇥(t ⇥ ), 8t 2 T \ T ⇥ , `(t) = `⇥(t). • ⌃ = ⌃ ⇥ . • Q = {M ⇥ [ M ⌦ : M ⇥ 2 Q ⇥ ^M⌦ 2 Q ⌦ }. • M 0 = M ⇥0 [ M ⌦0 .
Intuitively, the pattern ⌦ is applied as a filter on the system ⇥ by using the product ⇥ n ⌦. Any transition of label e in the system is synchronized with a transition of the pattern ⌦ labeled with the same event e. The product is asymmetric in the sense that ⇥ n ⌦ contains all the transitions of ⇥ whereas the transitions of ⌦ with such an event e have been replaced by a set of synchronized transitions with ⇥. Given the current marking of ⇥ n ⌦ and the occurrence of a new event e, two cases can happen. If the current marking of ⇥ n ⌦ enables a synchronized transition labeled with e, as it is prioritized, it will be triggered (the event e is therefore part of the pattern), otherwise only a transition from the system ⇥ can be triggered in ⇥ n ⌦ (the system keeps running by producing the event e but e is not part of the pattern).

With the help of this product, it is possible to identify in the system ⇥ the set of runs where the pattern ⌦ has occurred. The following results assert this intuition.

Let ⇥ ⌦ = ⇥n ⌦ and M ⇥⌦0 be the initial marking of ⇥ ⌦ . Let M ⇥0 (resp. M ⌦0 ) be the initial marking of ⇥ (resp. ⌦).

Theorem 1. Let ⇥ be the LLPPN of a system over the alphabet ⌃ and ⌦ be the LLPPN of a pattern:

L(⇥ n ⌦) = {⇢ 2 L(⇥) : ⇢ c ⌦}.
Theorem 1 is the fundamental property of the operator n. The set of runs that are characterized by the language L(⇥ n ⌦) are exactly the set of runs that match the pattern ⌦ [START_REF] Gougam | Diagnosability analysis of patterns on bounded labeled prioritized Petri nets[END_REF].

Implementation by Model-Checking

Through the Petri model of the system-pattern product (⇥ n ⌦), we have a model that determines for each possible run of the system if this evolution matches or not the pattern. This model can then be exploited to solve the diagnosis problem.

For an observable input sequence generated by the system ( ), the pattern diagnosis problem can be solved by determining the intersection between the system's evolutions that generate this input sequence and those that match the pattern. This intersection can be characterized through the synchronized product between the Petri net model of the system-pattern product and the Petri net model O of the observed input sequence ( ) produced by the system : ⇥ ⌦ ||O [START_REF] Hack | Petri net languages[END_REF].

Considering = e 1 , e 2 , . . . e k , the Petri net model O is an LLP P N defined by:

O = hP O , T O , A O , O , `, ⌃ O , Q O , M O0 i with: • P O = k+1 [ i=1 {p i } and T O = k [ i=1 {t i }; • A O = k [ i=1 {(p i , t i ), (t i , p i+1 )} • O = ? • `(t i ) = e i , 8t i 2 T and i 2 1..k; • ⌃ O = {e i . . . e k }; • Q O = {{p k+1 }} • M O0 = {p 1 }
Once the synchronized product ⇥ ⌦ ||O built, it is then possible to use model checking techniques to determine the results produced by the ⌦-diagnoser. Here we give a complete description of the implementation by Model-Checking.

Model checking problem

We propose here to implement the ⌦-diagnoser function by using the TINA toolkit of [START_REF] Bernard Berthomieu | The tool tina -construction of abstract state spaces for Petri nets and time Petri nets[END_REF] and [START_REF] Berthomieu | Model-checking bounded prioriterized time Petri nets[END_REF] that provides all the necessary tools to directly generate the Kripke structure on which the model-checking problem is defined [START_REF] Clarke | Model Checking[END_REF]. TINA actually generates enriched labeled Kripke structures from the analyzed Petri nets to check properties written in SE-LTL (State/Event Linear Temporal Logic) which extends LTL in the way presented here above.

A formula is a SE-LTL formula if it is a universally quantified formula The ⌦-diagnoser is based on two questions written in SE-LTL concerning the Petri net model ⇥ ⌦ ||O :

::= 8' such that ' ::= r | ¬' | ' _ ' || ' ^' | ' | ⇤' | ⌃' | 'U
1. Question ' CERTAIN : Are we sure that the pattern is recognized in each system's behaviour that produces the observations ? If the result is false the following question is asked to the model checker: 2. Question ' SAFE : Are we sure that the pattern is never recognized in each system's behaviour that produces the observations ? More formally, each question corresponds to a SE-LTL formula. Given M a marking of O k ⇥ ⌦ , M |⌦ denotes the restriction of M to ⌦ i.e. the marking of the pattern's places. In a similar way M |O denotes the restriction of M to O.

1. ' CERTAIN ⌘ ⇤((M |O 2 Q O ) ) (M |⌦ 2 Q ⌦ )), in
others words, is it always true (⇤) that if the system generates (M |O 2 Q O ) then ()) it matches the pattern 6 Case study

(M |⌦ 2 Q ⌦ ). 2. ' SAFE ⌘ ⇤((M |O 2 Q O ) ) (M |⌦ 6 2 Q ⌦ )),

Description of the system

The case study that we propose in this paper is a product transportation system. It is a two-level system composed of two product sites (namely sites 1 and 2) at level 2 where a set of products is stored and two assembly stations at level 1 that request products from level 2. A lift is used between the two levels. Each site has a conveyor belt to move a product from the site to the lift and each station also has a conveyor belt to get the products from the lift. Figure 6 presents the system model. Once a product is detected on the site 1, a

Product1 signal (noted P r1) is emitted. The product is then put in a box and becomes available. The box is then pushed and sent into the lift (action Push 1 that starts with event P 1 and ends with event EP 1 ). Site 2 behaves in a similar manner. A product from site 1 has a priority access to the lift.

In the Petri net model of Figure 6, this priority relation is indicated by a dashed edge between the transitions labeled Obs.

nS( 1) nT( 1) '

Monitoring of a single event class

We first present some experiments with a single event pattern ⌦ 1 as defined in Figure 3 with k = 1 and e = Req 1 . The aim of this first example is to show how the proposed framework is able to diagnose the occurrence of single events as the classical diagnosis methods do. The pattern ⌦ 1 represents the occurrence of the event Req 1 in the system. Table 1 shows the obtained results for the following scenario:

= Pr1 .Pr2 .D.U .D.ELReq2 .ERReq2 .ELReq1 .ERReq1 .U
In order to show the evolution of the diagnosis, any line of Table 1 represents one step of the scenario (i.e line 1 is Pr1 , line 2 is Pr1 .Pr2 , etc). The second column shows the number of states in the corresponding Kripke structure and the third column shows the number of transitions. The first two lines show a ⌦ 1 safe diagnosis. Diagnoses between lines 3 and 7 are ⌦ 1 ambiguous. Finally lines 8-10 show an ⌦ 1 certain diagnosis. These results can be intuitively explained as follows: Req 1 cannot occur before the lift goes down (lines 1-2) and as long as ELReq1 is not observed there is an ambiguity about the occurrence of Req 1 .

In classical diagnosis methods, events are sometimes partitioned into a set of fault classes. Here, also it is possible to represent this type of event class. With a pattern ⌦ 2 as represented in Figure 4, it is possible to represent one occurrence of event e 1 = Req 1 or one occurrence of event e 2 = Req 2 (two transitions t 1 and t 2 only), i.e. an event of class {Req 1 , Req 2 }. In this example, the set of accepting markings is composed of the markings {M 1 , M 2 } where Obs.

' 2 CERT. ? ' 2 SAF E ? ' 3 CERT. ? ' 3 SAF E ? Pr1 F T F T Pr2 F T F T D F F F T U T F F F D T F F F ELReq2 T F F F ERReq2 T F F F ELReq1 T F T F ERReq1 T F T F U T F T F
) (p 21 = 1 _ p 22 = 1)
). Table 2 shows the diagnosis results (left columns) on the same scenario as above for ⌦ 2 . When the first D is observed one of the Req may silently happen afterwards but not necessary hence the ambiguity. But as soon as we observe U , ⌦ 2 has certainly occurred.

Monitoring of the occurrence of multiple events

Another interesting pattern has the same structure as the one of Figure 4, it is the occurrence of both events Req ). This type of pattern is interesting if we want to supervise that a set of n specific types of event have occurred. Table 2 shows the diagnosis results (right columns) on the same scenario as above for this pattern. The diagnosis becomes certain (line 8) as soon as we both observe ELReq2 and ELReq1 .

Monitoring of a complex behaviour

The previous pattern examples show that the presented framework can model classical diagnosis problems where we look for the occurrence of a set of single events. However the purpose of this framework is to allow for the diagnosis of more complex behaviours. Back to our example, the global function of the system is to transfer items from conveyors at Level 2 to conveyors at Level 1. Figure 6 actually models the behaviour of the system but no single faulty event is present (any event that can happen is normal).

Figure 7 presents a complex behaviour that one might want to diagnose within the system of Figure 6. This pattern ⌦ 3 (m, n) models the consecutive occurrence of m = 3 events of type Req 1 that are not interleaved with occurrence of Req 2 events (the left conveyor keeps having its requests fulfilled while the right conveyor get stuck) and this pattern should occur n = 2 times (the pattern has a counter). This pattern is made up of normal events but defines a behaviour that could be unexpected (both conveyors at Level 1 should get items regularly).

