inference for mixtures of finite polya tree models mixtures of polya tree models provide a flexible alternative when a parametric model may only hold approximately i provide computational strategies for obtaining full semiparametric inference for mixtures of finite polya tree models given a standard parameterization including models that would be troublesome to fit using dirichlet process mixtures recommendations are put forth on choosing the level of a finite polya tree and model comparison is discussed several examples demonstrate the utility of finite polya tree modeling including data fit to generalized linear mixed models and several survival models

homoclinic solutions for a class of the second order hamiltonian systems abstract we study the existence of homoclinic orbits for the second order hamiltonian system q v q t q f t where q r n and v c 1 r r n r v t q k t q w t q is t periodic in t a map k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w is superlinear at the infinity and f is sufficiently small in l 2 r r n a homoclinic orbit is obtained as a limit of 2 kt periodic solutions of a certain sequence of the second order differential equations

existence of homoclinic orbits for second order hamiltonian systems without ar condition abstract the existence of homoclinic orbits is obtained for a class of the second order hamiltonian systems u t l t u t w t u t 0 t r by the local linking lemma where w t x is superquadratic and need not be nonnegative globally

semi supervised linear spectral unmixing using a hierarchical bayesian model for hyperspectral imagery this paper proposes a hierarchical bayesian model that can be used for semi supervised hyperspectral image unmixing the model assumes that the pixel reflectances result from linear combinations of pure component spectra contaminated by an additive gaussian noise the abundance parameters appearing in this model satisfy positivity and additivity constraints these constraints are naturally expressed in a bayesian context by using appropriate abundance prior distributions the posterior distributions of the unknown model parameters are then derived a gibbs sampler allows one to draw samples distributed according to the posteriors of interest and to estimate the unknown abundances an extension of the algorithm is finally studied for mixtures with unknown numbers of spectral components belonging to a know library the performance of the different unmixing strategies is evaluated via simulations conducted on synthetic and real data

toward a sparse bayesian markov random field approach to hyperspectral unmixing and classification recent work has shown that existing powerful bayesian hyperspectral unmixing algorithms can be significantly improved by incorporating the inherent local spatial correlations between pixel class labels via the use of markov random fields we here propose a new bayesian approach to joint hyperspectral unmixing and image classification such that the previous assumption of stochastic abundance vectors is relaxed to a formulation whereby a common abundance vector is assumed for pixels in each class this allows us to avoid stochastic reparameterizations and instead we propose a symmetric dirichlet distributionmodel with adjustable parameters for the common abundance vector of each class inference over the proposed model is achieved via a hybrid gibbs sampler and in particular simulated annealing is introduced for the label estimation in order to avoid the local trap problem experiments on a synthetic image and a popular publicly available real data set indicate the proposed model is faster than and outperforms the existing approach quantitatively and qualitatively moreover for appropriate choices of the dirichlet parameter it is shown that the proposed approach has the capability to induce sparsity in the inferred abundance vectors it is demonstrated that this offers increased robustness in cases where the preprocessing endmember extraction algorithms overestimate the number of active endmembers present in a given scene

joint bayesian model selection and estimation of noisy sinusoids via reversible jump mcmc in this paper the problem of joint bayesian model selection and parameter estimation for sinusoids in white gaussian noise is addressed an original bayesian model is proposed that allows us to define a posterior distribution on the parameter space all bayesian inference is then based on this distribution unfortunately a direct evaluation of this distribution and of its features including posterior model probabilities requires evaluation of some complicated high dimensional integrals we develop an efficient stochastic algorithm based on reversible jump markov chain monte carlo methods to perform the bayesian computation a convergence result for this algorithm is established in simulation it appears that the performance of detection based on posterior model probabilities outperforms conventional detection schemes

separation of non negative mixture of non negative sources using a bayesian approach and mcmc sampling this paper addresses blind source separation in the case where both the source signals and the mixing coefficients are non negative the problem is referred to as non negative source separation and the main application concerns the analysis of spectrometric data sets the separation is performed in a bayesian framework by encoding non negativity through the assignment of gamma priors on the distributions of both the source signals and the mixing coefficients a markov chain monte carlo mcmc sampling procedure is proposed to simulate the resulting joint posterior density from which marginal posterior mean estimates of the source signals and mixing coefficients are obtained results obtained with synthetic and experimental spectra are used to discuss the problem of non negative source separation and to illustrate the effectiveness of the proposed method

