transient turing patterns in a neural field model we investigate turing bifurcations in a neural field model with one spatial dimension for some parameter values the resulting turing patterns are stable while for others the patterns appear transiently we show that this difference is due to the relative position in parameter space of the saddle node bifurcation of a spatially periodic pattern and the turing bifurcation point by varying parameters we are able to observe transient patterns whose duration scales in the same way as type i intermittency similar behavior occurs in two spatial dimensions

wavelet like bases for the fast solutions of second kind integral equations a class of vector space bases is introduced for the sparse representation of discretizations of integral operators an operator with a smooth nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix to high precision a method is presented that employs these bases for the numerical solution of second kind integral equations in time bounded by o n log 2 n where n is the number of points in the discretization numerical results are given which demonstrate the effectiveness of the approach and several generalizations and applications of the method are discussed

turing instability and pattern formation in a two population neuronal network model abstract a two population firing rate model describing the dynamics of excitatory and inhibitory neural activity in one spatial dimension is investigated with respect to formation of patterns in particular stationary periodic patterns and spatiotemporal oscillations conditions for existence of spatially homogeneous equilibrium states are first determined and the stability properties of these equilibria are investigated it is shown that the nonlocal synaptic interactions may promote a finite bandwidth instability in a way analogous to diffusion effects in the classical turing instability for reaction diffusion equations and modulational instability in the theory of nonlinear waves in nonlocal defocusing kerr media our analysis relies on the wave number dependent invariants of the 2 2 matrix representing the spatially fourier transformed linearized evolution equations the generic picture which emerges is an instability consisting of a finite set of well separated unstable bands in wave number space gain bands the case with symmetrical exponentially decaying connectivity functions is investigated in detail allowing for a more comprehensive analysis of the gain band structure and in particular conditions for the excitation of a single gain band through a turing hopf bifurcation with the relative inhibition time constant as control parameter two typical situations emerge depending on the thresholds and inclinations of the sigmoidal firing rate functions i a single gain band is excited through a turing hopf bifurcation and the resulting state is a spatiotemporally oscillating pattern or ii the instability develops into a stationary periodic pattern i e a set of equidistant bumps the dependence of instability type on the inclinations of the firing rate function and the time constant are comprehensively investigated demonstrating for example that only stationary patterns can be generated for sufficiently small inhibitory time constants the nonlinear development of the gain band instabilities is further elucidated by direct numerical simulations

a characterization of matrix groups that act transitively on the cone of positive definite matrices abstract it is well known that the group of all nonsingular lower block triangular p p matrices acts transitively on the cone p of all positive definite p p matrices this result has been applied to obtain several major results in multivariate statistical distribution theory and decision theory here a converse is established if a matrix group acts transitively on p then its group algebra must be similar to the algebra of all lower block triangular p p matrices with respect to a fixed partitioning this implies the nonexistence of multivariate normal linear statistical models with unrestricted covariance structure that admit a transitive group action other than those classical models invariant under a full block triangular group

deterministic cca based algorithms for blind equalization of fir mimo channels in this paper a new deterministic technique for blind equalization of multiple input multiple output mimo channels is presented the proposed method relies on the channel order diversity of the finite impulse response mimo channel and it does not make any assumption about the spectra or the finite alphabet property of the sources assuming that the lengths of the single input multiple output channels are known or estimated a priori it is able to extract the sources up to a scale and rotation matrix ambiguity which only affects to those sources associated to single input multiple output simo channels with equal lengths unlike previously proposed techniques the method described in this paper obtains the equalizers and the best linear combination of their outputs in a single step and thus optimally the reformulation of the proposed method as a set of nested canonical correlation analysis problems is exploited to obtain efficient batch and adaptive equalization algorithms finally the performance of the proposed algorithms is evaluated by means of some simulation examples

using the bootstrap to select one of a new class of dimension reduction methods dimension reduction in a regression analysis of response y given a p dimensional vector of predictors x reduces the dimension of x by replacing it with a lower dimensional linear combination x of the x s without specifying a parametric model and without loss of information about the conditional distribution of y given x we unify three existing methods sliced inverse regression sir sliced average variance estimate save and principal hessian directions phd into a larger class of methods each method estimates a particular candidate matrix essentially a matrix of parameters we introduce broad classes of dimension reduction candidate matrices and we distinguish estimators of the matrices from the matrices themselves given these classes of methods and several ways to estimate any matrix we now have the problem of selecting a particular matrix and estimation method we propose bootstrap methodology to select among candidate matrices estimators and dimension and in particular we investigate l

stability and bifurcations in neural fields with finite propagation speed and general connectivity a stability analysis is presented for neural field equations in the presence of finite propagation speed along axons and for a general class of connectivity kernels and synaptic proper ties sufficient conditions are given for the stability of equilibrium solutions it is shown that the propagation delays play a significant role in nonstationary bifurcations of equilibria whereas the stationary bifurcations depend only on the connectivity kernel in the case of nonstationary bifurca tions bounds are determined on the frequencies of the resulting oscillatory solutions a perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns oscillations and traveling waves for high propagation speeds a simple method is derived that allows the determi nation of the bifurcation type by visual inspection of the fourier transforms of the kernel and its first moment results are numerically illustrated on a class of neurologically plausible systems with combinations of gaussian excitatory and inhibitory connections

