filtering and frequency interpretations of singular spectrum analysis new filtering and spectral interpretations of singular spectrum analysis ssa are provided it is shown that the variables reconstructed from diagonal averaging of reduced rank approximations to the trajectory matrix can be obtained from a noncausal convolution filter with zero phase characteristics the reconstructed variables are readily constructed using a two pass filtering algorithm that is well known in the signal processing literature when the number of rows in the trajectory matrix is much larger than number of columns many results reported in the signal processing literature can be used to derive the properties of the resulting filters and their spectra new features of the reconstructed series are revealed using these results two examples are used to illustrate the results derived in this paper

designing structured tight frames via an alternating projection method tight frames also known as general welch bound equality sequences generalize orthonormal systems numerous applications including communications coding and sparse approximation require finite dimensional tight frames that possess additional structural properties this paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems ieps which includes the frame design problem to apply this method one needs only to solve a matrix nearness problem that arises naturally from the design specifications therefore it is the fast and easy to develop versions of the algorithm that target new design problems alternating projection will often succeed even if algebraic constructions are unavailable to demonstrate that alternating projection is an effective tool for frame design the paper studies some important structural properties in detail first it addresses the most basic design problem constructing tight frames with prescribed vector norms then it discusses equiangular tight frames which are natural dictionaries for sparse approximation finally it examines tight frames whose individual vectors have low peak to average power ratio par which is a valuable property for code division multiple access cdma applications numerical experiments show that the proposed algorithm succeeds in each of these three cases the appendices investigate the convergence properties of the algorithm

on the moreau yosida regularization of the vector k norm related functions matrix optimization problems mops involving the ky fan k norm arise frequently from many applications in order to design algorithms to solve large scale mops involving the ky fan k norm we need to understand the first and second order properties of the moreau yosida regularization of the ky fan k norm function and the indicator function of the ky fan k norm ball according to the general theory on spectral functions in this paper we shall conduct a thorough study on the moreau yosida regularization of the vector k norm function and the indicator function of the vector k norm ball in particular we show that the proximal mappings associated with these two vector k norm related functions both admit fast and analytically computable solutions moreover we propose algorithms of low computational cost to compute the directional derivatives of these two proximal mappings and then completely characterize their frechet differentiability the work here thus builds the fundamental tools needed

information based detection of nonlinear granger causality in multivariate processes via a nonuniform embedding technique we present an approach framed in information theory to assess nonlinear causality between the subsystems of a whole stochastic or deterministic dynamical system the approach follows a sequential procedure for nonuniform embedding of multivariate time series whereby embedding vectors are built progressively on the basis of a minimization criterion applied to the entropy of the present state of the system conditioned to its past states a corrected conditional entropy estimator compensating for the biasing effect of single points in the quantized hyperspace is used to guarantee the existence of a minimum entropy rate at which to terminate the procedure the causal coupling is detected according to the granger notion of predictability improvement and is quantified in terms of information transfer we apply the approach to simulations of deterministic and stochastic systems showing its superiority over standard uniform embedding effects of quantization data length and noise contamination are investigated as practical applications we consider the assessment of cardiovascular regulatory mechanisms from the analysis of heart period arterial pressure and respiration time series and the investigation of the information flow across brain areas from multichannel scalp electroencephalographic recordings

a study of problems encountered in granger causality analysis from a neuroscience perspective granger causality methods were developed to analyze the flow of information between time series these methods have become more widely applied in neuroscience frequency domain causality measures such as those of geweke as well as multivariate methods have particular appeal in neuroscience due to the prevalence of oscillatory phenomena and highly multivariate experimental recordings despite its widespread application in many fields there are ongoing concerns regarding the applicability of granger causality methods in neuroscience when are these methods appropriate how reliably do they recover the system structure underlying the observed data what do frequency domain causality measures tell us about the functional properties of oscillatory neural systems in this paper we analyze fundamental properties of granger geweke gg causality both computational and conceptual specifically we show that i gg causality estimates can be either severely biased or of high variance both leading to spurious results ii even if estimated correctly gg causality estimates alone are not interpretable without examining the component behaviors of the system model and iii gg causality ignores critical components of a system s dynamics based on this analysis we find that the notion of causality quantified is incompatible with the objectives of many neuroscience investigations leading to highly counterintuitive and potentially misleading results through the analysis of these problems we provide important conceptual clarification of gg causality with implications for other related causality approaches and for the role of causality analyses in neuroscience as a whole

