finding maximum likelihood estimates of patterned covariance matrices by the em algorithm some patterned covariance matrices used to model multivariate normal data that do not have explicit maximum likelihood estimates can be viewed as submatrices of larger patterned covariance matrices that do have explicit maximum likelihood estimates in such cases the smaller covariance matrix can be viewed as the covariance matrix for observed variables and the larger covariance matrix can be viewed as the covariance matrix for both observed and missing variables the advantage of this perspective is that the em algorithm can be used to calculate the desired maximum likelihood estimates for the original problem two examples are presented

estimation of a covariance matrix using the reference prior estimation of a covariance matrix sum is a notoriously difficult problem the standard unbiased estimator can be substantially suboptimal we approach the problem from a noninformative prior bayesian perspective developing the reference noninformative prior for a covariance matrix and obtaining expressions for the resulting bayes estimators these expressions involve the computation of high dimensional posterior expectations which is done using a recent markov chain simulation tool the hit and run sampler frequentist risk comparisons with previously suggested estimators are also given and determination of the accuracy of the estimators is addressed

estimation of a covariance matrix with zeros summary we consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero we first present an algorithm which we call iterative conditional fitting for computing the maximum likelihood estimate of the constrained covariance matrix under the assumption of multivariate normality in contrast to previous approaches this algorithm has guaranteed convergence properties dropping the assumption of multivariate normality we show how to estimate the covariance matrix in an empirical likelihood approach these approaches are then compared via simulation and on an example of gene expression

identification of the number and locations of acoustic sources the identification of the number and locations of acoustic sources is a fundamental problem of engineering acoustics the method of multiple signal classification music has been widely used to solve this problem the main drawback of the music method is that the unavoidable measurement noise should be identical for all the channels of measurement system this requirement generally cannot be met in practical application in this work a method was proposed to identify the number and locations of multiple incoherent sources the proposed method was based on the traditional music method however without the basic assumption about the identity of the measurement noise besides a procedure was proposed to improve the calculation efficiency of identification the numerical and experimental results demonstrated that the identification accuracy of the proposed method was far better than that of the traditional music method the results also demonstrated that the proposed calculation procedure was very efficient only the data at a few frequencies should be used for identification

maximum likelihood estimation for linear gaussian covariance models summary we study parameter estimation in linear gaussian covariance models which are p dimensional gaussian models with linear constraints on the covariance matrix maximum likelihood estimation for this class of models leads to a non convex optimization problem which typically has many local maxima using recent results on the asymptotic distribution of extreme eigenvalues of the wishart distribution we provide sufficient conditions for any hill climbing method to converge to the global maximum although we are primarily interested in the case in which n p the proofs of our results utilize large sample asymptotic theory under the scheme n p 1 remarkably our numerical simulations indicate that our results remain valid for p as small as 2 an important consequence of this analysis is that for sample sizes n 14p maximum likelihood estimation for linear gaussian covariance models behaves as if it were a convex optimization problem

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

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

forward backward spatial smoothing techniques for coherent signal identification a spatial smoothing scheme is further investigated in the context of coherent signal classification it is shown that by making use of a set of forward and complex conjugated backward subarrays simultaneously it is always possible to estimate any k directions of arrival using at most 3k 2 sensor elements this is achieved by creating a smoothed array output covariance matrix that is structurally identical to a covariance matrix in some noncoherent situation by incorporating eigenstructure based techniques on this smoothed covariance matrix it then becomes possible to correctly identify all directions of arrival irrespective of their correlation

sparse estimation of a covariance matrix we suggest a method for estimating a covariance matrix on the basis of a sample of vectors drawn from a multivariate normal distribution in particular we penalize the likelihood with a lasso penalty on the entries of the covariance matrix this penalty plays two important roles it reduces the effective number of parameters which is important even when the dimension of the vectors is smaller than the sample size since the number of parameters grows quadratically in the number of variables and it produces an estimate which is sparse in contrast to sparse inverse covariance estimation our method s close relative the sparsity attained here is in the covariance matrix itself rather than in the inverse matrix zeros in the covariance matrix correspond to marginal independencies thus our method performs model selection while providing a positive definite estimate of the covariance the proposed penalized maximum likelihood problem is not convex so we use a majorize minimize approach in which we iteratively solve convex approximations to the original nonconvex problem we discuss tuning parameter selection and demonstrate on a flow cytometry dataset how our method produces an interpretable graphical display of the relationship between variables we perform simulations that suggest that simple elementwise thresholding of the empirical covariance matrix is competitive with our method for identifying the sparsity structure additionally we show how our method can be used to solve a previously studied special case in which a desired sparsity pattern is prespecified

