"The importance of big data does not revolve around how much data you have, but what you do with it. You can take data from any source and analyze it to find answers that enable 1) cost reductions, 2) time reductions, 3) new product development and optimized offerings, and 4) smart decision making. My thesis is mainly focused on high-dimensional image data analysis and computations. The image data I worked on are CT images of abdominal aortic aneurysm (AAA) and brain images of Alzheimer's Disease. I developed Bayesian calibration method for the former and Supervised learning with Markov Chain for the latter. Bayesian calibration has a long history within computer modeling in general. Bayesian calibration is an iterative process of updating uncertainty distributions on the calibration parameters in a way that is consistent with the observed data. Because of the advances in complex mathematical models and fast computer codes, Bayesian calibration of computer experiments are popular in the scientific research nowadays. As we know, compared to a computer model, a complex system in real life is expensive both in time and money to observe. Therefore, computer models can be a stand-alone tool or combined with (typically smaller) data from physical experiments or field observations. And Bayesian calibration is powerful in integrating all sources of uncertainties into the model definition and calibration procedure. For the AAA data, first I modeled only one patient (patient-specific prediction discussed in Chapter 2), and then built an advanced model which can incorporate all patients (multi- patients prediction in Chapter 3). In the process, semi-parametric functional data analysis, covariance modeling and Bayesian methods was highly practiced and used. The contributions are as follows. First, we formulate the Bayesian calibration of our AAA G&R; computation model taking into account model inadequacy, prior distributions of model parameters, measurement errors, and most importantly, longitudinal CT scan images. Next, we demonstrate how to achieve the proposed aims by solving the formulated Bayesian calibration problem using a simulation study and real data analysis. In particular, we compare and discuss the performance and computation time under different sampling cases for the computation model output data and (synthesized) patient data, both of which are synthesized by the G&R; computation. We apply our Bayesian calibration to the real CT data and validate our prediction, showing the usefulness of our approach to the computational science and medical communities in aiding decision making. For the Alzheimer's Disease data, the causes are currently being researched massively, but no definitive answers exist as yet. Genetic predisposition, abnormal protein deposits in the brain and environmental factors are suspected to play a role in the development of the disease. In Chapter 4, my main goal is to model the progression of Alzheimer's Disease by applying multi-state Markov model, and to investigate the significance of known risk factors like Age, ApoE4 and some brain structural volumetric variables getting from MRI like hippocampus, and at the same time, to predict the transitions between different clinical diagnosis states based transition rates and transition probabilities. We found that the model with age is not significant (p-value is 0.1733) according to the likelihood ratio test, while ApoE4 is a significant risk factor in our Markov model. Predictions based on transition rates and transition intensities were made and validated with the accuracy as high as 0.7849."--Pages ii-iii.
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