soliton dynamics in computational anatomy computational anatomy ca has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms among these geometric structures landmarks and image outlines in ca are shown to be singular solutions of a partial differential equation that is called the geodesic epdiff equation a recently discovered momentum map for singular solutions of epdiff yields their canonical hamiltonian formulation which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta the momentum map provides an isomorphism between landmarks and outlines for images and singular soliton solutions of the epdiff equation this isomorphism suggests a new dynamical paradigm for ca as well as new data representation

hierarchical geodesic models in diffeomorphisms hierarchical linear models hlms are a standard approach for analyzing data where individuals are measured repeatedly over time however such models are only applicable to longitudinal studies of euclidean data this paper develops the theory of hierarchical geodesic models hgms which generalize hlms to the manifold setting our proposed model quantifies longitudinal trends in shapes as a hierarchy of geodesics in the group of diffeomorphisms first individual level geodesics represent the trajectory of shape changes within individuals second a group level geodesic represents the average trajectory of shape changes for the population our proposed hgm is applicable to longitudinal data from unbalanced designs i e varying numbers of timepoints for individuals which is typical in medical studies we derive the solution of hgms on diffeomorphisms to estimate individual level geodesics the group geodesic and the residual diffeomorphisms we also propose an efficient parallel algorithm that easily scales to solve hgms on a large collection of 3d images of several individuals finally we present an effective model selection procedure based on cross validation we demonstrate the effectiveness of hgms for longitudinal analysis of synthetically generated shapes and 3d mri brain scans

inverse eigenvalue problems for extended hessenberg and extended tridiagonal matrices in inverse eigenvalue problems one tries to reconstruct a matrix satisfying some constraints given some spectral information here two inverse eigenvalue problems are solved first given the eigenvalues and the first components of the associated eigenvectors called the weight vector an extended hessenberg matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints the extended hessenberg matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal matrix containing the eigenvalues this inverse problem closely links to orthogonal rational functions the extended hessenberg matrix contains the recurrence coefficients given the nodes eigenvalues poles poles of the extended hessenberg matrix and a weight vector first eigenvector components determining the discrete inner product moreover it is also sort of the inverse of the rational arnoldi algorithm instead of using the rational arnoldi method to compute a krylov basis to approximate the spectrum we will reconstruct the orthogonal krylov basis given the spectral info in the second inverse eigenvalue problem we do the same but refrain from unitarity as a result we execute possibly non unitary similarity transformations on the diagonal matrix of eigenvalues to retrieve a non symmetric extended tridiagonal matrix the algorithm will be less stable but it will be faster as the extended tridiagonal matrix admits a low cost factorization of o n n equals the number of eigenvalues whereas the extended hessenberg matrix does not again there is a close link with orthogonal function theory the extended tridiagonal matrix captures the recurrence coefficients of bi orthogonal rational functions moreover it is again sort of inverse of the nonsymmetric lanczos algorithm given spectral properties we reconstruct the two basis krylov matrices linked to the nonsymmetric lanczos algorithm

statistics on diffeomorphisms via tangent space representations in this paper we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations the space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied specializing to the landmark matching problem of computational anatomy we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis pca in this setting with three dimensional face and hippocampus databases

orthogonal rational functions with real coefficients and semiseparable matrices when one wants to use orthogonal rational functions orfs in system identification or control theory it is important to be able to avoid complex calculations in this paper we study orfs whose numerator and denominator polynomial have real coefficients these orfs with real coefficients rorfs appear when the poles and the interpolation points appear in complex conjugate pairs which is a natural condition further we deduce that there is a strong connection between rorfs and semiseparable matrices

construction of a probabilistic model for impedance matrices impedance matrices allow for the coupling of domains with very different properties and possibly modeled with different methods this paper presents the construction of a probabilistic model for such matrices using a nonparametric method that allows for the consideration of data errors as well as model errors to enable the application of this method in the case of a domain for which no finite element model is available the identification of a hidden state variables model from the knowledge of the impedance matrix at a discrete set of frequencies is also described finally the construction of the probabilistic model of the impedance of a pile foundation on a layered unbounded soil illustrates the capabilities of the method

robust identification from band limited data consider the problem of identifying a scalar bounded input bounded output stable transfer function from pointwise measurements at frequencies within a bandwidth we propose an algorithm which consists of building a sequence of maps from data to models converging uniformly to the transfer function on the bandwidth when the number of measurements goes to infinity the noise level to zero and asymptotically meeting some gauge constraint outside error bounds are derived and the procedure is illustrated by numerical experiments

asymptotic normality of the posterior in relative entropy we show that the relative entropy between a posterior density formed from a smooth likelihood and prior and a limiting normal form tends to zero in the independent and identically distributed case the mode of convergence is in probability and in mean applications to code lengths in stochastic complexity and to sample size selection are discussed

