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Multiway data are becoming more and more common. While there are many approaches to extending principal component analysis (PCA) from usual data matrices to multiway arrays, their conceptual differences from the usual PCA, and the methodological implications of such differences remain largely unknown. This thesis aims to specifically address these questions. In particular, we clarify the subtle difference between PCA and singular value decomposition (SVD) for multiway data, and show that multiway principal components (PCs) can be estimated reliably in absence of the eigengaps required by the usual PCA, and in general much more efficiently than the usual PCs. Furthermore, the sample multiway PCs are asymptotically independent and hence allow for separate and more accurate inferences about the population PCs. The practical merits of multiway PCA are further demonstrated through numerical, both simulated and real data, examples.

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Canonical variate analysis (CVA) is a widely used method for analyzing group structure in multivariate data. It is mathematically equivalent to a one-way multivariate analysis of variance and often goes by the name of canonical discriminant analysis. Change over time is a central feature of many phenomena of interest to researchers. This dissertation extends CVA to longitudinal data. It develops models whose purpose is to determine what is changing and what is not changing in the group structure. Three approaches are taken: a maximum likelihood approach, a least squares approach, and a covariance structure analysis approach. All methods have in common that they hypothesize canonical variates which are stable over time. The maximum likelihood approach models the positions of the group means in the subspace of the canonical variates. It also requires modeling the structure of the within-groups covariance matrix, which is assumed to be constant or proportional over time. In addition to hypothesizing stable variates over time, one can also hypothesize canonical variates that change over time. Hypothesis tests and confidence intervals are developed. The least squares methods are exploratory. They are based on three-mode PCA methods such as the Tucker2 and parallel factor analysis. Graphical methods are developed to display the relationships between the variables over time. Stable variates over time imply a particular structure for the between-groups covariance matrix. This structure is modeled using covariance structure analysis, which is available in the SAS package Proc Calis. Methods related to CVA are also discussed. First, the least squares methods are extended to canonical correlation analysis, redundancy analysis, Procrustes rotation and correspondence analysis with longitudinal data. These least squares methods lend themselves equally well to data from multiple datasets. Lastly, a least squares method for the common principal components model is developed. / Ph. D.

This dissertation consists of three chapters describing the applications of factor models in different fields of asset pricing. The first chapter addresses the following issue: Prominent volatility-based factor pricing models focus exclusively on the second moment of asset returns, and hence, tend to identify volatile factors but with little risk premia. This chapter demonstrates that a simple asset return transform can arbitrarily upset the ranking of volatility-based factors, but not their prices of risks. Accordingly, we propose a new framework to identify factors based on their prices of risks, or the so-called principally priced risk factors (PPRFs). We construct these factors by generalizing the standard Sharpe ratio for a single asset to a set of assets, incorporating information from both the first and second moments of asset returns. The PPRF framework improves out-of-sample pricing performance in both equity and currency markets. The second chapter identifies the origins of covariance in institutional trading. Conceptually, we introduce two perspectives: the asset perspective, which prioritizes assets as the key market fundamentals, and the manager perspective, which prioritizes fund managers as the key market fundamentals that drive institutional trading covariance. Empirically, we establish that the asset perspective is the primary driver of covariance in institutional trading. Our analysis documents two further empirical patterns. First, returns stemming from the covariance in institutional trading from the asset perspective have higher volatility, offering valuable insights into the demand-based asset pricing literature. Second, the persistence in trading often breaks down during economic downturns, suggesting potential connections to the uncertainty-based business cycle literature. Finally, the third chapter examines the impact of changes in monetary policy rules on the asset valuations of firms with different profitability. I have the following two empirical findings. First, during periods of hawkish monetary policies, the 'profitability premium'— the expected extra return on investments in more profitable firms — tends to increase. Second, when analyzing the factors mediating this effect, changes in inflation expectations play a more significant role in influencing the profitability premium during transitions to a hawkish monetary regime, compared to the effects of real interest rate adjustments on production costs. These observations suggest a possible mechanism by which monetary policy may have different long-term effects on firms with different characteristics. / Doctor of Philosophy / This dissertation explores factor models in asset pricing across three chapters. The first chapter critiques volatility-based models that focus on asset return variance and introduces a new framework for identifying factors based on risk prices, enhancing pricing performance in equity and currency markets. The second chapter investigates the origins of covariance in institutional trading, emphasizing the asset perspective as the dominant influence and documenting higher volatility and breakdowns in trading persistence during economic downturns. The third chapter examines the effects of monetary policy changes on firm asset valuations, finding that hawkish policies increase the profitability premium, significantly influenced by shifts in inflation expectations rather than changes in real interest rates. These insights highlight the nuanced impacts of market fundamentals and monetary policy on asset pricing and firm profitability.

Principal Components Analysis is a standard tool in data analysis, widely used in data-rich fields such as computer vision, data mining, bioinformatics, and econometrics. For a set of vectors in n dimensions and a natural number k less than n, the method returns a subspace of dimension k whose average squared distance to that set is as small as possible. Besides saving computation by reducing the dimension, projecting to this subspace can often reveal structure that was hidden in high dimension. This thesis considers several novel extensions of PCA, which provably reveals hidden structure where standard PCA fails to do so. First, we consider Robust PCA, which prevents a few points, possibly corrupted by an adversary, from having a large effect on the analysis. When applied to learning noisy logconcave mixture models, the algorithm requires only slightly more separation between component means than is required for the noiseless case. Second, we consider Isotropic PCA, which can go beyond the first two moments in identifying ``interesting' directions in data. The method leads to the first affine-invariant algorithm that can provably learn mixtures of Gaussians in high dimensions, improving significantly on known results. Thirdly, we define the ``Subgraph Parity Tensor' of order r of a graph and reduce the problem of finding planted cliques in random graphs to the problem of finding the top principal component of this tensor.

Student Number : 9807515F - MSc Dissertation - School of Computational and Applied Mathematics - Faculty of Science / Statistical and algebraic techniques of subspace tracking were tested for filtering the earth’s response from airborne magnetic field gradients in order to discriminate the relatively small response (dipole) of objects on the earth’s surface, such as UXO. Filtering the data was not very effective with these methods but a subspace was found in the data for the magnitude of the magnetic moment of the dipole. This subspace is easily obtained using the singular value decomposition and can be used for an approximate location, without depth estimation, as well as the relative size of the dipole.

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Safe drinking water is paramount to a healthy society. Close to a hundred contaminants are regulated by the government. Utilities are using chloramines to disinfect water to reduce harmful byproducts that may present themselves with the use of chlorine alone. Using chlorine and ammonia to disinfect, ammonia oxidizing bacteria can present themselves in an unsuspecting utilities distribution network. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

This study is concerned in reducing high dimensionality problem of auxiliary variables in the calibration estimation with the presence of nonresponse. The calibration estimation is a weighting method assists to compensate for the nonresponse in the survey analysis. Calibration estimation using principal components (PCs) is new idea in the literatures. Principal component analysis (PCA) is used in reduction dimension of the auxiliary variables. PCA in calibration estimation is presented as an alternative method for choosing the auxiliary variables. In this study, simulation on the real data is used and nonresponse mechanism is applied on the sampled data. The calibration estimator is compared using different criteria such as varying the nonresponse rate and increasing the sample size. From the results, although the calibration estimation based on the principal components have reasonable outputs to use instead of the whole auxiliary variables for the means, the variance is very large compared with based on original auxiliary variables. Finally, we identified the principal component analysis is not efficient in the reduction of high dimensionality problem of auxiliary variables in the calibration estimation for large sample sizes.