P Meunier

E Vilermaux

Van Hove Singularities in Probability Density Functions of Scalars

Keywords: mixing, singularities, PDF Résumé mélange, singularités, PDF

A general theory for the Probability Density Function (PDF) of a scalar stirred in an axisymmetric time-dependent flow is derived. This theory reveals singularities, discontinuities and cusps occurring as soon as the spatial gradient of the scalar concentration vanishes somewhere in the field. These singularities are similar to the Van Hove singularities obtained in the density of vibration modes of a crystal. This feature, ubiquitous in convectiondiffusion problems, is documented experimentally for the mixing of a dye in a Lamb-Oseen vortex.

The interplay between molecular diffusion and simple deformation fields is a classical problem. It is solved in closed form in a variety of situations such as the saddle point flow, the simple shear in two dimensions [START_REF] Leveque | Les lois de la transmission de la chaleur par convection[END_REF][START_REF] Ranz | Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows[END_REF][START_REF] Moffatt | Transport effects associated with turbulence with particular attention to the influence of helicity[END_REF], in three dimensions [START_REF] Villermaux | Mixing in coaxial jets[END_REF], and in the axisymmetric point vortex [START_REF] Rhines | How rapidly is a passive scalar mixed within closed streamlines[END_REF][START_REF] Flohr | Accelerated scalar dissipation in a vortex[END_REF] or spreading vortex flow [START_REF] Marble | Mixing, diffusion and chemical reaction of liquids in a vortex field[END_REF][START_REF] Bajer | Accelerated diffusion in the centre of a vortex[END_REF]. In particular, simple situations are liable of a complete description of the overall Probability Density Function (PDF) of the scalar concentration from the Fourier equation [START_REF] Meunier | How vortices mix[END_REF].

Email address: meunier@irphe.univ-mrs.fr (P. Here, we discuss the condition under which the PDF presents a singularity. The phenomenon we analyse first is illustrated on Fig. 1. A lamellae of weakly diffusive dye (D = 10 -9 m 2 /s) with initial concentration c 0 and width s 0 ≈ 2 mm is deposited in the velocity field of an axisymmetric, time-dependent vortex. The azimuthal velocity profile is Gaussian, that of a Lamb-Oseen vortex [START_REF] Meunier | Analysis and optimization of the error caused by high velocity gradients in PIV[END_REF]

v θ (r, t) = Γ 2πr 1 -exp -r 2 a 2 0 + 4νt (1)
with a circulation Γ = 17 cm 2 /s and initial core size a 0 = 1.3 cm. The lamellae first rolls-up around the vortex (Fig. 1a), stretches and, conversely, gets thinner in its transverse direction. Diffusion is hasten by this stretching motion [START_REF] Meunier | How vortices mix[END_REF] and indeed, Fig. 1(b) shows that the dye concentration is smaller where the elongation is higher, i.e. at intermediate radii. Close to the vortex center in solid body rotation, and far from the vortex center where the velocity magnitude is small and so is the stretching rate, the dye still bears a concentration close to the initial one. The process goes on until at late stages (Fig. 1c), the dye has vanished in the diluting medium.

Compared to our first contribution on the topic [START_REF] Meunier | How vortices mix[END_REF], the set-up has been modified to allow the study of the late stages. Previously, the vortex would experience an axial destabilization 15 seconds after its formation due to the end effects. Here, the fluid is linearly stratified, stabilizing the vortex for 2 to 3 minutes. The dye is now injected by slowly translating a thin iron wire coated with fluoresceine, forming a vertical sheet of dye aligned with the vortex axis. A horizontal laser sheet allows the visualisation of a field crosssection, which is imaged from below (Fig. 1) using a Kodak 8 bits 2048 2 pixels CCD array providing a 12 pixels/mm resolution. While our theory was limited to a stationary point vortex v θ = Γ/2πr, we extend it here to the general case of a time-dependent axisymmetric flow as in Eq. 1. Computing the stretching of a length element dr and using incompressibility, the distance between two meterial points in the direction perpendicular to the lamellae (striation thickness) is s(r, t)

= s 0 [1 + (r t 0 dt∂(v θ /r)/∂r) 2 ] -1/2
, where s 0 is the initial lamellae thickness. In a frame of reference (O, x, y) tangent to the lamellae with y pointing outwards, the convection-diffusion equation for the dye concentration c amounts to a pure diffusion equation ∂c/∂τ = ∂ 2 c/∂ξ 2 , when the transverse coordinate y has been rescaled by the current striation thickness (ξ = y/s(t)) and time t has been rescaled by the current diffusion time (dτ = D dt/s 2 ). The dimensionless time τ is 4) and the dashed line to the approximation (6) using the maximal concentration.

τ (r, t) = Dt s 2 0 + Dr 2 s 2 0 t 0 t 0 ∂(v θ /r) ∂r dt 2 dt (2)
At late stages, the dye concentration has a Gaussian profile in the transverse direction, with a maximum

concentration c M (r, t) = c 0 erf(1/4 √ τ ) depending on τ only c(r, ξ, t) ≈ c 0 erf 1 4 τ (r) exp -6 ξ 2 1 + 24τ (r) (3)
The PDF of such a spatial profile is easily obtained by integration over r in the range where the maximum concentration c M (r) is larger than a given concentration level c:

P (c) = 2s 0 c M (r)>c |∂c/∂ξ| -1 dr /A
where A is a normalising constant equal to the surface area on which the PDF is calculated. Expressing ∂c/∂ξ = -12ξc/(1 + 24τ ), and inverting the function c(ξ), we find the general solution

P (c) = 2s 0 cA max 0, τ (r) + 1/24 log[erf(1/4 τ (r))] -log[c/c 0 ] 1/2 dr (4)
The function 'max(0,.)' stands for the integrand to vanish when the maximum concentration c M (r) is smaller than c; this allows the integration to be performed on the entire lamellae length at any time.

Defining the curvilinear abscissa σ along the lamellae (Fig. 1), this formula is extended to any twodimensional flow provided the stretching rate γ(σ, t) = -[ds(σ)/dt]/s(σ) of all the lamellae elements is known at any time along their Lagrangian trajectory. The integral is computed over σ instead of r for axisymmetric flows, giving

τ (σ, t) = D s 2 0 t 0 exp 2 t 0 γ(σ, t )dt dt (5)
Prediction ( 4) is plotted in Fig. 2 and shows good agreement with the measured distribution. The general solution (4) can be further simplified when τ (r) is rapidly varying with r. By a change of variable u = c M (r)/c, the integral is modified into

∞ 1 cdu τ + 1/24/ log(u)∂c M /∂r.
Since this function diverges for u = 1, the other terms can be replaced by their values in u = 1, which corresponds to c M (r) = c, giving τ + 1/24 dr. This ansatz, also plotted in Fig. 2, reproduces the shape of the PDF correctly, and is off by a factor 2 in amplitude. This discrepancy is a normalization artifact due to the fact that the PDF of the maximal concentration in Eq. 6 does not represent levels smaller than the minimum of c M (r). This PDF is thus restricted to c > 0.09c 0 , where the minimum of the maximal concentration over the dye lamellae is at that instant of time. This approximation is thus all the more valid that the lamellae is deposited initially far from the vortex center, in a region where τ (r) is systematically decreasing with r. There, this ansatz moreover predicts that P (c) ∼ t -3/4 c -3/2 , asymptotically [START_REF] Meunier | How vortices mix[END_REF].

P (c) ≈ τ + 1/24 ∂c M /∂r , (6)
These predictions seem to be good enough when τ (r) and thus c M (r) are monotonic functions of the radius. However, both predictions reveal a singularity when τ (r) has an extremum, since formulas ( 4) and ( 6) diverge when ∂c M /∂r vanishes. We now focus on these critical points.

Figure 2 shows that the theoretical PDF presents a singularity at c * /c 0 = 0.09. Before commenting on the experimental distribution, let us first recognize that such a phenomenon will obviously appear as soon as the gradient of the scalar ∂c/∂r vanishes somewhere in the concentration field. This was already noted by Van Hove [START_REF] Van Hove | The occurrence of singularities in the elastic frequency distribution of a crystal[END_REF] for the density of the elastic vibration frequencies of a crystal (density of modes). Depending on the shape of the dispersion relation ω(k) in the crystal, the density of modes D(ω) has a singularity where the group velocity ∂ω/∂k vanishes. It is in fact a very general feature, arising for any calculation of a density distribution. We recall here the different types of singularities, corresponding in our case to the different topologies of the concentration profile where these events occur. Let us examine first concentration fields in two dimensions.