covariance hypotheses which are linear in both the covariance and the inverse covariance both the covariance and the inverse covariance are products of models each of which consists of either i independent identically distributed random vectors which have a covariance with a real complex or quaternion structure or ii independent identically distributed random vectors with a parametrization of the covariance which is given by means of the clifford algebra the models i are well known for models ii we have found under the assumption that the distribution is nornal the exact distributions of the maximum likelihood estimates and the likelihood ratio test statistics 1 introduction the purpose of the present paper is to describe for the family of normal distributions the structure of those statistical hypotheses models which are linear in both the covariance and the inverse covariance in section 2 we show that under such a hypothesis the problem of maximum likelihood estimation of the covariance has an explicit solution and that these hypotheses are parametrized by jordan algebras quadratic subspaces based on the theory of jordan algebras it is then possible to give a canonical form for such hypotheses in section 3 we show that the jordan algebras are products of so called simple jordan algebras and in section 4 we prove that the hypotheses can be decomposed into products of hypotheses parametrized by simple jordan algebras the simple jordan algebras are classified according to their so called degree the only simple jordan algebras of degree 1 is the set of real numbers and corresponding to that we have the hypothesis under which the covariance is proportional to a known covariance in section 5 we give a canonical form for the hypotheses which are parametrized by simple jordan algebras of degree greater than 3 it tums out that these hypotheses are equivalent to those consisting of independent identically distributed random vectors which have a covariance with a real complex or quatemion structure and the distributions of the maximum likelihood estimates are given by wishart 1928 goodman 1963 and andersson 1975 respectively the statistical hypotheses which are parametrized by simple jordan algebras of degree 2 are new and in section 6 we give a complete solution to the problems of maximum likelihood inference it turns out that the distribution problems are not more complicated than those for the two dimensional normal distribution the mathematical structure of the hypotheses is closely related to the theory of spinors in quantum mechanics based on that theory we give a canonical form for such hypotheses in section 7

two dimensional direction estimation for a mixture of noncoherent and coherent signals this paper deals with the two dimensional 2 d direction of arrival doa estimation of a mixture of noncoherent including uncorrelated and partially correlated and coherent i e fully correlated narrowband signals impinging on a planar sensor array composed of two parallel uniform linear arrays ulas an oblique projection based approach for 2 d direction estimation opade is proposed by using some cross correlations between the received array data in the proposed opade the oblique projection is utilized to isolate the coherent signals from the noncoherent ones and the effect of additive noise is alleviated while the computationally intensive eigendecomposition is avoided and the estimated elevation and azimuth angles are paired automatically further an iterative alternating scheme is presented to improve the estimation accuracy of the oblique projector and hence that of the doas of coherent signals the cramer rao lower bound crb for the mixture of noncoherent and coherent signals is also derived explicitly where the prior knowledge of the signal correlation is incorporated into the 2 d doa estimation of noncoherent signals finally the effectiveness of the opade and the theoretical analysis are substantiated through numerical examples

the method of lines for the solutions of some nonlinear partial differential equations abstract in this paper we deal with the method of lines in the case of certain first order nonlinear partial differential equations as well as equations of the parabolic type a new aspect compared to earlier investigations is that the system we consider may contain partial and ordinary differential equations alike and in addition the r h s of the equations may involve some nonlinear operators the domain of definition of the solutions is bounded by time dependent functions our general results are demonstrated on a simple system which describes a biological model

