correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

outlier robust pca the high dimensional case principal component analysis plays a central role in statistics engineering and science because of the prevalence of corrupted data in real world applications much research has focused on developing robust algorithms perhaps surprisingly these algorithms are unequipped indeed unable to deal with outliers in the high dimensional setting where the number of observations is of the same magnitude as the number of variables of each observation and the dataset contains some arbitrarily corrupted observations we propose a high dimensional robust principal component analysis algorithm that is efficient robust to contaminated points and easily kernelizable in particular our algorithm achieves maximal robustness it has a breakdown point of 50 the best possible while all existing algorithms have a breakdown point of zero moreover our algorithm recovers the optimal solution exactly in the case where the number of corrupted points grows sublinearly in the dimension

a note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices recently johansson and johnstone proved that the distribution of the properly rescaled largest principal component of the complex real wishart matrix x x xtx converges to the tracy widom law as n p the dimensions of x tend to in some ratio n p 0 we extend these results in two directions first of all we prove that the joint distribution of the first second third etc eigenvalues of a wishart matrix converges after a proper rescaling to the tracy widom distribution second of all we explain how the combinatorial machinery developed for wigner random matrices in refs 27 38 and 39 allows to extend the results by johansson and johnstone to the case of x with non gaussian entries provided n p o p1 3 we also prove that max n1 2 p1 2 2 o p1 2 log p a e for general 0

bandwidth selection for high dimensional covariance matrix estimation the banding estimator of bickel and levina and its tapering version of cai zhang and zhou are important high dimensional covariance estimators both estimators require a bandwidth parameter we propose a bandwidth selector for the banding estimator by minimizing an empirical estimate of the expected squared frobenius norms of the estimation error matrix the ratio consistency of the bandwidth selector is established we provide a lower bound for the coverage probability of the underlying bandwidth being contained in an interval around the bandwidth estimate extensions to the bandwidth selection for the tapering estimator and threshold level selection for the thresholding covariance estimator are made numerical simulations and a case study on sonar spectrum data are conducted to demonstrate the proposed approaches supplementary materials for this article are available online

covariance matrix selection and estimation via penalised normal likelihood summary we propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix we reparameterise a covariance matrix through the modified cholesky decomposition of its inverse or the one step ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models the cholesky factor containing these regression coefficients is likely to have many off diagonal elements that are zero or close to zero penalised normal likelihoods in this situation with l1 and l2 penalities are shown to be closely related to tibshirani s 1996 lasso approach and to ridge regression adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the cholesky factor while because of its singularity the l1 penalty will set some elements to zero and produce interpretable models an algorithm is developed for computing the estimator and selecting the tuning parameter the proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix

central limit theorems for classical likelihood ratio tests for high dimensional normal distributions for random samples of size n obtained from p variate normal distributions we consider the classical likelihood ratio tests lrt for their means and covariance matrices in the high dimensional setting these test statistics have been extensively studied in multivariate analysis and their limiting distributions under the null hypothesis were proved to be chi square distributions as n goes to infinity and p remains fixed in this paper we consider the highdimensional case where both p and n go to infinity with p n y 0 1 we prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and variances we perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chisquare approximations for analyzing high dimensional data

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

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

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

confidence regions for entries of a large precision matrix we consider the statistical inference for high dimensional precision matrices specifically we propose a data driven procedure for constructing a class of simultaneous confidence regions for a subset of the entries of a large precision matrix the confidence regions can be applied to test for specific structures of a precision matrix and to recover its nonzero components we first construct an estimator for the precision matrix via penalized node wise regression we then develop the gaussian approximation to approximate the distribution of the maximum difference between the estimated and the true precision coefficients a computationally feasible parametric bootstrap algorithm is developed to implement the proposed procedure the theoretical justification is established under the setting which allows temporal dependence among observations therefore the proposed procedure is applicable to both independent and identically distributed data and time series data numerical results with both simulated and real data confirm the good performance of the proposed method

a bias correction method for indirect identification of closed loop systems a novel least squares like method is developed to deal with the bias problem in indirect identification of closed loop systems with coloured disturbances it is shown that combined with a bias correction procedure this method can give rise to consistent parameter estimates by fully and efficiently using the given information about the regulator simulation results are also included for illustration

