Antoine Cornuéjols
email: antoine@lri.fr

Andrée Tiberghien
email: andree.tiberghien@univ-lyon2.fr

Gérard Collet
email: gcollet@ac-grenoble.fr

A new mechanism for transfer between conceptual domains in scientific discovery and education

Confronted with problems or situations that do not yield to known theories and world views, the scientists and students are alike. Rarely are they able to directly build a model or a theory thereof. Rather, they must find ways to make sense of the circumstances using their current knowledge and adjusting what needs be in the process. This way of thinking, using past ways of perceiving the physical world to build new ones, does not follow a logical path and cannot be described as theory revision. Likewise, in many situations it is awkward, indeed often impossible, to resort to analogical reasoning to account for it. This paper presents a new mechanism, called 'tunnel effect', that may explain, in part, how scientists and students reason while constructing a new conceptual domain. 'Tunnel effect' is also contrasted with analogical reasoning.

Introduction

Ask for the first time a student to give an account of some experimental setting in terms of energy transformations. Observe the scientist trying to make sense of some baffling phenomena that do not yield to current theories. In both cases, to achieve the goal it is necessary foremost to discover a system of conceptual entities such that it becomes possible to interpret the situation by relating perceived features to these primitives, and then to make predictions and solve problems within the new theory. In both cases, it is rare, if it ever happens, that the new theory arises from scratch. Rather, the cognitive agent must rely on previous knowledge to perform the task. How is it that, sometimes with only one experiment acting as a trigger, it is possible to construct a new conceptual domain, like thermodynamics or electromagnetism, enabling to make sense of a wide range of phenomena, from a knowledge state that a priori is unsuited ? How is scientific discovery possible ? Or, in a more mundane way, how is it possible to learn new ways to interpret the world ? These questions are central for understanding learning, helping teaching and possibly assisting scientists. Yet, they have so far received only very partial answers. Philosophers, on the whole, prefer to study how theories can and must stand the test of experiments. Cognitive scientists, specially in machine learning, have centered their works on induction, which is of little help in that matter, devoting scant attention to the fundamental problem of interpretation, analogy making, and other mechanisms that deal with making sense of the world and allowing to take advantage of multiple sources of knowledge. This paper reports on a multidisciplinary head-on approach to the problem of learning new ways to interpret the world by relying on (and relating to) old ones. By studying how high school students address problems in conceptual domains that are new to them, we were led to analyze mechanisms that seemed to be at play in their segmenting the world, and constructing models of the situation, as well as the (re)conceptualization efforts that -sometimes-followed. In this paper, we focus on a reasoning mechanism that we hypothesize does explain the students behavior. We call it 'tunnel effect' for reasons that will be clarified later on. Like analogy, this mechanism allows the transfer of information from one conceptual domain to another one. Unlike analogy however, it does so without having to resort to two situations or cases, but only considers the one at hand, and it does not necessitate to specify beforehand a hierarchy of representation primitives in both domains (one being mostly unknown), nor to define how similarity between the two represented cases must be computed. In fact, it appears so natural that its scope covers a wide range of situations from metaphorical thinking to scientific discovery. Indeed it is possible to reanalyze scientific discoveries, like Maxwell's discovery of electromagnetism and of the ever since powerful concept of field theory, in a much simpler way than previous ones (e.g. [START_REF] Nerssessian | How do scientists think? Capturing the dynamics of conceptual changes in science[END_REF]). In the following, we start (section 2) by examining the problem at hand and how the traditional inferencing mechanisms deal with it. We then describe our experiments with students, our findings, and how they can be analyzed. The new reasoning mechanism we call tunnel effect is presented in section 3, and we show how it solves the problem defined in section 2. Section 4 contrasts tunnel effect with analogical reasoning underlying similarities and differences. Finally, section 5 is devoted to ongoing work and perspectives, as well as possible implications for educational sciences.

2 Definition of the problem : how to build new from old

Definition of the problem

To recap, the overall problem studied here can be described as the one of learning new ways of interpreting the world, making predictions and solving problems, when known conceptual universes turn out to be unsatisfactory. Two aspects of this definition must be stressed at once. First, even though the learning agent does not know, at time t, how to interpret the environment satisfactorily, he/she still must have some means, through his/her current knowledge to perceive and describe it. Otherwise, there is simply no possibility for the agent to even become aware of a problem. Second, if we admit that the agent is able to built some interpretation of the situations at hand, as we argued is necessary, then it follows that the whole problem rests on the satisfaction criteria : how is an agent to be satisfied, or dissatisfied, with one's interpretation ? How is it that satisfaction criteria may be defined a priori such that they specify what should be an interpretation in a yet to be discovered conceptual domain ? Upon examination, in order to be satisfactory, an interpretation or a model1 should :

(1) -allow to make predictions about the world which indeed correspond to what is observed, -solve problems about the potential set of phenomena studied, including providing explanations for them.

(2) -satisfy a set of "target constraints" that include deep beliefs akin to the themata of [START_REF] Holton | Thematic Origins of scientific thought. Kepler to Einstein[END_REF] and preconceptions about the world. Point (1) above corresponds to conditions of adequacy to the world. Point (2), target constraints, to conditions that result from the previous history of the agent, which includes social and cultural biases, as well as more scientific ones. For instance, while studying black body radiation, Planck was of course concerned that his theory would fit the experimental data (criterion of adequacy to the world), but also that it would give a picture of the world, which for him was deeply related to continuity, that allows to understand it (in particular how irreversible processes follow from conservative forces). He equally sets to himself that the theory should ensue only from the two first principles of thermodynamics (target constraints). The drama for Planck was that he had to abandon the continuity criterion in order to fulfill adequacy to the experimental data and sufficiency of the two first principles of thermodynamics (a criterion for which he was ready "to sacrifice every one of (his) previous convictions about physical laws" [START_REF] Planck | Letter from[END_REF]). However, in this case, we see how potent were the set of constraints deemed to be satisfied, and how they even forced a completely new vision of the physical world, not wished for at first, whereby quantum physics followed. Adequacy to the world and target constraints as defined above are therefore enough to specify target conceptual domains in a normative way. We are however interested in more, in that we look for a prescriptive way of building new conceptual domains, and even more precisely, we look for cognitive models of learning that could be amenable to computer simulations. In this framework, learning a new conceptual domain implies both the acquisition of new conceptual primitives with which it then becomes possible to segment the world differently, and the mastering of new control knowledge allowing to efficiently construct models of the environment. In this respect, there are three relevant subproblems :

• Pattern matching : how is the world to be matched against new, ill-known conceptual primitives ? • Models building : how ill-known conceptual primitives are combined in order to build models of the situation, and, in addition, how these models are articulated, or articulations, between interpretation domains, in particular the one in gestation and the currently more operational ones ? • Conceptualization and theory building : how a new conceptual domain comes to existence in interaction, both positive and negative, with existing ones ? These three issues may guide the analysis of human subjects understanding the world (cf. Ohlson, 1996), be they students, scientists or else. They are also key points to be answered in order to provide a machine intelligence account of the learning of new ways of interpreting the world.

