Nam Hoang
email: hoang_nam.ho@univ-lr.fr

Ho

Mourad Rabah
email: mourad.rabah@univ-lr.fr

Pascal Estraillier
email: pascal.estraillier@univ-lr.fr

Samuel Nowakowski
email: samuel.nowakowski@univ-lorraine.fr

A process for Trace-Based Criteria Weighting in Multiple Criteria Decision Making

Keywords: criteria weighting, interactive adaptive system, Multiple Criteria Decision Making, Naïve Bayes, traces

Decision-making has evolved as an interesting research problem for decision community. We consider a decision problem that takes into account several criteria called Multiple Criteria Decision Making (MCDM) in an interactive application for adaptive execution. In this paper, we present a method for automatically weighting criteria generation based on users' traces. In our method, we suggest a process that contains all steps describing alternately what is necessary to prepare in order to weight all criteria. We propose a modified method using Naïve Bayes network to exploit the traces (the past of users), there will be used as information for estimating the score of criteria. Experimental results are presented to illustrate a full process and an automatic generation of weighting criteria by a set of values.

I. INTRODUCTION

The context of our work is the decision making in Interactive Adaptive Systems (IAS). An IAS is a system that can adapt its execution according to users behaviours [START_REF] Brun | A taxonomy and evaluation of formalisms for the specification of interactive systems[END_REF], [START_REF] Pham | Handling the Misunderstanding in Interactions: Definition and Solution[END_REF]. The system checks users interactions and adapt application's scenario with respect to designer's constraints. Indeed, when users are in an interactive application, they perform actions to achieve a given goal. During the execution, it could happen that the system cannot continue according to the current scenario due to an ambiguity of user's actions during the interaction or if the current data are not sufficient to continue the logic of the execution. At this point, a decision support system is needed to resolve this situation. Decision-making must reach two conditions: one is to provide a solution that allows the user to continue the interactions and the other is to reach the objective of the application (designer's constraints). The main purpose of decision-making is to find the action that resolves the problem according to several application related criteria. These criteria may be numerous and heterogeneous. Making the decision must take into account all of them in order to choose among a set of possible solutions the best one. This is MCDM, the Multiple Criteria Decision Making [START_REF] Triantaphyllou | Multi-Criteria Decision Making : An Operations Research Approach[END_REF], [START_REF] Koksalan | Multiple Criteria Decision Making: From Early History to the 21st Century[END_REF].

Moreover, since we consider an interactive application with many interactions between system's actors. During application's execution, actors actions can generate traces [START_REF] Laflaquière | Trace-Based Framework for Experience Management and Engineering[END_REF], [START_REF] Clauzel | Enhancing synchronous collaboration by using interactive visualisation of modelled traces[END_REF]. These traces may contain valuable information and represent huge quantity of data to compute. Based on these traces, we can know users' past habits and abilities. This information helps the system to solve the adaptation problem. In this paper, we propose a method to use these traces for criteria weighting.

MCDM is the most well known branch of decisionmaking problem. So, there is a lot of works that focus on the strength of this approach in order to make the best decision. In [START_REF] Fishburn | Additive Utilities with Incomplete Product Set: Application to Priorities and Assignments[END_REF], [START_REF] Miller | Executive Decisions and Operations Research[END_REF] authors propose a typical decision matrix based computation. This decision matrix is a matrix in which elements indicate the performance of the alternative when it is evaluated according to each defined criterion. Others approaches have been detailed in [START_REF] Russell | Artificial Intelligence: A Modern Approach[END_REF], [START_REF] Angilella | Assessing non-additive utility for multicriteria decision aid[END_REF] based on utility function. In these methods, the decision maker has determined the weights of the decision criteria. The decision maker will estimate the score by himself. It does not rely on any factor or principle to compute. That is only a feeling guided weighting. The perception through feeling must be completed by perception through reason. It is necessary to have a method that can automatically estimate the score of each criterion in order to classify criteria according to their respective priority.

Therefore the criteria weighting evolve as a challenge for decision community. For example, authors determined criteria weights from partial ranking of the alternatives, individual criteria, or criteria pairs [START_REF] Marichal | Determination of weights of interacting criteria from a reference set[END_REF]. [START_REF] Hinloopen | Integration of ordinal and cardinal information in multi-criteria ranking with imperfect compensation[END_REF] integrate the assessment of the scores (cardinal input) and rankings (ordinal input) of the decision-makers' preference structure. Relative criteria importance is represented by a set of cardinal weights or ranks. The analytic hierarchy process (AHP) [START_REF] Saaty | How to make a decision: The analytic hierarchy process[END_REF] is based on decomposing a complex MCDM into a hierarchic structure. It contains three levels: the first is the goal of the problem, the second lists all necessary criteria to evaluate the goal and the final level indicates all of the existing solutions to solve the problem. A compared matrix is constructed in order to express the relative importance between two criteria. Each element of the matrix is estimated by a value indicating the relative importance ranging from 1 (equal importance) to 9 (extreme importance of one over the other). Therefore, from this matrix, we calculate an Eigen vector; this vector is the priority vector of criteria. Each value is the weight that corresponds to the criterion for every line of the matrix.

These methods are turning-points in decision-making community because they support a very clear model and a mathematical theory about the structure of MCDM and the criteria weighting. However the drawback is that it relies on the decision maker to determine the ranks or the relative importance between criteria. These values depend on decision maker's perception. It is not automatic and it depends extremely on the intuition of the decision maker.

In this paper, we propose an automatic weighting process of decision criteria for interactive adaptive systems. We use user traces as input data to score the criteria weights. The main idea of our proposition is the construction of a prediction model based on users' traces in order to estimate the ability of criteria achievement by the users. To build a prediction model, the data mining approach emerges as an efficient method to solve the problem. We apply Naïve Bayes network and adjust the results of the prediction step to get a score for each criterion. The obtained weights are aimed to be used in the decision-making process for system adaptation.

To illustrate our approach we have chosen an e-Learning case study. The considered e-Learning system is an IAS that has to adapt to users profile and progress. To make a decision, it uses users traces to predict future users behaviours and accordingly adapt the execution.

The paper is organized as follows: section II gives a process that indicates globally the trace-based MCDM for IAS; all of the steps of the proposed process are described in section III where we only focus on steps that concern criteria weighting; experimental results and discussion are presented in section IV, section V concludes the paper.

II. PROCESS FOR TRACE-BASED MCDM FOR IAS

Our proposed process for MCDM in interactive adaptive systems is a two phases process. The first phase describes how to estimate the score for each criterion. The second phase is devoted to decision-making. We introduce the complete process of MCDM in order to have a full overview of our methodology, but in this paper we will only consider the criteria weighting phase (step (i) to (v) below).

The full process consists in 8 steps:

-Objective identification (i): it defines the main objective the user wants to achieve. -Criteria identification (ii): all of criteria will be determined in order to evaluate these contributions to the achievement of the objective. -Data preparation (iii): mentioned data are users' traces based. All of traces created by the user will be collected and structured in a traces system. -Data extraction (iv): this step removes the irrelevant information and gets the most appropriated information to criteria examination.

-Criteria weighting (v): this step provides the weight of each criterion determined in (ii). Indeed, there are more criteria; they are not classified at the same level because each of them has a different relative importance of others. -Alternatives analysis (vi): it is also a necessary step in the decision because decision method requires a set of alternatives to give a best choice. -Decision-making (vii): this phase is the core of the decision. During the computation of this step, it can happen that one or more criteria are completed then, it could return to the step (v) to re-estimate the score for each criterion. -Results back up (viii) traces for next executions.

III. CRITERIA WEIGHTING DESCRIPTION

We will detail our criteria weighting approach. It is based on users' traces. We explain in this section how to exploit these traces in order to score decision criteria.

A. Objective identification

An objective is an outcome that the user wants to achieve in an application. In fact, one objective must satisfy several properties including five characteristics listed in the following acronym SMART (Specific, Measurable, Applicable, Realistic, Temporal) [14]. They allow the user to understand and facilitate the interpretation.

