Sylvain Chabanet
email: sylvain.chabanet@univ-lorraine.fr

Philippe Thomas
email: philippe.thomas@univ-lorraine.fr

Bril El-Haouzi
email: hind.el-haouzi@univ-lorraine.fr

CoreSelect: a new approach to select landmarks for dissimilarity space embedding

Keywords: Machine Learning, Proximity Learning, Surrogate Modeling, Sawmill Simulation

This paper studies an application of indefinite proximity learning to the prediction of baskets of products of logs in the sawmill industry. More precisely, it focuses on the usage of the dissimilarity space embedding framework to generate a set of features representing wood logs. According to this framework, data points are represented by a vector of dissimilarity measures toward a set of representative data points named landmarks. This representation can then be used to train any of the large variety of available ML models requiring structured features. However, this framework raises the problem of selecting these landmarks. A new method is proposed to select these landmarks which is compared with four other methods from the literature. Numerical experiments are run to compare these methods on a dataset from the Canadian sawmill industry. The data representations obtained are used to train random forests and neural networks ensemble models. Results demonstrate that both the Partition Around Medoids (PAM) method and the newly proposed CoreSelect methods lead to a small but significant reduction in the mean square error of the predictions.

INTRODUCTION

The process of sawing a wood log into lumber is diverging and in co-production. From a single log, a sawmill will obtain simultaneously several products with different dimensions and grades. In addition, the heterogeneity of shape and internal defects between logs make it difficult to anticipate what set of lumber would be obtained from sawing a log. All these factors greatly complicate production planning and control in this industry. Simulation tools have been, however, widely studied alone or in conjunction with other decision-support tools to alleviate these problems [START_REF] Chabanet | Toward digital twins for sawmill production planning and control: benefits, op-portunities, and challenges[END_REF]. Simulators can, in particular, be used to predict the set of lumber that would be obtained by sawing a specific log. This set of lumber is named the Basket of Products (BoP) of the log in the following of this study. To repeat this operation over all logs to be sawed, or at least a representative sample, allow to approximate the mix of product that would be obtained by a specific production plan.

Several authors, however, have mentioned the important computational time associated with saw-ing simulation models [START_REF] Morneau-Pereira | An optimization and simulation framework for integrated tactical planning of wood harvesting operations, wood allocation and lumber production[END_REF][START_REF] Wery | Simulation-optimisation based framework for sales and operations planning taking into account new products opportunities in a co-production context[END_REF]. This complicates their use for short-term decision problems which might still require several thousand simulation runs. To alleviate this problem, [START_REF] Morin | Machine learning-based metamodels for sawing simulation[END_REF], in particular, proposed to train machine learning surrogate models of these simulation models. These surrogate models, equivalently called metamodels, are machine learning models trained on past simulation results to predict the BoP of logs. These surrogate models are, therefore, approximations of the simulation models for specific sawmill configurations.

Many different sawmill simulation surrogate models have been studied in the literature. They can be distinguished, in particular, by the input considered to describe a log and predict its BoP. Some sawing simulation models like Optitek [START_REF] Goulet | Optitek: User's manual[END_REF] or SAWSIM1 can, indeed, process logs described by 3D scans of their surfaces. These scans are 3D point clouds providing information over the shape of the logs. Few ML models are, however, able to process this type of input directly. For this reason, [START_REF] Morin | Machine learning-based metamodels for sawing simulation[END_REF][START_REF] Morin | Machine learning-based models of sawmills for better wood allocation planning[END_REF] propose surrogate models making predictions from structured represen-tations of the logs based on know-how features commonly used in the industry. In particular, they use the length of the logs, their volumes, diameter at both extremities, curvature, and shrinking. [START_REF] Selma | An iterative closest point method for measuring the level of similarity of 3d log scans in wood industry. Service Orientation in Holonic and Multi-Agent Manufacturing: Proceedings of SOHOMA[END_REF] propose to use a dissimilarity function to compare pairs of logs from their scans and predict their BoP using a nearest-neighbors scheme. More precisely, they propose to use the Iterative Closest Point (ICP) dissimilarity which is a consequence of the ICP algorithm commonly used to align 3D shapes [START_REF] Besl | Method for registration of 3-d shapes[END_REF]. The idea of using pairwise ICP dissimilarity toward class medoids as features for other ML models was investigated, for example, by [START_REF] Chabanet | Medoid-based mlp: an application to wood sawing simulator metamodeling[END_REF] in the case of neural network surrogate models. Lastly, [START_REF] Martineau | Neural network architectures and feature extraction for lumber production prediction[END_REF] study several neural network surrogate models, including models based on the architecture pointnet [START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF], which is able to learn directly from 3D point clouds.

This study focuses on surrogate models predicting BoP from pairwise dissimilarities. Several previous studies have, in particular, proposed sawing simulation surrogate models able to predict the BoP of a log based on a vector of features composed of the dissimilarities of this log toward a set of representative logs from the model training dataset. This strategy is called the dissimilarity space embedding framework in the literature [START_REF] Duin | The dissimilarity representation for pattern recognition: a tutorial[END_REF]. For example, [START_REF] Chabanet | Medoid-based mlp: an application to wood sawing simulator metamodeling[END_REF] use such vectors as input to multi-layer perceptrons, while (Chabanet et al., 2021a) use a variant of a naïve Bayes classifier. These past studies, however, do not study alternative methods for the selection of the representative data points, also called landmarks, used to generate the vectors of dissimilarity features fed to the classifier.

The main contribution of this study is, therefore, the proposition of a novel method to select landmarks and its comparison with four other methods from the literature. The landmarks selected by these methods are used to train two types of ensemble models to predict BoP of logs.

The remainder of this article is organized as follows. Section 2 reviews the literature on indefinite proximity learning and formally introduces the dissimilarity space embedding framework. Section 3 presents the learning problem studied, the landmarks selection methods compared in this study as well as the dataset used during experiments. Experimental results are detailed in section 4. Lastly, section 5 concludes this study.

INDEFINITE PROXIMITY LEARNING

Non-metric proximity (similarity or dissimilarity) functions naturally arise in many fields to compare how alike two data items are. For example, the dynamic time warping dissimilarity [START_REF] Müller | Dynamic time warping. Information retrieval for music and motion[END_REF] is a popular method to compare time series, or the Jaccard dissimilarity [START_REF] Luo | Text document clustering based on neighbors[END_REF] has been used in many studies to compare text documents. Learning from these proximity functions can be an attractive alternative to learning from descriptive features. Most of the common methods proposed to learn from proximity function require specific properties such as symmetry or semi-definiteness which are not always respected in practice. Several families of methods have been, however, proposed by the literature to deal with the non-metric case [START_REF] Schleif | Indefinite proximity learning: A review[END_REF]. Many methods, for example, rely on applying transformations to the proximity matrix M that contains the pairwise proximity evaluation on the training dataset to make it positive semi-definite [START_REF] Munoz | From indefinite to positive semi-definite matrices[END_REF]. Once transformed, the matrix M can, then, be used to train kernel-based models like Support Vectors Machines (SVM). The out-of-sample extension of these methods, i.e, their extension to data points not in the training dataset to make new predictions, is, however, often not straightforward, and computationally costly [START_REF] Schleif | Indefinite proximity learning: A review[END_REF].

