Theses and Dissertations

DOI

https://doi.org/10.25772/7VBR-CA28

2020

Dissertation

Degree Name

Doctor of Philosophy

Department

Systems Modeling and Analysis

Edward L Boone

Darcy P Mays

James Mays

Yanjun Qian

Susan Simmons

Abstract

Although advances in modern computational algorithms have provided researchers the ability to work problems which were once too computationally complex to solve, problems with high computation or large parameter spaces still remain. Problems such as those involving Time Series can be such problems. Chapter 1 looks at the the use of Exponentially Weighted Moving Averages developed by \citep{holt2004forecasting, winters1960forecasting} which were thought to provide sufficient solutions to these Time Series. A discussion is provided which illustrates the shortcomings of the EWMA and how its infinite number of possible starting values provides the modeler with an endless number of possible solutions thereby providing no single viable solution. Additionally, a brief discussion involving methods proposed by \citet{box1968some} is introduced and its limitations discussed. This leads to the need for an improved model.

Chapter 2 examines the Dynamic Linear Model developed by \citet{harrison1999bayesian} as a solution. The Dynamic Linear Model involves updating future values using prior available information. By using these Bayesian updating methods the Dynamic Linear Model (DLM) is shown to provide a flexible approach to modeling time series problems in which the underlying process changes through time through both observational variances as well as system variances. The DLM illustrated here utilizes Bayesian statistical principles which assume specified \textit{a-priori} variance parameters for both the observation and system, yielding a Partial Bayesian Dynamic Linear Model. This process is also conducted when there are no specified \textit{a-priori} variance parameters for both the observation and system, and only a specified distribution is used, thereby yielding a Fully Bayesian Dynamic Linear Model. In both cases, these variance parameters are restricted such that the distribution of the unknown observation variance is strictly greater than the distribution of the unknown system variance. This restriction yields full model identifiability. This identifiability allows the utilization of conditional distribution properties on the process variances to evaluate both modeling effect and expected values. We then use these properties and their derivations along with Sampling Importance Resampling Methods for the low parameter dimension (Constant) Models and MCMC for more complicated models with higher parameter dimensions to improve the DLM forecasting abilities while addressing issues involved when sampling in high dimensional parameter spaces. Forecast flexibility is then examined using two sampling methods and these methods are combined with examples, demonstrating the FBDLM provides improved forecasting abilities. The forecast values for the PBDLM and its 95\% credible intervals are then compared with the forecast and 95\% credible intervals using the FBDLM.

Chapter 3 examines how The Dynamic Linear Model could be applied to Random Allocation Models with no covariates to create a Bayesian Adaptive Model which may be used to determine preferred treatment by reducing between group bias. Likewise, this Bayesian Adaptive Design is able to reduce unfavorable treatment assignment, by increasing the speed at which the more favorable treatment is identified. A sensitivity analysis is conducted whereby the mean treatment allocation to each treatment iss calculated and discussed. Finally a Bayesian power analysis is calculated with median and 95\% credible intervals calculated to determine decisive evidence in favor of the better treatment. This is then shown to aid clinical trial researchers who run into ethical issues of assigning too many less favorable treatments when determining the most appropriate treatment.

The ideas in Chapter 3 are then extended in Chapter 4 to include a single covariate such as gender or smoking. Similar to Chapter 3, this Bayesian Adaptive Design is able to reduce unfavorable treatment assignment, by increasing the speed at which the more favorable treatment are identified. A sensitivity analysis is conducted whereby the mean treatment allocation to each treatment is calculated and discussed. Finally a Bayesian power analysis is calculated with median and 95\% credible intervals calculated to determine decisive evidence in favor of the better treatment. This is then shown to aid clinical trial researchers who run into ethical issues of assigning too many less favorable treatments when determining the most appropriate treatment. We show a minimized treatment allocation budget and time to locate preferred treatment are available in the Bayesian Adaptive Design using the DLM. Furthermore, a sensitivity analysis is performed on mean and variance parameters and a Bayesian power analysis is conducted using Bayes Factor. Similar to Chapter 3, this power analysis is calculated to determine decisive evidence in favor of the better treatment.

Chapter 5 provides a general overview of the previous chapters. The importance and improvements using these methods is also discussed. Finally, future works are suggested.

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VCU Theses and Dissertations

11-6-2020

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