Current Search: Research Repository (x) » Statistics (x) » Patrangenaru, Victor (x)
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 Title
 Covariance on Manifolds.
 Creator

Balov, Nikolay H. (Nikolay Hristov), Srivastava, Anuj, Klassen, Eric, Patrangenaru, Victor, McGee, Daniel, Department of Statistics, Florida State University
 Abstract/Description

With ever increasing complexity of observational and theoretical data models, the sufficiency of the classical statistical techniques, designed to be applied only on vector quantities, is being challenged. Nonlinear statistical analysis has become an area of intensive research in recent years. Despite the impressive progress in this direction, a unified and consistent framework has not been reached. In this regard, the following work is an attempt to improve our understanding of random...
Show moreWith ever increasing complexity of observational and theoretical data models, the sufficiency of the classical statistical techniques, designed to be applied only on vector quantities, is being challenged. Nonlinear statistical analysis has become an area of intensive research in recent years. Despite the impressive progress in this direction, a unified and consistent framework has not been reached. In this regard, the following work is an attempt to improve our understanding of random phenomena on nonEuclidean spaces. More specifically, the motivating goal of the present dissertation is to generalize the notion of distribution covariance, which in standard settings is defined only in Euclidean spaces, on arbitrary manifolds with metric. We introduce a tensor field structure, named covariance field, that is consistent with the heterogeneous nature of manifolds. It not only describes the variability imposed by a probability distribution but also provides alternative distribution representations. The covariance field combines the distribution density with geometric characteristics of its domain and thus fills the gap between these two.We present some of the properties of the covariance fields and argue that they can be successfully applied to various statistical problems. In particular, we provide a systematic approach for defining parametric families of probability distributions on manifolds, parameter estimation for regression analysis, nonparametric statistical tests for comparing probability distributions and interpolation between such distributions. We then present several application areas where this new theory may have potential impact. One of them is the branch of directional statistics, with domain of influence ranging from geosciences to medical image analysis. The fundamental level at which the covariance based structures are introduced, also opens a new area for future research.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd1045
 Format
 Thesis
 Title
 High Level Image Analysis on Manifolds via Projective Shapes and 3D Reflection Shapes.
 Creator

Lester, David T. (David Thomas), Patrangenaru, Victor, Liu, Xiuwen, Barbu, Adrian G. (Adrian Gheorghe), Tao, Minjing, Florida State University, College of Arts and Sciences,...
Show moreLester, David T. (David Thomas), Patrangenaru, Victor, Liu, Xiuwen, Barbu, Adrian G. (Adrian Gheorghe), Tao, Minjing, Florida State University, College of Arts and Sciences, Department of Statistics
Show less  Abstract/Description

Shape analysis is a widely studied topic in modern Statistics with important applications in areas such as medical imaging. Here we focus on twosample hypothesis testing for both finite and infinite extrinsic mean shapes of configurations. First, we present a test for equality of mean projective shapes of 2D contours based on rotations. Secondly, we present a test for mean 3D reflection shapes based on the Schoenberg mean. We apply these tests to footprint data (contours), clamshells (3D...
Show moreShape analysis is a widely studied topic in modern Statistics with important applications in areas such as medical imaging. Here we focus on twosample hypothesis testing for both finite and infinite extrinsic mean shapes of configurations. First, we present a test for equality of mean projective shapes of 2D contours based on rotations. Secondly, we present a test for mean 3D reflection shapes based on the Schoenberg mean. We apply these tests to footprint data (contours), clamshells (3D reflection shape) and human facial configurations extracted from digital camera images. We also present the method of MANOVA on manifolds, and apply it to face data extracted from digital camera images. Finally, we present a new statistical tool called antiregression.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Lester_fsu_0071E_13856
 Format
 Thesis
 Title
 Nonparametric Data Analysis on Manifolds with Applications in Medical Imaging.
 Creator

Osborne, Daniel Eugene, Patrangenaru, Victor, Liu, Xiuwen, Barbu, Adrian, Chicken, Eric, Department of Statistics, Florida State University
 Abstract/Description

