Céphise Louison
email: cephise.louison@cea.fr

Fabien Ferlay

Delphine Keller

Daniel R Mestre

Operators' Accessibility Studies using Virtual Reality

Keywords: Virtual Reality, WEST project, Accessibility study, Avatar, Vibrotactile

The development of fusion plants is more and more challenging. Compared to previous fusion experimental devices, integration constraints, maintenance and safety requirements are key parameters in the ITER project. Components are designed in parallel and we must consider integration, assembly and maintenance issues, which might have a huge impact on the overall design. That also implies to consider the operator's feedback to assess the feasibility of accessibility or maintenance processes.

Virtual reality (VR) provides tools to optimize such integration. In 2010, the CEA IRFM decided to upgrade its design tools, by using VR during the life cycle (from design to operation) of a fusion component. The VR platform is intensively used in the design and assembly studies of WEST components. In particular, feasibility of the assembly scenario is assessed by the operators involving in the real assembly work.

To study this aspect, the use of static manikins is quite frequent in the industry. However, more complex studies, like the feasibility of assembly and maintenance tasks in complex and very confined environments, require enhanced features such as dynamic and biomechanically realistic virtual humans. We also study the contribution of tactile feedback to improve physical presence and interaction in the virtual environment (VE), which is very important for the validation of a given task's feasibility and the ergonomic evaluation of the posture and gesture of the operator. In particular, we show that adapted behavior in respect to physical elements of the VE can be obtained using a dynamic co-localized representation of the subject's body and a pseudo-haptic tactile feedback.

In this paper, we present integration studies involving operators and recent advances in the assessment of maintenance feasibility.

Introduction

Since 1990, industry has seen the interest of Virtual Reality (VR). By creating interactive and versatile environments and mock-ups, VR allows, inter alia, to evaluate the accessibility, feasibility and ergonomics problematic of assembly and maintenance tasks, in early stages of the design process. It could have a very positive impact, in terms of cost and time, by reducing the need for physical mock-ups and validating (or not) early designs. In the context of fusion plants, the integration constraints, maintenance and safety requirement are key parameters for the design. VR technologies are very efficient to optimize such integration, integration within parallel designing processes while taking into account assembly, maintenance and safety issues.

The role of operators in future facilities also needs to be studied. The human factor takes an important role, since the operator often has to interact with elements in a very tight and confined environment. Due to that, simulation in VR has to take into account this specificity. However, if visual rendering of a complex confined space is nowadays satisfying, haptic rendering remains difficult, if not impossible. This is a problem, since, to validate a maintenance task, the operator has to maneuver in a confined environment, to pay attention to his/her whole body and adopt a correct posture with respect to the environment (see Figure 1). Studies like the feasibility of assembly and maintenance tasks in complex and very confined environments require enhanced features such as dynamic and biomechanically realistic virtual humans (also called avatars) and need a substitute for the missing haptic feedback.

Approach

While obtaining a force feedback is relatively straightforward using a haptic force device, these systems cannot be applied to all types of scenarios. They are not satisfactory to interact with a complex Virtual Environment (VE), in particular when the whole body is involved. Two other interaction metaphors can be studied to "replace" haptic perception. The first one is the addition of a tangible interface, such as physical elements within the user's workspace, creating a mixed reality [START_REF] Milgram | A taxonomy of mixed reality visual displays[END_REF], [START_REF] Hoffmann | Physically touching virtual objects using tactile augmentation enhances the realism of virtual environments[END_REF]. However, that can be difficult to implement in the case of complex environment. The second one is lighter pseudo-haptic systems for representation of localized information such as contact points, without restriction of mobility (they can be wearable). This last metaphor can be implemented using vibrotactile interfaces. We propose to use vibrotactile feedback as a substitute for force feedback, to make users aware of impending or actual collisions in highly constrained spaces. This type of feedback has shown interesting result in the case of guidance [START_REF] Flores | Vibrotactile Guidance for Wayfinding of Blind Walkers[END_REF], [START_REF] Van Erp | Waypoint navigation with a vibrotactile waist belt[END_REF] and also in collision awareness [START_REF] Bloomfield | Virtual training via vibrotactile arrays[END_REF]- [START_REF] Louison | Vibrotactile Feedback for Collision Awareness[END_REF].

The democratization of VR HMD (Head Mounted Displays) such as the Oculus Rift and HTC Vive raise the problem of the representation of the subject's body. By default, the subject does not see his/her own body when using such a device. We propose to use a dynamic and co-localized representation of the subject's body to improve the operator's behavior regarding physical elements of the VE.

We experimented these factors (virtual body and pseudo-haptic feedback) in two experimentations that cover the two main kinds of situations we encounter: displacement in a confined environment and manual operation from an almost static position (limited displacement of the operator).

Experimental Study 1.1. Adaptive behavior

We realized a first experimentation to analyze the factors needed to obtain an optimal adaptive behavior. The lessons that can be learned from this simple scenario are intrinsically linked to some of our studies involving an operator's displacement within a confined environment. Our general hypothesis was that, when the subject wears a HMD, the presence of a (visual, co-localized) virtual self favors perceptual calibration of the body/environment relationships (and the processing of body-scaled information [START_REF] Warren | Visual guidance of walking through apertures: body-scaled information for affordances[END_REF]). Furthermore, in line with recent studies [START_REF] Louison | Vibrotactile Feedback for Collision Awareness[END_REF], we supposed that vibrotactile feedback (signaling, impending or actual collisions) might also contribute to behavioral calibration. Please refer to [START_REF] Mestre | The Contribution of a Virtual Self and Vibrotactile Feedback to Walking Through Virtual Apertures[END_REF] for details.

In this experimentation subjects (twenty male subjects voluntarily) were asked to walk straight from a starting point to a target position. The VE was composed of a three meters wide corridor. A sliding door was positioned in the middle of the corridor. The door could be opened with five different width apertures: 40, 50, 60, 70, 80cm (Figure . 2 C-D). We studied the instinctive behavior (affordance) of rotating our shoulders to pass through a narrow aperture; the rotation should depend of the aperture width.

Subjects were equipped with an HMD Oculus Rift DK2 device and a vibrotactile device that we developed previously [START_REF] Louison | Vibrotactile Feedback for Collision Awareness[END_REF]. Two vibrators were placed on the shoulders side. The subject's all-body position was obtained by an optical tracking system (ArtTrack®) (Figure 2, A) in order to record the subject's movements. These positional data could be connected to a real-time and co-localized virtual representation of the subject's body (Figure 2,B). We studied two aspects. The first one is the type of feedback. It could be only visual or augmented with vibrotactile stimulation. The vibrotactile feedback operates as a radar (similar to those in cars). When the subject approaches his shoulder from a virtual object (a side of the door in this case) a discontinuous vibration is sent to the corresponding actuator (with an intensity inversely proportional to the distance). The second condition is the presence (or not) of a virtual body also called "Avatar" (co-localized with the subject's body).

Figure 3:

The recording of all-body movements are used to build a co-localized avatar. The spheres' centers represent the vibrating actuators position on the subject's shoulder. Their size represents the spatial domain in which the actuators were active (that is when the subject shoulder's distance to a virtual object was inferior to their radius).

Results show that, in all conditions, subjects exhibited an adaptive behavior: rotating the shoulder to pass through a narrow aperture; in the same proportion as the outcome of previous studies [START_REF] Warren | Visual guidance of walking through apertures: body-scaled information for affordances[END_REF], [START_REF] Lepecq | Afforded actions as a behavioral assessment of physical presence in virtual environments[END_REF]. This result confirms that adaptive behavior, previously observed in natural conditions and in a CAVE (4 screens VR rooms) setup, is similar when participants wear a Head-Mounted Display (HMD).

The collisions percentage analysis between the subject's shoulders and the door shows interesting results. For small apertures, results show that both factors significantly affected the occurrence of collisions, these being minimal when both vibrotactile feedback and virtual body were present. The avatar and the vibrotactile feedback allow subjects to calibrate the perception of their body-environment relationships (using body-scaled information).

Feeling of contact

An important part of our studies involve manual operation with limited displacement of the body. The second experimentation was based on that kind of scenario. In this experimentation, we deepened the study of vibrotactile feedback. Our general hypothesis is vibrotactile feedback and especially multi-localized vibrator, contribute to enhance visuo-proprioceptive consistency. We use the term visuo-proprioceptive consistency to refer to the spatio-temporal coherence between the operator's and avatar's kinematics. This concept can be linked to the concept of kinematic symmetry proposed by McMahan [START_REF] Mcmahan | Exploring the Effects of Higher-Fidelity Display and Interaction for Virtual Reality Games[END_REF]. Spatio-temporal coherence describes a bi-directional link of interaction. The avatar must adopt the same posture as the user, being co-localized with the user. Conversely, the user must integrate the posture limitations imposed on the avatar by the virtual environment (VE). The experimentation is based on a study case from the CEA WEST assembly. Subjects (twenty eight subjects voluntarily) were asked to reach targets (semitransparent sphere) that appear in the VE with their right hand. We asked them to adopt the more fluid and natural movement as possible. The environment represents a part of a WEST facility. The VE was composed of a platform to access the back of an ICRH antenna (Figure 4 -B). We studied several Vibrotactile conditions: without activation of any actuator, with a single vibrator on the upper side of the hand or with ten vibrators along the arm (Figure 5). Subjects were equipped with an HMD Oculus Rift DK2 device and a vibrotactile device. Ten vibrators were placed all along his arm. Subject's all-body movements were tracked (Figure 4 -A) and used to connect to a realtime and co-localized morphological valid virtual representation of the subject's body (Avatar) (Figure 4 -B).

