Jean-François Bonnefon

Dominique Longin
email: dominique.longin@irit.fr

Manh-Hung Nguyen
email: .nguyen@irit.fr

A Logical Framework for Trust-Related Emotions

Keywords: Modal logic, emotions, trust, distrust

Emotion and trust are two important concerns for the elaboration of interaction systems that would be closer and more attractive to their users, in particular by endowing machines with the ability to predict, understand, and process emotions and trust. This paper attempts to construct a common logical framework for the representation of emotion and trust. This logical framework combines a logic of belief and choice, a logic of time, and a dynamic logic. Using this common framework, we identify formal relations between trust and emotions, for which we also provide behavioral validation.

Introduction

The rapidly growing field of affective computing aims at developing interaction systems that are closer and more attractive to their users, in particular by endowing machines with the ability to predict, understand, and process emotions (on the one hand), and trust (on the other hand). In this article, we introduce a unified logical approach to represent the cognitive structure of some emotions, of trust/distrust, and their relations at a formal level.

We formalize the concepts of emotions as well as trust/distrust based on cognitive models proposed by cognitive psychologists. Regarding emotions, we draw on cognitive theories (for more detail, see [START_REF] Scherer | Appraisal Processes in Emotion: Theory, methode, Research. Affective Science[END_REF]) which assume that emotions are closely tied to changes in beliefs and desires. We capitalize on psychological models that allow to recognize and distinguish emotions based on their decomposition in cognitive factors particularly the cognitive structure of emotion of Ortony et al. [START_REF] Ortony | The cognitive structure of emotions[END_REF], the cognitive patterns of emotion of Lazarus [START_REF] Lazarus | Emotion and Adaptation[END_REF] and the belief-desire theory of emotion (BDTE) [START_REF] Reisenzein | Emotions as metarepresentational states of mind: Naturalizing the belief-desire theory of emotion[END_REF][START_REF] Dretske | Naturalizing the Mind[END_REF]. Similarly, we attempt to adhere closely to cognitive definition of trust [START_REF] Castelfranchi | Social trust: A cognitive approach[END_REF] and distrust [START_REF] Castelfranchi | A Nonreductionist Approach to Trust[END_REF].

Although there are tight conceptual connections between emotion and trust [START_REF] Lahno | On the emotional character of trust[END_REF], and although there ware some separated formalization of the concepts of trust as the works of Herzig et al. [START_REF] Herzig | Prolegomena for a logic of trust and reputation[END_REF], and the concepts of emotions such as the works of Adam et al. [START_REF] Adam | A logical formalization of the OCC theory of emotions[END_REF] and Steunebrink et al. [START_REF] Steunebrink | Towards a quantitative model of emotions for intelligent agents[END_REF][START_REF] Steunebrink | A logic of emotions for intelligent agents[END_REF], there is not yet a common logic to represent them both. Our work aims at filling that gap by formally representing trust and emotions in a common logic; this common logic will enable us to lay bare the formal relations between trust and emotion. The logic we offer is a combination of the logic of beliefs and choices as the one of Herzig and Longin [START_REF] Herzig | C&L intention revisited[END_REF] (a refinement from Cohen and Levesque [START_REF] Cohen | Intention is choice with commitment[END_REF]), the logic of time (introduced by Arthur Prior [START_REF] Prior | Time and Modality[END_REF]), and dynamic logic introduced by Fischer and Ladner [START_REF] Michael | Propositional dynamic logic of regular programs[END_REF] and Harel et al. [START_REF] Harel | Dynamic Logic[END_REF]. This paper is organized as follows: Part 2 introduces the logical framework. Part 3 formalizes the cognitive structure of some emotions, Part 4 formalizes the cognitive structure of trust and distrust. Part 5 shows some formal relations in the effect of trust/distrust on the emotions, and provides behavioral validation for these relations.

2 Logical Framework Syntax. The syntactic primitives of our logic are as follows: a nonempty finite set of agents AGT = {i1, i2, . . . , in}, a nonempty finite set of atomic events EVT = {e1, e2, . . . , ep}, and a nonempty set of atomic propositions ATM = {p1, p2, . . .}. The variables i, j, k. . . denote agents. The expression i1:e1 ∈ AGT × EVT denotes an event e1 intentionally caused by agent i1 and e1 is thus called an "action". The variables α, β. . . denote such actions. The language of our logic is defined by the following BNF :

φ :=p | i:α-happens | ¬φ | φ ∨ φ | Xφ | X -1 φ | Gφ | Beli φ | Choicei φ | GrdI φ
where p ranges over ATM , i:α ranges over AGT × EVT , i:α-happens ranges over ATM for each i:α ∈ AGT × EVT , and I ⊆ AGT . The classical boolean connectives ∧ (conjunction), → (material implication), ↔ (material equivalence), ⊤ (tautology) and ⊥ (contradiction) are defined from ¬ (negation) and ∨ (disjunction).

i:α-happens reads "agent i is just about to perform the action α"; Xφ reads "φ will be true next instant"; X -1 φ reads "φ was true at the previous instant"; Gφ reads "henceforth, φ is true"; Beli φ reads "agent i believes that φ is true"; Choicei φ reads "agent i prefers that φ be true"; GrdI φ reads "φ is publicly grounded between the agents in group I".

Note 1

In the workshop, we used the concept of common belief (or mutual belief) MBeli,j. This concept is usually defined as follows:

MBeli,jϕ def = Beli ϕ ∧ Belj ϕ ∧ Beli Belj ϕ ∧ Belj Beli ϕ . . .
Thus, in KD45 logic for Bel , Beli MBeli,jϕ → MBeli,jϕ (and the same for Belj ). This is problematic for trust, because the fact that Beli MBeli,jϕ (i.e., the fact that the truster believes that there is a common belief about ϕ between i and j) be true is a sufficient condition. MBeli,jϕ is thus too strong because it necessarily implies that agent i cannot be wrong (when i believes that there is a common belief about ϕ). Maybe i is wrong, but it is sufficient.

Thus, we need an operator □ whose the meaning is "ϕ is okay for i and j" such that Beli □ϕ ̸ → □ϕ. This operator is the new operator of grounding we have introduced in the final version of our article.

This operator □ is the new operator GrdI where I ⊆ AGT . This operator is very close to the concept of group belief used in [START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF].

We define the following abbreviations:

i:α-done def = X -1 i:α-happens (Def i:α-done ) Happens i:α φ def = i:α-happens ∧ Xφ (DefHappens i:α ) Afteri:αφ def = i:α-happens → Xφ (DefAfter i:α ) Donei:αφ def = i:α-done ∧ X -1 φ (DefDone i:α ) Fφ def = ¬G¬φ (DefF) Goali φ def = Choicei FBeli φ (DefGoal i ) Intendi α def = Choicei Fi:α-happens (DefIntend i ) Capable i α def = ¬Afteri:α⊥ (DefCapable i ) Possibleiφ def = ¬Beli ¬φ (DefPossible i ) Awarenessiφ def = X -1 ¬Beli φ ∧ Beli φ (DefAwareness i )
i:α-done reads "agent i has done action α"; Happens i:α φ reads "agent i is doing action α and φ will be true next instant"; Afteri:αφ reads "φ is true after any execution of α by i"; Donei:αφ reads "agent i has done action α and φ was true at previous instant"; Fφ reads "φ will be true in some future instants"; Goali φ reads "agent i has the goal (chosen preference) that φ be true"; Intendi α reads "agent i intends to do α"; Capable i α reads "agent i is capable to do α"; Possibleiφ reads "agent i believes that it is possible φ"; Awarenessiφ reads "agent i has just experienced that φ is true".

Semantics.

For temporal operators, we use a semantics based on linear time described by a sequence (or story) of time points. (This semantics is very close to CTL* [START_REF] Clarke | Automatic verification of finite-state concurrent systems using temporal logic specifications[END_REF]) A frame F is a 4-tuples ⟨H, B, C , G ⟩ where: H is a set of stories that are represented as sequences of time points, where each time point is identified by an integer z ∈ Z, a time point z in a story h is called a situation < h, z >; B is the set of all Bi such that Bi(h, z) denotes the set of stories believed as being possible by the agent i in the situation < h, z >; C is the set of all Ci such that Ci(h, z) denotes the set of stories chosen by the agent i in the situation < h, z >; G is the set of all GI such that GI (h, z) denotes the set of stories which are publicly grounded in the group I of agents, in the situation < h, z >.

All the accessibility relations Bi are serial 1 , transitive 2 and euclidean 3 . This semantic is completely standard in epistemic logic (see [START_REF] Hintikka | Knowledge and Belief: An Introduction to the Logic of the Two Notions[END_REF][START_REF] Gochet | Epistemic Logic[END_REF]) All the accessibility relations GI are serial, transitive and euclidean (This is similar to the operator group grounding introduced by Gaudou et al. [START_REF] Gaudou | A Logical Framework for Grounding-based Dialogue Analysis[END_REF]). All the accessibility Ci are serial. Moreover, we impose for every z ∈ Z that: if h ′ ∈ Bi(h, z) then Ci(h, z) = Ci(h ′ , z). It means that if an agent believes that the world h' is possible from the world h, then the set of his/her preference worlds from h and h' are the same. In other terms, the worlds an agent prefers and the ones that agent believes that s/he prefers are the same (briefly, the agent is conscious about his/her preferences, and s/he prefers what s/he believes that s/he prefers).

A model M is a couple ⟨F, V⟩ where F is a frame and V is a function associating each atomic proposition p with the set V(p) of couple (h, z) where p is true. Truth conditions are defined as follows:

M, h, z |= p iff (h, z) ∈ V(p) M, h, z |= Xφ iff M, h, z + 1 |= φ M, h, z |= X -1 φ iff M, h, z -1 |= φ M, h, z |= Gφ iff M, h, z ′ |= φ for every z ′ ≥ z M, h, z |= Beli φ iff M, h ′ , z |= φ for every (h ′ , z) ∈ Bi(h, z) M, h, z |= Choicei φ iff M, h ′ , z |= φ for every (h ′ , z) ∈ Ci(h, z) M, h, z |= GrdI φ iff M, h ′ , z |= φ for every (h ′ , z) ∈ GI (h, z)
Other truth conditions are defined as usual.

Axiomatics. Due to our linear time semantics, the temporal operators satisfy the following principles:

i:α-happens ↔Xi:α-done

Xφ ↔¬X¬φ (2)

φ ↔XX -1 φ (3) φ ↔X -1 Xφ (4) Gφ ↔φ ∧ XGφ (5) G(φ → Xφ) →(φ → Gφ) (6)
Beli and Choicei operators are defined in a normal modal logic plus (D) axioms. 4 Thus, if □ represents a Beli operator or Choicei operator:

φ □φ (RN □ ) □(φ → ψ) →(□φ → □ψ) (K □ ) □φ →¬□¬φ (D □ )
For example, axiom D □ applied to operator Beli is DBel i , which is described as: Beli φ → ¬Beli ¬φ.

for every i ∈ AGT , B i (h, z) ̸ = ∅ if < h ′ , z > ∈ B i (h, z) and < h ′′ , z > ∈ B i (h ′ , z), then < h ′′ , z > ∈ B i (h, z) if < h ′ , z > ∈ B i (h, z) and < h ′′ , z > ∈ B i (h, z), then < h ′′ , z > ∈ B i (h ′ , z)
We use here the notation of modal logic axioms introduced by Chellas in [START_REF] Chellas | Modal Logic: an Introduction[END_REF].

(RN □ ) means that all theorems are believed (respectively: chosen) by every agent i; (K □ ) means that beliefs (respectively: choices) are closed under material implication for every agent i; (D □ ) means that beliefs (respectively: choices) of every agent i are rational: they cannot be contradictory.

The Beli operators satisfy the following principles of introspection:

Beli φ ↔Beli Beli φ (4Bel i ) ¬Beli φ ↔Beli ¬Beli φ (5Bel i )
that means that agent i is conscious of its beliefs and of its disbeliefs.

The following principle follows from the semantical constraint between belief accessibility relation and choice accessibility relation, and from axiom (D □ ) for Beli :

Choicei

φ ↔Beli Choicei φ (4BC ) ¬Choicei φ ↔Beli ¬Choicei φ (5BC )
that means that agent i is conscious of its choices and of its dischoices.