Over the past twenty years, there has been a rapid development in Nonparametric Statistical Analysis on Manifolds applied to Medical Imaging problems. In this body of work, we focus on two different medical imaging problems. The first problem corresponds to analyzing the CT scan data. In this context, we perform nonparametric analysis on the 3D data retrieved from CT scans of healthy young adults, on the SizeandReflection Shape Space of kads in general position in 3D. This work is a part...
Show moreOver the past twenty years, there has been a rapid development in Nonparametric Statistical Analysis on Manifolds applied to Medical Imaging problems. In this body of work, we focus on two different medical imaging problems. The first problem corresponds to analyzing the CT scan data. In this context, we perform nonparametric analysis on the 3D data retrieved from CT scans of healthy young adults, on the SizeandReflection Shape Space of kads in general position in 3D. This work is a part of larger project on planning reconstructive surgery in severe skull injuries which includes preprocessing and postprocessing steps of CT images. The next problem corresponds to analyzing MR diffusion tensor imaging data. Here, we develop a twosample procedure for testing the equality of the generalized Frobenius means of two independent populations on the space of symmetric positive matrices. These new methods, naturally lead to an analysis based on Cholesky decompositions of covariance matrices which helps to decrease computational time and does not increase dimensionality. The resulting nonparametric matrix valued statistics are used for testing if there is a difference on average between corresponding signals in Diffusion Tensor Images (DTI) in young children with dyslexia when compared to their clinically normal peers. The results presented here correspond to data that was previously used in the literature using parametric methods which also showed a significant difference.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5085
 Format
 Thesis
 Title
 Nonparametric Estimation of Three Dimensional Projective Shapes with Applications in Medical Imaging and in Pattern Recognition.
 Creator

Crane, Michael, Patrangenaru, Victor, Liu, Xiuwen, Huﬀer, Fred W., Sinha, Debajyoti, Department of Statistics, Florida State University
 Abstract/Description

This dissertation is on analysis of invariants of a 3D configuration from its 2D images in pictures of this configuration, without requiring any restriction on the camera positioning relative to the scene pictured. We briefly review some of the main results found in the literature. The methodology used is nonparametric, manifold based combined with standard computer vision reconstruction techniques. More specifically, we use asymptotic results for the extrinsic sample mean and the extrinsic...
Show moreThis dissertation is on analysis of invariants of a 3D configuration from its 2D images in pictures of this configuration, without requiring any restriction on the camera positioning relative to the scene pictured. We briefly review some of the main results found in the literature. The methodology used is nonparametric, manifold based combined with standard computer vision reconstruction techniques. More specifically, we use asymptotic results for the extrinsic sample mean and the extrinsic sample covariance to construct bootstrap confidence regions for mean projective shapes of 3D configurations. Chapters 4, 5 and 6 contain new results. In chapter 4, we develop tests for coplanarity. In chapter 5, is on reconstruction of 3D polyhedral scenes, including texture from arbitrary partial views. In chapter 6, we develop a nonparametric methodology for estimating the mean change for matched samples on a Lie group. We then notice that for k ≥ 4, a manifold of projective shapes of kads in general position in 3D has a structure of 3k − 15 dimensional Lie group (PQuaternions) that is equivariantly embedded in an Euclidean space, therefore testing for mean 3D projective shape change amounts to a one sample test for extrinsic mean PQuaternion Objects. The Lie group technique leads to a large sample and nonparametric bootstrap test for one population extrinsic mean on a projective shape space, as recently developed by Patrangenaru, Liu and Sughatadasa. On the other hand, in absence of occlusions, the 3D projective shape of a spatial configuration can be recovered from a stereo pair of images, thus allowing to test for mean glaucomatous 3D projective shape change detection from standard stereo pairs of eye images.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd4607
 Format
 Thesis
 Title
 Nonparametric Estimation of Three Dimensional Projective Shapes with Applications in Medical Imaging and in Pattern Recognition.
 Creator

Crane, Michael, Patrangenaru, Victor, Liu, Xiuwen, Huﬀer, Fred W., Sinha, Debajyoti, Department of Statistics, Florida State University
 Abstract/Description