Operational conceptual universes

Experimental setting

Target conceptual universe

Adequation to the world

Ba tter y

Wire Bulb

Model(t)

Wire ? Figure 1 depicts the main aspects of the overall problem as we see it. The cognitive agent tries to interpret its environment (for instance an experimental setting consisting in a battery connected to a bulb through two wires) in terms of a conceptual domain, in such a way as to satisfy both the adequacy to the world criterion (for instance correct predictions) and the target constraints (for instance, and roughly, to provide an interpretation in terms of energy transfers). At time t, the agent is not able to do that, but because per force he/she knows already a lot about the world, there are known (both operational and activable) conceptual domains that allow him/her to build models of the world, however unsatisfactory they may be in regards to the criteria. This implies that connections be established between an existing and compelling perception of the world, the associated knowledge and a theoretic model under construction, all forms of knowledge that are quite disparate. Furthermore, the new theory, because it is described using known terms (such as "heat" or "reservoir"), and the knowledge about objects and events involved in the perception of the situation, are related to other interpretative domains (such as knowledge pertaining to electricity). As a result, the discovery and acquisition of a new conceptual domain not only requires that connections be set up between the knowledge directly associated with the perception of the situation and the theory, but also that this be done in the context of other, diverse and more or less related, knowledge domains [START_REF] Tiberghien | Modelling as a basis for analysing teaching-learning situations[END_REF][START_REF] Tiberghien | Construction of prototypical situations in teaching the concept of energy[END_REF]. All of this characterizes nicely the challenge facing both the student and the scientist, as well as many other actors in situations of conceptual learning.

Ba tte ry Bu lb

Generator

Experiments in the physics of energy teaching

In order to study learning of new conceptual domains, we set up interpretation tasks in terms of a " new theory ". The idea was to force cognitive agents to learn a new way to interpret the world, and to study how they tend to do it. Concentrating on the three subproblems underlined above : segmenting, model building and reconceptualization, we specially focused our attention on pattern matching attempts between the " world " and the new theory, models built in the course of the task, difficulties encountered, dead-ends, and " repairs " that paved the way to the gradual mastering of a new conceptual system. Because a scientist or a student trying to come to terms with a set of ill-understood phenomena brings to his/her efforts a whole store of prior knowledge and beliefs, which constitutes a hard to delimit kind of meta control parameter, we organized experiments on human subjects in a way that allowed to control as precisely as possible the prior knowledge they would activate to bear on the problems that were given to them. In this way, it was possible to isolate and analyze the interpretation activity and problem-solving strategies from the activation processes that are responsible for summoning relevant knowledge sources in some context. More specifically, we performed experiments in physics teaching, and more precisely teaching a qualitative account of the physics of energy taught in high school classes around the age 16-17. The task involved small experimental settings that the students could experiment with, like small electrical circuits with masses and motors and so on, that were to be interpreted in terms of energy transfers and transformations along an "energy chain" starting and ending with an energy reservoir. The students work in pairs 2 . This experiment has been done in several contexts class and in Andrée Tiberghien's laboratory. We video-recorded several pairs of students and entirely transcribed their verbal productions. (For this paper 7 pairs were deeply analysed). interpretation. Likewise, the students had to discover the environment entity while there was no concrete, tangible, counterpart in the experimental setting. On the other hand, the task facing the students was easier than the one facing the scientists in that they did not have to "invent" the concepts necessary for the task. They were indeed provided beforehand with a declarative account of the target conceptual domain along with a lexicon of the authorized terms and icons that were to be used in their models of the situation (see figure 3). This is one way we were able to control the knowledge brought to bear by the students. In particular the students were asked to use primitives like reservoir, transformer, and transfer, for which they already had preconceptions (and how to do otherwise, except, may be, by some lengthy and convoluted paraphrase ?). These preconceptions helped them to understand the seed theory, but of course they could also hinder later proper conceptualization. The seed target domain also defined integrity rules that specified valid models, as, for instance, the "a complete energy chain starts and ends with a reservoir" rule. Together, the lexical entities used in the definition of the seed conceptual domain and the integrity rules constitute the target constraints for this particular task.

Theory (seed)

Energy can be characterized by: * its properties : One fact that emerged from this study was that out of 7 pairs of students, 6 produced the intermediate model of figure 4 (b) for the battery-bulb setting. They then departed from it to try to find alternatives, better suited models, meantime laboring over concepts like energy, transfers, and so on. This, in fact, did not strike us as worth of interest at first, so much it appeared to be expected. This intermediate model was after all none other than the classical circular electrical interpretation of the setting. Yet, upon reexamination, we were intrigued by the fact that this model, which acted as a powerful attractor, seemed also pivotal to enable further conceptual elaboration. Did the analysis of the why and how of this particular behavior could lead to a better understanding of the processes at play in the learning of new conceptual domains ? The rest of the paper is an answer to this.

3 The tunnel effect

Analysis of the experiment

In the experiment described in section 2.2, had the students been experts in the domain of energy chains, they should have produced the model of figure 4 (a). Instead, the vast majority produced at some point the model of figure 4 (b), which clearly looks like model (c) which corresponds to an interpretation of the experimental setting in terms of electricity. However, they were not committed to an electrical interpretation, but were genuinely seeking an interpretation that would satisfy the constraints that were provided to them, that is an interpretation in terms of energy chains (cf. figure 3). How to explain this discrepancy ? First of all, it is essential to realize that there is no such thing as an a priori unique, correct and complete representation of the reality. Any description of the world does depend on the subject's current understanding. However, there are descriptions that are very active in some situations and shared by a large set of individuals from the same cultural context. Thus, in our occidental culture of the turn of the third millennium, when someone sees some rather specific shape like in figure 5, he/she cannot escape to see a battery, even if on an exploratory mission on Mars. Hence, depending on the context, there are categories, properties, relations and so on that literally impose themselves in the foreground. They are accordingly the ones that are used when communicating about the situation. Consequently, and quite naturally, the students in our experiments use words like "the battery", "the wires", "the lamp" to point out things in the word. Not only do they use these words to communicate, they also think about the situation using the associated entities. For instance, suppose you are on an first exploratory mission to Mars, and you see a few steps ahead of you a shape like in figure 5. Not only will you describe it as a battery, but you would seem literally out of your mind if you did not also jump out of surprise and start considering all the implications, including the possibility of the existence of some extraterrestrial life forms knowing of electricity and having set foot on Mars. Referring to some set of perceptions using some word indeed goes far further than merely uttering a designation, it brings with it a whole lot of expectations, constraints on the world, and generally associations to other conceptual structures.

