on the covering number of a matroid element abstract let s be a nonempty finite set with cardinality m let m s i m be a matroid on s with no loops we present results connecting the structure of the bases of m k the k th power of m and the covering number of elements of s in m

the beta model maximum likelihood cramer rao bounds and hypothesis testing we study the maximum likelihood estimator in a setting where the dependent variable is a random graph and covariates are available on a graph level the model generalizes the well known beta model for random graphs by replacing the constant model parameters with regression functions cramer rao bounds are derived for special cases of the undirected beta model the directed beta model and the covariate based beta model the corresponding maximum likelihood estimators are compared with the bounds by means of simulations moreover examples are given on how to use the presented maximum likelihood estimators to test for directionality and significance finally the applicability of the model is demonstrated using temporal social network data describing communication among healthcare workers

joint estimation of porosity and saturation using stochastic rock physics modeling sediment porosity and saturation affect bulk modulus shearmodulus anddensity consequently estimatinghydrocarbon saturation and reservoir porosity from seismic data is a joint estimation problem uncertainty in porosity will lead toerrorsinsaturationprediction andviceversa porosityand saturation can be jointly estimated using stochastic rockphysicsmodelingandformalbayesianestimationmethodology knowledge of shear impedance reduces the uncertainty inporosityandthusalsoreducesuncertaintyinsaturationestimation this study investigates joint estimation of porosity and saturation by using rock physics stochastic modeling and bayesian estimation theory to derive saturation and porosity maps of expected pay sands in the field example the uncertainty in porosity quantified by the standard deviation stdassociatedwiththeposteriorprobabilitydensityfunctionpdf derivedfrominversionofseismicdataismuchless than the uncertainty in the derived saturation for a typical case the std associated with saturation is 24 while porositystdisabout1 34porosityunitsgivenseismic derived inversionattributeswithreasonableaccuracy comparisonof these numbers with prior estimates showed that inversion of seismic data decreased the uncertainty in porosity to 15 of the prior uncertainty while saturation uncertainty was only reduced to 92 of the prior uncertainty although these resultsmayvaryfromonelocationtoanother themethodology isgeneralandcanbeappliedtootherlocations

limits of sensing temporal concentration changes by single cells berg and purcell biophys j 20 193 1977 calculated how the accuracy of concentration sensing by single celled organisms is limited by noise from the small number of counted molecules here we generalize their results to the sensing of concentration ramps which is often the biologically relevant situation e g during bacterial chemotaxis we calculate lower bounds on the uncertainty of ramp sensing by three measurement devices a single receptor an absorbing sphere and a monitoring sphere we contrast two strategies simple linear regression of the input signal versus maximum likelihood estimation and show that the latter can be twice as accurate as the former finally we consider biological implementations of these two strategies and identify possible signatures that maximum likelihood estimation is implemented by real biological systems

robust mean squared error estimation of multiple signals in linear systems affected by model and noise uncertainties this paper is a continuation of the work in 11 and 2 on the problem of estimating by a linear estimator n unobservable input vectors undergoing the same linear transformation from noise corrupted observable output vectors whereas in the aforementioned papers only the matrix representing the linear transformation was assumed uncertain here we are concerned with the case in which the second order statistics of the noise vectors i e their covariance matrices are also subjected to uncertainty we seek a robust mean squared error estimator immuned against both sources of uncertainty we show that the optimal robust mean squared error estimator has a special form represented by an elementary block circulant matrix and moreover when the uncertainty sets are ellipsoidal like the problem of finding the optimal estimator matrix can be reduced to solving an explicit semidefinite programming problem whose size is independent of n

