Ivana Roche
email: ivana.roche@inist.fr

Nathalie Vedovotto
email: nathalie.vedovotto@inist.fr

Claire François
email: claire.francois@inist.fr

Dominique Besagni
email: dominique.besagni@inist.fr

Pascal Cuxac
email: pascal.cuxac@inist.fr

Marianne Hörlesberger
email: marianne.hoerlesberger@ait.ac.at

Dirk Holste
email: dirk.holste@ait.ac.at

Edgar L Schiebel
email: edgar.schiebel@ait.ac.at

Évaluation du potentiel d'applicabilité d'un projet de recherche : vers une méthodologie fondée sur l'analyse de contenu

Keywords: Processus de sélection, évaluation de projet, expertise, analyse de contenu, applicabilité, indicateur Selection process, project evaluation, expertise, content analysis, applicability, indicator Palabras clave: Proceso de selección, evaluación proyecto, trabajo experto, análisis de contenido, aplicabilidad, indicador

Introduction

La question que nous abordons dans ce travail est l'évaluation de l'applicabilité potentielle des travaux qu'un chercheur présente lors de conférences et/ou publie dans la littérature scientifique et technique. Nous nous sommes confrontés à cette problématique dans le cadre d'un projet européen [START_REF] Holste | [END_REF] où nous avons développé une méthodologie d'analyse fondée sur la modélisation de critères définis par l'ERC (European Research Council) et mis en oeuvre par leurs experts scientifiques lors du processus de sélection de projets de recherche en vue de leur financement.

Bien sûr, nous ne pouvons pas nous attendre à ce qu'un modèle numérique se substitue à l'expertise humaine ou à la reconnaissance de la communauté scientifique, toutes deux étant très difficiles à quantifier. Des modèles de ce type peuvent néanmoins servir à vérifier des décisions, fournir des informations complémentaires ou mettre en évidence des biais dans le processus de sélection [START_REF] Juznic | Scientometric indicators: peer review, bibliometric methods and conflict of interest[END_REF]. La nature et les objectifs des critères mis en oeuvre dans la sélection des projets de recherche sont très diversifiés : identité du porteur du projet, risque associé au projet, sujet peu conventionnel… Ceci est particulièrement évident dans le choix des indicateurs définis pour modéliser la stratégie, les missions et la politique scientifique des agences de financement pour établir une relation de cause à effet interprétable et utile. On observe des divergences entre les choix des experts et les propositions issues des indicateurs informétriques. Plusieurs études en ont approfondi les raisons sous-jacentes [START_REF] Juznic | Scientometric indicators: peer review, bibliometric methods and conflict of interest[END_REF][START_REF] Besselaar | Past performance, peer review and project selection: a case study in the social and behavioral sciences[END_REF][4].

Parmi les critères de l'ERC, nous trouvons ainsi l'applicabilité, caractérisée comme suit : "… may well be concerned with both new knowledge about the world and with generating potentially useful knowledge at the same time. Therefore, there is a much closer and more intimate connection between the resulting science and technology, with few of the barriers that arise when basic research and applied research are carried out separately." [START_REF]Frontier research: The European Challenge. High Level Expert Group Report[END_REF]. [START_REF] Stokes | Pasteur's Quadrant -Basic Science and Technological Innovation[END_REF] a introduit une approche permettant de distinguer entre recherche fondamentale et appliqué en définissant un schéma bidimensionnel appelé Quadrant de Pasteur. Il s'agit d'une caractérisation des travaux de recherche permettant de classer les relations existant entre les sciences fondamentales et l'innovation technologique. Les travaux de Louis Pasteur sont considérés comme un parfait exemple d'une recherche à la fois scientifiquement importante et susceptible d'applications pratiques. Le résultat est une caractérisation de trois types distincts de recherche : -la recherche purement fondamentale, illustrée par les travaux de Niels Bohr, physicien danois du début du 20ème siècle ; -la recherche purement appliquée, illustrée par les travaux de Thomas Edison, inventeur et industriel nord-américain ; -la recherche fondée sur la théorie mais à visée appliquée, décrite comme le Quadrant de Pasteur, scientifique français, chimiste et physicien de formation, pionnier de la microbiologie.

Donald Stokes

Dans cet article, nous définissons d'abord l'approche utilisée pour évaluer l'orientation appliqué d'un projet, puis nous décrivons le traitement des données. Pour conclure nous présentons et commentons les résultats obtenus.

Evaluation de l'orientation appliquée

Une solution classiquement adoptée pour déterminer l'orientation appliquée des travaux d'un chercheur amène à s'intéresser aux éventuels brevets au développement desquels il(elle) a pris part. [e.g. [START_REF] Glänzel | Patents cited in the scientific literature: An exploratory study of "reverse" citation relations[END_REF][START_REF] Moed | Handbook of quantitative science and technology research: The use of publication and patent statistics in studies of S&T systems[END_REF][START_REF] Glänzel | Publication activity, citation impact and bi-directional links between publications and patents in biotechnology[END_REF]. La soumission d'un brevet est souvent l'aboutissement d'une démarche de transfert de technologie et peut en effet être considérée comme l'accomplissement pratique de travaux de recherche présentant des caractéristiques résolument appliquées. Une autre possibilité est d'examiner directement les travaux publiés par le chercheur pour caractériser leur contenu en appliqué ou fondamental. Dans une étude précédente, nous avons modélisé ces deux critères à l'aide d'indicateurs calculés à partir des données affichées par le chercheur dans les documents consignés lors de la soumission de son projet, à savoir, le nombre de brevets auxquels il(elle) a contribué et les titres des périodiques où ses travaux, présentés dans son Curriculum Vitae, ont été publiés.

Le premier indicateur est fondé sur un simple dénombrement et présente des valeurs entières comprises dans l'intervalle [0, ∞[. Or, le nombre de brevets dans les Curriculum Vitae est souvent faible, ce qui produit un impact négatif sur la justesse de l'indicateur concerné. Le second consistait à calculer la part des publications du chercheur publiées dans des périodiques dont le contenu est catégorisé comme appliqué. Cet indicateur affiche des valeurs réelles variant de 0 à 1. Pour les deux indicateurs nous supposons alors que plus haute est leur valeur, plus affirmée est l'applicabilité des travaux du chercheur. Si cette approche paraît pragmatique, elle présente néanmoins quelques faiblesses dues en particulier à l'étape de catégorisation des périodiques. En effet, en appliquant à ces derniers une catégorisation binaire "appliquée ou fondamentale" il paraît a priori aisé de transposer automatiquement la catégorie du périodique à tous les articles que y sont publiés. La première difficulté consiste en la détermination des critères permettant d'obtenir la catégorisation des périodiques car, si beaucoup a été fait pour les caractériser selon un classement hiérarchique arborescent plus ou moins détaillé des domaines scientifiques, le problème reste entier lorsqu'il faut déterminer lesquels sont résolument appliqués ou fondamentaux.

De plus, la catégorie d'un périodique peut ne pas être unique mais dépendre du domaine scientifique du chercheur qui y publie ses travaux. Considérons, par exemple, le domaine de la Biologie comme a priori fondamental. Toutes les sources classées dans ce domaine reçoivent donc la catégorie "fondamentale" de même que tous les articles qui y sont publiés. Or cela ne correspond pas toujours à une réalité. En effet, si cela reste vrai pour les publications des biologistes dans une de ces sources, celle d'un informaticien qui apporterait un développement logiciel appliqué à la Biologie devrait, elle, recevoir la catégorie "appliquée".

Aussi, dans ce travail, nous nous affranchissons de cette étape de catégorisation en proposant un indicateur plus évolué, fondé sur une analyse de contenu opérée par un expert scientifique sur, d'une part, l'ensemble des publications qui citent au moins une des publications du chercheur et, d'autre part, l'ensemble des publications qui possèdent au moins une référence citée commune avec l'ensemble des références citées par le chercheur dans la bibliographie de son projet et sur lesquelles ce dernier est supposé trouver, en partie, ses fondements.

Concernant le premier ensemble, notre hypothèse est que, par la voie de la citation, les publications citantes expriment l'exploitation, à différents niveaux d'importance, des publications du chercheur et que, par conséquent, elles constituent une source d'information réelle et pragmatique sur l'utilisation des résultats de ses travaux passés dans le cadre de nouvelles recherches. L'analyse de contenu appliquée à cet ensemble nous donne le moyen d'apprécier l'applicabilité des travaux du chercheur au moment de la soumission de son projet. Pour le second ensemble, nous supposons que l'ensemble des publications ayant une ou plusieurs citations communes avec le projet du chercheur peuvent représenter le contexte scientifique dans lequel celui-ci vient également s'inscrire. L'analyse de contenu opérée sur cet ensemble nous permet de qualifier le degré d'application de ces travaux qui, avec le projet, se fondent sur un même ensemble de connaissances. Alors, par analogie, on associe au projet le même degré d'applicabilité. Finalement, la comparaison des résultats de ces deux analyses apporte une réponse à notre interrogation sur l'aspect évolutif du potentiel d'applicabilité des travaux d'un chercheur.

