covariate information matrix for sufficient dimension reduction building upon recent research on the applications of the density information matrix we develop a tool for sufficient dimension reduction sdr in regression problems called covariate information m

k means inverse regression li suggested the method of sliced inverse regression for dimension reduction in regressions with a univariate response in this article we extend that method to multivariate regressions by introducing a new way of performing the slicing this method applies for any number of response variables and may be particularly useful at the outset of an analysis before positing a multivariate model the emphasis is on application no new asymptotic theory is presented

using the bootstrap to select one of a new class of dimension reduction methods dimension reduction in a regression analysis of response y given a p dimensional vector of predictors x reduces the dimension of x by replacing it with a lower dimensional linear combination x of the x s without specifying a parametric model and without loss of information about the conditional distribution of y given x we unify three existing methods sliced inverse regression sir sliced average variance estimate save and principal hessian directions phd into a larger class of methods each method estimates a particular candidate matrix essentially a matrix of parameters we introduce broad classes of dimension reduction candidate matrices and we distinguish estimators of the matrices from the matrices themselves given these classes of methods and several ways to estimate any matrix we now have the problem of selecting a particular matrix and estimation method we propose bootstrap methodology to select among candidate matrices estimators and dimension and in particular we investigate l

testing predictor contributions in sufficient dimension reduction we develop tests of the hypothesis of no effect for selected predictors in regression without assuming a model for the conditional distribution of the response given the predictors predictor effects need not be limited to the mean function and smoothing is not required the general approach is based on sufficient dimension reduction the idea being to replace the predictor vector with a lower dimensional version without loss of information on the regression methodology using sliced inverse regression is developed in detail

sparse sufficient dimension reduction existing sufficient dimension reduction methods suffer from the fact that each dimension reduction component is a linear combination of all the original predictors so that it is difficult to interpret the resulting estimates we propose a unified estimation strategy which combines a regression type formulation of sufficient dimension reduction methods and shrinkage estimation to produce sparse and accurate solutions the method can be applied to most existing sufficient dimension reduction methods such as sliced inverse regression sliced average variance estimation and principal hessian directions we demonstrate the effectiveness of the proposed method by both simulations and real data analysis

a note on shrinkage sliced inverse regression summary we employ lasso shrinkage within the context of sufficient dimension reduction to obtain a shrinkage sliced inverse regression estimator which provides easier interpretations and better pre diction accuracy without assuming a parametric model the shrinkage sliced inverse regression approach can be employed for both single index and multiple index models simulation studies suggest that the new estimator performs well when its tuning parameter is selected by either the bayesian information criterion or the residual information criterion

a new approach to state observation of nonlinear systems with delayed output the article presents a new approach for the construction of a state observer for nonlinear systems when the output measurements are available for computations after a nonnegligible time delay the proposed observer consists of a chain of observation algorithms reconstructing the system state at different delayed time instants chain observer conditions are given for ensuring global exponential convergence to zero of the observation error for any given delay in the measurements the implementation of the observer is simple and computer simulations demonstrate its effectiveness

identifying regression outliers and mixtures graphically regressions in practice can include outliers and other unknown subpopulation structures for example mixtures of regressions occur if there is an omitted categorical predictor like gender or location and different regressions occur within each category the theory of regression graphics based on central subspaces can be used to construct graphical solutions to long standing problems of this type it is argued that in practice the central subspace automatically expands to incorporate outliers and regression mixtures thus methods of estimating the central subspace can be used to identify these phenomena without specifying a model examples illustrating the power of the theory are presented

groupwise dimension reduction via envelope method the family of sufficient dimension reduction sdr methods that produce informative combinations of predictors or indices are particularly useful for high dimensional regression analysis in many such analyses it becomes increasingly common that there is available a priori subject knowledge of the predictors for example they belong to different groups while many recent sdr proposals have greatly expanded the scope of the methods applicability how to effectively incorporate the prior predictor structure information remains a challenge in this article we aim at dimension reduction that recovers full regression information while preserving the predictor group structure built upon a new concept of the direct sum envelope we introduce a systematic way to incorporate the group information in most existing sdr estimators as a result the reduction outcomes are much easier to interpret moreover the envelope method provides a principled way to build a variety of prior structures into dimension reduc

