Visualizing and Interacting with Concept Hierarchies

Concept Hierarchies and Formal Concept Analysis are theoretically well grounded and largely experimented methods. They rely on line diagrams called Galois lattices for visualizing and analysing object-attribute sets. Galois lattices are visually seducing and conceptually rich for experts. However they present important drawbacks due to their concept oriented overall structure: analysing what they show is difficult for non experts, navigation is cumbersome, interaction is poor, and scalability is a deep bottleneck for visual interpretation even for experts. In this paper we introduce semantic probes as a means to overcome many of these problems and extend usability and application possibilities of traditional FCA visualization methods. Semantic probes are visual user centred objects which extract and organize reduced Galois sub-hierarchies. They are simpler, clearer, and they provide a better navigation support through a rich set of interaction possibilities. Since probe driven sub-hierarchies are limited to users focus, scalability is under control and interpretation is facilitated. After some successful experiments, several applications are being developed with the remaining problem of finding a compromise between simplicity and conceptual expressivity

Formal duality

We provide an overview of formal duality with an emphasis on the authors contributions. Every formally dual set can be obtained from a primitive formally dual set or, more generally from an irreducible formally dual set.Using several methods, including even set theory and the field-descent method, it is possible to obtain examples of primitive/irreducible formally dual sets as well as non-existence results. A graph search algorithm can be used for further investigation. Overall, primitive formally dual sets seem rare in cyclic groups, but occasionally exist in finite abelian groups.In dieser Dissertation geben wir einen Überblick über formale Dualität. Jede formal duale Menge kann von einer primitiven, oder allgemeiner von einer irreduziblen, formal dualen Menge, konstruiert werden. Methoden wie die 'even set' Theorie oder die 'field-descent' Methode, können genutzt werden, um Beispiele für primitive/irreduzible formal duale Mengen sowie nicht-Existenz Resultate zu erhalten. Weiterhin kann ein Suchalgorithmus genutzt werden. Primitive formal duale Mengen scheinen in zyklischen Gruppen selten zu sein, kommen aber gelegentlich in endlichen abelschen Gruppen vor

Optimal constraint-based decision tree induction from itemset lattices

International audienceIn this article we show that there is a strong connection between decision tree learning and local pattern mining. This connection allows us to solve the computationally hard problem of finding optimal decision trees in a wide range of applications by post-processing a set of patterns: we use local patterns to construct a global model. We exploit the connection between constraints in pattern mining and constraints in decision tree induction to develop a framework for categorizing decision tree mining constraints. This framework allows us to determine which model constraints can be pushed deeply into the pattern mining process, and allows us to improve the state-of-the-art of optimal decision tree induction

Visualizing Social Photos on a Hasse Diagram for Eliciting Relations and Indexing New Photos

International audienceSocial photos, which are taken during family events or parties, represent individuals or groups of people. We show in this paper how a Hasse diagram is an efficient visualization strategy for eliciting different groups and navigating through them. However, we do not limit this strategy to these traditional uses. Instead we show how it can also be used for assisting in indexing new photos. Indexing consists of identifying the event and people in photos. It is an integral phase that takes place before searching and sharing. In our method we use existing indexed photos to index new photos. This is performed through a manual drag and drop procedure followed by a content fusion process that we call ‘propagation'. At the core of this process is the necessity to organize and visualize the photos that will be used for indexing in a manner that is easily recognizable and accessible by the user. In this respect we make use of an Object Galois Sub-Hierarchy and display it using a Hasse diagram. The need for an incremental display that maintains the user's mental map also leads us to propose a novel way of building the Hasse diagram. To validate the approach, we present some tests conducted with a sample of users that confirm the interest of this organization, visualization and indexation approach. Finally, we conclude by considering scalability, the possibility to extract social networks and automatically create personalised albums

Efficient computation of rank probabilities in posets

As the title of this work indicates, the central theme in this work is the computation of rank probabilities of posets. Since the probability space consists of the set of all linear extensions of a given poset equipped with the uniform probability measure, in first instance we develop algorithms to explore this probability space efficiently. We consider in particular the problem of counting the number of linear extensions and the ability to generate extensions uniformly at random. Algorithms based on the lattice of ideals representation of a poset are developed. Since a weak order extension of a poset can be regarded as an order on the equivalence classes of a partition of the given poset not contradicting the underlying order, and thus as a generalization of the concept of a linear extension, algorithms are developed to count and generate weak order extensions uniformly at random as well. However, in order to reduce the inherent complexity of the problem, the cardinalities of the equivalence classes is fixed a priori. Due to the exponential nature of these algorithms this approach is still not always feasible, forcing one to resort to approximative algorithms if this is the case. It is well known that Markov chain Monte Carlo methods can be used to generate linear extensions uniformly at random, but no such approaches have been used to generate weak order extensions. Therefore, an algorithm that can be used to sample weak order extensions uniformly at random is introduced. A monotone assignment of labels to objects from a poset corresponds to the choice of a weak order extension of the poset. Since the random monotone assignment of such labels is a step in the generation process of random monotone data sets, the ability to generate random weak order extensions clearly is of great importance. The contributions from this part therefore prove useful in e.g. the field of supervised classification, where a need for synthetic random monotone data sets is present. The second part focuses on the ranking of the elements of a partially ordered set. Algorithms for the computation of the (mutual) rank probabilities that avoid having to enumerate all linear extensions are suggested and applied to a real-world data set containing pollution data of several regions in Baden-Württemberg (Germany). With the emergence of several initiatives aimed at protecting the environment like the REACH (Registration, Evaluation, Authorisation and Restriction of Chemicals) project of the European Union, the need for objective methods to rank chemicals, regions, etc. on the basis of several criteria still increases. Additionally, an interesting relation between the mutual rank probabilities and the average rank probabilities is proven. The third and last part studies the transitivity properties of the mutual rank probabilities and the closely related linear extension majority cycles or LEM cycles for short. The type of transitivity is translated into the cycle-transitivity framework, which has been tailor-made for characterizing transitivity of reciprocal relations, and is proven to be situated between strong stochastic transitivity and a new type of transitivity called delta*-transitivity. It is shown that the latter type is situated between strong stochastic transitivity and a kind of product transitivity. Furthermore, theoretical upper bounds for the minimum cutting level to avoid LEM cycles are found. Cutting levels for posets on up to 13 elements are obtained experimentally and a theoretic lower bound for the cutting level to avoid LEM cycles of length 4 is computed. The research presented in this work has been published in international peer-reviewed journals and has been presented on international conferences. A Java implementation of several of the algorithms presented in this work, as well as binary files containing all posets on up to 13 elements with LEM cycles, can be downloaded from the website http://www.kermit.ugent.be