robust linear channel estimation methods for per subcarrier transmit antenna selection the linear minimum mean squared error mse channel estimator for systems employing per subcarrier transmit antenna selection is developed frequency domain correlations after the selection process are shown to be approximated well using a simple function which makes near optimal channel estimation practically possible however the resultant estimators are not robust to errors in the assumed model in terms of the antenna to subcarrier assignment used at transmission as we motivate both analytically and via simulations this is an issue when the channels for conveying this information to the receiver are severely constrained we present windowed channel estimations and new robust estimation algorithms either individually or in combination as solutions to counter this sensitivity the robust estimators are developed based on the principles of 1 minimising the worst case mse over the set of possible models and 2 minimising the expected mse over the set of possible models the latter estimator is preferred due to the lower implementation complexity and better mse performance we also prove theoretical asymptotic in signal to noise ratio performances of the two estimators used with the proposed correlation model simulation results illustrate the performance gains and improved robustness offered by the developed estimators

active learning applied to rcs computations with nonuniform sampling using different objective functions an active learning framework is introduced to reduce the number of frequencies and angles one must consider for wideband monostatic scattering computations or measurements this method is used to optimally select those frequencies and angles that would be most informative resulting in nonuniform sampling and often a reduced number of points vis agrave vis uniform sampling in this paper we focus on jointly two dimensional optimal sampling in frequency and incident angle thetas for monostatic scattering the method consists of two basic steps one step involves estimation of model parameters using a least square ls algorithm the next step is to optimally choose the next point frequency and thetas for analysis by the computational model or experiment this new point is selected with the goal of reducing uncertainty in the parametric model quantified via the fisher information matrix iterating these two steps a sequence of numerical computations or measurements are performed each at the most informative point for learning the parameters of the associated simpler parametric model this idea is demonstrated here in the context of reducing the number of points frequencies and orientations at which a computational model must be employed and in order to avoid repeatedly gathering samplings at the edge of the input space an alternative objective function is applied which makes the actively selected points closer to a region of interest

maximum set estimators with bounded estimation error we consider the linear regression problem of estimating a deterministic parameter vector x from observations y hx w where h is known and w is additive noise we seek an estimator whose estimation error does not exceed a given maximum error for as wide a range of conditions as possible the maximum error is a design choice and is generally lower than the error provided by the well known least squares ls estimator we develop estimators guaranteeing the required error for as large a parameter set as possible and for as large a noise level as possible we discuss methods for finding these estimators and demonstrate that in many cases the proposed estimators outperform the ls estimator

minimax mse estimation of deterministic parameters with noise covariance uncertainties in this paper a minimax mean squared error mse estimator is developed for estimating an unknown deterministic parameter vector in a linear model subject to noise covariance uncertainties the estimator is designed to minimize the worst case mse across all norm bounded parameter vectors and all noise covariance matrices in a given region of uncertainty the minimax estimator is shown to have the same form as the estimator that minimizes the worst case mse over all norm bounded vectors for a least favorable choice of the noise covariance matrix an example demonstrating the performance advantage of the minimax mse approach over the least squares and weighted least squares methods is presented

lower bounds on the mean squared error of low rank matrix reconstruction we investigate the behavior of the mean square error mse of low rank matrix reconstruction and its special case matrix completion we first derive the constrained cramer rao bound crb on the mse matrix of any locally unbiased estimator and then analyze the behavior of the constrained crb when a subset of entries of the underlying matrix is randomly observed we design an alternating minimization procedure to compute the maximum likelihood estimator mle for the low rank matrix and demonstrate through numerical simulations that the performance of the mle approaches the constrained crb when the signal to noise ratio is high applying a chapman robbins type barankin bound allows us to derive lower bounds on the worst case scalar mse we demonstrate that the worst case scalar mse is infinite even if the model is identifiable however the infinite scalar mse is achieved only on a set of low rank matrices with measure zero we discuss the implications of these bounds and compare them with the empirical performance of the matrix lasso estimator and the existing bounds in the literature

linear minimax regret estimation of deterministic parameters with bounded data uncertainties we develop a new linear estimator for estimating an unknown parameter vector x in a linear model in the presence of bounded data uncertainties the estimator is designed to minimize the worst case regret over all bounded data vectors namely the worst case difference between the mean squared error mse attainable using a linear estimator that does not know the true parameters x and the optimal mse attained using a linear estimator that knows x we demonstrate through several examples that the minimax regret estimator can significantly increase the performance over the conventional least squares estimator as well as several other least squares alternatives

