Landrieu Loic

Guillaume Obozinski

Continuously indexed Potts models on unoriented graphs

This paper introduces an extension to undirected graphical models of the classical continuous time Markov chains. This model can be used to solve a transductive or unsupervised multi-class classification problem at each point of a network defined as a set of nodes connected by segments of different lengths. The classification is performed not only at the nodes, but at every point of the edge connecting two nodes. This is achieved by constructing a Potts process indexed by the continuum of points forming the edges of the graph.

We propose a homogeneous parameterization which satisfies Kolmogorov consistency, and show that classical inference and learning algorithms can be applied.

We then apply our model to a problem from geomatics, namely that of labelling city blocks automatically with a simple typology of classes (e.g. collective housing) from simple properties of the shape and sizes of buildings of the blocks. Our experiments shows that our model outperform standard MRFs and a discriminative model like logistic regression.

INTRODUCTION

Connections in networks typically have a length or weight that gives a measure of distance between the nodes connected, or the intensity of their interaction. This length information has been used to perform unsupervised or semisupervised classification on graphs based among others on graph partitioning algorithms (see e.g. [START_REF] Zhu | Introduction to semisupervised learning[END_REF]. When defining probabilistic graphical models on such networks, it is not clear how to take this distance into account naturally so that the interaction decreases with the distance. In this paper, we propose an unoriented counterpart of the continuous-time Markov process on a tree proposed by [START_REF] Holmes | An expectation maximization algorithm for training hidden substitution models[END_REF] which is naturally generalized to any unoriented graph.

In a continuous time Markov chain, a random state X t is associated with every point t ∈ R + . The generalization to a continuous tree mode considered by [START_REF] Holmes | An expectation maximization algorithm for training hidden substitution models[END_REF] is most simply described through its application in phylogenetics. The phylogenetic tree of a family of species is assumed given as a directed tree with branches of different lengths. The length of the branches measure the genetic distance between extant or extinct species. Branching nodes are associated with speciation events. Each point of each branch of the tree corresponds to the form taken by a species as it existed at one time in the past and the variable modeled as a random process and defined at each such point is typically a discrete trait of that species such as the nucleic acid among {A, C, T, G} at a certain position in the DNA. In the absence of speciation event, the state evolves like a continuous-time Markov chain, with time here being measure in terms of the genetic distance along an edge. When a branching occurs, the Markov chain is split into two identical states which continue to involve independently. For that process, if the edges of the trees are identified with line segments, there is a random variable X t associated with every point t of each of these segments. Since a tree is simply connected, removing point t will split the tree in at least two components, and with this model we have the fundamental Markov property that the subprocesses defined on each component are conditionally independent given X t .

We aim to extend these models in two ways. First, these continuously indexed processes are fundamentally oriented. This stems for the fact that the continuous Markov chain in this model is homogeneous, which implies that the conditional distributions forward in time are constant, a property which, while true forward, is not in general true backwards in time. This implies in particular that all marginals of the process on any finite set of points including at least all nodes of degree different than two is naturally parameterized as a product of conditionals p(x s |x t ) whose value depends on the graph only through the distance between s and t. We aim to propose natural parameterization for unoriented continuously indexed models with the same Markov property as the oriented trees. Second, the considered models are simply connected and we would like to propose an extension from weighted trees to general weighted graphs, where all edges are identified with a real segment of lengths equal to their weights, and which satisfy the Markov property in the sense that if a finite set of points A on these segments cuts the graph into several connected components, the processes on the two subgraphs are conditionally independent given (X a ) a∈A . The obtained models will be Potts models that take into account in a natural way the length of the edges and such that the interaction between two nodes decreases with the distance separating them.

After a discussion of related work, we first consider the simplest case of an unoriented continuous chain for which we propose an exponential family parametrization. Next, we show how this parametrization is naturally extended to general unoriented continuous graphs. We derive the marginal log-likelihood of different subsets of nodes, as well as the form of its gradients, and show that inference and learning in these models can be obtained with classical algorithms. We then extend the model and algorithms to the hidden Markov random field case where a feature vector is attached to a certain number node. In terms of experiments, we consider first a transductive classification problem from geomatics, which consists in assigning city blocks to different classes from simple buildings characteristics, while taking into account the distances between the blocks. Then we illustrate the possibility of using the model for transfer learning in order to refine predictions for city blocks from a new entirely unlabelled city.

RELATED WORK

The model we consider in this work can be viewed as an extension to undirected graphs of the continuous time Markov chain (CTMC). The continuous-time Markov chain [START_REF] Norris | Markov chains[END_REF]) is a fundamental model in probability and statistics for random variables that take values in a set of discrete states and that can transition at any point in continuous time from one state to another. Beyond its theoretical value, it has been applied directly in queuing theory, for the statistical modeling of chemical reactions and in genetics.

In genetics, CTMC models have been notably used to propose models of the evolution of DNA at the nucleotide level [START_REF] Nielsen | Statistical methods in molecular evolution[END_REF][START_REF] Durrett | Probability models for DNA sequence evolution[END_REF], with among several others, the celebrated Jukes-Cantor model. In this context, these models have been extended to directed trees, where the tree corresponds to a phylogeny of species or of proteins, and which has been used to estimate rate matrices or for genetic sequence alignment [START_REF] Von Bing | Modeling DNA base substitution in large genomic regions from two organisms[END_REF]. Like for CTMCs, the fact that these models are continuous arise from temporality, and the models derived are thus intrinsically oriented. For these CTMC on trees, [START_REF] Holmes | An expectation maximization algorithm for training hidden substitution models[END_REF] proposed an exponential family parametrization of the likelihood and showed that it was possible to design an EM algorithm to learn the rate matrices modeling the substitution of DNA bases over time, in a way that generalizes the classical EM algorithm on trees.

As is the case for the CTMC, continuously indexed processes arise typically as the limit of discretely indexed processes. Along these lines, Yaple and Abrams (2013) consider a continuum limit of the Ising model on a regular grid where the lengths of the edges are infinitesimal and use it to characterize the patterns of magnetic polarity in ferromagnetic materials through the resolution of integro-differential equations.

A different but also recent line of research combining ideas from the graphical models literature with stochastic processes is known under the name of continuous time Bayesian networks (CTBNs, [START_REF] Nodelman | Continuous time Bayesian networks[END_REF]. These are models of structured multivariate stochastic processes in time in which the interaction between the different components of the process can be modeled by a graphical model. These models are quite different than the continuous time tree models or the models we will propose in this paper in that, for CTBNs, the graphical model structure is somehow orthogonal to the direction of time which is the unique global oriented continuous variable for the process.

Last but not least, a common family of approaches which take into account the length of edges in a graph in the context of unsupervised or semi-supervised classification are the graph partitioning and related spectral clustering techniques (see e.g. Zhu and Goldberg, 2009, chap. 5). A review of these techniques is beyond the scope of this paper. We however discuss how these methods differ and are not directly comparable to ours in section 7.

NOTATIONS

All multinomial variables considered in the paper take values in K = {1, . . . , K} and are represented by the indicator vector x ∈ {0, 1} K whose sole non zero entry is x k when the multinomial is the kth state. We thus define X = {x ∈ {0, 1} K | k∈K x k = 1}. Given a vector x ∈ R K , Diag(x) is the diagonal matrix whose elements are the entries in x.

We use ⊙ (resp. ⊘) to denote the Hadamard product (resp. division), that is the entrywise multiplication (resp. division) of matrices.

We will denote nodes of graphical model with the sans-serif font a, b, and set of nodes with upper capitals of the same font: A, B.

CONTINUOUS GRAPH POTTS MODELS

An unoriented continuous chain model

To derive a parameterization of the model, we start with the case of an unoriented chain that we identify with the [0, l] segment, where without loss of generality l is an integer. We will denote by X a a multinomial random variable associated with the point a ∈ [0, l]. Before defining the process at any point of the segment, we model the joint distribution of the random variables X k for k an integer in {0, . . . , l}.

Denoting by x k ∈ {0, 1} K an instance of X k , and assuming that both unary and binary potentials are constant, the joint distribution of (X k ) k∈{0,1,...,l} can be written in multiplicative form as

p(x 0 , x 1 , . . . , x l ; U, h) ∝ l k=0 h ⊺ x k l-1 k=0 x ⊺ k U x k+1 ,
with h ∈ R K + * the vector of unary potential values and

U ∈ R K×K + *
the matrix of binary potential values. For reasons of symmetry and invariance along the chain, we assume that those parameters do not depend on the position k and that U = U ⊺ . Note that, while similar in spirit, the assumption that these parameters are constant is different from assuming that the Markov chain is homogeneous; we discuss this point in section 7. To get concise forms for the distributions induced on subsets of the X k s by marginalization, we introduce further H = Diag(h) and

W = H 1 2 U H 1 2
. If in particular we marginalize all variables except for the extreme points of the segment we then have

p(x 0 , x l ; W, h) ∝ x1••• x l-1 l-1 i=0 x ⊺ i U x i+1 l i=0 h ⊺ x i ∝ h ⊺ x 0 x ⊺ 0 H -1 2 W l H -1 2 x l h ⊺ x l ,
(See appendix for details). Similar calculations show that, for any sequence a 0 = 0 < a 1 < . . . < a m = l with a k ∈ {0, . . . , l}, denoting d j = d(a j , a j-1 ) = a ja j-1 the distances between consecutive nodes and A = {a 0 , • • • , a m }, we have:

p(x A ; W, h) ∝ m j=0 h ⊺ x aj m j=1 x ⊺ aj-1 H -1 2 W dj H -1 2 x aj .
By simply taking the logarithm of this expression we obtain a curved exponential family of distributions with loglikelihood

ℓ (x A ; θ) = m j=0 η ⊺ x aj + m-1 j=0 x ⊺ aj Λ(θ, d j ) x aj+1 -A(θ), (4.1) with ∀k ∈ K, η k = log(h k ), θ = (W, η), A the log- partition function and where Λ(θ, d) is defined entrywise by [Λ(θ, d)] kk ′ = log([H -1 2 W d H -1 2 ] kk ′ ).
It is now very natural to try and use this formula to extend the definition of the process to any sequence of points a 0 = 0 < a 1 < . . . < a m = l that are no longer restricted to take integer values. This requires however that for all for all s ≥ 0, W s should be a well defined real valued matrix with non-negative (or for learning purposes positive) entries. The fact that W is real symmetric and that all its powers should be real implies that it should have nonnegative eigenvalues. Since we can approximate a low rank matrix with a full rank matrix, we assume for convenience that all it eigenvalues are positive (any low rank matrix can be approximated by a full rank one). W is then a matrix exponential W = exp(Π). The fact that all its powers should have non-negative entries implies in particular that for any s, W s is completely positive 1 . We therefore need to characterize which conditions on Π are needed to obtain a valid W . Note that Π can be viewed as the counterpart of the rate matrix for CTMCs.

