domain: stat qfin

text:
Manase Bezara
email: m.bezara@istom.fr

Dr Salima Taibi
email: salima.taibi@unilasalle.fr

AN APPROACH OF PLS METHOD APPLIED TO MODEL THE RICE SELF-SUFFICIENCY OF PEASANT HOUSEHOLDS IN ATSINANANA MADAGASCAR

Keywords: peasant, multicolinearity, PLS method, model, self-sufficiency in rice C38; Q01

The aim of our study is to build a predictive model using data from survey studies. As the predictors were mostly qualitative variables, we applied an extension of PLS method. This work responds also to the need expressed by the Atsinanana agricultural region of Madagascar to strengthen the decision-making process in the fight against "no-self-sufficiency in rice" and the poverty in this locality.The study consists of constructing a simple scientific tool to improve the comprehension and to explain the links between permanent poverty and the main socio-economic factors: schooling, illiteracy, fragmentation of land, insecurity, isolation, demography) and to make effective the measures taken.The data is a random sample from a database collected in the 82 rural communes of Atsinanana by the students of the University of Toamasina and from the Campus Paysan project. We propose a Partial Least Square regression model of rice self-sufficiency.This study has helped to update the information on the subject and to deepen the knowledge on the issue. It is part of a search for improvement of statistical prediction methods by the PLS regression. We worked mainly with free software R.

Introduction

In this 21st century, the poverty rate is more than 70 % (Dabat et al., 2001). A situation has degenerated, as described in the 2016 World Bank report, which indicated for the principle Island, an alarming poverty rate of 92 %. Most of these poor people live in rural areas, depend on agriculture for their survival and are not self-sufficient in staple food, rice (UPDR- FAO-CIRAD, 2001).

In Madagascar, rural households face the scourge of widespread poverty. The enclaves of prosperity are rare. Studies have shown a certain correlation both between poverty and selfsufficiency in rice (Dabat et al., 2001).

In 2004, at the beginning of the Peasant Campus project *, the demand for rice for Malagasy national consumption indicated a gap of 200 thousand tons (campus paysan project). Then as a result of national effort, official data indicated an increase in production from 2007 to 2011, but the effort was not sustained. However, in 1970, Madagascar exported its rice production surpluses. Domestic supply satisfied domestic demand.

In 2017, the main problem in Madagascar was still the inadequacy of rice production, which appeared in several aspects: economic, political and religious. The records of the Ministry of Agriculture indicate that the country had 30 million hectares of uncultivated farmland in June 2017.

Experts pointed out that the country has potential in terms of rice production but it's handicapped by the lack of control over the means of production.

In December 2017, Madagascar was on the verge of a socio-political crisis related to rice. Domestic production was in sharp decrease. The market price has seen an extremal peak of 2450 Ar / kg = 0.78 euro, with a smic* equals 35 euros, and an average consumption 200 kg / year / Malagasian.

Questions have been raised: how to make Malagasy rice farming efficient? How to deal with this peasant hysteresis in terms of rice self-sufficiency?

The following two previous studies represent a real interest.

 The national agricultural census (RNA, 2005) establishes a diagnosis of the situation of farms in the Malagasy countryside. In fact, there is a permanent degradation of the areas resulting from parcels fragmentation, the use of hand tools such as spades, machetes and sickles for more than 90 % of farmers. We can observe little use of hitching equipment, so on the one hand, the problem of insecurity and theft operations and the decrease of rural areas has finally damaged the desire of investment.

 The analysis of rural households in Madagascar focuses on such issues as isolation and poverty and analyses the reasons why there is this rural Malagasy poverty, and how to relate the poverty level of the rural household with certain factors including isolation?

These questions represent the basis of our study focused mainly on rice farmers in the eastern region of Madagascar, formerly well-off and dynamic.

We collected, processed and analysed data on the peasantry of the Atsinanana region and sought to build a model explaining the rice self-sufficiency of rural households by projection on latent structures.

Data collection

We used data from two sources: We applied a probabilistic approach, the cluster sampling. From the 82 rural communes of the Atsinanana region, we drew a random sample of 3 communes from which we sent to all households practicing irrigated rice a questionnaire to be completed face-to-face.


We recorded an average return rate of about 35 % covering about 150 people, or around 30 households.

The variables measured are the income (or wealth) of the household in a year, the area harvested, the size of the household, the number of children (schooling, out-of-school), rice production, the level of self-sufficiency in rice, index of family literacy, landlockedness, isolation, security, ...).

Partial least square regression (PLS)

In fact, prediction problems often face the phenomenon of multicollinearity.

Correlations decrease the performances of modelling processes. Methods have been developed

to overcome this problem Breiman (2001), Taibi et al., (2010). Projection regression on latent structures or also by partial least squares is one of the most successful.

PLS is a method which performs to explain a set of Y target variables by a set of X predictors, using a construction-iterative process [START_REF] Wold | PLS-regression: a basic tool of chemometrics[END_REF]) that produces two sets of synthetic variables.

 One (tk)k, k < min (card (X), card(Y))+1 from predictive variables, two by two orthogonal.

 The other (un)n resulting from the predictive variables Y, on which the condition of orthogonality is not necessary.

The point of articulation of the iterative process is to group at each step the pairs (tk, uk) such as Max cov (tk , uk). These (tk, uk) we call "pls components" or latent structures.

Algorithm

Step 1: The first component pls, t1, is constructed from a specific linear combination of p centred explanatory variables xj.

The coefficients of the linear combination are chosen in order to better summarize the explanations for a better explanation of the target Y.

This implies:

      p k k k k p k k k y x y x x t 1 2 1 1 1 1 ) , ( cov ) , cov(   (1)
We perform a model of simple linear regression of y on t1

y = 2 1 ) , cov( 1 t y t  . t1 + y1 , y1 (2)
is the residual vector.

Then:

1 1 1 11 1 1 1 1 11 1 ..... . . ...... . . y x a x a y x c x c y p p p p           (3)
The translation to a matrix writing gives us.

1 1 1 1 1 1 1 1 1 1 . ; . ; . ; . .    c a t t t y c X t y X y X t t t t     (4)
Step 2: We construct the second component pls t2, uncorrelated to t1, which explains the residual y1 by a linear combination of xij regression residuals xj variables on t1.

The procedure is identical to that of step 1 to have (t2, w2, c2 , a2).

Step k: This procedure can be continued by iteration on the residuals. The number k of pls tk components to retain is determined by cross-validation.

Treatment and comments

Our comments are based on the indications of work of [START_REF] Tenenhaus | la regression PLS, théorie et pratique[END_REF]Tenenhaus ( , 2001) ) and Chavent et al. (2003), which will be focused on five pronunciations.

a-Total quality of regression and proportion of explained variance.

The first output of results points out the explicative performance of explicative and target variables in the building of latent variables, the number of which is limited by that of explicative variables.

Nevertheless, for targets, the calculated percentages express the explicative power of the model.

As part of our study, the first both constructed latent variables explain only 65.84 % of the information brought by all explicative variables. What implication for the four remaining latent variables with only 34.16 % of information. This first result also shows that the model is of a rather good quality 72.74 % changeability of targets. Therefore, our research question on poverty is explained by explicative variables. If both first latent variables are kept, this performance is still equal to 67.17 %.

Let us note that, the consideration of all latent variables identifies this performance with coefficient of determination R 2 of the linear regression. In our case, R 2 = 0.72.

b-Raw contribution in latent variables

The second output result, points out raw link (square correlation) of every attribute in latent variables. In our case, we see that four variables: enclosed-insecurity-alphabetisation-exploited area are strongly linked to the first latent variable. Their roles are therefore determinants in the explanation of this first Pls component.

Nevertheless, it's important to realise that at this level of the study, we can't state the real nature of the connection yet. We speak about absolute intensity of the link.

As for targets, we see that the variables: degree of poverty, family ease, family income, selfsufficiency in rice are explained well. Indicators in the output show us it: 77.55 % (degree of poverty), 68.48 % (family income), 64.75 % (self-sufficiency in rice).

Then, it was possible to quantify correspondence.

The second latent variable contains 11.4 % available information. The variables "age of the head of household, size of household are influential. For the target variable, rice-rowing/ ha is well explained. It is explicative of the poverty at the level of 11.92 %.

Concerning the third latent variable, the proportion of explained variance is 19.6 %. Some determination of the variable indication of schooling is pointed out. But third axis doesn't give enough explanation of poverty problem.

c-Sense of links and weight of variables

Loadings bring the sense of links in the logic of correlation sign. We think here that the upper absolute value 0.40 indicates a significant correlation. On the other hand, the weight of explicative variables indicates the role of each variable in the explanation of six Pls components. We can see that it's a duplication of X-loadings.

Variable importance in projection (VIP)

VIP indicator indicates the important level of explicative variables in the explanation of the target.

 VIP ≥ 1, high of explanation.

 VIP < 0.70 + coef( regress) ~ 0 , less importance.

In the case of our study, we have aggregated the VIP indicators as shown in Table 2. We can notice that, every column gives an account of the evolution of the role of corresponding attribute. The evolution of stocks in horizontal shows the influence combined by the Pls components. In our case, we kept 6 Pls components. We see easily across Pls component, the evolution and weight of 3 variables (Alphabetisation indicator, Enclosed and insecurity, Exploited area).

At the same time, the irrelevance of variable of schooling indicator is determined. This study has perhaps brought a partial explanation of the inflexible rate of descholarisation" recorded in Madagascar and it, in spite of efforts and plans of type «ept» (education for all people).

Model of prediction of self-sufficiency in Rice

We arrive at the level which allows us to establish a model of prediction by target and variables.

It can be presented in standardised coefficients form. The second case applies to the variables of origin without any prior transformation. We note that standardization allows an interpretation convenient. It is possible to put attribute themselves in comparison with others by level of standard deviation.  On predictive plan, if they augment for example the exploited area by 0.4826 ha, then the coefficient of self-sufficiency in rice grows for 0.670296*0.4239.