(i) First, consider a lamellae whose concentration profile presents a 2D saddle point: locally, the scalar concentration is equivalent to c(x, y) = c * + αx 2 -βy 2 with α and β positive. This case is shown schematically on Fig. 3(a) and can be obtained if the stretching is maximum somewhere along the lamellae. For c < c * the PDF is computed by integrating on a square surface of size

(R/ √ α, R/ √ β): P (c) = 4 R/ √ α 0
|∂c/∂y| -1 dx, and by expressing y as a function of c and x: P

(c) = 2Arcsinh(R/ √ c * -c)/ √ αβ.
The PDF is thus equivalent to -log(c * -c) close to the critical level c * . The same scaling can be obtained for c > c * and the PDF thus presents a logarithmic singularity at c = c * in that case. (ii) Second, consider the opposite case where the concentration has a local maximum on the lamellae, as can be realized if there is a minimum of stretching along the lamellae, as shown schematically on Fig. 3(b). The scalar concentration is locally equivalent to c(x, y) = c * -αx 2 -βy 2 . The PDF is equal to zero for c > c * and equal to π/ √ αβ for c < c * . The PDF is thus discontinuous as soon as the scalar concentration profile has a maximum. (iii) Third, the concentration profile presents a local minimum. This is not likely to happen in two dimensions since the topology is rather that of an elongated lamellae, with a transverse concave concentration profile. However, the PDF can be computed in this case also and is found to be discontinuous (0 for c < c * and π/ √ αβ for c > c * ). The PDF of a concentration field in two-dimensions can thus present logarithmic singularities or discontinuities at finite times i.e. as soon as the concentration profiles are affected by diffusion. One understands now why this divergence is rendered on the experimental distribution (Fig. 2) as a weak cusp only. This is primarily due to the finite size resolution of the digital images from which the concentration field is extracted. The logarithmic singularity comes from a very restricted area around a single point, the one with c = c * . The second reason comes from the finite binning of the concentration levels, which enlarges and smoothes the singularity. However, this singularity, which comes from a maximum of the stretching along the spiral does exist (as clearly seen on Fig. 1(b)) and would be all the more resolved that the uncertainty in the definition of the concentrations, and the spatial resolution are sharp enough.

Consider now three-dimensional flows, where the scalar is usually found in the form of sheets as a result of various turbulent or chaotic motions [START_REF] Cantwell | On the behavior of velocity gradient tensors invariats in direct numerical simulations of turbulence[END_REF][START_REF] Buch | Experimental study of the fine-scale structure of conserved scalar mixing in turbulent shear flows. part 1. sc 1[END_REF][START_REF] Fountain | Visualization of three-dimensional chaos[END_REF]. These are shown schematically in Fig. 4 lying in the (x, y) plane. A singularity in the corresponding concentration PDF may again appear when the concentration gradient ∂c/∂r vanishes. There are thus four different cases depending on whether the concentration profile has a maximum or a minimum in each direction.

(i) The first case corresponds to a concentration profile with a maximum of concentration: it is locally equivalent to c(x, y, z) = c * -αx 2 -βy 2 -γz 2 (with α, β and γ positive). This can be obtained if the stretching has a local minimum (see Fig. 4a). The PDF, computed on a volume of radius R is equal to π √ c * -c/ √ αβγ for c < c * and vanishes for c > c * . The PDF presents a cusp with an infinite (negative) slope for c < c * , as indicated on Fig. 4 (iv) Finally, the fourth case would correspond to a local minimum concentration as c(x, y, z) = c * +αx 2 + βy 2 + γz 2 . It is not likely to be very frequent in 3D flows where the scalar has rather the topology of a sheet with a transverse concave concentration profile. However, the PDF can be calculated in that case too and is equal to π √ c -c * / √ αβγ for c > c * and vanishes for c < c * . The PDF again presents a cusp with an infinite slope for c > c * . As in two dimensions and for the same reason, the PDF of a concentration field in three-dimensions can thus present a cusp with an infinite slope each time the gradient of the scalar vanishes. This algebraic divergence is weaker than the logarithmic one in two dimensions.

The singularities are prominent in simple fields, as the one generated by a Gaussian vortex in the present experiment, where the concentration gradient vanishes at one location in the field only. In a complex flow stirred randomly, the number of these singularities should increase with time as the scalar field is distorted by the underlying motion. Indeed, the more complex the spatial scalar distribution, the more likely it will contain saddle points and maxima. However, these singularities will occur at an increasing number of different critical concentration levels c * , at an increasing number of spatial locations in the flow [START_REF] Gibson | Fine structure of scalar fields mixed by turbulence. i. zero-gradient points and minimal gradient surfaces[END_REF]. The scalar PDF, initially presenting one, then two, then a large collection of singular cusps will progressively built-up in a continuous distribution, made of a continuous spectrum of singularities. Being rooted in the local topology of the field, these singularities are robust and may be found in 3D flows as well.

Fig. 1 .

 1 Fig. 1. Visualisation of the spiral formed by a dye lamellae wrapping around a Gaussian vortex. Images are obtained at t = 5 s, t = 50 s and t = 95 s and cover 10 × 10 cm 2 for a vortex of circulation Γ = 17 cm 2 s -1 .

Fig. 2 .

 2 Fig. 2. (a) Probability density function of the concentration c and (b) concentration field at t = 60 s. . The solid line corresponds to the full model (4) and the dashed line to the approximation (6) using the maximal concentration.

aFig. 3 .

 3 Fig. 3. Schematic of the singularities occurring in the scalar PDF of a 2D flow. (top) Pseudo-color plot of the scalar distribution c(x, y). (bottom) The PDF is computed using P (c) = 4 R/ √ α 0

Fig. 4 .

 4 Fig. 4. Schematic of the singularities occurring in the scalar PDF of a 3D flow. (left) Pseudo-color plot of the scalar distribution at the center of the 3D sheet. (right) The PDF is calculated using P (c) = 8 R/ √ α 0

  (a). (ii) The second case corresponds to a concentration profile locally equivalent to c(x, y, z) = c * + αx 2βy 2 -γz 2 , which is obtained if the stretching of the sheet is maximum along y and minimum along x (see Fig. 4b). The PDF is equal to π( √ γR -√ c -c * )/ √ αβγ for c > c * and to πR/ √ αβ for c < c * . The PDF presents a cusp with an infinite slope for c > c * , as indicated on Fig. 4(b). (iii) The third case corresponds to a concentration profile locally equivalent to c(x, y, z) = c * + αx 2 + βy 2 -γz 2 , which is obtained if the stretching of the sheet has a maximum (see Fig. 4c). The PDF is equal to π( √ γR -√ c * -c)/ √ αβγ for c < c * and to πR/ √ αβ for c > c * . The PDF presents a cusp with an infinite slope for c < c * , as indicated on Fig. 4(c).


Chloé Braud Coastal

Pascal Denis
email: pascal.denis@inria.fr

Learning Connective-based Word Representations for Implicit Discourse Relation Identification

We introduce a simple semi-supervised approach to improve implicit discourse relation identification. This approach harnesses large amounts of automatically extracted discourse connectives along with their arguments to construct new distributional word representations. Specifically, we represent words in the space of discourse connectives as a way to directly encode their rhetorical function. Experiments on the Penn Discourse Treebank demonstrate the effectiveness of these task-tailored representations in predicting implicit discourse relations. Our results indeed show that, despite their simplicity, these connective-based representations outperform various off-the-shelf word embeddings, and achieve state-of-the-art performance on this problem.

Introduction

A natural distinction is often made between explicit and implicit discourse relations depending on whether they are lexicalized by a connective or not, respectively. To illustrate, the Contrast relation in example (1a) is triggered by the connective but, while it is not overtly marked in example (1b). 1 Given the lack of strong explicit cues, the identification of implicit relations is a much more challenging and still open problem. The typically low performance scores for this task also hinder the development of text-level discourse parsers [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF][START_REF] Xue | The CoNLL-2015 shared task on shallow discourse parsing[END_REF]: implicit discourse relations 1 These examples are taken from documents wsj 0008 and wsj 0037, respectively, of the PDTB.

account for around half of the data for different genres and languages [START_REF] Prasad | The Penn Discourse Treebank 2.0[END_REF][START_REF] Sporleder | Using automatically labelled examples to classify rhetorical relations: An assessment[END_REF][START_REF] Taboada | Discourse markers as signals (or not) of rhetorical relations[END_REF][START_REF] Subba | An effective discourse parser that uses rich linguistic information[END_REF][START_REF] Soria | Lexical marking of discourse relations -some experimental findings[END_REF][START_REF] Versley | Linguistic tests for discourse relations in the TüBa-D/Z corpus of written German[END_REF].

(1) a. The house has voted to raise the ceiling to $3.1 trillion, but the Senate isn't expected to act until next week at the earliest.

b. That's not to say that the nutty plot of "A Wild Sheep Chase" is rooted in reality. It's imaginative and often funny.