Other authors, like [START_REF] Ong | Learning with non-positive kernels[END_REF], extend the theory of reproducing kernel Hilbert spaces, which underlie, for example, SVM, to reproducing Krein spaces. Learning algorithms in a Krein space can consider non-definite proximity matrices. This theory leads, in particular, to training models that are linear combinations of dissimilarities toward training data points.

Lastly, another general method, which is the one considered in this study, is the proximity (similarity or dissimilarity) space embedding method [START_REF] Duin | The dissimilarity representation for pattern recognition: a tutorial[END_REF]. Such a scheme first selects a small set of prototypes, called landmarks, in the training dataset. A point is, then, represented by the vector of dissimilarities toward these prototypes. More precisely, considering a training set D and a subset R = {r 1 , ..., r q } ⊆ D, a data point x is represented by:

D(x, R) = (d(x, r 1 ), ..., d(x, r q )), , (1)
where d denotes the proximity function. The points r 1 , ..., r q are the landmarks.

The use of dissimilarity space embedding has been, in particular, extensively studied in the context of labeled graph classification [START_REF] Livi | Optimized dissimilarity space embedding for labeled graphs[END_REF]. This framework has been, similarly, applied to time series classification. [START_REF] Jain | Time series classification in dissimilarity spaces[END_REF], for example, study a method to train SVM models for time series classification, using the time warping dissimilarity and a dissimilarity space embedding framework. Lastly, this framework has been used to predict BoP of logs based on the ICP dissimilarity in (Chabanet et al., 2021a;[START_REF] Chabanet | Medoid-based mlp: an application to wood sawing simulator metamodeling[END_REF].

This method has, in particular, two main advantages which motivate its choice in this study. The first advantage is that it does not restrict the choice of the ML model used. Data points are effectively embedded in a vector feature space. Therefore, any of the many and extensively studied ML models designed to learn in this classic setting can be used. For the specific problem studied in this paper, this allows, for example, to train Random Forests (RF) or multi-layer perceptrons to predict BoP from the ICP dissimilarity space. RF were, in particular, proven effective when trained on know-how features [START_REF] Morin | Machine learning-based models of sawmills for better wood allocation planning[END_REF]. The second advantage is that it allows the user to select the dimension of the proximity space. This is important because, while a larger proximity space might lead to better models, at least up to some point, it also means that more proximity function evaluations are required to embed new data points before making a prediction. Such evaluations can, however, be computationally expensive. The ICP dissimilarity, for example, is the result of an iterative optimization algorithm whose complexity is dependent on the number of points in the point clouds. Computational efficiency is, however, very important in the context of surrogate models.

Such a method, however, raises the problem of how to select landmarks. The best selection method, however, is dependent on the learning problem and dataset [START_REF] Pękalska | Prototype selection for dissimilarity-based classifiers[END_REF].

CASE STUDY

This study focuses on the usage of the dissimilarity space embedding framework to train sawing simulation surrogate models to predict baskets of products of logs from 3D scans of their surface. From a machine learning perspective, this problem can be modeled and has been modeled as either a classification problem or as a regression problem.

If the problem is modeled as a classification problem, every BoP present in the training dataset is associated with a class to be predicted. The main advantage of this method is that the surrogate model will always predict a feasible BoP. However, these training datasets can contain many different BoP, some of them appearing only once in the training dataset. It is If the problem is modeled as a (multi-output) regression problem, a BoP is modeled as a vector of size p where p is the number of standard products that can be sawed in the sawmill considered. The i th element of this vector correspond to the quantity of the i th type of lumber present in the BoP. While this eliminates the problem caused by rare and unobserved BoP, it also means that unfeasible BoP will be predicted. Typically, BoP predicted by regression model contains fractional quantities of product. It should be noticed, however, that these predictions are not designed to be used individually, but aggregated by batches of logs and fed to operational research models. For these reasons, in this study, the problem of predicting BoP of logs is modeled as a regression problem.

Dataset

The dataset used in this study originates from the Canadian forest-product industry. It contains information over 2219 pine, fir, and spruce wood logs. More precisely, each log has a 3D scan, a set of six know-how features, and a BoP obtained by simulating the sawing of the log with the software Optitek.

The 3D scans are point clouds, describing the surface of the logs. They are, therefore, constituted of an unordered list of points with three coordinates each. The number of points in a cloud varies from scan to scan and is, in particular, dependent on the length of the logs. The points are ordered in rough ellipsoids spanning the log surface. An example of such a scan is provided in figure 1.

In addition to these scans, each log is described by six know-how features: its length, diameters at both extremities, curvature, shrinking, and volume. These features are, in particular, used by [START_REF] Morin | Machine learning-based metamodels for sawing simulation[END_REF] to predict BoP of logs. Models trained to predict BoP Some of these features are used in this industry to classify logs in the log yard. This is, in particular, the case of the length and diameters. Logs from different clusters would have different possible BoP. This dataset, in particular, can be divided into four clusters, based on the length of the logs. Figure 2 presents the histogram of the length of logs, to which was fitted a mixture of four Gaussian distributions.

The sawmill model set to the simulator Optitek to generate this dataset was able to produce 47 types of products. The BoP are, therefore, modeled as vectors of dimension 47. In total, 870 different BoP are present in this dataset, of which 614 appear only once.

Landmarks selection

In this study, five methods will be compared to select landmarks to train dissimilarity features. These dissimilarity features will also be compared to the knowhow features introduced in the previous section.

The first and simplest method is to select landmarks at random in the training dataset. It is used as a baseline, for example, by [START_REF] Pękalska | Prototype selection for dissimilarity-based classifiers[END_REF] on binary and multi-class classification problems. While systematic methods perform, overall, better, the difference is sometimes small and depends on the number of landmarks selected.

