empirical bayes confidence intervals for selected parameters in high dimensional data modern statistical problems often involve a large number of populations and hence a large number of parameters that characterize these populations it is common for scientists to use data to select the most significant populations such as those with the largest t statistics the scientific interest often lies in studying and making inferences regarding these parameters called the selected parameters corresponding to the selected populations the current statistical practices either apply a traditional procedure assuming there were no selection a practice that is not valid or they use the bonferroni type procedure that is valid but very conservative and often noninformative in this article we propose valid and sharp confidence intervals that allow scientists to select parameters and to make inferences for the selected parameters based on the same data this type of confidence interval allows the users to zero in on the most interesting selected parameters without collecting more data the validity of

robust mse equalizer design for mimo communication systems in the presence of model uncertainties this paper considers a robust mean square error mse equalizer design problem for multiple input multiple output mimo communication systems with imperfect channel and noise information at the receiver when the channel state information csi and the noise covariance are known exactly at the receiver a minimum mean square error mmse equalizer can be employed to estimate the transmitted signal however in actual systems it is necessary to take into account channel and noise estimation errors we consider here a worst case equalizer design problem where the goal is to find the equalizer minimizing the equalization mse for the least favorable channel model within a neighborhood of the estimated model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and estimated channel models lagrangian optimization is used to convert this min max problem into a convex min min problem over a convex domain which is solved by interchanging the minimization order the robust mse equalizer and associated least favorable channel model can then be obtained by solving numerically a scalar convex minimization problem simulation results are presented to demonstrate the mse and bit error rate ber performance of robust equalizers when applied to the least favorable channel model

exact sample conditioned mse performance of the bayesian mmse estimator for classification error part i representation in recent years biomedicine has been faced with difficult high throughput small sample classification problems in such settings classifier error estimation becomes a critical issue because training and testing must be done on the same data a recently proposed error estimator places the problem in a signal estimation framework in the presence of uncertainty permitting a rigorous solution optimal in a minimum mean square error sense the uncertainty in this model is relative to the parameters of the feature label distributions resulting in a bayesian approach to error estimation closed form solutions are available for two important problems discrete classification with dirichlet priors and linear classification of gaussian distributions with normal inverse wishart priors in this work part i of a two part study we introduce the theoretical mean square error mse conditioned on the observed sample of any estimate of the classifier error including the bayesian error estimator for both bayesian models thus bayesian error estimation has a unique advantage in that its mathematical framework naturally gives rise to a practical expected measure of performance given an observed sample in part ii of the study we examine consistency of the error estimator demonstrate various mse properties and apply the conditional mse to censored sampling

minimax mse ratio estimation with signal covariance uncertainties in continuation to an earlier work we further consider the problem of robust estimation of a random vector or signal with an uncertain covariance matrix that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix while in the earlier work we developed and proposed a competitive minimax approach of minimizing the worst case mean squared error mse difference regret criterion here we study in the same spirit the minimum worst case mse ratio regret criterion namely the worst case ratio rather than difference between the mse attainable using a linear estimator ignorant of the exact signal covariance and the minimum mse mmse attainable by optimum linear estimation with a known signal covariance we present the optimal linear estimator under this criterion in two ways the first is as a solution to a certain semidefinite programming sdp problem and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure we then show that the linear minimax ratio regret estimator can also be interpreted as the mmse estimator that minimizes the mse for a certain choice of signal covariance that depends on the uncertainty region we demonstrate that in applications the proposed minimax mse ratio regret approach may outperform the well known minimax mse approach the minimax mse difference regret approach and the plug in approach where in the latter one uses the mmse estimator with an estimated covariance matrix replacing the true unknown covariance

bayesian analysis of covariance matrices and dynamic models for longitudinal data summary parsimonious modelling of the within subject covariance structure while heeding its positive definiteness is of great importance in the analysis of longitudinal data using the cholesky decomposition and the ensuing unconstrained and statistically meaningful reparameterisation we provide a convenient and intuitive framework for developing conditionally conjugate prior distributions for covariance matrices and show their connections with generalised inverse wishart priors our priors offer many advantages with regard to elicitation positive definiteness computations using gibbs sampling shrinking covariances toward a particular structure with considerable flexibility and modelling covariances using covariates bayesian estimation methods are developed and the results are compared using two simulation studies these simulations suggest simpler and more suitable priors for the covariance structure of longitudinal data

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

estimation in covariance components models abstract estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes the estimation of fixed and random effects when the variances and covariances are known is presented in bayesian terms point estimates of the unknown variances and covariances are computed using the em algorithm for maximum likelihood estimation from incomplete data the techniques are illustrated with data on law schools field mice and professional football teams

signal detection under mismatch corresp a binary detection problem of the neymann pearson type in which the probability density functions used are inaccurate versions of the true ones are considered the performance of the above suboptimal detection scheme as the number of observations increases is investigated a necessary and sufficient condition is given for the exponential convergence to zero of the two error probabilities as the number of observations increases the condition is in terms of an inequality between differences of asymptotic per sample informational divergence expressions

the generalized mandelbrot set perturbed by composing noise of additive and multiplicative adopting the experimental mathematics method combining complex variable function theory with computer aided drawing this paper researches on the structural characteristic and the fission evolution law of the generalized mandelbrot set generalized m set in short perturbed by composing noise of additive and multiplicative analyzes the effect of random perturbation to the generalized m set

physical meaning for mandelbrot and julia sets abstract we consider the dynamics of a kicked charged particle moving in a double well potential and a time dependent magnetic field in certain cases the stroboscopic dynamics reduces to the complex logistic map thus providing physical meaning for the mandelbrot set in other cases we obtain iterated function systems consisting of the inverse complex logistic map thus providing physical meaning for julia sets our approach can be generalized to complex mappings with a maximum of order q

optimal control of uncertain stochastic systems subject to total variation distance uncertainty this paper is concerned with optimization of uncertain stochastic systems in which uncertainty is described by a total variation distance constraint between the measures induced by the uncertain systems and the measure induced by the nominal system while the payoff is a linear functional of the uncertain measure robustness at the abstract setting is formulated as a minimax game in which the control seeks to minimize the payoff over the admissible controls while the uncertainty aims at maximizing it over the total variation distance constraint it is shown that the maximizing measure in the total variation distance constraint exists while the resulting payoff is a linear combination of l 1 and l infty norms further the maximizing measure is characterized by a linear combination of a tilted measure and the nominal measure giving rise to a payoff which is a nonlinear functional on the space of measures to be minimized over the admissible controls the abstract formulation and results are subsequ