The difficulty of this task lies in its dependence on a wide variety of linguistic factors, ranging from syntax, lexical semantics and also world knowledge [START_REF] Asher | Logics of Conversation[END_REF]. In order to deal with this issue, a common approach is to exploit hand-crafted resources to design features capturing lexical, temporal, modal, or syntactic information (Pitler et al., 2009;[START_REF] Park | Improving implicit discourse relation recognition through feature set optimization[END_REF]. By contrast, more recent work show that using simple low-dimensional word-based representations, either cluster-based or distributed (aka word embeddings), yield comparable or better performance [START_REF] Rutherford | Discovering implicit discourse relations through Brown cluster pair representation and coreference patterns[END_REF][START_REF] Braud | Comparing word representations for implicit discourse relation classification[END_REF], while dispensing with feature engineering. While standard low-dimensional word representations appear to encode relevant linguistic information, they have not been built with the specific rhetorical task in mind. A natural question is therefore whether one could improve implicit discourse relation identification by using word representations that are more directly related to the task. The problem of learning good representation for discourse has been recently tackled by [START_REF] Ji | Representation learning for text-level discourse parsing[END_REF] on the problem of text-level discourse parsing. Their approach uses two recursive neural networks to jointly learn the task and a transformation of the discourse segments to be attached. While this type of joint learning yields encouraging results, it is also computationally intensive, requiring long training times, and could be limited by the relatively small amount of manually annotated data available.

In this paper, we explore the possibility of learning a distributional word representation adapted to the task by selecting relevant rhetorical contexts, in this case discourse connectives, extracted from large amounts of automatically detected connectives along with their arguments. Informally, the assumption is that the estimated word-connective cooccurrence statistics will in effect give us an important insight to the rhetorical function of different words. The learning phase in this case is extremely simple, as it amounts to merely estimating cooccurrence frequencies, potentially combined with a reweighting scheme, between each word appearing in a discourse segment and its co-occurring connective. To assess the usefulness of these connectivebased representations,2 we compare them with pretrained word representations, like Brown clusters and other word embeddings, on the task of implicit discourse relation identification. Our experiments on the Penn Discourse Treebank (PDTB) [START_REF] Prasad | The Penn Discourse Treebank 2.0[END_REF] show that these new representations deliver improvements over systems using these generic representations and yield state-of-the-art results, and this without the use of other hand-crafted features, thus also alleviating the need for external linguistic resources (like lexical databases). Thus, our approach could be easily extended to resource-poor languages as long as connectives can be reliably identified on raw texts.

Section 2 summarizes related work. In Section 3, we detail our connective-based distributional word representation approach. Section 4 presents the automatic annotation of the explicit examples used to build the word representation. In Section 5, we describe our comparative experiments on the PDTB.

Related Work

Implicit discourse relation identification has attracted growing attention since the release of the PDTB, the first discourse corpus to make the distinction between explicit and implicit examples. Within the large body of research on this problem, we identify two main strands directly relevant to our work.

Finding the Right Input Representation

The first work on this task [START_REF] Marcu | An unsupervised approach to recognizing discourse relations[END_REF], which pre-dates the release of the PDTB, proposed a simple word-based representation: they use the Cartesian product of words appearing in the two segments. Given the knowledge-rich nature of the task, following studies attempted to exploit various hand-crafted resources and pre-processing systems to enrich their model with information on modality, polarity, tense, lexical semantics, and syntax, possibly combined with feature selection methods (Pitler et al., 2009;[START_REF] Lin | Recognizing implicit discourse relations in the Penn Discourse Treebank[END_REF][START_REF] Park | Improving implicit discourse relation recognition through feature set optimization[END_REF][START_REF] Biran | Aggregated word pair features for implicit discourse relation disambiguation[END_REF][START_REF] Junyi | Reducing sparsity improves the recognition of implicit discourse relations[END_REF]. Interestingly, [START_REF] Park | Improving implicit discourse relation recognition through feature set optimization[END_REF] concluded on the worthlessness of word-based features, as long as hand-crafted linguistic features were used. More recent studies however reversed this conclusion [START_REF] Rutherford | Discovering implicit discourse relations through Brown cluster pair representation and coreference patterns[END_REF][START_REF] Braud | Comparing word representations for implicit discourse relation classification[END_REF], demonstrating that word-based features can be effective provided they were not encoded using the sparse one-hot representation, but instead with a denser one (cluster based or distributed). This paper takes one step further by testing whether learning a simple task-specific, distributional word representation could lead to further improvements.

As noted, some previous work have also attempted to learn discourse-specific representation for the related problem of discourse parsing. Thus, [START_REF] Ji | Representation learning for text-level discourse parsing[END_REF] reports improvements on the RST Discourse Treebank [START_REF] Carlson | Building a discourse-tagged corpus in the framework of rhetorical structure theory[END_REF], by jointly learning a combination of the discourse units, represented by bag-of-words in a one-hot encoding, along with the sequence of actions of their shift-reduce parser. Our approach is attractively simpler, since training reduces to collecting frequency counts, and it can easily generate representations for unseen words without having to retrain the whole system.

Leveraging Explicit Discourse Data

Another line of work, also initiated in [START_REF] Marcu | An unsupervised approach to recognizing discourse relations[END_REF], propose to deal with the sparseness of the word pair representation by using additional data automatically annotated using discourse connectives. An appeal of this strategy is that one can easily identify explicit relations in raw data, as performance are high on this task (Pitler et al., 2009) and it is even possible to rely on simple heuristics [START_REF] Marcu | An unsupervised approach to recognizing discourse relations[END_REF][START_REF] Sporleder | Exploiting linguistic cues to classify rhetorical relations[END_REF][START_REF] Lan | Leveraging synthetic discourse data via multi-task learning for implicit discourse relation recognition[END_REF]. It has been shown, however, that using explicit examples as additional data for training an implicit relation classifier degrades performance, due to important distribution differences [START_REF] Sporleder | Using automatically labelled examples to classify rhetorical relations: An assessment[END_REF].

Recent attempts to overcome this issue involve domain adaptation strategies [START_REF] Braud | Combining natural and artificial examples to improve implicit discourse relation identification[END_REF][START_REF] Ji | Closing the gap: Domain adaptation from explicit to implicit discourse relations[END_REF], sample selection [START_REF] Rutherford | Improving the inference of implicit discourse relations via classifying explicit discourse connectives[END_REF][START_REF] Wang | Implicit discourse relation recognition by selecting typical training examples[END_REF], or multi-task algorithms [START_REF] Lan | Leveraging synthetic discourse data via multi-task learning for implicit discourse relation recognition[END_REF]. However, it generally involves longer training time since models are built on a massive amount of data, the strategy requiring a large corpus of explicit examples to overcome the noise induced by the automatic annotation strategy. In this paper, we circumvent this problem by using explicit data only for learning our word representations and not for estimating the parameters of our implicit classification model. Some aspects of the present work are similar to [START_REF] Biran | Aggregated word pair features for implicit discourse relation disambiguation[END_REF] in that they also exploit explicit data to compute co-occurrence statistics between word pairs and connectives. But the perspective is reversed, as they represent connectives in the contexts of co-occurring word pairs, with the aim of deriving similarity features between each implicit example and each connective. Furthermore, their approach did not outperform state-of-the-art systems.

The Connective Vector Space Model

Our discourse-based word representation model is a simple variant of the standard vector space model [START_REF] Peter | From frequency to meaning : Vector space models of semantics[END_REF]: that is, it represents individual words in specific co-occurring contexts (in this case, discourse connectives) that define the dimensions of the underlying vector space. Our specific choice of contexts was guided by two main considerations. On the one hand, we aim at learning word representations that live in a relatively lowdimensional space, so as to make learning a classification function over that space feasible. The number of parameters of that function grows proportionally with that of the input size. Although there is often a lack of consensus among linguists as to the exact definition of discourse connectives, they nevertheless form a closed class. For English, the PDTB recognizes 100 distinct connectives. On the other hand, we want to learn a vectorial representation that captures relevant aspects of the problem, in this case the rhetorical contribution of words. Adapting Harris (1954)'s famous quote, we make the assumption that words occurring in similar rhetorical contexts tend to have similar rhetorical meanings. Discourse connectives are by definition strong rhetorical cues. As an illustration, Pitler et al. (2009) found that connectives alone unambiguously predict a single relation in 94% of the PDTB level 1 data. By using connectives as contexts, we are thus linking each word to a relation (or a small set of relations), namely those that can be triggered by this connective. Note that for level 2 relations in the PDTB, the connectives are much more ambiguous (86.77% reported in [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF]), and it could be also the case if we expand the list of forms considered as connectives for English, or if we try to deal with other languages and domains. We however believe that the set of relations that can be triggered by a connective is limited (not all relations can be expressed by the same connective), and that one attractive feature of our strategy is precisely to keep this ambiguity.

