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Hotz, Thomas
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Hotz, Thomas
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Hotz, Thomas
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Hotz, T.
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2010Journal Article Research Paper [["dc.bibliographiccitation.firstpage","593"],["dc.bibliographiccitation.issue","4"],["dc.bibliographiccitation.journal","IEEE Transactions on Pattern Analysis and Machine Intelligence"],["dc.bibliographiccitation.lastpage","603"],["dc.bibliographiccitation.volume","32"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Hotz, Thomas"],["dc.contributor.author","Munk, Axel"],["dc.date.accessioned","2017-09-07T11:46:05Z"],["dc.date.available","2017-09-07T11:46:05Z"],["dc.date.issued","2010"],["dc.description.abstract","We propose an intrinsic multifactorial model for data on Riemannian manifolds that typically occur in the statistical analysis of shape. Due to the lack of a linear structure, linear models cannot be defined in general; to date only one-way MANOVA is available. For a general multifactorial model, we assume that variation not explained by the model is concentrated near elements defining the effects. By determining the asymptotic distributions of respective sample covariances under parallel transport, we show that they can be compared by standard MANOVA. Often in applications manifolds are only implicitly given as quotients, where the bottom space parallel transport can be expressed through a differential equation. For Kendall's space of planar shapes, we provide an explicit solution. We illustrate our method by an intrinsic two-way MANOVA for a set of leaf shapes. While biologists can identify genotype effects by sight, we can detect height effects that are otherwise not identifiable."],["dc.identifier.doi","10.1109/TPAMI.2009.117"],["dc.identifier.gro","3142944"],["dc.identifier.isi","000274548800003"],["dc.identifier.pmid","20224117"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/404"],["dc.language.iso","en"],["dc.notes.intern","WoS Import 2017-03-10"],["dc.notes.status","final"],["dc.notes.submitter","PUB_WoS_Import"],["dc.relation.issn","0162-8828"],["dc.title","Intrinsic MANOVA for Riemannian Manifolds with an Application to Kendall's Space of Planar Shapes"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.subtype","original"],["dspace.entity.type","Publication"]]Details DOI PMID PMC WOS2015Journal Article [["dc.bibliographiccitation.firstpage","177"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Annals of the Institute of Statistical Mathematics"],["dc.bibliographiccitation.lastpage","193"],["dc.bibliographiccitation.volume","67"],["dc.contributor.author","Hotz, Thomas"],["dc.contributor.author","Huckemann, Stephan"],["dc.date.accessioned","2017-09-07T11:50:30Z"],["dc.date.available","2017-09-07T11:50:30Z"],["dc.date.issued","2015"],["dc.description.abstract","This paper gives a comprehensive treatment of local uniqueness, asymptotics and numerics for intrinsic sample means on the circle. It turns out that local uniqueness as well as rates of convergence are governed by the distribution near the antipode. If the distribution is locally less than uniform there, we have local uniqueness and asymptotic normality with a square-root rate. With increased proximity to the uniform distribution the rate can be arbitrarily slow, and in the limit, local uniqueness is lost. Further, we give general distributional conditions, e.g., unimodality, that ensure global uniqueness. Along the way, we discover that sample means can occur only at the vertices of a regular polygon which allows to compute intrinsic sample means in linear time from sorted data. This algorithm is finally applied in a simulation study demonstrating the dependence of the convergence rates on the behavior of the density at the antipode."],["dc.identifier.doi","10.1007/s10463-013-0444-7"],["dc.identifier.gro","3147630"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/5093"],["dc.language.iso","en"],["dc.notes.status","final"],["dc.notes.submitter","chake"],["dc.relation.issn","0020-3157"],["dc.title","Intrinsic means on the circle: uniqueness, locus and asymptotics"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]Details DOI2008Journal Article [["dc.bibliographiccitation.firstpage","699"],["dc.bibliographiccitation.issue","4"],["dc.bibliographiccitation.journal","Journal of Multivariate Analysis"],["dc.bibliographiccitation.lastpage","714"],["dc.bibliographiccitation.volume","100"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Hotz, Thomas"],["dc.date.accessioned","2017-09-07T11:50:32Z"],["dc.date.available","2017-09-07T11:50:32Z"],["dc.date.issued","2008"],["dc.identifier.doi","10.1016/j.jmva.2008.08.008"],["dc.identifier.gro","3147646"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/5101"],["dc.notes.status","public"],["dc.notes.submitter","chake"],["dc.publisher","Elsevier