trace pursuit a general framework for model free variable selection abstractwe propose trace pursuit for model free variable selection under the sufficient dimension reduction paradigm two distinct algorithms are proposed stepwise trace pursuit and forward trace pursuit stepwise trace pursuit achieves selection consistency with fixed p forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality the screening consistency property of forward trace pursuit based on sliced inverse regression is established finite sample performances of trace pursuit and other model free variable selection methods are compared through numerical studies supplementary materials for this article are available online

sequential sufficient dimension reduction for large p small n problems summary we propose a new and simple framework for dimension reduction in the large p small n setting the framework decomposes the data into pieces thereby enabling existing approaches for n p to be adapted to n p problems estimating a large covariance matrix which is a very difficult task is avoided we propose two separate paths to implement the framework our paths provide sufficient procedures for identifying informative variables via a sequential approach we illustrate the paths by using sufficient dimension reduction approaches but the paths are very general empirical evidence demonstrates the efficacy of our paths additional simulations and applications are given in an on line supplementary file

dimension reduction for the conditional kth moment in regression the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model central subspaces are designed to capture all the information for the regression and to provide a population structure for dimension reduction here we introduce the central kth moment subspace to capture information from the mean variance and so on up to the kth conditional moment of the regression new methods are studied for estimating these subspaces connections with sliced inverse regression are established and examples illustrating the theory are presented

sampled data h8 state feedback control of systems with state delays in the present paper we adopt the second approach to the nite horizon sampled data state delay h 1 control problem note that in the case of state delays the lifting technique leads to complicated discrete system with in nite dimensional state variable we obtain the piecewise constant state feedback controllers by solving coupled rpdes between sampling and updating the terminal conditions at the sampling instants we derive an asymptotic approximation to the solution of these rpdes by expanding it in the powers of the largest delay the resulting approximation is obtained by solving uncoupled low order partial dia erential equations the performance of the system with the controller that has been obtained using the zero approximation the one that corresponds to zero delay is analysed when the open loop system possesses a non zero delay bounded real lemmas for systems with state delays and jumps are obtained 2 problem formulation

efficient sparse estimate of sufficient dimension reduction in high dimension abstractin this article we propose a new efficient sparse estimate ese in sufficient dimension reduction using distance covariance our method is model free and does not need any kernel function or slicing selection moreover it can naturally deal with multivariate response scenarios making it appealing in a modified sequential algorithm that targets the large p small n problems compared with screening procedures that only use marginal utility our method can extract more useful information from the data and is capable of determining the size of the selected submodel automatically while most of screening procedures cannot under mild conditions based on manifold theories and techniques it can be shown that our method would perform asymptotically as well as if the true irrelevant predictors were known which is referred to as the oracle property extensive simulation studies and two real data examples demonstrate the effectiveness and efficiency of the proposed approach it is remarkable that the a

h norm and invariant manifolds of systems with state delays abstract the problem of finding bounds on the h norm of systems with a finite number of point delays and distributed delay is considered sufficient conditions for the system to possess an h norm which is less or equal to a prescribed bound are obtained in terms of riccati partial differential equations rpde s we show that the existence of a solution to the rpde s is equivalent to the existence of a stable manifold of the associated hamiltonian system for small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary differential equations that govern the flow on the slow manifold of the hamiltonian system this leads to an algebraic finite dimensional criterion for systems with small delays

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

technical communique on the bounded real and positive real conditions for affine nonlinear state delayed systems and applications to stability in this paper bounded real conditions for affine nonlinear state delayed systems are derived using the concept of dissipativeness necessary and sufficient conditions for the system to be dissipative and to have finite l 2 gain also referred to as the bounded real condition are given the implications on the stability of the system and feedback interconnections of such systems are also considered finally an equivalent of the positive real lemma is derived and its implications on stability of the system and feedback interconnections of such systems are similarly discussed

convergence of the solution of a nonsymmetric matrix riccati differential equation to its stable equilibrium solution we consider the initial value problem for a nonsymmetric matrix riccati dierential equation where the four coecient matrices form an m matrix we show that for a wide range of initial values the riccati dierential equation has a global solution x t on 0 1 and x t converges to the stable equilibrium solution as t goes to infinity

bounded real criteria for linear time delay systems considers the problem of finding the h sub spl infin norm of a linear time varying system having a delayed state and with an output that comprises a linear combination of the current state as well as of the delayed state both delay independent and delay dependent bounded real criteria are derived which provide sufficient conditions for the system to possess an h sub spl infin norm which is less than or equal to a prescribed bound the stationary case for time invariant systems is also tackled and in this situation the proposed bounded real criteria are given in terms of linear matrix inequalities two examples are presented which compare the guaranteed bounds that are achieved by our results with the actual h sub spl infin norm of the system

principal hessian directions revisited abstract li has suggested the method of principal hessian directions for estimating plotting directions that capture curvature in regression functions this article revisits li s proposal offering a number of suggestions for improved application

on the degrees of freedom of the lasso we study the effective degrees of freedom of the lasso in the framework of stein s unbiased risk estimation sure we show that the number of nonzero coefficients is an unbiased estimate forthe degrees of freedom of the lasso a conclusion that requires no special assumption on the predictors in addition the unbiased estimator is shown to be asymptotically consistent with these results on hand various model selection criteria cp aic and bic are available which along with the lars algorithm provide a principled and efficient approach to obtaining the optimal lasso fit with the computational effort of a single ordinary least squares fit

quadratic vector equations in this chapter we aim to study in an unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses specific cases of these problems have been studied extensively in the past by several authors for references to the single equations and results we refer the reader to the following sections in particular section 2 3 many of the results appearing here have already been proved for one or more of the single instances of the problems resorting to specific characteristics of the problem in some cases the proofs we present here are mere rewritings of the original proofs with a little change of notation to adapt them to our framework but in some cases we are effectively able to remove some hypotheses and generalize the results by abstracting the specific aspects of each problem

a new algorithm for estimating the effective dimension reduction subspace the statistical problem of estimating the effective dimension reduction edr subspace in the multi index regression model with deterministic design and additive noise is considered a new procedure for recovering the directions of the edr subspace is proposed many methods for estimating the edr subspace perform principal component analysis on a family of vectors say 1 l nearly lying in the edr subspace this is in particular the case for the structure adaptive approach proposed by hristache et al 2001a in the present work we propose to estimate the projector onto the edr subspace by the solution to the optimization problem minimize maxl 1 l lt i a l subject to a am where am is the set of all symmetric matrices with eigenvalues in 0 1 and trace less than or equal to m with m being the true structural dimension under mild assumptions n consistency of the proposed procedure is proved up to a logarithmic factor in the case when the structural dimension is not larger than 4 moreover the stochastic error of the estimator of the projector onto the edr subspace is shown to depend on l logarithmically this enables us to use a large number of vectors l for estimating the edr subspace the empirical behavior of the algorithm is studied through numerical simulations

comments on bounded real criteria for linear time delay systems xiefu jiang et al point out some some misleading statements in the above mentioned paper shaked et al 1998 and relation results are given also wen shyong yu points out serious mistakes in the same paper and also untenable main results

synthesis of non rational controllers for linear delay systems this paper addresses the control of linear delay systems using non rational controllers the structure of the controller is chosen so as to copy the structure of the plant reproducing the delays in the state and in the output the resulting stabilization and performance design problems are entirely expressed as linear matrix inequalities although the design inequalities are based on delay independent stability conditions the overall design is delay dependent in the sense that the controller makes use of the delay parameter of the plant this parameter is assumed to be constant yet arbitrary using non rational controllers we overcome the main difficulty faced when designing rational controllers for linear delay systems which is to incorporate in the design problem the matrix multiplier used to prove stability with respect to the delayed part of the system we illustrate the paper with several examples and provide extensive comparisons with existent results

shrinkage inverse regression estimation for model free variable selection the family of inverse regression estimators that was recently proposed by cook and ven effective in dimension reduction hv transformina the hiah dimensional predictor ni has proven effective in dimension reduction by transforming the high dimensional predictor vector to its low dimensional projections we propose a general shrinkage estimation strategy for the entire inverse regression estimation family that is capable of simultaneous dimension reduction and variable selection we demonstrate that the new estimators achieve consistency in variable selection without requiring any traditional model meanwhile retaining the root n estimation consistency of the dimension reduction basis we also show the effectiveness of the new estimators through both simulation and real data analysis

optimal sufficient dimension reduction in regressions with categorical predictors though partial sliced inverse regression partial sir chiaromonte et al 2002 sufficient dimension reduction in regressions with categorical predictors ann statist 30 475 497 extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice when this condition fails partial sir may provide misleading results in this article we propose a new estimation method via a minimum discrepancy approach without this restriction our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi squared distribution it also gives us the ability to test predictor effects an asymptotic chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained

robust variable and interaction selection for logistic regression and general index models under the logistic regression framework we propose a forward backward method soda for variable selection with both main and quadratic interaction terms in the forward stage soda adds in predic

finite horizon h sub spl infin state feedback control of continuous time systems with state delays the finite horizon h sub spl infin control of time invariant linear systems with a finite number of point and distributed time delays is considered the controller is obtained by solving coupled riccati type partial differential equations the solutions to these equations and the resulting controllers are approximated by series expansions in powers of the largest delay unlike the infinite horizon case these approximations possess both regular and boundary layer terms the performance of the closed loop system under a memoryless zero approximation controller is analyzed

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

global existence and stability of solutions of matrix riccati equations we consider a matrix riccati equation containing two parameters c and the quantity c denotes the average total number of particles emerging from a collision which is assumed to be conservative i e 0 c 1 and 0 1 is an angular shift let s c 0 c 1 and 0 1 stability analysis for two steady state solutions xmin and xmax are provided in particular we prove that xmin is locally asymptotically stable for s 1 0 while xmax is unstable for s 1 0 for c 1 and 0 xmin xmax is neutral stable we also show that such equations have a global positive solution for c s provided that the initial value is small and positive

direct effects testing a two stage procedure to test for effect size and variable importance for correlated binary predictors and a binary response in applications such as medical statistics and genetics we encounter situations where a large number of highly correlated predictors explain a response for example the response may be a disease indicator and the predictors may be treatment indicators or single nucleotide polymorphisms snps constructing a good predictive model in such cases is well studied less well understood is how to recover the true sparsity pattern that is finding which predictors have direct effects on the response and indicating the statistical significance of the results restricting attention to binary predictors and response we study the recovery of the true sparsity pattern using a two stage method that separates establishing the presence of effects from inferring their exact relationship with the predictors simulations and a real data application demonstrate that the method discriminates well between associations and direct effects comparisons with lasso based methods demonstrate favourable performance of the proposed method

sufficient dimension reduction via inverse regression a family of dimension reduction methods the inverse regression ir family is developed by minimizing a quadratic objective function an optimal member of this family the inverse regression estimator ire is proposed along with inference methods and a computational algorithm the ire has at least three desirable properties 1 its estimated basis of the central dimension reduction subspace is asymptotically efficient 2 its test statistic for dimension has an asymptotic chi squared distribution and 3 it provides a chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors current methods like sliced inverse regression belong to a suboptimal class of the ir family comparisons of these methods are reported through simulation studies the approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in slic

memoryless h sup spl infin controllers for state delayed systems in this note a memoryless h sup spl infin controller for the state delayed system is presented the controller is a delay independent stabilizer for the state delayed system which also reduces the h sup spl infin norm of the closed loop transfer function from the disturbance to the controlled output to a prescribed level the controller can be obtained by solving a modified riccati equation

spl hscr sub spl infin filtering for continuous time linear systems with delay the problem of spl hscr sub spl infin filtering for continuous time linear systems with time delayed measurement is investigated the authors develop a methodology for designing linear filters which ensure a prescribed bound on the spl lscr sub 2 induced gain from the noise signals to the estimation error filtering problems for time varying systems over a finite horizon as well as stationary infinite horizon filtering for time invariant systems are tackled in the finite horizon case our estimation procedure entails an overdesign that stems from the last d seconds of the time interval 0 t where d is the delay length this overdesign becomes smaller as t increases and it vanishes in the infinite horizon case

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

sparse generalized eigenvalue problem optimal statistical rates via truncated rayleigh flow the sparse generalized eigenvalue problem gep plays a pivotal role in a large family of high dimensional statistical models including sparse fisher s discriminant analysis canonical correlation analysis and sufficient dimension reduction the sparse gep involves solving a non convex optimization problem most existing methods and theory in the context of specific statistical models that are special cases of the sparse gep require restrictive structural assumptions on the input matrices we propose a two stage computational framework to solve the sparse gep at the first stage we solve a convex relaxation of the sparse gep taking the solution as an initial value we then exploit a non convex optimization perspective and propose the truncated rayleigh flow method which we call rifle to estimate the leading generalized eigenvector we show that rifle converges linearly to a solution with the optimal statistical rate of convergence theoretically our method significantly improves on the existing literature by eliminating structural assumptions on the input matrices to achieve this our analysis involves two key ingredients a new analysis of the gradient based method on non convex objective functions and a fine grained characterization of the evolution of sparsity patterns along the solution path thorough numerical studies are provided to validate the theoretical results

sparse sliced inverse regression sliced inverse regression sir is an innovative and effective method for dimension reduction and data visualization of high dimensional problems it replaces the original variables with low dimensional linear combinations of predictors without any loss of regression information and without the need to prespecify a model or an error distribution however it suffers from the fact that each sir component is a linear combination of all the original predictors thus it is often difficult to interpret the extracted components by representing sir as a regression type optimization problem we propose in this article a new method called sparse sir that combines the shrinkage idea of the lasso with sir to produce both accurate and sparse solutions the efficacy of the proposed method is verified by simulation and a real data example is given

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

linear quadratic regulators with eigenvalue placement in a vertical strip a computational method is presented for finding the linear quadratic regulator such that the optimal closed loop system has eigenvalues lying within a vertical strip in the complex plane the proposed method is suitable for the two stage optimal design of two time scale systems

coordinate independent sparse sufficient dimension reduction and variable selection sufficient dimension reduction sdr in regression which reduces the dimension by replacing original predictors with a minimal set of their linear combinations without loss of information is very helpful when the number of predictors is large the standard sdr methods suffer because the estimated linear combinations usually consist of all original predictors making it difficult to interpret in this paper we propose a unified method coordinate independent sparse estimation cise that can simultaneously achieve sparse sufficient dimension reduction and screen out irrelevant and redundant variables efficiently cise is subspace oriented in the sense that it incorporates a coordinate independent penalty term with a broad series of model based and model free sdr approaches this results in a grassmann manifold optimization problem and a fast algorithm is suggested under mild conditions based on manifold theories and techniques it can be shown that cise would perform asymptotically as well as if the true irrelevant predictors were known which is referred to as the oracle property simulation studies and a real data example demonstrate the effectiveness and efficiency of the proposed approach

a convex formulation for high dimensional sparse sliced inverse regression summarysliced inverse regression is a popular tool for sufficient dimension reduction which replaces covariates with a minimal set of their linear combinations without loss of information on the conditional distribution of the response given the covariates the estimated linear combinations include all covariates making results difficult to interpret and perhaps unnecessarily variable particularly when the number of covariates is large in this paper we propose a convex formulation for fitting sparse sliced inverse regression in high dimensions our proposal estimates the subspace of the linear combinations of the covariates directly and performs variable selection simultaneously we solve the resulting convex optimization problem via the linearized alternating direction methods of multiplier algorithm and establish an upper bound on the subspace distance between the estimated and the true subspaces through numerical studies we show that our proposal is able to identify the correct covariates in the high dimensional setting

a semiparametric approach to dimension reduction we provide a novel and completely different approach to dimension reduction problems from the existing literature we cast the dimension reduction problem in a semiparametric estimation framework and derive estimating equations viewing this problem from the new angle allows us to derive a rich class of estimators and obtain the classical dimension reduction techniques as special cases in this class the semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact the common assumption of linearity and or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression the semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example this article has online supplementary material

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

correlation pursuit forward stepwise variable selection for index models summary a stepwise procedure correlation pursuit cop is developed for variable selection under the sufficient dimension reduction framework in which the response variable y is influenced by the predictors through an unknown function of a few linear combinations of them unlike linear stepwise regression cop does not impose a special form of relationship such as linear between the response variable and the predictor variables the cop procedure selects variables that attain the maximum correlation between the transformed response and the linear combination of the variables various asymptotic properties of the cop procedure are established and in particular its variable selection performance under a diverging number of predictors and sample size is investigated the excellent empirical performance of the cop procedure in comparison with existing methods is demonstrated by both extensive simulation studies and a real example in functional genomics