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

martingale transforms and complex uniform convexity les transformees de martingale et les operateurs integraux singuliers de calderon zygmund sont bornees comme des operateurs de l 2 l 1 dans l 2 l q quand 0 q 1 si y est un sous espace reflexif de l 1 alors l 1 y peut se renormer pour etre 2 complexe uniformement convexe on donne une nouvelle demonstration de la propriete de cotype 2 de l 1 h

finite size effects in learning and generalization in linear perceptrons most properties of learning and generalization in linear perceptrons can be derived from the average response function g we present a method for calculating g using only simple matrix identities and partial differential equations using this method we first rederive the known result for g in the thermodynamic limit of perceptrons of infinite size n which has previously been calculated using replica and diagrammatic methods we also show explicitly that the response function is self averaging in the thermodynamic limit extensions of our method to more general learning scenarios with anisotropic teacher space priors input distributions and weight decay terms are discussed finally finite size effects are considered by calculating the o 1 n correction to g we verify the result by computer simulations and discuss the consequences for generalization and learning dynamics in linear perceptrons of finite size

duality in testing multivariate hypotheses summary this paper derives a duality result for a general class of hypothesis testing problems in multivariate analysis utilizing the relationship between convex cones and their polar cones together with the properties of minimum norm problems between points and cones in euclidian space special cases of this result yield generalizations of a well known duality relation in multivariate equality constraints testing for example any multivariate inequality constraints test on the parameters of a multivariate normal random vector has an equivalent multivariate one sided test in terms of the vector of dual variables associated with the constraints also any combination multivariate inequality and equality constraints test has an equivalent combination multivariate one sided and two sided test in terms of the vector of dual variables associated with both sets of constraints

regression based prediction for two stage survey data with correlated normal errors abstract regression based prediction using the least squares estimates of the regression parameters is a popular practice however the application of such a method for a large scale survey data can be very misleading especially if the population under study has a particular structure 1 recent studies by hoque 2 and bhatti 3 have revealed the special structure of the population in multistage survey data in this paper to accommodate the special population structure we consider a heteroscedastic multiple regression model for the two stage survey data from a population in which the observations within each cluster or region are equi correlated but uncorrelated with the units of different clusters assuming that the errors follow a multivariate normal distribution the marginal likelihood estimate of the correlation parameter has been proposed and the prediction distribution for a set of future responses is obtained by using the structural relation of the model the prediction distribution is found to be student t whose number of degrees of freedom depends on the sample sizes and the dimension of the regression parameter

multiple window based minimum variance spectrum estimation for multidimensional random fields spectrum analysis is viewed as the problem of estimating the power of a process contained within narrow bands this view leads naturally to consideration of filter based methods for estimating spectra multiple window based estimators where the power in a band is estimated as the average of the powers estimated at the outputs of multiple filters are considered the filters are designed using a linearly constrained minimum variance criterion commonly employed in adaptive beamforming this results in filters that automatically adjust their sidelobes to minimize leakage of energy from outside the band of interest expressions for the bias and variance of the power estimates are derived assuming the sample covariance matrix estimate is used to estimate the data covariance matrix and that the data are independent and identically gaussian distributed these expressions lead to the definition of a performance factor that indicates the degree of variance reduction obtained via multiple window processing a technique for obtaining increased suppression of energy leaking through the filter sidelobes at the expense of the response fidelity to energy in the band is presented simulations are presented to illustrate the effectiveness of the technique

dynamic instabilities in scalar neural field equations with space dependent delays in this paper we consider a class of scalar integral equations with a form of space dependent delay these non local models arise naturally when modelling neural tissue with active axons and passive dendrites such systems are known to support a dynamic oscillatory turing instability of the homogeneous steady state in this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability the appropriate amplitude equations are found to be the coupled mean field ginzburg landau equations describing a turing hopf bifurcation with modulation group velocity of o 1 importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio temporal kernels defining the neural field equation of interest indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio temporal interactions we illustrate the predictive power of this form of analysis with comparison against direct numerical simulations paying particular attention to the competition between standing and travelling waves and the onset of benjamin feir instabilities

simultaneous estimation and prediction using the expected coverage measure criterion on utilise des methodes d invariance pour construire des procedures minimax dans les modeles lineaires multivariables

connection coefficients on an interval and wavelet solutions of burgers equation a definition of connection coefficients is introduced and techniques of computation are presented we use semi implicit time difference scheme to solve burgers equation by applying the evaluations of connection coefficients in calculating the integrals of the variational form comparisons of accuracy and robustness of numerical solutions are mentioned in the examples

the weakly nonlocal limit of a one population wilson cowan model abstract we derive the weakly nonlocal limit of a one population neuronal field model of the wilson cowan type in one spatial dimension by transforming this equation to an equation in the firing rate variable it is shown that stationary periodic solutions exist by appealing to a pseudo potential analysis the solutions of the full nonlocal equation obey a uniform bound and the stationary periodic solutions in the weakly nonlocal limit satisfying the same uniform bound are characterized by finite ranges of pseudo energy constants the time dependent version of the model is reformulated as a ginzburg landau khalatnikov type of equation in the firing rate variable where the maximum minimum points correspond stable unstable homogeneous solutions of the weakly nonlocal limit based on this formulation it is also conjectured that the stationary periodic solutions are unstable we implement a numerical method for the weakly nonlocal limit of the wilson cowan type of model based on the wavelet galerkin approach we perform some numerical tests to illustrate the stability of homogeneous solutions and the evolution of the bumps

a renormalisation group method i gaussian integration and normed algebras this paper is the first in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields fermion fields or both our immediate motivation is a specific model involving both boson and fermion fields which arises as a representation of the continuous time weakly self avoiding walk in this paper we define normed algebras suitable for a renormalisation group analysis and develop methods for performing analysis on these algebras we also develop the theory of gaussian integration on these normed algebras and prove estimates for gaussian integrals the concepts and results developed here provide a foundation for the continuation of the method presented in subsequent papers in the series

multiple bumps in a neuronal model of working memory we study a partial integro differential equation defined on a spatially extended domain that arises from the modeling of working or short term memory in a neuronal network the equation is capable of supporting spatially localized regions of high activity which can be switched on and off by transient external stimuli we analyze the effects of coupling between units in the network showing that if the connection strengths decay monotonically with distance then no more thanon e regionof high activity canpersist whereas if they decay inanoscillatory fashion then multiple regions can persist

boundary values of functions in vector valued hardy spaces and geometry on banach spaces abstract the spaces of boundary values of vector valued functions in hardy spaces defined by either holomorphic functions on the disk or harmonic functions with maximal function in l p are characterized in terms of vector valued measures of bounded p variation we extend to the case p 1 a characterization of the radon nikodym property based on the existence of limits at the boundary for harmonic functions with maximal function in l 1 in the case 0 p 1 we find the umd property as the necessary and sufficient condition to make the spaces defined by maximal function and by conjugate poisson kernel coincide

spatially structured activity in synaptically coupled neuronal networks ii lateral inhibition and standing pulses we consider the existence and stability of standing pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled networks of excitatory and inhibitory neurons in a single spatial domain assuming an arrangement of synaptic connections described by lateral inhibition previous formal arguments have demonstrated the existence of both stable and unstable standing pulses s amari biol cybern 27 1977 pp 77 87 these results have formed the basis for several recent hypotheses regarding the generation of sustained activity patterns in prefrontal cortex and other brain regions implicit in the lateral inhibition arrangement however is the assumption that the dynamics of inhibition are instantaneous here we present two arguments demonstrating the loss of stability of standing pulse solutions through a hopf bifurcation when more realistic inhibitory dynamics are considered the first argument parallels amari s formal presentation while the seco

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

a predictive approach to the analysis of designed experiments abstract viewing the analysis of designed experiments as a model selection problem we introduce the use of a predictive bayesian criterion in this context based on the predictive density of a replicate experiment pdre a calibration of the criterion is provided to assist in the model choice the relationships of the proposed criterion to other prevalent criteria such as aic bic and mallows s c p are given an information theoretic criterion based on the pdre s of two competing models is also introduced and compared with the usual f statistic for two nested models examples are given to illustrate the proposed methodology

some characterizations of the analytic radon nikodým property in banach spaces the analytic radon nikod m property for a complex banach space x is characterized in several ways analogous to the common characterizations of the radon nikod m property for example the banach space x has the analytic radon nikod m property if and only if any operator from l1 to x that factors through l1h01 is representable the characterizations are used to study complemented subspaces of l1h01 and to define the analytic radon nikod m property for real banach spaces

