a nonparametric estimation procedure for bivariate extreme value copulas summary a bivariate extreme value distribution with fixed marginals is generated by a onedimensional map called a dependence function this paper proposes a new nonparametric estimator of this function its asymptotic properties are examined and its small sample behaviour is compared to that of other rank based and likelihood based procedures the new estimator is shown to be uniformly strongly convergent and asymptotically unbiased through simulations it is also seen to perform reasonably well against the maximum likelihood estimator based on the correct model and to have smaller l1 l2 and l errors than any existing nonparametric alternative the n consistency of the proposed estimator leads to nonparametric estimation of tawn s 1988 dependence measure that may be used to test independence in small samples

nonsmooth algorithms and nesterov s smoothing technique for generalized fermat torricelli problems we present algorithms for solving a number of new models of facility location which generalize the classical fermat torricelli problem our first approach involves using nesterov s smoothing technique and the minimization majorization principle to build smooth approximations that are convenient for applying smooth optimization schemes another approach uses subgradient type algorithms to cope directly with the nondifferentiability of the cost functions convergence results of the algorithms are proved and numerical tests are presented to show the effectiveness of the proposed algorithms

modelling dependence within joint tail regions standard approaches for modelling dependence within joint tail regions are based on extreme value methods which assume max stability a particular form of joint tail dependence we develop joint tail models based on a broader class of dependence structure which provides a natural link between max stable models and weaker forms of dependence including independence and negative association this approach overcomes many of the problems that are encountered with standard methods and is the basis for a poisson process representation that generalizes existing bivariate results we apply the new techniques to simulated and environmental data and demonstrate the marked advantage that the new approach offers for joint tail extrapolation

modelling extreme multivariate events summary the classical treatment of multivariate extreme values is through componentwise ordering though in practice most interest is in actual extreme events here the point process of observations which are extreme in at least one component is considered parametric models for the dependence between components must satisfy certain constraints two new techniques for generating such models are presented aspects of the statistical estimation of the resulting models are discussed and are illustrated with an application to oceanographic data

rank based inference for bivariate extreme value copulas consider a continuous random pair x y whose dependence is characterized by an extreme value copula with pickands dependence function a when the marginal distributions of x and y are known several consistent estimators of a are available most of them are variants of the estimators due to pickands bull inst internat statist 49 1981 859 878 and cap era a foug eres and genest biometrika 84 1997 567 577 in this paper rank based versions of these estimators are proposed for the more common case where the margins of x and y are unknown results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank based estimators their finite and large sample performance is then compared to that of their known margin analogues as well as with endpoint corrected versions thereof explicit formulas and consistent estimates for their asymptotic variances are also suggested

l measure of non exchangeability for bivariate extreme value and archimax copulas in the class of bivariate extreme value copulas an upper bound is calculated for the measure of non exchangeability based on the l norm distance between a copula c and its transpose ct x y c y x copulas that are maximally non exchangeable with respect to are also determined moreover similar upper bounds are given respectively for the class of all ev copulas having a fixed upper tail dependence coefficient and for the larger class of archimax copulas

an m estimator for tail dependence in arbitrary dimensions consider a random sample in the max domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model a new estimator for the unknown parameter is defined as the value that minimizes the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts the minimization problem has with probability tending to one a unique global solution the estimator is consistent and asymptotically normal the spectral measures of the tail dependence models to which the method applies can be discrete or continuous examples demonstrate the applicability and the performance of the method

a consensus algorithm in cat 0 space and its application to distributed fusion of phylogenetic trees abstract based on convex analysis a novel consensus algorithm of dynamical points in a cat 0 space is developed in this paper in which the associated communication graph uniformly contains a directed spanning tree the proposed algorithm provides an efficient method of solving consensus problems in a general cat 0 space while having certain robustness against weak communication the application of the new algorithm to the distributed fusion of phylogenetic trees is shown with demonstrative case simulations together with a study on the algorithm s robustness and efficiency

markov chain models for threshold exceedances in recent research on extreme value statistics there has been an extensive development of threshold methods first in the univariate case and subsequently in the multivariate case as well in this paper an alternative methodology for extreme values of univariate time series is developed by assuming that the time series is markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions a new likelihood representation for threshold methods is presented which we apply to a markovian time series an important motivation for developing this kind of theory is the possibility of calculating probability distributions for functionals of extreme events we address this issue by showing how a theory of compound poisson limits for additive functionals can be combined with simulation to obtain numerical solutions for problems of practical interest the methods are illustrated by application to temperature data

on some nonregular tests for a modified weibull model summary certain tests on the parameters of a modified weibull distribution are considered nonregular asymptotic behaviour of likelihood based test statistics is observed competing tests based on extreme order statistics are investigated and comparisons are made of power functions