Table 3 presents some experimental results for a set of scenarios S 1 . . . S 14 where the occurrence of the pattern ⌦ 3 (m, n) is diagnosed. For each scenario, Table 3 shows the size of the observable sequence (column 2), the number m of Req 1 events in the pattern ⌦ 3 (m, n) (column 3) behaviouras well as the number n of occurrences (counter). The size of the involved Kripke structure as a number of states (nS( )) and transitions (nT ( )) for a given scenario is also given. The time here represents the computation time of all the diagnosis process (the computation of the Kripke structure and the verification of the questions). Scenarios S 1 . . . S 4 diagnose the occurrence of ⌦ 3 (5, 1) based on a certain amount of cycles (see Section 6.2). The sequence being composed of 10 events, S 1 diagnoses 2 cycles of , S 2 diagnoses 10 of them, etc. Based on , 5 consecutive occurrences of Req 1 can never happen that is why the result is always safe. Scenarios S 5 . . . S 7 diagnoses the occurrence of ⌦ 3 (m, 10), m = 2, 5, 10 so here we attempt to diagnose that m consecutive occurrences of Req 1 happen 10 times. For m = 5, 10, the result is safe for the same reason as above. For n = 2, there is an ambiguity. Indeed, consider two consecutive cycles of , once ELReq1 is observed in the first cycle, it is possible that one Req 1 has occurred just before ELReq1. Within the second cycle, before observing ELReq2, one second occurrence of Req 1 may have happened (followed by one occurrence of Req 2 ), so it is possible (but not certain) that two consecutive occurrences of Req 1 have occurred and, as there are 200 cycles, this can happen 10 times, hence the ambiguous result. Scenario S 8 is similar as S 5 but the observable sequence is a set of cycles 0 where any ELReq2, ERReq2 in have been replaced by ELReq1, ERReq1, the diagnosis is therefore certain. Scenario S 9 presents a scenario composed of 100 cycles of followed by 75 cycles of 0 followed by 100 cycles of followed by 75 cycles of 0 (i.e. 350 cycles): 10 consecutive occurrences of Req 1 happen more than 10 times in S 9 . The last five scenarios S 10 . . . S 14 rely on the fact that the initial state of the system is unknown (unknown states of the conveyors and the lift). These scenarios illustrate the fact that the method can be applied on an observation window [START_REF] Su | Diagnosis of des by independent windows[END_REF] that does not contain the first observations of the system.

Conclusion

This paper adresses the problem of patterns diagnosis by a generic framework for pattern modeling and model checking technics. The diagnosis method is designed in the framework of Petri nets (L-type Labeled Prioritized Petri Nets) and it can be viewed as a pattern matching based method. The construction of a Petri net machine capturing the matching of a pattern by the system is done by means of a specific Petri net product. Then a translation of the diagnosis problem into a model-checking problem allows an efficient resolution that is entrusted to the Petri net analyser TINA. The paper reports the results of experimenting this pattern diagnosis approach on an illustrative example. Future work could consolidate some aspects of the method. In particular, the expressivity of patterns that currently generate regular language may be extended to context-freelanguage. Another research direction is to extend the work to time systems. We currently investigate the use of Time Petri Nets to diagnose more interesting behaviours.

Figure 1 :

 1 Figure 1: System example ⇥ 1 with ⌃ o = {c, d}.

Figure 1

 1 Figure1presents an example of such a model ⇥ 1 . Bold events c and d are the observable events of the system whereas the events {a, b, e, f } = ⌃ u are the unobservable events. The dashed line represents the priority between the transitions of event b and a. It ensures that the transition labeled with b is always fired before the one labeled with a in a marking M such that M (p 0 ) 1 ^M (p 1 ) 1 (as the initial marking of the system).

Figure 2 :

 2 Figure 2: Two sequence patterns ⌦ a!e and ⌦ a!b with Q ⌦a!e = {M : M (pp 3 ) = 1} and Q ⌦ a!b = {M : M (pp 0 3 ) = 1}.

Figure 3 :

 3 Figure 3: Pattern (⌦ 1 ) about k occurrences of an event e with the final markings Q ⌦1 = {M : M (p 2 ) = k}.

Figure 4 :

 4 Figure 4: Pattern (⌦ 2 ) about the occurrence of n events in no fixed order with the final markings Q ⌦2 = {{p 21 , p 22 , . . . , p 2n }}.

Figure 5 :

 5 Figure 5: How ⇢ = bcaabcac matches = acc: ⇢ c .

  ' r ::= e | e 4 e e ::= p | a | c | e 5 e with p a place symbol, a a transition symbol, c 2 N, 4 2 {=, <, >, , } and 5 2 {+, , ⇤, /}. The operators (next), ⇤ (always), ⌃ (eventually) and U (until) have their usual LTL semantics.

1 and

 1 Req 2 where, as explained in Figure4, the only accepting marking is M (p 21 ) = 1 and M (p 22 ) = 1 and M (p) = 0 for the other two places p. In this case the ' CERT AIN question becomes: ' 3 CERT AIN = ⇤((p obs = 1) ) (p 21 = 1 ^p22 = 1)

Figure 7 :

 7 Figure 7: Pattern ⌦ 3 (3, 2): two occurrences of three consecutive requests Req 1 : accepting marking M , M (p 4 ) = 2.

  n others words, is it always true (⇤) that if the system generates (M |O 2 Q O ) then ()) it does not match the pattern(M |⌦ 6 2 Q ⌦ ).The complete and automated implementation of the method to solve the pattern diagnosis problem on a system⇥ is: 1. Compute ⇥ ⌦ = ⇥ n ⌦. 2.Compute O the LLPPN that represents the sequence of observations . 3. Compute the synchronisation of ⇥ ⌦ and O: ⇥ ⌦ ||O 4. Compute ' CERT AIN from ⇥ ⌦ ||O and check ' CERT AIN on the Kripke structure of ⇥ ⌦ ||O.

5. If the result is false, then compute ' SAF E from ⇥ ⌦ ||O and check ' SAF E on the Kripke structure of ⇥ ⌦ ||O. 6. ⌦ ( ) returns ⌦ certain if ' CERT AIN is true, else it returns ⌦ safe if ' SAF E , otherwise it returns ⌦ ambiguous.

Table 1 :

 1 Diagnosis results on the pattern ⌦ 1 .by P 1 and P 2 . After a product has been pushed into the lift, the lift goes down (action Down) to reach level 1 (signal D detects the end of the Down action). At this level the box is directed to the station that makes the request (either request Req 1 or request Req 2 ). If it is request Req 1 , then the box is pushed on the conveyor belt of station 1 (action PushReq 1 that starts with event Req 1 and ends with event EPReq 1 ). Then the conveyor belt performs a moveleft action (LeftReq 1 , ELReq 1 ) to deliver the box with the product and a move-right action to go back to initial position (RightReq 1 , ERReq 1 ). The behaviour of station 2 is similar except that its conveyor belt moves right first (RightReq 2 , ERReq 2 ) and then moves left (LeftReq 2 , ELReq 2 ). Once the box has been pushed on a conveyor belt, the lift goes up (action Up ending with the emission of signal U ) to level 2.

	Pr1	4	3	1 CERT AIN ? ' 1 SAF E ? F T
	Pr2	7	8	F	T
	D	12	13	F	F
	U	18	19	F	F
	D	23	25	F	F
	ELReq2	26	30	F	F
	ERReq2	31	37	F	F
	ELReq1	32	38	T	F
	ERReq1	33	39	T	F
	U	34	40	T	F

Table 2 :

 2 Diagnosis results on the pattern ⌦ 2 and ⌦ 3 . M 1 (p 21 ) = 1 and for any other place p M 1 (p) = 0 and M 2 (p 22 ) = 1 and for any other place p M 2 (p) = 0. In this example, the ' CERT AIN question can have the following form: let p obs denote the last place of the sequence of observations.

	' 2 CERT AIN = ⇤((p obs = 1)

Table 3 :

 3 Prod 1 Avail 1 Diagnosis results on the pattern ⌦ 3 .

			Prod 1		WaitUp		Prod 2
			Pr 1			Pr 2
					U
						Prod 2 Avail 2
			P 1		Up	P 2
			Push 1			Push 2
			EP 1			EP 2
					Down
			Wait1		D	Wait2
			Req 1		Req 2
					WaitDown
			PushReq 1			PushReq 2
	ERReq 1	EPReq 1			EPReq 2	ELReq 2
			LeftReq 1			RightReq 2
			ELReq 1		ERReq 2
			RightReq 1			LeftReq 2
			Figure 6: Case study: the product transportation system
	Scenario Size nb(Req 1 ) counter	nS( )	nT ( )	Time (ms)	Result
	S 1	20	5	1	92	107	11	safe
	S 2	200	5	1	1028	1223	11	safe
	S 3	2000	5	1	10388	12383	235	safe
	S 4	5000	5	1	25988	30983	1105	safe
	S 5	2000	2	10	107534	128379	1993	ambiguous
	S 6	2000	5	10	10388	12383	228	safe
	S 7	2000	10	10	10388	12383	239	safe
	S 8	2000	2	10	7201	8400	156	certain
	S 9	3500	10	10	17009	20266	577	certain
	S 10	1000	5	2	907987	2155290	15745	certain
	S 11	1000	5	10	1201381 2877901	21018	certain
	S 12	2000	10	8	2737559 6482932	65066	certain
	S 13	2000	20	8	5999626 14245740	244379	certain
	S 14	1750	20	8	4812356 11444801	148629	safe


Alain Berlinet

Abdallah Elamine

André Mas
email: mas@math.univ-montp2.fr

Local linear regression for functional data

Keywords: Functional Data, Regression model, Kernel, Mean square error, Small ball probability, Inverse problem

come

1 Introduction

The data and the model

In probability theory, random functions have been for quite a long time under the lights. The tremendous advances in computer science and the opportunity to deal with data collected at a high frequency make it now possible for statisticians to study models for "high-dimensional data". As a consequence many of them focused their attention on models for functional data i.e. models that are suited for curves, for instance spectral curves, growth curves or interest rates curves... Even if seminal articles on functional data analysis date back to more than 20 years (see [START_REF] Dauxois | Asymptotic theory for the principal component analysis of a random vector function : some applications to statistical inference[END_REF]), this area is currently going through a deep bustle. The book by [START_REF] Ramsay | Functional data analysis[END_REF] initiated a series of monographs on the subject : [START_REF] Bosq | Linear Processes in Function Spaces[END_REF], Ramsay and Silverman again (2002), [START_REF] Ferraty | Nonparametric Functional Data Analysis[END_REF].

