Alexandre Aussem
email: aaussem@univ-lyon1.fr

Sergio Rodrigues De Morais
email: sergio.rodrigues-de-morais@insa-lyon.fr

A Conservative Feature Selection Algorithm with Informatively Missing Data

Keywords: Réseaux Bayésiens, données manquantes, couverture de Markov, classification probabiliste, selection de variables Bayesian networks, missing data, Markov boundary, probabilistic classification, feature subset selection

Cet article présente un nouvel algorithme conservateur de sélection de variables avec données manquantes. Il est conservateur au sens où il fait une hypothèse au pire cas sur le processus de perte des données. Il s'applique en particulier aux données manquantes vérifiant l'hypothèse IM ("informatively missing"), i.e., quand les données manquantes ne peuvent être inférées à partir des seules données disponibles. L'algorithme est fondé sur la recherche de la couverture de Markov de la variable cible. Une évaluation empirique est menée sur plusieurs bases de données synthétiques et réelles pour évaluer son efficacité.

ABSTRACT. This papers introduces a novel conservative feature subset selection method with informatively missing data, i.e., when data is not missing at random but due to an unknown censoring mechanism. This is achieved in the context of determining the Markov blanket (MB) of the target variable in a Bayesian network. The method is conservative in the sense that it constructs the MB that reflects the worst-case assumption about the missing data mechanism, when the missing values cannot be inferred from the available data only. An application of the method on synthetic and real-world incomplete data is carried out to illustrate its practical relevance.

Introduction

A principled solution to the feature subset selection problem is to determine the Markov boundary (MB) of a target variable T [START_REF] Koller | Toward Optimal Feature Selection[END_REF][START_REF] Margaritis | Bayesian Network Induction via Local Neighborhoods[END_REF][START_REF] Pearl | Probabilistic Reasoning in Intelligent Systems : Networks of Plausible Inference[END_REF]. The Markov boundary of T , denoted by MB T , is defined as any minimal subset of V (the full set) that renders the rest of V independent of T . In recent years, they have been a growing interest in inducing the MB automatically from data. Very powerful correct, scalable and data-efficient constraint-based (CB) algorithms have been proposed recently, e.g., PCMB [START_REF] Peña | Scalable, Efficient and Correct Learning of Markov Boundaries under the Faithfulness Assumption[END_REF], IAMB [START_REF] Tsamardinos | Algorithms for Large Scale Markov Blanket Discovery[END_REF] or its variants : Fast-IAMB [START_REF] Yaramakala | Fast Markov Blanket Discovery[END_REF] and Inter-IAMB [START_REF] Yaramakala | Speculative Markov Blanket Discovery for Optimal Feature Selection[END_REF]. These methods yields compact MB by heeding independencies in the data. They systematically check the data for independence relations and use those relationships to infer necessary features in the MB. When no entry is missing in the database, the MB can be estimated efficiently with these methods. An excellent trade-off between time and quality of reconstruction is systematically obtained. Unfortunately, when the database is incomplete, i.e., some entries are reported as unknown, the simplicity and efficiency of these methods are lost.

Scoring approaches are capable of dealing with incomplete records in the database based on the expectation maximization (EM) principle [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]. The EM algorithm [START_REF] Friedman | Learning Belief Networks in the Presence of Missing Values and Hidden Variables[END_REF][START_REF] Friedman | The Bayesian Structural EM Algorithm[END_REF][START_REF] Francois | Learning the Tree Augmented Naive Bayes Classifier from incomplete datasets[END_REF][START_REF] Francois | Generation of Incomplete Test-Data using Bayesian Networks[END_REF] and Gibbs sampling [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF] are notorious solutions to handle incomplete data sets, but both methods assume implicitly that data are missing at random. Under this assumption, the missing values can be inferred from the available data. However, this assumption does not hold and it is hard, if not impossible, to test in practice. The decrease in accuracy my be severe with EM-based methods when the assumption is violated. More recently, Robust Bayesian Estimator (RBE) [START_REF] Ramoni | Robust Learning with Missing Data[END_REF] methods were proposed to learn conditional probability distributions from incomplete data sets without making any assumption about the missing data mechanism. The major feature of the RBE is to produce probability estimates that are robust with respect to different types of missing data. This robustness is achieved by providing probability intervals containing the estimates that can be learned from all completed data sets.

In this paper, we transpose Ramoni's ideas [START_REF] Ramoni | Robust Learning with Missing Data[END_REF] to the problem of finding a conservative Markov blanket of a variable with information missing entries. By conservative, we mean the MB that renders the rest of the variables independent of the target without making any assumption about the unknown censoring mechanism. The problem is addressed in this paper by maximizing the conditional dependence measure over all possible ways to fill the missing data. The idea is exemplified with the G2 statistic in this paper but its generalization is fairly easy. The intuition behind the method is similar to that of [START_REF] Ramoni | Robust Learning with Missing Data[END_REF]. The idea is that, when no information about the pattern of missing data is available, an incomplete database contains the set of all possible estimates and this paper provides a characterization of these constraints. This conservative test addresses the main shortcoming of CB methods with missing data : the difficulty of performing a classical (non bayesian) independence test when some entries are missing without making any assumption about the missing data mechanism.

The remainder of this paper describes our approach. Sections 2 and 3 establish some notation and reviews the background and motivation of the research. Section 4 describes the theoretical framework of the method, while Section 5 applies the method to synthetic and real incomplete data sets.

Constraint-based Markov blanket discovery

We denote the conditional independence of the variable X and Y given Z, in some distribution P with Ind P (X; Y |Z), dependence as Dep P (X; Y |Z). Constraint-Based (CB for short) learning methods systematically check the data for independence relations and use those relationships to infer necessary features to be included in the MB. They rely on a probabilistic association measure between X and Y conditionally on Z denoted by Assoc(X; Y |Z). In our implementation, we use a well-known CB method called IAMB [START_REF] Tsamardinos | Algorithms for Large Scale Markov Blanket Discovery[END_REF] along with a statistically oriented conditional independence test based on the G-test :

G = 2 m i=1 p j=1 q k=1 n(i, j, k) ln n(i, j, k)n(•, •, k) n(i, •, k)n(•, j, k) . [1]
where n(i, j, k) is the number of times simultaneously X = x i , Y = x j and Z = z k in the sample, that is, the value of the cell (i, j, k) in the contingency table .  A distribution P is said faithful with respect to G if the d-separations in the DAG identify all and only the conditional independencies in P . Suppose < G, P > satisfies the faithfulness condition, it may be shown that : (1) the set of parents, children and parents of children of X is the unique Markov boundary of X, and (2) X and Y are not adjacent in G iff ∃Z ∈ V \ {X ∪ Y } such that Ind P (X; Y |Z) (see [START_REF] Neapolitan | Learning Bayesian Networks[END_REF] for instance). Tsamardinos et al. prove in [19] that IAMB is correct under the faithfulness assumption.