This dissertation is on analysis of invariants of a 3D configuration from its 2D images in pictures of this configuration, without requiring any restriction on the camera positioning relative to the scene pictured. We briefly review some of the main results found in the literature. The methodology used is nonparametric, manifold based combined with standard computer vision re construction techniques. More specifically, we use asymptotic results for the extrinsic sample mean and the extrinsic...
Show moreThis dissertation is on analysis of invariants of a 3D configuration from its 2D images in pictures of this configuration, without requiring any restriction on the camera positioning relative to the scene pictured. We briefly review some of the main results found in the literature. The methodology used is nonparametric, manifold based combined with standard computer vision re construction techniques. More specifically, we use asymptotic results for the extrinsic sample mean and the extrinsic sample covariance to construct boot strap confidence regions for mean projective shapes of 3D configurations. Chapters 4, 5 and 6 contain new results. In chapter 4, we develop tests for coplanarity. In chapter 5, is on reconstruction of 3D polyhedral scenes, including texture from arbitrary partial views. In chapter 6, we develop a nonparametric methodology for estimating the mean change for matched samples on a Lie group. We then notice that for k '' 4, a manifold of projective shapes of kads in general position in 3D has a structure of 3k and #8722; 15 dimensional Lie group (PQuaternions) that is equivariantly embedded in an Euclidean space, therefore testing for mean 3D projective shape change amounts to a one sample test for extrinsic mean PQuaternion Objects. The Lie group technique leads to a large sample and nonparametric bootstrap test for one population extrinsic mean on a projective shape space, as recently developed by Patrangenaru, Liu and Sughatadasa [1]. On the other hand, in absence of occlusions, the 3D projective shape of a spatial configuration can be recovered from a stereo pair of images, thus allowing to test for mean glaucomatous 3D projective shape change detection from standard stereo pairs of eye images.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd7118
 Format
 Thesis
 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
 Testing for the Equality of Two Distributions on High Dimensional Object Spaces and Nonparametric Inference for Location Parameters.
 Creator

Guo, Ruite, Patrangenaru, Victor, Mio, Washington, Barbu, Adrian G. (Adrian Gheorghe), Bradley, Jonathan R., Florida State University, College of Arts and Sciences, Department...
Show moreGuo, Ruite, Patrangenaru, Victor, Mio, Washington, Barbu, Adrian G. (Adrian Gheorghe), Bradley, Jonathan R., Florida State University, College of Arts and Sciences, Department of Statistics
Show less  Abstract/Description

Our view is that while some of the basic principles of data analysis are going to remain unchanged, others are to be gradually replaced with Geometry and Topology methods. Linear methods are still making sense for functional data analysis, or in the context of tangent bundles of object spaces. Complex nonstandard data is represented on object spaces. An object space admitting a manifold stratification may be embedded in an Euclidean space. One defines the extrinsic energy distance associated...
Show moreOur view is that while some of the basic principles of data analysis are going to remain unchanged, others are to be gradually replaced with Geometry and Topology methods. Linear methods are still making sense for functional data analysis, or in the context of tangent bundles of object spaces. Complex nonstandard data is represented on object spaces. An object space admitting a manifold stratification may be embedded in an Euclidean space. One defines the extrinsic energy distance associated with two probability measures on an arbitrary object space embedded in a numerical space, and one introduces an extrinsic energy statistic to test for homogeneity of distributions of two random objects (r.o.'s) on such an object space. This test is validated via a simulation example on the Kendall space of planar kads with a VeroneseWhitney (VW) embedding. One considers an application to medical imaging, to test for the homogeneity of the distributions of Kendall shapes of the midsections of the Corpus Callosum in a clinically normal population vs a population of ADHD diagnosed individuals. Surprisingly, due to the high dimensionality, these distributions are not significantly different, although they are known to have highly significant VWmeans. New spread and location parameters are to be added to reflect the nontrivial topology of certain object spaces. TDA is going to be adapted to object spaces, and hypothesis testing for distributions is going to be based on extrinsic energy methods. For a random point on an object space embedded in an Euclidean space, the mean vector cannot be represented as a point on that space, except for the case when the embedded space is convex. To address this misgiving, since the mean vector is the minimizer of the expected square distance, following Frechet (1948), on an embedded compact object space, one may consider both minimizers and maximizers of the expected square distance to a given point on the embedded object space as mean, respectively antimean of the random point. Of all distances on an object space, one considers here the chord distance associated with the embedding of the object space, since for such distances one can give a necessary and sufficient condition for the existence of a unique Frechet mean (respectively Frechet 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 around the extrinsic mean ( antimean ) located at a smooth point, one derives the limit distribution of such estimators.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Guo_fsu_0071E_13977
 Format
 Thesis
 Title
 The Use of a MetaAnalysis Technique in Equating and Its Comparison with Several Small Sample Equating Methods.
 Creator