collaborative sparse regression using spatially correlated supports application to hyperspectral unmixing this paper presents a new bayesian collaborative sparse regression method for linear unmixing of hyperspectral images our contribution is twofold first we propose a new bayesian model for structured sparse regression in which the supports of the sparse abundance vectors are a priori spatially correlated across pixels i e materials are spatially organized rather than randomly distributed at a pixel level this prior information is encoded in the model through a truncated multivariate ising markov random field which also takes into consideration the facts that pixels cannot be empty i e there is at least one material present in each pixel and that different materials may exhibit different degrees of spatial regularity second we propose an advanced markov chain monte carlo algorithm to estimate the posterior probabilities that materials are present or absent in each pixel and conditionally to the maximum marginal a posteriori configuration of the support compute the minimum mean squared error estimates of the abundance vectors a remarkable property of this algorithm is that it self adjusts the values of the parameters of the markov random field thus relieving practitioners from setting regularization parameters by cross validation the performance of the proposed methodology is finally demonstrated through a series of experiments with synthetic and real data and comparisons with other algorithms from the literature

supervised nonlinear spectral unmixing using a postnonlinear mixing model for hyperspectral imagery this paper presents a nonlinear mixing model for hyperspectral image unmixing the proposed model assumes that the pixel reflectances are nonlinear functions of pure spectral components contaminated by an additive white gaussian noise these nonlinear functions are approximated using polynomial functions leading to a polynomial postnonlinear mixing model a bayesian algorithm and optimization methods are proposed to estimate the parameters involved in the model the performance of the unmixing strategies is evaluated by simulations conducted on synthetic and real data

joint segmentation of piecewise constant autoregressive processes by using a hierarchical model and a bayesian sampling approach we propose a joint segmentation algorithm for piecewise constant autoregressive ar processes recorded by several independent sensors the algorithm is based on a hierarchical bayesian model appropriate priors allow us to introduce correlations between the change locations of the observed signals numerical problems inherent to bayesian inference are solved by a gibbs sampling strategy the proposed joint segmentation methodology yields improved segmentation results when compared with parallel and independent individual signal segmentations the initial algorithm is derived for piecewise constant ar processes whose orders are fixed on each segment however an extension to models with unknown model orders is also discussed theoretical results are illustrated by many simulations conducted with synthetic signals and real arc tracking and speech signals

joint bayesian endmember extraction and linear unmixing for hyperspectral imagery this paper studies a fully bayesian algorithm for endmember extraction and abundance estimation for hyperspectral imagery each pixel of the hyperspectral image is decomposed as a linear combination of pure endmember spectra following the linear mixing model the estimation of the unknown endmember spectra is conducted in a unified manner by generating the posterior distribution of abundances and endmember parameters under a hierarchical bayesian model this model assumes conjugate prior distributions for these parameters accounts for nonnegativity and full additivity constraints and exploits the fact that the endmember proportions lie on a lower dimensional simplex a gibbs sampler is proposed to overcome the complexity of evaluating the resulting posterior distribution this sampler generates samples distributed according to the posterior distribution and estimates the unknown parameters using these generated samples the accuracy of the joint bayesian estimator is illustrated by simulations conducted on synthetic and real aviris images

existence of homoclinic solutions for a class of time dependent hamiltonian systems n 1 n n 2 where q s q q g r w g c r r r and l t g c r r 1 n is a positive definite symmetric matrix for all t g r the case where l t and w t q are either periodic in t or independent of t were studied by several authors see for instance ambrosetti and w x w x bertotti 1 coti zelati ekeland and sere 4 coti zelati and rabinowitz w x w x w x 5 rabinowitz 15 17 sere 19 and their references in this kind of problem the function l t plays an important role for example if l t s ai where a 0 and i denotes the identity matrix n n w x felmer and silva 8 have showed the existence of homoclinic orbits as the limit of subharmonic solutions provided that hs is autonomous see also w x w x rabinowitz and tanaka 18 serra tarallo and terracini 20 using an w x w x approach introduced by 4 see also 19 treated the hs problem

