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System identification studies how to construct mathematical models for dynamical systems from the input and output data, which finds applications in many scenarios, such as predicting future output of the system or building model based controllers for regulating the output the system. Among many other methods, convex optimization is becoming an increasingly useful tool for solving system identification problems. The reason is that many identification problems can be formulated as, or transformed into convex optimization problems. This transformation is commonly referred to as the convexification technique. The first theme of the thesis is to understand the efficacy of the convexification idea by examining two specific examples. We first establish that a l1 norm based approach can indeed help in exploiting the sparsity information of the underlying parameter vector under certain persistent excitation assumptions. After that, we analyze how the nuclear norm minimization heuristic performs on a low-rank Hankel matrix completion problem. The underlying key is to construct the dual certificate based on the structure information that is available in the problem setting.         Recursive algorithms are ubiquitous in system identification. The second theme of the thesis is the study of some existing recursive algorithms, by establishing new connections, giving new insights or interpretations to them. We first establish a connection between a basic property of the convolution operator and the score function estimation. Based on this relationship, we show how certain recursive Bayesian algorithms can be exploited to estimate the score function for systems with intractable transition densities. We also provide a new derivation and interpretation of the recursive direct weight optimization method, by exploiting certain structural information that is present in the algorithm. Finally, we study how an improved randomization strategy can be found for the randomized Kaczmarz algorithm, and how the convergence rate of the classical Kaczmarz algorithm can be studied by the stability analysis of a related time varying linear dynamical system.

Approved for public release; distribution is unlimited / This thesis describes a Bayesian method to determine the number of samples needed to estimate a proportion or probability with 95% confidence when prior bounds are placed on that proportion. It uses the Uniform [a,b] distribution as the prior, and develops a computer program and tables to find the sample size. Tables and examples are also given to compare these results with other approaches for finding sample size. The improvement that can be obtained with this method is fewer samples, and consequently less cost in Weapons Testing is required to meet a desired confidence size for a proportion or probability. / http://archive.org/details/bayesianmethodto00flor / Lieutenant Commander, Hellenic Navy

"We analyze mortality data from prostate, colon, lung, and all other types (called other cancer) to obtain age specific and age adjusted mortality rates for white males in the U.S. A related problem is to estimate the relative occurrences of these four types of cancer. We use Bayesian method because it permits a degree of smoothing which is needed to analyze data at a small area level and to assess the patterns. In the recent Atlas of the United States Mortality (1996) each type of cancer was analyzed individually. The difficulty in doing so is that there are many small areas with zero deaths. We conjecture that simultaneous analyses might help to overcome this problem, and at the same time to estimate the relative occurrences. We start with a Poisson model for the deaths, which produces a likelihood function that separates into two parts: a Poisson likelihood for the rates and a multinomial likelihood for the relative occurrences. These permit the use of a standard Poisson regression model on age as in Nandram, Sedransk and Pickle (1999), and the novelty is a multivariate logit model on the relative occurrences in which per capita income, the percent of people below poverty level, education (percent of people with four years of college) and two criteria pollutants, EPAPM25 and EPASO2, are used as covariates. We fitted the models using Markov chain Monte Carlo methods. We used one of the models to present maps of occurrences and rates for the four types. An alternative model did not work well because it provides the same pattern by age and disease. We found that while EPAPM25 has a negative effect on the occurrences, EPASO2 has a positive effect. Also, we found some interesting patterns associated with the geographical variations of mortality rates and the relative occurrences of the four cancer types."

Steam generators in nuclear power plants experienced varying degrees of under-deposit pitting corrosion. A probabilistic model to accurately predict pitting corrosion is necessary for effective life-cycle management of steam generators. This thesis presents an advanced probabilistic model of pitting corrosion characterizing the inherent randomness of the pitting process and measurement uncertainties of the in-service inspection (ISI) data obtained from eddy current (EC) inspections. A Bayesian method is developed for estimating the model parameters. The proposed model is able to estimate the number of actual pits, the actual pit depth as well as the maximum pit depth, which is the main interest of the pitting corrosion model. A MATLAB program of the Markov chain Monte Carlo technique is developed to perform the Bayesian estimations. Simulation experiments are performed to check the behavior of the Bayesian method. Results show that the MCMC algorithm is an effective way to estimate the model parameters. Also, the effectiveness and efficiency of Bayesian modeling are validated. A comprehensive case study is also presented on the in-service inspection data of pitting corrosion in a steam generator unit. The Weibull distribution is found to be an appropriate probability distribution for modeling the actual pit depth in steam generators.