Functional Data Analysis has drawn much attention and many of the classical multivariate data analysis techniques such as Principal Component Analysis, Correlation Analysis, ANOVA, Linear Discrimination were generalized to curves. But statistical inference gave and gives birth to many papers. Linear regression and autoregression for instance rise an interesting inverse problem (see [START_REF] Yao | Functional linear regression analysis for longitudinal data[END_REF], [START_REF] Müller | Generalized functional linear models[END_REF], [START_REF] Cai | Prediction in functional linear regression[END_REF], [START_REF] Cardot | CLT in functional linear models[END_REF], Mas (2007a)). Even more recently the case of nonparametric regression was introduced in Ferraty, Vieu (2003) then studied in [START_REF] Masry | Nonparametric regression estimation for dependent functional data: asymptotic normality[END_REF] and Ferraty, Mas, Vieu (2007) : a Nadaraya-Watson type estimator was proposed. This model is the starting point of our article.

In the sequel we will consider a sample drawn from random elements with values in an infinite dimensional vector space : X 1 , ..., X n . Here X i = X i (•) is a random function defined, say, on a compact interval of the real line [0, T ]. We will also assume once and for all that the X i 's take their values in a separable Hilbert space denoted H. This Hilbert space is endowed with an inner product •, • from which is derived the norm • . Such techniques as wavalets or splines yield reconstructed curves in the (Hilbert) Sobolev spaces :

W m,2 = f ∈ L 2 ([0, T ]) : f (m) ∈ L 2 ([0, T ])
where f (m) denotes the m th derivative of f . Further information on Sobolev spaces may be found in [START_REF] Adams | Sobolev Spaces[END_REF]. However for the sake of generality we will consider H as the sequence space l 2 and any vector x will be classically decomposed in a basis, say (e i ) i∈N so that :

x 2 = +∞ i=1
x, e i 2 .

We are given a sample (y i , X i ) 1≤i≤n ∈ (R × H) ⊗n of independent and identically distributed data. Let m be the regression function that maps H onto R.

The model is a classical non parametric regression model :

y i = m(X i ) + ε i 1 ≤ i ≤ n. (1)
or, with other symbols : m(x 0 ) = E (y|X = x 0 )

where y and X stand for random variables with the same distributions as y 1 and X 1 . The noise ε follows both assumptions :

E (ε|X) = 0, E ε 2 |X = σ 2 ε
and σ 2 ε does not depend on X. The issue of the expectation of X (should the X's be centered or not ?) is not crucial ; it will be addressed later on but for simplicity we assume that E (X) = 0. Let x 0 be a fixed and known point in H. We are aiming at estimating m (x 0 ).

In finite dimension, and more precisely when X i is a real-valued random variable, m(x 0 ) may be estimated by considering an affine approximation of m around x 0 :

m (x) ≈ m (x 0 ) + m ′ (x 0 ) (x -x 0 )
when x is close to x 0 . This approach leads us to finding a solution to the following minimization problem :

min a∈R,b∈R n i=1 (y i -a -b(x 0 -X i )) 2 K x 0 -X i h (2)
which is nothing but a weighted mean square program (weighted by the K ((x 0 -X i ) /h)'s). Here K is a kernel : a measurable positive function such that K = 1 and h = h n the bandwidth indexed by the sample size. Then a * , one of the two solutions of the display above is the estimate of m (x 0 ) . As a special case taking b = 0 comes down to the classical Nadaraya-Watson estimator. We refer the interested reader to [START_REF] Nadaraya | On estimating regression[END_REF] and Fan (1993) about this topic. The generalization of (2) to higher orders (namely approximating locally m by a polynomial) gives birth to the local polynomial estimate of m (x 0 ). We refer for instance to [START_REF] Chen | Linear minimax efficiency of local polynomial regression smoothers[END_REF] for a recent article. Convergence in probability and asymptotic normality of the kernel polynomial estimators for a density function, variable bandwith and local linear regression smoothers, were studied by Fan and Gijbels (1992). When x belongs to a Hilbert space, the principle remains the same. The function m is now approximated by : m (x) ≈ m (x 0 ) + ϕ (x 0 ) , x -x 0 where ϕ (x 0 ) ∈ H is the first order derivative of m at x 0 (the gradient in fact) and the local linear estimate of m at x 0 stems from the following adapted weighted least square program :

min a∈R,ϕ∈H n i=1 (y i -a -ϕ, X i -x 0 ) 2 K X i -x 0 h . (3)
At last the estimate m n (x 0 ) of m (x 0 ) is a * , solution of (3). We refer to Barrientos-Marin, Ferraty, Vieu (2007) for another approach. These authors consider a program simplified from the one above (they replace the functional paramater ϕ by a scalar one). But display (3) seems to be a true generalization of (2) since ϕ like b estimates the derivative of m.

Remark 1 Investigating higher orders approximations turns out to be especially uneasy in this functional setting. For instance a local quadratic estimate involves the second order derivative of m (the Hessian operator) which is a symmetric positive operator on H. The local linear method appears as a good trade-off between the complexity of the method and its accuracy.

However solving (3) is not easy in this framework. The aim of the present work is to provide a bound for the mean square error of the estimate a * of m (x 0 ) that is :

E [ m n (x 0 ) -m (x 0 )]
2 through a classical bias-variance decomposition. The paper is organized as follows : the two next subsections are devoted to pointing out the two main problems that arise from the model and that are symptomatic of the functional framework. The needed assumptions are commented, then the central result is given before the last section which contains all the mathematical derivations.

The estimate and the ill-posed problem

In order to go ahead we need to define two linear operators from H to H (the first is non-random, the second is random, based on the sample). The usual sup-norm for operator T will be denoted :

T ∞ = sup x∈B1 T x
where B 1 stands for the closed unit ball of H. From now on the reader should be familiar with basic notions related to the theory of bounded and unbounded linear operators on Hilbert space. A wide literature exists on this stopic which is central in the mathematical science. Some of our references are [START_REF] Weidman | Linear Operators in Hilbert Spaces[END_REF], [START_REF] Akhiezer | Theory of Linear Operators in Hilbert Spaces[END_REF], [START_REF] Dunford | Linear Operators[END_REF], Gohberg, Goldberg, Kaashoek (1991) amongst many others.

Definition 2

The theoretical local covariance operator of X at x 0 ∈ H associated with the kernel K is defined by :

Γ K = E K X 1 -x 0 h ((X 1 -x 0 ) ⊗ (X 1 -x 0 ))
and its empirical counterpart is :

Γ n,K = 1 n n k=1 K X k -x 0 h ((X k -x 0 ) ⊗ (X k -x 0 )) . (4)
Remark 3 In fact neither Γ K nor Γ n,K are truly covariance operators since the involved random elements are not centered, they could also be named "local second order moments operators". Also note that Γ K depends on h and h will depend on the sample size n. So the reader must keep in mind that the index n was dropped in the notation Γ K .

It is important to give some basic properties of these operators. We listed those which will be useful in the sequel :

• Γ K and Γ n,K are self -adjoint and trace-class hence compact whenever K has compact support.

• Both operators tend to zero when h does. Indeed :

Γ K ∞ ≤ E K X 1 -x 0 h X 1 -x 0 2 ≤ Ch 2
as will be shown in the section devoted to mathematical derivations. The operator Γ n,K also tends to 0 as a consequence of the strong law of large numbers for sequences of independent Banach valued random variables (see [START_REF] Ledoux | Probability in Banach Spaces. Isoperimetry and Processes[END_REF]).

• When Γ K is one to one its inverse exists. Sufficient conditions on K and on X for Γ K to be injective are not difficult to find but this interesting issue is slightly above the scope of this work. Then Γ -1 K is an unbounded linear operator acting from a dense domain of H onto H. It should be stressed that Γ -1 K is continuous at no point of its domain (it is nowhere continuous).

Imagine that the distribution of the data (namely of the couple (y, X)) is known. We could consider to solve, instead of (3) :

min a∈R,ϕ∈H E (y -a -ϕ, X -x 0 ) 2 K X -x 0 h . (5)
The first stumbling stone appears within the next Proposition.

Proposition 4 Even when the distribution of the data is known, the solution a * th of the "theoretical" program (5) exists only when Γ K is one to one. Then a * th is the solution of a linear inverse problem which involves the unbounded inverse (whenever it exists) of Γ K :

a * th = E (yK) -Γ -1 K E (yKZ) , E (KZ) E (K) -Γ -1 K E (KZ) , E (KZ) (6)
where, for the sake of shortness, we denoted :

Z (x 0 ) = Z = X -x 0 and K = K ( X -x 0 /h) .
The problem gets deeper when we go back to the original and empirical program (3) It turns out that the solution cannot be explicitely written since Γ n,K (which replaces now Γ K ) has no inverse because it has a finite rank. Its rank is clearly bounded by n. In other words the inverse Γ -1 n,K does not exist. A classical remedy consists in replacing Γ -1 n,K by a bounded operator Γ † n,K depending on n and such that Γ † n,K "behaves pointwise" like the inverse of Γ n,K . This inverse operator, which is not always the Moore-Penrose pseudo inverse, will be called the regularized inverse of Γ n,K . Several procedures could be carried out.

• Truncated spectral regularization : here this method matches the usual "Moore-Penrose" pseudo inversion hence Γ n,K Γ † n,K and Γ † n,K Γ n,K are both projection operators on H. In fact if the spectral decomposition of Γ n,K is Γ n,K = mn i=1 µ i,n (u i,n ⊗ u i,n ) where for all i (µ i,n , u i,n ) are the eigenvalues/eigenvectors of Γ n,K (we will always asssume that the positive µ i,n 's are arranged in decreasing order) :

Γ † n,K = Nn i=1 1 µ i,n (u i,n ⊗ u i,n ) , (7)
where N n ≤ m n .

• Penalization : Now Γ † n,K = (Γ n,K + α n S) -1 where α n is a (positive) sequence tending to zero and S is a known operator chosen so that Γ n,K + α n S has a bounded inverse. Here S may be taken to be the identity operator.

• Tikhonov regularization : It comes down here, since Γ n,K is symmetric, to taking :

Γ † n,K = Γ 2 n,K + α n I -1 Γ n,K .
The sequence α n is again positive and tends to zero.

Several other methods exist. The reader is referred for instance to [START_REF] Tikhonov | Solutions of Ill-posed problems, Winstons and sons, Washington[END_REF], [START_REF] Groetsch | Inverse Problems in the Mathematical Sciences[END_REF] or [START_REF] Engl | Regularization of inverse problems[END_REF].