minimax estimation of deterministic parameters in linear models with a random model matrix we consider the problem of estimating an unknown deterministic parameter vector in a linear model with a random model matrix with known second order statistics we first seek the linear estimator that minimizes the worst case mean squared error mse across all parameter vectors whose possibly weighted norm is bounded above we show that the minimax mse estimator can be found by solving a semidefinite programming problem and develop necessary and sufficient optimality conditions on the minimax mse estimator using these conditions we derive closed form expressions for the minimax mse estimator in some special cases we then demonstrate through examples that the minimax mse estimator can improve the performance over both a baysian approach and a least squares method we then consider the case in which the norm of the parameter vector is also bounded below since the minimax mse approach cannot account for a nonzero lower bound we consider in this case a minimax regret method in which we seek the estimator that minimizes the worst case difference between the mse attainable using a linear estimator that does not know the parameter vector and the optimal mse attained using a linear estimator that knows the parameter vector for analytical tractability we restrict our attention to the scalar case and develop a closed form expression for the minimax regret estimator

flags and equality of tensors abstract we state a necessary and sufficient condition for equality of decomposable symmetrized tensors when the symmetrizer is associated to an irreducible character of s p corresponding to the partition of p and the tensors are the images by this symmetrizer of the tensor product of a family of vectors of rank partition we make an application of this result to the representation of flags by projective points

on minimum variance unbiased estimation of clock offset in a two way message exchange mechanism for many applications distributed networks require the local clocks of the constituent nodes to run close to an agreed upon notion of time most of the widely used clock synchronization algorithms in such systems employ the sender receiver protocol based on a two way timing message exchange paradigm maximum likelihood estimator mle of the clock offset based on the timing message exchanges between two clocks was derived in d r jeske on maximum likelihood estimation of clock offset ieee trans commun vol 53 pp 53 54 jan 2005 when the fixed delays are symmetric and the variable delays in each direction assume an exponential distribution with an unknown mean herein the best linear unbiased estimate using order statistics blue os of the clock offset between two nodes is derived assuming both symmetric and asymmetric exponential network delays respectively the rao blackwell lehmann scheffe theorem is then exploited to obtain the minimum variance unbiased estimate mvue for the clock offset which it is shown to coincide with the blue os in addition it is found that the mvue of the clock offset in the presence of symmetric network delays also coincides with the mle finally in the presence of asymmetric network delays although the mle is biased it is shown to achieve lesser mean square error mse than the mvue in the region around the point where the bidirectional network link delays are symmetric and hence its merit as the most versatile estimator is fairly justified

robust wiener kolmogorov theory a minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete time wide sense stationary signals when knowledge of the signal and or noise spectra is inexact the solution is given under mild regularity conditions in terms of a least favorable pair of spectra thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily it is noted that this design method leads in particular to robust n step predictors robust causal filters and robust n lag smoothers the method of design is illustrated by a thorough development of the special case of one step noiseless prediction further solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise and numerical examples are given for this case which illustrate the effectiveness of this design

non parametric covariance methods for incidence density analyses of time to event data from a randomized clinical trial and their complementary roles to proportional hazards regression the principal response criteria for many clinical trials involve time to event variables usual methods of analysis for this type of response criterion include product limit estimators of cumulative survival for the treatment groups stratified logrank tests to compare treatments and proportional hazards regression models with treatment and relevant covariates when adjustment for covariates is of some importance the relative roles of these methods may be of some concern particularly for confirmatory clinical trials which must provide convincing findings to regulatory agencies unadjusted methods may have lower power but there are issues regarding adjustment for covariates that may be controversial these issues include applicability of proportional hazards assumptions whether the correct model has been specified and whether there is parallelism between treatments for relationships with covariates one way to address these issues is to use non parametric analysis of covariance strategies with extensions to log incidence density estimation the principal basis for this method is no association between covariates and treatment groups as provided by randomized assignment of patients to groups the background theory and strategies for computation are described for this method aspects of its application are illustrated for a clinical trial with two treatment groups and 722 patients the objective of analysis for this clinical trial is evaluation of treatment effects with and without adjustment for 22 a priori covariates and a stratification for three geographical regions

