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

a sensitivity tradeoff for plants with time delay in this paper we show that the sensitivity function of a scalar feedback system must satisfy an integral constraint when the open loop transfer function is strictly proper and contains a time delay the integral constraint is identical to the classical bode sensitivity integral which holds provided the open loop gain has greater than a one pole rolloff the design implications of the two constraints are somewhat different however and this difference is discussed the relation between open loop phase and a conflict between bandwidth and sensitivity properties is explored

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

dimension reduction and semiparametric estimation of survival models in this paper we propose a new dimension reduction method by introducing a nominal regression model with the hazard function as the conditional mean which naturally retrieves information from complete data and censored data as well moreover without requiring the linearity condition the new method can estimate the entire central subspace consistently and exhaustively the method also provides an alternative approach for the analysis of censored data assuming neither the link function nor the distribution hence it exhibits superior robustness properties numerical studies show that the method can indeed be readily used to efficiently estimate survival models explore the data structures and identify important variables

comments on robust tracking control for rigid robotic manipulators in the original paper by m zhihong and m palaniswami ibid vol 39 p 154 9 1994 a robust tracking control using a nominal feedback controller and a variable structure compensator for a rigid robotic manipulator with uncertain dynamics was presented it is assumed that the model uncertainty is bounded the author claims that this assumption is not necessarily correct and that the results obtained are therefore untenable

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

stability theory for adaptive systems method of averaging and persistency of excitation a method of averaging is developed for the stability analysis of linear differential equations with small time varying coefficients which do not necessarily possess an average value the technique is then applied to determine the stability of a linear equation which arises in the study of adaptive systems where the adaptive parameters are slowly varying the stability conditions are stated in the frequency domain which shows the relation between persistent excitation and unmodeled dynamics

logarithmic integrals interpolation bounds and performance limitations in mimo feedback systems we study performance limitation issues found in linear multivariable feedback systems our main contributions include bode and poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions these results characterize and quantify explicitly how open loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance the role of time delay is also studied in this context most notably we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros and in particular on how pole and zero directions are aligned the latter is characterized by angles measuring the mutual orientation between zero and pole directions and it is shown to play a crucial role in multivariable system design

global control with application to bifurcating power systems a multilevel nonlinear control scheme that is capable of system control over wide ranges of operating conditions is proposed the first control level confines the system operation to some reference segments in its state space each reference segment is further subdivided using the bifurcation analysis for establishing the bounds of the state space of the system with different control requirements the principles of global control lead to a hierarchical control structure which combines regional linear or nonlinear controllers smoothly the scheme is illustrated for control of a power system known to have diverse dynamical behavior the control will accomplish desired steady state fixed point or small oscillation operation of the power system for a wide range of loading levels

non linear dynamics in adaptive control chaotic and periodic stabilization abstract in this paper the non linear features of adaptive control algorithms are highlighted using a simple linear discrete time example based on classical design it is demonstrated that in the presence of undermodelling errors non linear phenomena in the feedback gain such as limit cycles and even chaos arise despite these complicated dynamics robust stabilization of the plant can still occur for the class of undermodelling errors considered the set of plants stabilizable by this adaptive controller is completely characterized

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

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

graphical tools for quadratic discriminant analysis sufficient dimension reduction methods provide effective ways to visualize discriminant analysis problems for example cook and yin showed that the dimension reduction method of sliced average variance estimation save identifies variates that are equivalent to a quadratic discriminant analysis qda solution this article makes this connection explicit to motivate the use of save variates in exploratory graphics for discriminant analysis classification can then be based on the save variates using a suitable distance measure if the chosen measure is mahalanobis distance then classification is identical to qda using the original variables just as canonical variates provide a useful way to visualize linear discriminant analysis lda so do save variates help visualize qda this would appear to be particularly useful given the lack of graphical tools for qda in current software furthermore whereas lda and qda can be sensitive to nonnormality save is more robust

bifurcation in model reference adaptive control systems abstract we study the effect of unmodeled dynamics on the performance of a simple model reference adaptive control mechanism exact stability boundaries are computed and different routes to chaotic behavior and global instability are described typically we observed a region of period doubling and hopf bifurcations similar to those observed in the ziegler nichols approach to pid control design the analysis applies to higher order and certain nonlinear systems as well

generalization of integral constraints on sensitivity to time delay systems extends results on integral constraints to time delay systems key technical issues in this extension include properties of nonminimum phase zeros the convergence of blaschke products and the poisson integral the results have implications for scalar and multivariable control system design as well as in filtering problems

the lasso method for variable selection in the cox model summary i propose a new method for variable selection and shrinkage in cox s proportional hazards model my proposal minimizes the log partial likelihood subject to the sum of the absolute values of the parameters being bounded by a constant because of the nature of this constraint it shrinks coe cients and produces some coe cients that are exactly zero as a result it reduces the estimation variance while providing an interpretable final model the method is a variation of the lasso proposal of tibshirani designed for the linear regression context simulations indicate that the lasso can be more accurate than stepwise selection in this setting