-Specific: a purpose's statement should be stated specifically and unambiguously. It is also important to indicate the formulation of an objective without negation. This means that the user should avoid targets containing "must not". The statement must focus the user on the essential target. -Measurable: an objective is measurable because it allows the achievement estimation of the target. -Applicable: it must concern the context. The objective is a part that relates to considered problem, it cannot exceed the scope of the problem.

-Realistic: when an objective is defined, it must be reachable. For an achievable objective, it can be determined how to make it succeed. -Temporal: a good objective is a target that satisfies the time constraints.

B. Criteria identification

A criterion is a principle or an element of reference that allows to judge, to estimate and to define something. In our case, the criteria are references to evaluate the achievement of the objective. In order to evaluate an objective, it must be based on one or more criteria. So it requires to identify all criteria that relate to the objective achievement [START_REF] Haarstrick | Multi-criteria decision making MCDM -A conceptual approach to optimal landfill monitoring[END_REF]. The criteria identification depends on each application and the intuition of the designer.

A criterion can be continuous or discrete. For example, the users' number in the application is calculated by a discrete value, the execution time of a task is represented by a continuous value.

In this paper, we assume that a problem, which contains n criteria, is {c1, c2,…, cn}.

C. Data preparation and extraction

A context is an observable activity because we observe all the information and traces in the past of the users. We present in this section how to structure a user profile [START_REF]IMS ePortfolio Best Practice and Implementation Guide[END_REF] combining the users' traces as shown in Fig. 2.

The information of the user is divided into four categories: Administration, Goals, Skills and Activities. We will describe in detail each category in the following:

-Administration: personal information (name, date of birth, gender), certification and diplomas, preferences and non-preferences about languages (English, French, Spanish…), technology. -Goals: the target that the user wishes to achieve.

-Skills: user's knowledge about personal expertise.

-Activities: the information recorded by the user. In our approach, data are a combination of basic information and the traces of the users. Knowing that, we now determine what information is usable to evaluate the users' criteria. The extracted information concerns skills, language, activities that contribute in evaluating the criteria accomplishment.

After extracting contributed information (attribute), we filter all the users who dealt with actual activity and we also observe the result of the criteria completion. This result is indicated by Success (user complete this criteria) or Failure (user cannot reach this criteria). This database will be used as a training database. The structure of the training database is described as follow: For new users, we will collect all information that concerns the attribute to determine the criteria achievement u (att 1 , att 2, ,…, att k ). Each actual user will estimate the probability to reach the defined criteria. It is the base for criteria weighting before making decision.

D. Criteria weighting

We will detail in this section an approach whereby the criteria are automatically weighted. In fact, in order to make decision engine working, we must first solve the problem by preparing these previous steps because all decision algorithms require a volume of information to process. One part of the data is the weight of criteria or in other words having a list of priority of the criteria. Indeed, one problem obviously contains many criteria that users want to achieve; perhaps there is a criterion that is not as important as the others. It will consider what the target users would reach early. If the criteria are not prioritized, the engine will assume that they have the same of importance level. This can influence the achievement of the criteria for the final objective. So it is the reason why we must supply to decision engine a list of the criteria weight to find a satisfying alternative. From the extracted data in the previous step, we will process and compute to obtain a priority list of criteria. The weight considered in our approach is a numerical value.

We now present the way to apply data mining in our approach. The goal is to use link relevant information to criteria, as shown in Table I, and then to predict the probability of criteria accomplishment. The users want to reach the defined objective, and this is evaluated by n criteria. Based on users' skills and traces, we can estimate what criteria the users can reach by computing the accomplishment probability. These probabilities represent the ability of criteria accomplishment. We realize that this problem is as a binary classification problem. For each user and for each criterion, we use the extracted data to predict if he can satisfy the criterion (Success) or not (Failure). Among classification methods, we make an analysis of these methods via the characteristic of our problem in order to choose the one that we can take as prediction algorithm. We analyse the algorithms that have been often used in the literature for data classification. We evaluate these methods by four characteristics relating our classification problem. First, we consider the data, because we must analyse the values of the users' traces and skills (both of continuous and discrete type) in order to estimate. Second, we need a method that runs fast during the execution because if the algorithm runs for a long time, it will influence the others phases in the system. Then, the next factor linked to our prediction problem is that the number of records in the training database because some methods require huge samples to train. The four following methods are possible candidates to build a prediction model: Naïve Bayes [START_REF] Domingos | On the optimality of the simple Bayesian classifier under zero-one loss[END_REF] [18], k Nearest Neighbors (k-NN) [START_REF] Tan | Introduction to data mining[END_REF], Decision tree C4.5 [START_REF] Quinlan | C4.5: Programs for Machine Learning[END_REF], Support Vector Machine (SVM) [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF]. Among these four methods, we decide to use Naïve Bayes algorithm because it is the most suitable for our problem. In fact, the data that we have seen and used in our case are of various types. Values are numeric, string (continuous or discrete). This is the reason why we do not use the SVM (Support Vector Machine) although the prediction quality of SVM is very high. In addition, the Bayes method is simple and easy to implement. Because of the small size of the training database, we cannot apply k-NN method. Between the two remaining methods, Naïve Bayes and decision tree C4.5, we realize that the decision tree must choose an attribute to partition data. If the selection attribute is not good, the quality of the tree will be affected. To conclude, we consider the Naïve Bayes method the appropriate method for our case.

The Naïve Bayes method is based on Bayes theorem. It is a classification technique based on conditional probability. A posteriori probability P(h/D) of a hypothesis h given a set of data D is given by Bayes theorem:

P h / D ( ) = P h ( ) . P D / h ( ) P D ( ) (1)
P(D/h) is called likelihood, and it represents that some data is produced under the assumption of this hypothesis h. P(h) is a priori probability of hypothesis h (probability before having data D). P(D) is a priori probability of data D.

If there is a user u that wants to start an application with n criteria, we will exploit his profile, his traces (k attributes) to predict the probability of achievement for each criterion. So a user can be represented by a record containing the values for each extracted attributes. We assume that we extract m attributes for one user. A criterion is labelled Success or Failure. There are two hypotheses in our case. In order to build a prediction model based on Naïve Bayes, we need to have our training base. Assume that we have a base of records as Table I; we calculate the likelihood of Success and Failure:

According to our statement, we focus on the failure probability. But in this context, we have more than one user; all of them want to finish the application by reaching all criteria. If many users can reach a criterion than the others in a set of criteria, it must not prioritize it over the others. We should focus on the criteria whose users' number of achievement is low in order to help users to finish all the criteria. So we are interested in the number of users who cannot reach the criteria, in other hands, the number of users who have a Failure probability. Then, we divide this number with the total of users who attend in the activity. Finally, we obtain a rate, which is the weight of criteria.

IV. EXPERIMENTATION

Experimentation has been carried out to illustrate the performance of the proposed method. First, we present the problem we want to solve. Our approach will be illustrated in this section. The considered context is a learning environment, where there are one teacher and several learners. During the execution i.e., the course, teacher and learners proceed sequentially by action. If there is a moment when the teacher or learners cannot decide to carry out an action, a decision engine will help them to choose the best action on the set of the possible actions. The best action is defined as an action that satisfies both the goals of the teacher and the learners. Evidently, the success of the course is evaluated by the contribution of the teacher and the learners via the criteria. So the decision engine must take into account all of the criteria in order to obtain the best solution. But what is priority law inside this set of criteria?

In the following, we will describe all the steps of criteria weighting procedure. The result of this experimentation is a list of weights.

A. Identification of the objective

We will now define the final objective at the end of this course. We use SMART standard in section III.A.

-Specific: the statement of this objective is understandable. -Measurable: deciding a course is completed or not belongs to the teacher. -Applicable: in education, is the result of a course applicable? -Realistic: the accomplishment of the course is an objective that can be reached. -Temporal: A course has to be feasible with the deadlines given by the teacher.

B. Criteria identification

We will now consider the criteria in which we evaluate the final objective. In fact, in the learning course, in order to determine if the course ends or not, we need to take into account learners and teacher both. If we consider only one actor, the evaluation of the course could not be exact. Thus, we propose to identify criteria for assessing the quality of a course. We identify three criteria:

-Does the course follow the sequence of actions defined by the teacher? -Is course's program respected? -The evaluation of teacher about the comprehension of learners.