rank analysis of covariance with right censored data a modification of the rank analysis of covariance suitable for randomly right censored dependent variate observations is proposed and studied the test suggested is based on an appropriate permutation distribution of the rank scores and for large samples may be conducted by using the chi square distribution the principal assumptions are that both censoring and covariate distributions do not depend on the treatment group

auditory motivated level crossing approach to instantaneous frequency estimation we address the problem of estimating the instantaneous frequency if of a phase signal using its level crossing lc information based on front end auditory processing motivation we show that the problem of if estimation using lc information can be cast in the framework of estimation from irregularly sampled data the formulation has the generality of estimating different types of if without the need for a quasistationary assumption we consider two types of if polynomial and bandlimited we use polynomial interpolating functions for the former and for the latter we propose a novel line plus sum of sines model the model parameters are estimated by linear regression considering the noisy case lc data for different levels is analyzed and methods for combining different estimators from lcs are discussed theoretical and extensive simulation results show that the performance of the zero crossing zc based if estimator and the level crossing based if estimator with smaller level values is better than those obtained with higher level values or their combinations the new technique reaches the crame spl acute r rao bound crb roughly above 4 db signal to noise ratio snr and its performance does not deteriorate rapidly with mismatch in the if order compared with the other techniques in the literature

low complexity lmmse based mimo ofdm channel estimation via angle domain processing in many situations multiple input multiple output with orthogonal frequency division multiplexing mimo ofdm channels tend to be spatially correlated due for example to limited scattering prior knowledge of this channel spatial correlation and the channel frequency correlation can be exploited by using the linear minimum mean square error lmmse technique however the complexity of the 2 d lmmse technique which fully utilizes both the channel spatial and frequency correlation is quite high to solve this problem this paper presents and analyzes several low complexity suboptimal approximated lmmse channel estimation techniques in the angle domain where the channel model lends itself to a physical interpretation the choice of angle domain techniques is largely dependent on the extent of channel stochastic information e g channel correlation or power that is available to the receiver nevertheless all the proposed angle domain techniques have much lower complexity compared to the 2 d lmmse technique further all the angle domain techniques improve over the conventional least square ls technique for all the typical mimo ofdm models under consideration more importantly our simulation results show that the angle domain quasi 1 d ql d lmmse technique can achieve similar performance compared to the 2 d lmmse technique for all typical mimo ofdm models with significantly lower complexity

robust hypothesis testing for bounded classes of probability densities corresp least favorable pairs of probability density functions and robust tests are obtained for classes of density functions specified as bands with upper and lower bounds the results are shown to extend huber s solution for the epsilon contaminated classes which is contained as a special case a minimum distance property of the least favorable density pair is also made explicit

minimax robust prediction of discrete time series we discuss a robust approach for predicting a weakly stationary discrete time series whose spectral density f is not exactly known we assume that we know that f mathfrak d where mathfrak d is a convex set of spectral densities fulfilling some not too stringent conditions we proof the existence of a most indeterministic density f0 in mathfrak d and we show that the classical optimal linear predictor for a time series with spectral density f0 is mini max robust in the sense that it minimizes the maximal possible prediction error

