the sensitivity of the stable lyapunov equation we present an analysis of the sensitivity of the solution of the lyapunov equation a x xa w where a is stable this analysis leads to a spectral norm bound on the relative perturbation of the solution which is optimal for a certain class of estimates and which is essentially equivalent to the frobenius norm bound obtained from the associated kronecker product system the latter bound can be expressed in terms of operatorname sep a a and is known to accurately reflect the sensitivity of the lyapunov problem but it is hard to interpret in terms of the original matrix a in contrast the spectral norm bound which we derive is directly related to the minimal l 2 damping of the dynamical system dot z az moreover this dynamical link with the sensitivity problem leads to a new method of systematically investigating the norm behavior of e at as well as providing a wealth of information about control theoretic aspects of dot z az when a is the closed loop state matrix

solution of lyapunov equations by alternating direction implicit iteration abstract in this report a new procedure is presented for solving the lyapunov matrix equation first the system is reduced to tridiagonal form with gaussian similarity transformations then the resulting system is solved with alternating direction implicit adi iteration a matrix commutation property essential for model problem convergence of adi iteration applied to elliptic difference equations is not needed for this application all stable lyapunov matrix equations are model adi problems

model reduction for discrete unstable systems based on generalized normal representations the input and output normal representations of stable systems are generalized for unstable systems these are then used to reduce the order of unstable discrete time linear multivariable systems resulting in reduced order models with the same number of unstable poles as the full order model and with an a priori upper bound on the reduction error the generalized normal representations are then modified to include a frequency weighting resulting in frequency weighted generalized normal representations these are used to derive frequency weighted reduced order models for unstable discrete systems with the same number of unstable poles as the full order model and with an a priori upper bound on the frequency weighted reduction error

a general class of lower bounds in parameter estimation a general class of bayesian lower bounds on moments of the error in parameter estimation is formulated and it is shown that the cramer rao the bhattacharyya the bobrovsky zakai and the weiss weinstein lower bounds are special cases in the class the bounds can be applied to the estimation of vector parameters and any given function of the parameters the extension of these bounds to multiple parameter is discussed

measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

approximate solution of large sparse lyapunov equations describes a simple method for efficiently estimating the dominant eigenvalues and eigenvectors of the solution to a lyapunov equation without first solving the equation explicitly the method is based on the power method and matrix vector multiplications and is particularly suitable for problems where those multiplications can be done efficiently such as where the coefficient matrices are large and sparse or low rank the same idea is directly applicable to balanced truncation order reduction of linear systems

brief accuracy enhancing methods for balancing related frequency weighted model and controller reduction we consider the solution of the balancing related frequency weighted model and controller reduction problems using accuracy enhanced numerical algorithms we propose first new stability enforcing choices of the frequency weighted grammians which can guarantee the stability of reduced models for two sided frequency weights then we show that for the frequency weighted controller reduction problems with standard stability and performance enforcing frequency weights the computation of the frequency weighted grammians can be done by solving reduced order lyapunov equations for both frequency weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their cholesky factors this allows the extension of the square root and balancing free accuracy enhancing techniques to the frequency weighted case

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

hilbert space embeddings and metrics on probability measures a hilbert space embedding for probability measures has recently been proposed with applications including dimensionality reduction homogeneity testing and independence testing this embedding represents any probability measure as a mean element in a reproducing kernel hilbert space rkhs a pseudometric on the space of probability measures can be defined as the distance between distribution embeddings we denote this as k indexed by the kernel function k that defines the inner product in the rkhs we present three theoretical properties of k first we consider the question of determining the conditions on the kernel k for which k is a metric such k are denoted characteristic kernels unlike pseudometrics a metric is zero only when two distributions coincide thus ensuring the rkhs embedding maps all distributions uniquely i e the embedding is injective while previously published conditions may apply only in restricted circumstances e g on compact domains and are difficult to check our conditions are straightforward and intuitive integrally strictly positive definite kernels are characteristic alternatively if a bounded continuous kernel is translation invariant on d then it is characteristic if and only if the support of its fourier transform is the entire d second we show that the distance between distributions under k results from an interplay between the properties of the kernel and the distributions by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies third to understand the nature of the topology induced by k we relate k to other popular metrics on probability measures and present conditions on the kernel k under which k metrizes the weak topology