Infinitesimal generator Π

To easily compute the matrix exponential we use the eigendecomposition of Π:

Π = P ⊺ ΣP, Σ = Diag(σ), P ⊺ P = P P ⊺ = I K (4.2)
and exponentiate its eigenspectrum 2 .

In the context of learning, it is natural to assume that the entries of W s are actually strictly positive so that the loglikelihood is always finite. The following lemma provides sufficient and necessary conditions on Π for the entries of exp (lΠ) to be either non negative or positive.

Lemma 1. For Π a square matrix, [exp (l Π)] i,j ≥ 0 ∀l ∈ R + and ∀i, j if and only if Π i,j ≥ 0 for all i = j. Similarly, [exp (l Π)] i,j > 0 for all i, j and ∀l ∈ R * + , if and only if the sequences u

(k) i,j k∈N with u (k) i,j = Π k i,j
is such that its first non-zero value exists and is strictly positive, for all i = j . This lemma is proved in the appendix.

It is easy to see from the proof of the lemma that Π i,j > 0 for i = j is a sufficient condition for [exp (l Π)] i,j to be positive for all i, j and for all l ∈ R * + . Note that the likelihood obtained in (4.1) is invariant by a multiplication of H or U and thus of W by a positive scalar, because of normalization. As a result it is also invariant by addition of a constant multiple of the identity matrix to Π or equivalently to σ. This means that the likelihood is invariant by addition of an arbitrary identical constant to all the eigenvalues (σ i ) i∈K . In particular, it is possible to choose this constant sufficient large to guarantee that the diagonal of Π is positive. This implies that it will be conveniently possible to parameterize the model by the entrywise logarithm of Π. [START_REF] Seber | A matrix handbook for statisticians[END_REF] p. 223). 2 One caveat of this parametrization is that if W is close to low rank, the corresponding eigenvalues in σ have to take large negative values. This could be addressed by working with (σ -1 k ) k∈K .

1 A ∈ R K×K is completely positive iff there exists B ∈ R K×m + with A = BB ⊺ (see e.g.

Existence of the process on the chain

Proposition 2. There exists a stochastic process (X a ) a∈[0,l] defined at all points of the segment [0, l] whose finite marginal on any finite set of points containing a 0 and a l is given by (4.1).

Proof. Let A = {a 0 , . . . , a m } and B = {b 0 , . . . , b n } two such sets with a 0 = b 0 = 0 and a m = b n = l. It is clear that using (4.1) to define a joint probability distribution on (X a ) a∈A∪B , the distribution obtained by marginalization of elements of A\B using the same type of derivation used in (4.1) is still of the form of (4.1). Since the same holds for B\A, we just showed that the collection of proposed marginals are consistent and by Kolmogorov's extension theorem (Chung and Speyer, 1998, chap. 6). This proves the existence of the process. It should be noted that, in a real graph, the segments connecting a node of degree two are essentially merged into a single segment by concatenation. We will call all nodes of degree different than two junction nodes. Conversely, identifying nodes and points in the real graph, any point that is not a junction node can actually be viewed as a degree two node.

Definition 3. Let S be the set of junction nodes. Given A a set of points on the real graph, we will call the induced discrete graph on A ∪ S, denoted by G A the graph with vertices A ∪ S and whose edges E A link the nodes that can be joined on the real graph by segments not containing elements of A ∪ S:

E A = {(a, b) | ] a, b [ ∩ (A ∪ S) = ∅}.
To distinguish them from S \ A, we will call the set of nodes in A observed nodes.

3 Real graphs extend the notion of real trees which have been introduced previously in the literature [START_REF] Chiswell | Introduction to Lambda Trees[END_REF] and are of interest notably in mathematical cladistics and to construct Brownian trees.

Towards a Potts model on real graphs

To extend the stochastic process previously defined to real graphs, we first define its marginals. In particular, given a set of points A = {a 0 , • • • , a m }, the marginal on A ∪ S is naturally defined as follows: let G A = (A ∪ S, E A ) be the induced discrete graph on A ∪ S, we propose to define the log-marginal distribution on (X a ) a∈A∪S as

ℓ (x A∪S ; θ) = a∈A∪S η ⊺ x a + (a,b)∈E A x ⊺ a Λ(θ, d ab ) x b -A(θ), (4.3)
with θ = (η, W ) which we reparametrize from now on with θ = (η, Π). If A does not contain S, then p (x A ) is obtained by marginalizing x S\A out in p (x A∪S ).

Existence of the process on a real graph

The existence of the process on a real graph is again proven using Kolmogorov's theorem:

Proposition 4. There exists a stochastic process (X a ) a∈G defined at all points of the real graph G with log-marginals on any set of nodes A containing the junction nodes given by Eq. ( 4.3).

Proof. Let A and B be two subsets of nodes on the real graph G, for which the distributions x A and x B are obtained by marginalizing S out of x A∪S and x B∪S in Eq. (4.3). We note that a node on an edge is conditionally independent of any node on a different edge given x S . Proposition 2 tells us that the marginals are consistent on each edge with fixed endpoints, from which we can deduce that the definition of the definition of the process on A ∪ S and B ∪ S provided in Eq. ( 4.3) is consistent since it is obtained by marginalization of the joint distribution at the nodes A∪B∪S. The process being consistent on A and A∪S by definition of p(x A ), and similarly on B and B ∪ S, we have proved Kolmogorov consistency between A and B which in turn proves the existence of the process on the real graph.

We will refer to the obtained process, illustrated on

INFERENCE

Probabilistic inference is an operation which is key to learning and making predictions in graphical models. In the case of our continuous graph G, if we consider any segment

[a, b] with a, b ∈ S and any a ′ , b ′ ∈ [a, b], it should be noted that p(x {a,a ′ ,b ′ ,b} ) = p(x {a ′ ,b ′ } |x {a,b} )p(x {a,b} )
where p(x {a,b} ) is computed as a clique marginal of p(x S ), and p(x {a ′ ,b ′ } |x {a,b} ) has a simple analytical expression given that reduces to the model on the segment. This implies that marginal distributions on any finite collection of nodes on the same edge can be computed efficiently provided the edge marginals of the induced model on S can be computed efficiently. In spite of the fact that the graph has uncountably many nodes, inference can thus be performed by any classical inference algorithm, i.e. the sum-product algorithm if the graph is a tree and typically approximate inference techniques otherwise, such as loopy belief propagation.

LEARNING

In this section, we focus on learning the model from data.

Since the process values are only observed at a finite number of points, we are somehow always in the situation where some nodes are unobserved. However, when all junctions nodes are observed the joint likelihood of a given set of nodes has the closed form expression of Eq. (4.3). Since this a curved exponential family, the log-likelihood is in general not a concave function of the parameters4 .

To avoid having to cope with positivity constraints, and given the rapid divergence of the likelihood on the boundary of the domain we parameterize the likelihood by η and the entrywise logarithm of Π, since given the remark following lemma 1, it possible to take Π positive entrywise.

For the CTMC directed tree, [START_REF] Holmes | An expectation maximization algorithm for training hidden substitution models[END_REF] consider the likelihood of the entire process, show that it has a canonical exponential family form with a small number of sufficient statistics and derive an EM algorithm based on this representation to learn the parameters. A similar exponential family form can be obtained for our process, with also a small number of sufficient statistics and in theory it is possible to construct a similar EM algorithm. Unfortunately, in our case the M-step of the algorithm would still require solving a convex optimization problem whose solution is not closed form. We therefore do not pursue further this approach or detail the corresponding canonical exponential family form of the process. We propose instead to optimize the likelihood using a gradient based method. We show that the gradient can be computed from the moments obtained by performing the probabilistic inference on the model in different settings. In the next sections (sections 6.1 -6.4), we derive the form of the gradient of the likelihood, first when all junction nodes are observed, then, when any set of nodes is observed, and finally, when some nodes are observed and another (typically larger) set of nodes emits observed vectors of features that are each conditionally independent given the state of associated node, as in a hidden Markov random field setting. Since computing the inference is typically intractable in graphs, we introduce a variational approximation in 6.5 that allows for faster (linear) computation. The proofs of lemmas and propositions presented can be found in the appendix.

