geodesic convexity and covariance estimation geodesic convexity is a generalization of classical convexity which guarantees that all local minima of g convex functions are globally optimal we consider g convex functions with positive definite matrix variables and prove that kronecker products and logarithms of determinants are g convex we apply these results to two modern covariance estimation problems robust estimation in scaled gaussian distributions and kronecker structured models maximum likelihood estimation in these settings involves non convex minimizations we show that these problems are in fact g convex this leads to straight forward analysis allows the use of standard optimization methods and paves the road to various extensions via additional g convex regularization

maximum likelihood estimation of a structured covariance matrix with a condition number constraint in this paper we deal with the problem of estimating the disturbance covariance matrix for radar signal processing applications when a limited number of training data is present we determine the maximum likelihood ml estimator of the covariance matrix starting from a set of secondary data assuming a special covariance structure i e the sum of a positive semi definite matrix plus a term proportional to the identity and a condition number upper bound constraint we show that the formulated constrained optimization problem falls within the class of maxdet problems and develop an efficient procedure for its solution in closed form remarkably the computational complexity of the algorithm is of the same order as the eigenvalue decomposition of the sample covariance matrix at the analysis stage we assess the performance of the proposed algorithm in terms of achievable signal to interference plus noise ratio sinr both for a spatial and a doppler processing the results show that interesting sinr improvements with respect to some existing covariance matrix estimation techniques can be achieved

regularized tyler s scatter estimator existence uniqueness and algorithms this paper considers the regularized tyler s scatter estimator for elliptical distributions which has received considerable attention recently various types of shrinkage tyler s estimators have been proposed in the literature and proved work effectively in the large p small n scenario nevertheless the existence and uniqueness properties of the estimators are not thoroughly studied and in certain cases the algorithms may fail to converge in this work we provide a general result that analyzes the sufficient condition for the existence of a family of shrinkage tyler s estimators which quantitatively shows that regularization indeed reduces the number of required samples for estimation and the convergence of the algorithms for the estimators for two specific shrinkage tyler s estimators we also proved that the condition is necessary and the estimator is unique finally we show that the two estimators are actually equivalent numerical algorithms are also derived based on the majorization minimization framework under which the convergence is analyzed systematically

an asymptotic analysis of composite beams with kinematically corrected end effects a finite element based beam analysis for anisotropic beams with arbitrary shaped cross sections is developed with the aid of a formal asymptotic expansion method from the equilibrium equations of the linear three dimensional 3d elasticity a set of the microscopic 2d and macroscopic 1d equations are systematically derived by introducing the virtual work concept displacements at each order are split into two parts such as fundamental and warping solutions first we seek the warping solutions via the microscopic 2d cross sectional analyses that will be smeared into the macroscopic 1d beam equations the variations of fundamental solutions enable us to formulate the macroscopic 1d beam problems by introducing the orthogonality of asymptotic displacements to six beam fundamental solutions the end effects of a clamped boundary are kinematically corrected without applying the sophisticated decay analysis method the boundary conditions obtained herein are applied to composite beams with solid and thin walled cross sections in order to demonstrate the efficiency and accuracy of the formal asymptotic method based beam analysis famba presented in this paper the numerical results are compared to those reported in literature as well as 3d fem solutions

group symmetric robust covariance estimation in this paper we consider tyler s robust covariance rm m estimator under group symmetry constraints we assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group referred to as group symmetry examples of group symmetric structures include circulant perhermitian and proper quaternion matrices we introduce a group symmetric version of tyler s estimator styler and provide an iterative fixed point algorithm to compute it the classical results claim that at least n p 1 sample points in general position are necessary to ensure the existence and uniqueness of tyler s estimator where p is the ambient dimension we show that the styler requires significantly less samples in some groups even two samples are enough to guarantee its existence and uniqueness in addition in the case of elliptical populations we provide high probability bounds on the error of the styler these too quantify the advantage of exploiting the symmetry structure finally these theoretical results are supported by numerical simulations

computations and evaluations of higher order theories for free vibration analysis of beams abstract this paper deals with higher order theories for the analysis of free vibration of beam structures refined theories are implemented by the application of the unified formulation by the first author which allows one to introduce any order expansions of the displacement unknowns over the beam sections the selection of the most appropriate theory is made by using a so called axiomatic asymptotic approach which permits one to retain only those terms of the displacement expansion which have been established to be significant with respect to an assigned control parameter the finite element method is used to provide numerical solutions various beam sections as well as boundary conditions are considered depending on the vibration modes bending torsion etc quite different theories are selected in general the number of the effective terms of the resultant theories is much lower than the full expansion case amount the nature of these terms can differ very much as different beam geometries and boundary conditions are considered it has been concluded that the method proposed appears to be suitable and convenient to establish the most appropriate beam theory for a given problem it leads in fact to the cheapest computational model for a given accuracy

classical refined and component wise analysis of reinforced shell structures this paper compares early and very recent approaches to the static analysis of reinforced shell wing structures early approaches were those based on the pure semimonocoque theory along with the beam assumptions of the euler bernoulli and timoshenko type the recent approaches are based on a hierarchical one dimensional formulation these are obtained by adopting various polynomial expansions of the displacement field above the cross section of the structure according to the unified formulation which was recently proposed by the first author two classes were developed in the unified formulation framework in the first class taylor expansion models were developed by exploiting n order taylor like polynomials classical beam theories euler bernoulli and timoshenko were obtained as special cases of taylor expansion in the second class lagrange expansion models were built by means of four and nine point lagrange type polynomials over the cross section of the wing the component wise approach was obtai

