Michel Grabisch
email: michel.grabisch@univ-paris1.fr

Agnieszka Rusinowska
email: rusinowska@gate.cnrs.fr

Measuring influence in command games ⋆

Keywords: JEL Classification: C7, D7 influence function, follower, influence index, command game, commandable player, Shapley-Shubik index, Banzhaf index, Coleman indices, König-Bräuninger index

In the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. For some influence functions we define the equivalent command games. We show that not for all influence functions the compatible command games exist. Moreover, we propose a more general definition of the influence index and show that under some assumptions, some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. Both the Shapley-Shubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between these two power indices lies in the weights for the influence indices. An example of the Confucian model of society is broadly examined.

Introduction

Studying an interaction and an influence among players in voting situations exposes many challenging problems to be solved. In traditional framework of power indices (see e.g., [1-6, 12, 15-21, 23], see also [START_REF] Machover | The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes[END_REF] for an overview), neither interaction nor influence among the voters is assumed. The concept of the interaction or cooperation among players in a cooperative game is studied, for instance, in [START_REF] Grabisch | An axiomatic approach to the concept of interaction among players in cooperative games[END_REF], where players in a coalition are said to exhibit a positive (negative) interaction when the worth of the coalition is greater (smaller) than the sum of the individual worths. The authors present an axiomatization of the interaction indices which are extensions of the Shapley and Banzhaf values.

Another approach is presented in [START_REF] Hu | On authority distributions in organizations: controls[END_REF][START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF], where the command structure of Shapley [START_REF] Shapley | A Boolean model of organization authority based on the theory of simple games[END_REF] is applied to model players' interaction relations by simple games. For each player, boss sets and approval sets are introduced. While the boss sets are defined as the 'sets of individuals that the player must obey, regardless of his own judgment or desires', the 'consent of the approval set is sufficient to allow the player to act, if he wishes' [START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF].

Based on the boss and approval sets, a simple game called the command game for a player is built. Furthermore, an equilibrium authority distribution is formulated to which the Shapley-Shubik index is applied.

Coming still from a different direction is an approach proposed by Hoede and Bakker [START_REF] Hoede | A theory of decisional power[END_REF], where the authors talk explicitly about an influence between players in a social network, and they define the Hode-Bakker index. Players, who are to make a yes-no decision, have their inclinations to say either 'yes' or 'no', but due to the influence by the others, they can decide differently from their inclinations. Formally speaking, to each possible configuration i of individual inclinations, the configuration Bi of decisions of all players is assigned, where B denotes the influence function. Then, based on Bi, a group decision either 'yes' or 'no' is made.

The model of Hoede and Bakker is the point of departure for our research on influence. As noticed in [START_REF] Grabisch | A model of influence in a social network[END_REF], despite the interest of the framework of the Hoede-Bakker index in a situation where influence exists, this index does not give a full description of the situation, in the sense that it hides the actual role of the influence function B. Many economic, political, real-life situations may be modeled as a social network with players influencing each other. It is therefore very important to study different ways of defining influence between agents, and to investigate useful tools that will fully measure and analyze the influence. Motivated by the importance of studying these issues in detail, we have decided to conduct a broad research on influence between players in a social network. In [START_REF] Grabisch | A model of influence in a social network[END_REF], we introduce and study weighted influence indices of a coalition on a player, and consider different influence functions. In [START_REF] Grabisch | Measuring influence among players with an ordered set of possible actions[END_REF] we extend the yes-no model to a multi-choice game, and assume that players have a totally ordered set of possible actions instead of just two actions 'yes' or 'no'. Each player has an inclination to choose one of the actions. Consequently, generalized influence indices are investigated.

What are the differences and advantages of the influence model in contrast to standard power indices? Roughly speaking, a power index of a player gives an answer to the question of effect the player has on the outcome in a voting situation, or differently speaking, to the question of decisiveness of that player. In a standard voting situation, once a proposal is submitted, voters cast votes, voting either 'yes' or 'no'. A vote configuration gives a possible result of voting, and the winning configurations are the ones that lead to the passage of the proposal. A player is said to be decisive if his vote coincides with the voting result, and if by changing his vote, he changes the result of voting. The framework of influence which we study is broader than voting games, where the standard power indices are defined. Our analysis starts 'one step earlier', before voting takes place. As mentioned above, in our model, voters are settled in a social network, they are inclined to make one of the possible decisions, but they may be influenced by the others, and consequently, they may vote differently from their original inclinations. One of the tools that we introduce in such a broad framework of influence, is the concept of influence index. This tool measures particularly the influence between players, i.e., the ability to make agents vote differently from their original inclinations, and as such, it is basically different from the concept of power index, which focuses on the effect that a player has on the outcome. Nevertheless, if we make some assumptions 'linking' a voting game to the influence model, standard power indices can also be expressed by our weighted influence indices. For instance, we may define a winning coalition in a voting game as a coalition of players with the same inclination such that a given agent always follows this coalition (assuming that all remaining players have a different inclination). This link will be precisely used in the sequel to relate command games and influence functions.

The aim of the present paper is to study the relation between the command games considered by Hu and Shapley [START_REF] Hu | On authority distributions in organizations: controls[END_REF][START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF], and the influence model investigated in [START_REF] Grabisch | Measuring influence among players with an ordered set of possible actions[END_REF][START_REF] Grabisch | A model of influence in a social network[END_REF]. We show that our framework of influence is more general than the framework of command games. In the influence model, many different influence functions may be proposed. Depending on how the influence function is chosen, the influence-function approach provides a much wider spectrum, compared to the command-game approach. In the present paper, we define several influence functions which capture the command structure proposed by Hu and Shapley. These influence functions, called command influence functions, explain the command games, by specifying how exactly players decide according to the command structure. The command influence functions that we define are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented command influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. So, the influence functions and the command games are equivalent if the set of commandable players is equal to the set of followers. For some influence functions introduced in [START_REF] Grabisch | A model of influence in a social network[END_REF], we define the equivalent command games. In particular, we show that not for all influence functions compatible command games exist.

Moreover, we propose a more general definition of the influence index and, as already mentioned, we show that under some assumptions, some power indices which can be used in the command games, can be expressed through the weighted influence indices under the considered command influence functions. It is interesting to remark that both the Shapley-Shubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between these two power indices lies in the weights for the influence indices.

The structure of the paper is the following. Definitions of power indices we refer to in the paper, and the framework of the command games are briefly presented in Section 2. The latter model of the relationship we search for, that is, the influence model, is summarized in Section 3. The core of the paper is presented in Sections 4 and 5, where the relation between the influence model and the command games is investigated. In Section 4, we define several influence functions which are equivalent or compatible with the command games. We show relations between the weighted influence indices under the command influence functions and two well-known power indices, i.e., the Shapley-Shubik index and the (absolute) Banzhaf index. In Section 5, we define the equivalent command games for some influence functions introduced in [START_REF] Grabisch | A model of influence in a social network[END_REF]. In Section 6, the relations between the influence model and the command games are illustrated by an example mentioned in [START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF], i.e., by the Confucian model of society. In Section 7, we conclude. The paper contains also an appendix, in which we show relations between the weighted influence indices and some other power indices, i.e., the Coleman indices and the König-Bräuninger index.

Basic definitions and notations

Power indices

First, we introduce several notations for convenience. Cardinality of sets S, T, . . . will be denoted by the corresponding lower case s, t, . . .. We omit braces for sets, e.g., {k, m}, N \ {j}, S ∪ {j} will be written km, N \ j, S ∪ j, etc.

A (0, 1)-game is a pair (N, v), where N = {1, ..., n} is the set of players, and a function v : 2 N → {0, 1} satisfying v(∅) = 0 is the characteristic function. A nonempty subset of N is called a coalition. A simple game is a (0, 1)-game such that v is not identically equal to 0 and is monotonic, i.e., v(S) ≤ v(T ) whenever S ⊆ T . A coalition S is winning if v(S) = 1, and is

loosing if v(S) = 0. A game is superadditive if v(S ∪ T ) ≥ v(S) + v(T ) whenever S ∩ T = ∅.
Let SG n denote the set of all simple superadditive n-person games. Player k is a swing in a winning coalition S if his removal from the coalition makes it loosing, i.e., if v(S) = 1 and v(S \ k) = 0. A minimal winning coalition is a winning coalition in which all players are swings.

A power index is a function φ :

SG n → R n which assigns to each (N, v) ∈ SG n a vector φ(N, v) = (φ 1 (N, v), ..., φ n (N, v)).
The Shapley-Shubik index [START_REF] Shapley | A method for evaluating the distribution of power in a committee system[END_REF] of player k ∈ N in a game (N, v) is defined by

Sh k (N, v) = S⊆N :k∈S (n -s)!(s -1)! n! (v(S) -v(S \ k)). (1)
The non-normalized Banzhaf index (the absolute Banzhaf index, [START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF]) of player k ∈ N in a game (N, v) is defined by

Bz k (N, v) = 1 2 n-1 S⊆N :k∈S (v(S) -v(S \ k)), (2)
and the normalized Banzhaf index Bz is given by

Bz k (N, v) = Bz k (N, v) j∈N Bz j (N, v)
.

The Coleman 'power of a collectivity to act' [START_REF] Coleman | Control of collectivities and the power of a collectivity to act[END_REF][START_REF] Coleman | Individual Interests and Collective Action: Selected Essays[END_REF] in a game (N, v) is defined by

A(N, v) = S⊆N v(S) 2 n . (3)
The Coleman index 'to prevent action' [START_REF] Coleman | Control of collectivities and the power of a collectivity to act[END_REF][START_REF] Coleman | Individual Interests and Collective Action: Selected Essays[END_REF] of player k ∈ N in a game (N, v) is defined by

Col P k (N, v) = S⊆N :k∈S (v(S) -v(S \ k)) S⊆N v(S) . (4)
The Coleman index 'to initiate action' [START_REF] Coleman | Control of collectivities and the power of a collectivity to act[END_REF][START_REF] Coleman | Individual Interests and Collective Action: Selected Essays[END_REF] of player k ∈ N in a game (N, v) is given by

Col I k (N, v) = S⊆N :k / ∈S (v(S ∪ k) -v(S)) 2 n -S⊆N v(S) . (5)
The König-Bräuninger inclusiveness index [START_REF] König | The inclusiveness of European decision rules[END_REF] of player k ∈ N in a game (N, v) is given by

KB k (N, v) = S⊆N :k∈S v(S) S⊆N v(S) . (6)

Command games

We recapitulate briefly the main concepts concerning the command games introduced by Hu and Shapley [START_REF] Hu | On authority distributions in organizations: controls[END_REF][START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF]. Let N = {1, ..., n} be the set of players (voters). For k ∈ N and S ⊆ N \ k:

-S is a boss set for k if S determines the choice of k; -S is an approval set for k if k can act with an approval of S.

It is assumed that any superset (in N \ k) of a boss set is a boss set.

For each k ∈ N , a simple game (N, W k ) is built, called the command game for k, where the set of winning coalitions is

W k := {S | S is a boss set for k} ∪ {S ∪ k | S
is a boss or approval set for k}.

We can recover the boss sets for k

Boss k = {S ⊆ N \ k | S ∈ W k } = W k ∩ 2 N \k
and the approval sets for k

App k = {S ⊆ N \ k | S ∪ k ∈ W k but S / ∈ W k }.
We have Boss k ∩ App k = ∅. In particular, if App k = 2 N \k , then k is called a free agent: he needs no approval (since ∅ ∈ App k ), and nobody can boss him (since

Boss k = ∅). If App k = ∅, then k is called a cog.
Given the command games {(N, W k ) | k ∈ N }, for any coalition S ⊆ N , the command function ω(S) is defined as the set of all members that are 'commandable' by S:

ω(S) := {k ∈ N | S ∈ W k }. (7)
In [START_REF] Hu | On authority distributions in organizations: controls[END_REF], it is shown that ω(∅) = ∅, ω(N ) = N , and ω(S) ⊆ ω(S ′ ) whenever S ⊂ S ′ .