model free variable selection computing power has encouraged the modelling of data sets of ever increasing size data mining applications in finance marketing and bioinformatics are obvious examples a limitation of nearly all existing variable selection methods is the need to specify the correct model before selection when the number of predictors is large model formulation and validation can be difficult or even infeasible on the basis of the theory of sufficient dimension reduction we propose a new class of model free variable selection approaches the methods proposed assume no model of any form require no nonparametric smoothing and allow for general predictor effects the efficacy of the methods proposed is demonstrated via simulation and an empirical example is given

on the interpretation of regression plots abstract a framework is developed for the interpretation of regression plots including plots of the response against selected covariates residual plots added variable plots and detrended added variable plots it is shown that many of the common interpretations associated with these plots can be misleading the framework also allows for the generalization of standard plots and the development of new plotting paradigms a paradigm for graphical exploration of regression problems is sketched

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

integral manifolds of slow adaptation conditions are given under which an integral manifold slow manifold exists for a nonlinear system representing a broad class of adaptive algorithms the parameter update equation restricted to the manifold is an exact description of the slow adaptation process which can be approximately analyzed by averaging stability properties of the slow motion in the manifold are extended off the manifold by a set of conditions under which the manifold is shown to be attractive this two time scale analysis provides a geometrical visualization of instability phenomena observed in adaptive systems and generalizes earlier local stability results

l 1 regularized least squares for support recovery of high dimensional single index models with gaussian designs it is known that for a certain class of single index models sims y f xp 1t 0 e support recovery is impossible when x n 0 ip p and a model complexity adjusted sample size is below a critical threshold recently optimal algorithms based on sliced inverse regression sir were suggested these algorithms work provably under the assumption that the design x comes from an i i d gaussian distribution in the present paper we analyze algorithms based on covariance screening and least squares with l1 penalization i e lasso and demonstrate that they can also enjoy optimal up to a scalar rescaled sample size in terms of support recovery albeit under slightly different assumptions on f and e compared to the sir based algorithms furthermore we show more generally that lasso succeeds in recovering the signed support of 0 if x n 0 and the covariance satisfies the irrepresentable condition our work extends existing results on the support recovery of lasso for the linear model to a more general class of sims

crossover frequency limitations in mimo nonminimum phase feedback systems this paper investigates limitations and design tradeoffs of the closed loop sensitivity performance of linear time invariant nonminimum phase uncertain multiple input multiple output plants with i inputs and m outputs where m spl les l it is shown that if rows i sub 1 i sub k of the plant transfer function form a k spl times l nonminimum phase transfer matrix and if the design is such that the sensitivity gain of k 1 rows among the rows i sub 1 i sub k of the closed loop transfer function is low then by necessity the sensitivity gain of the remaining row is high this sensitivity constraint is quantified with the help of the crossover frequency restriction of a specially constructed single input single output transfer function that includes the right half plane zeros and poles of the k spl times l transfer matrix

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

semiparametric receiver operating characteristic analysis to evaluate biomarkers for disease the receiver operating characteristic roc curve is a popular method for characterizing the accuracy of diagnostic tests when test results are not binary various methodologies for estimating and comparing roc curves have been developed one approach due to pepe uses a parametric regression model rocx u g h0 u 0x with the baseline function h0 u specified up to a finite dimensional parameter in this article we extend the regression models by allowing arbitrary nonparametric baseline functions we also provide asymptotic distribution theory and procedures for making statistical inference we illustrate our approach with dataset from a prostate cancer biomarker study simulation studies suggest that the extra flexibility inherent in the semiparametric method is gained with little loss in statistical efficiency

a simplified approach to analyze complementary sensitivity tradeoffs in continuous time and discrete time systems continuous time and discrete time complementary sensitivity bode integrals csbis are investigated via a simplified approach in this note for continuous time feedback systems with unbounded frequency domain the csbi weighted by 1 omega 2 is considered where this simplified method reveals a more explicit relationship between the value of csbi and the structure of the open loop transfer function with a minor modification of this method the csbi of discrete time system is derived and illustrative examples are provided compared with the existing results on csbi neither the cauchy integral theorem nor the poisson integral formula is used throughout the analysis and the analytic constraint on the integrand is removed

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

small amplitude chaos and ergodicity in adaptive control abstract the theory for continuous iterated maps on the real line is used to study the dynamics of a continuous family of one parameter adaptive systems we establish sufficient and necessary conditions for uniform asymptotic stability and ultimate boundedness exact bifurcation boundaries and sufficient conditions for the existence of chaotic attractors and ergodicity of the estimated parameter it is argued that the chaotic dynamics are structurally stable that they persist in higher dimensional adaptive control systems and that they do not necessarily lead to poor performance when the controller is modified to deal with the infinite drift and small divisor problems a novel modification involving a sign change in the adaptive law is proposed

local bifurcations and feasibility regions in differential algebraic systems the dynamics of a large class of physical systems such as the general power system can be represented by parameter dependent differential algebraic models of the form x spl dot f and 0 g typically such constrained models have singularities this paper analyzes the generic local bifurcations including those which are directly related to the singularity the notion of a feasibility region is introduced and analyzed it consists of all equilibrium states that can be reached quasistatically from the current operating point without loss of local stability it is shown that generically loss of stability at the feasibility boundary is caused by one of three different local bifurcations namely the saddle node and hopf bifurcations and a new bifurcation called the singularity induced bifurcation which is analyzed precisely here for the first time the latter results when an equilibrium point is at the singular surface under certain transversality conditions the change in the eigenstructure of the system jacobian at the equilibrium is established and the local dynamical structure of the trajectories near this bifurcation point is analyzed

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

local feedback stabilization and bifurcation control ii stationary bifurcation abstract local feedback stabilization and bifurcation control of nonlinear systems are studied for the case in which the critical linearized system possesses a simple zero eigenvalue sufficient conditions are obtained for local stabilizability of the equilibrium point at criticality and for local stabilizability of bifurcated equilibria these conditions involve assumptions on the controllability of the critical mode for the linearized system explicit stabilizing feedback controls are constructed the projection method of analysis of stationary bifurcations is employed this work complements an earlier study by the same authors systems control lett 7 1986 11 17 of stabilization and bifurcation control in the hopf bifurcation case of two pure imaginary eigenvalues of the linearized system at criticality

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

dimension reduction for censored regression data without parametric assumptions high dimensional regression analysis is already complex this is made even harder when data are subject to censoring in this article we seek ways of reducing the dimensionality of the regressor before applying nonparametric smoothing techniques if the censoring time is independent of the lifetime then the method of sliced inverse regression can be applied directly otherwise modification is needed to adjust for the censoring bias a key identity leading to the bias correction is derived and the root n consistency of the modified estimate is established patterns of censoring can also be studied under a similar dimension reduction framework some simulation results and an application to a real data set are reported 1 introduction survival data are often subject to censoring when this occurs the incompleteness of the observed data may induce a substantial bias in the sample several approaches have been suggested to overcome the associated difficulties in regression including the accelerated failure time model censored linear regression the cox proportional hazard model and many others survival analysis becomes even more intricate when the dimension of the regressor increases to apply any of the aforementioned methods users are required to specify a functional form which relates the outcome variables to the input ones however in reality knowledge needed for an appropriate model specification is often inadequate as a matter of fact the acquisition of such information may well turn out to be one of the primary goals of the study itself under such circumstances it seems preferable to have exploratory tools that rely less on such model specification this is the issue to be addressed in this article the dimension reduction approach of li 1991 will be extended to settings which allow for censoring in the data we shall offer methods of finding low dimensional projections of the data for visually examining the censoring pattern we shall show how censored regression data can still be analyzed without assuming the functional form a priori

constraints on sensitivity characteristics in linear multivariable discrete time control systems abstract this paper is concerned with the limitations on the sensitivity characteristics for linear multivariable discrete time control systems some integral type constraints and the lower bounds of the weighted h norm imposed by the unstable poles the unstable zeros and their directions of the open loop system or the given plant are developed by a factorization approach these constraints and bounds which are tighter than those in the previous work are also characterized by the state space representations the descriptions are closely related to a special type of the algebraic riccati equation and the relation between the sampling period and the sensitivity performance is discussed for digital control systems the constraints on the more general function mixed sensitivity function are investigated in a similar way

disturbance generated bifurcations in a simple adaptive system simulation evidence abstract we present complete characterization based on computer simulation of the possible dynamics exhibited by an adaptive control system where the first order plant with a single unknown pole has constant disturbance at the input the adaptive system is taken from the current adaptive control systems literature here we show that the system in fact undergoes bifurcations and exhibits rich dynamics as the constant disturbance is varied the system undergoes saddle node bifurcation subcritical hopf bifurcation and a saddle connection bifurcation the last bifurcation means that the system acquires a homoclinic orbit for a specific disturbance while the first two bifurcations are local the presence of the homoclinic orbit has the potential when periodically disturbed to generate nonlocal complicated behavior this latter behavior is often referred to as horseshoe chaos

local feedback stabilization and bifurcation control i hopf bifurcation abstract local bifurcation control problems are defined and employed in the study of the local feedback stabilization problem for nonlinear systems in critical cases sufficient conditions are obtained for the local stabilizability of general nonlinear systems whose linearizations have a pair of simple nonzero imaginary eigenvalues the conditions show in particular that generically these nonlinear critical systems can be stabilized locally even if the critical modes are uncontrollable the analysis also yields a direct method for computing stabilizing feedback controls use is made of bifurcation formulae which require only a series expansion of the vector field

disturbance instabilities in an adaptive system two types of disturbance instability of simple adaptive schemes are analyzed a drift equation shows how stable equilibria are forced to infinity by infrequent but persistent switches of a constant disturbance the modification proposed by p ioannou counteracts these instabilities

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

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