If we then look at the process by which the students built their interpretation of the experimental setting, they start by trying to recognize the matches between the outstanding categories they identify like "the battery", "the wires" and "the lamp" and the target concepts that have been given to them in the seed theory. They thus match without much trouble the battery with a reservoir, and the lamp with a transformer3 . Without entering into details here, this is rather easy because, on one hand, batteries and reservoirs share a lot of common properties like being subject to be full or empty, or to play a causal role in many situations, and, on the other hand, lamps transforms electricity into light (and heat), and thus is an instance of a transformer.

Martin (27) :

The reservoir, it will be the battery Sara ( 28 [START_REF] Megalakaki | Corpus de dialogues de trois groupe d'élèves résolvant trois problèmes mettant en jeu une activité de modélisation[END_REF] and [START_REF] Megalakaki | Expérience MARIANNE : corpus de dialogues de trois gourpes d'élèves résolvant une séquence de problèmes mettant en jeu une activité de modélisation[END_REF].

Now, when they come to consider the wires, it is clear that in the context of this experiment -a physics course and a typical example of an electrical circuit-, the most operational and highly activated interpretation domain is the electricity domain as it has been previously learned in the physics classes. Consequently, and again without entering into details, the wires are matched with means for energy transfers, and the electrical current is matched with transferred energy.

(See figure 6 for instances of exchanges between pairs of students solving the problem). This electrical counterpart to energy transfers allows to import the naturally inferenced (within the electricity interpretation domain) directions of the electrical currents and to make them also the directions for the energy transfers.

It is interesting to see that in this process, compelling interpretations of the world, related to batteries, lamps and electrical circuits have naturally been incorporated into the built model, they have also completely shaped it. Thus, in particular, the commanding electrical interpretation of the setting has led to the model (b) of figure 4 which is akin to the electrical interpretation of model (c). Thus the overall interpretation, even though it is based on foreign pieces of interpretation and is deeply of an electrical nature, fits most of the syntactical constraints for the satisfaction criteria, and can be mistaken for a valid energy chain interpretation of the setting. This is possible due to the fact that some target entities like reservoir or transformer match at least partially concrete4 entities like battery and lamp, and thanks to the apparent conformity to the syntactical constraints. This is also due to an apparent synonymy for the students between current and energy.

But if the interpretation activity stopped there, that would not lead to any further "discovery" by the cognitive agents, nor to the reconceptualization that is part of the learning of a new conceptual domain. There would simply be an electrical interpretation of the phenomenon disguised as an energy chain one. What is interesting is that, except in one case, all the students then embarked in further predictive activity based on the model they just came up with. They thus re-interpreted the model stemming from electrical considerations within the energy interpretation domain ! That is they suspended the underlying raison d'être of the model to interpret it within a new conceptual domain. This seemingly instantaneous autonomization of the model and reinterpretation within a new conceptual domain, is reminiscent of the tunnel effect in quantum physics whereby a particle can occasionally tunnel through barriers, or, more precisely, escape a potential well to enter another one without having enough energy to overcome the potential barrier between them. In our case, the passage from one interpretation domain to another one should go along with a dismantling of the first interpretation/model before reconstructing a new one in the new domain. Instead, the very same model becomes autonomous from its former source domain and is reinterpreted as such in a new one as if a tunnel had been drilled between the two domains allowing one to go from one to the other unnoticed and inconspicuously. Before analyzing how this is possible, it is interesting to see what happens next. Some students, "reading" the consequences of their model, predict that energy will come back to the battery, which they know, from their prior background knowledge, is not possible. These students are thus in front of a violation of the adequacy to the physical world criterion. Others (a minority) realize that their model does not satisfy the integrity rule of the seed theory according to which the initial and final reservoirs should be different. In any case, an intense reexamination of the model, its meaning and its justification takes place, leading to a better mastering of the target domain [START_REF] Cauzinille-Marmèche | Co-adaptation of students' knowledge domains when interpreting a physical situation in terms of a new theory[END_REF] provide an analysis of the "repair mechanisms" used by students to adapt their model).

The tunnel effect mechanism

In its broad lines, the tunnel effect is very simple to define. It involves two main stages : 1. A source (possibly composite) interpretation of the experimental setting disguises itself under the dresses of a legitimate model in the target interpretation domain. 2. This model, bare of its underlying justifications, is then re-interpreted within the target interpretation domain.

There might follow unforeseen predictive consequences of this model, possibly leading to adaptation and reconceptualisation in the target domain.

As an illustration, going back to the energy chain interpretation task, we can distinguish the two following stages : 1. From a multiplicity of considerations involving batteries, lamps, electrical wires (categories that are outstanding in the context of this task and are necessary for the purpose of description and communication among the students) and the provided seed theory with its reservoir, transfer, transformer and energy entities, the students built a model of the setting that can appear to be a valid model within the target interpretation domain. In particular, remember that under the label "energy" in this model, there is the association with electrical current that has allowed to endow these energy transfers with directions. 2. In a second stage, this model is then re-interpreted within the target domain of energy chains. Consequently, it is predicted that energy will flow back to the battery which is also the final reservoir of energy. And this prediction does not satisfy the students who, at this time, hold an energy point of view. This leads the students to challenge the current model.

Reservoir Transformer Transfers

Energy

Note that there really is reinterpretation taking place here. If not, that is if the label "energy" was still attached to the underlying meaning of electrical current, there would not be any problem with the energy (electrical current) going back to the battery. Indeed, one pair of students in our experiment was contented with its intermediate model and never reached the re-interpretation stage.

Note also how unlikely it is that the model of figure 7 would have been produced directly within the target interpretation domain. There is no a priori reason for the arrows going forth and back, and this arrangement violates the integrity rule that says that the initial reservoir should be different from the final one. Section 3.4 provides another illustration of tunnel effect in the case of the discovery of thermodynamics by Carnot, Joule, Thomson and Clausius.

To sum up somewhat boldly what has been said, a tunnel effect can occur each time a model (something that is expressed with symbols and has enough duration to be re-examined) becomes autonomous from its initial justifications and is liable to be re-interpreted in a new interpretation domain. Now, there are several questions that need to be considered in order to have a more operational specification of tunnel effects :

• How some (possibly composite) interpretation can disguise itself under a valid model within a target interpretation domain ? • What can be imported, and how, from the source interpretation to the re-interpretation domain once only the expressed model remains without its underlying raison d'être ? We study each of these questions in turn.

How a target interpretation domain can a priori, before being completely set up, specify what a valid model should look like ?