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

bayesian inference in hidden markov models through the reversible jump markov chain monte carlo method hidden markov models form an extension of mixture models which provides a flexible class of models exhibiting dependence and a possibly large degree of variability we show how reversible jump markov chain monte carlo techniques can be used to estimate the parameters as well as the number of components of a hidden markov model in a bayesian framework we employ a mixture of zero mean normal distributions as our main example and apply this model to three sets of data from finance meteorology and geomagnetism

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

lattice independent component analysis for functional magnetic resonance imaging we introduce a lattice independent component analysis lica unsupervised scheme to functional magnetic resonance imaging fmri data analysis lica is a non linear alternative to independent component analysis ica such that ica s statistical independent sources correspond to lica s lattice independent sources in this paper lica uses an incremental lattice source induction algorithm ilsia to induce the lattice independent sources from the input dataset the ilsia computes a set of strongly lattice independent vectors using properties of lattice associative memories regarding lattice independence and chebyshev best approximation the lattice independent sources constitute a set of affine independent vectors that define a simplex covering the input data lica carries out data linear unmixing based on the lattice independent sources basis therefore lica is a hybrid combination of a non linear lattice based component and a linear unmixing component the principal advantage over ica is that lica does not impose any probabilistic model assumptions on the data sources we compare lica with ica in two case studies firstly including simulated fmri data lica discovers the spatial location of meaningful sources with less ambiguity than ica secondly including real data from an auditory stimulation experiment lica improves over some state of the art ica variants discovering the activation patterns detected by statistical parametric mapping spm on the same data

nonlinear hyperspectral unmixing with robust nonnegative matrix factorization we introduce a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures the new model extends the commonly used linear mixing model by introducing an additional term accounting for possible nonlinear effects that are treated as sparsely distributed additive outliers with the standard nonnegativity and sum to one constraints inherent to spectral unmixing our model leads to a new form of robust nonnegative matrix factorization with a group sparse outlier term the factorization is posed as an optimization problem which is addressed with a block coordinate descent algorithm involving majorization minimization updates simulation results obtained on synthetic and real data show that the proposed strategy competes with the state of the art linear and nonlinear unmixing methods

monte carlo methods for adaptive sparse approximations of time series this paper deals with adaptive sparse approximations of time series the work is based on a bayesian specification of the shift invariant sparse coding model to learn approximations for a particular class of signals two different learning strategies are discussed the first method uses a gradient optimization technique commonly employed in sparse coding problems the other method is novel in this context and is based on a sampling estimate to approximate the gradient in the first approach we compare two monte carlo estimation techniques gibbs sampling and a novel importance sampling method the second approach is based on a direct sample estimate and uses an extension of the gibbs sampler used with the first approach both approaches allow the specification of different prior distributions and we here introduce a novel mixture prior based on a modified rayleigh distribution experiments demonstrate that all gibbs sampler based methods show comparable performance the importance sampler was found to work nearly as well as the gibbs sampler on smaller problems in terms of estimating the model parameters however the method performed substantially worse on estimating the sparse coefficients for large problems we found that the combination of a subset selection heuristic with the gibbs sampling approaches can outperform previous suggested methods in addition the methods studied here are flexible and allow the incorporation of additional prior knowledge such as the nonnegativity of the approximation coefficients which was found to offer additional benefits where applicable

homoclinic solutions for nonautonomous second order hamiltonian systems with a coercive potential we shall be concerned with the existence of homoclinic solutions for the second order hamiltonian system q vq t q f t where t r and q rn a potential v c1 r rn r is t periodic in t coercive in q and the integral of v 0 over 0 t is equal to 0 a function f r rn is continuous bounded square integrable and f 0 we will show that there exists a solution q0 such that q0 t 0 and q 0 t 0 as t although q 0 is not a solution of our system we are to call q0 a homoclinic solution it is obtained as a limit of 2kt periodic orbits of a sequence of the second order differential equations

existence of even homoclinic orbits for second order hamiltonian systems some existence theorems for even homoclinic orbits are obtained for a class of second order nonautonomous hamiltonian systems with symmetric potentials under a class of new superquadratic conditions a homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil boundary value problems which are obtained by the minimax methods

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

existence of homoclinic solution for the second order hamiltonian systems abstract an existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order hamiltonian systems u t l t u t w t u t 0 t r by the minimax methods in the critical point theory specially the generalized mountain pass theorem where l t is unnecessary uniformly positively definite for all t r and w t x satisfies the superquadratic condition w t x x 2 as x uniformly in t and need not satisfy the global ambrosetti rabinowitz condition

periodic solutions for a class of non autonomous hamiltonian systems abstract we consider the existence of nontrivial periodic solutions for a superlinear hamiltonian system h j u a t u h t u 0 u r 2 n t r we prove an abstract result on the existence of a critical point for a real valued functional on a hilbert space via a new deformation theorem different from the works in the literature the new deformation theorem is constructed under the cerami type condition instead of palais smale type condition in addition the main assumption here is weaker than the usual ambrosetti rabinowitz type condition 0 h t u u h t u 2 u r 0 this result extends theorems given by li and willem j math anal appl 189 1995 6 32 and li and szulkin j differential equations 112 1994 226 238

hyperspectral unmixing in presence of endmember variability nonlinearity or mismodeling effects this paper presents three hyperspectral mixture models jointly with bayesian algorithms for supervised hyperspectral unmixing based on the residual component analysis model the proposed general formulation assumes the linear model to be corrupted by an additive term whose expression can be adapted to account for nonlinearities nls endmember variability ev or mismodeling effects mes the nl effect is introduced by considering a polynomial expression that is related to bilinear models the proposed new formulation of ev accounts for shape and scale endmember changes while enforcing a smooth spectral spatial variation the me formulation considers the effect of outliers and copes with some types of ev and nl the known constraints on the parameter of each observation model are modeled via suitable priors the posterior distribution associated with each bayesian model is optimized using a coordinate descent algorithm which allows the computation of the maximum a posteriori estimator of the unknown model parameters the proposed mixture and bayesian models and their estimation algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inferences and the computational complexity when compared with the state of the art algorithms

densities spectral densities and modality this paper considers the problem of specifying a simple approximating density function for a given data set x 1 x n simplicity is measured by the number of modes but several different definitions of approximation are introduced the taut string method is used to control the numbers of modes and to produce candidate approximating densities refinements are introduced that improve the local adaptivity of the procedures and the method is extended to spectral densities

hierarchical bayesian sparse image reconstruction with application to mrfm this paper presents a hierarchical bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white gaussian noise our hierarchical bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate bayes priors we propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero the prior has hyperparameters that are tuned automatically by marginalization over the hierarchical bayesian model to overcome the complexity of the posterior distribution a gibbs sampling strategy is proposed the gibbs samples can be used to estimate the image to be recovered e g by maximizing the estimated posterior distribution in our fully bayesian approach the posteriors of all the parameters are available thus our algorithm provides more information than other previously proposed sparse reconstruction methods that only give a point estimate the performance of the proposed hierarchical bayesian sparse reconstruction method is illustrated on synthetic data and real data collected from a tobacco virus sample using a prototype mrfm instrument

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

joint segmentation of multivariate astronomical time series bayesian sampling with a hierarchical model astronomy and other sciences often face the problem of detecting and characterizing structure in two or more related time series this paper approaches such problems using bayesian priors to represent relationships between signals with various degrees of certainty and not just rigid constraints the segmentation is conducted by using a hierarchical bayesian approach to a piecewise constant poisson rate model a gibbs sampling strategy allows joint estimation of the unknown parameters and hyperparameters results obtained with synthetic and real photon counting data illustrate the performance of the proposed algorithm

bayesian estimation of linear mixtures using the normal compositional model application to hyperspectral imagery this paper studies a new bayesian unmixing algorithm for hyperspectral images each pixel of the image is modeled as a linear combination of so called endmembers these endmembers are supposed to be random in order to model uncertainties regarding their knowledge more precisely we model endmembers as gaussian vectors whose means have been determined using an endmember extraction algorithm such as the famous n finder n findr or vertex component analysis vca algorithms this paper proposes to estimate the mixture coefficients referred to as abundances using a bayesian algorithm suitable priors are assigned to the abundances in order to satisfy positivity and additivity constraints whereas conjugate priors are chosen for the remaining parameters a hybrid gibbs sampler is then constructed to generate abundance and variance samples distributed according to the joint posterior of the abundances and noise variances the performance of the proposed methodology is evaluated by comparison with other unmixing algorithms on synthetic and real images

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

emulation of numerical models with over specified basis functions abstractmathematical models are frequently used to explore physical systems but can be computationally expensive to evaluate in such settings an emulator is used as a surrogate in this work we propose a basis function approach for computer model emulation to combine field observations with a collection of runs from the numerical model we use the proposed emulator within the kennedy o hagan framework of model calibration a novel feature of the approach is the use of an over specified set of basis functions where number of bases used and their inclusion probabilities are treated as unknown quantities the new approach is found to have smaller predictive uncertainty and computational efficiency than the standard gaussian process approach to emulation and calibration along with several simulation examples focusing on different model characteristics we also use the method to analyze a dataset on laboratory experiments related to astrophysics

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

corrigendum on bayesian analysis of mixtures with an unknown number of components we regret to report that there is an error in equation 12 on p 740 concerning the birth death move for empty components the correct expression is a p k 1 p k 1 b k w y1 j 1y wj yk k 1 dk 1 k0 1 bk 1 g1 k wj 1y wj in the paper as printed the power of 1y wj in the nal factor the jacobian term was given as k instead of ky 1 the source of the error was neglect of the condition j wj 1 in computing the partial derivatives of wj with respect to wj note that expression 11 for the split combine move acceptance ratio is correct as printed having made the correction we repeated the calculations leading to all the numerical results reported in the paper as might be expected the e ects of the error are noticeable but small the maximum changes to any of the posterior probabilities p kjy presented in tables 1 and 2 and fig 6 are 0 015 0 011 and 0 020 respectively in each case the maximum discrepancy occurs near the mode of the distributions and so has little impact the error in fig 2 is within plotting accuracy in none of the gures is the visual impression altered and none of our qualitative conclusions are a ected we are grateful to tobias ryden of lund university for discovering this error and we apologize for any confusion that it has caused

bayesian curve fitting using mcmc with applications to signal segmentation we propose some bayesian methods to address the problem of fitting a signal modeled by a sequence of piecewise constant linear in the parameters regression models for example autoregressive or volterra models a joint prior distribution is set up over the number of the changepoints knots their positions and over the orders of the linear regression models within each segment if these are unknown hierarchical priors are developed and as the resulting posterior probability distributions and bayesian estimators do not admit closed form analytical expressions reversible jump markov chain monte carlo mcmc methods are derived to estimate these quantities results are obtained for standard denoising and segmentation of speech data problems that have already been examined in the literature these results demonstrate the performance of our methods

hyperspectral image unmixing using a multiresolution sticky hdp this paper is concerned with joint bayesian endmember extraction and linear unmixing of hyperspectral images using a spatial prior on the abundance vectors we propose a generative model for hyperspectral images in which the abundances are sampled from a dirichlet distribution dd mixture model whose parameters depend on a latent label process the label process is then used to enforces a spatial prior which encourages adjacent pixels to have the same label a gibbs sampling framework is used to generate samples from the posterior distributions of the abundances and the parameters of the dd mixture model the spatial prior that is used is a tree structured sticky hierarchical dirichlet process shdp and when used to determine the posterior endmember and abundance distributions results in a new unmixing algorithm called spatially constrained unmixing scu the directed markov model facilitates the use of scale recursive estimation algorithms and is therefore more computationally efficient as compared to standard markov random field mrf models furthermore the proposed scu algorithm estimates the number of regions in the image in an unsupervised fashion the effectiveness of the proposed scu algorithm is illustrated using synthetic and real data