generalized bayesian information criterion for source enumeration in array processing we investigate the problem of enumerating source signals impinging on an array of sensors in an information theoretic framework the conventional bayesian information criterion bic does not yield satisfactory performance for this problem because it only considers the density of the observations in order to remedy the limitations of the bic we propose a generalized bayesian information criterion gbic rule by incorporating the density of the sample eigenvalues or corresponding statistics such a density contains extra information and complements the density of the observations in constructing the gbic as a result two different expressions for the gbic are suggested simulation results validate the superiority of the proposed gbic over the conventional bic in terms of correctly determining the number of sources while their computational costs are comparable

roy s largest root test under rank one alternatives roy s largest root is a common test statistic in a variety of hy pothesis testing problems despite its popularity obtaining accurate tractable approximations to its distribution under the alternative has been a long standing open problem in multivariate statistics in this paper assuming gaussian observations and a rank one alternative also known as concen trated non centrality we derive simple yet accurate approximations for the distribution of roy s largest root test for five of the most common settings these include signal detection in noise multivariate analysis of variance and canonical correlation analysis our main result is that in all five cases roy s test can be approximated using simple combinations of standard uni variate distributions such as central and non central 2 and f our results allow approximate power calculations for roy s test as well as estimates of sample size required to detect given rank one effects by this test both of which are important quantities in hypothesis driven research

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

a rate of convergence result for the largest eigenvalue of complex white wishart matrices it has been recently shown that if x is an n n matrix whose entries are i i d standard complex gaussian and l1 is the largest eigenvalue of x x there exist sequences mn n and sn n such that l1 mn n sn n converges in distribution to w2 the tracy widom law appearing in the study of the gaussian unitary ensemble this probability law has a density which is known and computable the cumulative distribution function of w2 is denoted f2 in this paper we show that under the assumption that n n 0 we can find a function m continuous and nonincreasing and sequences n n and n n such that for all real s0 there exists an integer n s0 for which if n n n s0 we have with ln n l1 n n n n s s0 n n 2 3 p ln n s f2 s m s0 exp s the surprisingly good 2 3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution w2 is a good approximation to the empirical distribution of ln n in simulations an important fact from the point of view of e g statistical applications

witt groups of curves and surfaces we study witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two in both dimensions we determine both the classical witt group and balmer s shifted witt groups in the case of curves the results are supplemented with a complete description of the shifted grothendieck witt groups in a second step we analyse the relationship of witt groups of smooth complex curves and surfaces with their real topological k groups they turn out to be surprisingly close for all curves and for all projective surfaces of geometric genus zero the witt groups may be identified with the quotients of their even ko groups by the images of their complex topological k groups under realification

principles of multivariate analysis a user s perspective part i looking at multivariate data part ii samples populations and models part iii analysing ungrouped data part iv analysing grouped data part v analysing association among variables appendix some basic matrix theory a1 definitions a2 elementary arithmetic operations a3 determinants and inverses a4 quadratic forms a5 latent roots and vectors a6 matrix square root a7 partitioned matrices a8 vector differentiation references index

on the number of principal components in high dimensions we consider the problem of how many components to retain in the application of principal component analysis when the dimension is much higher than the number of observations to estimate the number of components we propose to sequentially test skewness of the squared lengths of residual scores that are obtained by removing leading principal components the residual lengths are asymptotically left skewed if all principal components with diverging variances are removed and right skewed if not the proposed estimator is shown to be consistent performs well in high dimensional simulation studies and provides reasonable estimates in a number of real data examples

model order selection based on information theoretic criteria design of the penalty information theoretic criteria itc have been widely adopted in engineering and statistics for selecting among an ordered set of candidate models the one that better fits the observed sample data the selected model minimizes a penalized likelihood metric where the penalty is determined by the criterion adopted while rules for choosing a penalty that guarantees a consistent estimate of the model order are known theoretical tools for its design with finite samples have never been provided in a general setting in this paper we study model order selection for finite samples under a design perspective focusing on the generalized information criterion gic which embraces the most common itc the theory is general and as case studies we consider a the problem of estimating the number of signals embedded in additive white gaussian noise awgn by using multiple sensors b model selection for the general linear model glm which includes e g the problem of estimating the number of sinusoids in awgn the analysis reveals a trade off between the probabilities of overestimating and underestimating the order of the model we then propose to design the gic penalty to minimize underestimation while keeping the overestimation probability below a specified level for the considered problems this method leads to analytical derivation of the optimal penalty for a given sample size a performance comparison between the penalty optimized gic and common aic and bic is provided demonstrating the effectiveness of the proposed design strategy