Caglak, Serdar, Paek, Insu, Patrangenaru, Victor, Almond, Russell G., Roehrig, Alysia D., Florida State University, College of Education, Department of Educational Psychology...
Show moreCaglak, Serdar, Paek, Insu, Patrangenaru, Victor, Almond, Russell G., Roehrig, Alysia D., Florida State University, College of Education, Department of Educational Psychology and Learning Systems
Show less  Abstract/Description

The main objective of this study was to investigate the improvement of the accuracy of small sample equating, which typically occurs in teacher certification/licensure examinations due to a low volume of test takers per test administration, under the NonEquivalent Groups with Anchor Test (NEAT) design by combining previous and current equating outcomes using a metaanalysis technique. The proposed metaanalytic score transformation procedure was called "metaequating" throughout this study....
Show moreThe main objective of this study was to investigate the improvement of the accuracy of small sample equating, which typically occurs in teacher certification/licensure examinations due to a low volume of test takers per test administration, under the NonEquivalent Groups with Anchor Test (NEAT) design by combining previous and current equating outcomes using a metaanalysis technique. The proposed metaanalytic score transformation procedure was called "metaequating" throughout this study. To conduct metaequating, the previous and current equating outcomes obtained from the chosen equating methods (ID (Identity Equating), CircleArc (CA) and Nominal Weights Mean (NW)) and synthetic functions (SFs) of these methods (CAS and NWS) were used, and then, empirical Bayesian (EB) and metaequating (META) procedures were implemented to estimate the equating relationship between test forms at the population level. The SFs were created by giving equal weight to each of the chosen equating methods and the identity (ID) equating. Finally, the chosen equating methods, the SFs of each method (e.g., CAS, NWS, etc.), and also the META and EB versions (e.g., NWEB, CAMETA, NWSMETA, etc.) were investigated and compared under varying testing conditions. These steps involved manipulating some of the factors that influence the accuracy of test score equating. In particular, the effect of test form difficulty levels, the groupmean ability differences, the number of previous equatings, and the sample size on the accuracy of the equating outcomes were investigated. The Chained Equipercentile (CE) equating with 6univariate and 2bivariate moments loglinear presmoothing was used as the criterion equating function to establish the equating relationship between the new form and the base (reference) form with 50,000 examinees per test form. To compare the performance of the equating methods, small numbers of examinee samples were randomly drawn from examinee populations with different ability levels in each simulation replication. Each pairs of the new and base test forms were randomly and independently selected from all available condition specific test form pairs. Those test forms were then used to obtain previous equating outcomes. However, purposeful selections of the examinee ability and test form difficulty distributions were made to obtain the current equating outcomes in each simulation replication. The previous equating outcomes were later used for the implementation of both the META and EB score transformation procedures. The effect of study factors and their possible interactions on each of the accuracy measures were investigated along the entirescore range and the cut (reduced)score range using a series of mixedfactorial ANOVA (MFA) procedures. The performances of the equating methods were also compared based on posthoc tests. Results show that the behaviors of the equating methods vary based on the each level of the group ability difference, test form difficult difference, and new group examinee sample size. Also, the use of both META and EB procedures improved the accuracy of equating results on average. The META and EB versions of the chosen equating methods therefore might be a solution to equate the test forms that are similar in their psychometric characteristics and also taken by new form examinee samples less than 50. However, since there are many factors affecting the equating results in reality, one should always expect that equating methods and score transformation procedures, or in more general terms, estimation procedures may function differently, to some degree, depending on conditions in which they are implemented. Therefore, one should consider the recommendations for the use of the proposed equating methods in this study as a piece of information, not an absolute guideline, for a rule of thumbs for practicing small sample test equating in teacher certification/licensure examinations.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Caglak_fsu_0071E_12863
 Format
 Thesis