Gradient of the likelihood on a segment

Given that the model is parameterized by exponentials of Π, the gradients involve the differential of the matrix exponential. We will therefore repeatedly use the function ψ l,Π with ψ l,Π (X) = P ⊺ (P XP ⊺ ) ⊙ Γ l,Π P, where Π = P Diag(σ)P ⊺ is the eigenvalue decomposition of Π and

[Γ l,Π ] i,j =    exp (l σ i ) -exp (l σ j ) σ i -σ j if σ i = σ j l exp(l σ j ) if σ i = σ j .
The function ψ is such that the gradient of x ⊺ exp (lΠ) y is ψ l,Π (xy ⊺ ). It is essentially switching to the spectral space of Π, where the gradient has a simple multiplicative form given by Γ and then maps the result back to the original space. With this function, we thus have Lemma 5. The gradient with respect to variable Π of the log-likelihood ℓ of x a and x b on a segment of length l whose end points are a and b can be written as

∇ Π ℓ (x a , x b ; θ) = ψ l,Π (x a x ⊺ b -E [X a X ⊺ b ]) ⊘ W l .

Gradient of the likelihood in a real graph

We now compute the gradient of the log-likelihood for the joint distribution of the nodes (x a ) a∈A , with the subset A containing the junction nodes S. We still denote G A = (A, E A ) the induced discrete graph of A on G. And since Π is identical for every edge, a direct application of the chain rule implies that:

Proposition 6. The gradient of the likelihoods are computed5 as

∇ Π ℓ (x A ; θ) = (a,b)∈E A ψ d ab ,Π (x a x ⊺ b -µ ab ) ⊘ W d ab ∇ η ℓ (x A ; θ) = a∈A (x a -µ a ) -1 2 (a,b)∈E A (x a -µ a + x b -µ b )
.

with µ a = E [X a ] and µ ab = E [X a X ⊺ b ].

Partially observed junction nodes

To learn from partially labelled data it is necessary to consider the likelihood of X B for B a set of nodes that does not necessarily contain S. Let B be a set of observed nodes, i.e. for which we know the states x B , and A a set of unobserved nodes containing S\B. We have the following distributions

ℓ (x A∪B ; θ) = a∈A∪B η ⊺ x a + (a,b)∈E A∪B x ⊺ a Λ(θ, d ab )x b -A A∪B (θ) ℓ (x A |x B ; θ) = a∈A∪B η ⊺ x a + (a,b)∈E A∪B x ⊺ a Λ(θ, d ab )x b -A A|B (θ, x B ) ,
We can rewrite the log-likelihood as follows [START_REF] Wainwright | Graphical models, exponential families, and variational inference[END_REF] :

ℓ (x B ; θ) = A A|B (Π, h, x B ) -A A∪B (Π, h) ,
and its gradient are therefore computed as Proposition 7.

∇ Π ℓ (x B ; θ) = a,b)∈E A∪B ψ d ab ,Π (µ ab|B -µ ab ) ⊘ W d ab ∇ η ℓ (x B ; θ) = a∈A∪B µ a|B -µ a -1 2 (a,b)∈E A∪B µ a|B -µ a + µ b|B -µ b . with µ ab|B = E [X a X ⊺ b |X B = x B ], µ a|B = E [X a |X B = x B ].

Hidden Markov

We consider a hidden Markov random field variant of our model in which some nodes have, in addition to the state variable, a feature vector with a state specific distribution.

More precisely, we envision to learn from data on a graph in which the states of a set of nodes B are observed and in which each node in a set A (with A ∩ B = ∅) provides an observed feature vectors y a which is conditionally independent of the rest of the graph given the corresponding node state x a . For simplicity, we assume that S ⊂ A ∪ B.

The joint and conditional distribution of observed and unobserved variables are very similar as above

ℓ (x A∪B , y A ; θ, κ) = a∈A∪B η ⊺ x a + a∈A log (p (y a |x a ; κ)) + (a,b)∈E A∪B x ⊺ a Λ(θ, d ab )x b -A A∪B (θ, κ) ℓ (x A |y A , x B ; θ, κ) = a∈A∪B η ⊺ x a + a∈A log (p (y a |x a ) ; κ) + (a,b)∈E A∪B x ⊺ a Λ(θ, d ab )x b -A A|B (θ, κ, x B , y A ) ,
which allows us to rewrite the likelihood of observations as

ℓ (x B , y A ) = A A|B (θ, κ, y A , x B ) -A A∪B (θ, κ).
Given that the model for p(y a |x a ) is Gaussian or at least an exponential family, when envisioning an EM algorithm to learn κ and θ, it is easy to see that the update for κ is closed form while that of θ is not. This motivates a variant of the EM algorithm which does not attempt to maximize with respect to both κ and θ simultaneously but which either maximizes the expected likelihood with respect to κ or maximizes it with respect to θ. The algorithm can then be summarized as an E-M1-E-M2 algorithm, where the Estep is the usual computation of expected sufficient statistics given current parameters, M1 solves for κ in closed form and M2 maximizes with respect to θ using gradient ascent6 .

Variational approximation

For graphs with cycles, since inference is intractable, we replace the likelihood by a pseudo-likelihood obtained using a variational approximation of the log-partition. Our variational approximation is the one associated with the entropy of Bethe (see, e.g. section 4.1 in [START_REF] Wainwright | Graphical models, exponential families, and variational inference[END_REF], but other choices would be possible. The main motivation behind this approximation is that the exact gradient of this pseudo-likelihood is directly obtained from the pseudo-moments given by loopy BP. In practice, damping needs to be used (see Wainwright and Jordan, 2008, chap. 7).

In term of complexity, the parametrization of CGMRF could suggest that inference is slower than in the discrete setting since the computation of the SVD of Π is required. However, since the number of states is typically much smaller than the number of nodes in the graph, the computational cost of the SVD is negligeable compared to the overall cost of the algorithm. Hence, inference in the CGMRF is just as hard as for any discrete MRF.

The log-likelihood is a curved exponential family and is in particular not a convex function of the parameters, while it is convex for a standard MRF. As a consequence the pseudo log-likelihood based on the variational approximation is also non-convex. We use gradient descent with a line-search based on the Wolfe conditions to find a local minimum (see Nocedal and Wright, 1999, chap. 3). Empirically the algorithm is not trapped in bad local minima but takes more iterations to converge than the MRF counterpart. Experiments showed that the training for CGMRFs was only two times longer than for regular MRFs.

DISCUSSION

In this section, we discuss more precisely features of CGM-RFs that are unique or common with other models and approaches existing in the literature.

First, we note that for a tree, our model is not equivalent to that of [START_REF] Holmes | An expectation maximization algorithm for training hidden substitution models[END_REF]. Their model uses a constant rate matrix (i.e. the Markov process is homogeneous) while we use constant infinitesimal potentials, which do not lead to a constant rate matrix on any orientation of the tree.

If the tree is just the segment [0, L], for s and t with 0 < s < t < L a CTMC is such that p(x t |x s ) only depends on ts and not on L. By contrast for our model log p (x t |x s ) depends also on Lt and Ls since log p

(x t |x s ) = x ⊺ s Λ (t-s) x t +x ⊺ t η+x ⊺ t Λ (L-t) 1-x ⊺ s Λ (L-s) 1
, where for simplicity we omitted the dependance in θ, and 1 is the constant vector equal to 1. See the appendix for an illustration and further discussion of the differences between the models.

Our model has in common with graph partitioning techniques and spectral clustering (SC) that the distance between nodes are taken into account. But there are several important differences: first, in SC, there is no model learning in the sense that no parameters are learned to optimize the model [START_REF] Bach | Learning spectral clustering, with application to speech separation[END_REF] who learn the metric for SC, are an exception). Second, our model captures that there could be different transition probabilities between different classes along the graph which is not possible in SC. Then, the main assumption in SC is that classes are separated by edges of smaller weights so that each class is as disconnected as possible. By contrast, our model authorizes (to some extent) transitions between classes on short edges and moreover permits that each class corresponds to several connected components. Our models extends naturally to a hidden Markov model that makes it possible to include feature vectors for some nodes and not for others, which is not possible with SC techniques.

Another graph-based approach to classification which is perhaps more related to ours is the work of [START_REF] Zhu | Semisupervised learning using Gaussian fields and harmonic functions[END_REF] on binary classification with harmonic functions. Indeed, the Gaussian field considered there is similar to the Potts model we obtain on the junction nodes. The approach of [START_REF] Zhu | Semisupervised learning using Gaussian fields and harmonic functions[END_REF] is however just concerned by inference and not by learning, but their approach could be extended both to multi-class classification and to perform learning of the parameters.

EXPERIMENTS

We present in this section experiments on real data. Synthetic experiments on the core model of the CGMRF (without hidden layer) can be found in section 6 of the appendix.

In geographic information systems, data is often aggregated either on regular grid or on cells corresponding to abstract administrative boundaries, which do not necessarily reflect the structure of a city. A fairly natural type of representation for urban environment is based on graphs and in particular weighted graphs which can encode a distance information.

We consider a problem from geomatics in which this type of representation could be beneficial and which consists in predicting building use in urban and peri-urban environments from a few annotations and simple building shape characteristics that can be extracted easily from aerial images. More precisely, we consider the transductive learning problem of assigning city blocks to one category from {individual housing, collective housing, industrial/commercial area}.

Building the city block continuous graph

A city can be divided into city blocks using its layout and road network as in Figure 8.1. Assuming that the blocks are given, we compute the Voronoi diagram of the block centroids and link together blocks with adjacent Voronoi cells. Edges are annotated with a proximity measure, in our case the distance between their respective closest buildings. This provides a continuous graph encapsulating the s tructure of the city. Each block is then annotated into one of three categories : individual residential, collective residential and industrial/commercial area. The blocks are annotated by hand using cadastral information, business registration codes, and resorting to Google street view images for ambiguous blocks (see Figure 8.1).

