La réduction des informations redondantes et/ou non-pertinentes dans la description de données est une étape importante dans plusieurs domaines scientifiques comme les statistiques, la vision par ordinateur, la fouille de données ou l’apprentissage automatique. Dans ce manuscrit, nous abordons la réduction de la taille des signatures des images par une méthode issue de l’Analyse Formelle de Concepts (AFC), qui repose sur la structure du treillis des concepts et la théorie des treillis. Les modèles de sac de mots visuels consistent à décrire une image sous forme d’un ensemble de mots visuels obtenus par clustering. La réduction de la taille des signatures des images consiste donc à sélectionner certains de ces mots visuels. Dans cette thèse, nous proposons deux algorithmes de sélection d’attributs (mots visuels) qui sont utilisables pour l’apprentissage supervisé ou non. Le premier algorithme, RedAttSansPerte, ne retient que les attributs qui correspondent aux irréductibles du treillis. En effet, le théorème fondamental de la théorie des treillis garantit que la structure du treillis des concepts est maintenue en ne conservant que les irréductibles. Notre algorithme utilise un graphe d’attributs, le graphe de précédence, où deux attributs sont en relation lorsque les ensembles d’objets à qui ils appartiennent sont inclus l’un dans l’autre. Nous montrons par des expérimentations que la réduction par l’algorithme RedAttsSansPerte permet de diminuer le nombre d’attributs tout en conservant de bonnes performances de classification. Le deuxième algorithme, RedAttsFloue, est une extension de l’algorithme RedAttsSansPerte. Il repose sur une version approximative du graphe de précédence. Il s’agit de supprimer les attributs selon le même principe que l’algorithme précédent, mais en utilisant ce graphe flou. Un seuil de flexibilité élevé du graphe flou entraîne mécaniquement une perte d’information et de ce fait une baisse de performance de la classification. Nous montrons par des expérimentations que la réduction par l’algorithme RedAttsFloue permet de diminuer davantage l’ensemble des attributs sans diminuer de manière significative les performances de classification. / In several scientific fields such as statistics, computer vision and machine learning, redundant and/or irrelevant information reduction in the data description (dimension reduction) is an important step. This process contains two different categories : feature extraction and feature selection, of which feature selection in unsupervised learning is hitherto an open question. In this manuscript, we discussed about feature selection on image datasets using the Formal Concept Analysis (FCA), with focus on lattice structure and lattice theory. The images in a dataset were described as a set of visual words by the bag of visual words model. Two algorithms were proposed in this thesis to select relevant features and they can be used in both unsupervised learning and supervised learning. The first algorithm was the RedAttSansPerte, which based on lattice structure and lattice theory, to ensure its ability to remove redundant features using the precedence graph. The formal definition of precedence graph was given in this thesis. We also demonstrated their properties and the relationship between this graph and the AC-poset. Results from experiments indicated that the RedAttsSansPerte algorithm reduced the size of feature set while maintaining their performance against the evaluation by classification. Secondly, the RedAttsFloue algorithm, an extension of the RedAttsSansPerte algorithm, was also proposed. This extension used the fuzzy precedence graph. The formal definition and the properties of this graph were demonstrated in this manuscript. The RedAttsFloue algorithm removed redundant and irrelevant features while retaining relevant information according to the flexibility threshold of the fuzzy precedence graph. The quality of relevant information was evaluated by the classification. The RedAttsFloue algorithm is suggested to be more robust than the RedAttsSansPerte algorithm in terms of reduction.