In [START_REF] Hu | On authority distributions in organizations: controls[END_REF][START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF], the authors define an authority distribution π = (π 1 , ..., π n ) over an organization (N, {(N, W k ) | k ∈ N }), and they create the power transition matrix of the organization, which is the stochastic matrix P = [P (j, k)] n j,k=1 such that

P (j, k) := Sh k (N, W j ), (8)
and Sh k (N, W j ) is the Shapley-Shubik index of player k in the command game for player j. Hence, P (j, k) represents the power of player k to "influence" player j. They use a Markov chain to describe the organization's long-run authority π. The authority distribution π is assumed to satisfy the authority equilibrium equation given by π = πP, i.e., π k = j∈N π j P (j, k), ∀k ∈ N.

3 The influence model

Direct versus opposite influence

In this section, the main concepts introduced in [START_REF] Grabisch | A model of influence in a social network[END_REF] are summarized. Let N := {1, ..., n} be the set of players (agents, actors, voters). The players are to make a yes-no decision. Each player has an inclination either to say 'yes' (denoted by +1) or 'no' (denoted by -1). Let i = (i 1 , ..., i n ) denote an inclination vector and I := {-1, +1} n be the set of all inclination vectors. In particular, (1, 1, . . . , 1) ∈ I and (-1, -1, . . . , -1) ∈ I will be denoted by 1 N and -1 N , respectively. Players may influence each other, and as a consequence of the influence, the final decision of a player may be different from his original inclination. Each inclination vector i ∈ I is therefore transformed into a decision vector Bi = ((Bi) 1 , ..., (Bi) n ), where B : I → I is the influence function. Let B denote the set of all influence functions. We are interested in measuring the influence that a coalition of players with the same inclination has on an agent outside that coalition. We distinguish between a direct influence and an opposite influence. In the first case, the player's inclination is different from the inclination of that coalition, but the player's decision is the same as the inclination of the coalition. In the latter case, the player's inclination coincides with the inclination of the coalition, but by a kind of reactive behavior, his decision is different from the inclination of the coalition. To formalize, we introduce for any S ⊆ N the set

I S := {i ∈ I | ∀k, j ∈ S [i k = i j ]}, (10)
and I k := I, for any k ∈ N . We denote by i S the value i k for some k ∈ S, i ∈ I S . For each S ⊆ N and j ∈ N , we define the set I S→j of all inclination vectors of potential direct influence of S on j, and the set I * S→j (B) of all inclination vectors of observed direct influence of S on j under given B ∈ B:

I S→j := {i ∈ I S | i j = -i S } I * S→j (B) := {i ∈ I S→j | (Bi) j = i S }.
Similarly, for each S ⊆ N and j ∈ N we introduce the set I op S→j of all inclination vectors of potential opposite influence, and the set I * op S→j (B) of all inclination vectors of observed opposite influence of a coalition on a player:

I op S→j := {i ∈ I S | i j = i S } I * op S→j (B) := {i ∈ I op S→j | (Bi) j = -i S }.
For each S ⊆ N and j ∈ N \ S, we introduce a set of weights α S→j i ∈ [0, 1], for all inclination vectors i ∈ I S→j (or I op S→j ). The weight α S→j i represents how much the inclination vector i is important when computing the influence of S over j (other intepretations are possible, e.g., probability of occurrence, etc.). We assume that for each S ⊆ N and j ∈ N \ S, there exists i ∈ I S→j such that α S→j i > 0, and there exists i ∈ I op S→j such that α S→j i > 0. Moreover, as stated in [START_REF] Grabisch | A model of influence in a social network[END_REF], we impose for simplicity the symmetry assumption that α S→j i depends solely on the number of players with the same inclination as players of S under i ∈ I S (including the players from S, but excluding player j). Apart from these assumptions, no other restrictions are needed. In particular, we do not impose any normalization condition, although such a condition could be added, if the weights convey some special meaning, e.g., probabilities, etc. Determining α S→j i depends on which inclination vectors (viewed as particular 'situations') we want to take into account or to emphasize (or which are more likely to occur) for computing the influence index. If, for example, we consider a model which fully takes into account any possibility of influence, all weights are always equal to 1. If, on the other hand, we take into account only those situations (i.e., inclination vectors) in which all agents outside S ∪ j have the inclination different from the inclination of S, we assign 1 to α S→j i under such i's, and 0 to α S→j i in all remaining cases. In [START_REF] Grabisch | A model of influence in a social network[END_REF] we consider also another way of sharing the ability to influence, in which, given S and j, two inclination vectors with equal numbers of players having the same inclination as the inclination of S are treated equally, but the maximal weight 1 is divided by the number of possible coalitions that could be 'responsible' for the influence.

Given B ∈ B, for each S ⊆ N , j ∈ N \ S, the weighted direct influence index of coalition S on player j under B is defined as

d α (B, S → j) := i∈I * S→j (B) α S→j i i∈I S→j α S→j i ∈ [0, 1],
and the weighted opposite influence index of coalition S on player j under B is defined as

d op α (B, S → j) := i∈I * op S→j (B) α S→j i i∈I op S→j α S→j i ∈ [0, 1].
When measuring direct influence of coalition S on player j, the point of departure is the set of all inclination vectors of potential direct influence of S on j, and consequently, in the denominator of the weighted direct influence index of S on j, we have the weighted sum over such inclination vectors. Furthermore, we consider only those inclination vectors of potential direct influence that are the vectors of observed direct influence, and in the numerator of the index we have the weighted sum over such inclination vectors of observed direct influence. The definition of the weighted opposite influence index is analogous, except that instead of inclination vectors of (potential or observed) direct influence, we consider inclination vectors of (potential or observed) opposite influence.

One of the key concepts in the influence model is the concept of a follower of a given coalition, that is, a voter who always follows the inclination of the coalition in question.

Let ∅ = S ⊆ N and B ∈ B. The set F B (S) of followers of S under B is therefore defined as F B (S) := {j ∈ N | ∀i ∈ I S [(Bi) j = i S ]}. ( 11
)
We put F B (∅) := ∅. We have, in particular,

d α (B, S → j) = 1, ∀j ∈ F B (S) \ S.
Assume F B is not identically the empty set. We define the kernel of B, that is, the set of 'true' influential coalitions. A set S belongs to the kernel of B if S has a non-empty set of followers under B, but any of its proper subsets has no follower.

K(B) := {S ∈ 2 N | F B (S) = ∅, and S ′ ⊂ S ⇒ F B (S ′ ) = ∅}. ( 12
)

A general case of influence

Next, we generalize the concept of influence to be able to comprise all imaginable cases of influence, in particular, the direct influence, the opposite influence, an influence of a coalition on its member, etc. The point of departure will be the set I S defined in [START_REF] Grabisch | A model of influence in a social network[END_REF]. In other words, we assume that a coalition may influence a player only in situations when all members of the coalition have the same inclination. Furthermore, an observed influence of a coalition S on a player j, which is now characterized by λ ∈ {-i j , i S , -i S , +1, -1}, takes place if (Bi) j = λ. Hence, the set I S→j,λ (B) of all inclination vectors of influence of S on j under B is defined as

I S→j,λ (B) := {i ∈ I S | (Bi) j = λ}.
Definition 1 Given B ∈ B, for each S ⊆ N , j ∈ N , the weighted influence index of coalition S on player j under B is defined as

ψ α,λ (B, S → j) := i∈I S→j,λ (B) α S→j i i∈I S α S→j i . ( 13
)
The set I S→j,λ (B) captures several kinds of decisions made by the influenced player j, depending on λ. In other words, we can consider different definitions of the influence. If, for instance, λ = i S , then by the influence of S we mean following the inclination of that coalition S, and consequently, when measuring the influence we record all the situations in which player j decides according to the inclination of coalition S. In such situations, for example, I S→j,λ=i S (B) = I S means that player j is a follower of coalition S under B, and the weighted influence index ψ α,λ=i S (B, S → j) = 1. While under λ = -i S we are interested in all the situations in which player j decides differently from the inclination of coalition S, setting λ = -i j means that an influenced player is defined as a player whose decision is different from his inclination. In particular, the weighted direct influence index is recovered as follows:

ψ α,λ (B, S → j) = d α (B, S → j) if λ = -i j and α S→j i = 0 for i j = i S ,
and the weighted opposite influence index is recovered as:

ψ α,λ (B, S → j) = d op α (B, S → j) if λ = -i j and α S→j i = 0 for i j = -i S .
The essence of both the direct influence and the opposite influence is that the decision of an influenced player is different from his original inclination (λ = -i j ). The only difference between these two types of influence lies in the inclination vectors we consider: while in the direct influence we observe only situations in which the inclinations of a player and an influencing coalition are different from each other (α S→j i = 0 for i j = i S ), in case of the opposite influence we take into account only situations in which the inclinations of a player and an influencing coalition are the same (α S→j i = 0 for i j = -i S ). The set of followers and the kernel are defined like in [START_REF] Hoede | A theory of decisional power[END_REF] and [START_REF] Holler | Power, luck and the right index[END_REF], respectively.

Influence functions

We recapitulate some influence functions defined in [START_REF] Grabisch | A model of influence in a social network[END_REF] that we refer to in the present paper.

(i) The Majority function -Let n ≥ t > ⌊ n 2 ⌋, and introduce for any i ∈ I the set i + := {k ∈ N | i k = +1}. The majority influence function Maj [t] ∈ B is defined by

Maj [t] i := 1 N , if |i + | ≥ t -1 N , if |i + | < t , ∀i ∈ I, (14)
which simply means that if a majority of players has an inclination +1, then all players decide +1, and if not, all players decide -1. We have for each S ⊆ N :

F Maj [t] (S) = N, if s ≥ t ∅, if s < t (15) K(Maj [t] ) = {S ⊆ N | |S| = t}.
(ii) The Guru function -Let k ∈ N be a particular player called the guru. The guru influence function

Gur [ e k] ∈ B is defined by (Gur [ e k] i) j = i e k , ∀i ∈ I, ∀j ∈ N, (16)
that is, when a guru exists, every player always follows the guru. We have for each S ⊆ N :

F Gur [ e k] (S) = N, if k ∈ S ∅, if k / ∈ S (17)
K(Gur [ e k] ) = { k}.
(iii) The identity function Id ∈ B is defined by

Idi = i, ∀i ∈ I, (18)
and simply depicts the absence of any influence. We have for each

S ⊆ N , F Id (S) = S. The kernel is K(Id) = {{k}, k ∈ N }. (iv)
The reversal function -Id ∈ B is simply the opposite of the identity function:

-Idi = -i, ∀i ∈ I. ( 19
)
We have for each S ⊆ N , F -Id (S) = ∅. The kernel is K(-Id) = ∅.

Command influence functions

We can apply our model of influence to the framework of command games. First, we propose several influence functions, called here the command influence functions that are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of that coalition under the command influence function. In a sense, the command influence functions explain the command games, specifying how exactly players decide according to their boss and approval sets.

A particular case of the influence functions which are compatible with the command games are the functions that are equivalent to the command games. A command influence function is equivalent to the set of command games if it is compatible with this set of command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games.

Definition 2 Let {(N, W k ) | k ∈ N }
) | k ∈ N } are compatible if ω(S) ⊆ F B (S), ∀S ⊆ N. ( 20
)
The influence function B and the set of command games

{(N, W k ) | k ∈ N } are equivalent if ω ≡ F B .
For each set of command games, a necessary condition for the existence of an influence function which is compatible with the command games is that for each player k, any two winning coalitions in the command game for k have at least one common member.

