Dimensionality reduction methods allow for the study of high-dimensional systems by producing low-dimensional descriptions that preserve the relevant structure and features of interest. For dynamical systems, attractors are particularly important examples of such features, as they govern the long-term dynamics of the system, and are typically low-dimensional even if the state space is high- or infinite-dimensional. Methods for reduction need to be able to determine a suitable reduced state space in which to describe the attractor, and to produce a reduced description of the corresponding dynamics. In the presence of a parameter space, a system can possess a family of attractors. Parameters are important quantities that represent aspects of the physical system not directly modelled in the dynamics, and may take different values in different instances of the system. Therefore, including the parameter dependence in the reduced system is desirable, in order to capture the model's full range of behaviour. Existing methods typically involve algebraically manipulating the original differential equation, either by applying a projection, or by making local approximations around a fixed-point. In this work, we take more of a geometric approach, both for the reduction process and for determining the dynamics in the reduced space. For the reduction, we make use of an existing secant-based projection method, which has properties that make it well-suited to the reduction of attractors. We also regard the system to be a manifold and vector field, consider the attractor's normal and tangent spaces, and the derivatives of the vector field, in order to determine the desired properties of the reduced system. We introduce a secant culling procedure that allows for the number of secants to be greatly reduced in the case that the generating set explores a low-dimensional space. This reduces the computational cost of the secant-based method without sacrificing the detail captured in the data set. This makes it feasible to use secant-based methods with larger examples. We investigate a geometric formulation of the problem of dimensionality reduction of attractors, and identify and resolve the complications that arise. The benefit of this approach is that it is compatible with a wider range of examples than conventional approaches, particularly those with angular state variables. In turn this allows for application to non-autonomous systems with periodic time-dependence. We also adapt secant-based projection for use in this more general setting, which provides a concrete method of reduction. We then extend the geometric approach to include a parameter space, resulting in a family of vector fields and a corresponding family of attractors. Both the secant-based projection and the reproduction of dynamics are extended to produce a reduced model that correctly responds to the parameter dependence. The method is compatible with multiple parameters within a given region of parameter space. This is illustrated by a variety of examples.

In this dissertation, dimensionality reduction for hyperspectral remote sensing imagery is investigated to alleviate practical application difficulties caused by high data dimension. Band selection and band clustering are applied for this purpose. Based on availability of object prior information, supervised, semi-supervised, and unsupervised techniques are proposed. To take advantage of modern computational architecture, parallel implementations on cluster and graphics processing units (GPU) are developed. The impact of dimensionality reduction on the following data analysis is also evaluated. Specific contributions are as below. 1. A similarity-based unsupervised band selection algorithm is developed to select distinctive and informative bands, which outperforms other existing unsupervised band selection approaches in the literature. 2. An efficient supervised band selection method based on minimum estimated abundance covariance is developed, which outperforms other frequently-used metrics. This new method does not need to conduct classification during band selection process or examine original bands/band combinations as do traditional approaches. 3. An efficient semi-supervised band clustering method is proposed, which uses class signatures to conduct band partition. Compared to traditional unsupervised clustering, computational complexity is significantly reduced. 4. Parallel GPU implementations with computational cost saving strategies for the developed algorithms are designed to facilitate onboard processing. 5. As an application example, band selection results are used for urban land cover classification. With a few selected bands, classification accuracy can be greatly improved, compared to the one using all the original bands or those from other frequently-used dimensionality reduction methods.

Manifold learning algorithms have been shown to be useful for many applications of numerical analysis. Unfortunately, existing algorithms often produce noisy results, do not scale well, and are unable to benefit from prior knowledge about the expected results. We propose a new algorithm that iteratively discovers manifolds by preserving the local structure among neighboring data points while scaling down the values in unwanted dimensions. This algorithm produces less noisy results than existing algorithms, and it scales better when the number of data points is much larger than the number of dimensions. Additionally, this algorithm is able to benefit from existing knowledge by operating in a semi-supervised manner.

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The improved spectral resolution of modern hyperspectral sensors provides a means for discriminating subtly different classes of on ground materials in remotely sensed images. However, in order to obtain statistically reliable classification results, the number of necessary training samples can increase exponentially as the number of spectral bands increases. Obtaining the necessary number of training signals for these high-dimensional datasets may not be feasible. The problem can be overcome by preprocessing the data to reduce the dimensionality and thus reduce the number of required training samples. In this thesis, three dimensionality reduction methods, all based on parametric projection pursuits, are investigated. These methods are the Sequential Parametric Projection Pursuits (SPPP), Parallel Parametric Projection Pursuits (PPPP), and Projection Pursuits Best Band Selection (PPBBS). The methods are applied to very high spectral resolution data to transform the hyperspectral data to a lower-dimension subspace. Feature extractors and classifiers are then applied to the lower-dimensional data to obtain target detection accuracies. The three projection pursuit methods are compared to each other, as well as to the case of using no dimensionality reduction preprocessing. When applied to hyperspectral data in a precision agriculture application, discriminating sicklepod and cocklebur weeds, the results showed that the SPPP method was optimum in terms of accuracy, resulting in a classification accuracy of >95% when using a nearest mean, maximum likelihood, or nearest neighbor classifier. The PPPP method encountered optimization problems when the hyperspectral dimensionality was very high, e.g. in the thousands. The PPBBS method resulted in high classification accuracies, >95%, when the maximum likelihood classifier was utilized; however, this method resulted in lower accuracies when the nearest mean or nearest neighbor classifiers were used. When using no projection pursuit preprocessing, the classification accuracies ranged between ~50% and 95%; however, for this case the accuracies greatly depended on the type of classifier being utilized.