Cette démarche nous permet, en outre, de considérer deux niveaux d'applicabilité : le premier correspondant à la simple utilisation des résultats publiés dans des nouveaux travaux s'y référant et le second amenant, via l'analyse de leur contenu, à la caractérisation de ces derniers à l'aune d'un critère ne comprenant que deux valeurs possibles : appliqué ou fondamental. Notre analyse de contenu produit une représentation cartographique de chacun des deux ensembles considérés, sous la forme d'une carte thématique organisant leur contenu en classes de publications similaires. L'examen de ces deux cartes de classes permet à des experts du domaine scientifique concerné d'y détecter la présence de thématiques présentant des caractéristiques appliquées et d'en évaluer l'importance relative. De plus, la comparaison des résultats obtenus pour chacune des deux cartes thématiques rend alors possible une caractérisation de l'évolution du degré d'applicabilité des travaux du chercheur. Quatre types de « parcours » scientifiques peuvent être observés. Deux montrent une réelle stabilité : les travaux passés et présents du chercheur sont et demeurent à visée soit appliquée, soit fondamentale. Les deux autres parcours sont, de notre point de vue, bien plus intéressants car ils montrent à proprement parler une transition : -soit vers l'applicatif : le chercheur passe du fondamental à l'appliqué ; -soit vers le fondamental : le chercheur, à l'origine appliqué, passe au fondamental. Nous avons représenté ces différents parcours à l'aide du Quadrant de Pasteur (cf. figure 1).

Si le parcours vers l'appliqué est aisément interprétable, celui qui montre une transition vers le fondamental appelle un questionnement : cette transition ne peut-elle être interprétée comme un simple passage obligé, nécessaire pour acquérir des fondements théoriques qu'ensuite le chercheur peut mettre en oeuvre dans le cadre de nouvelles futures applications ?

Méthodologie

Les données primaires sont directement extraites des documents déposés lors de la soumission du projet, en l'occurrence, le nom du chercheur et les références que celui-ci cite dans la bibliographie de son projet. A partir du nom du chercheur nous allons déterminer une liste de publications consistant en sa production publiée dans la littérature scientifique et technique et répertoriée dans une source de données bibliographiques permettant, en outre, d'accéder aux références citées. Cette liste obtenue, il est alors aisé de singulariser l'ensemble des publications qui citent au moins un des articles du chercheur. Ce premier corpus est considéré comme la représentation du contexte scientifique relatif aux publications citantes qui emploient, à plus ou moins grande échelle, les connaissances véhiculées par les travaux passés du chercheur. D'autre part, à l'exception des auto-citations, toutes les références citées par le chercheur dans la bibliographie de son projet font l'objet d'une recherche de leurs publications citantes respectives. L'ensemble de ces dernières constitue le second corpus qui nous considérons comme la représentation du contexte scientifique relatif aux publications citantes qui utilisent des connaissances communes correspondant aux références citées par le chercheur dans son projet. Une analyse de contenu est alors opérée sur chacun de ces deux corpus. Pour réaliser cette analyse, les notices bibliographiques du corpus sont reformatées et intégrées dans la station d'analyse de l'information Stanalyst. Une première étape de fouille de textes, appliquant des techniques fondées sur le traitement automatique des langues, opère une indexation assistée des notices en leur associant des mots-clés. Une approche classificatoire est ensuite appliquée à ce corpus de notices bibliographiques enrichies. L'outil logiciel employé, implémenté dans Stanalyst, met en oeuvre un algorithme de classification automatique non supervisée et non hiérarchique, la méthode des K-means axiales, inspirée du formalisme des cartes auto-adaptatives de Kohonen. Cette méthode emploie les mots-clés comme des indicateurs du contenu des notices bibliographiques qui, à leur tour, sont considérées comme des indicateurs des thématiques de recherche. Cette étape est suivie d'une analyse en composantes principales permettant le positionnement des classes sur une carte 2D. Les relations entre les classes sont ensuite utilisées pour la construction de réseaux thématiques qui vont constituer, si l'on s'accorde une métaphore géographique, une carte du domaine de recherche représenté par le corpus. L'expert procède alors, pour chacun des deux corpus, à l'analyse des résultats obtenus, à savoir, le contenu des classes ainsi que leur positionnement et leurs relations dans la carte de classes. Lors de cette analyse, l'expert doit appliquer une grille de lecture particulière, à même de l'amener à une évaluation du caractère majoritairement appliqué ou fondamental du contenu de chaque corpus. Il(elle) s'intéresse pour ce faire au contenu de chaque classe des deux cartes, examinant le titre et les mots-clés associés à chacune des notices afin d'évaluer le degré d'application de ces classes. Pour chaque classe, il(elle) : -détermine la proportion de sujets fondamentaux et appliqués, respectivement Pf et Pa (dont la somme est égale à 1) -pondère Pf avec la valeur -1, afin de tenir compte de l'impact négatif des sujets fondamentaux, puisque nous nous intéressons ici à la recherche appliquée détermine le "degré d'application" de chaque classe, égal à (Pa -Pf) et compris dans l'intervalle [-1, 1]. Une valeur égale à -1 correspond à une classe totalement fondamentale, une valeur égale à 1 à une classe totalement appliquée, et les valeurs intermédiaires indiquent des classes présentant ces deux caractéristiques. La somme des valeurs attribuées à chaque classe d'une carte fournit le "degré d'application" du corpus. Si cette valeur est négative, le corpus traite majoritairement de recherche fondamentale, et inversement, si cette valeur est positive, le corpus est majoritairement concerné par la recherché appliquée.

Cette étape d'expertise réalisée sur les deux corpus va ensuite permettre de les comparer et de déterminer si le parcours scientifique du chercheur est stablement localisé dans le périmètre de la recherche soit appliquée, soit fondamentale, ou bien s'il montre une transition entre les deux.

Résultats

Dans cette étude, pour illustrer la méthodologie présentée nous l'avons appliquée à un cas d'étude issu de la campagne 2009 d'appel à projets de l'ERC. Aussi, parmi les 25 grands domaines couverts par l'ERC, nous avons choisi celui de l'Ingénierie de la communication et des systèmes. Dans ce domaine, 31 projets ont été soumis et nous en avons choisi un parmi les quatre qui ont été sélectionnés par le panel d'experts de l'ERC en charge de ce domaine scientifique. Pour des raisons de confidentialité, nous ne pouvons pas afficher des données nominatives ou bien des informations permettant, par recoupement, de les retracer. Appelons donc le chercheur porteur du projet CHE. La recherche sur le WoS (Web of Science) à partir du nom du chercheur a ramené, après vérification de l'inexistence d'homonymies ou d'artefacts, 24 publications s'étalant de 2000 à 2009. Pour leur part, toujours selon le WoS, ces publications reçoivent des citations de 663 publications, que nous avons collectées et qui constituent notre premier corpus que, par la suite, nous appellerons FC. D'autre part, dans son projet, CHE présente une bibliographie citant au total 45 références. Parmi elles, 5 sont des auto-citations, que nous écartons, et 25 ne sont pas présentes dans le WoS. Les 15 références restantes, dont la date de publication est comprise dans l'intervalle [2000,2007], reçoivent des citations de 4612 publications, que nous avons extraites et qui vont former notre second corpus, appelé par la suite SC.

Les notices de FC et SC ont alors subi un reformatage leur permettant d'intégrer la plateforme d'analyse Stanalyst et ont été enrichies avec une indexation par des mots-clés obtenue grâce à l'application d'outils de text mining fondés sur des techniques de TAL (traitement automatique de langue). Cette étape d'indexation assistée a, de plus, bénéficié d'une expertise scientifique à deux niveaux : lors de la constitution du vocabulaire servant de référentiel terminologique et, d'autre part, lors de la validation du résultat final avec l'élimination de termes jugés trop génériques. La table 1 résume les caractéristiques principales des deux corpus indexés. Les cartes obtenues pour FC et SC ainsi que les résultats de l'expertise en termes de "niveau d'application" de chaque cluster sont présentés dans les figures 2 et 3. Par exemple, les classes "synchronization", "energy consumption", "interference", "channel estimation" et "monitoring" de la figure 2 obtiennent respectivement la valeur -1 ; -0,34 ; 0 ; 0,5 et 1.

Le score de "niveau d'application" calculé pour chaque corpus est : 4.17 pour FC et 8.15 pour SC. Ces valeurs montrent que les deux corpus sont appliqués, mais on observe une nette évolution entre FC et SC. En effet, en comparant les deux cartes, nous constatons que FC présente cinq classes considérées comme totalement fondamentales, alors que SC n'en comporte aucune. De plus, certaines classes présentes dans les deux cartes, par exemple "energy consumption", montrent une augmentation notable de leur "niveau d'application" dans SC.

Figure 1 .

 1 Figure 1. Schéma méthodologique d'un processus d'évaluation du potentiel d'applicabilité d'un projet de recherche

Figure 2 .

 2 Figure 2. Cartographie du premier corpus (FC), visualisant le degré d'application de chaque classe, sous forme de rectangles texturés (quadrillage= appliqué, hachures= fondamental)

Figure 3 .

 3 Figure 3. Cartographie du second corpus (SC), visualisant le degré d'application de chaque classe, sous forme de rectangles texturés (quadrillage= appliqué, hachures= fondamental)

Table 1 .