Our experiment in physics teaching is a special case where the target interpretation domain is provided ahead of time and outside any "project " from the students. Hence, the seed theory can be completely a priori specified. Here, it imposes various constraints on any candidate model of the world by imposing a formalism and some qualitative integrity rules (e.g. "an energy chain should start and end with a reservoir " ). In actual scientific discovery processes, the specifications for target models are set up a priori from preconceptions about the target domain. For instance, Maxwell in its search for a theory for electromagnetism was looking for models of the phenomena with continuous interactions rather than Newtonian like actions at a distance, and thus was looking for a formalism relying on the differential calculus. As we stressed already, Planck was adamant that his models would ultimately rest only on the two first principles of thermodynamics. Carnot, in thermodynamics, was looking for cyclic and reversible models of the steam engines in order to be able to establish their maximal efficiency. In any case, even though the target conceptual domain is per force not completely defined a priori, there always are minimal criteria that specify how a valid model should look like. These criteria may be erroneous, of a temporary nature, partial, but they always exist as a projection of what is believed to be fundamental in the target domain. In this way, many models, that may turn up to be unsatisfactory in face of measured phenomena or because of inconsistencies, may appear at first as legitimate candidates. Any model, whatever its underlying justification, that thus satisfies these general requirements is open to be interpreted within the target domain. And any model that was built at least partially outside the target interpretation domain, and that appears nonetheless legitimate in this domain can be said to be disguised.

What can be imported, and how, from the source interpretation to the re-interpretation domain ?

Once again, it is helpful to look at the energy chain task. At one point, the students are ready to associate electrical current, that they literally "see" in the experimental setting, with the target entity energy. They do this presumably because, at the notional level where current and energy share many properties like being fluid, circulating, being agents for causality, and so on. Once this association has been approved, then everywhere in the model that electrical current would appear, it is replaced with the label energy. But what is essential is that, at the same time, everywhere some properties of energy are needed (like its transfer direction) in the model, this is the properties associated with electrical current that are imported. And these are not thought upon and pondered, but on the contrary, they are smuggled in without further immediate checking. Hence the circular nature of the model of figure 7. It is important to realize that this phenomenon, which is central in what we call the tunnel effect in cognition, is ordinary. It happened when Carnot was equating the "caloric" with heat, and thereby introducing -smuggling in-its conservative property. It happened to Maxwell when he equated the ether (incompressible fluid) with a model for electromagnetic interactions, smuggling in the seeds for the difficulties faced in physics until Einstein's special relativity theory got rid of them (and of most of the smuggled in properties of ether). It happens all the time, and it happens unconsciously. This smuggling might turn out to be genial when it brings with it unexpected solutions to outstanding problems. It might also hinder further solution. We discuss these aspects in section 3.4 below. To sum up, each time entities from two different interpretation domains are matched, they can potentially bring with them in these associations further attached properties that are new to the other entity. And this can happen in both ways. For instance, we noted that energy transfers found themselves naturally endowed with directions as soon as energy was matched with electrical current. Likewise, in another task not presented here, one student matched reservoir with a weighting object, then to show that the weight could be filled up (!) by being lifted. An example of a property not to be found originally in the notion of weight (source domain), but really brought by the contextual match with reservoir (target domain).

Each time a new conceptual domain is learned (either by being taught or by discovery), it is unavoidable that, at the start, it is related and articulated to the current (situated or contextual) operational interpretation domains. (At the start, student cannot speak about "the reservoir" in the setting. They must use designations like "the battery" that are operational to them at this time). Because of this, when learners build models of the environment in the new target conceptual domain, they necessarily do this by matching entities and structures from the operational domains to the target one. What we argue here is that, most of the time, by doing this, there are hidden properties that are smuggled in these matching operations. These properties shape the model built in unchecked ways. This is only when re-interpretation occurs entirely within the target domain in construction that these hidden aspects may reveal themselves, thus bringing out unforeseen consequences in the target interpretation. This phenomenon that we call tunnel effect is therefore responsible for transferring information from the source domain(s) to the target one. Section 4 below discusses this information transfer property in comparison with analogical reasoning, the most well-known inference mechanism for transferring information between domains.

Tunnel effect as a way to decompose problem solving

One can see tunnel effect as a way to ease problem-solving in an ill-mastered conceptual domain. For instance, no students were able to solve directly the first energy chain task. The problem was simply too hard for them. On the other hand, 12 out of 14 produced the intermediate model of figure 7, which is arguably an electric model of the setting disguised as a legitimate model in the energy domain. If it is difficult for an agent to solve directly a problem in an ill-mastered domain, it might be easier to disguise an interpretation stemming from well-known domains into a legitimate candidate model in the target domain. The question then is of course that of seeing if that step (a kind of forgery, except it may be unconscious resulting from automatic inferencing in the source domain(s)) helps or hinders further resolution of the problem. It is difficult to answer this question in general, except that some version of the now famous nofree lunch theorem known in Machine Learning and Optimization Theory is likely to apply and state that, overall, tunnel effect must equally ease and hinder problem-solving in new domains depending on the context. However, there are reasons to think that tunnel effect may be a powerful help in problem-solving in some cases. Figure 8 suggests why. Thanks to tunnel effect, there are apparently more solutions to the interpretation problem, and hence more opportunities to find one of them. The problem then, if a fallacious solution has been found, is to be able to find a way towards the correct solution. We show in section 3.4 that this may be facilitated by the focus naturally provided by the processes underlying tunnel effect. In each schema, the horizontal axis stands for the space of potential models and the vertical axis stands for the quality of the model with regards to the world. Of course, depending on the interpretation domain (for instance centered on electricity or on energy exchanges), the same models may have different degrees of quality.

How tunnel effect activates further adaptation and conceptual learning

Two cases must be examined with respect to the opportunities for learning opened when a model has been obtained using tunnel effect :

1. The model obtained remains valid even after being re-interpreted in the target domain under construction.

2. The model turns out to be erroneous either when confronted with the world or because internal inconsistencies are discovered within the target interpretation domain. We study these two cases in turn.

The model remains valid.

This is what happened during the construction of thermodynamics by Carnot, Clapeyron, Thomson, Joule, Clausius and others [START_REF] Longair | Theoretical Concepts in Physics[END_REF]Science & Vie, 1994). Carnot, influenced by the theory of the caloric (an imponderable fluid with the property of being conserved and which he equated to heat) and by his father's work on the calculation of the efficiency of water mills, devised a cyclic and reversible model describing an ideal steam engine. Thanks to this model, he was able to demonstrate that there exists a maximal efficiency for steam engines, and that it depends on the difference of temperature between the hot source of heat (caloric) and the cold one. Later on, through a series of very meticulous experiments, Joule was able to show that heat was not a conservative quantity and was exchangeable with work. However, it turned out that Carnot's model was in fact neutral with respect to the caloric hypothesis and when reinterpreted in the context of the new theory about heat and work, still remained a very helpful tool for thought experiments, one which eventually lead to the discovery by Clausius of a special state function called entropy. We have here one instance of a model obtained through tunnel effect (its cyclic and reversible nature was deeply a result of the belief in the caloric theory even though this was never explicitly expressed by Carnot) which is still valid once the interpretation domain changes. The model by itself cannot therefore act as a trigger for re-evaluation of the target domain, and other symptoms must show. However, because it remains valid, it can help shape the new conceptual system and serve as a test bed for it, potentially through thought experiments as this was the case for Carnot's model in thermodynamics. 2. The model turns out to be erroneous when re-interpreted. In our energy chain experiments, this happened either when students realized that the model implied that the energy was flowing back to the battery (which they knew was incorrect), or when they discovered an inconsistency with the target integrity rule stating that the initial energy reservoir should be different from the final one. The natural question is then why is the model wrong in the investigated aspect ? A reexamination of the path that led to this conclusion in the model can then point towards one of two causes. First, the associations made between entities from the target domain and the source one(s) could be erroneous. For instance, many students question the association they made between electrical current and energy or between the wires and the transfers. This can lead to a differentiation process whereby the target entities gain autonomy with respect to the source ones. Second, the automatic inferencing process that determined the problematic aspect of the model can be disclosed and limitations for its range been set. This is what happened when some students realized that the circular nature of the electrical current did not carry to the energy entity. This inference was henceforth stopped when building a model. This short discussion convincingly shows in our opinion that tunnel effects, not only help finding models, even erroneous ones, but that they also provide guidelines for further re-examination and reconceptualisation when needed. This is however an issue that deserves much further work.