finite size effects and optimal test set size in linear perceptrons fluctuations in the test error are important in the learning theory of finite dimensional systems as they represent how well the test error matches the average test error by explicitly finding the variance of the test error due to randomness present in both the data set and algorithm for a linear perceptron of dimension n we are able to address such questions as the optimal test set size where exact results were not tractable a good approximation is given to the variance we find that the optimal test set size possesses a phase transition between linear and 2 3 power law scaling in the system size n

permutation inference for canonical correlation analysis canonical correlation analysis cca has become a key tool for population neuroimaging allowing investigation of associations between many imaging and non imaging measurements as age sex and other variables are often a source of variability not of direct interest previous work has used cca on residuals from a model that removes these effects then proceeded directly to permutation inference we show that a simple permutation test as typically used to identify significant modes of shared variation on such data adjusted for nuisance variables produces inflated error rates the reason is that residualisation introduces dependencies among the observations that violate the exchangeability assumption even in the absence of nuisance variables however a simple permutation test for cca also leads to excess error rates for all canonical correlations other than the first the reason is that a simple permutation scheme does not ignore the variability already explained by previous canonical variables here we propose solutions for both problems in the case of nuisance variables we show that transforming the residuals to a lower dimensional basis where exchangeability holds results in a valid permutation test for more general cases with or without nuisance variables we propose estimating the canonical correlations in a stepwise manner removing at each iteration the variance already explained while dealing with different number of variables in both sides we also discuss how to address the multiplicity of tests proposing an admissible test that is not conservative and provide a complete algorithm for permutation inference for cca

localized activity patterns in two population neuronal networks abstract we investigate a two population neuronal network model of the wilson cowan type with respect to existence of localized stationary solutions bumps and focus on the situation where two separate bump solutions one narrow pair and one broad pair exist the stability of the bumps is investigated by means of two different approaches the first generalizes the amari approach while the second is based on a direct linearization procedure a classification scheme for the stability problem is formulated and it is shown that the two approaches yield the same predictions except for one notable exception the narrow pair is generically unstable while the broad pair is stable for small and moderate values of the relative inhibition time at a critical relative inhibition time the broad pair is typically converted to stable breathers through a hopf bifurcation in our numerical example the broad pulse pair remains stable even when the inhibition time constant is three times longer than the excitation time constant thus our model results do not support the claim that slow excitation mediated by e g nmda receptors is needed to allow stable bumps

dimension reduction in binary response regression abstract the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model focusing on the central subspace we investigate such sufficient dimension reduction in regressions with a binary response three existing methods sliced inverse regression principal hessian direction and sliced average variance estimation and one new method difference of covariances are studied for their ability to estimate the central subspace and produce sufficient summary plots combining these numerical methods with the graphical methods proposed earlier by cook leads to a novel paradigm for the analysis of binary response regressions

wavelet galerkin solutions for one dimensional partial differential equations in this paper we describe how wavelets may be used to solve partial differential equations these problems are currently solved by techniques such as finite differences finite elements and multi grid the wavelet method however offers several advantages over traditional methods wavelets have the ability to represent functions at different levels of resolution thereby providing a logical means of developing a hierarchy of solutions furthermore compactly supported wavelets such as those due to daubechies1 are localized in space which means that the solution can be refined in regions of high gradient e g stress concentrations without having to regenerate the mesh for the entire problem in order to demonstrate the wavelet technique we consider the one dimensional counterpart of helmholtz s equation by comparison with a simple finite difference solution to this problem with periodic boundary conditions we show how a wavelet technique may be efficiently developed dirichlet boundary conditions are then imposed using the capacitance matrix method described by proskurowski and widlund2 and others the convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method at least for problems with simple geometries

hierarchical interaction models lauritzen and wermuth have proposed a class of models for mixed qualitative and continuous data defined by two properties that the continuous variables are normally distributed given the qualitative variables and that a set of conditional independence relations hold between specified pairs of variables the present paper examines an extension to this class called hierarchical interaction models a compact form for model representation is described and an estimation algorithm is given some properties of the models a compact form for model representation is described and an estimation algorithm is given some properties of the models concerning marginalization and conditioning are examined the class includes and generalizes hierarchical log linear models standard fixed effect anova multivariate anova and multivariate regression models

predictive inference on equicorrelated linear regression models beyond the customary analysis through the estimation and hypothesis testing about the parameters of the multiple regression models often a natural interest is to predict the responses for a given set of values of the predictors the main objective of this article is to obtain the prediction distribution for a set of future responses from a multiple linear regression model which follow equicorrelation structure it derives the marginal likelihood estimate for the equicorrelation parameter r and then uses the invariant differentials to compute the joint distribution of the unobserved but realized future errors the prediction distribution is derived by using the structural relation of the model the main finding of this paper is that the prediction distribution turned out to be a student t which depends only on the estimated r and is invariant to the degrees of freedom of the original student t distribution