aspects of nonnormality for iterative methods abstract recently new optimal krylov subspace methods have been discovered for normal matrices in light of this novel ways to quantify nonnormality are considered in connection with various families of matrices we use as a criterion how for a given matrix these iterative methods introduced can be employed via e g inexpensive matrix factorizations the unitary orbit of the set of binormal matrices provides a natural extension of normal matrices its elements yield polynomially normal matrices of moderate degree in this context several matrix nearness problems arise

aliasing of polynomial phase signal parameters aliasing of sampled monocomponent polynomial phase signals is investigated the results are applicable to general sampling schemes and it is shown that aliasing can be avoided by sampling the signal nonuniformly in time examples illustrate how this property can be exploited in doppler radar in order to avoid ambiguity in estimates of the kinematic parameters of the target

multiplicative error bound for balanced stochastic truncation model reduction the multiplicative error for the balanced stochastic truncation model reduction method is discussed and an h sub infinity norm error bound is presented the result together with a previous result of the authors syst contr lett vol 14 p 307 14 1990 establishes that the same error bound applies to the relative error and the multiplicative error

a jacobi type method for computing balanced realizations a new numerical scheme for computing balancing coordinate transformations in linear systems theory is presented the method is closely related to the jacobi method for diagonalizing symmetric matrices here the minimization of the sum of traces of the gramians by orthogonal and nonorthogonal jacobi type rotations is considered the algorithm is shown to be globally convergent to a balancing transformation that arranges the hankel singular values in a prescribed ordering local quadratic convergence of the algorithm is shown

computation of system balancing transformations and other applications of simultaneous diagonalization algorithms an algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization the algorithm requires no squaring up or unnecessary matrix products various algorithmic aspects are discussed in detail a key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product other contragredient transformation applications are also described it is further shown that a similar approach may be taken involving the generalized singular value decomposition to the classical simultaneous diagonalization problem these svd based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

multi level substructuring combined with model order reduction methods abstract in many fields of engineering problems linear time invariant dynamical systems lti systems play an outstanding role they result for instance from discretizations of the unsteady heat equation and they are also used in optimal control problems often the order of lti systems is a limiting factor since it becomes easily very large as a consequence these systems cannot be treated efficiently without model reduction algorithms in this paper a new approach for the combination of model order reduction methods and recent multi level substructuring mls techniques is presented similar multi level substructuring methods have already been applied successfully to huge eigenvalue problems up to several millions of degrees of freedom however the presented approach does not make use of a modal analysis like former algorithms instead the original system is decomposed in smaller lti systems which are treated with recent model reduction methods furthermore the error which is induced by this substructuring approach is analysed and numerical examples based on the oberwolfach benchmark collection are given in this paper

alternating direction implicit iteration for systems with complex spectra the alternating direction implicit adi iteration model problem has hitherto been the discretized dirichlet problem with an spd matrix splitting for which optimum iteration parameters are obtained as the solution due to w b jordan of a rational chebyshev minimax problem over the reals recently disclosed application to a class of sylvester equations bf ax bf xb bf c with bf a and bf b positive real matrices required generalization to complex spectra in the positive real half plane theory is developed here for applying jordan s parameters to complex domains which remain close to the real line and for predicting associated error reduction numerical studies are presented in support of this theory this analysis provided a basis for analysis of more general complex spectra which is reported in another paper

extended ziv zakai lower bound for vector parameter estimation the bayesian ziv zakai bound on the mean square error mse in estimating a uniformly distributed continuous random variable is extended for arbitrarily distributed continuous random vectors and for distortion functions other than mse the extended bound is evaluated for some representative problems in time delay and bearing estimation the resulting bounds have simple closed form expressions and closely predict the simulated performance of the maximum likelihood estimator in all regions of operation

a generalized adi iterative method the adi iterative method for the solution of sylvester s equationax xb c proceeds by strictly alternating between the solution of the two equations begin gathered left a delta k 1 i right x 2k 1 x 2k left b delta k 1 i right c hfill x 2k 2 left b tau k 1 i right left a tau k 1 i right x 2k 1 c hfill end gathered fork 0 1 2 herex o is a given initial approximate solution and the k and k are real or complex parameters chosen so that the computed approximate solutionsx k converge rapidly to the solutionx of the sylvester equation ask increases this paper discusses the possibility of solving one of the equations in the adi iterative method more often than the other one i e relaxing the strict alternation requirement in order to achieve a higher rate of convergence our analysis based on potential theory shows that this generalization of the adi iterative method can give faster convergence than when strict alternation is required

