The objective of this dissertation is to develop a suitable statistical methodology for functional data analysis. Modern advanced technology allows researchers to collect samples as functional which means the ideal unit of samples is a curve. We consider each functional observation as the resulting of a digitized recoding or a realization from a stochastic process. Traditional statistical methodologies often fail to be applied to this functional data set due to the high dimensionality. Functional hypothesis testing is the main focus of my dissertation. We suggested a testing procedure to determine the significance of two curves with order restriction. This work was motivated by a case study involving high-dimensional and high-frequency tidal volume traces from the New York State Psychiatric Institute at Columbia University. The overall goal of the study was to create a model of the clinical panic attack, as it occurs in panic disorder (PD), in normal human subjects. We proposed a new dimension reduction technique by non-negative basis matrix factorization (NBMF) and adapted a one-degree of freedom test in the context of multivariate analysis. This is important because other dimension techniques, such as principle component analysis (PCA), cannot be applied in this context due to the order restriction. Another area that we investigated was the estimation of functions with constrained restrictions such as convexification and/or monotonicity, together with the development of computationally efficient algorithms to solve the constrained least square problem. This study, too, has potential for applications in various fields. For example, in economics the cost function of a perfectly competitive firm must be increasing and convex, and the utility function of an economic agent must be increasing and concave. We propose an estimation method for a monotone convex function that consists of two sequential shape modification stages: (i) monotone regression via solving a constrained least square problem and (ii) convexification of the monotone regression estimate via solving an associated constrained uniform approximation problem.

Cette thèse introduit des nouveaux modèles de signaux musicaux informés par la physique des instruments. Alors que les communautés de l'acoustique instrumentale et du traitement du signal considèrent la modélisation des sons instrumentaux suivant deux approches différentes (respectivement, une modélisation du mécanisme de production du son, opposée à une modélisation des caractéristiques "morphologiques" générales du son), cette thèse propose une approche collaborative en contraignant des modèles de signaux génériques à l'aide d'information basée sur l'acoustique. L'effort est ainsi porté sur la construction de modèles spécifiques à un instrument, avec des applications aussi bien tournées vers l'acoustique (apprentissage de paramètres liés à la facture et à l'accord) que le traitement du signal (transcription de musique). En particulier nous nous concentrons sur l'analyse de musique pour piano, instrument pour lequel les sons produits sont de nature inharmonique. Cependant, l'inclusion d'une telle propriété dans des modèles de signaux est connue pour entraîner des difficultés d'optimisation, allant jusqu'à endommager les performances (en comparaison avec un modèle harmonique plus simple) dans des tâches d'analyse telles que la transcription. Un objectif majeur de cette thèse est d'avoir une meilleure compréhension des difficultés liées à l'inclusion explicite de l'inharmonicité dans des modèles de signaux, et d'étudier l'influence de l'apport de cette information sur les performances d'analyse, en particulier dans une tâche de transcription. / This thesis introduces new models of music signals informed by the physics of the instruments. While instrumental acoustics and audio signal processing target the modeling of musical tones from different perspectives (modeling of the production mechanism of the sound vs modeling of the generic "morphological'' features of the sound), this thesis aims at mixing both approaches by constraining generic signal models with acoustics-based information. Thus, it is here intended to design instrument-specific models for applications both to acoustics (learning of parameters related to the design and the tuning) and signal processing (transcription). In particular, we focus on piano music analysis for which the tones have the well-known property of inharmonicity. The inclusion of such a property in signal models however makes the optimization harder, and may even damage the performance in tasks such as music transcription when compared to a simpler harmonic model. A major goal of this thesis is thus to have a better understanding about the issues arising from the explicit inclusion of the inharmonicity in signal models, and to investigate whether it is really valuable when targeting tasks such as polyphonic music transcription.

