computing two dimensional poincar maps for hyperchaotic dynamics poincar map pm is one of the felicitous discrete approximation of the continuous dynamics to compute pm the discrete relation s between the successive point of interactions of the trajectories on the suitable poincar section ps are found out these discrete relations act as an amanuensis of the nature of the continuous dynamics in this article we propose a computational scheme to find a hyperchaotic pm hpm from an equivalent three dimensional 3d subsystem of a 4d or higher hyperchaotic model for the experimental purpose a standard four dimensional 4d hyperchaotic lorenz stenflo system hlss and a five dimensional 5d hyperchaotic laser model hlm is considered equivalent 3d subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3d subsystems the quantitative measurement of this comparison is made promising by recurrence quantification analysis rqa various two dimensional 2d poincar mas are computed for several suitable poincar sections for both the systems but only some of them are hyperchaotic in nature the hyperchaotic behavior is verified by positive values of both one dimensional 1d lyapunov exponent le i and 2d lyapunov exponent le ii at the end similarity of the dynamics between the hyperchaotic systems and their 2d hyperchaotic poincar maps hpm has been established through mean recurrence time mrt statistics for both of 4d hlss and 5d hlm and the best approximated discrete dynamics for both the hyperchaotic systems are found out

bubbling bifurcation loss of synchronization and shadowing breakdown in complex systems abstract complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to high dimensional chaos this paper focuses on a lattice of coupled logistic maps to investigate the relationship between the loss of chaos synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems in the neighborhood of the critical transition to strongly non hyperbolic behavior the system undergoes on off intermittency with respect to the synchronization manifold this has been confirmed by numerical diagnostics of synchronization and non hyperbolic behavior the latter using the statistical properties of finite time lyapunov exponents

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

bayesian sparse multiple regression for simultaneous rank reduction and variable selection we develop a bayesian methodology aimed at simultaneously estimating low rank and row sparse matrices in a high dimensional multiple response linear regression model we consider a carefully devised shrinkage prior on the matrix of regression coefficients which obviates the need to specify a prior on the rank and shrinks the regression matrix towards low rank and row sparse structures we provide theoretical support to the proposed methodology by proving minimax optimality of the posterior mean under the prediction risk in ultra high dimensional settings where the number of predictors can grow sub exponentially relative to the sample size a one step post processing scheme induced by group lasso penalties on the rows of the estimated coefficient matrix is proposed for variable selection with default choices of tuning parameters we additionally provide an estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space we exhibit the performance of the proposed methodology in an extensive simulation study and a real data example

bayesian factorizations of big sparse tensors it has become routine to collect data that are structured as multiway arrays tensors there is an enormous literature on low rank and sparse matrix factorizations but limited consideration of extensions to the tensor case in statistics the most common low rank tensor factorization relies on parallel factor analysis parafac which expresses a rank k tensor as a sum of rank one tensors in contingency table applications in which the sample size is massively less than the number of cells in the table the low rank assumption is not sufficient and parafac has poor performance we induce an additional layer of dimension reduction by allowing the effective rank to vary across dimensions of the table taking a bayesian approach we place priors on terms in the factorization and develop an efficient gibbs sampler for posterior computation theory is provided showing posterior concentration rates in high dimensional settings and the methods are shown to have excellent performance in simulations and several

nonmonotonic twist maps abstract we investigate a family of nonmonotonic radial twist maps constructed by unfolding the linear twist function we find several new modes of reconnection and derive conditions for them to occur reconnection and bifurcation of higher order resonances are studied in detail including vortex pairs and triplets the structural instability of some of these exotic figures is mitigated by the presence of chaotic separatrix layers invariant curves of involution pairs are utilized to locate and chart their often very complex metamorphoses

