This dissertation addresses three important open questions in the context of generating random vectors having discrete support. The first question relates to the "NORmal To Anything" (NORTA) procedure, which is easily the most widely used amongst methods for general random vector generation. While NORTA enjoys such popularity, there remain issues surrounding its efficient and correct implementation particularly when generating random vectors having denumerable support. These complications stem primarily from having to safely compute (on a digital computer) certain infinite summations that are inherent to the NORTA procedure. This dissertation addresses the summation issue within NORTA through the construction of easily computable truncation rules that can be applied for a range of discrete random vector generation contexts. The second question tackled in this dissertation relates to developing a customized algorithm for generating multivariate Poisson random vectors. The algorithm developed (TREx) is uniformly fast—about hundred to thousand times faster than NORTA—and presents opportunities for straightforward extensions to the case of negative binomial marginal distributions. The third and arguably most important question addressed in the dissertation is that of exact nonparametric random vector generation on finite spaces. Specifically, it is wellknown that NORTA does not guarantee exact generation in dimensions higher than two. This represents an important gap in the random vector generation literature, especially in view of contexts that stipulate strict adherence to the dependency structure of the requested random vectors. This dissertation fully addresses this gap through the development of Maximum Entropy methods. The methods are exact, very efficient, and work on any finite discrete space with stipulated nonparametric marginal distributions. All code developed as part of the dissertation was written in MATLAB, and is publicly accessible through the Web site https://filebox.vt.edu/users/pasupath/pasupath.htm. / Ph. D.

The 911 emergency response process is a core component of the emergency services critical infrastructure sector in the United States. Modeling and simulation of a complex stochastic system like the 911 response process enables policy makers and stakeholders to better understand, identify, and mitigate the impact of attacks/disasters affecting the 911 system. Modeling the 911 response process as a series of queue sub-systems will enable analysis into how CI failures impact the different phases of the 911 response process. Before such a model can be constructed, the probability distributions of the inter-arrivals of events into these various sub-systems needs to be identified. This research is a first effort into investigating the stochastic behavior of inter-arrival times of different events throughout the 911 response process. I use the methodology of input modeling, a statistical modeling approach, to determine whether the exponential distribution is an appropriate model for these inter-arrival times across a large dataset of historical 911 dispatch records.

The aim of this thesis is to examine optimization sensitivity in SCORE to the accuracy of particular input data models used in a simulation model of a production line. The purpose is to evaluate if it is sufficient to model input data using sample mean and default distributions instead of fitted distributions. An existing production line has been modeled for the simulation study. SCORE is based on maximizing any key performance measure of the production line while simultaneously minimizing the number of improvements necessary to achieve maximum performance. The sensitivity to the input models should become apparent the more changes required. The experiments concluded that the optimization struggles to obtain convergence when fitted distribution models were used. Configuring the input parameters to the optimization might yield better optimization result. The final conclusion is that the optimization is sensitive to what input data models are used in the simulation model.

The study of neural and behavioural mechanisms of perceptual decision making is often done by experimental tasks involving the categorization of sensory stimuli. Among the key perceptual tasks that decision neuroscience researchers use are motion discrimination paradigms that include tracking and specifying the net direction of a single dot or a group of moving dots. These motion discrimination paradigms, such as the random-dot motion task, allow the measurement of the participant's perceptual decision making abilities in multiple task difficulty levels by varying the amount of noise in the sensory stimuli. Computational models of perceptual decision making, such as the drift-diffusion model, are widely used to analyze the behavioural measurements from these motion discrimination experiments. However, the standard drift-diffusion model can only analyze the average measures like reaction times or the proportion of correct decisions to explain the behavioural data. In the past decade, an emerging computational modeling approach was introduced to analyze the choice behaviour based on precise noise patterns in the sensory stimuli. These computational models that use spatiotemporal stimulus details have shown promise in the single-trial analysis of motion discrimination behaviour. In this thesis, I further develop the advanced computational models of perceptual decision making that use spatiotemporal dynamics of motion stimuli to provide detailed explanations of perceptual choice behaviour. First, I demonstrate the usefulness of equipping an extended Bayesian Model, equivalent to the extended drift-diffusion model, with trial-wise stimulus information leading to a significantly better explanation of behavioural data from a single-dot tracking experiment. Second, I show that the extended drift-diffusion model constrained by spatiotemporal stimulus details can explain the consistent biased choice behaviour in response to stochastic motion stimuli. Based on this model-based analysis, I provide evidence that the source of the observed biased choice behaviour is the presence of subtle motion information in the sensory stimuli. These results further emphasize the effectiveness of using spatiotemporal details of stochastic stimuli in detailed model-based analyses of the experimental data and provide computational interpretations of the data related to underlying mechanisms of perceptual decision making.

Perceptual decision making can be described as a process of accumulating evidence to a bound which has been formalized within drift-diffusion models (DDMs). Recently, an equivalent Bayesian model has been proposed. In contrast to standard DDMs, this Bayesian model directly links information in the stimulus to the decision process. Here, we extend this Bayesian model further and allow inter-trial variability of two parameters following the extended version of the DDM. We derive parameter distributions for the Bayesian model and show that they lead to predictions that are qualitatively equivalent to those made by the extended drift-diffusion model (eDDM). Further, we demonstrate the usefulness of the extended Bayesian model (eBM) for the analysis of concrete behavioral data. Specifically, using Bayesian model selection, we find evidence that including additional inter-trial parameter variability provides for a better model, when the model is constrained by trial-wise stimulus features. This result is remarkable because it was derived using just 200 trials per condition, which is typically thought to be insufficient for identifying variability parameters in DDMs. In sum, we present a Bayesian analysis, which provides for a novel and promising analysis of perceptual decision making experiments.

Perceptual decision making can be described as a process of accumulating evidence to a bound which has been formalized within drift-diffusion models (DDMs). Recently, an equivalent Bayesian model has been proposed. In contrast to standard DDMs, this Bayesian model directly links information in the stimulus to the decision process. Here, we extend this Bayesian model further and allow inter-trial variability of two parameters following the extended version of the DDM. We derive parameter distributions for the Bayesian model and show that they lead to predictions that are qualitatively equivalent to those made by the extended drift-diffusion model (eDDM). Further, we demonstrate the usefulness of the extended Bayesian model (eBM) for the analysis of concrete behavioral data. Specifically, using Bayesian model selection, we find evidence that including additional inter-trial parameter variability provides for a better model, when the model is constrained by trial-wise stimulus features. This result is remarkable because it was derived using just 200 trials per condition, which is typically thought to be insufficient for identifying variability parameters in DDMs. In sum, we present a Bayesian analysis, which provides for a novel and promising analysis of perceptual decision making experiments.