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

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

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

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

optimal controllers for finite wordlength implementation when a controller is implemented by a digital computer with a d and d a conversion numerical errors can severely affect the performance of the control system there exist realizations of a given controller transfer function exhibiting arbitrarily large effects from computational errors assuming sufficient excitation of the system the problem of designing an optimal controller in the presence of both external disturbances and internal roundoff errors is solved the results reduce to the standard lqg controller when infinite precision computation is used for finite precision however the separation principle does not hold a penalty is also added to the cost function to penalize the sum of the wordlengths used to compute the fractional part of each state variable of the controller this sum can be used to represent the lower bound on computer memory needed for controller synthesis it measures controller complexity and is minimized penalized here

comments on robust fragile or optimal with reply the issue of fragile controllers produced by using popular robust and optimal control synthesis methods was raised by keel bhattacharyya in the above mentioned paper ibid vol 42 1997 the commenter points out that the controller fragility problem described in the paper is 1 well known in the literature 2 due to the poor choice of examples 3 can easily be fixed by changing the optimization criteria and 4 can be fixed by changing the implementation in reply keel bhattacharyya disagree with him on all counts and explain their argument

clutter subspace estimation in low rank heterogeneous noise context this paper addresses the problem of the clutter subspace projector csp estimation in the context of a disturbance composed of a low rank lr heterogeneous clutter modeled here by a spherically invariant random vector sirv plus a white gaussian noise wgn in such context the corresponding lr adaptive filters and detectors require less training vectors than classical methods to reach equivalent performance unlike classical adaptive processes which are based on an estimate of the noise covariance matrix cm the lr processes are based on a csp estimate this csp estimate is usually derived from a singular value decomposition svd of the cm estimate however no maximum likelihood estimator mle of the cm has been derived for the considered disturbance model in this paper we introduce the fixed point equation that mle of the csp satisfies for a disturbance composed of a lr sirv clutter plus a zero mean wgn a recursive algorithm is proposed to compute this solution numerical simulations validate the introduced estimator and illustrate its interest compared to the current state of art

covariance estimation in high dimensions via kronecker product expansions this paper presents a new method for estimating high dimensional covariance matrices the method permuted rank penalized least squares prls is based on a kronecker product series expansion of the true covariance matrix assuming an i i d gaussian random sample we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity for covariance matrices of low separation rank our results establish that prls has significantly faster convergence than the standard sample covariance matrix scm estimator the convergence rate captures a fundamental tradeoff between estimation error and approximation error thus providing a scalable covariance estimation framework in terms of separation rank similar to low rank approximation of covariance matrices the mse convergence rates generalize the high dimensional rates recently obtained for the ml flip flop algorithm for kronecker product covariance estimation we show that a class of block toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank r that ensures a given level of bias simulations are presented to validate the theoretical bounds as a real world application we illustrate the utility of the proposed kronecker covariance estimator for spatio temporal linear least squares prediction of multivariate wind speed measurements

complex elliptically symmetric distributions survey new results and applications complex elliptically symmetric ces distributions have been widely used in various engineering applications for which non gaussian models are needed in this overview circular ces distributions are surveyed some new results are derived and their applications e g in radar and array signal processing are discussed and illustrated with theoretical examples simulations and analysis of real radar data the maximum likelihood ml estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness and for convergence of the iterative fixed point algorithm are established specific ml estimators for several ces distributions that are widely used in the signal processing literature are discussed in depth including the complex t distribution k distribution the generalized gaussian distribution and the closely related angular central gaussian distribution a generalization of ml estimators the m estimators of the scatter matrix are also discussed and asymptotic analysis is provided applications of ces distributions and the adaptive signal processors based on ml and m estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for direction of arrival doa estimation and beamforming furthermore experimental validation of the usefulness of ces distributions for modelling real radar data is given