Remark 5 In all situations it should be noted that :

sup n Γ † n,K Γ n,K ∞ < +∞, sup n Γ n,K Γ † n,K ∞ < +∞.
All these regularizing methods may also be applied to Γ K as well and lead to Γ † K and this operator depends on n even if this index does not explicitely appear. One may then prove that for all x in the domain of

Γ -1 K , Γ † K x → Γ -1 K
x when n goes to infinity. In addition to the boundedness, the operator Γ † n,K is always selfadjoint and positive.

We are now in a position to propose an estimate for m (x 0 ). This estimate will depend on the chosen regularization technique applied to Γ n,K . We will see that, under suitable conditions on Γ † n,K the convergence of our estimate does not depend on the choice of Γ † n,K .

Proposition 6

The local linear estimate of m (x 0 ) is the real solution m n (x 0 ) of (3) :

m n (x 0 ) = n i=1 y i ω i,n n i=1 ω i,n , (8)
where

ω i,n = K X i -x 0 h 1 -X i -x 0 , Γ † n,K Z K,n and
Z K,n = 1 n n i=1 K X i -x 0 h (X i -x 0 ) .
The proof of the this Proposition is omitted since it stems from calculations similar to those carried out in the proof of Proposition 4.

It is easy to check that (8) is the empirical counterpart of [START_REF] Bosq | Linear Processes in Function Spaces[END_REF]. We finally see that m n (x 0 ) may be viewed as a linear combination of the outputs y 1 , ..., y n and may be expressed from a * th just by replacing expectations by sums. The reader may also compare our estimate with its one-dimensional counterpart (display 2.2 p.198 in Fan (1993)) and will also notice that the nice properties of the ω i,n 's in this setting do not hold anymore (see display 2.5 p. 198 in Fan (1993) and the lines below).

The next section is devoted to developing the framework as well as the assumptions needed to get our central result.

Assumptions and framework

In all the sequel we assume :

A 1 :
The kernel K is one-sided, defined on [0, 1], bounded and K (1) > 0. Besides K ′ is also defined on [0, 1] , is non-null and belongs to L 1 ([0, 1]).

We did not try to find minimal conditions on the kernel. However the assumption K (1) > 0 is rather rarely required in the non-parametric literature -to the authors' knowledge-and is essential here.

The small ball problem and the class Gamma

Consider the one-dimensional version of our model ( 1) and take X ∈ R with density f . Fan (1993) studied the minimax properties of the local linear estimate in this setting and gave the Mean Square Error (see Theorem 2 p.199). This MSE depends on f (x 0 ). Here appears the second major problem. When the data belong to an infinite-dimensional space, their density does not exist, in the sense that Lebesgue's measure -or any universal reference measure with similar properties-does not exist. Consequently we must expect serious troubles since it is plain that the density of the functional input X cannot be defined as easily as if it real or even multivariate. Once again this problem will not be managed by just letting the dimension tend to infinity and we should find a way to overcome this major concern.

It turns out that in many computations of expectations the problem mentioned above may be shifted to what is known in probability theory as small ball problems. Roughly speaking, if ϕ is a real valued function (we set x 0 = 0 for simplicity), E (ϕ ( X ) K ( X /h)) may be expressed by means of P ( X ≤ h) and ϕ only. We refer to Lemma 29 in the proof section for an immediate illustration. Instead of knowing and estimating a density we must now focus on P ( X ≤ h) for small h and everyone may understand why this function is often referred to as the "small ball probability associated with X". We propose such references as Li, Linde (1993), Kuelbs, Li, Linde (1994), Li, Linde (1999) as well as the monograph by [START_REF] Li | Gaussian processes : Inequalities, small ball probabilities and applications[END_REF] which provides an interesting state of the art in this area.

What can be said about the function P ( X ≤ h) ? Obviously, by Glivenko-Cantelli's theorem it will be easily estimated from the sample (the rate of convergence is non parametric). Besides it is not hard to see that, under suitable but mild assumptions, if X ∈ R p with density f : R p → R + , P ( X -x 0 ≤ h) ∼ h p f (x 0 ). But this fact leaves unsolved the question : what can be said when p → +∞ ?

In probability theory most of the small ball considerations focused on the case where X is the brownian motion, the brownian bridge or some known relatives. Several norms were investigated as well. Most of the theorems collected in the literature yield :

P ( X < h) ≍ C 1 h α exp - C 2 h β (9)
where α, β, C 1 and C 2 are positive constants. The symbol ≍ is sometimes replaced by the more precise ∼ . Another serious problem comes from the fact that the C ∞ function on the right in the display above has its derivates null at zero at all orders. Other results assess that, when x 0 belongs to the Reproducing Kernel Hilbert Space of X,

P ( X -x 0 < h) ∼ C x0 P ( X < h)
where C x0 does not depend on h but on x 0 and on the distribution of X. Two majors contributions will be found in Meyer-Wolf , Zeitouni (1993) and in Dembo, Meyer-Wolf , Zeitouni (1995). The authors give the exact asymptotic of P ( X l 2 ≤ h) when X is a l 2 -valued gaussian random element (by means of large deviation theory) :

X = (a 1 x 1 , a 2 x 2,.... ) (10)
with x i independent, N (0, 1)-distributed and

a 2 i < +∞. When a i = i -r (r > 1/2
) they obtain a formula similar to [START_REF] Chen | Linear minimax efficiency of local polynomial regression smoothers[END_REF]. Recently [START_REF] Mas | Local functional principal component analysis, to appear in Complex Analysis and Operator Theory[END_REF] derived the estimate when a i = exp (-ci) , c > 0 and got :

P ( X < h) ∼ C 1 [log (1/h)] -1/2 exp -C 2 [log (h)] 2 . (11)
A very strange fact is that both functions in ( 9) and ( 11) belong to a class of functions known in the theory of regular variations : the class Gamma introduced and studied by de [START_REF] De Haan | A form of regular variation and its application to the domain of attraction of the double exponential distribution[END_REF][START_REF] De Haan | Equivalence classes of regularly varying functions[END_REF]. This class arises in the theory of extreme values and is closely related to the domain of attraction of the double exponential distribution. It was initially introduced by de Haan as a "Form of Regular Variation". We provide now the definition of the class Gamma at 0, denoted Γ 0 .

Definition 7 A function f belongs to de Haan's class Γ 0 with auxiliary function ρ if f maps a positive neighborhood of 0 onto a positive neighborhood of 0, f (0) = 0, f is non decreasing and for all x ∈ R, and ρ (0) = 0 with :

lim s↓0 f (s + xρ (s)) f (s) = exp (x) (12)
In a recent manuscript, [START_REF] Mas | A representation theorem for gaussian small ball probabilities[END_REF] proved that, in the framework of Dembo, Meyer-Wolf , Zeitouni (1995), the small ball probability of any random element that may be defined like display [START_REF] Dauxois | Asymptotic theory for the principal component analysis of a random vector function : some applications to statistical inference[END_REF] belongs to the class Gamma. A work is in progress to prove that, under suitable assumptions on the auxiliary function, the reciprocal also holds. The auxiliary functions appearing in displays ( 9) and ( 11) may be easily computed. [START_REF] Mas | A representation theorem for gaussian small ball probabilities[END_REF] proved that ρ depends only on the sequence a (•) that defines X in [START_REF] Dauxois | Asymptotic theory for the principal component analysis of a random vector function : some applications to statistical inference[END_REF].

The next Proposition illustrates the Definition above and will be useful in the section devoted to the main results.

In all the sequel and especially within the proof section, C denotes a constant (which will vary from a theorem to another).

Proposition 8 When the small ball probability is defined by the right hand side of ( 9), the function ρ is :

ρ (s) = Cs 1+β (13)
with β > 0, and when the small ball probability is defined by the right hand side of [START_REF] Dembo | Exact behavior of gaussian semi-norms[END_REF], the function ρ is :

ρ (s) = C s |log s| (14)
Starting from all these considerations it seems reasonable to assume the following :

A 2 : Let F (h) = F x0 (h) = P ( X -x 0 ≤ h)
be the shifted small ball probability of X. We assume that F ∈ Γ 0 with auxiliary function ρ.

Gamma varying functions feature original properties and we give now one of them which will be useful later in the proof section. We refer to Proposition 3.10.3 and Lemma 3.10.1 p.175 in Bingham, Goldie, Teugels (1987).

Proposition 9 If F ∈ Γ 0 with auxiliary function ρ then for all x ∈ [0, 1[ , lim h→0 + F (hx) F (h) = 0 (15) lim h→0 ρ (h) h = 0 (16)
Assumption A 2 is central to tackle our problem since the mean square error, computed from our estimate actually depends on ρ. But additional assumptions should hold, especially on the distributions of the margins of X.

Assumptions on the marginal distributions

The next assumption essentially aims at simplifiying the technique of proof but could certainly be alleviated at the expense of more tedious calculations (see also [START_REF] Mas | Local functional principal component analysis, to appear in Complex Analysis and Operator Theory[END_REF]

and comments therein).

A 3 : There exists a basis (e i ) 1≤i≤n such that the margins ( X, e i ) 1≤i≤n are independent real random variables.

In all the sequel, f i = f i,x0 stands for the density of the real-valued random variable X -x 0 , e i . The behavior around 0 of the shifted density f i is crucial, like in the finite dimensional settting. It has to be smooth in a sense that is going to be made more clear now. Note that f i (0) is nothing but the density of the non-shifted random variable X, e i evaluated at x 0 , e i .

Let V 0 be a fixed neighborhood of 0, set

α i = sup u∈V0 f i (u) -f i (-u) u (f i (u) + f i (-u))
and assume that :

A 4 : +∞ i=1 α 2 i < +∞.
This assumption is close to those required in [START_REF] Mas | Local functional principal component analysis, to appear in Complex Analysis and Operator Theory[END_REF]. The next Proposition illustrates assumption A 4 in the important case when X is gaussian.

