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Browsing by Subject "Markov Chain Monte Carlo"
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Item Modern Monte Carlo Methods and Their Application in Semiparametric Regression(2021-05) Thomas, Samuel Joseph; Tu, Wanzhu; Boukai, Ben; Li, Xiaochen; Song, FengguangThe essence of Bayesian data analysis is to ascertain posterior distributions. Posteriors generally do not have closed-form expressions for direct computation in practical applications. Analysts, therefore, resort to Markov Chain Monte Carlo (MCMC) methods for the generation of sample observations that approximate the desired posterior distribution. Standard MCMC methods simulate sample values from the desired posterior distribution via random proposals. As a result, the mechanism used to generate the proposals inevitably determines the efficiency of the algorithm. One of the modern MCMC techniques designed to explore the high-dimensional space more efficiently is Hamiltonian Monte Carlo (HMC), based on the Hamiltonian differential equations. Inspired by classical mechanics, these equations incorporate a latent variable to generate MCMC proposals that are likely to be accepted. This dissertation discusses how such a powerful computational approach can be used for implementing statistical models. Along this line, I created a unified computational procedure for using HMC to fit various types of statistical models. The procedure that I proposed can be applied to a broad class of models, including linear models, generalized linear models, mixed-effects models, and various types of semiparametric regression models. To facilitate the fitting of a diverse set of models, I incorporated new parameterization and decomposition schemes to ensure the numerical performance of Bayesian model fitting without sacrificing the procedure’s general applicability. As a concrete application, I demonstrate how to use the proposed procedure to fit a multivariate generalized additive model (GAM), a nonstandard statistical model with a complex covariance structure and numerous parameters. Byproducts of the research include two software packages that all practical data analysts to use the proposed computational method to fit their own models. The research’s main methodological contribution is the unified computational approach that it presents for Bayesian model fitting that can be used for standard and nonstandard statistical models. Availability of such a procedure has greatly enhanced statistical modelers’ toolbox for implementing new and nonstandard statistical models.Item A semiparametric modeling framework for potential biomarker discovery and the development of metabonomic profiles(BioMed Central, 2008-01-23) Ghosh, Samiran; Grant, David F.; Dey, Dipak K.; Hill, Dennis W.; Mathematical Sciences, School of ScienceBackground The discovery of biomarkers is an important step towards the development of criteria for early diagnosis of disease status. Recently electrospray ionization (ESI) and matrix assisted laser desorption (MALDI) time-of-flight (TOF) mass spectrometry have been used to identify biomarkers both in proteomics and metabonomics studies. Data sets generated from such studies are generally very large in size and thus require the use of sophisticated statistical techniques to glean useful information. Most recent attempts to process these types of data model each compound's intensity either discretely by positional (mass to charge ratio) clustering or through each compounds' own intensity distribution. Traditionally data processing steps such as noise removal, background elimination and m/z alignment, are generally carried out separately resulting in unsatisfactory propagation of signals in the final model. Results In the present study a novel semi-parametric approach has been developed to distinguish urinary metabolic profiles in a group of traumatic patients from those of a control group consisting of normal individuals. Data sets obtained from the replicates of a single subject were used to develop a functional profile through Dirichlet mixture of beta distribution. This functional profile is flexible enough to accommodate variability of the instrument and the inherent variability of each individual, thus simultaneously addressing different sources of systematic error. To address instrument variability, all data sets were analyzed in replicate, an important issue ignored by most studies in the past. Different model comparisons were performed to select the best model for each subject. The m/z values in the window of the irregular pattern are then further recommended for possible biomarker discovery. Conclusion To the best of our knowledge this is the very first attempt to model the physical process behind the time-of flight mass spectrometry. Most of the state of the art techniques does not take these physical principles in consideration while modeling such data. The proposed modeling process will apply as long as the basic physical principle presented in this paper is valid. Notably we have confined our present work mostly within the modeling aspect. Nevertheless clinical validation of our recommended list of potential biomarkers will be required. Hence, we have termed our modeling approach as a "framework" for further work.