nested arrays in two dimensions part ii application in two dimensional array processing this paper explores the practical application of a new class of two dimensional arrays namely the two dimensional nested arrays in array processing problems like two dimensional direction of arrival estimation nested arrays constitute a class of two dimensional arrays with physical sensors on lattice s whose difference co array gives rise to a virtual two dimensional array with o mn elements although the number of physical sensors used is m n the structure of the two dimensional nested array was proposed in a companion paper and several issues related to the geometrical orientation of the difference co array in the two dimensional space were addressed in this paper the practical aspects of the application of the nested array in two dimensional array processing are considered which can enable us to exploit the increased degrees of freedom offered by the co array under the constraint of a fixed total number of physical sensors the optimal structure of the 2d nested array is solved which maximizes the number of elements in the virtual co array this provides closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors it is shown that one can obtain o n2 virtual elements in the co array using only n physical sensors to exploit the increased degrees of freedom large number of virtual sensors offered by the virtual array for two dimensional doa estimation of more sources than sensors a novel algorithm based on the concept of two dimensional spatial smoothing is proposed and several issues related to identifiability in two dimensional doa estimation problem which is a fundamental issue in two dimensions are addressed the validity of the proposed methods is verified through several numerical examples

detection of adaptive shifts on phylogenies by using shifted stochastic processes on a tree summary comparative and evolutive ecologists are interested in the distribution of quantitative traits between related species the classical framework for these distributions consists of a random process running along the branches of a phylogenetic tree relating the species we consider shifts in the process parameters which reveal fast adaptation to changes of ecological niches we show that models with shifts are not identifiable in general constraining the models to be parsimonious in the number of shifts partially alleviates the problem but several evolutionary scenarios can still provide the same joint distribution for the extant species we provide a recursive algorithm to enumerate all the equivalent scenarios and to count the number of effectively different scenarios we introduce an incomplete data framework and develop a maximum likelihood estimation procedure based on the expectation maximization algorithm finally we propose a model selection procedure based on the cardinal of effective scenarios to estimate the number of shifts and for which we prove an oracle inequality

vector sensor array processing for electromagnetic source localization the authors present a new approach for localizing electromagnetic sources using sensors where the output of each is a vector consisting of the complete six electric and magnetic field components two types of source transmissions are considered 1 single signal transmission sst and 2 dual signal transmission dst the model is given in terms of several parameters including the wave direction of arrival doa and state of polarization a compact expression is derived for the cramer rao bound crb on the estimation errors of these parameters for the multi source multi vector sensor model quality measures including mean square angular error msae and covariance of vector angular error cvae are introduced and their lower bounds are derived the advantage of using vector sensors is highlighted by explicit evaluation of the msae and cvae bounds for source localization with a single vector sensor a simple algorithm for estimating the source doa with this sensor is presented along with its statistical performance analysis

oblique projection based enumeration of mixed noncoherent and coherent narrowband signals in many application scenarios the signals impinging on the array of sensors comprise the noncoherent uncorrelated and or partially correlated signals and the coherent signals with several groups due to multipath propagation in this paper we consider the problem of estimating the number of noncoherent narrowband signals and that of coherent narrowband signals with multiple groups impinging on a planar sensor array composed of two parallel uniform linear arrays ulas and an oblique projection based enumerator for the mixed signals opems is proposed by utilizing the qr decomposition based ratio criterion qrrc for rank determination of a matrix in the opems the number of noncoherent signals and that of coherent signals in each group are estimated separately where only the elevation angles of noncoherent signals are estimated to isolate the coherent signals from the noncoherent ones and the computationally intensive and time consuming eigendecomposition procedure is avoided the consistency of the proposed opems is analyzed and its effectiveness is verified through numerical examples

automatic generation of evolutionary hypotheses using mixed gaussian phylogenetic models phylogenetic comparative methods are widely used to understand and quantify the evolution of phenotypic traits based on phylogenetic trees and trait measurements of extant species such analyses depend crucially on the underlying model gaussian phylogenetic models like brownian motion and ornstein uhlenbeck processes are the workhorses of modeling continuous trait evolution however these models fit poorly to big trees because they neglect the heterogeneity of the evolutionary process in different lineages of the tree previous works have addressed this issue by introducing shifts in the evolutionary model occurring at inferred points in the tree however for computational reasons in all current implementations these shifts are intramodel meaning that they allow jumps in 1 or 2 model parameters keeping all other parameters global for the entire tree there is no biological reason to restrict a shift to a single model parameter or even to a single type of model mixed gaussian phylogenetic models mgpms incorporate the idea of jointly inferring different types of gaussian models associated with different parts of the tree here we propose an approximate maximum likelihood method for fitting mgpms to comparative data comprising possibly incomplete measurements for several traits from extant and extinct phylogenetically linked species we applied the method to the largest published tree of mammal species with body and brain mass measurements showing strong statistical support for an mgpm with 12 distinct evolutionary regimes based on this result we state a hypothesis for the evolution of the brain body mass allometry over the past 160 million y

