Yuri Goegebeur
email: yuri.goegebeur@imada.sdu.dk

Armelle Guillou
email: armelle.guillou@math.unistra.fr

Jing Qin
email: qin@imada.sdu.dk

On kernel estimation of the second order rate parameter in multivariate extreme value statistics

Keywords: Multivariate extreme value statistics, second order parameter, stable tail dependence function

We introduce a flexible class of kernel type estimators of a second order parameter appearing in the multivariate extreme value framework. Such an estimator is crucial in order to construct asymptotically unbiased estimators of dependence measures, as e.g. the stable tail dependence function. We establish the asymptotic properties of this class of estimators under suitable assumptions. The behaviour of some examples of kernel estimators is illustrated by a simulation study in which they are also compared with a benchmark estimator of a second order parameter recently introduced in the literature.

Introduction

Measuring the strength of the dependence in the extremes is a challenging topic which has received considerable attention in the recent multivariate extreme value literature. Several tools can be used, either some coefficients of tail dependence or some functions, such as the Pickands Now, consider a sample of size n drawn from F and an intermediate sequence k = k n , i.e. k → ∞ as n → ∞ with k/n → 0. Let us denote X (j) k,n the k-th order statistic among n realisations of the margins X (j) , j = 1, ..., d. The empirical estimator of L is then given by

L k (x) = 1 k n i=1 1l {X (1)
i ≥X

(1) n-[kx 1 ]+1,n or ... or X (d) i ≥X

n-[kx d ]+1,n } .

Recently the poor performance of this empirical estimator in terms of bias has been emphasized in the literature. This bias-issue is common in extreme value statistics, e.g. it is also present in the univariate and the regression contexts, and generally it complicates the practical application of extreme value methods. All the contributions dealing with bias reduction in the multivariate framework (see [START_REF] Fougères | Bias correction in multivariate extremes[END_REF][START_REF] Beirlant | Bias-corrected estimation of stable tail dependence function[END_REF], Escobar-Bach et al., 2016) require the following second or third order conditions, depending on the type of asymptotic properties that one wants to establish.

Second order condition: There exist a positive function α such that α(t) → 0 as t → ∞ and a non null function M such that for all x with positive coordinates

lim t→∞ 1 α(t) t 1 -F F ← 1 (1 -t -1 x 1 ), ..., F ← d (1 -t -1 x d ) -L(x) = M (x), (2)
uniformly on any [0, T ] d for T > 0.

The second order condition implies that the function α is regularly varying at infinity of index ρ ≤ 0, i.e. α(ty)/α(t) → y ρ as t → ∞ for all y > 0, and M (x) is homogeneous of order 1 -ρ.

Third order condition: There exist a positive function β such that β(t) → 0 as t → ∞ and a non null function N such that for all x with positive coordinates

lim t→∞ 1 β(t) t 1 -F F ← 1 (1 -t -1 x 1 ), ..., F ← d (1 -t -1 x d ) -L(x) α(t) -M (x) = N (x), (3)
uniformly on any [0, T ] d for T > 0. Also N is not a multiple of M .

It can be shown that the third order condition implies that β is regularly varying at infinity of order ρ ≤ 0 and that N is homogeneous of order 1 -ρ -ρ .

The asymptotically unbiased estimators of the stable tail dependence function L proposed in the abovementioned recent papers depend on the second order parameter ρ, which has to be estimated from the sample. The problem of estimating second order parameters is also present in the univariate framework, but in that context, several estimators have already been proposed

in the literature and they perform reasonably well in practice, although not very stable as a function of the intermediate sequence k used to compute them. We refer for instance to Fraga an obstacle to overcome in order to obtain the bias-correction. Thus, although these papers introduce estimators for ρ, the performance of these estimators has not been studied in detail.

Until now, due to the difficulty of estimating this parameter ρ, it is often suggested in practice to replace it by a canonical value like e.g. -1 (see Escobar-Bach et al., 2016). However, a mis-specification of this parameter implies that from a theoretical perspective one loses the bias correction. Thus, similarly to the univariate context, it is an important challenge to be able to estimate ρ in the multivariate context. To reach this goal, we introduce in Section 2 a flexible class of kernel estimators for ρ, for which we derive the asymptotic normality under suitable assumptions, in particular the third order condition mentioned above. Our estimator depends on some tuning parameters that we have to select. For some specific values of them, our estimator encompasses the one proposed in [START_REF] Beirlant | Bias-corrected estimation of stable tail dependence function[END_REF]. In Section 3, the performance of our estimator is compared with a benchmark from the recent literature. All the proofs of the results are postponed to the appendix.

Kernel estimator and asymptotic properties

Motivated by the homogeneity property of the function L, we introduce the following scaled estimator

L k,a (x) := a -1 L k (ax)
for a positive parameter a. The basic building block for the ρ estimator will be the following kernel statistic

L k (x; K, ξ) := 1 k k j=1 K(a j ) L ξ k,a j (x)
where a j := j k+1 , j = 1, ..., k, ξ ∈ N, and K is a function defined on [0, 1] which is positive and such that 1 0 K(u)du = 1. This function is called a kernel function in the sequel. As a first step in the development of the estimator we derive the weak limit of L k (x), when properly normalised. Let {e 1 , . . . , e d } be the canonical basis vectors of R d . Theorem 1. Let X 1 , ..., X n be independent random vectors in R d with common joint df F and continuous marginal dfs F j , j = 1, ..., d. Assume that the third order condition (3) holds with negative indices ρ and ρ and that the first order partial derivatives of L, say ∂ j L, exist and that

∂ j L is continuous on the set of points {x ∈ R d + : x j > 0}.
Suppose further that the function M is continuously differentiable and N continuous. Assuming that the intermediate sequence k

satisfies √ kα(n/k) → ∞ and √ kα(n/k)β(n/k) → λ 1 ∈ R, we have √ k L k (x) -L(x) -α n k M (x) -α n k β n k N (x) d -→ Z L (x) in D([0, T ] d ) for every T > 0, where Z L (x) := W L (x) - d j=1 W L (x j e j )∂ j L(x),
with W L a continuous centered Gaussian process with covariance structure

E[W L (x)W L (y)] = µ{R(x) ∩ R(y)} where R(x) := {u ∈ R d + :
there exists j such that 0 ≤ u j ≤ x j } and µ is the measure defined as µ {R(x)} := L(x).

Note that we assume

√ kα(n/k)β(n/k) → λ 1 ∈ R, while Fougères et al. (2015) took λ 1 = 0.
Our assumption will allow us to study the estimation of ρ in a more general context. In particular we can make the bias term explicit, unlike [START_REF] Fougères | Bias correction in multivariate extremes[END_REF], where no information about the bias of the estimator of ρ is available.

