rupture solutions of an elliptic equation with a singular nonlinearity we obtain infinitely many non radial rupture solutions of the equation with by constructing infinitely many radially symmetric regular solutions of the equation on s n 1

sparse matrix graphical models matrix variate observations are frequently encountered in many contemporary statistical problems due to a rising need to organize and analyze data with structured information in this article we propose a novel sparse matrix graphical model for these types of statistical problems by penalizing respectively two precision matrices corresponding to the rows and columns our method yields a sparse matrix graphical model that synthetically characterizes the underlying conditional independence structure our model is more parsimonious and is practically more interpretable than the conventional sparse vector variate graphical models asymptotic analysis shows that our penalized likelihood estimates enjoy better convergent rates than that of the vector variate graphical model the finite sample performance of the proposed method is illustrated via extensive simulation studies and several real datasets analysis

parameter expansion to accelerate em the px em algorithm summary the em algorithm and its extensions are popular tools for modal estimation but ar e often criticised for their slow convergence we propose a new method that can often make em much faster the intuitive idea is to use a covariance adjustment to correct the analysis of the m step capitalising on extra information captured in the imputed complete data the way we accomplish this is by parameter expansion we expand the complete data model while preserving the observed data model and use the expanded complete data model to generate em this parameter expanded ei m px em algorithm shares the simplicity and stability of ordinary em but has a faster rate of convergence since its m step performs a more efficient analysis the px em algorithm is illustrated for the multivariate t distribution a random effects model factor analysis probit regression and a poisson imaging model

empirical bayes hierarchical models for regularizing maximum likelihood estimation in the matrix gaussian procrustes problem procrustes analysis involves finding the optimal superposition of two or more forms via rotations translations and scalings procrustes problems arise in a wide range of scientific disciplines especially when the geometrical shapes of objects are compared contrasted and analyzed classically the optimal transformations are found by minimizing the sum of the squared distances between corresponding points in the forms despite its widespread use the ordinary unweighted least squares ls criterion can give erroneous solutions when the errors have heterogeneous variances heteroscedasticity or the errors are correlated both common occurrences with real data in contrast maximum likelihood ml estimation can provide accurate and consistent statistical estimates in the presence of both heteroscedasticity and correlation here we provide a complete solution to the nonisotropic ml procrustes problem assuming a matrix gaussian distribution with factored covariances our analysis generalizes simplifies and extends results from previous discussions of the ml procrustes problem an iterative algorithm is presented for the simultaneous numerical determination of the ml solutions

decomposable graphical gaussian model determination we propose a methodology for bayesian model determination in decomposable graphical gaussian models to achieve this aim we consider a hyper inverse wishart prior distribution on the concentration matrix for each given graph to ensure compatibility across models such prior distributions are obtained by marginalisation from the prior conditional on the complete graph we explore alternative structures for the hyperparameters of the latter and their consequences for the model model determination is carried out by implementing a reversible jump markov chain monte carlo sampler in particular the dimension changing move we propose involves adding or dropping an edge from the graph we characterise the set of moves which preserve the decomposability of the graph giving a fast algorithm for maintaining the junction tree representation of the graph at each sweep as state variable we use the incomplete variance covariance matrix containing only the elements for which the corresponding element of the inverse is nonzero this allows all computations to be performed locally at the clique level which is a clear advantage for the analysis of large and complex datasets finally the statistical and computational performance of the procedure is illustrated by mean of both artificial and real datasets

sparse multi dimensional graphical models a unified bayesian framework abstractmulti dimensional data constituted by measurements along multiple axes have emerged across many scientific areas such as genomics and cancer surveillance a common objective is to investigate the conditional dependencies among the variables along each axes taking into account multi dimensional structure of the data traditional multivariate approaches are unsuitable for such highly structured data due to inefficiency loss of power and lack of interpretability in this article we propose a novel class of multi dimensional graphical models based on matrix decompositions of the precision matrices along each dimension our approach is a unified framework applicable to both directed and undirected decomposable graphs as well as arbitrary combinations of these exploiting the marginalization of the likelihood we develop efficient posterior sampling schemes based on partially collapsed gibbs samplers empirically through simulation studies we show the superior performance of our approach in compari

