universal microscopic correlation functions for products of independent ginibre matrices we consider the product of n complex non hermitian independent random matrices each of size n n with independent identically distributed gaussian entries ginibre matrices the joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single ginibre matrix but with a more complicated weight given by a meijer g function depending on n using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite n and fixed n they are given by the determinant of the corresponding kernel which we construct explicitly in the large n limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum after unfolding they are identical to that of the ginibre ensemble with n 1 and thus universal in contrast the microscopic correlations we find at the origin differ for each n 1 and generalize the known bessel law in the complex plane for n 2 to a new hypergeometric kernel 0fn 1

products of random matrices we derive analytic expressions for infinite products of random 2 x 2 matrices the determinant of the target matrix is log normally distributed whereas the remainder is a surprisingly complicated function of a parameter characterizing the norm of the matrix and a parameter characterizing its skewness the distribution may have importance as an uncommitted prior in statistical image analysis

probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

data free inference of the joint distribution of uncertain model parameters a critical problem in accurately estimating uncertainty in model predictions is the lack of details in the literature on the correlation or full joint distribution of uncertain model parameters in this paper we describe a framework and a class of algorithms for analyzing such missing data problems in the setting of bayesian statistics the analysis focuses on the family of posterior distributions consistent with given statistics e g nominal values confidence intervals the combining of consistent distributions is addressed via techniques from the opinion pooling literature the developed approach allows subsequent propagation of uncertainty in model inputs consistent with reported statistics in the absence of data

a two stage filter for smoothing multivariate noisy data on unstructured grids abstract experimental data as well as numerical simulations are very often affected by noise random fluctuations that distort the final output or the intermediate products of a numerical process in this paper a new method for smoothing data given on nonstructured grids is proposed it takes advantage of the smoothing properties of kernel weighted averaging and least squares techniques the weighted averaging is performed following a shepard like procedure where gaussian kernels are employed while least squares fitting is reduced to the use of very few basis functions so as to improve smoothness though at the price of interpolation accuracy once we have defined an n point grid and want to make a smoothed fit at a given grid point this method reduces to consideration of all m point stencils for a given value of m where m n that include that point and to make a least squares fitting for that particular point in each of those stencils finally the various results thus obtained are weight averaged the weights being inversely proportional to the distance of the point to the middle of the stencil then this process is repeated for all the grid points so as to obtain the smoothing of the input function though this method is generalized for the multivariate case one two and five dimensional test cases are shown as examples of the performance of this method

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

products of independent quaternion ginibre matrices and their correlation functions we discuss the product of independent induced quaternion 4 ginibre matrices and the eigenvalue correlations of this product matrix the joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single ginibre matrix but with a more complicated weight function we find the skew orthogonal polynomials corresponding to the weight function of the product matrix and use the method of skew orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size the radial behavior of the density of states is studied in the limit of large matrices and the macroscopic density is discussed the microscopic limit at the origin at the edge s and in the bulk is also discussed for the radial behavior of the density of states

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

maximum entropy and bayesian data analysis entropic prior distributions the problem of assigning probability distributions which reflect the prior information available about experiments is one of the major stumbling blocks in the use of bayesian methods of data analysis in this paper the method of maximum relative entropy me is used to translate the information contained in the known form of the likelihood into a prior distribution for bayesian inference the argument is inspired and guided by intuition gained from the successful use of me methods in statistical mechanics for experiments that cannot be repeated the resulting entropic prior is formally identical with the einstein fluctuation formula for repeatable experiments however the expected value of the entropy of the likelihood turns out to be relevant information that must be included in the analysis the important case of a gaussian likelihood is treated in detail

