<p dir="ltr">This dissertation addresses three progressively fundamental problems for functional data analysis: (1) To do efficient inference for the functional mean model accounting for within-subject correlation, we propose the refined and bias-corrected empirical likelihood method. (2) To identify functional subjects potentially from different populations, we propose the dominant-set based unsupervised clustering method using the similarity matrix. (3) To learn the similarity matrix from various similarity metrics for functional data clustering, we propose the modularity guided and dominant-set based semi-supervised clustering method.</p><p dir="ltr">In the first problem, the empirical likelihood method is utilized to do inference for the mean function of functional data by constructing the refined and bias-corrected estimating equation. The proposed estimating equation not only improves efficiency but also enables practically feasible empirical likelihood inference by properly incorporating within-subject correlation, which has not been achieved by previous studies.</p><p dir="ltr">In the second problem, the dominant-set based unsupervised clustering method is proposed to maximize the within-cluster similarity and applied to functional data with a flexible choice of similarity measures between curves. The proposed unsupervised clustering method is a hierarchical bipartition procedure under the penalized optimization framework with the tuning parameter selected by maximizing the clustering criterion called modularity of the resulting two clusters, which is inspired by the concept of dominant set in graph theory and solved by replicator dynamics in game theory. The advantage offered by this approach is not only robust to imbalanced sizes of groups but also to outliers, which overcomes the limitation of many existing clustering methods.</p><p dir="ltr">In the third problem, the metric-based semi-supervised clustering method is proposed with similarity metric learned by modularity maximization and followed by the above proposed dominant-set based clustering procedure. Under semi-supervised setting where some clustering memberships are known, the goal is to determine the best linear combination of candidate similarity metrics as the final metric to enhance the clustering performance. Besides the global metric-based algorithm, another algorithm is also proposed to learn individual metrics for each cluster, which permits overlapping membership for the clustering. This is innovatively different from many existing methods. This method is superiorly applicable to functional data with various similarity metrics between functional curves, while also exhibiting robustness to imbalanced sizes of groups, which are intrinsic to the dominant-set based clustering approach.</p><p dir="ltr">In all three problems, the advantages of the proposed methods are demonstrated through extensive empirical investigations using simulations as well as real data applications.</p>

Le mélanome est la forme la plus grave de cancer de la peau. Cette thèse a contribué au développement de deux approches différentes pour le diagnostic assisté par ordinateur du mélanome : approche analytique et approche comparative.L'approche analytique imite le comportement du dermatologue en détectant les caractéristiques de malignité sur la base de méthodes analytiques populaires dans une première étape, et en combinant ces caractéristiques dans une deuxième étape. Nous avons étudié l’impacte d’un système du diagnostic automatique utilisant des images dermoscopique de lésions cutanées pigmentées sur le diagnostic de dermatologues. L'approche comparative, appelé concept du Vilain Petit Canard (VPC), suppose que les naevus chez le même patient ont tendance à partager certaines caractéristiques morphologiques ainsi que les dermatologues identifient quelques groupes de similarité. VPC est le naevus qui ne rentre dans aucune de ces groupes, susceptibles d'être mélanome. / Melanoma is the most serious type of skin cancer. This thesis focused on the development of two different approaches for computer-aided diagnosis of melanoma: analytical approach and comparative approach. The analytical approach mimics the dermatologist’s behavior by first detecting malignancy features based on popular analytical methods, and in a second step, by combining these features. We investigated to what extent the melanoma diagnosis can be impacted by an automatic system using dermoscopic images of pigmented skin lesions. The comparative approach, called Ugly Duckling (UD) concept, assumes that nevi in the same patient tend to share some morphological features so that dermatologists identify a few similarity clusters. UD is the nevus that does not fit into any of those clusters, likely to be suspicious. The goal was to model the ability of dermatologists to build consistent clusters of pigmented skin lesions in patients.