In this experimentation, the avatar cannot pass through elements of the virtual environment. The avatar's members are blocked at the surface of the object, but nothing prevents the subject from continuing his movement. These situations create a visuoproprioceptive inconsistency: the avatar and the subject do not have any longer the same posture. Preliminary results from this experiment show that the vibrotactile feedback reduces visuo-proprioceptive inconsistencies. Also, subjective data (post-hoc questionnaire) show that the majority of subjects found that the extra information given by the multi-localized vibrotactile device is relevant. It allows a fast location of contact points without visual feedback. We also asked for participation in the experiment to CEA operators, who have several working years of experience with ToreSupra/WEST facility. Their feedback was positive and they estimated that the setup: VR device, colocalized avatar and multi-vibrotactile feedback is a very useful tool to evaluate accessibility.

Conclusion

A C B

Within the frame of the use of VR simulations for assessing complex assembly tasks performed for the WEST project, one approach mixing a dynamic and colocalized representation of the subject's body with a multi-localized vibrotactile feedback as substitute of contact tactile clue has been tested. Results show that these two factors allow calibrating the perception of the operator's body-environment relationships and enhance visuo-proprioceptive consistency. The extra information, offered by a localized vibrotactile feedback, allow a fast location of contact points without overloading the visual sense and preserve a visuo-proprioceptive coherence. We conclude that associating these two aspects to enhance the validity of assembly and maintenance simulations realized in virtual reality.

Figure 1 :

 1 Figure 1: Welding task simulation in Virtual Reality and with a real mockup.

Figure 2 :

 2 Figure 2: A. One subject in the physical environment. He wears the HMD and is equipped with body markers. B. A representation of the subject passing though the aperture. C. Subjective view inside the HMD.

Figure 4 :

 4 Figure 4: A. Subject in the experimental setup. He wears the HMD, and he is equipped with an ART target for the tracking of the head's translational movements. B. The virtual environment C. Subjective view inside the HMD.

Figure 5 :

 5 Figure 5: Vibrotactile arm suit device with ten actuators (left). Vibrators' position along the arm. In the condition with a single vibrator only the vibrator '0' is used (right).

Acknowledgements

The authors wish to thank the CRVM (www.crvm.eu) team for decisive assistance in the experimental process. They also acknowledge the contribution of Jean-Claude Lepecq from Aix-Marseille University to this line of research. We thank Vincent Perrot, Caroline Chabal and Yves Soulabaille from the CEA DEN for their involvement in the project. We also want to thank Laurent Chodorge, Claude Andriot and the CEA LIST team for the assistance with the XDE physics engine. Céphise Louison is supported by a doctoral grant form CEA.


Cristina Nica

Agnès Braud
email: agnes.braud@unistra.fr

Xavier Dolques

Marianne Huchard
email: huchard@lirmm.fr

Florence Le Ber

Exploring Temporal Data Using Relational Concept Analysis: An Application to Hydroecology

This paper presents an approach for mining temporal data, based on Relational Concept Analysis (RCA), that has been developed for a real world application. Our data are sequential samples of biological and physico-chemical parameters taken from watercourses. Our aim is to reveal meaningful relations between the two types of parameters. To this end, we propose a comprehensive temporal data mining process starting by using RCA on an ad hoc temporal data model. The results of RCA are converted into closed partially ordered patterns to provide experts with a synthetic representation of the information contained in the lattice family. Patterns can also be filtered with various measures, exploiting the notion of temporal objects. The process is assessed through some quantitative statistics and qualitative interpretations resulting from experiments carried out on hydroecological datasets.

Introduction

Exploring temporal datasets is a major challenge in current research and various methods have therefore been proposed since the 90's [START_REF] Agrawal | Mining sequential patterns[END_REF]. It is worth pointing out that temporal data are relational, so that relational methods [START_REF] Džeroski | Relational data mining[END_REF] can be useful to respect their relational structure, e.g. [START_REF] Ferreira | Exploring multi-relational temporal databases with a propositional sequence miner[END_REF]. In particular, Relational Concept Analysis (RCA, [START_REF] Rouane-Hacene | Relational concept analysis: Mining concept lattices from multi-relational data[END_REF]) allows to classify relational data and provides hierarchical results which facilitates the analysis step.

Based on these properties, we propose to use RCA for exploring sequential datasets from the hydroecological domain. These datasets were collected during the Fresqueau project3 that focused on methods for assessing the quality of watercourses. The collected data represent biological (Bio) and physico-chemical (PhC) samples taken at fixed points (river sites) and repeated in time. Both parameters are used by the experts to determine the quality of watercourses.

Therefore, a global assessment of the temporal relationship between PhC and Bio parameters is needed. To this end, preprocessings of the raw sequential data allow to build a qualitative temporal model that can be used to apply RCA on these data. The RCA result is a family of lattices that can be navigated by the users. The users can select relevant navigation paths through the lattices (starting from concepts in a main lattice) by applying measures of interest based on the concept extents, that can be linked to geographical information in our application. Furthermore, in order to help their analysis and to synthetize the results, we propose to transform those concepts within closed partially ordered patterns (cpo-patterns, [START_REF] Casas-Garriga | Summarizing sequential data with closed partial orders[END_REF]), i.e. directed acyclic graphs where vertices are labelled with information extracted from the concepts out of the family of lattices. Since concepts can be more or less general or specific, the extracted patterns can be classified within three types, according to the number of vertices that are labelled with general information. Then the users can choose to select and to navigate general or specific paths in the lattices.

The paper is structured as follows. Section 2 presents basic definitions and related work. Section 3 describes the hydroecological data and their preprocessing while the RCA process is detailed in Section 4. Section 5 introduces some measures of interest dealing with the temporal dimension of obtained concepts. Section 6 presents cpo-patterns in order to help the analysis. Section 7 describes and discusses the experimental results carried out on Fresqueau datasets. Section 8 concludes and gives a few perspectives of this work.

Basics and Related Work

Relational Concept Analysis (RCA, [START_REF] Rouane-Hacene | Relational concept analysis: Mining concept lattices from multi-relational data[END_REF]) extends Formal Concept Analysis (FCA [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]) to classify sets of objects described by attributes and relations, thus allowing to discover knowledge patterns and implication rules in relational datasets. RCA applies iteratively FCA on a Relational Context Family (RCF) that is constituted of a set K of object-attribute contexts and a set R of object-object contexts. K contains n object-attribute formal contexts

K i = (G i , M i , I i ) , i ∈ {1, ..., n}. R contains m object-object relational contexts R j = (G k , G l , r j ) , j ∈ {1, ..., m},
where G k , called the domain of the relation, and G l , called the range of the relation, are respectively the sets of objects of K k and K l , and r j ⊆ G k × G l , k, l ∈ {1, ..., n}. At each step, object-attribute contexts are extended with relational attributes taking the syntactic form qr j (C), where q is a quantifier, r j is a relation and C = (X, Y ) is a concept where X is a subset of objects from the range of r j . This paper uses the existential quantifier:

∃r j (C) is an attribute of o ∈ G k if r j (o) ∩ X = ∅.
RCA process consists in applying FCA first on each object-attribute context of an RCF, and then iteratively on each object-attribute context extended by the relational attributes created using the concepts from the previous step. The RCA result is obtained when the family of lattices of two consecutive steps are isomorphic and the contexts are unchanged.

RCA has been applied to various data, e.g. for software model analysis and re-engineering [START_REF] Arévalo | Building abstractions in class models: Formal concept analysis in a model-driven approach[END_REF]. To our knowledge, this is the first time that RCA is used to explore sequential datasets. There are, however, various related FCA approaches. [START_REF] Wolff | Temporal Concept Analysis[END_REF] introduced Temporal Concept Analysis where objects are characterized with a date and a state (i.e. a set of attributes). Data are merged into a single context, and the resulting concept lattice is analysed thanks to the date element in the concepts, so that temporal relations between concepts are actually revealed by the analyst. This approach has been used to analyse sequential data about crime suspects [START_REF] Poelmans | A Method based on Temporal Concept Analysis for Detecting and Profiling Human Trafficking Suspects[END_REF]. In our RCA approach, the temporal relation between dates is considered as an object-object relation and it links concepts from several lattices. In [START_REF] Ferré | The efficient computation of complete and concise substring scales with suffix trees[END_REF], sequential datasets are processed without involving any partial order. In [START_REF] Casas-Garriga | Summarizing sequential data with closed partial orders[END_REF], closed subsequences are mined and then grouped in a lattice similar to a concept lattice. In [START_REF] Buzmakov | On mining complex sequential data by means of FCA and pattern structures[END_REF], sequential data are mapped onto pattern structures whose projections are used to build a pattern concept lattice. The authors combine the stability of concepts and the projections of pattern structures in order to select relevant patterns.