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

a continuous description of intervertebral motion by means of spline interpolation of kinematic data extracted by videofluoroscopy in vivo analysis of intervertebral kinematics provides useful information about spinal disorders and performance of disk prostheses diagnosis of intervertebral instability is based on measurement of abnormal range of segmental motion in sagittal plane through functional flexion extension radiography however this concise measure does not take into account the progression of segmental motion in between flexion and extension extremes fluoroscopy can support analysis of intervertebral kinematics during patient s motion with an acceptable x ray dose a spline based method designed for a continuous time description of intervertebral motion extracted by videofluoroscopy is proposed fluoroscopic sagittal sequences of lumbar spine were processed by an automated method based on template matching to track vertebrae a smoothing spline interpolation of the estimated intervertebral kinematic data was performed and a continuous time description of segmental rotation and translation was obtained the smoothing parameter was chosen both to preserve motion and to reduce noise concise measurements were extracted by the continuous time kinematics and compared with standard clinical measurements of intervertebral sagittal rotation and translation the trajectory of instantaneous center of rotation never presented before for in vivo spinal segments was provided and compared with standard measurements of the finite center of rotation results showed a good agreement with standard clinical measurements on average absolute differences resulted 0 74 degree for sagittal rotation 0 59 mm for translation and 1 02 mm for the x and y position of center of rotation the proposed method offers an effective technique for the continuous time description of intervertebral motion maintaining standard clinical measurements for diagnosis of lumbar instability

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

data smoothing by cubic spline filters in this correspondence a digital filter that allows the computation of a smoothing cubic spline for equispaced data with a constant control parameter is proposed filters to compute its first and second derivatives are also presented derived from the classical matrix solution these filters offer an efficient way to calculate smoothed data and its derivatives especially when the length of data is long moreover these filters have been found to possess several interesting properties for instance the smoothing filter is a low pass filter with the maximum flatness property in addition a useful relation between the filter bandwidth and the control parameter is established which can be used for its optimal choice in practice the proposed filters can easily be implemented either with a recursive structure for off line processing or with a nonrecursive implementation for real time processing

complete diagrammatics of the single ring theorem using diagrammatic techniques we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices the formalism allows us to map two distinct areas of free random variables hermitian positive definite operators and non normal r diagonal operators we also rederive the haagerup larsen theorem and show how its recent extension to the eigenvector correlation function appears naturally within this approach

products of independent elliptic random matrices for fixed m 1 we study the product of m independent n times n elliptic random matrices as n tends to infinity our main result shows that the empirical spectral distribution of the product converges with probability 1 to the m th power of the circular law regardless of the joint distribution of the mirror entries in each matrix this leads to a new kind of universality phenomenon the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves our result also generalizes earlier results of gotze tikhomirov on the asymptotic spectrum of products of independent random matrices available at http arxiv org abs 1012 2710 and o rourke soshnikov j probab 16 81 2219 2245 2011 concerning the product of independent iid random matrices

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws

spectral relations between products and powers of isotropic random matrices we show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble is equal to the eigenvalue density of n th power of a single matrix from this ensemble in the limit when the size of the matrix tends to infinity using this observation one can derive the limiting density of the product of n independent identically distributed non hermitian matrices with unitary invariant measures in this paper we discuss two examples the product of n girko ginibre matrices and the product of n truncated unitary matrices we also provide evidence that the result holds also for isotropic orthogonal ensembles

nonparametric fixed interval smoothing with vector splines a computationally efficient algorithm for nonparametric smoothing of vector signals with general measurement covariances is presented this algorithm provides an alternative to the optimal smoothing algorithms that hinge on possibly inaccurate parametric state space models automatic procedures that use the measurements to determine how much to smooth are developed and compared this adaptation allows the data to speak for itself without imposing a gauss markov model structure a nonparametric approach to covariance estimation for the case of independently identically distributed i i d measurement errors is presented monte carlo simulations demonstrate the performance of the algorithm