optshrink an algorithm for improved low rank signal matrix denoising by optimal data driven singular value shrinkage the truncated singular value decomposition of the measurement matrix is the optimal solution to the representation problem of how to best approximate a noisy measurement matrix using a low rank matrix here we consider the unobservable denoising problem of how to best approximate a low rank signal matrix buried in noise by optimal re weighting of the singular vectors of the measurement matrix we exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the independent identically distributed gaussian noise case our analysis brings into sharp focus the shrinkage and thresholding form of the optimal weights the nonconvex nature of the associated shrinkage function on the singular values and explains why matrix regularization via singular value thresholding with convex penalty functions such as the nuclear norm will always be suboptimal we validate our theoretical predictions with numerical simulations develop an implementable algorithm optshrink that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries

an mdl framework for sparse coding and dictionary learning the power of sparse signal modeling with learned overcomplete dictionaries has been demonstrated in a variety of applications and fields from signal processing to statistical inference and machine learning however the statistical properties of these models such as underfitting or overfitting given sets of data are still not well characterized in the literature as a result the success of sparse modeling depends on hand tuning critical parameters for each data and application this work aims at addressing this by providing a practical and objective characterization of sparse models by means of the minimum description length mdl principle a well established information theoretic approach to model selection in statistical inference the resulting framework derives a family of efficient sparse coding and dictionary learning algorithms which by virtue of the mdl principle are completely parameter free furthermore such framework allows to incorporate additional prior information to existing models such as markovian dependencies or to define completely new problem formulations including in the matrix analysis area in a natural way these virtues will be demonstrated with parameter free algorithms for the classic image denoising and classification problems and for low rank matrix recovery in video applications however the framework is not limited to this imaging data and can be applied to a wide range of signal and data types and tasks

analysis of the performance and sensitivity of eigendecomposition based detectors a new framework is presented for the analysis of the performance of detection methods such as aic and mdl which are based on the eigenvalues of the sample covariance matrix it is shown that theoretical analysis of the probabilities of overestimation and underestimation can be much more conveniently carried out via a proposed particularly simple sequence of statistics also the breakdown of these detection methods in the presence of model nonidealities is explored by theory simulations and experimentation with real array data for example theoretical arguments are given to demonstrate the high degree of sensitivity of the detectors to unknown deviations of the noise from whiteness

solution approximation in infinite horizon linear quadratic control we consider the problem of choosing a discounted cost minimizing infinite stage control sequence under nonstationary positive semidefinite quadratic costs and linear constraints specific cases include the nonstationary lq tracker and regulator problems we show that the optimal costs for finite stage approximating problems converge to the optimal infinite stage cost as the number of stages grows to infinity under a state reachability condition we show that the set unions of all controls optimal to all feasible states for the finite stage approximating problems converge to the set of infinite stage optimal controls a tie breaking rule is provided that selects finite stage optimal controls so as to force convergence to an infinite horizon optimal control

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

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

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

estimation of the number of sources in unbalanced arrays via information theoretic criteria estimating the number of sources impinging on an array of sensors is a well known and well investigated problem a common approach for solving this problem is to use an information theoretic criterion such as minimum description length mdl or the akaike information criterion aic the mdl estimator is known to be a consistent estimator robust against deviations from the gaussian assumption and nonrobust against deviations from the point source and or temporally or spatially white additive noise assumptions over the years several alternative estimation algorithms have been proposed and tested usually these algorithms are shown using computer simulations to have improved performance over the mdl estimator and to be robust against deviations from the assumed spatial model nevertheless these robust algorithms have high computational complexity requiring several multidimensional searches in this paper which is motivated by real life problems a systematic approach toward the problem of robust estimation of the number of sources using information theoretic criteria is taken an mdl type estimator that is robust against deviation from assumption of equal noise level across the array is studied the consistency of this estimator even when deviations from the equal noise level assumption occur is proven a novel low complexity implementation method avoiding the need for multidimensional searches is presented as well making this estimator a favorable choice for practical applications