Dealing with missing entries

When the database is complete, that is, all entries are known, the computation of Assoc(X; Y |Z) is straightforward. Unfortunately, the expression cannot be computed when entries are reported as unknown. Suppose for simplicity that Z is empty and that we wish to estimate the maximum value of Assoc(X, Y ) from an incomplete database D = {X j , Y j } j=1,...,n in which some entries of the variable X j and Y j are unknown. These unknown entries give rise to multiple types of incomplete cases that are relevant to the estimation of n(i, j). For instance, Table I shows a database with several entries missing denoted by " ?". n(?, j) will for instance denote the number Case 1 2 3 4 5 6 7 8 X 1 2 1 ? 2 ? 1 1 Y 2 1 ? 1 2 ? ? 2

Tableau 1. Cases with some items missing.

Y = 1 Y = 2 Y =? X = 1 0 2 2 X = 2 1 1 0 X =? 1 0 1 Tableau 2. Extended contingency table.
of cases where X is missing and Y take value y j . n(?, j) = 1 in this example. The extended contingency table is shown in Table 2. The task is to estimate the n(i, j) values. The issues involved in estimating theses values from an incomplete data set D are better explained if we regard D as the result of a deletion process applied to a complete but unknown database D c . We define a consistent completion of D to be any complete database D c from which we can obtain D using some deletion process. The set of consistent completions D c is given by all databases in which the unknown entries are replaced by one of the possible values of the unobserved variables.

Deletion process

According to [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF], the assumptions about the missing data mechanisms may be classified into three categories :

-missing completely at random (MCAR) : the probability that an entry is missing is independent of both observed and unobserved values in the data set ;

-missing at random (MAR) : the probability that an entry is missing is a function of the observed values in the data set ;

-informatively missing (IM) : the probability that an entry is missing depends on both observed and unobserved values in the data set.

In order to specify the deletion processes, a dummy binary variable R i may be associated with each variable X i . When R i takes value '1', the entry X i = x i is not observed and vice-versa. When the probability distribution of each R i is independent of X 1 , . . . , X n , the data may be seen as MCAR. When this probability distribution is a function of the observed values in the data set, data are MAR. Now, when this probability distribution is a function of the observed and unobserved entries, data are IM. The expectation maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] and Gibbs

Y = 0 Y = 1 X = 1 0 4 X = 2 3 1 Tableau 3. Completed contingency table maximizing G.
sampling [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF] are well known solutions to handle incomplete data sets but they rely on the assumption that data are MAR. The problem is that MCAR and MAR assumptions are hard, if not impossible, to test. On the other hand, one cannot simply infer the missing entries from the observed ones anymore when the data is IM. Hence the need for a general approach dealing with the IM worst case censoring mechanism.

A conservative Markov blanket

The solution we propose is based on the idea that, even with no information on the missing data mechanism, an incomplete data set D constrains the set of estimates that can be induced from its consistent completions. Following this principle, we introduce the conservative statistical test with no assumptions about the missing data mechanism. It is conservative in the sense that makes always the worst case assumption by assuming dependency when independency cannot be guaranteed in all the distributions associated with the consistent data completions. Let Using ConsAssoc(X; Y |Z; D) yields a conservative test in the sense that it always take the worst-case assumption about the missing data mechanism to decide whether X and Y are conditionally independent. It is implicitly assumed in the definition that ConsAssoc(X; Y |Z; D c ) = Assoc(X; Y |Z; D c ) for any completion D c obtained from D. In general, whatever the way the missing database is completed, we would want an edge to mean a direct dependency when the CB algorithm is run on these data. As we know, the faithfulness entails this. The following theorem shows that a conservative Markov blanket can be obtained using IAMB(T, D, ConsAssoc) (i.e., IAMB run with the conservative test) as shown next :

Theorem 1 Suppose the independence tests are correct and that the learning database D c is an independent and identically distributed sample from a probability distribution P faithful to a DAG G. Suppose given an incomplete database D that was obtained from D c by some missing data mechanism. Then, IAMB(T, D, ConsAssoc) returns a conservative Markov blanket of X.

Proof : If X ∈ PC T (where PC T denote the set of parents and children in G) then X remains dependent on T conditioned on any set

Z ∈ U \ {X, Y }. Therefore, from definition 1, we have α < Assoc(X; Y |Z; D c ) ≤ ConsAssoc(X; Y |Z; D) for all Z ∈ U \ {X, Y }.
So X is necessarily in the output of IAMB(T, D) run with the conservative test. Now, if X ∈ MB T \PC T (where MB T denote the MB of T ), then X is spouse of T owing to the faithfulness assumption. So there is a variable Y ∈ PC T such that T ⊥ X|Z ∪ Y for all Z ∈ U \ {T, X}. Recall that IAMB works in two steps. In the first stage, candidate nodes are added sequentially to the current candidate MB set when they are not found independent on T conditioned on the current MB. The extra nodes are removed from MB in the second step. As Y will enter this set during the first stage, X will also enter but it will never leave this set in Phase II because T ⊥ X|{Z ∪ Y } for all Z ∈ U \ {T, X}. .

A conservative independence test

This section shows how to design practically a conservative test based on the G statistic. For sake of clarity, Z is supposed empty. The case Z non empty will be discussed in the sequel. As a shorthand, we note p j=1 n(i, j) = n(i, •) and m i=1 n(i, j) = n(•, j). Let n 0 (i, j) be the number of non missing cases in cell (i, j) and let x ij (resp. y ij and z ij ) be the number of additional cases that are affected to cell (i, j) owing to n(?, j) (resp. n(?, j) and n(?, ?)). The value that would be computed from the complete data set (if known) is

n(i, j) = n 0 (i, j) + x ij + y ij + z ij , ∀i, j.
[2]

The information conveyed by the incomplete cases impose several constraints on the variables x ij , y ij and z ij . In order to identify the maximum estimate of G, we have to consider the problem of maximizing G subject to equality and inequality constraints :

(P)                                                    Max G = 2 p i=1 m j=1 n(i, j) ln n(i, j) n(i, •)n(•, j)/n , subject to p j=1 x ij = n(i, ?), ∀i, m i=1 y ij = n(?, j), ∀j, m i=1 p j=1 z ij = n(?, ?), n(i, j) = n 0 (i, j) + x ij + y ij + z ij , ∀i, j, x ij ≥ 0, ∀i, j, y ij ≥ 0, ∀i, j. z ij ≥ 0, ∀i, j.
[3]