time sequence diffeomorphic metric mapping and parallel transport track time dependent shape changes abstract serial mri human brain scans have facilitated the detection of brain development and of the earliest signs of neuropsychiatric and neurodegenerative diseases monitoring disease progression and resolving drug effects in clinical trials for preventing or slowing the rate of brain degeneration to track anatomical shape changes in serial images we introduce new point based time sequence large deformation diffeomorphic metric mapping ts lddmm to infer the time flow of within subject geometric shape changes that carry known observations through a period its euler lagrange equation is generalized for anatomies whose shapes are characterized by point sets such as landmarks curves and surfaces the time dependent momentum obtained from the ts lddmm encodes within subject shape changes for the purpose of across subject shape comparison we then propose a diffeomorphic analysis framework to translate within subject deformation in a global template without incorporating across subject anatomical variations via parallel transport technique the analysis involves the retraction of the within subject time dependent momentum along the ts lddmm trajectory from each time to the baseline the translation of the momentum in a global template and the reconstruction of the ts lddmm trajectory starting from the global template

robust identification in h abstract we consider system identification in the banach space h in the framework proposed by helmicki jacobson and nett it is shown that there is no robustly convergent linear algorithm for identifying exponentially stable systems in the presence of noise which is not tuned to prior information about the unknown system or noise various nonlinear algorithms some closely related to one of gu and khargonekar are analysed and results on trigonometric interpolation used to provide new error bounds an application of these techniques to approximation is given and finally some numerical results are provided for illustration

inferential models a framework for prior free posterior probabilistic inference posterior probabilistic statistical inference without priors is an important but so far elusive goal fisher s fiducial inference dempster shafer theory of belief functions and bayesian inference with default priors are attempts to achieve this goal but to date none has given a completely satisfactory picture this article presents a new framework for probabilistic inference based on inferential models ims which not only provides data dependent probabilistic measures of uncertainty about the unknown parameter but also does so with an automatic long run frequency calibration property the key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter and the prediction of this auxiliary variable with a random set before conditioning on data here we present a three step im construction and prove a frequency calibration property of the im s belief function under mild conditions a corresponding optimality theory is develo

diffeomorphisms groups and pattern matching in image analysis in a previous paper it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group we show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images using lie group ideas we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub optimal gradient pattern matching its links with fluid models are established

generalizations of orthogonal polynomials we give a survey of recent generalizations of orthogonal polynomials that includes multidimensional matrix and vector orthogonal polynomials and multivariate versions multipole orthogonal rational functions variants and extensions of the orthogonality conditions multiple orthogonality most of these generalizations are inspired by the applications in which they are applied we also give a glimpse of these applications which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role moment problems numerical quadrature rational approximation linear algebra recurrence relations and random matrices

a new approach to the rational interpolation problem abstract we shall reformulate the classical newton pade rational interpolation problem nprip to take away several drawbacks of the newton pade approach a new recursive algorithm not reordering the interpolation points will be designed enabling us to give a parametrization of all solutions not only for the linearized rip but also for the proper rip this paper generalizes our earlier work where the pade approximation problem i e when all the interpolation points coincide at the origin was solved by reformulating it as a minimal partial realization problem van barel 1989 van barel and bultheel 1989

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

an approach to the conditional error rate principle with nuisance parameters summary in the presence of nuisance parameters the conditional error rate principle is difficult to apply because of the dependency of the conditional error function of the preplanned test on nuisance parameters to use the conditional error rate principle with nuisance parameters we propose to search among tests that guarantee overall error control for the test that maximizes a weighted combination of the conditional error rates over possible values of the nuisance parameters we show that the optimization problem that defines such a test can be solved efficiently by existing algorithms

orthonormal rational function vectors in this paper we develop a matrix framework to solve the problem of finding orthonormal rational function vectors with prescribed poles open image in new window with respect to a certain discrete inner product that is defined by a set of data points open image in new window and corresponding weight vectors wi j our algorithm for solving the problem is recursive and it is of complexity open image in new window if all data points are real or lie on the unit circle then the complexity is reduced by an order of magnitude

landmark matching via large deformation diffeomorphisms this paper describes the generation of large deformation diffeomorphisms spl phi spl omega 0 1 sup 3 spl rlhar2 spl omega for landmark matching generated as solutions to the transport equation d spl phi x t dt spl nu spl phi x t t t spl isin 0 1 and spl phi x 0 x with the image map defined as spl phi spl middot 1 and therefore controlled via the velocity field spl nu spl middot t t spl isin 0 1 imagery are assumed characterized via sets of landmarks x sub n y sub n n 1 2 n the optimal diffeomorphic match is constructed to minimize a running smoothness cost spl par l spl nu spl par sup 2 associated with a linear differential operator l on the velocity field generating the diffeomorphism while simultaneously minimizing the matching end point condition of the landmarks both inexact and exact landmark matching is studied here given noisy landmarks x sub n matched to y sub n measured with error covariances spl sigma sub n then the matching problem is solved generating the optimal diffeomorphism spl phi spl circ x 1 spl int sub 0 sup 1 spl nu spl circ spl phi spl circ x t t dt x where spl nu spl circ spl middot argmin sub spl nu spl middot spl int sub 1 sup 1 spl int sub spl omega spl par l spl nu x t spl par sup 2 dxdt spl sigma sub n 1 sup n y sub n spl phi x sub n 1 sup t spl sigma sub n sup 1 y sub n spl phi x sub n 1 conditions for the existence of solutions in the space of diffeomorphisms are established with a gradient algorithm provided for generating the optimal flow solving the minimum problem results on matching two dimensional 2 d and three dimensional 3 d imagery are presented in the macaque monkey