Before turning to the details of how we construct our distributional connective-based model, note that we decided to learn a unique representation for any individual word, irrespective of its position (with)in a particular segment. That is, we represent both arguments of a connective as a single bag of words. Other designs are of course possible: we could directly learn distinct word representation for left and right segment words, or even the pair of words [START_REF] Conrath | Unsupervised extraction of semantic relations using discourse[END_REF], to take into account the fact that some relations are oriented (e.g. Reason contains the cause in the first argument and Result in the second one). An obvious drawback of these more expressive representations is that they would need much more data to compute a robust estimate of the frequency counts. 12 Table 1: Illustrative example of association measures between connectives and words.

Building the Distributional Representation

Our discourse-based representations of words are obtained by computing a matrix of co-occurrence between the words and the chosen contexts. The frequency counts are then weighted in order to highlight relevant associations. More formally, we note V the set of the n words appearing in the arguments, and C the set of the m connective contexts. We build the matrix F, of size n × m, by computing the frequency of each element of V with each element of C. We note f i,j the frequency of the word w i ∈ V appearing in one argument of the connective c j ∈ C. We use two standard weighting functions on these raw frequencies: the normalized Term Frequency (TF), eq. ( 1), and the Positive Pointwise Mutual Information (PPMI), eq. ( 2), which is a version of the PMI where negative values are ignored (with p i,j the joint probability that the word w i appears with connective c j , and p i, * and p * ,j , relative frequency of resp. w i and c j ). These two measures are then normalized by multiplying the value by the Inverse Document Frequency (IDF) for a word w i , eq. ( 3), as in [START_REF] Biran | Aggregated word pair features for implicit discourse relation disambiguation[END_REF]. In the final matrices, the i th row corresponds to the m-dimensional vector for the i th word of V. The j th column is a vector corresponding to the j th connective.

TF i,j = f i,j n k=1 f k,j (1)
PPMI i,j = max(0, log p i,j p i, * p * ,j ) (2)
IDF i = log m m k=1 f i,k (3)
Table 1 illustrates the weighting of the words using the TF and the PPMI normalized with IDF. For instance, the presence of the negation "not" is positively linked to Contrast through but and while whereas it receives a null or a very small weight with the temporal connective before. The final vec-tor for this word, < 0.37, 0.18, 0.0 > with TF-IDF or < 0.36, 0.06, 0.0 > with PPMI-IDF, is intended to guide the implicit model toward a contrastive relation, thus potentially helping in identifying the relation in example (1b). In contrast, the word "week" is more likely to be found in the arguments of temporal relations that can be triggered by before but also while, an ambiguity kept in our representation whereas approaches based on using explicit examples as new training data generally choose to annotate them using the most frequent sense associated with the connective, often limiting themselves to the less ambiguous ones [START_REF] Marcu | An unsupervised approach to recognizing discourse relations[END_REF][START_REF] Sporleder | Using automatically labelled examples to classify rhetorical relations: An assessment[END_REF][START_REF] Lan | Leveraging synthetic discourse data via multi-task learning for implicit discourse relation recognition[END_REF][START_REF] Braud | Combining natural and artificial examples to improve implicit discourse relation identification[END_REF][START_REF] Rutherford | Improving the inference of implicit discourse relations via classifying explicit discourse connectives[END_REF]. Finally, a word occuring with all connectives, not discriminant, such as "reality" is associated with a null weight for all dimensions: it thus has no impact on the model. Since we have 100 connectives for the PDTB, the representation is already of quite low dimensionality. However, it has been shown (Turney and Pantel, 2010) that using a dimensionality reduction algorithm could help capturing the latent dimensions between the words and their contexts and reducing the noise. We thus also test versions with a reduction Components Analysis (PCA) [START_REF] Jolliffe | Principal component analysis[END_REF].

Using the Word-based Representation

So far, our distributional framework associates a word with a d-dimensional vector (where d ≤ m). We now need to represent a pair of arguments (i.e., the spans of text linked by a relation), modeled here as a pair of bags of words. Following [START_REF] Braud | Comparing word representations for implicit discourse relation classification[END_REF], we first sum all word vectors contained in each segment, thus obtaining a ddimensional vector for each segment. We then combine the two segment vectors to build a composite vector representing the pair of arguments, by ei-ther concatenating the two segment vectors (leading to a 2d-dimensional vector) or by computing the Kronecker product between them (leading to a d 2 -dimensional vector). Finally, these segmentpair representations will be normalized using the L 2 norm to avoid segment size effects. These will then be used as the input of a classification model, as described in Section 5. Given these combination schemes, it should be clear that despite the fact that each individual word receives a unique vectorial representation irrespective of its position, the parameters of the classification model associated with a given word are likely to be different depending of whether it appears in the left or right segment.

Automatic Annotation of Explicit Examples

In order to collect reliable word-connective cooccurrence frequencies, we need a large corpus where the connectives and their arguments have been identified. We therefore rely on automatic annotation of raw data, instead of using the relatively small amount of explicit examples manually annotated in the PDTB (roughly 18, 000 examples). Specifically, we used the Bllip corpus3 composed news articles from the LA Times, the Washington Post, the New York Times and Reuters and containing 310 millions of words automatically POS-tagged.

Identifying the Connectives and their Arguments

We have two tasks to perform: identifying the connectives and extracting their arguments. 4 Rather than relying on manually defined patterns to annotate explicit examples [START_REF] Marcu | An unsupervised approach to recognizing discourse relations[END_REF][START_REF] Sporleder | Using automatically labelled examples to classify rhetorical relations: An assessment[END_REF][START_REF] Rutherford | Improving the inference of implicit discourse relations via classifying explicit discourse connectives[END_REF], we use two binary classification models inspired by previous works on the PDTB (Pitler and Nenkova, 2009;[START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF]: the first one identifies the connectives and the second one localizes the arguments between inter-and intra-sentential, an heuristic being then used to decide on the exact boundaries of the arguments.

Discourse connectives are words (e.g., but, since)

or grammaticalized multi-word expressions (e.g., as soon as, on the other hand) that may trigger a discourse relation. However, these forms can also appear without any discourse reading, such as because in: He can't sleep because of the deadline. We thus need to disambiguate these forms between discourse and non discourse readings, a task that has proven to be quite easy on the PDTB (Pitler and Nenkova, 2009). This is the task performed by our first binary classifier: a pattern-matching is used to identify all potential connectives, and the model predicts if they have discourse reading in context.

We then need to extract the arguments of the identified connectives, that is the two spans of text linked by the connective. This latter task has proven to be extremely hard on the PDTB [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF][START_REF] Xue | The CoNLL-2015 shared task on shallow discourse parsing[END_REF] because of some annotation principles that make the possible types of argument very diverse. As first proposed in [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF], we thus split this task into two subtasks: identifying the relative positions of the arguments and delimiting their exact boundaries.

For an explicit example in the PDTB, one argument, called Arg2, is linked to the connective, and thus considered as easy to extract [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF]. The other argument, called Arg1, may be located at different places relative to Arg2 [START_REF] Prasad | The Penn Discourse Treebank 2.0[END_REF]: we call intra-sentential the examples where Arg1 is a clause within the same sentence as Arg2 (60.9% of the explicit examples in the PDTB), and intersentential the other examples, that is Arg1 is found in the previous sentence, in a non-adjacent previous sentence (9%) or in a following sentence (less than 0.1%). In this work, we build a localization model by only considering these two coarse casesthe example is either intra-or inter-sentential. Note that this distinction is similar to what has been done in [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF]: more precisely, these authors distinguish between "same-sentence" and "previous sentence" and ignore the cases where the Arg1 is in a following sentence. We rather choose to include them as being also inter-sentential. When the position of Arg1 has been predicted, an heuristic is in charge of finding the exact boundaries of the arguments.

Here, the problem is that in addition to the variety of locations, the annotators were almost free to choose any boundary for an argument in the PDTB: an argument can cover only a part of a sentence, an entire sentence or several sentences. Statistical approaches intended to solve this task lead for now to low performance even when complex sequential models are used, and they often rely on the syntactic configurations [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF][START_REF] Xue | The CoNLL-2015 shared task on shallow discourse parsing[END_REF]. We thus decided to define an heuristic to perform this task, following the simplifying assumptions also used in previous work since [START_REF] Marcu | An unsupervised approach to recognizing discourse relations[END_REF]. We assume that: (1) Arg1 is either in the same sentence as Arg2 or in the previous one, (2) an argument covers at most one sentence and (3) a sentence contains at most two arguments. As it can be deduced from (1), our final model ignores the finer distinctions one can make for the position of inter-sentential examples (i.e. we never extract Arg1 from a non-adjacent previous sentence or a following one).

