O Bernard

S Madani

A Sciandra

MODELLING THE RESPONSE OF THE MARINE MICRO ALGAE EMILIANIA HUXLEYI TO AN ELEVATION OF pCO 2

Keywords: coccolithophore, photosynthesis, calcification, model, pCO 2

Coccolithophorids are marine microalgae producing micro calcite plates. Under certain circonstances these microorganisms can form massive blooms which can be detected by satellites. These phytoplanktonic organisms play an important role in atmospheric CO 2 trapping through large scale precipitations of calcium carbonate in the oceans. However, recent experiments revealed that the associated fluxes that represent a sink of carbon may be slow down by an increase in atmospheric CO 2 concentration. In this paper we design models to account for the decrease in calcification and photosynthesis rates observed after an increase of pCO 2 in Emiliania huxleyi continuous photobioreactors. Since the involved mechanisms are still not completely clear, we developed several models, each of them based on a different hypothesis. These models are kept at a very general level, by maintaining the growth and calcification functions in a generic form, i.e. independent on the shape of these functions and on parameters values. The analysis is thus performed using these generic functions were the only hypothesis is an increase of these rates with respect to the regulating carbon species. As a result, each model responds differently to a pCO 2 elevation. Surprisingly, the only model whose behaviour is in agreement with the experimental results corresponds to a coupling of photosynthesis and calcification with carbonate as the regulating species, whereas bicarbonate is the substrate of these processes.

Introduction

Coccolithophorids are both autotrophic and calcifying phytoplanktonic species playing a major role in global biogeochemical cycling of elements. Emiliania huxleyi is an ubiquitous coccolithophorid (see Figure 1) that forms very spread and dense blooms detectable from space by satellites (see Figure 2), due to the highly scattering calcareous plates produced during growth.

Photosynthesis and calcification are biological processes which transform dissolved inorganic carbon (DIC) into respectively particulate organic and inorganic matter which, being denser than seawater, sink towards the oceanic floor. The export of particulate organic carbon (POC) from the surface oceanic layers, insured by the biological pump, constitutes a sink of carbon on a geological scale [START_REF] Falkowski | Evolution of the nitrogen cycle and its influence on the biological sequestration of co2 in the ocean[END_REF]:

6CO 2 + 12H 2 O -→ C 6 H 12 O 6 + 6O 2 + 6H 2 0 (1)
The DIC precipitation into particulate inorganic carbon (PIC), and more specifically into calcium carbonate, also constitutes a sink of carbon, even if calcification produces CO 2 :

Ca + 2 + 2HCO - 3 -→ CaCO 3 + CO 2 + H 2 O (2)
Being responsible for almost 70% of the biogenic carbonate precipitation in the oceans, these communities have been studied in the context of the predicted increase of CO 2 in the atmosphere resulting from the anthropogenic activity [START_REF] Houghton | The Science of Climate Change[END_REF]. Based on the accumulation rate of CO 2 observed in the atmosphere from the beginning of the industrial era, current models roughly predict a doubling of the partial pressure of the atmospheric CO 2 (pCO 2 ) at the end of the 21th century. Since the atmosphere tends to be in equilibrium with the superficial oceanic layers, the change in atmospheric CO 2 will have a direct effect on the CO 2 seawater concentration, and consequently on the carbonate system speciation. For example, [START_REF] Zeebe | CO 2 in seawater: equilibrium, kinetics, isotopes[END_REF] have calculated that, if the pCO 2 was doubled in the seawater, the concentration of carbonate should be reduced by 50 % approximately, and the pH by 0.35 units. The consequences are not only on the chemical properties but also on the biological processes. The reductions mentioned above, by diminishing the buffer capacity of sea water, will enhance the production of CO 2 via calcification. Under these conditions, the expected increase of pCO 2 for the next decades would exert a positive feedback on the production of CO 2 by the calcifying communities [START_REF] Frankignoulle | Marine calcification as a source of carbon dioxide: positive feedback of increasing atmospheric co2[END_REF]. Nevertheless, recent studies have shown that cellular calcification could be depressed consequently to a rise of pCO 2 in seawater. For example, by increasing pCO 2 in phytoplankton cultures by the means of strong acid additions, [START_REF] Riebesell | Reduced calcification of marine plankton in response to increased atmospheric co2[END_REF][START_REF] Zondervan | Effect of co2 concentration on the pic/poc ratio in the coccolithophore emiliania huxleyi grown under light-limiting conditions and different daylengths[END_REF] induced a significant decrease of the calcification rate and of the calcification/photosynthesis ratio in Gephyrocapsa oceanica, and, with a less extend, in Emiliania huxleyi. Therefore, an increase in atmospheric pCO 2 can have antagonistic effects on the production of CO 2 by calcification, so that consequences on the long term are difficult to predict. The objective of this paper is to derive a set of models to explain the carbon fixation mechanisms and to reproduce chemostat experiments published in [START_REF] Sciandra | Response of coccolithophorid emiliania huxleyi to elevated partial pressure of co2 under nitrogen limitation[END_REF] where pCO 2 was doubled to simulate the future evolution of atmospheric CO 2 . 2 Model design

Introduction

We have considered 2 classes of models, which suppose respectively that photosynthesis and calcification are either coupled (class CI) or uncoupled (class UI). There is a consensus to admit that HCO - 3 is the substrate of calcification, and that reaction (2) produces a CO 2 which can then be used for photosynthesis together with the HCO - 3 and/or CO 2 directly uptaken by the cell. In the proposed series of models we investigated several possibilities for the regulation of these processes. Indeed, we assume that the regulation is not necessarily driven by the substrate availability, but that it can result from another species concentration, among HCO - 3 , CO 2 and CO 2-

3

. Therefore, we examined the possibility that, for each class of models (CI or UI) these species can be regulating photosynthesis or calcification.

Name Meaning

Unit S 1 nitrogen mmol.L -1 D dissolved inorganic carbon (CO 2 ,HCO - 3 ,CO 2- 3 ) mmol.L -1 N particulate nitrogen mmol.L -1 Q internal quota of nitrate per unit of biomass / X biomass (organic carbon concentration) mmol.L -1 C calcium carbonate mmol.L -1 S 2 Calcium mmol.L -1
Table 1: Meaning of the state variables whose evolution is described by each of the 12 proposed models.

The 12 models use the same state variables (Table 1), which makes their comparison possible. The algal biomass in the chemostat (X) is represented by the POC concentration together with the particulate nitrogen concentration (N). The concentration of nitrate (N O 3 ) in the chemostat (S 1 ) is explicitly taken into account since it is the limiting nutrient for growth. The cellular quota of nitrogen (Q), is thus expressed as the N/X ratio. The dissolved inorganic carbon and calcium (Ca 2+ ) concentrations are denoted D and S 2 respectively. Finally, C is the PIC concentration, i.e. the calcite contained in attached as well as free coccoliths. The concentrations in the chemostat renewal medium are denoted with the subscript in .

The carbonate system

The carbonate alkalinity (CA) represents the electric charges carried in the carbonate system:

CA = [HCO - 3 ] + 2[CO 2- 3 ] (3)
The total alkalinity (TA) is defined by [START_REF] Zeebe | CO 2 in seawater: equilibrium, kinetics, isotopes[END_REF]:

T A = CA + [B(OH) - 4 ] + [OH -] -[H + ] (4)
Let us denote λ = T A -2S 2 . In a first approximation, the ions mainly contributing to λ depend on the salinity and remain constant. In the same way, we assume that

λ 0 = [B(OH) - 4 ] + [OH -] -[H + ] remains constant compared to CA. Finally we have CA = λ -λ 0 + 2S 2 .
In order to compute the [HCO - 3 ] and [CO 2- 3 ] concentrations, we use the dissociation constants of the carbon dioxide (K 1 ) and bicarbonate (K 2 ) (the proton concentration, [H + ] is denoted h):

K 1 = h[HCO - 3 ] [CO 2 ]
and

K 2 = h[CO 2- 3 ] [HCO - 3 ] (5)
The total dissolved inorganic carbon (D ) is defined as:

D = [HCO - 3 ] + [CO 2- 3 ] + [CO 2 ] (6)
Note that, in the considered pH range, we have

[HCO - 3 ] >> [CO 2- 3 ] >> [CO 2 ]
(see for example [START_REF] Zeebe | CO 2 in seawater: equilibrium, kinetics, isotopes[END_REF]). It follows that, in a first approximation, bicarbonate is the main carbon species in the bicarbonate system: We deduced from equations ( 3) and ( 6), in the considered pH range:

[HCO - 3 ] D (7
[CO 2- 3 ] CA -D = λ -λ 0 + 2S 2 -D (8)

The coupled models

In the CI class models, photosynthesis and calcification are coupled so that the carbon assimilation for the coccoliths (C) and for the biomass (X) can be represented as follows:

1-α α S 2 + 1 α D µ(.)X -→ 1-α α C + X (9)
where µ(.) represents the specific algal growth rate. The coupling between photosynthesis and calcification induces the same expression µ(.)X for the kinetics of these two processes. The constant α represents the proportion of the total up taken DIC which is allocated to photosynthesis.