The sound and complete axiomatization of GrdI operator is defined as the one of common belief operator (also called mutual belief), which is closed to the operator described in Walton and Krabbe [START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF], also introduced by Gaudou et al. [START_REF] Gaudou | A Logical Framework for Grounding-based Dialogue Analysis[END_REF]:

φ GrdI φ (RNGrd I ) GrdI (φ → ψ) →(GrdI φ → GrdI ψ) (KGrd I ) GrdI φ →¬GrdI ¬φ (DGrd I ) GrdI φ →GrdI GrdI φ (4Grd I ) ¬GrdI φ →GrdI ¬GrdI φ (5Grd I )
Axiom (RNGrd I ) means that every tautology is public ground. Axiom (KGrd I ) means that if φ is publicly grounded in I and that φ implies ψ then ψ is also publicly grounded in I. Axiom (DGrd I ) means that the set of grounded informations is consistent: it can not be the case that both φ and ¬φ are simultaneously grounded. The positive introspection axiom (4Grd I ) and negative introspection axiom (5Grd I ) account for the public character of GrdI . From these collective awareness results: if φ has (resp. has not) been grounded then it is established that φ has (resp. has not) been grounded.

Linear time semantics entail the following principles:

Gφ →Afteri:αφ (7)
Happens i:α φ →After j:β φ

Afteri:αφ ↔¬Happens i:α ¬φ

Axiom [START_REF] Clarke | Automatic verification of finite-state concurrent systems using temporal logic specifications[END_REF] describe the relationship between time and action: if henceforth φ is true then after every action α of every agent i, φ will be true. (Note that the converse is not valid: it is possible that φ be true after every action α of every agent i performed in a situation < h, z >, and that φ be false at time z ′ > z.)

As time is linear, actions are deterministic on a given history. Thus, axiom (8) reads: if agent i is just about to perform α after what φ will be true, then after every performance of every action β by every agent j, φ will be true. In other words, if action α leads to a time point where φ is true, then every action performed by every agent leads to this time point.

Finally, axiom [START_REF] De | The Rationality of Emotion[END_REF] means that Afteri:α and Happens i:α operators are dual operators. This property is fair with respect to dynamic logic [START_REF] Harel | Dynamic Logic[END_REF].

Formalization of the cognitive structure of emotion

In this section, we present the formalization of emotions, based on their cognitive structure as proposed by Ortony et al. [START_REF] Ortony | The cognitive structure of emotions[END_REF], Frijda [START_REF] Frijda | The Emotions[END_REF] as well as those of Reisenzei [START_REF] Reisenzein | Emotions as metarepresentational states of mind: Naturalizing the belief-desire theory of emotion[END_REF] and Scherer et al. [START_REF] Scherer | Appraisal Processes in Emotion : Theory, Methods, Research, chapter Appraisal Considered as a Process of Multilevel Sequential Checking[END_REF].

Joy/Distress. The cognitive structure of Joy consists of two main factors: (i) a proposition φ is desirable for agent i, and (ii) agent i just experienced that φ is the case. To formalize the first factor, we consider that agent i desiring φ means that i wants φ to be the case. So we formalize desire as a goal (chosen preference). Therefore, the first factor is potentially formalized as Goali φ, the second factor may be formalized as Beli φ.

However, we assume that emotion is triggered at the moment when all its factors are fulfilled, and that its intensity then decreases with time [START_REF] De | The Rationality of Emotion[END_REF][START_REF] Frijda | The Emotions[END_REF]. Accordingly, we include a time factor into most emotional formulas. Thus, the first factor of Joy in particular means that agent i now recalls that at the previous instant, s/he desired φ, until experiencing that φ was in fact true: Beli X -1 Goali φ. It means that in order to be joyful, agent i must keep in mind his desire in the previous instant. Hereafter, we add this analysis for almost emotional formulas. The second factor means that agent i has just experienced that φ is true and did not previously know it: Awarenessiφ.

The same analysis applies to Distress, except that in the first factor of Distress, φ is undesirable for agent i, which we assume to mean that agent i desired ¬φ: Beli X -1 Goali ¬φ. We accordingly formalize the concept of Joy and Distress: Definition 1 (Joy/Distress)

Joy i φ def = Beli X -1 Goali φ ∧ Awarenessiφ Distress i φ def = Beli X -1 Goali ¬φ ∧ Awarenessiφ
To illustrate the definition of Joy, we can say that an individual is joyful when he has just realized that he won the lottery (Awarenessman(win lottery)) with the trivial assumption that he had been desiring to win the lottery (Belman X -1 Goalman (win lottery)). In contrast, to illustrate the definition of Distress, we can say that an individual feels distress when she learns she has lost her job (Awarenesswoman(lost job)) assuming that she had the goal not to lose her job (Belwoman X -1 Goalwoman ¬(lost job)).

Hope/Fear. The cognitive structure of Hope consists of two factors: (i) a proposition φ is desirable for agent i, and (ii) agent i believes that φ may be true in the future. To formalize the first factor, we consider that φ is not true at the moment when i hopes for it: Goali φ.

We interpret the second factor, as meaning that among all of possible future worlds, agent i believes that there is at least one world in which φ will be the case. In other terms, agent i does not believe that φ will be false in all of possible future worlds: PossibleiFφ. If i believes that φ can never be the case in all of possible future worlds, then i has no ground for hope.

The same analysis applies to Fear, except that φ is now undesirable for agent i: Goali ¬φ.

We accordingly formalize the concept of Hope and Fear as follows:

Definition 2 (Hope/Fear)

Hope i φ def = Goali φ ∧ PossibleiFφ Fear i φ def = Goali ¬φ ∧ PossibleiFφ
For example, a debutante is hopeful about being asked to dance, for she thinks it is possible (Possible girl F(being asked to dance)) and this is what she wants (Goal girl (being asked to dance)). In contrast, an employee fears to be fired when he does not wish to be fired (Goal employee ¬(f ired)) but believes it is a possibility Possible employee F(to be fired)).

Satisfaction/Disappointment. The cognitive structure of Satisfaction consists of three factors: (i) agent i desired a proposition φ, (ii) agent i used to believe that φ might be true in the near future, and (iii) agent i now experiences that φ is really the case. The first two factors mean that now, agent keeps in mind that at the previous instant, s/he desired φ and believed that φ could be true in the future (Beli X -1 (Goali φ ∧ PossibleiFφ)) (cf. the analysis of the second factor of Hope). The last factor means that i now experiences that φ is true, but did not know it the previous instant (Awarenessiφ).

The difference in the case of Disappointment is agent recalls that, in the previous instant, s/he desired ¬φ instead of φ, and s/he believed that ¬φ was possibly true in the future (Beli X -1 (Goali ¬φ ∧ PossibleiF¬φ)). We formalize Satisfaction and Disappointment as follows: Definition 3 (Satisfaction/Disappointment)

Satisfaction i φ def = Beli X -1 (Goali φ ∧ PossibleiFφ) ∧ Awarenessiφ Disappointment i φ def = Beli X -1 (Goali ¬φ ∧ PossibleiF¬φ) ∧ Awarenessiφ
For example, when the debutante realizes that she is indeed asked to dance (Awareness girl (asked to dance)) she is satisfied. Were she not to be asked to dance (Awareness girl (not asked to dance)), she would feel disappointed.

We can point out the relations between Satisfaction, Disappointment and Hope:

Satisfaction i φ ↔Beli X -1 Hope i φ ∧ Awarenessiφ (10) Disappointment i φ ↔Beli X -1 Hope i ¬φ ∧ Awarenessiφ (11)
The relation between Satisfaction and Joy can be formalized as the following proposition:

Proposition 1 (Satisfaction implies Joy)

Satisfaction i φ →Joy i φ
That is, if we feel satisfaction about something, then we will also feel joy about it.

Fear-confirmed/Relief. The cognitive structure of Fear-confirmed consists of three factors: (i) a proposition φ was undesirable for agent i, (ii) agent i believed that φ might be true in the near future, and (iii) agent i now experiences that φ is really true.

We use the same analysis as for Satisfaction, except agent recalls that in the previous instant, ¬φ was desirable for agent i (Beli X -1 Goali ¬φ).

The difference in the case of Relief is agent recalls that, in the previous instant, s/he desired φ (Beli X -1 Goali φ), and believed that ¬φ might be true in the near future (Beli X -1 (Goali φ ∧ PossibleiF¬φ)). We formalize Fear-confirmed and Relief as:

Definition 4 (Fear-confirmed/Relief) FearConfirmed i φ def = Beli X -1 (Goali ¬φ ∧ PossibleiFφ) ∧ Awarenessiφ Relief i φ def = Beli X -1 (Goali φ ∧ PossibleiF¬φ) ∧ Awarenessiφ
For example, the employee's fear of being fired is confirmed when he learns that he is indeed about to be fired (Awareness employee (f ired)) which he had been afraid of (Bel employee X -1 (Goal employee ¬(f ired) ∧ Possible employee F(f ired))). In contrast, were he to learn that he is not going to be fired (Awareness employee (not fired)), he would feel relief.

We can also point out the relations between Fear-confirmed, Relief and Fear:

FearConfirmed i φ ↔Beli X -1 Fear i φ ∧ Awarenessiφ (12) Relief i φ ↔Beli X -1 Fear i ¬φ ∧ Awarenessiφ (13)
The relation between Fear-confirmed and Distress is stated in the following proposition:

Proposition 2 (Fear-confirmed implies Distress)

FearConfirmed i φ → Distress i φ
That is, if our fears about something are confirmed, then we feel distressed.

Formalization of Trust

We now present the formalization of trust and distrust based on the cognitive definition of Castelfranchi and colleagues [START_REF] Castelfranchi | Social trust: A cognitive approach[END_REF][START_REF] Castelfranchi | A Nonreductionist Approach to Trust[END_REF].

Trust. We formalize the concept of trust based on Castelfranchi and Falcone's definition [START_REF] Castelfranchi | Social trust: A cognitive approach[END_REF] of trust in action which says that agent i trusts agent j to ensure φ by performing action α if and only if agent i desires to achieve φ (Goali φ), and agent i expects that: (i) φ can be achieved by doing action α (Beli Afterj:αφ); (ii) agent j is able to perform action α (Beli Capable j α); and (iii) agent j has the intention to do such an action (Beli Intendj α).

However, these three factors are only necessary conditions, but not sufficient ones. For example, imagine that a robber wants to steal something located on the second floor of a mansion. There is a nurse on the first floor. The robber desires that the nurse stays where she is, because it makes his robbery possible. He also believes that it is possible that the nurse will stay where she is, and that it is actually her intention. Thus, the three conditions are satisfied, but we are reluctant nonetheless to say that the robber trusts the nurse to stay where she is in order to allow for his stealing, because there is no agreement between the nurse (trustee) and the robber (trustor). So here we need to add another condition for trust: an agreement between trustor and trustee that the trustee will perform such an action (GrdI trustee : α-happens), where I = {trustor, trustee}. We accordingly formalize the concept of trust as:

Definition 5 (Trust) Trusti,j(α, φ) def = Goali φ ∧ Beli Afterj:αφ ∧ Beli Capable j α∧ Beli Intendj α ∧ Grd {i,j} j:α-happens
For example, a boss trusts his secretary to prepare a report in order to present it at a company meeting because the boss desires the report (Goal boss (report)), and in his opinion, the report can be possibly ready after the secretary prepares it (Bel boss Af tersecretary:prepare(report)), the secretary has the ability and intention to prepare the report (Bel boss Capable secretary (prepare) ∧ Bel boss Intendsecretary (prepare))). It is clear that in the relation between the boss and his secretary, there is an agreement that the secretary will prepare the report in time (Grd boss,secretary secretary : prepare-happens).

Distrust. We also adopt the definition of distrust given by Castelfranchi et al. [START_REF] Castelfranchi | A Nonreductionist Approach to Trust[END_REF] which says that agent i distrusts agent j to ensure φ by performing action α if and only if agent i desires to achieve φ (Goali φ), and agent i believes that at least one of these conditions is fulfilled: (i) agent j is not in the capacity to do action α: Beli ¬Afterj:αφ, or (ii) agent j is able to do α but he has not intention to do α: PossibleiAfterj:αφ ∧ Beli ¬Intendj α . We accordingly formalize this concept as:

Definition 6 (Distrust) DisTrusti,j(α, φ) def = Goali φ ∧ (Beli ¬Afterj:αφ∨ (PossibleiAfterj:αφ ∧ Beli ¬Intendj α))
For example, in spite of desiring the report (Goal boss (report)), the boss does not trust a new employee to prepare it because he believes the new employee is unable to perform that task(Bel boss ¬After employee:prepare (report)).