According to these assumptions, once a connective is identified, knowing its localization is almost enough to identify the boundaries of its arguments. More precisely, if a connective is predicted as intersentential, then our heuristic picks the entire preceding sentence as Arg1, Arg2 being the sentence containing the connective, according to assumptions (1) and ( 2). If a connective is predicted as intrasentential, then the sentence containing the connective is split into two segments -according to (3) -, more precisely, the sentence is split around the connective using the punctuation and making it necessary to have a verb in each argument.

Settings We thus built two models using the PDTB: one to identify the discourse markers (connective vs not connective), and one to identify the position of the arguments with respect to the connective (inter-vs intra-sentential). The PDTB contains 18, 459 explicit examples for 100 connectives. For both models, we use the same split of the data as in [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF]. The test set contains 923 positive instances of connectives and 2, 075 negative instances, and 546 inter-sentential and 377 intrasentential examples. Both models are built using a logistic regression model optimized on the development set (see Section 5), and the same simple feature set [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF][START_REF] Johannsen | Disambiguating explicit discourse connectives without oracles[END_REF] without syntactic information. With C the connective, F the following word and P the previous one, our features are: C, P+C, C+F, C-POS5 , P-POS, F-POS, P-POS+C-POS and C-POS+F-POS.

Results

Our model identifies discourse connective with a micro-accuracy of 92.9% (macro-F 1 91.5%). These scores are slightly lower than the state-of-theart in micro-accuracy, but high enough to rely on this annotation. When applying our model to the Bllip data, we found 4 connectives that correspond to no examples. We thus have examples for only 96 connectives. For distinguishing between inter-and intra-sentential examples, we get a micro-accuracy of 96.1% (macro-F 1 96.0), with an F 1 of 96.7 for the intra-and 95.3 for the inter-sentential class, again close enough to the state-of-the-art [START_REF] Lin | A PDTB-styled end-to-end discourse parser[END_REF].

Coverage Using these models on Bllip, we are able to extract around 3 million connectives, along with their arguments. Our word representation has a large vocabulary (see [START_REF] Collobert | A unified architecture for natural language processing: Deep neural networks with multitask learning[END_REF] and hierarchical log-bilinear embeddings (HLBL) [START_REF] Mnih | Three new graphical models for statistical language modelling[END_REF] using the implementation in [START_REF] Turian | Word representations: A simple and general method for semi-supervised learning[END_REF], Hellinger PCA (H-PCA) [START_REF] Lebret | Word emdeddings through Hellinger PCA[END_REF] and our connective-based representation (Bllip).

Experiments

Our experiments investigate the relevance of our connective-based representations for implicit discourse relation identification, recast here as multiclass classification problem. That is, we aim at evaluating the usefulness of having a word representation linked to the task, compared to using generic word representations (either one-hot, cluster-based or distributed), and whether they encode all the information relevant to the task, thus comparing systems with or without additional hand-crafted features.

Data

The PDTB [START_REF] Prasad | The Penn Discourse Treebank 2.0[END_REF] is the largest corpus annotated for discourse relations, formed by newspaper articles from the Wall Street Journal. It contains 16, 053 pairs of spans of text annotated with one or more implicit relations. The relation set is organized in a three-level hierarchy. We focus on the level 1 coarse-grained relations and keep only the first relation annotated. We use the most spread split of the data, used in [START_REF] Rutherford | Discovering implicit discourse relations through Brown cluster pair representation and coreference patterns[END_REF][START_REF] Rutherford | Improving the inference of implicit discourse relations via classifying explicit discourse connectives[END_REF][START_REF] Braud | Comparing word representations for implicit discourse relation classification[END_REF] among others, that is sections 2-20 for training and 21-22 for testing. The other sections are used for development. The number of examples per relation is reported in Table 3. It can be seen that the dataset is highly imbalanced, with the relation Expansion accounting for more than 50% of the examples.

Settings

Feature Set Our main features are based on the words occurring in the arguments. We test simple baselines using raw tokens. The first one uses the Cartesian product of the tokens, a feature template, generally called "Word pairs", used in most of the previous study for this task as in [START_REF] Marcu | An unsupervised approach to recognizing discourse relations[END_REF]Pitler et al., 2009;[START_REF] Lin | Automatically evaluating text coherence using discourse relations[END_REF][START_REF] Braud | Comparing word representations for implicit discourse relation classification[END_REF][START_REF] Ji | Closing the gap: Domain adaptation from explicit to implicit discourse relations[END_REF]. It is the sparsest representation one can build from words, and it corresponds to using the combination scheme based on the Kronecker product to combine the one-hot vectors representing each word. We also report results with a less sparse version where the vectors are com-bined using concatenation.

We also compare our systems to previous approaches that make use of word based representations but not linked to the task. We implement the systems proposed in [START_REF] Braud | Comparing word representations for implicit discourse relation classification[END_REF] in multiclass, that is using the Brown clusters [START_REF] Peter | Classbased n-gram models of natural language[END_REF], the Collobert and Weston [START_REF] Collobert | A unified architecture for natural language processing: Deep neural networks with multitask learning[END_REF] and the hierarchical log-bilinear embeddings [START_REF] Mnih | Three new graphical models for statistical language modelling[END_REF] using the implementation in [START_REF] Turian | Word representations: A simple and general method for semi-supervised learning[END_REF] 7 , and the HPCA [START_REF] Lebret | Word emdeddings through Hellinger PCA[END_REF] 8 . We use the combination schemes described in Section 3 to build vector representations for pairs of segments. For these systems and ours, using the connectivebased representations, the dimensionality of the final model depends on the number of dimensions d of the representation used and on the combination scheme -the concatenation leading to 2d dimensions and the Kronecker product to d 2 .

All the word representations used -the off-theshelf representations as well as our connective-based representation (see Section 4) -are solely or mainly trained on newswire data, thus on the same domain as our evaluation data. The CnW embeddings we use in this paper, with the implementation in [START_REF] Turian | Word representations: A simple and general method for semi-supervised learning[END_REF], as well as the HLBL embeddings have been obtained using the RCV1 corpus, that is one year of Reuters English newswire. The H-PCA have been built on the Wikipedia, the Reuters corpus and the Wall street Journal. We thus do not expect any out-of-domain issue when using these representations.

Finally, we experiment with additional features proposed in previous studies and well described in (Pitler et al., 2009;[START_REF] Park | Improving implicit discourse relation recognition through feature set optimization[END_REF]: production rules9 , information on verbs (average verb phrases length and Levin classes), polarity [START_REF] Wilson | Recognizing contextual polarity in phrase-level sentiment analysis[END_REF], General Inquirer tags [START_REF] Stone | The General Inquirer: A Computer Approach to Content Analysis[END_REF], information about the presence of numbers and modals, and first, last and first three words. We concatenate these features to the ones built using word representations.

Model We train a multinomial multiclass logistic regression model. 10 In order to deal with the class imbalance issue, we use a sample weighting scheme where each instance has a weight inversely proportional to the frequency of the class it belongs to.

Parameters We optimize the hyper-parameters of the algorithm, that is the regularization norm (L1 or L2), and the strength of the regularization C ∈ {0.001, 0.005, 0.01, 0.1, 0.5, 1, 5, 10, 100}. When using additional features or one-hot sparse encodings over the pairs of raw tokens, we also optimize a filter on the features by defining a frequency cutoff t ∈ {1, 2, 5, 10, 15, 20}. We evaluate the unsupervised representations with different number of dimensions. We test versions of the Brown clusters with 100, 320, 1, 000 and 3, 200 clusters, of the Collobert and Weston embeddings with 25, 50, 100 and 200 dimensions, of the hierarchical logbilinear embeddings with 50 and 100 dimensions, and of the Hellinger PCA with 50, 100 and 200 dimensions. Finally, the distributional representations of words based on the connective are built using either no PCA -thus corresponding to 96 dimensions-, or a PCA 11 keeping the first k dimensions with k ∈ {2, 5, 10, 50}. 12 We optimize both the hyperparameters of the algorithm and the number of dimensions of the unsupervised representation on the development set based on the macro-F 1 score, the most relevant measure to track when dealing with imbalanced data.

Results

Our results are summarized in Table 4. Using our connective-based word representation allows improvements of above 2% in macro-F 1 over the baseline systems based on raw tokens (One-hot), the competitive systems using pre-trained representations (Brown and Embed.) and the state-of-the-art results in terms of macro-F 1 (R&X 15). These improvements demonstrate the efficiency of the representation for this task.

We found that using an unsupervised word representation generally leads to improvements over the 10 http://scikit-learn.org/dev/index.html. 11 Implemented in scikit-learn, applied with default settings. 12 Keeping resp. 11.3%, 36.6%, 56.2% or 95.3% of the variance of the data.