Following the quota model viewpoint [START_REF] Droop | Vitamin B12 and marine ecology. IV. the kinetics of uptake growth and inhibition in Monochrysis lutheri[END_REF][START_REF] Burmaster | The unsteady continuous culture of phosphate-limited Monochrysis lutheri Droop: experimental and theoritical analysis[END_REF][START_REF] Droop | 25 years of algal growth kinetics, a personal view[END_REF], we consider that growth and external N O 3 (S 1 ) uptake are uncoupled. The nitrogen mass transfer is thus given by:.

S 1 ρ(.)X -→ N ( 10
)
where N is the algal nitrogen and ρ(.) the external N O 3 uptake rate.

We can now straightforwardly (see e.g. [START_REF] Bastin | On-line estimation and adaptive control of bioreactors[END_REF]) derive from the mass transfer equations ( 9) and ( 10) the mass balance based model (d is the dilution rate):

Ṡ1 = d(S 1in -S 1 ) -ρ(S 1 )X (11) Q = ρ(S 1 ) -µ(Q, Dp)Q (12) Ẋ = -dX + µ(Q, Dp)X (13)
Ċ = -dC + 1 -α α µ(Q, Dp)X (14)
Ḋ = d(D in -D) - 1 α µ(Q, Dp)X -K L a(ψ(S 2 , D) -K H pCO 2 ) (15) Ṡ2 = d(S 2in -S 2 ) - 1 -α α µ(Q, Dp)X ( 16
)
where CO 2 = ψ(S 2 , D) is derived from equations ( 5), ( 8) and [START_REF] Houghton | The Science of Climate Change[END_REF].

Hypothesis 1

We assume that ρ(S 1 ) is an increasing function of the N O 3 concentration (S 1 ), and that µ(Q, Dp) is an increasing function of both the internal nitrogen quota (Q) and the DIC species (Dp) selected for its role in the regulation of growth and calcification.

It must be emphasised, at this level, that the qualitative analysis to determine the qualitative response to a pCO 2 increase, remains valid for any functions ρ(S 1 ) and µ(Q, Dp) with the considered qualitative properties.

The 3 models of the CI class, named CI-HCO - 3 , CI-CO 2 , CI-CO 2- 3 differ only by the inorganic carbon component Dp controlling photosynthesis and calcification.

The uncoupled models

This class of models supposes that calcification and photosynthesis are uncoupled, so that incorporation of inorganic carbon follows two different metabolic pathways: where µ(.) and σ(.) are the rate of photosynthesis and calcification, respectively. The process of nitrogen uptake is still assumed to be independent from carbon assimilation and follows thus reaction 10.

D µ(.)X -→ X S 2 + D σ(.)X -→ C (17)
The following equations gives the structure adopted for the UI class models whatever the considered regulating species for photosynthesis (Dp) and calcification (Dc):

Ṡ1 = d(S 1in -S 1 ) -ρ(S 1 )X (18) Q = ρ(S 1 ) -µ(Q, Dp)Q (19) Ẋ = -dX + µ(Q, Dp)X (20) Ċ = -dC + σ(Dc)X (21) Ḋ = d(D in -D) -µ(Q, Dp)X -σ(Dc)X -K L a(ψ(S 2 , D) -K H pCO 2 ) (22) Ṡ2 = d(S 2in -S 2 ) -σ(Dc)X (23) CI UI E HCO - 3 CO 2 CO 2- 3 HCO - 3 HCO - 3 HCO - 3 CO 2 CO 2 CO 2 CO 2- 3 CO 2- 3 CO 2- 3 X HCO - 3 CO 2 CO 2- 3 HCO - 3 CO 2 CO 2- 3 HCO - 3 CO 2 CO 2- 3 P D * Dp * n.m. Dc * n.m. X * Q * S * 1 n.m. C * S * 2 n.m. C * X * → → → pH * TA Table 2:
Qualitative variations of the state variables at steady-state after an elevation of pCO 2 for the 12 considered models, and for the experimental results of [START_REF] Sciandra | Response of coccolithophorid emiliania huxleyi to elevated partial pressure of co2 under nitrogen limitation[END_REF] (n.m.: not measured).

There are 9 possible combinations of Dp and Dc for this equation system, depending on the regulating carbon species for calcification and photosynthesis among CO 2 , HCO -

Study of the qualitative response of the models set

In order to determine which regulation mechanism is the most appropriate, we assess the qualitative model consistency with respect to experimental observations. We thus study the qualitative behaviour for both classes of models (coupled or uncoupled) and for all the considered potential regulating species. This analysis has been performed in the framework of general (increasing) functions µ(.), ρ(.) and σ(.). It is thus independent of the parameters values (except in some particular cases) and provide us with a diagnosis on the model adequacy. This analysis is performed at steady-state. It mainly consists in determining the evolution of the equilibrium of the model when the CO 2 partial pressure is increased. In other term, if x * i is a state variable of a given model at equilibrium, we study the sign of

∂x * i ∂pCO2 .
For sake of place limitation we only present the result of this analysis. The qualitative behaviour for each state variable in response to increased pCO 2 is presented in Table 2, depending on the carbon compounds driving the process. These results are also compared to the experiments outcomes from [START_REF] Sciandra | Response of coccolithophorid emiliania huxleyi to elevated partial pressure of co2 under nitrogen limitation[END_REF].

Numerical simulations

The parameters values and units are presented in Table 3. The computations were performed over 45 simulated days. The first 25 days were necessary to ensure that a steady-state had been reached for each variable. The partial pressure of carbon dioxide (pCO 2 ) was then instantly doubled (340 µatm to 580 µatm), with increase of bubbling (corresponding to a strong increase in K L a) during 30 minutes. and the numerical behaviour of the system was observed for a further 20 days. In Figure 3 model simulations are compared to experimental data for models CI-CO 2- 3 and UI-CO 2- 3 -HCO - 3 .

Discussion

The first outcome of this study is that the models where HCO - 3 would straightforwardly drive photosynthesis and calcification (model CI-HCO - 3 or UI-HCO - 3 -HCO - 3 ) predict increase of P OC and P IC as well as calcification and photosynthesis rates; all these are contradicted by the experiments. In the same way, model UI-CO 2 -HCO - 3 predicts a contradictory response. Indeed, all the models where only species CO 2 and/or HCO - 3 are involved have predictions that are opposite to experimental evidence. It is noteworthy that among these 6 models, the 2 CI models have exactly the same qualitative behaviour, and the same for the 4 UI models which behave identically. The difference between these 2 groups is their C/X predictions (constant for the CI models and increasing for the UI models). It follows thus that CO 2- 3 is required in the regulation process to explain the experimental observations. Actually, this result can be explained much more simply by a simple argue based on the growth rate. We base this reasoning on the assumption that the growth rate (or photosynthesis rate) is an increasing function of both the internal quota and a controlling inorganic carbon compound. The experimentally observed biomass decrease induced by the shift of pCO 2 leads to an increase in the internal quota (particulate nitrogen stays constant At this point, note that if pH is considered as a possible controlling factor, it can also induce the same conclusion. However pH has rarely a direct effect and often impacts the physico chemical equilibria. This is the reason why pH was not investigated explicitly in our model, but we considered the variables that may more directly affect the processes.

Among the considered model, only the models where CO 2- 3 regulates photosynthesis are compatible with the experimental decrease of POC and PIC. The 4 remaining models only differ by their predictions of the ratio PIC over POC. In fact models UI-CO 2- 3 -HCO - 3 and UI-CO 2- 3 -CO 2 have the same behaviour (C/X increases). These quantities are not affected by the pCO 2 change for model UI-CO 2- 3 while they are of opposite trend for model UI-CO 2- 3 -CO 2- 3 (C/X decreases). However, due to the noise in the data, the slight experimental increase of the PIC over POC ratio is not significant and cannot be used to select one of the 4 models. Note however that if another experiment could measure a significant increase of this ratio, the most appropriate model would be UI-CO

Conclusion

In this study we have investigated the coupled effect of a nitrogen limitation and of a pCO 2 increase. Based on several possible regulation mechanisms we have proposed 12 models. These models are generic since they only rely on qualitative hypotheses (except for quantitative simulations were some functions were chosen). The only possible set of models that can explain both a decrease of calcification and of photosynthesis are models were photosynthesis is regulated by CO 2- 3 (or pH). This seems to contradict the classical hypotheses, even if they were stated on the basis of nitrogen unlimited cultures. These unexpected results may also question the experimental protocols where computation of dissolved inorganic carbon is made using classical methods (see e.g. [START_REF] Zeebe | CO 2 in seawater: equilibrium, kinetics, isotopes[END_REF]) under the hypotheses of equilibrium with atmospheric CO 2 which is not a valid approximation for chemostat cultures with high biomass concentration.