From this definition, we can decompose the concept of distrust based only on the ability of trustee: [START_REF] Castelfranchi | Social trust: A cognitive approach[END_REF]. When i trusts j, i has a positive expectation about j's power and performance. Hope also implies some positive expectation. The greater the expectations, the deeper the trust; and, conversely, the deeper the disappointment when expectations are unrealized [START_REF] Bryce | Formalizing Civic Engagement: NGOs and the Concepts of Trust, Structure, and Order in the Public Policy Process[END_REF].

The first relation is formalized as follows:

Proposition 3 (Trust implies Hope)

Trusti,j(α, φ) →Hope i φ
This means that when we trust someone about an action that will bring some results, we are hopeful that the results will be obtained. For example, in a commercial transaction, when the buyer trusts his seller to send him a product after payment (Trust buyer,seller (send, receipt)), he will be hopeful that he will receive the product (Hope buyer receive product). This proposition will be proved by applying Lemma 1: if we believe that φ is true after every execution of action α, and that someone is able to do α, then we believe that there is at least a future world in which φ is true.

Lemma 1

Beli Afterj:αφ ∧ Beli Capable j α →PossibleiFφ Once we trust someone to do an action to bring us something, we hope for the positive result of the action. In case of success, we feel satisfaction (formalized as Proposition 4). Conversely, in case of failure, we feel disappointment (formalized as Proposition 5).

Proposition 4 (Successful Trust implies Satisfaction)

Beli Donej:αTrusti,j(α, φ) ∧ Awarenessiφ →Satisfactioniφ This means that when we believe that what we trusted has now occurred, we are satisfied about it. For example, when the boss trusted his secretary to prepare the report (Donesecretary:prepareTrust boss,secretary (prepare, having report)), and on the morning of the day after, he has received the report (Bel boss having report), then he is satisfied (Satisfaction boss having report). This proposition has a corollary which is deduced from Proposition 1 and 4: When we experience that what we trusted has really occurred, we will also feel joy about it.

Corollary 1

Beli Donej:αTrusti,j(α, φ) ∧ Awarenessiφ →Joy i φ Proposition 5 (Unsuccessful Trust implies Disappointment) Beli Donej:αTrusti,j(α, φ) ∧ Awarenessi¬φ →Disappointment i ¬φ This means that we feel disappointed if what we trusted does not in fact occur. For example, a businessman trusted his partner to arrive on time to negotiate a contract. The businessman feels disappointed if the partner has not yet arrived at the scheduled time.

Behavioral validation

Although the propositions that we proved in the previous section are intuitively plausible, some of them have not yet received behavioral validation from the field of experimental psychology. We decided to collect empirical data concerning three propositions in this article, related to the emotions that follow trust when it is confirmed (Proposition 4), and when it is unconfirmed (Proposition 5); and the emotions that follow distrust, when it is unconfirmed (Proposition 8) 5 .

Following the analysis in (Section 4) which argues that trust is the conjunction of the intention, the capacity, and the agreement of trustee, the presence of Agreement is intentionally fixed for the future test. We therefore operationalize Trust as the conjunction of Intention and Capacity, and Distrust as the three remaining cases. Participants to the survey read 8 different stories, following a 2 × 2 × 2 within-subject design. The variables manipulated in the stories were Intention (Yes/No), Capacity (Yes/No), and Outcome (Success/Failure). As an example, here is the story corresponding to Intention = Yes, Capacity = Yes, and Outcome = Success.

Mr. Boss is the marketing director of a big company. He needs an important financial report before a meeting tomorrow morning, but he has no time to write it because of other priorities. He asks Mr. Support to prepare it and put it on his desk before tomorrow morning.

• Mr. Boss believes that Mr. Support has the intention to prepare the report in time. • Mr. Boss believes that Mr. Support is able to prepare the report in time.

The morning after, Mr. Boss finds the report on his desk when he arrives. In your opinion, what does he feel?

In the condition Intention = No, "Mr. Boss believes that Mr. Support has the intention to prepare the report in time" was replaced with "Mr. Boss believes that Mr. Support has no intention to prepare the report in time." In the condition Capacity = No, "Mr. Boss believes that Mr. Support is able to prepare the report in time" was replaced with "Mr. Boss believes that Mr. Support is unable to prepare the report in time." Finally, in the condition Outcome = Failure, "Mr. Boss finds the report on his desk when he arrives" was replaced with "Mr. Boss does not find the report on his desk when he arrives."

After reading each story, participants rated the extent to which the main character would feel each of 7 emotions, which included our target emotions, satisfaction, disappointment, and relief; but also some emotions that we included for exploratory purposes, such as anger or thankfulness. Ratings used a 6-point scale anchored at Not at all and Totally.

A total of 100 participants took part in an online survey. The survey was offered in two languages, French (30% of the final sample) and Vietnamese (70%). Language was entered as a control variable in all statistical analyses, but added only a small overall main effect on participants' responses, and will not be discussed any further. Satisfaction. Unsurprisingly, the analysis of variance detected a huge effect of Outcome, F (1, 98) = 597, p < .001, accounting for most of the observed variance, η 2 p = .86. In other terms, Satisfaction is almost perfectly predicted by Outcome alone. The analysis, however, also detects a comparatively small interaction effect Outcome × Trust, F (1, 98) = 8.8, p < .01, η 2 p = .08, reflecting the fact that success is even more pleasant in case of trust. Table 1 shows that the biggest score of Satisfaction is in the case of Trust follows a Success: M = 4.9, SD < 1.5. The data are in line with what was expected from Proposition 4.

Satisfaction

Relief. The analysis detected main effects of Trust, F (1, 98) = 19.1, p < .001, η 2 p = .23; and Outcome, F (1, 98) = 127, p < .001, η 2 p = .80. However, these main effects were qualified by an interaction effect Trust × Outcome, F (1, 98) = 12.3, p < .001, η 2 p = .31. Table 1 shows that the score of Relief is especially high in the case of Success is obtained despite of Distrust: M = 3.6, SD < 1.9. This interaction reflects our expectation (Proposition 8).

Disappointment. The analysis detected main effects of Trust, F (1, 98) = 28.4, p < .001, η 2 p = .16; and Outcome, F (1, 98) = 389, p < .001, η 2 p = .56. However, these main effects were qualified by an interaction effect Trust × Outcome, F (1, 98) = 44.7, p < .001, η 2 p = .11. Table 1 shows that the score of Disappointment is especially high in the case of Failure is obtained despite of Trust: M = 4.6, SD < 1.7. This interaction reflects our expectation (Proposition 5).

Conclusion

This paper introduced a logical framework that can represent the cognitive structure of emotions, trust, and the formal relations between them. In other terms, it enables to represent the effect of trust (and distrust) on emotions. Furthermore, this logical framework respects the instantaneity of emotions that previous logics of emotions did not fulfill. Finally, the formal relations between emotion and trust laid bare by the logical framework were subjected to a behavioral validation following the methods of experimental psychology. The success of this behavioral validation gives strong support to our approach, which is shown to capture lay users' intuitions about trust-related emotion.

Although we have added time factor into almost emotional formulas, which enables to eliminate rightly emotion when the relevant event has passed a long time, but it have not yet helped us to represent the nature of continuous intensity of emotions. Additionally, this paper has formalized only the effect of trust/distrust on emotions but not yet the effect of emotions on trust/distrust. These current limitations are also the potential perspective for our future research.

Definition 7 (

 7 Distrust based on ability) C-DisTrusti,j(α, φ) def = Goali φ ∧ Beli ¬Afterj:αφ 5 Trust-Related Emotions 5.1 Formal Relations Trust and Hope. Trust and Hope have an important relation because they both feature a positive expectation

acknowledgement

  This work has been supported by the Agence Nationale de la Recherche (ANR), contract No. ANR-08-CORD-005-1, and by a doctoral scholarship awarded by the University of Toulouse, contract No. 26977-2007.

Table 1 :

 1 Mean and standard deviations of affective ratings, as a function of Trust and Outcome.Descriptive statistics are displayed in Table1. Participants' responses were analyzed by means of a repeated-measure analysis of variance, aimed at detecting statistically reliable effects of Trust and Outcome on our emotions of interest.

					Relief	Disappointment
		Trust	Distrust	Trust	Distrust	Trust	Distrust
	Success 4.9 (1.5) 4.6 (1.6) 2.8 (1.9) 3.6 (1.9) 1.1 (0.6) 1.3 (0.8)
	Failure	1.1 (0.5) 1.4 (1.0) 1.3 (1.0) 1.3 (0.9) 4.6 (1.7) 3.2 (1.4)

We could not test Proposition 7 for a linguistic reason: Neither in French nor in Vietnamese (the two languages used in our experiment) could we find an everyday term equivalent to 'fear confirmed'.

DisTrust and Fear. Distrust features a negative expectation, involving fear of the other [START_REF] Lewicki | Trust, trust development, and trust repair[END_REF][START_REF] Aghion | Regulation and Distrust*[END_REF]. We state the relation between Distrust based on ability and Fear as Proposition 6.

Proposition 6 (DisTrust implies Fear)

This means that if we distrust someone to do an action to bring us something then we fear that our desire might not be fulfilled. For example, the boss might distrust his assistant with the preparation of a report he needs, and more specifically distrusts him to finish the report by the next morning (DisTrust boss,assistant (f inish, report)). Therefore, he is fearful that he might miss the report the next morning (Fear boss ¬report). This proposition will be proved by applying Lemma 2: if we believe that someone is unable to do an action to bring about something, then we believe that there is at least a future world without the expected result of this action.

Lemma 2

Beli ¬Afterj:αφ →PossibleiF¬φ Once we distrust someone to do an action to bring about something, we experience fear. If the results are indeed negative, we feel fear-confirmed (formalized as Proposition 7). If, however the action is in fact successfully performed, we feel relief (formalized as Proposition 8).

Proposition 7 (Confirmation of DisTrust implies Fear-confirmed)

Beli Donej:αC-DisTrusti,j(α, φ) ∧ Awarenessi¬φ →FearConfirmedi¬φ If the boss realizes that his assistant really did not finish the report (Bel boss ¬report), he feels fear-confirmed (FearConfirmed boss ¬report). Combining the two Propositions 2 and 7, we arrive at a corollary: when we experience that what we distrusted has now happened, we feel distressed about it.

Corollary 2

Beli Donej:αC-DisTrusti,j(α, φ) ∧ Awarenessi¬φ →Distressi¬φ Proposition 8 (Non-confirmation of DisTrust implies Relief) Beli Donej:αC-DisTrusti,j(α, φ) ∧ Awarenessiφ →Reliefiφ If the boss discovers that his assistant did in fact finish the report (Bel boss report), he feels relieved (Relief boss report).


Jean-Michel Loubes

Anne-Franc ¸oise Yao
email: anne-francoise.yao@univmed.fr.

KERNEL INVERSE REGRESSION FOR SPATIAL RANDOM FIELDS

Keywords: Kernel estimator, Spatial regression, Random fields, Strong mixing coefficient, Dimension reduction, Inverse Regression

published or not. The documents may come

Introduction

Spatial statistics includes any techniques which study phenomenons observed on spatial subset S of R N , N ≥ 2 (generally, N = 2 or N = 3). The set S can be discret, continuous or the set of realization of a point process. Such techniques have various applications in several domains such as soil science, geology, oceanography, econometrics, epidemiology, forestry and many others (see for example [START_REF] Ripley | Spatial Statistics[END_REF], [START_REF] Cressie | Statistics for Spatial Data[END_REF] or [START_REF] Guyon | Random Fields on a Network -Modeling, Statistics, and Applications[END_REF] for exposition, methods and applications).

Most often, spatial data are dependents and any spatial model must be able to handle this aspect. The novelty of this dependency unlike the time-dependency, is the lack of order relation. In fact, notions of past, present and futur does not exist in space and this property gives great flexibility in spatial modelling.

In the case of spatial regression that interests us, there is an abundant literature on parametric models. We refer for example to the spatial regression models with correlated errors often used in economics (see e.g. Anselin and Florax [START_REF] Anselin | New Directions in Spatial Econometrics[END_REF], Anselin and Bera [START_REF] Anselin | Spatial Dependence in Linear Regression Models with an Introduction to Spatial Econometrics[END_REF], Song and Lee [START_REF] Song | A note on s 2 in a spatially correlated error component regression for panel data[END_REF]) or to the spatial Generalized Linear Model (GLM) study in Diggle et al. [START_REF] Diggle | Model-based geostatistics (with discussion)[END_REF] and Zhang [START_REF] Zhang | On estimation and prediction for spatial generalized linear mixed models[END_REF]. Recall also the spatial Poisson regression methods which have been proposed for epidemiological data (see for example Diggle [START_REF] Diggle | Statistical Analysis of Spatial Point Patterns[END_REF] or Diggle et al [START_REF] Diggle | Model-based geostatistics (with discussion)[END_REF]).