Besides, there exist various methods to explore qualitative sequential data. Indeed, sequential pattern mining is an active research area, in relation to the exponential growth of temporal and spatio-temporal databases. Sequential patterns have been introduced by [START_REF] Agrawal | Mining sequential patterns[END_REF] and used for different purposes. Such an approach has been developed within the Fresqueau project and focused on closed po-patterns, which were selected through various measures [START_REF] Fabrègue | Discriminant temporal patterns for linking physico-chemistry and biology in hydro-ecosystem assessment[END_REF]. Indeed, selecting relevant results is a main challenge for all approaches dealing with large datasets. In FCA, the most used measures for selecting relevant concepts are stability [START_REF] Kuznetsov | On stability of a formal concept[END_REF], probability and separation [START_REF] Klimushkin | Approaches to the selection of relevant concepts in the case of noisy data[END_REF]. Unfortunately, these measures are not able to take into account the specific structure of concepts built on temporal objects. We thus propose to use specific measures, as detailed in Section 5.

Context and Data Preprocessing

In the Fresqueau project, the analysed data cover various compartments such as physico-chemistry, hydrobiology, hydromorphology and land use (as described in [START_REF] Bimonte | Un système décisionnel pour l'analyse de la qualité des eaux de rivières[END_REF]). Here, we try to tackle the following issue by means of RCA: Can experts explain values of biological parameters from PhC values occuring in past months and thus improve the global assessment of the quality of watercourse ecosystems?

To answer this question we should mention that the quality of watercourses is determined by the Bio parameters (e.g. Standardised Global Biological Index (IBGN), Biological Index of Diatoms (IBD) and Fish Biotic Index (IPR)). Hence, the objects of interest from our work are the Bio samples and we want to assess, over a period of time, the impact of PhC macro-parameters (e.g. Nitrogen (AZOT), Phosphor (PHOS) and Particulate Matter (PAES)) on Bio ones.

Table 1(a) illustrates a small raw sequential dataset of Bio and PhC samples taken from a site (e.g. S1) corresponding to a river segment. A set of sites constitutes a geographical area. A data sequence is a chronologically ordered set of PhC samples with a Bio one at the end, all taken from the same site. This raw sequential dataset shows measurements made only for IBGN Bio parameter and for four PhC parameters namely Ammonium (N H + 4 ), Kjeldahl Nitrogen The raw sequential dataset contains only numerical values. For mining such data, we transform them by applying discretization and selection processes based on domain knowledge. The discretization aims at converting numerical values into qualitative ones. To this end, we use qualitative values for Bio and PhC parameters that are provided by the SEQ-Eau4 standard. Both types of parameters have five qualitative values, namely very good, good, medium, bad and very bad represented respectively by the colors blue, green, yellow, orange and red. In addition, SEQ-Eau standard groups PhC parameters into macro-parameters. For example, N H + 4 , N KJ and N O - 2 are grouped into AZOT macro-parameter. The selection process considers only relevant data by defining some constraints based on expert advice. For instance, the only analysed PhC samples are those taken within 4 months before a Bio parameter, from the same site.

Table 1(b) shows the preprocessed sequential dataset ready to be mined using RCA. This sequential dataset is obtained by applying the discretization and selection processes to the raw sequential dataset illustrated in Tab. 1(a). It is worth pointing out that the preprocessed sequential dataset is significantly small compared to the raw one thanks to the macro-parameters and the limited analysed period of time.

Temporal Relational Analysis

The sequential dataset is structured following the schema depicted in Fig. 1. The four rectangles represent the four sets of objects we manipulate: Bio samples, PhC samples, Bio parameters and PhC parameters. The links between Bio/PhC samples and PhC samples are defined by the temporal binary relation is preceded by (denoted by ipb). This temporal relation associates one sample to another one if the first sample is preceded in time by the second one, on the same site. There Following the temporal data model illustrated in Fig. 1, we build the RCF depicted in Tab. 2 for a small hydroecological sequential dataset. The tables KPHC (PhC parameters), KBIOS (Bio samples) and KPHCS (PhC samples) represent object-attribute contexts. There is no object-attribute context for Bio parameters because each dataset is restricted to one value of one parameter (here IBGN red). KBIOS and KPHCS have no column since the samples are only described using the qualitative relations. The tables RPHCS-ipb-PHCS, RBIOS-ipb-PHCS , RbPHC and RgPHC represent object-object contexts. In these object-object contexts, a row is an object from the domain of the relation, a column is an object from the range of the relation and a cross indicates a link between two objects. For example, RPHCS-ipb-PHCS defines the temporal relations (ipb) between PhC samples and has KPHCS both as domain and range. RbPHC defines the qualitative relations between PhC samples and PhC parameters that have the blue (b) qualitative value.

Figure 2 represents the family of concept lattices obtained by applying RCA on the RCF illustrated in Tab. 2. There are three lattices, one for each formal context: L KPHCS (PhC samples, Fig. 2(a)), L KPHC (PhC parameters, Fig. 2(b)) and L KBIOS (Bio samples, Fig. 2(c)). Each concept is represented by a box structured from top to bottom as follows: concept name, simplified intent and simplified extent. As said before, we have used the existential quantifier to build relational attributes. For instance, the intent of C KPHCS 2 from concept L KPHCS contains the relational attribute ∃RgPHC(C KPHC 1) inherited from concept C KPHCS 5. This relational attribute is common to all PhC samples that measure a green PHOS parameter, which represents the extent of concept C KPHC 1 shown in Fig. 2(b). The navigation amongst the lattices shown in Fig. 2 follows the concepts used to build relational attributes. For example, the aforementioned relational attribute ∃RgPHC(C KPHC 1) allows us to navigate from concept C KPHCS 2 out of L KPHCS to concept C KPHC 1 out of L KPHC .

Measures of Interest for Temporal Concepts

To analyse the results of the RCA process, experts start from a main lattice, here the lattice L KBIOS , and navigate through the relational attributes linking concepts of different lattices. Besides, since RCA process can produce a large number of interrelated concepts, depending on the dataset volume and characteristics, some interestingness measures are required to select relevant concepts from where to start the navigation.

Such measures should take into account the specificity of concepts built on temporal objects, whereas well-known measures (e.g. concept stability) fit basic concepts. For example, Fig. 3 depicts two concept extents where the temporal objects are the Bio samples. Both concepts -that we call temporal concepts -have the same number of Bio samples and they cover the same geographical area. If two Bio samples are deleted, following the idea of stability measure, one of the site S2 and one of S3, then both concepts still have the same number of Bio samples but they cover different river sites.

To overcome this limitation, we introduce below an approach based on the distribution of temporal concept extents. The main idea in our method states that a concept is relevant if it is frequent and related to many sites where Bio samples are evenly distributed amongst these sites. Accordingly, we try to find temporal concepts whose intents represent universally available regularities in the studied geographical area. In our example, both concepts have the same frequency (7 samples), but the distribution is different: Concept 1 is more relevant than Concept 2.

Let (X, Y ) be a formal concept of the main lattice, then its extent X is a set of temporal objects -or pairs -(Object, Date). If the value of Object is not identical for all the pairs, then the pairs can be grouped into categories by objects. We accordingly define X which represents the set of distinct objects Definition 2 (Support and Richness (ρ)). The support of a concept (X, Y ) corresponds to the number of pairs (Object, Date) out of X. Its richness, represented by ρ, is defined as the cardinality of X.

Definition 3 (Distribution index (IQV)). The distribution of a concept (X, Y ) describes the number of times each object out of X occurs in X and it is measured by the Index of Qualitative Variation (IQV, [START_REF] Frankfort-Nachmias | Social Statistics for a Diverse Society, chap. Measures of Variability[END_REF]). IQV is based on the ratio of observed differences in Xφ to the total number of possible differences within Xφ (ρ > 1).

IQV = ρ |X| 2 - ρ i=1 φ oi 2 |X| 2 (ρ -1)
(1)

If ρ = 1, IQV = 0.
Our choice of IQV stems from the observation that the objects of X do not have an intrinsic ordering. Thus, measuring their distribution using the IQV [START_REF] Frankfort-Nachmias | Social Statistics for a Diverse Society, chap. Measures of Variability[END_REF] seems interesting. The IQV ranges from 0 to 1. When all pairs of X contain the same object, there is no diversity and the IQV is 0. In contrast, when there are different objects and all pairs of Xφ have equal φ o , there is even distribution and the IQV is 1.