The second method is an algorithm providing a locally optimal solution to the k-medoid clustering problem. It is often named alternate in the literature. The k-medoid problem consist in finding a subset R = (r 1 , ..., r q ) in a dataset D = (x 1 , ..., x n ) so that R minimize:

q ∑ i=1 ∑ x∈C i d(x, r i ) , (2)
with x ∈ C i if d(x, r i ) = min r∈R d(x, r).
This method is named k-center in [START_REF] Pękalska | Prototype selection for dissimilarity-based classifiers[END_REF] and is, overall, the best-performing method on the classification problems it was tested on. While [START_REF] Pękalska | Prototype selection for dissimilarity-based classifiers[END_REF] apply it on a class-by-class basis, however, it is applied on the whole dataset here. This method is based on a k-medoids algorithm, i.e, a generalization of the well-known k-means algorithm for data clustering. The landmarks correspond to the cluster centers. The general principle of this algorithm is presented in algorithm 1.

Algorithm 1 Alternate

Input D = (x 1 , ..., x n ), set of training inputs Output R = (r 1 , ..., r q ), set of landmarks Initialize R = (r 1 , ..., r q ) at random or following an heuristic.

Initialize clusters C 1 , ...,C q so that D = j∈ 1,q C j while End condition is False do for x ∈ D do i ← max j∈ 1,q (d(x, r j )) set x into C i end for for i ∈ 1, q do r i ← argmin x∈C i (∑ x∈C i (d(x, r j )) end for end while

This algorithm is iterative. The complexity of one iteration is O(n 2 ) with n the size of the training dataset [START_REF] Schubert | Fast and eager k-medoids clustering: O (k) runtime improvement of the pam, clara, and clarans algorithms[END_REF]. Iterations are performed until some ending condition appends, either that a maximum number of iterations is reached or that the set of landmarks stops changing.

The third method is based on the Partitioning Around Medoids (PAM) algorithm [START_REF] Sarle | Finding groups in data: An introduction to cluster analysis[END_REF]. It solves the same k-medoid problem and is more computationally intensive but more accurate [START_REF] Schubert | Fast and eager k-medoids clustering: O (k) runtime improvement of the pam, clara, and clarans algorithms[END_REF]. PAM is constituted of two parts: Build, which initializes the centers of the clusters, and Swap, which swaps cluster centers with other data points in order to decrease the clustering cost defined in equation 2. An outline of the Swap procedure is presented in algorithm 2. Depending on the exact implementation, the complexity of one iteration ranges from O(n 2 ) to O(q 2 n 2 ) with n the size of the training dataset and q the number of clusters. The implementation used for this article is O(q(nq) 2 ) [START_REF] Maranzana | On the location of supply points to minimize transportation costs[END_REF].

The fourth method evaluated in this study is named Dselect and proposed by [START_REF] Kar | Similarity-based learning via data driven embeddings[END_REF]. This heuristic is based on the idea that landmarks should be as dissimilar from one another as possible. In particular, they introduced a heuristic, named Dselect, that iteratively selects new landmarks as the ones minimizing the average dissimilarity toward previously selected landmarks. A pseudocode of this Algorithm 2 PAM Swap Input D = (x 1 , ..., x n ), set of training inputs Output R = (r 1 , ..., r q ), set of landmarks R ← Build(D) Initialize clusters C 1 , ...,C q empty for x ∈ D do i ← max j∈ 1,q (d(x, r j )) set x into C i end for while cost decrease do for r ∈ R do for x ∈ D \ R do Compute cost change when swapping r and x end for end for perform the best swap end while heuristic is presented in algorithm 3. It is an iterative greedy algorithm. Contrary to k-medoids where all landmarks are reevaluated at every iteration, Dselect starts from an empty set of landmarks and adds them one by one until the set is completed. As such, the complexity of the whole algorithm is O(nq 2 ) with n the size of the training dataset and q the number of landmarks. It is, therefore, less costly than Alternate and PAM as long as the number of landmarks remains small in comparison with the dataset size.

Algorithm 3 Dselect

Input D = (x 1 , ..., x n ), set of training inputs Output R = (r 1 , ..., r q ), set of landmarks

r 1 ← random element from D for i ∈ 1, q do r i ← argmax x∈D\R ( 1 i ∑ r j ∈R d(x, r j )) end for
The fifth method evaluated is a variant of Dselect which is proposed in this study. This method is named CoreSelect. A pseudocode is given in algorithm 4. Like Dselect, it is a greedy algorithm. The difference is that at every iteration, the next landmark is selected as the data point that maximizes their minimal dissimilarity with previously selected landmarks. A motivation for this modification of Dselect is that, in the metric case, this strategy yields an approximate solution to a q-center problem [START_REF] Dyer | A simple heuristic for the p-centre problem[END_REF]:

min R=(r 1 ,...,r q )⊂X ∆(r 1 , ..., r q ) , (3)
with

∆(r 1 , ..., r q ) = max x∈X min R d(x, r i ) . (4)
More precisely, it ensures that the maximum distance between a point and the nearest selected landmark is at most twice that of the optimal solution. It should be noticed that this q-center problem is different from the one solved by the alternate and PAM algorithms. These algorithms minimize the sum of distances toward cluster centers. On the other hand, this method has been proven to be an approximate solution to the problem of minimizing the radiuses of the clusters. The complexity of this algorithm is the same as Dselect: O(nq 2 ).

Algorithm 4 CoreSelect

Input D = (x 1 , ..., x n ), set of training inputs Output R = (r 1 , ..., r q ), set of landmarks r 1 ← random element from D for i ∈ 1, q do r i ← argmax x∈D\R (min r j ∈R (d(x, r j )) end for

Surrogate models

Two ensemble models are used as sawing simulation surrogate models in this study. The first is the Random Forest (RF) algorithm [START_REF] Breiman | Random forests[END_REF]. The prediction of the forest is the average of the prediction of individual decision trees. Random forests were, in particular, selected for their good performances as sawmill simulator surrogate models in [START_REF] Morin | Machine learning-based metamodels for sawing simulation[END_REF][START_REF] Morin | Machine learning-based models of sawmills for better wood allocation planning[END_REF] on know-how features. An important characteristic of random forest is that, to lower the correlation between the base trees and further reduce the variance of the ensemble, trees are trained on bootstrap samples of the training dataset. In addition, every split of the tree is optimized on a random subsample of the available features. Hyperparameters for this model were selected by trial and error. In particular, the number of trees in the forest was set to 500, the total number of landmarks used for the dissimilarity space embedding was set to 100 and the fraction of the number of features considered to optimize each split was set to 10, except for the baseline using the know-how features. In this case, all six features are considered for each split selection.

The second ensemble model investigated in this study is an ensemble of small artificial neural network (ANN) models. These neural networks are feedforward models with a single hidden layer, trained with the Levenberg-Marquardt algorithm. Similarly to what was done in [START_REF] Chabanet | Medoid-based mlp: an application to wood sawing simulator metamodeling[END_REF], the activation function of the hidden layer is a hyperbolic tangent and the activation function of the output layer is a sigmoid. The sigmoid output, in particular, ensure that, after rescaling the predictions, the quantities of lumber predicted are always between 0 and the maximum quantity observed in the training dataset. The number of neurons in the hidden layer was set by trials and errors to 2, which is consistent with the results of [START_REF] Chabanet | Medoid-based mlp: an application to wood sawing simulator metamodeling[END_REF]. As for the random forest model, the number of landmarks was set to 100 and the number of weak learners to 500. The number of features used as input for each network was set to 10.

EXPERIMENTAL RESULTS

Experiments are run as follows. The dataset is, first, divided at random into a small training set of size 500, and an evaluation set of size 1719. For each landmark selection method, 100 landmarks are selected on the training set and used to embed the data in a dissimilarity space. The know-how features are also used to obtain a sixth representation of the data. For each representation, a random forest and an ANN ensemble are trained on the training set. The Mean Square Error (MSE) of the prediction is measured on the evaluation set. To average out the impact of the exact train-test separation of the dataset, this process is repeated 30 times.

Table 1 presents the number of landmarks selected by each method in each of the 4 length clusters defined in section 3.1. These numbers are averaged over 30 repetitions of the experiments. Different selection methods have different behavior. The number of landmarks selected by the random method in each cluster is, naturally, proportional to the size of each cluster. Therefore, the largest cluster, cluster 3, which represents 60% of the dataset, has, on average, approximately 60% of the landmarks. Similarly, cluster 4, which represents 22% of the dataset, contains, on average, approximately 22% of the landmarks. On the opposite, the smallest cluster, cluster 2, contains only 6.7% of the landmarks.

Both PAM and CoreSelect smooth slightly the distribution of the landmarks over the clusters. In particular, 40.0 and 35.9 landmarks are selected in average in cluster 3 by PAM and CoreSelect respectively. This is less than the amount selected by the random selection method. On the opposite, they select respectively 16.3 and 17.2 landmarks in cluster 2.

Dselect and Alternate have very different behavior. Dselect, in particular, mostly selects landmarks in the two extremal clusters, clusters 1 and 4. This might be explained, in the metric case, by the tendency of DSelect to select the next point far from the geometric median of the previously selected landmarks, which is the point minimizing the sum of distances toward the landmarks. On the opposite, Alternate selects most of the landmarks, 90 on average, from cluster 3 which is the largest cluster.

The MSE evaluated on the evaluation test for the different landmark selection methods and surrogate models are presented in table 2. Several facts have to be mentioned. First, the lowest MSE is obtained for the RF surrogate model and the PAM and CoreSelect landmark selection results. These two MSE cannot be said to be statistically different from these experiments. In particular, a paired student test over the MSE measured on the 30 repetitions of the experiment has a p-value of 0.41. It should be noticed, however, that CoreSelect has a lower computational cost than PAM. Comparing both these methods to the third best method, i.e., the RF model with random landmarks, yields p-values lower than 3 × 10 -5 in both cases. Therefore, both PAM and the newly proposed CoreSelect selection methods allow to improve upon the random baseline, as well as the know-how features. On the contrary, both the Dselect and Alternate selection methods show significantly worse MSE for the Random Forest model. P-values of the students test are 3 × 10 -10 and 5 × 10 -4 respectively. This might be due to the highly irregular dispersion of the landmarks across clusters.

In general, ANN ensemble surrogate models have higher MSE than RF surrogates. The impact of the various landmarks selection methods over the average MSE is, however, different. In particular, in this case, the method with the lowest MSE is PAM alone. This time, using CoreSelect does not lead to lower MSE than the random method. In particular, the p-value of a paired student test is, here, 0.32. Both Dselect and Alternate, however, lead to higher MSE than the random selection method.

Selection method

Selection Time Random 8.9 × 10 -5 (2.8 × 10 -4 ) Alternate 2.6 × 10 -2 (1.9 × 10 -3 ) PAM 15.7 (0.6) Dselect 2.0 × 10 -2 (6.7 × 10 -4 ) CoreSelect 2.3 × 10 -2 (6.8 × 10 -4 ) Table 3: Average and standard deviation over 100 repetitions of the time required by each method to select 100 landmarks in a subset of size 500 of the dataset.

To complement the previous experimental results, the computation times required by the selection methods with the implementation used for these experiments were estimated. Dselect and CoreSelect were implemented from scratch in Python using the numpy library. Alternate and PAM were implemented as wrapper around clustering functions from the scikitlearn-extra 2 library. All experiments were run on a computer with an intel Core i7 vPRO 10 th generation CPU at 2.70GHz. Table 3 presents the average times required by each landmark selection method, over a hundred new random subsets of size 500 of the whole dataset. Unsurprisingly, the fastest method is by far the random selection. Dselect, CoreSelect, and Alternate have very similar computational times, between 0.020 and 0.026 seconds in these experiments. On the opposite, PAM is very slow, as it needs, on average, 15.7 seconds to select the landmarks. Considering that the two best methods are, here CoreSelect and PAM which perform similarly with random forest models, Coreselect present a clear advantage in terms of time.

CONCLUSION

This article studies surrogate models for sawmill simulation. In particular, it focuses on the use of the dissimilarity space embedding framework to create a feature space used to train models and make predictions. Because this framework raises the question of the methods used to select the landmarks which form its core, five landmarks selection methods are compared on this method.

Numerical experiments were run using a dataset from the Canadian Sawmill industry to train RF and ANN ensemble models on data representations obtained from each method. Results were also compared with baselines obtained from know-how representation of the data points.

Among the combinations of landmarks selection methods and ML models evaluated, the lowest MSE was obtained for the RF model, with landmarks obtained from either the PAM or the newly proposed CoreSelect method. CoreSelect, however, has lower computational complexity than PAM.

Several limits to this study should, however, be mentioned and lead to future works. First, previous works have shown that the performances of these surrogate models can change widely from one sawmill to another, especially depending on the number of standard products they produce. Therefore, the experiments presented in this study should be repeated on other independent datasets. Similarly, the impact of the size of the training dataset and of the number of landmarks should be investigated in detail. Lastly, the reason why CoreSelect leads to lower MSE than the random baseline with the RF models but not with the ANN ensemble should be explored as it might lead to a deeper insight into the behavior of these models on dissimilarity spaces.

Figure 1 :

 1 Figure 1: Example of a 3D scan of a log also possible that the training dataset does not contain all of the possible BoP.If the problem is modeled as a (multi-output) regression problem, a BoP is modeled as a vector of size p where p is the number of standard products that can be sawed in the sawmill considered. The i th element of this vector correspond to the quantity of the i th type of lumber present in the BoP. While this eliminates the problem caused by rare and unobserved BoP, it also means that unfeasible BoP will be predicted. Typically, BoP predicted by regression model contains fractional quantities of product. It should be noticed, however, that these predictions are not designed to be used individually, but aggregated by batches of logs and fed to operational research models. For these reasons, in this study, the problem of predicting BoP of logs is modeled as a regression problem.

Figure 2 :

 2 Figure 2: Histogram of the length of the logs in the dataset from these descriptive features are, therefore, used as baselines in this study.Some of these features are used in this industry to classify logs in the log yard. This is, in particular, the case of the length and diameters. Logs from different clusters would have different possible BoP. This dataset, in particular, can be divided into four clusters, based on the length of the logs. Figure2presents the histogram of the length of logs, to which was fitted a mixture of four Gaussian distributions.The sawmill model set to the simulator Optitek to generate this dataset was able to produce 47 types of products. The BoP are, therefore, modeled as vectors of dimension 47. In total, 870 different BoP are present in this dataset, of which 614 appear only once.

Table 1 :

 1 Average number and standard deviation over 30 repetitions of the number of landmarks in each length cluster

	Selection method	Cluster 1	Cluster 2	Cluster 3	Cluster 4
	Random	12.0 (3.1)	6.7 (2.5)	59.9 (5.9)	21.5 (4.4)
	Alternate	3.3 (1.2)	3.0 (1.3)	90.2 (2.0)	3.5 (1.0)
	PAM		23.8 (2.9)	16.3 (2.6)	40.0 (2.4)	19.9 (2.2)
	Dselect		48.8 (0.5)	0.03 (0.2)	0.9 (0.6)	50.3 (0.6)
	CoreSelect	27.5 (3.0)	17.2 (3.0)	35.9 (3.1)	19.4 (1.8)
	Selection	Random	ANN
	method	Forest	ensemble
	Know-how	1.819 (0.035) 1.968 (0.020)
	Random	1.773 (0.027) 1.828 (0.020)
	Alternate	1.847 (0.035) 1.985 (0.041)
	PAM	1.762 (0.029) 1.823 (0.020)
	Dselect	1.781 (0.034) 1.855 (0.023)
	CoreSelect	1.763 1.830 (0.020)

Table 2 :

 2 Average and standard deviation over 30 experiment repetitions of the MSE obtained for each model on the evaluation set.

https://www.halcosoftware.com/ software-1-sawsim, last accessed May

https://scikit-learn-extra.readthedocs.io/ en/stable/install.html, last accessed in May 2023

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support of the ANR-20-THIA-0010-01 Projet LOR-AI (Lorraine Intellgence Artificielle) and the région Grand EST. We are also extremely grateful to FPInnovation, who gathered and processed the dataset used in this study.


Rania Abdelghani
email: rania.abdelghani@inria.fr

Edith Law

Chloé Desvaux

Pierre-Yves Oudeyer

Hélène Sauzèon

Keywords: Educational technologies, epistemic curiosity, metacognition

Curiosity-driven learning has shown significant positive effects on students' learning experiences and outcomes. But despite this importance, reports show that children lack this skill, especially in formal educational settings.

To address this challenge, we propose an 8-session workshop that aims to enhance children's curiosity through training a set of specific metacognitive skills we hypothesize are involved in its process. Our workshop contains animated videos presenting declarative knowledge about curiosity and the said metacognitive skills as well as practice sessions to apply these skills during a reading-comprehension task, using a web platform designed for this study (e.g. expressing uncertainty, formulating questions, etc).

We conduct a pilot study with 15 primary school students, aged between 8 and 10. Our first results show a positive impact on children's metacognitive efficiency and their ability to express their curiosity through question-asking behaviors.

Interactive environments for training children's curiosity through the practice of metacognitive skills : a pilot study

Curiosity plays a central role in enhancing motivation, learning and fostering academic achievement [START_REF] Litman | Curiosity and the pleasures of learning: Wanting and liking new information[END_REF][START_REF] Kang | The wick in the candle of learning: Epistemic curiosity activates reward circuitry and enhances memory[END_REF][START_REF] Jepma | Neural mechanisms underlying the induction and relief of perceptual curiosity[END_REF]. But despite its importance, there is evidence that curiosity decreases with formal schooling [START_REF] Engel | Childrens need to know: Curiosity in schools[END_REF].

In addressing this issue, studies such as in [START_REF] Jirout | Curiosity in schools. The new science of curiosity[END_REF] suggest that curiosity is a malleable skill that we can explicitly promote in classrooms via training specific information-seeking behaviors. Even further, and beyond practicing such behaviors, research such as [START_REF] Murayama | Process account of curiosity and interest: A reward-learning perspective[END_REF][START_REF] Loewenstein | The psychology of curiosity: A review and reinterpretation[END_REF][START_REF] Berlyne | Curiosity and learning[END_REF] suggest that training epistemic curiosity requires working on specific metacognitive skills that are involved in its process, such as the ability to evaluate and monitor one's own knowledge.

Following these findings, the present work proposes to test a curiosity training that focuses on four specific metacognitive skills that we hypothesize are involved in its triggering and maintaining mechanisms : identifying uncertainty, generating educated guesses about it, seeking new information to resolve it and assessing the value of this new information. In practice, this training contains a first part that proposes declarative knowledge about these metacognitive skills using animated videos. The second part contains practice sessions to help participants acquire procedural knowledge about these skills. They use a dedicated web application where they interact with conversational agents that give specific cues to help them apply the said skills during a reading-comprehension task.

2 Related work

Curiosity and the related metacognitive skills

Looking at the relationship between curiosity-driven learning and metacognition, we find four relevant articulations : 1) the activation of epistemic curiosity mobilizes the metacognitive ability to evaluate ones own knowledge and, more specifically, to be aware of their learning deficiencies and evaluate how close they feel to compensate for them. This is known as the information-gap theory introduced by Lowenstein in [START_REF] Loewenstein | The psychology of curiosity: A review and reinterpretation[END_REF]. 2) the sustainability of epistemic curiosity states is fostered when the reward offered by the subsequent gained information is anticipated using metacognition, i.e. predictive judgment or guessing [START_REF] Murayama | Process account of curiosity and interest: A reward-learning perspective[END_REF].

Indeed, making educated guesses about uncertainties before looking to resolve them can amplify feelings of curiosity and guide individuals in their information-seeking behaviors in order to find specific evidence to accept or reject their hypothesis [START_REF] Gruber | How curiosity enhances hippocampus-dependent memory: The prediction, appraisal, curiosity, and exploration (pace) framework[END_REF][START_REF] Gross | Cultivating an understanding of curiosity as a seed for creativity[END_REF]. 3) efficient curiosity-driven behaviors require metacognitive regulation skills : these latter are required for conscious information seeking, for evaluating the value of products from seeking for possible correction if unsatisfactory, and for maintaining sustained attention over long time of information search [START_REF] Murayama | Process account of curiosity and interest: A reward-learning perspective[END_REF]. And finally 4) the individuals' capacity to monitor and assess their own learning progress is also a driver for maintaining curiosity-driven learning. This is explained in the Learning Progress (LP) Theory [START_REF] Oudeyer | Intrinsic motivation systems for autonomous mental development[END_REF] suggesting that the learners become most curious when they work on tasks that provide them with an optimal level of learning progress, as perceived and evaluated by these learners themselves. Meaning that the faculty to accurately self-assess and monitor progress plays an important role in maintaining curiosity. See Appendix A for a view of the articulation between curiosity-driven learning and metacognition.

Technologies for training curiosity

Previous work has explored ways to elicit curiosity using new technology. For example, studies such as in [START_REF] Gordon | Can children catch curiosity from a social robot[END_REF][START_REF] Ceha | Expression of curiosity in social robots: Design, perception, and effects on behaviour[END_REF] showed that implementing interactions with social robots that exhibit curious behaviors had significant positive effects on children's own curiosity. Similarly, previous studies suggest high benefits for using conversational agents in education to support metacognitive strategies [START_REF] Grigoriadou | Historical text comprehension reflective tutorial dialogue system[END_REF] and higher-level thinking [START_REF] V. Aleven | Tutoring answer explanation fosters learning with understanding[END_REF]. Work in [START_REF] Alaimi | Pedagogical agents for fostering question-asking skills in children[END_REF][START_REF] Abdelghani | Conversational agents for fostering curiosity-driven learning in children[END_REF] investigated using such agents to help children practise question asking and question-guided learning and showed rather positive results on the exploratory behaviors and subsequent domain-knowledge learning progress.

Current study

Our present study offers a novel interdisciplinary approach which integrates psychological theories of curiosity with the design of novel educational technologies to provide a metacognitive training for curiosity. We propose an approach for testing these theories by operationalizing curiosity into a set of metacognitive exercises and studying their effects on students.

For this, we rely on the theories presented in the section above ( 2.1) and propose a training based on the four metacognitive skills we see involved in curiosity mechanisms : the ability to 1) identify learning uncertainties, 2) formulate educated guesses about the uncertainty, 3) lead organized information-searching behaviors to compensate for the knowledge gap and finally, 4) monitor the learning progress made through the information-seeking behavior. We call this curiosity training framework identify-guess-seek-assess.

In practice, our training consists of two parts : the first part presents declarative knowledge about curiosity and our four metacognitive skills of interest (identify, guess, seek, assess); presented in the form of animated videos. The second part of the workshop is designed with the aim to help children acquire procedural knowledge of the 4-step curiosity-based metacognitive loop. For this, participants will practice these skills explicitly during a reading-comprehension task using a web-based platform where they interact with conversational agents that help them apply each of these skills.

We conduct a pilot study to test our training with 15 primary school students. Our evaluation is based on assessing the accessibility of our content and the pre-post intervention effects on the participants' 1) metacognitive efficiency-which is a key facilitator for curiosity, 2) ability to ask curious questions about a task at hand and 3) their perception of the value of curiosity.

Methods

Experimental design and procedure

We propose a 8-session workshop, of 45 minutes each, as detailed in Figure 1.

The test sessions : sessions 1 and 8 in Figure 1 are dedicated to take the pre-and post-intervention measures related to curiosity and metacognition. Details about these measures can be found in AppendixD.

Figure 1: Timeline of the study and associated measures

The introductory sessions : During each of these sessions (2-5), we present a 4-minute animated video explaining a concept related to curiosity and/or metacognition that we created within the team. We introduce the four curiosity-related metacognitive skills during one of these videos (see Appendix B for more details about the content of these videos). We model the skills with 2D characters to make it easier for participants to remember their roles: a first controller to reflect the IDENTIFY skill, a detective to reflect the GUESS skill, a explorer to reflect the SEEK skill and a second controller to reflect the ASSESS skill. See Figure 1 for more details about the procedure and Appendix B for more details about the metacognitive characters.

The practice sessions : During sessions 6 & 7, children had to put into practice the skills they saw with the previous videos. More specifically, their task was to use a four-step learning loop (identify, guess, seek, assess) in order to acquire new and useful information about a given text. To do so, they used a web platform where they interacted with handcrafted conversational agents representing the said metacognitive characters. The agents guide participants into using these skills by reminding them of their roles : once participants finish reading the text, the first controller appears (representing the IDENTIFY skill) and guides them into identifying a knowledge gap. After validating this step, the detec-tive agent appears to guide them into formulating a hypothesis about it (GUESS skill), and so on. See Figure 2 for an example of the interface and AppendixC for examples of the agents' utterances and snaps of the interactions with all four agents. The agents had predefined scripted behaviors and did not give any feedback regarding children's inputs.

During the first two cycles, and in order to help children get familiar with the task, the agents gave a list of propositions along with the description of their roles (i.e a list of uncertainties for the IDENTIFY agent, of guesses for the GUESS agent and of questions for the SEEK agent); children were free to either use the agents' propositions or to enter their own input. During the 6 remaining tasks, children did not have this help automatically and had to explicitly ask for it if they were blocked.

In total, children had 8 texts, i.e. 8 learning cycles to complete following the four skills.

Participants

We recruited 15 CM1 students from a French primary school, aged between 8 and 10 years old, with 8 girls and 7 boys (see Appendix D for a detailed description of the participants' profiles).

The study was approved by the institute's ethics committee (certificate n2019-23) and only started after having all participants parents' signed consents which contained the study goals, procedure and the data collected.

5 Preliminary results

Accessibility of the workshop

Could children understand the content of the videos ?

During each of the 4 introductory sessions, we wrapped up the session with a 6-item quiz about the content presented in the video to assess how much children were able to understand it. Results suggested indeed a good accessibility with a mean score of M=75.99% and SD=23.62 for the four videos; details about the scores for each video can be found in Appendix E.1.

Were children able to use the four metacognitive skills to perform curiosity-driven learning ?

During the training with the web-based platform, children had to perform 8 curiosity-driven learning cycles using the four metacognitive skills. A complete cycle is one where children identify an uncertainty, generate a hypothesis for it, ask the relevant question about it and then decide whether or not they could find the answer they're looking for within a given list of propositions. Our results show that all children succeeded to complete at least four cycles during the training with a mean percentage of correct cycles completed of M=64.42% and SD=20.31%.

Pre-post intervention effects

How did the workshop affect children's metacognitive sensitivity ?

Children's metacognitive sensitivity reflects how accurate they are in judging their own level of knowledge, it is thus a facilitator for curiosity. See AppendixD for details about how we compute this measure.

As shown in Figure 3, results of the ANOVA test show indeed a significant enhancement in children's metacognitive efficiency (F(1,28)=8.2; p-value=0.007). Furthermore, and in investigating the role of our training on this enhancement, we run two ANCOVA tests with either score at the first (i.e. scores for videos) or the second (i.e. score of completion of learning cycles) part of the workshop as a co-variate. Results showed a significant effect for the understanding of the videos on children's metacognitive sensitivity (F(1,27)=20.67; p-value=0.0001) revealing that the higher declarative knowledge they gained from the videos, the more they had extensive gains in metacognitive sensitivity.

How did the workshop affect children's curious questionasking skills ?

Asking divergent questions is a primary expression of curiosity in children and can be seen as a behavioral measure for it [START_REF] Gallagher | A preliminary report on analyses of classroom interaction[END_REF][START_REF] Alaimi | Pedagogical agents for fostering question-asking skills in children[END_REF]. To measure children's ability to formulate these questions, we use an offline test and expert annotations as described in AppendixD, before and after the workshop.

Results showed a significant progress in this ability after the workshop, as shown in Figure 4. We also tested the effect of the training on this progress using ANCOVA tests. The tests reveal a significant effect of the performance during the practice sessions on the progress in time (F(1,27)=4.21; p-value=0.05) revealing that the higher they gained procedural knowledge from the practice sessions, the more extensive they had gains in question-asking performance.

How did the workshop affect children's perception of curiosity ?

Children's perception of curiosity is an important measure for us since it affects the individuals' curiosity-related behaviors. To assess this we use Post's standardized test [START_REF] Post | Development and validation of a questionnaire to measure primary school childrens images of and attitudes towards curiosity (the ciac questionnaire)[END_REF] (see details about the test in Appendix D).

Our results of the ANOVA test show no significant difference in this measure before and after the workshop (F(1,38)=0.37 with p-value=0.5).

Discussion

The results of this study show the general accessibility of the two parts of the training and significant effects on behavioral curiosity-related measures. More specifically, we see that the first part which gives declarative knowledge about curiosity and metacognitive skills affects children's metacognitive sensitivity. While the second part that helps gain procedural knowledge these concepts affects their ability to ask curiositydriven questions. Following Anderson and Krathwohl's taxonomy of learning [START_REF] Krathwohl | A revision of bloom's taxonomy: An overview[END_REF], these observations can be rather expected since metacognitive sensitivity require declarative knowledge about what one does and does not know while asking curious questions requires procedural knowledge (i.e. being aware of a missing information and then formulating a relevant question to reach it).

However, we see no enhancement in participants' perception of curiosity with this first sample. This observation can be explained by the nature (i.e. digital interaction) and the short duration of the intervention (8 days). It is indeed well documented in social attitude literature that reliable and sustainable attitudinal changes often require longer interventions and/or with more realistic social interaction with teacher or between children [START_REF] Cordova | Intrinsic motivation and the process of learning: Beneficial effects of contextualization, personalization, and choice[END_REF].

Conclusion, limitations and and future directions

In this work, we contribute to the promotion of metacognition-based approaches to foster curiosity, while proposing novel activities. Despite these encouraging first results, more participants need to be recruited in order to replicate the findings and confirm the efficiency of the intervention. In a further step, we aim to leverage new technologies (e.g. AI-based systems) in order to have more powerful conversational agents and more fluid interactions with children.

Acknowledgments

This work has been funded by the educational technologies start-up Ev-idenceB, the French National Association of Research and Technology (ANRT), and the ANR DeepCuriosity project. The authors thank Mr Benjamin Erben for helping create the content for the educational videos and Mrs Isabelle Coutand, the teacher who participated with her students in this study and the research team members who helped conduct the experiments in classes.

A Curiosity framework with relation to the four metacognitive skills

Taking inspiration from Murayama's curiosity-driven learning framework [START_REF] Murayama | Process account of curiosity and interest: A reward-learning perspective[END_REF], we link this aptitude with four essential metacognitive skills as seen in Figure 5.

B Pedagogical material for sessions 2-5 : videos content and metacognitive characters B.1 Pedagogical content for the animated videos

The pedagogical content for the animated videos presenting declarative knowledge about curiosity and metacognitive-related skills are summarized below.

• Session 1 : The goal is for children to understand what is motivation, its types and the importance of intrinsic motivation for learning. The videos introduces motivation as a fuel for learning : it gives the brain the goals it needs to stay engaged and active in the task. It introduces two types : extrinsic motivation that relies on external rewards from the environment such as money, etc and intrinsic motivation that is generated within the individual, following his interests, personal desires ... . We explain that the latter is more important for our brain to stay engaged in a task.

• Session 2 : The goal is for children to understand what is curiosity and its importance for learning. The video introduces curiosity as a form of intrinsic motivation. It presents its advantages following the three pillars for an efficient and enjoyable learning experience : 1) allows to modulate the individuals' memory, 2) guides their attention and 3) enhances their sense of self-competency.

• Session 3 : The goal is for children to understand what is metacognition and to introduce them to the four metacognitive skills affecting curiosity. The videos introduces metacognition as a set of skills that represents 'Thinking about thinking': it facilitates a longerlasting learning and avoids the 'Illusion of Knowledge' trap. The video also introduces the four curiosity-related metacognitive skills that are important to learning : Observing and identifying uncertainty, predicting and generating guesses, pursuing knowledge gaps through question-asking and exploration and evaluating newlyacquired information.

• Session 4 : The goal is for children to understand how to use metacognition to be more curious and be a better learner. The videos presents curiosity as a malleable skill that individuals can actually control, using their metacognitive skills. It emphasizes the need to use the observational skills to identify knowledge gaps that can poke curiosity. It also talks about how to control our own curiosity in order to learn tasks that are suited to our competency levels. And finally, the video encourages the use of the evaluation skills in order to be engage in further and longer-lasting learning.

B.2 Description of the metacognitive characters

The metacognitive characters reflect the four metacognitive skills involved in curiosity-driven learning; they were first introduced in the videos. They then accompany children in the practice sessions on the training platform. They were presented with the following roles :

• The controller : when learning new content, it observes the task, reflects on its previous knowledge and chooses which uncertainty or missing information to pursue : reflects the IDENTIFY skill.

• The detective : formulates educated guesses and makes predictions about the missing information to pursue : reflects the GUESS skill.

• The explorer : pursues the uncertainty by asking the relevant questions or by exploring the relevant resources : reflects the SEEK skill.

• The second controller : evaluates whether the inquiry resulted in learning progress with respect to the initial uncertainty : reflects the ASSESS skill.

C Material for sessions 6-7 : The training platform

Examples of the utterances of the agents representing the four curiosityrelated metacognitive skills can be seen in the platform snaps (Figures 678).