shotgun stochastic search for large p regression model search in regression with very large numbers of candidate predictors raises challenges for both model specification and computation for which standard approaches such as markov chain monte carlo mcmc methods are often infeasible or ineffective we describe a novel shotgun stochastic search sss approach that explores interesting regions of the resulting high dimensional model spaces and quickly identifies regions of high posterior probability over models we describe algorithmic and modeling aspects priors over the model space that induce sparsity and parsimony over and above the traditional dimension penalization implicit in bayesian and likelihood analyses and parallel computation using cluster computers we discuss an example from gene expression cancer genomics comparisons with mcmc and other methods and theoretical and simulation based aspects of performance characteristics in large scale regression model searches we also provide software implementing the methods

hyper markov laws in the statistical analysis of decomposable graphical models this paper introduces and investigates the notion of a hyper markov law which is a probability distribution over the set of probability measures on a multivariate space that i is concentrated on the set of markov probabilities over some decomposable graph and ii satisfies certain conditional independence restrictions related to that graph a stronger version of this hyper markov property is also studied our analysis starts by reconsidering the properties of markov probabilities using an abstract approach which thereafter proves equally applicable to the hyper markov case next it is shown constructively that hyper markov laws exist that they appear as sampling distributions of maximum likelihood estimators in decomposable graphical models and also that they form natural conjugate prior distributions for a bayesian analysis of these models as examples we construct a range of specific hyper markov laws including the hyper multinomial hyper dirichlet and the hyper wishart and inverse wishart laws these laws occur naturally in connection with the analysis of decomposable log linear and covariance selection models

partial regularity of stable solutions to the supercritical equations and its applications abstract we prove the partial regularity of stable solutions of supercritical elliptic equations as an application we prove that any smooth stable entire solution to supercritical equations with p in a suitable range is radially symmetric

two dimensional pattern matching with rotations the problem of pattern matching with rotation is that of finding all occurrences of a two dimensional pattern in a text in all possible rotations we prove an upper and lower bound on the number of such different possible rotated patterns subsequently given an m m array pattern and an n n array text over some finite alphabet we present a new method yielding an o n2m3 time algorithm for this problem

positive solutions of an elliptic equation with negative exponent stability and critical power abstract we study positive solutions of the equation u x u p in r n n 2 where p 0 2 and is a bounded or unbounded domain we show that there is a critical power p p c such that this equation with r n has no stable positive solution for p p c but it admits a family of stable positive solutions when 0 p p c if p p c min 0 we further show that this equation with b r 0 has no positive solution with finite morse index that has an isolated rupture at 0 and analogously it has no positive solution with finite morse index when r n b r among other results we also classify the positive solutions over b r 0 which are not bounded near 0

convergence rates and asymptotic standard errors for markov chain monte carlo algorithms for bayesian probit regression consider a probit regression problem in which y 1 y n are independent bernoulli random variables such that pr y i 1 x i t where x i is a p dimensional vector of known covariates that are associated with y i is a p dimensional vector of unknown regression coefficients and denotes the standard normal distribution function we study markov chain monte carlo algorithms for exploring the intractable posterior density that results when the probit regression likelihood is combined with a flat prior on we prove that albert and chib s data augmentation algorithm and liu and wu s px da algorithm both converge at a geometric rate which ensures the existence of central limit theorems for ergodic averages under a second moment condition although these two algorithms are essentially equivalent in terms of computational complexity results of hobert and marchev imply that the px da algorithm is theoretically more efficient in the sense that the asymptotic variance in the central limit theorem under the px da algorithm is no larger than that under albert and chib s algorithm we also construct minorization conditions that allow us to exploit regenerative simulation techniques for the consistent estimation of asymptotic variances as an illustration we apply our results to van dyk and meng s lupus data this example demonstrates that huge gains in efficiency are possible by using the px da algorithm instead of albert and chib s algorithm