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

random matrix ensembles with an effective extensive external charge recent theoretical studies of chaotic scattering have encountered ensembles of random matrices in which the eigenvalue probability density function contains a one body factor with an exponent proportional to the number of eigenvalues two such ensembles have been encountered an ensemble of unitary matrices specified by the so called poisson kernel and the laguerre ensemble of positive definite matrices here we consider various properties of these ensembles jack polynomial theory is used to prove a reproducing property of the poisson kernel and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the dyson circular ensemble for the laguerre ensemble the scaled global density is calculated exactly for all even values of the parameter while for random matrices with unitary symmetry the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class

the gauss bonnet chern mass under geometric flows the gauss bonnet chern mass was defined and studied by ge wang and wu adv math 266 84 119 2014 in this paper we consider the evolution of gauss bonnet chern mass along the ricci flow and the yamabe flow

novel parameter estimation of autoregressive signals in the presence of noise this paper proposes a new method for estimating the parameters of an autoregressive ar signal from observations corrupted by white noise the feature of the new method is that the observation noise variance estimate is converted into the only solution of a nonlinear equation to yield unbiased estimate of the ar parameters moreover a convergent newton iterative algorithm with a deterministic initial point is presented for efficient implementation of the proposed new estimation method as a result the proposed new method can minimize the error of estimating the variance of the observation noise since more accurate estimates of this observation noise variance can be attained at earlier stages the proposed method can achieve a good performance in estimating the ar signal parameters numerical results demonstrate that the proposed new algorithm is more effective in terms of accuracy and robustness against noise than conventional algorithms

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

a globally consistent nonlinear least squares estimator for identification of nonlinear rational systems this paper considers identification of nonlinear rational systems defined as the ratio of two nonlinear functions of past inputs and outputs despite its long history a globally consistent identification algorithm remains illusive this paper proposes a globally convergent identification algorithm for such nonlinear rational systems to the best of our knowledge this is the first globally convergent algorithm for the nonlinear rational systems the technique employed is a two step estimator though two step estimators are known to produce consistent nonlinear least squares estimates if a n consistent estimate can be determined in the first step how to find such a n consistent estimate in the first step for nonlinear rational systems is nontrivial and is not answered by any two step estimators the technical contribution of the paper is to develop a globally consistent estimator for nonlinear rational systems in the first step this is achieved by involving model transformation bias analysis noise variance estimation and bias compensation in the paper two simulation examples and a practical example are provided to verify the good performance of the proposed two step estimator

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

test for bandedness of high dimensional covariance matrices and bandwidth estimation motivated by the latest effort to employ banded matrices to estimate a high dimensional covariance we propose a test for being banded with possible diverging bandwidth the test is adaptive to the large p small n situations without assuming a specific parametric distribution for the data we also formulate a consistent estimator for the bandwidth of a banded high dimensional covariance matrix the properties of the test and the bandwidth estimator are investigated by theoretical evaluations and simulation studies as well as an empirical analysis on a protein mass spectroscopy data

a new approach to cholesky based covariance regularization in high dimensions summary in this paper we propose a new regression interpretation of the cholesky factor of the covariance matrix as opposed to the well known regression interpretation of the cholesky factor of the inverse covariance which leads to a new class of regularized covariance estimators suitable for high dimensional problems regularizing the cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator in particular one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of bickel levina 2008b which is not guaranteed to be positive definite we also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint and compare the numerical performance of several methods in simulations and on a sonar data example

sure information criteria for large covariance matrix estimation and their asymptotic properties consider n independent and identically distributed p dimensional gaussian random vectors with covariance matrix sigma the problem of estimating sigma when p is much larger than n has received a lot of attention in recent years yet little is known about the information criterion for covariance matrix estimation how to properly define such a criterion and what are the statistical properties we attempt to answer these questions in this paper by focusing on the estimation of bandable covariance matrices when p n but log p o n motivated by the deep connection between stein s unbiased risk estimation sure and akaike information criterion aic in regression models we propose a family of generalized sure sure c indexed by c for covariance matrix estimation where c is some constant when c is 2 sure2 provides an unbiased estimator of the frobenius risk of the covariance matrix estimator furthermore we show that by minimizing sure2 over all possible banding covariance matrix estimators we attain the minimax optimal rate of convergence under the frobenius norm and the resulting estimator behaves like the covariance matrix estimator obtained by the so called oracle tuning when the true covariance matrix is exactly banded we prove that by minimizing text sure log n we select the true bandwidth with probability tending to one therefore our analysis indicates that sure2 and text sure log n can be regarded as the aic and bayesian information criterion for large covariance matrix estimation respectively