Very few inference mechanisms have been proposed that deal with the transfer of information between different conceptual domains. Analogical reasoning is one of them -the most famous-, blending is another one [START_REF] Fauconnier | Conceptual integration networks[END_REF], and, we submit, tunnel effect is a contender too. A full comparative study of the three of them would be more than interesting, but is beyond the scope of this paper. However, we believe that a comparison with analogical reasoning might help to enlighten some characteristics of the tunnel effect as an inferencing mechanism. We will concentrate in each case on the conditions for a transfer between interpretation domains to occur, and on the information content that is transferred. According to the dominant view on analogy (e.g. (Falkenheimer et al., 1989;[START_REF] Greiner | Learning by understanding analogies[END_REF]), analogical reasoning involves the interpretation of two cases, -called the source case for the supposedly well-known one, and the target case for the one to be completed-, that may be interpreted within two different interpretation domains (e.g. the solar system as a source case and the supposedly ill-understood atom system as a target one). Each case is supposed to be represented as a graph of relations and nodes standing for primitive concepts. Analogical reasoning implies then that a best partial match be found between the two graphs, and, in a second step, that the part of the graph representing the source case with no counterpart in the target case representation be copied, translated and added to the target representation in order to fill the missing part. Many questions arise as to the principles that should govern both the matching operation, the translation and the transfer, not to speak about subsequent verification and adaptation. Deep concerns have also been expressed about the interpretation process of the two cases during analogy and the ensuing representation of the cases (e.g. (Hosftädter, 1995;[START_REF] Mitchell | Analogy-Making as Perception[END_REF]). It is important to note that both domains -the source and target-must be sufficiently well understood in order that the respective conceptual primitives be identified, put in hierarchy and potentially matched. This view of analogical reasoning thus prevents the consideration of a target domain that would be in gestation and of which conceptual primitives would be very uncertain. If we consider then the analogical inferencing mechanism as a kind of black box with inputs and outputs, the inputs consist in the source and target conceptual domains (the conceptual primitives and their relationships (including the said over-important hierarchies) and in the two cases (be they already represented as some would pretend is realistic or be they interpreted in the context of the analogy as others would insist is unavoidable). The black box then searches for one satisfying matching between the two cases (given as rigid representations or not) and computes the completion of the target case representation. The output or information gained in the operation consists therefore in the added features and properties of the target case. In contrast, tunnel effect only involves the interpretation of a single situation or case (e.g. an experimental setting or a set of phenomena). The input of the tunnel effect black box consists in the operational source interpretation domain(s), the target criteria that specifies the target interpretation domain (including preconceptions about some target entities, their properties and relationships), and the case (situation or set of phenomena) to be interpreted and understood in the target interpretation domain (e.g. the battery-lamp experiment to be interpreted in terms of energy exchanges, the electromagnetic interactions as measured in Faraday's experiments in terms of a theory in germ in Maxwell's head, or the steam engines in terms of heat and work and other related variables in the nascent thermodynamics). The black box then searches for a model of the case satisfying the target criteria. Because most target entities are not yet operational and interpretable directly in the world, they have to be translated in terms of the more operational interpretation domains given as inputs. In this translation process, submitted to the target criteria, and during model building, some aspects of the model may be automatically filled up through automatic inferencing within the source domain(s) (as is the case when the arrows for transfers are automatically specified when it is decided to translate energy transfer from the notion of electrical current). The output or information gained in the operation consists in the unexpected (because not planned) consequences of the model when interpreted within the target interpretation domain, or in the experimental setting if some target entities are already partially interpretable in the world (as is the case for "energy" for 16-17 years old students).

Analogy

Tunnel effect -Reconceptualization focuses on associated entities that led to inconsistencies in order to differentiate them -Progressive operationalisation of the new conceptual domain -Articulation with primitive perceptions about the world and with the source conceptual domain Table 1. A summary of the main features of analogical reasoning versus features of tunnel effect.

In both analogical reasoning and tunnel effect, the detection of discrepancies between the resulting model and the world or of other inconsistencies opens opportunities for learning. The difference lies in the fact that tunnel effect is intrinsically intended towards the process of building the domain interpretation domain (through the setting up of connections between this domain, the operational ones in the context and the world) whereas analogical reasoning is oriented towards the completion of some specific case with the help of another 'similar' one.

While failed analogies may lead to reconceptualisation in the target interpretation domain, this is much less direct than the learning that may occur when a tunnel effect has produced an unfit model of the world in the interpretation domain.

Conclusions

This paper takes seriously the idea that cognition may imply the existence (and coexistence) of several different interpretation universes, and that a specially important type of learning consists in acquiring new ways of interpreting the world or some aspects of it. In our study we focused on the passage from the currently operational interpretation domain(s) to a new target one when the attention of the cognitive agent is driven towards the interpretation and understanding of some phenomenon or set of phenomena.

In studying the type of conceptual learning at play when students are learning a new conceptual domain or when scientists are struggling to find new ways to account for the world, we discovered the pivotal role of intermediate expressed models. Indeed, when a new interpretation domain is learnt (i.e. new segmenting of the world and new inference rules), the new concepts and new rules are not yet settled nor directly interpretable in the world (think about the first time you heard of tensor calculus or of electrons). They have to be linked with known entities. Therefore, when a model is built in terms of target entities, it in fact refers to the world mostly through entities and relations belonging to the currently operational domain(s). Aspects of this model might thus be filled in thanks to automatic (and unchecked) inferences within the source domain(s). This is the basis for the tunnel effect. These added features, expressed in the model, when re-interpreted within the target domain may bring out unforeseen consequences. Tunnel effect is thus a special inference mechanism at play when models are built at the intersection (but not quite in fact) of some operational interpretation domain(s) -with its/their automatic inferencing capability-and a new ill-known one. Tunnel effect is ubiquitous, mostly unconscious and central in the learning of new conceptual domains. It has so far, to the best of our knowledge, not been described and studied. Tunnel effect eases the construction of models by providing inference mechanisms from the source domain(s) that make up for the as yet non-existent inference mechanisms of the target domain. In so doing, erroneous models might be obtained. These intermediate models can help or hinder reaching a later, more adapted, model. Even though we think we have strong arguments to the effect that tunnel effect can be a powerful guide for further reconceptualisation (see section 3.4), this is still a matter for research, specially in view to the fact that, in case favorable conditions could be identified, one could envision using well-guided tunnel effects to ease the teaching of scientific domains.