Example 10 Let X be a centered gaussian random element in H with Karhunen-Loève expansion :

X = +∞ k=1 λ k η k e k .
Here the λ k 's are the eigenvalues of the covariance operator of X, E (X ⊗ X) , the e k 's are the associated eigenvectors and the η k 's are real-valued random variables N (0, 1)-distributed. It is a well-known fact that X, e k = √ λ k η k are independent real gaussian random variables and A 3 holds. Then

f i (u) = 1 √ 2πλi exp -(u-x0,ei ) 2 2λi and sup u∈V0 |f i (u) -f i (-u)| u |f i (u) + f i (-u)| ≤ C
x 0 , e i λ i whenever x 0 , e i /λ i → 0 when i tends to infinity and A 4 holds if :

+∞ i=1 x 0 , e i 2 λ 2 i < +∞ ( 17
)
Example 11 We can also consider the family of densities indexed by the integer m :

f i (u) = C m √ λ i 1 1 + u-x0,ei √ λi 2m
where C m is a normalizing constant. We find :

α i ≤ C | x 0 , e i | λ m i
and assumption A 4 holds whenever the sequence

| x0,ei | λ m i i∈N ∈ l 2 .
Since the rate of decrease of the λ i 's is intimately related to the smoothness of the random function X, we may easily infer that A 4 should be interpreted as a smoothness condition on the function x 0 . In other words, the coordinates of x 0 in the basis e i should tend to zero at a rate which is significantly quicker than the eigenvalues of the covariance operator of X and hence that x 0 should be sufficiently smoother than X.

It should also be noted that, when the family of densities f i is not uniformly smooth enough in a neighborhood of 0, Assumption A 4 may fail. For instance, it is not hard to see that the α i 's are not even finite when f i is the density of a shifted Laplace random variable :

f i (u) = 1 2λ i exp - |u -x 0 , e i | λ i .
Remark 12 The issue of the expectation of the functional input X should be raised now. We assumed sooner that the X i 's are centered. But in practical situations we can expect µ = E (X) to be a non-null function. Then considering a new shift x 0 -µ instead of x 0 solves the problem. So we can always consider the centered version of X but we must take into account that any assumption made on x 0 should be valid for x 0 -µ. For instance [START_REF] Ferraty | Functional nonparametric statistics: a double infinite dimensional framework[END_REF] should be replaced by

+∞ i=1 x 0 -µ, e i 2 λ 2 i < +∞.

Smoothness of the regression function

In order to achieve our estimating procedure we cannot avoid to assume that the function m is regular. Since m is a mapping from H to R, its first order derivative is an element of L (H, R) , the space of bounded linear functionals from H to R which is nothing than H * ≃ H. As announced sooner m ′ (x 0 ) ∈ H.

The second order derivative belongs to

L (L (H, R) , R) ≃ L (H × H, R)
and is consequently a quadratic functional on H × H and may be represented by a symmetric positive linear operator from H to H (the Hessian operator). We will sometimes use abusive notations such as m ′′ (x 0 ) (u) , v below and throughout the proofs.

A 5 : The first order derivative of m at x 0 m ′ (x 0 ) is defined, non null and there exists a neighborhood V (x 0 ) of x 0 such that :

sup x ∈V(x0 ) m ′′ (x) ∞ < +∞.
This last display may be rewritten : for all u in H and all x in a neighborhood of x 0

m ′′ (x) u, u ≤ C u 2 .
Remark 13 Assumption A 5 assesses in a way that "the second order derivative of m in a neighborhood of x 0 is bounded".

Back to the regularized inverse

We need for immediate purpose to define a sequence involved in the rate of convergence of our estimate.

Definition 14 Let v (h) the positive sequence defined by :

v = v (h) = E K X -x 0 h X -x 0 ρ ( X -x 0 ) . (18)
It is plain that v tends to zero when h does. Since they will be used in the sequel we list now some results from [START_REF] Mas | Local functional principal component analysis, to appear in Complex Analysis and Operator Theory[END_REF]. They are collected in the next Proposition and consist in bounding thre norms of operators Γ K and Γ n,K Proposition 15 The following bound are valid

Γ K ∞ ≥ Cv (h) , (19)
Γ n,K -Γ K ∞ = O L 2 h 2 F (h) n . ( 20
)
Besides Γ K /v (h) may converge to a bounded operator, say S, that may be compact.

Before giving the main results we have to get back to the regularized inverse of Γ n,K . Indeed a bound on the norm of Γ † n,K may be derived. Under the assumption that h 2 F 1/2 (h) / n 1/2 v (h) → 0 we see that Γ n,K ∞ ≥ Cv (h) As a consequence of these facts we expect the norm Γ † n,K to diverge with rate at least 1/v (h) since :

0 < C < Γ n,K Γ † n,K ∞ ≤ Γ † n,K ∞ Γ n,K ∞ .
If the operator S mentioned in the Proposition above is compact, we may even be aware that the norm of v (h) Γ † n,K will tend to infinity since S -1 is unbounded whenever S -1 exists. All this leads us to considering the next and last assumption on Γ † n,K :

A 6 : There exists a sequence r n ↓ 0 such that max Γ † K ∞ , Γ † n,K ∞ ≤ 1 r n v (h) a.s.
Here the parameter r n just depends on the chosen regularizing method (penalization, Tikhonov, etc.) and may be viewed as a tuning parameter.

Remark 16

In fact as will be seen below the sequence r n may no tend to zero. But the situation when r n ↓ 0 is the most unfavorable one and we intend to investigate it with care. However r n v (h) always tend to zero and cannot be bounded below because of ( 19) and [START_REF] Gohberg | Classes of Linear Operators Vol I & II, Operator Theory : Advances and Applications[END_REF]. Besides if Γ K /v (h) converges to an operator with bounded inverse, the sequence r n can always be chosen constant.

Let us take some examples to illustrate the role of r n . We keep the notations of display [START_REF] Cai | Prediction in functional linear regression[END_REF] and of the lines below.

• Truncated spectral regularization : remind that

Γ † K = Nn i=1 1 µ i,n (u i,n ⊗ u i,n )
where (µ i,n , u i,n ) are the eigenelements of Γ K and Γ K ∞ = sup i {µ i,n } = µ 1,n (as announced sooner the eigenvalues are positive and arranged in a decreasing order). Hence

Γ † K ∞ = 1/ inf 1≤i≤Nn {µ i,n } = µ Nn,n then r n = µ Nn,n /µ 1,n ↓ 0 is the inverse of the conditioning index of operator Γ † K . • Penalization : Now Γ † n,K = (Γ n,K + α n I) -1 with Γ † K = mn i=1 1 µ i,n + α n (u i,n ⊗ u i,n )
and we can take r n = α n /µ 1,n . It is possible here to get r n ↑ +∞ by an accurate choice of α n and some information on µ 1,n .

• Tikhonov regularization :

Here Γ † K = Γ 2 K + α n I -1 Γ K and Γ † K = mn i=1 µ i,n µ 2 i,n + α n (u i,n ⊗ u i,n ) .
A choice for r n is here α n /µ 2 1,n and the same remark as above holds.

Statement of the results

The central result of this article is a bound on the Mean Square Error for the local linear estimate of the pointwise evaluation of the regression function at a fixed design. In the sequel the generic notation C stands for universal constants.

Theorem 17 Fix x 0 in H. When assumptions A 1 -A 6 hold and if nF (h) → +∞ :

E ( m n (x 0 ) -m (x 0 )) 2 ≤ C h 6 r 2 n + h 4 + h 2 nF (h) + v 2 (h) F 2 (h) + C nF (h) 1 + h 2 nr n v (h) + v (h) r n F (h)
.

where the first line arises from the bias of our estimate and the second stems from its variance.

Remark 18 If K is chosen to be the naive kernel, K (s) = 11 [0,1] (s) , assumption A 1 can be removed and the previous theorem remains valid.

Remark 19 It turns out that the variance term is decomposed into three. The first is (nF (h)) -1 and is classical (see [START_REF] Ferraty | Advances in nonparametric regression for functional variables[END_REF]). The two others stem directly from the underlying inverse problem and the sequence r n appears.

Note that we did not fix the issue of the sequence r n involved in the regularizing inverses Γ † K and Γ † n,K . Theorem 17 may be simplified under mild additional assumptions. Proposition 20 Taking r n ≍ h then

E ( m n (x 0 ) -m (x 0 )) 2 ≤ Ch 4 + C nF (h) 1 + h nv (h)
This Proposition is derived from Theorem 17 and Lemma 29.

Remark 21 Turning back to Proposition 8 and considering displays ( 13) and ( 14) it is not hard to see that both functions ρ are regularly varying at 0 with index 1 + β for the first and 1 for the second and hence that Proposition 20 holds. It should also be noted that from property [START_REF] Ferraty | Advances in nonparametric regression for functional variables[END_REF] in Proposition 9 that we can truly expect ρ to be of index larger than 1 whenever it is regularly varying at 0. This fact motivates the next Proposition.

Proposition 22 Under the assumptions of Theorem 17 and of Proposition 20, if the auxiliary function ρ is regularly varying at 0 with index g ≥ 1,

v (h) ≍ hρ (h) F (h) . ( 21
)
Then if ρ (s) ≥ Cs 4 in a neighborhood of 0, the mean square error becomes :

E ( m n (x 0 ) -m (x 0 )) 2 ≤ C h 4 + 1 nF (h)
and the rate of decrease of the Mean Square Error depends on h * given by

(h * ) 4 F (h * ) = 1 n . (22)
If ρ (s) /s 4 → 0 when s → 0 the above rate is damaged. For instance taking r n ≍ h the MSE becomes :

E ( m n (x 0 ) -m (x 0 )) 2 ≤ C h 4 + 1 n 2 F 2 (h) ρ (h) .
Remark 23 Display 21 was proved in [START_REF] Mas | Local functional principal component analysis, to appear in Complex Analysis and Operator Theory[END_REF]. In the first case (when ρ (s) ≥ Cs 4 ), since the bias term is here an O h 4 , the rate of convergence of our estimate outperfoms the one computed in Ferraty, Mas, Vieu (2007). The estimate was a classical Nadaraya-Watson kernel estimator whose bias was an O h 2 . Obviously the rate of convergence in the second case is damaged but even for very irregular processes such as Brownian motion or Brownian Bridge function ρ (s) is above s 2 or s 3 depending on the norms that are used. The interested reader is referred for instance to displays (20) and ( 22) in Mayer-Wolf, Zeitouni (1993) or Proposition 6.1 p.568 in Li, Shao (2002) but will have to carry out some additional computations. It seems reasonable to think that this unfavorable situation will rarely occur in a usual statistical context (with functions reconstructed on "smooth spaces"). However we prove just below that, even when ρ decays rapidly to 0, it is always possible to choose a regularizing method for Γ n,K that reaches the best rate of display [START_REF] De Haan | A form of regular variation and its application to the domain of attraction of the double exponential distribution[END_REF].

