Concentration Inequalities for Dependent Random Variables

Ph.DDOCTOR OF PHILOSOPH

The Control of Grain-Scale Mechanics on Channel form, Landscape Dynamics, and Climatic Perturbations in Gravel-Bedded Rivers

Landscapes evolve over millions of years, through the complex interplay of climate and tectonics. Mountains in particular represent a staggering range of spatial and temporal scales, challenging our ability to understand how the landscape is sculpted. Mountains do not simply disappear by bulk denudation. The key process of river incision results from the entrainment, displacement, and collision of coarse particles with the bed; a phenomenon known as bed load transport. This dissertation seeks to elucidate how bed load transport in natural rivers is driven by floods, to provide a mechanistic connection between climate and landscape evolution. Field surveys of coarse particle displacement and channel geometry are combined with hydrological time series, to study the interaction between floods and bed load dynamics, and their implications for channel form. Results from tagged cobbles demonstrate that mean particle displacement is proportional to applied fluid momentum in excess of the threshold of motion, while dispersion of tracers is superdiffusive due to the burial and excavation of cobbles. These field surveys reveal that particle motion remains in a state of partial transport for a diverse population of flows, and that particle sorting and transport distances closely match theory developed from small-scale laboratory experiments. Analysis of hydrological time series shows that the threshold of particle motion truncates the distribution of applied stress, resulting in thin-tailed distributions of forcing for flows above the threshold of motion. This analysis further shows that, because a coarse-grained river adjusts its geometry so that the flow at the banks is at the threshold of motion, the probability of experiencing larger stresses diminishes exponentially. Field surveys of channel geometry and particle size reveal that the geomorphological impacts of urbanization are reduced for coarse-grained channels adjusted to frequent sediment transport events. Taken together, these observations indicate that the threshold of particle motion represents a first-order control on the influence of climate on river dynamics, and the landscapes through which they flow

Misconceptions in rational numbers, probability, algebra, and geometry.

In this study, the author examined the relationship of probability misconceptions to algebra, geometry, and rational number misconceptions and investigated the potential of probability instruction as an intervention to address misconceptions in all 4 content areas. Through a review of literature, 5 fundamental concepts were identified that, if misunderstood, create persistent difficulties across content areas: rational number meaning, additive/multiplicative structures, absolute/relative comparison, variable meaning, and spatial reasoning misconceptions. Probability instruction naturally provides concrete, authentic experiences that engage students with abstract mathematical concepts, establish relationships between mathematical topics, and connect inter-related problem solving strategies. The intervention consisted of five probability lessons about counting principles, randomness, independent and dependent event probability, and probability distributions. The unit lasted approximately two weeks. This study used mixed methodology to analyze data from a randomly assigned sample of students from an untreated control group design with a switching replication. Document analysis was used to examine patterns in student responses to items on the mathematics knowledge test. Multiple imputation was used to account for missing data. Structural equation modeling was used to examine the causal structure of content area misconceptions. Item response theory was used to compute item difficulty, item discrimination, and item guessing coefficients. Generalized hierarchical linear modeling was used to explore the impact of item, student, and classroom characteristics on incorrect responses due to misconceptions. These analyses resulted in 7 key findings. (1) Content area is not the most effective way to classify mathematics misconceptions; instead, five underlying misconceptions affect all four content areas. (2) Mathematics misconception errors often appear as procedural errors. (3) A classroom environment that fosters enjoyment of mathematics and value of mathematics are associated with reduced misconception errors. (4) Higher mathematics self confidence and motivation to learn mathematics is associated with reduced misconception errors. (5) Probability misconceptions do not have a causal effect on rational numbers, algebra, or geometry misconceptions. (6) Rational number misconceptions do not have a causal effect on probability, algebra, or geometry misconceptions. (7) Probability instruction may not affect misconceptions directly, but it may help students develop skills needed to bypass misconceptions when solving difficult problems

Goodness-of-Fit Testing Using Cross-Validation Bayes Factors

Statistical methods for selecting between two competing models have a long and storied history from both the frequentist and Bayesian perspectives. That being said, there are known limitations that exist when using frequentist tests based on P-values for model selection. Therefore, we prefer to take a Bayesian approach to model selection that utilizes Bayes factors. In this research, we consider two different model selection problems: multivariate nonparametric goodness-of-fit and comparing two parametric models. For both problems, we propose intuitive and computationally simple model selection methods that take advantage of data splitting and cross-validation Bayes factors. Bayesian multivariate nonparametric goodness-of-fit is a difficult problem. The alternative model often requires an infinite-dimensional prior distribution that makes computation of the marginal likelihood complex. By applying data splitting, we are able to form a nonparametric alternative model using the familiar multivariate kernel density estimate and compute a cross-validation Bayes factor very easily. As for comparing two parametric models (either nested or non-nested), difficulties can arise when formulating prior distributions or approximating marginal likelihoods for either model. We can avoid both of these concerns by computing a prior-free cross-validation Bayes factor by using data splitting. These Bayes factors depend solely on computing maximum likelihood estimates and evaluating likelihood functions. In both scenarios, we show that our cross-validation Bayes factors are consistent at an exponential rate, regardless of which hypothesis is true. This includes the traditionally difficult case where the smaller of two nested parametric models is true. We also provide numerous simulation studies and real data analyses to explore performance and practical application of these methods

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282