estimation of a signal waveform from noisy data using low rank approximation to a data matrix an analysis and improvement of a data adaptive signal estimation algorithm are presented perturbation analysis of a reduced rank data matrix is used to reveal its statistical properties the obtained information is used for calculating the performance of the toeplitz restoration algorithm of d tufts et al 1982 this analysis leads to improvements of the methods and the predicted improvements are demonstrated by simulation and comparison with the cramer rao bounds

continuity of closest rank p approximations to matrices in signal processing the singular value decomposition and rank characterization of matrices play prominent roles the mapping which associates with any complex m n matrix x its closest rank p approximation x p need not be continuous when the pth and the p 1 st singular values of x are equal this mapping maps in fact a matrix to a set of matrices furthermore an example is given to show that large errors in computing x p can be expected when p is sufficiently close to p 1 it is finally shown that this mapping is closed in the sense of zangwill the property of closedness is an essential assumption of a global convergence proof for algorithms involving this mapping e g see 1

exploration of synergistic and redundant information sharing in static and dynamical gaussian systems to fully characterize the information that two source variables carry about a third target variable one must decompose the total information into redundant unique and synergistic components i e obtain a partial information decomposition pid however shannon s theory of information does not provide formulas to fully determine these quantities several recent studies have begun addressing this some possible definitions for pid quantities have been proposed and some analyses have been carried out on systems composed of discrete variables here we present an in depth analysis of pids on gaussian systems both static and dynamical we show that for a broad class of gaussian systems previously proposed pid formulas imply that i redundancy reduces to the minimum information provided by either source variable and hence is independent of correlation between sources and ii synergy is the extra information contributed by the weaker source when the stronger source is known and can either increase or decrease with correlation between sources we find that gaussian systems frequently exhibit net synergy i e the information carried jointly by both sources is greater than the sum of information carried by each source individually drawing from several explicit examples we discuss the implications of these findings for measures of information transfer and information based measures of complexity both generally and within a neuroscience setting importantly by providing independent formulas for synergy and redundancy applicable to continuous time series data we provide an approach to characterizing and quantifying information sharing amongst complex system variables

horizontal visibility graph transfer entropy hvg te a novel metric to characterize directed connectivity in large scale brain networks we propose a new measure horizontal visibility graph transfer entropy hvg te to estimate the direction of information flow between pairs of time series hvg te quantifies the transfer entropy between the degree sequences of horizontal visibility graphs derived from original time series twenty one rossler attractors unidirectionally coupled in the posterior to anterior direction were used to simulate 21 channel electroencephalography eeg brain networks and validate the performance of the hvg te we showed that the hvg te is robust to different levels of coupling strengths between the coupled rossler attractors a wide range of time delays different sample sizes the effects of noise and linear mixing and the choice of reference for eeg data we also applied hvg te to eeg data in 20 healthy controls and compared its performance to a recently introduces phase based te measure pte we found that compared with pte hvg te consistently detected stronger posterior to anterior information flow patterns in the alpha band 8 13hz eeg brain networks for three different references moreover in contrast to pte hvg te does not require an assumption on the periodicity of input signals therefore it can be more widely applicable even for non periodic signals this study shows that the hvg te is a directed connectivity measure to characterise the direction of information flow in large scale brain networks

computationally efficient maximum likelihood estimation of structured covariance matrices by invoking the extended invariance principle exip we present herein a computationally efficient method that provides asymptotic for large samples maximum likelihood aml estimation for structured covariance matrices and is referred to as the aml algorithm a closed form formula for estimating the hermitian toeplitz covariance matrices that makes aml computationally simpler than most existing hermitian toeplitz matrix estimation algorithms is derived although the aml covariance matrix estimator can be used in a variety of applications we focus on array processing our simulation study shows that aml enhances the performance of angle estimation algorithms such as music by making them very close to the corresponding cramer rao bound crb for uncorrelated signals numerical comparisons with several structured and unstructured covariance matrix estimators are also presented

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

a convergent composite mapping fourier domain iterative algorithm for 3 d discrete tomography the discrete tomography problem is to reconstruct a binary function defined on a discrete lattice from its sums in directions parallel to the axes of the lattice the problem has multiple solutions we wish to determine a solution close in the least squares sense to a specified starting function we formulate the problem in the fourier domain and use the agarwal cooley fast convolution or good thomas fast fourier transform to unwrap the multidimensional 2 d or 3 d problem into a long 1 d problem in which the given projection data specify some values of the dft of the 1 d sequence of binary values the z transform of the dft values evaluated on the unit circle yields these binary values the problem is now to determine the minimum perturbation of the unconstrained dft values of the starting function which cause a toeplitz matrix to lose rank this type of problem is well known in spectral estimation we obtain a new iterative algorithm which converges to a solution of the discrete tomography problem close to the starting function

normalizing the causality between time series recently a rigorous yet concise formula was derived to evaluate information flow and hence the causality in a quantitative sense between time series to assess the importance of a resulting causality it needs to be normalized the normalization is achieved through distinguishing a lyapunov exponent like one dimensional phase space stretching rate and a noise to signal ratio from the rate of information flow in the balance of the marginal entropy evolution of the flow recipient it is verified with autoregressive models and applied to a real financial analysis problem an unusually strong one way causality is identified from ibm international business machines corporation to ge general electric company in their early era revealing to us an old story which has almost faded into oblivion about seven dwarfs competing with a giant for the mainframe computer market

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

signal enhancement a composite property mapping algorithm a signal enhancement algorithm is developed that seeks to recover a signal from noise contaminated distorted measurements made on that signal this object is achieved by utilizing a set of properties which the signal is known or is hypothesized as possessing the measured signal is modified to the smallest degree necessary to sequentially possess each of the individual properties conditions for the algorithm s convergence are established in which the primary requirement is that a composite property mapping be closed this is a relatively unrestricted condition in comparison to that required of most existing signal enhancement algorithms

the change of similarity invariants under row perturbations the problem of the possible similarity classes of all the matrices obtained by small perturbations on some rows of a given complex square matrix is under consideration some necessary conditions on these similarity classes are provided the near converse problem is also studied under what conditions can a given collection of polynomials or partitions be the invariant factors or the weyr characteristic of a matrix obtained by perturbing some rows of a given matrix a complete answer is provided in some particular cases the obtained conditions can all be derived from the previously mentioned necessary conditions

a gradient system approach for hankel structured low rank approximation abstract rank deficient hankel matrices are at the core of several applications however in practice the coefficients of these matrices are noisy due to e g measurements errors and computational errors so generically the involved matrices are full rank this motivates the problem of hankel structured low rank approximation structured low rank approximation problems in general do not have a global and efficient solution technique in this paper we propose a local optimization approach based on a two levels iteration experimental results show that the proposed algorithm usually achieves good accuracy and shows a higher robustness with respect to the initial approximation compared to alternative approaches

low rank optimization with convex constraints the problem of low rank approximation with convex constraints which appears in data analysis system identification model order reduction low order controller design and low complexity modeling is considered given a matrix the objective is to find a low rank approximation that meets rank and convex constraints while minimizing the distance to the matrix in the squared frobenius norm in many situations this nonconvex problem is convexified by nuclear norm regularization however we will see that the approximations obtained by this method may be far from optimal in this paper we propose an alternative convex relaxation that uses the convex envelope of the squared frobenius norm and the rank constraint with this approach easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original nonconvex problem coincide a semidefinite programming representation of the convex envelope is derived which allows us to apply this approach to several known problems our example on optimal low rank hankel approximation model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear norm regularization and may outperform balanced truncation