Figure 1 .

 1 Figure 1. A view on the problem of learning a new conceptual domain.

Figure 3 .

 3 Figure3. A simplified version of the seed for the target conceptual domain given to the students. The left part presents the conceptual definitions for the target domain . The right part provides the symbols with which to express the model and the syntactic rules that should be satisfied.

Figure 4 .

 4 Figure 4. Three interpretations of the experimental setting of the left column.

Figure 5 .

 5 Figure 5. A shape that we almost necessarily designate by the category "battery".

Figure 6 .

 6 Figure6. Some instances of dialogs between pairs of students during the task (from[START_REF] Megalakaki | Corpus de dialogues de trois groupe d'élèves résolvant trois problèmes mettant en jeu une activité de modélisation[END_REF] and[START_REF] Megalakaki | Expérience MARIANNE : corpus de dialogues de trois gourpes d'élèves résolvant une séquence de problèmes mettant en jeu une activité de modélisation[END_REF].

Figure 7 .

 7 Figure 7. The intermediate model for the battery-bulb experimental setting.

Figure 8 .

 8 Figure8. A decomposition process facilitating problem-solving in an ill-mastered conceptual domain. In each schema, the horizontal axis stands for the space of potential models and the vertical axis stands for the quality of the model with regards to the world. Of course, depending on the interpretation domain (for instance centered on electricity or on energy exchanges), the same models may have different degrees of quality.

  No comparison is involved, only interpretation • Involves associations at the notional level between target entities and source ones • Associated with confusion at the notional level. No notion of similarity between constructs • There is transfer by reinterpretation of the model of which some aspects have been automatically filledin within the source interpretation domain(s). The built model gains autonomy and is reinterpreted in the target domain • New information is produced through automatic completion of the model within the source domain • The source domain(s) is(are) the most operational for interpretation in the current situation • Learning :

	• Two experimental settings or situations that are	• One experimental setting or situation only
	posited as analogs to each other
	• Interpretation takes place both in the source domain	• Interpretation takes place in the source domain
	and in the target domain (there are two situations to	subject to the target constraints and adequacy to the
	be interpreted).	world criterion.
	• Relies heavily on comparisons :	•
	• Implies complex pattern matching between the two
	case representations
	• Tightly associated with the notion of similarity
	between structures. One problem is to explain how
	this similarity is computed
	• There is transfer by matching, alignment and
	completion from the source to the target
	• New information is produced through the
	completion of the target case representation
	• Does not explain how the source is chosen
	• Learning is supposed to arise as :
	-learning of indexing scheme
	-generalization and abstraction from analog cases
	-not really new conceptualization, except by
	generalization

We will use both terms interchangeably in this paper, on the ground that we are ultimately interested in giving an artificial intelligence account of the discussed inference mechanisms, and that in AI one's interpretation is represented as a model (some symbolic construction or system) standing for the situation at hand.

Figure2. Above : one experimental setting involving a battery connected to a luminous bulb through two wires. Students were to produce an interpretation of this setting in terms of a chain of energy transfers and transformations starting and ending with an energy reservoir. Below : a correct interpretation, called target interpretation.On one hand, it is important to notice that the interpretation task was not trivial, even in the simplest of the experimental settings shown in figure 2. For instance, there were two wires from the battery to the bulb which satisfied the closed electrical circuit condition, but only one counterpart, standing for the transfer of energy under the form of electrical work, in the target2 In fact the students are given successively three tasks, only the first task is discueed in the paper. In the first task the experimental material is made up of a bulb, two wires, a battery. In the second task the experiment consists of an object hanging on a string which is completely rolled round the axle of a motor (working as a generator). A bulb is connected to the terminals of the motor. When the object is falling, the bulb shines (figure3). In the third task the experiments consists in a battery connected to an electrical motor. An object is hanging from a string, attached to the axle of the motor, which is completely unrolled at the beginning. A correct solution is given to the students after the first task.

We use a special font to indicate a conceptual target entity, while we use italics to indicate some entity considered at a notional level, that is outside a specific theory.

" Concrete entities " are of course no more concrete that conceptual ones. As[START_REF] Schrödinger | Ma Conception du Monde[END_REF] stated, " chairs and tables are, as much as state vectors (in quantum physics), intellectual constructs deemed to articulate our experience around a small set of invariants. Chairs and tables are theoretical entities to the same extent as state vectors. The only difference being that the theory in which chairs and tables are integrated is the one we had to build since our early youth in order to survive ". Thus concrete entities, like chairs and tables, refer to class of very familiar objects whereas, for the learner, energy has no direct correspondence in the physical world.


Pierre Lejeail

Vincent Lemaire

Antoine Cornuéjols

Adam Ouorou

Triclustering based outlier-shape score for time series in a fraud detection platform

This paper presents a triclustering based outlier-shape score for time series in the context of a fraud detection platform for wholesale traffic for a telecommunications carrier. We propose to use triclustering as an exploration module for outlier shape detection using whole time series. Three main steps compose this approach: (1) projection of data in a new space of time series related features (e.g. derivative), (2) estimation of the density of known normal data using a triclustering method (3) computation of an outlierness score quantifying the distance to the estimator from step (2). We conduct an evaluation of the methodology by focusing on its ability to separate data from different classes. Our preliminary results to assess this approach are very encouraging.

Introduction -Context

Telecommunications companies in different countries use a variety of international telecoms routes to send traffic to each other. In a "wholesale market", telecom carriers can obtain traffic to make up a shortfall, or send traffic on other routes, by trading with other carriers in the wholesale or carrier-to-carrier market. Minutes exchanges allow carriers to buy and sell call terminations (dispatch of a call to its destination). Prices in the wholesale market can change on a daily or weekly basis. A carrier will look for least cost routing function to optimize its trading on the wholesale market.

The quality of routes on the wholesale market can also vary, as the traffic may be going on an illegal route.

Fig. 1. A value chain providing a call termination between two users

A value chain (as the example in Fig. 1) exists between operators that provide the connection between two customers. But this value chain could be broken if a fraudster finds a way to generate communication without paying. For the last decades, fraud has been a growing concern in the telecommunication industry. In a 2017 survey [START_REF]CFCA: 2017 Global Fraud Loss Survey. Survey Results[END_REF], the CFCA3 estimates the annual global fraud loss of $30 billion (USD). Therefore, detecting and preventing, when possible, is essential in this domain. Regarding wholesale market, a list of fraud is known [START_REF]I3 Forum: I3f Fraud Classification[END_REF] by the operators and fraud detection platforms already exist.