This dissertation reported an experimental answer to the long-standing question of how three-dimensionality appears in wall-bounded magnetohydrodynamic flows and presented also an experimental study on the transition to turbulence in a confined, mostly quasi two-dimensional flow. Accordingly, it was shown the analysis of a vortex array with susceptibility to three-dimensionality, enclosed in a cubic container and a mostly, quasi two-dimensional vortex pair confined by the walls of a shallow, cylindrical container. Both containers were hermetically filled by a liquid metal fluid and subject to a constant, homogeneous magnetic field. The flow forcing was made by injecting constant electric current from one wall that intersects magnetic field lines (Hartmann wall). Flow characteristics and the presence of three-dimensionality were monitored by measuring electric potentials on either Hartmann walls that confined the liquid metal. A form of three-dimensionality termed as weak appeared through differential rotation along the axis of individual vortices, while a strong form manifested itself in vortices that do not extend from one to the other Hartmann wall. In the cubic container, this resulted into an array of novel, spectacular flow structures that were both steady and strongly three-dimensional, and, yielded to a frequency-selective breakdown of quasi two-dimensionality in chaotic and turbulent flow regimes. The mostly quasi two-dimensional flow in the shallow, cylindrical container was shown to undergo a sequence of supercritical bifurcations to turbulence triggered by boundary layer separations from the circular wall. For very high forcing, the flow reached a turbulent regime where the dissipation increased drastically. This was related to a possible transition from a laminar to a turbulent Hartmann layer.

Thesis introduces current research trends in clickstream analysis and proposes a new heuristic that could be used for dimensionality reduction of semantically enriched data in Web Usage Mining (WUM). Click-fraud and conversion fraud are identified as key prospective application areas for WUM. Thesis documents a conversion fraud vulnerability of Google Analytics and proposes defense - a new clickstream acquisition software, which collects data in sufficient granularity and structure to allow for data mining approaches to fraud detection. Three variants of K-means clustering algorithms and three association rule data mining systems are evaluated and compared on real-world web usage data.

The large amount of available data on a diverse range of human activities provides many opportunities for understanding, improving and revealing unknown patterns in them. Powerful automatic methods for extracting this knowledge from data are already available from machine learning and data mining. They, however, rely on the expertise of analysts to improve their results when those are not satisfactory. In this context, interactive multidimensional projections are a useful tool for the analysis of multidimensional data by revealing their underlying structure while allowing the user to manipulate the results to provide further insight into this structure. This manipulation, however, has received little attention regarding their influence on the mappings, as they can change the final layout in unpredictable ways. This is the main motivation for this research: understanding the effects caused by changes in these mappings. We approach this problem from two perspectives. First, the user perspective, we designed and developed visualizations that help reduce the trial and error in this process by providing the right piece of information for performing manipulations. Furthermore, these visualizations help explain the changes in the map caused by such manipulations. Second, we defined the effectiveness of manipulation in quantitative terms, then developed an experimental framework for assessing manipulations in multidimensional projections under this view. This framework is based on improving mappings using known evaluation measures for these techniques. Using the improvement of measures as different types of manipulations, we perform a series of experiments on five datasets, five measures, and four techniques. Our experimental results show that there are possible types of manipulations that can happen effectively, with some techniques being more susceptible to manipulations than others. / O grande volume de dados disponíveis em uma diversa gama de atividades humanas cria várias oportunidades para entendermos, melhorarmos e revelarmos padrões previamente desconhecidos em tais atividades. Métodos automáticos para extrair esses conhecimentos a partir de dados já existem em áreas como aprendizado de máquina e mineração de dados. Entretanto, eles dependem da perícia do analista para obter melhores resultados quando estes não são satisfatórios. Neste contexto, técnicas de projeção multidimensional interativas são uma ferramenta útil para a análise de dados multidimensionais, revelando sua estrutura subjacente ao mesmo tempo que permite ao analista manipular os resultados interativamente, estendendo o processo de exploração. Essa interação, entretanto, não foi estudada com profundidade com respeito à sua real influência nos mapeamentos, já que podem causar mudanças não esperadas no mapeamento final. Essa é a principal motivação desta pesquisa: entender os efeitos causados pelas mudanças em tais mapeamentos. Abordamos o problema de duas perspectivas. Primeiro, da perspectiva do usuário, desenvolvemos visualizações que ajudam a diminuir tentativas e erros neste processo provendo a informação necessária a cada passo da interação. Além disso, essas visualizações ajudam a explicar as mudanças causadas no mapeamento pela manipulação. A segunda perspectiva é a efetividade da manipulação. Definimos de forma quantitativa a efetividade da manipulação, e então desenvolvemos um arcabouço para avaliar manipulações sob a visão da efetividade. Este arcabouço é baseado em melhorias nos mapeamentos usando medidas de avaliação conhecidas para tais técnicas. Usando tais melhorias como diferentes formas de manipulação, realizamos uma série de experimentos em cinco bases de dados, cinco medidas e quatro técnicas. Nossos resultados experimentais nos dão evidências que existem certos tipos de manipulação que podem acontecer efetivamente, com algumas técnicas sendo mais suscetíveis a manipulações do que outras.