(P) is combinatorial problem with 3mp variables. If we relax the integrity constraints the problem may be solved using nonlinear programming techniques. From Karush-Kuhn-Tucker (KKT) Theorem, the problem of maximizing G subject to equality and inequality constraints is obtained by optimizing the Lagrange function L with respect to 3mp + m + p + 1 parameters subject to an extended condition set. Let x = (x 11 , . . . , x mp ) ∈ mp , y = (y 11 , . . . , y mp ) ∈ mp and z = (z 11 , . . . , z mp ) ∈ mp be a candidate solution in C. From the KKT conditions, if x is a local extrema of the M inf (x) subject to h i (x) = 0 for i = 1, . . . , n and g j (x) ≥ 0 j = 1, . . . , p, there exist λ = (λ 1 , . . . , λ n ) ∈ n and a positive vector

µ = (µ 1 , . . . , µ m ) ∈ m such that ∇f (x) + m i=1 λ i ∇h i (x) + p i=1 µ i ∇g i (x) = 0 and µ i g i (x) = 0, ∀i.
We need to express ∇G by taking the derivative of G with respect to x ij (y ij and z ij play a symmetric role),

1 2 ∂G ∂x ij = 1 2 ∂G ∂y ij = 1 2 ∂G ∂z ij = ln n(i, j) n(i, •)n(•, j)/n -1 [4]
Define λ x i and µ x i (resp. λ y i and µ y i ) and λ z and µ z i , the Lagrange multipliers associated to the constraints. From the KKT conditions, the maximum is obtained when

n(i, j) = e βij • n(i, •)n(•, j)/n, ∀i, j.
[5]

where the parameters β ij are solution of

   β ij = ln n -(λ x i + µ x ij ), β ij = ln n -(λ y j + µ y ij ), β ij = ln n -(λ z + µ z ij ).
[6] with µ x ij , µ y ij and µ z ij solution of

µ x ij x ij = 0, µ y ij y ij = 0, µ z ij z ij = 0, µ x ij ≥ 0, µ y ij ≥ 0, µ z ij ≥ 0. [7]
This is a nonlinear system that does not seem to admit a closed-form solution. β ij = 0 is interpreted as a local deviation of cell (i, j) to independence. β ij > 0 when n(i, j) is over-represented and vice-versa. The next theorem will help characterize G's maximum.

Theorem 2 If (x, y) is a solution of (P), then the following assertions hold :

-For all i, there is unique j such that x ij = 0 and ∀j = j , β ij ≤ β ij , -For all j, there is unique i such that y i j = 0, and ∀i = i , β ij ≤ β i j , -There is unique pair (i , j ) such that z i j = 0,and ∀(i, j) = (i , j ), (•, k). This is the minimum of G. From KKT conditions, the maximum is reached only when an inequality is active (at x = 0 or at x = min (n ij , n i,? )). The same results hold for G ijk (x) = G(n 11 , . . . , n ij -y, . . . , n kj + y, . . . , n pm ). The maximum of G requires all G ijk (x) to be locally maximized, otherwise G can be increased by permuting the missing cases across lines or columns. The maximum for G can be characterized further. Let i and j be such that x ij = 0 and y ij = 0.

β ij ≤ β i j , Proof : Consider G ijk (x) = G(n 11 , . . . , n ij -x, . . . , n ik + x, . . . , n pm ) for all (i, j, k) over the domain 0 ≤ x ≤ min (n ij , n i,? ). G (x) = 0 iff n(i, j)/n(•, j) = n(i, k)/n
β ij = ln n -λ x i = ln n -λ y i , since µ x ij = 0 and µ y ij = 0. Therefore, λ x i = λ y i .
For all k = i, β kj ≤ β ij and for all k = j, β ik ≤ β ij . This completes the proof.

Owing to Theorem 2, to maximize G, a single cell (i, j ) along each line i (resp. column j) of the contingency table should receive all the missing entries n(i, ?) (resp. n(?, j)). In addition, a single cell (i , j ) increases the most the G value. (i , j ) is such that β ij ≤ β i j for all (i, j). In order to avoid the burdensome numerical resolution of the optimization problem P, we propose in the next section a simple heuristic that greedily completes the missing entries for the test to be conservative.

Extension to conditional G-tests

While the previous section only discussed the case where Z is empty, Theorem 2 can easily be generalized to conditional tests ConsAssoc(X; Y |Z; D) although it is not discussed here for sake of conciseness. With the above result in mind, we devised a greedy heuristic termed GreedyGmax in order to approximate the maximum in order O(n) when n stands for the number of cells in the contingency table. The idea is to select sequentially the triple (i, j, k) that increases most G. From Theorem 2, we know the cell that will increase G the most is the one for which β (i,j,k) is the largest. Since the β (i,j,k) depend on the missing entries, they are unknown. Our idea is to estimate β (i,j,k) by β 0 (i,j,k) using the available data only in Equation 5and fill all the possible missing entries in the cell with the highest β 0 (i,j,k) value. This procedure is then repeated until the database is completed. The Greedymax heuristic is depicted in Algorithm 2. β 0 (i,j,k) are computed at line 5 and they are sorted in decreasing order at line 7.

Algorithm 1 GreedyGmax

Require: X, Y : testing variables ; Z : conditioning set ; D : an incomplete data set ; Ensure: Gmax : an upper bound for the G-statistic ;

1: for all i, j, k do 2:

Compute n o (i, j, k) 3: end for 4: for all i, j, k do 5:

β 0 (i,j,k) = n 0 (i, j, k)n 0 (•, •, k)/n 0 (i, •, k)n 0 (•, j, k) 6: end for 7: Cell=Sort {β 0 (i,j,k) } 8: idx = 1 9: repeat 10:
Fill Cell(idx) with as much as possible missing entries 11:

idx = idx + 1 12: until D is complete 13: Gmax = G-statistic on completed data
It should be noted that the method is not yet conservative strictly speaking, as the greedy maximization heuristic is not guaranteed to find the global maximum. We shall argue, however, that the probability of a completion mechanism to yield G > GreedyGmax is sufficiently small that it can be considered as zero in practice. For sake of illustration, Table 3 shows the completion obtained when GreedyGmax is applied to data shown in Table 1. In this simple example, the heuristic yields the maximum for G.

Experimental evaluation

This section reports the results of an experiment based on real-world data with missing entries. The aim of these experiments is to show that the MB returned by the conservative method can reveal interesting dependencies that may have been missed by standard approaches. This raises an interesting question : is a conservative MB learning algorithm useful in the context of feature selection ? In other word, can we design more efficient probabilistic classifiers by use of a conservative Markov blanket ? What are the missing rates and/or the missing mechanisms for which the methods works best ?

In this section, we consider the Interleaved Incremental Association Markov Boundary (Inter-IAMB) [START_REF] Tsamardinos | Algorithms for Large Scale Markov Blanket Discovery[END_REF][START_REF] Aliferis | HITON : a novel Markov Blanket algorithm for optimal variable selection[END_REF] as our reference Markov boundary discovery algorithm. Inter-IAMB is variant of IAMB that has been proposed to improve its data efficiency while still being correct under faithfulness assumptions. Inter-IAMB seeks directly the minimal subset of V (the full set) that renders the rest of V independent of T , then MB T as IAMB does (Tsamardinos et al., 2003). The key difference between IAMB and Inter-IAMB is that the shrinking phase is interleaved into the growing phase in Inter-IAMB.