C. Data preparation and extraction

We would like to extract information in the profile of learners who enrolled in a Data Mining course. Previously obtained data are used as training base to build a prediction model of the learners' achievement criteria. Suppose we have a base of learners' profiles. For each registered learner in learning environment, we have created a profile containing data as shown in Fig. 2. We only consider courses in the field of Probability, Mathematics, Algorithm, Database, Programming and English because these courses are the prerequisite for the Data Mining course.

For all the learners, we associate the results of the three criteria: time, sequence and level of understanding.

Then, we present the way to assess these six attributes. For probability and mathematics, we use the obtained grades. For algorithm and database, we use a scale from 0 (learner do not have knowledge about the domain) to 5 (learner is expert in the domain). We use levels to evaluate programming skill: Basic, Normal, Advanced, and Expert. The English speaking ability can be obtained with these levels: weak, intermediate and advanced. Table IV We now associate the result of each learner the criteria accomplishment in the database. It means, for each learner who have these requirements, we observe that he is on Success (S) or Failure (F) in the evaluation of three criteria: time, program and comprehension. Indeed, a learner has different skills and abilities. Between the result and skills, there is a very intimate relationship. If a learner has the best skills, it will be probably reach all the criteria of the teacher. If a skill is not good, it can be influenced by one or more criteria.

In order to have a training database for our experimentation, we will build a database of 1000 learners (1000 samples). That means there were 1000 learners have been attended in the Data mining course. When they have been finished the course, we observed the accomplishment for each determined criterion. For six attributes, we create data in respecting the discrete uniform distribution function. In fact, each attribute is obtained a value in a set of value; the probability of obtaining each value is equal. Due to the difficulty of criteria evaluation, we create a database by referencing the idea of teachers of data mining. They are many experiences in the domain and the learners' evaluation. So based on their ideas, we have the data in respecting the rule: if half of attributes are superior to the average thresholds (for two attributes about probability and mathematic: 10, for algorithm and database: 2, for programming: Normal and for English: Intermediate), the learner will reach the criterion of time. If the values of 4 or 5 attributes are superior to the average, the criterion of programming will be achieved. If six attributes are perfect, the learner will reach the criterion of comprehension. In addition, besides respecting theses rules, we also create data that symbolize the noise of the data. For example, a learner has a good evaluation in six attributes, but he does not concentrate on the course, so he cannot achieve this course. As another example, if a learner has only three attributes that satisfy with the average, but he works with a lot of effort, he can reach all the criteria of the course. We combine all possible cases in order to have a completed database. It contains normal and also noise data; the goal is to increase the robustness of the prediction model.

All records in the training base above is reserved in a database and each record contains six fields corresponding to the six attributes of learners and three fields are the result of each criterion. An example illustrates in Table V For new learners, we consider only six attributes to predict the probability of achievement for each criterion. The constructions of the prediction model in the phase of criteria weighting will estimate if the new learner can be reach the criteria of the course. The detailed description of weighting will be presented in the next section.

D. Criteria weighting

We develop in this section a process for ordering the priority of the criteria. In order to prioritize the criteria, we use the training database created above to construct the prediction model. We will predict results of achievement of the learners based on the skills and abilities of learners.

From 1000 samples, we test the predictive ability of the Naïve Bayes algorithm. The test protocol that we use is k-fold method. The main idea of this protocol is to divide randomly the initial base in 10 folds. Nine of the 10 folds used for the training base, the rest is used for the testing base. We repeat 10 times to have a variety of data. We calculate the prediction rate every once time. After making 10 times, we calculate the average of 10 times, this value shows the rate of correct prediction of the model or the performance of our prediction model.

We present in the following the performance of three prediction models with three criteria: the time prediction, the program prediction and the comprehension prediction. The Table VI details the performance of each model. For the criterion of time: the performance of prediction model is extremely high. We realize that the success rate decrease in two criteria program and comprehension. Because, the comprehension of learner is very difficult to evaluate, and the three prediction models that we build is an approximate model. They do not predict exactly all cases, but there is an error tolerance for each model, there are success and failure rate.

A learner has the information about the skills, the traces in the past as follows: Probability = 12, Math = 15, Algorithm = 2, Database = 1, Programming = Normal and English = Courant. We denote learner 1 (att 1 =12, att 2 =15, att 3 =2, att 4 =1, att 5 =Normal, att 6 =Courant). Using these values for three prediction models to estimate the probability of achievement of three criteria. For example, we estimate P(Success/learner 1 ) and P(Failure/learner 1 ) on the criterion of Time based on (2) and [START_REF] Triantaphyllou | Multi-Criteria Decision Making : An Operations Research Approach[END_REF] We apply the statement indicated in the section III.D. We arrange the priority of criteria by arranging the failure probability in the Table VII. We will get the priority and the weight of three criteria: Comprehension (0.969), Program (0.967), Time (0.001).

For this user, it requires to concentrate on the comprehension because of the program and the time criteria, he can be reached within difficulty. The probability of prediction of comprehension criterion is low, so it is very important to make comprehension in priority in order to help this user finish all of the criteria. This is the case of having one learner. If we want to weight the criteria for a class with many learners, how to process our problem? We will illustrate an example of a class that contains 15 learners who will participate in the Data Mining course as Table VIII. We consider only the total number of learners who do not reach the criteria. Based on these values, we divide by the total of learners; we obtain the weight of three criteria as follows: Time (1/15 = 0.067), Program (7/15 = 0.467) and Comprehension (9/15 = 0.6).

There are the weights of three criteria. They can be used in all of decision methods that require a list of criteria weights.

This experimentation illustrated how to estimate the criteria in MCDM. Our approach is applicable in interactive application under the hypothesis of having a traces base in order to build a prediction model for criteria accomplishment. If we consider a situation that we cannot collect or observe the traces, our approach is not applicable.

V. CONCLUSION

In this paper, we have presented a process to estimate the weight of criteria in multi criteria decision making for an interactive application. Firstly, we have identified firstly the objectives and the criteria that contribute to reach these objectives. To weight criteria, we use both user's profile and user's traces. The collected data is put in a database. We extract from the obtained database the pertinent information that is used to configure criteria accomplishment prediction model. We have used the Naïve Bayes algorithm to construct this model. Each defined criterion is predicted in order to determine the probability of criterion achievement. We focus only the criterion failure prediction in order to help users to increase the weight of criteria which finishing probability is low. This principle will help users to take into account criteria being difficult to achieve. Experimentation has been introduced to illustrate our approach by describing all the steps of the presented process.

Our work on criteria weighting is a first contribution to our methodology to make decisions in interactive adaptive systems. We will use our results to set the weights in decision methods. These weights are estimated using a prediction model, which is more precise than the weights obtained by the intuition of the decision maker.

Our future work is devoted to the problem of increasing the performance of our criteria prediction model. We will also try to build a database with real and consequent data in order to increase the robustness of the model by considering data containing noise as in reality.

Hoang Nam Ho is a PhD student at the research Laboratory L3i "Informatics, Image, Interactions", University of La Rochelle. His researches concern the adaptive execution in the context of interactive applications. He uses data analysis technologies, especially data mining and decision-making, in order to perform the adaptation during application execution.

Mourad Rabah obtained his PhD in computer science at LAAS-CNRS laboratory in Toulouse (France) in the dependability domain. Since 2002, he is an Associate Professor at University of La Rochelle within the L3i Laboratory. His current work deals with Interactive Adaptive Systems where the system adapts its execution to users interactions and behaviours. He explores the application of fault tolerant technics in order to improve application scenario structuring and adaptation decision-making in interactive systems.

Pascal Estraillier is a full professor at the Computer Sciences department and the research Laboratory L3i "Informatics, Image, Interactions" at the University of La Rochelle. He is also scientific adviser in the directorate "National Management of Research and Innovation" at the French Ministry of Research, in charge of ICT area. His researches concern the architecture of software components in distributed and cooperative systems. He applies its results on Multi-agent Paradigm and uses formal specification theories in order to validate the behaviour and the interactions between components and to manage interoperability constraints. The results are mainly applied to the games, serious game and e-learning domains in many research projects founded by the E.U, the french A.N.R, the regional council, and several companies (France Telecom, XD-Production,...).

