on the equivalence of time and frequency domain maximum likelihood estimation maximum likelihood estimation has a rich history it has been successfully applied to many problems including dynamical system identification different approaches have been proposed in the time and frequency domains in this paper we discuss the relationship between these approaches and we establish conditions under which the different formulations are equivalent for finite length data a key point in this context is how initial and final conditions are considered and how they are introduced in the likelihood function

noisy fir identification as a quadratic eigenvalue problem this correspondence describes a method for identifying fir models in the presence of input and output noise the proposed algorithm takes advantage of both the bias compensation principle and the instrumental variable method it is based on a nonlinear system of equations whose unknowns are the fir coefficients and the input noise variance this system allows mapping the noisy fir identification problem into a quadratic eigenvalue problem the identification problem is thus solved without requiring the use of iterative least squares algorithms the performance of the proposed approach has been tested and compared with that of other identification methods by means of monte carlo simulations

further relaxations of the semidefinite programming approach to sensor network localization recently a semidefinite programming sdp relaxation approach has been proposed to solve the sensor network localization problem although it achieves high accuracy in estimating the sensor locations the speed of the sdp approach is not satisfactory for practical applications in this paper we propose methods to further relax the sdp relaxation more precisely to relax the single semidefinite matrix cone into a set of small size semidefinite submatrix cones which we call a sub sdp ssdp approach we present two such relaxations although they are weaker than the original sdp relaxation they retain the key theoretical property and numerical experiments show that they are both efficient and accurate the speed of the ssdp is even faster than that of other approaches based on weaker relaxations the ssdp approach may also pave a way to efficiently solving general sdp problems without sacrificing the solution quality

a monte carlo method for computing the marginal likelihood in nondecomposable gaussian graphical models summary a centred gaussian model that is markov with respect to an undirected graph g is characterised by the parameter set of its precision matrices which is the cone m g of positive definite matrices with entries corresponding to the missing edges of g constrained to be equal to zero in a bayesian framework the conjugate family for the precision parameter is the distribution with wishart density with respect to the lebesgue measure restricted to m g we call this distribution the g wishart when g is nondecomposable the normalising constant of the g wishart cannot be computed in closed form in this paper we give a simple monte carlo method for computing this normalising constant the main feature of our method is that the sampling distribution is exact and consists of a product of independent univariate standard normal and chi squared distributions that can be read off the graph g computing this normalising constant is necessary for obtaining the posterior distribution of g or the marginal likelihood of the corresponding graphical gaussian model our method also gives a way of sampling from the posterior distribution of the precision matrix

errors in variables identification using maximum likelihood estimation in the frequency domain this paper deals with the identification of errors in variables eiv models corrupted by additive and uncorrelated white gaussian noises when the noise free input is an arbitrary signal not required to be periodic in particular a frequency domain maximum likelihood ml estimator is proposed and analyzed in some detail as some other eiv estimators this method assumes that the ratio of the noise variances is known the estimation problem is formulated in the frequency domain it is shown that the parameter estimates are consistent an explicit algorithm for computing the asymptotic covariance matrix of the parameter estimates is derived the possibility to effectively use lowpass filtered data by using only part of the frequency domain is discussed analyzed and illustrated

the frisch scheme in dynamic system identification abstract the use of the frisch scheme in the identification of linear dynamic systems is investigated in order to describe the whole family of models that can explain given input output noisy sequences unlike the algebraic case it is shown that in general only a single model is compatible with the data these results are first proposed for single input single output systems and then generalized to the multivariable case

finding positive matrices subject to linear restrictions abstract we characterize the existence of a positive definite l l matrix x the entries of which satisfy n nonhomogeneous linear conditions by the existence of a minimum for an associated function v smooth and strictly convex on r n if there exist solutions x 0 then lim x v x and the critical point x 0 of v can be approximated by the conjugate gradients method knowing x 0 provides by a simple analytic formula the unique solution x maximizing the entropy tr x ln x j 1 l j ln j where 1 l are the eigenvalues of x subject to the given restrictions related results are obtained in the semipositive definite case too

frequency domain system identification using arbitrary signals it is the common conviction that frequency domain system identification suffers from the drawback that it cannot handle arbitrary signals without introducing systematic errors this paper shows that it is possible to deal with nonperiodic signals without any approximation and under the same assumptions as in the time domain by estimating simultaneously some initial conditions and the system model parameters

connections between semidefinite relaxations of the max cut and stable set problems we describe links between a recently introduced semidefinite relaxation for the max cut problem and the well known semidefinite relaxation for the stable set problem underlying the lovasz s theta function it turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra we also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations

equivalent relaxations of optimal power flow several convex relaxations of the optimal power flow opf problem have recently been developed using both bus injection models and branch flow models in this paper we prove relations among three convex relaxations a semidefinite relaxation that computes a full matrix a chordal relaxation based on a chordal extension of the network graph and a second order cone relaxation that computes the smallest partial matrix we prove a bijection between the feasible sets of the opf in the bus injection model and the branch flow model establishing the equivalence of these two models and their second order cone relaxations our results imply that for radial networks all these relaxations are equivalent and one should always solve the second order cone relaxation for mesh networks the semidefinite relaxation and the chordal relaxation are equally tight and both are strictly tighter than the second order cone relaxation therefore for mesh networks one should either solve the chordal relaxation or the socp relaxation trading off tightness and the required computational effort simulations are used to illustrate these results

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

an upper bound for the permanent of a nonnegative matrix abstract let a be a fully indecomposable nonnegative matrix of order n with row sums r l r n and let s i equal the smallest positive element in row i of a we prove the permanental inequality per a i 1 n s i i 1 n r i s i and characterize the case of equality in 1984 donald elwin hager and salamon gave a graph theoretic proof of the special case in which a is a nonnegative integer matrix

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

the real positive definite completion problem for a simple cycle abstract we consider the question of whether a real partial positive definite matrix in which the specified off diagonal entries consist of a full n cycle has a positive definite completion this lies in contrast to the previously studied chordal case we give two solutions in one we describe about n 2 independent conditions on angles associated with a normalization of the data that are necessary and sufficient the second is more computational and allows presentation of all positive definite completions as well as answering the existence question

extremal bipartite matrices abstract let g be an undirected graph on vertices 1 n let m g be the convex cone of all positive semidefinite hermitian matrices a satisfying aij 0 if i j is not an edge of g in the case that g is a complete bipartite graph we characterize all extreme rays of m g in doing so we shall also find the maximum rank that an extreme matrix in m g can have

positive semidefinite completions of partial hermitian matrices abstract we classify the ranks of positive semidefinite completions of hermitian band matrices and other partially specified hermitian matrices with chordal graphs and specified main diagonals completing a partially specified matrix means filling in the unspecified entries

sub direct sums and positivity classes of matrices abstract it is well known that a direct sum is positive semidefinite if and only if each of the direct summands is positive semidefinite in fact it is also known that this statement remains true if positive semidefinite is replaced with doubly nonnegative completely positive totally nonnegative m matrix and p matrix etc for each of these classes we consider corresponding questions for a more general sum of two matrices of which the direct sum and ordinary sum are special cases

an efficient recursive total least squares algorithm for fir adaptive filtering an algorithm for recursively computing the total least squares tls solution to the adaptive filtering problem is described this algorithm requires o n multiplications per iteration to effectively track the n dimensional eigenvector associated with the minimum eigenvalue of an augmented sample covariance matrix it is shown that the recursive least squares rls algorithm generates biased adaptive filter coefficients when the filter input vector contains additive noise the tls solution on the other hand is seen to produce unbiased solutions examples of standard adaptive filtering applications that result in noise being added to the adaptive filter input vector are cited computer simulations comparing the relative performance of rls and recursive tls are described

large scale semidefinite programming via a saddle point mirror prox algorithm in this paper we first demonstrate that positive semidefiniteness of a large well structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked in a linear fashion to the matrix we derive also the dual counterpart of the outlined representation which expresses the possibility of positive semidefinite completion of a well structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix using the representations we then reformulate well structured large scale semidefinite problems into smooth convex concave saddle point problems which can be solved by a prox method developed in 6 with efficiency mathcal o epsilon 1 implementations and some numerical results for large scale lovasz capacity and maxcut problems are finally presented