1 s expansion for generalized dimensions in a hierarchical s state potts model we consider the s state potts model on the diamond hierarchical lattice for large s we show that the generalized dimensions dq of the density of zeros supported by the associated julia set are given by dq 1 q 1 s 2 3 4 log 2 o s 1 the information dimension d1 equals 1 to all orders

asymptotic and essentially singular solutions of the feigenbaum equation for suitably defined largen we express feigenbaum s equation as a singular schroder functional equation whose solution is obtained using a scaling ansatz in the limit of infiniten certain self consistency conditions on the scaled schroder solution lead to an essentially singular solution of feigenbaum s equation with a length scale factor of 0 0333 and a limiting feigenvalue of 30 50 in agreement with eckmann and wittwer s value of 0 0333831 and their conjectured estimate of 30

optimal classifiers with minimum expected error within a bayesian framework part ii properties and performance analysis in part i of this two part study we introduced a new optimal bayesian classification methodology that utilizes the same modeling framework proposed in bayesian minimum mean square error mmse error estimation optimal bayesian classification thus completes a bayesian theory of classification where both the classifier error and our estimate of the error may be simultaneously optimized and studied probabilistically within the assumed model having developed optimal bayesian classifiers in discrete and gaussian models in part i here we explore properties of optimal bayesian classifiers in particular invariance to invertible transformations convergence to the bayes classifier and a connection to bayesian robust classifiers we also explicitly derive optimal bayesian classifiers with non informative priors and explore relationships to linear and quadratic discriminant analysis lda and qda which may be viewed as plug in rules under gaussian modeling assumptions finally we present several simulations addressing the robustness of optimal bayesian classifiers to false modeling assumptions companion website http gsp tamu edu publications supplementary dalton12a

minimax robust deconvolution filters under stochastic parametric and noise uncertainties the author consider the design of robust deconvolution filters for linear discrete time systems with stochastic parameter and noise uncertainties it is assumed that some large but bounded uncertainties exist in the driving and measurement noise covariances as well as the second order statistics of stochastic parameters and initial conditions three kinds of minimax sensitivity criteria are used to develop the techniques to the synthesis of minimax deconvolution filters under uncertain linear stochastic systems their approach is based on saddle point theory and the sensitivity analysis of kalman filters the design algorithms give the recursive realization of the minimax deconvolution filters for the time varying uncertain systems under fairly general conditions for the time invariant uncertain case the existence and solutions of steady state deconvolution filters are further developed finally the utility of the minimax design approaches and the properties of the resulting minimax deconvolution filters are illustrated by a numerical example

mandelbrot set in coupled logistic maps and in an electronic experiment we suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps the idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into a set of two identical real one dimensional quadratic maps with a particular coupling hence dynamical behavior of similar nature may occur in coupled dissipative nonlinear systems which relate to the feigenbaum universality class to substantiate the feasibility of this concept we consider an electronic system which exhibits dynamical phenomena intrinsic to complex analytic maps experimental results are presented providing the mandelbrot set in the parameter plane of this physical system

timing estimation for a filtered poisson process in gaussian noise the problem of estimation of time shift of an inhomogeneous casually filtered poisson process in the presence of additive gaussian noise is discussed approximate expressions for the likelihood function the map estimator and the mmse estimator that becomes increasingly accurate as the per unit time density of superimposed filter responses becomes small are obtained the optimal map estimator takes the form of a cascade of linear and memoryless nonlinear components for smooth point process intensities the performance of the map estimator is studied via local bias and local variance a rate distortion type lower bound on the mse of any estimator of time delay is then derived by identification of a communications channel that accounts for the mapping from time delay to observation process results of numerical studies of estimator performance are presented based on the examples considered it is concluded 1 the small error mse of the nonlinear map estimator can be significantly better than the small error mse of the optimal linear estimator 2 the rate distortion lower bound can be significantly tighter than the poisson limited bounds determined in previous studies

misiurewicz point patterns generation in one dimensional quadratic maps abstract in a family of one dimensional quadratic maps misiurewicz points are unstable and the orbits of such a points are repulsive on the contrary the orbits of superstable periodic points are attractive here we study the patterns of the symbolic sequences of both misiurewicz and superstable periodic points and show that a misiurewicz point pattern can be obtained as the limit of the sum of a superstable periodic orbit pattern plus itself or some of its heredity transmitters repeated an infinite number of times inversely when a misiurewicz point pattern is given we also show that it is possible to find both the superstable periodic orbit pattern and the heredity transmitters that generate such a pattern