Figure 1 :

 1 Figure 1: Microscopic view of Emiliania huxleyi (×10000).

Figure 2 :

 2 Figure 2: Bloom of E.huxleyi detected from satellite (south of Cornwall, UK)

)

  Proceedings 5th MATHMOD Vienna, February 2006 (I.Troch, F.Breitenecker, eds.) Mathematical Modelling and Control of Chemical and Bio-chemical Processes 3 -3

  Proceedings 5th MATHMOD Vienna, February 2006 (I.Troch, F.Breitenecker, eds.) Mathematical Modelling and Control of Chemical and Bio-chemical Processes 3 -4

3 and CO 2 - 3 .

 23 For example, in the model denoted UI-HCO - 3 -CO 2 . Dp = [HCO - 3 ] and Dc = [CO 2 ]

1 )Figure 3 :

 13 Figure 3: Comparison between model CI-CO 2- 3 (-), UI-CO 2- 3 -HCO - 3 (. . . ) and data for POC (X), PIC (C) Nitrate (S 1 ) and TA. Organic carbon is simultaneously estimated by biovolume (light grey points).

1 )Figure 4 :

 14 Figure 4: Comparison between model CI-CO 2- 3 (-), UI-CO 2- 3 -HCO - 3 (. . . ) and data for the DIC species.

Table 3 :

 3 Estimates of the model parameters for model UI-CO 2- 3 -HCO -

	3

Table 4 :

 4 Parameters used to fit the model to the data if we keep in mind that, under chemostat steady state, the growth rate is constant (and equals d), this implies that the increase in internal quota must be compensated by a decrease of the controlling variable in order to maintain a constant growth rate. The only variable to decrease in response to the shift in the bicarbonate system is CO 2- 3 , explaining thus why only this model can provide an adequation with the experiments.

Proceedings 5th MATHMOD Vienna, February 2006 (I.Troch, F.Breitenecker, eds.) Mathematical Modelling and Control of Chemical and Bio-chemical Processes

Proceedings 5th MATHMOD Vienna, February 2006 (I.Troch, F.Breitenecker, eds.) Mathematical Modelling and Control of Chemical and Bio-chemical Processes 3 -9


Alexandre Bazin

Miguel Couceiro

Marie-Dominique Devignes

Amedeo Napoli

Steps Towards Causal Formal Concept Analysis

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The study of causal relations is central in applied science [START_REF] Kleinberg | Why: A guide to finding and using causes[END_REF]. Experimental protocols are often implemented in order to manipulate a potential cause (an object, state or process) in such a way that its effects can be inferred. Such a process produces what is called interventional data. However, when objects cannot be directly manipulated, one has to rely on purely observational data. Studying causality from data involves two related tasks:

discovering the causal structure, i.e., is there a causal relation between two sets of variables? inferring causal effects, i.e., how does the cause lead to the effect?

In this paper, we are interested in the discovery of the causal structure in observational data.

Inferring causal relations from observational data is a challenging task. Most of the approaches in the literature are probabilistic [START_REF] Hoyer | Nonlinear causal discovery with additive noise models[END_REF][START_REF] Janzing | Information-geometric approach to inferring causal directions[END_REF][START_REF] Joris M Mooij | Distinguishing cause from effect using observational data: methods and benchmarks[END_REF][START_REF] Peters | Causal discovery with continuous additive noise models[END_REF][START_REF] Sgouritsa | Inference of cause and effect with unsupervised inverse regression[END_REF]. Nonetheless, recent algorithmic approaches based on approximations of Kolmogorov complexity are becoming quite popular [START_REF] Budhathoki | MDL for causal inference on discrete data[END_REF][START_REF] Budhathoki | Origo: causal inference by compression[END_REF][START_REF] Blöbaum | Cause-effect inference by comparing regression errors[END_REF][START_REF] Marx | Identifiability of cause and effect using regularized regression[END_REF][START_REF] Osman Mian | Discovering fully oriented causal networks[END_REF]. The set of causal relations forms the causal structure. It can be presented to humans to help them understand a situation described by data and support decision making [START_REF] Eells | Rational decision and causality[END_REF] or it can be used to automatically select important variables in data [START_REF] Yu | Causality-based feature selection: Methods and evaluations[END_REF]. Most works on causality focus on univariate causal inference where the task consists in determining whether a causal relation exists between two variables x and y, and deciding its direction, i.e., the cause and the effect. The causal structure is then represented by a directed acyclic graph called the causal diagram.

In practice, however, effects often require the interaction of multiple causes to appear. We then talk of multivariate causal relations. In the multivariate case, we allow causal relations between sets X and Y of variables. We will use X c -→ Y to denote the fact that X causes Y . The corresponding causal diagrams have yet to be properly formalised and studied. In this paper we propose an approach based on Formal Concept Analysis to address and tackle this issue.

Formal Concept Analysis (FCA) [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF] is a lattice-theoretic mathematical framework that enables the representation of the underlying structure of data. Originally, FCA is applied to binary data, but several extensions have been proposed to manage other data types, e.g., numerical [START_REF] Ganter | Pattern structures and their projections[END_REF], fuzzy [START_REF] Bȇlohlávek | Concept lattices and order in fuzzy logic[END_REF], incomplete [START_REF] Li | Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction[END_REF], multidimensional [START_REF] Voutsadakis | Polyadic concept analysis[END_REF], relational [START_REF] Rouane-Hacene | Relational concept analysis: mining concept lattices from multi-relational data[END_REF] and sequential [START_REF] Buzmakov | On mining complex sequential data by means of fca and pattern structures[END_REF] data. FCA is particularly suited to the discovery and representation of rule patterns such as implications [START_REF] Ryssel | Fast algorithms for implication bases and attribute exploration using proper premises[END_REF][START_REF] Manuel Rodríguez-Jiménez | Negative attributes and implications in formal concept analysis[END_REF], association and decision rules [START_REF] Pasquier | Mining association rules using formal concept analysis[END_REF][START_REF] Zhai | Decision implication canonical basis: a logical perspective[END_REF], and link keys [START_REF] Atencia | Link key candidate extraction with relational concept analysis[END_REF].

In this paper, we propose an FCA-based framework that relies on causal inference to represent the causal structure of the variables in datasets. In Section 2, we briefly survey existing causal inference methods and recall basic FCA notions needed throughout the paper. We discuss properties of causal relations in Section 3, and use them to define a closure operator that will be the key to the proposed framework. In Section 4, we show how FCA structures can be used to represent causal structures, and provide algorithms for computing them. In Section 5, we illustrate our approach on the well-known public dataset Iris. We end the paper with a discussion on the usefulness of the framework for computing and representing causal relations in Section 6.

Preliminaries

Notation

Throughout the paper, we use calligraphic letters (A, B, . . .) to denote important sets and structures, whereas (sub)sets and individual elements are denoted by upper case letters (A, B, . . .) and by lower case letters (a, b, . . .), respectively. We may specify a set extensively, e.g., {a, b, c, d, e}, or in the simplified form abcde, especially to improve readability of figures.

Causal Inference

A causal relation between a single cause x and a single effect y is called univariate. The structure of a set of such relations is usually represented by a directed acyclic graph (DAG), called a causal diagram or causal network, in which the vertices are variables and the edges denote a causal relation between the cause and its effect. Recent efforts have been devoted to learning a causal diagram [START_REF] Glymour | Review of causal discovery methods based on graphical models[END_REF] from observational data, whether it be the complete diagram [START_REF] Spirtes | Constructing bayesian network models of gene expression networks from microarray data[END_REF][START_REF] Spirtes | Causation, prediction, and search[END_REF] or just parts of it [START_REF] Gao | Local causal discovery of direct causes and effects[END_REF]. Some approaches try to learn the skeleton of the diagram (is there a causal relation between these two variables?) first, and then to orient the edges (which of these variables is the cause?).

