No description available.

Dados de alta dimensão são tipicamente tratados como pertencentes a um único subespaço do espaço onde estão imersos. Entretanto, dados utilizados em aplicações reais estão usualmente distribuídos entre subespaços independentes e com dimensões distintas. Um objeto de estudo surge a partir dessa afirmação: como essa distribuição em subespaços independentes pode auxiliar tarefas de visualização? Por outro lado, se o dado parece estar embaralhado nesse espaço de alta dimensão, como visualizar seus padrões e realizar tarefas como classificação? Podemos, por exemplo, mapear esse dado num outro espaço utilizando uma função capaz de o desembaralhar, de modo que os padrões intrínsecos fiquem mais claros e, assim, facilitando nossa tarefa de visualização ou classificação. Essa Tese apresenta dois estudos que abordam ambos os problemas. Para o primeiro, utilizamos técnicas de subspace clustering para definir, quando existente, a estrutura de subespaços do dado e estudamos como essa informação pode auxiliar em visualizações utilizando projeções multidimensionais. Para o segundo problema, métodos de kernel, bastante conhecidos na literatura, são as ferramentas a nos auxiliar. Utilizamos a medida de similaridade do kernel para desenvolver uma nova técnica de projeção multidimensional capaz de lidar com dados imersos no espaço de características induzido implicitamente pelo kernel. / High-dimensional data are typically handled as laying in a single subspace of the original space. However, data involved in real applications are usually spread around in distinct subspaces which may have different dimensions. We would like to study how the subspace structure information can be used to improve visualization tasks. On the other hand, what if the data is tangled in this high-dimensional space, how to visualize its patterns or how to accomplish classification tasks? One could, for example, map the data in another high-dimensional space using amapping capable of untangle the data making the patterns clear, rendering the visualization or classification an easy task. This dissertation presents an study for both problems pointed out above. For the former, we use subspace clustering techniques to define, when it exists, a subspace structure, studying how this information can be used to support visualization tasks based on multidimensional projections. For the latter problem we employ kernel methods, well known in the literature, as a tool to assist visualization tasks. We use a similarity measure given by the kernel to develop acompletely new multidimensional projection technique capable of dealing with data embedded in the implicit feature space defined by the kernel.

No description available.

The proposed research is devoted to devising system identification and fault detection approaches and algorithms for a system characterized by nonlinear dynamics. Mathematical models of dynamical systems and fault models are built based on observed data from systems. In particular, we will focus on statistical subspace instrumental variable methods which allow the consideration of an appealing mathematical model in many control applications consisting of a nonlinear feedback system with nonlinearities at both inputs and outputs. Different solutions within the proposed framework are presented to solve the system identification and fault detection problems. Specifically, Augmented Subspace Instrumental Variable Identification (ASIVID) approaches are proposed to identify the closed-loop nonlinear Hammerstein systems. Then fast approaches are presented to determine the system order. Hard-over failures are detected by order determination approaches when failures manifest themselves as rank deficiencies of the dynamical systems. Geometric interpretations of subspace tracking theorems are presented in this dissertation in order to propose a fault tolerance strategy. Possible fields of application considered in this research include manufacturing systems, autonomous vehicle systems, space systems and burgeoning bio-mechanical systems.

Krylov subspace iterative methods provide an effective tool for reducing the solution of large linear systems to a size for which a direct solver may be applied. However, the problems of limited storage and speed are still a concern. Therefore, in this dissertation work, we present iterative Krylov subspace algorithms for non-Hermitian systems which do have fixed memory requirements and have favorable convergence characteristics. This dissertation describes three projects. The first project concerns short-term recurrence Krylov subspace methods for nearly-Hermitian linear systems. In 2008, Beckermann and Reichel introduced a short-term recurrence progressive GMRES algorithm for nearly-Hermitian linear systems. However, we have found this method to be unstable. We document the instabilities and introduce a different fixed-memory algorithm to treat nearly-Hermitian problems. We present numerical experiments demonstrating that the performance of this algorithm is competitive. The other two projects involve extending a strategy called Krylov subspace recycling, introduced by Parks and colleagues in 2005. This method requires more overhead than other subspace augmentation methods but offers the ability to recycle subspace information between cycles for a single linear system and recycle information between related linear systems. In the first project, we extend subspace recycling to the block Krylov subspace setting. A block Krylov subspace is a generalization of Krylov subspace where a single starting vector is replaced with a block of linearly independent starting vectors. We then apply our method to a sequence of matrices arising in a Newton iteration applied to fluid density functional theory and present some numerical experiments. In the second project, we extend the methods of subspace recycling to a family of linear systems differing only by multiples of the identity. These problems arise in the theory of quantum chromodynamics, a theory of the behavior of subatomic particles. We wish to build on the class of Krylov methods which allow the simultaneous solution of all shifted linear systems while generating only one subspace. However, the mechanics of subspace recycling complicates this situation and interferes with our ability to simultaneously solve all systems using these techniques. Therefore, we introduce an algorithm which avoids this complication and present some numerical experiments demonstrating its effectiveness. / Mathematics

This thesis consists of four chapters. Chapter 1 contains the preliminaries. We give the background, notation and some results needed for this work, and we describe our main results of this thesis. In Chapter 2 we will introduce a class of weighted Bergman spaces. We then will discuss some properties about the multiplication operator, Mz , on them. We also characterize the dual spaces of these weighted Bergman spaces. In Chapter 3 we will characterize the reducing subspaces of multiple weighted shifts. The reducing subspaces of the Bergman and the Dirichlet shift of multiplicity N are portrayed from this characterization. In Chapter 4 we will introduce the class of super-isometrically dilatable operators and describe their elementary properties. We then will discuss an equivalent description of the invariant subspace lattice for the Bergman shift. We will also discuss the interpolating sequences on the bidisk. Finally, we will examine a special class of super-isometrically dilatable operators. One corollary of this work is that we will prove that the compression of the Bergman shift on two compliments of two invariant subspaces are unitarily equivalent if and only if the two invariant subspaces are equal. / Ph. D.