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

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

the rate distortion function on classes of sources determined by spectral capacities the quantity sup a in cal r r a d is considered where cal r is a class of homogeneous n parameter sources and r a d denotes the single letter mean square error mse rate distortion function for the individual source a in particular the case in which the class cal r is specified in terms of spectral information is treated for general classes of spectral measures whose upper measures are capacities in the sense of choquet alternating of order two this type of class includes many common models for spectral uncertainty such as mixture models spectral band models and neighborhoods generated by variation and prohorov metrics it is shown that each such class contains a worst case source whose rate distortion function achieves the supremum over the class for each value of distortion this source is characterized as having a spectral density that is a derivative in the sense of huber and strassen of the upper spectral measure with respect to lebesgue measure on pi pi n moreover it is shown that the spectral measure of the worst case source is closest in a sense defined by directed divergence to lebesgue measure which corresponds to a memoryless source numerical results are presented for the particular case in which the source spectral measure is a mixture of a gauss markov spectrum and an unknown contaminating component

nonlinear estimation for a class of systems this paper considers nonlinear estimation problems for classes of models and employs relative entropy to describe the uncertainty classes two optimization problems are formulated on general banach spaces and their solutions are sought 1 when the transition probability between the signal to be estimated x and the measurement y or stochastic kernel is unknown and 2 when the joint probability induced by the random variables rvs x y is unknown for both problems the uncertainty is described by a relative entropy constraint between the unknown distribution and a fixed nominal distribution the results include existence of the optimal measures using weak convergence techniques and properties associated with the estimate of the true distribution classical examples are chosen to illustrate the applicability of the results

robust deconvolution of deterministic and random signals we consider the problem of designing an estimation filter to recover a signal x n convolved with a linear time invariant lti filter h n and corrupted by additive noise our development treats the case in which the signal x n is deterministic and the case in which it is a stationary random process both formulations take advantage of some a priori knowledge on the class of underlying signals in the deterministic setting the signal is assumed to have bounded weighted energy in the stochastic setting the power spectra of the signal and noise are bounded at each frequency the difficulty encountered in these estimation problems is that the mean squared error mse at the output of the estimation filter depends on the problem unknowns and therefore cannot be minimized beginning with the deterministic setting we develop a minimax mse estimation filter that minimizes the worst case point wise mse between the true signal x n and the estimated signal over the class of bounded norm inputs we then establish that the mse at the output of the minimax mse filter is smaller than the mse at the output of the conventional inverse filter for all admissible signals next we treat the stochastic scenario for which we propose a minimax regret estimation filter to deal with the power spectrum uncertainties this filter is designed to minimize the worst case difference between the mse in the presence of power spectrum uncertainties and the mse of the wiener filter that knows the correct power spectra the minimax regret filter takes the entire uncertainty interval into account and as demonstrated through an example can often lead to improved performance over traditional minimax mse approaches for this problem

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

bayesian correlation estimation we propose prior probability models for variance covariance matrices in order to address two important issues first the models allow a researcher to represent substantive prior information about the strength of correlations among a set of variables secondly even in the absence of such information the increased flexibility of the models mitigates dependence on strict parametric assumptions in standard prior models for example the model allows a posteriori different levels of uncertainty about correlations among different subsets of variables we achieve this by including a clustering mechanism in the prior probability model clustering is with respect to variables and pairs of variables our approach leads to shrinkage towards a mixture structure implied by the clustering we discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models including the evaluation of normalising constants that are functions of parameters of interest the normalising constants result from the restriction that the correlation matrix be positive definite we discuss examples based on simulated data a stock return dataset and a population genetics dataset

operating with external arguments in the mandelbrot set antenna the external argument theory of douady and hubbard allows us to know both the potential and the field lines in the exterior of the mandelbrot set nonetheless there are no explicit formulae to operate with external arguments and the external argument theory is difficult to apply in this paper we introduce some tools in order to obtain formulae to operate with external arguments in the mandelbrot set antenna thus we introduce the harmonic tool to calculate both the external arguments of the period doubling cascade hyperbolic components and the external arguments of the last appearance hyperbolic components likewise we introduce composition rules applied to external arguments that with the aid of the concept of heredity allows the calculation of all the external arguments that constitutes the family tree of a given external argument

image detection under low level illumination image detection under low light level conditions is treated as a hypothesis testing problem in which the observations are modeled as a shot noise process since computing the likelihood ratio for shot noise processes is not feasible the use of a one dimensional test statistic obtained by filtering and sampling the observations is proposed the filter is chosen to maximize a generalized signal to noise ratio the likelihood ratio for the one dimensional test statistic is evaluated numerically by inverting the corresponding characteristic function under each hypothesis

minimax linear smoothing for capacities the case of correlated signals and noise results of an earlier paper giving minimax linear smoothers for the problem of estimating a homogeneous signal field in an additive orthogonal noise field when both have uncertain spectral properties are extended to the case in which the signal and noise fields are arbitrarily correlated as before spectral uncertainty is modeled by assuming that the spectral measures of the signal and noise fields lie in classes of measures generated by two alternating choquet capacities it is demonstrated that this problem admits a general solution in terms of the huber strassen derivative between the capacities that generate the uncertainty sets and that the least favorable spectra for smoothing in orthogonal noise are also the least favorable marginal spectra for smoothing in correlated noise the resulting filter is seen to be a zonal filter that also arises as the solution to an analogous problem in nonparametric minimax hypothesis testing these new results extend the applicability of minimax robust smoothing techniques to application involving signal dependent noise phenomena such as multipath and clutter which are usually difficult to model precisely

posterior propriety and admissibility of hyperpriors in normal hierarchical models hierarchical modeling is wonderful and here to stay but hyperparameter priors are often chosen in a casual fashion unfortunately as the number of hyperparameters grows the effects of casual choices can multiply leading to considerably inferior performance as an extreme but not uncommon example use of the wrong hyperparameter priors can even lead to impropriety of the posterior for exchangeable hierarchical multivariate normal models we first determine when a standard class of hierarchical priors results in proper or improper posteriors we next determine which elements of this class lead to admissible estimators of the mean under quadratic loss such considerations provide one useful guideline for choice among hierarchical priors finally computational issues with the resulting posterior distributions are addressed

