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Solution of linear ill-posed problems using overcomplete dictionaries
Title
Solution of linear ill-posed problems using overcomplete dictionaries.
Creator
Gupta, Pawan, Pensky, Marianna, Swanson, Jason, Zhang, Teng, Foroosh, Hassan, University of Central Florida
Abstract / Description
In this dissertation, we consider an application of overcomplete dictionaries to the solution of general ill-posed linear inverse problems. In the context of regression problems, there has been an enormous amount of effort to recover an unknown function using such dictionaries. While some research on the subject has been already carried out, there are still many gaps to address. In particular, one of the most popular methods, lasso, and its variants, is based on minimizing the empirical...
Show moreIn this dissertation, we consider an application of overcomplete dictionaries to the solution of general ill-posed linear inverse problems. In the context of regression problems, there has been an enormous amount of effort to recover an unknown function using such dictionaries. While some research on the subject has been already carried out, there are still many gaps to address. In particular, one of the most popular methods, lasso, and its variants, is based on minimizing the empirical likelihood and unfortunately, requires stringent assumptions on the dictionary, the so-called, compatibility conditions. Though compatibility conditions are hard to satisfy, it is well known that this can be accomplished by using random dictionaries. In the first part of the dissertation, we show how one can apply random dictionaries to the solution of ill-posed linear inverse problems with Gaussian noise. We put a theoretical foundation under the suggested methodology and study its performance via simulations and real-data example. In the second part of the dissertation, we investigate the application of lasso to the linear ill-posed problems with non-Gaussian noise. We have developed a theoretical background for the application of lasso to such problems and studied its performance via simulations.
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Date Issued
2019
Identifier
CFE0007811, ucf:52345
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007811
Filtering Problems in Stochastic Tomography
Title
Filtering Problems in Stochastic Tomography.
Creator
Gomez, Tyler, Swanson, Jason, Yong, Jiongmin, Tamasan, Alexandru, Dogariu, Aristide, University of Central Florida
Abstract / Description
Distinguishing signal from noise has always been a major goal in probabilistic analysis of data. Such is no less the case in the field of medical imaging, where both the processes of photon emission and their rate of absorption by the body behave as random variables. We explore methods by which to extricate solid conclusions from noisy data involving an X-ray transform, long the mathematical mainstay of such tools as computed axial tomography (CAT scans). Working on the assumption of having...
Show moreDistinguishing signal from noise has always been a major goal in probabilistic analysis of data. Such is no less the case in the field of medical imaging, where both the processes of photon emission and their rate of absorption by the body behave as random variables. We explore methods by which to extricate solid conclusions from noisy data involving an X-ray transform, long the mathematical mainstay of such tools as computed axial tomography (CAT scans). Working on the assumption of having some prior probabilities assigned to various states a body can be found in, we introduce and make rigorous an understanding of how to condition these into posterior probabilities by using the scan data.
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Date Issued
2017
Identifier
CFE0006740, ucf:51839
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0006740
Nonparametric and Empirical Bayes Estimation Methods
Title
Nonparametric and Empirical Bayes Estimation Methods.
Creator
Benhaddou, Rida, Pensky, Marianna, Han, Deguang, Swanson, Jason, Ni, Liqiang, University of Central Florida
Abstract / Description
In the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet...
Show moreIn the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet series based empirical Bayes estimator constructed in Pensky and Alotaibi (2005). We propose an adaptive version of this estimator using Lepski's method and show that the estimator attains optimal convergence rates. The theory is supplemented by numerous examples. Our study of the functional deconvolution model expands results of Pensky and Sapatinas (2009, 2010, 2011) to the case of estimating an $(r+1)$-dimensional function or dependent errors. In both cases, we derive minimax lower bounds for the integrated square risk over a wide set of Besov balls and construct adaptive wavelet estimators that attain those optimal convergence rates. In particular, in the case of estimating a periodic $(r+1)$-dimensional function, we show that by choosing Besov balls of mixed smoothness, we can avoid the ''curse of dimensionality'' and, hence, obtain higher than usual convergence rates when $r$ is large. The study of deconvolution of a multivariate function is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Finally, we consider a multichannel deconvolution model with long-range dependent Gaussian errors. We do not limit our consideration to a specific type of long-range dependence, rather we assume that the eigenvalues of the covariance matrix of the errors are bounded above and below. We show that convergence rates of the estimators depend on a balance between the smoothness parameters of the response function, the smoothness of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the channels.
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Date Issued
2013
Identifier
CFE0004814, ucf:49737
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004814
Accelerated Life Model with Various Types of Censored Data
Title
Accelerated Life Model with Various Types of Censored Data.