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

the covering number of elements of a matroid and associated transformations abstract let s be a nonempty finite set with cardinality m let m s i m be a matroid on s let x be an element of s which is not a loop of m the covering number of x in m is the smallest positive integer s such that x is a coloop of the union of s copies of m we investigate relations between the covering number of the elements in m and in its minors applications to the study of the rank partition and associated transformation are presented

fundamental limits of linear equalizers diversity capacity and complexity linear equalizers les have been widely adopted for practical systems due to their low computational complexity however it is also well known that les provide inferior performance relative to a maximum likelihood equalizer mle or other near mles because les usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information more importantly unlike mle or near mles the performance of les has not been well quantified this hinders more general applications of les in wireless systems in this paper we reveal a fundamental link between a channel parameter orthogonality deficiency od of the channel matrix and the diversity and capacity of les we identify that when the od of channel matrix has an upper bound strictly less than 1 the same diversity order as that of mles is collected by les and the outage capacity loss relative to mles is also a constant over signal to noise ratio snr these results can be applied to designing a framework for hybrid equalizers furthermore by studying the statistical properties of the od and comparing the complexity of different equalizers we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix od the theoretical analysis is corroborated by computer simulations

the covering number of the elements of a matroid and generalized matrix functions abstract let s be a nonempty finite set with cardinality m let m be a matroid on s with no loops the covering number of an element x in s is the smallest positive integer k such that x is a coloop of the union of k copies of m we investigate connections between the structure of m and the values of the covering numbers of elements of s applications to the study of the rank partition and generalized matrix functions are presented

on posterior distributions for signals in gaussian noise with unknown covariance matrix a bayesian approach to estimate parameters of signals embedded in complex gaussian noise with unknown color is presented the study specifically focuses on a bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems by integrating out uncertainties regarding the noise color an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited several noninformative priors for the covariance matrix such as the reference prior the jeffreys prior and modifications to this are considered some of the priors result in analytical solutions whereas others demand numerical approximations in the linear signal model connections are made between the standard adaptive maximum likelihood aml estimate and a bayesian solution using the jeffreys prior with adjustments to the jeffreys prior correspondence to the regularized solution is also established this in turn enables a formal treatment of the regularization parameter simulations indicate that significant improvements compared to the aml estimator can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior the simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged

comparing between estimation approaches admissible and dominating linear estimators we treat the problem of evaluating the performance of linear estimators for estimating a deterministic parameter vector x in a linear regression model with the mean squared error mse as the performance measure since the mse depends on the unknown vector x a direct comparison between estimators is a difficult problem here we consider a framework for examining the mse of different linear estimation approaches based on the concepts of admissible and dominating estimators we develop a general procedure for determining whether or not a linear estimator is mse admissible and for constructing an estimator strictly dominating a given inadmissible method so that its mse is smaller for all x in particular we show that both problems can be addressed in a unified manner for arbitrary constraint sets on x by considering a certain convex optimization problem we then demonstrate the details of our method for the case in which x is constrained to an ellipsoidal set and for unrestricted choices of x as a by product of our results we derive a closed form solution for the minimax mse estimator on an ellipsoid which is valid for arbitrary model parameters as long as the signal to noise ratio exceeds a certain threshold

robust mean squared error estimation in the presence of model uncertainties we consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties where the vector x is known to satisfy a weighted norm constraint we first assume that the model is known exactly and seek the linear estimator that minimizes the worst case mean squared error mse across all possible values of x we show that for an arbitrary choice of weighting the optimal minimax mse estimator can be formulated as a solution to a semidefinite programming problem sdp which can be solved very efficiently we then develop a closed form expression for the minimax mse estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of mayer and willke with a specific choice of shrinkage factor that explicitly takes the prior information into account next we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst case mse across all possible values of x and all possible values of the model matrix as we show the robust minimax mse estimator can also be formulated as a solution to an sdp finally we demonstrate through several examples that the minimax mse estimator can significantly increase the performance over the conventional least squares estimator and when the model matrix is subject to uncertainties the robust minimax mse estimator can lead to a considerable improvement in performance over the minimax mse estimator