Based on the result of Theorem 1 we can now establish the weak limit of the kernel statistic

L k (x; K, ξ). Let I(K, r) := 1 0 K(u)u -r du. Theorem 2. Let K be a continuous kernel function satisfying          I (K, (ξ + 1)/2) < ∞, 1 k k j=1 K(a j ) = 1 + o 1 √ k , 1 k k j=1 K(a j )a -ρ j = I(K, ρ) + o 1 √ k .
Suppose that the conditions of Theorem 1 hold together with

√ kα 2 n k → λ 2 ∈ R. Then, we have √ k L k (x; K, ξ) -L ξ (x) -α n k ξL ξ-1 (x)M (x)I(K, ρ) -α n k β n k ξL ξ-1 (x)N (x)I(K, ρ + ρ ) -α 2 n k ξ(ξ -1) 2 L ξ-2 (x)M 2 (x)I(K, 2ρ) d -→ ξL ξ-1 (x) 1 0 K(u) 1 u Z L (ux)du in D([ε, T ] d ) for every ε > 0 and T > ε. Definition For (ξ 1 , ξ 2 ) ∈ N × N, a, r ∈ (0, 1), let ρ k (x; K, ξ 1 , ξ 2 ) := 1 - 1 log r log ∆ k,a (rx; K, ξ 1 , ξ 2 ) ∆ k,a (x; K, ξ 1 , ξ 2 ) ∧ 0 where ∆ k,a (x; K, ξ 1 , ξ 2 ) := 1 a ξ 1 L k (ax; K, ξ 1 ) 1 ξ 1 -L k (x; K, ξ 2 ) 1 ξ 2 .
The weak convergence of this estimator for ρ is established in the next theorem.

Theorem 3. Assume the conditions of Theorem 2 hold, M never vanishes except on the axes, and consider real numbers a and r both in (0, 1). Then we have

√ kα n k ρ k (x; K, ξ 1 , ξ 2 ) -ρ + I(K, ρ + ρ ) I(K, ρ) N (x) M (x) a -ρ-ρ -1 a -ρ -1 r -ρ -1 log r β n k + I(K, 2ρ) I(K, ρ) -I(K, ρ) ξ 1 -1 2 a -2ρ -ξ 2 -1 2 a -ρ -1 M (x) L(x) r -ρ -1 log r α n k d -→ - 1 (a -ρ -1)M (x)I(K, ρ) log r 1 a 1 0 K(u) Z L (uarx) u du - 1 0 K(u) Z L (urx) u du r 1-ρ - 1 a 1 0 K(u) Z L (uax) u du + 1 0 K(u) Z L (ux) u du , in D([ε, T ] d ),
for every ε > 0 and T > ε.

Note that in the particular case where ξ 1 = ξ 2 = 1, we recover the estimator proposed by • the logistic model, for which L(x, y) = (x 1/s + y 1/s ) s and ρ = -1. We set s = 1/3;

• the bivariate Pareto of type II model, called BPII(p), for which L(x, y) = x+y-(x -p +y -p ) -1/p . We set p = 4 which corresponds to ρ = -0.5;

• the bivariate Cauchy distribution, for which L(x, y) = (x 2 + y 2 ) 1/2 and ρ = -2.

In this simulation study, we compare our new class of kernel estimators for ρ with the benchmark estimator already proposed in the literature by [START_REF] Fougères | Bias correction in multivariate extremes[END_REF] and defined as

ρ k (x) := 1 - 1 log r log ∆ k,a (rx) ∆ k,a (x) ∧ 0 where ∆ k,a (x) := a -1 L k (ax) -L k (x)
, with advocated values r = a = 0.4 for practical use.

Our class of estimators depends on several tuning parameters that we have to select in practice.

An extensive simulation study leads us to the following conclusions:

• the performance of our estimator, whatever the distribution, is almost the same for all the pairs of ξ i , i = 1, 2, used. To keep the length of the paper under control, we only report the results when (ξ 1 , ξ 2 ) = (1, 1) (which coincides with the estimator introduced in Beirlant et al., 2016) and (4, 4);

• concerning the kernel, we tried also different types of families, all satisfying the conditions of our Theorem 2, but again we show only the results for the power kernel, K τ (t) := (τ + 1)t τ 1l {t∈[0,1]} , where I(K τ , r) = 1+τ 1+τ -r and we set τ = 2 and 10; • concerning a and r, as in [START_REF] Fougères | Bias correction in multivariate extremes[END_REF], the value 0.4 for both parameters seems overall best, although in some particular cases, a larger value, say 0.5, is better. corresponding to the mean value computed at k = 990 to represent the performance of this estimator. Since this mean value is sometimes far away from the true one, the corresponding MSE may not appear on the right-hand side of the figures in view of the scale used. The values τ = 2 or 10 seem to give always good results with a slightly better performance to τ = 10 for some distributions. When ρ = -1, our estimator has almost no bias and is very stable as a function of k. When ρ = -1/2 some bias appears, but this is expected since it is well-known in the extreme value literature, that bias occurs in case ρ ∈ (-1, 0], making estimation practically difficult. Now if ρ < -1, a bias also appears, whatever the value of τ , though for large values of k, which is the range to consider, the estimator gets close to the true value. This bias is not usual but due to the value of the pair (a, r) which is not appropriate to that framework. Indeed, as illustrated in Figure 2 for this distribution, a = r = 0.5 leads to less bias than in case a = r = 0.4, but an MSE more or less at the same level. On the contrary, for distributions with ρ = -1 the choice a = r = 0.5 leads to more bias and a larger MSE compared to a = r = 0.4.

This motivates why we essentially illustrate the performance of our estimator in case a = r = 0.4 which is overall the best choice.

To conclude, similarly to the univariate framework, no particular values of the tuning parameters are best for the whole parameter space. This has been already observed by Fraga [START_REF] Alves | A new class of semi-parametric estimators of the second order parameter[END_REF] and [START_REF] Goegebeur | Kernel estimators of the second order parameter in extreme value statistics[END_REF]. However, on the contrary to the univariate context, our class of estimators for ρ with the specific values of the parameters considered in this section seems to have nice bias properties with rather stable sample paths as a function of k, which alleviates the choice of k to some extent, and it outperforms the benchmark estimator, ρ k . Indeed, due to the large variability of ρ k in some cases, one has that ρ k is usually better in terms of MSE.