global optimization through rotation space search this paper introduces a new algorithmic technique for solving certain problems in geometric computer vision the main novelty of the method is a branch and bound search over rotation space which is used in this paper to determine camera orientation by searching over all possible rotations problems can be reduced to known fixed rotation problems for which optimal solutions have been previously given in particular a method is developed for the estimation of the essential matrix giving the first guaranteed optimal algorithm for estimating the relative pose using a cost function based on reprojection errors recently convex optimization techniques have been shown to provide optimal solutions to many of the common problems in structure from motion however they do not apply to problems involving rotations the search method described in this paper allows such problems to be solved optimally apart from the essential matrix the algorithm is applied to the camera pose problem providing an optimal algorithm the approach has been implemented and tested on a number of both synthetically generated and real data sets with good performance

3d discrete rotations using hinge angles in this paper we study 3d rotations on grid points computed by using only integers for that purpose we investigate the intersection between the 3d half grid and the rotation plane from this intersection we define 3d hinge angles which determine a transit of a grid point from a voxel to its adjacent voxel during the rotation then we give a method to sort all 3d hinge angles with integer computations the study of 3d hinge angles allows us to design a 3d discrete rotation and to estimate the rotation between a pair of digital images in correspondence

high dimensional semiparametric bigraphical models in multivariate analysis a gaussian bigraphical model is commonly used for modelling matrix valued data in this paper we propose a semiparametric extension of the gaussian bigraphical model called the nonparanormal bigraphical model a projected nonparametric rank based regularization approach is employed to estimate sparse precision matrices and produce graphs under a penalized likelihood framework theoretically our semiparametric procedure achieves the parametric rates of convergence for both matrix estimation and graph recovery empirically our approach outperforms the parametric gaussian model for non gaussian data and is competitive with its parametric counterpart for gaussian data extensions to the categorical bigraphical model and the missing data problem are discussed copyright 2013 oxford university press

computing upper and lower bounds of rotation angles from digital images rotations in the discrete plane are important for many applications such as image matching or construction of mosaic images we suppose that a digital image a is transformed to another digital image b by a rotation in the discrete plane there are many angles giving the rotation from a to b which we call admissible rotation angles from a to b for such a set of admissible rotation angles there exist two angles that achieve the lower and the upper bounds to find those lower and upper bounds we use hinge angles as used in nouvel and remila incremental and transitive discrete rotations in r reulke u eckardt b flash u knauer k polthier eds combinatorial image analysis lecture notes in computer science vol 4040 springer berlin 2006 pp 199 213 a sequence of hinge angles is a set of particular angles determined by a digital image in the sense that any angle between two consecutive hinge angles gives the identical rotation of the digital image we propose a method for obtaining the lower and the upper bounds of admissible rotation angles using hinge angles from a given euclidean angle or from a pair of corresponding digital images

on the cauchy problem for a reaction diffusion equation with a singular nonlinearity abstract we consider the following cauchy problem with a singular nonlinearity p u t u u x r n t 0 0 u t 0 c lb r n with n 3 and having a positive lower bound we find some conditions on the initial value such that the local solutions of p vanish in finite time meanwhile we obtain optimal conditions on for global existence and study the large time behavior of those global solutions in particular we prove that if 0 and n 3 x u s x 2 1 n 2 2 1 1 1 x 2 1 where u s is a singular equilibrium of p and 1 then p has a unique global classical solution u with u u s and u x t 1 1 1 1 1 1 1 t 1 1 on the other hand the structure of positive radial solutions of the steady state of p is studied and some interesting properties of the positive solutions are obtained moreover the stability and weakly asymptotic stability of the positive radial solutions of the steady state of p are also discussed

positive solutions of a semilinear elliptic equation with singular nonlinearity abstract this paper is devoted to the study of positive solutions of the semilinear elliptic equation u k x u p 0 x r n with n 3 and p 0 asymptotic behaviours of sky states and uniqueness of singular sky states are obtained via invariant manifold theory of dynamical systems the dirichlet problem in exterior domains is also studied it is proved that this problem has infinitely many positive solutions with fast growth