time irreversibility of financial time series based on higher moments and multiscale kullback leibler divergence abstract irreversibility is an important property of time series in this paper we propose the higher moments and multiscale kullback leibler divergence to analyze time series irreversibility the higher moments kullback leibler divergence hmkld can amplify irreversibility and make the irreversibility variation more obvious therefore many time series whose irreversibility is hard to be found are also able to show the variations we employ simulated data and financial stock data to test and verify this method and find that hmkld of stock data is growing in the form of fluctuations as for multiscale kullback leibler divergence mkld it is very complex in the dynamic system so that exploring the law of simulation and stock system is difficult in conventional multiscale entropy method the coarse graining process is non overlapping however we apply a different coarse graining process and obtain a surprising discovery the result shows when the scales are 4 and 5 their entropy is nearly similar which demonstrates mkld is efficient to display characteristics of time series irreversibility

a new algebra of toeplitz plus hankel matrices and applications we introduce a new simultaneously diagonalizable real algebra a of symmetrical centrosymmetrical matrices having a toeplitz plus hankel structure we give the corresponding orthonormal basis of eigenvectors which are alternately symmetrical and skewsymmetrical vectors an application is the construction of a symmetrical toeplitz plus centrosymmetrical hankel matrix of equal row sums having a prescribed real spectrum this matrix can be used as the starting matrix for symmetrical centrosymmetrical isospectral flows in particular for the isospectral flow corresponding to the construction of a regular toeplitz matrix having prescribed eigenvalues moreover if a is a noise representation of an unknown matrix in a of rank k then we give a procedure to approximate a by a matrix in a of rank k

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

a successive projection method it is of both theoretical and practical interest to find the projection of a point to the intersection of a finite number of closed convex sets by a sequence of projections to the individual sets successively in this paper we study such a method and analyze its convergence properties a main feature of the method is its capability to decompose the projection problem into several small ones for some structured sparse problems these small subproblems can be solved independently and the presented method has a potential use in parallel computation

rank reduction for modeling stationary signals rank reduction is developed as a general principle for trading off model bias and model variance in the analysis and synthesis of signals the principle is applied to three basic problems stationary time series modeling stationary time series whitening and vector quantization each problem brings its own surprises and insights

computing the radius of controllability for state space systems abstract in this paper we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair a b in order to do so we construct a sequence of structured matrices from given system matrices a and b controllability of the pair a b is equivalent to a condition on the null space dimension of an appropriate matrix in this sequence we show that the dimension of the reachability space is also related to the above condition further it is shown that the nearest structured low rank approximation slra of this structured matrix corresponds to a nearest uncontrollable system to the pair a b

multi scale symbolic transfer entropy analysis of eeg abstract from both global and local perspectives we symbolize two kinds of eeg and analyze their dynamic and asymmetrical information using multi scale transfer entropy multi scale process with scale factor from 1 to 199 and step size of 2 is applied to eeg of healthy people and epileptic patients and then the permutation with embedding dimension of 3 and global approach are used to symbolize the sequences the forward and reverse symbol sequences are taken as the inputs of transfer entropy scale factor intervals of permutation and global way are 37 57 and 65 85 where the two kinds of eeg have satisfied entropy distinctions when scale factor is 67 transfer entropy of the healthy and epileptic subjects of permutation 0 1137 and 0 1028 have biggest difference and the corresponding values of the global symbolization is 0 0641 and 0 0601 which lies in the scale factor of 165 research results show that permutation which takes contribution of local information has better distinction and is more effectively applied to our multi scale transfer entropy analysis of eeg

penalized maximum likelihood estimation of covariance matrices with linear structure in this paper a space alternating generalized expectation maximization sage algorithm is presented for the numerical computation of maximum likelihood ml and penalized ml pml estimates of the parameters of covariance matrices with linear structure for complex gaussian processes by using a less informative hidden data space and a sequential parameter update scheme a sage based algorithm is derived for which convergence of the likelihood is demonstrated to be significantly faster than that of an em based algorithm that has been previously proposed in addition the sage procedure is shown to easily accommodate penalty functions and a sage based algorithm is derived and demonstrated for forming pml estimates with a quadratic smoothness penalty

