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 Title
 Bayesian nonparametric estimation via Gibbs sampling for coherent systems with redundancy.
 Creator

Lawson, Kevin Lee., Florida State University
 Abstract/Description

We consider a coherent system S consisting of m independent components for which we do not know the distributions of the components' lifelengths. If we know the structure function of the system, then we can estimate the distribution of the system lifelength by estimating the distributions of the lifelengths of the individual components. Suppose that we can collect data under the 'autopsy model', wherein a system is run until a failure occurs and then the status (functioning or dead) of each...
Show moreWe consider a coherent system S consisting of m independent components for which we do not know the distributions of the components' lifelengths. If we know the structure function of the system, then we can estimate the distribution of the system lifelength by estimating the distributions of the lifelengths of the individual components. Suppose that we can collect data under the 'autopsy model', wherein a system is run until a failure occurs and then the status (functioning or dead) of each component is obtained. This test is repeated n times. The autopsy statistics consist of the age of the system at the time of breakdown and the set of parts that are dead by the time of breakdown. Using the structure function and the recorded status of the components, we then classify the failure time of each component. We develop a nonparametric Bayesian estimate of the distributions of the component lifelengths and then use this to obtain an estimate of the distribution of the lifelength of the system. The procedure is applicable to machinetest settings wherein the machines have redundant designs. A parametric procedure is also given.
Show less  Date Issued
 1994, 1994
 Identifier
 AAI9502812, 3088467, FSDT3088467, fsu:77272
 Format
 Document (PDF)
 Title
 On knotting of randomly embedded polygons in R(3).
 Creator

Diao, Yuanan., Florida State University
 Abstract/Description

A long circular polymer molecule can sometimes be treated as a polygon that is randomly embedded in R$\sp3$, especially when the molecule is very "thin". One is naturally led to the problem of "randomly embedded ngons in R$\sp3$" when considering the topological entanglement of these long polymer molecules., Call a randomly embedded ngon in R$\sp3$ a random loop. There are different mathematical models for a random loop, depending on point of view. We work in the freelyjointed case, with...
Show moreA long circular polymer molecule can sometimes be treated as a polygon that is randomly embedded in R$\sp3$, especially when the molecule is very "thin". One is naturally led to the problem of "randomly embedded ngons in R$\sp3$" when considering the topological entanglement of these long polymer molecules., Call a randomly embedded ngon in R$\sp3$ a random loop. There are different mathematical models for a random loop, depending on point of view. We work in the freelyjointed case, with no restrictions on angles between edges. If one assumes that each edge of the random loop with any orientation is a standard Gaussian random vector, then such a model is called a Gaussian random loop model (GRL). If one assumes that the length of each step is a constant, then such a model is called an equilateral random loop model (ERL). We prove the FrischWassermanDelbruck conjecture, which says the knotting probability of a random loop of n steps goes to 1 as n goes to infinity. More precisely, we prove that, when n is large enough, the knotting probability of a GRL (or ERL) of n steps exceeds 1 $$ exp($$n$\sp\varepsilon$), where $\varepsilon$ is a positive constant. The result in the GRL case is joint with N. Pippenger and D. W. Sumners.
Show less  Date Issued
 1990, 1990
 Identifier
 AAI9023915, 3162045, FSDT3162045, fsu:78243
 Format
 Document (PDF)
 Title
 The Oneand TwoSample Problem for Data on Hilbert Manifolds with Applications to Shape Analysis.
 Creator

Qiu, Mingfei, Patrangenaru, Victor, Liu, Xiuwen, Slate, Elizabeth H., Barbu, Adrian G. (Adrian Gheorghe), Clickner, Robert Paul, Paige, Robert, Florida State University, College...
Show moreQiu, Mingfei, Patrangenaru, Victor, Liu, Xiuwen, Slate, Elizabeth H., Barbu, Adrian G. (Adrian Gheorghe), Clickner, Robert Paul, Paige, Robert, Florida State University, College of Arts and Sciences, Department of Statistics
Show less  Abstract/Description

This dissertation is concerned with high level imaging analysis. In particular, our focus is on extracting the projective shape information or the similarity shape from digital camera images or Magnetic Resonance Imaging(MRI). The approach is statistical without making any assumptions about the distributions of the random object under investigation. The data is organized as points on a Hilbert manifold. In the case of projective shapes of finite dimensional configuration of points, we...
Show moreThis dissertation is concerned with high level imaging analysis. In particular, our focus is on extracting the projective shape information or the similarity shape from digital camera images or Magnetic Resonance Imaging(MRI). The approach is statistical without making any assumptions about the distributions of the random object under investigation. The data is organized as points on a Hilbert manifold. In the case of projective shapes of finite dimensional configuration of points, we consider testing a onesample null hypothesis, while in the infinite dimensional case, we considered a neighborhood hypothesis testing methods. For 3D scenes, we retrieve the 3D projective shape, and use the Lie group structure of the projective shape space. We test the equality of two extrinsic means, by introducing the mean projective shape change. For 2D MRI of midsections of Corpus Callosum contours, we use an automatic matching technique that is necessary in pursuing a onesample neighborhood hypothesis testing for the similarity shapes. We conclude that the mean similarity shape of the Corpus Callosum of average individuals is very far from the shape of Albert Einstein's, which may explain his geniality. Another application of our Hilbert manifold methodology is twosample testing problem for VeroneseWhitney means of projective shapes of 3D contours. Particularly, our data consisting comparing 3D projective shapes of contours of leaves from the same tree species.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Qiu_fsu_0071E_12922
 Format
 Thesis
 Title
 Statistical Analysis on Object Spaces with Applications.
 Creator

Yao, Kouadio David, Patrangenaru, Victor, Kercheval, Alec N., Liu, Xiuwen, Mio, Washington, Wang, Xiaoming, Florida State University, College of Arts and Sciences, Department of...
Show moreYao, Kouadio David, Patrangenaru, Victor, Kercheval, Alec N., Liu, Xiuwen, Mio, Washington, Wang, Xiaoming, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Most of the data encountered is bounded nonlinear data. The Universe is bounded, planets are sphere like shaped objects, and life growing on Earth comes in various shapes and colors that can hardly be represented as points on a linear space, and even if the object space they sit on is embedded in a Euclidean space, their mean vector can not be represented as a point on that object space, except for the case when such space is convex. To address this misgiving, since the mean vector is the...
Show moreMost of the data encountered is bounded nonlinear data. The Universe is bounded, planets are sphere like shaped objects, and life growing on Earth comes in various shapes and colors that can hardly be represented as points on a linear space, and even if the object space they sit on is embedded in a Euclidean space, their mean vector can not be represented as a point on that object space, except for the case when such space is convex. To address this misgiving, since the mean vector is the minimizer of the expected square distance, following Fr\'echet (1948), on a compact metric space, one may consider both minimizers and maximizers of the expected square distance to a given point on the object space as mean, respectively {\bf antimean} of a given random point. Of all distances on a object space, one considers here the chord distance associated with an embedding of the object space, since for such distances one can give a necessary and sufficient condition for the existence of a unique Fr\'echet mean (respectively Fr\'echet antimean). For such distributions these location parameters are called extrinsic mean (respectively extrinsic antimean), and the corresponding sample statistics are consistent estimators of their population counterparts. Moreover one derives the limit distribution of such estimators around a mean located at a smooth extrinsic antimean. Extrinsic analysis is thus a general framework that allows one to run object data analysis on nonlinear object spaces that can be embedded in a numerical space. In particular one focuses on VeroneseWhitney (VW) means and antimeans of 3D projective shapes of configurations extracted from digital camera images. The 3D data extraction is greatly simplified by an RGB based algorithm followed by the FaugerasHartleyGuptaChen 3D reconstruction method. In particular one derives two sample tests for face analysis based on projective shapes, and more generally a MANOVA on manifolds method to be used in 3D projective shape analysis. The manifold based approach is also applicable to financial data analysis for exchange rates.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Yao_fsu_0071E_13605
 Format
 Thesis
 Title
 Trend and VariablePhase Seasonality Estimation from Functional Data.
 Creator

Tai, LiangHsuan, Gallivan, Kyle A., Srivastava, Anuj, Wu, Wei, Klassen, E. (Eric), Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The problem of estimating trend and seasonality has been studied over several decades, although mostly using single time series setup. This dissertation studies the problem of estimating these components from a functional data point of view, i.e. multiple curves, in situations where seasonal effects exhibit arbitrary time warpings or phase variability across different observations. Rather than ignoring the phase variability, or using an offtheshelf alignment method to remove phase, we take...
Show moreThe problem of estimating trend and seasonality has been studied over several decades, although mostly using single time series setup. This dissertation studies the problem of estimating these components from a functional data point of view, i.e. multiple curves, in situations where seasonal effects exhibit arbitrary time warpings or phase variability across different observations. Rather than ignoring the phase variability, or using an offtheshelf alignment method to remove phase, we take a modelbased approach and seek Maximum Likelihood Estimators (MLEs) of the trend and the seasonal effects, while performing alignments over the seasonal effects at the same time. The MLEs of trend, seasonality, and phase are computed using a coordinate descent based optimization method. We use bootstrap replication for computing confidence bands and for testing hypothesis about the estimated components. We also utilize loglikelihood for selecting the trend subspace, and for comparisons with other candidate models. This framework is demonstrated using experiments involving synthetic data and three real data (Berkeley growth velocity, U.S. electricity price, and USD exchange fluctuation). Our framework is further applied to another biological problem, significance analysis of gene sets of timecourse gene expression data and outperform the stateoftheart method.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Tai_fsu_0071E_13816
 Format
 Thesis