<p>Data je karakterizacija raznih tipova specijalnih elemenata mreže, kao &scaron;to su kodistributivni, neutralni, skrativi, standardni, izuzetni, neprekidni, beskonačno distributivni i drugi i ti rezultati su primenjeni u strukturnim ispitivanjima algebri, posebno u mrežama kongruencija, podalgebri i slabih kongruencija algebri.&nbsp; Specijalni elementi su posebno proučavani i u bipolumrežama i dobijene su nove teoreme reprezentacije za bipolumreže. Ispitana je kolekcija svih mreža sa istim skupom i-nerazloživih elemenata, pokazano je da je ta kolekcija i sama mreža u odnosu na inkluziju i daju se karakterizacije te mreže.&nbsp; Re&scaron;avan je problem preno&scaron;enja mrežnih identiteta sa mreže podalgebri i kongruencija na mrežu slabih kongruencija. Proučavane su osobine svojstva preseka kongruencija i svojstva pro&scaron;irenja kongruencija i neke varijante tih svojstava u vezi sa mrežama slabih kongruencija. Date su karakterizacije mreže slabih kongruencija nekih posebnih klasa algebri i varijeteta, kao &scaron;to su unarne algebra, mreže, grupe, Hamiltonove algebra i druge.</p> / <p>A characterization of various types of special elements in lattices: codistributive,&nbsp; neutral, cancellable, standard, exceptional, continuous, infinitely distributive and others are given, and the results are applied in structural investigations in algebras, in particular in lattices of subalgebras, congruences and weak congruences. Special elements are investigated also in bi-semilattices and new representation theorems for bisemilattices are obtained. The collection of all lattices with the same poset of meet-irreducible elements is studied and it is proved that this collection is a lattice under inclusion and characterizations of this lattice is given.&nbsp; A problem of transferability of lattice identities from lattices of subalgebras and congruences to&nbsp; lattices of weak congruencse of&nbsp; algebras is solved. The congruence intersection property and the congruence extension property as well as various alternations of these properties are investigated in connection with weak congruence lattices. Characterizations of weak congruence lattices of special classes of algebras and varieties, as unary algebras, lattices, groups, Hamiltonian algebras and others are given.</p>

給定兩個隨機變數的條件機率矩陣A和B，相容性問題的主要課題包 含：(一)如何判斷他們是否相容？若相容，則如何檢驗聯合分配的唯一性 或找出所有的聯合分配；(二)若不相容，則如何訂定測量不相容程度的方 法並找出最近似聯合分配。目前的文獻資料有幾種解決問題的途徑，例 如Arnold and Press (1989)的比值矩陣法、Song et al. (2010)的不可約 化對角塊狀矩陣法及Arnold et al. (2002)的數學規劃法等，經由這些方法 的啟發，本文發展出創新的特徵向量法來處理前述的相容性課題。 當A和B相容時，我們觀察到邊際分配分別是AB′和B′A對應特徵值1的 特徵向量。因此，在以邊際分配檢驗相容性時，特徵向量法僅需檢驗滿足 特徵向量條件的邊際分配，大幅度減少了檢驗的工作量。利用線性代數中 的Perron定理和不可約化對角塊狀矩陣的概念，特徵向量法可圓滿處理相 容性問題(一)的部份。 當A和B不相容時，特徵向量法也可衍生出一個測量不相容程度的簡單 方法。由於不同的測量方法可得到不同的最近似聯合分配，為了比較其優 劣，本文中提出了以條件分配的偏差加上邊際分配的偏差作為評量最近似 聯合分配的標準。特徵向量法除了可推導出最近似聯合分配的公式解外， 經過例子的驗證，在此評量標準下特徵向量法也獲得比其他測量法更佳的 最近似聯合分配。由是，特徵向量法也可用在處理相容性問題(二)的部份。 最後，將特徵向量法實際應用在兩人零和有限賽局問題上。作業研究的 解法是將雙方採取何種策略視為獨立，但是我們認為雙方可利用償付值表 所提供的資訊作為決策的依據，並將雙方的策略寫成兩個條件機率矩陣， 則賽局問題被轉換為相容性問題。我們可用廣義相容的概念對賽局的解進 行分析，並在各種測度下討論賽局的解及雙方的最佳策略。 / Given two conditional probability matrices A and B of two random variables, the issues of the compatibility include: (a) how to determine whether they are compatible? If compatible, how to check the uniqueness of the joint distribution or find all possible joint distributions; (b) if incompatible, how to measure how far they are from compatibility and find the most nearly compatible joint distribution. There are several approaches to solve these problems, such as the ratio matrix method(Arnold and Press, 1989), the IBD matrix method(Song et al., 2010) and the mathematical programming method(Arnold et al., 2002). Inspired by these methods, the thesis develops the eigenvector approach to deal with the compatibility issues. When A and B are compatible, it is observed that the marginal distributions are eigenvectors of AB′ and B′A corresponding to 1, respectively. While checking compatibility by the marginal distributions, the eigenvector approach only checks the marginal distributions which are eigenvectors of AB′ and B′A. It significantly reduces the workload. By using Perron theorem and the concept of the IBD matrix, the part (a) of compatibility issues can be dealt with the eigenvector approach. When A and B are incompatible, a simple way to measure the degree of incompatibility can be derived from the eigenvector approach. In order to compare the most nearly compatible joint distributions given by different measures, the thesis proposes the deviation of the conditional distributions plus the deviation of the marginal distributions as the most nearly compatible joint distribution assessment standard. The eigenvector approach not only derives formula for the most nearly compatible distribution, but also provides better joint distribution than those given by the other measures through the validations under this standard. The part (b) of compatibility issues can also be dealt with the eigenvector approach. Finally, the eigenvector approach is used in solving game problems. In operations research, strategies adopted by both players are assumed to be independent. However, this independent assumption may not be appropriate, since both players can make decisions through the information provided by the payoffs for the game. Let strategies of both players form two conditional probability matrices, then the game problems can be converted into compatibility issues. We can use the concept of generalized compatibility to analyze game solutions and discuss the best strategies for both players in a variety of measurements.