orthogonal basis functions in discrete least squares rational approximation we consider a problem that arises in the field of frequency domain system identification if a discrete time system has an input output relation y z g z u z with transter function g then the problem is to find a rational approximation n for g the data given are measurements of input and output spectra in the frequency points zk u zk y zk nk 1 together with some weight the approximation criterion is to minimize the weighted discrete least squares norm of the vector obtained by evaluating g n in the measurement points if the poles of the system are fixed then the problem reduces to a linear least squares problem in two possible ways by multiplying out the denominators and hide these in the weight which leads to the construction of orthogonal vector polynomials or the problem can be solved directly using an orthogonal basis of rational functions the orthogonality of the basis is important because if the transfer function n is represented with respect to a nonorthogonal basis then this least squares problem can be very ill conditioned even if an orthogonal basis is used but with respect to the wrong inner product e g the lebesgue measure on the unit circle numerical instability can be fatal in practice we show that both approaches lead to an inverse eigenvalue problem which forms the common framework in which fast and numerically stable algorithms can be designed for the computation of the orthonormal basis

marginal inferential models prior free probabilistic inference on interest parameters the inferential models im framework provides prior free frequency calibrated and posterior probabilistic inference the key is the use of random sets to predict unobservable auxiliary variables connected to the observable data and unknown parameters when nuisance parameters are present a marginalization step can reduce the dimension of the auxiliary variable which in turn leads to more efficient inference for regular problems exact marginalization can be achieved and we give conditions for marginal im validity we show that our approach provides exact and efficient marginal inference in several challenging problems including a many normal means problem in nonregular problems we propose a generalized marginalization technique and prove its validity details are given for two benchmark examples namely the behrens fisher and gamma mean problems

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

system identification with generalized orthonormal basis functions a least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function where the expansion is in terms of recently introduced generalized basis functions the basis functions are orthogonal in h2 and generalize the pulse laguerre and kautz bases one of their important properties is that when chosen properly they can substantially increase the speed of convergence of the series expansion this leads to accurate approximate models with only a few coefficients to be estimated explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates

system identification using kautz models in this paper the problem of approximating a linear time invariant stable system by a finite weighted sum of given exponentials is considered system identification schemes using laguerre models are extended and generalized to kautz models which correspond to representations using several different possible complex exponentials in particular linear regression methods to estimate this sort of model from measured data are analyzed the advantages of the proposed approach are the simplicity of the resulting identification scheme and the capability of modeling resonant systems using few parameters the subsequent analysis is based on the result that the corresponding linear regression normal equations have a block toeplitz structure several results on transfer function estimation are extended to discrete kautz models for example asymptotic frequency domain variance expressions

robust identification and interpolation in h abstract we consider system identification in h in the framework proposed by helmicki jacobson and nett an algorithm using the jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems it is shown that this and similar identification algorithms can be successfully combined with a model reduction procedure to produce low order models connections with the nevanlinna pick interpolation problem are explored and an algorithm is given in which the identified model interpolates the given noisy data some numerical results are provided for illustration finally the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions

a generalized orthonormal basis for linear dynamical systems in many areas of signal system and control theory orthogonal functions play an important role in issues of analysis and design in this paper it is shown that there exist orthogonal functions that in a natural way are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l sub 2sup n to this end use is made of balanced realizations of inner transfer functions the orthogonal functions can be considered as generalizations of for example the pulse functions laguerre functions and kautz functions and give rise to an alternative series expansion of rational transfer functions it is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion i e to obtain a good approximation by retaining only a finite number of expansion coefficients consequences for identification of expansion coefficients are analyzed and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients

inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices abstract matrices represented as a sum of diagonal and semiseparable ones are considered here these matrices belong to the class of structured matrices which arises in numerous applications fast o n algorithms for their inversion were developed before under additional restrictions which were a source of instability our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms in this paper the case of semiseparable matrices of order one is considered

combining anatomical manifold information via diffeomorphic metric mappings for studying cortical thinning of the cingulate gyrus in schizophrenia spatial normalization is a crucial step in assessing patterns of neuroanatomical structure and function associated with health and disease errors that occur during spatial normalization can influence hypothesis testing due to the dimensionalities of mapping algorithms and anatomical manifolds landmarks curves surfaces volumes used to drive the mapping algorithms the primary aim of this paper is to improve statistical inference using multiple anatomical manifolds and large deformation diffeomorphic metric mapping lddmm algorithms we propose that combining information generated by the various manifolds and algorithms improves the reliability of hypothesis testing we used this unified approach to assess variation in the thickness of the cingulate gyrus in subjects with schizophrenia and healthy comparison subjects three different lddmm algorithms for mapping landmarks curves and triangulated meshes were used to transform thickness maps of the cingulate surfaces into an atlas coordinate system we then tested for group differences by combining the information from the three types of anatomical manifolds and lddmm mapping algorithms the unified approach provided reliable statistical results and eliminated ambiguous results due to surface mismatches subjects with schizophrenia had non uniform cortical thinning over the left and right cingulate gyri especially in the anterior portion as compared to