covariance subspace and intrinsic crame spl acute r rao bounds crame spl acute r rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists the frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail the set of subspaces called the grassmann manifold and the set of covariance positive definite hermitian matrices have no fixed coordinate system associated with them and do not possess a vector space structure both of which are required for deriving classical crame spl acute r rao bounds intrinsic versions of the crame spl acute r rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators in the example of covariance matrix estimation closed form expressions for both the intrinsic and flat bounds are derived and compared with the root mean square error rmse of the sample covariance matrix scm estimator for varying sample support k the accuracy bound on unbiased covariance matrix estimators is shown to be about 10 log 10 n k sup 1 2 db where n is the matrix order remarkably it is shown that from an intrinsic perspective the scm is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice the rmse of the standard method of estimating subspaces using the singular value decomposition svd is compared with the intrinsic subspace crame spl acute r rao bound derived in closed form by varying both the signal to noise ratio snr of the unknown p dimensional subspace and the sample support in the simplest case the crame spl acute r rao bound on subspace estimation accuracy is shown to be about p n p sup 1 2 1 2 snr sup 1 2 rad for p dimensional subspaces it is seen that the svd based method yields accuracies very close to the crame spl acute r rao bound establishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace the analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation

maximum likelihood estimation of compound gaussian clutter and target parameters compound gaussian models are used in radar signal processing to describe heavy tailed clutter distributions the important problems in compound gaussian clutter modeling are choosing the texture distribution and estimating its parameters many texture distributions have been studied and their parameters are typically estimated using statistically suboptimal approaches we develop maximum likelihood ml methods for jointly estimating the target and clutter parameters in compound gaussian clutter using radar array measurements in particular we estimate i the complex target amplitudes ii a spatial and temporal covariance matrix of the speckle component and iii texture distribution parameters parameter expanded expectation maximization px em algorithms are developed to compute the ml estimates of the unknown parameters we also derived the cramer rao bounds crbs and related bounds for these parameters we first derive general crb expressions under an arbitrary texture model then simplify them for specific texture distributions we consider the widely used gamma texture model and propose an inverse gamma texture model leading to a complex multivariate t clutter distribution and closed form expressions of the crb we study the performance of the proposed methods via numerical simulations

on locking free shear deformable beam finite elements abstract this is an elementary exposition of beam finite element models an overview of existing and new displacement finite element models of the euler bernoulli and shear deformation beam theories is presented a locking free finite element model using the form of the exact solution of the timoshenko beam theory is developed the element contains the euler bernoulli beam element as a special case and yields exact nodal values for the generalized displacements for constant material and geometric properties of beams the procedure is extended to develop a locking free finite element model of the simplified third order theory of beams the element contains the euler bernoulli and the locking free timoshenko beam finite elements as special cases the element stiffness matrix of the unified beam bending element is 4 4 and yet has the superconvergence character the advantage of this element is that it can be used to obtain exact nodal displacements and forces from the equilibrium equations of the element in a frame structural analysis using one finite element per a structural member this cannot be achieved with the traditional equal interpolation reduced integration element because the element exhibits shear locking for thin members unless two or more elements per a structural member are used

relative stability for ascending and positively homogeneous operators on banach spaces abstract for a closed convex cone k in a real banach space with a norm increasing on k and a selfmapping t of k which is continuous positively homogeneous and ascending it is shown that the nonlinear eigenvalue problem tx x has a unique solution x k 0 up to a scalar r which is relatively stable in the sense that for a suitable function c of k into r formula moreover an estimate for the speed of convergence is given

performance analysis of tyler s covariance estimator this paper analyzes the performance of tyler s m estimator of the scatter matrix in elliptical populations we focus on the non asymptotic setting and derive estimation error bounds depending on the number of samples n and the dimension p we show that under mild conditions the squared frobenius norm of the error of the inverse estimator decays like p 2 n with high probability

a distribution free m estimator of multivariate scatter the existence and uniqueness of a limiting form of a huber type m estimator of multivariate scatter is established under certain conditions on the observed sample these conditions hold with probability one when sampling randomly from a continuous multivariate distribution the existence of the estimator is proven by showing that it is the limiting point of a specific algorithm hence the proof is constructive for continuous populations the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal an important property of the estimator is that its asymptotic distribution is distribution free with respect to the class of continuous elliptically distributed populations this distribution free property also holds for the finite sample size distribution when the location parameter is known in addition the estimator is the most robust estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

concave perron frobenius theory and applications many important results of classical perron frobenius theory can be extended from linear selfmappings of the standard cone in finite dimensional real space to concave selfmappings of this cone this is in particular true for minima of linear mappings albeit the spectrum of these special concave mappings is more intricate than that for linear mappings as classical perron frobenius theory has numerous applications there are many new applications for its concave extension 1991 mathematical subject classification primary 39a11 15a48 secondary 47h07 47h12

exact maximum likelihood estimates for sirv covariance matrix existence and algorithm analysis in this paper we investigate the existence and the algorithm analysis of an adaptive scheme that has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non gaussian noise this latter has been modeled by spherically invariant random vector sirv which is the product c of the square root of a positive unknown random variable tau and an independent gaussian vector x c radic tau x a similar line of work was undertaken in the context of compound gaussian noise and this paper extends the previous results in the case of sirv modeled noise more precisely the fixed point estimate to be studied verifies a nonlinear algebraic equation e x f x the aim of this paper is twofold first we prove that e admits a unique solution x secondly we show that the corresponding iterative algorithm xn 1 f xn converges to x for every admissible initial condition