asymptotic power of sphericity tests for high dimensional data this paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity we establish the convergence under the null hypothesis and the alternative of the log ratio of the joint densities of the sample covariance eigenvalues to a gaussian process indexed by the norm of the perturbation when the perturbation norm is larger than the phase transition threshold studied in baik et al 2005 the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain when the norm is below the threshold the process is non degenerate so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous using the asymptotic theory of statistical experiments we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region in particular we show that the asymptotic power of the tracy widom type tests is trivial whereas that of the eigenvalue based likelihood ratio test is strictly larger than the size and close to the power envelope

nonparametric detection of signals by information theoretic criteria performance analysis and an improved estimator determining the number of sources from observed data is a fundamental problem in many scientific fields in this paper we consider the nonparametric setting and focus on the detection performance of two popular estimators based on information theoretic criteria the akaike information criterion aic and minimum description length mdl we present three contributions on this subject first we derive a new expression for the detection performance of the mdl estimator which exhibits a much closer fit to simulations in comparison to previous formulas second we present a random matrix theory viewpoint of the performance of the aic estimator including approximate analytical formulas for its overestimation probability finally we show that a small increase in the penalty term of aic leads to an estimator with a very good detection performance and a negligible overestimation probability

on information theoretic criteria for determining the number of signals in high resolution array processing an important problem in high resolution array processing is the determination of the number of signals arriving at the array information theoretic criteria provide a means to achieve this two commonly used criteria are the akaike information criterion aic and minimum descriptive length mdl criterion while the aic tends to overestimate even at a high signal to noise ratio snr the mdl criterion tends to underestimate at low or moderate snr by excluding irrelevant parameters a new log likelihood function has been chosen utilizing this new log likelihood function gives a set of more accurate estimates of the eigenvalues and in the establishment of modified information theoretic criteria which moderate the performance of the aic and the mdl criterion computer simulations confirm that the modified criteria have superior performance

patch based near optimal image denoising in this paper we propose a denoising method motivated by our previous analysis of the performance bounds for image denoising insights from that study are used here to derive a high performance practical denoising algorithm we propose a patch based wiener filter that exploits patch redundancy for image denoising our framework uses both geometrically and photometrically similar patches to estimate the different filter parameters we describe how these parameters can be accurately estimated directly from the input noisy image our denoising approach designed for near optimal performance in the mean squared error sense has a sound statistical foundation that is analyzed in detail the performance of our approach is experimentally verified on a variety of images and noise levels the results presented here demonstrate that our proposed method is on par or exceeding the current state of the art both visually and quantitatively

reachable controllable sets and stabilizing control of constrained linear systems we consider discrete time linear systems with constrained controls we present an algorithm for computing an external representation of the n step reachable set and controllable set it is an improvement of previous methods in particular it is shown that for constructing the set of initial states that can be driven to the origin asymptotically one can restrict the search to a polyhedral set in the unstable subspace of the autonomous system which yields a drastic reduction in the computational burden application to the synthesis of stabilizing feedback controllers is discussed

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

source enumeration via mdl criterion based on linear shrinkage estimation of noise subspace covariance matrix numerous methodologies have been investigated for source enumeration in sample starving environments for those having their root in the framework of random matrix theory the involved distribution of the sample eigenvalues is required instead of relying on the eigenvalue distribution this work devises a linear shrinkage based minimum description length ls mdl criterion by utilizing the identity covariance matrix structure of noise subspace components with linear shrinkage and gaussian assumption of the observations an accurate estimator for the covariance matrix of the noise subspace components is derived the eigenvalues obtained from the estimator turn out to be a linear function of the corresponding sample eigenvalues enabling the ls mdl criterion to accurately detect the source number without incurring significantly additional computational load furthermore the strong consistency of the ls mdl criterion for m n and m n c 0 is proved where m and n are the antenna number and snapshot number respectively simulation results are included for illustrating the effectiveness of the proposed criterion