Representation

Macro-F 1 Acc. use of raw tokens (One-hot), a conclusion in line with the results reported in [START_REF] Braud | Comparing word representations for implicit discourse relation classification[END_REF] for binary systems. However, contrary to their findings, in multiclass, the best results are not obtained using the Brown clusters, but rather the dense, real valued representations (Embed. and Bllip). Furthermore, concerning the combination schemes, the concatenation (⊕) generally outperforms the Kronecker product (⊗), in effect favoring lower dimensional models. More importantly, the distributional representations based on connectives (Bllip) allows performance at least similar or even better than those obtained with the other dense representations unconnected to the task (Embed.). While simply based on weighted co-occurrence counts, thus really easy and fast to build, these representations generally outperform the ones learned using neural networks (see CnW and HLBL in Figure 1). Besides, our second best representation is also distributional, namely HPCA (see Figure 1). These result are thus in line with the conclusions in [START_REF] Lebret | Word emdeddings through Hellinger PCA[END_REF] for other NLP tasks: distributional representations, while simpler to obtain, may allow similar results than distributed ones. Our best results with Bllip are obtained without the use of a dimensionality reduction method, thus keeping the 96 dimensions corresponding to the connectives identified in the raw data. Our new word representation like the other low-dimensional ones yield higher scores as one increases the number of dimensions (see Figure 1). This could be a limitation of our strategy, since the number of connectives in the PDTB is fixed. However, one could easily expand our model to include additional lexical elements that might have a rhetorical function such as modals or specific expressions such as one reason is.

We also tested the addition of hand-crafted features traditionally used for the task. We found that, either using a pre-trained word representation or our representation based on connectives, adding these features leads to small or even no improvements and suggest that these representations already encode the information provided by these features. This conclusion has however to be nuanced: when looking at the scores per relation reported in Table 5, the use of the connective based word representation alone allows the best performance for Temporal and Contingency, but the addition of new features dramatically increase the scores for Comparison showing that some information are missing for this relation. Moreover, this relation is the one taking the most advantage of the addition of explicit data in (Rutherford and Xue, 2015), demonstrating that these data could probably provide even more information than the ones we leverage through our representations.

Finally, our results are similar or even better than those reported in [START_REF] Rutherford | Improving the inference of implicit discourse relations via classifying explicit discourse connectives[END_REF] in terms of macro-F 1 . Our systems correspond however to a lower micro-accuracy. Looking at the scores per relation in Table 5, we found that we obtain better results for all the relations except Expansion, the most represented, which could explain the loss in accuracy. It is noteworthy that we generally obtain better results even without the additional features used in this work. Moreover, our systems requires lower training time (since we only train on implicit examples) and alleviate the need for the sample selection strategy used to deal with the distribution differences between the two types of data. 15" are the scores reported in [START_REF] Rutherford | Improving the inference of implicit discourse relations via classifying explicit discourse connectives[END_REF].

Conclusion

We presented a new approach to leverage information from explicit examples for implicit relation identification. We showed that building distributional representations linked to the task through connectives allows state-of-the-art performance and alleviates the need for additional features. Future work includes extending the representations to new contexts -such as the Alternative Lexicalization annotated in the PDTB, the modals or some adverbs -using more sophisticated weighting schemes (Lebret and [START_REF] Lebret | Word emdeddings through Hellinger PCA[END_REF] and testing this strategy for other languages and domains.

  TF-IDF PPMI-IDF Freq. TF-IDF PPMI-IDF Freq. TF-IDF PPMI-

Figure 1 :

 1 Figure 1: F1 scores on dev against the number of dimensions.

Table 2

 2

	) compared to exist-

Table 2 :

 2 Lexicon

coverage for Brown clusters

[START_REF] Peter | Classbased n-gram models of natural language[END_REF]

,

Collobert and Weston (CnW )

Table 3 :

 3 Number of examples in train, dev, test.

Table 4 :

 4 Results for multiclass experiments. R&X 15 are the

	One-hot ⊗	39.0	48.6
	One-hot ⊕	40.2	50.2
	Best Brown ⊗	37.5	50.6
	Best Brown ⊕	40.6	51.2
	Best Embed. ⊗	41.0	51.7
	Best Embed. ⊕	41.6	50.1
	Best dense + add feat.	40.8	51.2
	Bllip TF-IDF ⊗	41.4	51.0
	Bllip TF-IDF ⊕	40.1	50.0
	Bllip PPMI-IDF ⊗	38.9	48.2
	Bllip PPMI-IDF ⊕	42.2 *	52.5
	Best Bllip + add feat.	42.8 *	51.7
	R&X 15	40.5	57.1

scores reported in

[START_REF] Rutherford | Improving the inference of implicit discourse relations via classifying explicit discourse connectives[END_REF]

; One-hot: onehot encoding of raw tokens ; Brown and Embed.: pre-trained representations ; Bllip: connective based representation. * p ≤ 0.1 compared to One-hot ⊗ with t-test and Wilcoxon.

Table 5 :

 5 Scores per relation for multiclass experiments, "R&X

		Bllip PPMI-IDF ⊕ Bllip + add feat	R&X 15
	Rel	Prec	F 1	Prec	F 1	Prec	F 1
	Temp 23.0	29.9	23.7	27.9	38.5 14.7
	Cont	49.6	47.1	46.7	46.3	49.3 43.9
	Comp 35.9	27.7	35.0	34.3	44.9 34.2
	Exp	62.8	64.0	63.7	62.6	61.4 69.1

Available at https://bitbucket.org/chloebt/ discourse-data.

https://catalog.ldc.upenn.edu/ LDC2008T13

Note that contrary to studies using automatically annotated explicit examples as new training data, we do not need to annotate the relation triggered by the connective.

The connective POS is either the node covering the connective, or the POS of its first word if no such node exists.

 6 Training and development sets, only.

http://metaoptimize.com/projects/ wordreprs/

http://lebret.ch/words/

We use the gold standard parses provided in the Penn Treebank[START_REF] Marcus | Building a large annotated corpus of english: The Penn Treebank[END_REF].

Acknowledgements

We thank the three anonymous reviewers for their comments. Chloé Braud was funded by the ERC Starting Grant LOWLANDS No. 313695. Pascal Denis was supported by ERC Grant STAC No. 269427, and by a grant from CPER Nord-Pas de Calais/FEDER DATA Advanced data science and technologies 2015-2020.


Marta Avalos

L E Goff Mélanie

Joly Pierre

Jutand Marthe-Aline

Alioum Ahmadou

EVOLUTION OF TEACHING STRATEGIES IN A FRENCH ODL UNIVERSITY COURSE

The university course on statistical methods in health at the Bordeaux School of Public Health, University of Bordeaux, has been run as an Open and Distance Learning (ODL) program since 2004 on the basics of statistical reasoning in the health field. The course is mainly for professionals. In more than ten years, about 1,000 people have been trained with over a third coming from sub-Saharan Africa. The program aims to meet a growing demand for statistical training from professionals from the south whose mobility is limited. Each year a satisfaction survey is sent to students with a view to improving the program. Even though participation in the survey is anonymous and not compulsory, it is a valuable source of comments and ideas. These have led to innovative pedagogical practices such as "tutored exercises" with individual correction, the use of new statistical software, summary sheets and flipped classrooms. However, benchmarking of the program has shown that more could be done. Teaching strategies should evolve within the framework of distance learning in terms of content, form and interactivity. This article discusses the development of these new educational strategies from their inception as well as future projects.

CONTEXT

The Bordeaux School of Public Health at the University of Bordeaux (France) set up online "university diplomas" in 2001 in response to a request for proper education, training and adequate certification in public health from a French-speaking audience that could not be present for face-toface training for geographical or professional reasons. University diplomas are institution-specific degrees presented as single subject courses that reflect the strengths of individual universities and offer students opportunities to gain university-level education in specific fields. The Open and Distance Learning (ODL) program was born out of political will, the commitment of the teaching staff and thanks to financial opportunities that, among other things, allowed the establishment of an information technology department that manages the learning platform.

Currently, ten online French university courses are available (http://ead.isped.u-bordeaux2.fr). University courses in Spanish are also proposed. Among the online university courses, two concern the biostatistics track (Statistical methods in health and Statistical regression methods in epidemiology) and four the epidemiology track. Indeed, epidemiology and biostatistics are one of the strengths of the Bordeaux School of Public Health, which is recognized internationally. The courses are run entirely online except for the final session exam, which is conducted in accredited centres around the world such as universities, French embassies, French high schools or centres within the University Agency for French-speaking communities (AUF). In addition, the online master's program in Public Health at the University of Bordeaux became effective in fall 2007. The master's degree is a more advanced program that has exam sessions but also two-week face-to-face courses organised in a few accredited centres (currently the University of Bordeaux in France, the University of Abomey-Calavi in Benin, and the Institut Pasteur in French New Caledonia in the Pacific Ocean). The master's degree is offered online-and campus-based while only an online program is proposed for the ten university diplomas.