Unlike the parametric case, the spatial regression on nonparametric setting have been studied by a few paper: quote for example Biau and Cadre [START_REF] Biau | Nonparametric spatial prediction[END_REF], Lu and Chen [START_REF] Lu | Spatial kernel regression estimation: Weak concistency[END_REF], Hallin et al. [START_REF] Hallin | Local linear spatial regression[END_REF], Carbon et al. [START_REF] Carbon | Kernel regression estimation for random fields[END_REF], Tran and Yakowitz [START_REF] Tran | Nearest neighbor estimators for random fields[END_REF] and Dabo-Niang and Yao [START_REF] Dabo-Niang | Kernel regression estimation for continuous spatial processes[END_REF]. Their results show that, as in the i.i.d. case, the spatial nonparametric estimator of the regression function is penalized by the dimension of the regressor. This is the spatial counterpart of the well-known problem called "the curse of dimensionality". Recall that dimension reduction methods are classically used to overcome this issue. Observing an i.i.d. sample Z i = (X i , Y i ) the aim is to estimate the regression function m(x) = E(Y |X = x). In the dimension reduction framework, one assumes that there exist Φ an orthonormal matrix d × D, with D as small as possible, and g : R D → R, an unknown function such that the function m(.) can be written as (1.1) m(x) = g(Φ .X).

Model (1.1) conveys the idea that "less information on X" , Φ .X; provides as much information on m(.) as X. The function g is the regression function of Y given the D dimensional vector Φ.X. Estimating the matrix Φ and then the function g (by nonparametric methods) provides an estimator which converges faster than the initial nonparametric estimator. The operator Φ is unique under orthogonal transformation. An estimation of this latter is done through an estimation of his range Im(Φ T ) (where Φ T is the transpose of Φ) called Effective Dimensional Reduction space (EDR).

Various methods for dimension reduction exist in the literature for i.i.d observations. For example we refer to the multiple linear regression, the generalized linear model (GLM) in [START_REF] Brillinger | A Generalized Linear Model with "Gaussian" regressor variables[END_REF], the additive models (see e.g. Hastie and Tibshirani [START_REF] Hastie | Generalized additive models (with discussion)[END_REF]) deal with methods based on estimation of the gradient of the regression function m(.) developped in for example in [START_REF] Hirstache | Structure adaptative for dimension reduction[END_REF] or [START_REF] Xia | An adaptive estimation of dimension reduction space[END_REF].

In this paper, we focus on the inverse regression method, proposed by Li [START_REF] Li | Sliced inverse regression for dimension reduction, with discussions[END_REF]: if X is such that for all vector b in R d , there exists a vector B of R D such that E(b T X|Φ.X) = B T (Φ.X) (this latter condition is satisfied as soon as X is elliptically distributed), then, if Σ denotes the variance of X, the space Im(Σ -1 var(E(X|Y )) is included into the EDR space. Moreover, the two spaces coincide if the matrix Σ -1 var(E(X|Y )) is of full rank. Hence, the estimation of the EDR space is essentially based on the estimation of the covariance matrix of the inverse regression E(X|Y ) and Σ which is estimated by using a classical empirical estimator. In his initial version, Li suggested an estimator based on the regressogram estimate of E(X|Y ) but drawbacks of the regressogram lead other authors to suggest alternatives based on the nonparametric estimation of EX|Y , see for instance [START_REF] Hsing | Nearest-neighbor inverse regression[END_REF] or [START_REF] Zhu | Asymptotics for kernel estimate of sliced inverse regression[END_REF] which enable to recover the optimal rate of convergence in √ n.

This work is motivated by the fact that to our knowledge, there is no inverse regression method estimation for spatially dependent data under strong mixing condition. Note however that a dimension reduction method for supervised motion segmentation based on spatial-frequential analysis called Dynamic Sliced Inverse Regression (DSIR) has been proposed by Wu and Lu [START_REF] Wu | Supervised motion segmention by spatial-frequential analysis and dynamic sliced inverse regression[END_REF]. We propose here a spatial counterpart of the estimating method of [START_REF] Zhu | Asymptotics for kernel estimate of sliced inverse regression[END_REF] which uses kernel estimation of EX|Y . Other methods based on other spatial estimators of EX|Y will be the subject of futher investigation.

As any spatial model, a spatial dimension reduction model must take into account spatial dependency. In this work, we focus on an estimation on model (1.1) for spatial dependent data under strong mixing conditions. The spatial kernel regression estimation of EX|Y being studied in [START_REF] Biau | Nonparametric spatial prediction[END_REF][START_REF] Carbon | Kernel density estimation for random fields: The l 1 theory[END_REF][START_REF] Carbon | Kernel regression estimation for random fields[END_REF].

An important problem in spatial modelling is that of spatial prediction. The aim being reconstruction of a random field over some domain from a set of observed values. It is such a problem that interest us in the last part of this paper. More precisely, we will use the properties of the inverse regression method to build a dimension reduction predictor which corresponds to the nonparametric predictor of [START_REF] Biau | Nonparametric spatial prediction[END_REF]. It is an interesting alternative to parametric predictor methods such as the krigging methods (see e.g. [START_REF] Wackernagel | Multivariate Geostatistics[END_REF], [START_REF] Cressie | Statistics for Spatial Data[END_REF]) or spatial autoregressive model (see for example [START_REF] Cressie | Statistics for Spatial Data[END_REF]) since it does not requires any underlying model. It only requires the knowledge of the number of the neighbors. We will see that the property of the inverse regression method provides a way of estimating this number. This paper falls into the following parts. Section 2 provides some notations and assumptions on the spatial process, as well as some preliminar results on U-statistics. The estimation method and the consistency results are presented in Section 3. Section 4 uses this estimate to forecast a spatial process. Section 5 is devoted to Conclusion. Proofs and the technical lemmas are gathered in Section 6.

General setting and preliminary Results

2.1. Notations and assumptions. Throughout all the paper, we will use the following notations. For all b ∈ R d , b (j) will denote the j th component of the vector b; a point in bold i = (i 1 , ..., i N ) ∈ n ∈ (N * ) N will be referred to as a site, we will set

1 N = ( 1, ..., 1 N times ); if n = (n 1 , ..., n N ), we will set n = n 1 × ... × n N and write n → +∞ if min i=1,...,N n i → +∞ and n i n k < C for some constant C > 0. The symbol . will denote any norm over R d , u ∞ = sup x |u(x)| for some function u and C an arbitrary positive constant. If A is a set, let 1 A (x) = 1 if x ∈ A 0 otherwise . The notation W n = O p (V n ) (respectively W n = O a.s (V n )) means that W n = V n S n
for a sequence S n , which is bounded in probability (respectively almost surely).

We are interested in some R d × R-valued stationary and measurable random field

Z i = (X i , Y i ), i ∈ (N * ) N , (N, d ≥ 1
) defined on a probability space (Ω, A, P). Without loss of generality, we consider estimations based on observations of the process (Z i , i ∈ Z N ) on some rectangular set

I n = i = (i 1 , ..., i N ) ∈ Z N , 1 ≤ i k ≤ n k , k = 1, ..., N for all n ∈ (N * ) N .
Assume that the Z i 's have the same distribution as (X, Y ) which is such that:

• the variable Y has a density f . • ∀j = 1, ..., d each component X (j) of X, is such that the pair (X (j) , Y ) admits an unknown density f X (j) ,Y with respect to Lebesgue measure λ over R 2 and each X (j) is integrable.

Spatial dependency.

As mentionned above, our model as any spatial model must take into account spatial dependence between values at differents locations. Of course, we could consider that there is a global linear relationships between locations as it is generally done in spatial linear modeling, we prefer to use a nonlinear spatial dependency measure. Actually, in many circumstances the spatial dependency is not necessarly linear (see [START_REF] Arbia | Exploring nonlinear spatial dependency in the tails[END_REF]). It is, for example, the classical case where one deals with the spatial pattern of extreme events such as in the economic analysis of poverty, in the environmental science,... Then, it is more appropriate to use a nonlinear spatial dependency measure such as positive dependency (see [START_REF] Arbia | Exploring nonlinear spatial dependency in the tails[END_REF]) or strong mixing coefficients concept (see Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF]). In our case, we will measure the spatial dependency of the concerned process by means of α-mixing and local dependency measure.

Mixing condition :

The field (Z i ) is said to satisfy a mixing condition if:

• there exists a function X : R + → R + with X (t) ↓ 0 as t → ∞, such that whenever

S, S ′ ⊂ (N * ) N , α(B(S), B(S ′ )) = sup A∈B(S), B∈B(S ′ ) |P (B ∩ C) -P (B)P (C)| ≤ ψ(CardS, CardS ′ ) X (dist(S, S ′ )) (2.1)
where B(S)(resp. B(S ′ )) denotes the Borel σ-fields generated by (Z i , i ∈ S) (resp. (Z i , i ∈ S ′ )), Card S (resp. CardS ′ ) the cardinality of S(resp. S ′ ), dist(S, S ′ ) the Euclidean distance between S and S ′ , and ψ : N 2 → R + is a symmetric positive function nondecreasing in each variable. If ψ ≡ 1, then Z i is called strong mixing. It is this latter case which will be tackled in this paper and for all v ≥ 0, we have

α (v) = sup i, j∈R N , i-j =v α (σ (Z i ) , σ (Z j )) ≤ X (v).
• The process is said to be Geometrically Strong Mixing (GSM) if there exists a non-negative constant ρ ∈ [0, 1[ such that for all u > 0, α(u) ≤ Cρ u .

Remark. A lot of published results have shown that the mixing condition (2.1) is satisfied by many time series and spatial random processes (see e.g. Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF], Guyon [START_REF] Guyon | Random Fields on a Network -Modeling, Statistics, and Applications[END_REF], Rosenblatt [START_REF] Rosenblatt | Stationnary Sequences and Random Fields[END_REF], Doukhan [START_REF] Doukhan | Mixing -Properties and Examples[END_REF]). Moreover, the results presented in this paper could be extended under additional technical assumptions to the case, often considered in the literature, where ψ satisfies:

ψ(i, j) ≤ c min(i, j), ∀ i, j ∈ N,
for some constant c > 0.

In the following, we will consider the case where α(u) ≤ Cu -θ , for some θ > 0. But, the results can be easly extend to the GSM case.

Local dependency measure.

In order to obtain the same rate of convergence as in the i.i.d case, one requires an other dependency measure, called a local dependency measure. Assume that • For ℓ = 1, ..., d, there exits a constant ∆ > 0 such that the pairs (X (ℓ) i , X j ) and ((X

(ℓ) i , Y i ), (X (ℓ) j , Y j )) admit densities f i,j and g i,j , as soon as dist(i, j) > ∆, such that |f i, j (x, y) -f (x) f (y) | ≤ C, ∀x, y ∈ R |g i, j (u, v) -g (u) g (v) | ≤ C, ∀u, v ∈ R 2
for some constant C ≥ 0.

Remark. The link between the two dependency measures can be found in Bosq [START_REF] Bosq | Nonparametric Statistics for Stochastic Processes -Estimation and Prediction -2nd Edition[END_REF].

Note that if the second measure (as is name point out) is used to control the local dependence, the first one is a kind of "asymptotic dependency" control.

Results on U-statistics.