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

on the phases of a complex matrix abstract in this paper we define the phases of a complex sectorial matrix to be its canonical angles which are uniquely determined from a sectorial decomposition of the matrix properties of matrix phases are studied including those of compressions schur complements matrix products and sums in particular by exploiting a notion known as the compound numerical range we establish a majorization relation between the phases of the eigenvalues of ab and the phases of a and b this is then applied to investigate the rank of i a b with phase information of a and b which plays a crucial role in feedback stability analysis a pair of problems banded sectorial matrix completion and decomposition is studied the phases of the kronecker and hadamard products are also discussed

a generalized instrumental variable estimation method for errors in variables identification problems the errors in variables identification problem concerns dynamic systems whose input and output variables are affected by additive noise several estimation methods have been proposed for identifying dynamic errors in variables models in this paper it is shown how a number of common methods for errors in variables methods can be put into a general framework resulting into a generalized instrumental variable estimator give various computational aspects of give are presented and the asymptotic distribution of the parameter estimates is derived

the real positive semidefinite completion problem for series parallel graphs abstract we consider the partial real symmetric matrices x whose diagonal entries are equal to 1 and whose off diagonal entries are specified only on a subset of the positions the question is to determine whether x can be completed to a positive semidefinite matrix extending a result of barrett et al 3 we give a set of necessary conditions for x to be completable and show that these conditions are also sufficient if and only if the graph corresponding to the positions of the specified entries is series parallel i e has no k 4 minor

on the solvability of derived matrix problems including completions and duals abstract a class of m by n real complex matrices is semi algebraic sa if membership in it may be checked via a finite list of polynomial inequalities in the entries the real and imaginary parts of the entries real algebraic geometry is used to show that 1 very many familiar classes of matrices some perhaps surprising are sa and 2 that two important derived problems matrix completions and hadamard duals lead to an sa solution when they come from an sa class this means that at least there exists a finite solution to these problems though it may be difficult to find the methodology likely extends to other problems of interest

sparse sos relaxations for minimizing functions that are summations of small polynomials this paper discusses how to find the global minimum of functions that are summations of small polynomials small means involving a small number of variables some sparse sum of squares sos techniques are proposed we compare their computational complexity and lower bounds with prior sos relaxations under certain conditions we also discuss how to extract the global minimizers from these sparse relaxations the proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems numerical experiments are presented which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems lastly we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization

a decomposition methodology and a class of algorithms for the solution of nonlinear equations in this paper we develop a decomposition methodology to find a solution of a system of nonlinear equations the decomposition methodology is based on the matrix theoretic notion of nonzero transversal and the graph theoretic notion of minimal essential set this decomposition methodology gives rise to a class of three iterative methods each of which uses a derivative matrix whose order is equal to the cardinality of the minimal essential set we prove that the iterative methods are types of the newton method for the case of a system of linear equations we show that the iterative methods are combinations of the generalized gaussian elimination and the newton method

inheritance principles for chordal graphs abstract we prove two results concerning positive completions of partial positive matrices first we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph along a fixed increasing chordal sequence then a formula for the determinant of each completion is obtained in terms of the associated parameters as an application we obtained a proof of a conjecture of w w barrett c r johnson and l rodman regarding an inheritance principle for chordal graphs

positive semidefinite matrix completions on chordal graphs and constraint nondegeneracy in semidefinite programming let g v e be a graph in matrix completion theory it is known that the following two conditions are equivalent i g is a chordal graph ii every g partial positive semidefinite matrix has a positive semidefinite matrix completion in this paper we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to iii for any g partial positive definite matrix that has a positive semidefinite completion constraint nondegeneracy is satisfied at each of its positive semidefinite matrix completions

closed loop optimal experiment design solution via moment extension we consider optimal experiment design for parametric prediction error system identification of linear time invariant multiple input multiple output systems in closed loop when the true system is in the model set the optimization is performed jointly over the controller and the spectrum of the external excitation which can be reparametrized as a joint spectral density matrix the optimal solution consists of first computing a finite set of generalized moments of this spectrum as the solution of a semi definite program a second step then consists of constructing a spectrum that matches this finite set of optimal moments and satisfies some constraints due to the particular closed loop nature of the optimization problem this problem can be seen as a moment extension problem under constraints here we first show that the so called central extension always satisfies these constraints leading to a constructive procedure for the optimal controller and excitation spectrum we then show that one can construct a broader set of parametrized optimal solutions that also satisfy the constraints the additional degrees of freedom can then be used to achieve additional objectives

exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion a basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities the sparsity characterized with a chordal graph structure can be detected in the variable matrix or in a linear or nonlinear matrix inequality constraint of the problem we classify the sparsity in two types the domain space sparsity d space sparsity for the symmetric matrix variable in the objective and or constraint functions of the problem which is required to be positive semidefinite and the range space sparsity r space sparsity for a linear or nonlinear matrix inequality constraint of the problem four conversion methods are proposed in this framework two for exploiting the d space sparsity and the other two for exploiting the r space sparsity when applied to a polynomial semidefinite program sdp these conversion methods enhance the structured sparsity of the problem called the correlative sparsity as a result the resulting polynomial sdp can be solved more effectively by applying the sparse sdp relaxation preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems

geometry of maximum likelihood estimation in gaussian graphical models author s uhler caroline advisor s sturmfels bernd abstract algebraic statistics exploits the use of algebraic techniques to develop new paradigms and algorithms for data analysis the development of computational algebra software provides a powerful tool to analyze statistical models in part i of this thesis we use methods from computational algebra and algebraic geometry to study gaussian graphical models algebraic methods have proven to be useful for statistical theory and applications alike we describe a particular application to computational biology in part ii part i of this thesis investigates geometric aspects of maximum likelihood estimation in gaussian graphical models more generally we study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection in chapter 2 we examine these problems at the interface of statistics and optimization from the perspective of convex algebraic geometry and characterize the cone of all sufficient statistics for which the maximum likelihood estimator mle exists in chapter 3 we develop an algebraic elimination criterion which allows us to find exact lower bounds on the number of observations needed to ensure that the mle exists with probability one this is applied to bipartite graphs grids and colored graphs we also present the first instance of a graph for which the mle exists with probability one even when the number of observations equals the treewidth computational algebra software can be used to study graphs with a limited number of vertices and edges in chapter 4 we study the problem of existence of the mle from an asymptotic point of view by fixing a class of graphs and letting the number of vertices grow to infinity we prove that for very large cycles already two observations are sufficient for the existence of the mle with probability one part ii of this thesis describes an application of algebraic statistics to association studies rapid research progress in genotyping techniques have allowed large genome wide association studies existing methods often focus on determining associations between single loci and a specific phenotype however a particular phenotype is usually the result of complex relationships between multiple loci and the environment we develop a method for finding interacting genes i e epistasis using markov bases we test our method on simulated data and compare it to a two stage logistic regression method and to a fully bayesian method showing that we are able to detect the interacting loci when other methods fail to do so finally we apply our method to a genome wide dog data set and identify epistasis associated with canine hair length

completing a block diagonal matrix with a partially prescribed inverse abstract a complete solution of the matrix completion problem a b 1 c d is obtained in terms of solutions of a riccati type matrix equation several special cases are described in detail

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

a subspace approach to estimation of autoregressive parameters from noisy measurements this correspondence describes a method for estimating the parameters of an autoregressive ar process from a finite number of noisy measurements the method uses a modified set of yule walker yw equations that lead to a quadratic eigenvalue problem that when solved gives estimates of the ar parameters and the measurement noise variance

mihaly matrix completions moments and sums of hermitian squares by mihaly bakonyi and hugo j woerdeman princeton series in applied mathematics princeton university press 2011 xii 518 pp hardback isbn 978 0 691 12889 4

the symmetric m matrix and symmetric inverse m matrix completion problems abstract the symmetric m matrix and symmetric m 0 matrix completion problems are solved and results of johnson and smith linear algebra appl 290 1999 193 are extended to solve the symmetric inverse m matrix completion problem 1 a pattern i e a list of positions in an n n matrix has symmetric m completion i e every partial symmetric m matrix specifying the pattern can be completed to a symmetric m matrix if and only if the principal subpattern r determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete or in graph theoretic terms if and only if each component of the graph of r is a clique 2 a pattern has symmetric m 0 completion if and only if the pattern is permutation similar to a pattern that is block diagonal with each diagonal block either complete or omitting all diagonal positions or in graph theoretic terms if and only if every principal subpattern corresponding to a component of the graph of the pattern either omits all diagonal positions or includes all positions 3 a pattern has symmetric inverse m completion if and only if its graph is block clique and no diagonal position is omitted that corresponds to a vertex in a graph block of order 2 the techniques used are also applied to matrix completion problems for other classes of symmetric matrices

exploration of np hard enumeration problems by simulated annealing the spectrum values of permanents abstract the monte carlo optimisation technique simulated annealing has been used to sample the possible values of the permanent of fully indecomposable 0 1 matrices the basic idea in simulated annealing is that a markov chain explores the space of feasible solutions each of which has an associated energy here minus the permanent of the matrix the control parameter temperature is gradually reduced in order to focus the search on the low energy region of state space and eventually reaching the minimum although the minimum energy per se largest permanent is known the spectrum above the minimum is not so this importance sampling technique is quite efficient interest in permanents is based on the fact that it is complete for the class p of enumeration problems that is as hard as counting the number of accepting computations of any nondeterministic polynomial time turing machine the spectrum of possible values for the permanent of a 0 1 matrix with given topological dimension d is quite interesting but only partially understood the present work is an experimental study of this spectrum using the handle basis representation of directed graphs and simulated annealing to sample the possible values the results show that gaps exist in the spectrum of values of the permanent but also show the existence of values for d 8 and 9 not hitherto observed the new values in the spectrum of the permanent for d 8 is 72 and for d 9 are 119 132 136 138 and 140 further we found that the density of states grows exponentially with the permanent values the larger the value of the permanent the larger the number of neighbours and the larger the energy barriers between the permanent values of the neighbours

dualistic differential geometry of positive definite matrices and its applications to related problems abstract analysis on the set of positive definite matrices is involved in many engineering problems we develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it consequently various dualistic aspects of the set are elucidated next using the theory we derive new results and interesting geometrical interpretations for matrix approximation positive definite matrix completion and linear matrix inequality problems

positive p matrices completion problem with zeros in the inverse completion in this paper based on the completion problem of partial p matrix for 1 chordal graph we obtain the partial positive p matrix completions of k chordal graphs and k chordal block graphs under some conditions and solve partial positive p matrix completion problems with zeros in the inverse completion for k chordal graphs and k chordal block graphs k 2

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

on representing the positive semidefinite cone using the second order cone the second order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity second order cone formulations can also be solved more efficiently than semidefinite programming problems in general we consider the following question posed by lewis and glineur parrilo saunderson is it possible to express the general positive semidefinite cone using second order cones we provide a negative answer to this question and show that the 3 times 3 positive semidefinite cone does not admit any second order cone representation in fact we show that the slice consisting of 3 times 3 positive semidefinite hankel matrices does not admit a second order cone representation our proof relies on exhibiting a sequence of submatrices of the slack matrix of the 3 times 3 positive semidefinite cone whose second order cone rank grows to infinity

frequency domain subspace identification using nuclear norm minimization and hankel matrix realizations subspace identification techniques have gained widespread acceptance as a method of obtaining a low order model from data these are based on using the singular value decomposition as a means of estimating the underlying system order and extracting a basis for the extended observability space in the presence of noise rank determination becomes difficult and the low rank estimates lose the structure required for exact realizability furthermore the noise corrupts the singular values in a manner that is inconsistent with physical noise processes these problems are addressed by an optimization based approach using a nuclear norm minimization objective by using hankel matrices as the underlying data structure exact realizability of the low rank system models is maintained noise in the data enters the formulation linearly allowing for the inclusion of more realistic noise weightings a cumulative spectral weight is presented and shown to be useful in estimating models from data corrupted via noise a numerical example illustrates the characteristics of the problem

distributing the kalman filter for large scale systems this paper presents a distributed kalman filter to estimate the state of a sparsely connected large scale n dimensional dynamical system monitored by a network of n sensors local kalman filters are implemented on nl dimensional subsystems nl lt n obtained by spatially decomposing the large scale system the distributed kalman filter is optimal under an lth order gauss markov approximation to the centralized filter we quantify the information loss due to this lth order approximation by the divergence which decreases as l increases the order of the approximation l leads to a bound on the dimension of the subsystems hence providing a criterion for subsystem selection the approximated centralized riccati and lyapunov equations are computed iteratively with only local communication and low order computation by a distributed iterate collapse inversion dici algorithm we fuse the observations that are common among the local kalman filters using bipartite fusion graphs and consensus averaging algorithms the proposed algorithm achieves full distribution of the kalman filter nowhere in the network storage communication or computation of n dimensional vectors and matrices is required only nl lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors in other words knowledge of the state is itself distributed

bayesian precision and covariance matrix estimation for graphical gaussian models with edge and vertex symmetries summarygraphical gaussian models with edge and vertex symmetries were introduced by hojsgaard lauritzen 2008 who gave an algorithm for computing the maximum likelihood estimate of the precision matrix for such models in this paper we take a bayesian approach to its estimation we consider only models with symmetry constraints and which thus form a natural exponential family with the precision matrix as the canonical parameter we identify the diaconis ylvisaker conjugate prior for these models develop a scheme to sample from the prior and posterior distributions and thus obtain estimates of the posterior mean of the precision and covariance matrices such a sampling scheme is essential for model selection in coloured graphical gaussian models in order to verify the precision of our estimates we derive an analytic expression for the expected value of the precision matrix when the graph underlying our model is a tree a complete graph on three vertices or a decomposable graph on four vertices with various symmetries and we compare our estimates with the posterior mean of the precision matrix and the expected mean of the coloured graphical gaussian model that is of the covariance matrix we also verify the accuracy of our estimates on simulated data

matrices with banded inverses inversion algorithms and factorization of gauss markov processes the paper considers the inversion of full matrices whose inverses are l banded we derive a nested inversion algorithm for such matrices applied to a tridiagonal matrix the algorithm provides its explicit inverse as an element wise product hadamard product of three matrices when related to gauss markov random processes gmrp this result provides a closed form factored expression for the covariance matrix of a first order gmrp this factored form leads to the interpretation of a first order gmrp as the product of three independent processes a forward independent increments process a backward independent increments process and a variance stationary process we explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy we then consider the issue of approximating general nonstationary gaussian processes by gauss markov processes under two optimality criteria the kullback leibler distance and maximum entropy the problem reduces to approximating general covariances by covariance matrices whose inverses are banded our inversion result is an efficient algorithmic solution to this problem we evaluate the information loss between the original process and its gauss markov approximation

a graph theoretic upper bound on the permanent of a nonnegative integer matrix ii the extremal case abstract matrices a for which the upper bound per a 1 min c i 1 r i 1 holds with equality are characterized cases where the bound is achieved correspond to multigraphs with the property that there exists a unique path from any vertex to any disjoint cycle union this occurs precisely when some multigraph associated with a has the mastercycle property all cycles thread all branchpoints in the same circular order such multigraphs may also be characterized as a circular concatenation of certain acyclic multigraphs each having a unique source and sink this analysis yields two normal forms for the extremal matrices one based on a nested block decomposition and another based on an overlapping block decomposition the extremal cases are invariant under contractions yielding another characterization

p matrix completions under weak symmetry assumptions an n by n matrix is called a matrix if it is one of weakly sign symmetric positive nonnegative p matrix weakly sign symmetric positive nonnegative p0 1 matrix or fischer or koteljanskii matrix in this paper we are interested in matrix completion problems that is when a partial matrix has a matrix completion here we prove that a combinatorially symmetric partial positive p matrix has a positive p matrix completion if the graph of its specified entries is an n cycle in general a combinatorially symmetric partial matrix has a matrix completion if the graph of its specified entries is a 1 chordal graph this condition is also necessary for weakly sign symmetric p0 or p0 1 matrices