Proposition 1 Let {(N, W k ) | k ∈ N }
be a set of command games. If there exists an influence function which is compatible with this set of command games, then the following holds:

∀k ∈ N ∀S, S ′ ∈ W k [S ∩ S ′ = ∅]. (21)
Proof: Suppose that there exists an influence function B which is compatible with the set of command games {(N, W k ) | k ∈ N }, and the condition (21) does not hold. Hence, from [START_REF] Penrose | The elementary statistics of majority voting[END_REF], ω(S) ⊆ F B (S), for each S ⊆ N . Moreover, since [START_REF] Rae | Decision-rules and individual values in constitutional choice[END_REF] does not hold, there exist k ∈ N , and S, S ′ ∈ W k such that S ∩ S ′ = ∅. Since S, S ′ ∈ W k , we have k ∈ ω(S) and k ∈ ω(S ′ ), which by virtue of (20) implies k ∈ F B (S) and k ∈ F B (S ′ ). ¿From [START_REF] Hoede | A theory of decisional power[END_REF], (Bi) k = i S for each i ∈ I S , and (Bi)

k = i S ′ for each i ∈ I S ′ . Take i ∈ I S ∩ I S ′ such that i S = i S ′ . Hence, we have (Bi) k = i S = i S ′ = (Bi) k , a contradiction.
The context of the rest of this section is the following. First, for an arbitrary set of command games, we construct one compatible command influence function (Section 4.2), and two equivalent command influence functions (Sections 4.1 and 4.3). This means that our model of influence is broader than the framework of the command games. Moreover, while in [START_REF] Hu | On authority distributions in organizations: controls[END_REF][START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF] the authors apply the Shapley-Subik index to the command games, in the present paper we apply several power indices (i.e., the ones recapitulated in Section 2.1) to the command games. We show that these power indices coincide with some expressions of the weighted influence indices under the command influence functions. In this section, we present the relations concerning the Shapley-Shubik and the Banzhaf index, and in the Appendix we show the relations concerning the Coleman indices and the König-Bräuninger index.

An influence function with abstention

The first command influence function we propose is related to an extended model of influence (see [START_REF] Grabisch | Measuring influence among players with an ordered set of possible actions[END_REF]), in which players have an ordered set of possible actions. In this model, each player has an inclination either to say 'yes' (denoted by +1) or 'no' (denoted by -1), that is, I = {-1, +1} n , but he has three options for making his decision: 'yes', 'no', or 'to abstain' (denoted by 0), that is, B(I) ⊆ {-1, 0, +1} n . The command influence function is defined as follows.

Definition 3 Given a set of command games {(N, W k ) | k ∈ N }, the command influence function Com ∈ B is defined for each k ∈ N and i ∈ I by (Comi) k :=      +1, if {j ∈ N | i j = +1} ∈ W k -1, if {j ∈ N | i j = -1} ∈ W k 0, otherwise . (22)
By Proposition 1, Com is well defined. According to the command influence function Com, for each voter k and each inclination vector, if all players with the same inclination forms a winning coalition in his command game, the voter k follows the inclination of this winning coalition. Otherwise, that is, if none of the two coalitions with all members having the same inclination is winning, the voter k simply abstains.

Proposition 2 Let {(N, W k ) | k ∈ N } be a set of command games, and ω(S) be a command function as defined in [START_REF] Machover | The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes[END_REF]. We have

F Com (S) = ω(S), ∀S ⊆ N, (23)
where F Com (S) is the set of followers of S under the command influence function Com.

Proof:

F Com (∅) = ∅ = ω(∅)
. Take an arbitrary nonempty S ⊆ N . Suppose that F Com (S) ⊆ ω(S). Hence, there exists k ∈ F Com (S) such that k / ∈ ω(S), and therefore (Comi) k = i S for all i ∈ I S , and S / ∈ W k . Take i ∈ I S such that i S = +1, and i j = -1 for each j / ∈ S. Hence, (Comi) k = i S = +1, but since S / ∈ W k we have also either (Comi) k = -1 or (Comi) k = 0, a contradiction. Suppose now that ω(S) ⊆ F Com (S). Hence, there exists k ∈ ω(S) such that k / ∈ F Com (S). This means that S ∈ W k , and there exists i ∈ I S such that (Comi) k = i S . However, by [START_REF] Shapley | A Boolean model of organization authority based on the theory of simple games[END_REF]

, (Comi) k = i S which contradicts (Comi) k = i S .
Consequently, by virtue of ( 12) and ( 23), the kernel of Com is given by:

K(Com) = {S ∈ 2 N | ω(S) = ∅, and S ′ ⊂ S ⇒ ω(S ′ ) = ∅}.
Remark 1 By virtue of [START_REF] Shapley | A method for evaluating the distribution of power in a committee system[END_REF], for each set of command games {(N, W k ) | k ∈ N }, the command influence function Com ∈ B defined in ( 22) is equivalent to this set of command games.

Proposition 3 Let {(N, W j ) | j ∈ N } be a set of command games, and Com be the command influence function defined in [START_REF] Shapley | A Boolean model of organization authority based on the theory of simple games[END_REF]. Then for each j, k

∈ N Sh k (N, W j ) = ψ α (Sh) ,λ=i k (Com, k → j) -ψ α (Sh) ,λ=-i k (Com, k → j) (24)
where Sh k (N, W j ) is the Shapley-Shubik index of player k in the command game for j, ψ α,λ is the weighted influence index defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF], and for each i ∈ I α

(Sh)k→j i =    1 n( n-1 |i|-1 ) , if i k = +1 1 n( n-1 n-|i|-1 ) , if i k = -1 (25)
and

|i| := |{m ∈ N | i m = +1}|.
Proof: By a coalition S we mean the set of players with the same inclination i S , while all players outside S have the inclination -i S . Consequently, two coalitions are formed under each inclination vector: a coalition of players with positive inclination, and a coalition of players with negative inclination. We have: k ∈ S iff i k = i S . Moreover, by [START_REF] Shapley | A Boolean model of organization authority based on the theory of simple games[END_REF], given (N, W j ): v(S) = 1 iff S ∈ W j iff (Comi) j = i S for each i ∈ I S such that i m = -i S for each m / ∈ S. From (1), we have

Sh k (N, W j ) = S⊆N :k∈S (n -s)!(s -1)! n! (v(S) -v(S \ k)) = S⊆N :k∈S (n -s)!(s -1)! n! v(S) - S⊆N :k / ∈S (n -s -1)!s! n! v(S) = S⊆N :k∈S v(S) n n-1 s-1 - S⊆N :k / ∈S v(S) n n-1 n-s-1 = S∈W j :k∈S 1 n n-1 s-1 - S∈W j :k / ∈S 1 n n-1 n-s-1 = 1 2   i∈I + k α (Sh)k→j i - i∈I - k α (Sh)k→j i   with I + k := {i ∈ I | (Comi) j = i k }, I - k := {i ∈ I | (Comi) j = -i k }. Indeed, i∈I + k α (Sh)k→j i = i∈I + k :i k =+1 1 n n-1 |i|-1 + i∈I + k :i k =-1 1 n n-1 n-|i|-1
.

By definition of Comi, we have

[i ∈ I + k ∧ i k = +1] iff [i + := {m ∈ N | i m = +1} ∈ W j ∧ k ∈ i + ]. Similarly, [i ∈ I + k ∧ i k = -1] iff [i -:= {m ∈ N | i m = -1} ∈ W j ∧ k ∈ i -].
Denoting winning coalitions by S, we obtain

i∈I + k α (Sh)k→j i = S∈W j :k∈S 1 n n-1 s-1 + S∈W j :k∈S 1 n n-1 s-1 = 2 S∈W j :k∈S 1 n n-1 s-1 .
Similarly, we obtain

i∈I - k α (Sh)k→j i = 2 S∈W j :k / ∈S 1 n n-1 n-s-1 . i:i k =+1 α (Sh)k→j i = i:i k =+1 1 n n-1 |i|-1 = n |i|=1 1 n n-1 |i|-1 n -1 |i| -1 = 1 i:i k =-1 α (Sh)k→j i = i:i k =-1 1 n n-1 n-|i|-1 = n-1 |i|=0 1 n n-1 n-|i|-1 n -1 |i| = 1,
and therefore

i∈I α (Sh)k→j i = 2. Sh k (N, W j ) = 1 2   i∈I + k α (Sh)k→j i - i∈I - k α (Sh)k→j i   = i∈I + k α (Sh)k→j i i∈I α (Sh)k→j i - i∈I - k α (Sh)k→j i i∈I α (Sh)k→j i = ψ α (Sh) ,λ=i k (Com, k → j) -ψ α (Sh) ,λ=-i k (Com, k → j),
where ψ α,λ is the weighted influence index defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF], and α (Sh) is given in (25).

Proposition 4 Let {(N, W j ) | j ∈ N } be a set of command games, and Com be the command influence function defined in [START_REF] Shapley | A Boolean model of organization authority based on the theory of simple games[END_REF]. Then for each j, k

∈ N Bz k (N, W j ) = ψ α (Bz) ,λ=i k (Com, k → j) -ψ α (Bz) ,λ=-i k (Com, k → j) (26)
where Bz k (N, W j ) is the Banzhaf index of player k in the command game for j, ψ α,λ is the weighted influence index defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF] with

α (Bz)k→j i = 1, ∀i ∈ I. (27)
Proof: The proof is similar to the one of Proposition 3. From (2), we have

Bz k (N, W j ) = 1 2 n-1 S⊆N :k∈S (v(S) -v(S \ k)) = 1 2 n-1 S⊆N :k∈S v(S) - 1 2 n-1 S⊆N :k / ∈S v(S) = 1 2 n-1 |{i ∈ I | (Comi) j = i k }| 2 - |{i ∈ I | (Comi) j = -i k }| 2 = |{i ∈ I | (Comi) j = i k }| |I| - |{i ∈ I | (Comi) j = -i k }| |I| = ψ α (Bz) ,λ=i k (Com, k → j) -ψ α (Bz) ,λ=-i k (Com, k → j)
where ψ α,λ is the weighted influence index defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF], and α (Bz) is given in (27).

What we show in Propositions 3 and 4 is that under a certain assumption, both the Shapley-Shubik index and the Banzhaf index of player k in the command game for j are equal to the difference between the weighted influence index in which player j is said to be influenced by k if he follows k, and the weighted influence index in which the influence of k on j means that player j's decision is opposite to the inclination of k. Both weighted influence indices are measured under the command influence function defined in [START_REF] Shapley | A Boolean model of organization authority based on the theory of simple games[END_REF]. The difference between the results on these two power indices lies only in the weights for the influence indices: while for the Shapley-Shubik index the weights are defined in (25), the weights for the Banzhaf index are always equal to 1, as given in (27).

The results shown in Propositions 3 and 4 are not very surprising. The essential part of both the Shapley-Shubik and Banzhaf indices of player k is a marginal contribution (v(S)v(S \ k)) of k in any coalition S containing k. We assume that a player is a member of a coalition if his inclination coincides with the inclination of that coalition. Since a coalition S is winning (that is, v(S) = 1) in the command game for j if the player j decides according to the inclination of S, considering the marginal contributions of k means considering a difference between the inclination vectors under which the player j decides according to the inclination of k, and the inclination vectors under which j's decision is different from the inclination of k. This difference between the specific inclination vectors is captured by our weighted influence indices. In order to get the equity between the Shapley-Shubik index and the difference between the weighted influence indices, the weights need to be properly defined, and they are equal to the coefficients of the Shapley-Shubik index. Similar remark holds for the Banzhaf index.

An influence function in which a player without a boss follows himself

Next, we mention another command influence function which is compatible with the command games.

( Comi) k :=      +1, if {j ∈ N | i j = +1} ∈ W k -1, if {j ∈ N | i j = -1} ∈ W k i k , otherwise . ( 28
)
The command influence function Com is similar to the function Com defined in ( 22) with a difference that for each inclination vector, if none of the two coalitions with all members having the same inclination is winning, now instead of abstaining, the voter in question simply follows his own inclination. As a consequence, we do not have the equality between the sets ω(S) and F Com (S) (as we had before between ω(S) and F Com (S)), but the inclusion. For each coalition S, the set of followers of S under Com consists of all players that are commandable by S, and of those members k of S for which any winning coalition in the command game for k contains at least one player from S.

Proposition 5 Let {(N, W k ) | k ∈ N } be a set of command games, and ω(S) be a command function as defined in [START_REF] Machover | The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes[END_REF]. We have

F Com (S) = ω(S) ∪ {k ∈ S | ∀S ′ ∈ W k [S ∩ S ′ = ∅]}, ∀S ⊆ N, ( 29
)
where F Com (S) is the set of followers of S under the command influence function Com.

