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Algebraic Relations Via A Monte Carlo Simulation, Alison Elaine Becker

Theses and Dissertations

The conjugation action of the complex orthogonal group on the polynomial functions on $n \times n$ matrices gives rise to a graded algebra of invariants, $\mathcal{P}(M_n)^{O_n}$. A spanning set of this algebra is in bijective correspondence to a set of unlabeled, cyclic graphs with directed edges equivalent under dihedral symmetries. When the degree of the invariants is $n+1$, we show that the dimension of the space of relations between the invariants grows linearly in $n$. Furthermore, we present two methods to obtain a basis of the space of relations; we construct a basis using an idempotent of the group algebra …

A Deep Learning Approach To Uncertainty Quantification, Mst Afroja Akter

Mathematics & Statistics ETDs

In this thesis we consider ordinary differential equations (ODEs) with random parameters. We focus on Monte Carlo (MC) sampling for computing the statistics of some quantities of interest (QoIs) given by the solution of the ODE problems. We use the 4th order accurate Runge-Kutta (RK4) method as the deterministic ODE solver. We then develop a hybrid MC sampling method that combines RK4 with neural network models to efficiently compute the statistics of QoIs within a desired accuracy. We present several numerical examples to verify the accuracy and efficiency of the proposed hybrid method compared to classical MC sampling. The hybrid …

Adaptive Monte Carlo Sampling For Cloud And Microphysics Calculations, Thomas Franz-Peter Roessler

Theses and Dissertations

An important problem in large-scale modeling of the atmosphere is the parametrization of clouds and microphysics on subgrid scales. The framework Cloud Layers Unified By Binormals (CLUBB) was developed to improve the parametrization of subgrid variability. Monte Carlo sampling is used to couple the different physical processes, which improves the grid average of subgrid tendencies.

In this Thesis we develop an adaptive Monte Carlo sampling algorithm that re-uses sample points of the previous time step by re-weighting them according to the change of the underlying distribution. This process is called 'what-if sampling' and is an application of importance sampling. An …

Monte Carlo Approx. Methods For Stochastic Optimization, John Fowler

Pomona Senior Theses

This thesis provides an overview of stochastic optimization (SP) problems and looks at how the Sample Average Approximation (SAA) method is used to solve them. We review several applications of this problem-solving technique that have been published in papers over the last few years. The number and variety of the examples should give an indication of the usefulness of this technique. The examples also provide opportunities to discuss important aspects of SPs and the SAA method including model assumptions, optimality gaps, the use of deterministic methods for finite sample sizes, and the accelerated Benders decomposition algorithm. We also give a …

American Spread Option Models And Valuation, Yu Hu

Theses and Dissertations

Spread options are derivative securities, which are written on the difference between the values of two underlying market variables. They are very important tools to hedge the correlation risk. American style spread options allow the holder to exercise the option at any time up to and including maturity. Although they are widely used to hedge and speculate in financial market, the valuation of the American spread option is very challenging. Because even under the classic assumptions that the underlying assets follow the log-normal distribution, the resulting spread doesn't have a distribution with a simple closed formula. In this dissertation, we …

Analytical And Computational Methods For The Study Of Rare Event Probabilities In Dispersive And Dissipative Waves, Daniel S. Cargill

Dissertations

The main focus of this dissertation is the application of importance sampling (IS) to calculate the probabilities associated with rare events in nonlinear, large-dimensional lightwave systems that are driven by noise, including models for fiber-based optical communication system and mode-locked lasers. Throughout the last decade, IS has emerged as a valuable tool for improving the efficiency of simulating rare events in such systems. In particular, it has shown great success in simulating various sources of transmission impairments found in optical communication systems, with examples ranging from large polarization fluctuations resulting from randomly varying fiber birefringence to large pulse-width fluctuations resulting …

Bias In Monte Carlo Simulations Due To Pseudo-Random Number Generator Initial Seed Selection, Jack C. Hill, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

Pseudo-random number generators can bias Monte Carlo simulations of the standard normal probability distribution function with initial seeds selection. Five generator designs were initial-seeded with values from 10000HEX to 1FFFFHEX, estimates of the mean were calculated for each seed, the distribution of mean estimates was determined for each generator and simulation histories were graphed for selected seeds.

Impact Of Rank-Based Normalizing Transformations On The Accuracy Of Test Scores, Shira R. Solomon, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

The purpose of this article is to provide an empirical comparison of rank-based normalization methods for standardized test scores. A series of Monte Carlo simulations were performed to compare the Blom, Tukey, Van der Waerden and Rankit approximations in terms of achieving the T score’s specified mean and standard deviation and unit normal skewness and kurtosis. All four normalization methods were accurate on the mean but were variably inaccurate on the standard deviation. Overall, deviation from the target moments was pronounced for the even moments but slight for the odd moments. Rankit emerged as the most accurate method among all …

Modeling The Hydrolyzing Action Of Secretory Phospholipase A2 With Ordinary Differential Equations And Monte Carlo Methods, Zijun Lan Dozier

Theses and Dissertations

Although cell membranes normally resist the hydrolysis of secretory phospholipase A2, a series of current investigations demonstrated that the changes in lipid order caused by increased calcium has a relationship with the susceptibility to phospholipase A2. To further explore this relationship, we setup ordinary differential equations models, statistic models and stochastic models to compare the response of human erythrocytes to the hydrolyzing action of secretory phospholipase A2 and the relationship between the susceptibility of hydrolysis and the physical properties of secretory phospholipase A2. Furthermore, we use models to determine the ability of calcium ionophore to increased membrane susceptibility.

Teaching Random Assignment: Do You Believe It Works?, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

Textbook authors admonish students to check on the comparability of two randomly assigned groups by conducting statistical tests on pretest means to determine if randomization worked. A Monte Carlo study was conducted on a sample of n = 2 per group, where each participant’s personality profile was represented by 7,500 randomly selected and assigned scores. Independent samples t tests were conducted and the results demonstrated that random assignment was successful in equating the two groups on 7,467 variables. The students’ focus is redirected from the ability of random assignment to create comparable groups to the testing of the claims of …

You Think You’Ve Got Trivials?, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

Effect sizes are important for power analysis and meta-analysis. This has led to a debate on reporting effect sizes for studies that are not statistically significant. Contrary and supportive evidence has been offered on the basis of Monte Carlo methods. In this article, clarifications are given regarding what should be simulated to determine the possible effects of piecemeal publishing trivial effect sizes.

Monte Carlo Simulation Of The Game Of Twenty-One, Douglas E. Loer

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The purpose of this paper is to demonstrate the application of computer simulation to the game of Twenty-One to predict a player's expected return from the game. Twenty-One has traditionally been one of the most popular casino games and has attracted much effort to accurately estimate the house's true advantage. Probability theory has been tried, but the thousands of different combinations of cards possible in all hands throughout the entire pack make it practically impossible to apply probability theory without overlooking some possibilities. For this reason, Twenty-One is a perfect candidate for simulation. By blocking several simulations, normal theory can …