a class of algorithms for identification in h abs traet in this paper a class of algorithms for the problem of system identification in are investigated these algorithms are characterized by a two stage structure and involve a class of window functions some conditions in terms of properties of the window functions are derived which guarantee robust convergence of the algorithms identification errors are analyzed for several common window functions this leads to some insights into the trade off between the error induced by approximation and that due to noise

penalization bias reduction and default priors in logistic and related categorical and survival regressions penalization is a very general method of stabilizing or regularizing estimates which has both frequentist and bayesian rationales we consider some questions that arise when considering alternative penalties for logistic regression and related models the most widely programmed penalty appears to be the firth small sample bias reduction method albeit with small differences among implementations and the results they provide which corresponds to using the log density of the jeffreys invariant prior distribution as a penalty function the latter representation raises some serious contextual objections to the firth reduction which also apply to alternative penalties based on t distributions including cauchy priors taking simplicity of implementation and interpretation as our chief criteria we propose that the log f 1 1 prior provides a better default penalty than other proposals penalization based on more general log f priors is trivial to implement and facilitates mean squared error reduction and sensitivity analyses of penalty strength by varying the number of prior degrees of freedom we caution however against penalization of intercepts which are unduly sensitive to covariate coding and design idiosyncrasies copyright 2015 john wiley sons ltd

generalized fiducial inference for ultrahigh dimensional regression in recent years the ultrahigh dimensional linear regression problem has attracted enormous attention from the research community under the sparsity assumption most of the published work is devoted to the selection and estimation of the predictor variables with nonzero coefficients this article studies a different but fundamentally important aspect of this problem uncertainty quantification for parameter estimates and model choices to be more specific this article proposes methods for deriving a probability density function on the set of all possible models and also for constructing confidence intervals for the corresponding parameters these proposed methods are developed using the generalized fiducial methodology which is a variant of fisher s controversial fiducial idea theoretical properties of the proposed methods are studied and in particular it is shown that statistical inference based on the proposed methods will have correct asymptotic frequentist property in terms of empirical perform

large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the alzheimer type in large deformation diffeomorphic metric mapping lddmm the diffeomorphic matching of images are modeled as evolution in time or a flow of an associated smooth velocity vector field v controlling the evolution the initial momentum parameterizes the whole geodesic and encodes the shape and form of the target image thus methods such as principal component analysis pca of the initial momentum leads to analysis of anatomical shape and form in target images without being restricted to small deformation assumption in the analysis of linear displacements we apply this approach to a study of dementia of the alzheimer type dat the left hippocampus in the dat group shows significant shape abnormality while the right hippocampus shows similar pattern of abnormality further pca of the initial momentum leads to correct classification of 12 out of 18 dat subjects and 22 out of 26 control subjects

functional uniform priors for nonlinear modeling summary this article considers the topic of finding prior distributions when a major component of the statistical model depends on a nonlinear function using results on how to construct uniform distributions in general metric spaces we propose a prior distribution that is uniform in the space of functional shapes of the underlying nonlinear function and then back transform to obtain a prior distribution for the original model parameters the primary application considered in this article is nonlinear regression but the idea might be of interest beyond this case for nonlinear regression the so constructed priors have the advantage that they are parametrization invariant and do not violate the likelihood principle as opposed to uniform distributions on the parameters or the jeffrey s prior respectively the utility of the proposed priors is demonstrated in the context of design and analysis of nonlinear regression modeling in clinical dose finding trials through a real data example and simulation

the optimal template effect in hippocampus studies of diseased populations we evaluate the impact of template choice on template based segmentation of the hippocampus in epilepsy four dataset specific strategies are quantitatively contrasted the closest to average individual template the average shape version of the closest to average template a best appearance template and the best appearance and shape template proposed here and implemented in the open source toolkit advanced normalization tools ants the cross correlation similarity metric drives the correspondence model and is used consistently to determine the optimal appearance minimum shape distance in the diffeomorphic space determines optimal shape our evaluation results show that with respect to gold standard manual labeling of hippocampi in epilepsy optimal shape and appearance template construction outperforms the other strategies for gaining data derived templates our results also show the improvement is most significant on the diseased side and insignificant on the healthy side thus the importance of the template increases when used to study pathology and may be less critical for normal control studies furthermore explicit geometric optimization of the shape component of the unbiased template positively impacts the study of diseased hippocampi

computing large deformation metric mappings via geodesic flows of diffeomorphisms this paper examine the euler lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in dupuis et al 1998 and trouve 1995 in which two images i 0 i 1 are given and connected via the diffeomorphic change of coordinates i 0 1 i 1 where 1 is the end point at t 1 of curve t t 0 1 satisfying t v t t t 0 1 with 0 id the variational problem takes the form mathop arg text m in limits upsilon dot phi t upsilon t left dot phi right left int 0 1 left upsilon t right 2 text d t left i 0 circ phi 1 1 i 1 right l 2 2 right where v t v is an appropriate sobolev norm on the velocity field v t and the second term enforces matching of the images with l 2 representing the squared error norm in this paper we derive the euler lagrange equations characterizing the minimizing vector fields v t t 0 1 assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms we describe the implementation of the euler equations using semi lagrangian method of computing particle flows and show the solutions for various examples as well we compute the metric distance on several anatomical configurations as measured by 0 1 v t v dt on the geodesic shortest paths

robust bayesian variable selection in linear models with spherically symmetric errors this paper studies bayesian variable selection in linear models with general spherically symmetric error distributions we construct the posterior odds based on a separable prior which arises as a class of mixtures of gaussian densities the posterior odds for comparing among nonnull models are shown to be independent of the error distribution if this is spherically symmetric because of this invariance we refer to our method as a robust bayesian variable selection method we demonstrate that our posterior odds have model selection consistency and that our class of prior functions are the only ones within a large class which are robust in our sense

objective bayesian analysis of spatially correlated data spatially varying phenomena are often modeled using gaussian random fields specified by their mean function and covariance function the spatial correlation structure of these models is commonly specified to be of a certain form e g spherical power exponential rational quadratic or matern with a small number of unknown parameters we consider objective bayesian analysis of such spatial models when the mean function of the gaussian random field is specified as in a linear model it is thus necessary to determine an objective or default prior distribution for the unknown mean and covariance parameters of the random field we first show that common choices of default prior distributions such as the constant prior and the independent jeffreys prior typically result in improper posterior distributions for this model next the reference prior for the model is developed and is shown to yield a proper posterior distribution a further attractive property of the reference prior is that it can be used

worst case deterministic identification in h sub infinity the continuous time case results obtained by the authors 1991 worst case deterministic h sub infinity identification of discrete time plants are extended to continuous time plants the problem involves identification of the transfer function of a stable strictly proper continuous time plant from a finite number of noisy point samples of the plant frequency response the assumed information consists of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant an upper bound on the roll off rate of the plant and an upper bound on the noise level concrete plans of identification algorithms are provided for this problem explicit worst case deterministic error bounds for each algorithm establish that they are robustly convergent and essentially asymptotically optimal additionally these bounds provide an a priori computable h sub infinity uncertainty specification corresponding to the resulting identified plant transfer function as an explicit function of the plant and noise prior information and the data cardinality

evolutions equations in computational anatomy one of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs notably in the brain or the heart over the last decade our group has progressively developed several approaches for this problem all related to the riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act several important shape evolution equations that are now used routinely in applications have emerged over time our goal in this paper is to provide an overview of these equations placing them in their theoretical context and giving examples of applications in which they can be used we introduce the required theoretical background before discussing several classes of equations of increasingly complexity these equations include energy minimizing evolutions deriving from riemannian gradient descent geodesics parallel transport and jacobi fields

inference from intrinsic bayes procedures under model selection and uncertainty in this article we present a fully coherent and consistent objective bayesian analysis of the linear regression model using intrinsic priors the intrinsic prior is a scaled mixture of g priors and promotes shrinkage toward the subspace defined by a base or null model while it has been established that the intrinsic prior provides consistent model selectors across a range of models the posterior distribution of the model parameters has not previously been investigated we prove that the posterior distribution of the model parameters is consistent under both model selection and model averaging when the number of regressors is fixed further we derive tractable expressions for the intrinsic posterior distribution as well as sampling algorithms for both a selected model and model averaging we compare the intrinsic prior to other mixtures of g priors and provide details on the consistency properties of modified versions of the zellner siow prior and hyper g priors supplementary materials for this arti

discrete linearized least squares rational approximation on the unit circle abstract we already generalized the rutishauser gragg harrod reichel algorithm for discrete least squares polynomial approximation on the real axis to the rational case in this paper a new method for discrete least squares linearized rational approximation on the unit circle is presented it generalizes the algorithms of reichel ammar gragg for discrete least squares polynomial approximation on the unit circle to the rationale case the algorithm is fast in the sense that it requires order m computation time where m is the number of data points and is the degree of the approximant we describe how this algorithm can be implemented in parallel examples illustrate the numerical behavior of the algorithm

a new approach to the rational interpolation problem the vector case we generalize our earlier results on rational interpolation which were given in van barel and bultheel this journal 1990 for the scalar case and in bultheel and van barel 1990 for the vector case when all the interpolation points coincide to the case of vector data given at arbitrary points that may coincide or not this is the vector valued newton pade problem we give a recursive algorithm which has the important advantage over other algorithms that we do not need a reordering of the given interpolation data to overcome a singularity in the interpolation table not even in the nonnormal vector case it also generates all the information needed to give all solutions of the problem

bayesian duality and risk analysis on the statistical manifold of exponential family with censored data abstract information geometry has been attracted wide attentions in the past few decades this paper focuses on the bayesian duality on a statistical manifold derived from the exponential family with data from life tests based on life testing data the statistical manifold is constructed with a new cumulant generating function the bregman divergence between two parameter points is studied the dual coordinate system and dual function are obtained then the dualistic structure on the manifold is discussed the results show that the maximum likelihood estimate can be obtained by minimizing the bregman divergence induced from the dual function the bayesian analysis and prediction are investigated based on informative and non informative priors consider the gamma distribution as an example the closed form representations of the dual coordinate system and dual function are obtained a real data set is employed to illustrate the methodologies and experimental designs developed in this paper