Samuel

Figure 1 .

 1 Figure 1. Proposed process for MCDM problem.

Figure 2 .

 2 Figure 2. Modeling a user by different information tuples.

TABLE I .

 I DATA AND TRACES ASSOCIATED WITH CRITERIA

	Attribute 1	……	Attribute k	Criterion 1	… Criterion 2
	Val 1	……	Val k	Success	… Failure
	Val 2	……	Val l	Failure	… Success

  summarizes the attributes.

TABLE IV

 IV

		.
		DATA MINING COURSE ASSESSMENT
	Requirement	Assessment
	Probability (P) Mathematic (M)	[0, 20]
	Algorithm (A)	0 (have no knowledge) to 5 (expert in the
	Database (D)	domain)
	Programming (Prog)	Basic or Normal or Advanced or Expert
	English (E)	Weak or Intermediate or Advanced

  :

			TABLE V.
			EXAMPLE OF COMPLETED DATABASE
	P	M	A D Prog	E	Time	Pro-gram	Compre -hension
	19 15	1 3 basic	Advance	S	S	S
	11 10	1 0 basic	Weak	S	F	F

TABLE VI .

 VI PERFORMANCE OF PREDICTION MODEL ON THREE CRITERIA

	Prediction	Time	Program	Comprehension
	Success rate	99.5%	84.9%	76.3%
	Failure rate	0.5%	15.1%	23.7%

  . P(Success) et P(Failure) are calculated during the building of the prediction model on training base:

						6
	P Success / learner 1 (	) = P Success (	)× P att i / Success ( ∏	) / P learner 1 (	)
						i=1
	= P Success (	)× P 12 / Success (	)×...× P Courant / Success (	) / P learner 1 (	)
						6
	P Failure / learner 1 (	) = P Failure (	)× P att i / Failure ( ∏	) / P learner 1 (	)
						i=1
	= P Failure (	)× P 12 / Failure (	)×...× P Courant / Failure (	) / P learner 1 (	)
	Then, we use (4) and (5) to normalize
	P(Success/learner 1 ) and P(Failure/learner 1 ) as Table VII.

TABLE VIII

 VIII Naïve Bayes. The result of every time is the success or the failure of the learner for each criterion. The TableIXdescribes the detailed result for estimation.

				.
			ATTRIBUTES OF 15 LEARNERS
	Learner Proba	Math Algo	Data	Prog	English
	1	12	14	2	3	Normal Weak
	2	9	15	4	1	Basic	Intermediate
	3	6	9	3	1	Advance Weak
	4	14	10	1	1	Expert	Intermediate
	5	5	7	2	0	Normal Weak
	6	11	8	1	1	Advance Intermediate
	7	15	17	4	3	Advance Advance
	8	8	10	3	4	Normal Intermediate
	9	6	9	4	5	Expert	Intermediate
	10	14	12	3	2	Normal Weak
	11	16	11	1	1	Advance Advance
	12	9	16	2	3	Expert	Weak
	13	10	11	3	5	Basic	Intermediate
	14	7	9	1	1	Basic	Weak
	15	11	14	5	5	Normal Intemediate
		Each learner will be estimated by prediction model that
	is built by

TABLE IX .

 IX RESULT OF THE CRITERIA PREDICTION OF 15 LEARNERS

		Time		Program		Comprehension
	Learner	Success Failure Success Failure Success Failure
	1	x		x		x
	2	x			x		x
	3	x			x		x
	4	x			x		x
	5		x		x		x
	6	x		x			x
	7	x		x		x
	8	x			x		x
	9	x			x		x
	10	x		x		x
	11	x		x			x
	12	x		x		x
	13	x		x		x
	14	x			x		x
	15	x		x		x
	Total	14	1	8	7	6	9

  Nowakowski is an associate professor at University of Lorraine as member of LORIA (Lorraine Research Laboratory in Computer Science and its Applications), a research unit (UMR 7503) common to CNRS, the University of Lorraine and INRIA. His current research activities are focused on robust Control methods for recommender systems, topology and social web modelling. He is involved in research projects dealing with ePortfolio in higher education and adult education at regional and European levels. He is involved in research activities dealing with identity through digital technologies.

P Success / u ( ) = P Success ( )× P att 1 / Success ( )

P Failure / u ( ) = P Failure ( )× P att 1 / Failure ( )

The Table III is the result of conversion into probability by normalization given:

P Failure ( ) = P Failure / u ( ) P Success / u ( )

( )

Then, we propose a statement that is the new principle to solve our problem. It states: if a user has a tendency to reach a criterion in a set of criteria, the disregard of prioritizing it over the others can be justifiable. We should, instead, focus on the criteria whose achievement rates are low in order to allow the user to complete all the criteria.

After the computation of the percentage of each criterion, we consider the Failure probability. We classify these values in decreasing order; we will deduce a list of priority of criteria that corresponds to the order of the probability values. We associate for each criterion a Failure probability that indicates its weight.

In addition, our presentation below corresponds to a case with one user. In the case of many users who want to execute the application, how can we react? We also propose an adjusted method that can be solved in this case. For each user, we use the predictive model to estimate the achievement for each criterion (Success or Failure). We regroup the results as shown in Table III.


G Rückert

N Perry
email: bn.perry@i2m.u-bordeaux1.fr

S Sire

S Marya

Enhanced Weld Penetrations In GTA Welding with Activating Fluxes Case studies: Plain Carbon & Stainless Steels, Titanium and Aluminum

Keywords: ATIG, Weld penetrations, Steels, Titanium, Aluminum

Flux applications prior to the convention Gas Tungsten Arc Welding (GTAW) is known to improve weld penetrations and improve process competitiveness. This paper summarizes the investigations on aluminum, plain carbon steels, stainless steels and titanium. The importance of flux composition, homogeneity and profile of its application are shown to be primordial in determining the weld depth to width ratio of weld pools. The mechanisms that lead to improved penetrations along with some industrial case studies are presented.

Introduction

Gas Tungsten Arc Welding (GTA or TIG welding) has successfully been used ever since its invention as a reliable welding technique. Easy to implement in application with a comparatively low capital investment, the process yields cosmetic welds of excellent characteristics. However, due to low inherent weld penetrations at nominal currents, the process lacks productivity in comparison with other welding technologies on the market. For thickness over 3 mm, multi-pass welds with filler additions remain mandatory. Attempts to improve penetrations by increasing welding current or reducing weld speeds generally remain ineffective. In fact, the weld pool widens with insignificant effect on penetration. It seems thus important to look for alternative means to improve GTA performance.

Flux for TIG welding efficiency improvements: ATIG (Activated TIG) & FBTIG (Flux Bounded TIG). Theoretical studies supplemented with experimental investigations suggest that the weld penetration is dominated by surface tension temperature gradients over the pool [START_REF] Burgardt | Interaction between impurities and welding variable determining GTA weld shape[END_REF][START_REF] Mills | Factors affecting variable weld penetration[END_REF]. The presence of some surface-active species that increase the melt surface tension with temperature induces thermo-capillary flows deep into the weld and results in deeper penetrations. As such, sulfur and to some extent oxygen in the pool has been suggested to produce deep and narrow welds. Increasing the arc power density through its constriction is an alternative route to improve weld penetration [START_REF] Lucas | Activating flux increasing the performance and productivity of the TIG and plasma processes[END_REF][START_REF] Perry | Study and development of flux enhanced GTA penetrations in a commercial grade titanium[END_REF].