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

large dimensional behavior of regularized maronna s m estimators of covariance matrices robust estimators of large covariance matrices are considered comprising regularized linear shrinkage modifications of maronna s classical m estimators these estimators provide robustness to outliers while simultaneously being well defined when the number of samples does not exceed the number of variables by applying tools from random matrix theory we characterize the asymptotic performance of such estimators when the numbers of samples and variables grow large together in particular our results show that when outliers are absent many estimators of the regularized maronna type share the same asymptotic performance and for these estimators we present a data driven method for choosing the asymptotically optimal regularization parameter with respect to a quadratic loss robustness in the presence of outliers is then studied in the nonregularized case a large dimensional robustness metric is proposed and explicitly computed for two particular types of estimators exhibiting interesting differences depending on the underlying contamination model the impact of outliers in regularized estimators is then studied with interesting differences with respect to the nonregularized case leading to new practical insights on the choice of particular estimators

balanced realizations via gradient flow techniques abstract the task of finding a class of balanced minimal realizations is shown to be equivalent to finding limiting solutions of certain gradient flow differential equations by viewing such algebraic tasks in the context of calculus they are amenable to analog computational solutions or parallel processing machines perhaps even neural networks the convergence rates of the differential equations is exponential and consequentially convergence is rapid and numerical stability properties are attractive

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

optimal realizations of floating point implemented digital controllers with finite word length considerations the closed loop stability issue of finite precision realizations is investigated for digital controllers implemented in floating point arithmetic unlike the existing methods which only address the effect of the mantissa bits in floating point implementation to the sensitivity of closed loop stability the sensitivity of closed loop stability is analyzed with respect to both the mantissa and exponent bits of floating point implementation a computationally tractable fwl closed loop stability measure is then defined and the method of computing the value of this measure is given the optimal controller realization problem is posed as searching for a floating point realization that maximizes the proposed fwl closed loop stability measure and a numerical optimization technique is adopted to solve for the resulting optimization problem simulation results show that the proposed design procedure yields computationally efficient controller realizations with enhanced fwl closed loop stability performance

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

models of q algebra representations matrix elements of the q oscillator algebra this article continues a study of function space models of irreducible representations of q analogs of lie enveloping algebras motivated by recurrence relations satisfied by q hypergeometric functions here a q analog of the oscillator algebra not a quantum algebra is considered it is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and the corresponding matrix elements of the group operators on these representation spaces are computed this local approach applies to more general families of special functions e g with complex arguments and parameters than does the quantum group approach it is shown that the matrix elements themselves transform irreducibly under the action of the algebra q analogs of a formula are found for the product of two hypergeometric functions 1f1 and the product of a 1f1 and a bessel function they are interpreted here as expansions of the matrix elements of a group operator

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

generalized robust shrinkage estimator and its application to stap detection problem recently in the context of covariance matrix estimation in order to improve as well as to regularize the performance of the tyler s estimator 1 also called the fixed point estimator fpe 2 a shrinkage fixed point estimator has been originally introduced in 3 first this work extends the results of 4 5 by giving the general solution of the shrinkage fixed point algorithm secondly by analyzing this solution called the generalized robust shrinkage estimator we prove that this solution converges to a unique solution when the shrinkage parameter losing factor tends to 0 this solution is exactly the fpe with the trace of its inverse equal to the dimension of the problem this general result allows one to give another interpretation of the fpe and more generally on the maximum likelihood approach for covariance matrix estimation when constraints are added then some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor finally this work is applied to a space time adaptive processing stap detection problem on real stap data

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

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

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

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

hansen lommel orthogonality relations for jackson s q bessel functions abstract two hansen lommel orthogonality relations for jackson s q bessel functions are proved

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

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

addition formulas for q bessel functions the two dimensional euclidean algebra provides an algebraic setting for q bessel functions using a two variable realization of this algebra q analogs of the lommel and graf addition formulas for the q bessel functions are obtained

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

on q analogues of the fourier and hankel transforms for h exton s q analogue of the bessel function going back to w hahn in a special case but different from f h jackson s q bessel functions we derive hansen lommel type orthogonality relations which by a symmetry turn out to be equivalent to orthogonality relations which are q analogues of the hankel integral transform pair these results are implicit in the context of quantum groups in a paper by vaksman and korogodskii as a specialization we get q cosines and q sines which admit q analogues of the fourier cosine and fourier sine transforms

