Many techniques have been developed to analyze time series. Spectral analysis and cycle regression analysis represent two such techniques. This study combines these two powerful tools to produce two new algorithms; the spectral algorithm and the one-pass algorithm. This research encompasses four objectives. The first objective is to link spectral analysis with cycle regression analysis to determine an initial estimate of the sinusoidal period. The second objective is to determine the best spectral window and truncation point combination to use with cycle regression for the initial estimate of the sinusoidal period. The third is to determine whether the new spectral algorithm performs better than the old T-value algorithm in estimating sinusoidal parameters. The fourth objective is to determine whether the one-pass algorithm can be used to estimate all significant harmonics simultaneously.

Clustering is an unsupervised learning technique widely used to group data into homogeneous clusters. For many real-world data containing categorical values, existing algorithms are often computationally costly in high dimensions, do not work well on noisy data with missing values, and rarely provide theoretical guarantees on clustering accuracy. In this thesis, we propose a general categorical data encoding method and a computationally efficient spectral based algorithm to cluster high-dimensional noisy categorical (nominal or ordinal) data. Under a statistical model for data on m attributes from n subjects in r clusters with missing probability epsilon, we show that our algorithm exactly recovers the true clusters with high probability when mn(1-epsilon) >= CMr² log³M, with M=max(n,m) and a fixed constant C. Moreover, we show that mn(1- epsilon)² >= r *delta/2 with 0< delta <1 is necessary for any algorithm to succeed with probability at least (1+delta)/2. In case, where m=n and r is fixed, for example, the sufficient condition matches with the necessary condition up to a polylog(n) factor, showing that our proposed algorithm is nearly optimal. We also show our algorithm outperforms several existing algorithms in both clustering accuracy and computational efficiency, both theoretically and numerically. In addition, we propose a spectral algorithm with standardization to cluster mixed data. This algorithm is computationally efficient and its clustering accuracy has been evaluated numerically on both real world data and synthetic data.

[pt] A identificação algorítmica de matrizes de Jacobi a partir de variáveis espectrais é um tema tradicional de análise numérica. Uma nova representação, as coordenadas bidiagonais, naturalmente exigiu que fosse considerado um novo algoritmo. O algoritmo é apresentado e confrontado com as técnicas habituais. / [en] Algorithms relating Jacobi matrices and spectral variables are standard objects in numerical analysis. The recent discovery of bidiagonal coordinates led to the search of an appropriate algorithm for these new variables. The new algorithm is presented and compared with previous techniques.

Une étude numérique de la dynamique d'ondes internes de gravité en fluide stablement stratifié est menée. On décrit un algorithme pseudo-spectral<br />parallèle permettant d'intégrer les équations de Navier-Stokes sur une machine paralèele. En deux dimensions d'espace, on analyse la dynamique d'un<br />champ d'ondes internes propagatives, d'amplitude modérée et initialement plan et monochromatique. Le champ d'ondes est instable et déferle. Le déferlement produit une turbulence de petites échelles spatiales influencées par la stratification. L'étude<br />est étendue au cas tridimensionnel, plus réaliste. En trois dimensions, on étudie le même champ d'ondes internes, que l'on perturbe par un bruit infinitésimal ondulatoire tridimensionnel, mais on considère des ondes statiquement stables et<br />instables (grandes amplitudes). On montre que le déferlement d'une onde interne est un processus intrinsèquement tridimensionnel, y compris pour les ondes de faible amplitude. La tridimensionalisation du champ d'ondes s'opère dans les zones de l'espace où le champ de densité devient statiquement instable. L'effondrement gravitationnel d'une zone est de structure transverse au plan de propagation de l'onde. Les effets de la turbulence des petites échelles sur la production de la composante non propagatrice de l'écoulement, le mode de vorticité potentielle et la production d'un écoulement moyen, permet de conclure que seule une petite proportion de l'énergie mécanique initiale est convertie sous ses deux formes, la majeure partie étant dissipée par la dissipation visqueuse et conduction thermique. On reconsidère le mode de vorticiée potentielle par une approche Hamiltonienne non-canonique du fluide parfait stratifié. La dérivation d'un système de dynamique modifiée permet d'étudier la relaxation d'un écoulement stratifié, conservant sa vorticité potentielle et sa densité, vers un état stationnaire d'énergie minimale, correspondant au mode de vorticité potentielle.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-10-11T17:36:54Z No. of bitstreams: 1 DissVAK.pdf: 4608619 bytes, checksum: 3022034b9035e4e8caf1195902d24581 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T13:03:21Z (GMT) No. of bitstreams: 1 DissVAK.pdf: 4608619 bytes, checksum: 3022034b9035e4e8caf1195902d24581 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T13:03:27Z (GMT) No. of bitstreams: 1 DissVAK.pdf: 4608619 bytes, checksum: 3022034b9035e4e8caf1195902d24581 (MD5) / Made available in DSpace on 2016-10-21T13:03:34Z (GMT). No. of bitstreams: 1 DissVAK.pdf: 4608619 bytes, checksum: 3022034b9035e4e8caf1195902d24581 (MD5) Previous issue date: 2016-08-12 / Não recebi financiamento / Cluster analysis is the organization of a collection of patterns into clusters based on similarity which is determined by using properties of data. Clustering techniques can be useful in a variety of knowledge domains such as biotechnology, computer vision, document retrieval and many others. An interesting area of biology involves the concept of microRNAs (miRNAs) that are approximately 22 nucleotide-long non-coding RNA molecules that play important roles in gene regulation. Clustering miRNA sequences can help to understand and explore sequences belonging to the same cluster that has similar biological functions. This research work investigates and explores seven unsupervised clustering algorithms based on graphs that can be divided into three categories: algorithm based on region of influence, algorithm based on minimum spanning tree and spectral algorithm. To assess the contribution of the proposed algorithms, data from miRNA families stored in the online miRBase database were used in the conducted experiments. The results of these experiments were presented, analysed and evaluated using clustering validation indexes as well as visual analysis. / A análise de agrupamento é uma organização de coleção de padrões em grupos, baseando-se na similaridade das propriedades pertencentes aos dados. A técnica de agrupamento pode ser utilizado em muitas áreas de conhecimento como biotecnologia, visão computacional, recuperação de documentos, entre outras. Uma área interessante da biologia envolve o conceito de microRNAs (miRNAs), que são moléculas não-codificadas de RNA com aproximadamente 22 nucleotídeos e que desempenham um papel importante na regulação dos genes. O agrupamento de sequências de miRNA podem ajudar em sua exploração e entendimento, pois as sequências que pertencem ao mesmo grupo possuem uma função biológica similar. Esse trabalho explora e investiga sete algoritmos de agrupamentos não-supervisionados baseados em grafos que podem ser divididos em três categorias: algoritmos baseados em região de influência, algoritmos baseados em árvore spanning minimal e algoritmo espectral. Para avaliar a contribuição dos algoritmos propostos, os experimentos conduzidos utilizaram os dados das famílias de miRNAs disponíveis no banco de dados denominado miRBase. Os resultados dos experimentos foram apresentados, analisados e avaliados usando índices de validação de agrupamento e análise visual.