Nonnegative matrix factorization (NMF) is an important data-analysis problem which concerns factoring a given d n matrix A with nonnegative entries into matrices B and C where B and C are d k and k n with nonnegative entries. It has numerous applications including Object recognition, Topic Modelling, Hyper-spectral imaging, Music transcription etc. In general, NMF is intractable and several heuristics exists to solve the problem of NMF. Recently there has been interest in investigating conditions under which NMF can be tractably recovered. We note that existing attempts make unrealistic assumptions and often the associated algorithms tend to be not scalable. In this thesis, we make three major contributions: First, we formulate a model of NMF with assumptions which are natural and is a substantial weakening of separability. Unlike requiring a bound on the error in each column of (A BC) as was done in much of previous work, our assumptions are about aggregate errors, namely spectral norm of (A BC) i.e. jjA BCjj2 should be low. This is a much weaker error assumption and the associated B; C would be much more resilient than existing models. Second, we describe a robust polynomial time SVD-based algorithm, UTSVD, with realistic provable error guarantees and can handle higher levels of noise than previous algorithms. Indeed, experimentally we show that existing NMF models, which are based on separability assumptions, degrade much faster than UTSVD, in the presence of noise. Furthermore, when the data has dominant features, UTSVD significantly outperforms existing models. On real life datasets we again see a similar outperformance of UTSVD on clustering tasks. Finally, under a weaker model, we prove a robust version of uniqueness of NMF, where again, the word \robust" refers to realistic error bounds.

This thesis identifies and extends techniques that can be linked to the principle of maximum entropy (maxent) and applied to parameter estimation in machine learning and statistics. Entropy functions based on deformed logarithms are used to construct Bregman divergences, and together these represent a generalization of relative entropy. The framework is analyzed using convex analysis to charac- terize generalized forms of exponential family distributions. Various connections to the existing machine learning literature are discussed and the techniques are applied to the problem of non-negative matrix factorization (NMF).

Cette thèse est consacrée au démixage en imagerie hyperspectrale en particulier dans le cas où des objets de petite taille sont présents dans la scène. Les images hyperspectrales contiennent une grande quantité d’information à la fois spectrale et spatiale, et chaque pixel peut être vu comme le spectre de réflexion de la zone imagée. Du fait de la faible résolution spatiale des capteurs le spectre de réflexion observé au niveau de chaque pixel est un mélange des spectres de réflexion de l’ensemble des composants imagés dans le pixel. Une problématique de ces images hyperspectrales est le démixage, qui consiste à décomposer l’image en une liste de spectres sources, appelés endmembers, correspondants aux spectres de réflexions des composants de la scène d’une part, et d’autre part la proportion de chacun de ces spectres source dans chaque pixel de l’image. De nombreuses méthodes de démixage existent mais leur efficacité reste amoindrie en présence de spectres sources dits rares (c’est-à-dire des spectres présents dans très peu de pixels, et souvent à un niveau subpixelique). Ces spectres rares correspondent à des composants présents en faibles quantités dans la scène et peuvent être vus comme des anomalies dont la détection est souvent cruciale pour certaines applications.Nous présentons dans un premier temps deux méthodes de détection des pixels rares dans une image, la première basée sur un seuillage de l’erreur de reconstruction après estimation des endmembers abondants, la seconde basée sur les coefficients de détails obtenus par la décomposition en ondelettes. Nous proposons ensuite une méthode de démixage adaptée au cas où une partie des endmembers sont connus a priori et montrons que cette méthode utilisée avec les méthodes de détection proposées permet le démixage des endmembers des pixels rares. Enfin nous étudions une méthode de rééchantillonnage basée sur la méthode du bootstrap pour amplifier le rôle de ces pixels rares et proposer des méthodes de démixage en présence d’objets de petite taille. / This thesis is devoted to the unmixing issue in hyperspectral images, especiallyin presence of small sized objects. Hyperspectral images contains an importantamount of both spectral and spatial information. Each pixel of the image canbe assimilated to the reflection spectra of the imaged scene. Due to sensors’ lowspatial resolution, the observed spectra are a mixture of the reflection spectraof the different materials present in the pixel. The unmixing issue consists inestimating those materials’ spectra, called endmembers, and their correspondingabundances in each pixel. Numerous unmixing methods have been proposed butthey fail when an endmembers is rare (that is to say an endmember present inonly a few of the pixels). We call rare pixels, pixels containing those endmembers.The presence of those rare endmembers can be seen as anomalies that we want todetect and unmix. In a first time, we present two detection methods to retrievethis anomalies. The first one use a thresholding criterion on the reconstructionerror from estimated dominant endmembers. The second one, is based on wavelettransform. Then we propose an unmixing method adapted when some endmembersare known a priori. This method is then used with the presented detectionmethod to propose an algorithm to unmix the rare pixels’ endmembers. Finally,we study the application of bootstrap resampling method to artificially upsamplerare pixels and propose unmixing methods in presence of small sized targets.