enhanced puma for direction of arrival estimation and its performance analysis direction of arrival doa estimation is a problem of significance in many applications in practice due to the occurrence of coherent signals and or when the number of available snapshots is small it is a challenge to find doas accurately this problem is revisited here through a new enhanced principal singular vector utilization for modal analysis epuma doa estimation approach which improves the threshold performance by first generating p k doa candidates for k sources where p geq k and then judiciously selecting k of them the asymptotic variance of epuma is theoretically derived and numerical results are provided to validate the asymptotic analysis and illustrate the practical merits of epuma

detection of the number of signals using the benjamini hochberg procedure this paper presents a novel approach to detect multiple signals embedded in noisy observations from a sensor array we formulate the detection problem as a multiple hypothesis test to control the global level of the multiple test we apply the false discovery rate fdr criterion proposed by benjamini and hochberg compared to the classical familywise error rate fwe criterion the fdr controling procedure leads to a significant gain in power for large size problems in addition we apply the bootstrap technique to estimate the observed significance level required by the fdr controling procedure simulations show that the fdr controling procedure always provides higher probability of correct detection than the fwe controling procedure furthermore the reliability of the proposed test procedure is not affected by the gain in power of the test

a complete characterization of reachable sets for constrained linear time varying systems we consider linear time varying systems with linear constraints on both control and state variables the reachable set is a convex polyhedron that we completely characterize in terms of its boundary hyperplanes

an algorithm to find maximal state constraint sets for discrete time linear dynamical systems with bounded controls and states an algorithm to find a polyhedral approximant to the maximal state constraint set given state and input bounds is suggested for linear discrete time dynamical systems

spherical cap packing asymptotics and rank extreme detection we study the spherical cap packing problem with a probabilistic approach such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector when the set of unit vectors are themselves independently uniformly distributed we further develop the extreme value distribution limit of the maximal inner product which characterizes its uncertainty around the bound as applications of the above mentioned asymptotic results we derive 1 an asymptotic sharp universal uniform bound on the maximal spurious correlation as well as its uniform convergence in distribution when the explanatory variables are independently gaussian distributed and 2 an asymptotic sharp universal bound on the maximum norm of a low rank elliptically distributed vector as well as related limiting distributions with these results we develop a fast detection method for a low rank structure in high dimensional gaussian data without using the spectrum information

performance of statistical tests for single source detection using random matrix theory this paper introduces a unified framework for the detection of a single source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver the generalized maximum likelihood test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace using recent results from random matrix theory a practical way to evaluate the threshold and the p value of the test is provided in the asymptotic regime where the number k of sensors and the number n of observations per sensor are large but have the same order of magnitude the theoretical performance of the test is then analyzed in terms of receiver operating characteristic roc curve it is in particular proved that both type i and type ii error probabilities converge to zero exponentially as the dimensions increase at the same rate and closed form expressions are provided for the error exponents these theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix

on the use of order statistics for improved detection of signals by the mdl criterion the minimum description length mdl criterion for model selection has been successfully applied to the problem of estimating the number of sources in noise we show that by employing an order statistics approach in the existing solution a new processor with uniformly improved performance is constructed at the cost of greater complexity we prove that the resulting mdl based processor is consistent results of simulations of practical scenarios of interest demonstrate how significant the performance improvement can be

optimal shrinkage of singular values we consider the recovery of low rank matrices from noisy data by shrinkage of singular values in which a single univariate nonlinearity is applied to each of the empirical singular values we adopt an asymptotic framework in which the matrix size is much larger than the rank of the signal matrix to be recovered and the signal to noise ratio of the low rank piece stays constant for a variety of loss functions including mean square error mse square frobenius norm the nuclear norm loss and the operator norm loss we show that in this framework there is a well defined asymptotic loss that we evaluate precisely in each case in fact each of the loss functions we study admits a unique admissible shrinkage nonlinearity dominating all other nonlinearities we provide a general method for evaluating these optimal nonlinearities and demonstrate our framework by working out simple explicit formulas for the optimal nonlinearities in the frobenius nuclear and operator norm cases for example for a square low rank n by n matrix observed in white noise with level sigma the optimal nonlinearity for mse loss simply shrinks each data singular value y to sqrt y 2 4n sigma 2 or to 0 if y this optimal nonlinearity guarantees an asymptotic mse of 2nr sigma 2 which compares favorably with optimally tuned hard thresholding and optimally tuned soft thresholding providing guarantees of 3nr sigma 2 and 6nr sigma 2 respectively our general method also allows one to evaluate optimal shrinkers numerically to arbitrary precision as an example we compute optimal shrinkers for the schatten p norm loss for any p 0