deformable templates using large deformation kinematics a general automatic approach is presented for accommodating local shape variation when mapping a two dimensional 2 d or three dimensional 3 d template image into alignment with a topologically similar target image local shape variability is accommodated by applying a vector field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth globally positive definite jacobian smoothness is guaranteed without specifically penalizing large magnitude deformations of small subvolumes by constraining the transformation on the basis of a stokesian limit of the fluid dynamical navier stokes equations this differs fundamentally from quadratic penalty methods such as those based on linearized elasticity or thin plate splines in that stress restraining the motion relaxes over time allowing large magnitude deformations kinematic nonlinearities are inherently necessary to maintain continuity of structures during large magnitude deformations and are included in all results after initial global registration final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability

orthogonal rational functions and tridiagonal matrices we study the recurrence relation for rational functions whose poles are in a prescribed sequence of numbers that are real or infinite and that are orthogonal with respect to an hermitian positive linear functional we especially discuss the interplay between finite and infinite poles the recurrence relation will also be described in terms of a tridiagonal matrix which is a generalization of the jacobi matrix of the polynomial situation which corresponds to placing all the poles at infinity this matrix not only describes the recurrence relation but it can be used to give a determinant expression for the orthogonal rational functions and it also allows for the formulation of a generalized eigenvalue problem whose eigenvalues are the zeros of an orthogonal rational function these nodes can be used in rational gauss type quadrature formulas and the corresponding weights can be obtained from the first components of the corresponding eigenvectors

robust convergence of two stage nonlinear algorithms for identification in h two stage nonlinear algorithms for the problem of identification in are analyzed in terms of the fourier transform of the window function conditions characterizing the robust convergence of the two stage nonlinear algorithm are derived the sufficiency of the conditions holds for a general class of window functions while the necessity is proved for robust convergence of the first stage of the algorithm under the mild additional restriction that the window function is even symmetric some improved upper bounds for the worst case identification error are also obtained

a parallel algorithm for discrete least squares rational approximation a new method for discrete least squares linearized rational approximation is presented it generalizes the algorithm of rutishauser gragg harrod reichel for discrete least squares polynomial approximation to the rational case the algorithm is fast in the sense that it requires orderm computation time wherem is the number of data points and is the degree of the approximant we describe how this algorithm can be implemented in parallel

generalized fiducial inference a review and new results abstractr a fisher the father of modern statistics proposed the idea of fiducial inference during the first half of the 20th century while his proposal led to interesting methods for quantifying uncertainty other prominent statisticians of the time did not accept fisher s approach as it became apparent that some of fisher s bold claims about the properties of fiducial distribution did not hold up for multi parameter problems beginning around the year 2000 the authors and collaborators started to reinvestigate the idea of fiducial inference and discovered that fisher s approach when properly generalized would open doors to solve many important and difficult inference problems they termed their generalization of fisher s idea as generalized fiducial inference gfi the main idea of gfi is to carefully transfer randomness from the data to the parameter space using an inverse of a data generating equation without the use of bayes theorem the resulting generalized fiducial distribution gfd can

frequency domain identification with generalized orthonormal basis functions a method is considered for the identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain to this end use is made of the empirical transfer function estimate etfe identified from time domain data as a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions being generalizations of the classical pulse and laguerre and kautz types of bases an asymptotic analysis of the estimated models is provided and conditions for consistency are formulated explicit and transparent bias and variance expressions are established the latter ones also valid in a situation of undermodeling

an objective approach to prior mass functions for discrete parameter spaces we present a novel approach to constructing objective prior distributions for discrete parameter spaces these types of parameter spaces are particularly problematic as it appears that common objective procedures to design prior distributions are problem specific we propose an objective criterion based on loss functions instead of trying to define objective probabilities directly we systematically apply this criterion to a series of discrete scenarios previously considered in the literature and compare the priors the proposed approach applies to any discrete parameter space making it appealing as it does not involve different concepts according to the model supplementary materials for this article are available online

the formal definition of reference priors reference analysis produces objective bayesian inference in the sense that inferential statements depend only on the assumed model and the available data and the prior distribution used to make an inference is least informative in a certain information theoretic sense reference priors have been rigorously defined in specific contexts and heuristically defined in general but a rigorous general definition has been lacking we produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions the explicit expression can be used to derive new reference priors both analytically and numerically

numerically robust transfer function modeling from noisy frequency domain data using vector orthogonal polynomials as basis functions for the representation of the rational form of a linear time invariant system in frequency domain identification problems it is shown that the notorious numerical ill conditioning of these maximum likelihood problems can be overcome completely for the identification of high order 100 100 systems operating over a wide frequency band or even in the situation of over or undermodeling condition numbers less than ten are reported for real measurements

reference priors with partial information summary in this paper reference priors are derived for three cases where partial information is available if a subjective conditional prior is given two reasonable methods are proposed for finding the marginal reference prior if instead a subjective marginal prior is available a method for defining the conditional reference prior is proposed a sufficient condition is then given under which this conditional reference prior agrees with the conditional reference prior derived in the first stage of the reference prior algorithm of berger bernardo 1989 1992 finally under the assumption of independence a method for finding marginal reference priors is also proposed various examples are given to illustrate the methods

information geometry of generalized bayesian prediction using alpha divergences as loss functions in this paper the methods of information geometry are employed to investigate a generalized bayes rule for prediction taking alpha divergences as the loss functions optimality and asymptotic properties of the generalized bayesian predictive densities are considered we show that the bayesian predictive densities minimize a generalized bayes risk we also find that the asymptotic expansions of the densities are related to the coefficients of the alpha connections of a statistical manifold in addition we discuss the difference between two risk functions of the generalized bayesian predictions based on different priors finally using the non informative priors i e jeffreys and reference priors uniform prior and conjugate prior two examples are presented to illustrate the main results

diffeomorphic registration using geodesic shooting and gauss newton optimisation this paper presents a nonlinear image registration algorithm based on the setting of large deformation diffeomorphic metric mapping lddmm but with a more efficient optimisation scheme both in terms of memory required and the number of iterations required to reach convergence rather than perform a variational optimisation on a series of velocity fields the algorithm is formulated to use a geodesic shooting procedure so that only an initial velocity is estimated a gauss newton optimisation strategy is used to achieve faster convergence the algorithm was evaluated using freely available manually labelled datasets and found to compare favourably with other inter subject registration algorithms evaluated using the same data

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

unitary rank structured matrices in this paper we describe how one can represent a unitary rank structured matrix in an efficient way as a product of elementary unitary or givens transformations we also provide some basic operations for manipulating the representation such as the transition to zero creating form the transition to a unitary givens weight representation as well as an internal pull through process of the two branches of the representation finally we characterize how to determine the shift correction term to the rank structure and we provide some applications to this result

asymptotic inference for mixture models by using data dependent priors for certain mixture models improper priors are undesirable because they yield improper posteriors however proper priors may be undesirable because they require subjective input we propose the use of specially chosen data dependent priors we show that in some cases data dependent priors are the only priors that produce intervals with second order correct frequentist coverage the resulting posterior also has another interpretation it is the product of a fixed prior and a pseudolikelihood

eigenvalue computation for unitary rank structured matrices in this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps first we perform a reduction of the given matrix into hessenberg form next we compute the eigenvalues of this resulting hessenberg matrix via an implicit qr algorithm along the way we explain how the knowledge of a certain shift correction term to the structure can be used to speed up the qr algorithm for unitary hessenberg matrices and how this observation was implicitly used in a paper due to william b gragg we also treat an analogue of this observation in the hermitian tridiagonal case

group actions homeomorphisms and matching a general framework this paper constructs metrics on the space of images cal i defined as orbits under group actions cal g the groups studied include the finite dimensional matrix groups and their products as well as the infinite dimensional diffeomorphisms examined in trouve 1999 quaterly of applied math and dupuis et al 1998 quaterly of applied math left invariant metrics are defined on the product cal g times cal i thus allowing the generation of transformations of the background geometry as well as the image values examples of the application of such metrics are presented for rigid object matching with and without signature variation curves and volume matching and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another

optimally conditioned instrumental variable approach for frequency domain system identification accurate frequency domain system identification demands for reliable computational algorithms the aim of this paper is to develop a new algorithm for parametric system identification with favorable convergence properties and optimal numerical conditioning recent results in frequency domain instrumental variable identification are exploited which lead to enhanced convergence properties compared to classical identification algorithms in addition bi orthonormal polynomials with respect to a data dependent bi linear form are introduced for system identification hereby optimal numerical conditioning of the relevant system of equations is achieved this is shown to be particularly important for the class of instrumental variable algorithms for which numerical conditioning is typically quadratic compared to alternative frequency domain identification algorithms superiority of the proposed algorithm is demonstrated by means of both simulation and experimental results

conditional inferential models combining information for prior free probabilistic inference summary the inferential model im framework provides valid prior free probabilistic inference by focusing on predicting unobserved auxiliary variables but efficient im based inference can be challenging when the auxiliary variable is of higher dimension than the parameter here we show that features of the auxiliary variable are often fully observed and in such cases a simultaneous dimension reduction and information aggregation can be achieved by conditioning this proposed conditioning strategy leads to efficient im inference and casts new light on fisher s notions of sufficiency conditioning and also bayesian inference a differential equation driven selection of a conditional association is developed and validity of the conditional im is proved under some conditions for problems that do not admit a conditional im of the standard form we propose a more flexible class of conditional ims based on localization examples of local conditional ims in a bivariate normal model and a normal variance components model are also given

the fundamental role of general orthonormal bases in system identification the purpose of this paper is threefold firstly it is to establish that contrary to what might be expected the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure secondly it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros this is achieved by employing certain new results on generalized fourier series and the asymptotic properties of toeplitz like matrices in such a way that the new variance expressions presented here encompass pre existing ones as special cases via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification namely that orthonormal bases are much more than an implementational option offering improved numerical properties in fact they are an intrinsic part of estimation since as shown here orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them

simultaneous multi scale registration using large deformation diffeomorphic metric mapping in the framework of large deformation diffeomorphic metric mapping lddmm we present a practical methodology to integrate prior knowledge about the registered shapes in the regularizing metric our goal is to perform rich anatomical shape comparisons from volumetric images with the mathematical properties offered by the lddmm framework we first present the notion of characteristic scale at which image features are deformed we then propose a methodology to compare anatomical shape variations in a multi scale fashion i e at several characteristic scales simultaneously in this context we propose a strategy to quantitatively measure the feature differences observed at each characteristic scale separately after describing our methodology we illustrate the performance of the method on phantom data we then compare the ability of our method to segregate a group of subjects having alzheimer s disease and a group of controls with a classical coarse to fine approach on standard 3d mr longitudinal brain images we finally apply the approach to quantify the anatomical development of the human brain from 3d mr longitudinal images of pre term babies results show that our method registers accurately volumetric images containing feature differences at several scales simultaneously with smooth deformations

viability of methods for generating stable reduced order models various methods for generating stable reduced order models are shown to have a serious disadvantage in that the reduced model may approximate the nondominant poles of the system and hence lead to erroneous approximations an example is given to compare these methods with the standard pade approximation method for which this disadvantage does not exist

ordered group reference priors with application to the multinomial problem summary noninformative priors are developed using the reference prior approach for multiparameter problems in which there may be parameters of interest and nuisance parameters for a given grouping of parameters and ordering of the groups intuitively according to inferential importance an algorithm for determining the associated reference prior is presented the algorithm is illustrated on the multinomial problem with discussion of the variety and success of various groupings and ordering strategies

a unifying construction of orthonormal bases for system identification this paper develops a general and very simple construction for complete orthonormal bases for system identification this construction provides a unifying formulation of many previously studied orthonormal bases since the common fir and recently popular laguerre and two parameter kautz model structures are restrictive special cases of the construction presented here however in contrast to these special cases the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided

subspace restricted singular value decompositions for linear discrete ill posed problems the truncated singular value decomposition is a popular solution method for linear discrete ill posed problems these problems are numerically underdetermined therefore it can be beneficial to incorporate information about the desired solution into the solution process this paper describes a modification of the singular value decomposition that permits a specified linear subspace to be contained in the solution subspace for all truncations modifications that allow the range to contain a specified subspace or that allow both the solution subspace and the range to contain specified subspaces also are described

state space representation for arbitrary orthogonal rational functions abstract we first give a brief summary of the arguments that lead to the conclusion that the use of orthogonal rational functions in system identification should not be restricted to the takenaka malmquist basis because there are important cases where orthogonality with respect to a certain nonnegative weight function is required the main result however is a generalization of known state space descriptions for the takenaka malmquist basis to the more general case this is important for control problems

identification of rational approximate models in h sup spl infin using generalized orthonormal basis robust identification with fir models fails to be successful when the number of coefficients to be estimated becomes large caused by lightly damped modes of the plant or poles very close to the unit circle the paper presents a two stage algorithm to obtain a low order approximate model in frequency domain in a generalized orthonormal basis with guaranteed h sup spl infin error bound for deterministic linear time invariant stable systems the first stage being an l sup spl infin rational approximation and the second nonlinear step being an h sup spl infin rational approximation

objective priors for discrete parameter spaces this article considers the development of objective prior distributions for discrete parameter spaces formal approaches to such development such as the reference prior approach often result in a constant prior for a discrete parameter which is questionable for problems that exhibit certain types of structure to take advantage of structure this article proposes embedding the original problem in a continuous problem that preserves the structure and then using standard reference prior theory to determine the appropriate objective prior four different possibilities for this embedding are explored and applied to a population size model the hypergeometric distribution the multivariate hypergeometric distribution the binomial beta distribution and the binomial distribution the recommended objective priors for the first third and fourth problems are new

estimating a product of means bayesian analysis with reference priors abstract suppose that we observe x n 1 and independently y n 1 and are concerned with inference mainly estimation and confidence statements about the product of means this problem arises most obviously in situations of determining area based on measurements of length and width it also arises in other practical contexts however for instance in gypsy moth studies the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass approximately independent samples can be obtained for each mean see southwood 1978 noninformative prior bayesian approaches to the problem are considered in particular the reference prior approach of bernardo 1979 an appropriate reference prior for the problem is developed and relatively easily implementable formulas for posterior moments e g the posterior mean and variance and credible sets are derived comparisons with alternative noninf

dynamics of structures coupled with elastic media a review of numerical models and methods abstract this paper reviews issues and developments in the field of structure environment interaction problems in which the environment is an elastic body possibly unbounded it covers in particular the fields of soil structure interaction ground borne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium the general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled and the domain decomposition technique classically used for its description is put up extensions for the cases of nonlinear structure and uncertain environment are then discussed finally the ongoing research in the fields of unbounded interface moving interface and multiple interfaces are reviewed and summarized

default priors for bayesian and frequentist inference summary we investigate the choice of default priors for use with likelihood for bayesian and frequentist inference such a prior is a density or relative density that weights an observed likelihood function leading to the elimination of parameters that are not of interest and then a density type assessment for a parameter of interest for independent responses from a continuous model we develop a prior for the full parameter that is closely linked to the original bayes approach and provides an extension of the right invariant measure to general contexts we then develop a modified prior that is targeted on a component parameter of interest and by targeting avoids the marginalization paradoxes of dawid and co workers this modifies jeffreys s prior and provides extensions to the development of welch and peers these two approaches are combined to explore priors for a vector parameter of interest in the presence of a vector nuisance parameter examples are given to illustrate the computation of the priors