Steam generators in nuclear power plants experienced varying degrees of under-deposit pitting corrosion. A probabilistic model to accurately predict pitting corrosion is necessary for effective life-cycle management of steam generators. This thesis presents an advanced probabilistic model of pitting corrosion characterizing the inherent randomness of the pitting process and measurement uncertainties of the in-service inspection (ISI) data obtained from eddy current (EC) inspections. A Bayesian method is developed for estimating the model parameters. The proposed model is able to estimate the number of actual pits, the actual pit depth as well as the maximum pit depth, which is the main interest of the pitting corrosion model. A MATLAB program of the Markov chain Monte Carlo technique is developed to perform the Bayesian estimations. Simulation experiments are performed to check the behavior of the Bayesian method. Results show that the MCMC algorithm is an effective way to estimate the model parameters. Also, the effectiveness and efficiency of Bayesian modeling are validated. A comprehensive case study is also presented on the in-service inspection data of pitting corrosion in a steam generator unit. The Weibull distribution is found to be an appropriate probability distribution for modeling the actual pit depth in steam generators.

The current generation of climate models in the Coupled Model Intercomparison Project Phase 5 (CMIP5) is used to assess the future changes in daily precipitation and its extremes. The simple average of all the models, i.e. the multi-model ensemble mean (MMEM), has been widely used due to its simplicity and better performance than most individual models. Weighting techniques are also proposed to deal with the systematic biases within the models. However, both methods are designed to reduce the uncertainties for the study of climate mean state. They will induce problems when the climate extremes are of interest. We utilize a Bayesian weighting method to investigate the rainfall mean state and perform a probability density function based assessment of daily rainfall extremes. Satellite measurement is used to evaluate the short historical period. The weighting method can be only applied to regions rather than hemispheric scale, and thus three tropical regions including the Amazon, Congo, and Southeast Asia are studied. The method based on the Gamma distribution for daily precipitation is demonstrated to perform much better than the MMEM with respect to the extreme events. A use of the Kolmogorov-Smirnov statistic for the distribution assessment indicates the method is more applicable in three tropical wet regions over land mentioned above. This is consistent with previous studies showing the Gamma distribution is more suitable for daily rainfall in wet regions. Both methods provide consistent results. The three regions display significant changes at the end of the 21st century. The Amazon will be drier, while the Congo will not have large changes in mean rainfall. However, both of the Amazon and Congo will have large rainfall variability, implying more droughts and floods. The Amazon will have 7.5% more little-rain days (defined as > 0.5 mm/d) and 4.5 mm/d larger 95th percentile for 2092-2099, and the Congo will have 2.5% more little-rain days and 1 mm/d larger 95th percentile. Southeast Asia will be dryer in the western part and wetter in the eastern part, which is consistent with the different changes in the 5th percentile. It will also experience heavier rainfall events with much larger increases in the 95th percentile. The future changes, especially the increase in rainfall extremes, are very likely associated with the strengthening of hydrological cycle. / text

Indiana University-Purdue University Indianapolis (IUPUI) / The late-onset outcome issue is common in early phase dose- nding clinical trials. This problem becomes more intractable in phase I/II clinical trials because both toxicity and e cacy responses are subject to the late-onset outcome issue. The existing methods applying for the phase I trials cannot be used directly for the phase I/II trial due to a lack of capability to model the joint toxicity{e cacy distribution. We propose a conditional weighted likelihood (CWL) method to circumvent this issue. The key idea of the CWL method is to decompose the joint probability into the product of marginal and conditional probabilities and then weight each probability based on each patient's actual follow-up time. We further extend the proposed method to handle more complex situations where the late-onset outcomes are competing risks or semicompeting risks outcomes. We treat the late-onset competing risks/semi-competing risks outcomes as missing data and develop a series of Bayesian data-augmentation methods to e ciently impute the missing data and draw the posterior samples of the parameters of interest. We also propose adaptive dose- nding algorithms to allocate patients and identify the optimal biological dose during the trial. Simulation studies show that the proposed methods yield desirable operating characteristics and outperform the existing methods.

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My MSc thesis is focused on parametric designs of Phase I clinical trials, using the continual reassessment method. A parametric model with unknown parameters is assumed. The observations are either toxic or nontoxic. Observations of toxicities are used to update the posterior distribution. Dose selection for the next patient is based on the estimated toxicity probability. The objective is to identify the maximum tolerated dose to be used in Phase II clinical trials. We introduce a new class of parametric functions for the continual reassessment method. This class is formed with the cumulative distribution function of the normal distribution. The major advantage is that we can choose different normal distributions to model different toxicity probability functions. We conduct simulation studies and compare our new design with the existing parametric designs, and have found that our design performs better by choosing the appropriate values of the mean and variance. / October 2016