consistency and convergence rate of phylogenetic inference via regularization it is common in phylogenetics to have some perhaps partial information about the overall evolutionary tree of a group of organisms and wish to find an evolutionary tree of a specific gene for those organisms there may not be enough information in the gene sequences alone to accurately reconstruct the correct gene tree although the gene tree may deviate from the species tree due to a variety of genetic processes in the absence of evidence to the contrary it is parsimonious to assume that they agree a common statistical approach in these situations is to develop a likelihood penalty to incorporate such additional information recent studies using simulation and empirical data suggest that a likelihood penalty quantifying concordance with a species tree can significantly improve the accuracy of gene tree reconstruction compared to using sequence data alone however the consistency of such an approach has not yet been established nor have convergence rates been bounded because phylogenetics is a non standard inference problem the standard theory does not apply in this paper we propose a penalized maximum likelihood estimator for gene tree reconstruction where the penalty is the square of the billera holmes vogtmann geodesic distance from the gene tree to the species tree we prove that this method is consistent and derive its convergence rate for estimating the discrete gene tree structure and continuous edge lengths representing the amount of evolution that has occurred on that branch simultaneously we find that the regularized estimator is adaptive fast converging meaning that it can reconstruct all edges of length greater than any given threshold from gene sequences of polynomial length our method does not require the species tree to be known exactly in fact our asymptotic theory holds for any such guide tree

a mixture model for multivariate extremes this thesis is a contribution to multivariate extreme value statistics the tail of a multivariate distribution function is characterized by its spectral distribution for which we propose a new semi parametric model based on mixtures of dirichlet distributions to estimate the components of this model reversible jump monte carlo markov chain and em algorithms are developed their performances are illustrated on real and simulated data obtained using new representations of the extremal logistic and dirichlet models in parallel with the estimation of the spectral distribution extreme value statistic machinery requires the selection of a threshold in order to classify data as extreme or not this selection is achieved by a new method based on heuristic arguments it allows a selection independent of the dimension of the data its performance is illustrated on real and simulated data primal scientific interests behind a multivariate extreme value analysis reside in the estimation of quantiles of rare events and in the exploration of the dependence structure for which the estimation of the spectral measure is a means rather than an end these two issues are addressed for the first a monte carlo method is developed based on simulation of extremes it is compared with classical and new methods of the literature for the second one an original conditional dependence analysis is proposed which enlightens various aspects of the dependence structure of the data examples using real data sets are given in the last part the semi parametric model and the presented methods are extended to spatial extremes it is made possible by considering the spectral distribution as the distribution of a random probability an original viewpoint adopted throughout this thesis classical multivariate extremes are extended to extremes of random measures the application is illustrated on rainfall data in china

dependence modelling for spatial extremes current dependence models for spatial extremes are based upon max stable processes within this class there are few inferentially viable models available and we propose one further model more problematic are the restrictive assumptions that must be made when using max stable processes to model dependence for spatial extremes it must be assumed that the dependence structure of the observed extremes is compatible with a limiting model that holds for all events more extreme than those that have already occurred this problem has long been acknowledged in the context of finite dimensional multivariate extremes in particular when data display dependence at observable levels but are independent in the limit we propose a flexible class of models that is suitable for such data in a spatial context in addition we consider the situation where the extremal dependence structure may vary with distance we apply our models to spatially referenced significant wave height data from the north sea finding evidence that their extremal structure is not compatible with a limiting dependence model

elastic geodesic paths in shape space of parameterized surfaces this paper presents a novel riemannian framework for shape analysis of parameterized surfaces in particular it provides efficient algorithms for computing geodesic paths which in turn are important for comparing matching and deforming surfaces the novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape preserving transformations of surfaces the basic idea is to formulate a space of embedded surfaces surfaces seen as embeddings of a unit sphere in ir3 and impose a riemannian metric on it in such a way that the reparameterization group acts on this space by isometries under this framework we solve two optimization problems one given any two surfaces at arbitrary rotations and parameterizations we use a path straightening approach to find a geodesic path between them under the chosen metric second by modifying a technique presented in 25 we solve for the optimal rotation and parameterization registration between surfaces their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces we illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces

exact simulation of max stable processes max stable processes play an important role as models for spatial extreme events their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult algorithms based on finite approximations are often inexact and computationally inefficient we present a new algorithm for exact simulation of a max stable process at a finite number of locations it relies on the idea of simulating only the extremal functions that is those functions in the construction of a max stable process that effectively contribute to the pointwise maximum we further generalize the algorithm by dieker mikosch 2015 for brown resnick processes and use it for exact simulation via the spectral measure we study the complexity of both algorithms prove that our new approach via extremal functions is always more efficient and provide closed form expressions for their implementation that cover most popular models for max stable processes and multivariate extreme value distributions for simulation on dense grids an adaptive design of the extremal function algorithm is proposed