D Details for data collection procedure and measures

D.0.1 Profile measures

We were left with 15 participants, the description of their profiles can be found in Table 1.

For these profile measures, we used the following standardized questionnaires :

• Curiosity trait : To assess children's curiosity trait, their parents answered Litman's questionnaire of curiosity [START_REF] Litman | The measurement of curiosity as a feeling of deprivation[END_REF]. • Perception of curiosity : This is an important measure for us as we think that one of the relevant brakes that can keep children from asking questions in the classroom is their negative perception of this behavior. To assess children's perception of this, we use Post's validated CIAC questionnaire [START_REF] Post | Development and validation of a questionnaire to measure primary school childrens images of and attitudes towards curiosity (the ciac questionnaire)[END_REF]. This questionnaire contains items to assess children's general attitude towards curiosity and asking questions, their perception of its importance and their negative ideas about it, if any.

D.1 Behavioral curiosity-related measures

Metacognitive sensitivity : This is an important measure for us as recognizing uncertainties is a key step to engage in curious learning. We measure this using a general knowledge quiz where participants report their confidence levels in their answers. We then compute the difference between the 'hit rates' i.e. the number of times their confidence judgment corresponds to their response accuracy) and the 'false alarms' (i.e. number of times they incorrectly judge their answer) [START_REF] Fleming | Hmeta-d: Hierarchical bayesian estimation of metacognitive efficiency from confidence ratings[END_REF]. Ability to ask curious questions : Asking divergent questions is a primary expression of curiosity in children and can be seen as a be- havioral measure for it [START_REF] Gallagher | A preliminary report on analyses of classroom interaction[END_REF][START_REF] Alaimi | Pedagogical agents for fostering question-asking skills in children[END_REF]. Divergent questions bring new information and require more complex reasoning and process such as linking two ideas, making hypotheses, etc. To assess this aspect, we use an offline test inspired from creativity studies such as in [START_REF] Diakidoy | Domain specificity in creativity testing: A comparison of performance on a general divergentthinking test and a parallel, content-specific test[END_REF]. This test consists of giving participants a short text to read then asking them to write as many questions about it as they can, within 2 minutes. We then count the total number of the correct questions generated during the test and compute the percentage of the divergent questions amongst them using human experts annotations. A question is accepted if it its indeed, a question not a statement, and is semantically related to the text. If a question is repeated, it is only taken into account once.

A question is counted as divergent if its answer is explicitly stated in the text. This is based on Gallagher's classification [START_REF] Gallagher | A preliminary report on analyses of classroom interaction[END_REF]. Example: For the text in the screenshots above, the question What is the temperature of the magma? is considered to be divergent, whereas "Where is the mantle situated?" is not as the answer to it is explicitly stated in the text.

This process of data coding was performed by one of the researchers that led this study. All data was anonymized: the coder could only see the identifiers that children were given randomly at the beginning of the intervention.

D.2 Learning experience measures

Motivation : The participants motivation to use the proposed platform was assessed via two questionnaires: the general motivation scale [START_REF] Cordova | Intrinsic motivation and the process of learning: Beneficial effects of contextualization, personalization, and choice[END_REF] and Vallerand's types of motivation scale [START_REF] Vallerand | Construction et validation de l'chelle de motivation en ducation (eme) [construction and validation of the motivation toward education scale[END_REF].

The general motivation scale was used to investigate the potential short-term motivation. It contains one sub-scale for evaluating participants' motivation to reuse the platform in the future, one sub-scale for evaluating their perceived competence and one sub-scale for assessing the degree of preference with respect to a favourite school activity.

On the other hand, Vallerand's scale was used to probe intrinsic and extrinsic motivational mechanisms in the educational settings we had. It is composed of three sub-scales that differentiate: intrinsic motivation, extrinsic motivation and amotivation.

Task : The participants' behavior was also evaluated in terms of the subjective workload they experienced during the intervention. For this, we used the NASA-TLX workload multi-dimensional scale developed in [START_REF] Hart | Development of nasa-tlx (task load index): Results of empirical and theoretical research[END_REF]. Information about the intensity of six workload-related factors are used in order to estimate a reliable measure of workload: mental demand, physical demand, temporal demand, performance, effort and frustration.

E Additional results

E.1 Details about the scores of the four videos

The details of the accessibility of the four videos are like the following : video 1 concerning the definition of motivation and its types (M=71.79% and SD=21.48) ; video 2 concerning the definition of curiosity and its importance to learning (M=84.05% and SD=21.6) ; video 3 concerning the definition of metacognition and the four skills involved in curiosity (M=74% and SD=25.49) ; and video 4 concerning how metacognition helps curiosity and learning (M=74.13% and SD=23.62).

E.2 How motivated were children to do the training ?

Children's general motivation to use the platform was rather high (M=37.72 and SD=10.24) for a maximum score of 54. For the types of moti-vation, children were more intrinsically than extrinsically motivated to do the training (M=76.78% and SD=30.19 for intrinsic motivation and M=66.66% and SD=28.7 for extrinsic motivation). See Appendix D for details about the scales used to measures these data.

E.3 How demanding was the training for children ?

Results of the Nasa-tlx scale [START_REF] Hart | Development of nasa-tlx (task load index): Results of empirical and theoretical research[END_REF] revealed that participants did not perceive the tasks as very demanding : M=27.57 and SD=11.17 for a total possible score of 60. See details about this scale in Appendix D.

Figure 2 :

 2 Figure 2: Agent's behavior for the fourth metacognitive skill "AS-SESS".

Figure 3 :

 3 Figure 3: Children's ability to judge the accuracy of their answers correctly was significantly enhanced after the workshop. Their understanding of the videos during the first part of the workshop had a significant effect on this progress (F(1,27)=20.67; p-value=0.0001).

Figure 4 :

 4 Figure 4: Children's ability to ask curiosity-based questions during learning was significantly enhanced after the workshop. Their performance during the second part of the workshop had a significant effect on this progress (F(1,27)=4.21; p-value=0.05).

Figure 5 :

 5 Figure 5: Curiosity-driven learning framework and link with the metacognitive skills we propose to train during our intervention

Figure 6 :

 6 Figure 6: Agent's behavior for the first metacognitive skill "IDEN-TIFY".

Figure 7 :

 7 Figure 7: Agent's behavior for the second metacognitive skill "GUESS".

Figure 8 :

 8 Figure 8: Agent's behavior for the third metacognitive skill "SEEK".

Table 1 :

 1 Profile measures for the participants

	Measure	Mean ± std
	Age	9.05 ± 0.53
	Device use frequency	30.88 ± 6.66
	Curiosity trait	27.6 ± 4.9
	Perception of curiosity 34.05 ± 8.8