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

optimizing stability bounds of finite precision pid controller structures this paper investigates a recently derived lower bound stability measure for sampled data controller structures subject to finite wordlength fwl constraints the optimal realization of the digital pid controller with fwl considerations is formulated as a nonlinear optimization problem and an efficient strategy based on adaptive simulated annealing asa is adopted to solve this complex optimization problem a numerical example of optimizing the finite precision pid controller structure for a simulated steel rolling mill system is presented to illustrate the effectiveness of the proposed strategy

quantum symmetries of q difference equations a general method is presented to determine the symmetry operators of linear q difference equations it is applied to q difference analogs of the helmoltz heat and wave equations in diverse dimensions the symmetries are found to generate q deformations of classical lie algebras the method of separation of variables is used to obtain solutions which are seen to involve many basic special functions this allows the derivation of various identities and formulas for these q functions

brief paper an efficient controller structure with minimum roundoff noise gain based on a polynomial operator approach a new sparse controller structure is derived which is actually an improved version of the recently proposed structure li g 2004 a polynomial operator based dfiit structure for iir filters ieee transactions on circuits and systems ii 51 147 151 the performance of the proposed structure is analyzed by deriving the corresponding expression of roundoff noise gain and the problem of finding optimized sparse structures is solved efficiently with a genetic algorithm ga a numerical example is given which shows that the newly developed structure can achieve much better performance than some well known structures and particularly outperforms the traditional optimal fully parametrized realization greatly in terms of reducing roundoff noise and implementation complexity

generalised modal realisation as a practical and efficient tool for fwl implementation finite word length fwl effects have been a critical issue in digital filter implementation for almost four decades although some optimisations may be attempted to get an optimal realisation with regards to a particular effect for instance the parametric sensitivity or the round off noise gain the purpose of this article is to propose an effective one i e taking into account all the aspects based on the specialised implicit form a new effective and sparse structure named modal realisation is proposed this realisation meets simultaneously accuracy low sensitivity round off noise gain and overflow risk few and flexible computational efforts with a good readability thanks to sparsity and simplicity no tricky optimisation is required to obtain it as well two numerical examples are included to illustrate the modal realisation s interest

yet another basic analogue of graf s addition formula an identity involving basic bessel functions and al salam chihara polynomials is proved for which we recover graf s addition formula for the bessel function as the base q tends to 1 the corresponding product formula is derived some known identities for jackson s q bessel functions are obtained as limiting cases as special cases we prove identities for q charlier polynomials

joint covariance estimation with mutual linear structure we consider the problem of joint estimation of structured covariance matrices assuming the structure is unknown estimation is achieved using heterogeneous training sets namely given groups of measurements coming from centered populations with different covariances our aim is to determine the mutual structure of these covariance matrices and estimate them supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices we develop a new efficient algorithm discovering the structure and using it to improve the estimation our technique is based on the application of principal component analysis in the matrix space we also derive an upper performance bound of the proposed algorithm in the gaussian scenario and compare it with the cramer rao lower bound numerical simulations are presented to illustrate the performance benefits of the proposed method

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

bounds for a mixture of low rank compound gaussian and white gaussian noises we consider the problem of estimating the parameters of a low rank compound gaussian process in white gaussian noise this situation typically arises in radar applications where clutter is relevantly modeled as compound gaussian with a rank deficient covariance matrix of the speckle using a minimal and unconstrained parametrization of the problem we derive lower bounds for estimation of the parameters describing the covariance matrix first assuming the textures are deterministic the cramer rao bound is derived which enables one to assess the impact of the time varying textures on the estimation performance then considering the textures as random hybrid bounds are derived in addition we derive a lower bound for estimating the projector on the clutter subspace numerical simulations enable one to evaluate the impact of random time varying textures compared to the conventional case of constant texture i e of gaussian subspace signals in gaussian noise