conic geometric optimization on the manifold of positive definite matrices we develop geometric optimization on the manifold of hermitian positive definite hpd matrices in particular we consider optimizing two types of cost functions i geodesically convex g convex and ii log nonexpansive ln g convex functions are nonconvex in the usual euclidean sense but convex along the manifold and thus allow global optimization ln functions may fail to be even g convex but still remain globally optimizable due to their special structure we develop theoretical tools to recognize and generate g convex functions as well as cone theoretic fixed point optimization algorithms we illustrate our techniques by applying them to maximum likelihood parameter estimation for elliptically contoured distributions a rich class that substantially generalizes the multivariate normal distribution we compare our fixed point algorithms with sophisticated manifold optimization methods and obtain notable speedups

shrinkage algorithms for mmse covariance estimation we address covariance estimation in the sense of minimum mean squared error mmse when the samples are gaussian distributed specifically we consider shrinkage methods which are suitable for high dimensional problems with a small number of samples large p small n first we improve on the ledoit wolf lw method by conditioning on a sufficient statistic by the rao blackwell theorem this yields a new estimator called rblw whose mean squared error dominates that of lw for gaussian variables second to further reduce the estimation error we propose an iterative approach which approximates the clairvoyant shrinkage estimator convergence of this iterative method is established and a closed form expression for the limit is determined which is referred to as the oracle approximating shrinkage oas estimator both rblw and oas estimators have simple expressions and are easily implemented although the two methods are developed from different perspectives their structure is identical up to specified constants the rblw estimator provably dominates the lw method for gaussian samples numerical simulations demonstrate that the oas approach can perform even better than rblw especially when n is much less than p we also demonstrate the performance of these techniques in the context of adaptive beamforming

the rate of total entropy generation for riemann problems abstract under fairly broad conditions we establish a lower bound for the rate of total entropy generation associated with a similarity solution of a riemann problem for a hyperbolic system of conservation laws for weak riemann problems under more restrictive assumptions we show that dafermos entropy condition is equivalent to the classical condition

low complexity algorithms for low rank clutter parameters estimation in radar systems this paper addresses the problem of the clutter subspace projector estimation in the context of a disturbance composed of a low rank heterogeneous compound gaussian clutter and white gaussian noise in such a context adaptive processing based on an estimated orthogonal projector onto the clutter subspace instead of an estimated covariance matrix requires less samples than classical methods the clutter subspace estimate is usually derived from the eigenvalue decomposition of a covariance matrix estimate however it has been previously shown that a direct maximum likelihood estimator of the clutter subspace projector can be obtained for the considered context in this paper we derive two algorithms based on the block majorization minimization framework to reach this estimator these algorithms are shown to be computationally faster than the state of the art with guaranteed convergence finally the performance of the related estimators is illustrated on realistic space time adaptive processing for airborne radar simulations

regularized robust estimation of mean and covariance matrix under heavy tailed distributions in this paper the joint mean covariance estimation problem is considered under the scenario that the number of samples is small relative to the problem dimension the samples are assumed drawn independently from a heavy tailed distribution of the elliptical family which can model scenarios where the commonly adopted gaussian assumption is violated either because of the data generating process or the contamination of outliers under the assumption that prior knowledge of the mean and covariance matrix is available we propose a regularized estimator defined as the minimizer of a penalized loss function which combines the prior information and the information provided by the samples the loss function is chosen to be the negative log likelihood function of the cauchy distribution as a conservative representative of heavy tailed distributions and the penalty term is constructed with the prior being its global minimizer the resulting regularized estimator shrinks the mean and the covariance matrix to the prior target the existence and uniqueness of the estimator for finite samples are established under certain regularity conditions numerical algorithms are derived for the estimator based on the majorization minimization framework with guaranteed convergence and simulation results demonstrate that the proposed estimator achieves better estimation accuracy compared to the benchmark estimators

free vibration analysis of civil engineering structures by component wise models abstract higher order beam models are used in this paper to carry out free vibration analysis of civil engineering structures refined kinematic fields are developed using the carrera unified formulation cuf which allows for the implementation of any order theory without the need for ad hoc formulations the principle of virtual displacements in conjunction with the finite element method fem is used to formulate stiffness and mass matrices in terms of fundamental nuclei the nuclei depend neither on the adopted class of beam theory nor on the fem approximation along the beam axis this paper focuses on a particular class of cuf models that makes use of lagrange polynomials to discretize cross sectional displacement variables this class of models are referred to as component wise cw in recent works according to the cw approach each structural component e g columns walls frame members and floors can be modeled by means of the same 1d formulation a number of typical civil engineering structures e g simple beams arches truss structures and complete industrial and civil buildings are analyzed and cw results are compared to classical beam theories euler bernoulli and timoshenko refined beam models based on taylor like expansions of the displacements on the cross section and classical solid shell fem solutions from the commercial code msc nastran the results highlight the enhanced capabilities of the proposed formulation it is in fact demonstrated that cw models are able to replicate 3d solid results with very low computational efforts

validation of the variational asymptotic beam sectional analysis the computer program vabs variational asymptotic beam section analysis uses the variational asymptotic method to split a three dimensional nonlinear elasticity problem into a twodimensional linear cross sectional analysis and a one dimensional nonlinear beam problem this is accomplished by taking advantage of certain small parameters inherent to beam like structures vabs is able to calculate the one dimensional cross sectional stiffness constants with transverse shear and vlasov refinements for initially twisted and curved beams with arbitrary geometry and material properties several validation cases are presented first an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method and the solution is shown to be identical to that obtained from the theory of elasticity the shear center locations calculated by vabs for various cross sections agree well with those obtained from common engineering analyses comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model for numerical validation values of the one dimensional variables are extracted from an abaqus model and compared with results from a one dimensional beam analysis using cross sectional constants from vabs furthermore point wise three dimensional stress and strain fields are recovered using vabs and the correlation with the three dimensional results from abaqus is excellent finally classical theory is shown to be insufficient for general purpose beam modeling appropriate refined theories are recommended for some classes of problems