hybrid fm polynomial phase signal modeling parameter estimation and cramer rao bounds parameter estimation for a class of nonstationary signal models is addressed the class contains combination of a polynomial phase signal pps and a frequency modulated fm component of the sinusoidal or hyperbolic type such signals appear in radar and sonar applications involving moving targets with vibrating or rotating components a novel approach is proposed that allows us to decouple estimation of the fm parameters from those of the pps relying on properties of the multilag high order ambiguity function ml haf the accuracy achievable by any unbiased estimator of the hybrid fm pps parameters is investigated by means of the cramer rao lower bounds crlbs both exact and large sample approximate expressions of the bounds are derived and compared with the performance of the proposed methods based on monte carlo simulations

quasi ml period estimation from incomplete timing data given a noisy sequence of possibly shifted integer multiples of a certain period it is often of interest to accurately estimate the period with known integer regressors the problem is classical linear regression in many applications however the regressors are unknown integers and only loose bounds on the period are available examples include hop period and timing estimation wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band pulse repetition interval pri analysis and passive rotating beam radio scanning we study several pertinent period estimators our emphasis is on a quasi maximum likelihood approach developed herein and an earlier method based on the fourier transform of a dirac delta train representation of the data surprisingly both are capable of attaining the clairvoyant crame spl acute r rao bound at moderate signal to noise ratios snrs even for short e g 10 samples we carefully address parameter identifiability issues and corroborate our findings with extensive simulations

robust rls in the presence of correlated noise using outlier sparsity relative to batch alternatives the recursive least squares rls algorithm has well appreciated merits of reduced complexity and storage requirements for online processing of stationary signals and also for tracking slowly varying nonstationary signals however rls is challenged when in addition to noise outliers are also present in the data due to e g impulsive disturbances existing robust rls approaches are resilient to outliers but require the nominal noise to be white an assumption that may not hold in e g sensor networks where neighboring sensors are affected by correlated ambient noise prewhitening with the known noise covariance is not a viable option because it spreads the outliers to noncontaminated measurements which leads to inefficient utilization of the available data and unsatisfactory performance in this correspondence a robust rls algorithm is developed capable of handling outliers and correlated noise simultaneously in the proposed method outliers are treated as nuisance variables and estimated jointly with the wanted parameters identifiability is ensured by exploiting the sparsity of outliers which is effected via regularizing the least squares ls criterion with the l1 norm of the outlier vectors this leads to an optimization problem whose solution yields the robust rls estimates for low complexity real time operation a suboptimal online algorithm is proposed which entails closed form updates per time step in the spirit of rls simulations demonstrate the effectiveness and improved performance of the proposed method in comparison with the nonrobust rls and its state of the art robust renditions

mv pure estimator minimum variance pseudo unbiased reduced rank estimator for linearly constrained ill conditioned inverse problems this paper proposes a novel estimator named minimum variance pseudo unbiased reduced rank estimator mv pure for the linear regression model designed specially for the case where the model matrix is ill conditioned and the unknown deterministic parameter vector to be estimated is subjected to known linear constraints as a natural generalization of the gauss markov blue estimator the mv pure estimator is a solution of the following hierarchical nonconvex constrained optimization problem directly related to the mean square error expression in the first stage optimization under a rank constraint we minimize simultaneously all unitarily invariant norms of an operator applied to the unknown parameter vector in view of suppressing bias of the proposed estimator then in the second stage optimization among all pseudo unbiased reduced rank estimators defined as the solutions of the first stage optimization we find the one achieving minimum variance we derive a closed algebraic form of the mv pure estimator and show that well known estimators the gauss markov blue estimator the generalized marquardt s reduced rank estimator and the minimum variance conditionally unbiased affine estimator subject to linear restrictions are all special cases of the mv pure estimator we demonstrate the effectiveness of the proposed estimator in a numerical example where we employ the mv pure estimator to the ill conditioned problem of reconstructing a 2 d image subjected to linear constraints from blurred noisy observation this example demonstrates that the mv pure estimator outperforms all aforementioned estimators as it achieves smaller mean square error for all values of signal to noise ratio