Conclusions

The established PLS model allowed the leaders of the 82 communal localities to have a more rational approach to the problem of no-self-sufficiency in rice, a main source of rural poverty in the Atsinanana region, with a view to improving the decision-making performance. Three main

  factors have been identified: illiteracy, isolation, land fragmentation with some ambiguity about the behaviour of the schooling indicator. A judicious choice of communal samples, and an adapted technique of data collection the elaborated PLS model would have been otherwise efficient. The difficulties are many, insofar as our data are not able to point out particular problems related to each specific municipality, but also the inexistence of statistical archives that would have helped our work. Finally, the work triggered some form of awareness-raising process among all economic and strategic actors currently seeking to set up the Atsinanana Region Observatory of Rurality or the University of Toamasina's Multidisciplinary Laboratory. Bibliography 1. Breiman, L. (2001). Random Forests. Machine Learning, 45,pp. 5-32. 2. Chavent, M., Patouille, B., (2003), Calcul des coefficients de regression et du Press en regression PLS. Revue Modulad. Vol. 30, pp. 1-9. 3. Dabat, M., Jenn-Treyer, O., Razafimandimby, S., Bockel, M.H, (2008), l'histoire inachevée de la régulation du marché du riz à Madagascar. Economie rurale. 303-305, pp. 75-89. 4. Doob, J.L. (1953) Stochastic process, Wiley, New York., p. 654. 5. Hubert, P. J. (1981) Robust Statistics, Wiley, New York.

Table 1 Performance of the variables in explanation of latent factors Latent Factor

 1

		Input variables (X)	Target Variables (Y)
		Current X	Cumulative X	Current Y	Cumulative Y
		( %)	( %)	( %)	( %)
	1	43.442	43.442	55.256	55.256
	2	22.407	65.849	11.92	67.176
	3	19.599	85.448	1.152	68.329
	4	3.937	89.384	3.293	71.621
	5	5.402	94.786	0.699	72.32
	6	5.214	100	0.053	72.374

Source: author's calculations, R software

  boost its family income. The second Pls component is procreated by no opposition. A conjunction of the age of the head of household with the size of household for a better access to a proper rice-growing can also be seen.It is also possible to comment on the indicator weight. For target variables, weight reflects their correlation with Pls component (target scores) to be predicted. It allows to estimate what is explained well by the Pls component .We cannot claim that that the first Pls component explains the conjunction principally, family income, self-sufficiency in rice, and displays opposition with variable degree of poverty.

	in rice and
	analphabete
		Poorness( degree)
	enclosed	Poorness (self-sufficiency in rice).
		Poorness (family income).
	insecurity
	exploited area

At this level, we can comment Pls component more correctly. The first Pls component is formed by a conjunction of in alphabetisation and exploited area and this, in opposition to variables insecurity, is enclosed. So, a household taught to read and write, exploiting a proper area living in a village not enclosed without too much problem of security is fortunate enough to be self-sufficient

Table 2 VIP indicator

 2

	Input	Factor 1	Factor 2	Factor 3	Factor 4	Factor 5	Factor 6
	Age household	0.5551		0.7115	0.6949	0.7152	0.715
	Size household	0.5558	0.7045	0.7084	0.746	0.754	0.7537
	Ind.Alphab	1.3869	1.2582	1.2476	1.2193	1.2134	1.2142
	School Indic	0.4172	0.4367	0.5056	0.5517	0.5505	0.5503
	Exploited area	1.4506	1.4679	1.4562	1.4277	1.4209	1.4207
	Encl insecur	1.0868	1.0352	1.0292	1.0635	1.0609	1.0608

Source: author's calculations, R software

Table 3 Parameters of the predictive model Target variable(s)

 3

		Average		1.5909	0.3182	33.7273	3.0909
		Sd		0.8541	0.4239	7.1991	1.269
	Input	Average	Sd	Ricgro (t/ha)	SelfSufRice Income/P Degpoorness
	Age household	37.6818	13.4142	0.15661	-0.007493	0.17437	-0.04387
	Size household	4.5455	1.4385	0.062203	-0.108666	-0.564393	0.525006
	Ind.Alphab. 0.4118	0.2777	0.137124	0.343618	0.302385	-0.245969
	School Indic	0.3818	0.3049	-0.100946 -0.028718	0.437846	-0.335946
	Exploi area	0.73	0.4826	0.564254	0.670596	0.68931	-0.54884
	Encl.insecur 3.4091	1.2596	0.06307	0.023467	0.288747	-0.071989
	Source: author's calculations, R software
	Where the form of model :
	SelfSufRice = 0.670596. Exploi. area + 0.343618. Ind Alphab + 0.023467.
	Encl-Insecur -(0.108666.Sizehousehold + 0.028718.School Indic + 0.007493.Agehousehold)
	It is possible to see that:

 In order of importance variables exploited area , alphabetisation indicator , enclosedinsecurity, have impacts on variable self-sufficiency in rice.

 Variables age of the head of household, schooling indicator, number of children, destabilise the self-sufficiency in rice.

Corresponding author. E-mail address: m.bezara@istom.fr,

Corresponding author. E-mail address:salima.taibi@unilasalle.fr

Corresponding author. E-mail


Ethel Djeumen

Quentin Sourisseau

Sylvain Chataigner

Romain Créac'hcadec

Marie-Odette Quemere

Jean-Philippe Court

Firas Sayed

Creep investigations on steel fasteners adhesively bonded to steel plates

Keywords: creep, adhesive bonding, fasteners, Burger model

Creep investigations on steel fasteners adhesively

bonded to steel plates E. DJEUMEN a , Q. SOURISSEAU b , S. CHATAIGNER c , R. CREAC'HCADEC d ,M.O. QUÉMÉRÉ e , J.P. COURT f , F. SAYED AHMAD g a. IFSTTAR b. IFSTTAR quentin.sourisseau@ifsttar.fr c. IFSTTAR sylvain.chataigner@ifsttar.fr d. ENSTA Bretagne, IRDL, UMR CNRS 6027 romain.creachcadec@ensta-bretagne.fr e. COLD PAD moq@cold-pad.com f. COLD PAD jpc@cold-pad.com g. COLD PAD fsa@cold-pad.com

Résumé :

L'entreprise Cold Pad a développé une solution innovante de connecteurs collés pour structures métalliques permettant d'éviter les opérations de soudure. Ceci permet de limiter la pénibilité des opérations sur site et est particulièrement bien adapté aux environnements contraignants pour lesquels les travaux à chaud sont proscrits. En outre, ce procédé permet d'éviter une fragilisation locale de la structure métallique et limite les concentrations de contrainte qui tendent à diminuer la durée de vie en fatigue. Des questions se posent cependant concernant la durabilité de la solution proposée. L'étude présentée aborde un aspect particulier de ce sujet : le comportement au fluage.

Les études menées ont porté à la fois sur la réalisation d'investigations expérimentales et de travaux de modélisation. Nous décrirons dans un premier les investigations expérimentales réalisées à échelle réelle, pour différents niveaux et types de sollicitations. Les connecteurs peuvent en effet être soumis à deux sollicitations différentes : de la traction ou du cisaillement. Les investigations ont permis d'obtenir les modes de rupture des assemblages, d'estimer la répétabilité du procédé, d'obtenir les temps à rupture sous fluage mais également de mesurer pendant l'essai l'évolution de déplacements locaux pertinents. Ceux-ci ont révélé un comportement non-linéaire proche du modèle proposé par Burger. Ce modèle a ensuite été choisi pour mener des travaux de modélisation décrits dans la seconde partie. L'implémentation de la loi sera détaillée, puis les résultats des modélisations comparés aux résultats des investigations expérimentales. Enfin, on s'intéressera à l'évolution des profils de contrainte interne pendant les essais de fluage. Des investigations additionnelles sont en cours et devraient permettre d'améliorer la stratégie de modélisation utilisée.

Abstract :

Cold Pad has developed an innovative solution alternative to welding or bolting that relies on the use of adhesive bonding to connect mechanical fasteners to existing steel structures. This allows avoiding heavy on-site operations and is particularly well adapted to applications requiring cold working. In addition, such a process prevents from local steel material embrittlement, residual stresses creation, or geometrical stress concentration that may decrease the structure's life expectancy. There is however a legitimate concern regarding the durability of structural adhesive bonding that needs to be studied. The presented investigations were carried out on one particular aspect of durability: the creep behavior of such connections. For the considered applications (constant load), it may indeed be one of the main issues that governs the connected fastener mechanical capacity. In order to be able to investigate the creep behavior of the developed solution, both experimental and numerical investigations were carried out. The experimental investigations will first be described.

Those were led on real scale assembly, at different stress levels, and under different load situations. The fastener may indeed be submitted by either predominant tension load, or shear load. The investigations allowed obtaining failure modes, repeatability, time to failure data, but also, local displacements evolution with time. The results revealed a non-linear evolution of those displacements closed to a Burger law. This model was thus chosen for the numerical modelling that will be presented in the second part. The law implementation will first be described, and the numerical results will be compared to the obtained experimental measurements. In addition, the evolution of the internal stress state inside the adhesive layer will be described and discussed. The experimental investigations are still ongoing with different parameters that may affect the creep behavior of the adhesive. In addition, the modelling strategy is still under development in order to be able to describe the whole creep process and to better describe the assembly behavior.