estimation of parameters in the extended growth curve model via outer product least squares for covariance abstract in this paper estimation of parameters in the extended growth curve model without assumption of normality is presented via outer product least squares for covariance copls the copls estimator for covariance can be explicitly expressed and proved to follow a linear transformation of k 1 independent wishart distributions for a normal error matrix where k is the number of profiles of the growth curve then two stage generalized least squares gls estimators for regression coefficients are elaborately derived finally the copls estimator and the two stage gls estimators are shown to have desired properties both in finite and large samples simulation studies for finite sample sizes are provided to confirm that the copls estimator and the two stage gls estimators are alternative competitors with some evident merits to the existing maximum likelihood estimators in small samples a real data set is analyzed to demonstrate the proposed methodology

generalized collinearity diagnostics abstract working in the context of the linear model y x e we generalize the concept of variance inflation as a measure of collinearity to a subset of parameters in denoted by 1 with the associated columns of x given by x 1 the essential idea underlying this generalization is to examine the impact on the precision of estimation in particular the size of an ellipsoidal joint confidence region for 1 of less than optimal selection of other columns of the design matrix x 2 treating still other columns x 0 as unalterable even hypothetically in typical applications x 1 contains a set of dummy regressors coding categories of a qualitative variable or a set of polynomial regressors in a quantitative variable x 2 contains all other regressors in the model save the constant which is in x 0 if 2 v denotes the realized variance of and 2 u is the variance associated with an optimal selection of x 2 then the corresponding scaled dispersion ellipsoids to be compared are v x x v

the complex convexity of quasi normed linear spaces abstract a notion of uniform convexity is defined for quasi normed complex spaces by replacing norms of midpoints of segments in the space by norms of centers of complex discs in the space complex uniformly convex spaces c u c spaces always have cotype in banach lattices possessing cotype and being complex uniformly convex are the same duals of c algebras are c u c and in fact have cotype 2 the behavior of certain martingale difference sequences in c u c spaces is examined this leads to an isomorphic description of the spaces having the property

tensor products and statistics abstract a dictionary between operator based and matrix based languages in multivariate statistical analysis is proposed then this formulary is applied to asymptotic factorial analyses especially for giving asymptotic covariance matrices and operators in an explicit form finally we present the mathematical foundations on which are based the functional tools i e tensor products of linear spaces of vectors and of operators

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

wiener filters in canonical coordinates for transform coding filtering and quantizing canonical correlations are used to decompose the wiener filter into a whitening transform coder a canonical filter and a coloring transform decoder the outputs of the whitening transform coder are called canonical coordinates these are the coordinates that are reduced in rank and quantized in our finite precision version of the gauss markov theorem canonical correlations are in fact cosines of the canonical angles between a source vector and a measurement vector they produce new formulas for error covariance spectral flatness and entropy

approximation of jensen measures by image measures under holomorphic functions and applications we show that jensen measures defined on c n or more generally on a complex banach space x can be approximated by the image of lebesgue measure on the torus under x valued polynomials defined on c we give similar characterizations for jensen measures in terms of analytic martingales and hardy martingales the results are applied to approximate plurisubharmonic martingales by hardy martingales which enables us to give a characterization of the analytic radon nikodym property of banach spaces in terms of convergence of plurisubharmonic martingales thus solving a problem of g a edgar

on the quantification of damping model uncertainty abstract the characterization of uncertainty in the damping forces in a vibrating structure is of significant interest in structural dynamics the most common approach is to use a viscous damping matrix with uncertain coefficients the viscous damping is not the only damping model within the scope of linear analysis any model which makes the energy dissipation functional non negative is a possible candidate for a valid damping model there are many functional forms which can be used as damping models in this paper two approaches are proposed to quantify such model form uncertainty associated with the use of the viscous damping model not only the model parameters the first approach is based on an ensemble of equivalent damping functions and the second approach is based on the random matrix theory the results obtained from the two methods are compared using numerical examples it was observed that the random matrix theory can be used to quantify damping model uncertainty in some cases

the computation of wavelet galerkin approximation on a bounded interval this paper describes exact evaluations of various finite integrals whose integrands involve products of daubechies compactly supported wavelets and their derivatives and or integrals these finite integrals play an essential role in the wavelet galerkin approximation of differential or integral equations on a bounded interval

experimental randomization and the validity of normal theory inference abstract this article examines how randomization helps to validate model based probability statements the asymptotic validity of normal theory tests for means in multivariate linear models is established under general conditions numerical results are presented showing how randomization provides a limited degree of robustness in small experiments the conditional relevance of randomization is also discussed consider the following additive model for a comparative experiment y gu where y is a vector of responses is a vector of treatment effects u is a vector of unit effects and g is a permutation representing the assignment of treatments to units the vector gu is obtained by using g to permute the coordinates of u it is assumed that is fixed and g and u are random and independent the experimenter observes y and g but not or u and chooses the distribution of g that is the randomization strategy in this setup three probabilities may be considered conditional given g conditional g