determinantal formulae for matrix completions associated with chordal graphs abstract let a be a partial positive definite hermitian matrix and let g a be the undirected graph of the specified off diagonal entries of a when g a is a chordal graph we present an explicit determinantal formula for the unique determinant maximizing positive definite completion maximum entropy completion of a in order to do this we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree t of the intersection graph of its set of maximal cliques the formula for the maximum determinant is then a quotient of products of specified principal minors of a in which each minor in the numerator corresponds to a maximal clique of g a while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of t we next show that the denominator terms can be identified graph theoretically with the minimal vertex separators of g a and that these terms are independent of the particular proper spanning tree t then an alternative formula is given for the maximum determinant as an inclusion exclusion like ratio of principal minors associated with the maximal cliques of g a finally we show that the entries of the determinant maximizing completion may be calculated by solving a sequence of simple one variable maximization problems

maximum entropy kernels for system identification bayesian nonparametric approaches have been recently introduced in system identification scenario where the impulse response is modeled as the realization of a zero mean gaussian process whose covariance kernel has to be estimated from data in this scheme quality of the estimates crucially depends on the parametrization of the covariance of the gaussian process a family of kernels that have been shown to be particularly effective in the system identification framework is the family of diagonal correlated dc kernels maximum entropy properties of a related family of kernels the tuned correlated tc kernels have been recently pointed out in the literature in this technical note we show that maximum entropy properties indeed extend to the whole family of dc kernels the maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the dc kernel in particular we prove that the dc kernel admits a closed form factorization inverse and determinant these results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the dc estimator

a parameterization of positive definite matrices in terms of partial correlation vines abstract we present a parameterization of the class pd n of positive definite n n matrices using regular vines and partial correlations using a bijection from 1 1 n 2 c n c n is the class of n n correlation matrices with a clear probabilistic interpretation ann statist 30 4 2002 1031 we suggest a new approach to various problems involving positive definiteness

the inertia sets of symmetric tridiagonal sign patterns abstract we give a complete description of the possible inertias of real symmetric matrices with a given arbitrarily prescribed tridiagonal symmetric sign pattern the inertia set of such a pattern is a subset of z 2 given by a system of linear inequalities whose parameters are determined from the sign sequence of the pattern s diagonal entries

graphical gaussian models with edge and vertex symmetries summary we introduce new types of graphical gaussian models by placing symmetry restrictions on the concentration or correlation matrix the models can be represented by coloured graphs where parameters that are associated with edges or vertices of the same colour are restricted to being identical we study the properties of such models and derive the necessary algorithms for calculating maximum likelihood estimates we identify conditions for restrictions on the concentration and correlation matrices being equivalent this is for example the case when symmetries are generated by permutation of variable labels for such models a particularly simple maximization of the likelihood function is available

a direct proof for the matrix decomposition of chordal structured positive semidefinite matrices agler helton mccullough and rodman proved that a graph is chordal if and only if any positive semidefinite psd symmetric matrix whose nonzero entries are specified by a given graph can be decomposed as a sum of psd matrices corresponding to the maximal cliques this decomposition is recently exploited to solve positive semidefinite programming efficiently their proof is based on a characterization for psd matrix completion of a chordal structured matrix due to grone johnson sa and wolkowicz this note gives a direct and simpler proof for the result of agler et al which leads to an alternative proof of grone et al

a graph theoretic upper bound on the permanent of a nonnegative integer matrix i abstract let a be a fully indecomposable n n matrix with nonnegative integer entries then the permanent of a is bounded above by 1 min c i 1 r i 1 where c i and r i are the column and row sums of a the inequality results from a bound on the number of disjoint cycle unions in an associated multigraph this bound can improve via contractions

a unified framework for eiv identification methods when the measurement noises are mutually correlated in this paper the previously introduced generalized instrumental variable estimator give is extended to the case of errors in variables models where the additive input and output noises are mutually correlated white processes it is shown how many estimators proposed in the literature can be described as various special cases of a generalized instrumental variable framework it is also investigated how to analyze the common situation where some of the equations that define the estimator are to hold exactly and others to hold approximately in a least squares sense providing a detailed study of the accuracy analysis

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

frequency domain identification of fir models in the presence of additive input output noise this paper describes a new approach for identifying fir models from a finite number of measurements in the presence of additive and uncorrelated white noise in particular two different frequency