Proof: Suppose that F Com (S) ⊆ ω(S) ∪ {k ∈ S | ∀S ′ ∈ W k [S ∩ S ′ = ∅]} for a certain S ⊆ N .
Hence, there exists k ∈ F Com (S) such that k / ∈ ω(S), and either k / ∈ S or (k ∈ S, but there exists S ′ ∈ W k such that S ∩ S ′ = ∅). Consequently, since k ∈ F Com (S) and k / ∈ ω(S), we have ( Comi) k = i S for all i ∈ I S , and S / ∈ W k . Take i ∈ I S such that i S = +1, and i j = -1 for all j / ∈ S. Hence, ( Comi)

k = i S = +1. If k / ∈ S, then ( Comi) k = i S =
S ′ ∈ W k , S ∩ S ′ = ∅). Since k / ∈ F Com (S), there exists i ∈ I S such that ( Comi) k = -i S .
If k ∈ ω(S), then S ∈ W k , and therefore ( Comi) k = i S , a contradiction. Suppose then k ∈ S and for all S ′ ∈ W k , S ∩ S ′ = ∅. Hence, in particular,

i k = i S . If there is S ′ ∈ W k such that i ∈ I S ′ , then ( Comi) k = i S ′ = i S , a contradiction. If i / ∈ I S ′ for all S ′ ∈ W k , then ( Comi) k = i k = i S , a contradiction.
Remark 2 By virtue of (29), for each set of command games {(N, W k ) | k ∈ N }, the command influence function Com ∈ B defined in ( 28) is compatible with this set of command games.

Non-symmetric influence functions

Next, we propose a command influence function under which we treat a winning coalition as the set of potential yes-voters only.

(Comi) k := +1, if {j ∈ N | i j = +1} ∈ W k -1, if {j ∈ N | i j = +1} / ∈ W k , ∀i ∈ I, ∀k ∈ N. ( 30
)
According to definition (30), for each player k, if the set of players with positive inclination creates a winning coalition in the command game for k, then k decides 'yes', and if the set of players with positive inclination is not a winning coalition in the command game for k, then player k's decision is 'no'. Note that we could define another command influence function in which instead of looking at the positive inclination of the players, we would consider only the negative inclination of the agents, and a winning coalition would be the set of potential no-voters only. Since the results of these two cases are similar, we study only the command influence function defined by (30).

Proposition 6 Let {(N, W k ) | k ∈ N } be a set of command games satisfying [START_REF] Rae | Decision-rules and individual values in constitutional choice[END_REF], and ω(S) be a command function as defined in [START_REF] Machover | The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes[END_REF]. We have

F Com (S) = ω(S), ∀S ⊆ N, ( 31
)
where F Com (S) is the set of followers of S under the command influence function Com.

Proof: F Com (∅) = ∅ = ω(∅). Take an arbitrary nonempty S ⊆ N . Suppose that F Com (S) ⊆ ω(S). Hence, there exists k ∈ F Com (S) such that k / ∈ ω(S), and therefore (Comi) k = i S for all i ∈ I S , and S / ∈ W k . Take i ∈ I S such that i S = +1 and i m = -1 if m / ∈ S. Hence, (Comi) k = +1 and (Comi) k = -1, a contradiction. Suppose now that ω(S) ⊆ F Com (S). Hence, there exists k ∈ ω(S) such that k / ∈ F Com (S). This means that S ∈ W k and there is i ∈ I S such that (Comi) k = -i S . Hence, if i S = +1, then (Comi) k = -1 and (Comi) k = +1, because S ∈ W k implies S ′ ∈ W k for any S ′ ⊇ S, a contradiction. If i S = -1, then (Comi) k = +1, and also since S ∈ W k , and from [START_REF] Rae | Decision-rules and individual values in constitutional choice[END_REF]

, {j ∈ N | i j = +1} / ∈ W k . Hence (Comi) k = -1 and (Comi) k = +1, a contradiction.
Consequently, by virtue of ( 12) and ( 31), the kernel of Com is given by:

K(Com) = {S ∈ 2 N | ω(S) = ∅, and S ′ ⊂ S ⇒ ω(S ′ ) = ∅}.
Remark 3 By virtue of (31), for each set of command games {(N, W k ) | k ∈ N }, the command influence function Com ∈ B defined in ( 30) is equivalent to this set of command games.

Proposition 7 Let {(N, W j ) | j ∈ N } be a set of command games, and Com be the command influence function defined in (30). Then for each j, k

∈ N Sh k (N, W j ) = ψ e α (Sh) ,λ=i k (Com, k → j) -ψ b α (Sh) ,λ=-i k (Com, k → j) (32)
where Sh k (N, W j ) is the Shapley-Shubik index of player k in the command game for j, ψ α,λ is the weighted influence index defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF], and for each i ∈ I α

(Sh)k→j i = 1 n( n-1 |i|-1 ) , if i k = +1 0, if i k = -1 α (Sh)k→j i = 0, if i k = +1 1 n( n-1 n-|i|-1 ) , if i k = -1 (33)
and

|i| := |{m ∈ N | i m = +1}|.
Proof: We introduce a bijection f :

I → 2 N such that for each i ∈ I, f (i) = {k ∈ N | i k = +1}. Hence, k ∈ S iff i k = +1. Given (N, W j ), we have by (30): v(S) = 1 iff S = f (i) ∈ W j iff (Comi) j = +1. Hence, from (1)
, we have

Sh k (N, W j ) = S⊆N :k∈S (n -s)!(s -1)! n! (v(S) -v(S \ k)) = S⊆N :k∈S (n -s)!(s -1)! n! v(S) - S⊆N :k / ∈S (n -s -1)!s! n! v(S) = S⊆N :k∈S v(S) n n-1 s-1 - S⊆N :k / ∈S v(S) n n-1 n-s-1 = i∈I + k α (Sh)k→j i - i∈I - k α (Sh)k→j i
where

I + k := {i ∈ I | i k = +1 ∧ (Comi) j = +1}, I - k := {i ∈ I | i k = -1 ∧ (Comi) j = +1}.
We have:

i:i k =+1 α (Sh)k→j i = i:i k =+1 1 n n-1 |i|-1 = n |i|=1 1 n n-1 |i|-1 n -1 |i| -1 = 1 i:i k =-1 α (Sh)k→j i = i:i k =-1 1 n n-1 n-|i|-1 = n-1 |i|=0 1 n n-1 n-|i|-1 n -1 |i| = 1,
and therefore

Sh k (N, W j ) = i∈I + k α (Sh)k→j i - i∈I - k α (Sh)k→j i = i∈I + k α (Sh)k→j i i:i k =+1 α (Sh)k→j i - i∈I - k α (Sh)k→j i i:i k =-1 α (Sh)k→j i = i:(Comi) j =i k α (Sh)k→j i i∈I S α (Sh)k→j i - i:(Comi) j =-i k α (Sh)k→j i i∈I S α (Sh)k→j i = ψ e α (Sh) ,λ=i k (Com, k → j) -ψ b α (Sh) ,λ=-i k (Com, k → j)
where ψ α,λ is defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF], and α (Sh) and α (Sh) are given in (33).

Proposition 8 Let {(N, W j ) | j ∈ N } be a set of command games, and Com be the command influence function defined in (30). Then for each j, k

∈ N Bz k (N, W j ) = ψ e α (Bz) ,λ=i k (Com, k → j) -ψ b α (Bz) ,λ=-i k (Com, k → j) (34)
where Bz k (N, W j ) is the Banzhaf index of player k in the command game for j, ψ α,λ is the weighted influence index defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF], and for each i ∈ I α

(Bz)k→j i = 1, if i k = +1 0, if i k = -1 α (Bz)k→j i = 0, if i k = +1 1, if i k = -1. (35)
Proof: The proof is similar to the one of Proposition 7. From (2), we have

Bz k (N, W j ) = 1 2 n-1 S⊆N :k∈S (v(S) -v(S \ k)) = 1 2 n-1 S⊆N :k∈S v(S) - 1 2 n-1 S⊆N :k / ∈S v(S) = |{i ∈ I | i k = +1 ∧ (Comi) j = +1}| 2 n-1 - |{i ∈ I | i k = -1 ∧ (Comi) j = +1}| 2 n-1 = |{i ∈ I | i k = +1 ∧ (Comi) j = +1}| |{i ∈ I | i k = +1}| - |{i ∈ I | i k = -1 ∧ (Comi) j = +1}| |{i ∈ I | i k = -1}| = ψ e α (Bz) ,λ=i k (Com, k → j) -ψ b α (Bz) ,λ=-i k (Com, k → j)
where ψ α,λ is defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF], and α (Bz) and α (Bz) are given in (35).

As in the case of the command influence function with abstention, we ask the same question about the meaning of Propositions 7 and 8. As before, under a certain assumption, both the Shapley-Shubik index and the Banzhaf index of player k in the command game for j are equal to the difference between the weighted influence index in which player j is said to be influenced by k if he follows k, and the weighted influence index in which the influence of k on j means that player j's decision is opposite to the inclination of k. Both weighted influence indices are measured now under the command influence function defined in (30). Moreover, the weights are different from the ones defined for the command influence function with abstention, although closely related. Remark that α (Sh)k→j i and α (Sh)k→j i are identical when i k = +1, and the former is 0 if i k = -1. It means that for positive influence, situations where k votes 'no' are discarded, which is in accordance with the definition of Com (player j follows the winning coalition only if this coalition votes 'yes'). The situation for α (Sh)k→j i is dual. The same comment applies to the case of the Banzhaf index.

Command games defined for some influence functions

In this section, we work in the opposite direction than in Section 4, and for some influence functions defined in [START_REF] Grabisch | A model of influence in a social network[END_REF] we construct the equivalent command games. In particular, we show that not for all influence functions the compatible command games exist. For each set of the equivalent command games, apart from defining the winning coalitions, we describe also the boss sets and the approval sets.

Proposition 9 Let n ≥ t > ⌊ n
2 ⌋ and Maj [t] ∈ B be the majority function as defined in [START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF]

. Let {(N, W Maj [t] k ) | k ∈ N } be a set of command games given by W Maj [t] k = {S ⊆ N | s ≥ t}, ∀k ∈ N. (36)
The majority function Maj [t] is equivalent to the set of games {(N, W

Maj [t] k ) | k ∈ N }.
According to Proposition 9, the command games, in which winning coalitions for each player are the ones with the cardinality at least t, n ≥ t > ⌊ n 2 ⌋, are equivalent to Maj [t] . Proof: By virtue of (36), for each S ⊆ N , the command function ω Maj [t] (S) is given by

ω Maj [t] (S) = N, if s ≥ t ∅, if s < t which from (15) is equal to F Maj [t] (S).
Remark 4 Note that for the set of command games {(N, W

Maj [t] k ) | k ∈ N } defined by (36), we have for n > 2, n ≥ t > ⌊ n 2 ⌋, and k ∈ N Boss Maj [t] k = {S ⊆ N | s ≥ t ∧ k / ∈ S} App Maj [t] k = {S ⊆ N | s = t -1 ∧ k / ∈ S}.
In particular, for t = n, k ∈ N ,

Boss Maj [t] k = ∅, App Maj [t] k = N \ k.
Proposition 10 Let Gur [ e k] ∈ B be the guru function as defined in [START_REF] König | The inclusiveness of European decision rules[END_REF], with the guru

k ∈ N . Let {(N, W Gur [ e k] k
) | k ∈ N } be a set of command games given by

W Gur [ e k] k = {S ⊆ N | k ∈ S}, ∀k ∈ N. (37)
The guru function Gur [ e k] is equivalent to the set of games {(N, W

Gur [ e k] k ) | k ∈ N }.
According to Proposition 10, the command games, in which winning coalitions for each player are the coalitions containing a certain player k, are equivalent to the guru function Gur [ e k] with the guru k.

Proof: By virtue of (37), for each S ⊆ N , the command function ω Gur [ e k] (S) is given by

ω Gur [ e k] (S) = N, if k ∈ S ∅, if k / ∈ S
which from ( 17) is equal to F Gur [ e k] (S).

Remark 5 In the command games {(N, W

Gur [ e k] k
) | k ∈ N } defined by (37), the guru k is a free agent, and the remaining players are cogs, i.e.,

Boss Gur [ e k] e k = ∅, App Gur [ e k] e k = 2 N \ e k Boss Gur [ e k] k = {S ⊆ N | k ∈ S ∧ k / ∈ S}, App Gur [ e k] k = ∅, f or k = k.
Proposition 11 Let Id ∈ B be the identity function as defined in [START_REF] Owen | Values of games with a priori unions[END_REF]. Let {(N, W Id k ) | k ∈ N } be a set of command games given by

W Id k = {S ⊆ N | k ∈ S}, ∀k ∈ N. ( 38
)
The identity function Id is equivalent to the set of games {(N, W Id k ) | k ∈ N }. According to Proposition 11, the command games in which, for each player k, winning coalitions for k are the coalitions containing k, are equivalent to the identity function.

Proof: ¿From (38), we have ω Id (S) = S for each S ⊆ N . On the other hand, also F Id (S) = S for S ⊆ N . [START_REF] Owen | Modification of the Banzhaf-Coleman index for games with a priori unions[END_REF]. There is no set of command games such that the reversal function is compatible with these command games.

Proof: We know that F -Id (S) = ∅ for S ⊆ N . Hence, in particular, F -Id (N ) = ∅. But for any set of command games, we have ω(N ) = N , and therefore there is no command game compatible with -Id.

Example -The Confucian model

In [START_REF] Hu | On authority distributions in organizations: equilibrium[END_REF] the Confucian model of society is mentioned. We have four players in the society, i.e., N = {1, 2, 3, 4}, with the king (1), the man (2), the wife (3), and the child (4). The rules are as follows:

(i) The man follows the king; (ii) The wife and the child follow the man; (iii) The king should respect his people.

Let us define the command games for this example. By virtue of the rules (i) and (ii), we have immediately:

W 2 = {1,
App 2 = App 3 = App 4 = ∅,
which means that players 2, 3, and 4 are cogs. How can we translate the rule (iii) into the set W 1 of winning coalitions in the command game for player 1? We propose several interpretations of this rule, and consequently, several command games for player 1.

The command game with W 1 = {1234}

If W 1 = {1234}, then we get:

Boss 1 = ∅, App 1 = {234},
i.e., the king needs the approval of all his people.

ω(1) = ω(13) = ω(14) = ω(134) = {2}, ω(2) = ω(23) = ω(24) = ω(234) = {3, 4} ω(3) = ω(4) = ω(34) = ∅, ω(N ) = N, ω(12) = ω(123) = ω(124) = {2, 3, 4}.
The Shapley-Shubik index matrix is then:

P = [Sh k (N, W j )] n j,k=1 =     1 4 1 4 1 4 1 4 1 0 0 0 0 1 0 0 0 1 0 0     and from (9) one has            π 1 = 1 4 π 1 + π 2 π 2 = 1 4 π 1 + π 3 + π 4 π 3 = 1 4 π 1 π 4 = 1 4 π 1 π 1 + π 2 + π 3 + π 4 = 1
We have already calculated the Shapley-Shubik index matrix for this example. The Banzhaf index matrices are equal to:

Bz = [Bz k (N, W j )] n j,k=1 =     1 8 1 8 1 8 1 8
1 0 0 0 0 1 0 0 0 1 0 0

    Bz = [ Bz k (N, W j )] n j,k=1 =     1 4 1 4 1 4 1 4
1 0 0 0 0 1 0 0 0 1 0 0     .

6.2

The command game with W 1 = {123, 124, 134, 1234}

If W 1 = {123, 124, 134, 1234}, then we get:

Boss 1 = ∅, App 1 = {23, 24, 34, 234},
i.e., the king needs the approval of the majority of his people.

ω(1) = ω(13) = ω(14) = {2}, ω(134) = {1, 2}, ω(2) = ω(23) = ω(24) = ω(234) = {3, 4} ω(3) = ω(4) = ω(34) = ∅, ω(12) = {2, 3, 4}, ω(123) = ω(124) = ω(N ) = N.
The Shapley-Shubik index matrix is then:

P = [Sh k (N, W j )] n j,k=1 =     1 2 1 6 1 6 1 6 1 0 0 0 0 1 0 0 0 1 0 0     and from (9) one has            π 1 = 1 2 π 1 + π 2 π 2 = 1 6 π 1 + π 3 + π 4 π 3 = 1 6 π 1 π 4 = 1 6 π 1 π 1 + π 2 + π 3 + π 4 = 1
which gives the authority distribution:

π = 1 11 (6, 3, 1, 1).
As before, we apply now the command influence functions to the model. Table 2 presents the inclination and decision vectors under the three command influence functions.

Note that for W 1 = {123, 124, 134, 1234} the decision vectors under Com are the same as the decision vectors under Com for W 1 = {1234}.

The results are similar as before, that is:

F B (1) = {2}, F B (2) = {3, 4}, F B (12) = {2, 3, 4}, for B ∈ {Com, Com} F Com (1) = {1, 2}, F Com (2) = {3, 4}, F Com (12) = N.
In particular:

d α (B, 1 → 2) = d α (B, 2 → 3) = d α (B, 2 → 4) = 1, for B ∈ {Com, Com, Com}.
∈ I Comi Comi Comi (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, -1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, -1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, -1, 1, 1) (1, 1, -1, -1) (1, 1, -1, -1) (1, 1, -1, -1) (-1, 1, 1, 1) (0, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (1, -1, 1, -1) (0, 1, -1, -1) (1, 1, -1, -1) (-1, 1, -1, -1) (1, 1, -1, -1) (0, 1, 1, 1) (1, 1, 1, 1) (-1, 1, 1, 1) (-1, 1, 1, -1) (0, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (1, -1, -1, 1) (0, 1, -1, -1) (1, 1, -1, -1) (-1, 1, -1, -1) (-1, -1, 1, 1) (0, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, 1, -1, 1) (0, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (1, -1, -1, -1) (0, 1, -1, -1) (1, 1, -1, -1) (-1, 1, -1, -1) (-1, 1, -1, -1) (-1, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, 1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1)
The kernel is equal to:

K(B) = {{1}, {2}}, for B ∈ {Com, Com, Com}.
The Banzhaf index matrices are equal to:

Bz = [Bz k (N, W j )] n j,k=1 =     1 2 1 4 1 4 1 4 1 0 0 0 0 1 0 0 0 1 0 0     Bz = [ Bz k (N, W j )] n j,k=1 =     2 5 1 5 1 5 1 5 1 0 0 0 0 1 0 0 0 1 0 0     .
6.3 The command game with W 1 = {12, 13, 14, 123, 124, 134, 1234}

If W 1 = {12, 13, 14, 123, 124, 134, 1234}, then we get:

Boss 1 = ∅, App 1 = {2, 3
, 4, 23, 24, 34, 234}, i.e., the king needs the approval of at least one of his people.

ω(1) = {2}, ω(13) = ω(14) = ω(134) = {1, 2}, ω(2) = ω(23) = ω(24) = ω(234) = {3, 4} ω(3) = ω(4) = ω(34) = ∅, ω(12) = ω(123) = ω(124) = ω(N ) = N.
The Shapley-Shubik index matrix is then: We apply the command influence functions to the model. Table 3 presents the inclination and decision vectors under the command influence functions.

P = [Sh k (N, W j )] n j,k=1 =     3 4 1 12 1 12 1 12 1 0 0 0 0 1 0 0 0 1 0 0     and from (9) one has            π 1 = 3 4 π 1 + π 2 π 2 = 1 12 π 1 + π 3 + π 4 π 3 = 1 12 π 1 π 4 = 1 12 π 1 π 1 + π 2 + π 3 + π 4 = 1
∈ I Comi Comi Comi (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, -1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, -1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (1, -1, 1, 1) (1, 1, -1, -1) (1, 1, -1, -1) (1, 1, -1, -1) (-1, 1, 1, 1) (0, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (1, -1, 1, -1) (1, 1, -1, -1) (1, 1, -1, -1) (1, 1, -1, -1) (1, 1, -1, -1) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 1) (-1, 1, 1, -1) (-1, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (1, -1, -1, 1) (1, 1, -1, -1) (1, 1, -1, -1) (1, 1, -1, -1) (-1, -1, 1, 1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, 1, -1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (1, -1, -1, -1) (0, 1, -1, -1) (1, 1, -1, -1) (-1, 1, -1, -1) (-1, 1, -1, -1) (-1, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, 1) (-1, -1, 1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, 1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1) (-1, -1, -1, -1)
As before, under all three influence functions, each player (except the king) always follows the inclination of his boss sets. Moreover, under Com, the king almost always follows his inclination, except two cases of abstention when the inclination of all his people is different from his own inclination. Under the function Com, the king always follows his own inclination as in the previous two command games for player 1. Under the function Com, the king almost always follows his inclination, except one case when his inclination is positive while the inclination of all his people is negative.

We have also:

Conclusion

This paper presents a comparison between two models that deal with modeling players' interactions: the command games and the influence model. The link between the models is expressed by defining the influence functions that are compatible with or equivalent to the command games. An influence function and the command games are compatible if each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. An influence function and the command games are equivalent if they are compatible, and additionally each follower of a coalition under the command influence function is the commandable player for that coalition in the command game. Consequently, for each set of the command games, we define two equivalent influence functions and one compatible influence function. Moreover, for some influence functions we define the equivalent command games. Nevertheless, not for all influence functions the compatible command games exist. In the paper, we also show links between some power indices, which can be used in the command games, and the generalized weighted influence indices. We note that results concerning the Shapley-Shubik index and the Banzhaf index differ only in the weights for the influence indices. The concluding remark can be that the influence model is more general than the framework of the command games, and the concepts of the influence function and the influence index can capture the command structure.

A research agenda concerning our future work on the influence model contains several issues. In particular, we plan to introduce the authority distribution based on the influence indices. Moreover, we intend to introduce dynamic aspects into the model. We want to study the behavior of the series Bi, B 2 i, ..., B n i, ..., to find convergence conditions, to investigate the corresponding influence indices, and relations between the repeated influence model and the command games. We plan to analyze a generalized model of influence, in which each player has a continuum of options to choose (that is, a model in which the inclination or opinion of each player lies in an interval, say [0, 1], where each i k ∈ [0, 1] can be interpreted as player k's degree of inclination to say 'yes'). An important issue for future research concerns an axiomatic characterization of the influence indices. Furthermore, it would be interesting to test the new concepts and to run lab experiments concerning the influence between players. KB k (N, W j ) = ψ α,λ=i k (Com, k → j) ψ α,λ =0 (Com, k → j)

where A(N, W j ) is the Coleman power of a collectivity to act in the command game for player j, Col P k (N, W j ) is the Coleman index to prevent action of player k in the command game for j, and KB k (N, W j ) is the König-Bräuninger inclusiveness index of k in the command game for j. Moreover, ψ α,λ is the weighted influence index defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF] with

α k→j i = 1, ∀i ∈ I. (41)
Proof: (39) results immediately from (3). Let j ∈ N , and take an arbitrary k ∈ N . We have

A(N, W j ) = S⊆N v(S) 2 n = |{i ∈ I | (Comi) j = 0}| 2|I| = ψ α,λ =0 (Com, k → j) 2
where α is given in (41). We have

Definition 4

 4 Given a set of command games {(N, W k ) | k ∈ N }, the command influence function Com ∈ B is defined for each k ∈ N and i ∈ I by

  +1 and ( Comi) k = -1, contradiction. Suppose that k ∈ S and there is S ′ ∈ W k such that S ∩ S ′ = ∅. Hence, i S ′ = -1, and therefore ( Comi) k = +1 and ( Comi) k = -1, a contradiction. Suppose now that ω(S)∪{k ∈ S | ∀S ′ ∈ W k [S ∩S ′ = ∅]} ⊆ F Com (S) for a certain S ⊆ N . Hence, there exists k such that k / ∈ F Com (S), and either k ∈ ω(S) or (k ∈ S and for all

Definition 5

 5 Given a set of command games {(N, W k ) | k ∈ N }, the command influence function Com ∈ B is defined by

Remark 6

 6 In the command games {(N, W Id k ) | k ∈ N } defined by (38), all players are free agents, i.e., Boss Id k = ∅, App Id k = 2 N \k . Proposition 12 Let -Id ∈ B be the reversal function as defined in

F B ( 1 )

 1 = {2}, F B (2) = {3, 4}, F B (12) = N, for B ∈ {Com, Com} F Com (1) = {1, 2}, F Com (2) = {3, 4}, F Com (12) = N. d α (B, 1 → 2) = d α (B, 2 → 3) = d α (B, 2 → 4) = 1, for B ∈ {Com, Com, Com}. K(B) = {{1}, {2}}, for B ∈ {Com, Com, Com}.The Banzhaf index matrices are equal to:Bz = [Bz k (N, W j )] n j,k=1 = Bz k (N, W j )] n j,k=1 =

  S⊆N v(S) = |{i ∈ I | (Comi) j = 0}| 2 = 2 n-1 ψ α,λ =0 (Com, k → j) > 0. (42)By virtue of (2) and (4), we have for each j, k ∈ NCol P k (N, W j ) = 2 n-1 Bz k (N, W j ) S⊆N v(S) = Bz k (N, W j ) ψ α,λ =0 (Com, k → j)which together with (26) gives (40). Since we haveS⊆N :k∈S v(S) = |{i ∈ I | (Comi) j = i k }| 2 = 2 n-1 ψ α,λ=i k (Com, k → j),(43)hence from (6), (42), and (43), we haveKB k (N, W j ) = S⊆N :k∈S v(S) S⊆N v(S) = ψ α,λ=i k (Com, k → j) ψ α,λ =0 (Com, k → j) .Proposition 14 Let j ∈ N and (N, W j ) be the command game for j, and Com ∈ B be the command influence function defined by (30). ThenA(N, W j ) = ψ α,λ=+1 (Com, k → j), ∀k ∈ N(44)and for each k ∈ NCol P k (N, W j ) = ψ e α (Bz) ,λ=i k (Com, k → j)ψ b α (Bz) ,λ=-i k (Com, k → j) 2ψ α,λ=+1 (Com, k → j)(45)From (6), (46), and (47), we haveKB k (N, W j ) = S⊆N :k∈S v(S) S⊆N v(S)= ψ e α (Bz) ,λ=i k (Com, k → j) 2ψ α,λ=+1 (Com, k → j) .

  Boss 2 = {1, 13, 14, 134}, Boss 3 = {2, 12, 24, 124}, Boss 4 = {2, 12, 23, 123}

	12, 13, 14, 123, 124, 134, 1234}
	W 3 = {2, 12, 23, 24, 123, 124, 234, 1234}
	W 4 = {2, 12, 23, 24, 123, 124, 234, 1234}.
	Hence, we have

Table 2 .

 2 The inclination and decision vectors for W1 = {123, 124, 134, 1234} i

Table 3 .

 3 The inclination and decision vectors for W1 = {12, 13, 14, 123, 124, 134, 1234}

	i

which gives the authority distribution: π = 1 9 (4, 3, 1, 1).

Let us apply now the command influence functions to the model. Table 1 presents the inclination and decision vectors under the three command influence functions.

Table 1. The inclination and decision vectors for

Note that under B ∈ {Com, Com, Com} each player (except the king) always follows the inclination of his boss sets. Under the influence function Com, the king will say 'yes' / 'no' only if all players (including himself) have the positive / negative inclination. In the remaining cases, the king abstains. Under the influence function Com, the king always follows his own inclination, since 1234 is the unique winning coalition in the command game for player 1. Under the influence function Com, if all people of the king are against, his decision is also 'no', but if all his people are in favor (i.e., the king has the approval of his people), his decision is 'yes' only if his inclination is also positive. In case the people of the king are not unanimous, the king has no approval, and consequently he chooses 'no', even if his inclination is positive.

We have, in particular:

We have, d α (B, S → j) = 1 for each j ∈ F B (S) \ S. Hence, we have:

The kernel is equal to:

Appendix -relations between the weighted influence indices and the power indices

Next, we show relations between some power indices and the weighted influence indices under the command influence functions defined in Section 4.

Proposition 13 Let j ∈ N and (N, W j ) be the command game for j, and Com ∈ B be the command influence function defined by [START_REF] Shapley | A Boolean model of organization authority based on the theory of simple games[END_REF]. Then

and for each k ∈ N

where A(N, W j ) is the Coleman power of a collectivity to act in the command game for player j, Col P k (N, W j ) / Col I k (N, W j ) is the Coleman index to prevent action / the Coleman index to initiate action of player k in the command game for j, and KB k (N, W j ) is the König-Bräuninger inclusiveness index of k in the command game for j. Moreover, ψ α,λ is the weighted influence index defined in [START_REF] Hu | On authority distributions in organizations: controls[END_REF], α is given in (41), and α (Bz) , α (Bz) are given in (35).

Proof: (44) results immediately from (3). Let j ∈ N , and take an arbitrary k ∈ N . We have

where α is given in (41). We have

By virtue of ( 2) and ( 4), we have for each j, k ∈ N

which together with (34) gives (45). Moreover we note that

By virtue of ( 5), (47), (48), and (49), we have


Azzouz Dermoune
email: azzouz.dermoune@univ-lille1.fr

Mohammed Es Sebaiy
email: mohammedsebaiy@gmail.com

Jabrane Moustaaid
email: jabrane.mst@gmail.com

Interpolation and linear prediction of data -three kernel selection criteria

Keywords: Kernel interpolation, stochastic interpolation, linear algebra interpolation, cubic spline interpolation, climate change detection

Interpolation and prediction have been useful approaches in modeling data in many areas of applications. The aim of this paper is the prediction of the next value of a time series (time series forecasting) using the techniques in interpolation of the spatial data, for the two approaches kernel interpolation and kriging. We are interested in finding some sufficient conditions for the kernels and provide a detailed analyse of the prediction using kernel interpolation. Finally, we provide a natural idea to select a good kernel among a given family of kernels using only the data. We illustrate our results by application to the data set on the mean annual temperature of France and Morocco recorded for a period of 115 years (1901 to 2015).

Introduction

Interpolation and prediction have been useful approaches in modelling data in many areas of applications such as the prediction of the meteorological variables, surface reconstruction and Interpolation of spatial data [START_REF] Scheuerer | Interpolation of spatial data-a stochastic or a deterministic problem[END_REF] among many more. For more details see [START_REF] Chilès | How to adapt kriging to non-classical problems: three case studies[END_REF], [START_REF] Chilès | Geostatistics, Modeling Spatial Uncertainty[END_REF], [START_REF] Wendland | Scattered data approximation[END_REF] and [START_REF] Mardikis | Comparison of interpolation methods for the prediction of reference evapotranspiration-an application in greece[END_REF]. In this work we extend the results of Scheuerer [START_REF] Scheuerer | Interpolation of spatial data-a stochastic or a deterministic problem[END_REF] to the linear prediction approach of time series. We also cite the work of Dermoune et all [START_REF] Dermoune | Moustaaid. Parametrizations, weights, and optimal prediction[END_REF] where the parametrizations and the cubic spline were used as a model of prediction and we extend this results to the kernel interpolation framework.

Interpolation of spatial data is a very general mathematical problem and it's precise mathematical formulation as defined in [START_REF] Scheuerer | Interpolation of spatial data-a stochastic or a deterministic problem[END_REF] is to reconstruct a function f : T → R with T is is a domain in R d , based on its values at a finite set of data points X = {x 1 , . . . , x n } ⊂ T , the values f (x 1 ), . . . , f (x n ) assumed to be known. But, in our case we are interested in the time series forecasting problem we have T = {x 1 , . . . , x n , x n+1 } represent the time and the time series is f (x 1 ), . . . , f (x n ) with the unknown value is f (x n+1 ). In other words, we want to predict effectively the value f (x n+1 ) using the known values f (x 1 ), . . ., f (x n ). From [START_REF] Scheuerer | Interpolation of spatial data-a stochastic or a deterministic problem[END_REF] we have that both approaches kernel interpolation and kriging have the same approximant for the interpolation of spatial data problem, even with the different model assumption, a general overview in both approaches can be fond in [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF].

Linear prediction and kernel interpolation

Let R {x 1 ,...,x n+1 } be the Hilbert space of real functions on {x 1 , . . . , x n+1 } with inner product (., .) and norm N (.). The dual of R {x 1 ,...,x n+1 } is spanned by the point evaluation linear forms δ x : f → f (x), x ∈ {x 1 , . . . , x n+1 }, that is (R {x 1 ,...,x n+1 } ) * = (δ x 1 , . . . , δ x n+1 ).

Moreover, the dual norm N * is defined by

(N * (µ)) 2 = sup{|µ(f )| 2 : N (f ) ≤ 1},
for all µ ∈ (R {x 1 ,...,x n+1 } ) * . Now, for any function f ∈ R {x 1 ,...,x n+1 } and any sequence of real numbers (w 1 , . . . , w n ), we define the linear prediction of f

(x n+1 ) f (x n+1 ) = n i=1 w i f (x i ),
with the error

Err n (f ) := |f (x n+1 ) - n i=1 w i f (x i )|,
and the worst error in the unit ball w.r.t. the norm N (.)

Werr(f ) := sup{|f (x n+1 ) - n i=1 w i f (x i )| 2 : N (f ) ≤ 1} = (N * (δ x n+1 - n i=1 w i δ x i )) 2 . (1)
In the rest oh this paper, we endow the vector space R {x 1 ,...,x n+1 } with the scalar inner product

(f, f ) = (f, f ) K -1 = n+1 i=1 n+1 j=1 f (x i )f (x j )k (-1) (x i , x j ) = f K -1 f , with f = (f (x 1 ), . . . , f (x n+1 )) and K = [k(x i , x j ) : i, j = 1, . . . , n + 1]
is a fixed (n + 1) × (n + 1) symmetric positive definite matrix, with k (-1) (x i , x j ) denotes the (i, j) entry of K -1 . The norm defined by K is given by N (f ) = K -1/2 f , with • denotes the Euclidean norm.

Min-max prediction and kernel interpolation Definition (Min-max prediction). A linear prediction f

* (x n+1 ) of f (x n+1 ) is called min- max if f * (x n+1 ) = n i=1 w * i f (x i ), (2)
where (w * 1 , ..., w * n ) are given by the minimization of the Werr(f ) 1.

The following result give us the optimal weights associate to the min-max prediction w.r.t. to the norm K -1/2 • . Proposition 2.2. The the worst error in the unit ball, Werr(f ), w.r.t. to the norm K -1/2 • is equals

Werr(f ) = δ x n+1 - n i=1 w i δ x i ) 2 K 1/2 (3)
where • K 1/2 denotes the dual norm defined by the dual scalar inner product

(δ x i , δ x j ) K = k(x i , x j ), i, j = 1, . . . , n + 1.
Proof. From the general theory of reproducing kernel Hilbert spaces,see [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF][START_REF] Scheuerer | Interpolation of spatial data-a stochastic or a deterministic problem[END_REF], we have sup

K -1/2 f ≤1 {|f (x n+1 ) - n i=1 w i f (x i )| 2 } = sup K -1/2 f ≤1 {[K -1/2 f ] [K 1/2 (-w 1 , . . . , -w n , 1) (-w 1 , . . . , -w n , 1)K 1/2 ][K -1/2 f ]} = the largest eigenvalue of [K 1/2 (-w 1 , . . . , -w n , 1) (-w 1 , . . . , -w n , 1)K 1/2 ] = K 1/2 (-w 1 , . . . , -w n , 1) 2 = (-w 1 , . . . , -w n , 1)K(-w 1 , . . . , -w n , 1) = δ x n+1 - n i=1 w i δ x i ) 2 K 1/2 .
Corollary 2.3. The optimal weights of the min-max linear prediction of f (x n+1 ) are given by

w * = (w * 1 , . . . , w * n ) = [k(x n+1 , x 1 ), . . . , k(x n+1 , x n )][k(x i , x j ) : i, j = 1, . . . , n] -1 . ( 4
)
Proof. The optimal weights are given by the minimization

arg min{ δ x n+1 - n i=1 w i δ x i 2 K 1/2 : w 1 , . . . , w n ∈ R}, (5)
which is the solution of the system n j=1

w j k(x i , x j ) = k(x n+1 , x i ), i = 1, . . . , n, (6)
it follows easily that w * is given by 4.

Remarks 2.4.

1) The worst case linear prediction error in the ball with the radius r > 0 w.r.t. to the norm

K -1/2 • is equal to sup K -1/2 f ≤r {|f (x n+1 ) - n i=1 w i f (x i )| 2 } = r 2 (-w 1 , . . . , -w n , 1)K(-w 1 , . . . , -w n , 1) ,
as a result the optimal weights (4) do not depend on the radius of the ball.

2) The prediction using the spline interpolating w.r.t. the norm K -1/2 • (see, e.g., [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF]) defined by the minimizer :

S(f ) = arg min{ K -1/2 f : f (x 1 ), . . . , f (x n ) are fixed},
coincide with the prediction [START_REF] Dermoune | Moustaaid. Parametrizations, weights, and optimal prediction[END_REF].

3) The min-max prediction ( 2) is equal to

f * (x n+1 ) = n j=1 α * j k(x n+1 , x j ), (7)
where α * j , j = 1, ..., n is the solution of the system

n j=1 α * j k(x i , x j ) = f (x i ), i = 1, . . . , n, (8)
Now, we turn to the interpolation of the function f at the set {x 1 , . . . , x n } using span(k 1 , . . . , k n ) where k j denotes the j-th column of the matrices K. Then the interpolation of the function f equals

I(f ) = n j=1 α * j k j
with the weights α * are given by [START_REF] Mardikis | Comparison of interpolation methods for the prediction of reference evapotranspiration-an application in greece[END_REF]. The following Proposition gives the error of interpolation.

Proposition 2.5 (Interpolation error ). The error of interpolation, IErr, is given by

IErr(f ) := f (x n+1 ) -f * (x n+1 ) = [k (-1) n+1 f ][k(x n+1 , x n+1 ) - n i=1 w * i k(x i , x n+1 )]. (9)
Proof. First, observe that we can write the coordinates of f in the basis K as

f = n+1 j=1 [k (-1) j f ]k j .
with k (-1) j denotes the j-th row of K -1 . Therefore

I(f ) = n+1 j=1 [k (-1) j f ]I(k j ) = n j=1 [k (-1) j f ]k j + [k (-1) n+1 f ]I(k n+1 ),
because the interpolation of k j is exact for j = 1, . . . , n. Thus,

f -I(f ) = [K -1 n+1 f ](k n+1 -I(k n+1 )) = [k (-1) n+1 f ](0, . . . , 0, [k(x n+1 , x n+1 ) - n i=1 w * i k(x i , x n+1 )]) ,
which completes the proof.

Min-max linear prediction with constraint

In this section, we consider the optimization (5) under the constraint

n i=1 w i p k (x i ) = p k (x n+1 ), k = 1, . . . , q, (10)
where p 1 , . . . , p q ∈ R {x 1 ,...,x n+1 } are given. Solve the minimization (5) under the constraint ( 10) is equivalent to solve the system

n j=1 k(x i , x j )w j + q k=1 λ k p k (x i ) = k(x i , x n+1 ), i = 1, . . . , n, n j=1 w j p k (x j ) = p k (x n+1 ), k = 1, . . . , q, (11)
where λ 1 , . . ., λ q are the Lagrange multiplier. The solution is unique if the homogeneous system

n j=1 k(x i , x j )w j + q k=1 λ k p k (x i ) = 0, i = 1, . . . , n, n j=1 w j p k (x j ) = 0, k = 1, . . . , q,
has a unique solution w 1 = . . . = w n = 0, λ 1 = . . . = λ q = 0. This is equivalent to say that the columns (p k (x 1 ), . . . , p k (x n )) , k = 1, . . . , q, are linearly independent and that K is conditionally positive w.r.t. p 1 , . . ., p q , i.e. the system

n j=1 n j=1 k(x i , x j )w j w i = 0, n i=1 w i p k (x i ) = 0, k = 1, . . . , q,
has a unique solution w 1 = . . . = w n = 0. Observe that this is true if K is definite positive, but it is not necessary. Let w * 1 , . . ., w * n , λ * 1 , . . ., λ * q be the solution of the system [START_REF] Reinsch | Smoothing by spline functions[END_REF]. Then the optimal prediction under the constraint ( 10) is

f * (x n+1 ) = n i=1 w * i f (x i ). (12)

Constraint's parametrization

Now, let z (1) = (z

(1) 1 , . . . , z (1)
n ) be a particular solution of the system (10) and z 1 = (z 11 , . . . , z n1 ) , . . ., z n-q = (z 1n-q , . . . , z nn-q ) , n -q independent solutions of the corresponding homogeneous system. Then the general solution of the system (10) has the form

w = z (1) + n-q l=1 wl z l .
Let us consider the basis

B = [p 1 , . . . , p q , n i=1 z i1 k i , . . . , n i=1 z in-q k i , b n+1 ] (13) =: [b 1 , . . . , b n+1 ]
such that the expansion of f in the basis B is given by

f = q l=1 θ l f p l + n-q l=1 θ q+l f b q+l + θ n+1 f b n+1 , with θ n+1 = (-(z (1)
) , 1).

The unknows are the rows θ 1 , . . ., θ n , and the column b n+1 . They are solution of the system

δ(i = j) = q l=1 p l (x i )θ lj + n-q l=1 b q+l (x i )θ (q+l)j + b n+1 (x i )θ (n+1)j i, j = 1, . . . , n + 1. ( 14
)
The interpolation of any function g at the set {x 1 , . . . , x n } using span(b 1 , . . . , b n ) is given by the map

I(g) = n j=1 β j b j ,
with β is the solution of the system n j=1

β j b j (x i ) = g(x i ), i = 1, . . . , n.
Using similar arguments as in proposition 2.5, we can deduce the following result.

Proposition 2.6. The value

n j=1 z (1) j k(x j , x n+1 ) + I(f - n j=1 z (1)
j k j )(x n+1 )
coincides with f * (x n+1 ) given by [START_REF] De Boor | A practical guide to splines[END_REF]. In addition, the error is equal to

f (x n+1 ) -f * (x n+1 ) = f (x n+1 ) - n j=1 z (1)
j k(x j , x n+1 ) -I(f - n j=1 z (1)
j k j )(x n+1 ) = [b (-1) n+1 (f - n j=1 z (1)
j k j )][b n+1 (x n+1 ) -I(b n+1 )(x n+1 )].
If K is invertible and p l = k l with l = 1, . . . , n, then θ l = k (-1) l for l = 1, . . . , n, and the basis ( 14) is given by b l = k l with l = 1, . . . , n, and b n+1 =

k n+1 k(x n+1 ,x n+1 )-I(k n+1 )(x n+1 ) .

Constraint's effect on the kernel

From the notations above the general solution of the system (10) has the form

w = z (1) + n-q l=1 wl z l .
As a consequence the quadratic form

δ x n+1 - n i=1 w i δ x i 2 K = µ n+1-q - n-q l=1 wl µ l 2 K, with µ 1 = n i=1 z i1 δ x i , . . ., µ n-q = n i=1 z in-q δ x i , µ n+1-q = δ x n+1 -n i=1 z (1)
i δ x i , and the entries of the (n + 1 -q) × (n + 1 -q) kernel K are given by

k(l 1 , l 2 ) = (µ l 1 , µ l 2 ) K , l 1 , l 2 = 1, . . . , n + 1 -q.
Observe that K is positive definite if and only if the columns (p k (x 1 ), . . . , p k (x n )) , k = 1, . . . , q, are linearly independent and K is conditionally positive w.r.t. p 1 , . . ., p q . It follows that

sup{|f (x n+1 ) - n i=1 w i f (x i )| 2 : f K-1 f ≤ 1} = µ n+1-q - n-q l=1 wl µ l 2 K,
where f = ( f (1), . . . , f (n + 1 -q)) ∈ R n+1-q are defined by

f (1) = n i=1 z i1 f (x i ), . . . , f (n -q) = n i=1 z in-q f (x i ), f (n + 1 -q) = f (x n+1 ) - n i=1 z (1) i f (x i ).
The map f ∈ R {x 1 ,...,x n+1 } → f K-1 f is a semi kernel having the null space spanned by p 1 , . . ., p q . That being the case, the optimal weights w * are given by w * = arg min{ µ n+1-q -n-q l=1 wl µ l 2 K : w1 , . . . , wn-q ∈ R}, and then predict f (x n+1 ) is equal to

n i=1 z (1) i f (x i ) + n-q l=1 w * l ( n i=1 z il f (x i )).
The latter predictor coincides with [START_REF] De Boor | A practical guide to splines[END_REF]. Moreover, the spline

S( f ) = arg min f (n+1-q) { f K-1 f : f (1), . . . , f (n -q) are fixed} is such that S( f )(n + 1 -q) = f * (x n+1 ) - n i=1 z (1) i f (x i ) with f * (x n+1
) is the optimal prediction under the constraint [START_REF] De Boor | A practical guide to splines[END_REF].

From the expansion of f in the basis B = [b 1 , . . . , b n+1 ] (13), we can conclude the following result.

Proposition 2.7. If the weights w satisfy the constraint [START_REF] Schoenberg | Contributions to the problem of approximation of equidistant data by analytic functions, quart[END_REF], then

|f (x n+1 ) - n i=1 w i f (x i )| 2 = | n+1-q l=1 fl {b q+l (x n+1 ) - n i=1 w i b q+l (x i )}| 2 .
It follows that

sup{|f (x n+1 ) - n i=1 w i f (x i )| 2 : f K-1 f ≤ 1} = (b q+1 (x n+1 ) - n i=1 w i b q+1 (x i ), . . . , b n+1 (x n+1 ) - n i=1 w i b n+1-q (x i )) K-1 (b 1+q (x n+1 ) - n i=1 w i b 1+q (x i ), . . . , b n+1 (x n+1 ) - n i=1 w i b n+1 (x i ))
= (-w 1 , . . . , -w n , 1)R KR (-w 1 , . . . , -w n , 1) = (-w1 , . . . , -wn-q , 1) K(-w1 , . . . , -wn-q , 1)

= µ n+1-q - n-q l=1 wl µ l 2 K,
with the (n + 1)

× (n + 1 -q) matrix R = [b q+1 , . . . , b n+1 ].

Semi-kernel and constraint

Now, conversely we consider a semi-kernel Q on R {x 1 ,...,x n+1 } with the null space spanned by q functions p 1 , . . ., p q and let S(f ) = arg min{Q(f, f ) : f (x 1 ), . . . , f (x n ) are fixed} be the spline defined by the semi-norm Q, and

S(f )(x n+1 ) = n i=1 w * i f (x i ).
We consider a basis B = [b 1 , . . . , b n+1 ] such that b l = p l with l = 1, . . . , q and let (θ 1 , . . . , θ q , u 1 , . . . , u n+1-q ) be the coordinates of f , i.e.

f = q l=1 θ l p l + n+1-q l=1 u l b q+l .
It follows that

Q(f, f ) = n+1-q l 1 =1 n+1-q l 2 =1 u l 1 u l 2 Q(b q+l 1 , b q+l 2 ) = Q 1/2 u 2 ,
and the kernel Q =: [Q l 1 ,l 2 : l 1 , l 2 = 1, . . . , n + 1 -q] is invertible. If the weights w satisfy the constraint (10), then

|f (x n+1 ) - n i=1 w i f (x i )| 2 = | n+1-q l=1 u l {b l (x n+1 ) - n i=1 w i b l (x i )}| 2 , therefore, sup{|f (x n+1 ) - n i=1 w i f (x i )| 2 : Q(f, f ) ≤ 1} = sup{|f (x n+1 ) - n i=1 w i f (x i )| 2 : u Qu ≤ 1} = (b q+1 (x n+1 ) - n i=1 w i b q+1 (x i ), . . . , b n+1 (x n+1 ) - n i=1 w i b n+1-q (x i ))Q -1 (b 1+q (x n+1 ) - n i=1 w i b 1+q (x i ), . . . , b n+1 (x n+1 ) - n i=1 w i b n+1 (x i )) = (-w 1 , . . . , -w n , 1)RQ -1 R (-w 1 , . . . , -w n , 1) ,
where the (n + 1)

× (n + 1 -q) matrix R = [b q+1 , . . . , b n+1 ].

