Studies in the Analysis of Stochastic Processes

Wills, Peter E.

Graduate Thesis Or Dissertation

Studies in the Analysis of Stochastic Processes Public Deposited

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/wp988j816

Abstract

As the quantity of recorded data grows exponentially, the development of techniques for analysis of such data have become a popular topic throughout industrial and academic research. However, the practical application of modern analytical methods must eventually confront the challenge of noisy data. The successful application of such methods is predicated on the ability to understand, quantity, and anticipate the effects of random fluctuations in the inputs. This thesis is a compilation of three distinct projects, which are connected by the theme of understanding stochastic processes and extracting their essential information.In the first project, we analyze the efficacy of an approach for hypothesis testing and confidence interval construction, which can be seen as an application of test supermartingales [130]. This approach is flexible, and in particular can be applied in tandem with arbitrary stopping rules, so that the number of trials to be performed need not be fixed in advance of the experiment. The method was originally developed for application in tests of local realism, which is a foundational concept in quantum mechanics [162, 163].The present work examines the efficacy of the method when applied to parameter estimation based on repeated samples of a Bernoulli random variable. The relative simplicity of this scenario allows for benchmarking against an optimal technique (based on exact calculation of tail probabilities) and for rigorous analysis of the inversion problem necessary to generate confidence intervals based on our hypothesis tests. We show that our test supermartingale method attains an asymptotically optimal gain rate, which is the exponential rate of decay of the resulting p-value as the number of trials increases. We also show that the separation between the endpoint of a one-sided confidence interval and the true probability of success is O(√log(n)/n), while the optimal endpoint separation is O(1/√n). This O(√log n) difference can be viewed as the cost of the robustness against stopping rules. We propose an adaptive modification to our method which yields an O(1/√ n) endpoint separation when the number of trials is known in advance. This work resulted in the publication [156].The second project examines the effect of thermal perturbations on standing wave structures that arise in thin-film magnetics. These structures, known as magnetic droplet solitons, or simply droplets, have been observed experimentally [9, 106] and thoroughly studied through the lens of partial differential equations [24, 36, 98]. Our analysis extends the analysis begun in [24], using the machinery of stochastic partial differential equations to quantify the effects of physical sources of randomness on the linewidth of the observed droplet. We obtain an analytical expression for the linewidth based on the linearized equations of motion, and use numerical simulations to compare this to the linewidth generated by the full nonlinear model. Along the way, we uncover a deterministic regime of drift instability missed by previous analyses [23, 24]. This work resulted in the publication [155]The third project examines methods for pairwise comparison of graphs, focusing in particular on a method based on the effective graph resistance [48], referred to as the resistance-perturbation distance, or simply the resistance distance. Previous work [107] has established basic properties of the resistance distance; we continue that program by extending the method to be applicable to graphs of different sizes, and examining the efficacy of the method in detecting transitions in the community structure of a dynamic random graph model. We show that in order for the resistance distance to effective discern these transitions in community structure, the number of cross-community edges must be asymptotically dominated by the mean degree of the graph. In order to do this, we establish an a

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