Data descriptors and learning setting

A block is then described by the weighted average of characteristics of the buildings it contains, each building counting with a weight proportional to its volume. We tested 10 different building descriptors, found that floor area and height were the most discriminative, and that adding more descriptors actually decreases the performance of all tested algorithms.

We use the example of Sevran, a French city of 50 000 inhabitants north of Paris. We divided it into 461 blocks, We consider the transductive learning problem of predicting all block labels from a subset of labelled blocks. In our experiments, 7% of annotated labels, corresponding to 28 blocks, are used for training and the remaining are used for testing.

Competing algorithms

As baselines we consider two algorithms that do not take into account spatial information: a generative Gaussian mixture model and a logistic regression trained each using the 7% revealed labels. We also consider classical hidden MRFs, which cannot take into account the distance, and whose graph is either the same as for the CGMRF or a pruned graph in which all edges longer than a threshold (corresponding to the average city block radius) have been removed. The different graphs are illustrated on Figure 8.2. Note that the Gaussian mixture model does not take the graph structure into account, and can be interpreted as an edgeless MRF In all Markov models, we use Gaussian emissions to model the distribution of the building descriptors given the block label, which can conveniently be optimized in closed form. To train the CGMRF and MRF models we learn the parameter θ with the maximum likelihood principle following the approach presented in section 6.5.

Results analysis

For each model, we construct a precision-coverage curve, obtained by sorting the probabilistic predictions by increasing values of their entropies, and reported on Figure 8.3. The confidence bands represented corresponds to one standard error for the estimation of the mean precision.

We can see that enriching the simple Gaussian mixture model by adding a graph structure significantly improves the overall performance. Building a MRF using all the edges from the Voronoi proximity or only retaining a fraction of the shorter edges yields similar results, on par with logistic regression. Building a CMRF using the edges annotated with a distance measure leads to a performance which is significantly above all others based on estimated standard errors.

When making prediction for all unlabeled points from the 7% of revealed annotations, the different algorithms yield the following average precisions (over the 300 resamplings): for the Gaussian mixture model 88.0%, for logistic regression 92.5%, the full MRF 92.4%, the pruned MRF 91.6% and our CGMRF 94.0%. Both pruned MRF and full MRF outperform the simple Gaussian mixture model, but not logistic regression, even though their precision at intermediate coverage is higher. The misclassification error of the CGMRF is 20% smaller than that of logistic regression, 21.5% smaller than for the best MRF model, and 50.2% smaller than for the Gaussian mixture. The gain in precision is not only obtained in average since the misclassification error in the CMRF was lower than MRF and logistic regression in respectively 193 and 293 out of 300 experiments. Wilcoxon signed rank tests assigns respectively p-values of 7 • 10 -26 and 3 • 10 -24 to the common median hypothesis.

In this experiment, with 461 nodes and 2718 edges the inference takes less than 0.1s on a CPU at 3.3GHz. Learning requires usually around 50 calls to the inference step for the MRF (5s total), while it is closer to 100 for the CGMRF (10s total).

Transfer learning on another city

We now consider the problem of predicting block labels on a new unannotated city using partial annotation from a given city. More precisely, we train our model with 15% of revealed labels from Sevran, and consider several schemes to make predictions on the neighboring urban area formed by Pierrefitte-sur-Seine together with Stains, for a total of 63000 inhabitants and 583 blocks, for which both graph and features are available but no labels are revealed. We consider logistic regression and the Gaussian mixture model trained from the annotated blocks from Sevran as baselines, and test for each of the CGMRF and MRF the models learnt as follows:

• θ and κ are learnt on data from Sevran

• idem followed by a single EM-step on κ alone (E-M2) on the graph of Pierrefitte+Stains

• idem followed by an EM-step on θ (E-M1) and then an EM-step on κ (E-M2).

We use the 359 labelled blocks (out of 583) of the Pierrefitte/Stains conglomeration as a testing set and construct the precision-coverage curves reported on Figure 8.4 (see the appendix for a figure comaring more approaches). We observe that the CGMRF setting is superior to its competitors, and that the relearning step improves the performance. The MRFs does not perform as well, which can be explained by the initial prediction being inferior, and relearning degrades its performance. The setting where only one E-M2 step is performed yields in both cases results comprised between the two other settings.

CONCLUSION

In this paper, we constructed a Potts model over a continuous graph and showed how to compute the likelihood of several of its variants as well as the corresponding gradients, for the purpose of learning.

Our experiments on a problem from geomatics show that this model outperforms regular MRFs, and compares favorably with logistic regression which although discriminative does not leverage unlabelled data. Finally, we showed that the model can be used to perform transfer learning from a first partially labelled graph towards a new completely unlabeled graph.

4. 4

 4 Extending the model to graphs4.4.1 Real graphsTo extend the model we proposed on a segment to undirected trees and more generally to undirected graphs, we first define what we will call continuous graphs or real graphs 3 . Given a weighted graph G = (V, E) with the weight d ab associated with the edge (a, b) ∈ E, we define the associated real graph G as the space constructed as the union of line segments of lengths d ab associated with the edges (a, b) ∈ E and whose extreme points are respectively identified with the nodes a and b through an equivalence relation. Put informally, a real graph is the set of line segments that we usually draw to represent an abstract graph. For any pair of points a ′ , b ′ on the same segment [a, b], we will denote by d a ′ b ′ the length of that subsegment.

  Figure 4.1: (a) Representation of a real graph with a zoom that shows that edges are actually a continuum of nodes linked by infinestimal unoriented edges. (b) The induced discrete graph associated with the junction nodes in red and the observed nodes in blue.

Figure 4

 4 Figure4.2: (left) Toy example illustrating that the process is defined at all points of the continuous graph. For a model on three classes (red, green blue) each point of each edge is colored with the mixture of these three colors corresponding to the probability of observing each of the classes, given that all the circle nodes are observed with the given colors.

  Figure 4.2, as a continuous graph Potts model or continuous graph Markov random field (CGMRF).

Figure 8

 8 Figure 8.1: (left) Buildings and road network of Sevran. (middle) Division into city blocks. (right) City blocks with annotations. Blue: individual housing, cyan: collective housing, red: industrial/commercial area. (Best seen in color.)

Figure 8

 8 Figure 8.2: (left) continuous graph used to train the HCGMRF, the darker the edge the shorter the annoted distance, (middle) graph used for the HMRF including all edges or (right) with only edges shorter than a threshold.

Figure 8 . 3 :

 83 Figure 8.3: Precision coverage curves on Sevran. Averaged precision coverage curves for the inference for 300 random resamplings of 7% of revealed labels on the city of Sevran. (Best seen in color.)

It is however clearly concave when all edges are of the same length, because the constraint of equality of the parameters for all potentials is a convex constraint.

Note that a single spectral decomposition of Π allows to compute W d ab efficiently for all pairs (a, b).

Note that gradient ascent itself requires to perform some inference to recompute the log-partition function


Pierre-Emmanuel Arduin
email: pierre-emmanuel.arduin@utc.fr

On the Use of Cognitive Maps to Identify Meaning Variance

Keywords: cognitive map, mental model, graph theory, interaction, group decision making

Cognitive science, as well as psychology, considers that individuals use internal representations of the external reality in order to interact with the world. These representations are called mental models and are considered as a cognitive structure at the basis of reasoning, decision making, and behavior. This paper relies on a fieldwork realized as closely as possible from the respondents. We propose an approach based on graph theory in order to study the meanings given by several people to the same concept, and to identify those who give it the same meaning. The use of tools from graph theory combined with the study of cognitive maps leaded us to highlight the importance of interaction notably within group decision making. This idea, as well as the limits of our approach, are discussed at the end of this paper.

Introduction

Mental models are "personal, internal representations of external reality that people use to interact with the world around them" for Jones et al. in [START_REF] Jones | Mental models: An interdisciplinary synthesis of theory and methods[END_REF]. These representations are abstractions of the reality relying notably for an individual on his/her personal experiences, perceptions and understanding of the world that surrounds him/her. These representations may vary from one person to another, as well as the meanings they give to the same concept. For Daft and Weick, "Interpretation gives meaning to data, but it occurs before organizational learning and action." ( [START_REF] Daft | Toward a model of organizations as interpretation systems[END_REF], p. 286), that it the reason why we consider that identifying meaning variance could be useful for group decision making or negotiation. If authors as Hall et al. in [START_REF] Hall | Causal policy maps of managers: Formal methods for elicitation and analysis[END_REF] and Morgan in [START_REF] Morgan | Risk Communication: A Mental Models Approach[END_REF] focus on cognitive mapping as a way to improve management and to study communication breakdown risks, others as Quinn in [START_REF] Quinn | Finding culture in talk: a collection of methods, chapter How to reconstruct schemas people share[END_REF] and Langan-Fox et al. in [START_REF] Langan-Fox | Team mental models: Techniques, methods, and analytic approaches[END_REF] consider cognitive mapping as a way to elaborate a "shared or team mental model" ( [START_REF] Langan-Fox | Team mental models: Techniques, methods, and analytic approaches[END_REF], p. 242). For Eden and Ackermann in [START_REF] Eden | Group decision and negotiation in strategy making[END_REF], cognitive maps can be used for issue structuring in the context of a group decision support system.

In this article, we propose an approach in order to identify people who give the same meaning to the same concept through the use of cognitive maps and their analysis with tools of graph theory. The efficiency of the proposed approach relies notably on the fieldwork with the respondents: the meanings they give to the same concept have to be understood. After introducing background theories in the area of cognitive mapping, we present tools of graph theory in order to study cognitive maps. Our own approach is then explained and discussed, from the interviewing of the respondents to the identification of meaning variance through the use of their cognitive maps.

Background Theory and Assumptions

This study relies on the assumption that two individuals give the same meaning (respectively different meanings) to the same concept if and only if their cognitive maps related to this concept are practically similar (respectively different). Section 2.2 refines this idea of "quasi-similarity".

Mental models and their elicitation with cognitive maps are introduced in the first part of this section. Tools of graph theory which can be used to study cognitive maps are then presented in the second part of this section.

Mental models and their elicitation through cognitive maps

In 1943, Craik drew a parallel between machines functioning and human brain in [START_REF] Craik | The Nature of Explanation[END_REF]. This parallel leaded to mental models, which are functional representations of the reality: we use them to interact with the world. They are functional in the sense that they are simplified and incomplete.

Klayman and Ha, with the theory of "confirmation bias", suggest in [START_REF] Klayman | Hypothesis testing in rule discovery: Strategy, structure, and content[END_REF] that individuals seek information adapted to their actual understanding of the world. Indeed for Collins and Gentner in [START_REF] Collins | How people construct mental models[END_REF], when you explain a domain with which you are unfamiliar, you try to instantiate it on a familiar domain, that you think as being similar. You can for example use water flow to explain electrical current.

Cognitive maps are knowledge structures representing for an individual his/her assumptions and beliefs on the world for Kearney and Kaplan in [START_REF] Kearney | Toward a methodology for the measurement of the knowledge structures of ordinary people: The conceptual content cognitive map (3CM)[END_REF]. These assumptions and beliefs provide himself/herself a framework to interpret new information and to elaborate decisions regarding on new situations ( [START_REF] Kaplan | Environments for people, chapter Attention and fascination: The search for cognitive clarity[END_REF] and [START_REF] Kaplan | Cognition and environment: Functioning in an uncertain world[END_REF]). Eden sees cognitive maps as "model of thinking" ( [START_REF] Eden | On the nature of cognitive maps[END_REF], p. 261). Jones et al. highlight in [START_REF] Jones | Mental models: An interdisciplinary synthesis of theory and methods[END_REF] how cognitive mapping allows studying understanding similarities and differences between several people. Most of the procedures used to elicit mental models consider that they can be represented as a network of concepts and relations and rely on direct or indirect elicitation: Direct Elicitation The respondent is here asked to realize a representation of his/her mental model by using drawings, words, and symbols. In [START_REF] Kearney | Toward a methodology for the measurement of the knowledge structures of ordinary people: The conceptual content cognitive map (3CM)[END_REF], the authors introduce a method where the participants must identify the concepts that they consider as important for a given domain, and organize them visually / spatially in a way that represents their understanding of the given domain.

Indirect Elicitation For Carley and Palmquist in [START_REF] Carley | Extracting, representing, and analyzing mental models[END_REF], the representation of a mental model can be extracted from written documents or oral propositions.

Mental models elicitation has to focus not only on the concepts that are considered as important, but also on the way they are organized from a cognitive point of view and on their interactions. Cognitive mapping is the process for someone of realizing a cognitive map of his/her mental model related to a domain. It leads to understand how people understand a system. Group decision making and negotiation could be improved if individuals with different points of view are stimulated to work together. A shared understanding has to be identified and supported between the different stakeholders for Jones et al. in [START_REF] Jones | Mental models: An interdisciplinary synthesis of theory and methods[END_REF]. For Weick, a kind of "convergence" can be obtained across managers ( [START_REF] Weick | Sensemaking in Organizations[END_REF], p. 80). Decision making processes concern simultaneously individuals, groups, and societies. That is the reason why identifying meaning variance, i.e. differences of understanding, has to be considered as a source of improvements for negotiation and group decision making.

Analyzing cognitive maps with tools of graph theory

Ozesmi and Ozesmi use in [START_REF] Ozesmi | Ecological models based on peoples knowledge: a multi-step fuzzy cognitive mapping approach[END_REF] the elicitation of mental models by asking respondents to define important variables for a given system. These variables are then written on cards and the respondents have to organize them in a way that reflects their understanding of the system. Ozesmi and Ozesmi propose to use graph theory tools in order to explore the complexity of the constructed networks. The density is for example a way to study cognitive maps, by counting the nodes (N ) and the connections (C), the density (D) can be determined as follows:

D = C N 2
more the density is high, more the number of relations between the concepts in the cognitive map is high. The hierarchical index from Mac Donald, noted h and introduced in [START_REF] Macdonald | Trees and Networks in Biological Models[END_REF], can also be envisaged:

h = 12 (N -1)N (N + 1) v d -(v) -d -(v) N 2
where N is the total number of nodes and d -(v) the outdegree of the node v. The value of this index is 0 when the system is democratic and 1 when it is hierarchical. The interested reader can have a look at [START_REF] Ozesmi | Ecological models based on peoples knowledge: a multi-step fuzzy cognitive mapping approach[END_REF] or [START_REF] Langfield-Smith | Measuring differences between cognitive maps[END_REF] where the authors present these measures and others. They are notably interested in the most mentioned nodes and in the most central nodes. The firsts are those which are mentioned by the most important number of respondents, whereas the seconds are those which have, for a given respondent, the higher degree in his/her cognitive map. Our approach focuses on these two last measures, which allow rapid experimentations. In the future, they will be compared with others measures in order to justify the relevance of this choice.

Our Approach to Identify Meaning Variance

The study presented in this section has been realized with the participation and the agreement of five Computer Science students and five Management Science students at Paris-Dauphine University in 2013.

The first part of this section presents the way of conducting the interviews. The construction of the results is then explained in the second part of this section. Finally these results are discussed in the last part of this section. Every concept given by the students has been translated into English.

Conducting the interviews

First of all when you intend to conduct cognitive mapping interviews, you have to realize your own cognitive map of the studied concept. This precaution may prevent reactions you may have during the interviews and that would influence the ongoing cognitive mapping. "Information System" is the studied concept in this work.

The number of interviews has also to be considered. We limited our study to ten interviews not only for practical reasons, but also according to Ozesmi and Ozesmi that noticed in [START_REF] Ozesmi | Ecological models based on peoples knowledge: a multi-step fuzzy cognitive mapping approach[END_REF] how the number of new concepts can stagnate depending on the number of interviews (see figure 1). For Carley and Palmquist in [START_REF] Carley | Extracting, representing, and analyzing mental models[END_REF], this stagnation may result from a limited vocabulary for a given subject. Glaser and Strauss in [START_REF] Glaser | The Discovery of Grounded Theory: Strategies for Qualitative Research[END_REF] consider notably that a researcher "trying to reach saturation [...] maximizes differences in his groups in order to maximize the varieties of data bearing on the category" (p. 62). The interviews have all been realized individually and the relation between the researcher and the respondent has clearly been explained: this is not an exam, there is no evaluation. The average duration of an interview was 7'14". Post-it notes and a pen were given to the respondent and we began asking: "Imagine that you have to explain what an "Information System" is to someone who absolutely does not know what it is. What concepts or ideas do you need?".

The respondent was invited to write these concepts on as many post-it notes as necessary. The average time between the end of the question and the beginning of an answer was 10,29 seconds. Even when he/she thought he/she had finished, we asked our question again in order to ensure that no concept was forgotten. Then a large piece of paper was given to the respondent, who was invited to organize his/her post-it notes. New concepts could be added at every moment, and at the end we asked to the respondent if the production he/she realized corresponds to the representation for him/her of an "Information System", i.e. if this production corresponds to the meaning he/she gives to this concept.

Constructing the results from the interviews

The figure 2 represents the graph associated to the cognitive map generated by the first respondent for the concept of "Information System". The links between nodes are not necessarily causal as it may be the case in cognitive maps. The most mentioned nodes and those which are the most central are now going to be studied.

The firsts are determined by counting the number of occurrences of every node, so that within our sample of respondents "Data", "Human Resources" and "Information" are the most mentioned nodes (these results are presented more precisely in [START_REF] Arduin | Vers une métrique de la commensurabilité des schémas d'interprétation (English title: Towards a metric of the commensurability of interpretative frameworks)[END_REF]). The seconds are determined by regarding every cognitive map as an undirected graph. Post-it notes become nodes and their position as well as their relationships become edges. So that in the case of the first cognitive map, whose associated graph is presented figure 2, the most central nodes are "Information" and "Human Resources": these nodes are the most connected to the rest of the graph. Table 1 represents for our sample of respondents the most central nodes per respondent, regarding to the most mentioned nodes. From this table the figure 3 can be generated and highlights the respondents which have common central nodes regarding to the most mentioned nodes: the meaning they gave to the concept of "Information System" seems to be closer for them (in green) than for the others (in white). The reader may have noticed several respondents who have no central nodes regarding to the most mentioned nodes (respondents 2, 3, and 9). Indeed their most central nodes are not related to the most mentioned nodes. We say that the meaning they gave to the concept of "Information System" is unusual for the considered group.

Discussing the results

The major restriction of our approach is that it is reductive because it focuses only on the most mentioned and on the most central nodes. This restriction, induced by Ozesmi and Ozesmi in [START_REF] Ozesmi | Ecological models based on peoples knowledge: a multi-step fuzzy cognitive mapping approach[END_REF] constitutes nevertheless a strength of our approach: it is necessary in order to treat cognitive maps, which can rapidly lead to complex graphs. According to our experimentations, this restriction, even if it reduces cognitive maps semantic only to the most mentioned and the most central nodes, strongly accelerates the way of conducing that kind Fig. 3. Respondents whose central nodes are common regarding to the most mentioned nodes, the meaning they gave to the concept of "Information System" seems to be close of investigation. The results can easily be processed by concept-mapping and mind-mapping software for example. The reader may have noticed in the figure 3 that the respondents 1 and 6 are those who central nodes are common to the greater number of persons regarding to the most mentioned nodes. Table 1 shows how these respondents are those who have several central nodes. This observation leaded us to conclude that the meaning they gave to the concept of "Information System" covers several meanings given by the others respondents. We are now trying to establish a connection between the cognitive maps, i.e. the most central nodes for a respondent, and his/her academical background, i.e. if he/she is a Computer Science student or a Management Science student.

Conclusions and future works

In this paper we presented an approach in order to identify meaning variance between the members of a group by analyzing their cognitive maps through the use of graph theory tools. We began introducing background theories in the area of mental models and cognitive mapping. We presented then tools of graph theory which can be used to study cognitive maps. We finally explained and discussed our approach from the interviewing of the respondents to the identification of meaning variance through the use of their cognitive maps.

Individually and during face-to-face interviews, our approach has been thought as closely as possible from the respondents. A concept is given and the respondents are then asked to represent their understandings of this concept. These representations are analyzed with tools of graph theory in order to highlight the respondents who gave a similar meaning to the given concept. It is crucial to give to the respondents the time to realize their representations. The atmosphere must be relaxed and the respondents have to feel confident.

Nevertheless our approach is reductive in the data considered. Relying on Ozesmi and Ozesmi [START_REF] Ozesmi | Ecological models based on peoples knowledge: a multi-step fuzzy cognitive mapping approach[END_REF] notably, we proposed to focus only on the most mentioned nodes and on the most central nodes in order to analyze respondents' cognitive maps. This weakness is otherwise a strength of our approach: it makes feasible an analysis expensive to realize when the cognitive map as a whole is considered.

The conducted interviews lead us to consider not only the concepts identified by the respondents, but also their meanings. Only with ten students and during an academical case study, we observed so much that people can give different meanings to the same concept. We are now focusing on interactions during group decision making in order to observe, to identify, and to manage meaning variance through the use of cognitive maps.

Fig. 1 .

 1 Fig. 1. The number of new concepts depending on the number of interviews (source: Ozesmi and Ozesmi in [19])

Table 1 .

 1 The most central nodes per respondent (lines), regarding to the most mentioned nodes (rows)

				Organization
	Information
		Gathering	Processing	Storing	Disseminating
	Human Resources
	Fig. 2. The graph associated to the cognitive map generated by the first respondent
	for the concept of "Information System"
		Data	Human resources		Information
	Respondent 1		Human resources		Information
	Respondent 2
	Respondent 3
	Respondent 4		Human
	Respondent 5	Database
	Respondent 6	Data			Information
	Respondent 7		User
	Respondent 8 Data Structure
	Respondent 9
	Respondent 10			Dissemination of Information

Acknowledgments This paper has been prepared during the PhD of the author under the direction of Camille Rosenthal-Sabroux and Michel Grundstein. He is grateful to both of them for their support and advice. He would also like to thank the students who have accepted to take the time to be interviewed.


Pierre-Emmanuel Arduin

Julien Le Duigou

Diana Penciuc

Marie-Hélène Abel

Benoît Eynard

Heudiasyc / Roberval

Knowledge Sharing within Extended Enterprises: Case of Product Lifecycle Management systems

Keywords: Knowledge Sharing, Extended Enterprise, Tacit Knowledge, Semantic Interoperability, Product Lifecycle Management

When it is made explicit by someone, knowledge becomes information source of knowledge for someone else. Thus knowledge sharing cannot be reduced to information sharing. The aim of this paper is promote knowledge sharing, whether tacit or "explicited" by individuals within extended enterprises. Product Lifecycle Management (PLM) systems aim at an integrated management of all product-related information and processes within extended enterprises throughout the entire lifecycle of a product. In this paper, we propose (1) to outline a semantic interoperability between a collaborative platform and a Product Lifecycle Management (PLM) system, and (2) to highlight the conditions under which a piece of information shared through a PLM system may lead to one and only one interpretation.

Step (1) allows individuals to construct a shared understanding, supporting tacit knowledge sharing, whereas step (2) leads to ensure explicited knowledge sharing, i.e. knowledge that has been made explicit by someone within a certain context. PLM systems are strongly integrated within extended enterprises and their use will illustrate in this paper how our approach supports knowledge sharing. The conditions and limits of our approach, as well as its study within industrial fields, are discussed at the end of this paper.

Introduction

Working together within an extended enterprise is not as natural for people as working alone to pursue their own objectives. Within an extended enterprise, i.e. a network of firms collaborating in a project to achieve a common goal (Ross el al., 2006), systems interact in order to share information throughout a product lifecycle. Sharing knowledge cannot be reduced to sharing information: according to [START_REF] Tsuchiya | Improving knowledge creation ability through organizational learning[END_REF] knowledge can be regarded as the result of the interpretation by someone of information. So that authors as [START_REF] Walsham | Knowledge Management : The Benefits and Limitations of Computer Systems[END_REF] highlighted that the limit of computer systems is reached within extended enterprises when there is no awareness that the same information shared within the same extended enterprise through the same system may lead to different interpretations. The aim of this paper is (1) to outline a semantic interoperability between a collaborative platform and a Product Lifecycle Management (PLM) system, and (2) to highlight the conditions under which a piece of information shared through a PLM system may lead to one and only one interpretation.

Step (1) allows individuals to construct a shared understanding, supporting tacit knowledge sharing, i.e. knowledge that cannot be made explicit according to [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF], whereas step (2) leads to ensure explicited knowledge sharing, i.e. knowledge that has been made explicit by someone within a certain context. After remembering the vision of knowledge in the organization and the PLM model adopted in this work, the research proposition is introduced and tacit and explicited knowledge sharing will be explained. The perspectives of this work as well as its added value will finally be discussed at the end of this paper.

Background literature

Relying on the assumption that within extended enterprises individuals may interpret differently the same information, this work focuses on knowledge as being the result of the interpretation of information by someone according to [START_REF] Tsuchiya | Improving knowledge creation ability through organizational learning[END_REF]. The vision of knowledge in the organization adopted in this paper is introduced in the first parts of this section. Different models of PLM are then presented in the second part of this section.

A vision of knowledge in the organization

As the authors of this paper, we have got tacit knowledge, i.e. an individual cognitive construction, that we have structured into information during a process of sense-giving. As the readers of this paper, you have interpreted this information perceiving forms and colors, absorbed words, data, during a process of sensereading, possibly creating new tacit knowledge for you (see Fig. 1). Sense-giving and sense-reading processes are defined by [START_REF] Polanyi | Sense-giving and sense-reading[END_REF] as follows: "Both the way we endow our own utterance with meaning and our attribution of meaning to the utterances of others are acts of tacit knowing. They represent sense-giving and sense-reading within the structure of tacit knowing" (Polanyi, 1967, p. 301). Information is continuously interpreted during sense-reading processes. Within extended enterprises, information can be transmitted by speaking, writing or acting, and also through information systems, such as PLM systems. Knowledge can then be:

 explicited, i.e. it has been made explicit by someone within a certain context, it is socially constructed and can be supported by information technologies such as information. Individuals, as well as computers are "information processing systems" as said by Hornung (2009, p. 9),  tacit, it is not always articulated and cannot always be articulated, relying on [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF] notably:

"we can know more than we can tell". So that explicited knowledge is tacit knowledge that has been made explicit by someone within a certain context. It is information source of tacit knowledge for someone else. It is "what we know and can tell" answering to [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF] quoted above. The term "explicit knowledge" is often used (by Nonaka and[START_REF] Nonaka | The concept of «Ba»: building a foundation for knowledge creation[END_REF]Takeuchi, 1995, notably), whereas it does not reflect the dynamic of the sensegiving process as well as the term of "explicited knowledge". Indeed such process is attached to a certain person acting within a certain context. That is the reason why we prefer to use the expression "explicited knowledge", which clearly shows how every piece of information can be seen as a piece of knowledge that has been made explicit by someone within a certain context. One interprets information and creates a piece of tacit knowledge, which has a meaning for him/her. Within an extended enterprise, there may be someone who has received the same information and, interpreting it, has created a piece of tacit knowledge, which has a meaning for him/her. This meaning can differ from one person to another. Nevertheless, the use of a collaborative platform may lead to elaborate a shared understanding. We present in this paper how such platforms can be integrated in a PLM system. Now a few models of PLM are going to be presented.

Different models of Product Lifecycle Management

Product Lifecycle Management (PLM) aims at an integrated management of all product-related information and processes throughout the entire lifecycle for [START_REF] Saaksvuori | Product lifecycle management[END_REF] and [START_REF] Terzi | Product lifecycle management -from its history to its new role[END_REF]. The product data model is used to filter, to structure, to integrate and to control the voluminous information flow during the whole product lifecycle [START_REF] Eynard | UML based Specifications of PDM Product Structure and Workflow[END_REF]). Such model aims to structure product related information and to facilitate their reuse or their exchange. Nowadays, different types of product data models have been proposed depending on the industrial context or the lifecycle stage. The main objective of product data model is to support Product Data Model (PDM) functions of Product Lifecycle Management (PLM) throughout the product lifecycle [START_REF] Demoly | Product relationships management enabler for concurrent engineering and product lifecycle management[END_REF]. Le [START_REF] Duigou | Generic PLM system for SMEs: Application to an equipment manufacturer[END_REF] proposed a PLM structure supported by the French Technical Institute of Mechanical Industries (CETIM). The aim of this proposal was to provide a PLM solution. With this PLM system, they can get into an extended enterprise structure with measured investments and time. Based on a Product -Activity -Resource -Organization meta-data structure (Fig. 2), this proposal has to be aligned with the previous valuebased proposal, in order to be used to assess the product lifecycle model. Such PLM models should interoperate with others collaborative systems in order to elaborate a shared understanding. Indeed knowledge cannot always be made explicit and directly shared through PLM systems. In this paper, semantic interoperability of a PLM model with the model of the MEMORAe approach is outlined.

Such approach relies on ontologies in order to support the construction of a shared understanding notably through social interaction. It can be considered as good a way to ensure tacit and explicited knowledge sharing within extended enterprises.

Research proposition

PLM systems allow information sharing throughout a product lifecycle. Through semantic interoperability, we propose in this paper a way to share knowledge with such systems within extended enterprises.

The first part of this section presents the MEMORAe approach, which allows the elaboration of a shared understanding through the use of ontologies. We consider that such shared understanding ensures tacit knowledge sharing. The second part of this section explains how the semantic interoperability of an approach like MEMORAe with a PLM system may be realized in order to support tacit and explicited knowledge sharing through such PLM system.

The MEMORAe approach

In [START_REF] Abel | Competencies Management and Learning Organizational Memory[END_REF] the MEMORAe approach is presented as aiming to offer an alternative to the loss of competencies and knowledge in an organization. A competency is considered as a way to put into practice some knowledge within a specific context [START_REF] Abel | Competencies Management and Learning Organizational Memory[END_REF]. The MEMORAe approach offers an ontology-based learning organizational memory.

As explained by [START_REF] Davenport | Working Knowledge: How Organizations Manage What They Know[END_REF], transmitting information is not sufficient to share knowledge, due to the existence of individual interpretation in sense-giving and sense-reading processes [START_REF] Polanyi | Sense-giving and sense-reading[END_REF]. So that sharing information is not sufficient to share knowledge. Moreover even if authors as Ball et al. (1999) say that "more information is better", others as Berners-Lee et al. ( 2001) consider that a web of information with well-defined meaning facilitates collaborative work between computers and individuals. Ontologies could then clarify the structure of information source of knowledge. They are a way for sharing and for re-using knowledge, whether tacit or made explicit by someone within a certain context. Ontologies reflect a shared world view and for [START_REF] Domingue | Supporting ontology driven document enrichment within communities of practice[END_REF]. They can support communication and knowledge sharing through a community of practice. [START_REF] Wenger | Communities of Practice: Learning, Meaning, and Identity[END_REF] tools, documents, images, symbols, well-defined roles, specified criteria, codified procedures, regulations, and contracts that various practices make explicit for a variety of purposes" (Wenger, 1998, p. 47). With Web technologies, communities become present online, and for [START_REF] Grundstein | From capitalizing on Company's Knowledge to Knowledge Management[END_REF] an online community consists of people, a shared purpose, protocols and rules that guide interaction and computer systems. The use of a collaborative approach such as MEMORAe ensures the construction of a shared understanding within a community.

The MEMORAe project (see http://www.hds.utc.fr/memorae/ and Fig. 3) is a prototype showing ways to help users to capture knowledge. Entry points at the left of the screen provide direct access to a concept, whereas a history of the navigation is shown on the right of the screen. The part of the ontology describing current resources appears in the center of the screen and is framed by a short description of the current concept and by a list of shared resources related to the current concept (text or multimedia content) at the left of the screen. Within different spaces ("Tour Equipement" and "Atelier" in Fig. 3), individuals may have access to different resources and may be authorized to interact with different stakeholders (by chatting or through a wiki for example). By discovering new concepts, users can ensure that they interpret them correctly by browsing the entire screen of the MEMORAe project. The aim, in the MEMORAe approach, is to put into practice organizational learning through organizational memory, i.e. the explicit and persistent representation of knowledge and information in an organization, in order to facilitate access and re-use by members of the organization for their tasks for [START_REF] Dieng | Methods and tools for corporate knowledge management[END_REF], based on the synergy of three fundamental layers:  knowledge management to support capitalization,  Semantic Web to support sharing and interoperability,  Web 2.0 to support social processes. This approach is implemented in the MEMORAe project (Fig. 3) with the following functionalities:  explicited knowledge capitalization as a resource in a learning organization: classical resources (documents, videos, images, etc.), Web 2.0 resources, annotations, etc.,  management of communities: users and groups management, etc.,  work artifacts organization at an individual and a group level,  common-ontology sharing,  semantic-indexing of resources with concepts of the ontology,  resources and concepts annotation to support tacit knowledge explicitation,  customized user-access based on a selection of concepts relevant for a user profile (entry-points allowing to access to the resources indexed with user-interest concepts),  Web 2.0 tools enabling collaboration in diverse collaborative settings: semantic wiki, forum, chat etc.,  social networks to support information sharing within communities.

The ontology is an essential element of this approach as it serves at the same time the purpose of semantic indexing, common-reference enhancing collaboration trough a common vocabulary and interoperabilityenabler. Such ontology gives a context to capitalize explicited knowledge and provides advanced search mechanisms, when the ontology is expressed in a logic-based formalism. In a design context for example, communities are formed and have to closely work together. This fact generates particular collaboration needs, which have to be reflected in the ontology content and use: teams are created ad hoc, collaborators need to be aware of the human and geographical organization as well as the actual product development process and the reference solution. The context in which knowledge has been explicited, through a process of sense-giving in the sense of [START_REF] Polanyi | Sense-giving and sense-reading[END_REF], can be reconstructed based on the ontology and on the resources affected. Fig. 4 highlights a part of the domain ontology in the MEMORAe approach describing the organization: persons, groups, hierarchy and relationship between employees (managers or peers), the roles they play depending on the activity and on assigned responsibilities, division of the organization into departments and sites.

Towards tacit and explicited knowledge sharing

Collaborative working systems such as Computer-Supported Cooperative Work (CSCW) result of information technology whose implementation has been polarized around the individual user for [START_REF] Schmidt | Taking CSCW seriously: supporting articulation work[END_REF]. Individuals are users and they are individually using a system. [START_REF] Stockdale | An interpretive approach to evaluating information systems: A content, context, process framework[END_REF] notice that neglecting social activity leads to "meaningless conclusions", and [START_REF] Jordan | Ethnographic workplace studies and computer supported co-operative work[END_REF] insists when she reminds that knowledge is not only based on the group but is also tacit, embodied in individual interpretation: "We believe that there is yet another dimension that needs to be explored and that is the knowledge that is not only group-based but also tacit, implicit, embodied, and not articulated." (Jordan, 1996, p. 18).

Tacit and explicited knowledge sharing in PLM systems

The use of an approach as MEMORAe leads to collaboratively elaborate a shared understanding, so that tacit knowledge sharing is supported. Indeed tacit knowledge cannot always be made explicit according to [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF]. Collaboratively elaborate a shared understanding is a way to share tacit knowledge even if it cannot be made explicit. Additionally, within extended enterprises, PLM systems share information in every corner of the globe. Under certain conditions, a piece of information shared within a PLM system leads to one and only one interpretation. So that under certain conditions, sharing information within PLM systems is sufficient to share explicited knowledge. We are now going to explain these two major strengths of our proposal: (1) ensuring tacit knowledge sharing within a PLM system through the use of the MEMORAe approach, and (2) identifying the conditions allowing explicited knowledge to be shared through a PLM system. Within extended enterprises, knowledge is created and used by several actors, within several information systems and in every corner of the globe. The MEMORAe project (Fig. 3) is a platform relying on Web Semantic standards. It allows the creation of institutional or dynamic groups, which refer to:  institutional groups, which are created and leaded by a manager and include two or more collaborators,  dynamic groups, which are created by any member of the organization without requiring a validation from the hierarchy. It can be a community of interest around a specific topic, person or both. The opening of such a group to other members of the organization may be restricted within private groups. So, a shared environment is proposed and allows people, regardless where they are and regardless who they are, to elaborate a shared understanding through the use of ontologies. They can additionally actively participate in communities by creating them, by managing them, or by inviting relevant persons in them. This shared environment ensures tacit knowledge sharing and its integration by stakeholders: within such communities they all give the same meaning to the same information shared through the system. Explicited knowledge is tacit knowledge that has been made explicit by someone who created information within a certain context. It can be supported by information technology such as PLM. Nevertheless, contrarily to tacit knowledge, explicited knowledge is detached from its meaning, which depends of the person who is receiving it and of the context of its use. A piece of explicited knowledge EK is a piece of information created by a person P 1 from his/her tacit knowledge TK 1 within a certain context C 1 . That piece of explicited knowledge EK may be disseminated through an information system and it becomes then information source of tacit knowledge TK 2 for another person P 2 within a certain context C 2 . Note that even if EK is a raw object (it is information), TK 1 , C 1 , TK 2 , and C 2 may differ from one person to another, except when shared information is:  highly contextualized explicited knowledge, which refers to explicited knowledge attached to a context avoiding interpretation variance (in the sense of [START_REF] Arduin | On the Use of Cognitive Maps to Identify Meaning Variance[END_REF]. For example the sentence "the cutting speed on such material is x" may lead to different interpretations depending on the tool, the engine, the geometry of the piece, the lubrication, etc. Such piece of information is not highly contextualized and cannot lead to efficiently share explicited knowledge, because it may lead to different interpretations.  technical and unambiguous explicited knowledge, which refers to explicited knowledge detached from natural language ambiguity. It may not lead to different interpretations. For example "15°" is detached from natural language but it is ambiguous as it may lead to different interpretations: disseminated within a PLM system, "15°" may mean "a temperature of fifteen centigrade degrees" or "an angle of fifteen radius degrees". Such piece of information is not technical and unambiguous and cannot lead to efficiently share explicited knowledge, because it may lead to different interpretations. These two conditions under which explicited knowledge, i.e. information source of tacit knowledge for someone, can actually and efficiently be shared through PLM systems have been identified: there is no ambiguity, no discussion, and no need of explanations. The semantic interoperability of an approach like MEMORAe with a PLM system in order to support tacit and explicited knowledge sharing through such PLM system will now be outlined.

Interoperability between MEMORAe and PLM

To link a data model of PLM and MEMORAe, the proposed methodology is to use a model driven engineering approach to create a transformation from a model to the other. Model driven engineering is an approach that aims to use models all along the development cycle of a software passing from contemplative models to productive model. The model transformation allows creating a target model from a source model. The transformation is constituted of two steps: the specification of the transformation rules and the application of these rules to generate the target model. Several languages exist to describe these transformations, like QVT (Query/View/transformation) a standard from OMG and ATL (Atlas Transformation Language) that implement QVT in Eclipse. For the collaborative and human interaction domain, the main specific objects are:  cluster, to create and manage communities of practice,  annotations, to dynamically annotate the document. Fig. 5 illustrates the aimed semantic interoperability between MEMORAe and PLM models. At the moment of writing this article, we are focusing on the realization of such semantic interoperability through the use of ATL. Nevertheless, we already identified forthcoming results and perspectives of this work.

Discussing the results, the added value, and the perspectives of this work

Whereas information can efficiently be shared through information systems such as PLM systems, knowledge, whether tacit or made explicit by someone within a certain context, is extremely difficult to share. Information may be useful throughout a product lifecycle, nevertheless knowledge is a crucial resource whose absence can stop ongoing processes [START_REF] Grundstein | GAMETH: a constructivist and learning approach to identify and locate crucial knowledge[END_REF].

Through semantic interoperability, we proposed in this paper a way to share tacit knowledge within extended enterprises. The use of a collaborative platform allows stakeholders to elaborate a shared understanding. So that even if it cannot be made explicit [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF], tacit knowledge and its meaning are shared through information systems, when they interoperate with such collaborative platforms. By highlighting the conditions under which a piece of information may lead to one and only one interpretation within information systems, we also proposed in this paper a way to share explicited knowledge within extended enterprises. So that when it is disseminated, explicited knowledge and its meaning are shared through information systems, when these conditions are satisfied. Some semantic differences between MEMORAe and PLM models must now be highlighted. Indeed, they form research perspectives for this work that we need to be aware of. For example, whereas PLM models consider that a "product" is related to activities and resources, MEMORAe considers that a "product" is a resource able to produce information potentially source of knowledge for someone. On the same way, a "human" is a resource for PLM models, whereas it can also be a user for the MEMORAe model. Such semantic differences form a limitation of this work if and only if we are not aware of. So that they are now studied in order to be understood and managed.

During early experiments realized with students, it has been observed that the use of a collaborative platform, such as MEMORAe, facilitates and encourages interaction between stakeholders [START_REF] Abel | Competencies Management and Learning Organizational Memory[END_REF]. Observed students learned rapidly studied concepts and they all understood the same thing, whereas some of the studied concepts usually lead to different interpretations. This work must now rely on industrial fieldworks in order to identify whether the proposed approach will actually be useful within extended enterprises. At the time of writing this paper, it is not possible to present an industrial case study validating or invalidating the proposed approach. This constitutes also a weakness of this work, which is currently tested within several enterprises. The preliminary design phase in a product lifecycle can for example be facilitated when stakeholders share efficiently knowledge on the product to be designed.

Such phase is facilitated when stakeholders, regardless where they are and regardless who they are, share not only information on their wills about the envisaged product, but also knowledge they hold. They have the means to collaboratively elaborate a shared understanding of information disseminated through the PLM system. So that knowledge and its meaning are shared through such PLM system within extended enterprises.

Conclusions and perspectives

As pointed out along this paper, knowledge, whether tacit or made explicit by someone within a certain context, is a crucial resource which has to be shared within extended enterprises. This paper proposes (1) to outline a semantic interoperability allowing the integration of a collaborative platform within a PLM system, and (2) to highlight the conditions under which a piece of information shared through a PLM system may lead to one and only one interpretation.

The proposal aims at sharing tacit and explicited knowledge within extended enterprises. To do so, it is firstly proposed to make interoperate PLM with the MEMORAe approach. Such approach allows individuals to construct a shared understanding, which supports tacit knowledge sharing. Then even if it cannot be made explicit (according to [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF], tacit knowledge and its meaning can be shared through a PLM system using approaches such as MEMORAe. Secondly, conditions under which a piece of information shared within a PLM system may lead to one and only one interpretation are highlighted. Then not only information has been shared, but also explicited knowledge and its meaning have been shared through such PLM system. After remembering the vision of knowledge in the organization and the PLM model adopted in this work, the research proposition has been introduced and tacit and explicited knowledge sharing have been explained. The perspectives of this work as well as its added value have finally been discussed at the end of this paper. PLM systems have a real strength which is that they are strongly integrated and used within extended enterprises. They give individuals the ways to collect, to process and to share huge amounts of information. This work considers that when it has no meaning, information is useless for an organization. So that within extended enterprises, where individuals may be spread throughout the world, it is crucial to ensure that the same information shared through the same information system will have the same meaning for people, regardless where they are and regardless who they are.

Acknowledgement

This work was carried out and funded in the framework of the Labex MS2T. It was supported by the French Government, through the program "Investments for the future" managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).

Figure 1 :

 1 Figure 1: Knowledge sharing: explicited knowledge is transmitted and interpreted to create tacit knowledge

Figure 2 :

 2 Figure 2: Product-Activity-Resource-Organization meta-model (Le Duigou et al., 2012)

Figure 3 :Figure 4 :

 34 Figure 3: The main communication interface of the MEMORAe project web platform

Figure 5 :

 5 Figure 5: The aimed semantic interoperabilityThe closer PLM data model to MEMORAe ontology is the PLM data model from Le[START_REF] Duigou | Generic PLM system for SMEs: Application to an equipment manufacturer[END_REF]. They are both based on the same Product, Process, Resource, Organization main objects with a possible specialization at the application level (the company level). The differences allow extraction of the main objects specific to the PLM domain and to the collaborative and human interaction domain. For PLM specific domain, the main specific objects are:  the alternative and the optional objects, to construct different product configurations,  the versioned objects, to trace the history of an object. For the collaborative and human interaction domain, the main specific objects are:  cluster, to create and manage communities of practice,  annotations, to dynamically annotate the document. Fig.5illustrates the aimed semantic interoperability between MEMORAe and PLM models. At the moment of writing this article, we are focusing on the realization of such semantic interoperability through the use of ATL. Nevertheless, we already identified forthcoming results and perspectives of this work.

  presented what kind of practices is involved within a community of practice: "Such a concept of practice includes both the explicit and the tacit. It includes what is said and what is left unsaid; what is represented and what is assumed. It includes language,


Pierre-Emmanuel Arduin

Julien Le Duigou

Marie-Hélène Abel

Benoît Eynard

IMPACT OF INDIVIDUAL COGNITION ON PRODUCT LIFECYCLE MANAGEMENT SYSTEMS

Introduction

Product Lifecycle Management (PLM) aims at an integrated management of all product-related information and processes through the entire lifecycle for [START_REF] Terzi | Product lifecycle managementfrom its history to its new role[END_REF]. Within an extended enterprise, systems interact in order to share data and information throughout a product lifecycle. Beyond this computing point of view, individuals may be spread throughout the world and that is the reason why the same information on the same product shared through the same PLM system within the same extended enterprise may lead to different interpretations. Individual cognition, knowledge and particularly tacit knowledge as introduced by [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF] have to be considered during the design of such computing systems.

The aim of this work is to outline a semantic interoperability between a collaborative platform and a PLM system. Such interoperability allows individuals to construct a shared understanding, supporting tacit knowledge sharing, i.e. knowledge that cannot be made explicit. Our approach leads then to focus on the impact of individual cognition on PLM systems.

Research Background

Information is continuously interpreted through individual cognition during sense-reading processes according to [START_REF] Polanyi | Sense-giving and sense-reading[END_REF]. Within extended enterprises, information can be transmitted by speaking, writing or acting, and more generally, by information systems such as PLM systems. Knowledge can then be:

• explicited, i.e. it has been made explicit by someone within a certain context and it can be supported by information technologies. Individuals, as well as computers are "information processing systems" as said by Hornung (2009, p. 9), • tacit, it cannot always be articulated, relying on [START_REF] Polanyi | Personal Knowledge: Towards a Post Critical Philosophy[END_REF] notably: "we can know more than we can tell". However, it can be managed as explained by [START_REF] Arduin | From knowledge sharing to collaborative decision making[END_REF].

Le [START_REF] Duigou | Generic PLM system for SMEs: Application to an equipment manufacturer[END_REF] proposed a PLM model supported by the French Technical Institute of Mechanical Industries (CETIM). Such model should interoperate with a collaborative system in order to give individuals the means to elaborate a shared understanding. Through such interoperability, our approach focuses on the impact of individual cognition on PLM systems.

Research outlines

In [START_REF] Abel | Competencies management: Integrating Semantic Web and Technology Enhanced Learning Approaches for Effective Knowledge Management[END_REF] the MEMORAe approach is presented as aiming to offer an alternative to the loss of competencies and knowledge in an organization. Such approach offers an ontology-based learning organizational memory. Its use leads to collaboratively elaborate a shared understanding by focusing on individual cognition, so that tacit knowledge may be shared.

Our work aims at linking the PLM model of Le [START_REF] Duigou | Generic PLM system for SMEs: Application to an equipment manufacturer[END_REF] and MEMORAe. The proposed methodology is to use a model driven engineering approach to create a transformation from a model to the other. The model transformation allows creating a target model from a source model and is constituted of two steps: the specification of the transformation rules and the application of these rules to generate the target model. We are currently realizing such transformation and we are planning to study its implications and limits within industrial fields.