BV"],["dc.relation.issn","0047-259X"],["dc.title","Principal component geodesics for planar shape spaces"],["dc.type","journal_article"],["dc.type.internalPublication","unknown"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]Details DOI2013Journal Article [["dc.bibliographiccitation.firstpage","98"],["dc.bibliographiccitation.issue","1-2"],["dc.bibliographiccitation.journal","Journal of Mathematical Imaging and Vision"],["dc.bibliographiccitation.lastpage","106"],["dc.bibliographiccitation.volume","50"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Hotz, Thomas"],["dc.date.accessioned","2017-09-07T11:48:38Z"],["dc.date.available","2017-09-07T11:48:38Z"],["dc.date.issued","2013"],["dc.description.abstract","We survey some effects that singular strata may have in the positive curvature context of circles and shape spaces when conducting (semi-)intrinsic statistical analyses. Here, the analysis of data on a stratified space is based on statistical descriptors defined in a possibly different stratified space. E.g. in geodesic principal component analysis for shape spaces, shape data are described by generalized geodesics which naturally form a shape space of their own, different from the original one. In a general context, if the descriptors are obtained as generalized Fréchet means, under rather general circumstances, a strong law of large numbers is valid. If furthermore the descriptors are sufficiently well behaved, a classical central limit theorem can be adopted. One of the crucial conditions is that hitting of singular strata as well as of cut loci, if present, must be controlled. We review the statistical role of the cut locus of intrinsic means for circles as well as that of singular strata for shape spaces (occurring where the group action is degenerate) and conclude with an identification of potential research directions."],["dc.identifier.doi","10.1007/s10851-013-0462-3"],["dc.identifier.gro","3146940"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4719"],["dc.language.iso","en"],["dc.notes.status","final"],["dc.notes.submitter","chake"],["dc.relation.issn","0924-9907"],["dc.title","On Means and Their Asymptotics: Circles and Shape Spaces"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]Details DOI2010Conference Paper [["dc.bibliographiccitation.firstpage","39"],["dc.bibliographiccitation.lastpage","43"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Hotz, Thomas"],["dc.contributor.editor","Gusnanto, A."],["dc.contributor.editor","Mardia, K. V."],["dc.contributor.editor","Fallaize, C. J."],["dc.contributor.editor","Voss, J."],["dc.date.accessioned","2017-09-07T11:47:59Z"],["dc.date.available","2017-09-07T11:47:59Z"],["dc.date.issued","2010"],["dc.identifier.gro","3146825"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4629"],["dc.language.iso","en"],["dc.notes.status","public"],["dc.notes.submitter","chake"],["dc.publisher","Department of Statistics, University of Leeds"],["dc.publisher.place","Leeds"],["dc.relation.conference","The 29th Leeds Annual Statistical Research (LASR) Workshop"],["dc.relation.eventend","2010-07-08"],["dc.relation.eventlocation","Leeds"],["dc.relation.eventstart","2010-07-06"],["dc.relation.isbn","978-0-85316-293-3"],["dc.relation.ispartof","High-throughput Sequencing, Proteins and Statistics"],["dc.title","Geodesic and parallel models for leaf shape"],["dc.type","conference_paper"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]Details2013Journal Article [["dc.bibliographiccitation.firstpage","2238"],["dc.bibliographiccitation.issue","6"],["dc.bibliographiccitation.journal","The Annals of Applied Probability"],["dc.bibliographiccitation.lastpage","2258"],["dc.bibliographiccitation.volume","23"],["dc.contributor.author","Hotz, Thomas"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Le, Huiling"],["dc.contributor.author","Marron, J. S."],["dc.contributor.author","Mattingly, Jonathan C."],["dc.contributor.author","Miller, Ezra"],["dc.contributor.author","Nolen, James"],["dc.contributor.author","Owen, Megan"],["dc.contributor.author","Patrangenaru, Vic"],["dc.contributor.author","Skwerer, Sean"],["dc.date.accessioned","2017-09-07T11:50:34Z"],["dc.date.available","2017-09-07T11:50:34Z"],["dc.date.issued","2013"],["dc.identifier.doi","10.1214/12-aap899"],["dc.identifier.gro","3147638"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/5096"],["dc.notes.status","public"],["dc.notes.submitter","chake"],["dc.publisher","Institute of Mathematical Statistics"],["dc.relation.issn","1050-5164"],["dc.title","Sticky