Recent improvements in high performance fluorescent sensors and scientific CMOS cameras enable optical imaging of neural networks at a much larger scale. Our lab has demonstrated the ability of wide-field calcium-imaging (using GCaMP6f) to capture the concurrent dynamic activity from hundreds to thousands of neurons over millimeters of brain tissue in behaving mice. The expansiveness of the neuronal network captured by the system requires innovation in data analysis methods. This thesis explores data analysis techniques to extract dynamics of hippocampal neural network containing a large number of individual neurons recorded using GCaMP6, while mice were learning a classical eye puff conditioning behavior. GCaMP6 fluorescence signals in each neuron is first considered one dimension, and each dataset thus contains hundreds to thousands dimensions. To understand the network structure, we first performed dimension reduction technique to examine the low-dimension evolution of the neural trajectory using Gaussian Process Factor Analysis, which smooths across dimensions, while extracting the low dimension representation. Because of the slow time course of GCaMP6 signals, the Factor Analysis was biased to the long lasting decay phase of the signal that does not represent neural activities. We found that it is critical to first estimate the spike train inference prior to application of dimension reduction, such as using the Fast Nonnegative Deconvolution method. While the low-dimension presentation described intriguing features in the neural trajectories that paralleled the learning behavior of the animal, to further quantify the network changes we directly examined the network in the high dimension space. We calculated the changes in the distance of the network trajectory over time in the high dimension space without any filtering, and compared across different phases of the behavioral states. We found that the speed of the trajectory in the high dimension space is significantly higher when animal learned the task, and the trajectory travelled much further away from baseline during the delay phase of the conditioning behavior. Together, these results demonstrate that dimension reduction analysis technique and the network trajectory within the non-reduced high dimension space can capture evolving features of neural networks recorded using calcium imaging. While this thesis concerns the hippocampal dynamics during learning, such data analysis techniques are expected to be broadly applicable to other behaviorally relevant networks.

Given the heterogeneity of symptoms allowed in the diagnosis of psychotic disorders, as well as other challenges of categorical diagnosis (e.g., First et al., 2002; Krueger, 1999), the increased specificity brought by dimensional ratings of underlying features is often important. Models using the factorial structure of psychotic symptoms perform as good as or better than traditional categorical models (Allardyce, Suppes, & Van Os, 2007). DSM-5 has provided such a system of ratings to aid clinicians, the Clinician Rated Dimensions of Psychosis Symptom Severity Scale (PSS; APA, 2013). In this approach, the clinician rates symptom severity in eight domains which emphasize traditional psychotic symptomatology, cognition, and mood. Given its accessibility and the support of the DSM-5, it is possible that the measure could achieve wide use. However, little is known about the measure and the challenges of applying it in clinical settings. This study is a conceptual analysis of the conceptual foundation of the PSS, including its psychometric properties, applications, and demonstrated validity. It is also compared to the widely used Brief Psychiatric Rating Scale – Revised (BPRS-R). The PSS is more concise that other measures, and five of the PSS domains parallel the DSM-5's "Key Features That Define the Psychotic Disorders" (p. 87-88) (although the brief instructions of the PSS differ at times from DSM-5 definitions, and little in the way of definition is offered in the PSS itself). In contrast, no rationale is given for adding the remaining three domains. The dimensional model of the PSS has similarities to the factor structure typically found for symptomatology in psychotic disorder, but a number of important differences are noted. The data required for making ratings is never defined, although the only mention of data that might be helpful for rating one of the domains depends upon extensive testing. Although anchors for the ratings might, at first glance, appear to be given in the PSS, in fact, they offer almost nothing beyond the adjectives of "equivocal," "mild," "moderate," and "severe." Finally, we found that very little research exists on the PSS, no field trial was done, psychometric properties are largely unknown, and normative data is unavailable. The PSS is brief and provides a quick way to rate the severity of the five key features of psychosis required by DSM-5 diagnoses. Thus, it can work as a quick quantification of these features. Beyond this its utility is unknown, and it appears to lack the specificity of other rating scales, such as the BPRS-R.