convergent sequences of dense graphs i subgraph frequencies metric properties and testing we consider sequences of graphs gn and define various notions of convergence related to these sequences left convergence defined in terms of the densities of homomorphisms from small graphs into gn right convergence defined in terms of the densities of homomorphisms from gn into small graphs and convergence in a suitably defined metric in part i of this series we show that left convergence is equivalent to convergence in metric both for simple graphs gn and for graphs gn with nodeweights and edgeweights one of the main steps here is the introduction of a cut distance comparing graphs not necessarily of the same size we also show how these notions of convergence provide natural

estimating the probability of a rare event let x 1 y 1 x 2 y 2 x n y n be a random sample from a bivariate distribution function f which is in the domain of attraction of a bivariate extreme value distribution function g a subset c of r 2 is given which contains none of the observations we shall give an asymptotic confidence interval for pr x i y i c under certain conditions

a class of multivanate failure time distributions summary a class of continuous multivariate lifetime distributions is proposed the dependence between individuals in a group is modelled by a group specific quantity which can be interpreted as an unobserved covariate common to the individuals in the group and assumed to follow a positive stable distribution it is possible to include covariates in the model and discuss whether the dependence is still present after specific covariates are taken into account if the conditional hazards are proportional then the hazards in the marginal distributions are also proportional but with different constants of proportionality also the hazard for the minimum in a group is proportional to the marginal hazards if the conditional distributions given the group quantity are weibull then the marginal distributions are also weibull this class can be used to test the hypothesis of independence of litter mates in the proportional hazards model

bivariate extreme value theory models and estimation summary bivariate extreme value distributions arise as the limiting distributions of renormalized componentwise maxima no natural parametric family exists for the dependence between the marginal distributions but there are considerable restrictions on the dependence structure we consider modelling the dependence function with parametric models for which two new models are presented tests for independence and discriminating between models are also given the estimation procedure and the flexibility of the new models are illustrated with an application to sea level data extreme value theory has recently been an area of much theoretical and practical work univariate theory is a well documented area whereas bivariate multivariate extreme value theory has until recently received surprisingly little attention in the multivariate case no natural parametric family exists for the dependence structure so this must be modelled in some way in the analysis of environmental extreme value data there is a need for models of dependence between extremes from different sources for example at various sea ports or at various points of a river in this paper we consider bivariate extreme value distributions we assume without loss of generality that we have exponential marginal distributions with unit means the class of bivariate exponential distributions in which we are interested satisfy a strong stability relation exponential variables x y satisfy the stability relation if and only if w min ax by is also exponentially distributed for all a b 0 pickands 1981 therefore the models we will consider have particular application in reliability and survival analysis one approach to modelling the dependence structure is via parametric models this requires a flexible family of models which satisfy certain constraints models are of two kinds either differentiable or nondifferentiable all nondifferentiable models give distri butions which are singular with nonzero probability concentrated on a certain subspace the differentiable models have densities but the existing models are symmetric which leads to the variables being exchangeable here we present two new asymmetric differenti able models which have increased flexibility properties of the differentiable models are examined estimation of the parametric models has previously been by ad hoc methods because there is a nonregular estimation problem when the margins are independent for the

fisher information for a multivariate extreme value distribution summary explicit algebraic formulae for the fisher information matrix of the multivariate extreme value distribution with generalised extreme value margins and logistic dependence structure are given

toward a theory of statistical tree shape analysis to develop statistical methods for shapes with a tree structure we construct a shape space framework for tree shapes and study metrics on the shape space this shape space has singularities which correspond to topological transitions in the represented trees we study two closely related metrics on the shape space ted and qed qed is a quotient euclidean distance arising naturally from the shape space formulation while ted is the classical tree edit distance using gromov s metric geometry we gain new insight into the geometries defined by ted and qed we show that the new metric qed has nice geometric properties that are needed for statistical analysis geodesics always exist and are generically locally unique following this we can also show the existence and generic local uniqueness of average trees for qed ted while having some algorithmic advantages does not share these advantages along with the theoretical framework we provide experimental proof of concept results on synthetic data trees as well as small airway trees from pulmonary ct scans this way we illustrate that our framework has promising theoretical and qualitative properties necessary to build a theory of statistical tree shape analysis

object oriented data analysis sets of trees object oriented data analysis is the statistical analysis of populations of complex objects in the special case of functional data analysis these data objects are curves where standard euclidean approaches such as principal component analysis have been very successful recent developments in medical image analysis motivate the statistical analysis of populations of more complex data objects which are elements of mildly non euclidean spaces such as lie groups and symmetric spaces or of strongly non euclidean spaces such as spaces of tree structured data objects these new contexts for object oriented data analysis create several potentially large new interfaces between mathematics and statistics this point is illustrated through the careful development of a novel mathematical framework for statistical analysis of populations of tree structured objects

