Background Methods
5.2 List of publications
Scientific Conferences
• Alfred Baroulier, Mehdi Douch, Paul Messinesi, Laurent Le Brusquet, Gisela Lechuga, and Arthur Tenenhaus. Modèle de durée pour données multi-voie. 49èmes Journées de Statistique de la SFdS (JdS), 2017.
• Laurent Le Brusquet, Arthur Tenenhaus, and Gisela Lechuga. Analyse discrim-inante multivoie sparse. 48èmes Journées de Statistique de la SFdS (JdS), 2016.
• Laurent Le Brusquet, Arthur Tenenhaus, Gisela Lechuga, Vincent Perlbarg, Louis Puy-basset, and Damien Galanaud. Une pénalité de groupe pour des don-nées multivoie de grande dimension. 47èmes Jourdon-nées de Statistique de la SFdS (JdS), 2015.
• Arthur Tenenhaus, Laurent Le Brusquet, and Gisela Lechuga. Multiway reg-ularized generalized canonical correlation analysis. 47èmes Journées de Statistique de la SFdS (JdS), 2015.
• Laurent Le Brusquet, Gisela Lechuga, and Arthur Tenenhaus. Régression lo-gistique multivoie. 46èmes Journées de Statistique de la SFdS (JdS), 2014.
• Gisela Lechuga, Laurent Le Brusquet, Vincent Perlbarg, Louis Puybasset, Damien Galanaud, and Arthur Tenenhaus. Discriminant Analysis for Multiway Data. PLS 2014, Paris, France, 2014.
Book Chapter
• Gisela Lechuga, Laurent Le Brusquet, Vincent Perlbarg, Damien Galanaud, Louis Puybasset and Arthur Tenenhaus. Discriminant Analysis for Multiway Data. Springer Proceedings in Mathematics & Statistics. 2015.
Journals
• Gisela Lechuga, Laurent Le Brusquet, Vincent Perlbarg, Louis Puybasset, Damien Galanaud, and Arthur Tenenhaus. Discriminant Analysis for Multiway Data. 2017 - submitted.
Workshops
• Gisela Lechuga, Laurent Le Brusquet, Vincent Perlbarg, Louis Puybasset, Damien Galanaud and Arthur Tenenhaus. Fisher Discriminant Analysis and Logis-tic Regression extended to Multiway Data. Women in Machine Learning. Montreal, Canada December 2014.
Résumé
Dans cette thèse nous proposons d’étendre des méthodes statistiques classiques telles que l’analyse discriminante, la régression logistique, la régression de Cox, et l’analyse canonique généralisée régularisée au contexte des données multivoie et multibloc.
D’abord, nous decrivont le cas classique ou chaque individu I est décrit par plusieurs instances K de la même variable J , par example des mesures répétées à travers le temps ou à travers différentes modalités. Les données ont ainsi naturellement une structure tensorielle I × J × K qui doit être modifié afin d’analyser les données avec les méth-odes dans leur forme habituelle. Cette modification entraîne la perte de la structure des données et complique l’interpretabilité des résultats. Contrairement à leur formu-lation standard, nous proposons d’imposer une contrainte structurelle afin de respecter la structure d’origine. L’intérêt de cette contrainte est double:
• d’une part elle permet une étude séparée de l’influence des variables J et de l’influence des modalités K, conduisant ainsi à une interprétation facilitée des modèles car les variables d’intérêt sont rapidement identifiés.
• D’autre part, elle permet de restreindre le nombre de coefficients à estimer. On passe de J × K à J + K paramètres à estimer ce qui limite à la fois la complexité calculatoire et le phénomène de sur-apprentissage.
Vu que la taille du probléme explose avec le nombre de modalités, des stratégies pour gérer les problèmes liés a la grande dimension sont également discutées. On utilise des formulations duales et différentes formes de régularisation qui considérent elles aussi la structure multivoie des données. De plus, il est possible de rajouter des contraintes parsimonieuses. Dans le cas de l’analyse discriminante multivoie (MFDA) on étudie les cas ou les classes sont déséquilibrés.
Les différentes méthodes sont illustrées sur deux jeux de données réelles: (i) des données de spectroscopie ou le tenseur est composé de 13 individus pour lesqueles 750 longueurs d’onde on été mesurés à travers 7 profondeurs et (ii) des données d’imagerie par résonance magnétique multi-modales, avec lesquelles on cherche á prédire le rétab-lissement à long terme de patients ayant souffert d’un traumatisme cranien dans ces données le tenseur est constitué de 143 patients pour lesquels 4 differentes modalités d’images volumetriques avec 900, 000 variables on été obtenus.
Nous avons effectué une comparaison entre les méthodes standard et la formula-tion multivoie. Les résultats obtenus sont cohérent à travers les différentes méthodes. Dans ces deux cas les méthodes multivoie offrent de bons résultats quand ont com-pare les résultats obtenus avec les approches standards. Le principaux avantage étant l’amélioration de l’interpretabilité des résultats quand on considére la contrainte struc-turelle. Également les taux de bonne classification reste similaire pour les deux cas. On demontre ainsi les avantages obtenus quand une contrainte de Kronecker est imposé aux vecteurs de poids.
Dans le cas des données raman, ont retrouve que la profondeur 7 et les variables autour du spectre relatifs a l’eau sont les plus discriminantes. Pour les données coma la version multivoie semble fournir des informations plus concrètes sur l’emplacement des voxels discriminants en identifiant des zones spécifiques de la substance blanche sans aucun apriori donné. Ce résultat est encore amélioré lors de l’utilisation d’une pénalité de groupe qu’on considére dans la région de la matière blanche.
La mise en oeuvre des méthodes présentées dans ce travail démontre l’intérêt et l’importance de développer des modèles avec des contraintes structurelles. Cette thèse ouvre également la porte à l’intégration de notre méthode à d’autres modèles linéaires. Il serait intéressant de continuer à explorer la généralisation d’un plus grand nombre d’algorithmes et de développer plus de résultats théoriques sur la convergence des algo-rithmes.
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