singular behavior of an electrostatic elastic membrane system with an external pressure abstract we analyze nonnegative solutions of the nonlinear elliptic problem u f x u 2 p where 0 and p 0 are constants on a bounded domain of r n n 1 with a dirichlet boundary condition this equation models an electrostatic elastic membrane system with an external pressure p 0 where 0 denotes the applied voltage first we completely address the existence and nonexistence of positive solutions the classification of all possible singularities at x 0 for nonnegative solutions u x satisfying u 0 0 is then analyzed for the special case where b 1 0 r 2 and f x x with 2 in particular we show that for some u x admits only the isotropic singularity at x 0 and otherwise u x may admit the anisotropic singularity at x 0 when u x admits the isotropic singularity at x 0 the refined singularity of u x at x 0 is further investigated depending on whether p 0 by applying fourier analysis

simulation of hyper inverse wishart distributions in graphical models summary we introduce and exemplify an efficient method for direct sampling from hyperinverse wishart distributions the method relies very naturally on the use of standard junction tree representation of graphs and couples these with matrix results for inverse wishart distributions we describe the theory and resulting computational algorithms for both decomposable and nondecomposable graphical models an example drawn from financial time series demonstrates application in a context where inferences on a structured covariance model are required we discuss and investigate questions of scalability of the simulation methods to higher dimensional distributions the paper concludes with general comments about the approach including its use in connection with existing markov chain monte carlo methods that deal with uncertainty about the graphical model structure

some liouville theorems for henon type elliptic equations abstract we investigate here the nonlinear elliptic equations u x e u and u x u p 1 u with 2 p 1 and n 2 in particular we prove some liouville type theorems for weak solutions with finite morse index in the low dimensional euclidean spaces or half spaces

parameter expansion for data augmentation abstract viewing the observed data of a statistical model as incomplete and augmenting its missing parts are useful for clarifying concepts and central to the invention of two well known statistical algorithms expectation maximization em and data augmentation recently liu rubin and wu demonstrated that expanding the parameter space along with augmenting the missing data is useful for accelerating iterative computation in an em algorithm the main purpose of this article is to rigorously define a parameter expanded data augmentation px da algorithm and to study its theoretical properties the px da is a special way of using auxiliary variables to accelerate gibbs sampling algorithms and is closely related to reparameterization techniques we obtain theoretical results concerning the convergence rate of the px da algorithm and the choice of prior for the expansion parameter to understand the role of the expansion parameter we establish a new theory for iterative conditional sampling under the tra

marginal likelihood from the gibbs output abstract in the context of bayes estimation via gibbs sampling with or without data augmentation a simple approach is developed for computing the marginal density of the sample data marginal likelihood given parameter draws from the posterior distribution consequently bayes factors for model comparisons can be routinely computed as a by product of the simulation hitherto this calculation has proved extremely challenging our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density this simple identity holds for any parameter value an estimate of the posterior density is shown to be available if all complete conditional densities used in the gibbs sampler have closed form expressions to improve accuracy the posterior density is estimated at a high density point and the numerical standard error of resulting estimate is derived the ideas are applied to probit regression and finite mixture models