Remark 24 It may be fruitful for practical purposes to comment on formula [START_REF] De Haan | A form of regular variation and its application to the domain of attraction of the double exponential distribution[END_REF]. First we see that when X ∈ R d , F (h) ∼ Ch d then the rate of convergence in mean square turns out to be n -2/(4+d) which is the optimal rate of convergence for a twice-differentiable regression function (see [START_REF] Stone | Optimal global rates of convergence for nonparametric regression[END_REF]). When the small ball probability belongs to the class Γ 0 , this rate will depend on ρ. We know that the term F (h) will always tend to 0 quicker than h 4 and will consequently determine the choice of h. The situation is consequently more intricate than in the multivariate setting. However following the example of displays ( 9) and ( 11) we get repectively

h * n = C (log n) -1/β h * n = C (log n) -1/2
where β ≤ 3 when ρ (s) ≥ Cs 4 . Finally the rate of decrease of the mean square error is a O (log n)

-c
where c > 1.

The last Proposition is devoted to dealing with the situation described along Remark 16 : when r n does not tend to zero. This cannot happen when the regularizing method is the spectral truncation but may occur when either a penalization or a Tikhonov method are applied. We remind that we cannot avoid the condition r n v (h) ↓ 0. We start from Theorem 17.

Proposition 25 When assumptions A 1 -A 6 hold, if nF (h) → +∞, when the regularizing method allows to do so, taking r (h) = 1/ρ (h) provides :

E ( m n (x 0 ) -m (x 0 )) 2 ≤ Ch 4 + C 1 nF (h)
.

Obviously r n v (h) tends to 0. If the chosen method is penalization such that Γ † n,K = (Γ n,K + α n S)

-1
it suffices to take α n = h * F (h * ) to achieve our goal. The proof of this Proposition is easy hence omitted.

Remark 26

The rate obtained at display [START_REF] De Haan | A form of regular variation and its application to the domain of attraction of the double exponential distribution[END_REF] issued from Proposition 22 should be compared with the minimax rate obtained by Fan (1993) for scalar inputs. The MSE was then Ch 4 + C/ (nh) . We see that, replacing F (h) by h (which is logic if we consider the remark about the multivariate case just below display [START_REF] Chen | Linear minimax efficiency of local polynomial regression smoothers[END_REF] in the section devoted to the small ball problems), both formulas match. This fact leads us to another interesting issue : does this rate inherit the optimal (minimax) properties found by Fan in his article ? This question goes beyond the scope of this article. Besides not much has been done until now about optimal estimation for functional data -to the authors' knowledge. But there is no doubt that this issue will be addressed in the next future.

Conclusion

Obviously this article could be the starting point for other issues such as almost sure or weak convergence of the estimate. Almost all practical aspects were left out on purpose : they will certainly give birth to another article. However the main goal of this essentially theoretic work was to underline the rather large scope of our study. We had to seek several ideas in such various areas as probability theory, functional analysis, statistical theory of extremes, inverse problems theory. Finally it turns out that it is possible to get, in the functional setting, almost the same rate of decay for the bias as in the case of scalar inputs.

The variance involves the small ball probability evaluated at h, the selected bandwidth. A drawback arises with the necessity to introduce a new parameter : the regularizing sequence r n , which depends on the sample size (more precisely on the bandwidth h). We give no clue to find out in practical situations the bandwidth h but we guess that the ever wider literature on functional data will quickly overcome this problem by adapting classical methods such as cross-validation for instance. Another major practical concern relies in the estimation of the unknown auxiliary function ρ. Several tracks already appear to address this issue. One may think of adapting some techniques from extreme theory. After all ρ characterizes the extreme behaviour of X like tail indices for Weibull or Pareto distributions. The only difference stems from the fact that ρ is a function and not just a real number. The other idea lies in the article by [START_REF] Mas | A representation theorem for gaussian small ball probabilities[END_REF] where the auxiliary function ρ is explicitely linked with the eigenvalues of the ordinary covariance operator of X. From the estimation of these eigenvalues (which is a basic procedure) it should be possible to propose a consistent estimation of the auxiliary function as a by-product.

Proofs

For the sake of clarity we begin with an outline of the proofs. The following bias-variance decomposition for m n (x 0 ) -m (x 0 ) holds :

m n (x 0 ) -m (x 0 ) = n i=1 y i ω i,n n i=1 ω i,n -m (x 0 ) = n i=1 (y i -m (x 0 )) ω i,n n i=1 ω i,n = n i=1 (y i -m (X i )) ω i,n n i=1 ω i,n + n i=1 (m (X i ) -m (x 0 )) ω i,n n i=1 ω i,n
.

We denote :

T b,n = n i=1 (m (X i ) -m (x 0 )) ω i,n n i=1 ω i,n , (23)
T v,n = n i=1 (y i -m (X i )) ω i,n n i=1 ω i,n (24)
=

n i=1 ω i,n ε i n i=1 ω i,n
where ε was defined at display (1). Here T b,n is a bias term and T v,n is a variance term. Finally we get :

E [ m n (x 0 ) -m (x 0 )] 2 = ET 2 b,n + ET 2 v,n + 2E (T b,n T v,n ) (25)
and since

E (T b,n T v,n ) = E (T b,n E (T v,n |X 1 , ..., X n )) = 0
computing the mean square error of m n (x 0 ) comes down to computing ET 2 b,n and ET 2 v,n which will be done later.

The proof section is tiled into two subsections. The first one is devoted to giving preliminary results as well as Lemmas. In the second the main results are derived.

Preliminary results

We assume that assumptions A 1 -A 6 hold once and for all. The next two Lemmas are given for further purposes. Their proofs are omitted. The interested reader will find them in [START_REF] Mas | Local functional principal component analysis, to appear in Complex Analysis and Operator Theory[END_REF].

Lemma 27 If f belongs to the class Γ 0 with auxiliary function ρ, then for all p ∈ N,

1 0 t p √ 1 -t 2 f s 1 -t 2 dt ∼ s→0 2 p-1 2 Γ p + 1 2 f (s) ρ (s) s p+1 2
.

For any x =

x k e k in H and for i ∈ N set x 2 =i = k =i x 2 k . We denote f =i the density of X -x 0 =i . We need to compute both densities f X-x0 (density of X -x 0 ) and f X-x0,ei , X-x0 (density of the couple ( X -x 0 , e i , X -x 0 )).

Lemma 28

We have :

f X-x0,ei , X-x0 (u, v) = v √ v 2 -u 2 f i (u) f =i v 2 -u 2 11 {v≥|u|} , (26)
f X-x0 (v) = v 1 -1 f i (vt) √ 1 -t 2 f =i v 1 -t 2 dt. (27)
Besides if f X-x0 and f =i are Γ-varying for all i then they have all ρ as auxiliary function.

We begin with more specific computational Lemmas.

Lemma 29 Let ϕ be a positive real valued function, bounded on [0, 1] and regularly varying at 0 with index g ≥ 1 and let p ∈ N :

EK p X -x 0 h ϕ ( X -x 0 ) ∼ h→0 K p (1) ϕ (h) F (h) . (28)
As important special cases we mention :

EK X -x 0 h ∼ K (1) F (h) , EK 2 X -x 0 h ∼ K 2 (1) F (h) , E X -x 0 m K X -x 0 h ∼ K (1) F (h) h m .
Proof :

We prove (28) when p = 1 and denote P Xi-x0 /h the distribution of the random variable

X i -x 0 /h. Since EK X -x 0 h ϕ ( X -x 0 ) = 1 0 K (u) ϕ (hu) dP Xi-x0 /h (u) , and from K (u) ϕ (hu) = K (1) ϕ (h) - 1 u [K (s) ϕ (hs)]
′ ds we get :

EK X -x 0 h ϕ ( X -x 0 ) = K (1) ϕ (h) dP Xi-x0 /h (u) - 0≤u≤s≤1 [K (s) ϕ (hs)] ′ dP Xi-x0 /h (u)
Applying Fubini's Theorem we get :

EK X -x 0 h ϕ ( X -x 0 ) = K (1) ϕ (h) F (h) -[K (s) ϕ (hs)] ′ F (hs) ds = K (1) ϕ (h) F (h) (1 -R h ) with R h = 1 0 K ′ (s) ϕ (hs) + K (s) hϕ ′ (hs) ϕ (h) F (hs) F (h) ds
Since F is gamma-varying at 0, display [START_REF] Gijbels | Variable bandwith and local linear regression smoothers[END_REF] in Proposition 9 tells us that F (hs) /F (h) → 0 as h → 0. As ϕ is regularly varying at 0 with, say, index g ≥ 1, ϕ (hs) /ϕ (h) → s g as h goes to 0. Remind also that K ′ is integrable. We deal with

h ϕ ′ (hs) ϕ (h) = h ϕ ′ (h) ϕ (h) ϕ ′ (hs) ϕ ′ (h)
Now in Bingham, Goldie and Teugels (1987), the definition of regular variation is given p.18. From Theorem 1.7.2b p.39 we deduce that ϕ ′ is regularly varying with index g -1 ≥ 0 hence that :

lim h→0 ϕ ′ (hs) ϕ ′ (h) = s g-1
uniformly with respect to s ∈ ]0, 1] and by the direct part of Karamata's Theorem p.28 (take g = ρ -1, σ = 0 and f = ϕ ′ ) that :

lim sup h→0 h ϕ ′ (h) ϕ (h) → g
which means that hϕ ′ (hs) /ϕ (h) converges pointwise to gs g-1 (which is integrable with respect to Lebesgue's measure). Then we can apply Lebesgue's dominated convergence theorem and Proposition 9 (see display [START_REF] Gijbels | Variable bandwith and local linear regression smoothers[END_REF]) to get R h → 0 as h → 0. This last step leads to the announced result.