granger causality and transfer entropy are equivalent for gaussian variables granger causality is a statistical notion of causal influence based on prediction via vector autoregression developed originally in the field of econometrics it has since found application in a broader arena particularly in neuroscience more recently transfer entropy an information theoretic measure of time directed information transfer between jointly dependent processes has gained traction in a similarly wide field while it has been recognized that the two concepts must be related the exact relationship has until now not been formally described here we show that for gaussian variables granger causality and transfer entropy are entirely equivalent thus bridging autoregressive and information theoretic approaches to data driven causal inference

low rank update of singular values the notion of a low rank update arises in many important applications this paper deals with the inverse problem of updating a rectangular matrix by additive low rank matrices so as to reposition the associated singular values the setting is analogous to the classical pole assignment problem where eigenvalues of a square matrix are relocated precise and easy to check necessary and sufficient conditions under which the problem is solvable are completely characterized generalizing some traditional weyl inequalities for singular values the constructive proof makes it possible to compute such a solution numerically a pseudo algorithm is outlined

robust estimation of structured covariance matrices in the context of the narrowband array processing problem robust methods for accurately estimating the spatial correlation matrix using a priori information about the matrix structure are developed by minimizing the worse case asymptotic variance robust structured maximum likelihood type estimates of the spatial correlation matrix in the presence of noises with probability density functions in the in contamination and kolmogorov classes are obtained these estimates are robust against variations in the noise s amplitude distribution the kolmogorov class is demonstrated to be the natural class to use for array processing applications and a technique is developed to determine exactly the size of this class performance of bearing estimation algorithms improves substantially when the robust estimates are used especially when nongaussian noise is present a parametric structured estimate of the spatial correlation matrix that allows direct estimation of the arrival angles is also demonstrated

approximation of linear operators in the 2 norm abstract the problems of approximating linear operators in the 2 induced norm which are 1 finite dimensional unstructured and 2 infinite dimensional structured hankel have been solved the solutions of these two problems exhibit striking similarities these similarities suggest the search of a unifying framework for the approximation of linear operators in the 2 induced norm

total least squares for affinely structured matrices and the noisy realization problem structured rank deficient matrices arise in many applications in signal processing system identification and control theory the author discusses the structured total least squares stls problem which is the problem of approximating affinely structured matrices i e matrices affine in the parameters by similarly structured rank deficient ones while minimizing an l sub 2 error criterion it is shown that the optimality conditions lead to a nonlinear generalized singular value decomposition which can be solved via an algorithm that is inspired by inverse iteration next the author concentrates on the so called l sub 2 optimal noisy realization problem which is equivalent with approximating a given data sequence by the impulse response of a finite dimensional time invariant linear system of a given order this can be solved as a structured total least squares problem it is shown with some simple counter examples that classical algorithms such as the steiglitz mcbride 1965 iterative quadratic maximum likelihood and cadzow s 1988 iteration do not converge to the optimal l sub 2 solution despite misleading claims in the literature

blind deconvolution of covariance matrix inverses for autoregressive processes abstract matrix c can be blindly deconvoluted if there exist matrices a and b such that c a b where denotes the operation of matrix convolution we study the problem of matrix deconvolution in the case where matrix c is proportional to the inverse of the autocovariance matrix of an autoregressive process we show that the deconvolution of such matrices is important in problems of hankel structured low rank approximation hslra in the cases of autoregressive models of orders one and two we fully characterize the range of parameters where such deconvolution can be performed and provide construction schemes for performing deconvolutions we also consider general autoregressive models of order p where we prove that the deconvolution c a b does not exist if the matrix b is diagonal and its size is larger than p

expanding the transfer entropy to identify information circuits in complex systems we propose a formal expansion of the transfer entropy to put in evidence irreducible sets of variables which provide information for the future state of each assigned target multiplets characterized by a large contribution to the expansion are associated to the informational circuits present in the system with an informational character which can be associated to the sign of the contribution for the sake of computational complexity we adopt the assumption of gaussianity and use the corresponding exact formula for the conditional mutual information we report the application of the proposed methodology on two electroencephalography eeg data sets