optimal bayesian estimation in random covariate design with a rescaled gaussian process prior in bayesian nonparametric models gaussian processes provide a popular prior choice for regression function estimation existing literature on the theoretical investigation of the resulting posterior distribution almost exclusively assume a fixed design for covariates the only random design result we are aware of van der vaart and van zanten 2011 assumes the assigned gaussian process to be supported on the smoothness class specified by the true function with probability one this is a fairly restrictive assumption as it essentially rules out the gaussian process prior with a squared exponential kernel when modeling rougher functions in this article we show that an appropriate rescaling of the above gaussian process leads to a rate optimal posterior distribution even when the covariates are independently realized from a known density on a compact set the proofs are based on deriving sharp concentration inequalities for frequentist kernel estimators the results might be of independent interest

on some inequalities for the incomplete gamma function let p 1 be a positive real number we determine all real numbers p and p such that the inequalities formula math formula math are valid for all x 0 and we determine all real numbers a and b such that log 1 e ax x e t t log 1 e bx hold for all 0

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

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

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

an improved algorithm for computing topological entropy a new algorithm is presented for computing the topological entropy of a unimodal map of the interval the accuracy of the algorithm is discussed and some graphs of the topological entropy which are obtained using the algorithm are displayed

wavelike patterns in one dimensional coupled map lattices we investigate the existence of wavelike solution for the logistic coupled map lattices for which the spatiotemporal periodic patterns can be predicted by a simple two dimensional mapping the existence of such wavelike solutions is proved by the implicit function theorem with constraints we also examine the stabilities of these wave solutions under perturbations of uniform small deformation type we show that in some specific cases these perturbations are completely general the technique used in this paper is also applicable to investigate other space time regular patterns

the bayesian lasso the lasso estimate for linear regression parameters can be interpreted as a bayesian posterior mode estimate when the regression parameters have independent laplace i e double exponential priors gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances a connection with the inverse gaussian distribution provides tractable full conditional distributions the bayesian lasso provides interval estimates bayesian credible intervals that can guide variable selection moreover the structure of the hierarchical model provides both bayesian and likelihood methods for selecting the lasso parameter slight modifications lead to bayesian versions of other lasso related estimation methods including bridge regression and a robust variant

the hide skeldon acheson dynamo revisited hide et al hide skeldon acheson 1996 proc r soc a452 1369 1395 introduced a nonlinear system of three coupled ordinary differential equations to model a self exciting faraday disk homopolar dynamo a very small selection of its possible behaviours was presented in that paper subsequent studies have extended the system to incorporate the effects of a nonlinear motor an external battery and magnetic field the coupling of two or more identical dynamos together among other things in this paper we return to the original model with a view to perform a more extensive analysis of the hide et al dynamo for the first time we present bifurcation transition diagrams so that the two examples of chaotic dynamo action shown in figure 9 in that paper can now be placed into context we exhibit the coexistence of multiple attractors and also identify the lowest order unstable periodic orbits pertaining to some specific cases

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

fast sampling with gaussian scale mixture priors in high dimensional regression we propose an efficient way to sample from a class of structured multivariate gaussian distributions the proposed algorithm only requires matrix multiplications and linear system solutions its computational complexity grows linearly with the dimension unlike existing algorithms that rely on cholesky factorizations with cubic complexity the algorithm is broadly applicable in settings where gaussian scale mixture priors are used on high dimensional parameters its effectiveness is illustrated through a high dimensional regression problem with a horseshoe prior on the regression coefficients other potential applications are outlined

fast bayesian factor analysis via automatic rotations to sparsity abstractrotational post hoc transformations have traditionally played a key role in enhancing the interpretability of factor analysis regularization methods also serve to achieve this goal by prioritizing sparse loading matrices in this work we bridge these two paradigms with a unifying bayesian framework our approach deploys intermediate factor rotations throughout the learning process greatly enhancing the effectiveness of sparsity inducing priors these automatic rotations to sparsity are embedded within a pxl em algorithm a bayesian variant of parameter expanded em for posterior mode detection by iterating between soft thresholding of small factor loadings and transformations of the factor basis we obtain a dramatic accelerations b robustness against poor initializations and c better oriented sparse solutions to avoid the prespecification of the factor cardinality we extend the loading matrix to have infinitely many columns with the indian buffet process ibp prior the factor dimen

modeling of chaotic vibrations in symmetric vocal folds the chaos mechanism of above range phonation was examined in symmetrically modeled vocal folds by using the traditional two mass model assumption the poincare map technique was used to display chaotic attractors this method provided an effective description of irregular vocal fold movements the power spectrum lyapunov exponent and kaplan yorke dimension were employed to describe chaotic vibrations in the vocal fold model these nonlinear dynamic analyses suggested that for the positive lyapunov exponent chaotic attractors contribute to irregular vocal fold vibrations descriptions of complicated irregular vibrations of the vocal fold yielded evidence of chaos to investigate the effects of independent parameters such as subglottal pressure coupling stiffness and phonation neutral area bifurcation diagrams based on the poincare map were discussed the results confirmed that the dynamics of the two mass model was strongly influenced by independent parameters nonlinear dynamic methods were expecte

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

semiglobal stabilization of nonlinear systems using fuzzy control and singular perturbation methods singular perturbation methods based on a lyapunov approach are implemented for the derivation of sufficient conditions for the semiglobal stabilization with output tracking of nonlinear systems having internal dynamics the key idea of this approach lies in the decomposition of the full system into a subsystem exhibiting fast dynamics and to a subsystem with slow dynamics resulting in a substantial simplification of the stability analysis this two time scale behavior is achieved by implementing a properly designed fuzzy logic controller the key element of the proposed model based composite controller for the construction of the model based controller a new identification technique based on both classical and fuzzy tools is proposed results are presented for the semiglobal stabilization of a continuous stirred tank reactor which is a process with high nonlinear behavior

bayes and empirical bayes multiplicity adjustment in the variable selection problem this paper studies the multiplicity correction effect of standard bayesian variable selection priors in linear regression our first goal is to clarify when and how multiplicity correction happens automatically in bayesian analysis and to distinguish this correction from the bayesian ockham s razor effect our second goal is to contrast empirical bayes and fully bayesian approaches to variable selection through examples theoretical results and simulations considerable differences between the two approaches are found in particular we prove a theorem that characterizes a surprising aymptotic discrepancy between fully bayes and empirical bayes this discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical bayes estimate indeed even at the extreme when the empirical bayes estimate converges asymptotically to the true variable inclusion probability the potential for a serious difference remains

validity of numerical trajectories in the synchronization transition of complex systems we investigate the relationship between the loss of synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems in the neighborhood of the critical transition to strongly nonhyperbolic behavior the system undergoes on off intermittency with respect to the synchronization state there are potentially severe consequences of these facts on the validity of the computer generated trajectories obtained from dynamical systems whose synchronization manifolds share the same nonhyperbolic properties

complete poincare sections and tangent sets trying to extend a local definition of a surface of a section and the corresponding poincare map to a global one one can encounter severe difficulties we show that global transverse sections often do not exist for hamiltonian systems with two degrees of freedom as a consequence we present a method to generate the so called w section which by construction will be intersected by almost all orbits depending on the type of tangent set in the surface of the section we distinguish five types of w sections the method is illustrated by a number of examples most notably the quartic potential and the double pendulum w sections can also be applied to higher dimensional hamiltonian systems and to dissipative systems

elastic net regression modeling with the orthant normal prior the elastic net procedure is a form of regularized optimization for linear regression that provides a bridge between ridge regression and the lasso the estimate that it produces can be viewed as a bayesian posterior mode under a prior distribution implied by the form of the elastic net penalty this article broadens the scope of the bayesian connection by providing a complete characterization of a class of prior distributions that generate the elastic net estimate as the posterior mode the resulting model based framework allows for methodology that moves beyond exclusive use of the posterior mode by considering inference based on the full posterior distribution two characterizations of the class of prior distributions are introduced a properly normalized direct characterization which is shown to be conjugate for linear regression models and an alternate representation as a scale mixture of normal distributions prior distributions are proposed for the regularization parameters resulting in an infi

computational techniques for nonlinear dynamics in multiple friction oscillators this paper presents some numerical techniques used to investigate the nonlinear dynamics of a class of mechanical systems affected by dry friction forces they allow for the exact detection of the points which separate slip phases from stick phases in this way the correct dynamic equations are integrated during the different motion phases finally a method is described to perform the numerical computation of the lyapunov exponents in such non smooth systems

accurate numerical integration of state space models for aeroelastic systems with free play results show the importance of accurately locating the switching point between linear subdomains when numerically integrating a piecewise linear system of equations henon s method locates the switching point to a high degree of accuracy in one integration step while eliminating the need for a specified tolerance

predictability in spatially extended systems the predictability problem is discussed in turbulent fluids and in spatially extended systems we perform a numerical analysis of the spatial propagation of a small perturbation initially localized at a point of a one dimensional lattice of coupled maps where the interactions can be either local or non local in space in both cases the nonlinear terms in the dynamics of the perturbation cannot be neglected and the predictability problem cannot be reduced to the linearized evolution equations only for non local interactions is the predictability time proportional to the inverse lyapunov exponent lambda as in dynamical systems with a few degrees of freedom for local interactions it is proportional to the size of the system and is practically independent of the chaoticity degree lambda

bernstein von mises theorems for gaussian regression with increasing number of regressors this paper brings a contribution to the bayesian theory of nonparametric and semiparametric estimation we are interested in the asymptotic normality of the posterior distribution in gaussian linear regression models when the number of regressors increases with the sample size two kinds of bernstein von mises theorems are obtained in this framework nonparametric theorems for the parameter itself and semiparametric theorems for functionals of the parameter we apply them to the gaussian sequence model and to the regression of functions in sobolev and c classes in which we get the minimax convergence rates adaptivity is reached for the bayesian estimators of functionals in our applications

stability and accuracy of periodic flow solutions obtained by a pod penalty method we develop a new penalty method to derive low dimensional galerkin models for fluid flows with time dependent boundary conditions we then outline a procedure based on bifurcation analysis in selecting the proper values of the penalty parameter s that yield asymptotically stable periodic solutions of the highest possible accuracy we illustrate this new approach by studying flow past a circular cylinder using direct numerical simulation dns data and a wave structure interaction problem using particle image velocimetry piv data

the vanishing twist in the restricted three body problem abstract this paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three body problem near the lagrangian equilibrium l 4 a twistless torus is created near the tripling bifurcation of the short period family decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3 10 and 2 7 this scenario is studied by numerically integrating the regularised hamiltonian flow and finding rotation numbers of invariant curves in a two dimensional poincare map to corroborate the numerical results the birkhoff normal form at l 4 is calculated to eighth order truncating at this order gives an integrable system and the rotation numbers obtained from the birkhoff normal form agree well with the numerical results a global overview for the mass ratio 4 3 is presented by showing lines of constant energy and constant rotation number in action space

a method for accurate computation of the rotation and the twist numbers of invariant circles a method is proposed for accurate evaluation of the rotation and the twist numbers of invariant circles in two degrees of freedom hamiltonian systems or two dimensional symplectic maps the method uses the recurrence of orbits to overcome the problems usually arising because of the multivalued character of the angles due to modulo 2 pi that have to be added in order to evaluate the above numbers furthermore best convergent demoninators q n of these numbers can be estimated and we show that under a proper treatment of the sequences of q n iterations the accuracy is of the order of 1 q n 4 c 2001 elsevier science b v all rights reserved

the stability of the lagrange triangular point and a theorem of arnold deprit and deprit bartholome 5 computed the normal form for the hamiltonian of the planar restricted three body problem at the lagrange equilateral triangle equilibrium point up to terms of order 4 this normalization was carried out for all mass ratios p smaller than routh s critical mass ratio p1 i 1 j 9 except for two values p2 and pi at p2 and u the ratio of the linearized frequencies is 1 2 and 1 3 respectively and so low order resonance terms appear this normalization was carried out in order to apply the kam theory to prove the stability of this equilibrium point from the coefficients of the normal form they computed a quantity d defined below which when non zero establishes the stability of the equilibrium point they found that d 0 for 0 cp p u p2 p and p the value u does not correspond to resonance but is simply a point where the quantity d4 changes sign in this paper we shall establish the full stability of the lagrange equilibrium point in the planar restricted three body problem even in the case when u p we do this by computing by machine the normal form up to terms of order 6 and then applying a theorem of arnold 3 which establishes the stability of the equilibrium even in degenerate cases russman lo announced a theorem which implies that the equilibrium is isoenergetic stable when p p by isoenergetic stable he means that the system is stabie only on the energy surface defined by the energy at the equilibrium itself unfortunately there is not even an indication of the proof arnold announced his theorem in a doklady note with only a sketch of a proof the sketch indicates that his proof requires completely redoing all

another look at the saddle centre bifurcation vanishing twist abstract in the saddle centre bifurcation a pair of periodic orbits is created out of nothing in a hamiltonian system with two degrees of freedom it is the generic bifurcation with multiplier one we show that out of nothing should be replaced by out of a twistless torus more precisely we show that invariant tori of the normal form have vanishing twist right before the appearance of the new orbits vanishing twist means that the derivative of the rotation number with respect to the action for constant energy vanishes we explicitly derive the position of the twistless torus in phase and in parameter space near the saddle centre bifurcation the theory is applied to the area preserving henon map

high dimensional covariance matrix estimation in approximate factor models the variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems classical methods of estimating the covariance matrices are based on the strict factor models assuming independent idiosyncratic components this assumption however is restrictive in practical applications by assuming sparse error covariance matrix we allow the presence of the cross sectional correlation even after taking out common factors and it enables us to combine the merits of both methods we estimate the sparse covariance using the adaptive thresholding technique as in cai and liu j amer statist assoc 106 2011 672 684 taking into account the fact that direct observations of the idiosyncratic components are unavailable the impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied

is one dimensional poincare map sufficient to describe the chaotic dynamics of a three dimensional system study of continuous dynamical system through poincare map is one of the most popular topics in nonlinear analysis this is done by taking intersections of the orbit of flow by a hyper plane parallel to one of the coordinate hyper planes of co dimension one naturally for a 3d attractor the poincare map gives rise to 2d points which can describe the dynamics of the attractor properly in a very special case sometimes these 2d points are considered as their 1d projections to obtain a 1d map however this is an artificial way of reducing the 2d map by dropping one of the variables sometimes it is found that the two coordinates of the points on the poincare section are functionally related this also reduces the 2d poincare map to a 1d map this reduction is natural and not artificial as mentioned above in the present study this issue is being highlighted in fact we find out some examples which show that even this natural reduction of the 2d poincare map is not always justified because the resultant 1d map may fail to generate the original dynamics properly this proves that to describe the dynamics of the 3d chaotic attractor the minimum dimension of the poincare map must be two in general

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

sparse bayesian time varying covariance estimation in many dimensions dynamic covariance estimation for multivariate time series suffers from the curse of dimensionality this renders parsimonious estimation methods essential for conducting reliable statistical inference in this paper the issue is addressed by modeling the underlying co volatility dynamics of a time series vector through a lower dimensional collection of latent time varying stochastic factors furthermore we apply a normal gamma prior to the elements of the factor loadings matrix this hierarchical shrinkage prior effectively pulls the factor loadings of unimportant factors towards zero thereby increasing parsimony even more we apply the model to simulated data as well as daily log returns of 300 sp it is implemented in the r package factorstochvol author s abstract

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

high dimensional sparse factor modeling applications in gene expression genomics we describe studies in molecular profiling and biological pathway analysis that use sparse latent factor and regression models for microarray gene expression data we discuss breast cancer applications and key aspects of the modeling and computational methodology our case studies aim to investigate and characterize heterogeneity of structure related to specific oncogenic pathways as well as links between aggregate patterns in gene expression profiles and clinical biomarkers based on the metaphor of statistically derived factors as representing biological subpathway structure we explore the decomposition of fitted sparse factor models into pathway subcomponents and investigate how these components overlay multiple aspects of known biological activity our methodology is based on sparsity modeling of multivariate regression anova and latent factor models as well as a class of models that combines all components hierarchical sparsity priors address questions of dimension reduction and multiple co

expandable factor analysis bayesian sparse factor models have proven useful for characterizing dependence in multivariate data but scaling computation to large numbers of samples and dimensions is problematic we propose expandable factor analysis for scalable inference in factor models when the number of factors is unknown the method relies on a continuous shrinkage prior for efficient maximum a posteriori estimation of a low rank and sparse loadings matrix the structure of the prior leads to an estimation algorithm that accommodates uncertainty in the number of factors we propose an information criterion to select the hyperparameters of the prior expandable factor analysis has better false discovery rates and true positive rates than its competitors across diverse simulation settings we apply the proposed approach to a gene expression study of ageing in mice demonstrating superior results relative to four competing methods

study of chaos in hamiltonian systems via convergent normal forms abstract we use moser s normal forms to study chaotic motion in two degree hamiltonian systems near a saddle point besides being convergent they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity both aspects combined allowed a precise computation of the homoclinic interaction of stable and unstable manifolds in the full phase space rather than just the poincare section the formalism was applied to the henon heiles hamiltonian producing strong evidence that the region of convergence of these normal forms extends over that orginally established by moser

pseudo orbit shadowing in the family of tent maps on montre que les applications en tente ont la propriete d ombrage pour presque tous les parametres s mais elles n ont pas cette propriete pour un ensemble dense non denombrable des parametres on montre que pour une application tente chaque pseudo orbite peut etre approchee par une orbite actuelle d une application en tente avec une boucle peut etre legerement plus grande

factor modeling for high dimensional time series inference for the number of factors this paper deals with the factor modeling for high dimensional time series based on a dimension reduction viewpoint under stationary settings the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix and is therefore applicable when the dimension of time series is on the order of a few thousands asymptotic properties of the proposed method are investigated under two settings i the sample size goes to infinity while the dimension of time series is fixed and ii both the sample size and the dimension of time series go to infinity together in particular our estimators for zero eigenvalues enjoy faster convergence or slower divergence rates hence making the estimation for the number of factors easier in particular when the sample size and the dimension of time series go to infinity together the estimators for the eigenvalues are no longer consistent however our estimator for the number of the factors which is based on the ratios of the estimated eigenvalues still works fine furthermore this estimation shows the so called blessing of dimensionality property in the sense that the performance of the estimation may improve when the dimension of time series increases a two step procedure is investigated when the factors are of different degrees of strength numerical illustration with both simulated and real data is also reported

on the numerical computation of poincare maps abstract this note describes a method for finding simply and accurately the intersections of a numerically integrated trajectory with a surface of section

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

harmonic balance approach for an airfoil with a freeplay control surface the nonlinear aeroelastic response of a two dimensional airfoil including a control surface with freeplay placed in an incompressible flow is studied the model equations are formulated as a set of first order ordinary differential equations first the dynamic response is investigated by a time integration method and the time integration results are used for the verification of the harmonic balance results the interesting hysteresis phenomenon and the effect of initial conditions of the subcritical bifurcation are presented a higher order harmonic balance method is then derived to investigate the high harmonics of the airfoil motions the harmonic balance prediction is verified by comparison to the results from a numerical time marching integration and also by comparison to results from a previous experiment nomenclature a nondimensional distance of the elastic axis from the midchord with respect to the semichord b damping submatrix b semichord ch stiffness per unit span of wing in deflection c k generalized theodorsen function c c torsional stiffness per unit span of wing around a and of aileron around c c nondimensional distance of the control surface aileron hinge line from the midchord with respect to the semichord

chaotic saddles at the onset of intermittent spatiotemporal chaos in a recent study rempel and chian phys rev lett 98 014101 2007 it has been shown that nonattracting chaotic sets chaotic saddles are responsible for intermittency in the regularized long wave equation that undergoes a transition to spatiotemporal chaos stc via quasiperiodicity and temporal chaos in the present paper it is demonstrated that a similar mechanism is present in the damped kuramoto sivashinsky equation prior to the onset of stc a spatiotemporally chaotic saddle coexists with a spatially regular attractor after the transition to stc the chaotic saddle merges with the attractor generating intermittent bursts of stc that dominate the post transition dynamics

stability of synchronous chaos and on off intermittency in coupled map lattices in this paper we consider the stability of synchronous chaos in lattices of coupled n dimensional maps for global coupling we derive explicit conditions for computing the parameter values at which the synchronous chaotic attractor becomes unstable and bifurcates into asynchronous chaos in particular we show that after the bifurcation one generally observes on off intermittency a process in which the entire system evolves nearly synchronously but chaotically for long periods of time which are interrupted by brief bursts away from synchrony for nearest neighbor coupled systems however we show that the stability of the synchronous chaotic state is a function of the system size in particular for large systems we will not be able to observe synchronous chaos we derive a condition relating the local map s largest lyapunov exponent to the maximal system size under which one can still observe synchronous chaos and on off intermittency other issues related to the characterization of on off intermittent signals are also discussed

on a coupled logistic map with large strength abstract in this paper a class of diffusive coupled logistic map with large strength is considered depending on the parameter of the logistic family and the strength of coupling we show the existence of a forward invariant curve which determines in some sense the boundary of the set whose points have unbounded orbit

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

computing accurate poincare maps we present a numerical method particularly suited for computing poincare maps for systems of ordinary differential equations the method is a generalization of a stopping procedure described by henon physica d 5 1982 412 and it applies to a wide family of systems

the horseshoe estimator for sparse signals this paper proposes a new approach to sparsity called the horseshoe estimator the horseshoe is a close cousin of other widely used bayes rules arising from for example double exponential and cauchy priors in that it is a member of the same family of multivariate scale mixtures of normals but the horseshoe enjoys a number of advantages over existing approaches including its robustness its adaptivity to dierent sparsity patterns and its analytical tractability we prove two theorems that formally characterize both the horseshoe s adeptness at large outlying signals and its super ecient rate of convergence to the correct estimate of the sampling density in sparse situations finally using a combination of real and simulated data we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full bayesian model averaging approach using a discrete mixture prior to model signals and noise

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

chaotic sub dynamics in coupled logistic maps abstract we study the dynamics of laplacian type coupling induced by logistic family f x x 1 x where 0 4 on a periodic lattice that is the dynamics of maps of the form f x y 1 e f x e f y 1 e f y e f x where e 0 determines strength of coupling our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal in analytical way we detect some regions of parameters for which a horseshoe is present and using simulations global attractors and invariant sets are depicted

aeroelastic dynamic response and control of an airfoil section with control surface nonlinearities abstract nonlinearities in aircraft mechanisms are inevitable especially in the control system it is necessary to investigate the effects of them on the dynamic response and control performance of aeroelastic system in this paper based on the state dependent riccati equation method a state feedback suboptimal control law is derived for aeroelastic response and flutter suppression of a three degree of freedom typical airfoil section with the control law designed nonlinear effects of freeplay in the control surface and time delay between the control input and actuator are investigated by numerical approach a cubic nonlinearity in pitch degree is adopted to prevent the aeroelastic responses from divergence when the flow velocity exceeds the critical flutter speed for the system with a freeplay the responses of both open and closed loop systems are determined with runge kutta algorithm in conjunction with henon s method this method is used to locate the switching points accurately and efficiently as the system moves from one subdomain into another the simulation results show that the freeplay leads to a forward phase response and a slight increase of flutter speed of the closed loop system the effect of freeplay on the aeroelastic response decreases as the flow velocity increases the time delay between the control input and actuator may impair control performance and cause high frequency motion and quasi periodic vibration