For the sake of shortness we will sometimes set :

Z = X -x 0 , K = K ( X -x 0 /h)
and :

Z K,n = 1 n n i=1 Z i K i = 1 n n k=1 (X i -x 0 ) K ( X i -x 0 /h) .
The next lemma is a crucial.

Lemma 30

We have :

E [ZK] 2 = E K X -x 0 h (X -x 0 ) 2 ≤ Cv 2 (h) .
Remark 31 We can measure the sharpness of the previous bound. Indeed a very simple inequality would give by Lemma 29 :

E [ZK] 2 ≤ (E ZK ) 2 = (E Z K) 2 ∼ Ch 2 F 2 (h)
whereas in view of [START_REF] Ferraty | Nonparametric Functional Data Analysis[END_REF] and -when ρ is regularly varying at 0 with positive index-of Lemma 29,

E K X -x 0 h X -x 0 ρ ( X -x 0 ) 2 ≤ h 2 ρ 2 (h) F 2 (h) .
So the bound was improved by a rate of ρ 2 (h) = o h 2 .

Proof :

Computations here are quite similar but however distinct from those carried out in [START_REF] Mas | Local functional principal component analysis, to appear in Complex Analysis and Operator Theory[END_REF]. We start with projecting E K X-x0 h (X -x 0 ) on the basis (e i ) i∈N mentioned in A 3 and compute thanks to Lemma 28 :

E K X -x 0 h X -x 0 , e i = K v h uv √ v 2 -u 2 f i (u) f =i v 2 -u 2 11 {h≥v≥|u|} dudv = h 0 vK v h v -v u √ v 2 -u 2 f i (u) f =i v 2 -u 2 du dv. (29)
Now we deal with

v -v u √ v 2 -u 2 f i (u) f =i v 2 -u 2 du = v 0 u √ v 2 -u 2 (f i (u) -f i (-u)) f =i v 2 -u 2 du hence v -v u √ v 2 -u 2 f i (u) f =i v 2 -u 2 du ≤ sup 0≤u≤v≤h 1 u f i (u) -f i (-u) f i (u) + f i (-u) v 0 u 2 √ v 2 -u 2 (f i (u) + f i (-u)) f =i v 2 -u 2 du ≤ α i v -v u 2 √ v 2 -u 2 f i (u) f =i v 2 -u 2 du.
As a consequence of the preceding lines we get

E K X -x 0 h X -x 0 , e i ≤ α i E K X -x 0 h X -x 0 , e i 2 leading to E K X -x 0 h (X -x 0 ) 2 ≤ +∞ i=1 α 2 i E K X -x 0 h X -x 0 , e i 2 2 ≤ sup i E K X -x 0 h X -x 0 , e i 2 2 +∞ i=1 α 2 i ≤ Cv (h) 2 .
Lemma 32 Both following bounds hold :

E Z K,n -EZ K,n 2 ≤ C h 2 F (h) n , E Z K,n -EZ K,n 4 ≤ C h 4 F 2 (h) n 2 .
Proof : We may see Z K,n -EZ K,n as an array of n independent centered random element with values in a Hilbert space. Denote : U = ZK -E (ZK) .

Then Z K,n -EZ K,n = (1/n) n k=1 U k . We limit ourselves to proving the second display in the Lemma, which is the most technical. It is a slightly painful but however quite simple calculation to get :

E Z K,n -EZ K,n 4 ≤ C 1 n 3 E U 1 4 + 1 n 2 E U 1 2 2 + 1 n 2 E U 1 , U 2 U 1 , U 2 ≤ C 1 n 3 E U 1 4 + 2 n 2 E U 1 2 2
where the last line stems from the first by Cauchy-Schwarz inequality. We do not want to go too deeply into steps that may be easily deduced and we hope the reader will agree that, due to the n 3 denominator the first term on the right in the display above may be neglected with respect to the second (even if 4 ). We turn to :

E U 1 2 2 ≤ E U 1
E U 2 = E Z 2 K 2 -E [ZK] 2 .
It follows from Lemma 29 and Lemma 30 that :

E [ZK] 2 = o E Z 2 K 2
hence that :

E U 2 ∼ E Z 2 K 2 ∼ Ch 2 F (h)
which finishes the proof of Lemma 32.

Lemma 33

We have :

Eω 2 1,n ≤ C F (h) + h 2 F (h) nr n v (h) + v (h) r n .
Proof : Developping ω 2 1,n we get :

ω 2 1,n = K 2 1 + Γ † n,K Z K,n , K 1 Z 1 2 -2K 2 1 Γ † n,K Z K,n , Z 1 ≤ 2K 2 1 + 2 Γ † n,K Z K,n , K 1 Z 1 2 .
We deal essentially with the second term since by Lemma 29 we know that EK 2 1 = O (F (h)) . We have :

E Γ † n,K Z K,n , K 1 Z 1 2 ≤ CE Γ † n,K Z K,n , K 1 Z 1 2
where C is here nothing but sup s K (s) . Since the expectation in the above display we bay rewritten :

E Γ † n,K Z K,n , K i Z i 2
for all i we also have :

E Γ † n,K Z K,n , K 1 Z 1 2 ≤ C n n i=1 E Γ † n,K Z K,n , K i Z i 2 = CE 1 n n i=1 K i Γ † n,K Z K,n , Z i Γ † n,K Z K,n , Z i = CE Γ n,K Γ † n,K Z K,n , Γ † n,K Z K,n since for all u in H 1 n n i=1 K i u, Z i u, Z i = Γ n,K (u) , u .
At last

E Γ † n,K Z K,n , K 1 Z 1 2 ≤ CE Γ † n,K Γ n,K Γ † n,K Z K,n , Z K,n .
We set S n = Γ † n,K Γ n,K Γ † n,K (S n is a positive symmetic operator) and notice that :

S n ∞ ≤ C Γ † n,K ∞ ≤ C r n v (h) a.s. because sup n Γ n,K Γ † n,K ∞
< +∞. Our last inequality becomes :

E Γ † n,K Z K,n , K 1 Z 1 2 ≤ CE S n Z K,n , Z K,n = CE S 1/2 n Z K,n 2 ≤ C E S 1/2 n Z K,n -EZ K,n 2 + E S 1/2 n EZ K,n 2 ≤ C r n v (h) E Z K,n -EZ K,n 2 + C r n v (h) E (ZK) 2 .
We invoke Lemma 32 and Lemma 30 to bound both terms in the preceding display. At last we get :

E Γ † n,K Z K,n , K 1 Z 1 2 ≤ C h 2 F (h) nr n v (h) + v (h) r n
which yields the desired result.

Lemma 34 When nF (h) → +∞,

n i=1 K i nK (1) F (h) -1 L 2 → 0.
where

L 2
→ denotes convergence in mean square. Proof :

n i=1 K i nK (1) F (h) -1 = n i=1 (K i -EK i ) nK (1) F (h) + EK i K (1) F (h) -1.
By Lemma 29 the second term tends to zero. We deal with the first one. We note that :

E (K i -EK i ) 2 = EK 2 i -(EK i ) 2 ∼ K 2 (1) F (h)
by Lemma 29 again. Straightforward computations give :

n i=1 (K i -EK i ) nK (1) F (h) = O L 2 1 nF (h)
hence the conclusion.

Lemma 35

We have :

E Γ † n,K Z K,n , Z K,n 2 ≤ C h 4 F 2 (h) nr 2 n v 2 (h) + v 2 (h) r 2 n .
Proof : Since Γ † n,K is a positive operator, its square root exists and :

Γ † n,K Z K,n , Z K,n = Γ † n,K 1/2 Z K,n 2 ≤ C Γ † n,K 1/2 Z K,n -EZ K,n 2 + Γ † n,K 1/2 EZ K,n 2
.

Then :

Γ † n,K Z K,n , Z K,n 2 ≤ C Γ † n,K 1/2 Z K,n -EZ K,n 4 + Γ † n,K 1/2 EZ K,n 4 ≤ C Γ † n,K 2 ∞ Z K,n -EZ K,n 4 + EZ K,n 4 .
From Lemma 30 and Lemma 32 we get :

E Γ † n,K Z K,n , Z K,n 2 ≤ C h 4 F 2 (h) nr 2 n v 2 (h) + v 4 (h) r 2 n v 2 (h) .

Derivation of the main results

We start with a short and simple intermezzo about optimization in Hilbert spaces.

Proof of Proposition 4 : Consider the program :

min a∈R,ϕ∈H E (y -a -ϕ, X -x 0 ) 2 K X -x 0 h .
Simple computations lead to :

E (a, ϕ) = E (y -a -ϕ, X -x 0 ) 2 K X -x 0 h = C + a 2 EK + Γ K ϕ, ϕ -2aE (yK) -2 E (yZK) , ϕ + 2a E (ZK) , ϕ .
Obviously E (a, ϕ) is positive strictly convex and lim a, ϕ →+∞ E (a, ϕ) = +∞ hence E (a, ϕ) has a single minimum (see [START_REF] Rockafellar | Convex Analysis[END_REF] for further information about the minimization of convex functions). It is also differentiable for all (a, ϕ) in R × H. We compute its gradient :

∇E (a, ϕ) = 2aEK -2E (yK) + 2 E (ZK) , ϕ 2Γ K ϕ -2E (yZK) + 2aE (ZK)
from which we get the solutions (a * , ϕ * ) :

a * EK + E (ZK) , ϕ * = E (yK) Γ K ϕ * = E (yZK) -a * E (ZK) .
We see from the second line that ϕ * is not uniquely defined if Γ K is not one to one. Taking ϕ * = Γ -1 K (E (yZK) -a * E (ZK)) we get m n (x 0 ) as announced.

The forthcoming Lemma assesses that the random denominator of our estimate may be replaced by a non-random one.