Appendix: Proofs of the results

Proof of Theorem 1. Let U (j) i , j = 1, ..., d, be uniform random variables defined as

U (j) i := 1 -F j (X (j) i ) for j = 1, ..., d. Introduce now V k (x) := 1 k n i=1 1l {U (1) i ≤kx 1 /n or ... or U (d) i ≤kx d /n} .
Then, we can use the following decomposition

L k (x) -L(x) -α n k M (x) -α n k β n k N (x) = V k n k U (1)
[kx 1 ],n , ...,

n k U (d) [kx d ],n - n k 1 -F F ← 1 1 -U (1)
[kx 1 ],n , ..., F ← d 1 -U (d) [kx d ],n + n k 1 -F F ← 1 1 -U (1)
[kx 1 ],n , ..., F ← d 1 -U (d) [kx d ],n -L n k U (1)
[kx 1 ],n , ...,

n k U (d) [kx d ],n -α n k M n k U (1)
[kx 1 ],n , ...,

n k U (d) [kx d ],n -α n k β n k N n k U (1)
[kx 1 ],n , ...,

n k U (d) [kx d ],n +L n k U (1)
[kx 1 ],n , ...,

n k U (d) [kx d ],n -L(x) +α n k M n k U (1)
[kx 1 ],n , ...,

n k U (d) [kx d ],n -M (x) +α n k β n k N n k U (1)
[kx 1 ],n , ...,

n k U (d) [kx d ],n -N (x) =: A 1,k (x) + A 2,k (x) + A 3,k (x) + A 4,k (x) + A 5,k (x).
We have to study each term separately. According to the proof of Proposition 1 in Fougères et al. ( 2015), we have

√ kA 1,k (x) d -→ W L (x) (4)
√ kA 3,k (x) d -→ - d j=1 W L (x j e j )∂ j L(x) (5)
√ k n k U (j) [kx],n -x + W L (xe j ) -→ 0 a.s. (6)
Under the assumptions on the function M , this implies that sup

0≤x 1 ,...,x d ≤T √ k M n k U (1)
[kx 1 ],n , .

[kx d ],n -M (x) + d j=1 W L (x j e j )∂ j M (x) → 0 a.s. Consequently sup 0≤x 1 ,...,x d ≤T √ kA 4,k (x) P -→ 0. .., n k U (d)
Similarly using the continuity of the function N and (6), we have sup

0≤x 1 ,...,x d ≤T N (x) -N n k U (1)
under the assumption that √ kα( n k )β( n k ) → λ 1 . The last term that we have to consider is A 2,k (x). According to the third order condition and (6), we have

√ kA 2,k (x) = o √ k α n k β n k = o(1) a.s. ( 9
)
uniformly on [0, T ] d . Combining (4), ( 5), ( 7), ( 8) and ( 9), Theorem 1 follows.

Proof of Theorem 2. From Theorem 1, and by using the Skorohod construction, we have

L k (x) = L(x) + α n k M (x) + α n k β n k N (x) + Z L (x) √ k + o 1 √ k ,
where the o-term is a.s. uniform on [0, T ] d . Using again the homogeneity of the functions L, M, N, we deduce that

L k,a j (x) = L(x) + α n k a -ρ j M (x) + α n k β n k a -ρ-ρ j N (x) + 1 √ ka j Z L (a j x) + 1 a j o 1 √ k .
Now, by a Taylor series expansion, we have

1 k k j=1 K(a j ) L ξ k,a j (x) = L ξ (x) 1 k k j=1 K(a j ) + α n k ξL ξ-1 (x)M (x) 1 k k j=1 K(a j )a -ρ j +α n k β n k ξL ξ-1 (x)N (x) 1 k k j=1 K(a j )a -ρ-ρ j + ξL ξ-1 (x) √ k 1 k k j=1 K(a j ) 1 a j Z L (a j x) + R (1) n (x) + R (2) n (x) + o 1 √ k ,
where

R (1) n (x) := ξ(ξ -1) 2 L ξ-2 (x) 1 k k j=1 K(a j )y 2 j , R (2) n (x) := ξ(ξ -1)(ξ -2) 6 1 k k i=1
K(a j ){L(x) + θy j } ξ-3 y 3 j , with θ ∈ (0, 1),

y j := α n k a -ρ j M (x) + α n k β n k a -ρ-ρ j N (x) + 1 √ ka j Z L (a j x) + 1 a j o 1 √ k .
First we establish the convergence to zero of

∆ k := 1 k k j=1 K(a j ) Z L (a j x) a j - 1 0 K(u) Z L (ux) u du .
To this aim, it is instructive to realise that {W L (ax)/ L(x); 0 ≤ a ≤ 1} is a Wiener process.

Using the fact that the sample paths of a Wiener process are a.s. Hölder continuous of order γ < 1/2, and by Lévy's global modulus of continuity, one has for any ε > 0, T > ε and

δ ∈ (0, 1/2) that sup x∈[ε,T ] d ,(a,b)∈[0,1] 2 W L (ax) √ L(x) -W L (bx) √ L(x) |a -b| 1/2-δ ≤ C a.s. , (10)
where C is a positive constant.

We have, with ãj being a value between a j and a j+1 , j = 1, . . . , k, that

∆ k ≤ 1 k + 1 k j=1 K(a j ) Z L (a j x) a j -K(ã j ) Z L (ã j x) ãj + 1/(k+1) 0 K(u) Z L (ux) u du + 1 k + 1 1 k k j=1 K(a j ) Z L (a j x) a j =: T 1 + T 2 + T 3 .
Concerning T 1 , we easily establish

T 1 ≤ 1 k + 1 k j=1 K(a j ) 1 a j |Z L (a j x) -Z L (ã j x)| + 1 k + 1 k j=1 K(a j ) 1 a j -K(ã j ) 1 ãj |Z L (ã j x)| =: T 1,1 + T 1,2 .
Now, by the definition of Z L (x) and using the fact that ∂ i L(x) is homogeneous of order zero, we obtain the inequality

T 1,1 ≤ 1 k + 1 k j=1 K(a j ) 1 a j |W L (a j x) -W L (ã j x)| + d i=1 ∂ i L(x) k + 1 k j=1 K(a j ) 1 a j |W L (a j x i e i ) -W L (ã j x i e i )| =: T 1,1,1 + T 1,1,2 .
Using (10), we have for some small ι > 0,

T 1,1,1 ≤ C (k + 1) ι 1 k + 1 k j=1 K(a j )a -1/2-δ-ι j ,
where C > 0 is a constant, and hence we conclude T

1,1,1 = o(1) a.s. uniformly in x ∈ [ε, T ] d .
Using the continuity of ∂ i L(x), we obtain in the same way T 1,1,2 = o(1) a.s. uniformly in

x ∈ [ε, T ] d .
For T 1,2 we write

T 1,2 ≤ 1 k + 1 k j=1 K(a j ) 1 a j - 1 ãj |Z L (ã j x)| + 1 k + 1 k j=1 |K(a j ) -K(ã j )| 1 ãj |Z L (ã j x)| =: T 1,2,1 + T 1,2,2 .
Again using (10), and with arguments similar to those used for T 1,1 , we obtain T 1,2,1 = o(1) a.s. Proof of Theorem 3. The result can be established by using Theorem 2, the Skorohod representation, the homogeneity of the functions L, M, N , (10), and several Taylor series expansions.