closed form grammians and model reduction for flexible space structures the author derives analytical expressions for the grammians of a model given in modal coordinates for the dynamics of a flexible space structure fss these expressions are then used as the basis for a new algorithm to compute a dominant reduced order model for such a system in an efficient manner the resulting reduced order model is obtained directly rather than by truncating a balanced model of the entire system thus giving rise to significant efficiency gains for the typical fss case where the reduced order model is of very much lower order than the original system

algorithms for model reduction of large dynamical systems three algorithms for the model reduction of large scale continuous time time invariant linear dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described they rely on low rank approximations to the controllability and observability gramians which can efficiently be computed by adi based iterative low rank methods the first two model reduction methods are closely related to the well known square root method and schur method which are balanced truncation techniques the third method is a heuristic balancing free technique the performance of the model reduction algorithms is studied in numerical experiments

coprime factors reduction methods for linear parameter varying and uncertain systems abstract we present a generalization of the coprime factors model reduction method of meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models a complete derivation of coprime factorizations for this class of systems is also given the reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured l 2 induced stability constraint required in the standard nonfactored case reduction error bounds in the l 2 induced norm of the factorized mapping are given

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

balanced truncation of linear time varying systems in this paper balanced truncation of linear time varying systems is studied in discrete and continuous time based on relatively basic calculations with time varying lyapunov equations inequalities we are able to derive both upper and lower error bounds for the truncated models these results generalize well known time invariant formulas the case of time varying state dimension is considered input output stability of all truncated balanced realizations is also proven the method is finally successfully applied to a high order model

equivalence of kernel machine regression and kernel distance covariance for multidimensional phenotype association studies summary associating genetic markers with a multidimensional phenotype is an important yet challenging problem in this work we establish the equivalence between two popular methods kernel machine regression kmr and kernel distance covariance kdc kmr is a semiparametric regression framework that models covariate effects parametrically and genetic markers non parametrically while kdc represents a class of methods that include distance covariance dc and hilbert schmidt independence criterion hsic which are nonparametric tests of independence we show that the equivalence between the score test of kmr and the kdc statistic under certain conditions can lead to a novel generalization of the kdc test that incorporates covariates our contributions are 3 fold 1 establishing the equivalence between kmr and kdc 2 showing that the principles of kmr can be applied to the interpretation of kdc 3 the development of a broader class of kdc statistics where the class members are statistics corresponding to different kernel combinations finally we perform simulation studies and an analysis of real data from the alzheimer s disease neuroimaging initiative adni study the adni study suggest that snps of flj16124 exhibit pairwise interaction effects that are strongly correlated to the changes of brain region volumes

a case study in model reduction of linear time varying systems in this paper the balanced truncation procedure is applied to time varying linear systems both in continuous and in discrete time the methods are applied to a linear approximation of a diesel exhaust catalyst model the reduced order systems are obtained by using certain projections instead of direct balancing an approximate zero order hold discretization of continuous time systems is described and a new a priori approximation error bound for balanced truncation in the discrete time case is obtained the case study shows that there are several advantages to work in discrete time it gives simpler implementation with fewer computations

kernel methods for measuring independence we introduce two new functionals the constrained covariance and the kernel mutual information to measure the degree of independence of random variables these quantities are both based on the covariance between functions of the random variables in reproducing kernel hilbert spaces rkhss we prove that when the rkhss are universal both functionals are zero if and only if the random variables are pairwise independent we also show that the kernel mutual information is an upper bound near independence on the parzen window estimate of the mutual information analogous results apply for two correlation based dependence functionals introduced earlier we show the kernel canonical correlation and the kernel generalised variance to be independence measures for universal kernels and prove the latter to be an upper bound on the mutual information near independence the performance of the kernel dependence functionals in measuring independence is verified in the context of independent component analysis

analysis of the variance threshold of kay s weighted linear predictor frequency estimator a theoretical approximation for the variance of kay s weighted linear predictor frequency estimator is derived from this expression an inequality describing the variance threshold of the estimator is found the window weights are then optimized to improve the variance numerical simulations demonstrate that the variance approximations are valid for medium to high signal to noise ratios or for large numbers of samples