Dans ce mémoire, une méthode subspace fitting (SF) destinée à l’identification des paramètres mécaniques et l’évaluation de l’état de santé de structures vibrantes, est présentée. La méthode SF s’attache à extraire, à partir des méthodes d’identification par sous-espaces (4SID), une matrice d’observabilité du système et de la corréler, au sens de la norme, à une matrice d’observabilité théorique. L’originalité de ce travail est de construire la matrice d’observabilité théorique sur la base d’un modèle éléments finis (EF) de la structure considérée. En ajustant les paramètres inconnus du modèle EF, les propriétés mécaniques de la structure vibrante sont identifiées. Les coûts de calcul d’une telle procédure sont réduits en considérant une méthode de réduction de modèle basée sur la position des excitations et des capteurs. La méthode est évaluée pour l’identification des fréquences propres d’une structure vibrante. Des applications numériques et expérimentales s’attachent à montrer la pertinence d’une telle approche. En particulier, il est mis en évidence que la méthode SF permet d’identifier précisément les fréquences propres d’une structure, pour des niveaux de bruit importants. / In this thesis, a subspace fitting (SF) method is presented for the identification of mechanical parameters and assessment of the health condition of vibrating structures. The SF method attempts to extract, from subspace identification methods (4SID), a system observability matrix of the system and correlate them with a theoretical observability matrix. The originality of this work is to obtain the theoretical observability matrix from a finite element model (EF) of the structure. By adjusting unknown parameters of the FE model, the mechanical properties of the vibrating structure are identified. Computational costs of such a procedure are reduced by considering a model reduction method based on the excitations and sensors location. The method is evaluated for the identification of natural frequencies of a vibrating structure. Numerical and experimental applications are assessed to show the relevance of such an approach. In particular, it is highlighted that the SF method can accurately identify the natural frequencies of a structure to high noise levels.

High dimensional, low sample, data have singular sample covariance matrices,rendering them impossible to analyse by regular canonical correlation (CC). Byusing random subspace method (RSM) calculation of canonical correlation be-comes possible, and a Monte Carlo analysis shows resulting maximal CC canreliably distinguish between data with true correlation (above 0.5) and with-out. Statistics gathered from RSMCCA can be used to model true populationcorrelation by beta regression, given certain characteristic of data set. RSM-CCA applied on real biological data however show that the method can besensitive to deviation from normality and high degrees of multi-collinearity.

In this dissertation, we study iterative methods for computing eigenvalues and exponentials of large matrices. These types of computational problems arise in a large number of applications, including mathematical models in economics, physical and biological processes. Although numerical methods for computing eigenvalues and matrix exponentials have been well studied in the literature, there is a lack of analysis in inexact iterative methods for eigenvalue computation and certain variants of the Krylov subspace methods for approximating the matrix exponentials. In this work, we proposed an inexact inverse subspace iteration method that generalizes the inexact inverse iteration for computing multiple and clustered eigenvalues of a generalized eigenvalue problem. Compared with other methods, the inexact inverse subspace iteration method is generally more robust. Convergence analysis showed that the linear convergence rate of the exact case is preserved. The second part of the work is to present an inverse Lanczos method to approximate the product of a matrix exponential and a vector. This is proposed to allow use of larger time step in a time-propagation scheme for solving linear initial value problems. Error analysis is given for the inverse Lanczos method, the standard Lanczos method as well as the shift-and-invert Lanczos method. The analysis demonstrates different behaviors of these variants and helps in choosing which variant to use in practice.

In traversing the diverse field of biometric security and face recognition techniques, this investigation explores a rather rare comparative study of three of the most popular Appearance-based Face Recognition projection classes, these being the methodologies of Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA) and Independent Component Analysis (ICA). Both the linear and kernel alternatives are investigated along with the four most widely accepted similarity measures of City Block (L1), Euclidean (L2), Cosine and the Mahalanobis metrics. Although comparisons between these classes can become fairly complex given the different task natures, the algorithm architectures and the distance metrics that must be taken into account, an important aspect of this study is the completely equal working conditions that are provided in order to facilitate fair and proper comparative levels of evaluation. In doing so, one is able to realise an independent study that significantly contributes to prior literary findings, either by verifying previous results, offering further insight into why certain conclusions were made or by providing a better understanding as to why certain claims should be disputed and under which conditions they may hold true. The experimental procedure examines ten algorithms in the categories of expression, illumination, occlusion and temporal delay; the results are then evaluated based on a sequential combination of assessment tools that facilitate both intuitive and statistical decisiveness among the intra and inter-class comparisons. In a bid to boost the overall efficiency and accuracy levels of the identification system, the ‘best’ categorical algorithms are then incorporated into a hybrid methodology, where the advantageous effects of fusion strategies are considered. This investigation explores the weighted-sum approach, which by fusion at a matching score level, effectively harnesses the complimentary strengths of the component algorithms and in doing so highlights the improved performance levels that can be provided by hybrid implementations. In the process, by firstly exploring previous literature with respect to each other and secondly by relating the important findings of this paper to previous works one is also able to meet the primary objective in providing an amateur with a very insightful understanding of publicly available subspace techniques and their comparable application status within the environment of face recognition.