Lemma 36 When both

h 4 nr 2 n v 2 (h) and v(h)
rnF (h) tend to zero, the following holds :

n i=1 ω i,n nK (1) F (h) -1 L 2 → 0. Proof : n i=1 ω i,n = n i=1 K i -n Γ † n,K Z K,n , Z K,n hence n i=1 ω i,n nK (1) F (h) -1 = n i=1 K i nK (1) F (h) -1 - Γ † n,K Z K,n , Z K,n F (h) .
From Lemmas 34 and 35 we deduce that the announced Lemma 36 holds.

Variance term

We study first (see 24) :

T v,n = n i=1 (y i -m (X i )) ω i,n n i=1 ω i,n = n i=1 ε i ω i,n n i=1 ω i,n . It is plain that ET v,n = 0. Denote T v,n = n i=1 ε i ω i,n nK (1) F (h)
. We have :

T v,n -T v,n = T v,n F (h) -1 nK(1) n i=1 ω i,n F (h) .
We begin with a Proposition. By Lemma 36 just above we know that T v,n ∼ T v,n in L 2 sense i.e.

T v,n T v,n L 2 → 1.
.

Proposition 37

We have :

ET 2 v,n ≤ C 1 nF 2 (h) F (h) + h 2 F (h) nr n v (h) + v (h) r n .
Proof : As announced above it suffices to prove the Proposition for T v,n .

E T 2 v,n = E n i=1 ε i ω i,n nK (1) F (h) 2 = 1 n 2 K 2 (1) F 2 (h) E    E   n i=1 ε i ω i,n 2 |X 1 , ..., X n      = 1 n 2 K 2 (1) F 2 (h) E E n i=1 ε 2 i ω 2 i,n |X 1 , ..., X n since for i = j E [(ε i ω i,n ε j ω j,n ) |X 1 , ..., X n ] = ω i,n ω j,n E [(ε i ε j ) |X 1 , ..., X n ] = 0. Hence : E T 2 v,n = 1 n 2 K 2 (1) F 2 (h) σ 2 ε E n i=1 ω 2 i,n = σ 2 ε E ω 2 1,n nK 2 (1) F 2 (h) . By Lemma 33, E ω 2 1,n ≤ C F (h) + h 2 F (h) nr n v (h) + v (h) r n
from which we deduce the Proposition. Now we turn to the bias term.

Bias term

Remember that we have to deal with :

T b,n = n i=1 (m (X i ) -m (x 0 )) ω i,n n i=1 ω i,n .
Copying what was done above with T v,n , we know that we can focus on :

T b,n = n i=1 (m (X i ) -m (x 0 )) ω i,n nK (1) F (h)
via Lemma 36. For each i there exists c i ∈ B (x 0 , h) such that :

m (X i ) -m (x 0 ) = m ′ (x 0 ) , Z i + 1 2 m ′′ (c i ) (Z i ) , Z i .
with Z i = X i -x 0 . We deal with the first and second order derivatives separatedly :

T b,n = T b,n,1 + T b,n,2 with T b,n,1 = n i=1 m ′ (x 0 ) , Z i ω i,n nK (1) F (h) , T b,n,2 = 1 2 n i=1 m ′′ (c i ) (Z i ) , Z i ω i,n nK (1

) F (h) .

Proposition 38 We have :

E T 2 b,n,1 ≤ C h 2 nF (h) + C v 2 (h) F 2 (h) .

Proof of the Proposition :

We first see that :

n i=1 m ′ (x 0 ) , X i -x 0 ω i,n = n i=1 m ′ (x 0 ) , Z i K i 1 -Z i , Γ † n,K Z K,n = n i=1 m ′ (x 0 ) , Z i K i - n i=1 m ′ (x 0 ) , Z i K i Z i , Γ † n,K Z K,n = n m ′ (x 0 ) , Z K,n -n Γ n,K m ′ (x 0 ) , Γ † n,K Z K,n = n m ′ (x 0 ) , I -Γ n,K Γ † n,K Z K,n and
T b,n,1 = m ′ (x 0 ) , I -Γ n,K Γ † n,K Z K,n K (1

) F (h) .

Then we split into two terms :

m ′ (x 0 ) , I -Γ n,K Γ † n,K Z K,n = I -Γ n,K Γ † n,K m ′ (x 0 ) , Z K,n -EZ K,n + I -Γ n,K Γ † n,K m ′ (x 0 ) , EZ K,n .
The L 2 norm of the first is bounded by Ch F (h) /n (see Lemma 32) and the L 2 norm of the second is bounded by Cv (h) (see Lemma 30). This finishes the proof of Proposition 38.

We turn to T b,n,2 and cut it into two parts :

T b,n,2 = 1 2 n i=1 m ′′ (c i ) (Z i ) , Z i K i nK (1) F (h) - 1 2 n i=1 m ′′ (c i ) (Z i ) , Z i K i Z i , Γ † n,K Z K,n nK (1) F (h) = R bn1 + R bn2 .
The two forthcoming Propositions aim at giving a bound for the mean square norm of R bn1 and R bn2 .

Proposition 39 We get :

ER 2 bn1 ≤ C h 4 nF (h) + h 4 .

Proof of the Proposition :

It is plain to see that for all i and when Assumption A 5 holds :

0 ≤ m ′′ (c i ) (Z i ) , Z i K i ≤ sup x ∈V(x0 ) m ′′ (x) ∞ Z i 2 K i hence that : 0 ≤ R bn1 ≤ C 2 n i=1 Z i 2 K i nK (1) F (h)
It follows that :

0 ≤ R 2 bn1 ≤ C n i=1 Z i 2 K i 2 n 2 F 2 (

h) .

Then :

0 ≤ ER 2 bn1 ≤ C F 2 (h) E 1 n n i=1 K i Z i 2 2 = C F 2 (h)   1 n E K 2 i Z i 4 + 1 n 2 1≤i =j≤n E K i Z i 2 K j Z j 2   ≤ C F 2 (h) 1 n EK 2 i Z i 4 + EK i Z i 2 2 ≤ C F 2 (h) h 4 F (h) n + h 4 F 2 (h) = C h 4 nF (h) + h 4 .
We turn to R bn2 .

Proposition 40 We have :

ER 2 bn2 ≤ C h 6 r 2 n .
Dealing with R bn2 is a bit more complicated. We have :

-2R bn2 = 1 K (1) F (h) Γ † n,K Z K,n , 1 n n i=1 m ′′ (c i ) (Z i ) , Z i K i Z i .
The next operation consists in replacing Z K,n by its expectation. Like above in the proof of Proposition 38 as well as in the proof of Lemmas 33 and 35 we can add and subtract EZK from Z K,n . Once again we decide not to go through details here for the sake of shortness and clarity. Finally since the remaining involving Z K,n -EZ K,n tends to zero quicker in mean square, we can focus on :

4R 2 bn2 ≤ C F 2 (h) Γ † n,K 2 ∞ EKZ 2 1 n n i=1 m ′′ (c i ) (Z i ) , Z i K i Z i 2 . ( 30
)
At last we have to deal with :

E 1 n n i=1 m ′′ (c i ) (Z i ) , Z i K i Z i 2 .
Easy computations give :

1 n n i=1 m ′′ (c i ) (Z i ) , Z i K i Z i 2 = 1 n 2 n i=1 m ′′ (c i ) (Z i ) , Z i 2 K 2 i Z i 2 + 2 n 2 i<j m ′′ (c i ) (Z i ) , Z i m ′′ (c j ) (Z j ) , Z j K i Z i , K j Z j . (31)
We take expectations now and apply assumption A 5 to the first sum :

1 n 2 E n i=1 m ′′ (c i ) (Z i ) , Z i 2 K 2 i Z i 2 ≤ C 1 n 2 E n i=1 Z i 4 K 2 i Z i 2 = C n E K 2 i Z i 6 ≤ C n h 6 F (h) .
Since h 6 F (h) /n tends to zero at a rate much quicker than the next term we do not let it appear in the Proposition.

We fix i and j in [START_REF] Mas | A representation theorem for gaussian small ball probabilities[END_REF] and take expectation :

E m ′′ (c i ) Z i , Z i m ′′ (c j ) Z j , Z j K i Z i , K j Z j = E [ m ′′ (c i ) Z i , Z i K i Z i ] , E [ m ′′ (c j ) Z j , Z j K j Z j ] = E [ m ′′ (c) Z, Z KZ] 2 .
By assumption A 5 we get :

|E m ′′ (c i ) Z i , Z i m ′′ (c j ) (Z j ) , Z j K i Z i , K j Z j | ≤ (E m ′′ (c) Z, Z KZ ) 2 ≤ C E K Z 3 2
≤ Ch 6 F 2 (h) .

Finally with (30) at hand we have :

ER 2 bn2 ≤ C F 2 (h) v 2 v 2 r 2 n C n h 6 F (h) + Ch 6 F 2 (h) . ≤ C h 6 r 2 n
since nF (h) → +∞. At last we finish with the proof of the main Theorem which is considerably alleviated by all that was done above.

Proof of Theorem 17, Proposition 20 and Proposition 22 :

The proof of the Theorem stems from display [START_REF] Ledoux | Probability in Banach Spaces. Isoperimetry and Processes[END_REF], Propositions 37,[START_REF] Rockafellar | Convex Analysis[END_REF]39 and 40. Collecting these previous results we have :

E ( m n (x 0 ) -m (x 0 )) 2 ≤ C 1 nF 2 (h) F (h) + h 2 F (h) nr n v (h) + v (h) r n + C h 6 r 2 n + h 4 + h 2 nF (h) + v 2 (h) F 2 (h) . First from v (h) ≤ h 2 F (h) ,
we see that the first line above will be an O (1/ (nF (h))) whenever h 2 /r n and h 2 / (nr n v (h)) are bounded. We turn to the second line. The term is at least h 2 / (nF (h)) may be removed because it can be neglected with repect to the variance term. In order to reach an O h 4 for the bias we have to bound h 2 /r 2 n and 1/ h 2 nF (h) .

At last summing up all what was done above comes down to taking r n ≍ h, and n•min v (h) /h, h 2 F (h) ≥ C > 0.

Following the results of [START_REF] Mas | Local functional principal component analysis, to appear in Complex Analysis and Operator Theory[END_REF] this last inequality comes down, when ρ is regularly varying at 0 with positive index : nF (h) • min ρ (h) , h 2 ≥ C > 0 And Theorem 17 is proved.