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

in sample tests of predictive ability a new approach abstract this paper presents evidence linking in sample tests of predictive content and out of sample forecast accuracy our approach focuses on the negative effect that finite sample estimation error has on forecast accuracy despite the presence of significant population level predictive content we derive in sample tests that assess whether a variable has predictive content and whether this content is estimated precisely enough to improve forecast accuracy our tests are asymptotically non central chi square or non central normal we provide a convenient bootstrap for computing critical values in monte carlo and empirical analysis we examine the effectiveness of our testing procedure

analytic martingale convergence abstract let x be a complex banach space then every l 1 bounded analytic martingale in x converges if and only if x has the analytic radon nikodym property

canonical coordinates and the geometry of inference rate and capacity canonical correlations measure the cosines of principal angles between random vectors these cosines multiplicatively decompose concentration ellipses for second order filtering and additively decompose the information rate for the gaussian channel more over they establish a geometrical connection between error covariance error rate information rate and principal angles there is a limit to how small these angles can be made and this limit determines the channel capacity

geometry and multiple direction estimation abstract we consider the problem of the estimation of multiple target directions given the output of a phased array of identical elements the natural mathematical setting for this problem is the grassmannian manifold in order to understand the problems considered here a knowledge of exterior forms lie groups and differential geometry is essential we review some of the essential ideas relevant to the problem including the maurer cartan equations algebraic varieties in projective space and the invariant volume integral for symmetric spaces our purpose is to study probability measures on grassmannians in order to pose and solve the bayes problem on grassmannians

bayesian nonparametric clustering for positive definite matrices symmetric positive definite spd matrices emerge as data descriptors in several applications of computer vision such as object tracking texture recognition and diffusion tensor imaging clustering these data matrices forms an integral part of these applications for which soft clustering algorithms k means expectation maximization etc are generally used as is well known these algorithms need the number of clusters to be specified which is difficult when the dataset scales to address this issue we resort to the classical nonparametric bayesian framework by modeling the data as a mixture model using the dirichlet process dp prior since these matrices do not conform to the euclidean geometry rather belongs to a curved riemannian manifold existing dp models cannot be directly applied thus in this paper we propose a novel dp mixture model framework for spd matrices using the log determinant divergence as the underlying dissimilarity measure to compare these matrices and further using the connection between this measure and the wishart distribution we derive a novel dpm model based on the wishart inverse wishart conjugate pair we apply this model to several applications in computer vision our experiments demonstrate that our model is scalable to the dataset size and at the same time achieves superior accuracy compared to several state of the art parametric and nonparametric clustering algorithms

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

estimating the structural dimension of regressions via parametric inverse regression a new estimation method for the dimension of a regression at the outset of an analysis is proposed a linear subspace spanned by projections of the regressor vector x which contains part or all of the modelling information for the regression of a vector y on x and its dimension are estimated via the means of parametric inverse regression smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model no restrictions are placed on the distribution of the regressors the estimate of the dimension of the regression is based on optimal estimation procedures a simulation study shows the method to be more powerful than sliced inverse regression in some situations

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

eigenvalues distribution for products of independent spherical ensembles we consider the product of independent spherical ensembles by the special structure of eigenvalues as a rotation invariant determinant point process we show that the empirical spectral distribution of the product converges with probability one to a non random distribution and the limiting eigenvalue distribution is a power of spherical law we also present an interesting correspondence between the eigenvalues of three classes of random matrix ensembles and zeros of random polynomials

exploiting the hamiltonian structure of a neural field model abstract we study the unexpected disappearance of stable homoclinic orbits in regions of parameter space in a neural field model with one spatial dimension the usual approach of using numerical continuation techniques and local bifurcation theory is insufficient to explain the qualitative change in the model s behaviour the lack of robustness of the model to small perturbations in parameters is surprising and the phenomenon may be of broader significance than just our model by exploiting the hamiltonian structure of the time independent system we develop a numerical technique with which we discover that a small separate solution curve exists for a range of parameter values as the firing rate function steepens the small curve causes the main curve to break and stable homoclinic orbits are destroyed in a region of parameter space numerically we use level set analysis to find that a codimension one heteroclinic bifurcation occurs at the terminating ends of the solution curves by replacing the firing rate function with a step function we show analytically that the bifurcation is related to the value of the firing threshold we also show the existence of heteroclinic orbits at the breakpoints using a travelling front analysis in the time dependent system