intrinsically optimal bayesian robust filtering when designing optimal filters it is often unrealistic to assume that the statistical model is known perfectly the issue is then to design a robust filter that is optimal relative to an uncertainty class of processes robust filter design has been treated from minimax best worst case performance and bayesian best average performance perspectives heretofore the bayesian approach has involved finding a model specific optimal filter one that is optimal for some model in the uncertainty class lifting this constraint we optimize over the full class from which the original optimal filters were obtained for instance over all linear filters by extending the original characteristics determining the filter such as the power spectral density to effective characteristics that apply across the uncertainty class we demonstrate for both linear and morphological filtering that an intrinsically optimal bayesian robust filter can be represented in the same form as the standard solution to the optimal filter except via the effective characteristics solutions for intrinsic bayesian robust filters are more transparent and intuitive than solutions for model specific filters and also less tedious because effective characteristics push through the spectral theory into the bayesian setting whereas solutions in the model specific case depend on grinding out the optimization

additive perturbed generalized mandelbrot julia sets abstract adopting the experimental mathematics method combining complex variable function theory with computer aided drawing this paper researches on the structural characteristic and the fission evolution law of additive perturbed generalized mandelbrot julia sets generalized m j sets in short the corresponding relationship between point coordinates in generalized m set and the general structure of generalized j sets has been founded qualitatively and the physical meaning of the generalized m j sets has been expounded the following conclusions are deduced 1 chaotic patterns of fractal structure of generalized j sets may emerge out of double periodic bifurcation which shows that brownian movement can be chaotic 2 experimental evidence of li yorke theorem is given out 3 the additive perturbed generalized m set contains abundant information on the construction of generalized j sets 4 resemble logistic map in the process of a series of double periodic bifurcation coming into chaos generalized j sets also present self similarity in parameter space

quantifying the objective cost of uncertainty in complex dynamical systems real world problems often involve complex systems that cannot be perfectly modeled or identified and many engineering applications aim to design operators that can perform reliably in the presence of such uncertainty in this paper we propose a novel bayesian framework for objective based uncertainty quantification uq which quantifies the uncertainty in a given system based on the expected increase of the operational cost that it induces this measure of uncertainty called mocu mean objective cost of uncertainty provides a practical way of quantifying the effect of various types of system uncertainties on the operation of interest furthermore the proposed uq framework provides a general mathematical basis for designing robust operators and it can be applied to diverse applications including robust filtering classification and control we demonstrate the utility and effectiveness of the proposed framework by applying it to the problem of robust structural intervention of gene regulatory networks an important application in translational genomics

an rkhs approach to robust l sup 2 estimation and signal detection the application of rkhs reproducing kernel hilbert space theory to the problems of robust signal detection and estimation is investigated it is shown that this approach provides a general and unified framework in which to analyze the problems of l sup 2 estimation matched filtering and quadratic detection in the presence of uncertainties regarding the second order structure of the random processes involved

effect of noise on the dynamics of a complex map at the period tripling accumulation point as shown recently o b isaeva et al phys rev e 64 055201 2001 the phenomena intrinsic to dynamics of complex analytic maps under appropriate conditions may occur in physical systems we study scaling regularities associated with the effect of additive noise upon the period tripling bifurcation cascade generalizing the renormalization group approach of crutchfield et al phys rev lett 46 933 1981 and shraiman et al phys rev lett 46 935 1981 originally developed for the period doubling transition to chaos in the presence of noise the universal constant determining the rescaling rule for the intensity of the noise in period tripling is found to be gamma 12 206 640 9 em leader numerical evidence of the expected scaling is demonstrated

robust matched filters the problem of designing robust matched filters for situations in which there is uncertainty in the signal structure or noise statistics is considered two general aspects of this problem are treated first maximin robust designs are characterized for a general hilbert space formulation of the matched filtering problem and explicit solutions are given for two intuitively appealing models for uncertainty these results are seen to generalize earlier results for specific matched filtering problems the second general aspect treated is the application of the theoretical maximin results to the particular problem of designing filters to combat uncertain nonlinear channel distortion channel distortion is modeled by considering an unknown received signal which may differ in l 2 norm from the transmitted or nominal signal by no more than a fixed amount the effect of the channel distortion is seen to be equivalent to that of adding a white noise to the channel whose spectral height depends on the degree of distortion general expressions are developed for the determination of the robust filter and its performance and numerical results are presented for the case of baseband detection in gauss markov noise

hierarchical poisson regression modeling abstract the poisson model and analyses here feature nonexchangeable gamma distributions although exchangeable following a scale transformation for individual parameters with standard deviations proportional to means a relatively uninformative prior distribution for the shrinkage values eliminates the ill behavior of maximum likelihood estimators of the variance components when tested in simulation studies the resulting procedure provides better coverage probabilities and smaller risk than several other published rules and thus works well from bayesian and frequentist perspectives alike the computations provide fast accurate density approximations to individual parameters and to structural regression coefficients the computer program is publicly available through statlib

nonconjugate bayesian estimation of covariance matrices and its use in hierarchical models abstract the problem of estimating a covariance matrix in small samples has been considered by several authors following early work by stein this problem can be especially important in hierarchical models where the standard errors of fixed and random effects depend on estimation of the covariance matrix of the distribution of the random effects we propose a set of hierarchical priors hps for the covariance matrix that produce posterior shrinkage toward a specified structure here we examine shrinkage toward diagonality we then address the computational difficulties raised by incorporating these priors and nonconjugate priors in general into hierarchical models we apply a combination of approximation gibbs sampling possibly with a metropolis step and importance reweighting to fit the models and compare this hybrid approach to alternative markov chain monte carlo methods our investigation involves three alternative hps the first works with the spectral decomposition of the covariance matrix an