Let (X n , n ≥ 1) be a sequence of real-valued random variables with the same distribution as F . Let the functional:

Θ(F ) = R m h(x 1 , x 2 , ..., x m )dF (x 1 )...dF (x m ),
where m ∈ N, h(.) is some measurable function, called the kernel and F is a distribution function from some given set of distribution function. Without loss of generality, we can assume that h(.) is invariable by permutation. Otherwise, the transformation

1 m! 1≤i 1 =i 2 =.
.. =im≤n h(x i 1 , ..., x im ) will provide a symmetric kernel. A U-statistic with kernel h(.) of degree m based on the sample (X i , 1 ≤ i ≤ n) is a statistic defined by:

U n = (n -m)! n! 1≤i 1 =i 2 =... =im≤n h(X i 1 , ..., X im ) It is said to be an m-order U-statistic. Let h 1 (x 1 ) = R m-1 h(x 1 , x 2 , ..., x m ) m j=2 dF (x j ). The next Lemma is a consequence of Lemma 2.6 of Sun & Chian [30]. Lemma 2.1.
Let (X n , n ≥ 1) be a stationary sequence of strongly mixing random variables. If there exists a positive number δ and δ ′ (0

< δ ′ < δ) verifying γ = 6(δ-δ ′ ) (4+δ)(2+δ ′ ) > 1 such that (2.2) ||h(X 1 , ..., X m )|| 4+δ < ∞, (2.3)
R m |h(x 1 , ..., x m )| 4+δ m j=1 dF (x j ) < ∞, and α(n) = O(n -3(4+δ ′ )/(2+δ ′ ) ) . Then, U n = Θ(F ) + 2 n n i=1 (h 1 (X i ) -Θ(F )) + O p ( 1 n ).
To give strong consistency results, we need the following law of the iterated logarithm of U-statistics: Lemma 2.2. (Sun & Chian, [START_REF] Sun | Limiting behavior of the perturbed empirical distribution functions evaluated at u-statistics for strongly mixing sequences of random variables[END_REF]) Under the same conditions of the previous lemma, we have

U n -Θ(F ) = 2 n n i=1 (h 1 (X i ) -Θ(F )) + O a.s log log n n .
Remark 2.3.

• In the following, we are dealing with a kernel h(.) = K( . hn ) which depends on n. Actually, it is a classical approach to use U-statistics result to get some assymptotic results of kernel estimators, in the i.i.d case, we refer for example Härdle and Stoker [START_REF] Härdle | Investigating smooth multiple regression by method of average derivatives[END_REF]. In fact, the dependence of h n on n does not influence the asymptotical results presented here.

Estimation of the covariance of Inverse Regression Estimator

We suppose that one deals with a random field (Z i , i ∈ Z N ) which, corresponds, in the spatial regression case, to observations of the form

Z i = (X i , Y i ), i ∈ Z N , (N ≥ 1) at different locations of a subset of R N , N ≥ 1 with
some dependency structure. Here, we are particularly interested with the case where the locations take place in lattices of R N . The general continuous case will be the subject of a forthcoming work.

We deal with the estimation of the matrix Σ e = varE(X|Y ) based on the observations of the process: (Z i , i ∈ I n ) ; n ∈ (N * ) N . In order to ensure the existence of the matrix Σ = var X and Σ e = var E(X|Y ), we assume that E||X|| 4 < ∞. For sake of simplicity we will consider centered process so EX = 0.

To estimate model (1.1), as previously mentioned, one needs to estimate the matrix Σ -1 Σ e . On the one hand, we can estimate the variance matrix Σ by the empirical spatial estimator, whose consistency will be easily obtained. On the other hand, the estimation of the matrix Σ e is delicate since it requires the study of the consistency of a suitable estimator of the (inverse) regression function of X given Y :

r(y) = ϕ(y) f (y) if f (y) = 0; EY if f (y) = 0 where ϕ(y) = R x (i) f X (i) ,Y (x (i) , y)dx, 1 ≤ i ≤ d , y ∈ R.
An estimator of the inverse regression function r(.), based on (Z i , i ∈ I n ) is given by

r n (y) = ϕn(y) fn(y) if f n (y) = 0, 1 n i∈In Y i if f n (y) = 0, with for all y ∈ R, f n (y) = 1 nh n i∈In K y -Y i h n ϕ n (y) = 1 nh n i∈In X i K y -Y i h n ,
where f n is a kernel estimator of the density, K : R d → R is a bounded integrable kernel such that K (x) dx = 1 and the bandwidth h n ≥ 0 is such that lim n→+∞ h n = 0.

The consistency of the estimators f n and r n has been studied by Carbon et al [START_REF] Carbon | Kernel density estimation for random fields: The l 1 theory[END_REF]. To prevent small-valued density observations y, we consider the following density estimator:

f e,n (y) = max(e n , f n (y))
where (e n ) is a real-valued sequence such that lim n→∞ e n = 0. Then, we consider the corresponding estimator of r r e,n (y) = ϕ n (y) f e,n (y) .

Finally, for X = 1 n i∈In X i we consider the estimator of Σ e :

Σ e,n = 1 n r e,n (Y i ) r e,n (Y i ) T -X X T .
We aim at proving the consistency of the empirical variance associated to this estimator.

Remark. Here, we consider as estimator of the density f , f e,n = max(e n , f n ), to avoid small values. There are other alternatives such as f e,n = f n +e n or f e,n = max{(f n -e n ), 0}.

Weak consistency.

In the following, for a fixed η > 0 and a random variable Z in R d , we will use the notation

Z η = E(||Z|| η ) 1/η .
In this section, we will make the following technical assumptions

(3.1) r(Y ) f (Y ) 4+δ 1 < ∞, for some δ 1 > 0 and (3.2) r(Y ) f (Y ) 1 {f (Y )≤en} 2 = O 1 n1+δ 2 .
for some 1 > δ > 0.

These assumptions are the spatial counterparts of respectively r(Y ) 4+δ < ∞ and

r(Y ) 1 {f (Y )≤en} 2 = O 1 n 1 4 +δ
needed in the i.i.d case. We also assume some regularity conditions on the functions: K(.), f (.) and r(.):

• The kernel function K(.) : R → R + is a k-order kernel with compact support and satisfying a Lipschitz condition |K (x) -K (y)| ≤ C|x -y| • f (.) and r(.) are functions of C k (R) (k ≥ 2) such that sup y |f (k) (y)| < C 1 and sup y ||ϕ (k) (y)|| < C 2 for some constants C 1 and C 2 , Set Ψ n = h k n + √ log n √ nhn . Theorem 3.1. Assume that α(t) ≤ Ct -θ , t > 0, θ > 2N and C > 0. If E(||X||) < ∞ and ψ(.) = E(||X|| 2 |Y = .) is continuous. Then for a choice of h n such that nh 3 n (log n) -1 → 0 and nh θ 1 n (log n) -1 → ∞ with θ 1 = 4N +θ θ-2N , then, we get Σ e,n -Σ e = O p h k n + Ψ 2 n e 2 n Corollary 3.2. Under Assumptions of Theorem 3.1 with h ≃ n -c 1 , e n ≃ n -c 2 for some positive constants c 1 and c 2 such that c 2 k + 1 4k < c 1 < 1 2 -2c 2 , we have Σ e,n -Σ e = o p 1 √ n .
Corollary 3.3. (Central limit theorem) Under previous assumptions, we have

√ n (Σ e,n -Σ e ) L → Λ
where Λ is a zero-mean gaussian on the space of d-order matrix with covariance var r(Y )r(Y ) T .

Strong consistency.

Here we study the case where the response, Y takes values in some compact set. We replace the assumption r(Y )

f (Y ) 1 {f (Y )≤en} 2 = O 1 n 1 2 +δ by E exp r(Y ) 1 {f (Y )≤en} =
O n-ξ for some ξ > 0. : E exp γ||X|| < ∞ for some constant γ > 0.

Theorem 3.4. If (Z u ) is GSM, for a choice of h n such that nh 3 n (log n) -1 → 0 and n h n (log n) -2N -1 → ∞. Assume also that inf S f (y) > 0 for some compact set S, then under the Assumptions of Lemma 2.1, we have:

Σ e,n -Σ e = O a.s h k n + Ψ 2 n e 2 n .
Corollary 3.5. Under previous Assumptions, with

h n ≃ (n) -c 1 , e n ≃ n-c 2 for some positive constants c 1 and c 2 such that c 2 k + 1 4k ≤ c 1 < 1 2 -2c 2 , we get Σ e,n -Σ e = o a.s log log n n .
As mentionned previously, the eigenvectors associated with the positive eigenvalues of Σ -1

n Σ e, n provide an estimation of the EDR space. Classically, weak and strong consistency results concerning the estimation of the EDR space are obtained by using the previous consistency respectively of the Σ and Σ e and the theory of perturbation as for example in [START_REF] Zhu | Asymptotics for kernel estimate of sliced inverse regression[END_REF].

4. Spatial inverse methode for spatial prediction 4.1. Prediction of a spatial process. Let (ξ n , n ∈ (N * ) N ) be a R-valued strictly stationary random spatial process, assumed to be observed over a subset O n ⊂ I n (I n is a rectangular region as previously defined for some n ∈ (N * ) N ). Our aim is to predict the square integrable value, ξ i 0 , at a given site i 0 ∈ I n -O n . In practice, one expects that ξ i 0 only depends on the values of the process on a bounded vicinity set (as small as possible) V i 0 ⊂ O n ; i.e that the process (ξ i ) is (at least locally) a Markov Random Field (MRF) according to some system of vicinity.

Here, we will assume (without loss of generality) that the set of vicinity (V j , j ∈ (N * ) N ) is defined by V j of the form j + V (call vicinity prediction in Biau and Cadre [START_REF] Biau | Nonparametric spatial prediction[END_REF]). Then it is well known that the minimum mean-square error of prediction of ξ i 0 given the data in

V i 0 is E(ξ i 0 |ξ i , i ∈ V i 0 )
and we can consider as predictor any d-dimensional vector (where d is the cardinal of V) of elements of V i 0 concatenated and ordered according to some order. Here, we choose the vector of values of (ξ n ) which correspond to the d-nearest neighbors: for each i ∈ Z N , we consider that the predictor is the vector

ξ d i = (ξ i(k) ; 1 ≤ k ≤ d) where i(k) is the k-th nearest neighbor of i.
Then, our problem of prediction amounts to estimate :

m(x) = E(ξ i 0 |ξ d i 0 = x).
For this purpose we construct the associated process:

Z i = (X i , Y i ) = (ξ d i , ξ i ), i ∈ Z N
and we consider the estimation of m(.) based on the data (Z i , ∈ O n ) and the model (1.1). Note that the linear approximation of m(.) leads to linear predictors. The available literature on such spatial linear models (we invite the reader think of the kriging method or spatial auto-regressive method) is relatively abundant, see for example, Guyon [START_REF] Guyon | Random Fields on a Network -Modeling, Statistics, and Applications[END_REF], Anselin and Florax [START_REF] Anselin | New Directions in Spatial Econometrics[END_REF], Cressie [START_REF] Cressie | Statistics for Spatial Data[END_REF], Wackernagel [START_REF] Wackernagel | Multivariate Geostatistics[END_REF]. In fact, the linear predictor is the optimal predictor (in mimimun mean square error meaning) when the random field under study is Gaussian. Then, linear techniques for spatial predicition, give unsatisfactory results when the the process is not Gaussian. In this latter case, other approaches such as log-normal kriging or the trans-Gaussian kriging have been introduced. These methods consist in transforming the original data into a Gaussian distributed data. But, such methods lead to outliers which appear as an effect of the heavy-tailed densities of the data and cannot be delete. Therefore, a specific consideration is needed. This can be done by using, for example, a nonparametric model. That is what is proposed by Biau and Cadre [START_REF] Biau | Nonparametric spatial prediction[END_REF] where a predictor based on kernel methods is developped. But, This latter (the kernel nonparametric predictor) as all kernel estimator is submitted to the so-called dimension curse and then is penalized by d (= card(V)), as highlighted in Section 1. Classically, as in Section 1, one uses dimension reduction such as the inverse regression method, to overcome this problem. We propose here an adaptation of the inverse regression method to get a dimension reduction predictor based on model (1.1):

(4.1)

ξ i = g(Φ.ξ d i ).
Remark 4.1.

(1) To estimate this model, we need to check the SIR condition in the context of prediction i.e: X is such that for all vector b in R d , there exists a vector B of

R D such that E(b T X|Φ.X) = B T (Φ.X)
, is verify if the process (ξ i ) is a spatial elliptically distributed process such as Gaussian random field. (2) In the time series forecasting problem, "inverse regression" property can be an "handicap", since then, one needs to estimate the expectation of the "future" given the "past". So, the process under study must be reversible. The flexibility that provide spatial modelling overcome this default since as mentioned in the introduction, the notion of past, present and future does not exist.

At this stage, one can use the method of estimation of the model (1.1) given in Section 1 to get a predictor. Unfortunately (as usually in prediction problem) d is unknown in practice. So, we propose to estimate d by using the fact that we are dealing both with a Markov property and inverse regression as follows.

Estimation of the number of neighbors necessary for prediction.