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

a new mass for asymptotically flat manifolds abstract in this paper we introduce a mass for asymptotically flat manifolds by using the gauss bonnet curvature we first prove that the mass is well defined and is a geometric invariant if the gauss bonnet curvature is integrable and the decay order satisfies n 4 3 motivated by an elegant idea of lam 35 we then show a positive mass theorem for this new mass for asymptotically flat graphs over r n moreover we obtain various penrose type inequalities in this case in these penrose type inequalities we establish relationship between the new mass and various geometric integrals

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

bias correction in least squares identification abstract the least squares method in system identification leads generally to biased parameter estimates a conceptually simple modification is to estimate the bias and to compute compensated parameter estimates when white output noise is the only disturbance this principle compensated least squares cls can readily be used to obtain consistent estimates the main purpose of the paper is to investigate the accuracy of the parameter estimates obtained when the cls method is used it is proved that the estimates are asymptotically gaussian distributed an explicit expression for the covariance matrix is given it is also shown that a commonly used instrumental variable method gives asymptotically better accuracy than the compensated least squares method

identification of closed loop systems with low order controllers a bias correction method without noise modelling was recently proposed for indirect identification of closed loop systems a key assumption used in deriving this method required that the order of the controller is not less than that of the open loop plant a significant feature of the results presented in this paper is the removal of this assumption to enable the method to be applicable to unbiased identification of closed loop systems with low order controllers

testing mutual independence in high dimension via distance covariance summary we introduce an l2 type test for testing mutual independence and banded dependence structure for high dimensional data the test is constructed on the basis of the pairwise distance covariance and it accounts for the non linear and non monotone dependences among the data which cannot be fully captured by the existing tests based on either pearson correlation or rank correlation our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal it exhibits excellent finite sample performance in our simulation studies even when the sample size is small albeit the dimension is high and is shown to identify non linear dependence in empirical data analysis successfully on the theory side asymptotic normality of our test statistic is shown under quite mild moment assumptions and with little restriction on the growth rate of the dimension as a function of sample size as a demonstration of good power properties for our distance covariance based test we further show that an infeasible version of our test statistic has the rate optimality in the class of gaussian distributions with equal correlation

asymptotic behavior of support vector machine for spiked population model for spiked population model we investigate the large dimension n and large sample size m asymptotic behavior of the support vector machine svm classification method in the limit of n m at fixed m n we focus on the generalization performance by analytically evaluating the angle between the normal direction vectors of svm separating hyperplane and corresponding bayes optimal separating hyperplane this is an analogous result to the one shown in paul 2007 and nadler 2008 for the angle between the sample eigenvector and the population eigenvector in random matrix theorem we provide not just bound but sharp prediction of the asymptotic behavior of svm that can be determined by a set of nonlinear equations based on the analytical results we propose a new method of selecting tuning parameter which significantly reduces the computational cost a surprising finding is that svm achieves its best performance at small value of the tuning parameter under spiked population model these results are confirmed to be correct by comparing with those of numerical simulations on finite size systems we also apply our formulas to an actual dataset of breast cancer and find agreement between analytical derivations and numerical computations based on cross validation

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

tests for covariance matrix with fixed or divergent dimension testing covariance structure is of importance in many areas of statistical analysis such as microarray analysis and signal processing conventional tests for finite dimensional covariance cannot be applied to high dimensional data in general and tests for high dimensional covariance in the literature usually depend on some special structure of the matrix in this paper we propose some empirical likelihood ratio tests for testing whether a covariance matrix equals a given one or has a banded structure the asymptotic distributions of the new tests are independent of the dimension

comments on on a least squares based algorithm for identification of stochastic linear systems the bias eliminated least squares estimator proposed in the paper by zheng see ibid vol 46 p 1631 8 1998 is shown to be identical to a simple instrumental variable estimator using delayed input values as instruments

the gbc mass for asymptotically hyperbolic manifolds the paper consists of two parts in the first part by using the gauss bonnet curvature which is a natural generalization of the scalar curvature we introduce a higher order mass the gauss bonnet chern mass m mathbb h k for asymptotically hyperbolic manifolds and show that it is a geometric invariant moreover we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding penrose type inequality for this mass and weighted alexandrov fenchel inequalities in the hyperbolic space mathbb h n in the second part we establish these weighted alexandrov fenchel inequalities in mathbb h n for any horospherical convex hypersurface sigma begin aligned int sigma v sigma 2k 1 d mu ge c n 1 2k 1 omega n 1 left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned where sigma j is the j th mean curvature of sigma subset mathbb h n v cosh r and sigma is the area of sigma as an application we obtain an optimal penrose type inequality for the new mass defined in the first part begin aligned m k mathbb h ge frac 1 2 k left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned for asymptotically hyperbolic graphs with a horizon type boundary sigma provided that a dominant energy type condition tilde l k ge 0 holds both inequalities are optimal

closed loop identification revisited identification of systems operating in closed loop has long been of prime interest in industrial applications the problem offers many possibilities and also some fallacies and a wide variety of approaches have been suggested many quite recently the purpose of the current contribution is to place most of these approaches in a coherent framework thereby showing their connections and display similarities and differences in the asymptotic properties of the resulting estimates the common framework is created by the basic prediction error method and it is shown that most of the common methods correspond to different parameterizations of the dynamics and noise models the so called indirect methods e g are indeed direct methods employing noise models that contain the regulator the asymptotic properties of the estimates then follow from the general theory and take different forms as they are translated to the particular parameterizations we also study a new projection approach to closed loop identification with the advantage of allowing approximation of the open loop dynamics in a given and user chosen frequency domain norm even in the case of an unknown nonlinear regulator

comparing some classes of bias compensating least squares methods three different classes of bias compensating least squares identification methods are compared and shown to be identical it is also discussed how the user parameters in the classes can be chosen to achieve optimal accuracy of the parameter estimates

testing super diagonal structure in high dimensional covariance matrices abstract the covariance matrices are essential quantities in econometric and statistical applications including portfolio allocation asset pricing and factor analysis testing the entire covariance under high dimensionality endures large variability and causes a dilution of the signal to noise ratio and hence a reduction in the power we consider a more powerful test procedure that focuses on testing along the super diagonals of the high dimensional covariance matrix which can infer more accurately on the structure of the covariance we show that the test is powerful in detecting sparse signals and parametric structures in the covariance the properties of the test are demonstrated by theoretical analyses simulation and empirical studies

penalized discriminant analysis fisher s linear discriminant analysis lda is a popular data analytic tool for studying the relationship between a set of predictors and a categorical response in this paper we describe a penalized version of lda it is designed for situations in which there are many highly correlated predictors such as those obtained by discretizing a function or the grey scale values of the pixels in a series of images in cases such as these it is natural efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients both for improved prediction performance and interpretability we cast the classification problem into a regression framework via optimal scoring using this our proposal facilitates the use of any penalized regression technique in the classification setting the technique is illustrated with examples in speech recognition and handwritten character recognition

correlation tests and linear spectral statistics of the sample correlation matrix testing the independence of the entries of multidimensional gaussian observations is a very important problem in statistics with a number of applications in signal processing radar cognitive radio seismography and multiple other fields typically the problem is formulated as a binary hypothesis test whereby the presence of correlation is declared when the value of a certain statistic is higher than a certain predetermined threshold most of the statistics for correlation tests are constructed from the sample correlation matrix also known as sample coherence matrix in signal processing which is defined as a power normalized version of the sample covariance matrix in this paper correlation tests constructed from linear spectral statistics lss of the sample correlation matrix are analyzed under the asymptotic framework where both sample size and observation dimension become large but comparable in magnitude a central limit theorem clt is established on this class of statistics which is valid for generally correlated gaussian observations results show that lss asymptotically fluctuate as gaussian random variables under both the hypotheses with an asymptotic mean and variance that can be established for each particular test in particular this general clt can be used to establish the asymptotic behavior of two of the most important correlation test statistics namely the generalized likelihood ratio test and the frobenius norm test under both null and alternative hypotheses as a by product it is established that lss of sample covariance and sample correlation matrices have exactly the same first order behavior but quite different asymptotic fluctuations in the second order regime in both the cases the lss asymptotically behave as gaussian random variables although with quite different asymptotic means and variances

deterministic parallel analysis an improved method for selecting factors and principal components factor analysis and principal component analysis are used in many application areas the first step choosing the number of components remains a serious challenge our work proposes improved methods for this important problem one of the most popular state of the art methods is parallel analysis pa which compares the observed factor strengths with simulated strengths under a noise only model the paper proposes improvements to pa we first derandomize it proposing deterministic pa which is faster and more reproducible than pa both pa and deterministic pa are prone to a shadowing phenomenon in which a strong factor makes it difficult to detect smaller but more interesting factors we propose deflation to counter shadowing we also propose to raise the decision threshold to improve estimation accuracy we prove several consistency results for our methods and test them in simulations we also illustrate our methods on data from the human genome diversity project where they significantly improve the accuracy

chern s magic form and the gauss bonnet chern mass in this note we use chern s magic form phi k in his famous proof of the gauss bonnet theorem to define a mass for asymptotically flat manifolds it turns out that the new defined mass is equivalent to the one that we introduced recently by using the gauss bonnet chern curvature l k moreover this equivalence implies a simple proof of the equivalence between the adm mass and the intrinsically defined mass via the ricci tensor which was reconsidered by miao tam proc am math soc 144 753 761 2016 and herzlich ann henri poincare 17 12 3605 3617 2016 very recently

geometric invariance of mass like asymptotic invariants we study coordinate invariance of some asymptotic invariants such as the arnowitt deser misner mass or the chru ciel herzlich momentum given by an integral over a boundary at infinity when changing the coordinates at infinity some terms in the change of integrand do not decay fast enough to have a vanishing integral at infinity but they may be gathered in a divergence thus having vanishing integral over any closed hypersurface this fact could only be checked after direct calculation and was called a curious cancellation we give a conceptual explanation thereof

hypothesis testing on linear structures of high dimensional covariance matrix this paper is concerned with test of significance on high dimensional covariance structures and aims to develop a unified framework for testing commonly used linear covariance structures we first construct a consistent estimator for parameters involved in the linear covariance structure and then develop two tests for the linear covariance structures based on entropy loss and quadratic loss used for covariance matrix estimation to study the asymptotic properties of the proposed tests we study related high dimensional random matrix theory and establish several highly useful asymptotic results with the aid of these asymptotic results we derive the limiting distributions of these two tests under the null and alternative hypotheses we further show that the quadratic loss based test is asymptotically unbiased we conduct monte carlo simulation study to examine the finite sample performance of the two tests our simulation results show that the limiting null distributions approximate their null distributions quite well and the corresponding asymptotic critical values keep type i error rate very well our numerical comparison implies that the proposed tests outperform existing ones in terms of controlling type i error rate and power our simulation indicates that the test based on quadratic loss seems to have better power than the test based on entropy loss

on a least squares based algorithm for identification of stochastic linear systems a new form of bias eliminated least squares bels algorithm is developed to identify transfer function parameters of a linear time invariant system irrespective of noise dynamics unlike the bels estimator previously presented the main feature with the developed algorithm is that the transfer function parameters are consistently estimated in such a direct way that there is no need to prefilter observed data or to deal with a high order augmented system this greatly simplifies implementation of the bels based algorithms and reduces numerical efforts whereas a desirable estimation accuracy can still be achieved two simulation examples are presented that clearly illustrate the good performances of the developed algorithm including its superiority over one type of simple instrumental variable method

covariance regression analysis abstractthis article introduces covariance regression analysis for a p dimensional response vector the proposed method explores the regression relationship between the p dimensional covariance matrix and auxiliary information we study three types of estimators maximum likelihood ordinary least squares and feasible generalized least squares estimators then we demonstrate that these regression estimators are consistent and asymptotically normal furthermore we obtain the high dimensional and large sample properties of the corresponding covariance matrix estimators simulation experiments are presented to demonstrate the performance of both regression and covariance matrix estimates an example is analyzed from the chinese stock market to illustrate the usefulness of the proposed covariance regression model supplementary materials for this article are available online

a unified framework for bias compensation based methods in correlated noise case this technical note presents a unified framework for bias compensation principle bcp based methods applied for identification of linear systems subject to correlated noise by introducing a non singular matrix and an auxiliary vector uncorrelated with the noise the unified framework is established since there are rich possibilities of the choices of the introduced matrix and vector the proposed unified framework is very flexible it can be verified that the existing bcp based methods are special cases of the achieved result it also shows that the unified framework can be used for deriving new or simplified versions of the bcp type methods