Returning to our example (Fig. 3), both concepts have support

|X 1 | = |X 2 | = 7 and richness ρ 1 = ρ 2 = 3. For Concept 1 the distribution is IQV 1 = 3[7 2 -(3 2 +3 2 +1 2 )] 7 2 (3-1)
= 0.91 and for Concept 2 IQV 2 = 0.67. Hence, Concept 1 is computed as more relevant than Concept 2 since its objects (Bio samples) are better distributed amongst the sites.

CPO-patterns for Helping Expert Analysis

Since our aim is to facilitate the analysis work, we propose, in addition to the selection of relevant concepts, to convert those concepts into cpo-patterns. Indeed cpo-patterns are structures with a graphical representation easy to read and understand (e.g. Fig. 4). The expert can choose a cpo-pattern that highlights interesting, surprising knowledge, and deepen the analysis by exploring the area in the lattice surrounding the corresponding concept. Thus, starting from the family of lattices built using RCA, we extract cpo-patterns following the approach proposed in [START_REF] Nica | Extracting Hierarchies of Closed Partially-Ordered Patterns using Relational Concept Analysis[END_REF]. It is worth pointing out that there is a cpo-pattern for each concept out of the lattice corresponding to the objects of interest for the study, i.e. L KBIOS in our work.

Formally, let I = {I 1 , I 2 , ..., I m } be a set of items. An itemset IS is a non empty, unordered, set of items, IS = (I j 1 ...I j k ) where I j i ∈ I. Let IS be the set of all itemsets built from I. A sequence S is a non empty ordered list of itemsets, S = IS 1 IS 2 ...IS p where IS j ∈ IS. The sequence S is a subsequence of another sequence S = IS 1 IS 2 ...IS q , denoted as S s S , if p ≤ q and if there are integers j 1 < j 2 < ... < j k < ... < j p such that IS 1 ⊆ IS j 1 , IS 2 ⊆ IS j 2 , ..., IS p ⊆ IS j p . Sequential patterns have been defined by [START_REF] Agrawal | Mining sequential patterns[END_REF] as frequent Fig. 4: Hybrid cpo-pattern: each vertex corresponds to a set of parameter values, edges represent the temporal relation, e.g. IBGN red is preceded by PHOS red that is preceded by ? red ; this notation means that a PhC parameter with a red quality has been measured. subsequences found in a sequence database. A po-pattern is a directed acyclic graph G = (V, E, l). V is the set of vertices, E is a set of directed edges such that E ⊆ V × V, and l is a labelling function mapping each vertex to an itemset. A partial order can be defined on G as follows: for all {u, v} ∈ V 2 , u < v if there is a directed path from u to v. However, if there is no directed path from u to v, these elements are not comparable. Each path of the graph is a sequential pattern as defined before. The set of paths in G is denoted by P G . A po-pattern is associated to the set of sequences S G that contains all paths of P G . Furthermore, let G and G be two po-patterns with P G and P G their sets of paths. G is a sub po-pattern of G , denoted by

G g G , if ∀M ∈ P G , ∃M ∈ P G such that M s M . A po-pattern G is closed, denoted cpo-pattern, if there exists no po-pattern G such that G ≺ g G with S G = S G .
As described in [START_REF] Nica | Extracting Hierarchies of Closed Partially-Ordered Patterns using Relational Concept Analysis[END_REF], thanks to the hierarchical structure of the RCA results, more or less accurate cpo-patterns are extracted. Based on their accuracy, three types of cpo-patterns could be defined: abstract, hybrid and concrete. Firstly, the abstract cpo-pattern represents an imprecise common trend of the analysed data. Secondly, the hybrid one, depicted in Fig. 4, corresponds to a more or less accurate common trend of the analysed data. Finally, the concrete cpo-pattern designates an accurate common trend of the analysed data.

Experiments and Discussion

The experiments are carried out on a MacBook Pro with a 2.9 GHz Intel Core i7, 8GB DDR3 RAM running OS X 10.9.5. RCA is applied using the RCAExplore5 tool. For the extraction and selection of cpo-patterns we have developed an algorithm in Java 8 based on Java Collections Framework and Lambda Expressions.

Three sequential datasets (each dataset concerns only one Bio parameter having the yellow quality) from the Fresqueau project are analysed: IBD yellow , IP R yellow and IBGN yellow . These datasets are interesting since the yellow quality of watercourses represents a median area between good ecological status and bad ecological status of watercourses. Other quality values have also been analysed but are not presented here. The objective is to extract more or less accurate cpo-patterns representing frequent PhC trends of watercourses common in many sites. To this end, the datasets are preprocessed and temporally modelled as described in Sections 3 and 4. The temporal relational analysis relies on the IceBerg algorithm [START_REF] Stumme | Efficient data mining based on formal concept analysis[END_REF], which result is a concept lattice of frequent closed itemsets. A 10% threshold is used only for the input of Bio samples (it corresponds to the lattice of Bio samples that covers the objects of interest from our work). The choice of this value allows us to focus on the cpo-patterns that describe many sites.

Table 3 shows some quantitative statistics regarding the temporal relational analysis and the extraction of cpo-patterns. The results in Output column show that the number of extracted concepts for the IBGN dataset is about 3 times smaller than the number of extracted concepts for the IPR and IBD datasets. This reveals greater heterogeneity in IPR and IBD datasets in contrast with IBGN. Consequently, cpo-patterns linking PhC and IBGN Bio parameters represent more examples and will provide more reliable forecasts of the yellow quality of watercourses.

The CPO-patterns columns represent the different types of extracted cpopatterns and illustrate their quite large number that has to be reduced. To this end, we select relevant cpo-patterns based on the support, richness and distribution of the associated concepts (see Section 5). Figure 5 shows three scatter-plots (for the three sets of extracted concrete cpo-patterns in Tab. 3) of the distribution index (IQV) with respect to the support. The diameter of the circles is proportional to the richness. The user can first explore a few selected cpo-patterns based on high thresholds for these measures. Then he/she can follow the cpo-pattern hierarchy to deepen the analysis, as described below, or select more cpo-patterns based on lower thresholds. For example, by defining two thresholds θ IQV = 0.98 and θ Support = 25, the top-6 (IBGN), the top-26 (IBD) and the top-30 (IPR) best distributed and most frequent cpo-patterns are selected. Focusing on IBD, if the thresholds are e.g. θ IQV = 0.98 and θ Support = 20, 52 cpo-patterns are selected. These cpo-patterns cover various numbers of sampling sites, and thus more or less extensive geographical areas. To select greater or smaller areas, the cpo-patterns are ranked by analysing the diameter of the circles.

The qualitative interpretation of the extracted cpo-patterns was performed by an hydroecologist. In Fig. 6 is an interesting excerpt from the main lattice of IBGN yellow dataset. This group of cpo-patterns is subsumed by the abstract cpo-pattern of C KBIOS 868 (support = 28) that represents the less accurate common trend: often before yellow IBGN are sampled simultaneously a green PhC parameter and another yellow PhC parameter. Figure 6 also emphasizes the well-known correspondence between MOOX (organic matter pollutions) quality classes and IBGN ones: a yellow MOOX appears in the yellow IBGN cpo-pattern, which is associated to C KBIOS 595. The concepts C KBIOS 720, C KBIOS 550 and C KBIOS 400 highlight the impact of phosphorus pollution (PHOS) on macroinvertebrates (IBGN) that is a lesser-known fact.

Moreover, in Fig. 6 two benefits of exploring sequential data by means of RCA are observed. The first one is the generalisation order regarding the structure of the extracted cpo-patterns. For example, the structure of C KBIOS 400 cpopattern is more specific than the structure of its ancestor cpo-patterns, i.e. there exist a projection from its ancestor cpo-patterns into C KBIOS 400 cpo-pattern. The second benefit is the generalisation of items. For instance, the C KBIOS 550 cpo-pattern reveals the rule {PAES green , PHOS yellow } → {IBGN yellow } that is a specialisation of the rule revealed by the C KBIOS 720 cpo-pattern, that is {? green , PHOS yellow } → {IBGN yellow }. These properties are useful for the expert who can navigate from specific to general patterns or vice versa.

Conclusion

We have introduced an original approach for exploring temporal data using RCA. Given a hydroecological dataset, where data represent Bio or PhC samples measured at a given time in a certain site, we find hierarchies of more or less general cpo-patterns that summarize the impact of PhC parameters on Bio ones. A comprehensive process for mining sequential datasets has been proposed: 1) preprocessing of the raw data based on domain knowledge, 2) relational analysis of the preprocessed data based on an original temporal data model, 3) selection of temporal concepts using the distribution, the richness and the support measures, and 4) extraction of cpo-patterns by navigating amongst temporal concepts (step detailed in [START_REF] Nica | Extracting Hierarchies of Closed Partially-Ordered Patterns using Relational Concept Analysis[END_REF]). Our method has been applied to sequential datasets from the Fresqueau project.