 1 Caractéristiques des deux corpus indexés

	Références dans les classes


Ivana Roche
email: ivana.roche@inist.fr

Nathalie Vedovotto
email: nathalie.vedovotto@inist.fr

Dominique Besagni
email: dominique.besagni@inist.fr

Claire François
email: claire.francois@inist.fr

Marianne Hörlesberger
email: marianne.hoerlesberger@ait.ac.at

Edgar Schiebel
email: edgar.schiebel@ait.ac.at

How to address the role of a journal as a knowledge transfer vector between disciplines? A case study relying on citations analysis

Extended abstract

Science and technology (ST) journals are a usual diffusion support for scientific results, in which researchers submit their works to peer evaluation and supply them to reuse by a scientific audience. In this diffusion process, each journal plays the role of a knowledge transfer vector. If we hypothesize that a journal has an observed audience perimeter either limited to its own scope or bridging the gap between its scope and other disciplines, then we address the issue: can we formalize a typology in order to characterize its knowledge transfer vector role? Can the analysis of ex ante citations (bibliographies of the articles published in the journal) and ex post citations (reuse of the journal articles) help us to detect how wide the perimeter of the journal audience is?

The approach we undertake in this work is directly inspired on the methodology we developed within the framework of the European project DBF1 (Hörlesberger et al., 2013), the goal of which was to support the selection process of research projects submitted for financing to the ERC (European Research Council). We have produced an analytical methodology based on the informetric modeling of criteria used by ERC scientific experts. For this purpose, indicators were elaborated in accord with the strategic definition of frontier research by the ERC. In particular, an indicator was devised to characterize any project that "…stands at the forefront of creating new knowledge… is responsible for fundamental discoveries… achieves occasional revolutionary breakthroughs…." (EC, 2005), that we defined as innovativeness. To this end, we studied the evolution of a scientific domain, using clustering techniques that generate a representation of the publication scientific landscape based on its extracted terminology, with a diachronic analysis of clustering results (Roche et al., 2011).

In the present work, we do not focus anymore on the evolution of a particular scientific field but on the comparative analysis of two landscapes built from a given ST journal: the former consists of the cited references, extracted from the bibliography, of all its publications in a given year; the latter consists of any article, published in any journal, that cites articles of the studied journal published in the said given year. As with a "spot the difference" game, we aim at answering the question: is there a difference between those two landscapes in term of scientific field representation? Specifically: does the second landscape include only fields already present in the first one, or conversely, display new fields? This diachronic analysis will help us define if the journal plays its role of knowledge transfer vector either exclusively within the restricted perimeter of its original scope or also beyond the frontier defined by this perimeter.

In the following we present the operated methodology, the extracted data and the tools used. The results of two case studies performed on two different journals are then discussed.

Methodology and data

All data are extracted from the Scopus database, which characteristics in terms of cited and citing articles fit perfectly our needs. For a given journal J, we choose a publication year Y with enough hindsight for the published articles to have been cited in more recent published papers. We search for all the corresponding articles and we extract the references cited in their bibliographies and available in the Scopus database. This first corpus, called BIB and represented as a green ellipse in fig. 1, can be considered as an image of the landscape of the scientific foundations for the works published in journal J. In a second time, we look for all the publications that cite the articles published during year Y in journal J. This second corpus, called CIT and represented as an orange ellipse in fig. 1, can be considered as an image of the scientific landscape of the works that are based on the results published in journal J during year Y. Our methodology is developed under these assumptions.

Figure 1 -Methodological schema of the evaluation process of the knowledge transfer vector role of a journal

A content analysis is then performed on both corpora. In order to perform it, each corpus of bibliographic records is formatted and integrated in two information analysis tools: Stanalyst (Polanco et al., 2001) and VOSviewer (Van Eck and Waltman, 2010). In both cases, a data mining step, based on automatic language processing techniques, is used to obtain an assisted indexing of the records by assigning at them keywords. A clustering is then applied to this enriched corpus. Thematic networks emerge from the relations between clusters and, according to a geographical metaphor, build a map of the corpus research landscape, as shown in figure 1. A scientific expert performs an analysis of the clusters in terms of the scientific fields which they deal with, and of their relative position and relations in the maps. Finally, the comparison of the analyses obtained for both corpora allows us to determine if they show thematic differences. Especially, the expert is asked to look for the occurrence of new scientific fields in CIT, with regards to BIB.

Case study

We choose two journals to implement our methodology, "Electronics Letters" (EL) and "Neural networks" (NN), and the publication year Y=2010. The number of extracted records for both corpora is shown in table 1. Figure 2 shows the cluster maps obtained with VOSviewer for NN. Initial results indicate that EL and NN play diversely their role of knowledge transfer vector. Indeed, EL seems to act autarkically, since its BIB and CIT cluster maps show a great similarity in terms of thematic fields. Conversely, as shown in fig. 2, NN seems to play its role of knowledge transfer vector in an interdisciplinary way, as its CIT map shows scientific fields that are not highlighted in its BIB map. For instance, a sub-network of keywords dealing with cellular automata and optimization, absent from the BIB map, emerges in the CIT one. This expansion of the original scientific perimeter of NN could bring us to conclude that this journal has a wider scientific audience than EL. This led us to discuss the relevance of a complementary comparative study of maps, by the yardstick of an applicability criterion, in order to refine our analysis. More detailed results will be presented in the full paper.

Figure 2 -

 2 Figure 2 -Cluster maps obtained for Neural Networks, 2010. Left: cited articles (BIB corpus), right: citing articles (CIT corpus)

  EC -European Commission (2005). Frontier research: The European Challenge. High Level Expert Group Report, EUR 21619. Hörlesberger M., Roche I.,Besagni D., Scherngell T., Francois C., Cuxac P., Schiebel E., Zitt M., and Holste D. (2013). A concept for inferring 'frontier research' in grant proposals, Scientometrics (to appear) Polanco X., François C., Royauté J., Besagni D., Roche I.(2001). STANALYST: An integrated environment for clustering and mapping analysis on science and technology, 8 th ISSI conference, Sydney, Australia, Proceedings Vol. 2, 871-873 Roche I.,Ghribi M., Vedovotto N., François C., Besagni D., Cuxac P., Holste D., Hörlesberger M., Schiebel E. (2011). Detecting domain dynamics: Association Rule Extraction and diachronic clustering techniques in support of expertise, 1 st Global TechMining conference, Atlanta, USA. Van Eck, N.J.,Waltman, L. (2010). Software survey: VOSviewer, a computer program for bibliometric mapping. Scientometrics, 84(2), 523-538

Table 1 -Size of the extracted corpora of cited (BIB) and citing articles (CIT), for both journals (publication year = 2010)

 1

This work was partially inspired by DBF (Development and Verification of a Bibliometric Model for the Identification of Frontier Research), a Coordination and Support Action of the IDEAS specific programme of the European Research Council (ERC). The authors wish to acknowledge this contribution.

CollNet'13/ROCETAL/Extended Abstract


Pascal Cuxac
email: pascal.cuxac@inis.fr

Jean-Charles Lamirel
email: jean-charles.lamirel@loria.fr

Analysis of evolutions and interactions between science fields: the cooperation between feature selection and graph representation

This paper presents the application of a new method of feature selection for the analysis of the evolution and interaction of scientific domains. The query of bibliographic databases provides a corpus of scientific publications in different fields. Every scientific field is considered as a class issued from a machine learning process, either supervised or unsupervised, and each document is represented by a bag of words. Therefore, it is possible to select the most significant words of every class (domain). We then represent the words-classes relationships by a graph whose edges are weighted by a function of contrast. Thus we highlight the specific words for each area and those that are multidisciplinary. In addition, the joint analysis of two periods of time allows us to appreciate the evolution of the scientific fields.

Introduction

On the one hand, the development of dynamic information analysis methods, like incremental clustering and novelty detection techniques, is becoming a central concern in a bunch of applications whose main goal is to deal with large volume of textual information which is varying over time.

The purpose of the analysis and diachronic mapping is to track, for a given domain, changes in contexts (sub-themes) and the evolution of vocabularies and actors that materialize these changes in terms of appearances, disappearances, divergence or convergence. The applications relate to very various and highly strategic domains, including web mining, technological and scientific survey.

In order to identify and analyze the emergence, or to detect changes in the data, we have previously proposed two different and complementary approaches:

1. performing static classifications at different periods of time and analyze changes between these periods (time step approach or diachronic analysis); 2. developing methods of classification that can directly track the changes: incremental clustering methods (incremental clustering) and novelty detection methods (incremental supervised classification).

On the other hand, the concept of transdisciplinarity is often discussed in conjunction with its facets that are interdisciplinarity, multidisciplinarity, pluridisciplinarity [START_REF] Zaman | Multidisciplinarity, Interdisciplinarity and Transdisciplinarity: Theoretical Approaches and Implications for the Strategy of Post-Crisis Sustainable Development[END_REF]. As noted by Alvargonzalez (Alvargonzalez, 2011), the terms multidisciplinarity, interdisciplinarity, and transdisciplinarity are often used interchangeably. However these concepts can be defined as follows (Do Espirito Santo 1979):  Interdisciplinarity: interaction among different disciplines;  Multidisciplinarity: juxtaposition of various disciplines (without apparent connection between them);  Pluridisciplinarity: juxtaposition of various disciplines more or less related;  Transdisciplinarity: a common system for a set of disciplines.