granger causality analysis with nonuniform sampling and its application to pulse coupled nonlinear dynamics the granger causality gc analysis is an effective approach to infer causal relations for time series however for data obtained by uniform sampling i e with an equal sampling time interval it is known that gc can yield unreliable causal inference due to aliasing if the sampling rate is not sufficiently high to solve this unreliability issue we consider the nonuniform sampling scheme as it can mitigate against aliasing by developing an unbiased estimation of power spectral density of nonuniformly sampled time series we establish a framework of spectrum based nonparametric gc analysis applying this framework to a general class of pulse coupled nonlinear networks and utilizing some particular spectral structure possessed by these nonlinear network data we demonstrate that for such nonlinear networks with nonuniformly sampled data reliable gc inference can be achieved at a low nonuniform mean sampling rate at which the traditional uniform sampling gc may lead to spurious causal inference

structured low rank approximation this paper concerns the construction of a structured low rank matrix that is nearest to a given matrix the notion of structured low rank approximation arises in various applications ranging from signal enhancement to protein folding to computer algebra where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system the task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix this paper addresses some of the theoretical and numerical issues involved in the problem two procedures for constructing the nearest structured low rank matrix are proposed the procedures are flexible enough that they can be applied to any lower rank any linear structure and any matrix norm in the measurement of nearness the techniques can also be easily implemented by utilizing available optimization packages the special case of symmetric toeplitz structure using the frobenius matrix norm is used to exemplify the ideas throughout the discussion the concept rather than the implementation details is the main emphasis of the paper

nearest matrix with prescribed eigenvalues and its applications consider an n n matrix a and a set consisting of k n prescribed complex numbers lippert 2010 in a challenging article studied geometrically the spectral norm distance from a to the set of matrices whose spectra included specified set and constructed a perturbation matrix with minimum spectral norm such that a had in its spectrum this paper presents an easy practical computational method for constructing the optimal perturbation by improving and extending the methodology necessary definitions and lemmas of previous related works also some conceivable applications of this issue are provided

spectral operators of matrices the class of matrix optimization problems mops has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important ap plications involving structured low rank matrices this trend can be credited to some extent to the exciting developments in the emerging field of compressed sensing the lowner operator which gen erates a matrix valued function by applying a single variable function to each of the singular values of a matrix has played an important role for a long time in solving matrix optimization problems however the classical theory developed for lowner operators has become inadequate in these recent applications the main objective of this paper is to provide some necessary theoretical foundations for designing numerical methods for solving the mop this goal is achieved by introducing and conducting a thorough study on a new class of matrix valued functions coined as spectral operators of matrices several fundamental properties of spectral operators including the well definedness continuity di rectional differentiability frechet differentiability locally lipschitzian continuity order b ouligand differentiability 0 1 order g semismooth 0 1 and the characterization of clarke s generalized jacobian are systematically studied

information transfer in continuous processes we discuss a recently proposed quantity called transfer entropy which uses time series data to measure the amount of information transferred from one process to another in order to understand its foundation merits and limitations we review some aspects of information theoretic functionals while for symbol sequences these measures have an intuitive interpretation their application to continuous state processes and in particular their estimation from finite data sets is problematic for mutual information finite length scale estimates converge from below and can thus be used to reject the assumption that the observed processes are independent however mutual information does not provide any directional information conversely transfer entropy does resolve the directionality of information exchange but no similar monotonic convergence seems to hold thus only in the case of zero transfer entropy in one direction we can reliably infer an asymmetry of the information exchange

radial basis function approach to nonlinear granger causality of time series we consider an extension of granger causality to nonlinear bivariate time series in this frame if the prediction error of the first time series is reduced by including measurements from the second time series then the second time series is said to have a causal influence on the first one not all the nonlinear prediction schemes are suitable to evaluate causality indeed not all of them allow one to quantify how much knowledge of the other time series counts to improve prediction error we present an approach with bivariate time series modeled by a generalization of radial basis functions and show its application to a pair of unidirectionally coupled chaotic maps and to physiological examples

