Mathematics and Statistics, Department ofQueen's University Informationhttp://hdl.handle.net/1974/62020-09-19T02:48:28Z2020-09-19T02:48:28ZMultitaper Statistical Tests for the Detection of Frequency-Modulated SignalsBlanchette, Kianhttp://hdl.handle.net/1974/281182020-09-15T15:00:33ZMultitaper Statistical Tests for the Detection of Frequency-Modulated Signals
Blanchette, Kian
Detection of periodic signals in noise is an important problem in many scientific fields and there exist tools in the multitaper spectrum estimation and harmonic analysis framework for doing so, for example, the Harmonic F statistic. However, the Harmonic F statistic can lose effectiveness under certain types of frequency modulation, when the signal to noise ratio is low, and when the background spectrum is highly coloured. In his 2009 paper, "Polynomial Phase Demodulation in Multitaper Analysis," Thomson proposed methods for dealing with time series data (specifically, solar data) where these problems are present. In this thesis we propose an extension of this work to deal with the detection of frequency modulated signals. The method uses the Slepian sequences as projection filters to reconstruct the series based on a 2W band around a given carrier frequency and then tests the
instantaneous frequency series for a low-degree polynomial form in that band using the Slepians combined with an associated family of polynomials in a variance ratio test statistic. Under the null hypothesis that there are no sinusoidal signals with polynomial frequency modulation at the given carrier frequency, these test statistics are approximately distributed according to an F distribution with degrees of freedom depending on the number of tapers used and the degree of the polynomial being tested. We compare several such test statistics via simulation studies and apply them
to a solar time series from the GOLF SoHO instrument.
Mean Squared Error For Optimal Designs in the Polynomial and Spline RegressionPeters, Hollishttp://hdl.handle.net/1974/281152020-09-13T07:02:45ZMean Squared Error For Optimal Designs in the Polynomial and Spline Regression
Peters, Hollis
This thesis aims to further expand the existing optimal design theory for polynomial and spline regression models. We will show that the classical minimax design in polynomial regression models leads to the recently introduced stronger notion of R-optimality.
Furthermore, we aim to clarify the notion of optimal design theory for spline regression models. We will describe various types of cardinal splines and discuss related to them various theorems within the context of optimal design.
Force-free Plasma Equilibria and Differential Operators Connecting Certain EquationsPeng, Yuyanghttp://hdl.handle.net/1974/281142020-09-15T14:50:27ZForce-free Plasma Equilibria and Differential Operators Connecting Certain Equations
Peng, Yuyang
Differential operators connecting a linear case of the Grad--Shafranov equation and the axisymmetric Klein--Gordon equation are found. Differential operators linking another linear case of the Grad--Shafranov equation and the axisymmetric Helmholtz equation are also given. Infinite families of exact solutions to the Grad--Shafranov equation of both types and their corresponding force-free plasma equilibria are derived. General constructions of transforms among differential equations are presented.
Normal Order of Certain Arithmetic Functions and New Analogues of the Erd\H{o}s-Kac TheoremKar, Arpitahttp://hdl.handle.net/1974/280622020-09-01T14:36:37ZNormal Order of Certain Arithmetic Functions and New Analogues of the Erd\H{o}s-Kac Theorem
Kar, Arpita
In 1917, Hardy and Ramanujan planted the seeds of Probabilistic Number Theory when they defined `normal order' of an arithmetic function. This was nurtured further by Erd\H{o}s and Kac and many other great minds who infused further improvements to the probabilistic theory of additive functions. A systematic generalisation of this study to non-additive functions was initiated by Ram Murty and Kumar Murty in 1984. In this thesis, we study the theorem of Hardy and Ramanujan more deeply from a different perspective where we view the normal order results as estimates on sizes of certain exceptional sets. Using these results, we study the normal order of various non-additive functions like $\Omega(\phi(p+a))$ and $\Omega(\tau(p+a))$, thus generalising the results of Murty and Murty to shifts of prime arguments. We also study the distribution of these functions. Here $\Omega(n)$ counts the number of prime factors of a natural number $n$, with multiplicity, $\phi(n)$ denotes the Euler totient function and $\tau(n)$ denotes the Ramanujan tau function. Here and throughout the thesis, we denote $p$ as a prime. Finally, we study the moments of Dirichlet characters and establish normal order results in that direction as well.