The minimum admission requirements for the university course in Statistical methods in health are an undergraduate degree or at least five years of relevant work experience in the health sector. The university course Statistical regression methods in epidemiology and the master's degree are more selective with specific admission requirements. The subject courses programs are mostly included in the educational program of the Masters in Public Health. Thus, validating the single subject courses implies automatic validation of statistical or epidemiological courses in the interdisciplinary first year of the Masters.

STRUCTURE, FUNCTIONING, AUDIENCE

The university course on statistical methods in health at the Bordeaux School of Public Health, University of Bordeaux aims to teach the basics in statistical reasoning and methods in the health field. The course, which has been run since 2004, allows students to develop their ability to conduct data analysis as end-users.

The teaching period runs from mid-October to late May. The program involves 100 hours of teaching, plus personal study/work time (exercises, graded and non-graded assignments, and review for the final exam). This corresponds to a weekly work load of 4 to 5 hours. The course contains four modules:

-Descriptive statistics, introduction to general terminology -Introduction to the fundamental concepts: random variable, basic discrete and continuous probability laws, population, sample and sampling fluctuation concepts, sampling distributions, central limit theorem, point estimation, confidence interval -General principles of statistical hypothesis testing, one and two sample models, main parametric and non-parametric tests -Introduction to simple linear regression and one-factor ANOVA Each module is structured in three parts. First, the course introduces concepts, which are illustrated by several examples from health studies. Definitions and calculation methods are introduced as the need arises in a convenient, self-contained and intuitive way. Second, written exercises are proposed. Finally, lab exercises can be solved using software. Introductions to recommended statistical software and MS Excel spreadsheets are also provided.

The course schedule including the online availability of documents, theoretical courses, exercise corrections, assignment corrections and deadlines for graded and non-graded assignments, past years' exams etc., is announced at the beginning of the academic year. News and updates are posted using the news column or are distributed by email. The Bordeaux School of Public Health online degrees are offered using an in-house customised learning management system called Plei@de that is an intuitive and ergonomic platform. Like other learning platforms, for example Moodle or Dokeos, it provides technical support for exchange between teachers and students: repository for course material or web pages, upload of assignments, data files, quizzes, forums, technical assistance, space to share or store files, online resources and tools, student marks, etc. Plei@de offers a high degree of fluidity in the interaction between students and teachers. The use is simpler than Moodle in that sense that it offers, in the same space, educational, technical and administrative information. The platform also provides user statistics: number of visits, number of finished exercises and corresponding scores, downloaded documents, uploaded assignments, date of last access, etc. This can be used as a tool to assess the availability, motivation and serious-mindedness of students.

The final grade for a course is determined by a weighted average of the marks students receive on all of their course assignments and exams. To pass the course, students must have a weighted average of 50% or better on the exam and the graded assignments. There are two planned assignments (one for the first two modules and another for the last two modules) that are weighted equally and a final exam that accounts for two thirds of the grade. The final exam is taken in specific centres all around the world, where the presence of the student is required to ascertain his/her identity. After a decade, more than nine hundred students have participated, the number of students per year varying between 60 and 100. More than one-half of students are from Africa, the most numerous coming from sub-Saharan Africa (taken as the country of residence). Approximately a third of the students come from Metropolitan France (the part of France situated in Europe). In fact, a partnership has been established with the AUF which finances several students per year. Between 45 and 90% of the students per year pass the course, and the annual dropout rate (taken as the number of students absent for the final exam) varies between 12 and 30%. Table 1 presents differences in dropout rates and pass rates according to the region of residence for the last three years. Dropout rates are highly variable from year to year and intra-group variability is higher than intergroup variability. Metropolitan France residents group showed the highest pass rates; however the potential for confounding is obvious.

Table 1: Annual dropout rate, taken as the number of students absent for the final exam to N, the total of participants, and annual pass rate, taken as the number of students having a weighted average of 50% or better on the exam and the graded assignments to n, the number of participants present for the final exam. Rates are computed for participants coming from sub-Saharan Africa, Metropolitan France, and other regions for the last three years ( 83) 76% ( 67) 21% ( 95) 67% ( 75)

The vast majority of the students are in the health sector. For instance, in 2014-2015, only 13% of the students came from non-medical professions, 53% came from medicine and the remaining percentage came from dentistry, midwifery, pharmacy, nursing or associated health professions. Some of them are already experienced health professionals while others are still studying and not yet qualified.

INSTRUCTIONAL DESIGN, EVOLUTION

Once the purely online approach was adopted, the pedagogical thinking had to change entirely. We could not limit ourselves to downloadable educational content but had to explore the best possible ways of supervising or advising students throughout their gradual learning process. We had to take into account the traditional constraints associated with e-learning student profiles: e.g. professionally active adults who want to acquire an advanced degree or new professional skills but do not have time to attend regular classes. In addition, constraints specific to the targeted audience had to be considered, i.e. professionals in the public health sector located in different countries worldwide with variable internet connection quality and living in different time zones. We also realised we needed to develop interactive web pages. Animations could be used to reinforce conceptual understanding by illustrating particular points and active learning that made the course dynamic. However, downloadable documents (handouts, exercises, etc.) were required because of unequal access to Internet.

A flexible schedule, freedom of action, reduced indirect training costs and a cooperative learning environment are all advantages of online teaching compared to traditional face-to-face classes. However, factors such as student isolation may lead to a high dropout rate [START_REF] Packham | E-learning and Retention: key Factors Influencing Student Withdrawal[END_REF][START_REF] Park | Factors Influencing Adult Learners' Decision to Drop Out or Persist in Online Learning[END_REF]) and a high failure rate in distance learning. The most conventional ways to overcome this problem are forums, which we offered as part of each teaching module, or the use of email for direct contact with the teacher. In general, forum discussions dealt with misunderstanding of mathematical formulas or concepts, understanding the use and sense of statistical calculations and reasoning, particular problems when using statistical software, questions on particular steps when resolving exercises, etc. Forums are also conducive for exchange between students. Students can answer each other's questions, share difficulties, and give explanations and advice, which is beneficial for all students concerned [START_REF] Hall | Improving Self-efficacy in Statistics: Role of Self-explanation & Feedback[END_REF].

To assess the quality and effectiveness of our courses and understand what has not worked in order to change it, a systematic annual evaluation is conducted. Mills and Raju, 2011 summarized effective practices for teaching statistics online, and in particular, they recommend conducting evaluations to monitor the teaching and learning process. The satisfaction survey that is sent to students contributes to improving the program. Questions relate to learning objectives (for each module and software training), interactivity (utility, effectiveness), visual impact, technical functionalities (interface and navigation effectiveness) and training tools, assistance quality (technological, pedagogical or administrative problems), time (real availability, real personal study/work time), habits and environment (internet accessibility, material preferences). [START_REF] Tudor | Teaching Introductory Statistics Online -Satisfying the Students[END_REF] showed the impact of interactivity on student satisfaction.

Since participation in the survey is not compulsory, a selection bias effect is more than likely, so caution is required when interpreting these results [START_REF] Zumrawia | What Response Rates are needed to make Reliable Inferences from Student Evaluations of Teaching? Educational Research and Evaluation[END_REF]. Roughly, students responding to survey questions are globally satisfied. Since participation in the survey is anonymous, comparisons in reactions to course innovations or in survey responses between different groups (according to residence region, dropping out, passing the course, etc.) are not possible. The interest is to gather comments and ideas that have led to the implementation of new pedagogical practices. First, the forum alone cannot meet the expectations of all students. As in a classroom, shy or unsure students may be reluctant to openly formulate their questions on a forum. After two years of experience, we decided to introduce the notion of "tutored exercises" which are non-graded assignments in every module that lead to personalized feedback. They allow students to train, to evaluate themselves and to receive personalized correction so that they can identify potential errors or misunderstandings and improve their writing. By providing personalized feedback that guides students towards improvement, student motivation is fostered throughout the year and the feeling of isolation is somewhat reduced. Furthermore, teachers receive quick feedback on learners'' difficulties. In fact, there is a close relationship between doing "tutored exercises" and results in graded assignments and at the final exam (Figure 1). This relationship should be interpreted cautiously since further research is required to establish whether doing "tutored exercises" has a positive effect on motivation by keeping students involved and active or whether it is simply a marker of motivation. Examples used in the course, "tutored exercises", graded assignment and final exam questions are based on real publicly available data or real data with restricted use. Also, simulated data based on health studies are used. Real data with restricted use came from studies conducted in the Bordeaux School of Public Health, e.g. the intima-media thickness measurement study [START_REF] Mercié | Evaluation of cardiovascular risk factors in HIV-1 infected patients using carotid intima-media thickness measurement[END_REF] or the PAQUID cohort of individuals aged 65 years or older followed from 1988 until present to study the effects of different environmental, behavioural, and social vectors of age-related medical conditions and diseases [START_REF] Lemeshow | Illustration of analysis taking into account complex survey considerations: the association between wine consumption and dementia in the PAQUID study[END_REF]. Simulated data are based on epidemiological research articles or on studies conducted in the Bordeaux School of Public Health that are not available for educational purposes, such as data coming from the i-Share cohort aimed to gain better knowledge and understanding of students' state of health over a period of at least 10 years [START_REF] Guichard | Association Between Headaches and Tinnitus in Young Adults: Cross-Sectional Study[END_REF].