Note that we suppose that the underline process is a stationary Markov process with respect to the d-neighbors system of neighborhood, so the variables ξ i(k) and ξ i are independent as soon as k > d and

E(ξ i(k) |ξ i = y) = 0 (since (ξ i ) is a stationary zero mean process).
Futhermore since our estimator (of model (1.1)) is based on estimation of

E(X|Y = y) = E(ξ d i |ξ i = y) = (E(ξ i(k) |ξ i = y); 1 ≤ k ≤ d)
, that allows us to keep only the neighbors ξ i(k) for which E(ξ i(k) |ξ i = y) = 0. Then, an estimation of d is obtained by estimation of argmin k E(ξ i(k) |ξ i = y) = 0. We propose the following algorithm to get this estimator.

Algorithm for estimation of d, the number of neighbors.

(1) Initialization: specify a parameter δ > 0 (small) and fix a site j 0 ; set k = 1.

(2) compute r To get the predictor, we suggest the following algorithm:

(k) n (y) = i∈On,V j 0 ⊂On ξ i(k) K hn (y-ξ i ) i∈On,V j 0 ⊂On K hn (y-ξ i ) , the kernel estimate of r (k) (y) = E(X (k) |Y = y) (3) if |(r
(1) compute

r * n (y) = i∈On,V i 0 ⊂On ξ d i K hn (y -ξ i ) i∈On,V i 0 ⊂On K hn (y -ξ i ) (2) compute Σ e,n = 1 n i∈On,V i 0 ⊂On r * e,n (Y i ) r * e,n (Y i ) T -X X T .
(3) Do the principal component analisys of Σ -1 n Σ e, n both to get a basis of Im(Σ -1 n Σ e, n ) and estimation of the D, the dimension of Im(Φ) as suggested in the next remark (4) compute the predictor: ξi 0 = g * n (Φ * n .X i 0 ). based on data (Z i , i ∈ O n ); where g * n is the kernel estimate:

g * n (x) = i∈On,V i 0 ⊂On ξ i K hn Φ * n (x -ξ d i ) i∈On,V i 0 ⊂On K hn Φ * n (x -ξ d i ) ∀x ∈ R d . Remark 4.2.
(1) The problem of estimation of D in step ( 4) is a classical problem in dimension reduction problems. Several ways exist in the literature. One can for example use the eigenvalues representation of the matrix Σ -1 n Σ e, n , the measure of distance between spaces as in Li [START_REF] Li | Sliced inverse regression for dimension reduction, with discussions[END_REF] or the selection rule of Ferré [START_REF] Ferré | Determination of the dimension in sir and related methods[END_REF].

(2) Consitency on the convergence of ξi 0 to ξ i 0 can be obtained by sketching both result of Section 3 and results Biau and Cadre [START_REF] Biau | Nonparametric spatial prediction[END_REF].

Conclusion

In this work, we have proposed two dimension reduction methods for spatial modeling. The first one is a dimension reduction for spatial regression. It is a natural extension of the idea of Li [START_REF] Li | Sliced inverse regression for dimension reduction, with discussions[END_REF] (called Inverse Regression method) for spatially dependent variables under strong mixing condition. Then, on one hand, we can say that is a good alternative to spatial linear regression model since the link between the variables X and Y is not necessarly linear. Futhermore, as raises Li [START_REF] Li | Sliced inverse regression for dimension reduction, with discussions[END_REF], any linear model can be seen as a particular case of model (1.1) with g being the identity function and D = 1. On the other hand, as in the i.i.d. case, it requieres less data for calculus than spatial non-parametric regression methods.

The second method that we have studied here deals with spatial prediction modelling. Indeed, it is more general than kriging method were the gaussian assumption on the X is needed. Here, we requier that X belongs to a larger class of random variables (that obey to Li [START_REF] Li | Sliced inverse regression for dimension reduction, with discussions[END_REF]'s condition recalled in the introduction). Futhermore, our spatial prediction method has the ease of implementation property of the inverse regression methods. Then, for example, it allows to estimate the number of neighbors need to predict. That cannot do the non-parametric prediction method of Biau and Cadre [START_REF] Biau | Nonparametric spatial prediction[END_REF].

We have presented here the theoretical framework of our techniques. The next step is to apply them on real data. It is the subjet of works under development.

Proofs and Technical Results

Deviation Bounds .

To show the strong consistency results, we will use the following Bernstein type deviation inequality:

Lemma 6.1. Let ( ζ v , v ∈ N N ) be a zero-mean real-valued random spatial process such that each v ∈ (N * ) N there exists c > 0 verifying (6.1) E |ζ v | k ≤ k! c k-2 E |ζ v | 2 , ∀ k ≥ 2
for some constant c > 0. Let S n = v∈In ζ v . Then for each r ∈ [1, +∞] and each n ∈ (N * ) N and q ∈ (N * ) N such that 1 ≤ q i ≤ n i 2 and each ε > 0, (6.2)

P (|S n | > nε) ≤ 2 N +1 exp - qε 2 4( M 2 2 + 2 N cε) +2 N ×q×11 1 + 4c p N M 2/r 2 ε r 2r+1 α([p]) 2r/(2r+1)
where M 2 2 = sup v∈In Eζ 2 v .

Remark 6.2. Actually, this result is an extension of Lemma 3.2 of Dabo-Niang and Yao [START_REF] Dabo-Niang | Kernel regression estimation for continuous spatial processes[END_REF] for bounded processes. This extension is necessary since in the problem of our interessed, assuming the boundness of the processes amounts to assume that the X i 's are bounded. It is a restrictive condition which (generally) is incompatible with the cornerstone condition of the inverse regression (if X is elliptically distributed for example).

We will use the following lemma to get the weak consistency and a law of iterated of the logarithm as well as for the matrix Σ (as we will see immediately) than for the matrix Σ e (see the proofs of results of Section 3). Lemma 6.3. Let {X n , n ∈ N N } be a zero-mean stationary spatial process sequence, of strong mixing random variables.

(1) If E||X|| 2+δ < +∞ and α(n)

δ 2+δ < ∞, for some δ > 0. Then, 1 n i∈In X i = O p 1 n . (2) If E||X|| 2+δ < +∞ and α(n) δ 2+δ < ∞, for some δ > 0. Then, √ n( 1 n i∈In X i )/σ → N (0, 1) with σ 2 = i∈Z N cov(X k , X i ) (3) If E exp γ||X|| < ∞ for some constant γ > 0, if for all u > 0 , α(u) ≤ aρ -u , 0 < ρ < 1 or α(u) = C.u -θ , θ > N then, 1 n i∈In X i = o a.s log log n n . Remark 6.4.
• The first result is obtained by using covariance inequality for strong mixing processes (see Bosq [START_REF] Bosq | Nonparametric Statistics for Stochastic Processes -Estimation and Prediction -2nd Edition[END_REF]). Actually, it suffices to enumerate the X i 's into an arbitrary order and sketch the proof in Theorem 1.5 of Bosq [START_REF] Bosq | Nonparametric Statistics for Stochastic Processes -Estimation and Prediction -2nd Edition[END_REF]. • The law of the iterated of the logarithm holds by applying the previous Lemma 6.1 with ε = η log log n n , η > 0 and q = n log log n + 1.

Consistency of the inverse regression.

In Section 3, we have seen that the results are based on consistency results of the function r(.) which are presented now under some regularity conditions on the functions: K(.), f (.) and r(.).

• The kernel function K(.) : R → R + is a k-order kernel with compact support and satisfying a Lipschitz condition

|K (x) -K (y)| ≤ C|x -y| • f (.) and r(.) are functions of C k (R) (k ≥ 2) such that sup y |f (k) (y)| < C 1 and sup y ||ϕ (k) (y)|| < C 2 for some constants C 1 and C 2 ,
we have convergence result:

Lemma 6.5. Suppose α(t) ≤ Ct -θ , t > 0, θ > 2N and C > 0. If nh 3 n (log n) -1 → 0, nh θ 1 n (log n) -1 → ∞ with θ 1 = 4N +θ θ-2N , then (1)
(see, [START_REF] Carbon | Kernel density estimation for random fields: The l 1 theory[END_REF])

(6.3) sup y∈R |f n (y) -f (y)| = O p (Ψ n ) . (2) Furthermore, if E(||X||) < ∞ and ψ(.) = E(||X|| 2 |Y = .) is continuous, then (6.4) sup y∈R ||ϕ n (y) -ϕ(y)|| = O p (Ψ n ) .
Remark 6.6. Actually, only the result (6.3) is shown in Carbon et al [START_REF] Carbon | Kernel density estimation for random fields: The l 1 theory[END_REF] but the result (6.4) is easily obtained by noting that for all ε > 0,

P(sup y∈R ||ϕ n (y)-Eϕ n (y)|| > ε) ≤ E||X|| a n +P(sup y∈R ||ϕ n (y)-Eϕ n (y)|| > ε, ∀i, ||X i || ≤ a n ) with a n = η (log n) 1/4 , η > 0. Lemma 6.7. If (Z u ) is GSM, nh 3 n (log n) -1 → 0 and n h n (log n) -2N -1 → ∞, then (6.5
)

sup y∈R |f n (y) -f (y)| = O a.s (Ψ n ) .
Furthermore, if E (exp γ ||X||) < ∞ for some γ > 0 and ψ(.

) = E(||X|| 2 |Y = .) is continuous, then (6.6) sup y∈R ||ϕ n (y) -ϕ(y)|| = O a.s (Ψ n ) .
Remark. The equality (6.5) is due to Carbon et al [START_REF] Carbon | Kernel density estimation for random fields: The l 1 theory[END_REF]. The proof of the equality (6.6) is obtained applying Lemma 6.1 and sketching the proofs of Theorem 3.1 and 3.3 of Carbon et al [START_REF] Carbon | Kernel density estimation for random fields: The l 1 theory[END_REF]. Then it is omitted.

We will need the following lemma and the spatial block decomposition: Lemma 6.8. (Bradley's Lemma in Bosq [START_REF] Bosq | Parametric rates of nonparametric estimators and predictors for continuous time processes[END_REF]) Let (X, Y ) be an R d × R-valued random vector such that Y ∈ L r (P ) for some r ∈ [1, +∞]. Let c be a real number such that ||Y + c|| r > 0 and ξ ∈ (0, ||Y + c|| r ]. Then there exists a random variable Y * such that:

(1)

P Y * = P Y and Y * is independent of X, (2) P (|Y * -Y | > ξ) ≤ 11 (ξ -1 ||Y + c|| r ) r/(2r+1) × [α (σ(X), σ(Y ))] 2r/(2r+1) .

Spatial block decomposition.

Let

Y u = ζ v=([u i ]+1, 1≤i≤N ) , u ∈ R N .
The following spatial blocking idea here is that of Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF] and Politis and Romano [START_REF] Politis | Nonparametric resampling for homogeneous strong mixing random fields[END_REF].

Let

∆ i = i 1 (i 1 -1) ... i N (i N -1) Y u du . Then, S n = n 1 0 ... n N 0 Y u du = 1 ≤ i k ≤ n k k = 1, ...N ∆ i .
So, S n is the sum of 2

N P N q 1 × q 2 × • • • × q N terms ∆ i .
And each of them is an integral of Y u over a cubic block of side p. Let consider the classical block decomposition:

U(1, n, j) = (2j i +1)p k i =2j i p+1, 1≤i≤N ∆ k , U(2, n, j) = (2j i +1)p k i =2j i p+1, 1≤i≤N -1 2(j N +1)p k N =(2j N +1)p+1 ∆ k , U(3, n, x, j) = (2j i +1)p k i =2j i p+1, 1≤i≤N -2 2(j N-1 +1)p k N-1 =(2j N-1 +1)p+1 (2j N +1)p k N =2j N p+1 ∆ k , U(4, n, j) = (2j i +1)p k i =2j i p+1, 1≤i≤N -2 2(j N-1 +1)p k N-1 =(2j N-1 +1)p+1 2(j N +1)p k N =(2j N +1)p+1
∆ k , and so on. Note that

U(2 N -1 , n, j) = 2(j i +1)p k i =(2j i +1)p+1, 1≤i≤N -1 (2j N +1)p k N =2j N p+1 ∆ k . Finally, U(2 N , n, j) = 2(j i +1)p k i =(2j i +1)p+1, 1≤i≤N ∆ k . So, (6.7) S n = 2 N i=1 T (n, i),
with T (n, i) = q l -1 j l =0, l=1,...,N U(i, n, j). If n i = 2pt i , i = 1, ..., N, for all set of integers t 1 , ..., t N , then a term, say T n, 2 N + 1 containing all the ∆ k 's at the end, and not included in the blocks above, can be added (see Tran [31] or Biau and Cadre [START_REF] Biau | Spatial kernel density estimation[END_REF]). This extra term does not change the result of previous proof.