Fluxes have long been used to finely tune weld metal chemistry, ensure arc stability and shielding in welding industry. However the application of flux coatings in GTAW is still a rare practice. Some of the reasons being lack of information on flux composition, its role in weld formation, probable loss of weld cosmetic and weld instability with variations in coating thickness. Fluxes can intervene in modifying drastically the weld formation in multiple ways notably by modifying surface tension, its temperature gradients and constricting the arc that would increase power density and consequently the weld penetration. The arc constriction itself is possibly attributed to:

• modified charge distribution created by elements with large electron cross-sections such as Oxygen, halides and fluorides; • the creation of a conducting channel around the joint following flux vaporization in this hotter central region of the arc and to some extent; • indirect effect of weld pool deformation in case of low surface tension melts. The central depression with curved sidewalls is propitious for arc localization and subsequent better heat exchange with the weld pool.

The above listed changes can intervene all together or one may be more predominant than others. From theoretical considerations, it may be postulated that one of the requirements for the flux to be effective is that its melting temperature should be close to that of the material and it should not form stable compounds with the molten pool that would badly affect weld cosmetic. Authors have thus investigated the effect of different coatings on plain carbon steels, stainless steels, titanium, and aluminum. Both continuous and gapped coatings have been explored and possible mechanisms that contribute to improved penetrations (>7-10mm) are examined [START_REF] Rückert | Optimizing the design of silica coating for productivity gains during the TIG welding of a 304L stainless steel[END_REF][START_REF] Sire | New perspectives in GTA welding of carbon steels by the use of silica[END_REF][START_REF] Sire | On the development of a new flux bounded TIG process (FBTIG) to enhance weld penetrations in aluminum 5086[END_REF]9]. GTAW (or TIG), which when applied in conjunction with flux coating is designated as ATIG or activated TIG. Preliminary developments were first made in former Soviet Union and gained interest in western countries later when scientific information circulated by open exchanges. In ATIG process, the flux covers only a small proportion of the surface facing electrical around the joint. Flux coating may continuously cover the joint as in Fig. 1.a or discontinuous over a small width around the joint as in Fig. 1.b, the former is ATIG and the latter is called FBTIG or Flux bounded TIG. For better process stability, it is essential to ensure that it does not seep inside the joint and that its application is as uniform as possible. The flux is manually applied by brush and before application the flux ingredients are mixed in a solvent mostly acetone. After application, the flux is easily dried and is prerequisite for welding.

The paper will present a summary of results on the above-mentioned materials and examine future prospects. Some ongoing or proposed applications would be discussed. Results and Discussion ATIG. One of the consequences, besides important increase of weld penetration observed on cut cross sections, can be seen directly on the arc profile and its intensity. Figure 2, shows variation of arc profile and its digitalized image as it goes from the non covered or TIG to the covered flux (ATIG arc). Soon as it enters the flux covered side, arc starts to increase in intensity and higher hear intensity zone is drawn close to the surface (Fig. 2.b) and at stabilization, plasma ejection from the weld pool with depressions in its free surface are seen (Fig. 2.c). These changes become more significant as the welding current is increases. The most significant observation is related to the depression of the weld pool, hardly visible in TIG at higher currents, but very prominent at higher ATIG currents. This suggests a confined arc on surfaces with flux coatings, implying a significant increase in power density that would generate deeper weld pools. In case of titanium weld penetrations exceeding 7 mm have been obtained by a suitable combination of arc current and travel speed.

The flux developed by Perry et al. [START_REF] Perry | Study and development of flux enhanced GTA penetrations in a commercial grade titanium[END_REF][START_REF] Perry | Etude et développement de flux solides pour le soudage ATIG appliqué aux alliages de titane et aux aciers inoxydables[END_REF] for titanium alloys is composed of a blend of fluorides, though mono component MgF2 can be seen to be equally efficient except a minor change in weld cosmetic. In order to see if the increase in weld penetration is purely related to arc as seen from brighter arcs, bead on plate experiments with a defocused laser beam (Fig. 3) equally showed a subsequent increase in weld penetration when the surface was covered with a flux layer (Fig. 3b). In a mixed configuration (Fig. 3.c), penetrations became higher on non-coated side, which strongly indicates a surface tension driven flow as laser is in no way deflected as arc would be by an asymmetric disposition of a non-conducting flux reported later on here. This lends support that at least in case of flux proposed by Perry et al; surface tension did play some role in increasing weld penetrations via Marangoni type convection currents. However, the extent of each one of the possible mechanisms is difficult to ascertain. With manual flux application, it is very difficult to ensure its uniform application. Rückert et al. [START_REF] Rückert | Optimizing the design of silica coating for productivity gains during the TIG welding of a 304L stainless steel[END_REF] investigated the effect of flux thickness in case of a 304 stainless steel where a SiO2 coating did increase weld penetrations by 50 to 75 percent in comparison to pure TIG at equivalent process parameters. The optimum flux thickness depends on the weld current and was found to vary from 40 to 70 microns as the current was increased from 100 to 150A. Beyond this optimum thickness, a rapid decline in weld penetrations is observed (Figure 4). With thicker coatings, the ATIG weld penetrations would not be significantly different from those observed on uncovered surfaces. The thickness effect explains the reason of variable bead penetrations observed when the flux is applied manually.

Silica is equally effective as activating TIG flux for plain carbon steels. As shown in Figure 5 for plain carbon steel, whatever the current level, the depth to width ratio with flux is superior for fluxcovered plates. At a given current, weld penetrations are virtually twice those with conventional TIG. Silica is a highly resistive material and its positive role in increasing weld penetration needs some further comments (Figure 5). Due to its higher resistance, it is expected that silica might destabilize the arc as it progresses from the uncovered to the coating side. This is effectively observed and schematically represented in Figure 6. As the arc progresses from the bare to the covered side, weld pool is seen to change from semi circular (Fig. 6.a) to a more triangular pool (Fig. 6.d) i.e., the pool as seen from the front is narrowed in width. Weld points realized according to the schematics of the Figure 6 supplement these results from video observations. As the length of the arc into the coating increases, the molten pool is seen to become narrower, but more elongated opposite to the welding direction. These observations suggest that higher electric resistance of the coating tends to repulse the arc towards the bare plate side. By extension, in ATIG welding, where arc continuously moves in the flux coating, the pool would be equally elongated as the back of the pool becomes uncovered by earlier passage of the arc. Arc tends to move to points where the resistance is lowest as further suggested by observations reported in Figure 8. Here, flux is applied only one side of the joint and arc tends to deviate towards the bare side. Observation of cross sections reveals comparatively higher penetration on the bare side, which in first instance may look contrary to the fact that ATIG penetrations are better than TIG at equivalent process parameters. In fact as the arc moves towards the bare side (Fig. 7.b) more heat is generated in the bare side and this possibly contributes to higher penetrations. Nevertheless, surface tension variations (lower on silica side) over the pool would also generate Marangoni flow away to the non-coated side. This heat transfer would generate wider and deeper pool on the non-coated side Arc deviation away from flux is an important indicator of how its inherent tendency to choose the least resistive path may be implemented to increase its power density or constriction. In case of FBTIG developed by Sire et al. [START_REF] Sire | New perspectives in GTA welding of carbon steels by the use of silica[END_REF], a non-coated space around the joint, as in Figure 1.b, is managed such that the arc remains confined to this zone. Figure 9 suggests that a free surface in between the flux coatings gives equally well weld penetrations. A closer look of the etched sections further shows narrower heat affected zones with FBTIG (see Fig 9 .b). Results reported by Ruckert [START_REF] Rückert | Optimizing the design of silica coating for productivity gains during the TIG welding of a 304L stainless steel[END_REF] on the effect of coating thickness on weld penetration in ATIG and FBTIG showed that the latter is much more immune to coating thickness. In ATIG, before the arc stabilizes to create a conducting channel by fusion of the non-conducting coating, a large part of the energy is consumed just to create that conducting passage. Hence, the coating thickness becomes an important variable of the ATIG process. In FBTIG, this passage is pre generated and so the process performance is less sensitive to coating thickness. FBTIG or flux bounded TIG. FBTIG is just another variant of activated TIG and seemingly more suited for manual application of flux coatings as more immune to coating thickness. Further in ATIG, it's pretty difficult to keep the flux out of the joint. Figure 9 shows on an Al 1050 plate, that FBTIG in AC mode gives very deep penetrations with SiO2 coating. In fact, because of the inherent aluminum oxide layer, DC TIG welding is inappropriate with argon shielding gas. With AC TIG, usually at nominal currents indicated for Figure 9 are very small. Besides current and welding speed, the gap in between the coatings on the two sides of the joint is an important process parameter. As this gap increases, the difference between TIG and FBTIG decreases as anticipated. Arc becomes less constrained as the gap between the coating increases [START_REF] Sire | On the development of a new flux bounded TIG process (FBTIG) to enhance weld penetrations in aluminum 5086[END_REF]. Figure 10 shows that with FBTIG, arc is highly constrained with high intensity channel and the weld pool is depressed. The arc penetrates in to the joint and all this contributes to increased weld penetrations.