optimal robust classifiers qualitatively a filter is said to be robust if its performance degradation is acceptable for distributions close to the one for which it is optimal that is the one for which it has been designed this paper adapts the signal processing theory of optimal robust filters to classifiers the distribution class conditional distributions to which the classifier is to be applied is parameterized by a state vector and the principle issue is to choose a design state that is optimal in comparison to all other states relative to some measure of robustness a minimax robust classifier is one whose worst performance over all states is better than the worst performances of the other classifiers defined at the other states a bayesian robust classifier is one whose expected performance is better than the expected performances of the other classifiers the state corresponding to the bayesian robust classifier is called the maximally robust state minimax robust classifiers tend to give too much weight to states for which classification is very difficult and therefore our effort is focused on bayesian robust classifiers whereas the signal processing theory of robust filtering concentrates on design with full distributional knowledge and a fixed number of observation variables features design via training from sample data and feature selection are so important for classification that robustness optimality must be considered from these perspectives in particular for small samples in this context for a given sample size we will be concerned with the maximally robust state feature pair all definitions are independent of the classification rule however applications are only considered for linear and quadratic discriminant analysis for which there are parametric forms for the optimal discriminants

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

period doubling in maps with a maximum of order z abstract we consider the scaling behaviour in period doubling systems exemplified by the one dimensional map n 1 1 n z which has a maximum of order z z 1 the feigenbaum scaling factors and are studied as functions of z and more generally the scaling functions 1 and a in particular using the universal functions g x and h x we establish the inequality

lower bounds for parametric estimation with constraints a chapman robbins form of the barankin bound is used to derive a multiparameter cramer rao cr type lower bound on estimator error covariance when the parameter theta in r sup n is constrained to lie in a subset of the parameter space a simple form for the constrained cr bound is obtained when the constraint set theta sub c can be expressed as a smooth functional inequality constraint it is shown that the constrained cr bound is identical to the unconstrained cr bound at the regular points of theta sub c i e where no equality constraints are active on the other hand at those points theta in theta sub c where pure equality constraints are active the full rank fisher information matrix in the unconstrained cr bound must be replaced by a rank reduced fisher information matrix obtained as a projection of the full rank fisher matrix onto the tangent hyperplane of the full rank fisher matrix onto the tangent hyperplane of the constraint set at theta a necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable and examples for which the bound is achieved are presented in addition to providing a useful generalization of the cr bound the results permit analysis of the gain in achievable mse performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization

intrinsically bayesian robust kalman filter an innovation process approach in many contemporary engineering problems model uncertainty is inherent because accurate system identification is virtually impossible owing to system complexity or lack of data on account of availability time or cost the situation can be treated by assuming that the true model belongs to an uncertainty class of models in this context an intrinsically bayesian robust ibr filter is one that is optimal relative to the cost function in the classical sense and the prior distribution over the uncertainty class in the bayesian sense ibr filters have previously been found for both wiener and granulometric morphological filtering in this paper we derive the ibr kalman filter that performs optimally relative to an uncertainty class of state space models introducing the notion of bayesian innovation process and the bayesian orthogonality principle we show how the problem of designing an ibr kalman filter can be reduced to a recursive system similar to the classical kalman recursive equations except with effective counterparts such as the effective kalman gain matrix after deriving the recursive ibr kalman equations for discrete time we use the limiting method to obtain the ibr kalman bucy equations for continuous time finally we demonstrate the utility of the proposed framework for two real world problems sensor networks and gene regulatory network inference

robust matched filters for optical receivers the problem of designing optical receivers that are robust against uncertainty in the statistics of the observation process in photodetection is investigated in particular a modification in the design of the postdetection matched filter is proposed to account for possible uncertainty in the rate function of the incident light the rate of the dark current and in the statistics of the additive noise present at the input to the optical receiver this design is based on a game theoretic approach in which a filter is sought that has the maximum worst case output signal to noise ratio possible over the class of allowable statistics that is the design criterion is maximin signal to noise ratio a general characterization of maximin robust matched filters for observed poisson processes is presented in this context and specific solutions for several useful uncertainty models are obtained numerical results are presented for a specific example to illustrate the performance of the proposed technique

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

brief paper an analysis of the effects of spectral uncertainty on wiener filtering a representative selection of results from an extensive study of the performance of wiener filtering under spectral uncertainty is presented for a variety of spectral uncertainty models the wiener filter is shown to have undesirable sensitivity to even small deviations from those signal and noise spectral densities which were used to design the filter performance of a robust filter designed to have the best possible performance when the uncertainty is worst is also examined in most cases the robust filter s insensitivity to spectral uncertainty makes it preferable to the traditional wiener filter

performance bounds for mismatched decision schemes with poisson process observations abstract this paper develops a framework for analyzing the performance loss in fixed time interval decision algorithms that are based on observations of time inhomogeneous poisson processes when some parameters characterizing the observation process are not known exactly key to the development is the formulation of an analytically computable performance metric which can be used in lieu of the true but intractable error probabilities the proposed metric is obtained by identifying analytical upper bounds on the error probabilities in terms of the uncertain parameters using these tools it is shown that performance degrades gracefully as long as the true values of the parameters remain within a neighborhood of the nominal values used in decision making the results find direct application to problems of detecting illicit nuclear materials in transit

minimax linear smoothing for capacities abstract minimax linear smoothers are considered for the problem of estimating a homogeneous signal field in an additive orthogonal noise field a minimax game with the quadratic mean estimation error as an objective function is used to formulate this problem uncertainty in signal and noise field spectra is modeled using general nonparametric classes of measures proposed by huber and strassen for the problem of minimax hypothesis testing these classes which are described in terms of choquet alternating capacities of order 2 include the conventional models for spectral uncertainty and admit a general solution to the minimax linear smoothing problem author

noise perturbed generalized mandelbrot sets adopting the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing in this paper on the structure characteristics and the discontinuity evolution law of the additive noise perturbed generalized mandelbrot sets m sets was studied on the influence of stochastic perturbed parameters of the structure of generalized m sets was analyzed the physical meaning of the additive noise perturbed generalized m sets was expounded

strange objects in the complex plane julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems technically they are the closure of the set of unstable cycles of analytic maps physically they are sets of points which lead to chaotic behavior the mapf z z2 p is analyzed for smallp where the julia set is a closed curve and for largep where the julia set is completely disconnected in both cases the hausdorff dimension is calculated in perturbation theory and numerically an expression for the rate at which points escape from the neighborhood of the julia set is derived and tested in a numerical simulation of the escape

performance bounds for discrimination problems with uncertain statistics corresp decision designs that are insensitive to modeling uncertainty are developed for the chernoff bounds on the performance of binary hypothesis testing problems these designs are based on observations with statistical uncertainty modeled by using general classes generated by 2 alternating capacities the results are illustrated for the two cases of independent identically distributed observations with uncertainty in the probability distribution and discrete time stationary gaussian observations with spectral uncertainty and they are applicable to several other cases as well for the chernoff upper bounds on the error probabilities a robust decision design based on the i kel hood ratio test between a least favorable pair of probability distributions or spectral measures respectively is derived it is then shown that for all elements in the uncertainty class this choice of likelihood ratio guarantees the exponential convergence of the aforementioned chernoff bounds to zero as the number of observations or the length of the observation interval increases

error probability bounds for nuclear detection a collection of static and mobile radiation sensors is tasked with deciding within a fixed time interval whether a moving target carries radioactive material formally this is a problem of detecting weak time inhomogeneous poisson signals target radiation concealed in another poisson signal naturally occurring background radiation each sensor locally processes its observations to form a likelihood ratio which is transmitted once at the end of the decision interval to a fusion center the latter combines the transmitted information to optimally in the neyman pearson sense decide whether the measurements contain a radiation signal or just noise we provide a set of analytically derived upper bounds for the probabilities of false alarm and missed detection which are used to design threshold tests without the need for computationally intensive monte carlo simulations these analytical bounds couple the physical quantities of interest to facilitate planning the motion of the mobile sensors for minimizing the probability of missed detection the network reconfigures itself in response to the target motion to allow more accurate collective decisions within the given time interval the approach is illustrated in numerical simulations and its effectiveness demonstrated in experiments that emulate the statistics of nuclear emissions using a pulsed laser

bayesian regression with network prior optimal bayesian filtering perspective the recently introduced intrinsically bayesian robust filter ibrf provides fully optimal filtering relative to a prior distribution over an uncertainty class of joint random process models whereas formerly the theory was limited to model constrained bayesian robust filters for which optimization was limited to the filters that are optimal for models in the uncertainty class this paper extends the ibrf theory to the situation where there are both a prior on the uncertainty class and sample data the result is optimal bayesian filtering obf where optimality is relative to the posterior distribution derived from the prior and the data the ibrf theories for effective characteristics and canonical expansions extend to the obf setting a salient focus of the present study is to demonstrate the advantages of bayesian regression within the obf setting over the classical bayesian approach in the context of linear gaussian models

inference for multivariate normal hierarchical models this paper provides a new method and algorithm for making inferences about the parameters of a two level multivariate normal hierarchical model one has observed j p dimensional vector outcomes distributed at level 1 as multivariate normal with unknown mean vectors and with known covariance matrices at level 2 the unknown mean vectors also have normal distributions with common unknown covariance matrix a and with means depending on known covariates and on unknown regression coefficients the algorithm samples independently from the marginal posterior distribution of a by using rejection procedures functions such as posterior means and covariances of the level 1 mean vectors and of the level 2 regression coefficient are estimated by averaging over posterior values calculated conditionally on each value of a drawn this estimation accounts for the uncertainty in a unlike standard restricted maximum likelihood empirical bayes procedures it is based on independent draws from the exact posterior distributions unlike gibbs sampling the procedure is demonstrated for profiling hospitals based on patients responses concerning p 2 types of problems non surgical and surgical the frequency operating characteristics of the rule corresponding to a particular vague multivariate prior distribution are shown via simulation to achieve their nominal values in that setting

maximum entropy and robust prediction on a simplex the related problems of finite length robust prediction and maximizing spectral entropy over a simplex of covariance matrices are considered general properties of iterative solutions of these problems are developed and monotone convergence proofs are presented for two algorithms that provide such solutions the analogous problems for simplexes of spectral densities are also considered

a probabilistic approach to multivariable robust filtering and open loop control a new approach to robust filtering prediction and smoothing of discrete time signal vectors is presented linear time invariant filters are designed to be insensitive to spectral uncertainty in signal models the goal is to obtain a simple design method leading to filters which are not overly conservative modeling errors are described by sets of models parameterized by random variables with known covariances these covariances could either be estimated from data or be used as robustness tuning knobs a robust design is obtained by minimizing the spl hscrsub 2 norm or equivalently the mean square estimation error averaged with respect to the assumed model errors a polynomial solution based on an averaged spectral factorization and a unilateral diophantine equation is derived the robust estimator is referred to as a cautious wiener filter it turns out to be only slightly more complicated to design than an ordinary wiener filter the methodology can be applied to any open loop filtering or control problem in particular we illustrate this for the design of robust multivariable feedforward regulators decoupling and model matching filters

