discrete time lq optimal control problems for infinite markov jump parameter systems optimal control problems for discrete time linear systems subject to markovian jumps in the parameters are considered for the case in which the markov chain takes values in a countably infinite set two situations are considered the noiseless case and the case in which an additive noise is appended to the model the solution for these problems relies in part on the study of a countably infinite set of coupled algebraic riccati equations icare conditions for existence and uniqueness of a positive semidefinite solution to the icare are obtained via the extended concepts of stochastic stabilizability ss and stochastic detectability sd which turn out to be equivalent to the spectral radius of certain infinite dimensional linear operators in a banach space being less than one for the long run average cost ss and sd guarantee existence and uniqueness of a stationary measure and consequently existence of an optimal stationary control policy furthermore an extension of a lyapunov equation result is derived for the countably infinite markov state space case

sparse additive models summary we present a new class of methods for high dimensional non parametric regression and classification called sparse additive models our methods combine ideas from sparse linear modelling and additive non parametric regression we derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size sparse additive models are essentially a functional version of the grouped lasso of yuan and lin they are also closely related to the cosso model of lin and zhang but decouple smoothing and sparsity enabling the use of arbitrary non parametric smoothers we give an analysis of the theoretical properties of sparse additive models and present empirical results on synthetic and real data showing that they can be effective in fitting sparse non parametric models in high dimensional data

high dimensional semiparametric scale invariant principal component analysis we propose a new high dimensional semiparametric principal component analysis pca method named copula component analysis coca the semiparametric model assumes that after unspecified marginally monotone transformations the distributions are multivariate gaussian coca improves upon pca and sparse pca in three aspects i it is robust to modeling assumptions ii it is robust to outliers and data contamination iii it is scale invariant and yields more interpretable results we prove that the coca estimators obtain fast estimation rates and are feature selection consistent when the dimension is nearly exponentially large relative to the sample size careful experiments confirm that coca outperforms sparse pca on both synthetic and real world data sets

forest density estimation we study graph estimation and density estimation in high dimensions using a family of density estimators based on forest structured undirected graphical models for density estimation we do not assume the true distribution corresponds to a forest rather we form kernel density estimates of the bivariate and univariate marginals and apply kruskal s algorithm to estimate the optimal forest on held out data we prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest for graph estimation we consider the problem of estimating forests with restricted tree sizes we prove that finding a maximum weight spanning forest with restricted tree size is np hard and develop an approximation algorithm for this problem viewing the tree size as a complexity parameter we then select a forest using data splitting and prove bounds on excess risk and structure selection consistency of the procedure experiments with simulated data and microarray data indicate that the methods are a practical alternative to gaussian graphical models

control of linear switched systems with receding horizon modal information we provide an exact solution to two performance problems one of disturbance attenuation and one of windowed variance minimization subject to exponential stability considered are switched systems whose parameters come from a finite set and switch according to a language such as that specified by an automaton the controllers are path dependent having finite memory of past plant parameters and finite foreknowledge of future parameters exact convex synthesis conditions for each performance problem are expressed in terms of nested linear matrix inequalities the resulting semidefinite programming problem may be solved offline to arrive at a suitable controller a notion of path by path performance is introduced for each performance problem leading to improved system performance non regular switching languages are considered and the results are extended to these languages two simple physically motivated examples are given to demonstrate the application of these results

a convex approach to the mixed cal h _ 2 cal h _ infty control problem for discrete time uncertain systems this paper considers cal h 2 cal h infty control problems involving discrete time uncertain linear systems the uncertain domain is supposed to be convex bounded which naturally covers as a particular case the important class of interval matrices the cal h infty guaranteed cost control problem solved for this class of uncertain systems under no matching conditions can be stated as follows determine a state feedback gain if one exists such that the hi norm of a given transfer function remains bounded by a prespecified level for all possible models in the same context problems on the determination of the smallest cal h infty upper bound and the minimization of an cal h 2 cost upper bound subject to cal h infty constraints are also addressed the results follow from the fact that those problems are convex in the particular parametric space under consideration some examples illustrate the theory

high dimensional semiparametric bigraphical models in multivariate analysis a gaussian bigraphical model is commonly used for modelling matrix valued data in this paper we propose a semiparametric extension of the gaussian bigraphical model called the nonparanormal bigraphical model a projected nonparametric rank based regularization approach is employed to estimate sparse precision matrices and produce graphs under a penalized likelihood framework theoretically our semiparametric procedure achieves the parametric rates of convergence for both matrix estimation and graph recovery empirically our approach outperforms the parametric gaussian model for non gaussian data and is competitive with its parametric counterpart for gaussian data extensions to the categorical bigraphical model and the missing data problem are discussed copyright 2013 oxford university press

robust stability and stabilization of uncertain discrete time markovian jump linear systems this note deals with robust stability and control of uncertain discrete time linear systems with markovian jumping parameters systems with polytopic type parameter uncertainty in either the state space model matrices or in the transition probability matrix of the markov process are considered this note develops methods of robust stability analysis and robust stabilization in the mean square sense which are dependent on the system uncertainty the design of both mode dependent and mode independent control laws is addressed the proposed methods are given in terms of linear matrix inequalities numerical examples are provided to demonstrate the effectiveness of the derived results

joint estimation of multiple graphical models from high dimensional time series in this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions we assume that the data are collected from n subjects each of which consists of t possibly dependent observations the graphical models of subjects vary but are assumed to change smoothly corresponding to a measure of closeness between subjects we propose a kernel based method for jointly estimating all graphical models theoretically under a double asymptotic framework where both t n and the dimension d can increase we provide the explicit rate of convergence in parameter estimation it characterizes the strength one can borrow across different individuals and the impact of data dependence on parameter estimation empirically experiments on both synthetic and real resting state functional magnetic resonance imaging rs fmri data illustrate the effectiveness of the proposed method

brief robust one step receding horizon control of discrete time markovian jump uncertain systems this paper proposes a receding horizon control scheme for a set of uncertain discrete time linear systems with randomly jumping parameters described by a finite state markov process whose jumping transition probabilities are assumed to belong to some convex sets the control scheme for the underlying systems is based on the minimization of the worst case one step finite horizon cost with a finite terminal weighting matrix at each time instant this robust receding horizon control scheme has a more general structure than the existing robust receding horizon control for the underlying systems under the same design parameters the proposed controller is obtained using semidefinite programming

direct estimation of the area under the receiver operating characteristic curve in the presence of verification bias the area under a receiver operating characteristic roc curve auc is a commonly used index for summarizing the ability of a continuous diagnostic test to discriminate between healthy and diseased subjects if all subjects have their true disease status verified one can directly estimate the auc nonparametrically using the wilcoxon statistic in some studies verification of the true disease status is performed only for a subset of subjects possibly depending on the result of the diagnostic test and other characteristics of the subjects because estimators of the auc based only on verified subjects are typically biased it is common to estimate the auc from a bias corrected roc curve the variance of the estimator however does not have a closed form expression and thus resampling techniques are used to obtain an estimate in this paper we develop a new method for directly estimating the auc in the setting of verification bias based on u statistics and inverse probability weighting ipw closed form expressions for the estimator and its variance are derived we also show that the new estimator is equivalent to the empirical auc derived from the bias corrected roc curve arising from the ipw approach

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

high dimensional semiparametric gaussian copula graphical models ing the spearman s rho and kendall s tau we prove that the nonparanormal skeptic achieves the optimal parametric rates of convergence for both graph recovery and parameter estimation this result suggests that the nonparanormal graphical models can be used as a safe replacement of the popular gaussian graphical models even when the data are truly gaussian besides theoretical analysis we also conduct thorough numerical simulations to compare the graph recovery performance of dierent estimators under both ideal and noisy settings the proposed methods are then applied on a largescale genomic dataset to illustrate their empirical usefulness the r package huge implementing the proposed methods is available on the comprehensive r archive network http cran r project org

on control of discrete time state dependent jump linear systems with probabilistic constraints a receding horizon approach in this article we consider a receding horizon control of discrete time state dependent jump linear systems a particular kind of stochastic switching systems subject to possibly unbounded random disturbances and probabilistic state constraints due to the nature of the dynamical system and the constraints we consider a one step receding horizon using inverse cumulative distribution function we convert the probabilistic state constraints to deterministic constraints and obtain a tractable deterministic receding horizon control problem we consider the receding horizon control law to have a linear state feedback and an admissible offset term we ensure mean square boundedness of the state variable via solving linear matrix inequalities off line and solve the receding horizon control problem on line with control offset terms we illustrate the overall approach applied on a macroeconomic system

structural similarity and difference testing on multiple sparse gaussian graphical models we present a new framework on inferring structural similarities and differences among multiple high dimensional gaussian graphical models ggms corresponding to the same set of variables under distinct experimental conditions the new framework adopts the partial correlation coefficients to characterize the potential changes of dependency strengths between two variables a hierarchical method has been further developed to recover edges with different or similar dependency strengths across multiple ggms in particular we first construct two sample test statistics for testing the equality of partial correlation coefficients and conduct large scale multiple tests to estimate the substructure of differential dependencies after removing differential substructure from original ggms a follow up multiple testing procedure is used to detect the substructure of similar dependencies among ggms in each step false discovery rate is controlled asymptotically at a desired level power results are proved which demonstrate that our method is more powerful on finding common edges than the common approach that separately estimates ggms the performance of the proposed hierarchical method is illustrated on simulated datasets

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

brief constrained quadratic state feedback control of discrete time markovian jump linear systems in this paper we consider the quadratic optimal control problem of a discrete time markovian jump linear system subject to constraints on the state and control variables it is desired to find a state feedback controller which may also depend on the jump variable that minimizes a quadratic cost and satisfies some upper bounds on the norms of some random variables related to the state and control variables of the system the transition probability of the markov chain and initial condition of the system may belong to appropriate convex sets we obtain an approximation for the optimal solution of this problem in terms of linear matrices inequalities so that convex programming can be used for numerical calculations examples are presented to illustrate the usefulness of the developed results

quasi min max mpc algorithms for lpv systems in this paper a new model predictive controller mpc is developed for polytopic linear parameter varying lpv systems we adopt the paradigm used in gain scheduling and assume that the time varying parameters are measured on line but their future behavior is uncertain and contained in a given polytope at each sampling time optimal control action is computed by minimizing the upper bound on the quasi worst case value of an infinite horizon quadratic objective function subject to constraints on inputs and outputs the mpc algorithm is called quasi because the first stage cost can be computed without any uncertainty this allows the inclusion of the first move u k k separately from the rest of the control moves governed by a feedback law and is shown to reduce conservatism and improve feasibility characteristics with respect to input and output constraints proposed optimization problems are solved by semi definite programming involving linear matrix inequalities it is shown that closed loop stability is guaranteed by the feasibility of the linear matrix inequalities a numerical example demonstrates the unique features of the mpc design

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

discrete time markovian jump linear quadratic optimal control this paper is concerned with the optimal control of discrete time linear systems that possess randomly jumping parameters described by finite state markov processes for problems having quadratic costs and perfect observations the optimal control laws and expected costs to go can be precomputed from a set of coupled riccati like matrix difference equations necessary and sufficient conditions are derived for the existence of optimal constant control laws which stabilize the controlled system as the time horizon becomes infinite with finite optimal expected cost

the nonparanormal semiparametric estimation of high dimensional undirected graphs recent methods for estimating sparse undirected graphs for real valued data in high dimensional problems rely heavily on the assumption of normality we show how to use a semiparametric gaussian copula or nonparanormal for high dimensional inference just as additive models extend linear models by replacing linear functions with a set of one dimensional smooth functions the nonparanormal extends the normal by transforming the variables by smooth functions we derive a method for estimating the nonparanormal study the method s theoretical properties and show that it works well in many examples

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data