statistics for near independence in multivariate extreme values we propose a multivariate extreme value threshold model for joint tail estimation which overcomes the problems encountered with existing techniques when the variables are near independence we examine inference under the model and develop tests for independence of extremes of the marginal variables both when the thresholds are fixed and when they increase with the sample size motivated by results obtained from this model we give a new and widely applicable characterisation of dependence in the joint tail which includes existing models as special cases a new parameter which governs the form of dependence is of fundamental importance to this characterisation by estimating this parameter we develop a diagnostic test which assesses the applicability of bivariate extreme value joint tail models the methods are demonstrated through simulation and by analysing two previously published data sets

mri based medial axis extraction and boundary segmentation of cranial nerves through discrete deformable 3d contour and surface models this paper presents a segmentation technique to identify the medial axis and the boundary of cranial nerves we utilize a 3 d deformable one simplex discrete contour model to extract the medial axis of each cranial nerve this contour model represents a collection of two connected vertices linked by edges where vertex position is determined by a newtonian expression for vertex kinematics featuring internal and external forces the latter of which include attractive forces toward the nerve medial axis we exploit multiscale vesselness filtering and minimal path techniques in the medial axis extraction method which also computes a radius estimate along the path once we have the medial axis and the radius function of a nerve we identify the nerve surface using a two simplex deformable model which expands radially and can accommodate any nerve shape as a result the method proposed here combines the benefits of explicit contour and surface models while also achieving a cornerstone for future work that will emphasize shape statistics static collision with other critical structures and tree shape analysis

clustering tree structured data on manifold tree structured data usually contain both topological and geometrical information and are necessarily considered on manifold instead of euclidean space for appropriate data parameterization and analysis in this study we propose a novel tree structured data parameterization called topology attribute matrix t a matrix so the data clustering task can be conducted on matrix manifold we incorporate the structure constraints embedded in data into the non negative matrix factorization method to determine meta trees from the t a matrix and the signature vector of each single tree can then be extracted by meta tree decomposition the meta tree space turns out to be a cone space in which we explore the distance metric and implement the clustering algorithm based on the concepts like frechet mean finally the t a matrix based clustering tambac framework is evaluated and compared using both simulated data and real retinal images to illustrate its efficiency and accuracy

regional modelling of extreme storms via max stable processes asymptotic models for extremes of random processes often form the basis for estimating the extremal behaviour of environmental phenomena most such phenomena have a spatial dimension and the aim of this paper is to develop a procedure for modelling in continuous space the spatial dependence within extreme events a principal objective in the analysis as with other current research on extremes is to base inference on as much of the available data as possible the modelling procedures are justified on simulated data and subsequently applied to a series of rainfall data

spatial extremes models for the stationary case the aim of this paper is to provide models for spatial extremes in the case of stationarity the spatial dependence at extreme levels of a stationary process is modeled using an extension of the theory of max stable processes of de haan and pickands probab theory related fields 72 1986 477 492 we propose three one dimensional and three two dimensional models these models depend on just one parameter or a few parameters that measure the strength of tail dependence as a function of the distance between locations we also propose two estimators for this parameter and prove consistency under domain of attraction conditions and asymptotic normality under appropriate extra conditions

principal components analysis in the space of phylogenetic trees phylogenetic analysis of dna or other data commonly gives rise to a collection or sample of inferred evolutionary trees principal components analysis pca cannot be applied directly to collections of trees since the space of evolutionary trees on a fixed set of taxa is not a vector space this paper describes a novel geometrical approach to pca in tree space that constructs the first principal path in an analogous way to standard linear euclidean pca given a data set of phylogenetic trees a geodesic principal path is sought that maximizes the variance of the data under a form of projection onto the path due to the high dimensionality of tree space and the nonlinear nature of this problem the computational complexity is potentially very high so approximate optimization algorithms are used to search for the optimal path principal paths identified in this way reveal and quantify the main sources of variation in the original collection of trees in terms of both topology and branch lengths the approach is illustrated by application to simulated sets of trees and to a set of gene trees from metazoan animal species

the logarithm map its limits and frechet means in orthant spaces the first part of the paper studies the expression for and the properties of the logarithm map on an orthant space which is a simple stratified space with the aim of analysing frechet means of probability measures on such a space in the second part we use these results to characterise frechet means and to derive various of their properties including the limiting distribution of sample frechet means

relative optimality conditions and algorithms for treespace frechet means recent interest in treespaces as well founded mathematical domains for phylogenetic inference and statistical analysis for populations of anatomical trees has motivated research into efficient and