central limit theorems on open books"],["dc.type","journal_article"],["dc.type.internalPublication","unknown"],["dc.type.peerReviewed","no"],["dspace.entity.type","Publication"]]Details DOI2010Journal Article [["dc.bibliographiccitation.firstpage","1"],["dc.bibliographiccitation.journal","Statistica Sinica"],["dc.bibliographiccitation.lastpage","100"],["dc.bibliographiccitation.volume","20"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Hotz, Thomas"],["dc.contributor.author","Munk, Axel"],["dc.date.accessioned","2019-07-10T08:13:39Z"],["dc.date.available","2019-07-10T08:13:39Z"],["dc.date.issued","2010"],["dc.description.abstract","In this paper, we illustrate a new approach for applying classical statistical methods to multivariate non-linear data. In two examples occurring in the statistical study of shape of three dimensional geometrical objects, we illustrate that the current methods of PCA by linear Euclidean approximation are unsuitable if such data in non-linear spaces fall into regions of high curvature, or if they have a large spread. In the following we give an overview of the background of relevant previous work, and an introduction to the building blocks of our work."],["dc.identifier.fs","582259"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?gs-1/7238"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/61303"],["dc.language.iso","en"],["dc.notes.intern","Merged from goescholar"],["dc.relation.orgunit","Fakultät für Mathematik und Informatik"],["dc.rights","Goescholar"],["dc.rights.uri","https://goescholar.uni-goettingen.de/licenses"],["dc.subject.ddc","510"],["dc.title","Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric lie group actions"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.version","published_version"],["dspace.entity.type","Publication"]]Details2008Journal Article Research Paper [["dc.bibliographiccitation.firstpage","1507"],["dc.bibliographiccitation.issue","9"],["dc.bibliographiccitation.journal","IEEE Transactions on Pattern Analysis and Machine Intelligence"],["dc.bibliographiccitation.lastpage","1519"],["dc.bibliographiccitation.volume","30"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Hotz, Thomas"],["dc.contributor.author","Munk, Axel"],["dc.date.accessioned","2017-09-07T11:48:14Z"],["dc.date.available","2017-09-07T11:48:14Z"],["dc.date.issued","2008"],["dc.description.abstract","Quadratic differentials naturally define analytic orientation fields on planar surfaces. We propose to model orientation fields of fingerprints by specifying quadratic differentials. Models for all fingerprint classes such as arches, loops, and whorls are laid out. These models are parameterized by a few geometrically interpretable parameters that are invariant under euclidean motions. We demonstrate their ability in adapting to given observed orientation fields, and we compare them to existing models using the fingerprint images of the NIST Special Database 4. We also illustrate that these models allow for extrapolation into unobserved regions. This goes beyond the scope of earlier models for the orientation field as those are restricted to the observed planar fingerprint region. Within the framework of quadratic differentials, we are able to analytically verify Penrose's formula for the singularities on a palm [19]. Potential applications of these models are the use of their parameters as indexes of large fingerprint databases, as well as the definition of intrinsic coordinates for single fingerprint images."],["dc.identifier.doi","10.1109/TPAMI.2007.70826"],["dc.identifier.gro","3143245"],["dc.identifier.isi","000257504400001"],["dc.identifier.pmid","18617711"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/738"],["dc.notes.intern","WoS Import 2017-03-10"],["dc.notes.status","final"],["dc.notes.submitter","PUB_WoS_Import"],["dc.publisher","Ieee Computer Soc"],["dc.relation.issn","0162-8828"],["dc.title","Global models for the orientation field of fingerprints: An approach based on quadratic differentials"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.subtype","original"],["dspace.entity.type","Publication"]]Details DOI PMID PMC WOS2010Journal Article [["dc.bibliographiccitation.firstpage","84"],["dc.bibliographiccitation.journal","Statistica Sinica"],["dc.bibliographiccitation.lastpage","100"],["dc.bibliographiccitation.volume","20"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Hotz, Thomas"],["dc.contributor.author","Munk, Axel"],["dc.date.accessioned","2017-09-07T11:48:02Z"],["dc.date.available","2017-09-07T11:48:02Z"],["dc.date.issued","2010"],["dc.description.abstract","A general framework is laid out for principal component analysis (PCA) on quotient spaces that result from an isometric Lie group action