Mathematics and Statistics
http://hdl.handle.net/11122/974
2019-04-20T05:26:22ZThe treatment of missing data on placement tools for predicting success in college algebra at the University of Alaska
http://hdl.handle.net/11122/9762
The treatment of missing data on placement tools for predicting success in college algebra at the University of Alaska
Crawford, Alyssa
This project investigated the statistical significance of baccalaureate student placement tools such as tests scores and completion of a developmental course on predicting success in a college level algebra course at the University of Alaska (UA). Students included in the study had attempted Math 107 at UA for the first time between fiscal years 2007 and 2012. The student placement information had a high percentage of missing data. A simulation study was conducted to choose the best missing data method between complete case deletion, and multiple imputation for the student data. After the missing data methods were applied, a logistic regression with fitted with explanatory variables consisting of tests scores, developmental course grade, age (category) of scores and grade, and interactions. The relevant tests were SAT math, ACT math, AccuPlacer college level math, and the relevant developmental course was Devm /Math 105. The response variable was success in passing Math 107 with grade of C or above on the first attempt. The simulation study showed that under a high percentage of missing data and correlation, multiple imputation implemented by the R package Multivariate Imputation by Chained Equations (MICE) produced the least biased estimators and better confidence interval coverage compared to complete cases deletion when data are missing at random (MAR) and missing not at random (MNAR). Results from multiple imputation method on the student data showed that Devm /Math 105 grade was a significant predictor of passing Math 107. The age of Devm /Math 105, age of tests, and test scores were not significant predictors of student success in Math 107. Future studies may consider modeling with ALEKS scores, and high school math course information.
Master's Project (M.S.) University of Alaska Fairbanks, 2014
2014-05-01T00:00:00ZAnalyzing tree distribution and abundance in Yukon-Charley Rivers National Preserve: developing geostatistical Bayesian models with count data
http://hdl.handle.net/11122/9735
Analyzing tree distribution and abundance in Yukon-Charley Rivers National Preserve: developing geostatistical Bayesian models with count data
Winder, Samantha
Species distribution models (SDMs) describe the relationship between where a species occurs and underlying environmental conditions. For this project, I created SDMs for the five tree species that occur in Yukon-Charley Rivers National Preserve (YUCH) in order to gain insight into which environmental covariates are important for each species, and what effect each environmental condition has on that species' expected occurrence or abundance. I discuss some of the issues involved in creating SDMs, including whether or not to incorporate spatially explicit error terms, and if so, how to do so with generalized linear models (GLMs, which have discrete responses). I ran a total of 10 distinct geostatistical SDMs using Markov Chain Monte Carlo (Bayesian methods), and discuss the results here. I also compare these results from YUCH with results from a similar analysis conducted in Denali National Park and Preserve (DNPP).
Master's Project (M.S.) University of Alaska Fairbanks, 2018
2018-05-01T00:00:00ZToward an optimal solver for the obstacle problem
http://hdl.handle.net/11122/9727
Toward an optimal solver for the obstacle problem
Heldman, Max
An optimal algorithm for solving a problem with m degrees of freedom is one that computes a solution in O (m) time. In this paper, we discuss a class of optimal algorithms for the numerical solution of PDEs called multigrid methods. We go on to examine numerical solvers for the obstacle problem, a constrained PDE, with the goal of demonstrating optimality. We discuss two known algorithms, the so-called reduced space method (RSP) [BM03] and the multigrid-based projected full-approximation scheme (PFAS) [BC83]. We compare the performance of PFAS and RSP on a few example problems, finding numerical evidence of optimality or near-optimality for PFAS.
Master's Project (M.S.) University of Alaska Fairbanks, 2018
2018-04-01T00:00:00ZReliability analysis of reconstructing phylogenies under long branch attraction conditions
http://hdl.handle.net/11122/9726
Reliability analysis of reconstructing phylogenies under long branch attraction conditions
Dissanayake, Ranjan
In this simulation study we examined the reliability of three phylogenetic reconstruction techniques in a long branch attraction (LBA) situation: Maximum Parsimony (M P), Neighbor Joining (NJ), and Maximum Likelihood. Data were simulated under five DNA substitution models-JC, K2P, F81, HKY, and G T R-from four different taxa. Two branch length parameters of four taxon trees ranging from 0.05 to 0.75 with an increment of 0.02 were used to simulate DNA data under each model. For each model we simulated DNA sequences with 100, 250, 500 and 1000 sites with 100 replicates. When we have enough data the maximum likelihood technique is the most reliable of the three methods examined in this study for reconstructing phylogenies under LBA conditions. We also find that MP is the most sensitive to LBA conditions and that Neighbor Joining performs well under LBA conditions compared to MP.
Master's Project (M.S.) University of Alaska Fairbanks, 2018.
2018-05-01T00:00:00ZStreetlight Halos
http://hdl.handle.net/11122/9705
Streetlight Halos
Tape, Walter
The book treats streetlight halos, that is, atmospheric halos whose light source is nearby, rather than being the sun. Also see the elegant simulations of streetlight halos by Nicolas Lefaudeux at
http://opticsaround.blogspot.com/2013/07/la-simulation-des-halos-divergent.html
2010-01-01T00:00:00ZThe linear algebra of interpolation with finite applications giving computational methods for multivariate polynomials
http://hdl.handle.net/11122/9343
The linear algebra of interpolation with finite applications giving computational methods for multivariate polynomials
Olmsted, Coert D.
Linear representation and the duality of the biorthonormality relationship express the linear algebra of interpolation by way of the evaluation mapping. In the finite case the standard bases relate the maps to Gramian matrices. Five equivalent conditions on these objects are found which characterize the solution of the interpolation problem. This algebra succinctly describes the solution space of ordinary linear initial value problems. Multivariate polynomial spaces and multidimensional node sets are described by multi-index sets. Geometric considerations of normalization and dimensionality lead to cardinal bases for Lagrange interpolation on regular node sets. More general Hermite functional sets can also be solved by generalized Newton methods using geometry and multi-indices. Extended to countably infinite spaces, the method calls upon theorems of modern analysis.
Thesis (Ph.D.) University of Alaska Fairbanks, 1988
1988-01-01T00:00:00ZControl And Inverse Problems For One Dimensional Systems
http://hdl.handle.net/11122/9014
Control And Inverse Problems For One Dimensional Systems
Mikhaylov, Victor S.
The thesis is devoted to control and inverse problems (dynamical and spectral) for systems on graphs and on the half line. In the first part we study the boundary control problems for the wave, heat, and Schrodinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. The exact controllability in L2-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. The null controllability for the heat equation and exact controllability for the Schrodinger equation in arbitrary time interval are obtained. In the second part we consider the in-plane motion of elastic strings on a tree-like network, observed from the 'leaves.' We investigate the inverse problem of recovering not only the physical properties, i.e. the 'optical lengths' of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed. In the third part of the thesis we consider Inverse dynamical and spectral problems for the Schrodinger operator on the half line. Using the connection between dynamical (Boundary Control method) and spectral approaches (due to Krein, Gelfand-Levitan, Simon and Remling), we improved the result on the representation of so-called A---amplitude and derive the "local" version of the classical Gelfand-Levitan equations.
Thesis (Ph.D.) University of Alaska Fairbanks, 2009
2009-01-01T00:00:00ZControl Theoretic Approach To Sampling And Approximation Problems
http://hdl.handle.net/11122/9002
Control Theoretic Approach To Sampling And Approximation Problems
Bulanova, Anna S.
We present applications of some methods of control theory to problems of signal processing and optimal quadrature problems. The following problems are considered: construction of sampling and interpolating sequences for multi-band signals; spectral estimation of signals modeled by a finite sum of exponentials modulated by polynomials; construction of optimal quadrature formulae for integrands determined by solutions of initial boundary value problems. A multi-band signal is a function whose Fourier transform is supported on a finite union of intervals. The approach used in Chapter I is based on connections between the sampling and interpolation problem and the problem of the controllability of a dynamical system. We prove that there exist infinitely many sampling and interpolating sequences for signals whose spectra are supported on a union of two disjoint intervals, and provide an algorithm for construction of such sequences. There exist numerous methods for solving the spectral estimation problem. In Chapter II we introduce a new approach to this problem based on the Boundary Control method, which uses the connection between inverse problems of mathematical physics and control theory for partial differential equations. Using samples of the signal at integer moments of time we construct a convolution operator regarded as an input-output map of a linear discrete dynamical system. This system can be identified, and the exponents and amplitudes of the signal can be found from the parameters of the system. We show that the coefficients of the signal can be recovered by solving a generalized eigenvalue problem as in the Matrix Pencil method. Our method allows to consider signals with polynomial amplitudes, and we obtain an exact formula for these amplitudes. In the third chapter we consider an optimal quadrature problem for solutions of initial boundary value problems. The problem of optimization of an error functional over the set of solutions and quadrature weights is a problem of optimal control of partial differential equations. We obtain estimates for the error in quadrature formulae and an optimality condition for quadrature weights.
Thesis (Ph.D.) University of Alaska Fairbanks, 2009
2009-01-01T00:00:00ZAn investigation into the effectiveness of simulation-extrapolation for correcting measurement error-induced bias in multilevel models
http://hdl.handle.net/11122/8825
An investigation into the effectiveness of simulation-extrapolation for correcting measurement error-induced bias in multilevel models
Custer, Christopher
This paper is an investigation into correcting the bias introduced by measurement errors into multilevel models. The proposed method for this correction is simulation-extrapolation (SIMEX). The paper begins with a detailed discussion of measurement error and its effects on parameter estimation. We then describe the simulation-extrapolation method and how it corrects for the bias introduced by the measurement error. Multilevel models and their corresponding parameters are also defined before performing a simulation. The simulation involves estimating the multilevel model parameters using our true explanatory variables, the observed measurement error variables, and two different SIMEX techniques. The estimates obtained from our true explanatory values were used as a baseline for comparing the effectiveness of the SIMEX method for correcting bias. From these results, we were able to determine that the SIMEX was very effective in correcting the bias in estimates of the fixed effects parameters and often provided estimates that were not significantly different than those from the estimates derived using the true explanatory variables. The simulation also suggested that the SIMEX approach was effective in correcting bias for the random slope variance estimates, but not for the random intercept variance estimates. Using the simulation results as a guideline, we then applied the SIMEX approach to an orthodontics dataset to illustrate the application of SIMEX to real data.
Master's Project (M.S) University of Alaska Fairbanks, 2015
2015-04-01T00:00:00ZEffect of filling methods on the forecasting of time series with missing values
http://hdl.handle.net/11122/8795
Effect of filling methods on the forecasting of time series with missing values
Cheng, Mingyuan
The Gulf of Alaska Mooring (GAK1) monitoring data set is an irregular time series of temperature and salinity at various depths in the Gulf of Alaska. One approach to analyzing data from an irregular time series is to regularize the series by imputing or filling in missing values. In this project we investigated and compared four methods (denoted as APPROX, SPLINE, LOCF and OMIT) of doing this. Simulation was used to evaluate the performance of each filling method on parameter estimation and forecasting precision for an Autoregressive Integrated Moving Average (ARIMA) model. Simulations showed differences among the four methods in terms of forecast precision and parameter estimate bias. These differences depended on the true values of model parameters as well as on the percentage of data missing. Among the four methods used in this project, the method OMIT performed the best and SPLINE performed the worst. We also illustrate the application of the four methods to forecasting the Gulf of Alaska Mooring (GAK1) monitoring time series, and discuss the results in this project.
Master's Project (M.S.) University of Alaska Fairbanks, 2014
2014-12-01T00:00:00Z