improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates the problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered the focus is on finite sample size situations whereby the number of observations is limited and comparable in magnitude to the observation dimension using tools from random matrix theory and assuming a certain eigenvalue splitting condition new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived that are shown to be consistent in a more general asymptotic setting than the traditional one indeed these estimators are proven to be consistent not only when the sample size increases without bound for a fixed observation dimension but also when the observation dimension increases to infinity at the same rate as the sample size numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios where the observation dimension and the sample size are comparable in magnitude

state transitions and the continuum limit for a 2d interacting self propelled particle system we study a class of swarming problems wherein particles evolve dynamically via pairwise interaction potentials and a velocity selection mechanism we find that the swarming system undergoes various changes of state as a function of the self propulsion and interaction potential parameters in this paper we utilize a procedure which connects a class of individual based models to their continuum formulations and determine criteria for the validity of the latter h stability of the interaction potential plays a fundamental role in determining both the validity of the continuum approximation and the nature of the aggregation state transitions we perform a linear stability analysis of the continuum model and compare the results to the simulations of the individual based one c 2007 elsevier b v all rights reserved

condition number regularized covariance estimation estimation of high dimensional covariance matrices is known to be a difficult problem has many applications and is of current interest to the larger statistics community in many applications including so called the large p small n setting the estimate of the covariance matrix is required to be not only invertible but also well conditioned although many regularization schemes attempt to do this none of them address the ill conditioning problem directly in this paper we propose a maximum likelihood approach with the direct goal of obtaining a well conditioned estimator no sparsity assumption on either the covariance matrix or its inverse are are imposed thus making our procedure more widely applicable we demonstrate that the proposed regularization scheme is computationally efficient yields a type of steinian shrinkage estimator and has a natural bayesian interpretation we investigate the theoretical properties of the regularized covariance estimator comprehensively including its regularization path and proceed to develop an approach that adaptively determines the level of regularization that is required finally we demonstrate the performance of the regularized estimator in decision theoretic comparisons and in the financial portfolio optimization setting the proposed approach has desirable properties and can serve as a competitive procedure especially when the sample size is small and when a well conditioned estimator is required

self organization in systems of self propelled particles we investigate a discrete model consisting of self propelled particles that obey simple interaction rules we show that this model can self organize and exhibit coherent localized solutions in one and in two dimensions in one dimension the self organized solution is a localized flock of finite extent in which the density abruptly drops to zero at the edges in two dimensions we focus on the vortex solution in which the particles rotate around a common center and show that this solution can be obtained from random initial conditions even in the absence of a confining boundary furthermore we develop a continuum version of our discrete model and demonstrate that the agreement between the discrete and the continuum model is excellent

obstacle and predator avoidance in a model for flocking abstract the modeling and investigation of the dynamics and configurations of animal groups is a subject of growing attention in this paper we present a deterministic continuum model of flocking and use it to investigate the linearized response of an infinite extent flock to an obstacle or an attacking predator we show that the steady state flock response is in the form of density disturbances that resemble mach cones whose configuration is determined by the anisotropic propagation of waves through the flock we analytically and numerically test relations that predict the mach wedge angles disturbance heights and wake widths we find that these expressions are insensitive to many of the parameters of the model

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

a multiple testing approach to the regularisation of large sample correlation matrices abstract this paper proposes a regularisation method for the estimation of large covariance matrices that uses insights from the multiple testing m t literature the approach tests the statistical significance of individual pair wise correlations and sets to zero those elements that are not statistically significant taking account of the multiple testing nature of the problem the effective p values of the tests are set as a decreasing function of n the cross section dimension the rate of which is governed by the nature of dependence of the underlying observations and the relative expansion rates of n and t the time dimension in this respect the method specifies the appropriate thresholding parameter to be used under gaussian and non gaussian settings the m t estimator of the sample correlation matrix is shown to be consistent in the spectral and frobenius norms and in terms of support recovery so long as the true covariance matrix is sparse the performance of the proposed m t estimator is compared to a number of other estimators in the literature using monte carlo experiments it is shown that the m t estimator performs well and tends to outperform the other estimators particularly when n is larger than t

shrinkage algorithms for mmse covariance estimation we address covariance estimation in the sense of minimum mean squared error mmse when the samples are gaussian distributed specifically we consider shrinkage methods which are suitable for high dimensional problems with a small number of samples large p small n first we improve on the ledoit wolf lw method by conditioning on a sufficient statistic by the rao blackwell theorem this yields a new estimator called rblw whose mean squared error dominates that of lw for gaussian variables second to further reduce the estimation error we propose an iterative approach which approximates the clairvoyant shrinkage estimator convergence of this iterative method is established and a closed form expression for the limit is determined which is referred to as the oracle approximating shrinkage oas estimator both rblw and oas estimators have simple expressions and are easily implemented although the two methods are developed from different perspectives their structure is identical up to specified constants the rblw estimator provably dominates the lw method for gaussian samples numerical simulations demonstrate that the oas approach can perform even better than rblw especially when n is much less than p we also demonstrate the performance of these techniques in the context of adaptive beamforming

bayesian inference in models with euclidean structures abstract when a parameter space is endowed with a euclidean structure there is a euclidean measure called the structural or geometric prior which is an obvious candidate to represent ignorance concerning an unknown parameter the rationale for using this prior is strengthened when the unknown parameter is the canonical parameter of an exponential family then the structural prior is the uniform prior for the canonical parameter it is shown also that euclidean structures can be combined in different ways with group structures the structural or geometric prior is then the product of the invariant prior for the group structure and the structural or geometric prior for the euclidean structure

