We discuss the numerical solution of large-scale sparse projected discrete-time periodic Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

In this thesis, we study two methods for inference of parameters in the accelerated failure time model with right censoring data. One is the Wald-type method, which involves parameter estimation. The other one is empirical likelihood method, which is based on the asymptotic distribution of likelihood ratio. We employ a monotone censored data version of Kendall estimating equation, and construct confidence intervals from both methods. In the simulation studies, we compare the empirical likelihood (EL) and the Wald-type procedure in terms of coverage accuracy and average length of confidence intervals. It is concluded that the empirical likelihood method has a better performance. We also compare the EL for Kendall’s rank regression estimator with the EL for other well known estimators and find advantages of the EL for Kendall estimator for small size sample. Finally, a real clinical trial data is used for the purpose of illustration.

Type 2 diabetes (T2DM) is a growing concern worldwide, particularly among Aboriginal Canadians. Diet has been associated with diabetes risk, and dietary pattern analysis (DPA) provides a method in which whole dietary patterns may be explored in relation to disease. Factor analysis (FA) and reduced rank regression (RRR) of data from the Sandy Lake Health and Diabetes Project identified patterns associated with incident T2DM at follow-up. A RRR-derived pattern characterized by tea, hot cereal, and peas, and low intake of high-sugar foods and beef was positively associated with diabetes; however, the relationship was attenuated with adjustment for age and other covariates. A FA-derived pattern characterized by processed foods was positively associated with incident T2DM in a multivariate model (OR=1.38; CIs: 1.02, 1.86 per unit), suggesting intake of processed foods may predict T2DM risk.

The primary objective of this study was to assess the relationship between dietary patterns with diabetes and mortality risk from all-cause and cardiovascular disease while controlling for the confounding effects of fitness. The secondary objective was to examine the combined effects of dietary patterns and fitness on chronic disease and mortality risk. Participants consisted of 13,621 men and women from the Aerobics Center Longitudinal Study who completed a standardized medical examination and 3-day diet record between 1987 and 1999. Reduced rank regression was used to identify dietary patterns that were predictive of unfavorable profiles of cholesterol, white blood cell count, glucose, mean arterial pressure, HDL-cholesterol, uric acid, triglycerides, and body mass index. One primary dietary pattern emerged, which was labeled the “Unhealthy Eating Index”. This pattern was characterized by a large consumption of processed meat, red meat, white potato products, non-whole grains, added fat, and a small consumption of non-citrus fruits. After adjustment for covariates, the odds ratio for diabetes and the hazard ratio for all-cause mortality were 2.55 (95% confidence interval: 1.81-3.58) and 1.40 (1.02-1.91) in the highest quintile of the Unhealthy Eating Index when compared to the lowest quintile, respectively. After controlling for fitness, these risk estimates were reduced by 51.6% and 55.0%. The Unhealthy Eating Index was not a significant predictor of cardiovascular disease mortality before or after controlling for fitness. Examining the combined effects of dietary patterns and fitness revealed that both variables were independent predictors of diabetes (Ptrend <0.0001), while fitness (Ptrend <0.0001) but not unhealthy eating (Ptrend=0.071) significantly predicted all-cause mortality risk. These results suggest that both diet and fitness must be considered when studying disease. / Thesis (Master, Community Health & Epidemiology) -- Queen's University, 2009-07-08 07:11:06.809

Let $a \in \mathbb{Z}$ be a non-zero integer. Let $p$ be a prime such that $p \nmid a$. Define the index of $a$ modulo $p$, denoted $i_{a}(p)$, to be the integer $i_{a}(p) := [(\mathbb{Z}/p\mathbb{Z})^{\ast}:\langle a \bmod{p} \rangle]$. Let $N_{a}(x) := \#\{p \le x:i_{a}(p)=1\}$. In 1927, Emil Artin conjectured that \begin{equation*} N_{a}(x) \sim A(a)\pi(x) \end{equation*} where $A(a)>0$ is a constant dependent only on $a$ and $\pi(x):=\{p \le x: p\text{ prime}\}$. Rewrite $N_{a}(x)$ as follows: \begin{equation*} N_{a}(x) = \sum_{p \le x} f(i_{a}(p)), \end{equation*} where $f:\mathbb{N} \to \mathbb{C}$ with $f(1)=1$ and $f(n)=0$ for all $n \ge 2$.\\ \indent We examine which other functions $f:\mathbb{N} \to \mathbb{C}$ will give us formul\ae \begin{equation*} \sum_{p \le x} f(i_{a}(p)) \sim c_{a}\pi(x), \end{equation*} where $c_{a}$ is a constant dependent only on $a$.\\ \indent Define $\omega(n) := \#\{p|n:p \text{ prime}\}$, $\Omega(n) := \#\{d|n:d \text{ is a prime power}\}$ and $d(n):=\{d|n:d \in \mathbb{N}\}$. We will prove \begin{align*} \sum_{p \le x} (\log(i_{a}(p)))^{\alpha} &= c_{a}\pi(x)+O\left(\frac{x}{(\log x)^{2-\alpha-\varepsilon}}\right) \\ \sum_{p \le x} \omega(i_{a}(p)) &= c_{a}^{\prime}\pi(x)+O\left(\frac{x\log \log x}{(\log x)^{2}}\right) \\ \sum_{p \le x} \Omega(i_{a}(p)) &= c_{a}^{\prime\prime}\pi(x)+O\left(\frac{x\log \log x}{(\log x)^{2}}\right) \end{align*} and \begin{equation*} \sum_{p \le x} d(i_{a}) = c_{a}^{\prime\prime\prime}\pi(x)+O\left(\frac{x}{(\log x)^{2-\varepsilon}}\right) \end{equation*} for all $\varepsilon > 0$.\\ \indent We also extend these results to finitely-generated subgroups of $\mathbb{Q}^{\ast}$ and $E(\mathbb{Q})$ where $E$ is an elliptic curve defined over $\mathbb{Q}$. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-08-03 10:45:47.408