Many authors are interested in graph representation for estimating the interdisciplinarity of science. The goal is usually to determine whether a paper (or journal) is "interdisciplinary" or not. Using subject categories of Currents Contents, Adams et al. [START_REF] Adams | Bibliometric analysis of interdisciplinary research: Report to the Higher Education Funding Council for England[END_REF] define "interdisciplinarity index" based on the cited references and the "Shannon diversity index". Similarly, [START_REF] Porter | Is science becoming more interdisciplinary? Measuring and mapping six research fields over time[END_REF], using the subject categories of Web of Science, define the NAFKI interdisciplinarity metrics and relies on the representation method developed by Leysdesdorff [START_REF] Leydesdorff | Betweenness centrality as an indicator of the interdisciplinarity of scientific journals[END_REF]. On their own side, Leysdesdorff and al. [START_REF] Leydesdorff | A global map of science based on the ISI subject categories[END_REF] uses the ISI subject categories included in Science Citation Index and builds graphs using citing and cited dimensions, in order to map different scientific fields. The calculation of "betweenness centrality" allows to measure interdisciplinarity [START_REF] Leydesdorff | Betweenness centrality as an indicator of the interdisciplinarity of scientific journals[END_REF].

Van [START_REF] Van Raan | The interdisciplinary nature of science: theoretical framework and bibliometricempirical approach[END_REF] presents the interdisciplinary nature of science as an interaction of socio-economic problems, scientifically interesting problems, and interdisciplinarity. He uses bibliometric approaches to highlight the interdisciplinarity. Some work focuses on authors: Schummer [START_REF] Schummer | Multidisciplinarity, Interdisciplinarity, and patterns of research collaboration in nanoscience and nanotechnology[END_REF] is interested in collaboration between researchers (or institutions) for addressing the multidisciplinary field in nanoscience. Klein emphasizes that identifying experts is crucial, because they form an interdisciplinary appropriate epistemic community [START_REF] Klein | Evaluation of Interdisciplinary and Transdisciplinary Research: A Literature Review[END_REF].

We present hereafter an original word-based approach using feature maximization metric [START_REF] Lamirel | A new feature selection and feature contrasting approach based on quality metric: application to efficient classification of complex textual data[END_REF]) in order to detect significant differences between two time periods for the same scientific field, but also to detect transdisciplinary terms that are markers of crossdomain scientific collaborations. We show that our approach is also applicable to the authors (i.e. actors) allowing quickly highlighting those that are "bridges" between scientific fields.

Unlike common approaches based on graph analysis [START_REF] Porter | Is science becoming more interdisciplinary? Measuring and mapping six research fields over time[END_REF] [START_REF] Sayama | Characterizing Interdisciplinarity of Researchers and Research Topics Using Web Search Engines[END_REF], we are tackling the problem using a classification of documents (in scientific fields) in combination with a selection of features (index keywords) associated with document classes. Only then, we construct a graph visualizing the interaction between keywords and classes using links weighted by values of contrast defining the strength of the relation between these latter. Feature selection and links contrasting are based on the feature maximization metric (F-max) that has been already successfully used in an unsupervised context.

Feature selection

Since the 1990s, advances in computing and storage capacity allow the manipulation of very large data: it is not uncommon to have description space of several thousand or even tens of thousands of variables. One might think that classification algorithms are more efficient if there are a large number of variables. However, the situation is not as simple as this. The first problem that arises is the increase in computation time. Moreover, the fact that a significant number of variables are redundant or irrelevant to the task of classification significantly perturbs the operation of the classifiers. In addition, as soon as most learning algorithms exploit probabilities, probability distributions can be difficult to estimate in the case of the presence of a very high number of variables. The integration of a variable selection process in the framework of the classification of high dimensional data becomes thus a central challenge.

In the literature, three types of approaches for variable selection are mainly proposed: the integrated (embedded) approaches, the "wrapper" methods and the filter approaches. An exhaustive overview of the state-of-the-art techniques in this domain has been achieved by many authors, like Ladha and al. [START_REF] Ladha | Feature selection methods and algorithms[END_REF], Bolón-Canedo and al. (Bolón-Canedo, Sanchez-Maroño and Alonso-Betanzos, 2012), [START_REF] Guyon | An introduction to variable and feature selection[END_REF] or Daviet [START_REF] Daviet | Class-Add, une procédure de sélection de variables basée sur une troncature kadditive de l'information mutuelle et sur une classification ascendante hiérarchique en prétraitement[END_REF]. For an overview of these methods, you might refer to the previous articles, as well as to [START_REF] Lamirel | A new feature selection and feature contrasting approach based on quality metric: application to efficient classification of complex textual data[END_REF].

Our approach

To clarify the principle of our approach, which we named GRAFSEL, we follow four steps that are schematically presented in Figure 1: -We query a bibliographic database (PASCAL is used in our context) to produce corpora for each selected scientific fields (following classification codes or subject category for example) and/or time period; -The bibliographic records of each corpus are assigned to a class that represents the scientific field and/or the period. Doing so, we build up a classification mixing topics and time periods; -The records being represented their associated keywords, we select keywords related to each scientific field and/or period and compute the strength of the relations (i.e. contrast) between keywords and scientific fields and/or periods exploiting the feature maximization metric (F max) shortly described after; -The last step is to build the graph highlighting the relationships between the fields and the keywords by weighting the links of the graph with the formerly obtained contrast values. Feature maximization for variable selection -Feature maximization metric principles in unsupervised learning Feature maximization (F-max) is an unbiased cluster quality metrics that exploits the properties of the data associated to each cluster without prior consideration of clusters profiles. This metrics has been initially proposed in [START_REF] Lamirel | New classification quality estimators for analysis of documentary information: application to patent analysis and web mapping[END_REF]. Its main advantage is to be independent altogether of the clustering methods and of their operating mode.

Consider a set of data D represented with a set of features F, and a set of clusters C resulting from a clustering method. Feature maximization promotes clusters with maximum Feature F-measure. The Feature F-measure of a feature f associated to a cluster c is defined as the harmonic mean of Feature Recall and Feature Precision indexes which in turn are defined as:

, (1) (2)
where represents the weight of the feature f for data d and F c represent the set of features occurring in the data associated to the cluster c.

Two important application of the feature maximization metric are related to the estimation of the overall clustering quality and to incremental clustering [START_REF] Lamirel | A new approach for automatizing the analysis of research topics dynamics: application to optoelectronics[END_REF].

-Adaptation of feature maximization metric for feature selection in supervised learning Taking into consideration the basic definition of feature maximization metric presented in the former section, its exploitation for the task of feature selection in the context of supervised learning becomes a straightforward process, as soon as this generic metric can apply on data associated to a class as well as to those associated to a cluster. The feature maximizationbased selection process can thus be defined as a parameter-free class-based process in which a class feature is characterized using both its capacity to discriminate a given class from the others ( and its capacity to accurately represent the class data ( .

The set S c of features that are characteristic of a given class c belonging to an overall class set C results in:

(

where and .

and C /f represents the restriction of the set C to the classes in which the feature f is represented.

Finally, the set of all the selected features S C is the subset of F defined as:

(4)

Features that are judged relevant for a given class are the features whose representation is altogether better than their average representation in all the classes including those features and better than the average representation of all the features, as regard to the feature F-measure metric.

In the specific framework of the feature maximization process, a contrast enhancement step can be exploited complementary to the former feature selection step. The role of this step is to fit the description of each data to the specific characteristic of its associated class which have been formerly highlighted by the feature selection step [START_REF] Lamirel | A new feature selection and feature contrasting approach based on quality metric: application to efficient classification of complex textual data[END_REF]). In the case of our metric, it consists in modifying the weighting scheme of the data specifically to each class by taking into consideration the information gain provided by the Feature F-measures of the features, locally to that class.

Thanks to the former strategy, the information gain provided by a feature in a given class is proportional to the ratio between the value of the Feature F-measure of this feature in the class and the average value of the Feature F-measure of the said feature on all the partition.

Experimental results

-The datasets

We present the first results obtained from a corpus of bibliographic records extracted from the PASCAL database. Our experimental corpus included documents from the following six scientific fields: Physics, Geology, Electronics, Medicine (diagnosis techniques), Information Science and Linguistics, for the years 1995 and 2012 (respectively 61 109 and 64 036 scientific papers), distributed as shown in table 1.

-The results

The F-max feature selection method implemented here allows significantly reducing the number of features (words) in order to keep only the most important words (representative) of each class (Tab. 2). Of course, we thus eliminate words that could be found in several classes (domains), but our goal is to focus on the "words of specialties" by forgetting the more general words such as "analysis", "study", "method" or "model". Figure 2 shows the results obtained by taking into account altogether the whole corpus and the two time periods considered. For reasons of readability, we separate the main groups. All of the following graphs are obtained with a force-directed algorithm (spring algorithm).

After having computed a global graph showing general interactions between the fields, we have then separated the fields into two main groups in which strong interaction and/or evolution are more liable to occur.