semiparametric regression of multidimensional genetic pathway data least squares kernel machines and linear mixed models we consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least squares kernel machines lskms this unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way this semiparametric model also makes it possible to test for the overall genetic pathway effect we show that the lskm semiparametric regression can be formulated using a linear mixed model estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software both the regression coefficients of the covariate effects and the lskm estimator of the genetic pathway effect can be obtained using the best linear unbiased predictor in the corresponding linear mixed model formulation the smoothing parameter and the kernel parameter can be estimated as variance components using restricted maximum likelihood a score test is developed to test for the genetic pathway effect model variable selection within the lskm framework is discussed the methods are illustrated using a prostate cancer data set and evaluated using simulations

numerical solution of the lyapunov equation by approximate power iteration abstract we present the approximate power iteration api algorithm for the computation of the dominant invariant subspace of the solution x of large order lyapunov equations ax xa t q 0 without first computing the matrix x itself the api algorithm is an iterative procedure that uses krylov subspace bases in computing estimates of matrix vector products x v in a power iteration sequence application of the api algorithm requires that a a t x admits a good low rank solution then api provides an orthogonal basis of a subspace that closely approximates the dominant x invariant subspace of corresponding dimension analytical convergence results are also presented

on distance based permutation tests for between group comparisons summary permutation tests based on distances among multivariate observations have found many applications in the biological sciences two major testing frameworks of this kind are multiresponse permutation procedures and pseudo f tests arising from a distance based extension of multivariate analysis of variance in this paper we derive conditions under which these two frameworks are equivalent the methods and equivalence results are illustrated by reanalyzing an ecological data set and by a novel application to functional magnetic resonance imaging data

balancing related methods for minimal realization of periodic systems we propose balancing related numerically reliable methods to compute minimal realizations of linear periodic systems with time varying dimensions the first method belongs to the family of square root methods with guaranteed enhanced computational accuracy and can be used to compute balanced minimal order realizations an alternative balancing free square root method has the advantage of a potentially better numerical accuracy in case of poorly scaled original systems the key numerical computation in both methods is the solution of nonnegative periodic lyapunov equations directly for the cholesky factors of the solutions for this purpose a numerically reliable computational algorithm is proposed to solve nonnegative periodic lyapunov equations with time varying dimensions

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

a note on closed loop balanced truncation closed loop balanced truncation is a model reduction method that aims at preserving the closed loop properties of the original controller we show that this method is equivalent to frequency weighted balanced truncation with certain weighting and that for observer based controllers a significant amount of computation can be saved because of the observer state feedback structure of the controller

robust model reduction of uncertain systems maintaining uncertainty structure this article presents a new approach for or robust model reduction for uncertain systems based on an optimisation algorithm applied directly in the space of reduced order model parameters the proposed approach can be applied to both polytopic or affine parameter dependent models as well as for continuous or discrete time systems the developed procedure is capable of computing both fixed reduced order model as well as uncertain reduced order model keeping the original uncertainty structure the resulting reduced order model is guaranteed to be robust stable

a reflection on the implicitly restarted arnoldi method for computing eigenvalues near a vertical line abstract in this article we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted arnoldi method the extension to computing eigenvalues closest to a vertical line is straightforward by incorporating a shift without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis in a recent publication meerbergen and spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to hopf bifurcations which is of interest for the stability of dynamical systems the novel method is based on inverse iteration inverse power method applied on a lyapunov like eigenvalue problem to reduce the computational overhead significantly a projection was added this method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane we will prove in this paper that the combination of inverse iteration with the projection step is equivalent to sorensen s implicitly restarted arnoldi method utilizing well chosen shifts

frequency interval model reduction of bilinear systems in this technical note a new method for model reduction of bilinear systems is presented the method is developed in particular for many applications in which one is interested to approximate a system in a given frequency interval to this end new generalized frequency interval gramians are introduced for bilinear systems it is shown that these gramians are the solutions to the so called frequency interval generalized lyapunov equations the conditions for these equations to be solvable are derived and an algorithm is proposed to solve such equations iteratively the method is further illustrated with the help of an illustrative example the numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians

extremal probabilities for gaussian quadratic forms denote by q an arbitrary positive semidefinite quadratic form in centered gaussian random variables such that e q 1 we prove that for an arbitrary x 0 inf q p q x p 2 n n x where n 2 is a chi square distributed rv with n n x degrees of freedom n x is a non increasing function of x n 1 iff x x 1 1 5364 n 2 iff x x 2 x 1 where x 2 1 2989 etc n x rank q a similar statement is not true for the supremum if 1 x 2 and z 1 z 2 are independent standard gaussian rv s then sup0 1 2p z 1 2 1 z 2 2 x is taken not at 0 or at 1 2 but at 0 x 1 2 where x is a continuous increasing function from 1 0 to 2 1 2 e g 1 5 15 applications of our theorems include asymptotic quantiles of u and v statistics signal detection and stochastic orderings of integrals of squared gaussian processes

a posteriori error bounds for discrete balanced truncation balanced truncation of discrete linear time invariant systems is an automatic method once an error tolerance is specified and yields an a priori error bound which is why it is widely used in engineering for simulation and control we present some new insight into this method we derive a discrete version of antoulas s mathcal h 2 norm error formula cite p 218 ant05 and show how to adapt it to some special cases this error bound is an a posteriori computable upper bound for the mathcal h 2 norm of the error system defined as the system whose transfer function corresponds to the difference between the transfer function of the original system and the transfer function of the reduced system the main advantage of our results is that we use the information already available in the balanced truncation algorithm in order to compute the mathcal h 2 norm instead of computing one gramian of the corresponding error system there is always a computational restriction on solving high dimensional stein equations for gramians the a posteriori bound gives insight into the quality of the reduced system and can be used to solve many problems accompanying the order reduction operation

the sylvester equation and approximate balanced reduction abstract the purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems the resulting approach is completely automatic once an error tolerance is specified and also yields an error bound this is to be contrasted with existing projection methods namely pade via lanczos pvl and rational krylov which do not satisfy these properties our approach is based on the computation and approximation of the cross gramian of the system the cross gramian is the solution of a sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system it is well suited to large scale problems because there are no matrix factorizations of the large sparse system matrix only matrix vector products are required

outlier probability of generalized chirpogram based estimators for multicomponent polynomial phase signals outliers are often encountered by estimators that have a nonlinear nature in this paper the probability of outliers is derived concerning the generalized chirpogram gc based estimators for polynomial phase signals pps to derive the approximate theoretical expressions we develop an approach that is based on the extrema theory of complex gaussian envelope and the asymptotic poisson model the semi aliasing effect of pps is also considered in order to predict the outliers more accurately then expressions of the probability of outliers are presented for several typical cases of pps including single tone and chirp signals and their two component versions the gc with the hamming windowing is also incorporated numerical examples of computer simulations are also provided to show the accuracy of the theoretical expressions

statistical analysis of the product high order ambiguity function the high order ambiguity function haf was introduced for the estimation of polynomial phase signals pps embedded in noise since the haf is a nonlinear operator it suffers from noise masking effects and from the appearance of undesired cross terms and possibly spurious harmonics in the presence of multicomponent mc signals the product haf phaf was then proposed as a way to improve the performance of the haf in the presence of noise and to solve the ambiguity problem in this correspondence we derive a statistical analysis of the phaf in the presence of additive white gaussian noise awgn valid for high signal to noise ratio snr and a finite number of data samples the analysis is carried out in detail for single component pps but the multicomponent case is also discussed error propagation phenomena implicit in the recursive structure of the phaf based estimator are explicitly taken into account the analysis is validated by simulation results for both single and multicomponent ppss

complexity reduction through a schur based decomposition for reachability analysis of linear time invariant systems this article presents a method for complexity reduction in reachability analysis and safety preserving controller synthesis via schur based decomposition the decomposition results in either decoupled or weakly coupled lower dimensional subsystems reachable sets computed independently for each subsystem are back projected and intersected to yield an overapproximation of the actual reachable set moreover applying this technique to a class of unstable lti systems we show that when certain eigenvalue and state constraint conditions are satisfied further reduction of complexity is possible evaluating our method for a variety of examples we demonstrate that significant reduction in the computational costs can be achieved this technique has considerable potential utility for use in conjunction with computationally intensive reachability tools

poisson models and mean squared error for correlator estimator of time delay a method for modeling large errors in correlation based time delay estimation is developed in terms of level crossing probabilities the level crossing interpretation for peak ambiguity leads directly to an exact expression for the probability of large error involving the hazard function associated with the level crossing process two models for the distribution of the error over the level crossing time yield approximations to the mean square error mse that involve the low order