  Alves et al. (2003), Ciuperca and Mercadier (2010) and Goegebeur et al. (2010), among others. On the contrary, in the multivariate context this topic is still in its infancy. We are only aware of the estimators proposed by Fougères et al. (2015) and Beirlant et al. (2016). However, these papers are mainly focused on bias-corrected estimation of L, and the estimation of ρ is only

  [START_REF] Beirlant | Bias-corrected estimation of stable tail dependence function[END_REF] but we provide its asymptotic normality in Theorem 3 under more general assumptions which allow us to exhibit the bias of our estimator. Also, we provide here the result in terms of weak convergence of a stochastic process in x, whereas Beirlant et al. (2016) only provide a pointwise convergence result. Remark that our asymptotic limit does not depend on the values of ξ 1 and ξ 2 .3 Simulation studyTo assess the performance of our class of estimators in practice, we simulate B = 1000 samples of size n = 1000 from several distributions originally proposed by[START_REF] Fougères | Bias correction in multivariate extremes[END_REF] and used in[START_REF] Beirlant | Bias-corrected estimation of stable tail dependence function[END_REF]. To keep the length of the paper reasonable, we only include the following ones:

Figure 1

 1 Figure 1 displays the results for the logistic and BPII(4) distributions, in case of the particular choices a = r = 0.4, (ξ 1 , ξ 2 ) = (1, 1) (first and third rows) or (4, 4) (second and fourth rows), whereas Figure 2 shows similar plots for the bivariate Cauchy distribution in case a = r = 0.4 and 0.5 and the same pairs for (ξ 1 , ξ 2 ). The estimators for ρ were calculated at x = (0.5, 0.5). Also other values of x have been tried, but it turns out that the estimation results are not very sensitive with respect to the choice of this position. Each time, our class of estimators is computed with the power kernel with τ = 2 and 10. The left panel of the figures represents the mean of the estimators over the B samples as a function of k, whereas the right panel is the mean squared error (MSE), also as a function of k. On the left panel, the horizontal reference line (solid black line) indicates the true value of ρ. Due to the high volatility of the estimator ρ k , we follow the advices of Fougères et al. (2015) by drawing a horizontal line (solid grey line)

  in D([0, T ] d ) for every T > 0. Now according to the proof of Theorem 7.2.2 in de Haan and Ferreira (2006), we have sup x∈[0,T ]

[kx 1 ] 1

 11 ,...,x d ≤T √ kA 5,k (x) -→ 0 a.s.

,

  uniformly in x ∈ [ε, T ] d . Concerning T 1,2,2 , we use the fact that K is uniformly continuous over [0, 1] and obtain, for k large enoughT 1,2,2 ≤ ω k + 1 k j=1 1 ãj |Z L (ã j x)|,where ω > 0 can be chosen arbitrary small, and hence, again using (10) we have T 1,2,2 = o(1)a.s. uniformly in x ∈ [ε, T ] d .Also T 2 and T 3 can be analysed in a similar way, and allow us toconclude ∆ k = o(1) a.s. uniformly in x ∈ [ε, T ] d as k → ∞.After tedious calculations one can establish, provided ξ ≥ 2,R (1) n (x) = α 2 n k ξ(ξ -1) 2 L ξ-2 (x)M 2 (x)I(K, 2ρ) + o P 1 √ k ,where theo P term is uniform in x ∈ [ε, T ] d . Concerning R(2)n (x), the cases ξ = 1 and ξ = 2, are trivial since R(2)n (x) is exactly zero. In case ξ = 3, some straightforward but tedious computations lead to R(2)n (x) = o P 1 √ k , uniformly in x ∈ [ε, T ] d .Finally, if ξ ∈ N such that ξ > 3, then for n sufficiently large, we can use the following bound,|L(x) + θy j | ξ-3 ≤ (L(x) + |y j |) ξ-3where C is independent from j and x, which allows us to deduce that R |y j | 3 = o P 1 √ k uniformly in x ∈ [ε, T ] d . This achieves the proof of Theorem 2.

Acknowledgement

The authors thank the editor and the reviewers for their helpful comments which led to improvements of our first version of the paper. This work was supported by a research grant (VKR023480) from VILLUM FONDEN and an international project for scientific cooperation (PICS-6416).


Julien Henriet

Christophe Lang

Laurent Philippe

Femto-St Disc -Equipe Cartoon

AI-VT: an educative system based on artificial intelligence and designed to produce varied and consistent training lessons

Keywords: Educative system, sports training, artificial intelligence, distributed case-based reasoning, multi-agent system

AI-Virtual Trainer (AI-VT

) is an educative system based on Artificial Intelligence (AI) that proposes varied lessons to trainers. The agents of this multiagent system apply case-based reasoning to build solutions by analogy. However, as required by the field, AI-VT never proposes the same lesson twice, whereas the same objective may be posed many times consecutively. The adaptation process of AI-VT delivers an ordered set of exercises adapted to the objectives and sub-objectives chosen by trainers. This process has been enriched by including the notion of distance between exercises: the proposed tasks are not only appropriate but are hierarchically ordered. With this new version of the system, students are guided towards their objectives via an underlying theme. Finally, the agents responsible for the different parts of lessons collaborate with each other according to a dedicated protocol and decision-making policy since no exercise must appear more than once in the same lesson. The results prove that AI-VT, however perfectible, meets the requirements of this field.

Introduction

Motivation is a key aspect in teaching. Repetitivity causes students to be bored and to "turn themselves off" with respect to lessons. Nevertheless, repetitivity is of the utmost importance in acquiring skills. In addition, teaching cannot be reduced to a set of exercises. The exercises must be sorted according to complexity and each exercise must prepare students for the next one. At the same 1 time, it is necessary to treat all the aspects of a field, to introduce originality and diversity, and to adapt the exercise level and nature to the students' already acquired skills. This is particularly true in sports, where trainers must propose varied exercises and consistent lessons while having to train for the same skill over a given number of weeks.

This paper presents AI-Virtual Trainer (AI-VT), our education system based on Artificial Intelligence (AI). AI-VT is a Multi-Agent System (MAS) that uses Case-Based Reasoning (CBR) to provide consistent lessons with widely differing progressions. CBR is a problem solving method that adapts solutions from similar known problems in order to solve new problems [START_REF] Kolodner | Case-based reasoning[END_REF]. CBR is widely employed in e-learning systems and Intelligent Tutoring Systems (ITS) [START_REF] Graesser | Intelligent tutoring systems. Am Psych Ass[END_REF]. J. L. Kolodner [START_REF] Kolodner | Case-based reasoning-inspired approaches to education[END_REF] distinguished between two types of CBR-inspired approaches to education: Goal-Based Scenarios [START_REF] Schank | The design of goal-based scenarios. Taylor & Francis[END_REF] where learners achieve missions in simulated worlds thus confronting themselves with the real world, and Learning By Design [START_REF] Kolodner | Case-based learning aids[END_REF] in which learners design and build working devices to obtain feedback. CBR is actually well-suited to the latter type of system [START_REF] Jamsandekar | Online Learning -CBR approach[END_REF], as well as to other tools from AI [START_REF] Biswas | Learning by teaching: A new agent paradigm for educational software[END_REF] and Distributed AI (DAI) systems [START_REF] Gisolfi | Designing complex systems within distributed architectures: an intelligent tutoring systems perspective[END_REF] such as Genetic Algorithm (GA) [START_REF] Tan | Personalized course generation and evolution based on genetic algorithms[END_REF], Artificial Neural Network (ANN) [START_REF] Baylari | Design a personalized e-learning system based on item response theory and artificial neural network approach[END_REF] and MAS [START_REF] Rishi | Agent based student modeling in distributed CBR based intelligent tutoring system[END_REF]. A. Baylari and G. A. Montazer focused on the adaptation of tests to obtain a personalised estimation of a student's level [START_REF] Baylari | Design a personalized e-learning system based on item response theory and artificial neural network approach[END_REF]. They used an ANN in order to correlate student answers to the tests and the exercises proposed by teachers. The CBR and GA based e-learning system proposed by Huang et al. also provides lessons taking into account the curriculum and the incorrect response patterns of a pre-test given to the learner [START_REF] Huang | Constructing a personalized e-learning system based on genetic algorithm and case-based reasoning approach[END_REF]. O. P. Rishi et al. designed an ITS based on agents and a CBR system [START_REF] Rishi | Distributed case based reasoning for intelligent tutoring system: an agent based student modeling paradigm[END_REF] in which a Personal Agent is responsible for determining student level. A Teaching Agent then determines the educative strategy with the help of CBR regarding the description of the transmitted student level. Finally, a Course Agent provides and revises the lessons and exercises corresponding to the strategy proposed by the system with the help of a tutor. All these tools provided by AI, would nevertheless produce exactly the same exercises and lessons to train for a single given skill, or would propose a large set of exercises as an answer to the diversity constraint, and leave the teachers or the students to choose the most adapted exercises themselves. In this particular domain, repetitive activities are a drawback, yet lesson planning is a process based on adaptation of past experiences.

Many studies have highlighted the advantages of applying these AI concepts to sports training and physical activities [START_REF] Bartlett | Artificial intelligence in sports biomechanics: new dawn or false hope?[END_REF], [START_REF] Owusu | AI and computer-based methods in performance evaluation of sporting feats: an overview[END_REF], [START_REF] Balague | Overview of complex systems in sport[END_REF], [START_REF] Fister | Computational intelligence in sports: Challenges and opportunities within a new research domain[END_REF]. According to their classifications, AI-VT is an expert and knowledge-based system that aims at generating plans for training sessions according to agreed training strategies. Like MOPET [START_REF] Buttussi | MOPET: A context-aware and user-adaptative wearable system for fitness training[END_REF], MAS are often used in order to collect and analyse realtime data capted by devices worn by athletes. Mathematical models analyse movements and help athletes to optimise their techniques [START_REF] Mcnitt-Gray | Using technology and engineering to facilitate skill acquisition and improvements in performance[END_REF]. Ambient intelligence systems have also been used in order to adapt training sessions [START_REF] Vales-Alonso | Ambient intelligence systems for personalized sport training[END_REF]: J. Vales-Alonso et al. present a prototype for cross country running scenarios that maintains the heart rates of runners within target ranges. This system's decision tool is based on spline interpolations that take environmental parame-ters (temperature, track hardness) and the athlete's heart rates as parameters in order to select the next track in the running scenario. Mathematical models based on past experiences (polynomial interpolators) have been developed by I. Rygula and J. Cholewa in order to optimise the parameters of training loads [START_REF] Rygula | Application of a mathematical model in control of sport training[END_REF]. Maszczyk et al. explored the possibility to use ANN in order to predict performance of javelin throwers [START_REF] Maszczyk | A neural network model approach to athlete selection[END_REF]. A. Erdogan et al. also used ANN interpolations in order to predict the heart rate at the onset of blood lactate accumulation due to external parameters [START_REF] Erdogan | Non-invasive detection of the anaerobic threshold by a neural network model of the heart rate-work rate relationship[END_REF]. Actually, each interpolation tool of these systems (ANN, spline and polynomial) optimises one particular aspect of the athlete's training, whereas AI-VT is designed to deal with many objectives and many abilities simultaneously.

The particular fields of sports training and physical education must take into account the component exercises in lessons in order to help maintain motivation among athletes/students. Many exercises are stored in the AI-VT knowledge base. Each exercise may be linked to one or many sub-objectives (or to none at all), and each sub-objective is linked to one or many objectives. A lesson has a single objective divided into sub-objectives, with each sub-objective divided into exercises. Each agent in this system is in charge of proposing exercises for one sub-objective and must take into consideration the exercises proposed by the other agents. This introduces another difficulty since an exercise must not be proposed more than once in the same lesson. This new version of AI-VT introduces the notion of distance and order of exercises in the resulting lesson. As highlighted by the results of our previous investigations [START_REF] Henriet | Collaborative CBR-based agents in the preparation of varied training lessons[END_REF], each new exercise must be as close as possible to the preceding and following ones. An underlying theme must be evident and hierarchical difficulty must be taken into consideration in the sorted set of exercises. Thus, solutions proposed by all the agents must be chosen collectively, taking into account the training history (the previous lessons proposed to the athlete during the season) as well as the solutions proposed by all the other agents. Therefore, AI-VT agents are collaborative entities.

In the next section, we present the purpose, the requirements, the knowledge structures and the methods involved our distributed system AI-VT. System implementation and performance are then presented and analysed.

Materials and Methods

In the first part of this section, we detail the requirements and the lesson structure of our distributed system. The distributed architecture and the data flows are examined in the second part. Finally, in the third part, we present how a lesson is designed theoretically.

Lesson structure

We considered physical activities for which training lessons are guided by the objectives [START_REF] Tyler | Statistical Methods for Utilizing Personal Judgements to Evaluate Activities for Teacher-Training Curricula[END_REF]. Trainers have divided the courses into cycles of two to seven weeks, with a minimum of two lessons per week. One specific skill is chosen for each cycle, and the same objective can consequently be assigned to a tenth of consecutive lessons. The chosen objective is then divided into elementary abilities that have to be mastered by the student/athlete and the mastery of each skill becomes a single sub-objective. Indeed, each skill may be shared by more than one objective and the mastery of each skill is a time consuming process that is reached through the repetition of exercises. The delay required for mastering a single skill depends on the student. Some will learn faster than others and thus the trainer must adapt each lesson to the level of the group. Consequently, a lesson is a sorted set of exercises carefully chosen by the teacher. Some exercises are more complicated or fundamental than others. A lesson usually begins with simple exercises that prepare the student for the next ones. Each exercise must be accessible on the basis of the preceding one and lead the student to the following one. This implies that a certain complexity must be associated with each exercise and that calculated distances between exercises must be established.

Consequently, AI-VT must:

(i) propose pertinent sub-objectives and exercises in function of the objective decided by the trainer and the level the students have reached via their previous lessons, (ii) ensure that no exercise is proposed more than once during a given lesson and that the lessons in the same training cycle (to reach a single objective) are varied, (iii) build a consistent course such that it begins with the simplest exercise to then continue with a list of exercises sufficiently related to the preceding and following ones, which implies that the system is capable of replacing each exercise initially proposed by another one that fits better without any intervention from the trainer, and

(iv) help the trainer to determine the importance of each sub-objective and exercise and the time to be devoted to them.

System architecture and communication model

MAS constitute a paradigm designed to handle distributed systems. They are the product of AI research and reflect its limits: a single AI representing the behaviour of a unique entity cannot deal with collective behaviour. Thus the idea of intelligence distribution appears but cannot be called DAI. In a MAS, an agent is a physical or abstract entity having certain specific characteristics: perception of its environment (including itself and the other agents), the capability to act (upon itself or upon the environment) and autonomy in its decisions and actions.

As explained in the previous section, the choice of sub-objectives regarding an objective takes place via an autonomous process, as does the determination of exercises regarding a sub-objective, or of any other exercises chosen and their priority level. The initial choice of exercises regarding a sub-objective must be an autonomous process in order to ensure a wise and free selection of the most adapted exercises. These processes can be undertaken simultaneously, coming after the determination of sub-objectives. Each process is based on specific rules and reasoning. In addition, each one must interact with the other processes and take their choices into account. Thus, each process must be autonomous, make decisions, infer changes from the choices made by the others, be aware of its environment, communicate and interact with the others. Consequently, we can call them agents. As shown in Figure 1, the system is composed of four types of agent: the trainer agent, the student agents, the Objective Agent (OA) which is responsible for choosing the sub-objectives regarding an objective requested by the trainer, and the exercise agents. Each of these agents is responsible for proposing the exercises the best suited to a given sub-objective. A single sub-objective is proposed by the OA or by another Exercise Agent (EA). Each EA must also take into account the choices made by the other EAs: each exercise may be done only once during the entire lesson. Thus, the choices of the EAs are shared and a decision policy is designed and tested.

To share the memory among the EAs, we have chosen the Pilgrim protocol which is an efficient and secured protocol for concurrent, cooperative and collaborative works with shared memory [START_REF] Garcia | Towards an optimistic management of concurrency: a probabilistic study of the pilgrim protocol[END_REF]. The agents are dispatched over a ring and a token is exchanged. The originality of this protocol is the fact that the token carries each stored modification. Each agent has a copy of the shared memory (i.e. the set of exercises proposed by the other agents). In this protocol, the token is a structured entity that is transmitted from agent to agent and dispatched over a logic ring in an order determined at the beginning of their cooperation. When an agent wants to modify the shared memory, it can indicate its intention by placing a reservation flag above the token as soon as it receives it, on condition that the token has no other flag above it. The token continues its course over the ring and when the agent again receives the token, it is allowed to rewrite the modifications. Thus, all the other agents receive these modifications during the token's next revolution, after which all the agents have exactly the same version of the shared memory, even if two or more agents wanted to modify it at the same time. The decision policy is implemented over each EA that is able to modify its set of proposed exercises, if one or more of its exercises are identical to any of those found in another set.

Consequently, AI-VT is composed of two types of CBRs: CBR that proposes sub-objectives regarding objectives, and CBRs that propose exercises regarding sub-objectives. The next two subsections present these two types of CBRs.

Determination of sub-objectives

The OA is responsible for choosing the set of sub-objectives and their duration. Once the lesson has been chosen by the trainer, and after having analysed any additional student needs, the OA follows the CBR approach to make these choices in function of the sub-objectives already achieved and according to the student's degree of assimilation.

For the OA, a case is a set comprised of two parts: a problem and a solution. Each problem part is composed of an objective O, and the solution part of a set of (SO, D SO,O ) where SO is a sub-objective and D SO,O the duration required to reach this SO regarding O. Thus, formally, a source case s is expressed as s = (O, {SO, D S SO,0 }) The durations D SO,O are initialised by the trainer at the beginning of the season and updated by AI-VT after the lesson taking into account the remarks made by the trainer. Since student levels of expertise rise, we can consider that durations decrease and thus call them 'remaining durations'.

First, all the sub-objectives associated to O are retrieved. The adaptation phase consists of computing the duration of each sub-objective. We assumed that these durations are somehow linked to the importance of practising each sub-objective regarding the given objective. Consequently, the adaptation module sorts the set of sub-objectives according to D SO,O (descending order). Then the proposed durations are calculated according to the number of sub-objectives the trainer wants to work on.

Following the lesson, the trainer evaluates the students' acquired level of mastery. Before the lesson begins, each selected sub-objective is transmitted to one EA that will have to associate the corresponding exercises. After the lesson, each sub-objective duration is modified in proportion to the evaluation from 0 to 10 of the students' level for the proposed sub-objective.

Finally, during the capitalisation phase, the system subtracts the durations from all the durations of the source cases for which the practiced sub-objectives appear. Table 1 presents two source cases of the tests performed by one of the trainers (an aikido teacher) who evaluated AI-VT. The trainer chose 3 different sub-objectives. Since the trainer chose the objective become capable of 'Using a grip' for this lesson, the OA recalled case 1. The adaptation process then sorted the sub-objectives according to their durations and proposed the three first sub-objectives, allocating D SO = 60 3 = 20 minutes to each sub-objective. Consequently, after capitalisation, the new durations were those reported in the last column of Table 1. Thus, the less assimilated sub-objectives ('Relaxing despite a grip' and 'Pivoting around a grip') became the most immediate ones. We also note that, as required for the system specification, when the same objective ('Using a grip') was selected again, another set of sub-objectives (composed of the less assimilated ones and others) were selected. Thus, as required, the proposed solutions changed even if the same objective was requested again later.

Selection of exercises

This subsection presents how the exercises are chosen regarding the selected sub-objectives. The different steps of the exercise selection process are reported in Figure 2.

Case model

For this part of AI-VT, the problem part of a case is composed of a sub-objective while the solution part contains a set of exercises with a specified duration to be spent on them. Though, exercise durations are constant, as for sub-objectives the total remaining duration required for each exercise must be specified and will decrease from one lesson to the next. Consequently, as presented above, each source case is noted: σ = (SO, {(EX, CD σ EX , RD σ EX,SO )}). Each source case σ contains the exercises possible regarding SO. Assuming Card(Sol σ ) is the number of exercises of the solution part of σ, the target case τ i (i.e. the part of the lesson that will be proposed) taken into account by the EA EA i is noted:

Sub-objective

List of exercises

τ i = (SO i , n∈{1..Card(Solσ)} {(EX n , CD τi EXn , RD τi EXn,SOi )}).

Retrieval phase

EA 1 is the EA initiated by the trainer. Hence, its role also consists of initiating as many EAs as required, and creating the token of the Pilgrim protocol. Each EA i then retrieves the source case corresponding to SO i .

