QSpace Collection:http://hdl.handle.net/1974/7582014-08-30T20:15:37Z2014-08-30T20:15:37ZOn Malliavin Calculus and Concentration InequalitiesTreilhard, Johnhttp://hdl.handle.net/1974/122702014-07-07T20:54:52Z2014-07-07T04:00:00ZTitle: On Malliavin Calculus and Concentration Inequalities
Authors: Treilhard, John
Abstract: We prove new abstract results concerning concentration inequalities and density estimates for Malliavin differentiable random variables. The efficacy of these results are demonstrated by practical computations, such as the calculation of novel concentration inequalities for $Z = \max_{1 \leq i \leq n} N_i - E\left[ \max_{1 \leq i \leq n} N_i \right]$ where the $\{N_i\}_{i=1, ..., n}$ are Normal random variables, and $\int_0^1 B_s^4 ds - \frac{3}{4H+1}$ where $\{B_s, s \in [0,1] \}$ is a fractional Brownian motion with Hurst parameter $H$, as well as the derivation of non-asymptotic confidence intervals for the Hurst parameter of a fractional Brownian motion.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2014-07-03 23:31:55.9672014-07-07T04:00:00ZArithmetic problems around the ABC conjecture and connections with logicPasten, Hectorhttp://hdl.handle.net/1974/121232014-04-29T05:10:00Z2014-04-28T04:00:00ZTitle: Arithmetic problems around the ABC conjecture and connections with logic
Authors: Pasten, Hector
Abstract: The main theme in this thesis is the ABC conjecture. We prove some partial results towards it and we find new applications of this conjecture, mainly in the context of B\"uchi's n squares problem (which has consequences in logic related to Hilbert's tenth problem) and squarefree values of polynomials. We also study related topics, such as arithmetic properties of additive subgroups of Hecke algebras, function field and meromorphic value distribution, and undecidability of the positive existential theories over languages of arithmetic interest.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2014-04-28 10:47:54.0642014-04-28T04:00:00ZForecasting and Non-Stationarity of Surgical Demand Time SeriesMoore, Ianhttp://hdl.handle.net/1974/86192014-02-05T05:58:45Z2014-02-04T05:00:00ZTitle: Forecasting and Non-Stationarity of Surgical Demand Time Series
Authors: Moore, Ian
Abstract: Surgical scheduling is complicated by naturally occurring, and human-induced variability in the demand for surgical services. We used time series methods to detect, model and forecast these behaviors in surgical demand time series to help improve the scheduling of scarce surgical resources.
With institutional approval, we studied 47,752 surgeries undertaken at a large academic medical center over a six-year time frame. Each daily sample in this time series represented the aggregate total hours of surgeries worked on a given day. Linear terms such as periodic cycles, trends, and serial correlations explained approximately 80 percent of the variance in the raw data. We used a moving variance filter to help explain away a large share of the heteroscedastic behavior mainly attributable to surgical activities on specific US holidays, which we defined as holiday variance.
In the course of this research, we made a thoughtful attempt to understand the time series structure within our surgical demand data. We also laid a foundation, for further development, of two time series techniques, the multiwindow variance filter and cyclostatogram that can be applied not only to surgical demand time series, but also to other time series problems from other disciplines. We believe that understanding the non-stationarity, in surgical demand time series, may be an important initial step in helping health care managers save critical health care dollars.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-02-09 11:55:42.4942014-02-04T05:00:00ZON THE CONVERGENCE AND APPLICATIONS OF MEAN SHIFT TYPE ALGORITHMSAliyari Ghassabeh, Younesshttp://hdl.handle.net/1974/83652013-10-01T22:56:21Z2013-10-01T04:00:00ZTitle: ON THE CONVERGENCE AND APPLICATIONS OF MEAN SHIFT TYPE ALGORITHMS
Authors: Aliyari Ghassabeh, Youness
Abstract: Mean shift (MS) and subspace constrained mean shift (SCMS) algorithms are non-parametric, iterative methods to find a representation of a high dimensional data set on a principal curve or surface embedded in a high dimensional space. The representation of high dimensional data on a principal curve or surface, the class of mean shift type algorithms and their properties, and applications of these algorithms are the main focus of this dissertation. Although MS and SCMS algorithms have been used in many applications, a rigorous study of their convergence is still missing. This dissertation aims to fill some of the gaps between theory and practice by investigating some convergence properties of these algorithms. In particular, we propose a sufficient condition for a kernel density estimate with a Gaussian kernel to have isolated stationary points to guarantee the convergence of the MS algorithm. We also show that the SCMS algorithm inherits some of the important convergence properties of the MS algorithm. In particular, the monotonicity and convergence of the density estimate values along the sequence of output values of the algorithm are shown. We also show that the distance between consecutive points of the output sequence converges to zero, as does the projection of the gradient vector onto the subspace spanned by the D-d eigenvectors corresponding to the D-d largest eigenvalues of the local inverse covariance matrix. Furthermore, three new variations of the SCMS algorithm are proposed and the running times and performance of the resulting algorithms are compared with original SCMS algorithm. We also propose an adaptive version of the SCMS algorithm to consider the effect of new incoming samples without running the algorithm on the whole data set. As well, we develop some new potential applications of the MS and SCMS algorithm. These applications involve finding straight lines in digital images; pre-processing data before applying locally linear embedding (LLE) and ISOMAP for dimensionality reduction; noisy source vector quantization where the clean data need to be estimated before the quanization step; improving the performance of kernel regression in certain situations; and skeletonization of digitally stored handwritten characters.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-09-30 18:01:12.9592013-10-01T04:00:00Z