on a complete Riemannian manifold. If the quotient is a manifold, geodesics on the quotient can be lifted to horizontal geodesics on the original manifold. Thus, PCA on a manifold quotient can be pulled back to the original manifold. In general, however, the quotient space may no longer carry a manifold structure. Still, horizontal geodesics can be well-defined in the general case. This allows for the concept of generalized geodesics and orthogonal projection on the quotient space as the key ingredients for PCA. Generalizing a result of Bhattacharya and Patrangenaru (2003), geodesic scores can be defined outside a null set. Building on that, an algorithmic method to perform PCA on quotient spaces based on generalized geodesics is developed. As a typical example where non-manifold quotients appear, this framework is applied to Kendall’s shape spaces. In fact, this work has been motivated by an application occurring in forest biometry where the current method of Euclidean linear approximation is unsuitable for performing PCA. This is illustrated by a data example of individual tree stems whose Kendall shapes fall into regions of high curvature of shape space: PCs obtained by Euclidean approximation fail to reflect between-data distances and thus cannot correctly explain data variation. Similarly, for a classical archeological data set with a large spread in shape space, geodesic PCA allows new insights that have not been available under PCA by Euclidean approximation. We conclude by reporting challenges, outlooks, and possible perspectives of intrinsic shape analysis."],["dc.identifier.fs","582260"],["dc.identifier.gro","3146828"],["dc.identifier.purl","https://resolver.sub.uni-goettingen.de/purl?gs-1/7496"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/4632"],["dc.language.iso","en"],["dc.notes.intern","mathe"],["dc.notes.status","public"],["dc.notes.submitter","chake"],["dc.rights.access","openAccess"],["dc.subject","Extrinsic mean; forest biometry; geodesics; intrinsic mean; Lie group actions; non-linear multivariate statistics; orbifolds; orbit spaces; principal component analysis; Riemannian manifolds; shape analysis"],["dc.subject.ddc","510"],["dc.title","Rejoinder - Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric lie group actions"],["dc.type","journal_article"],["dc.type.internalPublication","unknown"],["dc.type.peerReviewed","no"],["dc.type.version","published_version"],["dspace.entity.type","Publication"]]Details2010Journal Article Research Paper [["dc.bibliographiccitation.firstpage","127"],["dc.bibliographiccitation.journal","Journal of the Royal Statistical Society. Series C, Applied statistics"],["dc.bibliographiccitation.lastpage","143"],["dc.bibliographiccitation.volume","59"],["dc.contributor.author","Hotz, Thomas"],["dc.contributor.author","Huckemann, Stephan"],["dc.contributor.author","Munk, Axel"],["dc.contributor.author","Gaffrey, D."],["dc.contributor.author","Sloboda, Branislav"],["dc.date.accessioned","2017-09-07T11:46:42Z"],["dc.date.available","2017-09-07T11:46:42Z"],["dc.date.issued","2010"],["dc.description.abstract","We analyse the shapes of star-shaped objects which are prealigned. This is motivated from two examples studying the growth of leaves, and the temporal evolution of tree rings. In the latter case measurements were taken at fixed angles whereas in the former case the angles were free. Subsequently, this leads to different shape spaces, related to different concepts of size, for the analysis. Whereas several shape spaces already existed in the literature when the angles are fixed, a new shape space for free angles, called spherical shape space, needed to be introduced. We compare these different shape spaces both regarding their mathematical properties and in their adequacy to the data at hand; we then apply suitably defined principal component analysis on these. In both examples we find that the shapes evolve mainly along the first principal component during growth; this is the 'geodesic hypothesis' that was formulated by Le and Kume. Moreover, we could link change-points of this evolution to significant changes in environmental conditions."],["dc.identifier.gro","3142997"],["dc.identifier.isi","000273320300007"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/463"],["dc.notes.intern","WoS Import 2017-03-10"],["dc.notes.status","final"],["dc.notes.submitter","PUB_WoS_Import"],["dc.publisher","Wiley-blackwell Publishing, Inc"],["dc.relation.issn","0035-9254"],["dc.title","Shape spaces for prealigned star-shaped objects-studying the growth of plants by principal components analysis"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.peerReviewed","yes"],["dc.type.subtype","original"],["dspace.entity.type","Publication"]]Details WOS