Stochastic approach

The statistical counterpart to the kernel interpolation is known as kriging (see e.g. [START_REF] Scheuerer | Interpolation of spatial data-a stochastic or a deterministic problem[END_REF]). It is based on the modeling assumption that (f (x 1 ), . . . , f (x n ), f (x n+1 )) is a realization of random vector Y x 1 , . . ., Y x n+1 over the same probability space (Ω, F, P). To predict Y x n+1 known Y x 1 , . . ., Y xn we need the mean and the covariance matrix of the random vector (Y x 1 , . . . , Y x n+1 ).

We assume that the mean (m(x 1 ), . . . , m(x n+1 )) (also called the trend) and the covariance function

k(x i , x j ) = cov(Y x i , Y x j ) of the random vector (Y x 1 , . . . , Y x n+1 ) exist. If Y x 1 , .
. ., Y xn are assumed to be known, then the best linear unbiased predictor (BLUP) of Y x n+1 is given by

Ŷx n+1 = n i=1 w * i Y x i ,
where the weights w * i are the solution of the following optimization problem

min{var(Y x n+1 - n i=1 w i Y x i ) : w 1 , . . . , w n ∈ R, n i=1 w i m(x i ) = m(x n+1 )}. ( 15
)
If the mean function m is modeled as

m(x i ) = q k=1 β k p k (x i ) : i = 1, . . . , n + 1,
and if we consider the weights such that n i=1 w i p l (x i ) = p l (x n+1 ), l = 1, . . . , q, then the optimal predictor

f (x n+1 ) = n i=1 w * i f (x i ) of f (x n+1
) in stochastic sense coincides with the interpolation [START_REF] De Boor | A practical guide to splines[END_REF].

Three kernel selection criteria

Kernel interpolation and prediction approaches are based on the knowledge of a symmetric positive definite matrix K and the trend p 1 , . . ., p q . To apply kernel interpolation it amounts to the assumption that one knows the degree of smoothness of the function f . In the context of partial differential equations, the function f belongs to some Sobolev space. In stochastic approach the covariance matrix and the trend are chosen using the maximum likelihood method or the Bayesian method.

Here we propose three natural criteria to compare two kernels K (1) and K (2) . Known f (x 1 ), . . ., f (x r ), we predict f (x r+1 ) using the kernel [k (l) (i, j) : i, j = 1, . . . , r], and we obtain the predictor f (l) (x r+1 ), with l = 1, 2, and r = 2, . . ., n -1. We propose the following three criteria to measure the performance of the Kernel K (l) :

1) M SP E(l) =:

n-1 j=1 |f (x j+1 )-f (l) (x j+1 )| 2 n-1
. We say that K (1) is better than

K (2) w.r.t. the MSPE criterion if M SP E(1) < M SP E(2). 2) M AXP E(l) =: max{|f (x j+1 ) -f (l) (x j+1 )| : j = 1, . . . , n -1}. We say that K (1) is better than K (2) w.r.t. the MAXPE criterion if M AXP E(1) < M AXP E(2).
3) We say that K (1) is statistically better than K (2) if

n-1 j=1 1 [|f (x j+1 )-f (1) (x j+1 )|<|f (x j+1 )-f 2) (x j+1 )|] n -1 > 1/2.
These criteria were also used in [?].

Application

In the climate change problem we are interested in the mean temperature f (t) at the time t.

The data are the years taken into account t 1 < . . . < t n+1 and the mean temperature f (t 1 ), . . ., f (t n ), and we are interested in the prediction of f (t n+1 ). We recall that arg min{

t n+1 t 1 |g (t)| 2 dt : g(t 1 ) = f (t 1 ), . . . , g(t n+1 ) = f (t n+1 ) are fixed}
is the natural C 2 cubic spline s which interpolates the points (t i , f (t i )), i = 1, . . . , n + 1. See [START_REF] Schoenberg | Contributions to the problem of approximation of equidistant data by analytic functions, quart[END_REF][START_REF] Reinsch | Smoothing by spline functions[END_REF]. We assume that f (t 1 ), . . ., f (t n+1 ) are the values of a natural C 2 cubic spline. We are going to predict f (t n+1 ) using three kernels, and we need some notations.

Kernel and semikernels using cubic splines

Let S = S 3 (t 1 , . . . , t n+1 ) be the set of C 2 cubic splines having the knots

t 1 < • • • < t n+1 .
Note that every element s ∈ S is a C 2 map on [t 1 , t n+1 ] and is a polynomial of degree three on each interval [t i , t i+1 ) for i = 1,. . . , n. More precisely, let

p 1 = s(t 1 ), . . . , p n+1 = s(t n+1 ), q 1 = s (t 1 ), . . . , q n+1 = s (t n+1 ), u 1 = s (t 1 ), . . . , u n+1 = s (t n+1 ), v 1 = s (t 1 +), . . . , v n = s (t n +)
be respectively the values of s and its derivatives up to order three at the knots. We have for every i = 1, . . . , n,

s(t) = p i + q i (t -t i ) + (t -t i ) 2 u i /2 + (t -t i ) 3 v i /6, t ∈ [t i , t i+1 ).
The following constraint for h i = t i+1 -t i , i = 1, . . . , n ensures the hypothesis that s is C 2 :

p i + q i h i + u i h 2 i /2 + v i h 3 i /6 = p i+1 , (16)
q i + u i h i + v i h 2 i /2 = q i+1 , ( 17
) v i = s (3) (t i ) = (u i+1 -u i )/h i . ( 18
)
It is well known (see [?]) that S has the dimension n + 3, and the set of natural spline S nat has the dimension n + 1. Hence an element s ∈ S (respectively s ∈ S nat ) is completely defined by n + 3 (respectively n + 1 parameters) independent parameters. Now we need to parametrize the set S in order to define properly an element s ∈ S. A parametrization of S is a one-to-one linear map ). The subscript notation 002 is justified by the fact that

Θ : s ∈ S → θ ∈ R n+3 .
p 1 = s(t 1 ) = s (0) (t 1 ), p 2 = s(t 2 ) = s (0) (t 2 ), u 1 = s (t 1 ) = s (2) (t 1 ), . . . , u n+1 = s (t n+1 ) = s (2) (t n+1 ).
See [START_REF] Dermoune | Moustaaid. Parametrizations, weights, and optimal prediction[END_REF][START_REF] Dermoune | Parametrizations, fixed and random effects[END_REF][START_REF] Dermoune | Estimation of noisy cubic spline using a natural basis[END_REF] Observe that span(b 002 1 , b 002 2 ) = span(1, t) with the column 1 = (1, . . . , 1) , t = (t 1 , . . . , t n+1 ) . We can show that

t n+1 t 1 |s (t)| 2 dt = n i=1 t i+1 t i |u i + t(u i+1 -u i )/h i | 2 dt = n i=1 (u 2 i + u i u i+1 + u 2 i+1 )h i /3 = (u 2 , . . . , u n )Q(u 2 , . . . , u n ) , (19)
with Q is a known n -1 × n -1 invertible matrix see [START_REF] Dermoune | Moustaaid. Parametrizations, weights, and optimal prediction[END_REF]. We also recall that

(u 2 , . . . , u n ) = U(p 1 , . . . , p n+1 ) ,
with U is a known n -1 × n + 1 matrix see [START_REF] Dermoune | Moustaaid. Parametrizations, weights, and optimal prediction[END_REF]

Now we propose the following predictors for f (t n+1 ). 0) We assume that s is Gaussian centred with the covariance matrix

K (0) = (Q (0) ) -1 with Q (0) is defined by t n+1 t 1 |s(t)| 2 dt = s Q (0) s.
1) We consider the spline S(f ) = arg min{(f (t 1 ), . . . , f (t n+1 ))P(f (t 1 ), . . . , f (t n+1 )) : f (t 1 ), . . . , f (t n ) are fixed},(23) defined by the kernel P (21) and the predictor f * (t n+1 ) = S(f )(t n+1 ) of f (t n+1 ). We assume that s is Gaussian with the mean p 1 b 002 1 (t) + p 2 b 002 2 (t) = β 1 1 + β 1 t and the covariance matrix K (1) = RQ -1 R with the kernel Q is given by (20). The predictor f (1) (t n+1 ) of f (t n+1 ) (12) using the kernel K (1) coincides with S(f )(t n+1 ).

2) We assume that s is Gaussian with the mean p 1 b 002 1 (t) + p 2 b 002 2 (t) = β 1 1 + β 1 t and the covariance matrix K (2) = RQR .

Let f (i) (t n+1 ) be the predictor of f (t n+1 ) (12) using the kernel K (i) with i = 0, 1, 2. Using real data, we are going to compare these three predictors.

Real data Application

As application in the climate change area we are interested in the annual mean temperature observed in France and Morocco from 1901 to 2015, the data are presented in Figure 1. We illustrate the importance of the kernel choice by considering the kernels K (0) , K (1) , K (2) . The three kernel selection criteria are presented in Table 1. The mean annual temperature of the year 2015 and 2016 (i.e. f (i) (t n ) and f (i) (t n+1 ), n = 114) are given in Tables (2, 3), as for Figure 2 it shows the splines of the predictors f (0) , f (1) , f (2) and the true temperature. The w * 1 , . . ., w * n of (12) for the kernels K (0) , K (1) , K (2) are presented in Figure 3. K (0) with 0.8288288 for K (1) and 0.8288288 for K (2) and 0.8468468 for K (2) Table 1: The three kernel selection criteria for the kernels K (0) ,K (1) , K (2) using Morroco and France data.

K (2) K (0) K (1) K (

  Defining a parametrization Θ is equivalent to the existence of the basis B = (b 1 , . . . , b n+3 ) of S such that, for all s ∈ S, s = n+3 i=1 θ i b i = Bθ. The parametrization Θ 002 = (p 1 , p 2 , u 1 , . . . , u n+1 ) defines the basis B 002 = (b 002 1 , . . . , b 002 n+3

Figure 1 :

 1 Figure 1: Mean annual temperatures in France and Morocco from 1901 to 2015.

Figure 2 :

 2 Figure 2: The splines of the predictors f (0) , f (1) , f (2) and the true temperature.

  for more details. It follows for s ∈ S nat that

	n
	s = p 1 b 002 1 + p 2 b 002 2 +	u i b 002 2+i ,
	i=2
	and then s = (s(t 1 ), . . . , s(t n+1 ) is given by
	n
	s = [b 002 1 (t), b 002 2 (t)](p 1 , p 2 ) +	R(u 2 , . . . , u n ) ,
	i=2
	Here the column b 002 i (t) = (b 002 i (t 1 ), . . . , b 002 i (t n+1 )) , with i = 4, . . . , n + 2, and the n +
	1 × n + 1 matrix
	R = [b 002 4 (t), . . . , b 002 n+2 (t)].

  . Therefore (u 2 , . . . , u n )Q(u 2 , . . . , u n ) = (p 1 , . . . , p n+1 )U QU(p 1 , . . . , p n+1 ) (20) =: (p 1 , . . . , p n+1 )P(p 1 , . . . , p n+1 ) (21)

Table 2 :

 2 The predictors f (i) (t n ), n = 114 (the mean annual temperature of the year 2015). Prediction 12.91553 12.54049 11.40698 17.86737 18.99740 18.49113

	2)

Table 3 :

 3 The predictors f (i) (t n+1 ), n = 114 (the mean annual temperature of the year 2016).

Remark 5.1. Table 1 shows that the kernel K (0) wins against K (1) and K (2) with respect to the three kernel selection criteria. 12) for the kernels K (0) , K (1) , K (2) .

Concluding remarks

The numerical results shows the three kernel selection criteria are stable, form Table 1 we have that the best kernel among the three kernels is K (0) w.r.t. all the three criteria for both France and Morocco data. Moreover, the representation of the splines (Figure 2) shows that too.

From Table 1 and Figure 2 we have that the kernel K (1) wins against K (2) . Considering the second derivative (u 2 , . . . , u n ) as Gaussian with the covariance matrix Q -1 is a good stochastic modelization, at least is better than the assumption that (u 2 , . . . , u n ) as Gaussian with the covariance matrix Q. Equivalently measuring the worst error in the unit ball using the norm Q 1/2 u is better than the norm Q -1/2 u .