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

robust least squares estimation with a relative entropy constraint given a nominal statistical model we consider the minimax estimation problem consisting of finding the best least squares estimator for the least favorable statistical model within a neighborhood of the nominal model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and nominal models for a gaussian nominal model and a finite observations interval or for a stationary gaussian process over an infinite interval the usual noncausal wiener filter remains optimal however the worst case performance of the filter is affected by the size of the neighborhood representing the model uncertainty on the other hand standard causal least squares estimators are not optimal and a characterization is provided for the causal estimator and the corresponding least favorable model the causal estimator takes the form of a risk sensitive estimator with an appropriately selected risk sensitivity coefficient

a competitive minimax approach to robust estimation of random parameters we consider the problem of estimating in the presence of model uncertainties a random vector x that is observed through a linear transformation h and corrupted by additive noise we first assume that both the covariance matrix of x and the transformation h are not completely specified and develop the linear estimator that minimizes the worst case mean squared error mse across all possible covariance matrices and transformations h in the region of uncertainty although the minimax approach has enjoyed widespread use in the design of robust methods we show that its performance is often unsatisfactory to improve the performance over the minimax mse estimator we develop a competitive minimax approach for the case where h is known but the covariance of x is subject to uncertainties and seek the linear estimator that minimizes the worst case regret namely the worst case difference between the mse attainable using a linear estimator ignorant of the signal covariance and the optimal mse attained using a linear estimator that knows the signal covariance the linear minimax regret estimator is shown to be equal to a minimum mse mmse estimator corresponding to a certain choice of signal covariance that depends explicitly on the uncertainty region we demonstrate through examples that the minimax regret approach can improve the performance over both the minimax mse approach and a plug in approach in which the estimator is chosen to be equal to the mmse estimator with an estimated covariance matrix replacing the true unknown covariance we then show that although the optimal minimax regret estimator in the case in which the signal and noise are jointly gaussian is nonlinear we often do not lose much by restricting attention to linear estimators

robust competitive estimation with signal and noise covariance uncertainties robust estimation of a random vector in a linear model in the presence of model uncertainties has been studied in several recent works while previous methods considered the case in which the uncertainty is in the signal covariance and possibly the model matrix but the noise covariance is assumed to be completely specified here we extend the results to the case where the noise statistics may also be subjected to uncertainties we propose several different approaches to robust estimation which differ in their assumptions on the given statistics in the first method we assume that the model matrix and both the signal and the noise covariance matrices are uncertain and develop a minimax mean squared error mse estimator that minimizes the worst case mse in the region of uncertainty the second strategy assumes that the model matrix is given and tries to uniformly approach the performance of the linear minimum mse estimator that knows the signal and noise covariances by minimizing a worst case regret measure the regret is defined as the difference or ratio between the mse attainable using a linear estimator ignorant of the signal and noise covariances and the minimum mse possible when the statistics are known as we show earlier solutions follow directly from our more general results however the approach taken here in developing the robust estimators is considerably simpler than previous methods

on the robust discrimination of poisson random counting measures corresp statistical decisions are considered between two hypotheses which consist of classes of poisson distributions for random counting measures each poisson distribution is generated by an intensity measure on a general observation space the classes are specified by choquet capacity bounds on the intensity measures this problem was first posed and studied by geraniotis and poor a minimax neyman pearson result for error probability performance is the main contribution recursive computation of the minimax test statistic is also investigated

robust wiener filtering with non parametric spectral uncertainty in this article robust least squares filtering problems are considered for non parametric multivariate spectral uncertainty defined by the so called spectral band and generalised moment constraints its major aim is to provide a basis for computing approximate solutions to worst case wiener filtering optimisation problems involving causal filters and multivariate signals it hinges upon associating upper and lower bounds on the minimum worst case performance achievable with causal filters with linear cost linear matrix inequality lc lmi constraint optimisation problems on the basis of a lagrangean duality formulation for the worst case least squares performance of a given filter upper bounds on it are obtained as the optimal values of lc lmi problems then for linearly parameterised classes of filter transfer functions a causal filter which optimises such an upper bound on worst case performance can also be obtained from an lc lmi optimisation problem to estimate the amount of conservatism incur

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

fast em type implementations for mixed effects models the mixed effects model in its various forms is a common model in applied statistics a useful strategy for fitting this model implements em type algorithms by treating the random effects as missing data such implementations however can be painfully slow when the variances of the random effects are small relative to the residual variance in this paper we apply the working parameter approach to derive alternative em type implementations for fitting mixed effects models which we show empirically can be hundreds of times faster than the common em type implementations in our limited simulations they also compare well with the routines in s plus and stata in terms of both speed and reliability the central idea of the working parameter approach is to search for efficient data augmentation schemes for implementing the em algorithm by minimizing the augmented information over the working parameter and in the mixed effects setting this leads to a transfer of the mixed effects variances into the regression slope parameters we also describe a variation for computing the restricted maximum likelihood estimate and an adaptive algorithm that takes advantage of both the standard and the alternative em type implementations

bounds on the unstable eigenvalue for period doubling bounds are given for the unstable eigenvalue of the period doubling operator for unimodal maps of the interval these bounds hold for all types of behaviour x r of the interval map near its critical point they are obtained by finding cones in function space which are invariant under the tangent map to the doubling operator at its fixed point

research on fractal structure of generalized m j sets utilized lyapunov exponents and periodic scanning techniques abstract in order to show details of fractal structure of mandelbrot set precisely lyapunov exponents and periodic scanning techniques have been brought forward by shirriff and welstead this paper generalizes these two techniques and puts forward periodicity orbit search and comparison technique which can be used to discuss the relationship of the generalized mandelbrot julia sets the generalized m j sets adopting the techniques mentioned above and the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing this paper researches on the structure topological inflexibility and the discontinuity evolution law of the generalized m j sets generated from the complex mapping z z c r and explores structure and distributing of periodicity petal and topological law of periodicity orbits of the generalized m sets and finds that the generalized m set contains abundant information of structure of the generalized j sets by founding the whole portray of the generalized j sets based on the generalized m set qualitatively furthermore the physical meaning of the generalized m j sets have been expounded

on robust wiener signal estimation a minimax formulation is considered for the problem of designing robust linear causal estimators of continuous time signals in noise it is shown that the minimax estimation problem can generally be reduced to finding a least favorable pair of spectra in particular robust predictors filters and finite lag smoothers are shown to be just the optimal predictors filters and smoothers respectively for a least favorable pair this result can often greatly simplify the design of robust estimators

on numerical approximations of the area of the generalized mandelbrot sets in the present work the area of the generalized mandelbrot sets is defined as the double limit of the areas of the plotted generalized mandelbrot sets in a given square lattice using the finite escape algorithm while the lattice resolution and the number of iteration counts used to plot them tends to infinity the asymptotic behavior of the areas of the generalized mandelbrot sets in terms of their degree growth is investigated finally numerical approximations of the area of the mandelbrot set are proposed by using tools from regression analysis

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

on the invariant sets of a family of quadratic maps the julia setb for the mappingz z 2 is considered where is a complex parameter for 2 a new upper bound for the hausdorff dimension is given and the monic polynomials orthogonal with respect to the equilibrium measure onb are introduced a method for calculating all of the polynomials is provided and certain identities which obtain among coefficients of the three term recurrence relations are given a unifying theme is the relationship betweenb and chains which is explored for 1 4 2 and for with 1 4 with the aid of the bottcher equation thenb is shown to be a holder continuous curve for 1 4

on the p point uncertainty class corresp the p point uncertainty class also known as sakrison s class b is considered although the p point class is not a 2 alternating choquet capacity class it is shown how many of the robust signal processing and communication theory results which have been developed for capacity classes can be applied to the p point class by embedding it in a slightly larger extended p point class

the generalized m sets from a class complex mapping system the method constructing the generalized mandelbrot sets the generalized m sets for positive integer index number generated from the composition of two simple complex mapping which was put forward by shirriff was expanded based on a class simple complex mapping system expanded by the author a series of the generalized m sets for real index number were constructed using the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing the structural characteristics and evolutions of the generalized m sets were studied the results show that the generalized m sets for integer index number have symmetry and fractal feature while the generalized m sets for decimal index number have discontinuity and collapse and their evolutions depend on the choice of the principal range of the phase angle and the author sets forth the physical meaning for the generalized m sets

minimax robust discrete time matched filters the problem of designing finite length discrete time matched filters is considered for situations in which exact knowledge of the input signal and or noise characteristics is not available such situations arise in many applications due to channel distortion incoherencies nonlinear effects and other modeling uncertainties in such cases it is often of interest to design a minimax robust matched filter i e a nonadaptive filter with an optimum level of worst case performance for the expected uncertainty class this problem is investigated here for three types of uncertainty models for the input signal namely the mean absolute mean square and maximum absolute distortion classes and for a wide generality of norm deviation models for the noise covariance matrix some numerical examples illustrate the robustness properties of the proposed designs

empirical bayes confidence sets for the mean of a multivariate normal distribution abstract through the use of an empirical bayes argument a confidence set for the mean of a multivariate normal distribution is derived the set is a recentered sphere is easy to compute and has uniformly smaller volume than the usual confidence set an exact formula for the coverage probability is derived and numerical evidence is presented which shows that the empirical bayes set uniformly dominates the usual set in coverage probability

fast low rank bayesian matrix completion with hierarchical gaussian prior models the problem of low rank matrix completion is considered in this paper to exploit the underlying low rank structure of the data matrix we propose a hierarchical gaussian prior model where columns of the low rank matrix are assumed to follow a gaussian distribution with zero mean and a common precision matrix and a wishart distribution is specified as a hyperprior over the precision matrix we show that such a hierarchical gaussian prior has the potential to encourage a low rank solution based on the proposed hierarchical prior model we develop a variational bayesian matrix completion method which embeds the generalized approximate massage passing technique to circumvent cumbersome matrix inverse operations simulation results show that our proposed method demonstrates superiority over some state of the art matrix completion methods

a system of alternately excited coupled non autonomous oscillators manifesting phenomena intrinsic to complex analytical maps a feasible model is introduced that manifests phenomena intrinsic to iterative complex analytical maps such as the mandelbrot set and julia sets the system is composed of two alternately excited coupled oscillators the idea is based on a turn by turn transfer of the excitation from one subsystem to another s p kuznetsov example of a physical system with a hyperbolic attractor of the smale williams type phys rev lett 95 2005 144101 accompanied with appropriate non linear transformation of the complex amplitude of the oscillations in the course of the process analytical and numerical studies are performed special attention is paid to an analysis of the violation of the applicability of the slow amplitude method with the decrease in the ratio of the period of the excitation transfer to the basic period of the oscillations the main effect is the rotation of the mandelbrot like set in the complex parameter plane one more effect is the destruction of subtle small scale fractal structure of the set due to the presence of non analytical terms in the complex amplitude equations

minimax discrimination for observed poisson processes with uncertain rate functions the problem of robust design is considered in the context of testing hypotheses concerning the rate function of an observed point process designs that are insensitive to uncertainty in the rate functions are developed by applying a minimax formulation to two different measures of signal to noise ratio uncertainty in the rate is modeled by using general classes of rate measures generated by choquet 2 alternating capacities and solutions are characterized for this case by a radon nikodym type derivative between such classes it is shown that for uncertainty within capacity classes the robust decision design developed for the signal to noise ratio is also robust in a weaker sense for the chernoff upper bounds on the error probabilities furthermore the use of such a test guarantees the exponential convergence of these bounds to zero with increasing length of the observation interval for all rates in the uncertainty class