soft computing approach to feature extraction based on both wavelet theory and fuzzy theory a soft computing system scs is proposed for feature extraction of signals the proposed scs approach possesses the advantages of soft decision making on wavelet coefficients for feature extraction adaptive selectivity of mapping factors to coarse to fine resolution and compact form of feature representation with the scs feature extractor fuzzy sets are used to provide a robust representation for signal information and wavelet transform is used to decompose a signal into detail and approximation signals at a given resolution the detail and approximation signals are inputted to the proposed scs to extract signal features at that resolution level the sensitivity in feature extraction of the proposed approach can be adapted by tuning the fuzzy sets for the detail and approximation signals at different resolutions the signal can be examined and suitable features can be extracted examples of both one dimensional signals and two dimensional fingerprint images are used to illustrate the proposed soft computing approach for feature extraction and pattern recognition the results show that the extracted features are sensitive enough to distinguish the similar signal from the different ones and are robust enough to tolerate noise corruption

a fast algorithm for signal extraction influence and cross validation in state space models summary a fast algorithm is developed for computing the conditional mean and variance of the signal given the observations in a signal plus noise model the resulting recursions can be applied immediately to provide new and efficient formulae for smoothing part or all of the state vector the ideas of studentized residuals and leverage from regression analysis are generalized to state space models and the new algorithm is used to compute the various measures the results are also applied to obtain a new efficient algorithm for polynomial spline smoothing suppose observations are generated by a gaussian signal plus noise process with the signal described by a state space model such a model for the signal occurs often in practice for example when the signal is the output of a stochastic difference or differential equation this paper presents a new algorithm for signal extraction that is the computa tion of the conditional mean and variance of the signal given the observations the usual method for signal extraction is to use the kalman filter see for example anderson moore 1979 p 105 followed by a smoothing step using an algorithm such as the fixed interval smoothing algorithm anderson moore 1979 p 187 our approach also uses the kalman filter but for the smoothing step introduces a new set of recursions which are more efficient than those for the fixed interval smoothing algorithm as for the fixed interval smoothing algorithm recursions for estimating the signal and its variance are carried out separately so that considerable additional savings can be made if the signal estimate alone is required the development uses ideas introduced by ansley kohn 1987a a remarkable property of the new recursions is that they can be applied immediately

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

summing free unitary random matrices i use quaternion free probability calculus an extension of free probability to non hermitian matrices which is introduced in a succinct but self contained way to derive in the large size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices weighted by complex numbers in the case of circular unitary ensemble summands i write them in terms of two master equations which i then solve and numerically test in four specific cases i conjecture a finite size extension of these results exploiting the complementary error function i prove a central limit theorem and its first subleading correction for independent identically distributed zero drift unitary random matrices

singular values of products of ginibre random matrices multiple orthogonal polynomials and hard edge scaling limits akemann ipsen and kieburg recently showed that the squared singular values of products of m rectangular random matrices with independent complex gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of meijer g functions we show that this point process can be interpreted as a multiple orthogonal polynomial ensemble we give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel which allows us to find its scaling limits at the origin hard edge the limiting kernels generalize the classical bessel kernels for m 2 they coincide with the scaling limits found by bertola gekhtman and szmigielski in the cauchy laguerre two matrix model which indicates that these kernels represent a new universality class in random matrix theory

nonparametric transformations for both sides of a regression model one way to model heteroscedasticity and skewness of the error distribution in regression is to transform both sides of the regression equation if it is possible to transform the regression equation to result in normally distributed errors then we can obtain more efficient parameter estimates and valid prediction intervals one problem with this approach is that the choice of transformation is usually restricted to the power or shifted power family often there is no scientific basis for this model and the limited flexibility of this parametric family may miss important features of the distribution a more comprehensive approach is to estimate the transformation by using nonparametric methods based on maximizing a penalized likelihood function this maximization problem leads naturally to a spline estimate for the log derivative of the transformation an algorithm for computing the estimate is given along with results on the existence and uniqueness of the estimate this nonparametric method is illustrated by using a non normally distributed fisheries data set