The main benefits of our approach are as follows. Using RCA produces hierarchical concepts, while cpo-patterns synthetize complex navigation paths, both facilitating the expert analysis. Furthermore, the proposed measures on temporal concepts are useful to select relevant information in our application.

In the future, we plan to apply our approach on other relational datasets. This will require to deeply investigate the behaviour of our measures and maybe to find other methods for selecting the extracted cpo-patterns.

Fig. 1 :

 1 Fig. 1: The modelling of the hydroecological sequential dataset.

Fig. 2 :

 2 Fig. 2: The family of concept lattices obtained by applying RCA on the RCF given in Tab.2. The * symbol represents all the relational attributes of KPHCS.

Fig. 3 :

 3 Fig. 3: Bio samples distribution by sites for two concept extents.

Fig. 5 :

 5 Fig. 5: Concrete cpo-patterns by distribution index, support and richness of the associated concepts.

Fig. 6 :

 6 Fig. 6: Excerpt from a hierarchy of cpo-patterns (IBGN yellow ).

Table 1 :

 1 Small example of raw and corresponding preprocessed sequential dataset. Nitrite (N O - 2 ) and Orthophosphate (P O 3- 4 ). For instance, 0.043 mg/l of N H + 4 is measured on 01/04, i.e. January 2004, for the site S1. An IBGN score of 8/20 is measured on September 2004 for the same site.

		(a) Raw Sequential Data		(b) Preprocessed Sequential Data
	Site Date N H + 4 N KJ N O -2 P O 3-4	IBGN Site Date AZOT PHOS IBGN
		08/05 -	-	-	-	10		08/05	-	-	Yellow
	S1	06/05 0.004 09/04 -	-0.012 0.035 ---	-8	S1	06/05 Blue Green 09/04 --	-Orange
		08/04 -	1.414 -	-	-		08/04 Green	-	-
		01/04 0.043 0.146 0.421	-	-
	(N KJ),

Table 2 :

 2 RCF composed of object-attribute contexts: KPHC, KBIOS and KPHCS; temporal object-object contexts: RBIOS-ipb-PHCS and RPHCS-ipb-PHCS; qualitative object-object contexts: RbPHC and RgPHC.

	object-attribute contexts					object-object contexts
	KPHC AZOT × AZOT PHOS × PHOS	KBIOS S1 20/01 S1 28/12 S2 30/02	KPHCS S1 17/01 S1 10/01 S1 25/12 S2 28/02 S2 20/02	RBIOS-ipb-PHCS S1 20/01 S1 28/12 S2 30/02	S1 17/01 × × S1 10/01	S1 25/12 ×	S2 28/02 × × S2 20/02	RPHCS-ipb-PHCS S1 17/01 S1 10/01 S1 25/12 S2 28/02 S2 20/02	S1 17/01	S1 10/01 ×	S1 25/12	S2 28/02	S2 20/02 ×	RbPHC S1 17/01 S1 10/01 × AZOT S1 25/12 × S2 28/02 S2 20/02 ×	PHOS	RgPHC S1 17/01 × × AZOT PHOS S1 10/01 S1 25/12 × S2 28/02 × × S2 20/02 ×

Table 3 :

 3 The results of mining the Fresqueau datasets. Bio and PhC Samples are the number of analysed samples; Output is the number of concepts from the main lattice (L KBIOS ) and the lattice of PhC samples (L KPHCS ); CPO-patterns is the number of the extracted cpo-patterns; Execution Time in seconds.

		Datasets	RCA		Extraction		Execution Time
	Index Quality	Samples Bio PhC LKBIOS LKPHCS Concrete Abstract Hybrid Output CPO-patterns	RCA & Extraction
	IPR			35699 39605	433	3388	31877	593
	IBD	yellow 80 194	32146 20947	503	1444	30198	115
	IBGN			9414 11580	305	815	8293	32

http://engees-fresqueau.unistra.fr/presentation.php?lang=en

http://rhin-meuse.eaufrance.fr/IMG/pdf/grilles-seq-eau-v2.pdf

http://dolques.free.fr/rcaexplore


Cristina Nica
email: cristina.nica@engees.unistra.fr

Agnès Braud
email: agnes.braud@unistra.fr

Xavier Dolques
email: xavier.dolques@engees.unistra.fr

Marianne Huchard
email: huchard@lirmm.fr

Florence Le Ber
email: florence.leber@engees.unistra.fr

Extracting Hierarchies of Closed Partially-Ordered Patterns using Relational Concept Analysis

This paper presents a theoretical framework for exploring temporal data, using Relational Concept Analysis (RCA), in order to extract frequent sequential patterns that can be interpreted by domain experts. Our proposal is to transpose sequences within relational contexts, on which RCA can be applied. To help result analysis, we build closed partially-ordered patterns (cpo-patterns), that are synthetic and easy to read for experts. Each cpo-pattern is associated to a concept extent which is a set of temporal objects. Moreover, RCA allows to build hierarchies of cpo-patterns with two generalisation levels, regarding the structure of cpo-patterns and the items. The benefits of our approach are discussed with respect to pattern structures.

Introduction

Different approaches have been designed to explore datasets containing relational data [START_REF] Džeroski | Relational data mining[END_REF]. Relational Concept Analysis (RCA, [START_REF] Rouane-Hacene | Relational concept analysis: Mining concept lattices from multi-relational data[END_REF]) classifies sets of objects described by attributes and relations, allowing the discovery of knowledge patterns and implication rules in relational datasets. RCA has been applied to various data, e.g. for software model analysis and re-engineering [START_REF] Arévalo | Building abstractions in class models: Formal concept analysis in a model-driven approach[END_REF][START_REF] Dolques | Fixing Generalization Defects in UML Use Case Diagrams[END_REF]. The RCA result is a family of interconnected concept lattices, where each lattice can have a huge number of concepts. Consequently, in order to facilitate the analysis step of the RCA output some special procedures for selecting relevant concepts or facilitating the navigation are compulsory.

This paper focuses on exploring qualitative temporal data using RCA, relying on its capability of classifying relational data and its hierarchical results which facilitate the analysis step. Although there exist well-known methods for mining temporal data, our aim is to explore the benefits of RCA, that structures the temporal patterns in a lattice allowing the navigation amongst them. We present a theoretical framework for this purpose, that is based on a temporal data model emphasizing a main lattice (i.e. containing the objects of interest) from the RCA results. Any concept of the main lattice corresponds to a set of sequential patterns that is synthesized into a closed partially-ordered pattern (cpo-pattern, [START_REF] Casas-Garriga | Summarizing sequential data with closed partial orders[END_REF]) to ease the analysis step. Indeed, such patterns are compact, contain the same information as the sets of sequential patterns they synthesize, and are easy to interpret. On this basis, we proceed as follows (Fig. 1). Firstly, we apply RCA on a relational context family containing the temporal data. Secondly, we extract the sequential patterns starting from the main lattice concepts, and build the cpo-patterns. Furthermore, thanks to the hierarchical structure of the RCA output, more or less general patterns can appear, allowing the exploration of the space of cpo-patterns from common to particular trends or vice versa without extra processing being required. The result obtained can be compared to pattern structures [START_REF] Ganter | Pattern structures and their projections[END_REF], that allow to build lattices on various data and especially graphs. As far as we know, this is still an open question to connect graph pattern structures and the set of graphs that can be built from the set of interconnected lattices of specific instantiations of RCA (like we do). To answer the question in the general case, we must consider the variety of scaling operators. Here we focus on the existential scaling operator.

The paper is structured as follows. Section 2 gives some theoretical background of our work. Section 3 introduces RCA relying on a simple temporal example. Section 4 details the RCA properties allowing to extract and to organise cpo-patterns into a hierarchy, and discusses our approach with respect to pattern structures. In addition, the whole method is described based on the same simple example. Section 5 presents related work. Section 6 concludes and gives a few perspectives of this work.

Background

Our approach relies both on sequential patterns and formal concept analysis domains.

Sequences, Sequential Patterns and PO-patterns

Let I = {I 1 , I 2 , ..., I m } be a set of items. An itemset IS is a non empty, unordered, set of items, IS = (I j 1 ...I j k ) where I j i ∈ I. Let IS be the set of all itemsets built from I. A sequence S is a non empty ordered list of itemsets, S = IS 1 IS 2 ...IS p where IS j ∈ IS. The sequence S is a subsequence of another sequence S = IS 1 IS 2 ...IS q , denoted as S s S , if p ≤ q and if there are integers j 1 < j 2 < ... < j k < ... < j p such that IS 1 ⊆ IS j 1 , IS 2 ⊆ IS j 2 , ..., IS p ⊆ IS j p .