Proof of Lemma 6.1.

Using (6.7) it suffices to show that (6.8)

P |T (n, i)| > nε 2 N ≤ 2 exp - ε 2 4v 2 (q) q +q×11 1 + 4C p N M 2/r 2 ε r/(2r+1) α([p]) 2r/(2r+1)
for each 1

≤ i ≤ 2 N .
Without loss of generality we will show (6.8) for i = 1. Now, we enumerate (as it is often done in this case) in arbitrary way the q = q 1 × q 2 × • • • × q N terms U(1, n, j) of sum of T (n, 1) that we call W 1 , ..., W q. Note that the U(1, n, j) are measurable with respect to the σ-field generated by Y u with u such that 2j i p ≤ u i ≤ (2j i + 1)p, i = 1, ..., N.

These sets of sites are separated by a distance at least p and since for all m = 1, ..., q there exists a j(m) such that W m = U(1, n, j(m)) which have the same distribution as

W * m , E|W m | r = E|W * m | r = E (2j 1 (m)+1)p 2j 1 (m)p ... (2j N (m)+1)p 2j N (m)p Y u du r , r ∈ [1, +∞].
Noting that

(2j k (m)+1)p 2j k (m)p Y u du = [2j k (m)p]+1 2j k (m)p Y u du + [(2j k (m)+1)p] v k =[2j k (m)p]+2 ζ v + 2j k (m)+1)p [(2j k (m)+1)p] Y u du = ([2j k (m)p] + 1 -2j k (m)p) ζ (v, v k =[2j k (m)p]+1) + [(2j k (m)+1)p] v k =[2j k (m)p]+2 ζ v + ((2j k (m) + 1)p -[(2j k (m) + 1)p]) ζ (v, v k =[(2j k (m)+1)p]+1) = [(2j k (m)+1)p]+1 v k =[2j k (m)p]+1 w(j, v) k ζ v
and |w(j, v) k | ≤ 1 ∀k = 1, ..., N, we have by using Minkovski's inequality and 6.1 one get (6.9)

E W m p N r ≤ c r-2 r! M 2 2 , ∀r ≥ 2.
Then, using recursively the version of Bradley's lemma gives in Lemma 6.8 we define independent random variables W * 1 , ..., W * q such that for all r ∈ [1, +∞] and for all m = 1, ..., q, W * m has the same distribution with W m and setting ω r r = p rN c r-2 M 2 2 , we have:

P (|W m -W * m | > ξ) ≤ 11 ||W m + ω r || r ξ r/(2r+1) α([p]) 2r/(2r+1) ,
where, c = δω r p and ξ = min nε 2 N+1 q , (δ -1)ω r p N = min εp N 2 , (δ -1)ω r p N for some δ > 1 specified below. Note that for each m,

||W m + c|| r ≥ c -||W m || r ≥ (δ -1)ω r p N > 0 so that 0 < ξ < ||W m + c|| r as required in Lemma 6.8.
Then, if δ = 1 + ε 2ωr ,

P (|W m -W * m | > ξ) ≤ 11 1 + 4ω r ε r/(2r+1) α([p]) 2r/(2r+1)
and

P q m=1 |W m -W * m | > nε 2 N +1 ≤ q × 11 1 + 4ω r ε r/(2r+1) α([p]) 2r/(2r+1) .
Now, note that Inequality (6.9) also leads (by Bernstein's inequality) to :

P q m=1 W * m > nε 2 N +1 ≤ 2 exp - nε 2 N+1 2 4 q m=1 E W 2 m + cnp N 2 N+1 ε Thus P (|T (n, 1)| > nε 2 N ) ≤ 2exp - qε 2 4( M 2 2 +2 N cε) + q × 11 1 + 4c p N M 2/r 2 ε r/(2r+1) α([p]) 2r/(2r+1)
Then, since q = q 1 × ... × q N and n = 2 N p N q, we get inequality (6.8)

(r en (Y i ) -r(Y i )) (r en (Y i ) -r(Y i )) T , S n, 2 = 1 n i∈In r(Y i ) (r en (Y i ) -r(Y i )) T
and

S n, 3 = 1 n i∈In (r en (Y i ) -r(Y i )) r(Y i ) T .
Note that S T n, 3 = S n, 2 , hence we only need to control the rate of convergence of the first two terms S n, 1 and S n, 2

We will successively prove that

S n,1 = O p Ψ 2 n e 2 n , and
S n,2 = O p Ψ 2 n e 2 n
+ h k n this latter will immediately implies that

S n,3 = O p Ψ 2 n e 2 n + h k n .
• Control on S n, 1

Since for each y ∈ R : (6.12) ren (y)r(y) = r(y) f en (y) (f (y)f en (y)) + 1 fen (y) (ϕ n (y)ϕ(y)) and (6.13)

f (y) -f en (y) = f (y) -f n (y) + (f n (y) -e n )1 {fn(y)<en} , for each i ∈ (N * ) N r en (Y i ) -r(Y i ) ≤ r(Y i ) e n ||f n -f || ∞ + 2 r(Y i ) 1 {fn(Y i )<en} + ϕ n -ϕ ∞ e n .
and

r en (Y i ) -r(Y i ) 2 ≤ 3 r(Y i ) 2 ||f n -f || 2 ∞ e 2 n + 4 r(Y i ) 2 1 {fn(Y i )<en} + ||ϕ n -ϕ|| 2 ∞ e 2 n .
Using the following inequality (see Ferré and Yao [START_REF] Ferré | Smoothed functional inverse regression[END_REF] for details):

(6.14) 1 {fn(Y i )<en} ≤ 1 {f (Y i )<en} + ||f n -f || 2 ∞ e 2 n ,
and by results on Lemmas 6.3 and 6.5, we have: • Control on S n, 2 .

S n,1 ≤ C n i∈In r(Y i ) 2 1 {f (Y i )<en} + O p Ψ 2 n e 2 n , C > 0. Now, noting that 1 n i∈In r(Y i ) 2 1 {f (Y i )<en} ≤ e 2 n 1 n i∈In r(Y i ) 2 f (Y i ) 2 1 {f (Y i )<en} , we have (since E ||r(Y i )|| 2 f (Y i ) 2 1 {f (Y i )<en} = O 1 n1+δ by assumption): (6.15) 1 n i∈In r(Y i ) 2 1 {f (Y i )<en} = O p e 2 n n1+δ and
S n,1 = O p e 2 n n1+δ + Ψ 2 n e 2 n because of Assumption E r(Y ) 2 f (Y ) 2 1 {f (Y )<en} = O 1 n1+δ . Now, since Ψ n = h k n + log
Noting that :

1 fe n = 1 f + f -fen fen f + fen -fe n fen fen = 1 f + f -en fen f 1 {f <en} + fen -fe n fen fe n
, with fen = max{f, e n }, we have:

S n, 2 = 1 n i∈In r(Y i )r(Y i ) T f en (Y i ) (f (Y i ) -f en (Y i )) + r(Y i ) f en (Y i ) (ϕ n (Y i ) -ϕ(Y i )) T = 1 n i∈In r(Y i )r(Y i ) T f (Y i ) (f (Y i ) -f n (Y i )) + r(Y i ) f (Y i ) (ϕ n (Y i ) -ϕ(Y i )) T + R n 1 + R n 2 .
where

R n 1 (Y i ) = r(Y i ) r(Y i ) T (f (Y i ) -f n (Y i )) + (ϕ n (Y i ) -ϕ(Y i )) T 1 f (Y i ) 1 {f (Y i )<en} + fen (Y i ) -f en (Y i ) fen (Y i )f en (Y i ) and R n 2 = r(Y i )r(Y i ) T f en (Y i ) (f n (Y i ) -f en (Y i )) .
Futhermore :

• since for all y ∈ R we have

R n1 ≤ 1 n i∈In ||r(Y i )|| ||ϕ n -ϕ|| ∞ + ||r(Y i )|| 2 ||f n -f || ∞ 1 f (Y i ) 1 {f <en} + ||f n -f || ∞ e 2
n (6.17)

• on the other hand we have

R n 2 ≤ 1 n i∈In ||r(Y i )|| 2 |f n (Y i ) -e n | f en (Y i ) 1 {fn(Y i )<en} ≤ 2 n i∈In ||r(Y i )|| 2 1 {fn(Y i )<en} . because for all y ∈ R , |f n (y) -f en (y)| = |f n (y) -e n | 1 {fn(y)<en} ≤ 2e n 1 {fn(y)<en} .
Then, it follows from (6.14 and 6.15) that:

R n 2 = O p e 2 n n1+δ + Ψ 2 n e 2 n
as for S n 1 , we deduce:

R n 2 = O p Ψ 2 n e 2 n
Now, observious that,

1 n i∈In ||r(Y i )|| 2 f (Y i ) 1 {f (Y i )<en} ≤ e n 1 n i∈In ||r(Y i )|| 2 f (Y i ) 2 1 {f (Y i )<en} ,
we have (as previously):

(6.18) 1 n i∈In ||r(Y i )|| 2 f (Y i ) 1 {f (Y i )<en} = O p e n n1+δ .
Moreover, since E r(Y ) 2 f (Y ) 2 1 {f (Y )<en} = O 1 n1+δ
, we also have:

(6.19) 1 n i∈In ||r(Y i )|| f (Y i ) 1 {f (Y i )<en} = O p 1 n1+δ 2
So combining (6.17), (6.18) and (6.19), we get:

R n 1 = O p e n Ψ n n1+δ + Ψ n n1+δ 2 + Ψ 2 n e 2 n = O p Ψ n n1+δ 2 + Ψ 2 n e 2
n, 2 = 1 n i=1 r(Y i )r(Y i ) T f (Y i ) (f n (Y i ) -f (Y i )) (1)
and

S (2) n, 2 = 1 n i=1 r(Y i ) f (Y i ) (ϕ n (Y i ) -ϕ(Y i )) T .
To finish, we are going to show that S

, 2 = O p (h k n + 1 nh n ) S (2) n, 2 = O p (h k n + 1 nh n ) Note that: S (1) n, 2 = 1 n i=1 τ (Y i ) f (Y i ) - 1 h n V n (1) n
where τ (.) is a function defined by τ (y) = r(y) r(y) T f (y)

for y ∈ R and

V n = 1 n2 i,j∈In τ (Y i ) K hn (Y i -Y j )
is a second-order Von Mises functional statistic which associated U-statistic is:

U n = 1 2n(n -1) i=1 j =i [τ (Y i ) + τ (Y j )]K hn (Y i -Y j ). Since: V n = U n + O p ( 1 n ), S (1)
n, 2 = 1 n i=1 τ (Y i ) f (Y i ) - 1 h n U n + O p 1 nh n .
We apply Lemma 2.1 with, m = 2, h(y 1 , y

2 ) = [τ (y 1 ) + τ (y 2 )] K hn (y 1 -y 2 ) h 1 (y) = 1 2 [τ (y).f * K hn (y) + (τ.f ) * K hn (y)] , and
Θ(F ) = E (h 1 (Y )) = E (τ (y).f * K hn (y)) . Since ||h(Y 1 , Y 2 )|| 4+δ ≤ C.||τ (Y )|| 4+δ < ∞,
by assumption (3.1) then,

U n = Θ(F ) + 2 n i (h 1 (Y i ) -Θ(F )) + O p ( 1 n).
and

S ( 1
)
n, 2 = 1 n i=1 τ (Y i ) f (Y i ) - Θ(F ) h n - 2 n i h 1 (Y i ) h n - Θ(F ) h n + O p 1 nh n = 1 n i=1 τ (Y i ) f (Y i ) - f * K hn (Y i ) h n + Θ(F ) -(τ.f ) * K hn (y i ) h n + O p 1 nh n .
Since f and r(.) belongs to C k (R), we get,

f * K hn (y) h n -f (y) ∞ = O(h k n ) and (τ.f ) * K hn (y) h n -(τ.f )(y) ∞ = O(h k n ).
Then, we have To complete the proof, we will use Lemma 6.3. To this aim, it suffices to choose θ = δ with δ > 2N then E||X|| 4+δ < ∞ and k α(k) 2 +c 1 +2c 2 log n this latter tend to zero as soon as

c 2 k + 1 4k ≤ c 1 < 1 2 -2c 2
The proof is obtained by sketching the proof of Corolary 3.2 and using the law of the iterated logarithm recalled in Lemma 6.3.

n

  (y)| > δ, then k = k + 1 and continue with Step 2; otherwise terminate and d = k. Then, we can compute a predictor based on d = k: 4.3. The dimension reduction predictor.