local and global stability for the solutions of a nonlinear renewal equation existence uniqueness and stability of the stationary solutions of the problem ua a t ut a t m a u a t u a t u 0 t 0 b a u a t u a t da u a 0 u0 a 0 a t 0 with m b and given are investigated together with their ecological implications

stochastic mckendrick von foerster models with applications abstract a newly presented mckendrick von foerster model with a stochastically perturbed mortality rate is examined a transformation method converting the model with non local boundary conditions into a system of stochastic functional differential equations is offered the method could be viewed as analogous to the one which is widely used for such type of deterministic problems the derived stochastic functional differential equations yield multiple classic population models with naturally born stochasticity including delayed nicholson s blowflies general recruitment and models with cannibalism which by itself could be objects of future analysis and applications

sparsity based two dimensional doa estimation for coprime array from sum difference coarray viewpoint this paper addresses the issue of two dimensional 2 d direction of arrival doa estimation with coprime planar arrays cppas via sparse representation our work differs from the partial spectral search approach 25 which suppresses the phase ambiguity by searching the common peaks of two subarrays we focus on the coprime property of cppa where the sparse array extension model with sum difference coarray sdca is derived for larger degrees of freedom dofs besides to optimize the selection of regularization parameter we also construct a new sparse representation algorithm by estimating the errors between the signal and noise parts further an iterative scheme is presented to transform the 2 d grids searching to several times of 1 d searching where the initial values are obtained by extracting one difference coarray from sdca so the proposed method can achieve aperture extension high estimation performance and low computational complexity besides the sparse array extension model for multiple input multiple output radars is discussed and the cramer rao bound for 2 d doa estimation with cppa is also derived in detail finally simulation results demonstrate the effectiveness of proposed method compared to the state of the art methods

stability of a nonlinear age dependent population model containing a control variable a very general nonlinear model of age dependent population growth which includes most of earlier presented extensions of the lotka von foerster model is considered a criterion for local asymptotic stability of equilibrium solutions of the model is given the conclusions of this paper are compared with analogous results for some important special cases considered by other authors

on spatial smoothing for two dimensional direction of arrival estimation of coherent signals we present an analysis of a spatial smoothing scheme extended for the estimation of two dimensional 2 d directions of arrival doas of coherent signals using a uniform rectangular array the uniform rectangular array is divided into overlapping rectangular subarrays by the extended scheme which is referred to as the 2 d spatial smoothing scheme the analysis shows that when the extended preprocessing scheme is used in conjunction with the eigenstructure technique the size of the subarrays should be at least k 1 spl times k 1 and the number of the subarrays must be no less than k spl times k in order to guarantee the decorrelation of spl kappa coherent signals for all possible scenarios the minimum size of the total uniform rectangular array is thus shown to be 2k spl times 2k instead of using a uniform rectangular array a minimal subarray structure incorporated with a minimal subarray grouping is also devised for resolving the 2 d doas of k coherent signals the number of sensor elements of the minimal total array is then k sup 2 4k 2 instead of 4k sup 2

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

development and analysis of a two stage beamformer for multiple correlated interferers using rectangular array a two stage two dimensional 2 d beamformer is proposed for signal enhancement in an environment of multiple correlated interferers using a rectangular array in the first stage subarray beamformers are constructed which exhibit reliable interference cancellation using difference preprocessed and spatially smoothed data a secondary combining of all possible subarray beamformers is then performed to fully exploit the array aperture it is shown that compared to the conventional one stage optimum beamformer the two stage beamformer performs equally well and requires a much lower complexity of implementation when a suitable subarray size is chosen

some mathematical aspects of modelling cell population dynamics abstract the purpose of this paper is to discuss different types of models of cell population dynamics based on differential equations and to point out the connection between them from the starting point of the mckendrick von foerster type model we derive lotka volterra type integral equations and some delaying differential equations we establish a connection between the original partial differential equation model and the takahashi ordinary differential equations model based on compartmental analysis by means of a numerical procedure named the method of lines