Clustering is actively used in several research fields, such as pattern recognition, machine learning and data mining. This dissertation focuses on clustering algorithms in the data mining area. Clustering algorithms can be applied to solve the unsupervised learning problem, which deals with finding clusters in unlabeled data. Most clustering algorithms require the number of cluster centers be known in advance. However, this is often not suitable for real world applications, since we do not know this information in most cases. Another question becomes, once clusters are found by the algorithms, do we believe the clusters are exactly the right ones or do there exist better ones? In this dissertation, we present two new Swarm Intelligence based approaches for data clustering to solve the above issues. Swarm based approaches to clustering have been shown to be able to skip local extrema by doing a form of global search, our two newly proposed ant clustering algorithms take advantage of this. The first algorithm is a kernel-based fuzzy ant clustering algorithm using the Xie-Beni partition validity metric, it is a two stage algorithm, in the first stage of the algorithm ants move the cluster centers in feature space, the cluster centers found by the ants are evaluated using a reformulated kernel-based Xie-Beni cluster validity metric. We found when provided with more clusters than exist in the data our new ant-based approach produces a partition with empty clusters and/or very lightly populated clusters. Then the second stage of this algorithm was applied to automatically detect the number of clusters for a data set by using threshold solutions. The second ant clustering algorithm, using chemical recognition of nestmates is a combination of an ant based algorithm and a consensus clustering algorithm. It is a two-stage algorithm without initial knowledge of the number of clusters. The main contributions of this work are to use the ability of an ant based clustering algorithm to determine the number of cluster centers and refine the cluster centers, then apply a consensus clustering algorithm to get a better quality final solution. We also introduced an ensemble ant clustering algorithm which is able to find a consistent number of clusters with appropriate parameters. We proposed a modified online ant clustering algorithm to handle clustering large data sets. To our knowledge, we are the first to use consensus to combine multiple ant partitions to obtain robust clustering solutions. Experiments were done with twelve data sets, some of which were benchmark data sets, two artificially generated data sets and two magnetic resonance image brain volumes. The results show how the ant clustering algorithms play an important role in finding the number of clusters and providing useful information for consensus clustering to locate the optimal clustering solutions. We conducted a wide range of comparative experiments that demonstrate the effectiveness of the new approaches.

京都大学 / 0048 / 新制・課程博士 / 博士(薬科学) / 甲第22651号 / 薬科博第123号 / 新制||薬科||13(附属図書館) / 京都大学大学院薬学研究科薬科学専攻 / (主査)教授 石濱 泰, 教授 緒方 博之, 教授 馬見塚 拓, 教授 山下 富義 / 学位規則第4条第1項該当 / Doctor of Pharmaceutical Sciences / Kyoto University / DFAM

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23314号 / 情博第750号 / 新制||情||128(附属図書館) / 京都大学大学院情報学研究科知能情報学専攻 / (主査)教授 山本 章博, 教授 鹿島 久嗣, 教授 河原 達也 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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