a unifying discussion of correlation analysis for complex random vectors the assessment of multivariate association between two complex random vectors is considered a number of correlation coefficients based on three popular correlation analysis techniques namely canonical correlation analysis multivariate linear regression and partial least squares are reviewed and connected to performance measures in signal processing and communications such as mean squared estimation error mutual information and signal to noise ratio snr for complex data there are three types of correlation coefficients which account for rotational reflectional and total i e rotational and reflectional dependencies between two random vectors these three types are defined and analyzed for different correlation coefficients and a numerical example is given it is often required to compare two complex random vectors in a lower dimensional subspace for the large class of increasing schur convex correlation coefficients it is shown that the low rank approximations of two random vectors maximizing a particular correlation coefficient are determined only by the constraints imposed on the correlation analysis technique in this context the correlation spread is defined as a normalized measure of how much of the overall correlation is contained in a low dimensional subspace

proof of the conjectures of h uhlig on the singular multivariate beta and the jacobian of a certain matrix transformation uhlig proposes two conjectures the first concerns the jacobian of the transformation y b x b where b is the matrix m x m and m and x y belong to the class of positive semidefinite matrices of the order of m x m of rank n m s m n the second is concerned with the singular multivariate beta distribution this article seeks to prove the two conjectures the latter result is then extended to the case of the singular multivariate f distribution and the respective density functions are located for the nonzero positive eigenvalues of the singular beta and f matrices

two channel constrained least squares problems solutions using power methods and connections with canonical coordinates the problem of two channel constrained least squares cls filtering under various sets of constraints is considered and a general set of solutions is derived for each set of constraints the solution is determined by a coupled asymmetric generalized eigenvalue problem this eigenvalue problem establishes a connection between two channel cls filtering and transform methods for resolving channel measurements into canonical or half canonical coordinates based on this connection a unified framework for reduced rank wiener filtering is presented then various representations of reduced rank wiener filters in canonical and half canonical coordinates are introduced an alternating power method is proposed to recursively compute the canonical coordinate and half canonical coordinate mappings a deflation process is introduced to extract the mappings associated with the dominant coordinates the correctness of the alternating power method is demonstrated on a synthesized data set and conclusions are drawn

a note on roy s largest root the name roy s largest root and similar names are used in practice to label two different but functionally related statistics one proportional to anf and the other a squared canonical correlation this note presents the logic that leads to the two formulations states which statistic some popular statistical packages use and shows the possible source of this inconsistency in the original work of roy 1953 and heck 1960

canonical coordinates for transform coding of noisy sources based on the additive white quantization noise model linear transform coders are derived for gaussian sources corrupted by noise there are two alternative design objectives minimizing the trace of the error correlation matrix and thus minimizing the mean squared error or minimizing the determinant of the error correlation matrix and thus maximizing information rate it is shown that a solution to both problems is to first transform the noisy observations into canonical coordinates quantize and apply a wiener filter in this coordinate system and then transform the result back to the original coordinates canonical coordinates are uncorrelated and quantization and wiener filtering are applied to each component independently the type of canonical coordinate system depends on the design objective quantization in half canonical coordinates minimizes the mean squared error and quantization in full canonical coordinates maximizes information rate finally it is also demonstrated in this paper that majorization is the fundamental principle underlying proofs of optimal transform coding

on module maps and bilinear forms defined on shift invariant subspaces of lp x abstract in this paper we prove a lifting of the commutant theorem on l p t x when x is a banach space of type 2 and p 2 we also obtain a generalization of sarason s lifting theorem then we study some modular versions of hilbertian factorizations for module maps between invariant subspaces of l p t x the last part of the paper is devoted to a representation theorem for invariant subspaces of h 2 x which are isomorphic to a hilbert space

eigenvalue density of linear stochastic dynamical systems a random matrix approach abstract eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics when uncertainties are considered the eigenvalue problem becomes a random eigenvalue problem in this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the wishart random matrices an analytical expression involving the stieltjes transform is derived for the density of the eigenvalues when the dimension of the corresponding random matrix becomes asymptotically large the mean matrices and the dispersion parameters associated with the mass and stiffness matrices are necessary to obtain the density of the eigenvalues in the frameworks of the proposed approach the applicability of a simple eigenvalue density function known as the mar enko pastur mp density is investigated the analytical results are demonstrated by numerical examples involving a plate and the tail boom of a helicopter with uncertain properties the new results are validated using an experiment on a vibrating plate with randomly attached spring mass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework

linear models based on noisy data and the frisch scheme we address the problem of identifying linear relations among variables based on noisy measurements this is a central question in the search for structure in large data sets often a key assumption is that measurement errors in each variable are independent this basic formulation has its roots in the work of charles spearman in 1904 and of ragnar frisch in the 1930s various topics such as errors in variables factor analysis and instrumental variables all refer to alternative viewpoints on this problem and on ways to account for the anticipated way that noise enters the data in the present paper we begin by describing certain fundamental contributions by the founders of the field and provide alternative modern proofs to certain key results we then go on to consider a modern viewpoint and novel numerical techniques to the problem the central theme is expressed by the frisch kalman dictum which calls for identifying a noise contribution that allows a maximal number of simultaneous linear relations a

sufficient forecasting using factor models we consider forecasting a single time series when there is a large number of predictors and a possible nonlinear effect the dimensionality was first reduced via a high dimensional approximate factor model implemented by the principal component analysis using the extracted factors we develop a novel forecasting method called the sufficient forecasting which provides a set of sufficient predictive indices inferred from high dimensional predictors to deliver additional predictive power the projected principal component analysis will be employed to enhance the accuracy of inferred factors when a semi parametric approximate factor model is assumed our method is also applicable to cross sectional sufficient regression using extracted factors the connection between the sufficient forecasting and the deep learning architecture is explicitly stated the sufficient forecasting correctly estimates projection indices of the underlying factors even in the presence of a nonparametric forecasting function the proposed method extends the sufficient dimension reduction to high dimensional regimes by condensing the cross sectional information through factor models we derive asymptotic properties for the estimate of the central subspace spanned by these projection directions as well as the estimates of the sufficient predictive indices we further show that the natural method of running multiple regression of target on estimated factors yields a linear estimate that actually falls into this central subspace our method and theory allow the number of predictors to be larger than the number of observations we finally demonstrate that the sufficient forecasting improves upon the linear forecasting in both simulation studies and an empirical study of forecasting macroeconomic variables

analysis of nonlocal neural fields for both general and gamma distributed connectivities abstract this work studies the stability of equilibria in spatially extended neuronal ensembles we first derive the model equation from statistical properties of the neuron population the obtained integro differential equation includes synaptic and space dependent transmission delay for both general and gamma distributed synaptic connectivities the latter connectivity type reveals infinite finite and vanishing self connectivities the work derives conditions for stationary and nonstationary instabilities for both kernel types in addition a nonlinear analysis for general kernels yields the order parameter equation of the turing instability to compare the results to findings for partial differential equations pdes two typical pde types are derived from the examined model equation namely the general reaction diffusion equation and the swift hohenberg equation hence the discussed integro differential equation generalizes these pdes in the case of the gamma distributed kernels the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges as a novel finding we obtain turing instabilities in fields with local inhibition lateral excitation while wave instabilities occur in fields with local excitation and lateral inhibition numerical simulations support the analytical results

testing for nodal dependence in relational data matrices relational data are often represented as a square matrix the entries of which record the relationships between pairs of objects many statistical methods for the analysis of such data assume some degree of similarity or dependence between objects in terms of the way they relate to each other however formal tests for such dependence have not been developed we provide a test for such dependence using the framework of the matrix normal model a type of multivariate normal distribution parameterized in terms of row and column specific covariance matrices we develop a likelihood ratio test lrt for row and column dependence based on the observation of a single relational data matrix we obtain a reference distribution for the lrt statistic thereby providing an exact test for the presence of row or column correlations in a square relational data matrix additionally we provide extensions of the test to accommodate common features of such data such as undefined diagonal entries a nonzero mean multipl

passive detection of correlated subspace signals in two mimo channels in this paper we consider a two channel multiple input multiple output passive detection problem in which there is a surveillance array and a reference array the reference array is known to carry a linear combination of broadband noise and a subspace signal of known dimension but unknown basis the question is whether the surveillance channel carries a linear combination of broadband noise and a subspace signal of the same dimension but unknown basis which is correlated with the subspace signal in the reference channel we consider a second order detection problem where these subspace signals are structured by an unknown but common p dimensional random vector of symbols transmitted from sources of opportunity and then received through unknown m times p matrices at each of the m element arrays the noises in each channel have spatial correlation models ranging from arbitrarily correlated to independent with identical variances we provide a unified framework to derive the generalized likelihood ratio test for these different noise models in the most general case of arbitrary noise covariance matrices the test statistic is a monotone function of canonical correlations between the reference and surveillance channels