Introduction

Steel structures in service, especially in harsh environments, may be locally affected by fatigue and corrosion and may therefore need reinforcement or repair. In addition, most often there is some need during the structure service life to connect additional equipment, secondary structures or distribution systems. In the case of steel structures, this is carried out using bolted or welded assembly that may be detrimental regarding operation issues (e.g. hot works such as welding in an explosive and flammable environment) and may induce heavy on-site constraints for the structure exploitation. In regards with these issues, adhesive bonding solution may appear as a good solution, provided that the robustness and the durability of the assembly is confirmed. The firm Cold Pad, Paris, France is currently developing an adhesively bonded fastener to provide such a solution for offshore applications [START_REF] Court | Bonded assembly and bonding method[END_REF]. They are also developing an overall solution that includes an application disposal ensuring precise control of the adhesive joint quality during application, and water tightness of the joint to prevent any durability issue linked to wet environment [START_REF] Hashim | Adhesive bonding of thick steel adherends for marine structures[END_REF] (Figure 1). Durability of adhesively bonded joint covers several damage mechanisms that may be induced by the environment or the applied load [START_REF] Kinloch | The science of adhesion -Part 2: Mechanics and mechanisms of failure[END_REF]. In the case of the studied bonded fasteners, as water tightness is ensured, creep appears to be one of the main durability issues that needs to be addressed [START_REF] Godzimirski | Numerical analysis of long-lasting strength of adhesive bonds[END_REF]. Most often, experimental creep studies of adhesively bonded joints are carried out at the scale of the adhesive alone [START_REF] Broughton | Creep testing of adhesive joints analysis of creep rupture data[END_REF]. Yet, it is well known that stress states at full scale of the bonded assembly are strongly non-uniform [START_REF] Cognard | Numerical analysis of the stress distribution in singlelap shear tests under elastic assumptionsapplication to the optimisation of the mechanical behaviour[END_REF]. It was thus decided for this study to realize experimental investigations at the scale of the bonded fastener, and to develop in parallel a finite element model. This model takes into account the creep behaviour of the adhesive in order to be able to visualize the internal stress state of the joint during the test and to compare some local measurements with numerical results. A short literature review on the main available creep models will first be introduced. Then, the developed experimental investigations will be described. Their aim was to investigate the two main loadings that may encounter the developed bonded fasteners: tension loading perpendicular to the bonded surface, and shear loading parallel to the bonded surface. After a presentation of the obtained results, the developed finite model will be introduced and the obtained results analysed. The last part will be dedicated to the comparison of the numerical model with the experimental results.

Literature review on creep modelling

Creep is the response of a material to a constant stress [START_REF] Salençon | Viscoélasticité pour le calcul des structures, l'École polytechnique Edition[END_REF]. The creep is characterized by three stages as illustrated in Figure 2 delimited by the evolution of the strain rate: primary stage with a decreasing strain rate, secondary stage with a constant strain rate and tertiary stage with an increasing strain rate until the failure. Mathematically, the creep strain ε (t) should be noted in the form of equation ( 1) [START_REF] Salençon | Viscoélasticité pour le calcul des structures, l'École polytechnique Edition[END_REF].

ε (t) = σ 0 J(t,T,σ 0 )
(1) J being creep compliance that depends mainly on time t, temperature T and applied stress σ 0 [START_REF] Billon | Propriétés et comportement mécanique des polymères thermoplastiques[END_REF]. Two main categories of modelling strategies exist in the literature: the empirical or semi-empirical approaches and the rheological ones. Main models will be described in the two following parts.

Semi-empirical models

Various mathematical forms have been empirically proposed to describe the mechanical behaviour under creep loading. The simplest is a linear function of the stress but non-linear (logarithmic for example) when depending on time. The most commonly used expression of creep compliance is a power law given by equation ( 2) [START_REF] Penny | Design for creep[END_REF][START_REF] Da Silva | Handbook of adhesion technology[END_REF]. J(t) = J 0 + J 1 t n

(2) where (J 0 ; J 1 ; n) are material parameters of the model.

Other empirical models exist. An example is that of Bailey Norton which allows to take into account the nonlinear behaviour and the influence of the temperature. The expression of creep compliance is then given by equation (3) [START_REF] Penny | Design for creep[END_REF]. J (t, σ, T) = C 0 σ C1 t C2 exp(-C T /T)

(3) (C 0 ;C 1 ;C 2 ;C T ) being the material constants of the model. Another example of a nonlinear model is given by equations ( 4) [START_REF] Dean | Modelling creep in toughened epoxy adhesives for finite element analysis[END_REF]. This empirical model is based on the fact that increasing the stress level has the effect of reducing the relaxation time t 0 . J(t, σ) = J 0 exp(-t/t 0 ) m (4) t 0 = A exp(-α σ 0 2 ) m where (A; α) are material parameters of the model.

Rheological models

The creep behaviour can also be described by a rheological approach with basic mechanical systems. Creep compliance is obtained by summing up the compliances of the Kelvin-Voigt and Maxwell models (equation 7) [START_REF] Godzimirski | Numerical analysis of long-lasting strength of adhesive bonds[END_REF].

J(t) = 1/E 1 + t/ɳ 1 + 1/E 2 [1 -exp (-E 2 / ɳ 2 .t)]
(7) where (E 1 ; E 2 ) are the spring parameters and (ɳ 1 ; ɳ 2 ) the viscous ones.

3D creep

The concept of creep can be generalized in 3D. The creep compliance J is in this case a tensor of order 4. The response to 3D creep is of the form of equation ( 8) [START_REF] Salençon | Viscoélasticité pour le calcul des structures, l'École polytechnique Edition[END_REF]. ε (t) = J : σ 0 (8) In the hypothesis of an isotropic linear viscoelastic behaviour, the relation ( 8) takes the form of equation ( 9) [START_REF] Salençon | Viscoélasticité pour le calcul des structures, l'École polytechnique Edition[END_REF].

ε (t) = (1 + ν(t)) J(t)σ 0 -ν(t)J(t)tr(σ 0 )I (9)
where J(t) and ν(t) are respectively the uniaxial creep compliance and the Poisson's ratio determined from a simple tensile creep experiment carried out in any direction. In general, the Poisson's ratio doesn't depend on time, ν(t) = ν [START_REF] Thai | Modélisation micromécanique et simulation numérique du fluage des bétons avec prise en compte de l'endommagement et des effets thermo-hydriques[END_REF].

It was decided to focus in the presented work on the first two stages of creep relying on the use of the presented Burger rheological model and to make a hypothesis of an isotropic behaviour.

Experimental investigations on adhesively bonded fasteners

The led investigations have been carried out at full scale of the developed adhesively bonded fasteners. Preliminary quasi-static tests have been realized to determine the ultimate capacities of the assemblies for two solicitations: tension loading (perpendicular to the bonded surface) and shear loading (parallel to the bonded surface). Then, creep investigations have been carried out for the two solicitations.

Description of the tested specimens

The tested steel fasteners have been developed by Cold Pad, Paris, France and are called C-Claw TM [START_REF] Court | Bonded assembly and bonding method[END_REF]. Two different fasteners were tested: one dedicated to a shear dominant solicitation mode: fastener S, and the other one dedicated to a tension dominant solicitation mode: fastener T. A photo of both studied fasteners T and S is given in Figure 4. Both consist in a circular steel plate with a connected steel stud (i.e. threaded rod). The circular steel plate geometry has been adapted to the applied load. These fasteners have been bonded to square 20 mm thick steel plates. The studied adhesive is a commercial methacrylate adhesive whose main characteristics are given in table 1 where E are respectively the Young's modulus and RT, the Tensile strength. The surface preparation has been carried out in agreement with the adhesive specifications. Two specific test settings have been developed to be able to apply the selected solicitation using a hydraulic test machine both in quasistatic and creep situations (Figure 5).

Tension test protocol

The tensile test was performed with the tensile fastener bonded to a 20mm thick plate bolted to a tensile frame. The assembly was connected to a hydraulic test machine via the tensile frame for the upper grip and a connection sleeve for the lower grip as shown in Figure 6. Alignment of the frame and the connection sleeve was essential to macroscopically ensure a load in pure tension and limit the cleavage of the assembly. The loading was applied at a speed of 1.2 kN/s. Three LVDT sensors placed at 120° measured out-of-plane displacements during the test as shown in Figure 5. It allowed obtaining indication on the symmetry of the test, but also on the long-term behaviour during creep tests.

Shear test protocol

The shear test in principle was identical to that in tension. The shear fastener was bonded to a 20mm thick plate bolted to the shear frame. Figure 7 presents the assembly used for the realization of shear tests. The assembly was connected to the loading machine via a shear frame for the upper grip and the connection plate for the lower grip. For the same reason as in tension, the alignment of the shear frame and the connection plate had to be achieved to limit cleavage of the assembly. Loading was done at the same speed of 1.2kN/s. A sensor makes it possible to measure the displacement in the plane (sensor 1) and three sensors placed at 120° made it possible to measure the out-of-plane displacement (sensors 2, 3, 4) during the test as illustrated in Figure 7.

Preliminary quasi static investigations

For both solicitations, preliminary quasi static investigations up to failure were carried out to determine the ultimate capacities of the bonded assemblies. Three tests for each configuration were realized at 25 °C. Similar capacities were obtained for both configurations.

Results of tensile creep tests

It was decided to carry out two creep tests in tension at an applied load corresponding respectively to 59 % and 69 % of the ultimate quasi static load capacity.

Results of shear creep tests

The shear creep tests were carried out for closed values of the percentage of the applied load in comparison with the quasi static ultimate load capacity: 59 and 67 % of the ultimate load capacity in shear.

4

Modelling of the tests

Development of models

For both solicitations, finite element analyses were carried out using the commercial Software Abaqus, Paris, France. To model the tensile test, it was possible to use a 2D axisymmetric model (Figure 10) by making the assumption that the bolted zones were sufficiently far from the adhesive layer. To further simplify, the assumption was made that the support parts (tensile frame, sleeve) were dimensionally stable and did not need to be taken into account in the model. The behaviours of the adhesive, the plate and the tensile fastener were assumed to be isotropic linear elastic. In a bonded assembly, the accuracy of the stress state is dependent on the number of elements in the thickness of the adhesive [START_REF] Cognard | Influence of the geometry of coaxial adhesive joints on the transmitted load under tensile and compression loads[END_REF]. A convergence study of the mesh allowed obtaining the necessary mesh refinement: 12 linear rectangular elements were used in the thickness of the adhesive to obtain representative results. On the whole, the assumptions made for the shear calculation were the same as in tension. Only the plate, the adhesive and the non-cylindrical part of the fastener were modelled. It was necessary to use a 3D model as the loading was not axisymmetric. The 3D shear model had a plane of symmetry shown in Figure 12 below. This plane of symmetry made it possible to model only half of the total geometry. The boundary conditions used for the shear calculation were shown in Figure 11.

Study of stress distribution and 2D/3D comparison

As preliminary investigations, the modelling was realized in elastic situation taking into account the obtained average quasi static ultimate capacities for both solicitations. For both cases, several stresses are studied: shear (SXY) and peel stresses (SYY), von Mises equivalent stress (SVM) and hydrostatic pressure (p).

For the case of tension load, Figure 12 presents the normalised stress states (in comparison with the ultimate von Mises stress) along the normalised bonded length (on one radius, as the assembly is axisymmetric). For each studied stress component, three positions in the adhesive layer were studied: top of the adhesive closed to the fastener, middle of the adhesive and bottom of the adhesive closed to the steel plate. It can be observed that the position in the adhesive thickness does not seem to have a great influence on the studied results. In addition, it can be highlighted that although the macroscopically applied load is a tensile stress, the stress state in the adhesive is not simple. The multiaxial stress state in the adhesive is explained by the non-regular geometry of the tensile fastener. In the middle of the adhesive layer, there seems to be mostly tensile stresses in the adhesive, whereas, far from the centre, shear stress seems to be predominant. Interestingly, the geometry of C-Claw TM seems to avoid obtaining stress concentrations at the free edges of the assembly in tension.

For the case of the shear load, the results in terms of normalised stresses (relatively with the maximum von Mises stress) along the main diameter (oriented along the load axis) are given in Figure 13. Again, there is no large difference between the different studied adhesive thickness locations. The obtained stress state is also multiaxial with a non-uniform shear stress and an equilibrated peel stress profiles (due to the applied moment). Interestingly, it can also be checked that the geometry of C-Claw TM seems to avoid obtaining stress concentrations at the free edges of the assembly in shear.

SVM

Determination of creep parameters from the measured displacements

The creep law used to model the creep behaviour of the adhesive is Burger's law. This is the most used law for polymeric materials. Moreover, the observation of the evolution of displacements measured during creep tests seems to be rather close to the Burger model.

The identification of Burger's law parameters is done by hand for both assemblies assuming a uniaxial tensile stress for tensile creep test and uniaxial shear stress for shear creep test. The used stress for identification is the average tensile and shear stress along the adhesive respectively for tensile and shear creep test. The obtained parameters are given in the table 3. The model is applied to the adhesive material in Abaqus software using a subroutine especially developed for the application.

Study of the Von Mises stress distribution during creep

Tension load

Figure 16 shows the comparison between the mean creep displacement measured during the tests (mean displacement of the three sensors placed at 120°) and the creep displacement determined numerically for the two tested loads. The differences between experimental and numerical curves were negligible. It was possible to model with good accuracy the creep displacements for the first two stages at different levels of loads. Yet, it must be highlighted that it was necessary to modify Burger's parameters and that they seem consequently to be dependent on the load level. Again, the used modelling approach seems to be well suited to the modelling of the creep of the assembly. Similarly, to the tension case, the Burger parameters had to be modified indicating a dependency of these parameters to the applied load. As far as out-of-plane displacements were concerned, there seemed to be large differences between the experimental measures and the model (

Shear load

Conclusions

If the interest in using adhesively bonded fasteners exists for steel structures, there have been few studies presenting experimental investigations led at the scale of the fastener. The presented study is one of those and aimed at initiating investigations on the creep behaviour of such fastener. The presented work investigated C-Claw TM solution developed by Cold Pad, Paris, France with a methacrylate adhesive system [START_REF] Court | Bonded assembly and bonding method[END_REF]. After a specific development on the test protocol, two main load situations that may be encountered on site were studied: tension and shear. Preliminary investigations allowed determining the ultimate capacities and the repeatability of the assembly process. A modelling work was then carried out using commercial finite element software Abaqus, Paris, France with specific development dedicated to the integration of Burger's isotropic creep model. This work aimed at first in analysing the initial stress state in both load situations (tension and shear). It was proved that for both cases, stress is not uniform and is actually composed of peel and shear components (or pressure and von Mises stresses) depending on the fastener's geometry and the applied load.

Creep modelling was then carried out with an initial determination of Burger's parameters from the obtained experimental results. A good agreement was obtained between the measured average main displacements (out of plane in the case of tension and in plane in the case of shear), and the numerically obtained ones provided that Burger's parameters were dependent on the applied load. The stress redistribution during creep was also highlighted for both load situations during stages 1 and 2. This in agreement with other studies led on different applications [START_REF] Roy | Nonlinear viscoelastic analysis of adhesively bonded joints[END_REF][START_REF] Su | Two-dimensional creep analysis of structural adhesive joints[END_REF][START_REF] Houhou | Analysis of the nonlinear creep behaviour of FRP-concrete bonded assemblies[END_REF][START_REF] Reza | The effect of viscoelasticity on creep behaviour of double-lap adhesively bonded joints[END_REF]. In the case of shear load, a closer study of the measured displacements allowed determining large differences for the out-of-plane displacement. This seems to indicate that there may be an anisotropic behaviour of the adhesive that is not currently taken into account in the modelling. Additional investigations are under progress to improve Burgers parameters identification, and study the possible implementation of anisotropic creep behaviour for stages 1 and 2. There is also a strong need for a model that would consider stage 3 to be able to model entirely creep process of such a bonded assembly [START_REF] Zehsaz | Parametric study of the creep failure of double lap adhesively bonded joints[END_REF].

Fig. 1 :

 1 Fig. 1: Photo of the solution developed by Cold Pad (application disposal on the left and adhesively bonded fastener on the right)

Fig. 2 :

 2 Fig. 2: Creep stages.

Figure 3 .

 3 a and Figure3.b show the rheological formulation and the representation of elasticity by the spring (E) and the viscosity by the dashpot (ɳ), where σ is the stress in the system and ε is the strain. Kelvin-Voigt's model combines spring and dashpot in parallel (Figure3.c). It is a solid model at infinity[START_REF] Salençon | Viscoélasticité pour le calcul des structures, l'École polytechnique Edition[END_REF]. The expression of creep compliance is given by equation[START_REF] Broughton | Creep testing of adhesive joints analysis of creep rupture data[END_REF].J(t) =1/E [1-exp(E/ɳ .t)](5) ɳ/E being the characteristic time of the model.

Fig. 3 :

 3 Fig. 3: Rheological creep models.

Fig. 4 :

 4 Fig. 4: Photo of studied C-Claw TM steel fastener T on the left and S on the right.

Table 1 :Fig. 5 :

 15 Fig. 5: Scheme of the two tested solicitation modes: Tension and Shear loadings.

Fig. 6 :

 6 Fig. 6: Protocol of the tensile test.

Fig. 7 :

 7 Fig. 7: Protocol of shear test.

Figure 8 . 8 :

 88 a and Figure8.b shows the evolution of the experimentally measured out-of-plane displacements for the tensile creep tests for these two respective loads. The test at 59% of the ultimate load capacity lasted 10 hours 36 minutes before being interrupted. Only the first two creep stages are visible in Figure8. The test at 69% of the ultimate load capacity lasted 3 hours 16 minutes before failure occurred. The displacement measured by the sensor 3 made it possible to distinguish the three creep stages. The first stage lasted about 30 minutes and the second stage 2.5 hours before the final stage started (stage 3). Interestingly, in both cases, it can be noticed that measured displacements are highly asymmetric. There seems to be a localisation of the phenomenon. Obviously, creep phenomenon was more pronounced when the applied load is higher. a) 59 % T ult b) 69 % T ult Fig. Measured displacements during tensile creep investigations at T = 25°C

Figure 9 . 9 :

 99 a and Figure 9.b shows the results of the measured displacements for these two shear creep tests. Only two displacements are presented: sensor 1 represents the displacement in the direction parallel to the bonded surface, sensor 3 represents the peel displacement in the extremity of the assembly. The sensors 2 and 4 measured very low displacements and are therefore not plotted here. a) 59 % S ult b) 67 % S ult Fig. Creep shear at T = 25°C. The test at 59 % of the ultimate load capacity in shear S lasted 15 hours before being interrupted. Only the first two creep stages were measured. The in-plane displacement of the shear fastener measured by the sensor 1 was 221 µm (5 times the static displacement during the loading). The out-of-plane displacement of the fastener measured by the sensor 3 has reached 74 µm (6 times the static displacement). Only the first two stages were measured. The test at 67 % of the ultimate load capacity S lasted 25 minutes before failure of the assembly for an in-plane displacement measured by sensor 1 equal to 380 µm and an out-of-plane displacement (sensor 3) of 230 µm. The three creep stages were measured in this case.

Fig. 10 :

 10 Fig. 10: Scheme of the studied 2D axisymmetric model.

Fig. 11 :

 11 Fig. 11: Boundary conditions for the shear model.

Fig. 12 :

 12 Fig. 12: Stress state for 2D axisymmetric model.

Fig. 13 :

 13 Fig. 13: Stress state along the adhesive for the shear model.

Figure 14 illustrates

 14 Figure 14 illustrates the phenomenon of stress redistribution for the tensile model that occurs due to the initial non-uniform distribution of stresses along the adhesive. Figure 14 b represents the von Mises stress state along the radius of the bonded surface during creep. Figure 14 c shows the average tensile displacement during creep and the localization of the three studied time moments. Interestingly, the stress redistribution with time allows decreasing stress concentration effect, and more particularly the ultimate von Mises stress by 23 %.

Figure 15 Fig. 14 :Fig. 15 :

 151415 Figure15illustrates the phenomenon of stress redistribution in the case of the shear test. This redistribution was less pronounced in shear than in tension because, initially, the distribution is more uniform in shear than in tension. The maximum von Mises stress after creep was reduced by 14%.

Fig. 16 :Fig. 17 :

 1617 Fig. 16: Experimental and numerical comparison of out-of-plane displacement for tensile creep test at 25°C.

Figures 17 and 18

 18 Figures 17 and 18 represent the comparison of experimental and numerical results of out-of-plane (sensor 3, Figure 17.b and Figure 18.b) and in-plane (sensor 1, Figure 17.a and Figure 18.a) displacements for shear creep tests with the two studied load levels. The difference between the experimental curve and the numerical one for the creep displacement in the plane was negligible

Figure 17 .Fig. 18 :

 1718 Fig. 18: Experimental and numerical comparison of creep displacements (in plan and out-of-plane) for shear test at 67 % and 25°C.