Adaptation phase

The adaptation phase is the most important. Indeed, as highlighted by the analysis of our previous version of the system, the retrieval of accurate exercises is not sufficient. AI-VT must propose a consistent and a progressive list of exercises. In order tosupply this drawback of our previous version of AI-VT, we have introduced now the notion of distance between exercises. Consequently, the new elaboration of a course now follows five stages. The first stage is based on the remaining durations of the techniques stored in the ontology. We assume that the exercises with the highest remaining durations are the most important to practice, or must be practiced regularly to reach a sub-objective. For this reason, the exercises are decreasingly ordered according to their remaining duration and each one is added until the sum of constant durations reaches the practise time required to attain sub-objective SO i , i.e. ( n CD τi EXn,SOi ) ≥ D T,i SOi,O . Before placing its list of exercises on the token, EA i verifies whether each exercise has been selected with a higher or equal duration by another EA. If so, EA i removes its exercise and replaces it with the next unselected one. EA i may not have enough remaining exercises for the duration period and thus places a LackOf Exercise instead.

The second stage is based on the distances between the exercises. Indeed, the trainers are invited to initialise the distances between each exercise and every other one. At the end of this stage, AI-VT replaced each LackOf Exercise by the unselected exercise closest to the one initially selected (proposed by another EA).

The third stage is based on the difficulty of the exercises also set by the trainer during the initialisation process of AI-VT. As a result of this stage, the system proposes an ordered list of exercises of increasing complexity. Within each sub-objective, the associated exercises are also ordered by increasing complexity.

During the fourth stage, the distances are considered once again. Each EA must verify that the distance between each exercise it has proposed and the previous one is lower than a threshold set by the trainer. If this is not the case, the exercise must be exchanged with the closest unused exercise that respects this distance constraint. During this process of choice, the distance between the new exercise and the previous one in the lesson, and the distance between this new exercise and the next one in the lesson must both be lower than the required threshold. This process may take a long time since all the EAs manipulate their exercises simultaneously.

Finally, during the fifth stage, each EA tries to switch the places of their consecutive exercises in order to optimise the distances between the consecutive exercises. During this stage, it is strictly forbidden to propose other exercises. Only their order can be changed. Hence, an exercise cannot be associated to another sub-objective. If another order that minimises the distances between two consecutive exercises is found, it is retained.

The adaptation phase requires the EAs to communicate with each other. They must associate ordered lists of exercises with sub-objectives sharing their solutions according to the Pilgrim protocol which has been enriched so that each stage ends with the acknowledgement of all the EAs. Each EA must wait for the acknowledgement of the others before going on to the next stage of the phase.

Revision phase

Before and after the lesson, the trainer can create, modify, replace, change the order of or remove exercises and durations. At the end of the course, the trainer evaluates the students' level. The same computations as those applied to the sub-objectives regarding the objective are applied to the remaining exercise durations: ∀i, ∀n ∈ {1..

N EX i }, CD τi EXn,SOi = CD τi EXn × levelEX n 10
.

Capitalisation phase

All the new exercises are added to the corresponding case and the new remaining durations are stored in the sub-objective/exercise ontology: ∀SO, ∀EX, RD σ EX,SO = max(0, (RD σ EX,SO -CD τ EX,SO )).

Results

As a means to evaluate AI-VT, four aikido trainers initialised the system for the same objective and requested ten consecutive lessons. At the end of each lesson, they evaluated the solution proposed by AI-VT and gave a mark (out of 20 points) regarding four aspects (each one out 5 points):

(i) the consistency of the techniques (exercises) proposed regarding the subobjectives, (ii) the variety of techniques, (iii) the progressivity of the techniques proposed, and

(iv) the connections established between the different techniques (prior and/or following).

A score of five points means the lesson meets its goals perfectly. Zero means that it was not satisfactory at all. The marks are reported in Figure 3. Actually, all the lessons proposed were very satisfying for all the trainers since all the marks vary from 12 to 16 points, with the mean mark totalling 14.65 and the median mark equal to 15. The lowest marks were given to lessons for which the techniques substituted by AI-VT were not a good match to the given subobjective. For example, one of the sub-objectives of Trainer #2 was 'Using a back side grip'; sometimes AI-VT proposed techniques that were close but front side grabs. This implies that a semantic element must be introduced in order to ensure that one exercise can be substituted for another and still correspond to the associated sub-objective.

Table 2 shows the sequence of the lessons that AI-VT proposed to one of the trainers (Trainer #1). He initialised the system with four sub-objectives and ten techniques associated with each of the different sub-objectives. Some of the techniques were associated with two sub-objectives. This trainer asked for ten lessons, each containing three sub-objectives, and for each sub-objective to be practiced through two aikido techniques.

The orders of appearance of each sub-objective are reported in the first part of Table 2. The sub-objective that appears with the lowest frequency is actually the one that was associated with the simplest techniques ('Making the partner lose balance' appears 6 times). Thus, this technique was always proposed first among the lessons and the trainer gave the students their highest marks. Consequently, this sub-objective quickly became less important than the others during AI-VT's retrieval phase. The sub-objective 'Relaxing despite 12 a grip' is the most frequent one (AI-VT proposed it 9 times out of 10) owing to the low student level: according to the trainer, this technique is actually one of the most difficult skills to master. The other two sub-objectives appeared 7 and 8 times; since only four sub-objectives were to be distributed over ten lessons, each required three sub-objectives. Nevertheless, the lessons were not repetitive since AI-VT never proposed the same ordered set of exercises (nor the same set of unsorted exercises) twice. The techniques proposed by the system are shown in the second part of Table 2: the numbers correspond to the order of appearance of each technique in the corresponding lesson. The names of the aikido techniques are not fully reported, only their acronyms composed of the first letters of the words. In order to quantify the diversity of the solutions proposed by AI-VT, we calculated the Shannon index [START_REF] Shannon | A mathematical theory of communication[END_REF], a value widely used in ecology in order to quantify species diversity. Considering that AI-VT delivered 10 lessons, each having 6 aikido techniques selected out of 19 possible techniques, and assuming N i is the number of times of appearance of technique i, the Shannon index H is computed according to the equation: H = -

10 i=1 Ni 19 × ln Ni 19 .
The Shannon entropy is therefore equal to H = 6.77 and quantifies the uncertainty associated with the prediction of retrieving each technique in one of the generated lessons. In order to quantify the lesson diversity, it is necessary to divide H by H max = -ln(p) where p is the probability that each technique would have been chosen for each lesson, assuming that each technique would have been chosen in each lesson with the same probability (which is an erroneous assumption considering the way AI-VT chooses techniques): H max = -ln( 19 ) = 0.4033 is a quantity closer to 0 than 1, it proves the high level of diversity of the sets of techniques of this sequence of lessons.

In the first column of Table 2, we added '(Syst.)' after the acronym of the techniques that the trainer did not associate with any of the sub-objectives: these techniques were added by AI-VT itself during the adaptation phase. This table shows that all the techniques initially associated with the sub-objectives were proposed at least three times, and AI-VT proposed 9 new techniques from the aikido catalogue that were not initially associated with any of the trainer's defined sub-objectives. The trainer found that these new techniques were always pertinent choices for reaching the sub-objectives and compatible with the neighbouring techniques. It is noteworthy that the most frequently proposed techniques (6 times) are initially proposed by the trainer ('URD SU ') and also by the system, initially ('RD IN '), but not by the trainer. This is due to the fact that these two techniques are very close to much of the initial set of techniques proposed by the trainer, and thus can easily be added to every lesson with no impact on the lesson consistency. The other techniques that are proposed 4 times or more ('KTD IO', 'AHKD IU', 'AHKD IO', 'KRD UKN', 'KTD TN', 'KTD SIN' ) are considered by the trainer to be fundamental exercises, and the fact that they were often proposed in the AI-VT lessons reassured him. In contrast, the techniques that appeared only once or twice were not initially chosen by the trainer; AI-VT substituted other techniques for them. These choices were nevertheless pertinent according to the trainer and regarding the sub-objectives to be reached.

Discussion

AI-VT is a distributed system that meets the requirements of the application domain. Though tested in the context of aikido training, it could easily be adapted to other sports. The system can propose varied and consistent training lessons for many weeks even if the same objective is requested several times consecutively. The consistency of the set of exercises proposed for each lesson is guaranteed by the introduction of exercise complexity and distances between the different techniques. Hence, these distances allow the system to propose exercises that were not initially chosen by the trainers and to sort the exercises in each course by giving them an underlying theme. Like other approaches [START_REF] Cordier | Engineering and learning of adaptation knowledge in case-based reasoning[END_REF], [START_REF] Dufour-Lussier | Automatic case acquisition from texts for process-oriented case-based reasoning[END_REF] and [START_REF] Henriet | An iterative precision vector to optimise the CBR adaptation of EquiVox[END_REF], ours establishes a link between the adaptation and capitalisation of revisions of CBR-systems. Indeed, we have designed a way to use the remarks made by users during the revision phase in order to enhance the accuracy of the adaptation process of CBR-systems. In addition, the introduction of remaining durations is of the utmost importance since it allows AI-VT to build varied solutions by analogy and thus to never propose the same lesson twice. Another originality of AI-VT resides in the fact that its propositions are groupings of solutions from CBR-based agents. These agents must collaborate closely, share memory and take into account the solutions proposed by the other agents.

However, many facets of AI-VT may be improved. Firstly, regarding the subobjectives, the distances between the exercises are not always sufficiently well considered to guarantee the consistency of the proposed exercises. Even though the exercises proposed by AI-VT were close to the exercises initially proposed by the trainer, they were not always consistent with respect to certain very specific sub-objectives. Consequently, we must introduce some semantics into the system so that it can substitute exercises that correspond to the meaning of the associated sub-objective. In addition, with each exercise AI-VT should also provide consistent instructions which depend both on the exercise and the sub-objective to be reached. Thus, when the system substitutes an exercise initially chosen by the trainer for one or its own choosing the trainer's first set of instructions is no longer consistent. This particular feature is linked to the previous one since it must be based on the sub-objective's meaning and the exercise's characteristics.

Finally, one limit to our system lies in the fact that it must be initialized every year at the beginning of the season or whenever no remaining duration is left. At that time, the trainer must produce all of the initial values. Further investigation will likely lead us to propose a process based on experience in order to compute initial values. Like other approaches, we will evaluate the possibility of basing these computations on pre-tests that evaluate student levels [START_REF] Huang | Constructing a personalized e-learning system based on genetic algorithm and case-based reasoning approach[END_REF] and [START_REF] Tan | Personalized course generation and evolution based on genetic algorithms[END_REF].

Conclusion

We have designed a distributed system based on artificial intelligence and that is capable of delivering varied lessons to trainers. AI-VT meets the requirements of the application domain and can easily be adapted to many areas in the field of sports education. Indeed, we have proved that AI-VT can propose ten consecutive lessons aimed at the same objective, each one different from the others even if the process of choice is based on analogy with capitalised courses and experience. The agents of AI-VT collaborate in order to provide the most suitable exercises for each part of each lesson. A collaboration protocol based on a token ring has been adapted and used, as well as a policy capable of merging and arbitrating between concurrent solutions proposed by the collaborative agents during the adaptation process. In addition, we have introduced a distance between the exercises and associated a complexity to each exercise in order to guarantee the progressivity and the existence of an underlying theme common to all of the exercises on the ordered list of exercises composing the lesson. This study proves the consistency of the collegial solutions proposed through its evaluation by four trainers.

The system could nevertheless be improved if it could take into consideration the exact meanings of the sub-objectives to be reached during each proposed lesson. Consequently, future investigation will focus on the system's ability to perceive and interpret sub-objective meanings in order to provide students with wisely devised instructions and exercises, even when the system replaces an exercise by the closest one during its adaptation phase. Finally, we will examine ways in which to introduce processes such as the pre-testing of students in order to reduce the system initialisation process that must be performed by the trainer.

Figure 1 :

 1 Figure 1: Communication model of AI-Virtual Trainer.

Figure 2 :

 2 Figure 2: Process of exercise selection.

Figure 3 :

 3 Figure 3: Trainer's evaluations of the generated lessons.

Table 1 :

 1 Example of modifications of durations after a lesson.

	Source Objective Sub-objectives	Initial	Trainer's	Stored
	case			duration	mark	duration
				(min.)	(points)	(min.)
	1	Using a	Breaking the partner's	90	7/10	90 -20×7 10 = 76
		grip	posture
			Relaxing despite a grip	90	3/10	90 -20×3 10 = 84
			Making the partner	80	4/10	80 -20×4 10 = 72
			loose balance
			Pivoting around a grip	80	-	80
	2	Breaking	Breaking a single grip	90	-	90
		a grip	Relaxing despite a grip	70	3/10	70 -20×3 10 = 64

Table 2 :

 2 Lessons proposed by AI-VT to Trainer #1.

	Skill				Order of appearance in Lesson
		#1 #2 #3 #4 #5 #6 #7 #8 #9 #10
	Making the partner
	lose balance	1	1		1		1		1		1
	Breaking the
	partner's posture	2	2	1		1	2	1		1
	Relaxing despite
	a grip	3	3	3	3	3		3	3	3	3
	Pivoting around
	a grip			2	2	2	3	2	2	2	2
	Aikido technique
	KTD IO	1			2		2		2		2
	AHKD IU	2	2				1		1
	AHKD IO	3	1		1						1
	KRD KNIO (Syst.)	4					3
	KRD UKN	5	6	6				5		5
	KRD AO	6						6		6
	KRD IN (Syst.)		3
	KTD TN		4	2		1		1		1
	KRD KH		5	5			4
	KTD SIN			1		2		2		2
	KTD UKNO			3				4			6
	RD KNS (Syst.)			4							5
	RD IN (Syst.)				3	3		3	3	3	3
	URD SU				4	4	6		4	4	4
	AHKD KNIO (Syst.)				5	5			5
	KRD KSNIO (Syst.)				6
	KRD IO (Syst.)					6
	AHKD IN (Syst.)						5
	KRD KSNYO (Syst.)								6

Acknowledgements

The author wishes to thank John Olsen for his help with the English language, and the trainers Hervé Guénard, Pascal Sobrino, Romuald Coutot and Paolo Atzori for their evaluations of the system.