Fig. 2. The Orange platform for fraud detection

One of these platforms, shown in in Fig. 2, has been realized by Orange (as a wholesale operator). This platform contains modules which exploit the information given by the expert (a scoring module, for example, which is a classifier), others explore the data to interact with the expert(s) by finding new patterns, including (but not only) malevolent ones. This goal is achieved by knowledge discovery and active learning techniques. Those exploration modules are responsible for adapting to the constant evolution of fraudster's behaviors under (in the case of this platform) the constraint of the limited time that experts can afford to spend in this exploration. This platform share similarities with the one presented by Veeramachaneni et al. in [START_REF] Veeramachaneni | AIˆ2: Training a Big Data Machine to Defend[END_REF]. Both platforms combine a supervised model for predictions with unsupervised models for exploration of unknown pattern, and takes into account the user feedbacks in the learning phase.

In this paper, we are interested in introducing a preliminary work on the knowledge discovery module and we propose a new methodology for the exploration of unlabeled data in an unbalanced data sets. To do this we suggest a new approach to find anomalous behaviors where individuals are represented as time series with a fixed length. By anomalous4 behavior, we mean a behavior different from the behaviors known as normal.

Outlier shape detection

Time series in wholesale fraud detection

Wholesale call record details (call duration, timestamps of the call, destination,...) are grouped by calling phone numbers. Thus, each phone number is represented by multivariate time series. Depending on the number of calls made, time series can have various numbers of points. Observations are kept on a fixed duration, W , (e.g. 6 hours), meaning that all time series have the same "length" (W ). The fraud detection platform must be used in a batch protocol and on past telecommunications (past calls) since the billing is not performed online5 between the operators concerned by a given call. The aim is to find abnormal behavior time series using comparison to a time series database used before to train a model. To inform the other operators that the calls related to those time series are potential frauds, the expert is responsible to assign or not a fraudulent character label to those time series. As we are interested by the global shape of the time series, our method tries to find the abnormality using the entire time series (on the whole length W ) rather than searching for outlier points/subsequences in the time series. Therefore, we propose an outlier shape detection algorithm for the whole time series rather than an outlier points detection algorithm.

Related works

Outlier detection has been extremely studied in the last decades, see for instance the Aggarwal's book [START_REF] Aggarwal | Outlier Detection[END_REF]. More specifically, for temporal data, there are surveys like the one of Gupta and al. in [START_REF] Gupta | Outlier detection for temporal data: A survey[END_REF]. Figure 3, from [START_REF] Gupta | Outlier detection for temporal data: A survey[END_REF], shows a perspective on the state of the art with respect to the data type aspect.

The methodology described in this paper regards time series and thus the red square of Figure 3. As explained in Section 2.1, the wholesale setting implies to work with outlier shape detection algorithms for the whole time series. The related state of the art is therefore in the blue square of the figure: "Direct Detection Time Series Outliers" and the "Unsupervised Discriminative Approaches"6 . The latter uses three key elements: a time series on a give representation, a clustering method and a similarity measure.

Suggested approach of outlierness score

Our intent is to investigate a new suggestion concerning two of the three key elements of the "Direct Detection Time Series Outliers" family methods: the representation and the clustering method. We propose in a preliminary work a methodology based on the estimation of the density of normal events using a triclustering method, on a panel of time series representations and then to compare distances between this density and new individuals. The farther an individual is from the 'normal behavior' distribution, the more likely it is anomalous. The main steps of this methodology are:

1. Transform each time series in several representations; 2. Estimate the density of this set of individuals using a triclustering method; 3. Choose the best representation of time series; 4. Compute an indicator of distance to find unknown behaviors to this density.

Time series representations

In the context of classification, the study of Gay and al. [START_REF] Gay | Feature Extraction over Multiple Representations for Time Series Classification[END_REF] indicates that the choice of representation could have an important impact on the results. In our case it is impossible to know in advance which representation will be the best for density estimation vs. outliers detection. Therefore, we conduct, in Section 4, experiments using a "pool" of representations. The representation used in this first approach are the same as Gay and al. in [START_REF] Gay | Feature Extraction over Multiple Representations for Time Series Classification[END_REF]: the original representation, the Fourier transform coefficients, the first derivative, the second derivative, the cumulated integral, the double cumulated integral and the autocorrelation function. Each of those representations captures different information of time series. Other representations will be used in future works.

Estimation of time series density

We suggest to use a triclustering approach on the following dimensions: customers (C), timestamps (T ) and a given variable (X) (for example the call duration) 7 .

The input data set D is considered to be a collection of N time series, denoted S i with i ∈ 1, . . . , N . The key idea of triclustering with time series is that a curve can be seen as a set of points is that a curve i can be represented as a set of triplets, i.e. S i = {(C i , T j , X ij ) ∀j ∈ 1, . . . , m i }. The total number of data points is denoted by m = ∑ N i=1 m i . No assumptions are made on the time series; they could have different number of points and different timestamp values. A triclustering model aims at jointly discretizing the numerical variables T , X and at making a partition of time series by grouping the values of the categorical variable C. The objective of such models is to estimate P (X, T, C), the joint probability of C, T and X. The output of the model gives groups of time series which are characterized by a bivariate discretization which estimates P (X, T |C).

Amongst the state of the art of the triclustering methods we decided to use the MODL one [START_REF] Guigourès | Utilisation des modèles de co-clustering pour l'analyse exploratoire des données[END_REF]. MODL estimates the constant piecewise joint density of C, T and X with a multidimensional data grid. The optimal grid M is found by a bottom-up greedy optimization of a Bayesian criterion which makes a trade-off between accuracy and robustness of the model:

cost (M ) = -log (p (M ) p (D|M ))
From the information theory point of view, this criterion can be interpreted as the encoding length of the grid plus the length of the dataset D, knowing M , according to the Minimum Description Length principle developed by Rissanen in [START_REF] Rissanen | Modeling by shortest data description[END_REF]. This method is advantageous for unsupervised learning since it is free of user parameters.

Indication about the representation used

The MODL method gives a criterion named "level" (a compression rate value between 0, for a null compression gain, and 1, for a perfect compression gain) which indicates the quality of the density estimation made. One triclustering will be made per representation. Ideally, the triclustering maximizing the quality of the density estimator (value of the level) will be the best, i.e. it would also most likely provide the best separation between known normal time series and outlier time series.

Outlier score

The found partitioning gives the relationship between clusters of behaviors and groups of variables and time intervals. Thus, the outlier behavior of a customer can be linked to a specific interval of time and a set of variables. For a given new time series (not present when training the model) the purpose will be to compute how far this time series appears with respect to the model trained.