Inferring the direction of a causal relation between x and y is a difficult task, and authors usually rely on some assumptions. The most common one is the absence of confounders [START_REF] Janzing | Causal inference using the algorithmic markov condition[END_REF][START_REF] Sun | Causal reasoning by evaluating the complexity of conditional densities with kernel methods[END_REF][START_REF] Zhang | On the identifiability of the post-nonlinear causal model[END_REF], i.e., variables z that cause both x and y. Also, many current approaches are based on additive noise models [START_REF] Hoyer | Nonlinear causal discovery with additive noise models[END_REF][START_REF] Hu | Causal inference and mechanism clustering of a mixture of additive noise models[END_REF], which assume that, if x causes y, then the value of y is a function of the value of x plus some additive noise: y = f (x) + n, for some function f, and where n is independent of x. Under this assumption, state of the art methodologies [START_REF] Hoyer | Nonlinear causal discovery with additive noise models[END_REF][START_REF] Janzing | Information-geometric approach to inferring causal directions[END_REF][START_REF] Joris M Mooij | Distinguishing cause from effect using observational data: methods and benchmarks[END_REF][START_REF] Peters | Causal discovery with continuous additive noise models[END_REF][START_REF] Sgouritsa | Inference of cause and effect with unsupervised inverse regression[END_REF] assume that the marginal distribution P (x) of the cause x and the conditional distribution P (y|x) of the effect y given the cause x are independent.

In contrast, algorithmic approaches are based on the Markov condition [START_REF] Janzing | Causal inference using the algorithmic markov condition[END_REF] and the idea that it is algorithmically easier to compute the effect from the cause than the other way around. The Kolmogorov complexity K(s) of a finite binary string s is the length of the shortest program for which a universal Turing machine outputs s and halts. In this setting, x is said to cause y when K(P (x))+ K(P (y|x)) << K(P (y)) + K(P (x|y)). As the Kolmogorov complexity is not computable, approximations are used to infer causal directions in Boolean [START_REF] Budhathoki | MDL for causal inference on discrete data[END_REF][START_REF] Budhathoki | Origo: causal inference by compression[END_REF] and numerical [START_REF] Blöbaum | Cause-effect inference by comparing regression errors[END_REF][START_REF] Marx | Identifiability of cause and effect using regularized regression[END_REF][START_REF] Osman Mian | Discovering fully oriented causal networks[END_REF] settings.

Probabilistic [START_REF] Chen | Nonlinear causal discovery for high dimensional data: A kernelized trace method[END_REF][START_REF] Janzing | Telling cause from effect based on highdimensional observations[END_REF][START_REF] Zscheischler | Testing whether linear equations are causal: a free probability theory approach[END_REF] and algorithmic [START_REF] Vreeken | Causal inference by direction of information[END_REF] approaches have been also proposed for the multivariate case, when causal relations are between sets of variables X and Y . However, this has received much less attention than the univariate case. In the experimental example presented in Section 5, we use ergo [START_REF] Vreeken | Causal inference by direction of information[END_REF], an algorithmic approach for inferring causal directions in the multivariate case by approximating the Kolmogorov complexity using cumulative and Shannon entropies.

Formal Concept Analysis

Formal Concept Analysis (FCA) is a mathematical framework based on lattice theory that aims at representing the information lying in binary datasets in terms of concept hierarchies and rules. Datasets are formalised as formal contexts that are triples (G, M, R) in which G is a set of objects, M is a set of attributes and R ⊆ G × M is a binary relation between objects and attributes. An object g is said to be described by an attribute m when (g, m) ∈ R. Given a formal context (G, M, R), we can define the two following derivation operators:

a b c d e 1 × × 2 × × × 3 × × × 4 × × 5 × ×
-• : 2 G → 2 M , G → G = {m ∈ M | ∀g ∈ G, (g, m) ∈ R}, and -• : 2 M → 2 G , M → M = {g ∈ G | ∀m ∈ M, (g, m) ∈ R}.
This pair of derivation operators forms a Galois connection, and so both compositions • : 2 G → 2 G and • : 2 M → 2 M constitute closure operators on G and M, respectively. We say that a formal context is object-reduced if, for each object o, there is no pair of objects p and q such that {o} = {p} ∩ {q} . We define attribute-reduced contexts analogously.

A pair

(G, M ) ∈ 2 G × 2 M is called a formal concept of K if M = G and G = M .
In this case, G is called the extent and M the intent of the formal concept (G, M ). The set of all the formal concepts of a formal context together with the partial order of extents induced by the inclusion of sets of objects or, equivalently, by the reversed inclusion of intents, forms a complete lattice called the concept lattice of the formal context.

Definition 1. ( Implications) Let K = (G, M, R) be a formal context. An implication is a pair (X, Y ) of attribute sets, often written in rule form X → Y . An implication X → Y holds in K when Y ⊆ X , i.e.
, the objects described by the attributes in X are also described by the attributes in Y . Let I K denote the set of all the implications that hold in K.

Implications capture certain regularities in the description of objects. For instance, in the formal context depicted in Figure 1, implications {b, c} → {d} and {e} → {d} hold, whereas {a} → {b} does not. The number of implications in a formal context (G, M, R) can grow exponentially with |G|, |M| and |R|.

Hence, even relatively small formal contexts can give rise to an exponential number of implications [START_REF] Sergei | Counting pseudo-intents and #Pcompleteness[END_REF]. Whether to present to a human analyst or to use in a computation, it is advantageous to avoid "redundant" implications. Two main approaches to reduce implication sets are usually considered, which may or may not entail loss of information. When loss of information is acceptable, one can use the many interestingness measures that have been proposed in the literature [START_REF] Geng | Interestingness measures for data mining: A survey[END_REF]. When information loss is to be avoided, we can make use of so-called implication bases.

Definition 2. ( Implication bases) Let K be a formal context. A subset I ⊆ I K is said to be an implication base of K if every implication in I K can be derived from those of I through Armstrong's axioms:

-If Y ⊆ X, then X → Y . -If X → Y , then X ∪ Z → Y ∪ Z for all Z. -If X → Y and Y → Z, then X → Z.
As every valid implication of a formal context K can be derived from an implication basis of K, implication bases contain the same information as the whole implication set I K . Different implication bases, with different properties, have been studied [START_REF] Guigues | Familles minimales d'implications informatives résultant d'un tableau de données binaires[END_REF][START_REF] Ryssel | Fast algorithms for implication bases and attribute exploration using proper premises[END_REF][START_REF] Bertet | The multiple facets of the canonical direct unit implicational basis[END_REF]. Here, we present the two best-known. Definition 3. ( Logical closure) Let I be an implication set. The logical closure I(X) of an attribute set X by I is the smallest superset of X such that

(A → B ∈ I and A ⊆ I(X)) implies B ⊆ I(X).
When I is an implication base, I(X) = X for all X ⊆ M. If K is the context depicted in Figure 1, then I({e}) = I K ({e}) = {d, e}. Note that the only difference between the logical closure and pseudo-closure resides in the strictness of the inclusion of premises, and it is not difficult to verify that the logical pseudo-closure is a closure operator. For instance, if K is the context depicted in Figure 1, then I K ({e}) = {e} because no implications in which the premise is a proper subset of {e} holds. Definition 5. ( Pseudo-intent) Let K = (G, M, R) be a formal context. An attribute set P ⊆ M is called a pseudo-intent if P = P and P = I K (P ).

Note that the attribute sets closed under the logical pseudo-closure are either intents or pseudo-intents. Thus the set of pseudo-intents can be obtained by computing the logical pseudo-closure I K (.) of attribute sets [START_REF] Bazhanov | Optimizations in computing the Duquenne-Guigues basis of implications[END_REF][START_REF] Bazin | Computing the Duquenne-Guigues basis: An algorithm for choosing the order[END_REF][START_REF] Kuznetsov | On the intractability of computing the Duquenne-Guigues base[END_REF][START_REF] Distel | On the complexity of enumerating pseudo-intents[END_REF]. Definition 6. ( Canonical base) The canonical base of a formal context is

{P → P | P is a pseudo-intent}
The notion of canonical base is also referred to as the Duquenne-Guigues base [START_REF] Guigues | Familles minimales d'implications informatives résultant d'un tableau de données binaires[END_REF]. It is the smallest implication base of a formal context in terms of number of implications, and it constitutes an optimal compression of the information contained in the implications or, equivalently, in the formal context. The canonical base of the formal context depicted in Figure 1 is:

-{e} → {d, e} -{a, d} → {a, b, c, d, e} -{b, c} → {b, c, d} -{c, d} → {b, c, d} -{b, c, d, e} → {a, b, c, d, e}
Definition 7. ( Proper premise) Let K be a formal context and a an attribute. An attribute set X is a proper premise of a if X → {a} holds, X = {a}, and, for all Y ⊂ X, Y → {a} does not hold.