robust covariance matrix estimation in heterogeneous low rank context this paper addresses the problem of robust covariance matrix cm estimation in the context of a disturbance composed of a low rank lr heterogeneous clutter plus an additive white gaussian noise the lr clutter is modeled by a spherically invariant random vector with assumed high clutter to noise ratio in such a context adaptive process should require less training samples than classical methods to reach equivalent performance as in a full rank clutter configuration the main issue is that classical robust estimators of the cm cannot be computed in the undersampled configuration to overcome this issue the current approach is based on regularization methods nevertheless most of these solutions are enforcing the estimate to be well conditioned while in our context it should be lr structured this paper therefore addresses this issue and derives an algorithm to compute the maximum likelihood estimator of the cm for the considered disturbance model several relaxations and robust generalizations of the result are discussed performance is finally illustrated on numerical simulations and on a space time adaptive processing for airborne radar application

multivariate generalized gaussian distribution convexity and graphical models we consider covariance estimation in the multivariate generalized gaussian distribution mggd and elliptically symmetric es distribution the maximum likelihood optimization associated with this problem is non convex yet it has been proved that its global solution can be often computed via simple fixed point iterations our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions our second contribution is a generalized framework for structured covariance estimation under sparsity constraints we show that the optimizations can be formulated as convex minimization as long the mggd shape parameter is larger than half and the sparsity pattern is chordal these include for example maximum likelihood estimation of banded inverse covariances in multivariate laplace distributions which are associated with time varying autoregressive processes

recursive estimation of the covariance matrix of a compound gaussian process and its application to adaptive cfar detection adaptive detection of signals embedded in gaussian or non gaussian noise is a problem of primary concern among radar engineers we propose a recursive algorithm to estimate the structure of the covariance matrix of either a set of gaussian vectors that share the spectral properties up to a multiplicative factor or a set of spherically invariant random vectors sirvs with the same covariance matrix and possibly correlated texture components we also assess the performance of an adaptive implementation of the normalized matched filter nmf relying on the newly introduced estimate in the presence of compound gaussian clutter dominated disturbance in particular it is shown that a proper initialization of the recursive procedure leads to an adaptive nmf with the constant false alarm rate cfar property and that it is very effective to operate in heterogeneous environments of relevant practical interest

shear stresses in prismatic beams with arbitrary cross sections in this paper the approximate computation of shear stresses in prismatic beams due to saint venant torsion and bending using the finite element method is investigated the shape of the considered cross sections may be arbitrary furthermore the basic co ordinate system lies arbitrarily to the centroid and not necessarily in principal directions for numerical reasons dirichlet boundary conditions of the flexure problem are transformed into neumann boundary conditions introducing a conjugate stress function based on the weak formulation of the boundary value problem isoparametric finite elements are formulated the developed procedure yields the relevant warping and torsion constants copyright 1999 john wiley sons ltd

interpolation and extrapolation of toeplitz matrices via optimal mass transport in this work we propose a novel method for quantifying distances between toeplitz structured covariance matrices by exploiting the spectral representation of toeplitz matrices the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain this may then be interpreted in the covariance domain suggesting a natural way of interpolating and extrapolating toeplitz matrices such that the positive semidefiniteness and the toeplitz structure of these matrices are preserved the proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise finally we illustrate how this approach can be used for several applications in signal processing in particular we consider interpolation and extrapolation of toeplitz matrices as well as clustering problems and tracking of slowly varying stochastic processes

knowledge aided covariance matrix estimation and adaptive detection in compound gaussian noise let x be a signal to be sparsely decomposed over a redundant dictionary a i e a sparse coefficient vector s has to be found such that x as it is known that this problem is inherently unstable against

a new rectangular beam theory abstract a new theory for beams of rectangular cross section which includes warping of the cross sections is presented in the present work by satisfying the shear free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required it is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in timoshenko beam theory provided that for the timoshenko equation the shear coefficient is taken to be 5 6 this value lies within the range of values 0 822 0 870 appearing in the literature for the beam of rectangular cross section results for two typical static examples are given for both the new theory and timoshenko beam theory these results are compared with the solutions of the comparable problems in the linear theory of elasticity for the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while timoshenko beam theory underestimates the shear correction term by 20 for the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution

a general theory for the accurate stress analysis of homogeneous and laminated composite beams abstract this paper has a twofold purpose initially it presents a general four degrees of freedom beam theory g4dofbt which takes into consideration the effects of both transverse shear and normal deformation on the basis of this new theory it then proposes a method suitable for the accurate stress analysis of either homogeneous or laminated composite beams subjected to arbitrary edge boundary conditions the new beam theory involves two general shape functions each of which is associated with one of the two unknown displacement components upon assigning simple particular forms to these shape functions most of the well known classical and variationally consistent refined beam models may be obtained as particular cases the new method for the accurate stress analysis of beam type structures is based on the specification of a new pair of shape functions these are obtained by introducing the stress distributions caused by the assumed g4dofbt displacement field into the appropriate equations of three dimensional elasticity which are subsequently solved for simply supported edges this is considered to provide an excellent choice of both shape functions as the method then yields the exact elasticity solution presented by pagano pagano n j 1969 exact solution for composite laminates in cylindrical bending j comp mat 3 398 411 for the cylindrical bending problem of simply supported infinite strips two particular examples are considered to show the potential of the present analysis these are dealing with stress analysis of homogeneous or laminated composite beams having one edge rigidly clamped and the other edge either guided or free of external tractions

