Mathematics and Statistics, Department of
http://hdl.handle.net/1974/6
Queen's University Information2020-08-10T11:31:21ZAdvances in the Detection and Characterization of Nonstationary Processes: An Application to Riometers
http://hdl.handle.net/1974/27915
Advances in the Detection and Characterization of Nonstationary Processes: An Application to Riometers
Marshall, Francois
Spectral-correlation detectors are used to determine if a stochastic temporal process contains nonstationary components from both the classes of spectrally-correlated and generalized almost-cyclostationary processes. The established multitaper spectral-correlation detectors are derived using solely the like-taper, cross-frequency, rotational and reflectional correlations between the eignecoefficient variables for the discrete Fourier transform of the temporal process. While this assumption is reasonable for the null hypothesis of zero frequency-offset spectral correlation, it is shown in this work to be less reliable for an alternative hypothesis of nonzero frequency-offset spectral correlation. Novel spectral-correlation detectors are introduced which account for the cross-taper, frequency-offset correlations between the eigencoefficient variables as well as the total correlation between them. In the asymptotic limit of increasing record size, these detectors are invariant to linear transfer functions. The test power and the robustness of the asymptotic quantile estimates to the process finitedimensional distributions is explored. The voltage series of a relative ionospheric opacity meter is considered for analysis. This instrument is sensitive to opacity fluctuations in the ionospheric D-region, and so it is used to monitor and forecast space-weather conditions. Results suggest that a complex-Gaussian approximation for the eigencoefficient variables is accurate. In this work, novel theory is introduced to help justify this distributional behaviour. The theory reveals a class of processes which can be used to assess the model uncertainty of the different spectral-correlation detectors, but it does not provide convergence rates over the record size. Therefore, a simulation study has been carried out to determine possible record sizes for which holds the aforementioned complex-Gaussian behaviour. For the considered riometer voltage series, results of this simulation study suggest that a 54-day record length at 120-second sampling period might well be sufficient for the complex-Gaussian behaviour to hold. Throughout the analysis, interesting periodic components are found in the voltage process.
2020-06-25T21:47:03ZOptimization Policies for Polya Contagion Networks
http://hdl.handle.net/1974/27901
Optimization Policies for Polya Contagion Networks
Harrington, Greg
This thesis investigates optimization policies for resource distribution in network epidemics, using a model that derives from the classical Polya process. The basic mechanics of this model, called the network Polya process, are based on a modified urn sampling scheme that accounts for both temporal and spatial contagion between neighbouring nodes in a network. We present various infection metrics and use them to develop two optimization problems, one which takes place upon initialization and one which occurs continually as the network Polya process develops. We frame these problems as both one-sided and two-sided resource allocation problems with fixed budgets, and analyze a suite of potential policies. Due to the complexity of these problems, we introduce effective proxy measures for the average infection rate in each case. We also prove that the two-sided infection-curing game on the so-called expected network exposure admits a Nash equilibrium. In both the curing and initialization scenarios we introduce heuristic policies that primarily function on the basis of limiting the number of targeted nodes within a particular network setup. Simulations are run for large-scale networks to compare performance of our heuristics to provably convergent gradient descent algorithms run on the simplified proxy measures.
2020-06-18T15:49:05ZSheaves of Phi-Principal Parts
http://hdl.handle.net/1974/27559
Sheaves of Phi-Principal Parts
Smirnov, Ilia
This thesis studies sheaves of phi-principal parts, a modification of the well-known sequence of sheaves of principal parts. In Chapter 1, we construct the sheaves of phi-principal-parts, and undertake their general study. Then, in Chapter 2, we apply these sheaves to the problem of counting how many Weierstrass points (counted according to their Weierstrass weight) are absorbed into a singularity in a curve degeneration, and, in Chapter 3, we apply these sheaves to the problem of counting lines in a linear system in a Grassmannian.
2020-01-22T18:30:39ZAccelerated Convergence of Saddle-Point Dynamics
http://hdl.handle.net/1974/26476
Accelerated Convergence of Saddle-Point Dynamics
McCreesh, Michael
In this thesis, a second-order continuous-time saddle-point dynamics is introduced that mimics Nesterov's accelerated gradient flow dynamics. We study the convergence properties of this dynamics using a family of time-varying Lyapunov functions. In particular, we study the convergence rate of the dynamics for classes of strongly convex-strongly concave functions. For a class of quadratic strongly convex-strongly concave functions and under appropriate assumptions, this dynamics achieves global asymptotic convergence; in fact, further conditions lead to an accelerated convergence rate. We also provide conditions for both local asymptotic convergence and local accelerated convergence of general strongly convex-strongly concave functions.
2019-08-19T20:52:41Z