Suppose now that there is a partial order on the items, (I, ≤). Then the order on itemsets is defined as follows: IS ⊆ H IS if ∀I j ∈ IS, ∃I j ∈ IS , I j ≤ I j and j = k → j = k . The order on sequences is defined accordingly.

Sequential patterns have been defined by [START_REF] Agrawal | Mining sequential patterns[END_REF] as frequent subsequences found in a sequence database. A sequential pattern is associated to a support, i.e. the number of sequences containing the pattern. Formally, the support of a sequential pattern M extracted from a sequence database D S is defined as:

Support(M ) = |{S ∈ D S |M s S}| (1)
Partially ordered patterns, po-patterns, have been introduced by [START_REF] Casas-Garriga | Summarizing sequential data with closed partial orders[END_REF], to synthesise sets of sequential patterns. Formally, a po-pattern is a directed acyclic graph G = (V, E, l). V is the set of vertices, E is a set of directed edges such that E ⊆ V × V, and l is a labelling function mapping each vertex to an itemset. With such a structure, we can determine a strict partial order on vertices u and v such that u = v, i.e. u < v if there is a directed path from u to v. However, if there is no directed path from u to v, these elements are not comparable. Each path of the graph is a sequential pattern as defined before. The set of paths in G is denoted by P G . A po-pattern is associated to the set of sequences S G that contain all paths of P G . Following Eq. 1, the support of a po-pattern is defined as:

Support(G) = |S G | = |{S ∈ D S |∀M ∈ P G , M s S}| (2)
Furthermore, let G and G be two po-patterns with P G and P G their sets of paths. G is a sub po-pattern of G, denoted by G g G, if ∀M ∈ P G , ∃M ∈ P G such that M s M . A po-pattern G is closed, denoted cpo-pattern, if there exists no po-pattern G such that G ≺ g G with S G = S G .

FCA and Pattern Structures

Formal Concept Analysis (FCA, [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]) considers a formal context which is a set of objects described by attributes, and builds from it a concept lattice used to analyse the objects. Concisely, a formal context K is a 3-tuple (G, M, I), where G is a set of objects, M a set of attributes, and I the incidence relation,

I ⊆ G×M . C = (X, Y ) where X = {g ∈ G|∀m ∈ Y, (g, m) ∈ I} and Y = {m ∈ M |∀g ∈ X, (g, m)
∈ I} is a formal concept built from K. X and Y are respectively the extent and the intent of the concept. Let C K be the set of all formal concepts that can be built on K. Let C 1 = (X 1 , Y 1 ) and C 2 = (X 2 , Y 2 ) be two concepts from C K , the concept generalisation order C K is here defined by

C 1 C K C 2 if and only if X 1 ⊆ X 2 (which is equivalent to Y 2 ⊆ Y 1 ). L K =(C K , C K )
is the concept lattice built from K. We denote by (L K ) the concept from L K whose extent has all the objects, and by ⊥(L K ) the concept from L K whose intent has all the attributes.

FCA is designed to deal with binary contexts, whereas attributes can be of various forms, e.g. intervals, multi-valued, etc. To generalise the FCA approach, [START_REF] Ganter | Pattern structures and their projections[END_REF] proposed to use pattern structures. A pattern structure (G, (D, ), δ) gives a description of a set of objects G by a set of descriptions (patterns) in D, which is provided with a similarity operation , such that (D, ) is a meet-semilattice. δ : G → D maps objects to their description and should verify that {δ(g)|g ∈ G} is a complete subsemilattice of (D, ). Patterns can be of different types, such as vectors of intervals [START_REF] Kaytoue-Uberall | Two complementary classification methods for designing a concept lattice from interval data[END_REF], sequences [START_REF] Buzmakov | FCA and pattern structures for mining care trajectories[END_REF] or labelled graphs [START_REF] Ganter | Pattern structures and their projections[END_REF].

Relational Analysis of Temporal Data

We propose a general temporal modelling of sequential datasets that allows the assessment of relationships between qualitative temporal objects. Here, we use a toy example from the medical domain, which is illustrated in Fig. 2, to explain our general approach.

In this example, we study the symptoms (S), e.g. fever, cough and fatigue, that indicate the presence of viruses (V), e.g. influenza and hepatitis, in patients. The symptoms and viruses are detected by medical examinations (ME) and viral tests (VT), respectively. These physical examinations are identified by temporal objects (Object, Date) where: Object represents the patient, and Date designates the time when the physical examination was done. A patient can do several medical examinations and viral tests. Symptoms, viruses, medical examinations and viral tests are sets of objects. Viral tests are linked to viruses by some qualitative binary relations has virus differentiated by the type of diagnosed virus, e.g. A, B or C. Similarly, medical examinations are linked to symptoms by qualitative relations has symptom (mS or hS) differentiated by the type of identified symptoms, e.g. moderate or high. Viral tests/medical examinations and medical examinations are linked by a temporal binary relation is preceded by (ipb) that associates a viral test/medical examination to a medical examination if the viral test/medical examination is preceded in time by the medical examination. There is no temporal binary relation between viral tests since our aim is to study the symptoms that prognosticate distinct types of viruses. In the following, based on the relational character of the toy example, we apply RCA in order to mine these qualitative temporal data.

RCA extends the purpose of FCA to relational data. RCA applies iteratively FCA on a Relational Context Family (RCF). An RCF comprises a set K of object-attribute contexts and a set R of object-object contexts. K contains n object-attribute formal contexts

K i = (G i , M i , I i ) , i ∈ {1, ..., n}. R contains m object-object relational contexts R j = (G k , G l , r j ) , j ∈ {1, ..., m}
, where G k that we call the domain of the relation and G l that we call the range of the relation are respectively the sets of objects of K k and K l , and r j ⊆ G k × G l , with k, l ∈ {1, ..., n}. New attributes, called relational attributes, extend the formal contexts by using object-object relations and already created concepts. A relational attribute takes the syntactic form qr j (C), where q is a quantifier, r j is a relation and C is a concept whose extent contains objects from the range of r j . This paper uses the existential quantifier which, for a relational context

R j = (G k , G l , r j ), creates a relation ∃r j between an object o ∈ G k and a concept C = (X, Y ) of the lattice L K l if r j (o) ∩ X = ∅.
RCA process consists in applying FCA first on each object-attribute context of an RCF, and then iteratively on each object-attribute context extended by the relational attributes created using the concepts from the previous step. The RCA result is obtained when the families of lattices of two consecutive steps are isomorphic and the contexts are unchanged.

Henceforth, we try to answer the following question by means of RCA: Can outbreaks of Influenza A virus be recognised assessing the symptoms, e.g. FEVER and COUGH, felt by patients?

Firstly, the RCA input (RCF) is built by following the temporal data model illustrated in Fig. 2. Table 1 depicts an example of RCF on medical sequential data collected during the last year. The three tables from the left hand side represent object-attribute contexts: KS (symptoms), KVT (viral tests) and KME (medical examinations). For example, KME has no column, i.e. a medical examination is described using qualitative binary relations, and the rows represent medical examinations identified by pairs such as P1 10/01, that is a medical examination done by patient P1 on 10th of January. There is no object-attribute context of viruses due to the set of viruses that contains only 1 and the result is given in Fig. 3. There is a lattice for each object-attribute context: L KME (medical examinations), L KS (symptoms) and L KVT (viral tests). L KME and L KVT are modified during the iterative steps due to the temporal and qualitative object-object contexts that have the domain KME and KVT, respectively. Each concept is represented by a box structured from top to bottom as follows: concept name, simplified intent, simplified extent. The representation of each lattice is simplified as every attribute/object is top-down/bottom-up inherited. Thus, an attribute/object is shown only in the highest/lowest concept where it appears. For example, the intent of concept CKME 4 from L KME contains the relational attributes ∃RmS(CKS 3) and ∃RhS(CKS 3) inherited from concepts CKME 10 and CKME 9, respectively; the extent contains the objects P1 25/12 and P2 11/07 inherited respectively from concepts CKME 3 and CKME 2. The arrows represent the generalisation order. The navigation amongst these lattices follows the concepts used to build relational attributes, e.g. the aforementioned ∃RhS(CKS 3) allows us to navigate from the concept CKME 4 from L KME to concept CKS 3 out of L KS .

Extracting CPO-patterns from RCA Result

We focus here on lattices built on temporal objects. Furthermore, let us recall that we consider a main lattice, which contains the objects of interest (in the toy example, the viral tests allowing the diagnosis on a patient). In addition, concepts in these lattices contain two types of relational attributes: 1) qualitative relational attributes and 2) temporal relational attributes. The range of a qualitative attribute is a concept which represents parameters (e.g. virus, symptom). The range of a temporal attribute is a concept that represents temporal observations (e.g. medical examinations).