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

renormalizing rectangles and other topics in random matrix theory we consider random hermitian matrices made of complex or realm n rectangular blocks where the blocks are drawn from various ensembles these matrices haven pairs of opposite real nonvanishing eigenvalues as well asm n zero eigenvalues form n these zero eigenvalues are kinematical in the sense that they are independent of randomness we study the eigenvalue distribution of these matrices to leading order in the large n m limit in which the rectangularity r m n is held fixed we apply a variety of methods in our study we study gaussian ensembles by a simple diagrammatic method by the dyson gas approach and by a generalization of the kazakov method these methods make use of the invariance of such ensembles under the action of symmetry groups the more complicated wigner ensemble which does not enjoy such symmetry properties is studied by large n renormalization techniques in addition to the kinematical function spike in the eigenvalue density which corresponds to zero eigenvalues we find for both types of ensembles that if r 1 is held fixed asn then nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin this separation arises because the nonzero eigenvalues are repelled macroscopically from the origin finally we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes a behavior governed by airy functions asr 1 the lobes come closer and the airy oscillatory behavior near the endpoints that are close to zero breaks down we interpret this breakdown as a signal thatr 1 drives a crossover to the oscillation governed by bessel functions near the origin for matrices made of square blocks

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

an iterative approach to two dimensional laplacian smoothing with application to image restoration abstract smoothing methods are increasingly important in modern graphically oriented data analysis there are several fast and reliable univariate cross validated smoothing algorithms but comparable methods are not available in higher dimensions this article develops an iterative approach to two dimensional laplacian spline smoothing based on tensor product cubic b splines estimating equations are developed and solved iteratively using two line symmetric successive over relaxation with conjugate gradient acceleration an asymptotic approximation for the generalized cross validation score is described the resulting algorithm is naturally suited to two dimensional scatterplot smoothing but has potential value for image restoration as well this is illustrated on a multicolor image example along with an image restoration an uncertainty assessment based on the entropy of the marginal posterior pixel value distribution is produced this analysis particularly highlights the statistical uncertainty in resol

multiplication law and s transform for non hermitian random matrices we derive a multiplication law for free non hermitian random matrices allowing for an easy reconstruction of the two dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles we define the corresponding non hermitian s transform being a natural generalization of the voiculescu s transform in addition we extend the classical hermitian s transform approach to deal with the situation when the random matrix ensemble factors have vanishing mean including the case when both of them are centered we use planar diagrammatic techniques to derive these results

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

large n limit for the product of two coupled random matrices for a pair of coupled rectangular random matrices we consider the squared singular values of their product which form a determinantal point process we show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem this vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure we carry out the steepest descent analysis for a 4 times 4 matrix valued riemann hilbert problem which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified bessel functions a careful study of the vector equilibrium problem combined with this asymptotic analysis ultimately leads to the aforementioned convergence result for the limiting mean distribution an explicit form of the associated spectral curve as well as local sine meijer g and airy universality results for the squared singular values considered

a note on data smoothing by cubic spline filters after a previous correspondence see ibid vol 46 p 2790 6 oct 1998 was published the author found out that several of the key results had already been reported by unser aldroubi and eden see ibid vol 41 p 821 33 and p 834 48 feb 1993 the author regrets this oversight on his part and wishes to acknowledge the prior contribution of unser aldroubi and eden they have shown in fact that the smoothing spline and the computation of the derivatives for equally spaced data can be performed by digital filtering they have also given the corresponding filters and discussed some of their properties in addition recursive implementation of these filters has been proposed and their computational efficiency over the conventional matrix solutions has been demonstrated thus the author s contribution is mainly in establishing some interesting properties of the smoothing filter especially the maximum flatness property of the filter as well as the relationship between the cutoff frequency of the filter and the smoothing control parameter moreover the author s approach to derive the smoothing filter directly from the matrix solution can always be seen as an alternative to the method proposed by unser et al the author has also stressed that the recursive smoothing filter gives the same result as the matrix solution except for the two extremities of the data