Table 3 :

 3 Burger's law parameters

	Applied load	E 1 (MPa)	η 1 (MPa.h)	E 2 (MPa)	η 2 (MPa.h)
	Tension (59%)	1200	5000	400	300
	Tension (69%)	1200	750	220	30
	Shear (59%)	1200	1700	40	35
	Shear (67%)	1200	19.83	100	2.083


Erva Nihan Kandemir
email: erva.nihan.kandemir@ens.psl.eu

Jill-Jênn Vie
email: vie@jill-jenn.net

Adam Hegel

Sanchez Ayte

Olivier Palombi
email: olivier.palombi@univ-grenoble-alpes.fr

Franck Ramus
email: franck.ramus@ens.psl.eu

Adam Sanchez
email: adam.sanchez@uness.fr

Adaptation of the Multi-Concept Multivariate Elo Rating System to Medical Students' Training Data

Keywords: Human-centered computing → User models;, Applied computing → E-learning knowledge tracing, Elo-based learning model, logistic regression

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Over the last decade, as a result of the increasing use of educational technology involving significant amounts of data and learning analytics, personalized learning has gained increasing popularity. This has led a number of research groups to study the adaptation of personalized learning into educational technologies, leveraging insights from learning analytics [START_REF] Aljawarneh | Data Science for Analyzing and Improving Educational Processes[END_REF][START_REF] Lee | Learning Analytics: Current Trends and Innovative Practices[END_REF]. Adaptive Learning Systems (ALSs) [START_REF] Lee | Adaptive Instructional Systems[END_REF] achieve personalized learning experiences by using users' prior interactions and by adjusting the learning content to match individual preferences and requirements. Research consistently highlights the effectiveness of ALSs when compared to non-adaptive systems, resulting in improved learning outcomes [START_REF] Lindsey | Improving Students' Long-Term Knowledge Retention Through Personalized Review[END_REF][START_REF] Ma | Intelligent Tutoring Systems and Learning Outcomes: A Meta-Analysis[END_REF][START_REF] Behzad Tabibian | Enhancing Human Learning via Spaced Repetition Optimization[END_REF][START_REF] Vanlehn | The Relative Effectiveness of Human Tutoring, Intelligent Tutoring Systems, and Other Tutoring Systems[END_REF] and a positive impact on student motivation [START_REF] Abdi | A multivariate ELO-based learner model for adaptive educational systems[END_REF], engagement [START_REF] Papoušek | Impact of Adaptive Educational System Behaviour on Student Motivation[END_REF], and comprehension [START_REF] Alamri | Using Personalized Learning as an Instructional Approach to Motivate Learners in Online Higher Education: Learner Self-Determination and Intrinsic Motivation[END_REF]. Today, prominent ALS platforms like Duolingo, and ALEKS deliver high-quality learning materials and personalized instruction to millions of users worldwide.

With all the benefits listed above, this adaptive method in online education requires effectively monitoring users' learning paths, a procedure called knowledge tracing. This knowledge-tracing process involves creating learner models based on student performance and interactions with the system to represent their ability levels and the difficulty of educational materials.

Background

Various models have been designed to monitor and predict students' evolving knowledge levels over time [START_REF] Brooks | Predictive Modelling in Teaching and Learning[END_REF]. These models can be broadly classified into four categories: Markov process models [START_REF] Corbett | Knowledge Tracing: Modeling the Acquisition of Procedural Knowledge[END_REF][START_REF] Gweon | Tracking student progress in a game-like learning environment with a monte carlo bayesian knowledge tracing model[END_REF], logistic models [START_REF] Benoît Choffin | DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills[END_REF][START_REF] Lindsey | Improving Students' Long-Term Knowledge Retention Through Personalized Review[END_REF][START_REF] Rasch | Studies in Mathematical Psychology: I. Probabilistic Models for Some Intelligence and Attainment Tests[END_REF][START_REF]Handbook of Modern Item Response Theory[END_REF], deep knowledge tracing models [START_REF] Chan | Clickstream knowledge tracing: Modeling how students answer interactive online questions[END_REF][START_REF] Ghosh | Context-aware attentive knowledge tracing[END_REF][START_REF] Piech | Deep Knowledge Tracing[END_REF][START_REF] Sherry Ruan | Variational deep knowledge tracing for language learning[END_REF][START_REF] Shin | Saint+: Integrating temporal features for ednet correctness prediction[END_REF], and rating systems [START_REF] Abdi | A multivariate ELO-based learner model for adaptive educational systems[END_REF]. The first three classes of these models are well-established and already extensively documented in existing literature [START_REF] Ghodai Abdelrahman | Knowledge tracing: A survey[END_REF]. Although they exhibit strong prediction capabilities when evaluating student performance, they are not without limitations, particularly when deploying them in online educational environments, where they often demand intricate parameter estimation and calibration procedures, typically relying on large datasets. This complexity can impede the development of adaptive systems, making them more challenging, time-consuming, and resource-intensive to create.

An intriguing alternative is the utilization of rating systems, offering a computationally more economical approach. Rating Systems, particularly the Elo Rating system [START_REF] Elo | The Rating of Chessplayers: Past and Present[END_REF], have been widely applied in educational technologies [START_REF] Attali | A Ranking Method for Evaluating Constructed Responses[END_REF][START_REF] Klinkenberg | Computer Adaptive Practice of Maths Ability Using a New Item Response Model for On-the-Fly Ability and Difficulty Estimation[END_REF][START_REF] Papoušek | Should We Give Learners Control Over Item Difficulty?[END_REF][START_REF] Pelánek | Applications of the Elo Rating System in Adaptive Educational Systems[END_REF]. Originally created for ranking chess players, the Elo rating system has been repurposed for educational settings, treating users and learning materials as opponents. In the educational context, it predicts ratings for users and questions, serving as an assessment of user ability and question difficulty.

In this framework, each user 𝑢 is associated with a global ability parameter denoted as 𝜃 𝑢 . Similarly, for each question 𝑖, there exists a question parameter 𝜃 𝑖 reflecting the difficulty level of that question. The probability of a user 𝑢 correctly attempting a multiple-choice question 𝑖, denoted as Pr(𝑎 𝑢𝑖 = 1|𝜃 𝑢 , 𝜃 𝑖 ), can be expressed as a logistic function of the difference between the user and question parameters:

Pr(𝑎 𝑢𝑖 = 1|𝜃 𝑢 , 𝜃 𝑖 ) = 𝜎 (𝜃 𝑢 -𝜃 𝑖 ) = 1 1 + 𝑒 -(𝜃 𝑢 -𝜃 𝑖 )
After a user 𝑢 attempts question 𝑖, both the user's ability and the question's difficulty undergo updates that are proportional to the difference between the estimated probability and the actual outcome. These updates are defined by the following formulas for question difficulty (𝜃 𝑖 ) and for user ability (𝜃 𝑢 ):

𝜃 𝑖 := 𝜃 𝑖 + 𝐾 (Pr(𝑎 𝑢𝑖 = 1|𝜃 𝑢 , 𝜃 𝑖 ) -𝑎 𝑢𝑖 ) 𝜃 𝑢 := 𝜃 𝑢 + 𝐾 (𝑎 𝑢𝑖 -Pr(𝑎 𝑢𝑖 = 1|𝜃 𝑢 , 𝜃 𝑖 )) (1)
Here, 𝑎 𝑢𝑖 represents the actual outcome of the attempt of the user 𝑢 on question 𝑖, and 𝐾 is a constant value that determines the degree of update sensitivity based on the user's most recent attempt. The choice of the constant 𝐾 in the update rule plays a pivotal role in shaping estimation dynamics. If 𝐾 is set too low, the estimation process progresses too slowly, leading to prolonged uncertainty in skill assessment and potential failure to reach correct values. Conversely, if 𝐾 is set too high, the system is unstable, heavily influenced by recent attempts, and thus provides erratic evaluations.

In light of this formulation, the classical Elo rating system in education reveals its iterative nature, refining user and question parameters after each interaction.

While the Elo rating system does not provide statistically guaranteed estimations, in contrast to well-calibrated logistic models, numerous studies have explored the accuracy of the Elo rating system and compared its performance to state-of-the-art models. For instance, study [START_REF] Wauters | Item Difficulty Estimation: An Auspicious Collaboration Between Data and Judgment[END_REF] found that the IRT-Rasch model version, proportion correct, and the Elo rating system, increasingly correlate with the true difficulty parameter as sample size increases. Another study [START_REF] Papoušek | Adaptive Practice of Facts in Domains with Varied Prior Knowledge[END_REF] compared Elo rating's question difficulty estimates with those obtained through the joint maximum likelihood method (JMLE) and observed nearly identical outcomes. Studies using simulated data [START_REF] Antal | On the Use of Elo Rating for Adaptive Assessment[END_REF][START_REF] Pelánek | Application of Time Decay Functions and the Elo System in Student Modeling[END_REF][START_REF] Pelánek | Applications of the Elo Rating System in Adaptive Educational Systems[END_REF] have also concluded that Elo rating systems perform similarly to the Rasch model, making them suitable for systems requiring real-time user knowledge adaptation without the need for extensive question pretesting on large sample sizes.

However, integrating the Elo rating algorithm into a real-time educational context presents challenges, especially in the initial stages when student abilities and question difficulties are unknown and are set to zero. Given the iterative nature of the model, these initial estimates are assumed to gradually correct themselves with each attempt. While starting the model from scratch is standard practice, to produce reliable estimates the system requires a substantial number of responses, typically at least 100 attempts [START_REF] Pelánek | Applications of the Elo Rating System in Adaptive Educational Systems[END_REF]. Furthermore, uncertain initial estimates can exert a lasting influence on subsequent updates, potentially resulting in persistent inaccuracies. This issue is especially pronounced for questions and users with a limited number of attempts within the educational platform, as they may have fewer opportunities for correction.

Goals of the present study

In this study, we have two primary objectives: firstly, to assess whether the Elo rating system meets the requirements of a complex real-life scenario with specific challenges, and secondly, to address the issue of imprecise early-stage estimations in the online application of the Elo rating system. To mitigate the uncertainty associated with initial Elo rating estimations, we used data collected in the previous year.

In brief, our learning platform is open to all French medical students throughout their studies to provide training for their medical studies (about 8600 students per year over 10 years, with one important national exam at the end of the 6 th year). The challenges raised by this particular learning context are multiple:

• The knowledge corpus is huge, as it encompasses all medical knowledge taught in French universities. • This corpus is structured into knowledge components that are themselves very large: 362 distinct topics or subcategories (themes/areas) of medical knowledge, spread over 31 medical specialties. • The corpus of questions is also huge (∼1,500,000 questions), such that a given student takes only a tiny fraction of available questions (usually drawn at random), almost never takes the same question twice, and such that a given question is only taken by a tiny fraction of students (outside exams). Thus, the matrix is extremely large and sparse. • Use of the training platform is optional, with some students using it intensively on a daily basis, and others doing most of their training outside the platform. • Students are based in 42 universities which cover the same curriculum from 1 st to 6 th year, but with different material and in a different order.

Despite these difficulties, the fact that all medical students from all universities can train and take exams on the same platform should make it possible and desirable to model their progress and use this modeling to provide them with an adaptive training program. Yet the first step is to show that a sufficiently reliable modeling of their progress is possible under the present conditions.

Thus, the main research question guiding this study is to examine the boundaries of the Elo rating system within a particularly challenging real-life scenario that encompasses various complexities. This evaluation involves a comparison of its accuracy against the widely accepted logistic regression model. Additionally, we seek to explore the potential advantages of initializing the Elo system with results obtained from logistic regression applied to data from preceding years. This study employed an observational research design, leveraging ecological data from the existing BNE (Banque nationale d'entraînement) digital learning platform.

BNE Platform

The BNE digital learning system serves as an online platform extensively utilized by over 8,800 medical students across all French universities annually. This platform is used by all medical faculties to administer exams. Exam questions are then added to the question bank, together with additional questions designed by professors for training purposes. This platform therefore holds a very large set of multiple-choice questions covering 31 medical specialties that are made available to students for training. For medical students, the platform is a valuable resource to prepare for the ECN (Épreuves Classantes Nationales) final exam. This final exam usually takes place in June of the sixth academic year and significantly influences the choice of students' medical specialization.

To enhance the pedagogical engagement of medical students on the platform, a notable feature allows learners to tailor their training experience. They can choose from various question types and medical specialties, enabling them to simulate and practice for a wide range of medical exams according to their preferences and needs.

BNE data set Overview

Within this section, we describe the BNE data set for the educational year 2020-2021, representing the most recent accessible data sourced from the BNE platform during our analysis. Additionally, we describe the usage patterns observed on the platform during this year, providing valuable insights into its structure and functionality.

From the BNE platform, direct access to the official ECN exam is unavailable. However, we do have access to a mock exam, typically conducted in mid-March (specifically on March 15 th , 16 th , and 17 th , 2021, for the 2020-2021 academic year). This mock exam closely mimics the format of the actual ECN final exam. For the purpose of testing student prediction models on our complex data set, we focused our analysis on the educational year of 2020-2021, specifically targeting 6 th -year users who participated in the mock final ECN exam. This selection aligns with the core objective of our study, which is to assess the models' performance through external validation using the mock final exam.

In the following sections, we will describe the data related to the six-month training period spanning from September 16, 2020 to March 14, 2021, leading up to the mock final exam, distinct from the data associated with the mock exam itself, on March 15 th , 16 th , and 17 th , 2021.

Training Period data set. Table 1 offers a comprehensive overview of our training period data set's key characteristics, in-

cluding the total number of users (8,616), questions (357,317), medical specialties [START_REF] Behzad Tabibian | Enhancing Human Learning via Spaced Repetition Optimization[END_REF], and attempts [START_REF] Pelánek | Applications of the Elo Rating System in Adaptive Educational Systems[END_REF]772,424).

Additionally, the specialty per question variable indicates the average number of specialty tags associated with each question. Here, each of the 31 medical specialties serves as a distinct knowledge component (KC). These knowledge components are much larger The questions exhibit a spectrum of dependence on medical specialty knowledge for their solution. While a considerable portion of questions rely on ability in a single medical specialty, many questions require knowledge spanning multiple specialties.

than is usually defined in the literature. In this context, when a question 𝑖 is tagged with a specific specialty 𝑠, the likelihood of correctly answering question 𝑖 hinges upon the user's specialtyspecific ability 𝑠. Conversely, we assess a user's ability in specialty 𝑠 by evaluating their ability to successfully tackle questions tagged with the same specialty 𝑠. The table reveals that the average number of specialties associated with each question (1.58) is greater than 1, underscoring that certain questions require knowledge spanning multiple medical specialties for accurate responses. In this context, our data set can be characterized as a multi-knowledge component data set (or multi-specialty data set in the specific context of BNE). For a more in-depth exploration of this distribution, Figure 1 provides a detailed breakdown of the count of questions requiring varying numbers of specialties.

The user-question sparsity in Table 1 indicates the proportion of missing values in the user-question interaction matrix. The table shows that our data set is substantial and exhibits significant sparsity (Sparsity (user, question) = 0.99). There are vastly more available questions than any user can take, and different users will take different questions (usually by random draw).

Lastly, the Attempts per User variable unveils the average number of attempts made by the same user on individual questions. This indicates whether students frequently revisit questions they have previously encountered. With a value of 1.05, it is evident that during the training period, students almost never re-attempt questions they have already attempted.

Figure 2 provides a detailed description of the platform's usage patterns across 31 available medical specialties during the 2020-2021 educational year. As mentioned earlier, the platform provides users with the option to create their own training sessions by selecting specific medical specialties and question types they want to study. This results in varying levels of popularity among the specialties. The figure reveals that while most students engage with questions from all specialties, there is considerable variability in both the quantity of available questions and the number of questions taken within each specialty. Here, it's worth noting that all the mock exam questions were entirely new, distinct from those encountered during the training period. Therefore, the data from the mock final exam does not provide a direct test of the knowledge of specific questions taken in the training period, but rather a test of students' ability to generalize what they have learned during courses and training to new questions in the same medical specialties, mirroring the format of the official ECN exam, which emphasizes generalization rather than memorization of specific knowledge.

Elo Rating: Model Extensions for Adaptation to the BNE data set

The standard iterative formulation of the Elo rating system, which computes user and question-related factors, has been previously described in the related literature [START_REF] Antal | On the Use of Elo Rating for Adaptive Assessment[END_REF][START_REF] Attali | A Ranking Method for Evaluating Constructed Responses[END_REF][START_REF] Klinkenberg | Computer Adaptive Practice of Maths Ability Using a New Item Response Model for On-the-Fly Ability and Difficulty Estimation[END_REF][START_REF] Papoušek | Should We Give Learners Control Over Item Difficulty?[END_REF][START_REF] Papoušek | Adaptive Practice of Facts in Domains with Varied Prior Knowledge[END_REF][START_REF] Pelánek | Application of Time Decay Functions and the Elo System in Student Modeling[END_REF][START_REF] Pelánek | Applications of the Elo Rating System in Adaptive Educational Systems[END_REF]. To optimize its adaptability for educational contexts, the Elo rating system has undergone numerous extensions. In this section, we indicate the specific modifications we have applied to tailor the model to our BNE data set.

2.3.1

Incorporating the guessing behavior into the Elo rating system.

In numerous studies employing the Elo rating as a prediction model for multiple-choice questions, researchers take into account the guessing rate when calculating the probability of correctness [START_REF] Papoušek | Adaptive Practice of Facts in Domains with Varied Prior Knowledge[END_REF][START_REF] Pelánek | Applications of the Elo Rating System in Adaptive Educational Systems[END_REF].

In such instances, the probability of a user 𝑢 attempting a multiple-choice question 𝑖 with 𝑛 opt choices correctly, denoted as Pr(correct | 𝜃 𝑢 , 𝜃 𝑖 ), can be expressed as follows:

Pr(𝑎 𝑢𝑖 = 1 | 𝜃 𝑢 , 𝜃 𝑖 ) = Pr(guessing | 𝑛 opt ) + 1 -Pr(guessing | 𝑛 opt ) 1 + 𝑒 -(𝜃 𝑢 -𝜃 𝑖 )
Within the BNE question pool, questions are divided into two main types: single-and multiple-answer questions. For singleselection questions (unique answer questions), 𝑃 (guessing|𝑛 opt ) is straightforward, equating to 1/𝑛 opt . For multiple choice questions, 𝑃 (guessing|𝑛 opt ) is the inverse of the sum of the possible ways to select any number of answers 𝑘 from the available options:

Pr(guessing | 𝑛 opt ) = 1 𝑛 opt 𝑘=1 𝑛 opt 𝑘 2.
3.2 Decreasing Uncertainty. The dynamics of the Elo rating system in educational settings are crucial for accurately assessing the skills and abilities of students and questions. The challenge lies in managing evolving uncertainties, which are inherently dynamic.

When new students or questions are introduced to the platform, our information on their true abilities or difficulties is limited, meaning high uncertainty. Consequently, during this initial phase, it is essential for the model to make significant updates to its estimates. As more data accumulates, students engage in multiple attempts, and questions are extensively attempted by a set of students, and the model should naturally become more certain about its estimation of ability levels or difficulty levels. In such cases, the model should reduce the update parameter as confidence in the estimates grows.

In order to meet this challenge, recent applications of the Elo rating system in educational contexts [START_REF] Abdi | A multivariate ELO-based learner model for adaptive educational systems[END_REF] have replaced the fixed constant 𝐾 in Equation 1 with a dynamic uncertainty function. This function, denoted as 𝑈 (𝑛), is defined as:

U (𝑛) = 𝑎 1 + 𝑏𝑛
where 𝑎 is the constant hyper-parameter determining the starting value; 𝑏 is the constant hyper-parameter determining the slope of changes; 𝑛 is the number of prior attempts of the user or question parameter.

The exact parameter values, as highlighted by [START_REF] Pelánek | Applications of the Elo Rating System in Adaptive Educational Systems[END_REF], carry relatively less weight, as different choices for 𝑎 and 𝑏 tend to yield remarkably similar outcomes. In our case, we set 𝑎 = 1 and 𝑏 = 0.5 for both question and user attempts. These values were determined through an optimization process using grid search. However, it is important to mention that our model consistently delivered stable performance, and the precise choice of parameter values had only a negligible effect on the results. Moreover, in keeping with the central aim of learner models, which is to effectively track shifts in user abilities, we have introduced a lower bound for the uncertainty function applied to user ability. By incorporating this lower bound of 0.03 into the user uncertainty function, we ensured that user ability updates persist even after a considerable number of attempts. With our current values of 𝑎 and 𝑏, this lower bound applies after 65 attempts.

2.3.3

Multi-tag Knowledge Component Extension. As previously described, in our BNE data one question may be tagged by multiple specialties. To account for the ability of users in each of the tagged medical specialties separately we used the multi-concept extended version of the Elo rating introduced by [START_REF] Abdi | A multivariate ELO-based learner model for adaptive educational systems[END_REF]. The difference is that, instead of having only one global user ability parameter 𝜃 𝑢 , we estimated user ability 𝜃 𝑢𝑠 for each specialty 𝑠.It is important to note that, given the absence of information regarding the relative importance of tagged specialties for each question in the data, we adopted a straightforward approach as outlined in [START_REF] Abdi | A multivariate ELO-based learner model for adaptive educational systems[END_REF]. Specifically, we computed the mean ability 𝜆 𝑢𝑖 of student 𝑢 on question 𝑖 by averaging user 𝑢's abilities across all medical specialties 𝑠 1 , . . . , 𝑠 𝛿 tagging question 𝑖, assigning equal weight to each specialty in this calculation.

𝜆 𝑢𝑖 = 1 𝛿 𝛿 ∑ 𝑘=1 𝜃 𝑢𝑠 𝑘
Furthermore, not all specialties may have the same average difficulty level. In order to alleviate this, we define and estimate distinct parameters for question difficulty (𝑑 𝑖 ) and specialty difficulty (𝜃 𝑠 ). We denote by 𝜇 𝑖 the sum of question difficulty 𝑑 𝑖 and the average of difficulties of all skills 𝑠 1 , . . . , 𝑠 𝛿 involved in question 𝑖:

𝜇 𝑖 = 𝑑 𝑖 + 1 𝛿 𝛿 ∑ 𝑘=1 𝜃 𝑠 𝑘
Thus the probability of answering correctly becomes:

Pr(𝑎 𝑢𝑖 = 1 | 𝜆 𝑢𝑖 , 𝜇 𝑖 ) = 𝑝 (𝜆 𝑢𝑖 , 𝜇 𝑖 ) ≜ 𝜎 (𝜆 𝑢𝑖 -𝜇 𝑖 )
The update of the question difficulty 𝑑 𝑖 remains the same. However, now the update on the user skill parameters 𝜃 𝑢𝑠 occurs on each tagged specialty separately, and the update for specialty difficulty 𝜃 𝑠 follows a similar pattern as the updates for item difficulty:

𝑑 𝑖 := 𝑑 𝑖 + 𝑈 (𝑛) (𝑝 (𝜆 𝑢𝑖 , 𝜇 𝑖 ) -𝑎 𝑢𝑖 ) 𝜃 𝑢𝑠 := 𝜃 𝑢𝑠 + 𝑈 (𝑛) (𝑎 𝑢𝑖 -𝑝 (𝜃 𝑢𝑠 , 𝑑 𝑖 + 𝜃 𝑠 )) 𝜃 𝑠 := 𝜃 𝑠 + 𝑈 (𝑛) (𝑝 (𝜃 𝑢𝑠 , 𝑑 𝑖 + 𝜃 𝑠 ) -𝑎 𝑢𝑖 ) .
It's important to note that while updating 𝜃 𝑢𝑠 and 𝜃 𝑠 , the prediction formula operates at the specialty level for each tagged specialty, just like in [START_REF] Abdi | A multivariate ELO-based learner model for adaptive educational systems[END_REF], although the 𝑑 𝑖 update is based on question-level prediction.

By utilizing the Elo rating system along with the aforementioned extensions, it becomes possible to estimate three critical aspects: user ability in each specialty, questions' individual difficulty, and specialties' global difficulty.

Data Preparation Process

Before starting to train the Elo rating model and Logistic regression over the 2020-2021 data set, we performed a series of pre-processing steps on the combined data from the training period data set and the mock ECN final exam data set. These steps were carried out in the following order:

• Removal of duplicated rows: 267 rows out of 29,900,533 were removed. Figure 3 offers a visual depiction of the distribution of answers across students, questions, and specialties in both the training period and the mock final exam data sets after the pre-processing.

Information Encoding & Initialization of Elo Ratings via Logistic Regression Outputs

As previously mentioned, in the Elo rating system, initial estimates for both questions and users are typically set to 0, which can lead to high uncertainty. To address this, an alternative approach is to use the logistic regression outcomes of the previous year's data as informed initial values for initializing Elo ratings, rather than starting from scratch. With this approach, the Elo rating algorithm is anticipated to converge faster and more accurately, providing a "head start" that conserves computational resources and leads to more precise estimates.

To prepare the extensive BNE dataset for logistic regression modeling, we employed a one-hot vector encoding method inspired by [START_REF] Vie | Knowledge Tracing Machines: Factorization Machines for Knowledge Tracing[END_REF]. This technique transformed each attempt in the original data set into a sparse vector containing all relevant information. In our adaptation of this approach, we aimed to closely align our logistic regression model with the principles of Elo rating estimations, while also incorporating all the aforementioned extensions we applied for the Elo rating system. To achieve this, we included the one-hot encoding of user-specialties interaction, question, and specialty for each attempt. With this approach, the logistic regression was able to estimate users' ability in each specialty, the difficulty of individual questions, and the overall difficulty of each specialty.

To implement the initialization approach, an essential step involves comparing the logistic regression model against the Elo rating system, utilizing data from the same year. This step aimed to ensure that, before employing the logistic regression model on the previous year's data and utilizing its outcomes for initialization purposes, the model aligned with the Elo framework, generating compatible and consistent results.

Subsequently, we applied the logistic regression model to the data from the 2019-2020 educational year, utilizing the outcome estimates as initial values for the Elo rating process applied to the 2020-2021 data. Our data set for the academic year 2019-2020 (spanning from September 15, 2019, to March 1, 2020) includes 400,774 distinct questions and 25,978 unique users. However, after filtering data to retain questions and users with enough attempts (cf. above), only 47,579 questions and 8,239 users were shared between 2019-2020 and 2020-2021.

As a result, we initialized the question difficulty and student ability values for the 2020-2021 academic year using the estimates obtained from the logistic regression model applied to the 2019-2020 data, whenever these values were available. In cases where values were not present in the 2019-2020 data set for a particular question or student, we initialized their 2020-2021 values to zero.

In order to allow the uncertainty function to be able to further update those initialized values, without entirely destabilizing the estimations, we set the initial number of previous attempts to 50, which seemed a reasonable compromise between the actual number of attempts (>100 which would make any update negligible) and 0 (which would underweight the previous history).

Performance Evaluation Metrics

In our performance evaluation, we examined the effectiveness of two variants of the Elo rating system on the training data set. One variant initialized all ability and difficulty values to 0, while the other initialized values based on logistic regression from the previous year. We also compared these Elo variants with logistic regression on the entire training data set. We used several key statistics, including Area Under the Curve (AUC), Root Mean Squared Error (RMSE), and Accuracy (ACC), to assess the prediction performance of these models first on the training period and second on the mock exam.

First, to comprehensively evaluate the prediction capabilities of the Elo rating system throughout its iterations, mirroring its real-world use within the platform, and understand the impact of initializing estimates via logistic regression, we compared these three models during the training period. We calculated the AUC, ACC, and RMSE scores for each training period day from September 16, 2020, to March 14, 2021, providing insights into how these models adapted and remained robust over time.

Subsequently, we turned our attention to evaluating these models' ability to predict performance on the mock exam using the same metrics: AUC, RMSE, and ACC. However, the mock exam posed a unique challenge as it featured entirely new questions that were not part of the training period. To address this challenge, we needed to estimate the difficulty of these mock exam questions. Our approach involved combining the entire training data set with a random selection of 60% of user data from the mock exam data set as the train set while designating the remaining 40% of user data from the mock exam data set as the test set. This allowed us to create a training set that encompassed all attempts, including those from the mock exam, for 60% of users. For the remaining 40% of users, we included only their attempts from the training period in the train set. By doing so, when we ran the learner models on the training set, we obtained estimates of ability for all students and difficulty for all questions, which in turn enabled us to measure the models' prediction ability on the mock exam.

Following this data division process, the training subset comprised a substantial 24,152,933 entries, involving 8,616 unique users and 74,971 unique questions. In parallel, the test subset consisted of 1,268,752 entries, encompassing 3,447 unique users and 372 unique questions.

To assess the prediction ability of the models on the mock exam, after executing the models on the specified training set and obtaining difficulty estimates for all questions and ability estimates for all students, we evaluated its prediction performance on the test set. This evaluation capitalized on the stabilized estimates derived from the comprehensive training data set.

RESULTS

Correlation between the Estimates

Figure 4 illustrates a notable positive correlation between the estimates of question difficulty (𝑟 = 0.97) and user ability (𝑟 = 0.86) derived from the Elo rating and logistic regression models for the same-year data. This strong positive correlation clearly indicates that both models converged toward similar final estimations regarding question difficulty and user ability. Figure 5 shows the question difficulty and student ability estimations using the logistic regression in the two successive years. Specifically, for shared questions, we observe a robust positive correlation of 0.98, indicating that the difficulty levels of these questions remain relatively consistent over time. However, when it comes to students' abilities, the correlation, albeit positive at 0.54, is not as strong. This suggests that students' abilities in various specialties have undergone some changes over the years, as we anticipated. Table 2 shows the prediction performance on the mock exam for the three models: Elo rating initialized at 0, Elo rating initialized historical data, and logistic regression. These results reveal that the three models show highly similar prediction accuracy, with a slightly better performance on the Elo rating model initiated with historical data over other models.

Prediction Accuracy

DISCUSSION

This research is an initial step in integrating the multi-concept Elo rating system into our medical training platform in order to achieve real-time estimates of user performance. The results demonstrated the Elo rating system's comparable prediction power to the logistic regression models, confirming its suitability for this specific data set. The Elo rating system offers several significant advantages in the context of our medical training platform. Firstly, the multi-concept Elo rating system excels in estimating concept-level competencies, which is crucial for tailoring adaptive learning experiences in our data in which questions mostly require knowledge from multiple medical specialties. Secondly, it stands out as a computationally efficient and cost-effective option especially in real-time estimations, compared to logistic regression models which require processing a vast amount of previously collected data.

Given our primary objective of identifying the most effective prediction model for online applications in our platform, and considering the challenges associated with logistic regression in terms of online adaptability, our focus naturally shifted towards a more detailed comparison of the two versions of the Elo rating system: one starting from scratch at the beginning of the year, and one with difficulty and ability values initialized based on the previous year's data.

Our findings underscore that while the overall performance in predicting mock exam results remains very similar regardless of the initialization approach, a distinct advantage emerges for initialization based on historical data, particularly during the initial phases of iteration. This may be important in scenarios that demand accurate estimations from the outset, such as real-time or online applications. This approach of initializing the model with historical data enhances the model's ability to produce quicker and more precise estimates, thereby enhancing the reliability of personalized learning environments utilizing Elo rating systems.

Unique Characteristics of the Data set

Our data set has very broad knowledge components, made of 31 distinct medical specialties, each of which is a huge corpus of information. Moreover, this platform prioritizes comprehension over memorization: questions are hardly ever repeated twice. Students have to generalize their knowledge while attempting the questions. This departure from purely memory-based learning provides an excellent chance to assess the model's performance in dealing with non-repeated inputs. While this limited exposure to questions challenges standard prediction models used for repeated question iterations, it also allows us to evaluate the model's flexibility in settings where students rely on wider conceptual knowledge rather than memorized responses. Additionally, our platform allows students to personalize their own training experiences. They have the option of selecting the medical specialty in which they want to train and the type of questions, resulting in unique interaction patterns for each student. This adds another layer of complexity to our data set. In addition to these complexities, we also lacked control over learning occurring outside the platform.

Despite these challenges, it is remarkable that the Elo rating system has achieved significant prediction power for assessing the accuracy of students' future responses, with about 73.7% accuracy and 0.81 AUC. This demonstrates the model's adaptability and endurance in circumstances that deviate from standard educational data. In addition, the Elo rating system has shown good online prediction accuracy during training, right from the start when initializing with historical data, and after about 15-20 days when starting from scratch. With the overarching goal of transforming our medical training platform into a personalized and adaptive format through knowledge tracing methods, we carefully considered the data set's characteristics. The Elo rating model stood out as a prime choice due to its simplicity, rapid parameter estimation, and real-time knowledge assessment capabilities. Its straightforwardness, coupled with widespread use in applications like online games and chess, makes it easily explainable compared to more complex models. An exemplar of transparency in a multivariate Elo version is evident in the study by [START_REF] Abdi | A multivariate ELO-based learner model for adaptive educational systems[END_REF]. The study demonstrates the feasibility of making the algorithm transparent to students, a practice that not only heightened their motivation to engage with the platform but also enhanced their trust in the recommendations provided. The implemented extensions further enhance adaptability to our data set's unique characteristics, offering optimization along with advantages in suitability and transparency. Notably, the multi-concept Elo rating model, in contrast to its single-concept version, acknowledging the non-transitive nature of skills, provides a realistic representation of learners' capabilities, crucial for accommodating interdependencies within medical specialties, potentially involving prerequisites. Thus, the multi-concept Elo rating model emerges as a fitting and transparent knowledge-tracing method for our complex data set.

Limitations and Further Work

One major limitation is that our tested models (logistic regression and Elo rating) do not consider the natural forgetting of knowledge over time, which is well-documented in human memory research dating back to Ebbinghaus in 1885 [START_REF] Ebbinghaus | Memory: A Contribution to Experimental Psychology[END_REF]. Incorporating learning and forgetting curves into prediction models, as shown in research like DAS3H [START_REF] Benoît Choffin | DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills[END_REF] and MV-Glicko [START_REF] Abdi | Modelling Learners in Adaptive Educational Systems: A Multivariate Glicko-Based Approach[END_REF], which builds upon the multivariate-Elo rating system [START_REF] Abdi | A multivariate ELO-based learner model for adaptive educational systems[END_REF], can improve the models' prediction accuracy.

Our study also suggests several promising future research directions. One area of focus is improving learning models to better assess question difficulty and student ability, especially for topics requiring knowledge of multiple concepts. This involves determining the importance of each concept in problem-solving and investigating how these concepts interact during the learning process. Additionally, comparing the Elo rating system with the Glicko rating model within our data set could provide insights into the role of learning and forgetting curves in student performance estimation.

Beyond the aforementioned research areas, a critical future direction involves implementing the Elo rating system for online recommendations regarding specialties and question difficulty. However, this endeavor presents challenges in platforms where questions often have multiple specialty tags. As underscored by the research findings of [START_REF] Cepeda | Spacing Effects in Learning: A Temporal Ridgeline of Optimal Retention[END_REF], the premature revisiting of knowledge can exert detrimental effects on long-term memory. Consequently, during the recommendation phase, it becomes imperative to not only select questions aligned with the student's current needs but also to ensure that these questions do not encompass specialties that may not be relevant to the student's current stage of learning. To address this, we can consider a new approach that calculates students' abilities based on combinations of specialties, rather than individual ones. This new strategy could substantially enhance the model's efficacy when suggesting questions that align with students' learning requirements and are pertinent to their current learning stage. Such an approach would ensure that students are presented with a tailored set of questions that optimally support their progress while avoiding the unnecessary revisiting of topics that might hinder long-term retention-a crucial consideration for the success of an adaptive learning platform.

CONCLUSION

In conclusion, our study underscores the remarkable adaptability of the Elo rating system to the intricate challenges posed by a large, sparse, and multifaceted data set, where questions are tagged with multiple knowledge components. The Elo rating system, along with its enhanced version that leverages historical data for initial estimations, has exhibited a commendable level of prediction accuracy.

These results offer reassurance regarding the Elo system's robustness and versatility, emphasizing its capability to provide reasonable predictive value even in complex situations. This insight is crucial for the broader learning analytics community, providing confidence in the effectiveness of the Elo rating system as a predictive model in educational settings. Overall, the study contributes to the ongoing discourse on learning analytics methodologies, offering practical insights and encouraging further exploration of the Elo rating system's applicability in diverse learning scenarios.

Figure 1 :

 1 Figure 1: Number of questions that require knowledge on any given number of medical specialties.The questions exhibit a spectrum of dependence on medical specialty knowledge for their solution. While a considerable portion of questions rely on ability in a single medical specialty, many questions require knowledge spanning multiple specialties.

Figure 2 :

 2 Figure 2: Overview of the use of the BNE Platform during the 2020-2021 educational year. The blue bars represent the count of unique users per medical specialty in the data set. The orange bars represent the count of unique questions available in the platform for each specialty. In addition, the overlaid line plot illustrates 'Attempts per Specialty,' the total number of user attempts on questions within each specialty during the educational year 2020-2021.

Figure 3 :

 3 Figure 3: Number of Attempts by Each User and to Each Question across the 31 Medical Specialties. The top left box plot shows the distributions of the number of attempts by each user across the 31 specialties. The bottom left box plot depicts the number of attempts to questions in each specialty. During the mock exam, all students took identical questions, resulting in quasi-uniform numbers of attempts given by students and received by questions (top and bottom right plots).

Figure 4 :

 4 Figure 4: Comparing Logistic Regression and Elo Rating outcomes for question difficulty (left) and user ability across 31 specialties (right) in the same 2020-2021 dataset. Scatter plots illustrate the alignment, with 𝑦 = 𝑥 lines for reference. The left plot displays Logistic Regression difficulty estimates on the 𝑥-axis and Elo Rating estimates on the 𝑦-axis. On the right, the plot contrasts user ability estimates, with Logistic Regression on the 𝑥-axis and Elo Rating on the 𝑦-axis. Sample sizes (𝑁 ) are included in each plot.

Figure 5 :

 5 Figure 5: Comparing Logistic Regression Outcomes. Estimated question difficulty (left) and user ability on each of 31 specialties (right) in the two successive education years (2019-2020 and 2020-2021) using the Logistic Regression model. 𝑦 = 𝑥 lines are given for reference.

Figure 6

 6 Figure6presents a visual representation of the RMSE and AUC scores over 180 consecutive training days for each model. Findings indicate a substantial prediction accuracy advantage at the beginning of the training year when initializing the ability and difficulty values based on the data from the previous year. This advantage is reflected in an initial boost in average accuracy (+0.016 AUC) and a reduction in the average error (-0.008 RMSE) during the initial 30 days of training. However, it's worth noting that the initial disparity between the two model versions diminishes rapidly and becomes less than 1 point within a few days. By the end of the training period, the advantage of initializing with historical data becomes nearly negligible, with only a marginal improvement in average accuracy (+0.002 AUC) and a minimal reduction in average error (-0.002 RMSE) observed during the last 30 days of training.Table2shows the prediction performance on the mock exam for the three models: Elo rating initialized at 0, Elo rating initialized historical data, and logistic regression. These results reveal that the three models show highly similar prediction accuracy, with a slightly better performance on the Elo rating model initiated with historical data over other models.

Figure 6 :

 6 Figure 6: Comparing Models' Prediction Performance During Training. RMSE (top) and AUC (bottom) values as a function of training days, for two versions of the Elo rating system. Shaded regions around the mean lines represent the 95% confidence intervals calculated using the standard error. A secondary 𝑦-axis on the right side illustrates the absolute difference between the two models with the green line plot.

Table 1 :

 1 BNE data set Summary

	Data	Period	Users	Questions	Specialties	Specialties per Question	Attempts	Sparsity (User, Question)	Attempts per User
	Training Period data set 16.09.2020-14.03.2021 8,616 357,317 31 1.58 26,772,424 0.99 1.05
	Mock Final Exam data set	15-17.03.2021	8,616	372	28 1.71 3,172,546 0.01	1

Table 2 :

 2 Comparing Models' Prediction Performance on Mock Exam

	Model
	Elo rating initialized at 0	0.419	0.812	0.737	0.717	0.685	0.701
	Elo rating initialized with historical data	0.418	0.813	0.738	0.720	0.684	0.702
	Logistic regression	0.419	0.811	0.736	0.714	0.689	0.701

RMSE (↓) AUC (↑) ACC (↑) Precision (↑) Recall (↑) F1 (↑)

ACKNOWLEDGMENTS

This research was funded by Agence Nationale de la Recherche, grants ANR-17-EURE-0017, ANR-10-IDEX-0001-02 and ANR-21-CE28-0030.