Figure 3 shows the graph obtained with all the selected fields of the second group which includes Physics, Electronics and Medicine. Looking to such graph, we must keep in mind that all phenomena that we visualize describe the corpus as a whole and cannot thus be interpreted in an absolute way by isolating a class from the others, because the nodes (keywords) of the graph are issued from the overall classification of the data (it is the same for the weights of the links). Referring to figure 2, we can however observe several interesting scenarios occurring in a parallel way in our experimental data:

-Evolution of scientific fields: the resulting graph highlights that, compared to other fields, the field of Medicine (and especially the "technical diagnosis" discipline) individualizes well into two groups for each analyzed period. Indeed, on the one side, we can observe a dense cloud of keywords (M-M) that are common to both periods but have a relatively weak link with the M95 and M12 classes. On the other side, keywords like "Sleep Disorder", "Hypercholesterolemia", "Forensic aspect", "Radiosurgery" are specific terms in 2012 (i.e. belong to M12 class), while keywords like "Arteriography", "Angina pectoris", "Heart valve", "Lymphocytic leukemia" likely belong to 1995 (M15 class). In an alternative way, keywords clouds related to Physics (and to a lesser extent the ones related to Electronics) remains homogeneous, whatever is the considered period, indicating less important temporal changes;

-Transdisciplinarity: there are small groups of keywords with share relationship with several different main clouds. These are transdisciplinary vocabularies reflecting cooperation between scientific fields or practical applications of new technologies: M-E keywords linking Medicine to Electronics, M-P keywords linking Medicine and Physics, E-P keywords linking Electronics and Physics, and finally, M-E-P keywords linking the three fields.

Conclusion

We have shown from a simple example that our fully unsupervised method allows a non-expert user to view the terms used in various disciplines and their temporal evolution. In a complementary way, with such method, it is also easy for the said user to distinguish the terms common to several topics. Depending on the case, the original graph might be too dense for clear viewing, but it is then easy to select a subgraph on which the analysis can be conducted. The originality of our GRAFSEL approach comes from the fact that the nodes of the graph result of the combination of a classification and a feature selection processes, which are applied in a first step, and the links of the graph result from a feature contrasting process, which is applied in a second step on the selected features (i.e. nodes). Hence, unlike the commonly used methods, we do not build here a graph of words but a graph of relationships between words and classes, each class representing a main domain under consideration. In such a way, another interesting application of our approach may be the detection of the authors which represent "bridges" between scientific fields. In figure 4, we see that in 18 years the landscape of the authors has been considerably renewed and, in addition, we can identify groups of people who are "Transmitters of Knowledge" between scientific domains or between periods of the same domains.

Last but not least, supervised as well as unsupervised classification methods can be used in this process.

Fig. 1 :

 1 Fig. 1: Principle of the GRAFSEL approach.

FigFig. 3 :

 3 Fig. 2: Global words-classes graph (G=Geology;P=Physics;E=Electronics;M=Medicine;S=Information-Science; L=Linguistics;12=2012;95=1995).

Fig. 4 :

 4 Fig. 4: Authors-classes graph for a subgroup of 3 scientific fields (P=Physics; E=Electronics; M=Medicine; 12=2012; 95=1995).

Tab. 2: Number of keywords before and after F-max feature selection

	Domain	Original keywords Selected keywords (%) selected
	Electronic_1995 (E15)	12813	1273	9.94
	Electronic_2012 (E12)	11706	2193	18.73
	Geology_1995 (G95)	16124	1613	10.00
	Geology_2012 (G12)	13768	1856	13.48
	Information science_1995 (S95)	4915	1163	23.66
	Information science_2012 (S12)	2338	365	15.61
	Linguistic_1995 (L95)	20186	322	1.59
	Linguistic_2012 (L12)	29093	322	2.72
	Medicine_1995 (M95)	10138	886	8.74
	Medicine_2012 (M12)	10326	1037	10.04
	Physics_1995 (P95)	12268	1051	8.57
	Physics_2012 (P12)	15397	1894	12.30


Hassan Bakouch
email: hbakouch@kau.edu.sa

Maher Kachour
email: maher.kachour@idraclyon.com

Saralees Nadarajah

An extended Poisson distribution

Keywords: 2000 Mathematics Subject Classification 62G30, 62E10 Estimation, Moments, Poisson distribution, Skellam distribution

The Poisson distribution is extended over the set of all integers. The motivation comes from the many reflected versions of the gamma distribution, the continuous analog of the Poisson distribution, defined over the entire real line. Various mathematical properties of the extended Poisson distribution are derived. Estimation procedures by the methods of moments and maximum likelihood are also derived with their performance assessed by simulation. Finally, a real data application is illustrated.

Introduction

It is well known that the Poisson distribution is the discrete analog of the gamma distribution. The gamma distribution is a popular model for continuous data. Many extensions of the gamma distribution have been proposed in the literature for improved modeling, see [START_REF] Johnson | Continuous Univariate Distributions[END_REF][START_REF] Johnson | Continuous Univariate Distributions[END_REF]. These extensions include reflected versions of the gamma distribution defined over the entire real line. The first reflected gamma distribution was proposed by [START_REF] Borghi | Sobre una distribución de frecuencias[END_REF]. This distribution was defined by taking a mirror image of the probability density function of the gamma distribution along the y axis. Several other reflected gamma distributions have been proposed since [START_REF] Borghi | Sobre una distribución de frecuencias[END_REF]: [START_REF] Ali | Skew-symmetric reflected distributions[END_REF] Each of the reflected gamma distributions including the Laplace distribution has received many real data applications. But data are never continuous. Data are discrete by nature. So, it would be useful to have reflected versions of the Poisson distribution. We are not aware of any reflected version of the Poisson distribution proposed in the literature. Another motivation is that the distribution is under-dispersed, equi-dispersed and over-dispersed, see Section 2.5.

The aim of this paper is to propose the first reflected version of the Poisson distribution, referred to as the extended Poisson (E-Po) distribution. We say that a random variable X ∼ E-Po(p, λ) with parameters λ > 0 and 0 ≤ p ≤ 1 if its probability mass function is defined as

P(X = k) =                        e -λ , if k = 0, pe -λ λ k k! , if k = 1, 2, . . .,
(1 -p)e -λ λ |k| |k|! , if k = . . . , -2, -1.

(

) 1
In other words, for all x ∈ Z, we have

P (X = x) = pP (Y = x) 1 {x≥0} + (1 -p)P (Y = |x|) 1 {x≤0} , (2)
where Y is a Poisson random variable with parameter λ > 0. Note that -If p = 1, then X has the same distribution than that of Y ; -If p = 0, then X has the same distribution than that of -Y .

Note also that if

X ∼ E-Po(p, λ) then -X ∼ E-Po(1 -p, λ) and |X| is a Poisson random variable with parameter λ.
The contents of this paper are organized as follows. Various mathematical properties of the E-Po distribution are derived in Section 2. These include the cumulative distribution function (Section 2.1), the quantile function (Section 2.2), the survival function (Section 2.3), the failure rate function (Section 2.3), the reverse failure rate function (Section 2.3), the conditional moments (Section 2.4), the probability generating function (Section 2.5), the moments (Section 2.5), the Shannon entropy (Section 2.6), the mean deviation about the mean (Section 2.7), the mean deviation about the median (Section 2.7), the distribution of sums and differences (Section 2.8). Estimation procedures by the method of moments (Section 3.1) and the method of maximum likelihood (Section 3.2) as well as a simulation study (Section 3.3) to assess their performance are given in Section 3. Finally, Section 4 illustrates practical usefulness of the E-Po distribution.

Mathematical properties

Cumulative distribution function

Let F X (x) = P(X ≤ x) denote the cumulative distribution function and ⌊•⌋ the floor function. Furthermore, for all x ≥ 0 recall that

e -λ ⌊x⌋ k=0 λ k k! = Γ (⌊x + 1⌋, λ) ⌊x⌋! ,
where Γ (•, •) denotes the incomplete gamma function. Suppose now that x < 0. Then, we have

F X (x) = ⌊x⌋ k=-∞ P(X = k) = (1 -p)e -λ ⌊x⌋ k=-∞ λ |k| |k|! = (1 -p)e -λ ∞ k=-⌊x⌋ λ k k! = (1 -p) 1 - Γ (-⌊x⌋, λ) (-⌊x⌋ -1)! . Now if 0 ≤ x < 1, then F X (x) = P(X = 0) + F X (-1) = e -λ + (1 -p) (1 -Γ (1, λ)) = (1 -p) + pe -λ .
Finally, for x ≥ 1, we have

F X (x) = ⌊x⌋ k=-∞ P(X = k) = F X (0) + ⌊x⌋ k=1 P(X = k) = (1 -p) + pe -λ + pe -λ ⌊x⌋ k=1 λ k k! = (1 -p) + pe -λ ⌊x⌋ k=0 λ k k! = (1 -p) + p Γ (⌊x + 1⌋, λ) ⌊x⌋! .

Quantile function

Letting Q(a, x) = Γ (a, x)/Γ (a) denote the regularized incomplete gamma function, we can rewrite the cdf of X as

F X (x) =                        (1 -p) [1 -Q (-⌊x⌋, λ)] , if x < 0, (1 -p) + pe -λ , if 0 ≤ x < 1, (1 -p) + pQ (⌊x + 1⌋, λ) , if x ≥ 1.
Hence, the quantile function of X can be expressed as

F -1 X (q) =                                    -Q -1 1 - q 1 -p , λ , if q < 1 -p [1 -Q(0, λ)], 0, if 1 -p [1 -Q(0, λ)] ≤ q < (1 -p) + pe -λ , Q -1 q -1 + p p , λ -1, if q ≥ (1 -p) + pe -λ ,
where Q -1 (a, λ) denotes the inverse function of Q(a, λ). This quantile function can be used to simulate variates of the EP distribution. The same simulation can be performed perhaps more easily by using (2).

R X (x) = P(X > x) = 1 -F X (x) =              pR Y (x), if x ≥ 0, pR Y (-x -1) + F Y (-x -1), if x ≤ -1, k X (x) = P(X = x | X ≥ x) = P(X = x) R X (x -1) =                          k Y (x), if x ≥ 1, e -λ p(1 -e -λ ) + e -λ , if x = 0, (1 -p)k Y (-x) p R Y (-x) R Y (-x-1) + F Y (-x) R Y (-x-1) , if x ≤ -1, and
τ X (x) = P(X = x | X ≤ x) = P(X = x) F X (x) =                          τ Y (x) 1 + (1-p) pF Y (x) , if x ≥ 1, e -λ pe -λ + (1 -p) , if x = 0, k Y (-x), if x ≤ -1,
respectively. Figures and show respectively the behaviour of the failure rate and the reverse failure functions of X, for different values of λ and p

Conditional moments

For lifetime models, it is of interest to know the conditional measure E{(Xx) k |X > x}, k = 1, 2, ..., which is important in prediction and can be used to measure the reliability of systems. For all x ∈ Z, the kth-order conditional moment is

c k X (x) = E (X -x) k | X ≥ x = ∞ j=x (j -x) k P(X = j) R X (x -1) .
In particular, for k = 1, 2, we have

c 1 X (x) = ∞ j=x R X (j) R X (x -1) =                        c 1 Y (x), if x ≥ 1, p(1 -e -λ ) p(1 -e -λ ) + e -λ c 1 Y (1) + 1 , if x = 0, -1 j=x R X (j) + c 1 X (0) p + (1 -p)e -λ R X (x -1) , if x ≤ -1, and
c 2 X (x) = 2 ∞ j=x jR X (j) R X (x -1) -(2x -1)c 1 X (x) =                        c 2 Y (x), if x ≥ 1, p(1 -e -λ ) p(1 -e -λ ) + e -λ c 2 Y (1) + c 1 Y (1) + c 1 X (0), if x = 0, Q(x) -(2x -1)c 1 X (x), if x ≤ -1,
where

Q(x) = p C 1 Y (1) + C 2 Y (1) R Y (0) + 2 -1 j=x jR X (j) R X (x -1)
and c k Y (x) be the kth-order conditional moment of Y . Note that c 1 X is called the mean residual life function of X, and the variance residual life function of X is

β X (x) = V (X -x | X ≥ x) = c 2 X (x) -c 1 X (x) 2 .

Probability generating function and Moments

Several interesting characteristics of a distribution can be studied by its probability generating function and moments, so we get them to the E-Po distribution. Define

G X,+ (s) = pP(X = 0) + ∞ k=1 s k P(X = k) = pG Y (s),
and

G X,-(s) = (1 -p)P(X = 0) + -1 k=-∞ s |k| P(X = k) = (1 -p)G Y (1/s),
where

G Y (s) = e -λ(1-s) is the pgf of Y , a Poisson random variable with parameter λ. Then, the pgf of X is G X (s) = e -λ pe sλ + (1 -p)e λ s . (3)
Fors = -∞, the pgf (3) is given by G X (-∞) = (1 -p)e -λ . This quantity is known as the survival fraction and it plays an important role in long-term survival analysis. For more details, see [START_REF] Rodrigues | On the unification of the long-term survival models[END_REF].

The characteristic function of X follows from (3) as

ϕ X (t) = E e itX = pe -λ(1-e it ) + (1 -p)e -λ(1-e -it )
,

where i = √ -1. For any r ∈ N \ {0}, the rth-order descending factorial moment is µ X (r) = E [X(X -1) . . . (X -(r -1))] = pG (r) Y (1) + (-1) r (1 -p)G (r) Y ′ (1), where G Y ′ (s) = s r-1 G Y (s). For any k ∈ N \ {0}, the kth-order moment of X is E X k =                        M (k) Y (0), if k even, (2p -1)M (k) Y (0), if k odd, where M Y (t) = E e tY = G Y e t = e -λ(1-e t ) .
Thus, the first four moments of X are

E(X) = (2p -1)λ, E X 2 = λ(1 + λ), E X 3 = (2p -1)λ 1 + 3λ + λ 2 , E X 4 = λ 1 + 7λ + 6λ 2 + λ 3 .
The corresponding variance, coefficient of variation, skewness and kurtosis are

V(X) = λ + 4p(1 -p)λ 2 , CV(X) = σ X λ(2p -1)
,

Skewness(X) = E (X -E(X)) 3 σ 3 X = λ 3 (1 -3(2p -1) + (2p -1) 3 ) σ 3 X + 6λ 2 (1 -p) σ 3 X + λ σ 3 X and Kurtosis(S) = E (X -E(X)) 4 σ 4 X -3 = λ 4 (1 + 2(2p -1) 2 -3(2p -1) 4 ) σ 4 X + 6λ 3 (1 -(2p -1) 2 ) σ 4 X + λ 2 (7 -4(2p -1) 2 ) σ 4 X + λ σ 4 X -3,
where

σ X = λ + 4p(1 -p)λ 2 .
The index of dispersion is defined as the ratio of the variance to the mean, i.e.,

ID = V(X) E(X) = 1 + 4p(1 -p)λ (2p -1)
.

Note that ID = -1, for p = 0; ID < 0, for 0 < p < 1 2 ; ID = 1, for p = 1 and ID > 0, for 1 2 < p < 1. So, the E-Po distribution is equi-dispersed if p = 0 or p = 1. On the other hand, one can see that ID < 1 ⇔ λ < -1 2p which is not possible since λ > 0 and ID > 1 ⇔ λ > -1 2p which is always real for the same reason. Similary, we have that ID > -1 ⇔ λ > - 1 2(1-p) which is always real. Thus, one can deduce that E-Po distribution is over-dispersed if 1 2 < p < 1 and under-dispersed if 0 < p < 1 2 .

Entropy

Entropy is used to measure the randomness of systems and it is widely used in various sciences such as physics. In this section, we give the Shannon entropy, which is the most popular entropy, defined by E[-log p(X)], where p(k) = P(X = k) denotes the probability mass function. Using ( 1), we obtain

E [-log p(X)] = -log p(0)P(X = 0) - ∞ k=1 log p(k)P(X = k) - -1 k=-∞ log p(k)P(X = k) = λe -λ + p [λ -log p] e -λ ∞ k=1 λ k k! -pe -λ log λ ∞ k=1 λ k (k -1)! +pe -λ ∞ k=1 log k! λ k k! +(1 -p) [λ -log(1 -p)] e -λ -1 k=-∞ λ -k (-k)! -(1 -p)e -λ log λ -1 k=-∞ (-k) λ -k (-k)! +(1 -p)e -λ -1 k=-∞ log(-k)! λ -k (-k)! = λ -[p log p + (1 -p) log(1 -p)] 1 -e -λ -λ log λ +e -λ ∞ k=1 log k! λ k k! . (4)
Note that ( 4) is a closed form expression expect for the single infinite summation.

Mean deviations

The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and median. These are known as the mean deviation about the mean and the mean deviation about the median -defined by

δ 1 (X) = E [|k -E(X)|] = ∞ k=-∞ |k -E(X)| P(X = k) (5)
and

δ 2 (X) = E [|k -Median(X)|] = ∞ k=-∞ |k -Median(X)| P(X = k), (6)
respectively. The measures can be calculated using the general relationship that

δ(X) = E [|k -m|] = ∞ k=-∞ |k -m| P(X = k) = ∞ k=⌊m⌋+1 (k -m)P(X = k) + ⌊m⌋ k=-∞ (m -k)P(X = k) = ∞ k=⌊m⌋+1 kP(X = k) - ⌊m⌋ k=-∞ kP(X = k) -m + 2m ⌊m⌋ k=-∞ P(X = k) = 2 ∞ k=⌊m⌋+1 kP(X = k) -E(X) -m + 2mF X (m), (7)
where E(X) = (2p -1)λ and F X (•) is given by Section 2.1. Substituting e -λ ⌊x⌋ k=0 λ k k! , the summation term in [START_REF] Inusah | A discrete analogue of the Laplace distribution[END_REF] can be expressed as

∞ k=⌊m⌋+1 kP(X = k) = pe -λ ∞ k=⌊m⌋+1 λ k (k -1)! = pλe -λ ∞ k=⌊m⌋ λ k k! =                          pλ + pλe -λ -1 k=⌊m⌋ λ k k! , if ⌊m⌋ ≤ -1, pλ, if ⌊m⌋ = 0, pλ 1 - Γ (⌊m⌋, λ) ⌊m -1⌋! , if ⌊m⌋ ≥ 1. (8)
Combing ( 7) and ( 8), we have an expression for δ(X). Expressions for the mean deviations in ( 5) and ( 6) can be obtained as particular cases.