diffuse multipath exploitation for adaptive radar detection we deal with the problem of detecting point like targets in diffuse multipath environments modeling the target echo as the superposition of a deterministic signal with an unknown scaling factor due to the direct path plus a zero mean complex circular symmetric gaussian random vector with an unknown covariance matrix accounting for the echoes from the glistening surface we devise a constrained generalized likelihood ratio test glrt for the resulting hypothesis testing problem enforcing the primary data covariance matrix due to both interference and multipath echoes to belong to a neighborhood of the secondary data sample covariance matrix remarkably the proposed decision scheme ensures the desirable constant false alarm rate cfar property with respect to the unknown parameters of the interference the performance assessment conducted on simulated data in terms of detection probability also in comparison with existing solutions highlights the effectiveness of the new approach to cope with diffuse multipath phenomena

an alternative to em for gaussian mixture models batch and stochastic riemannian optimization we consider maximum likelihood estimation for gaussian mixture models gmm s this task is almost invariably solved in theory and practice via the expectation maximization em algorithm em owes its success to various factors of which is its ability to fulfill positive definiteness constraints in closed form is of key importance we propose an alternative to em grounded in the riemannian geometry of positive definite matrices using which we cast gmm parameter estimation as a riemannian optimization problem surprisingly such an out of the box riemannian formulation completely fails and proves much inferior to em this motivates us to take a closer look at the problem geometry and derive a better formulation that is much more amenable to riemannian optimization we then develop riemannian batch and stochastic gradient algorithms that outperform em often substantially we provide a non asymptotic convergence analysis for our stochastic method which is also the first to our knowledge such global analysis for riemannian stochastic gradient numerous empirical results are included to demonstrate the effectiveness of our methods

statistical analysis of the nonhomogeneity detector for non gaussian interference backgrounds we derive the nonhomogeneity detector nhd for non gaussian interference scenarios and present a statistical analysis of the method the non gaussian interference scenario is assumed to be modeled by a spherically invariant random process sirp we present a method for selecting representative homogeneous training data based on our statistical analysis of the nhd for finite sample support used in covariance estimation in particular an exact theoretical expression for the nhd test statistic probability density function pdf is derived performance analysis of the nhd is presented using both simulated data and measured data from the multichannel airborne radar measurement mcarm program a performance comparison with existing nhd approaches is also included

refined one dimensional formulations for laminated structure analysis this paper proposes one dimensional formulations based on hierarchical expansions of the unknown displacement variables for the analysis of multilayered structures made of anisotropic composite layers the hierarchical technique shows variable kinematic properties and it is based on the carrera unified formulation two different classes of refined theories are proposed the first expands the unknown variables in terms of power polynomials of the cross sectional coordinates it consists of a taylor like expansion the second class of onedimensional theories uses lagrange polynomials lagrange expansion and subdomain discretizations of the cross section and it leads to only pure displacements as the unknown variables taylor like expansion is used to develop equivalent single layer formulations and lagrange expansion is used to construct both equivalent single layer and layerwise descriptions the finite element method is employed to develop numerical applications using the carrera unified formulation finite element matrices are obtained in terms of a few fundamental nuclei that are formally independent of all the considered one dimensional formulations a number of numerical examples are given concerning on beams plates and more complex structures comparisons with results from plate and solid models are provided the following has been concluded 1 the proposed formulation represents a reliable compact and accurate method to develop refined one dimensional models 2 the present one dimensional models are very effective at detecting both global and local responses of composite structures 3 shell and solidlike results are obtained with a significant reduction in the computational costs

signal detection in complex structured para normal noise in this paper we address target detection in correlated non gaussian noise we introduce a new class of multivariate complex valued distributions that allows us to specify flexible non gaussian marginals as well as correlation between the variables while preserving circular symmetry for noise belonging to this class we study the fundamental problem of signal detection under different settings and develop the needed generalized likelihood ratio tests we also consider estimation of the noise parameters and derive the maximum likelihood formulations we evaluate the merit of our approach using numerical simulations on synthetic and real data and clearly demonstrate the advantage of using both correlations and non gaussianity

existence in the large for riemann problems for systems of conservation laws on obtient un theoreme d existence pour le probleme de riemann pour des systemes non lineaires de lois de conservation on utilise l inegalite d entropie pour obtenir une estimation a priori des forces des chocs et des ondes de refraction formant une solution il s ensuit l existence d une telle solution par une application de la theorie du degre de dimension finie

on estimation of covariance matrices with kronecker product structure the estimation of signal covariance matrices is a crucial part of many signal processing algorithms in some applications the structure of the problem suggests that the underlying true covariance matrix is the kronecker product of two valid covariance matrices examples of such problems are channel modeling for multiple input multiple output mimo communications and signal modeling of eeg data in applications it may also be that the kronecker factors in turn can be assumed to possess additional linear structure the maximum likelihood ml method for the associated estimation problem has been proposed previously it is asymptotically efficient but has the drawback of requiring an iterative search for the maximum of the likelihood function two methods that are fast and noniterative are proposed in this paper both methods are shown to be asymptotically efficient the first method is a noniterative variant of a well known alternating maximization technique for the likelihood function it performs on par with ml in simulations but has the drawback of not allowing for extra structure in addition to the kronecker structure the second method is based on covariance matching principles and does not suffer from this drawback however while the large sample performance is the same it performs somewhat worse than the first estimator in small samples in addition the cramer rao lower bound for the problem is derived in a compact form the problem of estimating the kronecker factors and the problem of detecting if the kronecker structure is a good model for the covariance matrix of a set of samples are related therefore the problem of detecting the dimensions of the kronecker factors based on the minimum values of the criterion functions corresponding to the two proposed estimation methods is also treated in this work

toward fast transform learning this paper introduces a new dictionary learning strategy based on atoms obtained by translating the composition of k k convolutions with s s sparse kernels of known support the dictionary update step associated with this strategy is a non convex optimization problem we propose a practical formulation of this problem and introduce a gauss seidel type algorithm referred to as alternative least square algorithm for its resolution the search space of the proposed algorithm is of dimension ks ks which is typically smaller than the size of the target atom and much smaller than the size of the image moreover the complexity of this algorithm is linear with respect to the image size allowing larger atoms to be learned as opposed to small patches the conducted experiments show that we are able to accurately approximate atoms such as wavelets curvelets sinc functions or cosines for large values of k the proposed experiments also indicate that the algorithm generally converges to a global minimum for large values of k k and s s

a higher order lq decomposition for separable covariance models abstract we develop a higher order generalization of the lq decomposition and show that this decomposition plays an important role in likelihood based estimation and testing for separable or kronecker structured covariance models such as the multilinear normal model this role is analogous to that of the lq decomposition in likelihood inference for the multivariate normal model additionally this higher order lq decomposition can be used to construct an alternative version of the popular higher order singular value decomposition for tensor valued data we also develop a novel generalization of the polar decomposition to tensor valued data

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

covariance structure maximum likelihood estimates in compound gaussian noise existence and algorithm analysis recently a new adaptive scheme conte 1995 gini 1997 has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non gaussian noise this latter has been modeled by compound gaussian noise which is the product c of the square root of a positive unknown variable tau deterministic or random and an independent gaussian vector x c radictaux because of the implicit algebraic structure of the equation to solve we called the corresponding solution the fixed point fp estimate when tau is assumed deterministic and unknown the fp is the exact maximum likelihood ml estimate of the noise covariance structure while when tau is a positive random variable the fp is an approximate maximum likelihood aml this estimate has been already used for its excellent statistical properties without proofs of its existence and uniqueness the major contribution of this paper is to fill these gaps our derivation is based on some likelihood functions general properties like homogeneity and can be easily adapted to other recursive contexts moreover the corresponding iterative algorithm used for the fp estimate practical determination is also analyzed and we show the convergence of this recursive scheme ensured whatever the initialization

uniqueness failure for entropy solutions of hyperbolic systems of conservation laws we consider a nonlinear system of conservation laws which is strictly hyperbolic genuinely nonlinear in the large equipped with a convex entropy function and global riemann invariants nevertheless for such a system for dimension five it is shown that uniqueness of the similarity solution of a riemann problem satisfying the entropy condition can fail

unified framework to regularized covariance estimation in scaled gaussian models we consider regularized covariance estimation in scaled gaussian settings e g elliptical distributions compound gaussian processes and spherically invariant random vectors asymptotically in the number of samples the classical maximum likelihood ml estimate is optimal under different criteria and can be efficiently computed even though the optimization is nonconvex we propose a unified framework for regularizing this estimate in order to improve its finite sample performance our approach is based on the discovery of hidden convexity within the ml objective we begin by restricting the attention to diagonal covariance matrices using a simple change of variables we transform the problem into a convex optimization that can be efficiently solved we then extend this idea to nondiagonal matrices using convexity on the manifold of positive definite matrices we regularize the problem using appropriately convex penalties these allow for shrinkage towards the identity matrix shrinkage towards a diagonal matrix shrinkage towards a given positive definite matrix and regularization of the condition number we demonstrate the advantages of these estimators using numerical simulations

on convergence of kronecker graphical lasso algorithms this paper studies iteration convergence of kronecker graphical lasso kglasso algorithms for estimating the covariance of an i i d gaussian random sample under a sparse kronecker product covariance model and mse convergence rates the kglasso model originally called the transposable regularized covariance model by allen transposable regularized covariance models with an application to missing data imputation ann appl statist vol 4 no 2 pp 764 790 2010 implements a pair of ell 1 penalties on each kronecker factor to enforce sparsity in the covariance estimator the kglasso algorithm generalizes glasso introduced by yuan and lin model selection and estimation in the gaussian graphical model biometrika vol 94 pp 19 35 2007 and banerjee model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data j mach learn res vol 9 pp 485 516 mar 2008 to estimate covariances having kronecker product form it also generalizes the unpenalized ml flip flop ff algorithm of dutilleul the mle algorithm for the matrix normal distribution j statist comput simul vol 64 pp 105 123 1999 and werner on estimation of covariance matrices with kronecker product structure ieee trans signal process vol 56 no 2 pp 478 491 feb 2008 to estimation of sparse kronecker factors we establish that the kglasso iterates converge pointwise to a local maximum of the penalized likelihood function we derive high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity our results establish that kglasso has significantly faster asymptotic convergence than glasso and ff simulations are presented that validate the results of our analysis for example for a sparse 10 thinspace 000 times 10 thinspace 000 covariance matrix equal to the kronecker product of two 100 times 100 matrices the root mean squared error of the inverse covariance estimate using ff is 2 times larger than that obtainable using kglasso for sample size of n 100

cramer rao lower bounds on covariance matrix estimation for complex elliptically symmetric distributions this paper introduces the cramer rao lower bounds crlbs for the scatter matrix of complex elliptically symmetric distributions and compares them to the performance of the constrained ml estimators in the particular cases of complex gaussian generalized gaussian gg and t distributed observation vectors numerical results confirm the goodness of the ml estimators and the advantage of taking into proper account a constraint on the matrix trace for small data size the work is completed with the comparison with the performance of tyler s matrix estimator that shows a very robust behavior in almost all the analyzed cases and with the crlbs for the complex angular elliptical distributions whose tyler s estimator is the ml one

robust shrinkage estimation of high dimensional covariance matrices we address high dimensional covariance estimation for elliptical distributed samples which are also known as spherically invariant random vectors sirv or compound gaussian processes specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples large p small n we start from a classical robust covariance estimator tyler 1987 which is distribution free within the family of elliptical distribution but inapplicable when n p using a shrinkage coefficient we regularize tyler s fixed point iterations we prove that for all n and p the proposed fixed point iterations converge to a unique limit regardless of the initial condition next we propose a simple closed form and data dependent choice for the shrinkage coefficient which is based on a minimum mean squared error framework simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy tailed samples finally as a real world application we demonstrate the performance of the proposed technique in the context of activity intrusion detection using a wireless sensor network

polarimetric detection in compound gaussian clutter with kronecker structured covariance matrix in this paper we consider polarimetric adaptive detection in compound gaussian clutter whose covariance matrix cm has a kronecker structure we derive the cramer rao bound of a kronecker structured cm estimate and analyze the constant false alarm rate property of the adaptive subspace matched filter detector that uses kronecker structured estimates we provide a general expression for the average signal to clutter ratio loss scrl as a function of the mean square error of the covariance estimate the aforementioned expression is helpful in determining how many samples are required in order to achieve a desired average scrl level in practical scenarios based on that expression we show that the required sample size can be effectively reduced by exploiting the kronecker structure of the clutter cm we also derive the asymptotic detection performance of the adaptive subspace matched filter the analysis of scrl and detection performance can be extended to more general scenarios especially when the maximum likelihood estimate of the structured cm involves solving fixed point equations numerical simulations validate the merits of the proposed methods

a covariance shaping framework for linear multiuser detection a new class of linear multiuser receivers referred to as the covariance shaping multiuser csmu receiver is proposed for suppression of interference in multiuser wireless communication systems this class of receivers is based on the recently proposed covariance shaping least squares estimator and is designed to minimize the total variance of the weighted error between the receiver output and the observed signal subject to the constraint that the covariance of the noise component in the receiver output is proportional to a given covariance matrix so that we control the dynamic range and spectral shape of the output noise some of the well known linear multiuser receivers are shown to be special cases of the csmu receiver this allows us to interpret these receivers as the receivers that minimize the total error variance in the observations among all linear receivers with the same output noise covariance and to analyze their performance in a unified way we derive exact and approximate expressions for the probability of bit error as well as the asymptotic signal to interference noise ratio in the large system limit we also characterize the spectral efficiency versus energy per information bit of the csmu receiver in the wideband regime finally we consider a special case of the csmu receiver equivalent to a mismatched minimum mean squared error mmse receiver in which the channel signal to noise ratio snr is not known precisely using our general performance analysis results we characterize the performance of the mismatched mmse receiver we then treat the case in which the snr is known to lie in a given uncertainty range and develop a robust mismatched mmse receiver whose performance is very close to that of the mmse receiver over the entire uncertainty range

refined theory of composite beams the role of short wavelength extrapolation the present paper presents an asymptotically correct beam theory with nonclassical sectional degrees of freedom the basis for the theory is the variational asymptotical method a mathematical technique by which the three dimensional analysis of composite beam deformation can be split into a linear two dimensional cross sectional analysis and a nonlinear one dimensional beam analysis the elastic constants used in the beam analysis are obtained from the cross sectional analysis which also yields approximate closed form expressions for three dimensional distributions of displacement strain and stress such theories are known to be valid when a characteristic dimension of the cross section is small relative to the wavelength of the deformation however asymptotically correct refined theories may differ according to how they are extrapolated into the short wavelength regime thus there is no unique asymptotically correct refined theory of higher order than classical euler bernoulli like theory different short wavelength extrapolations can be obtained by changing the meaning of the theory s one dimensional variables numerical results for the stiffness constants of a refined beam theory and for deformations from the corresponding one dimensional theory are presented it is shown that a theory can be asymptotically correct and still have non positive definite strain energy density which is completely inappropriate mathematically and physically a refined beam theory which appropriately possesses a positive definite strain energy density and agrees quite well with experimental results is constructed by using a certain short wavelength extrapolation

deformation of short composite beam using refined theories high order flexural theories for short laminated composite beams subjected to mechanical and thermal loading are presented the formulation allows for warping of the cross section of the beam and eliminates the need for using arbitrary shear correction coefficients as in other theories based on higher order shear deformation theories the governing equations are obtained using the principle of virtual work pvw the justification for use of higher order shear deformation theories is established for short and composite beams where cross sectional warping is predominant

a refined one dimensional rotordynamics model with three dimensional capabilities abstract this paper evaluates the vibration characteristics of various rotating structures the present methodology exploits the one dimensional carrera unified formulation 1d cuf which enables one to go beyond the kinematic assumptions of classical beam theories according to the component wise cw approach lagrange like polynomial expansions le are here adopted to develop the refined displacement theories the le elements make it possible to model each structural component of the rotor with an arbitrary degree of accuracy using either different displacement theories or localized mesh refinements hamilton s principle is used to derive the governing equations which are solved by the finite element method the cuf one dimensional theory includes all the effects due to rotation namely the coriolis term spin softening and geometrical stiffening the numerical simulations have been performed considering a thin ring discs and bladed deformable shafts the effects of the number and the position of the blades on the dynamic stability of the rotor have been evaluated the results have been compared when possible with the 2d and 3d solutions that are available in the literature cuf models appear very practical to investigate the dynamics of complex rotating structures since they provide 2d and quasi 3d results while preserving the computational effectiveness of one dimensional solutions

robust estimation of structured covariance matrix for heavy tailed elliptical distributions this paper considers the problem of robustly estimating a structured covariance matrix with an elliptical underlying distribution with a known mean in applications where the covariance matrix naturally possesses a certain structure taking the prior structure information into account in the estimation procedure is beneficial to improving the estimation accuracy we propose incorporating the prior structure information into tyler s m estimator and formulating the problem as minimizing the cost function of tyler s estimator under the prior structural constraint first the estimation under a general convex structural constraint is introduced with an efficient algorithm for finding the estimator derived based on the majorization minimization mm algorithm framework then the algorithm is tailored to several special structures that enjoy a wide range of applications in signal processing related fields namely sum of rank one matrices toeplitz and banded toeplitz structure in addition two types of non convex structures i e the kronecker structure and the spiked covariance structure are also discussed where it is shown that simple algorithms can be derived under the guidelines of mm the algorithms are guaranteed to converge to a stationary point of the problems furthermore if the constraint set is geodesically convex such as the kronecker structure set then the algorithm converges to a global minimum numerical results show that the proposed estimator achieves a smaller estimation error than the benchmark estimators at a lower computational cost

dynamic response of thin walled structures by variable kinematic one dimensional models abstract this paper investigates the accuracy capabilities of using variable kinematic modeling in compact and thin walled beam like structures with dynamic loadings carrera unified formulation cuf is employed to introduce refined one dimensional 1d models with a variable order of expansion for the displacement unknowns over the beam cross section classical euler bernoulli and timoshenko beam theories are obtained as particular cases of these variable kinematic models while a higher order expansion permits the detection of in plane cross section deformation since it leads to shell like solutions finite element fe method is used to provide numerical results and the newmark method is implemented as a time integration scheme some assessments with closed form solutions are discussed and comparisons with shell type results obtained with commercial fe software are made further analyses address both compact and thin walled cross sections in particular the case of a deformable thin walled cylinder loaded by time dependent internal forces is discussed the results clearly show that finite elements which are formulated in the cuf framework do not introduce additional numerical problems with respect to classical beam theories comparisons with elasticity solutions prove that the present 1d cuf model offers an accuracy in analyzing thin walled structures which is typical of shell or three dimensional models with a remarkable reduction in the computational cost required

beam theory for strongly orthotropic materials abstract this paper presents a displacement based model for orthotropic beams under plane linear elasticity based on the only kinematic assumption of transverse inextensibility any given axial and transverse loading as well as boundary conditions at the beam ends are considered the solution is decomposed into the principal and the residual part corresponding to the interior and the boundary problems which are obtained by series expansions of polynomial functions and eigenfunctions respectively it is proved that the proposed one dimensional theory gives both interior and boundary exact two dimensional elasticity solutions for strongly orthotropic materials i e for ratio between shear modulus and axial young modulus approaching zero for isotropic and orthotropic materials the accuracy of the beam model is also analysed and compared with that of other theories in particular the complementary energy error of the interior solution with respect to two dimensional elasticity is evaluated the asymptotic estimate of the characteristic decay length of end effects given in choi and horgan j appl mech asme 44 424 430 1977 by two dimensional analysis is reobtained and the range of validity of boundary solution is discussed the numerical results presented agree very well with exact and finite element solutions even in the neighbourhood of clamped cross sections where the solution is mainly governed by the boundary problem

tyler s covariance matrix estimator in elliptical models with convex structure we address structured covariance estimation in elliptical distributions by assuming that the covariance is a priori known to belong to a given convex set e g the set of toeplitz or banded matrices we consider the general method of moments gmm optimization applied to robust tyler s scatter m estimator subject to these convex constraints unfortunately gmm turns out to be non convex due to the objective instead we propose a new coca estimator a convex relaxation which can be efficiently solved we prove that the relaxation is tight in the unconstrained case for a finite number of samples and in the constrained case asymptotically we then illustrate the advantages of coca in synthetic simulations with structured compound gaussian distributions in these examples coca outperforms competing methods such as tyler s estimator and its projection onto the structure set

regularized m estimators of scatter matrix in this paper a general class of regularized m estimators of scatter matrix are proposed that are suitable also for low or insufficient sample support small n and large p problems the considered class constitutes a natural generalization of m estimators of scatter matrix maronna 1976 and are defined as a solution to a penalized m estimation cost function using the concept of geodesic convexity we prove the existence and uniqueness of the regularized m estimators of scatter and the existence and uniqueness of the solution to the corresponding m estimating equations under general conditions unlike the non regularized m estimators of scatter the regularized estimators are shown to exist for any data configuration an iterative algorithm with proven convergence to the solution of the regularized m estimating equation is also given since the conditions for uniqueness do not include the regularized versions of tyler s m estimator necessary and sufficient conditions for their uniqueness are established separately for the regularized tyler s m estimators we also derive a simple closed form and data dependent solution for choosing the regularization parameter based on shape matrix matching in the mean squared sense finally some simulations studies illustrate the improved accuracy of the proposed regularized m estimators of scatter compared to their non regularized counterparts in low sample support problems an example of radar detection using normalized matched filter nmf illustrate that an adaptive nmf detector based on regularized m estimators are able to maintain accurately the preset cfar level