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

multivariate environmental statistics preface multivariate environmental statistics in agriculture v barnett a study of the relationship between diversity indices of benthic communities and heavy metal concentrations of northwest atlantic sediments n c bolgiano r n reid g p patil multivariate non normal statistics in site characterization and evaluation j h carson jr a k gupta spatial prediction in a multivariate setting n cressie needs and developments in multivariate environmental monitoring statistics g t flatman certain multivariate considerations in ranked set sampling and composite sampling designs s d gore et al relating two sets of variables in environmental studies r h green statistical analysis of biological monitoring data p guttorp using entropy in the redesign of an environmental monitoring network p guttorp et al graphical techniques for enhancing the utility of multivariate environmental statistics w a huber carcinogenicity tests involving multiple tumor sites r l kodell e o george errors in variables analysis of extrapolation procedures in environmental toxicology e linder g p patil g suter ii a unified linear model for estimation with composite sample data g lovison change point methods for spatial data i b macneill v k jandkyala a review of computer intensive multivariate methods in ecology b f j manly spatial temporal statistical analysis of multivariate environmental monitoring data k v mardia c r goodall correspondence analysis applied to environmental data sets d e myers a meta network approach to higher order spatial constructs and scale effects w l myers estimation of trend in chesapeake bay water quality data n k nagaraj s l brunenmeister the complexities and scenarios of ecosystem analysis l orloci multivariate methods in nuclear waste remediation needs and applications b a pulsipher omnibus robust procedures for assessment of multivariate normality and detection of multivariate outliers a singh multivariate assessment of trend in environmental variables e p smith s rheem g i holtzman multivariate chemometrics a case study applying and developing receptor models for the 1990 el paso winter pm10 receptor modeling scoping study c h spiegelman s dattner weighted averaging partial least squares regression wa pls definition and comparison with other methods for species environment calibration c j f ter braak et al multivariate aspects of adaptive cluster sampling s k thompson approximating multivariate distributions in stochastic models of insect population dynamics t e wehrly j h matis g w otis

properties of positive solutions to nonlocal problems with negative exponents in unbounded domains abstract we study nonlinear nonlocal problems involving the fractional p laplacian operator and nonlinearities with negative exponents in the whole space or in an unbounded epigraph domain the symmetry and monotonicity results are proved by using a direct method of moving planes

finite morse index steady states of van der waals force driven thin film equations abstract we study positive solutions of the equation u u p 1 in r n n 2 where p 0 and is a bounded or unbounded domain we show that there is a number p c p c n 0 such that this equation with r n has no stable positive solution for p p c we further show that there is a critical power p c p c n such that if p p c this equation with b r 0 has no positive solution with finite morse index that has an isolated rupture at 0 if 0 p p c this equation with b r 0 has a positive solution with finite morse index that has an isolated rupture at 0 and it has no positive solution with finite morse index when r n b r provided p max p c p c

a bayesian analysis of the multinomial probit model with fully identified parameters we present a new prior and corresponding algorithm for bayesian analysis of the multinomial probit model our new approach places a prior directly on the identified parameter space the key is the specification of a prior on the covariance matrix so that the 1 1 element if fixed at 1 and it is possible to draw from the posterior using standard distributions analytical results are derived which can be used to aid in assessment of the prior

bayesian analysis of matrix normal graphical models we present bayesian analyses of matrix variate normal data with conditional independencies induced by graphical model structuring of the characterizing covariance matrix parameters this framework of matrix normal graphical models includes prior specifications posterior computation using markov chain monte carlo methods evaluation of graphical model uncertainty and model structure search extensions to matrix variate time series embed matrix normal graphs in dynamic models examples highlight questions of graphical model uncertainty search and comparison in matrix data contexts these models may be applied in a number of areas of multivariate analysis time series and also spatial modelling

compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity we study the branch of semistable and unstable solutions i e those whose morse index is at most 1 of the dirichlet boundary value problem u f x 1 u 2 on a bounded domain n which models among other things a simple electrostatic microelectromechanical system mems device we extend the results of 11 relating to the minimal branch by obtaining compactness along unstable branches for 1 n 7 on any domain and for a large class of permittivity profiles f we also show the remarkable fact that powerlike profiles f x x can push back the critical dimension n 7 of this problem by establishing compactness for the semistable branch on the unit ball also for n 8 and as long as as a byproduct we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for in a natural range in all these results the conditions on the space dimension and on the power of the profile are essentially sharp 2007 wiley periodicals inc

parameterization and bayesian modeling progress in statistical computation often leads to advances in statistical modeling for example it is surprisingly common that an existing model is reparameterized solely for computational purposes but then this new configuration motivates a new family of models that is useful in applied statistics one reason why this phenomenon may not have been noticed in statistics is that reparameterizations do not change the likelihood in a bayesian framework however a transformation of parameters typically suggests a new family of prior distributions we discuss examples in censored and truncated data mixture modeling multivariate imputation stochastic processes and multilevel models