low rank approximation of the symmetric positive semidefinite matrix in this paper we consider the low rank approximation of the symmetric positive semidefinite matrix which arises in machine learning quantum chemistry and inverse problem we first characterize the feasible set by x yy t email protected r n x k and then transform low rank approximation into an unconstrained optimization problem finally we use the nonlinear conjugate gradient method with exact line search to compute the optimal low rank symmetric positive semidefinite approximation of the given matrix numerical examples show that the new method is feasible and effective

on the applicability of the lead lag ratio in causality assessment abstract within the large set of metrics that have been proposed to assess the presence of a causality relationship between time series the lead lag ratio llr has recently attracted increasing attention especially in the econophysics community it is based on the analysis of the asymmetry between the correlations obtained when one time series is shifted back or ahead in time with respect to the second one or in other words on the assessment of whether one time series is leading the other in spite of its popularity undoubtedly due to its conceptual and computational simplicity no empirical validation of the llr has hitherto been proposed we here take a first step towards this validation by presenting results corresponding to a large set of analyses on synthetic time series and comparing them with those yielded by the celebrated granger causality metric we show how in spite of behaving similarly to the granger causality in some specific situations the llr presents important drawbacks among which its unreliableness and the absence of a statistical significance test stand out that discourage its use in real world analyses

neural complexity a graph theoretic interpretation one of the central challenges facing modern neuroscience is to explain the ability of the nervous system to coherently integrate information across distinct functional modules in the absence of a central executive to this end tononi et al proc nat acad sci usa 91 5033 1994 proposed a measure of neural complexity that purports to capture this property based on mutual information between complementary subsets of a system neural complexity so defined is one of a family of information theoretic metrics developed to measure the balance between the segregation and integration of a system s dynamics one key question arising for such measures involves understanding how they are influenced by network topology sporns et al cereb cortex 10 127 2000 employed numerical models in order to determine the dependence of neural complexity on the topological features of a network however a complete picture has yet to be established while de lucia et al phys rev e 71 016114 2005 made the first attempts at an analytical account of this relationship their work utilized a formulation of neural complexity that we argue did not reflect the intuitions of the original work in this paper we start by describing weighted connection matrices formed by applying a random continuous weight distribution to binary adjacency matrices this allows us to derive an approximation for neural complexity in terms of the moments of the weight distribution and elementary graph motifs in particular we explicitly establish a dependency of neural complexity on cyclic graph motifs

an introduction to a class of matrix cone programming in this paper we define a class of linear conic programming which we call matrix cone programming or mcp involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone mcp has recently been found to have many important applications for example in nuclear norm relaxations of affine rank minimization problems in order to make the defined mcp tractable and meaningful we must first understand the structure of these epigraphs so far only the epigraph of the frobenius matrix norm which can be regarded as a second order cone has been well studied here we take an initial step to study several important properties including its closed form solution calm bouligand differentiability and strong semismoothness of the metric projection operator over the epigraph of the l 1 l infty spectral or operator and nuclear matrix norm respectively these properties make it possible to apply augmented lagrangian methods which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming to this class of mcp problems the work done in this paper is far from comprehensive rather it is intended as a starting point to call for more insightful research on mcp so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come

unfoldings and the rank one approximation of the tensor abstract the rank one approximation plays an important role in the study of large scale tensors and the existent finite methods for computing the rank one approximation are generally based on the matricization of the given tensor in this paper exploiting the segmentation of indices of the tensor we present a dynamical strategy for the matricization of the tensor and design a numerical method to compute the rank one approximation of the given tensor more importantly different from the common techniques of regarding a tensor as a matrix and using the matrix singular value decomposition as the approximation tool we transform the given tensor into an order 3 tensor and introduce the projection technique to the calculation of the rank one approximation of the given tensor the complexity analysis and numerical experiments show that the computational cost of our algorithm is no more than that of the existing algorithms numerical results show that compared with the existent finite algorithms our algorithm can provide a more accurate rank one approximation for more than 93 examples

marcinkiewicz integrals with rough homogeneous kernels of degree zero in product domains abstract the l2 boundedness of the marcinkiewicz integrals in product domains with component wise homogeneous kernels which belong to a certain orlicz space and satisfy the cancellation property is studied