optimum realizations of sampled data controllers for fwl sensitivity minimization the problem of an optimal finite word length state space realization of a digital controller is investigated the closed loop to be considered consists of a continuous time plant a discrete time controller a sampler a zero order hold and an antialiasing filter an effective algorithm is proposed to find the optimal sampled data controller realization minimizing the sensitivity of the closed loop performance with respect to coefficient errors in the state variable matrices of the controller realization in order to get a tractable problem a two step procedure is to be used very fast sampling at a multiple of the sampling frequency followed by blocking or lifting to obtain a single rate system the procedure allows consideration of the system s intersample behaviour

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

bounds for maximum likelihood regular and non regular doa estimation in k distributed noise we consider the problem of estimating the direction of arrival of a signal embedded in k distributed noise when secondary data which contains noise only are assumed to be available based upon a recent formula of the fisher information matrix fim for complex elliptically distributed data we provide a simple expression of the fim with the two data sets framework in the specific case of k distributed noise we show that under certain conditions the fim for the deterministic part of the model can be unbounded while the fim for the covariance part of the model is always bounded in the general case of elliptical distributions we provide a sufficient condition for unboundedness of the fim accurate approximations of the fim for k distributed noise are also derived when it is bounded additionally the maximum likelihood estimator of the signal doa and an approximated version are derived assuming known covariance matrix the latter is then estimated from secondary data using a conventional regularization technique when the fim is unbounded an analysis of the estimators reveals a rate of convergence much faster than the usual t 1 simulations illustrate the different behaviors of the estimators depending on the fim being bounded or not

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

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

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

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

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 quantum algebra approach to basic multivariable special functions generalizations to a base q of the appell and lauricella hypergeometric functions are studied within the framework of the representation theory of quantum algebras of type an this allows deriving various identities and formulae involving these q functions

convergence and fluctuations of regularized tyler estimators this paper studies the behavior of regularized tyler estimators rtes of scatter matrices the advantages of these estimators are twofold first they guarantee by construction a good conditioning of the estimate and second being derivatives of robust tyler estimators they inherit their robustness properties notably their resilience to outliers nevertheless one major problem that poses the use of rtes is represented by the question of setting the regularization parameter while a high value of is likely to push all the eigenvalues away from zero it comes at the cost of a larger bias with respect to the population covariance matrix a deep understanding of the statistics of rtes is essential to come up with appropriate choices for the regularization parameter this is not an easy task and requires working under asymptotic regimes wherein the number of observations n and or their size n increase together first asymptotic results have recently been obtained when n and n are large and commensurable interestingly no results concerning the regime of n going to infinity with n fixed exist this motivates our work in particular we prove in this paper that the rtes converge to a deterministic matrix when n with n fixed which is expressed as a function of the theoretical covariance matrix we also derive the fluctuations of the rtes around this limit and establish that these fluctuations converge in distribution to a multivariate gaussian distribution with parameters depending on the population covariance and the regularization coefficient

optimal finite wordlength linear quadratic regulation the authors consider the design of an lqg regulator implemented in fixed point arithmetic which is optimized with respect to its state wordlength and structure the approach extends previous design methods so that arithmetic errors are directly included the optimal finite controller gains are computed as a function of the arithmetic wordlength subject to an l sub 2 scaling constant on the controller states the design method used permits the cost for the near optimal structure to be developed so that this can then be used as a basis for comparison to other less complex structures

optimal design of large dimensional adaptive subspace detectors this paper addresses the design of adaptive subspace matched filter asmf detectors in the presence of a mismatch in the steering vector these detectors are coined as adaptive in reference to the step of utilizing an estimate of the clutter covariance matrix using training data of signal free observations to estimate the clutter covariance matrix we employ regularized covariance estimators that by construction force the eigenvalues of the covariance estimates to be greater than a positive scalar rho while this feature is likely to increase the bias of the covariance estimate it presents the advantage of improving its conditioning thus making the regularization suitable for handling high dimensional regimes in this paper we consider the setting of the regularization parameter and the threshold for asmf detectors in both gaussian and compound gaussian clutters in order to allow for a proper selection of these parameters it is essential to analyze the false alarm and detection probabilities for tractability such a task is carried out under the asymptotic regime in which the number of observations and their dimensions grow simultaneously large thereby allowing us to leverage existing results from random matrix theory simulation results are provided in order to illustrate the relevance of the proposed design strategy and to compare the performances of the proposed asmf detectors versus adaptive normalized matched filter anmf detectors under mismatch scenarios

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

more on the q oscillator algebra and q orthogonal polynomials properties of certain q orthogonal polynomials are connected to the q oscillator algebra the wall and q laguerre polynomials are shown to arise as matrix elements of q exponentials of the generators in a representation of this algebra a realization is presented where the continuous q hermite polynomials form a basis of the representation space various identities are interpreted within this model in particular the connection formula between the continuous big q hermite and the continuous q hermite polynomials is thus obtained and two generating functions for these last polynomials are derived algebraically

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

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

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 search algorithm for a class of optimal finite precision controller realization problems with saddle points with game theory we review the optimal digital controller realization problems that maximize a finite word length fwl closed loop stability measure for a large class of these optimal fwl controller realization problems which have saddle points a minimax based search algorithm is derived for finding a global optimal solution the algorithm consists of two stages in the first stage the closed form of a transformation set is constructed which contains global optimal solutions in the second stage a subgradient approach searches this transformation set to obtain a global optimal solution this algorithm does not suffer from the usual drawbacks associated with using direct numerical optimization methods to tackle these fwl realization problems furthermore for a small class of optimal fwl controller realization problems which have no saddle point the proposed algorithm also provides useful information to help solve them

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

summation formulae for the product of the q kummer functions from e q 2 using the representation of eq 2 on the non commutative space zz qz z q 0 summation formulae for the product of two three and four q kummer functions are derived

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

models of q algebra representations tensor products of special unitary and oscillator algebras this paper begins a study of one and two variable function space models of irreducible representations of q analogs of lie enveloping algebras motivated by recurrence relations satisfied by q hypergeometric functions the algebras considered are the quantum algebra uq su2 and a q analog of the oscillator algebra not a quantum algebra in each case a simple one variable model of the positive discrete series of finite and infinite dimensional irreducible representations is used to compute the clebsch gordan coefficients it is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a lie algebra to its lie group and the corresponding matrix elements of the group operators on these representation spaces are computed it is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations it is also demonstrated that general q hypergeometric functions can occur as

models of q algebra representations q integral transforms and addition theorems oscillator lie algebras with respect to countable bases of eigenfunctions of the car tan subalgebras and he computed the summation identities for bessel functions and laguerre polynomials associated with the addition theorems for these matrix elements he also studied matrix elements of the pseudo euclidean and pseudo oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the cartan subalgebras of these lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for bessel functions and for confluent hypergeometric functions here we work out q analogs of these results in which the usual exponential function mapping from the lie algebra to the lie group is replaced by the q exponential mappings e4 and e this study of representations of the euclidean quantum algebra and the q oscillator algebra not a quantum algebra leads to summation integral transform and q integral transform identities for q analogs of the bessel and confluent hypergeometric functions extending the results of vile in for the q 1 case

q gamma and q beta functions in quantum algebra representation theory integral representations and addition formulas for the q generalizations of the gamma and beta functions are obtained by studying function space models of a simple quantum algebra

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

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

a unified closed loop stability measure for finite precision digital controller realizations implemented in different representation schemes a computationally tractable unified finite word length closed loop stability measure is derived which is applicable to fixed point floating point and block floating point representation schemes both the dynamic range and precision of an arithmetic scheme are considered in this new unified measure for each arithmetic scheme the optimal controller realization problem is defined and a numerical optimization approach is adopted to solve it numerical examples are used to illustrate the design procedure and to compare the optimal controller realizations in different representation schemes

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