Creator
Pridemore, Kathryn, Pensky, Marianna, Mikusinski, Piotr, Swanson, Jason, Nickerson, David, University of Central Florida
Abstract / Description
The Accelerated Life Model is one of the most commonly used tools in the analysis of survival data which are frequently encountered in medical research and reliability studies. In these types of studies we often deal with complicated data sets for which we cannot observe the complete data set in practical situations due to censoring. Such difficulties are particularly apparent by the fact that there is little work in statistical literature on the Accelerated Life Model for complicated types...
Show moreThe Accelerated Life Model is one of the most commonly used tools in the analysis of survival data which are frequently encountered in medical research and reliability studies. In these types of studies we often deal with complicated data sets for which we cannot observe the complete data set in practical situations due to censoring. Such difficulties are particularly apparent by the fact that there is little work in statistical literature on the Accelerated Life Model for complicated types of censored data sets, such as doubly censored data, interval censored data, and partly interval censored data.In this work, we use the Weighted Empirical Likelihood approach (Ren, 2001) to construct tests, confidence intervals, and goodness-of-fit tests for the Accelerated Life Model in a unified way for various types of censored data. We also provide algorithms for implementation and present relevant simulation results.I began working on this problem with Dr. Jian-Jian Ren. Upon Dr. Ren's departure from the University of Central Florida I completed this dissertation under the supervision of Dr. Marianna Pensky.
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Date Issued
2013
Identifier
CFE0004913, ucf:49613
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004913
Functional Data Analysis and its application to cancer data
Title
Functional Data Analysis and its application to cancer data.
Creator
Martinenko, Evgeny, Pensky, Marianna, Tamasan, Alexandru, Swanson, Jason, Richardson, Gary, University of Central Florida
Abstract / Description
The objective of the current work is to develop novel procedures for the analysis of functional dataand apply them for investigation of gender disparity in survival of lung cancer patients. In particular,we use the time-dependent Cox proportional hazards model where the clinical information isincorporated via time-independent covariates, and the current age is modeled using its expansionover wavelet basis functions. We developed computer algorithms and applied them to the dataset which is...
Show moreThe objective of the current work is to develop novel procedures for the analysis of functional dataand apply them for investigation of gender disparity in survival of lung cancer patients. In particular,we use the time-dependent Cox proportional hazards model where the clinical information isincorporated via time-independent covariates, and the current age is modeled using its expansionover wavelet basis functions. We developed computer algorithms and applied them to the dataset which is derived from Florida Cancer Data depository data set (all personal information whichallows to identify patients was eliminated). We also studied the problem of estimation of a continuousmatrix-variate function of low rank. We have constructed an estimator of such functionusing its basis expansion and subsequent solution of an optimization problem with the Schattennormpenalty. We derive an oracle inequality for the constructed estimator, study its properties viasimulations and apply the procedure to analysis of Dynamic Contrast medical imaging data.
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Date Issued
2014
Identifier
CFE0005377, ucf:50447
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005377
Estimation and clustering in statistical ill-posed linear inverse problems
Title
Estimation and clustering in statistical ill-posed linear inverse problems.
Creator
Rajapakshage, Rasika, Pensky, Marianna, Swanson, Jason, Zhang, Teng, Bagci, Ulas, Foroosh, Hassan, University of Central Florida
Abstract / Description
The main focus of the dissertation is estimation and clustering in statistical ill-posed linear inverse problems. The dissertation deals with a problem of simultaneously estimating a collection of solutions of ill-posed linear inverse problems from their noisy images under an operator that does not have a bounded inverse, when the solutions are related in a certain way. The dissertation defense consists of three parts. In the first part, the collection consists of measurements of temporal...
Show moreThe main focus of the dissertation is estimation and clustering in statistical ill-posed linear inverse problems. The dissertation deals with a problem of simultaneously estimating a collection of solutions of ill-posed linear inverse problems from their noisy images under an operator that does not have a bounded inverse, when the solutions are related in a certain way. The dissertation defense consists of three parts. In the first part, the collection consists of measurements of temporal functions at various spatial locations. In particular, we studythe problem of estimating a three-dimensional function based on observations of its noisy Laplace convolution. In the second part, we recover classes of similar curves when the class memberships are unknown. Problems of this kind appear in many areas of application where clustering is carried out at the pre-processing step and then the inverse problem is solved for each of the cluster averages separately. As a result, the errors of the procedures are usually examined for the estimation step only. In both parts, we construct the estimators, study their minimax optimality and evaluate their performance via a limited simulation study. In the third part, we propose a new computational platform to better understand the patterns of R-fMRI by taking into account the challenge of inevitable signal fluctuations and interpretthe success of dynamic functional connectivity approaches. Towards this, we revisit an auto-regressive and vector auto-regressive signal modeling approach for estimating temporal changes of the signal in brain regions. We then generate inverse covariance matrices fromthe generated windows and use a non-parametric statistical approach to select significant features. Finally, we use Lasso to perform classification of the data. The effectiveness of theproposed method is evidenced in the classification of R-fMRI scans
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Date Issued
2019
Identifier
CFE0007710, ucf:52450
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007710
Bayesian Model Selection for Classification with Possibly Large Number of Groups
Title
Bayesian Model Selection for Classification with Possibly Large Number of Groups.
Creator
Davis, Justin, Pensky, Marianna, Swanson, Jason, Richardson, Gary, Crampton, William, Ni, Liqiang, University of Central Florida
Abstract / Description
The purpose of the present dissertation is to study model selection techniques which are specifically designed for classification of high-dimensional data with a large number of classes. To the best of our knowledge, this problem has never been studied in depth previously. We assume that the number of components p is much larger than the number of samples n, and that only few of those p components are useful for subsequent classification. In what follows, we introduce two Bayesian models...
Show moreThe purpose of the present dissertation is to study model selection techniques which are specifically designed for classification of high-dimensional data with a large number of classes. To the best of our knowledge, this problem has never been studied in depth previously. We assume that the number of components p is much larger than the number of samples n, and that only few of those p components are useful for subsequent classification. In what follows, we introduce two Bayesian models which use two different approaches to the problem: one which discards components which have "almost constant" values (Model 1) and another which retains the components for which between-group variations are larger than within-group variation (Model 2). We show that particular cases of the above two models recover familiar variance or ANOVA-based component selection. When one has only two classes and features are a priori independent, Model 2 reduces to the Feature Annealed Independence Rule (FAIR) introduced by Fan and Fan (2008) and can be viewed as a natural generalization to the case of L (>) 2 classes. A nontrivial result of the dissertation is that the precision of feature selection using Model 2 improves when the number of classes grows. Subsequently, we examine the rate of misclassification with and without feature selection on the basis of Model 2.
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Date Issued
2011
Identifier
CFE0004097, ucf:49091
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004097
Analysis of Employment and Earnings Using Varying Coefficient Models to Assess Success of Minorities and Women
Title
Analysis of Employment and Earnings Using Varying Coefficient Models to Assess Success of Minorities and Women.
Creator
Goedeker, Amanda, Pensky, Marianna, Song, Zixia, Swanson, Jason, Huang, Hsin-Hsiung, University of Central Florida
Abstract / Description
The objective of this thesis is to examine the success of minorities (black, and Hispanic/Latino employees) and women in the United States workforce, defining success by employment percentage and earnings. The goal of this thesis is to study the impact gender, race, passage of time, and national economic status reflected in gross domestic product have on the success of minorities and women. In particular, this thesis considers the impact of these factors in Science, Technology, Engineering...
Show moreThe objective of this thesis is to examine the success of minorities (black, and Hispanic/Latino employees) and women in the United States workforce, defining success by employment percentage and earnings. The goal of this thesis is to study the impact gender, race, passage of time, and national economic status reflected in gross domestic product have on the success of minorities and women. In particular, this thesis considers the impact of these factors in Science, Technology, Engineering and Math (STEM) industries. Varying coefficient models are utilized in the analysis of data sets for national employment percentages and earnings.
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Date Issued
2016
Identifier
CFE0006458, ucf:51425
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0006458
HJB Equation and Statistical Arbitrage applied to High Frequency Trading
Title
HJB Equation and Statistical Arbitrage applied to High Frequency Trading.
Creator
Park, Yonggi, Yong, Jiongmin, Swanson, Jason, Richardson, Gary, Shuai, Zhisheng, University of Central Florida
Abstract / Description
In this thesis we investigate some properties of market making and statistical arbitrage applied to High Frequency Trading (HFT). Using the Hamilton-Jacobi-Bellman(HJB) model developed by Guilbaud, Fabien and Pham, Huyen in 2012, we studied how market making works to obtain optimal strategy during limit order and market order. Also we develop the best investment strategy through Moving Average, Exponential Moving Average, Relative Strength Index, Sharpe Ratio.
Date Issued
2013
Identifier
CFE0004907, ucf:49628
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004907