an iterative svd krylov based method for model reduction of large scale dynamical systems abstract in this paper we propose a model reduction algorithm for approximation of large scale linear time invariant dynamical systems the method is a two sided projection combining features of the singular value decomposition svd based and the krylov based model reduction techniques while the svd side of the projection depends on the observability gramian the krylov side is obtained via iterative rational krylov steps the reduced model is asymptotically stable matches certain moments and solves a restricted h 2 minimization problem we present modifications to the proposed approach for employing low rank gramians in the reduction step and also for reducing discrete time systems several numerical examples from various disciplines verify the effectiveness of the proposed approach it performs significantly better than the q cover a yousouff r e skelton covariance equivalent realizations with applications to model reduction of large scale systems in c t leondes ed control and dynamic systems vol 22 academic press 1985 pp 273 348 a yousouff d a wagie r e skelton linear system approximation via covariance equivalent realizations journal of math anal and appl 196 1985 91 115 and the least squares s gugercin a c antoulas model reduction of large scale systems by least squares linear algebra appl 415 2 3 2006 290 321 methods that have a similar projection structure to the proposed method also in terms of both the h 2 and h error measures its performance is comparable to or sometimes better than that of balanced truncation moreover the method proves to be robust with respect to the perturbations due to usage of approximate gramians

model reduction via schur decomposition a simple procedure for model reduction based upon a schur decomposition of a given system is presented the obtained partitioned system allows the combination of singular perturbation and aggregation approaches

sor for ax xb c we consider a new approach to the block sor method applied to linear systems of equations which can be written as a matrix equation ax xb c such systems arise for example from finite difference discretizations of seperable elliptic boundary value problems on rectangular domains on one hand this gives us an iterative method for the solution of such matrix equations e g lyapunov s matrix equation where b at and on the other hand the problem of choosing appropriate parameters for the block sor method can be written in a more compact form which may be helpful especially for non self adjoint problems i e if a and b are nonsymmetric using this technique we determine under more general assumptions than those of chin and manteuffel the optimal sor parameters for the model problem of a convection diffusion equation

estimating the doa and the polarization of a polynomial phase signal using a single polarized vector sensor this paper introduces a novel algorithm to estimate the direction of arrival doa and the polarization of a completely polarized polynomial phase signal of an arbitrary degree the algorithm utilizes a polarized vector sensor comprising a spatially collocated six component electromagnetic vector sensor a dipole triad or a loop triad this esprit based algorithm is based on a time invariant matrix pencil pair derived from the time delayed data sets collected by a single polarized vector sensor the high order difference function of the signal s phase constructs the invariant factor used in the esprit algorithm the steering vector is estimated from the signal subspace eigenvector of the data correlation matrix following which the closed form doa and polarization can be obtained given the degree of the polynomial phase signal this approach resolves the two dimensional azimuth elevation angle and the polarization of the source and requires neither a priori knowledge of the polynomial phase signal s coefficients nor a priori knowledge of the polynomial phase signal s frequency spectrum the efficacy of the proposed algorithm is verified by monte carlo simulations estimation accuracies of the doa and the polarization parameters are evaluated by the closed form cramer rao bounds which are independent of the polynomial coefficients the degree of the polynomial phase signal and the azimuth angle of the source

time interval model reduction of bilinear systems in this paper a new method for model reduction of bilinear systems is presented the proposed technique is from the family of gramian based model reduction methods the method uses time interval generalised gramians in the reduction procedure rather than the ordinary generalised gramians and in such a way that it improves the accuracy of the approximation within the time interval for which the method is applied the time interval generalised gramians are the solutions to the generalised time interval lyapunov equations the conditions for these equations to be solvable are derived and an algorithm is proposed to solve these equations iteratively the method is further illustrated with the help of two illustrative examples the numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians

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 barankin bound and threshold behavior in frequency estimation this article presents the barankin 1949 bound as a fundamental statistical tool for the understanding of the threshold effect associated with the estimation of the frequency of a sinusoid in additive white gaussian noise it is shown that the threshold effect takes hold whenever the barankin bound departs significantly from the cramer rao bound in terms of the signal to noise ratio snr and the data length t the quantity snr spl times t ln t is shown to be a good indicator for deciding whether the snr is above threshold or not

increasing power for voxel wise genome wide association studies the random field theory least square kernel machines and fast permutation procedures imaging traits are thought to have more direct links to genetic variation than diagnostic measures based on cognitive or clinical assessments and provide a powerful substrate to examine the influence of genetics on human brains although imaging genetics has attracted growing attention and interest most brain wide genome wide association studies focus on voxel wise single locus approaches without taking advantage of the spatial information in images or combining the effect of multiple genetic variants in this paper we present a fast implementation of voxel and cluster wise inferences based on the random field theory to fully use the spatial information in images the approach is combined with a multi locus model based on least square kernel machines to associate the joint effect of several single nucleotide polymorphisms snp with imaging traits a fast permutation procedure is also proposed which significantly reduces the number of permutations needed relative to the standard empirical method and provides accurate small p value estimates based on parametric tail approximation we explored the relation between 448 294 single nucleotide polymorphisms and 18 043 genes in 31 662 voxels of the entire brain across 740 elderly subjects from the alzheimer s disease neuroimaging initiative adni structural mri scans were analyzed using tensor based morphometry tbm to compute 3d maps of regional brain volume differences compared to an average template image based on healthy elderly subjects we find method to be more sensitive compared with voxel wise single locus approaches a number of genes were identified as having significant associations with volumetric changes the most associated gene was grin2b which encodes the n methyl d aspartate nmda glutamate receptor nr2b subunit and affects both the parietal and temporal lobes in human brains its role in alzheimer s disease has been widely acknowledged and studied suggesting the validity of the approach the various advantages over existing approaches indicate a great potential offered by this novel framework to detect genetic influences on human brains

krylov subspace methods for the sylvester equation abstract we describe galerkin and minimal residual algorithms for the solution of sylvester s equation ax xb c the algorithms use krylov subspaces forwhich orthogonal bases are generated by the arnoldi process for certain choices of krylov subspaces the computation of the solution splits into the solution of many independent subproblems this makes the algorithms suitable for implementation on parallel computers

on the decay rate of hankel singular values and related issues this paper investigates the decay rate of the hankel singular values of linear dynamical systems this issue is of considerable interest in model reduction by means of balanced truncation for instance since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error the decay rate involves a new set of invariants associated with a linear system which are obtained by evaluating a modified transfer function at the poles of the system these considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two lyapunov equations the related problem of determining the decay rate of the eigenvalues of the solution to one lyapunov equation will also be addressed very often these eigenvalues like the hankel singular values are rapidly decaying this fact has motivated the development of several algorithms for computing low rank approximate solutions to lyapunov equations however until now conditions assuring rapid decay have not been well understood such conditions are derived here by relating the solution to a numerically low rank cauchy matrix determined by the poles of the system bounds explaining rapid decay rates are obtained under some mild conditions

the modelling and vibration control of beams with active constrained layer damping abstract the finite element method fem is combined with the golla hughes mctavish ghm model of viscoelastic materials vem to model a cantilever beam with active constrained layer damping treatments this approach avoids time consuming iteration in solving modal frequencies modal damping ratios and responses but the resultant finite element fe model has too many degrees of freedom d o f s from the point of view of control nor is it observable and controllable a new model reduction procedure is proposed an iterative dynamic condensation is performed in the physical space and guyan condensation is taken as an initial iteration approximation a reduced order model rom of suitable size emerges but it is still not observable and controllable accordingly a robust model reduction method is then employed in the state space a numerical example proves that this procedure reduces the model and assures the stability controllability and observability of the final reduced order model from finally a controller is designed by linear quadratic gaussian lqg method based on the from the vibration attenuation is evident

a schur method for balanced truncation model reduction it is shown that a not necessarily balanced state space realization of the moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product pq of the reachability and controllability grammians two specific methods for computing these bases are proposed one based on the ordered schur decomposition of pq and the other based on the cholesky factors of p and q the algorithms perform reliably even for nonminimal models

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

approximated gramians and balanced realization of lightly damped flexible structures this paper presents a method of approximating the balanced realization for lightly damped flexible structures with nonmodal damping the damped system is treated as a perturbation from the undamped system and the controllability and observability gramians are given as a power series in the perturbation scaling factor the approximation utilizes the special structure of second order mechanical systems to obtain closed form expressions for the series coefficient matrices the series form leads also to an approximated balancing algorithm which gives the balancing transformation explicitly

rational matrix gcds and the design of squaring down compensators a state space theory a state space construction for rational matrix greatest common divisors gcds of rational transfer matrices is given it is shown how the gcd results can be used to solve the problem of designing stable minimum phase squaring down compensators for multi variable plants one application is a direct state space construction for such compensators and a state space solution to fat plant h infinity control problems the results make use of the concepts of strongly observable systems and maximally observable systems and build upon the concepts introduced in the state space gcd extraction results for polynomial matrices of l m silverman and p van dooren 1979

thin plate regression splines discuss the production of low rank smoothers for d greater than or equal to 1 dimensional data which can be fitted by regression or penalized regression methods the smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the minimum possible perturbation of the thin plate spline smoothing problem given the dimension of the basis used to construct the smoother by making use of lanczos iteration the basis change and truncation are computationally efficient the smoothers allow the use of approximate thin plate spline models with large data sets avoid the problems that are associated with knot placement that usually complicate modelling with regression splines or penalized regression splines provide a sensible way of modelling interaction terms in generalized additive models provide low rank approximations to generalized smoothing spline models appropriate for use with large data sets provide a means for incorporating smooth functions of more than one variable into non linear models and improve the computational efficiency of penalized likelihood models incorporating thin plate splines given that the approach produces spline like models with a sparse basis it also provides a natural way of incorporating unpenalized spline like terms in linear and generalized linear models and these can be treated just like any other model terms from the point of view of model selection inference and diagnostics

a nonparametric approach for multiple change point analysis of multivariate data change point analysis has applications in a wide variety of fields the general problem concerns the inference of a change in distribution for a set of time ordered observations sequential detection is an online version in which new data are continually arriving and are analyzed adaptively we are concerned with the related but distinct offline version in which retrospective analysis of an entire sequence is performed for a set of multivariate observations of arbitrary dimension we consider nonparametric estimation of both the number of change points and the positions at which they occur we do not make any assumptions regarding the nature of the change in distribution or any distribution assumptions beyond the existence of the th absolute moment for some 0 2 estimation is based on hierarchical clustering and we propose both divisive and agglomerative algorithms the divisive method is shown to provide consistent estimates of both the number and the location of change points under standard

generalized reduced rank latent factor regression for high dimensional tensor fields and neuroimaging genetic applications we propose a generalized reduced rank latent factor regression model grrlf for the analysis of tensor field responses and high dimensional covariates the model is motivated by the need from imaging genetic studies to identify genetic variants that are associated with brain imaging phenotypes often in the form of high dimensional tensor fields grrlf identifies from the structure in the data the effective dimensionality of the data and then jointly performs dimension reduction of the covariates dynamic identification of latent factors and nonparametric estimation of both covariate and latent response fields after accounting for the latent and covariate effects grllf performs a nonparametric test on the remaining factor of interest grrlf provides a better factorization of the signals compared with common solutions and is less susceptible to overfitting because it exploits the effective dimensionality the generality and the flexibility of grrlf also allow various statistical models to be handled in a unified framework and solutions can be efficiently computed within the field of neuroimaging it improves the sensitivity for weak signals and is a promising alternative to existing approaches the operation of the framework is demonstrated with both synthetic datasets and a real world neuroimaging example in which the effects of a set of genes on the structure of the brain at the voxel level were measured and the results compared favorably with those from existing approaches

characterization of threshold for single tone maximum likelihood frequency estimation this paper presents a simple and direct approach to understanding the threshold effect associated with maximum likelihood estimation of the frequency of a single complex tone motivation for the approach stemming from known results in the field of phase locked loops is given it is shown both theoretically and experimentally that the onset of threshold can be directly characterized by a single easily computed parameter namely the cramer rao bound on the phase estimation error variance

equivalence of distance based and rkhs based statistics in hypothesis testing we provide a unifying framework linking two classes of statistics used in two sample and independence testing on the one hand the energy distances and distance covariances from the statistics literature on the other maximum mean discrepancies mmd that is distances between embeddings of distributions to reproducing kernel hilbert spaces rkhs as established in machine learning in the case where the energy distance is computed with a semimetric of negative type a positive definite kernel termed distance kernel may be defined such that the mmd corresponds exactly to the energy distance conversely for any positive definite kernel we can interpret the mmd as energy distance with respect to some negative type semimetric this equivalence readily extends to distance covariance using kernels on the product space we determine the class of probability distributions for which the test statistics are consistent against all alternatives finally we investigate the performance of the family of distance kernels in two sample and independence tests we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels and that other choices from this family can yield more powerful tests

eigenvalue decay bounds for solutions of lyapunov equations the symmetric case abstract we present two new bounds for the eigenvalues of the solutions to a class of continuous and discrete time lyapunov equations these bounds hold for lyapunov equations with symmetric coefficient matrices and right hand side matrices of low rank they reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix