asymptotics of r md and lad estimators in linear regression models with long range dependent errors this paper establishes the uniform closeness of a weighted residual empirical process to its natural estimate in the linear regression setting when the errors are gaussian or a function of gaussian random variables that are strictly stationary and long range dependent this result is used to yield the asymptotic uniform linearity of a class of rank statistics in linear regression models with long range dependent errors the latter result in turn yields the asymptotic distribution of the jaeckel 1972 rank estimators the paper also studies the least absolute deviation and a class of certain minimum distance estimators of regression parameters and the kernel type density estimators of the marginal error density when the errors are long range dependent

master symmetries for differential difference equations of the volterra type abstract it is demonstrated that in the case of integrable differential difference equations similar to the well known volterra equation unlike the korteweg de vries and nonlinear schrodinger equations there are many instances in which local master symmetries can be found those master symmetries are new interesting examples of local evolution chains explicitly depending on the time and discrete variable and integrable in a special sense the examples are constructed by a direct and elementary approach which enables one to get new integrable equations using miura type transformations

on the asymptotic expansion of the empirical process of long memory moving averages let x n i 1 a i e n i where the e i are iid with mean 0 and finite fourth moment and the a i are regularly varying with index where 1 2 1 so that x n has long range dependence this covers an important class of the fractional arima process for r 0 let y n r n n 1 1 j1 jr r s 1 a js e n js y n 0 n 2 n r var y n r and f r the rth derivative of the distribution function of x n the y n r are uncorrelated and are stochastically decreasing in r for any positive integer p 2 1 1 it is shown under mild regularity conditions that with probability 1 n n 1 i x n x p r 0 1 r f r x y n r o n n p uniformly for all x r o 1 2 1 2 p 1 2 this generalizes a host of existing results and provides the vehicle for a number of statistical applications

density estimation under long range dependence dehling and taqqu established the weak convergence of the empirical process for a long range dependent stationary sequence under gaussian subordination we show that the corresponding density process based on kernel estimators of the marginal density converges weakly with the same normalization to the derivative of their limiting process the phenomenon which carries on for higher derivatives and for functional laws of the iterated logarithm is in contrast with independent or weakly dependent situations where the density process cannot be tight in the usual function spaces with supremum distances

on the area of excursion sets of spherical gaussian eigenfunctions the high frequency behaviour for random eigenfunctions of the spherical laplacian has been recently the object of considerable interest also because of strong motivation arising from physics and cosmology in this paper we are concerned with the high frequency behaviour of excursion sets in particular we establish a uniform central limit theorem for the empirical measure i e the proportion of spherical surface where spherical gaussian eigenfunctions lie below a level z our proofs borrow some techniques from the literature on stationary long memory processes in particular we expand the empirical measure into hermite polynomials and establish a uniform weak reduction principle entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion as a result we establish a functional central limit theorem the limiting process is fully degenerate

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

asymptotic expansion of m estimators with long memory errors this paper obtains a higher order asymptotic expansion of a class of m estimators of the one sample location parameter when the errors form a long memory moving average a suitably standardized difference between an m estimator and the sample mean is shown to have a limiting distribution the nature of the limiting distribution depends on the range of the dependence parameter if for example 1 3 then a suitably standardized difference between the sample median and the sample mean converges weakly to a normal distribution provided the common error distribution is symmetric if 0 1 3 then the corresponding limiting distribution is nonnormal this paper thus goes beyond that of beran who observed in the case of long memory gaussian errors that m estimators t n of the one sample location parameter are asymptotically equivalent to the sample mean in the sense that var t n var x n 1 and t n x n o p var x n

linear integral equations and nonlinear difference difference equations abstract in this paper we present a systematic method to obtain various integrable nonlinear difference difference equations and the associated linear integral equations from which their solutions can be inferred it is argued that these difference difference equations can be regarded as arising from bianchi identities expressing the commutativity of backlund transformations applying an appropriate continuum limit we first obtain integrable nonlinear differential difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations as special cases we treat the difference difference versions and the differential difference versions of the korteweg de vries equation the modified korteweg de vries equation the nonlinear schrodinger equation the isotropic classical heisenberg spin chain and the complex and real sine gordon equation