Proposition

In this section we give some useful properties of RCA results that help the extraction process of cpo-patterns. A temporal object in the extent of a concept (c) LKVT Fig. 3: The concept lattice family obtained by applying RCA on the RCF given in Table 1. The * symbol represents all the relational attributes of KME from the main lattice is associated to a temporal sequence (e.g. the sequence of medical examinations before the final viral test). A concept extent represents thus a set of sequences. Navigating the relational attributes allows to reveal sequential patterns contained in all these sequences, as explained below.

Property 1. Each temporal relational attribute in a concept intent of the main lattice allows to extract at least one sequential pattern. On the contrary, if there is no temporal relational attribute in a concept intent, this concept represents no sequential pattern.

Indeed, let C be a concept of the main lattice and ∃t(C 1 ) a temporal relational attribute of its intent. If C 1 intent contains a qualitative attribute ∃q(C 2 ) then C 1 reveals an itemset of qualitative values (e.g. Fever high ); if C 1 intent contains a temporal attribute ∃t(C 2 ) then C 1 leads to another itemset in the sequential pattern, depending on C 2 intent. The temporal relational attributes reveal the position of the itemset (in the sequential pattern). If C 1 intent contains no temporal attribute, the sequential pattern is finished.

A naive approach to extract sequential patterns out of a concept intent takes into account all its qualitative and temporal relational attributes. Nevertheless, some properties of the RCA results can be used to improve the extraction process. In the following, properties 2 and 3 are introduced to reduce redundancy by considering only the relational attributes pointing the most specific concepts, and to prune temporal relational attributes that can be deduced by transitivity.

Property 2. Let C 1 = (X 1 , Y 1 ) ∈ L K and C 2 = (X 2 , Y 2 ) ∈ L K be two concepts such that C 1 C K C 2 . Let C = (X,
Y ) be a concept which intent has two relational attributes ∃r(C 1 ) and ∃r(C 2 ) (derived from the same relation r). Then ∃r(C 1 ) → ∃r(C 2 ).

Proof. ∃r(C 1 ) ∈ intent(C) ↔ ∀o ∈ X, r(o)∩X 1 = ∅. Since C 1 C K C 2 , X 1 ⊆ X 2 , and thus r(o) ∩ X 2 = ∅ ↔ ∃r(C 2 ) ∈ intent(C).
Hence, the relational attributes are ordered and ∃r(C 2 ) is redundant in the interpretation of C. Property 3. Let t be the temporal relation. Let C, C 1 and C 2 be three concepts such that {∃t(C 1 ), ∃t(C 2 )} ⊆ intent(C), and ∃t(C 2 ) ∈ intent(C 1 ). Then ∃t(C 2 ) ∈ intent(C) can be deduced from ∃t(C 1 ) ∈ intent(C).

Proof. The property 3 is directly obtained from the transitivity of the t relation.

In order to facilitate the analysis step the sets of sequential patterns extracted from the RCA result are then converted into cpo-patterns. We use therefore the pruning and merging steps proposed by [START_REF] Fabrègue | Mining closed partially ordered patterns, a new optimized algorithm[END_REF]. Property 4. Let C be a concept of the main lattice whose intent contains at least one temporal attribute. Then C can be associated to a cpo-pattern that summarises the set of sequential patterns deriving from C. Conversely, the cpopattern is associated to the extent of C and its support is |extent(C)|.

Proof. A set of sequential patterns can be transformed into po-patterns [START_REF] Casas-Garriga | Summarizing sequential data with closed partial orders[END_REF]. A popattern associated to a concept is closed since the corresponding set of sequences is maximal or equivalently the concept extent is maximal. Property 5. The set of cpo-patterns associated to the main lattice is ordered according to the inclusion on extents. This order corresponds to the subsumption on graphs g (cf. Section 2).

Proof. Let G and G be two cpo-patterns with P G and P G their sets of paths. Suppose

G (resp. G ) is associated to a concept C = (X, Y ) (resp. C = (X , Y )) and X ⊆ X . Then Y ⊆ Y ↔ ∀a ∈ Y , a ∈ Y . Then ∀M ∈ P G , ∃M ∈ P G , M s M → G g G.

Characterising CPO-patterns

To characterise the items in the extracted sequential patterns, the qualitative relational attributes are analysed. To this end, we define two types of relational attributes, depending on the generality or specificity of the concept they point to.

Definition 1 (Vague/Defined Relational Attribute). The relational attribute ∃r(C 1 ), where

C 1 is a concept of lattice L K , is called Vague if C 1 ≡ (L K ), respectively it is Defined if C 1 ≺ C K (L K ).
Relying on the taxonomy of items revealed by RCA, we define three types of items that disclose abstract and concrete information from the analysed data as follows:

let C be a concept whose intent has no qualitative relational attribute, e.g. CKME 0, see Fig. 3. Then the extracted item is an abstract item, denoted by "? ? ". The abstract item describes a collection of objects which point out the occurrence of dissimilar parameters having dissimilar qualities; let C be a concept whose intent contains a vague qualitative relational attribute ∃quality( ) (e.g. CKME 10). The extracted item is an abstract qualitative item, denoted by "? quality ". The abstract qualitative item describes a collection of objects which point out the occurrence of dissimilar parameters having the same quality; let C be a concept whose intent contains a defined qualitative relational attribute ∃quality(C 1 ), with extent(C 1 ) = {value} (e.g. CKME 7). The extracted item is a concrete qualitative item, denoted by "value quality ". The concrete qualitative item describes a collection of objects which point out the occurrence of the same concrete parameter having the same quality.

These types of items allow us to define the partial order (I, ≤) on the extracted items. For every value from the parameter set and every quality from the qualitative relations the relation ≤ is defined as follows:

value quality ≤ ? quality -? quality ≤ ? ? value quality ≤ ? ? value quality ≤ value quality .

The various extracted items allow us to introduce three new types of cpopatterns.

Definition 2 (Abstract/Hybrid/Concrete cpo-pattern). A cpo-pattern is as follows:

-Abstract if it contains only abstract and/or abstract qualitative items; -Hybrid if it contains abstract and/or abstract qualitative items and concrete qualitative items; -Concrete if it contains only concrete qualitative items.

Hybrid patterns can be characterised using a measure of precision referred to as accuracy.

Definition 3 (Accuracy(υ)).

Let G be a cpo-pattern and I G the multiset of items labelling the nodes of G (∀I ∈ I G , I ∈ I). Let I c G be the subset of I G containing the concrete qualitative items. The accuracy of G is defined as the ratio of the number of items in I c G to the total number of items in I G .

υ(G) = |I c G | |I G | ∈ [0, 1] (3)
If G is abstract, υ(G) = 0; if G is concrete υ(G) = 1.

RCA based CPO-patterns vs. Pattern Structures

In a previous work [START_REF] Fabrègue | Mining closed partially ordered patterns, a new optimized algorithm[END_REF], cpo-patterns were directly extracted from sequences, using a sequential pattern mining algorithm. In these cpo-patterns, there is no order on the set of items I, i.e. there are only concrete items. The resulting set of cpo-patterns, D, with the intersection operation on graphs allows to build a pattern structure (G, (D, ), δ) where G is the set of objects described by the cpo-patterns, through the δ relation.

The resulting lattice can be compared to the hierarchy of cpo-patterns built in our RCA-based approach. Firstly, in our approach, the hierarchy is built directly from the RCA result. Secondly our approach produces a hierarchy of items and thus more general cpo-patterns than in [START_REF] Fabrègue | Mining closed partially ordered patterns, a new optimized algorithm[END_REF]. Finally, the RCA-approach allows both to navigate along the sequences and to synthesize them within cpo-patterns.

Implementation with the Toy Example

To illustrate our method, let us examine the concept CKVT 6 of L KV T (Fig. 3(c)). Its intent contains four temporal relational attributes, the most specific being ∃RVT-ipb-ME(CKME 7) and ∃RVT-ipb-ME(CKME 9). On the contrary, the intents of concepts CKME 7 and CKME 9 (Fig. 3(a)) contain no temporal relational attribute. Accordingly, the chains of concepts starting from CKVT 6 are (CKVT 6)(CKME 7) and (CKVT 6)(CKME 9) as shown in Fig. 4. From these chains two sequential patterns can be extracted, denoted respectively S1 CKVT 6 and S2 CKVT 6 .