All the modules now start with a video presentation of the teacher who concisely summarises the course content. This provides learners with eye contact with the teacher. We recently introduced virtual classrooms by means of the Adobe Connect web conferencing software using a flipped classroom style. In general, students appreciate real-time contact and oral communication to ask questions (instead of writing questions using the forum, which needs precision and involves noninstant feedback). Furthermore, discussions are more fruitful and students' errors of interpretation or understanding are identified more quickly and easily. At least one virtual class per module and statistical software are planned and announced from the beginning of the academic year. Different Internet connections or equipment quality, time zones and availability are still a problem with synchronous online classes in which all students need to be online at the same specific time in order to participate. A brief summary of questions and discussions is posted on the forum especially for absent participants.

Initially, EpiInfo, an easy-to-use database and statistical analysis program widely used by epidemiologists, and MS Excel spreadsheet were the only programs taught in this university course. While they are well suited to our audience, neither EpiInfo nor MS Excel spreadsheet allow the implementation of all the methods taught in this course. R (R Core Team, 2013) offers greater scope and its fast-growing popularity makes its use mandatory. In recent years, the introduction to the R module has been improved and virtual classes allow direct explanations. It has become more and more difficult to propose multiple software solutions to resolve data analysis exercises. Consequently, we now focus on R mastery. However, students' results are still unsatisfactory so considerable effort needs to be made by students to master this software. As a result of students' requests or remarks concerning their habits, we introduced changes to the presentation. First, summary sheets were provided for each module. In addition, the redundant content available as web pages and power point slides was deleted and only downloadable pdf handouts were kept. Web pages were still useful for the animations they contained. This led to a simplified presentation. The homepage is now animated and frequently renewed to encourage users to return regularly. Multiple-choice questions (MCQ) have been added to each module with partial immediate feedback that encourages students to search for the right answers in the course. However, compared to popular e-learning platforms like Moodle which allowing for fast developments and the integration of new functionalities, our platform lacks modernity. Since Moodle has been widely adopted by the University of Bordeaux, we now need to create a link between the platforms in order to capitalise on what Moodle has to offer, e.g. the wide variety and potential of its evaluation tools.

Figure 2 shows annual dropout rates from 2004 to 2016 and summarizes innovations adopted over time. Also, organizational modifications are indicated. For example, the suppression of the second session exam (because of the organisational burden and small effectiveness) and the graded assignments using a MCQ system that improved timing of notification of results to students. In these graded assignments students are expected to solve each MCQ by conducting data analysis using software. In general, the relationship between innovations adopted and dropout rates are not conclusive and confounding is more than likely. However, adoption of "tutored exercises" shows a positive effect on the decrease of dropout rates.

CONCLUSION

Distance learning has become a great tool for gaining access to knowledge and diploma courses. It particularly encourages the professional development and improvement of public health in Southern countries while contributing to maintaining the integration of their health managers in them. Since its inception, the educational strategy of the Statistical methods in health course has not been limited to the launch of downloadable educational content but also included exploration of the best possible ways to supervise and counsel students throughout their gradual learning process. Thanks to feedback from annual training satisfaction surveys sent to students, we have introduced several changes, e.g. "tutored exercises", virtual classes, new statistical software training and supplementary MCQ. Because of the importance of provision to Southern countries, it would be useful to analyse feedback by region of residence. The assessment of R courses is still a challenge for students attracted by the course. Furthermore, it is difficult to keep abreast with other learning platforms such as Moodle that are developing innovative tools at a fast pace thanks to the heavy financial investments made in them. Nevertheless, after a decade, we now have sound experience in ODL, even though this sector requires constant re-evaluation, re-adaptation and re-thinking of the best educational strategies and learning tools.

Figure 1 :

 1 Figure 1: Average marks (the higher the better) of graded assignments and final exam per number of "tutored exercises" sent among the four "tutored exercises" (course in Statistical methods in health, Bordeaux School of Public Health, University of Bordeaux, France, 2014-2015, 83 students). n indicates the number of participants in the group.

Figure 2 :

 2 Figure 2: Dropout rates and innovations adopted over time from 2004 to 2016 (course in Statistical methods in health, Bordeaux School of Public Health, University of Bordeaux, France).

  course in Statistical methods in health, Bordeaux School of Public Health, University of Bordeaux, France, 2013-2016).

		2013-2014	2014-2015	2015-2016
		Dropout	Pass rate	Dropout	Pass rate	Dropout	Pass rate
	Residence	rate (N)	(n)	rate (N)	(n)	rate (N)	(n)
	Sub-Saharan Africa	8% (49) 56% (45)	24% (34) 65% (26)	21% (38) 50% (30)
	Metropolitan France	26% (38) 75% (28)	20% (30) 92% (24)	12% (42) 81% (37)
	Other regions	13% (15) 62% (13)	11% (19) 71% (17)	47% (15)	63% (8)
	All	16% (102) 63% (86)	19% (


Thomas Pietrzak
email: thomas.pietrzak@univ-lille1.fr

Towards the Full Experience of Playing Drums on a Virtual Drum Kit

A key concept of drums kits is the ability to customize elements and their placement. It is part of the creative process which influences the playing style. We discuss elements of drum kits customization, existing haptic interactive systems for drums, and technologies which would make it possible to implement a virtual drum set which provides the full experience of playing drums.

MOTIVATIONS

Drums is a particular musical instrument in the sense that each musician builds his own drum set to create his own sound. Each element produces a particular sound, takes a significant space, and costs a significant amount of money. Few musicians can afford a drum kit with tens of drums and cymbals like Terry Bozzio 1 or Mike Portnoy 2 . Electronic kits use pads and piezo sensors with a MIDI expander, and produces music similarly to synthesizers. Recent kits replaced pads with drumheads, which gives a more realistic and enjoyable haptic sensation. These kits give make it possible to customize the sound of each drum element. However the haptic feedback is not customizable, and these kits still need as many physical elements as virtual elements. A virtual drum kit would make it possible to build any kit possible.

HAPTIC DRUMS

Research about haptic drum focused on the emulation of a drum sound [START_REF] Peter | Hap-Kit: A haptic interface for a virtual drum-kit[END_REF]. Authors emulate a stroke on a drum head with a mechanical apparatus. However the user just presses a button to trigger the mechanical actuation. This does not reproduce the experience of playing drums. Later studies presented a drum emulation system [START_REF] Berdahl | A Physically Intuitive Haptic Drumstick[END_REF], which uses a PHANToM [START_REF] Massie | The PHANToM Haptic Interface: A Device for Probing Virtual Objects[END_REF] haptic device. There is more interaction, but it is still far away from the experience of playing a full drum set. The Ubiquitous Drums project is a step forward in this direction [START_REF] Smus | Ubiquitous Drums: a Tangible, Wearable Musical Interface[END_REF]. Input 1 http://terrybozzio.com/about-terry/kit-setup/the-big-kit/ 2 https://www.mikeportnoy.com/drums/mirage/ Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). sensors on the hands and feet make it possible to play drums without drum kits. However in this case there is no haptic feedback, and no actual drum kit.

CIRMMT Symposium on Force

To summarize, to our knowledge, there is no full haptic virtual drum kit which provides a full drumming experience with extensive customization of the drum elements.

TECHNOLOGIES

Technology is a key limitation for this application. The actuation apparatus has to react as fast as extreme limbs movements, and resist to arm strength. Typical robotic arms hardly reach this level of performance [START_REF] Massie | The PHANToM Haptic Interface: A Device for Probing Virtual Objects[END_REF]. We envision using alternative technologies such as a SPIDAR [START_REF] Murayama | SPIDAR G&G: A Two-Handed Haptic Interface for Bimanual VR Interaction[END_REF] or Electro-Muscle Stimulation [START_REF] Tamaki | PossessedHand: Techniques for Controlling Human Hands Using Electrical Muscles Stimuli[END_REF]. Besides, haptic drum kit have no visual aspect. However this is critical for actual play. Thankfully recent advances in VR headsets enables crating the sensation of being surrounded with the drum kit.

  Feedback and Music, dec 2016, Montreal, Canada © 2016 Copyright held by the owner/author(s).