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

statistical analysis of the performance of information theoretic criteria in the detection of the number of signals in array processing the performances of the akaike 1974 information criterion and the minimum descriptive length criterion methods are examined the events which lead to erroneous decisions are considered and on the basis of these events the probabilities of error for the two criteria are derived the probabilities of the first two events are derived based on the asymptotic distribution of the sample eigenvalues of an estimated hermitian matrix it is further shown that the probabilities of missing and false alarm for these two criteria can be evaluated to a close approximation although the derivation of the probabilities of error is based on an asymptotic analysis the results are confirmed to be in very close agreement with computer simulation results

large system analysis of a glrt for detection with large sensor arrays in temporally white noise this paper addresses the behaviour of a classical multi antenna glrt test that allows to detect the presence of a known signal corrupted by a multi path propagation channel and by an additive temporally white gaussian noise with unknown spatial covariance matrix the paper is focused on the case where the number of sensors m is large and of the same order of magnitude as the sample size n a context which is modeled by the large system asymptotic regime m n in such a way that m n c for c 0 the purpose of this paper is to study the behaviour of a glrt statistics in this regime and to show that the corresponding theoretical analysis allows to accurately predict the performance of the test when m and n are of the same order of magnitude

independent component analysis based source number estimation and its comparison for mechanical systems abstract it has been challenging to correctly separate the mixed signals into source components when the source number is not known a priori in this paper we propose a novel source number estimation based on independent component analysis ica and clustering evaluation analysis we investigate and benchmark three information based source number estimations akaike information criterion aic minimum description length mdl and improved bayesian information criterion ibic all the above methods are comparatively studied in both numerical and experimental case studies with typical mechanical signals the results demonstrate that the proposed ica based source number estimation with nonlinear dissimilarity measures performs more stable and robust than the information based ones for mechanical systems

detection of the number of signals a predicted eigen threshold approach a novel method for detecting the number of signals incident upon an array of sensors is described the method is based on finding upper thresholds for the observed eigenvalues of the correlation matrix of the array output the asymptotic normality of the multifold eigenvalue estimate is used to derive a prediction formula for the thresholds due to the fact that the detection error of the method can be controlled by a parameter t the performance of the method is superior to the mdl minimum description length under low snr signal to noise ratio and superior to an aic akaike information criterion under high snr the distribution functions of both missing and false alarm errors are evaluated so that the performance of the method can be analyzed and so that t can be chosen simulation results are presented to confirm the analysis

performance analysis of an improved music doa estimator this paper addresses the statistical performance of subspace doa estimation using a sensor array in the asymptotic regime where the number of samples and sensors both converge to infinity at the same rate improved subspace doa estimators were derived termed as g music in previous works and were shown to be consistent and asymptotically gaussian distributed in the case where the number of sources and their doa remain fixed in this case which models widely spaced doa scenarios it is proved in the present paper that the traditional music method also provides doa consistent estimates having the same asymptotic variances as the g music estimates the case of doa that are spaced of the order of a beamwidth which models closely spaced sources is also considered it is shown that g music estimates are still able to consistently separate the sources while this is no longer the case for the music ones the asymptotic variances of g music estimates are also evaluated

constrained controllability of discrete time systems the constrained controllability of the discrete time system xk 1 a k xk b k u k is considered where the control uk is termed admissible if it satisfies specified magnitude constraints constrained controllability is concerned with the existence of an admissible control which steers the state x to a given target set from a specified initial state conditions for checking constrained controllability to a given target set from a specified initial state are presented these conditions involve solving finite dimensional optimization problems and can be checked via numerical computation in addition conditions for checking global constrained controllability to a given target set are presented a system is globally constrained controllable if for every initial state there exists an admissible control that steers the system to the target if a given discrete time system is constrained controllable it may be desirable to obtain an admissible control that steers the system to the target from a specified initial s