Distribution of sums and differences

It is well known that the sum of independent Poisson random variables follows a Poisson distribution. It is also well known that the difference of two independent Poisson random variables follows a Skellam distribution (Skellam, 1946). But these properties do not hold for E-Po random variables, see Section 2.5. Here, we derive the distributions of the sum and the difference of two independent E-Po random variables and get their moments.

Suppose X 1 ∼ E-Po (p 1 , λ 1 ) and X 2 ∼ E-Po (p 2 , λ 2 ) are independent random variables. Since X ∼ E-Po(p, λ) implies -X ∼ E-Po(1 -p, λ), we only consider the distribution of the sum X 1 + X 2 = S say. For k > 0, we have

P(S = k) = P(X = 0)P(Y = k) + P(X = k)P(Y = 0) + k-1 j=1 P(X = j)P(Y = k -j) + -1 j=-∞ P(X = j)P(Y = k -j) + -1 j=-∞ P(Y = j)P(X = k -j) = p 2 e -λ1-λ2 λ k 2 k! + p 1 e -λ1-λ2 λ k 1 k! + p 1 p 2 e -λ1-λ2 k-1 j=1 λ j 1 λ k-j 2 j!(k -j)! + (1 -p 1 ) p 2 e -λ1-λ2 -1 j=-∞ λ -j 1 λ k-j 2 (-j)!(k -j)! +p 1 (1 -p 2 ) e -λ1-λ2 -1 j=-∞ λ k-j 1 λ -j 2 (k -j)!(-j)! = p 2 e -λ1-λ2 λ k 2 k! + p 1 e -λ1-λ2 λ k 1 k! +p 1 p 2 e -λ1-λ2 k-1 j=1 λ j 1 λ k-j 2 j!(k -j)! + (1 -p 1 ) p 2 e -λ1-λ2 λ k 2 k! [ 0 F 1 (; k + 1; λ 1 λ 2 ) -1] +p 1 (1 -p 2 ) e -λ1-λ2 λ k 1 k! [ 0 F 1 (; k + 1; λ 1 λ 2 ) -1] , (9)
where 0 F 1 (; a; x) denotes a hypergeometric function defined by

0 F 1 (; a; x) = ∞ k=0 1 (a) k x k k! , where (f ) k = f (f + 1) • • • (f + k -1)
denotes the ascending factorial. Similar calculations for k = 0 show that

P(S = 0) = e -λ1-λ2 + (1 -p 1 ) p 2 e -λ1-λ2 [ 0 F 1 (; 1; λ 1 λ 2 ) -1] +p 1 (1 -p 2 ) e -λ1-λ2 [ 0 F 1 (; 1; λ 1 λ 2 ) -1] . (10)
For k < 0, we have

P(S = k) = (1 -p 2 ) e -λ1-λ2 λ -k 2 (-k)! + (1 -p 1 ) e -λ1-λ2 λ -k 1 (-k)! + (1 -p 1 ) (1 -p 2 ) e -λ1-λ2 k-1 j=1 λ -j 1 λ j-k 2 (-j)!(j -k)! + (1 -p 1 ) (1 -p 2 ) e -λ1-λ2 -1 j=-∞ λ -j 1 λ j-k 2 (-j)!(j -k)! + (1 -p 1 ) (1 -p 2 ) e -λ1-λ2 -1 j=-∞ λ -j 2 λ j-k 1 (-j)!(j -k)! = (1 -p 2 ) e -λ1-λ2 λ -k 2 (-k)! + (1 -p 1 ) e -λ1-λ2 λ -k 1 (-k)! + (1 -p 1 ) (1 -p 2 ) e -λ1-λ2 k-1 j=1 λ -j 1 λ j-k 2 (-j)!(j -k)! + (1 -p 1 ) (1 -p 2 ) e -λ1-λ2 λ -k 2 (-k)! 1 + λ 1 λ 2 -k + (1 -p 1 ) (1 -p 2 ) e -λ1-λ2 λ -k 1 (-k)! 1 + λ 2 λ 1 -k = (1 -p 2 ) e -λ1-λ2 λ -k 2 (-k)! + (1 -p 1 ) e -λ1-λ2 λ -k 1 (-k)! + (1 -p 1 ) (1 -p 2 ) e -λ1-λ2 k-1 j=1 λ -j 1 λ j-k 2 (-j)!(j -k)! + 2 (1 -p 1 ) (1 -p 2 ) e -λ1-λ2 (-k)! (λ 1 + λ 2 ) k . (11)
To find the distribution of the difference X 1 -X 2 , just replace p 2 by 1 -p 2 in (9), [START_REF] Rodrigues | On the unification of the long-term survival models[END_REF] and [START_REF] Skellam | The frequency distribution of the difference between two Poisson variates belonging to different populations[END_REF].

The moments of X 1 + X 2 and X 1 -X 2 can be calculated using the facts that

E (X 1 + X 2 ) k = k m=0 k m E [X m 1 ] E X k-m 2 and E (X 1 -X 2 ) k = k m=0 k m (-1) k-m E [X m 1 ] E X k-m 2 ,
respectively.

Estimation and inference with simulation

Let X 1 , . . . , X n denote a sample of i.i.d. observations from the EP distribution with parameters p and λ. Here, we consider estimation of the parameters by the method of moments and the method of maximum likelihood with inference of the distribution parameters. Also, simulation results on the behavior of estimators are presented.

Method of moments

Since E (|X|) = λ, a method of moments estimator of λ is

λ n = 1 n n i=1 |X i |. (12)
Since P(X = 0) = e -λ , another method of moments estimator of λ is

λ n = -log 1 n n i=1 1 {Xi=0} . ( 13
)
The method of moments estimator for p can be obtained using moments of X given in Section 2.5: the ones corresponding ( 12) and ( 13) are

p n = 1 2       n i=1 X i nλ n + 1       (14)
and

p n = 1 2       n i=1 X i n λ n + 1       , (15)
respectively. The exact sampling distributions of λ n and λ n follow by noting that

n i=1 |X i | ∼ Poisson(nλ) (16)
and

n i=1 1 {Xi=0} ∼ Binomial n, e -λ , (17)
respectively. The exact sampling distributions of p n and p n are difficult to find because the sum of independent E-Po random variables does not have a closed form distribution function, see Section 2.8.

Method of maximum likelihood

Set x = (x 1 , . . . , x n ) and θ = (λ, p). The likelihood function is

L (x, θ) = e -nλ λ n i=1 |xi| n i=1 |x i |! n i=1 p1 {xi≥0} + (1 -p)1 {xi≤0} . The log likelihood function is log L (x, θ) = -nλ + n i=1 |x i | log λ - n i=1 log |x i |! + n i=1 log p1 {xi≥0} + (1 -p)1 {xi≤0} .
The maximum likelihood estimator θ n = λ n , p n is defined as θ n = argmax θ log L (x, θ) .Thus, the maximum likelihood estimator of λ is

λ n = 1 n n i=1 |x i |. ( 18
)
That of p is

p n = n i=1 1 {xi>0} n i=1 1 {xi>0} + n i=1 1 {xi<0} . ( 19
)
The exact sampling distribution of λ n follows from (16). The exact sampling distribution of p n is difficult to obtain. It can be approximated by asymptotic normality.

An exact 100(1 -α) percent confidence interval for λ based on ( 16) is

1 2n χ 2 2 n i=1 |xi|,α/2 < λ < 1 2n χ 2 2 n i=1 |xi|,1-α/2 , (20)
where χ 2 ν,a is the 100a percentile of a chisquare random variable with ν degrees of freedom. An exact 100(1 -α) percent confidence interval for λ based on (17) is

-log I -1 1-α/2 1 + n i=1 1 {Xi=0} , n - n i=1 1 {Xi=0} (21) < λ < -log I -1 α/2 n i=1 1 {Xi=0} , n + 1 - n i=1 1 {Xi=0} , (22)
where I -1 x (a, b) is the inverse function of the incomplete beta function ratio defined by

I x (a, b) = x 0 t a-1 (1 -t) b-1 dt/B(a, b), where B(a, b) = 1 0 t a-1 (1 - t) b-1 dt denotes the beta function.
Approximate confidence intervals can be based on asymptotic normality of the maximum likelihood estimators. As n → ∞, √ n λ n -λ, p n -p approaches a bivariate normal random vector with zero means and variancecovariance matrix V say. Standard calculations show that

E - ∂ 2 log L ∂λ 2 = 1 λ 2 E n i=1 |x i | = n λ , E - ∂ 2 log L ∂λ∂p = 0, E - ∂ 2 log L ∂p 2 = n 1 -e -λ p(1 -p) .
Hence,

V λ n ≈ λ n n , Cov λ n , p n ≈ 0, V ( p n ) ≈ p(1 -p) n (1 -e -λ )
.