The original G-test is applied using the available case analysis technique, i.e., using for the estimation of Ind(X, Y |Z) only the instances where X, Y and Z are non missing. The significance level for the independence tests is 0.01. We compare the accuracy of Inter-IAMB with the standard G-test versus Inter-IAMB with the conservative G-test based on GreedyGmax(X, Y |Z). In our implementation, Inter-IAMB considers both tests to be reliable when the number of instances in D is at least ten times the number of degrees of freedom and skips it otherwise. Skipping the test means the variables are assumed to be independent without actually performing the test.

Limits of the conservative test

Before we assess the benefits of this approach, let us first gauge its limits on a simple example. Consider two discrete variables X and Y with 5 modalities each. 200 databases were generated, each containing 2000 independent and identically distributed samples. Figure 1 plots the average and standard deviation of the p-values obtained from the conservative Gmax test between 2 independent variables as a function of the ratio of missing data. The missing mechanism is MCAR. As may be seen, both variables are considered dependent for a significance level of 0.01 as soon as the ratio of missing data is superior to (say) 3%. So the present approach will surely be inefficient for larger missing rates.

Synthetic data

This section reports the results of an experiment on a toy problem. The aim is to evaluate the effectiveness of the Gmax approach when the data is informatively missing. The toy Bayesian network applied for this experiment is composed of four random variables as depicted in Figure 2. All variables were subject to a deletion process. We associated several pairs (i, j) of variable with a dummy variable R ij that took on one of the two values 0 and 1 with some probability. As seen in Fig. 2, the original graph was augmented by the 4 variables R AB , R AD , R BC , R CD . The dummy variables have all the same probability table. They were chosen such that : when the parents of the dummy variable have different values, no data is removed. Otherwise, the entry for both parents have MR% chances to be removed. Since the distribution of the variables R ij depends of the unobserved values in the data set, all values removed with this process is IM. This deletion process was repeated with ten times with different missing rates. As may be seen, the conservative approach returns lesser and lesser false negatives as the missing rate increases. The price to pay in terms of false positives nodes seems affordable for the missing rates that we are considering.

Real data

For the second experiment we used data on Congressional Voting Records, available from the Machine Learning Repository at the University of California, Irvine (Blake, Keogh, Merz, 1998). The data set describes for each of the 435 member of the US House of Representative on the 16 key issues during the 1984. Hence, the data set consists of 435 cases on 16 binary attributes and two classes that represent the party affiliation. There are 289 values reported as unknown. Although these missing entries amount to 4% of the data set, the number of incomplete cases is 203, more than 45% of the total. An important feature of this data is that unknown entries, and hence what member of the US House of Representative did not vote on, can be predictive. We have applied InterIAMB in this data set as a solution to the feature subset selection for probabilistic classification. The table below shows the results of classification using the variables selected by InterIAMB by applying the G statistic and the GreedyGmax heuristic. One can see the same algorithm provides a better feature selection for probabilistic classification when applying the GreedGmax heuristic. Standard InterIAMB outputs 4 variables : adoption-of-the-budget-resolution, physicianfee-freeze, anti-satellite-test-ban and synfuels-corporation-cutback. InterIAMB with Gmax outputs 5 variables : the same 4 except anti-satellite-test-ban plus 2 others : synfuels-corporation-cutback and education-spending. It is interesting to note here that the first MB is not included in the second as we would expect from Theorem 1. This comes from an unwanted side-effect in practice : the cascading effect of early test errors causes errors to be present in the output MB. Therefore, it is not surprising. As may be seen, the Gmax technique slightly outperforms in terms of classification accuracy by about 2% on average.

Real data : Nasopharyngeal Carcinoma epidemiological data

In this section, we apply the method on Nasopharyngeal Carcinoma (NPC) epidemiological data that was made available to us by the International Agency for Research on Cancer (IARC) [START_REF] Feng | Dietary risk factors for nasopharyngeal carcinoma in Maghrebian countries[END_REF][START_REF] Aussem | Nasopharyngeal Carcinoma Data Analysis with a Novel Bayesian Network Skeleton Learning[END_REF]. This database is not yet available in the public domain. The original data is made up from 1289 instances and 321 discrete features to evaluate the risk of factors of NPC. The variables have 2 or 3 modalities. Among the 321 variables, we selected only the ones that have less than Tableau 5. NPC data : 10-fold cross-validation results with a naive Bayes classifier using the features selected by Inter-IAMB.

3% missing entries. This reduces the problem to 145 variables. The data was collected during a multi-center case-control study that has been undertaken in 2004 by the IARC) in the Maghreb (Morocco, Algeria and Tunisia), the endemic region of North Africa. Patients were interviewed according to a specific questionnaire. As all case control studies in epidemiology, half of population is comprised of individuals that are disease positive (the cases), and the other half (the control group) come from the same population that gave rise to the cases. As observed in Table 5, the Gmax technique yields here again a slight improvement in terms of overall accuracy. The false negative rate is important here as it represents a failure in detecting the disease. The latter was reduced by 4%.

Conclusion

In this paper, we discussed a conservative constraint-based Markov blanket learning method from incomplete data. The method is conservative in the sense that it constructs a Markov blanket that reflects the worst-case assumption about the missing data mechanism. An application of the method on synthetic and real-world incomplete data was carried out to illustrate its practical relevance. The method was shown to yield a benefit for databases with no more than 5% missing data. Future substantiation

  Assoc(X; Y |Z; D c ) be the value of Assoc(X; Y |Z) evaluated on complete set D c . Definition 1 Let Assoc(X; Y |Z) be an conditional association measure. ConsAssoc(X; Y |Z) is called a conservative association measure with respect to Assoc(X; Y |Z) if, for all incomplete database D, ConsAssoc(X; Y |Z; D) ≥ Assoc(X; Y |Z; D c ) for every consistent data completion D c obtained from D.

Figure 1

 1 Figure 1. p-values obtained from the conservative Gmax test between 2 independent variables as a function of the ratio of missing data.

Figure 2 .

 2 Figure 2. Benchmark with extra dummy variables. The probability tables are shown. Values represent percentages.

Figure 3 .

 3 Figure 3. Average missing and extra variables for learning Markov boundaries of all variables of the toy problem in Fig.2. In black : the conservative InterIAMB using the GreedyGmax heuristic. In grey : the standard InterIAMB. All results are averaged over 10 runs.

  Tableau 4. Congressional Voting Records : 10-fold cross-validation results with a naive Bayes classifier using the features selected by Inter-IAMB.

		Gtest	Gmax
	Correct classification rate 92.41% 94.25%
	True positive rate	0.94	0.95
	True negative rate	0.88	0.92
	False positive rate	0.077	0.113
	False negative rate	0.045	0.052
	Kappa statistic	0.8391	0.8786
		Gtest	Gmax
	Correct classification rate 0.6059 0.6160
	True positive rate	0.6340 0.6717
	True negative rate	0.5760 0.5568
	False positive rate	0.4240 0.4432
	False negative rate	0.3660 0.3283
	Kappa statistic	0.2102 0.2292

30/2008

JFRB -Lyon -29 et 30/2008


How do ecological networks can be implemented under the current laws and regulations?

Land planning is ruled by many laws and regulations from building permits to river protection How to use what exist to implement sustainable ecological networks? How to have multiple land use at a site? What are the novel legal tools that would be useful?

How does DIVA work?

Like all programs of the ministry, DIVA aims at providing scientific evidences to define policies and management strategies on topics relevant to environmental maangement This is done with 1) A call for proposals 2) A selection of projects 3) A continuous animation of the research teams 4) Work on selected topics involving all teams

My presentation

A diversity of landscapes within a zone

A diversity of landscape elements A diversity of land use and management

Biogeographic and landscape diversity

A diversity of habitat mosaics

A diversity of species

France, as all western Europe is made of cultural landscapes

Protected areas must be managed to maintain their characteristics, there is not such thing as "wilderness"

Ecological corridors go through farmland, productive forest, periurban areas

Nature protection policies in the European Union and France

Two main policies

Natura 2000: an environmental policy

Agri-environment schemes and cross compliance within the Common Agricultural Policy (CAP)

A set of habitats of ecological value that must be maintained by the member states

Policies that aim at counteracting the negative effects of agriculture on the environment, especially biodiversity

  From nature reserves to ecological networks A paradigm shift in terms of Nature Conservation Nature reserves/ protected areas: they are designed, set up according to a pre-existing list of threatened species, high ecological value habitats. They are based on observation Ecological networks: they are designed, set up according to an estimated (modeled) capacity to permit the movement of a range of species, not (or rarely) on actual observation of movement, they are based on the best current ecological theory

	From nature reserves to ecological networks
	A new French law (Grenelle 2) to promote ecological
		networks
	to
	Foster species movement between protected areas or
	biodiversity hotspots	squirrel
			Burel
		Sarrazin
	Sustain ecosystem services (pollination, water purification,
	vulture flood and erosion control, pest control etc.
	Bretagnolle	Little bustard	Dragon fly

http://ec.europa.eu/environment/nature/natura2000/index_en.htm http://ec.europa.eu/agriculture/envir/biodiv/index_en.htm

From nature reserves to ecological networks Nature protection in reserves is not sufficient for several reasons

Most species live outside reserves

Research on policies and the social integration of networks

The questions:

How do ecological networks can be implemented under the current laws and regulations?

How do the management requirements to maintain the corridor function fit the current land use systems?

What are the scientific information flow from science to policy makers and land managers?

What is the involvement of the stakeholders and the general public during the process of designing ecological networks?

What are the interactions between rural and peri-urban landscapes?

Urban extension in a rural landscape How to integrate those landscape features in their new environment?

How do the management requirements to maintain the corridor function fit the current land use systems?

Agricultural landscapes and practices are highly diverse

How do the management requirements to maintain the corridor function fit the current land use systems?

The maintenance or restoration of ecological networks may require several types of changes

In type land cover (shift to permanent grassland from annual crops)

In the localization of crops Crop management (date of harvesting, ban of the use of pesticide) Etc.

Farming system scholars conduct surveys among farmers to understand their systems and to evaluate the possible changes and their technical, economical and social feasibility

What is the involvement of the stakeholders and the general public during the process of designing ecological networks?

Ecological networks are implemented on private land What are the scientific information flow from science to policy makers and land managers?

The development of evidence based policies in Europe Specially for nature conservation http://www.cebc.bangor.ac.uk/

Thank you for your attention

The scales of planning: from region to municipalities


François Guerrin

Integrated Modeling of Agricultural Production Systems: Achievements and Remaining Issues

Keywords: Simulation modeling, Hybrid dynamical system, Activity representation, Situated action, Operations management, Agricultural production systems, Environmental assessment

Improving the sustainability of agriculture has become crucial to deal with tomorrow's challenges such as supplying food to a continuously growing world population while mitigating its environmental impacts (e.g. climate changes). Recycling organic wastes to substitute chemical fertilizers for various organic ones (e.g. sewage sludge, household refuses, plant residues, livestock manures, agro-food industrial wastes) is one of the ways towards this end. Addressing this calls for the coordinated use of heterogeneous knowledge on both the biophysical (i.e. organic products, soils, crops) and managerial (i.e. farmers' practices) components of the whole production systems. Computer models, encompassing various pieces of that knowledge, are built to represent these systems as linked production and consumption units spread over a territory. These models are used for simulating management scenarios and assessing their performances against agronomical and environmental criteria. This paper describes our main achievements: (i) a methodology for modeling and analyzing material flows on a territory scale; (ii) a conceptual modeling framework of farming systems; (iii) a way of representing human activity in farming systems based on the 'situated action' theory. It points also out two remaining issues: (iv) assessing simulated management scenarios; (v) using models with stakeholders to support their management practices.

INTRODUCTION

The research discussed in this paper is focused on simulation modeling of agricultural production systems considered at two organization levels: single farms (individual management) and organized sets of farms (collective management). The aim of this research can be rephrased as designing simulation models to help design management policies of farming systems (and conversely). These models are conceived with farming systems agronomists to help evaluate farming systems management. They allow the dynamics of the various material flows (namely, biomass) operating within the production systems in interaction with the farming practices to be simulated. Two modeling approaches have been favored until now: hybrid dynamical systems, encompassing both continuous and discrete variables, and multi-agent systems.

Two research issues of unequal importance, the second being tackled since only recently, are dealt with: • Finding representational structures (i.e. conceptual and formal frameworks) to make operational the available knowledge: designing the model is here the focus; • Finding tools (i.e. computer models and the way to use them) to support agricultural stakeholders: designing management policies is here the focus.

In terms of models, the main achievements are the following:

• Material flow dynamic simulation models, based on the analysis of agricultural practices [START_REF] Aubry | A conceptual model of animal wastes management at the farm scale. The case of the Reunion Island[END_REF], to reason about various cases of livestock waste management: single farms (MAGMA model [START_REF] Guerrin | Magma: A model to help manage animal wastes at the farm level[END_REF]); groups of farms (BIOMAS multi-agent system [START_REF] Courdier | Agent-based simulation of complex systems: Application to collective management of animal wastes[END_REF]); collective waste treatment plant supplied by multiple farms (APPROZUT model [START_REF] Guerrin | Simulation of stock control policies in a two-stage production system. Application to pig slurry management involving multiple farms[END_REF]); collective manure application plan considering the interaction between the individual (single farms) and collective (groups of farms) levels of management (COMET model [START_REF] Paillat | Simulation de la faisabilité d'un plan d'épandage de lisier de porc et conséquences sur les émissions gazeuses au stockage et à l'épandage[END_REF]). • Simulation of flow networks using timed automata and model-checking [START_REF] Hélias | Using timed automata and model-checking to simulate material flow in agricultural production systems. Application to animal waste management[END_REF]. • Joint representation of farming practices and biophysical flows within dairy farms (GAMEDE model [START_REF] Vayssières | Gamede: A global activity model for evaluating the sustainability of dairy enterprises. Part I. Whole-farm dynamic model[END_REF]) • Modeling framework of human activity at operations level with generic aim [START_REF] Guerrin | Dynamic simulation of action at operations level[END_REF].

This paper provides details about the principal methodological findings:

• A methodology for modeling and analyzing material flows on a territory scale (Section 2); • A conceptual modeling framework of farming systems (Section 3) illustrated on three models among those enumerated above (Section 4) ; • A way of representing human activity in farming systems based on the 'situated action' theory (Section 5).

It also discusses two important issues that still remain incompletely resolved:

• How to assess simulated management scenarios? (Section 6); • How to use our models with stakeholders to support their management? ( Section 7).

The perspectives open to the different sides of this work in the coming years are pointed out.

MODELING AND ANALYZING MATERIAL FLOWS ON A TERRITORY SCALE: THE 'MAFATE' APPROACH

Beyond the development of the simulation models enumerated in Section 1, one of the main achievements is the formalization of the approach which actually constituted the driving thread of the research done in partnership with systems agronomists. This approach, termed 'Mafate' [START_REF] Guerrin | Mafate: modelling and analysing matter flows on a territory scale[END_REF], encompasses several steps yielding the following outcomes:

1. Farm surveys, covering the diversity of management situations found in the considered territory; 2. Farm typology, defining the main farming types and characterizing both their structure and management policies; 3. Conceptual models, synthesizing the knowledge gained on farming practices from surveys; 4. Computer models, designed to simulate the interaction between the material flows and the farming practices at both 'individual' (intra-farms) and 'collective' (interfarms) levels of organization; 5. Simulation outputs of management alternatives checked by experts (e.g. agronomists, technical staff, skilled farmers) according to agricultural and environmental criteria; 6. Model validation as virtual experiment tools in relation with agricultural stakeholders.

Steps 1 and 3 are deemed essential prior to constructing flow management models in order to account for actual farming practices, identify and explicitly describe actual management constraints and strategies.

Step 2 is also very useful to take into account the diversity of situations found in the region considered. Performing simulations with the models (step 5), a long but interesting task, is mandatory to analyze dynamically (in contrast with widespread static methodologies as Life-cycle analysis) the functioning of the systems represented. Important work remain to be done, on the one hand, on multi-criteria assessment of simulated management strategies (step 5), on the other hand, on the use of models in management situations with agricultural stakeholders (step 6). These two issues, still incompletely resolved, are detailed below (Sections 5 and 6).

A CONCEPTUAL FRAMEWORK FOR MODELING AGRICULTURAL SYSTEMS

The ambition was to design a modeling framework with the following aims:

• Representing agricultural production systems on different temporal and spatial scales; • Integrating the various pieces of knowledge available on these systems; • Simulating the dynamics of interactions between management practices and material flows; • Assessing the impact of these practices on the systems' viability and sustainability; • Designing management strategies to improve the systems' performance against various criteria.

The material and work flow models that have been developed (cf. Section 1), the recent efforts of model generalization (extension of waste management to whole-farm operations in GAMEDE [START_REF] Vayssières | Gamede: A global activity model for evaluating the sustainability of dairy enterprises. Part I. Whole-farm dynamic model[END_REF]; generic simulation of action [START_REF] Guerrin | Dynamic simulation of action at operations level[END_REF]) and the design of a comprehensive approach ranging from the acquisition of knowledge to model building and simulation to support agricultural stakeholders (cf. Section 2) go in this direction.

These experiences allowed an understanding of the representation of agricultural production systems considered at different levels of organization on various temporal and spatial scales (farm, group of farms, agro-food supply chain) to emerge. According to this understanding, an 'Action-Flow-Stock' ontology has been devised [START_REF] Guerrin | Modelling agricultural production systems using an action-flow-stock ontology[END_REF]. According to this ontology, agricultural production systems are represented as a set of stocks connected by flows of materials controlled by the farming activities (Fig. 1). Two types of flows are distinguished: "workable" flows, which take place only if there is human intervention, and "biophysical" flows, which take place even in the absence of human intervention. These flows interact through human activity, which aims to guide the biophysical flows, among which those leading to the "products" of the system, by the workable flows it generates. The management of the production system can then be seen as the control of a set of stocks by the activities of the operator (i.e. the farmer and farm workforce). These activities stem from the confrontation between encountered situations and strategies: implementing strategies helps create new situations; the experience gained by this implementation can, in turn, change strategies.

The relevance of this conceptual framework, derived by generalization of livestock effluents management models listed in Section 1 (i.e. MAGMA, BIOMAS, APPROZUT), has been verified, on the one hand, at the level of individual farm operation [START_REF] Vayssières | Gamede: A global activity model for evaluating the sustainability of dairy enterprises. Part I. Whole-farm dynamic model[END_REF], on the other hand, at the level of collective management:

• Simulation of a hog slurry collective application plan in Brittany (Western France) using the COMET model [START_REF] Guerrin | Combining individual and collective management of animal manure to reduce environmental impacts on a territory scale[END_REF][21]; • Draft modeling [START_REF] Guerrin | Modelling animal farming systems as supply chains[END_REF] and life-cycle analysis of the Reunion Island swine sector described as a supply chain.

The coupling of workflow management models with mechanistic models of biophysical processes may, however, be problematic when the data necessary to the setting of the latter are missing or when their generic feature is not guaranteed in the local situation investigated. To represent these processes, we thus moved towards the synthesis of expert knowledge in the form of simple empirical rules or formulae validated locally. An example of such an empirical coupling is provided by the GAMEDE model [START_REF] Vayssières | Gamede: A global activity model for evaluating the sustainability of dairy enterprises. Part I. Whole-farm dynamic model[END_REF].

EXAMPLES OF MODELS BASED ON THE ACTION-FLOW-STOCK ONTOLOGY MAGMA: Livestock effluent management at farm level

The MAGMA model [START_REF] Guerrin | Magma: A model to help manage animal wastes at the farm level[END_REF] addresses the case of livestock effluent management within a farm. Two types of units are involved in such a "distribution" (i.e., one-to-many) configuration (Fig. 2): livestock enterprises producing animal wastes and consumption units, such as crop plots or waste treatment plants, where effluents are spread or supplied. Simulating MAGMA allows management strategies of livestock effluents to be assessed with respect to several indicators: environmental (nitrogen losses due to stock overflowing, fallow land spreading, over-fertilization of crops); agronomical (nitrogen applied to crops); economical (working time, vehicle mileages…) and organizational (frequency and temporal distribution of spreading actions). MAGMA has been used to analyze waste management policies in livestock farms in Reunion Island, such as that described in [START_REF] Paillat | Simulation de stratégies de gestion de matières organiques avec le modèle Magma. Application au cas d'une exploitation type[END_REF].

APPROZUT: Supply of treatment plant by multiple farms

The APPROZUT model [START_REF] Guerrin | Simulation of stock control policies in a two-stage production system. Application to pig slurry management involving multiple farms[END_REF] deals with the case of simulating a two-stage production system where the first stage is a set of pig farms producing slurry scattered over a territory and the second is a unique collective treatment plant where slurry is brought in a many-to-one fashion (Fig. 3). Policy assessment is mainly done in terms of organization and logistics. Approzut has been used to analyze a project of pig slurry treatment involving 51 pig farms located in a remote mountainous cirque in Reunion Island where available agricultural land was too scarce to spread raw slurry.

COMET: Mixed distribution and supply configuration

COMET [START_REF] Paillat | Simulation de la faisabilité d'un plan d'épandage de lisier de porc et conséquences sur les émissions gazeuses au stockage et à l'épandage[END_REF] essentially results from coupling together the MAGMA and APPROZUT logistic models yielding the distribution/supply configuration displayed on Fig. 4. It also includes sub-models simulating biophysical processes used as environmental assessment criteria (e.g. the STAL model [START_REF] Morvan | Vers une prévision opérationnelle des flux de N résultant de l'épandage de lisier : paramétrage d'un modèle dynamique de simulation des transformations de l'azote des lisiers (STAL)[END_REF], which simulates ammonia emissions at spreading).

... COMET has been used to jointly simulate individual manure spreading within single pig farms and the functioning of a collective spreading plan aimed at transferring manure surpluses from livestock farms to land loaned in remote crop farms in Brittany (Western France). The alternate use of dynamic simulation with COMET and static life-cycle analysis allowed the whole functioning of this case-study to be thoroughly assessed [START_REF] Guerrin | Combining individual and collective management of animal manure to reduce environmental impacts on a territory scale[END_REF].

A CONCEPTUAL SHIFT: FROM PLANNED ACTION TO SITUATED ACTION

The confrontation of action representation in the flow management models based on the Action-Flow-Stock ontology [START_REF] Guerrin | Modelling agricultural production systems using an action-flow-stock ontology[END_REF] with the ontology of agricultural production systems devised by Martin-Clouaire and Rellier [START_REF] Martin-Clouaire | Modelling and simulating work practices in agriculture[END_REF] led to question the paradigm of 'planned action' in favor of the theory of 'situated action' [START_REF] Suchman | Plans and situated actions: The problem of human-machine communication[END_REF].

The management problems at the operational level are, indeed, typically formulated in terms of planning and decision. This is the very Western conception that actions necessarily result from deliberations made with representations (plans) to decide in response to previously established intentions. The study of many domains, however, shows that a very large part of human activity is non-deliberative or, even, reactive in nature; it takes place in interaction with the local situations in which each agent is involved [3][14]. Therefore, the theory of "situated action" alleges there is no need of representing explicitly the activity to be performed; plans, although they may be used to guide action, never determine it completely.

The action modeling framework already drafted [START_REF] Guerrin | Dynamic simulation of action at operations level[END_REF] has the ambition to contribute to this situated action theory. The first reason is the construction of models. If the goal is to represent detail of agricultural, large and complex production systems, basing any action on a comprehensive and coherent plan appears elusive, due to the complexity of planning itself. This challenge is also justified from a theoretical point of view, except to enter an infinite recursion loop: if any action is planned, then so is planning, and planning for planning also, and so on... The other reason is linked with the usefulness of the models. If the objective is to evaluate production systems, it is by representing as better as possible what is actually done, and not what should be done (i.e. tasks specified by the plan), that can allow the impacts of activity to be measured and, in turn, the mutual influence of the context, thus modified, on the activity itself to be appreciated. Taking an a priori defined plan of action as essential determining factor would be similar to taking a static referent in an inherently dynamic environment to generate a process which is, also, dynamic.

In contrast, taking action as a focal point, the present approach is designed to meet Checkland's wishes: "...modelling purposeful human activity systems as sets of linked activities which together could exhibit the emergent property of purposefulness" [START_REF] Checkland | Soft systems methodology: A 30-year retrospective[END_REF].

It is, hence, the operational level that must the models represent in being primarily focused on action rather than on decision and planning. However, it is at the strategic level that these models must be used to assist researchers in experimenting the systems and, possibly, stakeholders in their decision processes, in keeping with Mc Cown's view [START_REF] Mccown | Changing systems for supporting farmers' decisions: problems, paradigms and prospects[END_REF]. In other words, if the model must represent the action of virtual agents at the operational level, its use must contribute to the decision-making of real actors at the strategic level. These are currently the main research objectives:

• Develop an ontology for representing systems of activities at the operational level by a minimum and consistent set of concepts; • Formalize this framework to build simulation models of agricultural production systems; • Analyze with these models these systems operation viewed as the interaction between biophysical processes and human activities; • Infer practical lessons to help manage these systems.

In this perspective, the concepts relevant to describing the coordination between actors, the spatial location of activities, the physical structure of the work setting and the relationship between the concepts of agent and action shall be specified. This is part of an ongoing PhD thesis project supervised by the author.

THE ISSUE OF ASSESSING FARMING SYSTEM MANAGEMENT

Any management requires the assessment of the system's performance it relates to. The comparison of management policies, so far, was based only on a few indicators calculated by the flow models: agronomic (e.g. nitrogen applied relatively to crop needs), environmental (e.g. nitrogen excess, ammonia and methane emissions), economic (e.g. working time, distance traveled by vehicles) or organizational (e.g. temporal distribution of activity, robustness to disturbances). These indicators take into account only two dimensions: technical, measured in terms of efficiency, and environmental, measured in terms of risk, taking nitrogen as main criterion. The technical dimension assesses the system at the level where it is represented. If it qualifies its viability in the short term, it does only little in appreciation of its contribution to sustainable development in the long term. The environmental dimension concerns the system outputs on a scale that encompasses it immediately (i.e. the impact on its immediate environment). Environmental risk is addressed only as "hazard" (occurrence of a risk factor) and ignores the sensitivity and the particular nature of the receiving environment. In both cases, the assessment is performed with a normative view.

To address these problems, we must distinguish between two questions:

• How to evaluate the technical performance of production systems? • How to evaluate their actual or potential environmental impacts?

In the first case, modeling biophysical flow is needed to simulate their interactions with the workable flows. This does not imply to represent all mechanisms in detail but, at least, to have a robust approximation of their evolution. To do this, the knowledge on the biophysical processes is synthesized by expressions for linking, as simply as possible, the causes and effects without going into the details of the underlying mechanisms (cf. Section 3). In the second case, comparing different management strategies is needed. The issue of sustainable development, which has become the essential assessment criterion, leads now to think the impacts of these systems in terms of risk (proven or alleged) on other time and space scales (often larger) than the ones on which they were previously considered. Hence, the interest in overall assessment approaches ("from cradle to grave"), such as life cycle analysis (LCA), which allows this comparison (although statically) through standardized indicators representative of different categories of impacts. An example of alternatively combining LCA with simulation modeling in a comprehensive approach to assess and help improve the design of a collective manure management plan by a group of farmers has been realized recently [12] [15].

These preliminary results are far, however, from exhausting the subject of environmental assessment which deserves to be rethought in the light of the objectives: what has to be assessed, for which purposes, with which actors? The goal of assessing the sustainability of farming systems striving to adapt to multiple change factors requires also defining the relevant space and time scales to be accounted for. The choice of the 'scale of representation' of a production system becomes, thus, a central issue for modeling, along with the methods of up-or downscaling the current models as soon as an extension or reduction of scope necessary to embrace larger or finer scales is sought. This questioning is a research perspective.

THE ISSUE OF USING MODELS FOR MANAGEMENT SUPPORT

The main question is: How to use simulation models to help stakeholders evaluate and design management strategies of production systems? This issue calls to other more specific questions related to:

• The ways of using the models: Which users? What situations? What modes of interaction? • The engineering of simulation likely to facilitate users' learning: Which cases to simulate? What scenarios? Which protocol? How to capitalize the knowledge gained through simulations?

Dealing with these questions was first attempted in the period 2004-2007, unfortunately with too little achievements. If a first experience of participatory simulation had been made to assist in the choice of the treatment process for pig manure in the locality of Grand Ilet in Reunion [START_REF] Médoc | A multi-modelling approach to help agricultural stakeholders design animal wastes management strategies in the Reunion Island[END_REF], it was using a GIS and a spreadsheet model developed by fellow agronomists. The dynamic simulation models listed in Section 1, although quite used by these colleagues, have not yet been tested truly to design management strategies with "real" agricultural actors. When it could have been the case, actually, the projects aborted prematurely for unexpected reasons: in Grand Ilet (with APPROZUT), the actionresearch dynamics that had been initially launched by researchers was interrupted once the folder had been assigned to one of the institutional partners; in Brittany (with COM-ET), the collective manure application project was stopped due to the opposition of residents, not accounted for in the model...! The phase shift between the researchers' and the actors' time explains, in part, this state of affairs. However, deeper causes must also be sought in our inability to correctly grasp the social games of players in these organizational or political processes. In these contexts, beyond a purely technical rationality, one might ask if actual decision still requires the support of a model. It seems not.

Nevertheless, the work with the MAGMA [START_REF] Paillat | Simulation de stratégies de gestion de matières organiques avec le modèle Magma. Application au cas d'une exploitation type[END_REF], APPROZUT [START_REF] Guerrin | A simulation approach to evaluate supply policies of a pig slurry treatment plant by multiple farms[END_REF] and COMET [START_REF] Guerrin | Combining individual and collective management of animal manure to reduce environmental impacts on a territory scale[END_REF] models allowed the way for a simulation approach to design management policies of production systems to be paved. The protocol was designed with an experimental logic: (i) construction of a base scenario corresponding to the current situation, (ii) assessment and analysis of the scenario through simulation, (iii) introduction of gradual changes for designing iteratively new scenarios. This dimming of the changes introduced in the simulation scenarios corresponds, from the point of view of operations management, to challenging firstly very short-term operational choices, then medium-term tactical decisions, and, finally, longer-term strategic decisions. The objective of this approach is not only to understand why farmers do what they do, but, above all, to understand their rooms for maneuver.

The production of documents allowing the user's approach to be represented and the knowledge gained by simulation to be capitalized is, for now, manually performed in a paper form. Using more sophisticated tools (e.g. mind maps, concept maps) to better organize this multimedia information (texts, graphics, data, etc.) should be considered in relation with the model users. If the simulation of actual cases of farms is interesting in view of advising individual farmers, reasoning on farm types can be useful for the purpose of supporting agricultural advisors or professional and public policy-makers to develop general scope alternatives at a micro-regional level. However, a too short experience in trying to elucidate the place of models in a decision process and finding the way to capitalize the knowledge gained from simulations has led, eventually, to consider cooperating with "real" researchers in management science, ergonomics or knowledge engineering to tackle these issues that are far from trivial.

SUMMARY

A way to improve the sustainability of agriculture is to design new management policies of agricultural production systems based on the integration of heterogeneous knowledge on their biophysical and human components. Simulation models, representing those systems as productive units spread over a territory, have been designed to assess these systems performance against agronomical and environmental criteria and, so doing, help design new management policies. Beyond the various simulation models realized to date, the main achievements were pointed out: a comprehensive approach and a conceptual framework for modeling and analyzing material flows on a territory scale; the challenge of the 'situated action' theory to represent human action in farming systems. Two incompletely resolved issues were also pinpointed: assessing the impacts of management policies at various scales and setting the practical ways to use simulation modeling with agricultural stakeholders. The research avenues that are thought of were also underlined: complete a generic modeling framework of human activity in agricultural systems, namely, by introducing the spatial dimension of action in addition to its temporal one; decide on the relevant temporal and spatial scales for assessing the sustainability of these systems and the related representational scale of the models used to simulate them; find the practical ways to use the models with agricultural stakeholders in decision making and capitalize the knowledge gained from practicing simulation.

Figure 1 .

 1 Figure 1. Action-Flow-Stock representation of a production unit (PU).

FarmFigure 2 .

 2 Figure 2. Distribution configuration in the MAGMA model to represent organic waste management within farms (PU: livestock enterprise; CU: consumption unit).

Figure 3 .

 3 Figure 3. Supply configuration in the APPROZUT model (PU: livestock farm; CU: single waste treatment plant).

Figure 4 .

 4 Figure 4. Mixed distribution/supply configuration in the COMET model (PU: livestock farm encompassing also crops; CUs: intermediate storage; CU: crop farm).

  DeMset 2011, International Conference on Design and Modeling in Science, Education, and Technology, Nov. 29th-Dec. 2nd, 2011 -Orlando, FL, USA