In words, a proper premise is a minimal attribute set that implies another attribute.

Definition 8. ( Proper premise base)

The base of proper premises [START_REF] Ryssel | Fast algorithms for implication bases and attribute exploration using proper premises[END_REF] of a formal context is defined by

{X → X | X is a proper premise}
The base of proper premises is the implication base with the smallest premises, and it constitutes the optimal compression of the information contained in the implications when the size of the premises is a concern. For example, the base of proper premises of Figure 1 is:

-{e} → {d, e} -{b, c} → {b, c, d} -{c, d} → {b, c, d} -{a, d} → {a, b, c, d, e} -{c, e} → {a, b, c, d, e} -{a, e} → {a, b, c, d, e} -{a, b, c} → {a, b, c, d, e}
It should be noticed that minimal pseudo-intents are necessarily proper premises.

Causal Closure

Works on modelling or discovering causal relations often make a number of assumptions on the nature of the causal relation: univariate or multivariate, absence of confounders, linearity of the function in the Additive Noise Models... Following the same tracks, we assume that causal relations have some desirable (and generally accepted) properties that resemble those of implications. In the remainder of this paper, we assume that all variables belong to a set V.

For every X, Y, Z ⊆ V, suppose that the following five properties hold: Property 1. A set of variables causes itself:

X c - → X.
Property 2. Causality is transitive:

X c - → Y ∧ Y c - → Z ⇒ X c - → Z.
In words, if X directly causes Y and Y directly causes Z, then X indirectly causes Z.

Property 3. If X causes a set of variables Y , then it causes all of its subsets:

X c - → Y ⇒ X c - → Z, ∀Z ⊆ Y.
In particular, X causes {y} for all y in Y .

Property 4. All supersets of X cause its effects:

X c - → Y ⇒ X ∪ Z c - → Y.
Note that this property is not incompatible with possible notions of negation as c -→ only denotes the presence of a causal relation and not the causal effect itself, which can be different between

X c - → Y and X ∪ Y c - → Y .
Property 5. A variable causes the union of its effects:

X c - → Y ∧ X c - → Z ⇒ X c - → Y ∪ Z. Properties 1 and 3 imply that if Y ⊆ X, then X c - → Y . Properties 1, 3 and 4 imply that if X c - → Y , then X ∪ Z c - → Y ∪ Z.
Since Property 2 is transitivity, these five properties imply Armstrong's axioms (see Definition 2). Note that transitivity, despite intuitive, is disputed [START_REF] Mcdonnell | Transitivity and proportionality in causation[END_REF] and counterexamples have been presented. Halpern [START_REF] Joseph Y Halpern | Sufficient conditions for causality to be transitive[END_REF] identified conditions under which causality is transitive.

Proposition 1. Consider the operator ξ : 2 V → 2 V defined by ξ(X) = X c - →Y Y.
Then ξ is a closure operator.

Proof. By Property 1, we have that X ⊆ ξ(X) so ξ(.) is extensive. From Properties 1, 2 and 3, we have that X ⊆ Y ⇒ ξ(X) ⊆ ξ(Y ), for every X, Y ⊆ V, and thus ξ(.) is monotone. Furthermore, by Properties 1 and 2 we also have that ξ(ξ(X)) = ξ(X), for every X ⊆ V, and hence ξ is idempotent. This shows that ξ is indeed a closure operator.

We refer to ξ(X) as the causal closure of X. The causal closure of X is made of X and all of its direct and indirect effects. When all causal relations are known, the causal closure can be computed by applying ξ successively until a fixpoint is reached. However, inferring causal relations from a dataset may be costly, which prevents us from testing whether X c -→ Y for all possible Y . As we will now observe, Property 3 ensures that the computation of ξ(X) can be simplified.

Proposition 2. Let X be a variable set and y a variable. Then,

ξ(X) = {y | X c - → {y}}
Proof. This follows from the fact that

X c - → Y ⇒ X c - → Z, ∀Z ⊆ Y .
Hence, computing ξ(X) directly from a dataset requires |V \ X| causal tests.

Reconstructing the FCA Trinity with Causality

In the previous section we showed that causal relations give rise to a closure operator ξ. Let V be the set of variables in a dataset. Suppose that K c is a formal context whose set of attributes is V, and that an implication X → Y holds if and only if X c -→ Y . As causal relations respect Armstrong's axioms, I Kc (X) = ξ(X), for all X ⊆ V, and any implication base of K c enables the derivation of all the causal relations. Thus, by using the closure operator ξ as if it were the closure operator induced by the (unknown) formal context K c , it is possible to apply the FCA machinery to compute representations of the causal relations in the dataset. FCA offers many structures to represent the content of the closure operator and, thus, of the causal relations: formal contexts, concept lattices, AOC-posets [START_REF] Godin | Building and maintaining analysis-level class hierarchies using galois lattices[END_REF], and implication bases. Each of them provides its unique perspective, but some are easier to handle when the closure operator is thought of as a black box.

In this section, we go through these structures and discuss both their usefulness as representations of causal relations and their ease of use. We illustrate the various notions and results that we present on the following set of causal relations from a dataset with five variables:

-v 1 causes v 2 : {v 1 } c - → {v 2 } -v 3 causes v 4 : {v 3 } c - → {v 4 } -v 1 and v 4 together cause v 5 : {v 1 , v 4 } c - → {v 5 }

The Causal Canonical Base

The canonical base of K c is the smallest implication base of K c , that is, it corresponds to the smallest set of causal relations that allow the derivation of all the causal relations in the dataset. We call it the causal canonical base of the dataset. It is a set of multivariate causal relations of the form X → ξ(X) (analogue to the implications X → X in classical FCA) in which X contains all the variables it causes except for those specified by the rule X → ξ(X) itself. In our running example, the causal canonical base is

-{v 1 } → {v 2 } -{v 3 } → {v 4 } -{v 1 , v 2 , v 4 } → {v 5 }
As the smallest rule-based representation of causal relations, the causal canonical base is most useful when the number of rules is a concern: when visualising the structure or when storing it for computer use. This base is also arguably the easiest representation to compute as existing algorithms for computing pseudoclosed sets make use of the closure operator as a black box. For instance, the next closure [START_REF] Ganter | Finding all closed sets: A general approach[END_REF] algorithm (Algorithm 1) can be used. This algorithm relies on the fact that the sets of variables closed under ξ are either closed under ξ or premises of the causal relations in the causal canonical base. It enumerates the elements of this closure system and, for each one, checks whether it is closed under ξ or a premise. In order to avoid computing the same element twice, it enumerates the sets in the lectic order: given an arbitrary ordering of variables, A ≤ B if and only if the smallest element in

(A ∪ B) \ (A ∩ B) is in B.
Algorithm 2 uses the Next procedure (Algorithm 2) that takes a set of variables as input and returns the set immediately succeeding it in the lectic order. When applied to the running example, Algorithm 1 starts with

V = ∅. Assuming that v 1 < v 2 < v 3 < v 4 < v 5 ,
I = ∅; 3 V = ∅; 4 while V = V do 5 if V = ξ(V ) then 6 I = I ∪ {V → ξ(V )}; 7 V = Next(V);
8 return I Different algorithms can be chosen to compute the logical closure of a set (see [START_REF] Bazhanov | Comparing performance of algorithms for generating the duquenne-guigues basis[END_REF]). The time complexity of Algorithm 2 is at most linear with respect to the complexity of the chosen algorithm, since Algorithm 2 performs at most |V| logical closures. The naïve logical closure algorithm is quadratic in the number of rules and, in this case, the complexity of Algorithm 2 would be quadratic in Algorithm 2: Next Input: Variable set V ∈ V, causal relation set I Output: The next variable set closed under ξ in the lectic order 1 for every variable x ∈ V \ V in decreasing order do

2 W = I({v ∈ V | v < x} ∪ {x}); 3 if min(W \ V ) = x then 4 return W ;
the size of the causal canonical base. Algorithm 1 calls Algorithm 2 as many times as there are sets closed under ξ , which is 2 |V| in the worst case.

Causal Proper Premises

The base of proper premises of K c is the implication base with the smallest premises. It is made of rules of the form X → ξ(X) in which X is a minimal (called a sufficient) cause of the variables y ∈ ξ(X) \ X. We call this base the causal sufficiency base. In our running example, the causal sufficiency base is

-{v 1 } → {v 2 } -{v 3 } → {v 4 } -{v 1 , v 3 } → {v 5 } -{v 1 , v 4 } → {v 5 }

Sufficient causes are particularly important because they allow us to pinpoint

what can be acted on to predict or modify the future. For instance, if the association of high blood pressure and sedentary lifestyle is found to be a sufficient cause of the development of a disease, correcting either of those will certainly help preventing the disease. The causal sufficiency base is therefore interesting as it readily contains all the sufficient causes of all the variables. However, it is more difficult to compute than the causal canonical base. Indeed, computing proper premises is usually seen as computing the minimal transversals of hypergraphs constructed from the formal context [START_REF] Ryssel | Fast algorithms for implication bases and attribute exploration using proper premises[END_REF]. In our case, we only have a closure operator and the formal context is unknown. Thus, computing the proper premises of the causal relation requires the use of a well-known FCA algorithm tailor-made for when the underlying formal context is unknown: attribute exploration [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF].

Attribute exploration is an algorithm for computing an implication base of a formal context that is only known to an expert. It works by repeatedly asking the expert whether an implication holds and, if not, to provide a counterexample in the form of an object whose description invalidates the implication. The algorithm thus produces an implication base and a formal context that contains the same information as the underlying, unknown formal context. In our case, we use the version proposed in [START_REF] Ryssel | Fast algorithms for implication bases and attribute exploration using proper premises[END_REF] for proper premises and the expert is played by the causal closure operator ξ in Algorithm 3.

Q ∈ T with I {Q → Q } do 14 if Q ⊆ ξ(Q) then 15 I = I ∪ {Q → Q }; 16 else 17 G = G ∪ {oi}; 18 R = R ∪ {(oi, x) | x ∈ ξ(Q)}; 19 if a ∈ o i then 20 E = E ∪ {V \ o i }; 21 i = i + 1; 22 return I
Initially, we suppose that we have no information about the unknown formal context, i.e., the causal relations. So we start with an empty implication set I and the trivial formal context K = ({o 1 }, V, ∅) with a single object (lines 2-5). The algorithm then enumerates implications Q → Q that hold in K and checks whether

Q c - → Q by computing ξ(Q). If Q → Q does not hold in the underlying context because Q does not cause Q , the counterexample is a new object o such that o = ξ(Q).
The proper premises of an attribute a in the context K are the minimal transversals of the hypergraph

H ∈ K,a = {o \ {a} | a ∈ o }.
where o = M \ o . Thus, the implications are enumerated as follows.

For each attribute a (line 6), the algorithm computes H ∈ K,a (line 7) and then incrementally computes its set of minimal transversals T using Berge's multiplication algorithm [START_REF] Berge | Hypergraphs[END_REF] (lines 9-11). For each transversal, and thus each proper premise Q (line 12), the validity of Q → ξ(Q) is tested (line 13). If the causal relation holds, Q → Q is added to the set of implications I (lines [START_REF] Distel | On the complexity of enumerating pseudo-intents[END_REF][START_REF] Eells | Rational decision and causality[END_REF].

If it does not hold, a new object o i whose description is ξ(Q) is created (lines [START_REF] Ganter | Pattern structures and their projections[END_REF][START_REF] Ganter | Finding all closed sets: A general approach[END_REF][START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF][START_REF] Gao | Local causal discovery of direct causes and effects[END_REF][START_REF] Geng | Interestingness measures for data mining: A survey[END_REF].

When applied on the running example, Algorithm 3 starts with the trivial context K = ({o 1 }, V, ∅). Suppose that the first variable to be considered is a = v 1 . As the context is empty but still contains an object, the hypergraph H ∈ K,v1 contains a single hyperedge, {v 2 , v 3 , v 4 , v 5 } and its set of minimal transversals is {{v 2 }, {v 3 }, {v 4 }, {v 5 }}. We have that {v 3 } = {v 1 , v 2 , v 3 , v 4 , v 5 } and I {v 3 } → {v 1 , v 2 , v 3 , v 4 , v 5 }. Hence, the algorithm enters the second while loop (line 13) for Q = v 3 . As {v 3 } ⊆ ξ({v 3 }) = {v 3 , v 4 }, a new counterexample is created and added to the context. This counterexample o i is such that o i = ξ({v 3 }). The algorithm then continues until all five variables have been considered, at which point it outputs the set I of all causal relations in the causal sufficiency base.

Causal Intent Lattice and Causal Intent Context

The lattice made of the sets of variables closed under ξ and ordered by inclusion, is another representation of the causal structure. Closed sets C are the maximal elements of their equivalence classes C ≡ that contain all the variable sets that have the same direct and indirect causes as C. We call them causal intents. The causal intent lattice corresponding to our running example is depicted in Fig. 3. The set {v 3 , v 4 } is a causal intent as it is the maximal element of the equivalence class that contains {v 3 } and {v 3 , v 4 }. The fact that {v 3 } belongs to the same equivalence class as {v 3 , v 4 } means that v 3 causes v 4 . As such, the causal relations can be inferred from the causal intent lattice. However, despite the similarities between the graphical representations of causal intent lattices and causal diagrams [START_REF] Glymour | Review of causal discovery methods based on graphical models[END_REF], the causal relations do not directly correspond to edges in the lattice. For instance, the edge between {v 2 , v 5 } and {v 2 , v 4 , v 5 } does not mean that {v 2 , v 5 } c -→ {v 4 }. Similarly, the edge between {v 2 } and {v 1 , v 2 } does not mean that {v 2 } c -→ {v 1 }. Instead, the two closed sets {v 2 } and {v 1 , v 2 } together with the fact that {v 1 } is not closed mean that all the objects described by v 1 are also described by v 2 while some objects are described by v 2 and not by v 1 . Hence, it means that {v 1 } c -→ {v 2 }. The formal context induced by the closure operator ξ (and for which the causal intent lattice is the intent lattice) is called the causal context. The causal context whose objects are ∧-irreducible is called the reduced causal context. The causal context of our running example is depicted in Fig 4.

A Concrete Example

The FCA based framework that we proposed to computing the causal structure requires the ξ operator. In practice, it is hard to infer all the causal relations from an observational dataset. Thus we have to rely on approximations provided by the various approaches presented in Subsection 2.2.

In our framework we suppose that a variable set X causes a variable y if and only if X and {y} are sufficiently correlated and that X by ergo, the multivariate causal relation inference approach proposed in [START_REF] Vreeken | Causal inference by direction of information[END_REF]. The correlation between the variable sets X and {y}, denoted by corr(X, {y}), is measured using Linear Canonical Correlation Analysis [START_REF] Hotelling | Relations between two sets of variates[END_REF] and we consider that two subsets X and Y are sufficiently correlated if corr(X, {y}) > 0.8.

c - → {y} is inferred ∅ {v2} {v4} {v5} {v1, v2} {v2
v1 v2 v3 v4 v5 × × × × × × × × × × × × × × × × × × ×
The Iris dataset is well known in pattern recognition and machine learning. It describes 150 flowers of the iris genus with the values of five variables sepal_length, sepal_width, petal_length, petal_width and class. The first four are numerical variables while the class is nominal with three possible values.

In the Iris dataset, both the causal canonical base and the causal sufficiency base contain the same rules:

-{petal_width} → {petal_length, class} -{petal_length} → {class} -{sepal_length} → {petal_length, petal_width, class}
We observe that the class is caused by petal_width, petal_length and sepal_length but not by sepal_width. Additionally, petal_length appears to be a direct cause of class, itself caused by petal_width, itself caused by sepal_length.

The corresponding reduced causal context is depicted in Fig. 5 and the causal intent lattice in Fig. 6. The causal relations can be read from the lattice, e.g., {petal_length} c -→ {class} can be inferred from the fact that {petal_length} belongs to the equivalence class of {petal_length, class}. Hence, from the Iris dataset, we infer that the classes of irises are directly "caused" by the petal length, itself caused by both the petal width and the sepal length.

sepal_length sepal_width petal_length petal_width class × × × × × × × × × × × × × ×

Discussion and Conclusion

In this paper, we adapted the FCA framework to compute and represent (multivariate) causal relations in datasets. Inference of (multivariate) causal relations is a challenging problem, for which the current state of the art approaches only manage to obtain satisfactory results in very specific settings. Our framework uses causal inference as a black box, and it can integrate by design any existing (or future) approaches to causal inference. As FCA can be extended to different data types, causal inference can also be adapted to different data types. This not only includes Boolean or numerical data, but also more complex data such as sequences. However, one must be careful when extending the causal framework in this paper. The properties of causality discussed in Section 3 are necessary for defining the underlying closure operator, and some of them, e.g., transitivity, do not necessarily hold. Also, our empirical studies used a combination of ergo and linear canonical correlation analysis for inferring causal relations. Using different approaches for measuring correlation and orienting the causal direction [START_REF] Chen | Nonlinear causal discovery for high dimensional data: A kernelized trace method[END_REF][START_REF] Janzing | Telling cause from effect based on highdimensional observations[END_REF][START_REF] Zscheischler | Testing whether linear equations are causal: a free probability theory approach[END_REF] could have produced different causal relations and thus a different lattice and a different causal context. The algorithms' performance for univariate causal inference was evaluated on real and synthetic datasets [START_REF] Joris M Mooij | Distinguishing cause from effect using observational data: methods and benchmarks[END_REF] for which the true causal relations are known. In the multivariate case, there are very few datasets with ground truth, and they only contain single cause and effect examples. This is clearly not suited to evaluate an approach conceived to representing multiple multivariate causal relations. Furthermore, it is not clear how to generate synthetic data according to multivariate causal relations. Having in mind the comparison of algorithmic approaches, we think that the next milestone in the study of causal inference is the development of new approaches for generating synthetic datasets with multivariate causal relations. In general, structures with cycles such as {{a} c -→ {b}, {b} c -→ {a, c}}, should also be taken into account. However, the existing causal inference approaches cannot deal with such relations. Being able to generate datasets containing such structures would thus pave the way to new (multivariate) causal inference approaches.

Fig. 1 .

 1 Fig. 1. A crosstable representing a formal context with five objects ({1, 2, 3, 4, 5}) and five attributes ({a, b, c, d, e}).

Fig. 2 .

 2 Fig. 2. Concept lattice of the formal context depicted in Fig. 1.

Definition 4 .

 4 ( Logical pseudo-closure) Let I be an implication set. The logical pseudo-closure I (X) of an attribute set X by I is the smallest superset of X such that (A → B ∈ I and A ⊂ I (X)) implies B ⊆ I (X).

Algorithm 1 :

 1 the algorithm then enumerates the sets {v 5 }, {v 4 } and {v 4 , v 5 }. As all of them are closed under ξ, no causal relation is added to I. Then, the algorithm reaches {v 3 }. As ξ({v 3 }) = {v 3 , v 4 }, the causal relation {v 3 } → {v 4 } is added to the implication set I. The algorithm continues until it reaches {v 1 , v 2 , v 3 , v 4 , v 5 }, at which point it stops and outputs I, which contains the three causal relations in the causal canonical base. Causal next closure for implications Input: Dataset with variables V Output: The causal canonical base of the dataset 1 begin 2

Algorithm 3 : 2 I = ∅; 3 G = {o1}; 4 R = ∅; 5 K 6 i = 2 ; 7 for a ∈ V do 8 E = H ∈ K,a ; 9 T = {∅}; 10 while there exists E ∈ E do 11 T 12 E

 3234562789101112 Causal attribute exploration. Input: Dataset with variables V Output: The causal sufficiency base of the dataset 1 begin = (G, V, R); = min(T ∨ {{a} | a ∈ E}); = E \ {E}; 13 while there exists

Fig. 3 .

 3 Fig. 3. Causal intents lattice of the running example.

Fig. 4 .

 4 Fig. 4. The reduced causal context of the running example.

Fig. 5 .

 5 Fig. 5. The reduced causal context of the Iris dataset.

Fig. 6 .

 6 Fig. 6. Causal intent lattice of the Iris dataset.

Acknowledgements

This work was supported partly by the french PIA project "Lorraine Université d'Excellence", reference ANR-15-IDEX-04-LUE, by TAILOR, a project funded by EU Horizon2020 research and innovation programme under GA No 952215, and by the Inria ProjectLab "Hybrid Approaches for Interpretable AI" (HyAIAI).


Alain Pocheau

Brice Saint--Michel

Marc Georgelin

Sylvain Deville

Solidification de suspensions : des forces thermomoléculaires aux effets granulaires

La solidification de suspensions, i.e. d'un mélange de particules et de fluide, se rencontre dans de nombreuses situations apparemment très diverses : les sols gelés, l'industrie alimentaire, la cryobiologie, ou la fabrication de matériaux biocomposites. Nous l'étudions ici dans un système modèle de solidification dirigée en lames minces permettant la visualisation du phénomène in situ.

En amont du front de solidification, nous observons la formation d'une couche dense de particules qui conditionne les évolutions du système. Son apparition résulte d'un force de répulsion, appelée force thermomoléculaire, induite par le front sur les particules. Son origine est électrostatique et sa portée nanométrique. Elle conduit les particules à être repoussées par le front de croissance jusqu'à ce que de nouvelles forces d'origine visqueuses permettent d'équilibrer la répulsion. Sa taille, fonction des paramètres de croissance et de l'ordre du millimètre, est physiquement reliée aux effets nanoscopiques de la force thermomoléculaire.

Après avoir documenté la densité et les tailles de cette couche de particules puis modélisé les forces et les équilibres en jeu dans le système, nous obtenons une première mesure indirecte des pressions thermomoléculaires dans ce système. L'étude des basses vitesses montre un changement de régime induit par friction des particules aux plaques par un analogue de l'effet Janssen des matériaux granulaires. Enfin, les variations de comportements induits par un changement d'épaisseur des plaques conduisent à modéliser la transition entre un ordre cristallin du réseau de particules près des plaques et un désordre loin de celles--ci. Au total, la compréhension de la physique de cette couche dense de particules permet d'envisager l'étude et la modélisation de sa dynamique couplée à celle du milieu dans des régimes d'intérêts généraux.

Résultats

La suspension est composée de particules de polystyrène de 3 microns de diamètre, de densité 1,05, diluées dans l'eau à une fraction volumique de φ 0 =10 ou 20%. Les températures imposées sont de 20°C sur un gap de 10mm. Les échantillons ont une partie utile de 45*100 mm et la visualisation est réalisée par réflexion ou par transmission. Par réflexion, elle permet d'observer, près des plaques, la couche dense de particules séparant en régime de croissance le front de solidification et la suspension (Fig. 1).

Fig. 1 : Image par réflexion de la couche dense de particule accumulées devant le front de solidification. Sa partie grise correspond à des arrangements désordonnés de particules et les spots brillants à des arrangements cristallins. φ 0 =20%, V=3 µm/s. Au début de la croissance à vitesse V, les particules sont repoussées par le front et s'accumulent dans cette couche dont la hauteur h(t) croît linéairement, suite au flux φ 0 V de particules venant de la suspension. Son taux de croissance permet alors une mesure indirecte de sa fraction volumique φ. A une épaisseur d'échantillon de 125 microns, sa valeur φ=0.63 correspond à celle d'un amas compact désordonné de particules (random close packing). L'analyse en régime stationnaire du système montre que le saut de fraction volumique de φ 0 (la suspension) à φ (la couche dense) s'accompagne de la formation dans celle--ci d'un différentiel de vitesse entre le fluide, v f , et les particules, v p (Fig. 2). Cela conduit dans ce milieux poreux à des effets visqueux induits par une vitesse de Darcy U=V(φ 0 -φ)/φ. Ceux--ci s'accumulent le long de la couche et se transmettent par contact entre particules à celles voisines du front. Ces dernières se trouvent ainsi poussées à entrer dans la phase solide. Une autre force les y aide aussi : la force de lubrification induite dans le nanofilm séparant ces particules du front. En faisant le bilan de ces deux forces de capture et de la force thermomoléculaire de répulsion, on obtient une relation liant la hauteur h de la couche à sa vitesse de Darcy U, paramétrée par sa perméabilité. Fig. 3 (droite) : Evolution de la hauteur h de la couche dense avec l'inverse de la vitesse de Darcy U pour un échantillon de 125 µm. La ligne représente l'ajustement de la modélisation sur les données.

Les évolutions de h en fonction de 1/U montrent une partie linéaire près de l'origine suivie d'une inflexion vers une croissance bien plus lente (FIg. 3). La relation précédente permet de retrouver la tendance linéaire et d'en déduire une première mesure indirecte de la pression thermomoléculaire induite par le front sur les particules.

L'inflexion observée en figure 3 est induite par les forces de friction entre plaques et particules. Ces forces, nulles à la frontière couche/suspension et maximales au niveau du front, présentent une augmentation exponentielle le long de la couche par un mécanisme analogue à l'effet Janssen selon lequel les matériaux granulaires se trouvent être soutenus par la friction aux parois dans les silos. Elles conduisent ici à une augmentation des forces de poussées des particules vers le front qui finit par supplanter les forces de friction visqueuse, conduisant ainsi à une quasi-stagnation de h à faible U.

L'analyse des courbes h(1/U) à différentes épaisseurs d'échantillons montre une évolution des pente à l'origine. Nous la corrélons à une modification de la fraction volumique des particules près des plaques, suite à l'ordre cristallin induit par leur planéité et leur impénétrabilité.

Perspectives

Au total, la compréhension de la physique de la couche dense de particules formée au devant du front de solidification de suspensions permet d'envisager l'étude et la modélisation de sa dynamique, en lien avec les régimes observés en physique des sols ou en fabrication de matériaux.

Fig. 2

 2 Fig. 2 (gauche) : Vitesse du fluide, v f , et des particules, v p , dans la suspension, la couche dense de particules et la phase solide. La vitesse de la phase solide est notée v s .

Remerciements : Ces travaux ont été soutenus par le Conseil de Recherche européen (European Research Council) sous le septième programme--cadre européen FP7 (FP7/2007--2013)/ERC grant agreement 278004.


Le Meur Mathieu

Boussafir Mohammed

Le Milbeau Claude

Debure Mathieu

Robinet Jean-Charles

& Lerouge

ORGANIC CHARACTERIZATION OF THE "TÉGULINES CLAY FORMATION", EAST PARIS-BASIN, FRANCE: A BULK AND MOLECULAR INVESTIGATION

Setting, material and methods

Several cores and trenches were performed between October and December 2017 in the "Brienne-le-Chateau area" (Aube department, France). Several techniques, ranging from bulk to molecular scales were used in order to get a global characterization of the OM fraction. Rock-Eval (RE) analysis permit us to get several parameters such as the Total Organic Carbon (TOC) content, Hydrogen Index (HI) or the Oxygen Index (OI). The quantitative organic petrography (palynofacies) consists on the microscopic observation of isolated OM, permitting the identification and the quantification of the organic compounds present in a constant amount of sediments (1 gram) after the elimination of carbonate and silicate by hydrochloric and hydrofluoric treatments. Finally, in order to get information at molecular scale, the labile fraction (lipids) of sediments was extracted from dried sediments, fractionated into neutral, acidic and polar compounds, derivatized using BSTFA and analyzed by GCMS.

Conclusion:

This study permitted to characterize the OM oxidation profile of the Tégulines Clay formation and to compare it with previous mineralogical, geological and petrophysical data. The organic investigation showed, for the Trench TPH1-1 oxidative conditions except for the surface sample. The AUB230 and AUB1010 borehole showed a gradual oxidation profile along the core and for the AUB230 core some preserved areas. The AUB240 borehole revealed more preserved conditions. All the samples were characterized by low content in OM. This low content was associated with more refractory conditions, supposing a limited influence of the OM on the transfer and reactivity of radionuclides.

Introduction

The Tégulines Clay formation (Paris Basin, France) is currently under investigation as a potential host for sub-surface geological repository of low-level long-lived radioactive waste. Lerouge et al. (2018, Doi.org/10.1155/2018/1606753) characterized the chemical, mineral, petrological, and textural properties of this rock highlighting a potential reactivity of the organic matter (OM) due to weathering processes. In order to get a global overview of the potential interaction of the Teguline Clays with radionucleides, a OM study is needed. Our objective was to fully characterize the quantity and quality of the OM of this marine clay formation. Indeed, the refractory or reactive character of OM is an indispensable data for evaluating the overall geochemical reactivity of the rock formation and its long-term evolution.

For further information, here is the flashcode of my researchgate webpage. E.mail adress :m.lemeur@hotmail.fr -For all samples of the AUB230 borehole (except at 6.10, 67.45 and 68.45 m deep) the most represented cluster is C4 which represent the more stable organic compounds. The presence of an unimodal Gaussian S2 curve with higher contribution of C4 cluster suggests the presence of more stable OM.

-The deconvolution of the S2 curve shows a multilobed shape and suggests the presence of different kinds of OM or the presence of OM with different states of degradation. Between 10 and 55 m, the S2 curve shape shows more contribution of the C4 cluster and suggests more homogenous conditions with an intermediate oxidation state.

-All the S2 peak deconvolution for the AUB240 borehole samples are similar. The C1 cluster is also represented revealing more preserved conditions.

-The TPH1-1 sample reveals a predominance of preserved ligno-cellulosic fragments, slightly degraded ligno-cellulosic fragments and divers particles.

-In the borehole AUB230, at 6.10 m, the palynofacies is dominated by grAOM.

There is the presence of pyrite with small particles (between 10 and 20 µm).

In addition, the div category decreases. The palynofacies at this depth is characterized by small particles (around 1 µm) revealing more oxidative conditions.

-At 25 m, the proportion of gAOM and pyrite increase. At the end of the borehole, (55-68m), the proportion of grAOM is the highest and the gAOM proportion is also high with bigger size and pyrite inclusion showing more preserved conditions. The palynofacies shows higher contribution of grAOM and ogAOM with pyrite inclusion. This feature also shows more preserved conditions.

-From a general overview, the TOC values show very poor organic content. The TPH 1-1 trench shows extremely low TOC values except for the surface sample (0 -30 cm). The rest of the trench does not contain any analyzable OM. The TPH 1-1 sample (0-30) is characterized by high HI and OI values. These values are typical of soils and reveals immature OM.

-The AUB1010 borehole shows a S2 deconvolution results comparable to the AUB230 borehole. The top of the profile shows more immature OM (contribution of C1).

-The AUB230 borehole is characterized by low content in labile compounds. Between 6 m to 21 m, the yields decrease suggesting more oxidated conditions. At 25 m, the yield increases suggesting more preserved conditions at this depth. The labile OM yields at the end of the borehole vary between 0.7 and 1 mg.g -1 showing more preserved conditions.

-The CPI is high for the trench sample and decrease after to values around 1. The CPI values are high at the top, suggesting more preserved OM more degraded conditions between 6 and 21 m and again, more preserved conditions at the end of the borehole.

-The Ref-FAs, are more present at the end of the borehole revealing more preserved conditions. In addition, a-C15:0, i-C15:0 are present in the samples at 6 and 9 m, revealing more bacterial contribution.

-The AUB240 borehole showed homogeneous palynofacies content and is characterized by the presence in majority of grAOM and ogAOM. There is also the presence of pyrite in inclusion in the ogAOM suggesting more preserved conditions.

-The AUB1010 borehole palynofacies shows small particles (1 µm) for all samples suggesting more oxidative conditions

Results 1 Fig. 1

 11 Fig. 1 (a) Geological map of the east part of the Paris Basin showing the studied area; (b) Geological map of the studied area studied showing the location of the AUB240, AUB230 and AUB1010 boreholes selected for the organic matter study.; (c) Major mineralogy of Tégulines clay from the AUB230 and AUB1010 boreholes deduced from XRD data treatment, compared with previous data of AUB111 and AUB121 boreholes (Lerouge et al., 2018).

Fig. 3

 3 Fig. 3 Hydrogen Index (HI, mg HC.g-1TOC), Oxygen Index (OI, mg CO2.g-1TOC, values +80 for more clarity) and Total Organic Carbon (TOC, %) in relation with depth for the AUB230 borehole (The sample TPH 1-1 0-30 was also incorporated). The grey part correspond to the superficial formation (TPH 1-1)

Fig. 5

 5 Fig. 5 Deconvolution of multilobed S2 signals from RE pyrolysis for a TPH11 030 (r2 = 0.996), b AUB230 6.10 (r2 = 0.994), c AUB230 25.32 (r2 = 0.997), d AUB230 67.45 (r2 = 0.998) and e AUB240 39 (r2 = 0.999). Green curves are the rebuilt S2 signal from deconvolution.

Fig. 7

 7 Fig. 7 Distribution of the different particles in TPH1-1 0-30 sample and Images of the particles encountered, scale: 100 µm


Olivier Dennler

Samuel Blanquart

François Coste

Catherine Belleannée

Nathalie Theret

Phyloge

Phylogenetic Functional Module Characterization of the ADAMTS / ADAMTS like Protein Family

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Methodology

Results

Conclusion

Objective

❖

  Multi-genes (e.g., paralogs, orthologs) ❖ Multi-domains ❖ Partial knowledge / experimental data [1] => A new method to identify functional protein regions ❖ Sequence conservation and phylogenomic inference ❖ Segment without a priori protein sequences in modules ❖ Protein-protein interactions (PPI) [3] and modules evolution [2] on a common gene phylogeny ❖ Changes in module composition occurring at the same time than protein-protein interactions ❖ Module: Ungapped conserved sequence regions, shared by at least 2 sequences ❖ Hypothesis: During evolution, modules gained at the same time than a phenotype, are involved in this phenotype Phylogenetic Functional Module Characterization of the ADAMTS / ADAMTS-like Protein Family ❖ Pipeline ❖ Correlate conserved sequence modules and phenotypes evolution ❖ Identify the co-appearance of conserved modules and phenotypes.