computing medians and means in hadamard spaces the geometric median as well as the frechet mean of points in a hadamard space are important in both theory and applications surprisingly no algorithms for their computation are hitherto known to address this issue we use a splitting version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function which extends a recent result of bertsekas math program 129 2011 pp 163 195 into hadamard spaces the method is quite robust and not only does it yield algorithms for the median and the mean but also it applies to various other optimization problems we moreover show that another algorithm for computing the frechet mean can be derived from the law of large numbers due to sturm ann probab 30 2002 pp 1195 1222 in applications computing medians and means is probably most needed in tree space which is an instance of a hadamard space invented by billera holmes and vo

statistical graph space analysis the sample mean is one of the most fundamental concepts in statistics properties of the sample mean that are well defined in euclidean spaces become unclear in graph spaces this paper proposes conditions under which the following properties are valid existence uniqueness and consistency of means the midpoint property necessary conditions of optimality and convergence results of mean algorithms the theoretical results address common misconceptions about the graph mean in graph edit distance spaces serve as a first step towards a statistical analysis of graph spaces and result in a theoretically well founded mean algorithm that outperformed six other mean algorithms with respect to solution quality on different graph datasets representing images and molecules highlightsa first step towards a theory of statistical graph space analysis is proposed mmm algorithm is proposed that outperformed six other mean algorithms necessary conditions of optimality are proved convergence of mmm algorithm is shown basic statistical and geometrical properties are shown

bivariate extreme value distributions an application of the gibbs sampler to the analysis of floods the quantitative definition of the magnitude of a flood episode exclusively in terms of its peak rate of discharge is often insufficient in common with most episodic hydrological phenomena floods are intrinsically multivariate random events characterized not only by their peak flow but also by their volume and the duration of discharge above critical thresholds a complete definition in terms of these three component random variables has not found its way into standard analytical practice this may be partly because the theory of multivariate extremes is not straightforward and partly because no intuitively attractive way of presenting the component variables in a probabilistic framework has been put forward we propose that the gibbs sampler a member of the new generation of computer intensive methods in computational statistics bypasses the theoretical difficulties we demonstrate the gibbs sampler and provide graphical presentations of the results for the annual flood hydrograph on the mekong river at vientiane laos we extend the results to evaluate the potential distribution of flood damage to the city

ordered multivariate extremes multivariate extreme value models and associated statistical methods are developed for vector observations whose components are subject to an order restriction the approach extends the multivariate threshold methodology of coles and tawn joe and co workers and smith and co workers the results are illustrated by an analysis of extreme rainfalls of different durations and by a study of the problem of linking a long series of daily rainfall extremes with a partially overlapping shorter series of hourly extremes

generalized principal component analysis gpca this paper presents an algebro geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points we represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point when the number of subspaces is known we show that these polynomials can be estimated linearly from data hence subspace segmentation is reduced to classifying one point per subspace we select these points optimally from the data set by minimizing certain distance function thus dealing automatically with moderate noise in the data a basis for the complement of each subspace is then recovered by applying standard pca to the collection of derivatives normal vectors extensions of gpca that deal with data in a high dimensional space and with an unknown number of subspaces are also presented our experiments on low dimensional data show that gpca outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k subspaces and expectation maximization we also present applications of gpca to computer vision problems such as face clustering temporal video segmentation and 3d motion segmentation from point correspondences in multiple affine views

modelling multivariate extreme value distributions summary multivariate extreme value distributions arise as the limiting joint distribution of normalized componentwise maxima minima no parametric family exists for the depen dence between the margins this paper extends to more than two variables the models and results for the bivariate case obtained by tawn 1988 two new families of physically motivated parametric models for the dependence structure are presented and are illus trated with an application to trivariate extreme sea level data in the analysis of environmental extreme value data there is often need to study joint inter site extreme behaviour examples are joint flooding at various sea ports or at various rain gauges in this paper the class of multivariate extreme value distributions is considered the marginal distributions are all extreme value distributions without loss of generality we follow pickands s 1981 approach in transforming the margins to unit mean exponential distributions the resulting class of p variable multivariate exponential distributions satisfies a strong stability relation de haan resnick 1977 and pickands 1981 show that the resulting dependence structure is uniquely specified up to an arbitrary finite positive measure on p 1 variables therefore the bivariate dependence structure is univariate only when p 3 as considered here does the problem become really multivari ate in addition unlike in the multivariate normal distribution the multivariate depen dence structure is generally only restricted and not uniquely specified by the dependence structure of all possible pairs much has been written about the representations of multivariate extremes see resnick 1987 for a good review on the other hand models and estimation for such distributions have received little attention these aspects are considered here as no finite parametric family exists for the general dependence structure it must be modelled in some way thus true multivariate models are required using the most flexible bivariate model of tawn 1988 as a basis multivariate extensions are generated a model derived in the discrete choice literature will also be introduced and extended the resulting families of models are physically motivated flexible and possess hierarchical properties

modelling extremes of the areal rainfall process risk assessment for many hydrological structures requires an estimate of the extremal behaviour of the rainfall regime within a specified catchment region in most cases it is the spatially aggregated rainfall which is the key process though in practice only pointwise rainfall measurements from a network of sites over the region are available in this paper we address the problem of making inferences about the extremal properties of the aggregated process from the pointwise data working within the usual extreme value paradigm a model is derived in which the resulting distribution is determined by the marginal tail behaviour and spatial dependence at extreme levels of the process data collected from a region in the south west of england are used to illustrate the procedure

joint segmentation and shape regularization with a generalized forward backward algorithm this paper presents a method for the simultaneous segmentation and regularization of a series of shapes from a corresponding sequence of images such series arise as time series of 2d images when considering video data or as stacks of 2d images obtained by slicewise tomographic reconstruction we first derive a model where the regularization of the shape signal is achieved by a total variation prior on the shape manifold the method employs a modified kendall shape space to facilitate explicit computations together with the concept of sobolev gradients for the proposed model we derive an efficient and computationally accessible splitting scheme using a generalized forward backward approach our algorithm treats the total variation atoms of the splitting via proximal mappings whereas the data terms are dealt with by gradient descent the potential of the proposed method is demonstrated on various application examples dealing with 3d data we explain how to extend the proposed combined approach to shape fields which for instance arise in the context of 3d t imaging modalities and show an application in this setup as well

likelihood based inference for max stable processes the last decade has seen max stable processes emerge as a common tool for the statistical modeling of spatial extremes however their application is complicated due to the unavailability of the multivariate density function and so likelihood based methods remain far from providing a complete and flexible framework for inference in this article we develop inferentially practical likelihood based methods for fitting max stable processes derived from a composite likelihood approach the procedure is sufficiently reliable and versatile to permit the simultaneous modeling of marginal and dependence parameters in the spatial context at a moderate computational cost the utility of this methodology is examined via simulation and illustrated by the analysis of united states precipitation extremes

backward backward splitting in hadamard spaces abstract the backward backward algorithm is a tool for finding minima of a regularization of the sum of two convex functions in hilbert spaces we generalize this setting to hadamard spaces and prove the convergence of an error tolerant version of the backward backward method

estimation of quantiles of the maximum of n observations summary given independent and identically distributed random variables from an unknown distribution f we want to estimate quantiles of fn using asymptotic extreme value theory two parametric approximations to fn are considered ahd approximate maximum likelihood estimates are obtained conditions for consistency and asymptotic normality of these estimates are given and properties of the estimates are obtained through a small simulation study

matrix geometric means based on shuffled inductive sequences abstract we propose a new deterministic sequence converging to the least squares mean of n symmetric positive definite spd matrices by least squares mean we refer to the riemannian barycenter with respect to the natural metric also known as the trace metric or affine invariant metric on the set p n of n n spd matrices in some papers this mean is also referred to as the karcher mean the sequence we propose belongs to the family of inductive sequences obtained by letting a point take successive steps towards the data points with step lengths progressively diminishing to zero we use the word inductive since each point of those sequences can be seen as the inductive mean of a well chosen set of points containing multiple replications of the data points we show that visiting the data points in a cyclic order usually results in a slow convergence and remedy this weakness by reordering repeatedly the data points using a shuffling algorithm we prove the convergence of the resulting inductive sequence to the least squares mean of the data in the general framework of npc spaces complete metric spaces with non positive curvature which includes the set p n we also illustrate numerically on p n that some points of this sequence are fairly accurate estimates of the least squares mean while being considerably cheaper to compute and perform well as initializers for state of the art algorithms to reduce further the computation time we finally exploit inductive sequences to produce a family of parallelizable algorithms for estimating the least squares mean

principal component analysis and the locus of the frechet mean in the space of phylogenetic trees evolutionary relationships are represented by phylogenetic trees and a phylogenetic analysis of gene sequences typically produces a collection of these trees one for each gene in the analysis analysis of samples of trees is difficult due to the multi dimensionality of the space of possible trees in euclidean spaces principal component analysis is a popular method of reducing high dimensional data to a low dimensional representation that preserves much of the sample s structure however the space of all phylogenetic trees on a fixed set of species does not form a euclidean vector space and methods adapted to tree space are needed previous work introduced the notion of a principal geodesic in this space analogous to the first principal component here we propose a geometric object for tree space similar to the formula see text th principal component in euclidean space the locus of the weighted frechet mean of formula see text vertex trees when the weights vary over the formula see text simplex we establish some basic properties of these objects in particular showing that they have dimension formula see text and propose algorithms for projection onto these surfaces and for finding the principal locus associated with a sample of trees simulation studies demonstrate that these algorithms perform well and analyses of two datasets containing apicomplexa and african coelacanth genomes respectively reveal important structure from the second principal components

ozone concentrations a robust analysis of multivariate extremes data on ozone concentrations at four monitoring sites in central canada are examined univariate extreme value methods do not allow for the required inference and multivariate methods exploiting the joint dependence of the data are necessary families of multivariate extreme value copulas with a flexible dependence structure and closed form cumulative distribution functions have recently been derived this article shows that maximum likelihood estimates of these parametric models may be obtained numerically and that their use can lead to increased efficiency in the estimation of margins we do not however have any assurance that all data are well modeled by these distributions because the space of multivariate extreme value copulas is infinite dimensional and data on ozone levels are prone to outliers this article addresses the robust methods required for proper analysis

geostatistics of extremes we describe a prototype approach to flexible modelling for maxima observed at sites in a spatial domain based on fitting of max stable processes derived from underlying gaussian random fields the models we propose have generalized extreme value marginal distributions throughout the spatial domain consistent with statistical theory for maxima in simpler cases and can incorporate both geostatistical correlation functions and random set components parameter estimation and fitting are performed through composite likelihood inference applied to observations from pairs of sites with occurrence times of maxima taken into account if desired and competing models are compared using appropriate information criteria diagnostics for lack of model fit are based on maxima from groups of sites the approach is illustrated using annual maximum temperatures in switzerland with risk analysis proposed using simulations from the fitted max stable model drawbacks and possible developments of the approach are discussed

strong law of large numbers for the l1 karcher mean abstract sturm s strong law of large numbers in cat 0 spaces has been an influential tool to study the geometric mean or also called karcher barycenter of positive definite matrices it provides an easily computable stochastic approximation based on inductive means convergence of a deterministic version of this approximation has been proved by holbrook providing his nodice theorem for the karcher mean of positive definite matrices the karcher mean has also been extended to the infinite dimensional case of positive operators on a hilbert space by lawson lim and then to probability measures with bounded support by the second author however the cat 0 property of the space is lost and one defines the mean as the unique solution of a nonlinear operator equation on a convex banach finsler manifold the formulations of sturm s strong law of large numbers and holbrook s nodice approximation are natural and both conjectured to converge however all previous techniques of their proofs break down due to the banach finsler nature of the space in this paper we prove both conjectures by establishing the most general l 1 form of sturm s strong law of large numbers and holbrook s nodice theorem in the operator norm by developing a stochastic discrete time resolvent flow for the karcher barycenter using its wasserstein contraction property

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

a second order nonsmooth variational model for restoring manifold valued images we introduce a new nonsmooth variational model for the restoration of manifold valued data which includes second order differences in the regularization term while such models were successfully applied for real valued images we introduce the second order difference and the corresponding variational models for manifold data which up to now only existed for cyclic data the approach requires a combination of techniques from numerical analysis convex optimization and differential geometry first we establish a suitable definition of absolute second order differences for signals and images with values in a manifold employing this definition we introduce a variational denoising model based on first and second order differences in the manifold setup in order to minimize the corresponding functional we develop an algorithm using an inexact cyclic proximal point algorithm we propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machin

bivariate threshold methods for extremes summary an extension of the threshold method for extreme values is developed to consider the joint distribution of extremes of two variables the methodology is based on the point process representation of bivariate extremes both parametric and nonparametric models are considered the simplest case to handle is that in which both marginal distributions are known for the more realistic case in which the marginal distributions are unknown a mixed parametric nonparametric method is proposed the techniques are illustrated with data on sulphate and nitrate levels taken from a major study of acid rain

generalized median graph computation by means of graph embedding in vector spaces the median graph has been presented as a useful tool to represent a set of graphs nevertheless its computation is very complex and the existing algorithms are restricted to use limited amount of data in this paper we propose a new approach for the computation of the median graph based on graph embedding graphs are embedded into a vector space and the median is computed in the vector domain we have designed a procedure based on the weighted mean of a pair of graphs to go from the vector domain back to the graph domain in order to obtain a final approximation of the median graph experiments on three different databases containing large graphs show that we succeed to compute good approximations of the median graph we have also applied the median graph to perform some basic classification tasks achieving reasonable good results these experiments on real data open the door to the application of the median graph to a number of more complex machine learning algorithms where a representative of a set of graphs is needed

