Local Kernels and the Geometric Structure of Data

We introduce a theory of local kernels, which generalize the kernels used in the standard diffusion maps construction of nonparametric modeling. We prove that evaluating a local kernel on a data set gives a discrete representation of the generator of a continuous Markov process, which converges in the limit of large data. We explicitly connect the drift and diffusion coefficients of the process to the moments of the kernel. Moreover, when the kernel is symmetric, the generator is the Laplace-Beltrami operator with respect to a geometry which is influenced by the embedding geometry and the properties of the kernel. In particular, this allows us to generate any Riemannian geometry by an appropriate choice of local kernel. In this way, we continue a program of Belkin, Niyogi, Coifman and others to reinterpret the current diverse collection of kernel-based data analysis methods and place them in a geometric framework. We show how to use this framework to design local kernels invariant to various features of data. These data-driven local kernels can be used to construct conformally invariant embeddings and reconstruct global diffeomorphisms

Kalman-Takens filtering in the presence of dynamical noise

The use of data assimilation for the merging of observed data with dynamical models is becoming standard in modern physics. If a parametric model is known, methods such as Kalman filtering have been developed for this purpose. If no model is known, a hybrid Kalman-Takens method has been recently introduced, in order to exploit the advantages of optimal filtering in a nonparametric setting. This procedure replaces the parametric model with dynamics reconstructed from delay coordinates, while using the Kalman update formulation to assimilate new observations. We find that this hybrid approach results in comparable efficiency to parametric methods in identifying underlying dynamics, even in the presence of dynamical noise. By combining the Kalman-Takens method with an adaptive filtering procedure we are able to estimate the statistics of the observational and dynamical noise. This solves a long standing problem of separating dynamical and observational noise in time series data, which is especially challenging when no dynamical model is specified

Predictors of student course evaluations.

This dissertation explored the relationship between student, course, and instructor-level variables and student course ratings. The selection of predictor variables was based on a thorough review of the extensive body of existing literature on student course evaluations, spanning from the 1920\u27 s to the present day. The sample of student course ratings examined in this study came from the entirety of student course evaluations collected during the fall 2010 and spring 2011 semesters at the College of Education and Human Development at a large metropolitan university in the southern United States. The student course evaluation instrument is composed of 19 statements concerning the instructor\u27s teaching ability, preparation, grading, the course text and organization to which the student rates their agreement with the statement on a 5 point Likert-type scale ranging from 1 Strongly Disagree , Poor , or Very Low to 5 Strongly Agree , Excellent or Very High . In order to assess the relationship between the student, course, and instructor-level variables and the student course rating, hierarchical linear modeling (HLM) analyses were conducted. Most of the variability in student course rating was estimated at the student-level and this was reflected in the fact that most of the statistically significant relationships were found at the student-level. Prior student course interest and the amount of student effort were statistically significant predictors of student course rating in all of the regression models. These findings were supported by previous studies and provide further evidence of such relationships. Additional HLM analyses were conducted to assess the relationship between student course rating and final course grade. Results of the HLM analyses indicated that student course rating was a statistically significant predictor of student course grade. This finding is consistent with the existing literature which posits a weak positive relationship between expected course grade and student course rating

ORGANIC FARMING IN DENMARK-PRODUCTIVITY, TECHNICAL CHANGE AND MARKET EXIT

This paper attempts to quantitatively measure the change in the productivity of Danish organic farming in recent years. Based on a translog production frontier framework the technical and scale efficiency on farm level is analysed by following a time trends as well as a general index model specification. We further try to analyse the significance of subsidies for promoting long term growth in organic production by estimating a bootstrapped bivariate probit model with respect to factors influencing the probability of organic market exit. The results revealed significant differencies in the organic farms' technical efficiencies, no significant total factor productivity growth and even a slightly negative rate of technical change in the period investigated. We found evidence for a positive relationship between subsidy payments and an increase in farm efficiency, technology improvements and a decreasing probability of organic market exit which was also confirmed for off farm income.Organic Farming, Total Factor Productivity, Market Exit, Agribusiness, Productivity Analysis,

Digital History and Reconstruction of the Political Geography of the Yucatan Peninsula

https://scholarworks.moreheadstate.edu/student_scholarship_posters/1004/thumbnail.jp

The equations of nature and the nature of equations

Systems of ${N}$ equations in ${N}$ unknowns are ubiquitous in mathematical modeling. These systems, often nonlinear, are used to identify equilibria of dynamical systems in ecology, genomics, control, and many other areas. Structured systems, where the variables that are allowed to appear in each equation are pre-specified, are especially common. For modeling purposes, there is a great interest in determining circumstances under which physical solutions exist, even if the coefficients in the model equations are only approximately known. The structure of a system of equations can be described by a directed graph ${G}$ that reflects the dependence of one variable on another, and we can consider the family ${\mathcal{F}(G)}$ of systems that respect ${G}$. We define a solution ${X}$ of ${F(X) = 0}$ to be robust if for each continuous ${F^*}$ sufficiently close to ${F}$, a solution ${X^*}$ exists. Robust solutions are those that are expected to be found in real systems. There is a useful concept in graph theory called "cycle-coverable". We show that if ${G}$ is cycle-coverable, then for "almost every" ${F\in\mathcal{F}(G)}$ in the sense of prevalence, every solution is robust. Conversely, when ${G}$ fails to be cycle-coverable, each system ${F\in\mathcal{F}(G)}$ has no robust solutions. Failure to be cycle-coverable happens precisely when there is a configuration of nodes that we call a "bottleneck," a criterion that can be verified from the graph. A "bottleneck" is a direct extension of what ecologists call the Competitive Exclusion Principle, but we apply it to all structured systems