To do this we use the similarity criterion which has been defined to assess the proximity between a new time series and the clusters of a MODL coclustering model in [START_REF] Guigourès | Discovering patterns in time-varying graphs: a triclustering approach[END_REF]. This criterion evaluates the decrease of the level (Section 3.3), when a time series is merged within a particular cluster. This decrease can be linked to the impact on the intra-cluster inertia of adding a new individual to its nearest cluster. Therefore, the less typical an individual from its nearest cluster, the higher the score.

Experimental setting and results

This section presents the data sets, the validation protocol and the results of the proposed methodology. The implementation of our method is based on the tool Khiops [START_REF] Boullé | Khiops: outil d'apprentissage supervisé automatique pour la fouille de grandes bases de données multi-tables, available at www.khiops.com[END_REF] and on R scripts for the automation of the methodology.

Our fundamental assumption is that a time series assigned to an unknown class is more likely to be anomalous than if it has been recognized as a member of a known 'normal class'. In the following experiments, we test this hypothesis by using supervised data sets where we emulate the normal/abnormal aspect of the behaviors by choosing a class as 'normal behaviors' and another as 'abnormal behaviors'. Our goal is to answer three questions :

1. Is a triclustering useful to find outlier-shape? 2. Can we find a good time series representation to detect outliers and help the triclustering? 3. Is the quality of density estimation of the triclustering correlated with the obtained results?

Protocol

As often in cyber criminality contexts, it is difficult to share data and results for confidential reason. Therefore, we experimented our method on 9 data sets from the UCR [START_REF] Chen | The ucr time series classification archive[END_REF], offering a wide variety in terms of number and length of time series. Each chosen dataset is composed of two classes of time series. As our model learns on a training dataset only composed of expected normal data, the traintest setting is not the one predefined in [START_REF] Chen | The ucr time series classification archive[END_REF]. We use the following protocol to build our train-test setting:

1. Choose the majority class as normal data; 2. Choose the minority class as anomalous data; 3. Draw 70% of normal data as train data; 4. Keep remaining data as test data.

The evaluation of the score provided by the method for each representation is made with the AUC on test data. This indicator measures how well the score sort the individuals of the different classes, i.e. a good AUC that individuals from the minority class tend to have the highest scores.

Results and discussion

Performances in term of AUC (rounded to the second decimal) are reported in Table 1 for each representation (TS: original time series, DV: first derivated, ddv: second derivated, IV: cumulated integral, IIV: double cumulated integral, PWS: Fourier transform, ACF: auto-correlations). The best result for each dataset is written in bold. We detail below the results in view to the three questions questions set in Section 4. Is a triclustering useful to find outlier-shape? The answer seems positive: At least one AUC is better than random for each dataset (except phalanx with an AUC for double cumulated integral of 0.55).

Dataset

Can we find a good time series representation to detect outliers and help the triclustering? The answer is clearly yes. The last column of the Table 1 indicates that we may find in the (small) pool of representation a representation which performs well. But no representation emerges as better that all others as the results of Gay and al. in [START_REF] Gay | Feature Extraction over Multiple Representations for Time Series Classification[END_REF].

Is the quality of density estimation of the triclustering correlated with the obtained results? For reasons of space limitation, we cannot give the Table of the level values (see Section 3.2) for all representations and data sets, but contrary to intuition, we do not find a positive correlation between the level value and the AUC. Thus the answer is negative. It appears that the method produces clusters of time series that are too specific in some cases. For example, the original representation of computers (training of 175 time series) is partitioned in 174 clusters. Despite the fact that this behavior does not necessarily imply bad results (e.g. the same problem is observed in WormsTwoClass), it is not desirable and may induce a greater variability and worst results than what could be expected.

Discussion : The experiments provide results that are consistent with the assumption that one should observe a larger score in general for classes which were not used in training. Those results are encouraging enough to pursue our efforts in this direction. In fact, by choosing the right threshold (which is a challenge by itself) for predictions one could obtain results nearly as good as in classification (e.g. comparing our results for Wafer, SonyAIBORobotSurface and ItalyPowerDemand with Gay and al. [START_REF] Gay | Feature Extraction over Multiple Representations for Time Series Classification[END_REF], our AUC implies an accuracy, with a threshold well-chosen, nearly as good as with supervised models). Several challenges were revealed by our results. First, there is, for the moment, no way to find a good representation of the data without using a test data set which will not be available in many production applications. This is an important issue since different representations of the same dataset can lead to significant differences between AUC (e.g. ItalyPowerDemand AUC varies from 0.92 to 0.43).

Conclusion

This paper has presented a fraud detection platform for telecom wholesale traffic and introduced a preliminary work on an exploration module. The methodology used is based on outlier detection techniques to detect anomalous unlabeled results. The main key features of this module are: (1) use of a non-parametric method, (2) based on the ability to learn an estimator of the data density and (3) consuming a minimal amount of the expert time. The results are encouraging. They are close to those of supervised models provided by Gay et al. Works still needs to be done, notably to find a way to select/build a good representation for anomaly detection without knowing labels.

Fig. 3 .

 3 Fig. 3. A perspective on the state of the art for temporal data

Table 1 .

 1 Description of time series data sets and comparison of different reprentations. (L=Length). The column "Best gives" the best results per line for the AUC.

		Train Test L	ROC AUC
			TS DV DDV IV IIV PWS ACF Best
	Wafer	4481 2134 152 0.98 1.00 1.00 0.99 0.99 0.98 0.99 1.00
	ECG5000	2043 1201 140 0.99 0.98 0.78 0.93 0.91 0.90 0.99 0.99
	Computers	175 325 720 0.32 0.51 0.44 0.60 0.54 0.50 0.64 0.64
	MoteStrain	479 793 84 0.96 0.95 0.84 0.85 0.66 0.61 0.82 0.96
	PhalangesOutlinesCorrect 1188 567 80 0.67 0.64 0.65 0.62 0.55 0.71 0.63 0.71
	ItalyPowerDemand	384 712 24 0.90 0.68 0.76 0.60 0.43 0.64 0.92 0.92
	SonyAIBORobotSurface	384 377 70 0.54 0.64 0.61 0.87 0.79 0.58 0.66 0.87
	Phalanx	235 641 80 0.40 0.30 0.31 0.41 0.55 0.52 0.44 0.55
	WormsTwoClass	105 153 900 0.77 0.67 0.56 0.78 0.72 0.35 0.72 0.78

Communications Fraud Control Association

In this paper, we also use the term 'outlier' as a synonym.

Fraud disputes between operators are resolved at billing time[START_REF]I3 Forum: I3f Fraud Classification[END_REF].

To save space, we do not give details on this part of the state of the art (see[START_REF] Gupta | Outlier detection for temporal data: A survey[END_REF] for more details), but in a potential extended version paper our method will be compared with competitive approaches. The reader may also note that our method uses density estimation and therefore the state of the art on density estimation could be of interest as well.