a dependence measure for multivariate and spatial extreme values properties and inference we present properties of a dependence measure that arises in the study of extreme values in multivariate and spatial problems for multivariate problems the dependence measure characterises dependence at the bivariate level for all pairs and all higher orders up to and including the dimension of the variable necessary and sufficient conditions are given for subsets of dependence measures to be self consistent that is to guarantee the existence of a distribution with such a subset of values for the dependence measure for pairwise dependence these conditions are given in terms of positive semidefinite matrices and non differentiable positive definite functions we construct new nonparametric estimators for the dependence measure which unlike all naive nonparametric estimators impose these self consistency properties as the new estimators provide an improvement on the naive methods both in terms of the inferential and interpretability properties their use in exploratory extreme value analyses should aid the identification of appropriate dependence models the methods are illustrated through an analysis of simulated multivariate data which shows that a lack of self consistency is frequently a problem with the existing estimators and by a spatial analysis of daily rainfall extremes in south west england which finds a smooth decay in extremal dependence with distance

a conditional approach for multivariate extreme values with discussion summary multivariate extreme value theory and methods concern the characterization estimation and extrapolation of the joint tail of the distribution of a d dimensional random variable existing approaches are based on limiting arguments in which all components of the variable become large at the same rate this limit approach is inappropriate when the extreme values of all the variables are unlikely to occur together or when interest is in regions of the support of the joint distribution where only a subset of components is extreme in practice this restricts existing methods to applications where d is typically 2 or 3 under an assumption about the asymptotic form of the joint distribution of a d dimensional random variable conditional on its having an extreme component we develop an entirely new semiparametric approach which overcomes these existing restrictions and can be applied to problems of any dimension we demonstrate the performance of our approach and its advantages over existing methods by using theoretical examples and simulation studies the approach is used to analyse air pollution data and reveals complex extremal dependence behaviour that is consistent with scientific understanding of the process we find that the dependence structure exhibits marked seasonality with ex tremal dependence between some pollutants being significantly greater than the dependence at non extreme levels

a multivariate distribution with weibull connections a simple form of multivariate distribution is defined which for certain parameter values has weibull marginals the distribution has a single parameter governing association between the variates which may be positive negative or zero the simple forms for marginal and conditional distributions hazard functions and densities make it attractive for practical application and interpretation

diagnostics for dependence within time series extremes summary the analysis of extreme values within a stationary time series entails various assumptions concerning its long and short range dependence we present a range of new diagnostic tools for assessing whether these assumptions are appropriate and for identifying structure within extreme events these tools are based on tail characteristics of joint survivor functions but can be implemented by using existing estimation methods for extremes of univariate independent and identically distributed variables our diagnostic aids are illustrated through theoretical examples simulation studies and by application to rainfall and exchange rate data on the basis of these diagnostics we can explain characteristics that are found in the observed extreme events of these series and also gain insight into the properties of events that are more extreme than those observed

maximum empirical likelihood estimation of the spectral measure of an extreme value distribution consider a random sample from a bivariate distribution function f in the max domain of attraction of all extreme value distribution function g this g is characterized by two extreme value indices and a spectral measure the latter determining the tail dependence structure of f a major issue in multivariate extreme value theory is the estimation of the spectral measure 1 p with respect to the l p norm for every p is an element of 1 infinity a nonparametric maximum empirical likelihood estimator is proposed for phi p the main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized asymptotic normality of the estimators is proved under conditions that allow for tail independence moreover the conditions are easily verifiable as we demonstrate through a number of theoretical examples a simulation study shows a substantially improved performance of the new estimators two case studies illustrate how to implement the methods in practice

non parametric bayesian inference on bivariate extremes the tail of a bivariate distribution function in the domain of attraction of a bivariate extreme value distribution may be approximated by the one of its extreme value attractor the extreme value attractor has margins that belong to a three parameter family and a dependence structure which is characterised by a probability measure on the unit interval with mean equal to one half called spectral measure inference is done in a bayesian framework using a censored likelihood approach a prior distribution is constructed on an in nite dimensional model for this measure the model being at the same time dense and computationally manageable a trans dimensional markov chain monte carlo algorithm is developed and convergence to the posterior distribution is established in simulations the bayes estimator for the spectral measure is shown to compare favorably with frequentist nonparametric estimators an application to a data set of danish re insurance claims is provided

models for the extremes of markov chains the modelling of extremes of a time series has progressed from the assumption of independent observations to more realistic forms of temporal dependence in this paper we focus on markov chains deriving a class of models for their joint tail which allows the degree of clustering of extremes to decrease at high levels overcoming a key limitation in current methodologies theoretical aspects of the model are examined and a simulation algorithm is developed through which the stochastic properties of summaries of the extremal txhaviour of the chain are evaluated the approach is illustrated through a simulation study of extremal events of gaussian autoregressive processes and an application to temperature data