<p>&nbsp / </p> <p align="left">The groups we consider in this study belong to the class <font face="F30">X</font><font face="F25" size="1"><font face="F25" size="1">0 </font></font><font face="F15">of all finitely generated groups with finite commutator subgroups.</font></p>

Many manual biomedical annotation tasks can be categorized as instances of the typical multi-label classification problem where several categories or labels from a fixed set need to assigned to an input instance. MeSH term assignment to biomedical articles and diagnosis code extraction from medical records are two such tasks. To address this problem automatically, in this thesis, we present a way to utilize latent associations between labels based on output label sets. We used random indexing as a method to determine latent associations and use the associations as a novel feature in a learning-to-rank algorithm that reranks candidate labels selected based on either k-NN or binary relevance approach. Using this new feature as part of other features, for MeSH term assignment, we train our ranking model on a set of 200 documents, test it on two public datasets, and obtain new state-of-the-art results in precision, recall, and mean average precision. In diagnosis code extraction, we reach an average micro F-score of 0.478 based on a large EMR dataset from the University of Kentucky Medical Center, the first study of its kind to our knowledge. Our study shows the advantages and potential of random indexing method in determining and utilizing implicit relationships between labels in multi-label classification problems.

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Low-dimensional signal structures naturally arise in a large set of applications in various fields such as medical imaging, machine learning, signal, and array processing. A ubiquitous low-dimensional structure in signals and images is sparsity, and a new sampling theory; namely, compressive sensing, proves that the sparse signals and images can be reconstructed from incomplete measurements. The signal recovery is achieved using efficient algorithms such as \ell_1-minimization. Recently, the research focus has spun-off to encompass other interesting low-dimensional signal structures such as group-sparsity and low-rank structure. This thesis considers low-rank matrix recovery (LRMR) from various structured-random measurement ensembles. These results are then employed for the in depth investigation of the classical blind-deconvolution problem from a new perspective, and for the development of a framework for the efficient sampling of correlated signals (the signals lying in a subspace). In the first part, we study the blind deconvolution; separation of two unknown signals by observing their convolution. We recast the deconvolution of discrete signals w and x as a rank-1 matrix wx* recovery problem from a structured random measurement ensemble. The convex relaxation of the problem leads to a tractable semidefinite program. We show, using some of the mathematical tools developed recently for LRMR, that if we assume the signals convolved with one another live in known subspaces, then this semidefinite relaxation is provably effective. In the second part, we design various efficient sampling architectures for signals acquired using large arrays. The sampling architectures exploit the correlation in the signals to acquire them at a sub-Nyquist rate. The sampling devices are designed using analog components with clear implementation potential. For each of the sampling scheme, we show that the signal reconstruction can be framed as an LRMR problem from a structured-random measurement ensemble. The signals can be reconstructed using the familiar nuclear-norm minimization. The sampling theorems derived for each of the sampling architecture show that the LRMR framework produces the Shannon-Nyquist performance for the sub-Nyquist acquisition of correlated signals. In the final part, we study low-rank matrix factorizations using randomized linear algebra. This specific method allows us to use a least-squares program for the reconstruction of the unknown low-rank matrix from the samples of its row and column space. Based on the principles of this method, we then design sampling architectures that not only acquire correlated signals efficiently but also require a simple least-squares program for the signal reconstruction. A theoretical analysis of all of the LRMR problems above is presented in this thesis, which provides the sufficient measurements required for the successful reconstruction of the unknown low-rank matrix, and the upper bound on the recovery error in both noiseless and noisy cases. For each of the LRMR problem, we also provide a discussion of a computationally feasible algorithm, which includes a least-squares-based algorithm, and some of the fastest algorithms for solving nuclear-norm minimization.

Type 2 diabetes (T2DM) is a growing concern worldwide, particularly among Aboriginal Canadians. Diet has been associated with diabetes risk, and dietary pattern analysis (DPA) provides a method in which whole dietary patterns may be explored in relation to disease. Factor analysis (FA) and reduced rank regression (RRR) of data from the Sandy Lake Health and Diabetes Project identified patterns associated with incident T2DM at follow-up. A RRR-derived pattern characterized by tea, hot cereal, and peas, and low intake of high-sugar foods and beef was positively associated with diabetes; however, the relationship was attenuated with adjustment for age and other covariates. A FA-derived pattern characterized by processed foods was positively associated with incident T2DM in a multivariate model (OR=1.38; CIs: 1.02, 1.86 per unit), suggesting intake of processed foods may predict T2DM risk.