inner statistical inference ii for any linear model there is according to an invariance argument a uniquely defined prior representing ignorance which will be called the inner prior it is shown that the corresponding posterior probabilities are weighted averages of conditional confidence levels this frequency interpretation compares favorably with the usual interpretation of confidence levels as unconditional probabilities of coverage of particular interest to users of statistical methods is that inner posterior intervals are always shorter than the corresponding confidence intervals and sometimes dramatically so

studies in the history of probability and statistics xlviii the bayesian contributions of ernest lhoste summary the contributions of ernest lhoste are largely unknown outside france and even within that country are not well known his important contributions were in the two areas of the development of prior distributions that represent little or no information and a sophisticated posterior analysis for normal and binomial populations his results are similar to those of haldane 1948 and jeffreys 1961 but they appeared much earlier in 1923 he gave a great deal of thought to how to represent vague prior knowledge and his results represent a significant and unique contribution to bayesian ideas in the early part of the 20th century

random matrix derived shrinkage of spectral precision matrices there has been much research on shrinkage methods for real valued covariance matrices and their inverses precision matrices in spectral analysis of p vector valued time series complex valued spectral matrices and precision matrices arise and good shrinkage methods are often required most notably when the estimated complex valued spectral matrix is singular as an improvement on the ledoit wolf lw type of spectral matrix estimator we use random matrix theory to derive a rao blackwell estimator for a spectral matrix its inverse being a rao blackwellized estimator for the spectral precision matrix a random matrix method has previously been proposed for complex valued precision matrices it was implemented by very costly simulations we formulate a fast completely analytic approach moreover we derive a way of selecting an important parameter using predictive risk methodology we show that both the rao blackwell estimator and the random matrix estimator of the precision matrix can substantially outperform the inverse of the lw estimator in a time series setting our new methodology is applied to eeg derived time series data where it is seen to work well and deliver substantial improvements for precision matrix estimation

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

estimation of a covariance matrix under stein s loss on decrit la technique generale de stein pour ameliorer l estimateur sans biais invariant le meilleur et l estimateur minimax de la matrice de covariance normale

covariance regularized regression and classification for high dimensional problems in recent years many methods have been developed for regression in high dimensional settings we propose covariance regularized regression a family of methods that use a shrunken estimate of the inverse covariance matrix of the features in order to achieve superior prediction an estimate of the inverse covariance matrix is obtained by maximizing its log likelihood under a multivariate normal model subject to a constraint on its elements this estimate is then used to estimate coefficients for the regression of the response onto the features we show that ridge regression the lasso and the elastic net are special cases of covariance regularized regression and we demonstrate that certain previously unexplored forms of covariance regularized regression can outperform existing methods in a range of situations the covariance regularized regression framework is extended to generalized linear models and linear discriminant analysis and is used to analyze gene expression data sets with multiple class and survival outcomes

interaction ruling animal collective behavior depends on topological rather than metric distance evidence from a field study numerical models indicate that collective animal behavior may emerge from simple local rules of interaction among the individuals however very little is known about the nature of such interaction so that models and theories mostly rely on aprioristic assumptions by reconstructing the three dimensional positions of individual birds in airborne flocks of a few thousand members we show that the interaction does not depend on the metric distance as most current models and theories assume but rather on the topological distance in fact we discovered that each bird interacts on average with a fixed number of neighbors six to seven rather than with all neighbors within a fixed metric distance we argue that a topological interaction is indispensable to maintain a flock s cohesion against the large density changes caused by external perturbations typically predation we support this hypothesis by numerical simulations showing that a topological interaction grants significantly higher cohesion of the aggregation compared with a standard metric one

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

high dimensional volatility matrix estimation via waveletsand thresholding we propose a locally stationary linear model for the evolution of high dimensional financial returns where the time varying volatility matrix is modelled as a piecewise constant function of time we introduce a new wavelet based technique for estimating the volatility matrix which 10 combines four ingredients a haar wavelet decomposition variance stabilization of the haar coefficients via the fisz transform prior to thresholding a bias correction and extra time domain thresholding soft or hard under the assumption of sparsity we demonstrate the interval wise consistency of the proposed estimators of the volatility matrix and its inverse in the operator norm with rates which adapt to the features of the target matrix we also propose a version of 15 the estimators based on the polarization identity which permits a more precise derivation of the thresholds we discuss the practicalities of the algorithm including parameter selection and how to perform it online a simulation study shows the benefits of the method which is illustrated using a stock index portfolio

continuum modeling of the equilibrium and stability of animal flocks abstract groups of animals often tend to arrange themselves in flocks that have characteristic spatial attributes and temporal dynamics using a dynamic continuum model for a flock of individuals we find equilibria of finite spatial extent where the density goes continuously to zero at a well defined flock edge and we discuss conditions on the model that allow for such solutions we also demonstrate conditions under which as the flock size increases the interior density in our equilibria tends to an approximately uniform value motivated by observations of starling flocks that are relatively thin in a direction transverse to the direction of flight we investigate the stability of infinite planar sheet flock equilibria we find that long wavelength perturbations along the sheet are unstable for the class of models that we investigate this has the conjectured consequence that sheet like flocks of arbitrarily large transverse extent relative to their thickness do not occur however we also show that our model admits approximately sheet like pancake shaped three dimensional ellipsoidal equilibria with definite aspect ratios transverse length scale to flock thickness determined by anisotropic perceptual response characteristics of the flocking individuals and we argue that these pancake like equilibria are stable to the previously mentioned sheet instability