Relying on the aforementioned types of items, the itemsets of S1 CKVT 6 and S2 CKVT 6 can be defined. Both extracted chains of concepts begin with CKVT 6 that by default represents the concrete qualitative item "Influenza A ". The intent of concept CKME 7 contains both types of qualitative relational attributes: the vague ∃RmS(CKS 3) and the defined ∃RmS(CKS 2). CKS 2 ≺ C KS CKS 3 (Fig. 3(b)), therefore the interpretation of CKME 7 is based on ∃RmS(CKS 2) that represents the concrete qualitative item "FEVER moderate ". Accordingly, the extracted sequential pattern is S1 CKVT 6 = (Influenza A )(FEVER moderate ) . Following the same principle, the intent of CKME 9 has only the vague qualitative relational attribute ∃RhS(CKS 3) that represents the abstract qualitative item "? high ". Consequently, Fig. 4: The set of concept intents/extents used to interpret the concept CKVT 6 the extracted sequential pattern is S2 CKVT 6 = (Influenza A )(? high ) . The full set of concept intents and extents navigated by our approach is shown in Fig. 4. The cpo-pattern summarising the two sequential patterns is depicted in Fig. 5 (b). Fig. 5: An excerpt from the hierarchy of cpo-patterns obtained for the RCF out of Table 1. The (d) cpo-pattern is extracted by adding new medical examinations to the RCF Following our method, i.e. extracting one cpo-pattern for each concept of the main lattice (Fig. 3(c)), a hierarchy of cpo-patterns is obtained. In Fig. 5, the abstract, hybrid and concrete cpo-patterns represent respectively the intent of concepts CKVT 0, CKVT 6 and CKVT 7. The abstract cpo-pattern (Fig. 5(a)) subsumes a group of cpo-patterns that share the less accurate common trend: often before Influenza A virus patients felt in any order a high symptom and another moderate symptom. The hybrid cpo-pattern (Fig. 5(b)) subsumes a subgroup of the aforementioned group, i.e. it is a specialisation of the abstract pattern (Fig. 5(a)). This subgroup encapsulates individual cpo-patterns that share the more or less accurate common trend: less often before Influenza A virus patients felt in any order moderate FEVER and another high symptom. The concrete cpopattern (Fig. 5(c)) is another specialisation of the abstract pattern (Fig. 5(a)). This pattern subsumes a subgroup of individual cpo-patterns that share the accurate common trend: less often before Influenza A virus patients felt in any order moderate COUGH and high FEVER. The concrete cpo-pattern depicted in Fig. 5(d) is extracted if new medical examinations are added to the RCF from Table 1. This pattern highlights the two generalization levels of the extracted hierarchy. Firstly, the structure of this pattern is more specific than the structure of the ancestors of the pattern, i.e. this pattern contains more vertices and more edges. Secondly, the generalisation of items is illustrated. For instance, the pattern reveals the rule {COUGH moderate , FEVER high } ⇒ {Influenza A } that is a specialisation of the rule {FEVER high } ⇒ {Influenza A } revealed by Fig. 5(c).

Related Work

To our knowledge, this is the first time that RCA is used to explore sequential datasets. There are, however, various related FCA approaches, e.g. [START_REF] Wolff | Temporal Concept Analysis[END_REF] introduced Temporal Concept Analysis where objects are characterised with a date and a state (i.e. a set of attributes). Data are merged into a single context, and the resulting concept lattice is analysed thanks to the date element in the concepts, so that temporal relations between concepts are actually revealed by the analyst. This approach has been used to analyse sequential data about crime suspects [START_REF] Poelmans | A Method based on Temporal Concept Analysis for Detecting and Profiling Human Trafficking Suspects[END_REF]. In our RCA approach, the temporal relation between dates is considered as an object-object relation and it links concepts from several lattices. In [START_REF] Ferré | The efficient computation of complete and concise substring scales with suffix trees[END_REF], sequential datasets are processed without involving any partial order. In [START_REF] Casas-Garriga | Summarizing sequential data with closed partial orders[END_REF], closed subsequences are mined and then grouped in a lattice similar to an FCA lattice. In [START_REF] Buzmakov | On mining complex sequential data by means of FCA and pattern structures[END_REF], sequential data are mapped onto pattern structures whose projections are used to build a pattern concept lattice. The authors combine the stability of concepts and the projections of pattern structures in order to select relevant patterns.

Besides, there exist various methods to explore qualitative sequential data. Indeed, sequential pattern mining is an active research area, in relation to the exponential growth of temporal and spatio-temporal databases. Sequential patterns have been introduced by [START_REF] Agrawal | Mining sequential patterns[END_REF] and used for different purposes, e.g. classification [START_REF] Cheng | Discriminative frequent pattern analysis for effective classification[END_REF] or prediction [START_REF] Wang | Sequential pattern mining for protein function prediction[END_REF]. Such an approach has been developed within the Fresqueau project and focused on cpo-patterns, which were selected through various measures [START_REF] Fabrègue | Discriminant temporal patterns for linking physico-chemistry and biology in hydro-ecosystem assessment[END_REF]. Sequential pattern mining approaches are more efficient from a scalability point of view, but RCA enables to deepen the result analysis by navigating within the lattice family. Moreover it reveals a taxonomy from the data, that can be used to organise the cpo-patterns. Such results can be related to [START_REF] Srikant | Mining Sequential Patterns: Generalizations and Performance Improvements[END_REF], where generalized sequential patterns are extracted based on a user-given taxonomy. RCA allows to discover this taxonomy. Some authors proposed to combine RCA and pattern structures. In [START_REF] Codocedo-Henriquez | Contributions to indexing and retrieval using Formal Concept Analysis[END_REF] RCA is adapted to integrate a description of G 1 , a set of source objects with descriptors (coming from a pattern structure (G 1 , (D, ), δ)) and relational attributes to a set of concepts on a target formal context (G 2 , M 2 , I 2 ). The relational attributes are built for a relation r ⊆ G 1 × G 2 and the usual scaling operators, like ∃ or ∀∃. This is formalized as a "heterogeneous pattern structure". An application to Information Research domain is described, where source objects are documents, descriptors are vectors of intervals of LV values, target objects are terms grouped into concepts when they have same meaning (represented by a synset), and the relation r connects documents to their included terms. LV are Latent Variables that abstract hidden topics spread over the documents. Finally, there is no extraction of complex graph patterns as in our case, since there is only one relation and descriptors in the initial pattern structure are interval vectors.

Conclusion

In this paper, we have presented a theoretical framework for exploring temporal data using RCA. Our work proposes a comprehensive process for exploring sequential datasets which spans: 1) the relational analysis step that relies on a temporal data model which allows to emphasize the objects of interest in the study and 2) the extraction step of cpo-patterns from the RCA result.

The result is a hierarchy of cpo-patterns associated to sets of temporal objects, that can be compared to a lattice of pattern structures. With respect to pattern structures, the proposed approach, thanks to RCA, allows both to navigate along the sequences and to build a set of cpo-patterns including various levels of generalisation.

Our method was applied to sequential datasets, dealing with biological and physico-chemical parameters sampled in waterbodies [START_REF] Berrahou | A qualityaware spatial data warehouse for querying hydroecological data[END_REF]. Data were collected from french databases during the ANR 11 MONU 14 Fresqueau project. Results showing the effectiveness of our approach on large datasets are presented in [START_REF] Nica | L'analyse relationnelle de concepts pour la fouille de données temporelles -Application à l'étude de données hydroécologiques[END_REF] and are available at a website 3 .

In the future, we will test our approach on other relational data such as spatial data, where various relations can be considered together (topology, distance, orientation).

Fig. 1 :

 1 Fig. 1: Analysis process schema

  Influenza A virus. The four tables from the right hand side represent object-object contexts: RVT-ipb-ME (viral test ipb medical examination), RME-ipb-ME (medical examination ipb medical examination), RmS (medical examination detects a moderate symptom) and RhS (medical examination detects a high symptom). For instance, RVT-ipb-ME has viral tests as rows and medical examinations as columns. A cross indicates a link between objects, e.g. the cell identified by the viral test P1 20/01 and the medical examination P1 17/01 contains a cross since both are undergone by patient P1 and 20/01 is preceded in time by 17/01.

Fig. 2 :

 2 Fig. 2: The modelling of the medical sequential dataset

Table 1 :

 1 RCF composed of object-attribute contexts: KS, KVT and KME; temporal object-object contexts: RVT-ipb-ME and RME-ipb-ME; qualitative object-object contexts: RmS and RhS.Secondly, RCA is applied on the RCF shown in Table

	Object-attribute contexts					Object-object contexts
	KS FEVER × FEVER COUGH × COUGH	KVT P1 20/01 P1 28/12 P2 30/02 P2 15/07	KME P1 17/01 P1 10/01 P1 25/12 P2 28/02 P2 20/02 P2 11/07	RVT-ipb-ME P1 20/01 × × P1 17/01 P1 10/01 P1 28/12 P2 30/02 P2 15/07	P1 25/12 ×	P2 28/02 × × P2 20/02	P2 11/07 ×	RME-ipb-ME P1 17/01 P1 10/01 P1 25/12 P2 28/02 P2 20/02	P1 17/01	P1 10/01 ×	P1 25/12	P2 28/02	P2 20/02 ×	P2 11/07	RmS P1 17/01 × × FEVER COUGH P1 10/01 P1 25/12 × P2 28/02 × × P2 20/02	RhS P1 17/01 P1 10/01 × FEVER P1 25/12 × P2 28/02 P2 20/02 ×	COUGH
								P2 11/07							P2 11/07 ×	P2 11/07 ×

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