natural spline functions their associated eigenvalue problem we consider the problem of studying the behaviour of the eigenvalues associated with spline functions with equally spaced knots we show that they are o left frac i 2m n right i 1 ldots n m wherem is the order of the spline andn the number of knots this result is of particular interest to prove optimality properties of the generalized cross validation method and had been conjectured by craven and wahba in a recent paper

generating random density matrices we study various methods to generate ensembles of random density matrices of a fixed size n obtained by partial trace of pure states on composite systems structured ensembles of random pure states invariant with respect to local unitary transformations are introduced to analyze statistical properties of quantum entanglement in bi partite systems we analyze the distribution of schmidt coefficients of random pure states such a distribution is derived in the case of a superposition of k random maximally entangled states for another ensemble obtained by performing selective measurements in a maximally entangled basis on a multi partite system we show that this distribution is given by the fuss catalan law and find the average entanglement entropy a more general class of structured ensembles proposed containing also the case of bures forms an extension of the standard ensemble of structureless random pure states described asymptotically as n by the marchenko pastur distribution

non hermitian random matrix models free random variable approach using the standard concepts of free random variables we show that for a large class of non hermitian random matrix models the support of the eigenvalue distribution follows from their hermitian analogs using a conformal transformation we also extend the concepts of free random variables to the class of non hermitian matrices and apply them to the models discussed by ginibre girko elliptic ensemble j ginibre j math phys bold 6 1440 1965 v l girko ital spectral theory of random matrices in russian nauka moscow 1988 and mahaux weidenm umlt u ller chaotic resonance scattering c mahaux and h a weidenm umlt u ller ital shell model approach to nuclear reactions north holland amsterdam 1969 copyright ital 1997 ital the american physical society

finite elements for stochastic media problems we consider discretization methods for stochastic partial differential equations which are used to model media with stochastic properties the wick product formulation of the stochastic pde is used and it is shown how it may be discretized both spatially and in the stochastic part a stochastic variational principle serves as the guideline for the discretization procedure it is shown that both the projection onto the wiener chaos and the hermite transform lead to the same fully discrete equations the application examples show that there are other advantageous bases besides the karhunen loeve expansion and that the methods may be used in a variety of settings

commutative law for products of infinitely large isotropic random matrices ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left and right multiplication by an arbitrary unitary matrix we show that the multiplication of large isotropic random matrices is spectrally commutative and self averaging in the limit of infinite matrix size n the notion of spectral commutativity means that the eigenvalue density of a product abc of such matrices is independent of the order of matrix multiplication for example the matrix abcd has the same eigenvalue density as adcb in turn the notion of self averaging means that the product of n independent but identically distributed random matrices which we symbolically denote by aaa has the same eigenvalue density as the corresponding power a n of a single matrix drawn from the underlying matrix ensemble for example the eigenvalue density of abccabc is the same as that of a 2 b 2 c 3 we also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues 0 we show that the singularities at the origin of the eigenvalue density and of the singular value density are in one to one correspondence in the limit n the eigenvalue density of an isotropic random matrix has a power law singularity at the origin s with a power se 0 2 when and only when the density of its singular values has a power law singularity with a power s 4 s these results are obtained analytically in the limit n we supplement these results with numerical simulations for large but finite n and discuss finite size effects for the most common ensembles of isotropic random matrices

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

bayesian confidence intervals for smoothing splines abstract the frequency properties of wahba s bayesian confidence intervals for smoothing splines are investigated by a large sample approximation and by a simulation study when the coverage probabilities for these pointwise confidence intervals are averaged across the observation points the average coverage probability acp should be close to the nominal level from a frequency point of view this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error these properties are independent of the bayesian assumptions used to derive this confidence procedure and they explain why the acp is accurate for functions that are much smoother than the sample paths prescribed by the prior this analysis accounts for the choice of the smoothing parameter bandwidth using cross validation in the case of natural splines an adaptive method for avo

an efficient method for calculating smoothing splines using orthogonal transformations a procedure for calculating the mean squared residual and the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m 1 using an orthogonal factorization is presented the procedure substantially overcomes the problem of ill conditioning encountered by a recently developed method which employs a cholesky factorization but still requires only orderm 2 n operations and ordermn storage

eigenvalue statistics for product complex wishart matrices the eigenvalue statistics for complex wishart matrices where is equal to the product of r complex gaussian matrices and the inverse of s complex gaussian matrices are considered in the case r s the exact form of the global density is computed the averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function for finite n the eigenvalue probability density function is computed and is shown to be an example of a biorthogonal ensemble a double contour integral form of the corresponding correlation kernel is derived which allows the hard edge scaled limit to be computed the limiting kernel is given in terms of certain meijer g functions and is identical to that found in the recent work of kuijlaars and zhang in the case s 0 properties of the kernel and corresponding correlation functions are discussed

induced ginibre ensemble of random matrices and quantum operations a generalization of the ginibre ensemble of non hermitian random square matrices is introduced the corresponding probability measure is induced by the ensemble of rectangular gaussian matrices via a quadratization procedure we derive the joint probability density of eigenvalues for such an induced ginibre ensemble and study various spectral correlation functions for complex and real matrices and analyse universal behaviour in the limit of large dimensions in this limit the eigenvalues of the induced ginibre ensemble cover uniformly a ring in the complex plane the real induced ginibre ensemble is shown to be useful to describe the statistical properties of evolution operators associated with random quantum operations for which the dimensions of the input state and the output state do differ

product of ginibre matrices fuss catalan and raney distributions squared singular values of a product of s square random ginibre matrices are asymptotically characterized by probability distributions p s x such that their moments are equal to the fuss catalan numbers of order s we find a representation of the fuss catalan distributions p s x in terms of a combination of s hypergeometric functions of the type s f s 1 the explicit formula derived here is exact for an arbitrary positive integer s and for s 1 it reduces to the marchenko pastur distribution using similar techniques involving the mellin transform and the meijer g function we find exact expressions for the raney probability distributions the moments of which are given by a two parameter generalization of the fuss catalan numbers these distributions can also be considered as a two parameter generalization of the wigner semicircle law

the signal extraction approach to nonlinear regression and spline smoothing abstract this article shows how to fit a smooth curve polynomial spline to pairs of data values yi xi prior specification of a parametric functional form for the curve is not required the resulting curve can be used to describe the pattern of the data and to predict unknown values of y given x both point and interval estimates are produced the method is easy to use and the computational requirements are modest even for large sample sizes our method is based on maximum likelihood estimation of a signal in noise model of the data we use the kalman filter to evaluate the likelihood function and achieve significant computational advantages over previous approaches to this problem

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

smoothing noisy data with spline functions a procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m 1 fitted ton distinct not necessarily equally spaced or uniformly weighted data points is presented the procedure requires orderm 2 n operations and therefore permits efficient orderm 2 n calculation of statistics associated with a polynomial smoothing spline including the generalized cross validation the method is a significant improvement over an existing method which requires ordern 3 operations

data smoothing and interpolation using eighth order algebraic splines a new type of algebraic spline is used to derive a filter for smoothing or interpolating discrete data points the spline is dependent on control parameters that specify the relative importance of data fitting and the derivatives of the spline a general spline of arbitrary order is first formulated using matrix equations we then focus on eighth order splines because of the continuity of their first three derivatives desirable for motor and robotics applications the spline s matrix equations are rewritten to give a recursive filter that can be implemented in real time for lengthy data sequences the filter is lowpass with a bandwidth that is dependent on the spline s control parameters numerical results including a simple image processing application show the tradeoffs that can be achieved using the algebraic splines

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

universality conjecture and results for a model of several coupled positive definite matrices the paper contains two main parts in the first part we analyze the general case of p geq 2 matrices coupled in a chain subject to cauchy interaction similarly to the itzykson zuber interaction model the eigenvalues of the cauchy chain form a multi level determinantal point process we first compute all correlations functions in terms of cauchy biorthogonal polynomials and locate them as specific entries of a p 1 times p 1 matrix valued solution of a riemann hilbert problem in the second part we fix the external potentials as classical laguerre weights we then derive strong asymptotics for the cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin as a result we obtain a new family of universality classes for multi level random determinantal point fields which include the bessel nu universality for 1 level and the meijer g universality for 2 level our analysis uses the deift zhou nonlinear steepest descent method and the explicit construction of a p 1 times p 1 origin parametrix in terms of meijer g functions the solution of the full riemann hilbert problem is derived rigorously only for p 3 but the general framework of the proof can be extended to the cauchy chain of arbitrary length p

random graph states maximal flow and fuss catalan distributions for any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems each edge of the graph represents a bipartite maximally entangled state each vertex represents a random unitary matrix generated according to the haar measure which describes the coupling between subsystems dividing all subsystems into two parts one may study entanglement with respect to this partition a general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented our technique relies on weingarten calculus and flow problems we analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free poissonian marchenko pastur distribution we derive a discrete family of generalized fuss catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues

spectral density of a wishart model for nonsymmetric correlation matrices the wishart model for real symmetric correlation matrices is defined as mathsf w mathsf aa t where matrix mathsf a is usually a rectangular gaussian random matrix and mathsf a t is the transpose of mathsf a analogously for nonsymmetric correlation matrices a model may be defined for two statistically equivalent but different matrices mathsf a and mathsf b as mathsf ab t the corresponding wishart model thus is defined as mathbf c mathsf ab t mathsf ba t we study the spectral density of mathbf c for the case when mathsf a and mathsf b are not statistically independent the ensemble average of such nonsymmetric matrices therefore does not simply vanishes to a null matrix in this paper we derive a pastur self consistent equation which describes spectral density of mathbf c at large matrix dimension we complement our analytic results with numerics

spectral density of generalized wishart matrices and free multiplicative convolution we investigate the level density for several ensembles of positive random matrices of a wishart like structure w xx where x stands for a non hermitian random matrix in particular making use of the cauchy transform we study the free multiplicative powers of the marchenko pastur mp distribution mp s which for an integer s yield fuss catalan distributions corresponding to a product of s independent square random matrices x x 1 x s new formulas for the level densities are derived for s 3 and s 1 3 moreover the level density corresponding to the generalized bures distribution given by the free convolution of arcsine and mp distributions is obtained we also explain the reason of such a curious convolution the technique proposed here allows for the derivation of the level densities for several other cases

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

multiplying unitary random matrices universality and spectral properties in this paper we calculate in the large n limit the eigenvalue density of an infinite product of random unitary matrices each of them generated by a random hermitian matrix this is equivalent to solving the unitary diffusion generated by a hamiltonian random in time we find that the average eigenvalue density is universal and depends only on the second moment of the generator of the stochastic evolution we find indications of critical behaviour eigenvalue spacing scaling like 1 n3 4 close to for a specific critical evolution time tc

a fast monte carlo cross validation procedure for large least squares problems with noisy data we propose a fast monte carlo algorithm for calculating reliable estimates of the trace of the influence matrixa involved in regularization of linear equations or data smoothing problems where is the regularization or smoothing parameter this general algorithm is simply as follows i generaten pseudo random valuesw 1 w n from the standard normal distribution wheren is the number of data points and letw w 1 w n t ii compute the residual vectorw a w iii take the normalized inner product w t w a w w t w as an approximation to 1 n tr i a we show both by theoretical bounds and by numerical simulations on some typical problems that the expected relative precision of these estimates is very good whenn is large enough and that they can be used in practice for the minimization with respect to of the well known generalized cross validation gcv function this permits the use of the gcv method for choosing in any particular large scale application with only a similar amount of work as the standard residual method numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools