Abstract

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 functions at various spatial locations. In particular, we study the 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 interpret the 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 from the 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 the proposed method is evidenced in the classification of R-fMRI scans

Notes

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Graduation Date

2019

Semester

Summer

Advisor

Pensky, Marianna

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0007710

URL

http://purl.fcla.edu/fcla/etd/CFE0007710

Language

English

Release Date

August 2019

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

Included in

Mathematics Commons

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