limit theorems for functionals of mixing processes with applications to u statistics and dimension estimation in this paper we develop a general approach for investigating the asymptotic distribution of functional xn f zn k k z of absolutely regular stochastic processes zn n z such functional occur naturally as orbits of chaotic dynamical systems and thus our results can be used to study probabilistic aspects of dynamical systems we first prove some moment inequalities that are analogous to those for mixing sequences with their help several limit theorems can be proved in a rather straightforward manner we illustrate this by re proving a central limit theorem of ibragimov and linnik then we apply our techniques to u statistics matrix equation with symmetric kernel h r r r we prove a law of large numbers extending results of aaronson burton dehling gilat hill and weiss for absolutely regular processes we also prove a central limit theorem under a different set of conditions than the known results of denker and keller as our main application we establish an invariance principle for u processes un h h indexed by some class of functions we finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

on the bahadur representation of sample quantiles for dependent sequences we establish the bahadur representation of sample quantiles for linear and some widely used nonlinear processes local fluctuations of empirical processes are discussed applications to the trimmed and winsorized means are given our results extend previous ones by establishing sharper bounds under milder conditions and thus provide new insight into the theory of empirical processes for dependent random variables

the generalized symmetry method for discrete equations the generalized symmetry method is applied to a class of completely discrete equations including the adler bobenko suris list assuming the existence of a generalized symmetry we derive a few integrability conditions suitable for testing and classifying the equations of this class those conditions are used at the end to test for integrability discretizations of some well known hyperbolic equations

on weighted u statistics for stationary processes a weighted u statistic based on a random sample x1 xn has the form un p 1 i jn wiijk xi xj where k is fixed symmetric measurable function and the wi are symmetric weights a large class of statistics can be expressed as weighted u statistics or variations thereof this paper establishes the asymptotic normality of un when the sample observations come from a non linear time series and linear processes msc 2000 subject classifications primary 60f05 secondary 60g10

semiparametric estimation from time series with long range dependence abstract this paper studies the behaviour in the presence of long memory time series dependence of semaparametric averaged derivative statistics which are useful in statistical inference on index models they were shown to be asomptotically normal under weak dependence conditions by robinson 1989 and under serial independence by powell et al 1989 we find that an element of long range dependence can lead either to a nonnormal limiting distribution or else to a normal one with a limiting variance which differs from that which obtains in case of weak dependence implying that inferences incorrectly based on weak dependence assumptions will be invalid

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

approximate normality of high energy hyperspherical eigenfunctions abstract the berry heuristic has been a long standing ansatz about the high energy i e large eigenvalues behaviour of eigenfunctions see 7 roughly speaking it states that under some generic boundary conditions these eigenfunctions exhibit gaussian behaviour when the eigenvalues grow to infinity our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to gaussianity focussing on the hyperspherical case i e for eigenfunctions of the laplace beltrami operator on the normalized d dimensional sphere also known as spherical harmonics some applications to non gaussian models are also discussed

the hierarchy of multi soliton solutions of the derivative nonlinear schrodinger equation we provide a relatively simple approach to backlund transformations for the derivative nonlinear schrodinger equation by iteration it leads to compact n soliton formulae both with asymptotically vanishing and non vanishing amplitudes the phenomenology of these solutions is discussed and illustrated in some detail

an empirical central limit theorem for intermittent maps we prove an empirical central limit theorem for the distribution function of a stationary sequence under a dependence condition involving only indicators of half line we show that the result applies to the empirical distribution function of iterates of expanding maps with a neutral fixed point at zero as soon as the correlations are summable

a general asymptotic scheme for inference under order restrictions limit distributions for the greatest convex minorant and its derivative are considered for a general class of stochastic processes including partial sum processes and empirical processes for independent weakly dependent and long range dependent data the results are applied to isotonic regression isotonic regression after kernel smoothing estimation of convex regression functions and estimation of monotone and convex density functions various pointwise limit distributions are obtained and the rate of convergence depends on the self similarity properties and on the rate of convergence of the processes considered

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

enhancing low rank subspace clustering by manifold regularization recently low rank representation lrr method has achieved great success in subspace clustering sc which aims to cluster the data points that lie in a union of low dimensional subspace given a set of data points lrr seeks the lowest rank representation among the many possible linear combinations of the bases in a given dictionary or in terms of the data itself however lrr only considers the global euclidean structure while the local manifold structure which is often important for many real applications is ignored in this paper to exploit the local manifold structure of the data a manifold regularization characterized by a laplacian graph has been incorporated into lrr leading to our proposed laplacian regularized lrr laplrr an efficient optimization procedure which is based on alternating direction method of multipliers admm is developed for laplrr experimental results on synthetic and real data sets are presented to demonstrate that the performance of lrr has been enhanced by using the manifold regularization

the empirical process of a short range dependent stationary sequence under gaussian subordination consider the stationary sequencex 1 g z 1 x 2 g z 2 whereg is an arbitrary borel function andz 1 z 2 is a mean zero stationary gaussian sequence with covariance functionr k e z 1 z k 1 satisfyingr 0 1 and 1 r k m where withi denoting the indicator function andf the continuous marginal distribution function of the sequence x n the integerm is the hermite rank of the family i g x f x x r letf n be the empirical distribution function ofx 1 x n we prove that asn the empirical processn 1 2 f n f converges in distribution to a gaussian process in the spaced

strong invariance principles for sequential bahadur kiefer and vervaat error processes of long range dependent sequences in this paper we study strong approximations invariance principles of the sequential uniform and general bahadur kiefer processes of long range dependent sequences we also investigate the strong and weak asymptotic behavior of the sequential vervaat process that is the integrated sequential bahadur kiefer process properly normalized as well as that of its deviation from its limiting process the so called vervaat error process it is well known that the bahadur kiefer and the vervaat error processes cannot converge weakly in the i i d case in contrast to this we conclude that the bahadur kiefer and vervaat error processes as well as their sequential versions do converge weakly to a dehling taqqu type limit process for certain long range dependent sequences

strong approximation results for the empirical process of stationary sequences we prove a strong approximation result for the empirical process associated to a stationary sequence of real valued random variables under dependence conditions involving only indicators of half lines this strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero as soon as the correlations decrease more rapidly than n 1 for some positive this shows that our conditions are in some sense optimal

explicit auto backlund relation through gauge transformation application of gauge transformation for generating explicit auto backlund relations is demonstrated for a number of integrable systems e g kdv sg nls dnls mixed dnls modified dnls lle etc this method turns out to be much simpler effective and straightforward using backlund relations all conserved quantities are expressed in a novel derivative free form which simplifies the calculation of important soliton characteristics

slowly decaying correlations testing normality nuisance parameters abstract slowly decaying serial correlations can cause goodness of fit tests for a distribution to reject the null hypothesis with probability tending to one with increasing sample size when the null distribution is completely specified simple null hypothesis the problem is actually worse than for the case where there are nuisance parameters to be estimated composite null hypothesis in particular we consider here testing normality we discuss limit theorems and propose correction terms to be incorporated in certain goodness of fit statistics to improve the rates of convergence for the simple hypothesis case we show how the problem is solved automatically for the composite null hypothesis case when the nuisance parameters are estimated from the data simulations illustrate the results

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

quad equations and auto backlund transformations of nls type systems treating an integrable quad equation along with its two generalized symmetries as a compatible system allows one to construct an auto backlund transformation for solutions of the related nls type system a fixed periodic reduction of the quad equation yields a quasi periodic reduction of its generalized symmetries that turns them into differential constraints compatible with the nls type system

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

the empirical process of some long range dependent sequences with an application to u statistics let x j infty j 1 be a stationary mean zero gaussian process with covariances r k ex k 1 x 1 satisfying r 0 1 and r k k d l k where d is small and l is slowly varying at infinity consider the two parameter empirical process for g x j bigg f n x t frac 1 n sum lbrack nt rbrack j 1 lbrack 1 g x j leq x p g x 1 leq x rbrack infty x infty 0 leq t leq 1 bigg where g is any measurable function noncentral limit theorems are obtained for f n x t and they are used to derive the asymptotic behavior of some suitably normalized von mises statistics and u statistics based on the g x j s the limiting processes are structurally different from those encountered in the i i d case

asymptotic properties of u processes under long range dependence let x i i geq 1 be a stationary mean zero gaussian process with covariances rho k pe x 1 x k 1 satisfying rho 0 1 and rho k k d l k where d is in 0 1 and l is slowly varying at infinity consider the u process u n r r in i defined as u n r frac 1 n n 1 sum 1 leq i neq j leq n 1 g x i x j leq r where i is an interval included in rset and g is a symmetric function in this paper we provide central and non central limit theorems for u n they are used to derive the asymptotic behavior of the hodges lehmann estimator the wilcoxon signed rank statistic the sample correlation integral and an associated scale estimator the limiting distributions are expressed through multiple wiener ito integrals

stein malliavin approximations for nonlinear functionals of random eigenfunctions on s d abstract we investigate stein malliavin approximations for nonlinear functionals of geometric interest for random eigenfunctions on the unit d dimensional sphere s d d 2 all our results are established in the high energy limit i e as the corresponding eigenvalues diverge in particular we prove a quantitative central limit theorem for the excursion volume of gaussian eigenfunctions this goal is achieved by means of several results of independent interest concerning the asymptotic analysis for the variance of moments of gaussian eigenfunctions the rates of convergence in various probability metrics for hermite subordinated processes and quantitative central limit theorems for arbitrary polynomials of finite order or general square integrable nonlinear transforms some related issues were already considered in the literature for the 2 dimensional case s 2 our results are new or improve the existing bounds even in these special circumstances proofs are based on the asymptotic analysis of moments of all order for gegenbauer polynomials and make extensive use of the recent literature on so called fourth moment theorems by nourdin and peccati

on photometric issues in 3d visual recognition from asingle 2d image we describe the problem of recognition under changing illumination conditions and changing viewing positions from a computational and human vision perspective on the computational side we focus on the mathematical problems of creating an equivalence class for images of the same 3d object undergoing certain groups of transformations mostly those due to changing illumination and briefly discuss those due to changing viewing positions the computational treatment culminates in proposing a simple scheme for recognizing via alignment an image of a familiar object taken from a novel viewing position and a novel illumination condition on the human vision aspect the paper is motivated by empirical evidence inspired by mooney images of faces that suggest a relatively high level of visual processing is involved in compensating for photometric sources of variability and furthermore that certain limitations on the admissible representations of image information may exist the psychophysical observations and the computational results that follow agree in several important respects such as the same apparent limitations on image representations

rank optimized logistic matrix regression toward improved matrix data classification while existing logistic regression suffers from overfitting and often fails in considering structural information we propose a novel matrix based logistic regression to overcome the weakness in the proposed method 2d matrices are directly used to learn two groups of parameter vectors along each dimension without vectorization which allows the proposed method to fully exploit the underlying structural information embedded inside the 2d matrices further we add a joint norm on two parameter matrices which are organized by aligning each group of parameter vectors in columns this added co regularization term has two roles enhancing the effect of regularization and optimizing the rank during the learning process with our proposed fast iterative solution we carried out extensive experiments the results show that in comparison to both the traditional tensor based methods and the vector based regression methods our proposed solution achieves better performance for matrix data classifications

mr ntd manifold regularization nonnegative tucker decomposition for tensor data dimension reduction and representation with the advancement of data acquisition techniques tensor multidimensional data objects are increasingly accumulated and generated for example multichannel electroencephalographies multiview images and videos in these applications the tensor objects are usually nonnegative since the physical signals are recorded as the dimensionality of tensor objects is often very high a dimension reduction technique becomes an important research topic of tensor data from the perspective of geometry high dimensional objects often reside in a low dimensional submanifold of the ambient space in this paper we propose a new approach to perform the dimension reduction for nonnegative tensor objects our idea is to use nonnegative tucker decomposition ntd to obtain a set of core tensors of smaller sizes by finding a common set of projection matrices for tensor objects to preserve geometric information in tensor data we employ a manifold regularization term for the core tensors constructed in the tucker decomposition an algorithm called manifold regularization ntd mr ntd is developed to solve the common projection matrices and core tensors in an alternating least squares manner the convergence of the proposed algorithm is shown and the computational complexity of the proposed method scales linearly with respect to the number of tensor objects and the size of the tensor objects respectively these theoretical results show that the proposed algorithm can be efficient extensive experimental results have been provided to further demonstrate the effectiveness and efficiency of the proposed mr ntd algorithm

nonparametric regression under long range dependent normal errors we consider the fixed design regression model with long range dependent normal errors and show that the finite dimensional distributions of the properly normalized gasser miiller and priestley chao estimators of the regression function converge to those of a white noise process furthermore the distributions of the suitably renormalized maximal deviations over an increasingly finer grid converge to the gumbel distribution these results contrast with our previous findings for the corresponding problem of estimating the marginal density of long range dependent stationary sequences

image representation using laplacian regularized nonnegative tensor factorization tensor provides a better representation for image space by avoiding information loss in vectorization nonnegative tensor factorization ntf whose objective is to express an n way tensor as a sum of k rank 1 tensors under nonnegative constraints has recently attracted a lot of attentions for its efficient and meaningful representation however ntf only sees euclidean structures in data space and is not optimized for image representation as image space is believed to be a sub manifold embedded in high dimensional ambient space to avoid the limitation of ntf we propose a novel laplacian regularized nonnegative tensor factorization lrntf method for image representation and clustering in this paper in lrntf the image space is represented as a 3 way tensor and we explicitly consider the manifold structure of the image space in factorization that is two data points that are close to each other in the intrinsic geometry of image space shall also be close to each other under the factorized basis to evaluate the performance of lrntf in image representation and clustering we compare our algorithm with nmf ntf ncut and gnmf methods on three standard image databases experimental results demonstrate that lrntf achieves better image clustering performance while being more insensitive to noise

closed form expressions for integrals of traveling wave reductions of integrable lattice equations we give a method to calculate closed form expressions in terms of multi sums of products for integrals of ordinary difference equations which are obtained as traveling wave reductions of integrable partial difference equations important ingredients are the staircase method a non commutative vieta formula and certain splittings of the lax matrices the method is applied to all equations of the adler bobenko suris classification with the exception of q4

learning locality constrained collaborative representation for robust face recognition abstract the models of low dimensional manifold and sparse representation are two well known concise models that suggest that each data can be described by a few characteristics manifold learning is usually investigated for dimension reduction by preserving some expected local geometric structures from the original space into a low dimensional one the structures are generally determined by using pairwise distance e g euclidean distance alternatively sparse representation denotes a data point as a linear combination of the points from the same subspace in practical applications however the nearby points in terms of pairwise distance may not belong to the same subspace and vice versa consequently it is interesting and important to explore how to get a better representation by integrating these two models together to this end this paper proposes a novel coding algorithm called locality constrained collaborative representation lccr which introduce a kind of local consistency into coding scheme to improve the discrimination of the representation the locality term derives from a biologic observation that the similar inputs have similar codes the objective function of lccr has an analytical solution and it does not involve local minima the empirical studies based on several popular facial databases show that lccr is promising in recognizing human faces with varying pose expression and illumination as well as various corruptions and occlusions