Conclusions

Fluxes based on fluorides for Titanium and oxides (mostly silica) for plain carbon steels, stainless steels and aluminum have been tested in ATIG and FBTIG and mechanisms that contribute to enhanced weld penetrations are examined. In ATIG, a continuous flux coating, 15mm wide is applied around the joint, whereas in FBTIG, an uncovered strip, 2 to 4 mm wide, and centered around the joint is managed. Fluxes are non-conducting in solid state and become conducting in liquid phase. This implies that in ATIG, arc needs to first melt the flux. Vaporized flux would modify electronic current density. Afterwards, ATIG arc operates on the created bare zone still surrounded by a non-melted flux coating. At this stage, difference between ATIG and FBTIG processes nearly vanishes. From energy considerations, FBTIG is more efficient as arc energy does not have to undercover the flux coating to pave way for a conducting channel on the metal surface.

When the width of the uncovered zone in FBTIG is increased, arc becomes freer and weld penetrations are reduced to that of TIG process. The prerequisite of the creation of a conducting channel in flux coating implies that the coating thickness would become a critical parameter as identified here. Whatever the flux initial disposition, the video stills underline constricted highly intense arcs in ATIG and FBTIG configurations along with depressions in the weld pool. That alone is propitious for enhanced weld penetrations. All flux coatings tested here under are shown to produce improved weld penetrations when a defocused laser beam is used to simulate weld pool formation. This, along with asymmetric bead on plate weld profiles on specimens with coatings confined to one side of the ATIG arc, suggest a contribution from surface tension driven Marangoni type flow in the weld pool. However, the contribution from arc constriction and surface tension gradient driven flows has not been quantified. Further investigations are needed to elucidate some aspects described in this paper.

Fig. 1 .

 1 Fig.1. Flux disposition in ATIG (a) and FBTIG (b)

Fig. 2 .Fig. 3 .

 23 Fig. 2. Arc views and intensity profiles in going from TIG to ATIG, 100A, 15 cm/min, T-40 [4,5]

Fig. 4 6 ]Fig. 5

 465 Fig.4 Effect of coating thickness (silica) on penetration, 304L steel [6]

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig.6. Still video captures with different arc positions (a to d) and schematic representation of weld pool (e to g) [7]

Fig. 9 .Fig. 11

 911 Fig.9. Comparative of weld pool penetrations on etched cross sections in ATIG(a) and FBTIG (b) configurations; 304 Steel, 150 A, 15 cm/min [7]


Jean-Luc Schwartz
email: jean-luc.schwartz@gipsa-lab.grenoble-inp.fr

Kaisa Tiippana
email: 2kaisa.tiippana@helsinki.fi

Tobias Andersen

Disentangling unisensory from fusion effects in the attentional modulation of McGurk effects: a Bayesian modeling study suggests that fusion is attention-dependent

Keywords: McGurk effect, attention, FLMP, modeling

The McGurk effect has been shown to be modulated by attention. However, it remains unclear whether attentional effects are due to changes in unisensory processing or in the fusion mechanism. In this paper, we used published experimental data showing that distraction of visual attention weakens the McGurk effect, to fit either the Fuzzy Logical Model of Perception (FLMP) in which the fusion mechanism is fixed, or a variant of it in which the fusion mechanism could be varied depending on attention. The latter model was associated with a larger likelihood when assessed with a Bayesian Model Selection criterion. Our findings suggest that distraction of visual attention affects fusion by decreasing the weight of the visual input.

Introduction

While it had been initially claimed since [START_REF] Mcgurk | Hearing lips and seeing voices[END_REF] [START_REF] Mcgurk | Hearing lips and seeing voices[END_REF] that the McGurk effect (conflicting visual speech altering the auditory speech percept due to audio-visual fusion) was automatic and not under the control of attention, it appeared later that instruction to attend more to audition or to vision might bias perception [START_REF] Massaro | Perceiving Talking Faces[END_REF]. More recently, [START_REF] Tiippana | Visual attention modulates audiovisual speech perception[END_REF] [START_REF] Tiippana | Visual attention modulates audiovisual speech perception[END_REF] showed that if attention is distracted from visual speech by the presentation of a concurrent visual stimulus (a leaf superimposed on the speaking face), the role of visual speech decreases in fusion, so that the McGurk effect gets weaker. The authors modeled their data with the Fuzzy Logical Model of Perception (FLMP) [START_REF] Massaro | Perceiving Talking Faces[END_REF], which provided a good fit as assessed with the root mean square error (RMSE). Since the FLMP entails a fixed integration rule, a good fit of the model suggests that the attentional effect acts on the visual input, rather than on fusion. However, they also noted that in the experimental data there was little evidence of unisensory visual attentional effects. This discrepancy is possible because of the non-linearity of the FLMP, which allows small, statistically non-significant differences in visual response probabilities to cause large, significant changes in audiovisual response probabilities. Tiippana et al. concluded that this discrepancy prevented them from being able to determine whether the attentional manipulation influenced unisensory processing or fusion based on FLMP fits. [START_REF] Schwartz | Bayesian model selection: The 0/0 problem in the Fuzzy-Logical Model of Perception[END_REF] [START_REF] Schwartz | Bayesian model selection: The 0/0 problem in the Fuzzy-Logical Model of Perception[END_REF] argued that the good fits of the FLMP to McGurk data might be due to over-fitting. He showed that the error function of the FLMP has a very steep slope in that area of parameter space, which models the McGurk illusion. This means that a small change in the parameters can cause a large change in the model likelihood. The model is, in other words, very flexible in that its parameters can be nudged to accommodate almost any data set, particularly for conflicting auditory and visual inputs, as in the McGurk effect. This is the hallmark of over-fitting. The problem with over-fitting is that, although the model fits well, it generalizes poorly: The model with parameters fit to one data set does a poor job in describing another, very similar dataset. In order to overcome this difficulty, one needs to take the entire likelihood function into account rather than just its maximum. This is the principle of the Bayesian Model Selection (BMS) criterion. This criterion involves computation of the global likelihood of a model considering a set of experimental data, which is computationally complex, but Schwartz introduced the so-called Laplace approximation (BMSL), which appears to be easy to implement and compute.

In a later study, Schwartz [START_REF] Schwartz | A reanalysis of McGurk data suggests that audiovisual fusion in speech perception is subjectdependent[END_REF] introduced a variation of the FLMP, the weighted FLMP (WFLMP), in which inputs from audition and vision are weighted. He compared the two models using various criteria: the RMSE, the RMSE corrected for the number of free parameters and the BMSL. He found that all measures favored the WFLMP. Closer inspection revealed that the RMSE based measures always favored the model with more free parameters, which could be due to overfitting. The BMS did not show this behavior indicating that it is less influenced by over-fitting.

In addition to good fits and ability to generalize a good model should also add to our qualitative understanding of the underlying cognitive processes. Schwartz showed that the WFLMP did that since its weights provided a meaningful indicator of how much individual observers relied on audition versus vision. This issue is similar to Tiippana's question whether an irrelevant visual object can distract visual attention and thereby decrease the weight of visual information in audiovisual speech perception. Therefore, Schwartz' approach might help resolve the paradox that Tiippana et al. encountered. Hence, in the current study, we examine this issue by comparing data fits provided by various implementations of the WFLMP vs. the normal FLMP, by minimizing the BMSL criterion to Tiippana et al.'s data.

Methods

Experimental data

In [START_REF] Tiippana | Visual attention modulates audiovisual speech perception[END_REF] [START_REF] Tiippana | Visual attention modulates audiovisual speech perception[END_REF], subjects (n=14) recognized consonants /k/, /p/ and /t/ in /eCe/ context, presented in extended factorial design in two conditions: attend Face and attend Leaf. In the latter, subjects attended to a leaf floating across the talker's face instead of the face. The data consisted of response distributions to 15 stimuli (3 auditory A, 3 visual V, 9 audiovisual AV where 3 were congruent A=V and 6 incongruent A≠V i.e. McGurk stimuli) in 2 conditions (Face and Leaf) for 5 response categories: /k/, /p/, /t/, combination (combination of A and V consonants) and 'other' (Fig. 1).

Figure 1 -Summarized results of the experimental data in [START_REF] Tiippana | Visual attention modulates audiovisual speech perception[END_REF].

Models

Two models were fitted to the data. Firstly, the FLMP:

P R i | A,V ( ) = P R i | A ( )P R i |V ( ) P R j | A ( ) P R j |V ( ) j "
where R i and R j are response categories, A and V are auditory and visual stimuli, P(R i |A), P(R i |V) and P(R i |A,V) are auditory, visual and audiovisual response probabilities, respectively. Considering the Face and Leaf conditions, the equations are:

P R i | A face ,V face ( ) = P R i | A face ( ) P R i |V face ( ) P R j | A face ( ) P R j |V face ( ) j " P R i | A leaf ,V leaf ( ) = P R i | A leaf ( ) P R i |V leaf ( ) P R j | A leaf ( ) P R j |V leaf ( ) j "
The second model was the WFLMP:

P R i | A,V ( ) = P R i | A ( ) " A P R i |V ( ) " V P R j | A ( ) " A P R j |V ( ) " V j #
where λ A and λ V are factors used to weight the auditory and visual inputs in the computation of the audiovisual responses (see other introductions of weights inside the FLMP in [START_REF] Schwarzer | Modeling face identificaton processing in children and adults[END_REF]).

For each condition (Face or Leaf), we define a λ value between 0 and 1, and compute λ A and λ V from λ by: λ A = λ / (1 -λ) and λ V = (1 -λ) / λ. Therefore the weighted model WFLMP needs two more parameters than FLMP (λ Face and λ Leaf ).

Criteria for models assessment

The assessment criteria applied were RMSE, corrected RMSE and BMSL, which all give smaller values, the better the model fit. Let us consider a speech perception experiment for categorization of speech stimuli involving n E experimental conditions E j , and in each condition, n C possible responses corresponding to different phonetic categories C i . In most papers comparing models in the field of audiovisual speech perception, the tool used to compare models is the fit estimated by the root mean square error RMSE, computed by taking the squared distances between observed and predicted probabilities of responses, averaging them over all categories C i and all experimental conditions E j , and taking the square root of the result:

RMSE = P j R i | A,V ( ) " p j R i | A,V ( ) [ ] 2 i, j # n E n C
where observed probabilities are in lower case and model probabilities in upper case.

Considering that two models might differ in their number of degrees of freedom, Massaro (1998) proposes to apply a correction factor k/(k-f) to RMSE, where k is the number of data points and f the number of degrees of freedom of the model. This provides the second criterion, the corrected RMSE:

RMSE cor = k k " f ( )

RMSE

The third criterion used here is the Laplacian approximation to the Bayesian Model Selection (BMSL) criterion. If D is a set of k data points d i , and M a model with parameters Θ, the best fit is the maximum of the likelihood of the model given the data set, that is the value of Θ maximizing L(Θ|M)= P(D|Θ,M). However, the maximally likely parameter set is not the only possible parameter set. By assessing models by comparing only their maximum likelihoods we ignore the possibility that this is not the true underlying model. BMS estimates the likelihood integrated over all parameter values [START_REF] Pitt | When a good fit can be bad[END_REF]:

BMS = 4log L 5 | M ( ) $ P 5 | M ( )d5
Bayesian Model Selection has already been applied to the comparison of AV speech perception models, including FLMP [START_REF] Massaro | Bayes factor of model selection validates FLMP[END_REF][START_REF] Pitt | Flexibility versus generalizablity in model selection[END_REF]. The computation of BMS through this equation is complex. It involves the estimation of an integral, which generally requires use of numerical integration techniques, typically Monte-Carlo methods. However, Jaynes (1995, ch. 24, [START_REF] Jaynes | Probability Theory: The logic of Science[END_REF]) proposes an approximation of the total likelihood based on an expansion of log(L) around the maximum likelihood point θ:

log L ( ) ( ) ¡ log L 6 ( ) ( ) C 1 2 76 ( ) T & 2 log L ( ) & 2 [ ] ¢ 76 ( )
where [∂ 2 log(L) / ∂Θ 2 ] θ is the Hessian matrix of the function log(L) computed at the position of the parameter set θ providing the maximal likelihood L max of the considered model. This leads to the so-called Laplace approximation of the BMS criterion [START_REF] Kass | Bayes factor[END_REF]:

BMSL = "log L max ( ) " m 2 log 2# ( ) + log V ( ) " 1 2 log $ ( )
where V is the total volume of the space occupied by parameters Θ, m is its dimension, that is the number of free parameters in the considered model, and Σ is defined by:

" #1 a $ 2 log L @ ) $£ 2 [ ] &
The preferred model considering the data D should minimize the BMSL criterion. There are in fact three kinds of terms in the computation of BMSL. Firstly, the term -log(L max ) is directly linked to the maximum likelihood of the model, more or less accurately estimated by RMSE: the larger the maximum likelihood, the smaller the BMSL criterion. Then, the two following terms are linked to the dimensionality and volume of the considered model. Altogether, they result in the handicapping of models that are too "large" (that is, models with a too high number of free parameters). Finally, the fourth term provides a term favoring models with a large value of det(Σ). Indeed, if det(Σ) is large, this means that the determinant of the Hessian matrix of log(L) is small, which expresses that the likelihood L does not vary too quickly around its maximum value L max . BMSL has the double interest to be easy to compute, and easy to interpret in terms of fit and stability. Furthermore, if the amount of available data is much greater than the number of parameters involved in the models to compare (that is, the dimension m of the Θ space) the probability distributions become highly peaked around their maxima, and the central limit theorem shows that the approximation of BMS by BMSL becomes quite reasonable. [START_REF] Kass | Bayes factor[END_REF] [START_REF] Kass | Bayes factor[END_REF] suggest that the approximation should work well for a sample size greater than 20 times the parameter size m. In our case, the ratio will be from 24 (in the model with the highest number of parameters, that is 50) to 100 (for the model with 12 parameters) and even 600 (for the smallest model with 2 parameters).

Varying the number of free parameters

The number of free parameters in most model comparison studies in AV speech perception is generally kept fixed to the "natural number of degrees of freedom" of the model, that is the number of free parameters necessary to implement the model in its most extensive definition. Care is generally taken to check that the models have basically the same number of degrees of freedom, otherwise the RMSE correction previously described could be applied. Notice that this correction looses some sense if a parameter is introduced with no effect on the model likelihood (a "useless parameter") while BMSL naturally discards useless parameters.

Of course, completely useless parameters generally do not exist, since this would correspond to some kind of misconception of the model. However, it is important to assess the possibility that some parameters are not really useful in the model behavior. For example, while all model comparisons generally involve a subject-by-subject assessment -and it will also be the case here -it may be interesting to test if some parameters could not in fact be similar from one subject to the other. The same could be done from one experimental condition to the other. Therefore, we systematically tested various implementations of the models to compare, with a progressively increasing number of fixed parameters and thus a decreasing number of free parameters, in order to attempt to determine the true number of degrees of freedom of the model, that is the number of free parameters really useful, and providing the highest global likelihood of the model knowing the data. Our basic assumption is that it is under the condition of true number of degree of freedom that models can be really assessed and compared in sound conditions.