finite sample approximation results for principal component analysis a matrix perturbation approach principal component analysis pca is a standard tool for dimensional reduction of a set of n observations samples each with p variables in this paper using a matrix perturbation approach we study the nonasymptotic relation between the eigenvalues and eigenvectors of pca computed on a finite sample of size n and those of the limiting population pca as n oo as in machine learning we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample pca and population pca under a spiked covariance model in addition we also consider the relation between finite sample pca and the asymptotic results in the joint limit p n with p n c we present a matrix perturbation view of the phase transition phenomenon and a simple linear algebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit moreover our analysis also applies for finite p n where we show that although there is no sharp phase transition as in the infinite case either as a function of noise level or as a function of sample size n the eigenvector of sample pca may exhibit a sharp loss of tracking suddenly losing its relation to the true eigenvector of the population pca matrix this occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise whose eigenvector points in a random direction

source enumeration in array processing using a two step test we investigate the problem of enumerating source signals impinging on an array of sensors given noisy limited observations the assumption of a large sample size is indispensable for most of the existing approaches so that they suffer significant performance degradation when only a small number of samples is available random matrix theory can capture the asymptotic distributions of the eigenvalues accurately and provide good approximations even for a finite sample size in this paper we take into account the relationship and distributions of the noise and signal eigenvalues based on random matrix theory and derive a simple two step test numerical simulations demonstrate that it is capable of correctly determining the number of sources in the case of small sample sizes

spectrum sensing in the presence of multiple primary users we consider multi antenna cooperative spectrum sensing in cognitive radio networks when there may be multiple primary users a detector based on the spherical test is analyzed in such a scenario based on the moments of the distributions involved simple and accurate analytical formulae for the key performance metrics of the detector are derived the false alarm and the detection probabilities as well as the detection threshold and receiver operation characteristics are available in closed form simulations are provided to verify the accuracy of the derived results and to compare with other detectors in realistic sensing scenarios

detection of signals by information theoretic criteria general asymptotic performance analysis detecting the number of sources is a well known and a well investigated problem in this problem the number of sources impinging on an array of sensors is to be estimated the common approach for solving this problem is to use an information theoretic criterion like the minimum description length mdl or the akaike information criterion aic although it has been gaining much popularity and has been used in a variety of problems the performance of information theoretic criteria based estimators for the unknown number of sources has not been sufficiently studied yet in the context of array processing the performance of such estimators were analyzed only for the special case of gaussian sources where no prior knowledge of the array structure if given is used based on the theory of misspecified models this paper presents a general asymptotic analysis of the performance of any information theoretic criterion based estimator and especially of the mdl estimator in particular the performance of the mdl estimator which assumes gaussian sources and structured array when applied to gaussian sources is analyzed in addition it is shown that the performance of a certain mdl estimator is not very sensitive to the actual distribution of the source signals however appropriate use of prior knowledge about the array geometry can lead to significant improvement in the performance of the mdl estimator simulation results show a good fit between the empirical and the theoretical results

flexible detection criterion for source enumeration in array processing two new procedures are proposed using random matrix theory to evaluate theoretically the performance of the bayesian information criterion bic for source enumeration in array processing the proposed procedures can predict precisely the estimation results of the bic also they shed light on the behavior of the bic in order to remedy the limitation of the bic that is the bic tends to underestimate the number of sources by one in most cases an extra parameter is introduced into the bic formulation which results in a new criterion for source enumeration referred to as the flexible detection criterion fdc by carefully choosing this parameter the fdc is capable of substantially reducing the probability of underestimation which is validated by theoretical analysis and simulations note that the fdc is consistent and has low computational complexity as the bic

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

model selection for sinusoids in noise statistical analysis and a new penalty term detection of the number of sinusoids embedded in noise is a fundamental problem in statistical signal processing most parametric methods minimize the sum of a data fit likelihood term and a complexity penalty term the latter is often derived via information theoretic criteria such as minimum description length mdl or via bayesian approaches including bayesian information criterion bic or maximum a posteriori map while the resulting estimators are asymptotically consistent empirically their finite sample performance is strongly dependent on the specific penalty term chosen in this paper we elucidate the source of this behavior by relating the detection performance to the extreme value distribution of the maximum of the periodogram and of related random fields based on this relation we propose a combined detection estimation algorithm with a new penalty term our proposed penalty term is sharp in the sense that the resulting estimator achieves a nearly constant false alarm rate a series of simulations support our theoretical analysis and show the superior detection performance of the suggested estimator