Note that λ n and p n are asymptotically independent. So, an approximate 100(1 -α) percent confidence interval for λ is

λ n -z α/2 λ n √ n < λ < λ n + z α/2 λ n √ n , (23)
where z a is the 100a percentile of a standard normal random variable. An approximate 100(1 -α) percent confidence interval for p is

p n -z α/2 1 √ n p n (1 -p n ) 1 -e -λn < p < p n + z α/2 1 √ n p n (1 -p n ) 1 -e -λn . (24)

A simulation study

Here, we assess the finite sample performance of the point and interval estimators in Sections 3.1 and 3.2. The assessment of the performance of the estimators is based on a simulation study:

1. generate ten thousand samples of size n from the E-Po distribution. The representation (2) was used to generate samples. 2. compute (13), ( 14), (15), ( 18) and ( 19) for the ten thousand samples, say λ n,i , p n,i , p n,i , λ n,i , p n,i for i = 1, 2, . . . , 10000.

compute the biases and mean squared errors given by bias

1 (n) = 1 10000 10000 i=1 λ n,i -λ , bias 2 (n) = 1 10000 10000 i=1 p n,i -p , bias 3 (n) = 1 10000 10000 i=1 ( p n,i -p) , bias 4 (n) = 1 10000 10000 i=1 λ n,i -λ , bias 5 (n) = 1 10000 10000 i=1 ( p n,i -p) , MSE 1 (n) = 1 10000 10000 i=1 λ n,i -λ 2 MSE 2 (n) = 1 10000 10000 i=1 p n,i -p 2 , MSE 3 (n) = 1 10000 10000 i=1 ( p n,i -p) 2 , MSE 4 (n) = 1 10000 10000 i=1 λ n,i -λ 2 , MSE 5 (n) = 1 10000 10000 i=1 ( p n,i -p) 2 .
4. compute the coverage probabilities and coverage lengths given by CP 1 (n) = 1 10000

10000 i=1 I {ℓ 1,i < λ < u 1,i } , CP 2 (n) = 1 10000 10000 i=1 I {ℓ 2,i < λ < u 2,i } , CP 3 (n) = 1 10000 10000 i=1 I {ℓ 3,i < λ < u 3,i } , CP 4 (n) = 1 10000 10000 i=1 I {ℓ 4,i < p < u 4,i } , CL 1 (n) = 1 10000 10000 i=1 (u 1,i -ℓ 1,i ) , CL 2 (n) = 1 10000 10000 i=1 (u 2,i -ℓ 2,i ) , CL 3 (n) = 1 10000 10000 i=1 (u 3,i -ℓ 3,i ) , CL 4 (n) = 1 10000 10000 i=1 (u 4,i -ℓ 4,i ) ,
where I{•} denotes the indicator function, (ℓ 1,i , u 1,i ) are the confidence limits of (20) for the i th simulated sample, (ℓ 2,i , u 2,i ) are the confidence limits of (21) for the ith simulated sample, (ℓ 3,i , u 3,i ) are the confidence limits of (23) for the i th simulated sample and (ℓ 4,i , u 4,i ) are the confidence limits of (24) for the ith simulated sample.

We repeated these steps for n = 10, 11, . . . , 100 with λ = 1 and p = 1/2, so computing the five biases, the five mean squared errors, the four coverage probabilities and the four coverage lengths for n = 10, 11, . . . , 100.

Figure 3 shows how the five biases vary with respect to n. Figure 4 shows how the five mean squared errors vary with respect to n. Figure 5 shows how the four coverage probabilities vary with respect to n. Figure 6 shows how the four coverage lengths vary with respect to n. The broken line in Figure 3 corresponds to the biases being zero. The broken line in Figure 5 corresponds to the nominal coverage probability of 0.95.

The following observations can be drawn from the Figures (3, 4, 5, and 6): the biases appear largest for the estimators, (13) and (18); the biases appear smallest for the estimators, ( 14) and (15); the biases for each estimator either decrease or increase to zero as n → ∞; the mean squared errors appear largest for the estimator, (13); the mean squared errors appear smallest for the estimators, ( 19) and ( 14); the mean squared errors for each estimator decrease to zero as n → ∞; the coverage probabilities for the confidence intervals, (20), ( 21) and ( 23), appear reasonably close to the nominal level for all n; the coverage probabilities for the confidence interval ( 24) approach the nominal level with increasing n and they appear reasonably close to the nominal level for all n ≥ 50; the coverage lengths appear largest for the confidence interval, (21); the coverage lengths appear smallest for the confidence interval, (24); the coverage lengths for each confidence interval decrease to zero as n → ∞. These observations are for only one choice for (λ, p), namely that (λ, p) = (1, 1/2). But the results were similar for other choices.

An application to real data

The students number of a specific (test) group from the Bachelor program (first year) at IDRAC International Management school (Lyon, France) who followed 60 sessions of courses in marketing covering the period, from 1/9/2012 to 1/4/2013 are time dependent. But after taking difference of every 2 consecutive sessions the resulting data set is a random sample of size 59. Thus, this data set represents the change in number of students between two sessions of course. Figure 7 illustrate the plots of the students number of the specific group and the difference of every two consecutive sessions, respectively. Descriptive statistics for the original data and the difference are presented in following table. The runs test was applied to both the original data set and the differenced one. The results of the test show that the number of students of the specific group is not a random sample (p-value ≃ 0), while after differencing the resulting data set is a random sample (p-value = 0.7077). Now, in order to fit the data, we propose our extended Poisson distribution. First, we estimated the distribution parameters by using ( 18 (O -E) 2 E = 6.557832 < χ 2 6,0.95 = 12.5916 indicating that the extended Poisson distribution fits the data well. Moreover, in order to assess the goodness of fit we followed also a computational approach. Explicitly, we seek to compare the quality adjustment for the observed data of Skellam (S) distribution [START_REF] Skellam | The frequency distribution of the difference between two Poisson variates belonging to different populations[END_REF], Extended Poisson (E-Po), Extended Binomial (EB) distribution [START_REF] Alzaid | An Extended Binomial Distribution with Applications[END_REF], and discrete analogue of the Laplace (DAL) distribution [START_REF] Inusah | A discrete analogue of the Laplace distribution[END_REF]. Indeed, using the data, we estimate parameter values associated to each distiribution, we simulated 1000 series of length 59 from each distribution and we kept the expected relative frequencies for each series. The reported frequencies are the mean over the 1000 series. Results are represented in the following table (see also Figure 7)

Relative frequency of students number and fitted distributions.

Modalities

Observed One can see that the extended Poisson distribution is the more appropriate to fit the data, compared to the other distributions. x

The failure rate function

The behaviour of the failure rate, with lambda = 0.9 and p = 0.87 x

The failure rate function

The behaviour of the failure rate, with lambda = 2.33 and p = 0.08 Fig. 1 The behaviour of the failure rate function with different value of λ and p.

  defined the skew reflected gamma distribution; Ali et al. (2008) defined skewed reflected distributions generated by a reflected gamma kernel; Ali et al. (2009) defined skewed reflected gamma distributions generated by a Laplace kernel; Ali et al. (2010) defined skewed inverse reflected gamma distributions. One should also mention that the Laplace distribution is the reflected version of the exponential distribution.

  Figures

Fig. 2

 2 Fig.2The behaviour of the reverse failure rate function with different value of λ and p.

Fig. 3

 3 Fig.3From top to bottom and from left to right : Biases of (18), (19), (13), (16) and (17) versus n = 10, 11, . . . , 100.

Fig. 4

 4 Fig.4From top to bottom and from left to right : Mean squared errors of (18), (19), (13), (16) and (17) versus n = 10, 11, . . . , 100.

Fig. 5

 5 Fig. 5 From top to bottom and from left to right : Coverage probabilities of (20), (22), (23) and (24) versus n = 10, 11, . . . , 100.

Fig. 6

 6 Fig. 6 From top to bottom and from left to right : Coverage lengths of (20), (22), (23) and (24) versus n = 10, 11, . . . , 100.

  2.3 Failure functionsLet Y denote a Poisson random variable with parameter λ. Let F Y , R Y , k Y , and τ Y denote, respectively, the cumulative distribution function, survival function, failure rate function, and the reverse failure function of Y . Then, the survival function, failure rate function, and the reverse failure function of X can be expressed as

  ) and (19). It follows that λn = 2.457627 and pn = 0.509434. The Pearson Chi-square test is performed to test the fitting. The null hypothesis is that the sample comes from the extended Poisson distribution and the alternative hypothesis is that sample does not come from extended Poisson distribution. The results of Pearson Chi-square test are given in the following table.

	Pearson chi-square test for extended Poisson distribution
	Modalities	Observed	Expected	(O -E) 2 /E
	≤ -4	5	6.245263	0.24829698
	-3	3	6.132144	1.59981959
	-2	11	7.485444	1.65014952
	-1	7	6.091603	0.13546275
	0	6	5.678383	0.01821597
	1	6	6.325895	0.01678936
	2	4	7.773346	1.83166164
	3	8	6.367995	0.41825400
	≥ 4	8	6.899926	0.63918202
	Total	59	59	6.557832