The logic guiding these progressive constraints decreasing the number of free parameters is that if the FLMP exploits the available free parameters to adapt its behavior to any pattern of experimental data (see [START_REF] Schwartz | Bayesian model selection: The 0/0 problem in the Fuzzy-Logical Model of Perception[END_REF]), it will have both a very low RMSE (even corrected) and a high BMSL (poor global likelihood), and hence it is necessary to take out as many free parameters as possible to really assess the regularity and consistency of subjects' behavior.

We compared six variants of the FLMP and WFLMP models. The baseline model is the 48-parameter FLMP_48, fitting 48 values for each subject, i.e. values of P(R i /A Face ), P(R i /V Face ), P(R i /A Leaf ) and P(R i /V Leaf ) for the three auditory and the three visual stimuli and 4 responses (/k/, /p/, /t/, "comb", the fifth one "other" being provided by normalization to 1). The corresponding WFLMP_50 model uses 50 parameters per subject, that is, the 48 previous ones plus λ Face and λ Leaf .

Then we tested five variants involving various decreases of the number of free parameters:

In FLMP_36, we assumed that since visual but not auditory attention was manipulated in the experiments, there is no difference between Face and Leaf conditions for auditoryonly responses, so that all values of P(R i /A Face ) and P(R i /A Leaf ) are equal, fixing 12 free parameters. WFLMP_38 is the same plus λ Face and λ Leaf .

In FLMP_24, we further assumed that responses to the auditory stimuli do not differ between subjects due to nearperfect recognition (96-100% correct), so that P(R i /A Face ) and P(R i /A Leaf ) are both equal to each other and the same for all subjects. This was done through a Round Robin technique, in which a given parameter for one subject is estimated from the mean value taken by the parameter in the whole corpus excluding the current subject from the computation. This technique, classical and computationally simple, separates the data used to estimate the parameter from the data used to test the model. The parameter is therefore not free because it is not adjusted to accommodate the test data. Instead, it is fixed by independent data. This reduced the number of free parameters by 12 leaving 12 free parameters per subject for P(R i /V Face ) and P(R i /V Leaf ) each. WFLMP_26 is the same plus λ Face and λ Leaf .

In FLMP_12, we further assumed that visual responses are the same in the Face and Leaf conditions, fixing 12 more parameters. WFLMP_14 is the same plus λ Face and λ Leaf .

FLMP_16 is a variant of FLMP_12 in which we added four free parameters enabling responses to visual /t/ differ between Face and Leaf conditions since this was the only statistically significant difference for visual-only stimuli in the experimental data. WFLMP_18 is the same plus λ Face and λ Leaf .

Finally, WFLMP_2 is a variant of WFLMP in which we assumed that all subjects have the same auditory-only and visual-only responses. That is, P(R i /A Face ) is equal to P(R i /A Leaf ), and P(R i /V Face ) is equal to P(R i /V Leaf ) and their values are identical for all subjects. The only free parameters are now λ Face and λ Leaf . Of course, there is no FLMP counterpart, since there would be no free parameter.

Results

The models were first assessed in terms of the RMSE (Fig. 2), which decreases as the number of parameters increases, as can be seen in Fig. 2 for both FLMP and WFLMP. However, the RMSE is lowest and almost the same for the two model variants with the highest number of parameters: FLMP_48/WFLMP_50 and FLMP_36/WFLMP_38. The latter variant is almost as good as the former despite the large decrease in parameters since in the data there is almost no difference in auditory-only responses between Face and Leaf conditions. However, the difference between FLMP and WFLMP becomes quite large for FLMP_16/WFLMP_18 and other variants with fewer parameters, so that WFLMP variants provide smaller RMSE values than the corresponding FLMP variants due to the two additional parameters. Correcting for the degrees of freedom does not change the pattern much, as can be seen by comparing Fig. 3 showing the corrected RMSE with Fig. 2. Again, there are four equally good models, FLMP_48, FLMP_36, WFLMP_50 and WFLMP_38.

Figure 3 -Corrected RMSE for FLMP (blue) and WFLMP (red).

Other details as in Fig. 2.

When assessing the models using the BMSL, the pattern is very different (Fig. 4). The BMSL decreases as the number of parameters decreases for all paired FLMP/WFLMP variants. It might seem puzzling that FLMP_12/WFLMP_14 have the lowest BMSL since here it is assumed that there is no difference between Face and Leaf conditions for visual responses, even though Tiippana et al. showed that there was a statistically significant difference for visual /t/. Probably these differences are of a second order compared with the basic phenomenon captured by the FLMP that audiovisual responses are well modeled by a multiplicative process.

The main finding here is that the best FLMP variant with 12 parameters is significantly poorer than the best WFLMP variant with 14 parameters, as shown by a Wilcoxon signed rank test. Finally, the rise of the BMSL curve for WFLMP_2 shows that BMSL is not just driven by the trend to decrease with the number of degrees of freedom: there is indeed a minimum, here for 14 degrees of freedom, and for the WFLMP rather than the FLMP. In the same vein, adding two free parameters from FLMP to WFLMP can lead to either a BMSL increase (for the first variants with too many free parameters) or a BMSL decrease for the last variants with fewer parameters.

Figure 4 -BMSL for FLMP (blue) and WFLMP (red).

The red ellipse marks the best variants. Other details as in Fig. 2. The red ellipsis marks the best configurations for FLMP and WFLMP

Discussion

This modeling work enables us to really assess the informational content of the provided experimental material in sound terms, relating unisensory and multisensory data in a coherent way thanks to Bayesian Model Selection. It appears that the WFLMP is associated with a larger global likelihood than FLMP, which suggests that in these data, there is indeed a modulation of fusion by attentional distraction, inducing a decrease in fusion per se, possibly superimposed with a modification in unisensory visual performance. This adds to the growing literature on attentional modulation of fusion in the McGurk effect (e.g. [START_REF] Alsius | Audiovisual integration of speech falters under high attention demands[END_REF] and also the paper submitted by Nahorna et al., in the present conference). This work extends methodological developments by the authors [START_REF] Schwartz | Bayesian model selection: The 0/0 problem in the Fuzzy-Logical Model of Perception[END_REF][START_REF] Schwartz | A reanalysis of McGurk data suggests that audiovisual fusion in speech perception is subjectdependent[END_REF][START_REF] Andersen | Modeling of audiovisual speech perception in noise[END_REF][START_REF] Andersen | Using the fuzzy logical model of perception in measuring integration of audiovisual speech in humans[END_REF] by confirming the superiority of the BMS approach to the RMSE approach in model assessment. However, while the BMS seems to provide an efficient criterion for model assessment and comparison, other tools could be used in future experiments, including cross-validation which provides a functional way to assess stability of the best fit, probably coherent with the BMS. But most importantly, the BMS (with its BMSL easy-to-compute approximation) together with the variable-degrees-of-freedom technique, should be used to re-assess various audiovisual fusion experiments on e.g. attentional [START_REF] Tuomainen | Audio-visual speech perception is special[END_REF], developmental [START_REF] Sekiyama | Issues in the development of auditory-visual speech perception: Adults, infants and children[END_REF] or cross-linguistic [START_REF] Sekiyama | McGurk effect in non-English listeners: Few visual effects for Japanese subjects hearing Japanese syllables of high auditory intelligibility[END_REF][START_REF] Massaro | Bimodal speech perception: An examination across languages[END_REF] effects on audiovisual in speech perception.

Figure 2 -

 2 Figure 2 -RMSE for the FLMP (in blue) and WFLMP (in red) for the corresponding degrees of freedom (48-50 for FLPM-48 and WFLMP-50, etc; 2 for WFLMP-2)

  Each plot provides mean response proportions for the[START_REF] Schwartz | A reanalysis of McGurk data suggests that audiovisual fusion in speech perception is subjectdependent[END_REF] response categories: /k/, /p/, /t/, combination (C) and 'other' (O). Error bars indicate the standard error of the mean across subjects. Plots are arranged so that rows indicate the auditory stimulus (none, /k/, /p/ or /t/) and columns indicate the visual stimulus (none, /k/, /p/ or /t/). Grey lines denote the attend Face condition and black lines the attend Leaf condition

Acknowledgements

We thank Prof. Mikko Sams for his collaboration in ref. [3] and other related previous work.