n n hn and en n 1+δ 2 ≤

 2 C log n n hn (for n large and C > 0 an arbitrary constante), we have:

1 fen (y)fe n (y) ≤ 1 e 2 n

 12 and by several calculus we also have fen (y)f en (y) ≤ |f (y)f n (y)| and then || fenf en || ∞ ≤ ||f nf || ∞ , we also have one hand:

6 . 4 . 6 . 5 . 1 2E 1 2 2 ≤ 1 √

 64651121 δ δ+4 < ∞; hence we have: Σ e,n -Σ e = O p ( 1 n). which ends the proof. Proof of corollary 3.2.The proof is achieved by replacingh n ≃ n-c 1 and e n ≃ n-c 2 with c 2 k + 1 4k < c 1 < 1 2 -2c 2 on equality (6.21) Proof of corrollary 3.3.Chosing h n ≃ n-c 1 and e n ≃ n-c 2 where c 2 k + 1 4k < c 1 < 1 2 -2c 2 on equality (6.21), one gets Σ e, n -Σ e = Σ e, n -Σ e + o p ( 1 √ n ) and the central limit theorem for spatial data and Slusky's theorem completes the proof.Proof of Theorem 3.4. Let v n = log log n n note that since Y take place on a compact set,1 f is bounded and replace the assumptionr(Y ) f (Y ) 1 {f (exp r(Y ) 1 {f (Y )≤en} = O n-ξ for some ξ > 0. Then, i ) 1 {f (Y i )≤en} > ε v nand because of Minskovski's inequality: for all k ∈ N * , E1 n i∈In r(Y i ) 1 {f (Y i )≤en} k ≤ r(Y ) 1 {f (Y )≤en} k k, we can say that with using the argumentE exp r(Y ) 1 {f (Y )≤en} = O n-ξ : i ) f (Y i ) 1 {f (Y i )≤en} > ε v n ≤ E exp r(Y ) 1 {f (Y )≤en} expfor some C 1 > 0. ≤ C 1 exp -ξ log nε n log log n 1 C 1 expmin(ξ , ε) log n + √ log n √ log log n ≤ C 1 expmin(ξ , ε) log n 1 + 1 (log n) log log n as n → +∞, expmin(ξ , ε) log n 1 + (log n) log log n ≃ n-C 2 where c 2 is positive constant. So, 1 n i∈In r(Y i ) f (Y i ) 1 {f (Y i )≤en} = o a.s is complet by using Lemma 6.7 and sketching the proof of Theorem 3.1 Proof of Corollary 3.5. If moreover we chose h n ≃ n-c 1 and e n ≃ n-c 2 where c 2

  the proof is completed by noting thatP (|S n | > nε) ≤ 2 N P (|T (n, i)| > nε 2 N ).6.3. Proof of the Theorem 3.1. We will prove the desired result on Σ e, n -Σ -Σ e = Σ e, n -Σ e,n + Σ e,n -Σ e .-Σ e, n = S n, 1 + S n, 2 + S n, 3

	To this aim, we set :
	(6.11) Σ e, n with 1 S n,1 = n i∈In
						e using
	an intermediate matrix	Σ e,n =	1 n	i∈In	r(Y i )r(Y i ) T .
	Start with the following decomposition
	Σ e, n We first show that:
	(6.10)	Σ e, n -Σ e, n = O p	1 n1 2 +δ +	Ψ 2 n n e 2	.

  Then, equality,(6.16), and (6.20) lead to (6.10).Recall that Ψn = h k n + log n n hn .Then, the fact that there exist a real A > 0 such that ∀ n > A,1 n hn < log -Σ e = Σ e,n -Σ e + O p Ψ 2

	n n hne 2 n	and :
	(6.20)				S n,2 = O p	Ψ 2 n n e 2	+ h k n
	Finally, using equality (6.10) one has;
	(6.21)		Σ e, n n n e 2	+ h k n .
	1 n i∈In	τ (Y i ) f * K hn (Y i ) h n	=	1 n i∈In	(τ.f )(Y i ) + O p (h k n ),
	1 n i∈In	(τ.f ) * K hn (Y i ) h n	=	1 n i∈In	(τ.f )(Y i ) + O(h k n )
					Θ(F ) h n	= E((τ.f )(Y )) + O(h k n ).
	Finally:	S	(1) n, 2 = O p (h k n +	1 n +	1 nh n	) = O p (h k n +	1 nh n	).
	By using similar arguments and applying Lemma 2.1 with m = 3, one also gets
					S	(2) n, 2 = O p (h k n +	1 nh n	).
	So,				S n,2 = O p	Ψ 2 n e 2 n	+ h k n +	1 nh n


Benoit Gaudou
email: bgaudou@ifi.edu.vn

Andreas Herzig
email: andreas.herzig@irit.fr

Dominique Longin
email: dominique.longin@irit.fr

Logical formalization of social commitments: Application to agent communication languages (short version)

Keywords: Agent communication languages, modal logic, grounding, commitments, BDI logic, speech act theory

Most agent Communication Languages are no longer defined in terms of the agents' mental attitudes, but in terms of social commitments. However, such social approaches have two drawbacks. First, the notion of commitment does not have a clear and unambiguous characterization. Second, commitments are completely unrelated to the agents' reasoning. We remedy this situation by combining a BDI logic with a logic of what is publicly grounded between agents.

Introduction

Initially, Agent Communication Languages (ACLs) such as the FIPA-ACL standard [START_REF]FIPA Communicative Act Library Specification[END_REF] had semantics in terms of mental states like belief and intention. However, such a basis lacks verifiability and requires strong hypotheses on agents like sincerity and cooperativity. In order to overcome these problems more recent semantics such as [START_REF] Fornara | Agent communication and artificial institutions[END_REF] among others are based on commitments. However, some drawbacks remain. First, the notion of commitment does not have a clear and unambiguous characterization. Second, commitments are completely unrelated to agents' reasoning abilities and in particular to their mental attitudes. We remedy this situation by means of a reductionist logical characterization of social commitments. Contrarily to e.g. Castelfranchi [START_REF] Castelfranchi | Commitment: from individual intentions to groups and organizations[END_REF] and as explained in section 2, we agree that individual mental attitudes are not enough to characterize social commitments: we have to combine a logic of individual mental attitudes (viz. individual beliefs and intentions) with a logic accounting for the social and public feature of social commitments. To meet this goal, we use the logic of grounding introduced in [START_REF] Gaudou | Grounding and the expression of belief[END_REF]: it extends a BDI-like logic by a modal operator of what is publicly established in a group of agents, as opposed to private mental attitudes (section 3). We then show that it provides a new semantics for commitment-based ACLs by recasting the approach of [START_REF] Fornara | Agent communication and artificial institutions[END_REF] (section 4).

What are commitments?

Commitment has various senses in the AI literature. First we have to distinguish internal commitment and social commitment: the term 'internal commitment' refers to the persistence of an agent's choice to perform an action or to hold some proposition, while 'social commitment' refers to a particular relation between agents. This paper focusses on the latter; however, internal commitment is accounted for via the notion of intention (defined in section 4).

Castelfranchi reduces social commitment of the debtor i to the creditor j w.r.t. the action α using mutual knowledge: "i and j mutually know that i intends to do α and this is j's goal, and that as for α j has specific rights on i (j is entitled by i to α)." [1, p. 3]. Unlike [START_REF] Fornara | Agent communication and artificial institutions[END_REF], we agree with Castelfranchi that social commitment is not primitive. In the sequel we discuss his definition in detail.

First, it requires mutual knowledge on i's intention, which logically implies that i's intention actually holds (under the hypothesis of introspection on intentions). But as Castelfranchi says himself, the actual intention of i to perform the action α is neither necessary nor sufficient for his social commitment to do α. We therefore to weaken his definition by substituting mutual knowledge with the notion of grounding capturing only the public feature of i's intention. Note that the public ground of an attitude does not imply that this attitude actually holds.

Second, the action α to which i is committed should be a goal of agent j. We argue that this hypothesis is also too strong. Compare this to speech act theory ([6, p. 182-183]) and in particular consider commissive acts. On the one hand, a promise creates a commitment of the speaker toward the hearer and requires what is promised to be good for the hearer, but on the other hand a threat has the same social result (creation of a commitment) but requires in contrary the object of the threat not to be good for the hearer. We therefore have to distinguish what we call desirable social commitment (when the performance of α is a goal of the creditor) from undesirable social commitment (when the performance of α is not a goal of the creditor). Social commitment à la Castelfranchi only covers desirable social commitments. We do not need to define social commitment in such a restrictive way. Hence we do not specify in our definition of social commitment whether the creditor j has the goal that α be performed or not.

Finally, while we agree that the normative aspect of social commitments is important in general, we focus in this paper on the informal commitments involved in ACLs, where the deontic part of commitment can be omitted: after a promise a social commitment is incurred, but rights and duties often stay implicit and undefined.

To conclude, we consider that a social commitment to do α can be characterized as a public ground of both agents on the intention of one agent to perform an action. Castelfranchi argues that commitments are created by communicative acts. This suggests to consider speech act theory and to formalize not only commitment as an effect of speech acts, but also speech acts creating and managing them.

In commitment-based ACL semantics, commitments are not static but may be in various states. Each speech act is characterized by the commitment it produces, or more precisely by the changes in commitment states it induces. A model in terms of a commitment life cycle is introduced in [START_REF] Fornara | Agent communication and artificial institutions[END_REF].

The grounding logic

The logic of [START_REF] Gaudou | Grounding and the expression of belief[END_REF] captures what is publicly grounded in a group of agents. It extends the logic of intention developed in [START_REF] Herzig | C&L intention revisited[END_REF] with a modal operator Grd , where the formula Grd I ϕ reads "ϕ is publicly grounded for the group I". We have shown that the logic provides a new semantics to FIPA ACL [START_REF]FIPA Communicative Act Library Specification[END_REF] and also captures Walton and Krabbe's notion of propositional social commitment [START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF] in persuasive dialogue games.

We refer to [START_REF] Gaudou | Grounding and the expression of belief[END_REF] for the details of the (fairly standard) possible worlds semantics and its axiomatics, and here only give the reading of relevant operators. In the sequel, i, j denote agents and α an action. The expression (i:α) denotes that i is the author of action α.

It is supposed that what is publicly grounded is consistent, public for every subgroup of I, and is formed by a joint agreement of each member. For the sake of simplicity (and also notational economy) we identify Grd {i} with the usual belief operator [START_REF] Herzig | C&L intention revisited[END_REF]: Grd {i} ϕ means that i believes ϕ. We write Grd i ϕ for convenience.

Int i ϕ reads "agent i has the intention that ϕ holds". We suppose that Int i is defined from belief, choice and time operators (cf. [START_REF] Herzig | C&L intention revisited[END_REF]). Done α ϕ expresses that the action α has been performed before which ϕ held. Therefore Done α ⊤ means that α has just been performed. Formula Int i Done α ⊤ expresses that i has the intention to do action α.

Formalization of commitments

In the sequel C (i, j , α) represents a social commitment of i toward j to do action α. Following the discussion of section 2 we define a social commitment to do α as the debtor's publicly grounded intention to perform α:

Definition. C (i, j , α) def = Grd {i,j} Int i Done (i:α) ⊤
Many speech acts can lead to incur such a commitment. Here we only characterize promises, which are a spontaneous commitment of an agent and therefore produce a social commitment whose debtor is the speaker. We impose as precondition of Promise that the speaker stays consistent, i.e. that he has not expressed yet that he doesn't want to perform the action. The semantics is provided in terms of their Feasibility Precondition (FP) their and Intentional Effect (IE). FP describes the condition under which an action can be performed, and IE describes the illocutionary intention that, when recognized by the addressee of the act, identifies the performed speech act.

Definition. i, Promise, j, Done (i:α) ⊤ FP: ¬Grd {i,j} ¬Int i Done (i:α) ⊤ IE: C (i, j , α)

Perspectives

The aim of this paper was to use logical framework to describe social approaches of ACLs. In particular we formalized the notion of social commitment. Due to space restrictions only a part of our work has been presented here. In particular, our logical formalism accounts for the whole life cycle of Colombetti et al.'s approach: on the one hand we can give a formal characterization of each of its possible states, on the other hand we can provide a semantics for every speech acts leading to the commitments dynamics. Moreover, we concentrate on commitments to do an action, but very interesting links can be done between such commitments and propositional commitment as defined in detail in [START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF].