from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

a behavioral characterization of the positive real and bounded real characteristic values in balancing in passivity preserving and bounded realness preserving model reduction by balanced truncation an important role is played by the so called positive real pr and bounded real br characteristic values both for the positive real as well as the bounded real case these values are defined in terms of the extremal solutions of the algebraic riccati associated with the system more precisely as the square roots of the eigenvalues of the product matrix obtained by multiplying the smallest solution with the inverse of the largest solution of the riccati equation in this paper we will establish a representation free characterization of these values in terms of the behavior of the system we will consider positive realness and bounded realness as special cases of half line dissipativity of the behavior we will then show that both for the pr and the br case the characteristic values coincide with the singular values of the linear operator that assigns to each past trajectory in the input output behavior its unique maximal supply extracting future continuation we will explain that the term singular values should be interpreted here in a generalized sense since in our setup the future behavior is only an indefinite inner product space c 2010 elsevier b v all rights reserved

system structure and singular control the general continuous time linear quadratic control problem is considered it is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem

in praise of simplicity not mathematistry ten simple powerful ideas for the statistical scientist ronald fisher was by all accounts a first rate mathematician but he saw himself as a scientist not a mathematician and he railed against what george box called in his fisher lecture mathematistry mathematics is the indispensable foundation of statistics but for me the real excitement and value of our subject lies in its application to other disciplines we should not view statistics as another branch of mathematics and favor mathematical complexity over clarifying formulating and solving real world problems valuing simplicity i describe 10 simple and powerful ideas that have influenced my thinking about statistics in my areas of research interest missing data causal inference survey sampling and statistical modeling in general the overarching theme is that statistics is a missing data problem and the goal is to predict unknowns with appropriate measures of uncertainty

input output structure of linear differential algebraic systems the problem of describing the input output structure of general linear differential algebraic systems is addressed explicit formulas are given in terms of the original parameters for systems with an arbitrary amount of redundancy these formulas allow one to establish whether the system determines an input output relation at all and if so they describe the rank of the transfer matrix and its pole zero structure at infinity the formulas may be seen as generalizations of a number of classical results on the input output structure of standard state space systems and descriptor systems satisfying certain constraints for the derivation the pencil representation rather than the descriptor representation is used

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

transformations of linear systems under external equivalence abstract we consider systems of linear differential and algebraic equations in which some of the variables are distinguished as external variables two systems are called equivalent if the set of solutions for the external variables is the same for both systems we give an operational form for this definition of equivalence i e we describe a set of system transformations having the property that two systems are equivalent if and only if they can be taken into each other by transformations from that set next an algorithm is described to transform a given system in general form to a system in minimal state space form this algorithm differs from existing methods in that it first takes the equations to first order form so that each subsequent step can be formulated and interpreted in state space terms we also compute the structure indices in terms of a state space description in nonminimal form and use this to prove the minimality of the end result of the algorithm finally an application is shown to the problem of ill posedness of feedback connections

bounded real and positive real balanced truncation using σ normalised coprime factors in this article we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix by meyer 1990 3 to balanced truncation with preservation of half line dissipativity special cases are preservation of positive realness and bounded realness we consider a half line dissipative input output system with quadratic supply rate given by a nonsingular symmetric matrix sigma with the property that its positive signature is equal to the number of input components of the system the transfer matrix of such system allows a sigma normalised right coprime factorisation we associate with such factorisation two lyapunov equations one of which is nonstandard involving the indefinite matrix sigma balancing will be based on making the unique solutions of these two lyapunov equations equal and diagonal the diagonal elements will be called the hankel sigma singular values because their squares are the nonzero eigenvalues of the composition of the graph hankel operator multiplication by sigma and the adjoint graph hankel operator this method of balanced truncation will be shown to preserve stability minimality and half line dissipativeness we will characterize the classical positive real and bounded real characteristic values in terms of the new hankel sigma singular values finally we will derive one step error bounds for the special case of balanced truncation of bounded real systems c 2009 elsevier b v all rights reserved

kolmogorov equations in infinite dimensions well posedness and regularity of solutions with applications to stochastic generalized burgers equations we develop a new method to uniquely solve a large class of heat equations so called kolmogorov equations in infinitely many variables the equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper lyapunov type functions this way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally lipschitz drifts apart from general analytic interest the main motivation is to apply this to uniquely solve martingale problems in the sense of stroock varadhan given by stochastic partial differential equations from hydrodynamics such as the stochastic navier stokes equations in this paper this is done in the case of the stochastic generalized burgers equation uniqueness is shown in the sense of markov flows

a bayesian justification of cox s partial likelihood summary in this paper we establish both naive and formal bayesian justifications of cox s 1975 partial likelihood and its various modifications we extend the original work of kalbfieisch 1978 who showed that the partial likelihood is a limiting marginal posterior under noninformative priors for baseline hazards we extend the result to scenarios with timedependent covariates and time varying regression parameters we establish results for continuous time as well as grouped survival data in addition we present a bayesian justification of a modified partial likelihood for handling ties we also present tools for simplification of the gibbs sampling algorithm for implementing partial likelihood based bayesian inference in various practical applications

reed solomon list decoding from a system theoretic perspective in this paper the sudan guruswami approach to list decoding of reed solomon rs codes is cast in a system theoretic framework with the data a set of trajectories or time series is associated which is then modeled as a so called behavior in this way a connection is made with the behavioral approach to system theory it is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the sudan guruswami approach the concept of weighted row reduced is introduced and used to achieve minimality two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given

extensions of the penalized spline of propensity prediction method of imputation little and an 2004 statistica sinica 14 949 968 proposed a penalized spline of propensity prediction pspp method of imputation of missing values that yields robust model based inference under the missing at random assumption the propensity score for a missing variable is estimated and a regression model is fitted that includes the spline of the estimated logit propensity score as a covariate the predicted unconditional mean of the missing variable has a double robustness dr property under misspecification of the imputation model we show that a simplified version of pspp which does not center other regressors prior to including them in the prediction model also has the dr property we also propose two extensions of pspp namely stratified pspp and bivariate pspp that extend the dr property to inferences about conditional means these extended pspp methods are compared with the pspp method and simple alternatives in a simulation study and applied to an online weight loss study conducted by kaiser permanente

to model or not to model competing modes of inference for finite population sampling finite population sampling is perhaps the only area of statistics in which the primary mode of analysis is based on the randomization distribution rather than on statistical models for the measured variables this article reviews the debate between design based and model based inference the basic features of the two approaches are illustrated using the case of inference about the mean from stratified random samples strengths and weakness of design based and model based inference for surveys are discussed it is suggested that models that take into account the sample design and make weak parametric assumptions can produce reliable and efficient inferences in surveys settings these ideas are illustrated using the problem of inference from unequal probability samples a model based regression analysis that leads to a combination of design based and model based weighting is described

a graph theoretic method for computing a formal representation of the largest controlled invariant subspace of a structured system the paper discusses a method to compute a kernel representation of the largest controlled invariant subspace in the kernel of the output mapping of a structured system the graph of the structured system is used to make the relevant equations and then a formula manipulation package can be used to compute a condensed kernel representation or even an image representation of the largest controlled invariant subspace

a complement about the structure of linear systems the aim of this paper is to establish new relationships between the steps of the well known basic algorithms i s a and c i s a it is suggested that this new structural information will provide a better understanding of the structural modifications and more particularly the increase of the infinite zero structure due to the use of non invertible input transformations

closed form reml estimators and sample size determination for mixed effects models for repeated measures under monotone missingness we derive the closed form restricted maximum likelihood estimator and kenward roger s variance estimator for fixed effects in the mixed effects model for repeated measures mmrm when the missing data pattern is monotone as an important application of the analytic result we present the formula for calculating the power of treatment comparison using the wald t test with the kenward roger adjusted variance estimate in mmrm it allows adjustment for baseline covariates without the need to specify the covariate distribution in randomized trials a simple two step procedure is proposed to determine the sample size needed to achieve the targeted power the proposed method performs well for both normal and moderately non normal data even in small samples n 20 in simulations an antidepressant trial is analyzed for illustrative purposes copyright 2017 john wiley sons ltd

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

on behavior homomorphisms and system equivalence in this paper we introduce the concept of a behavior homomorphisms and isomorphisms and use it to present a uni ed approach to the study of equivalence of di erent behavior representations concentrating on a special representation we call normalized arma narma representation c 2001 elsevier science b v all rights reserved

semiparametric regression for repeated outcomes with nonignorable nonresponse abstract we consider inference about the parameter indexing the conditional mean of a vector of correlated outcomes given a vector of explanatory variables when some of the outcomes are missing in a subsample of the study and the probability of response depends on both observed and unobserved data values that is nonresponse is nonignorable we propose a class of augmented inverse probability of response weighted estimators that are consistent and asymptotically normal can for estimating when the response probabilities can be parametrically modeled and a can estimator exists the proposed estimators do not require full specification of a parametric likelihood and their computation does not require numerical integration our estimators can be viewed as an extension of generalized estimating equation estimators that allows for nonignorable nonresponse we show that our class essentially consists of all can estimators of we also show that the asymptotic variance of the optimal estimator in our

converting high order linear pdes to first order abstract a straightforward way requiring no calculations is offered to convert a system of high order constant coefficient linear pdes into a system of first order pdes

equivalence of rational representations of behaviors this article deals with the equivalence of representations of behaviors of linear differential systems in general the behavior of a given linear differential system has many different representations in this paper we restrict ourselves to kernel and image representations two kernel representations are called equivalent if they represent one and the same behavior for kernel representations defined by polynomial matrices necessary and sufficient conditions for equivalence are well known in this paper we deal with the equivalence of rational representations i e kernel and image representations that are defined in terms of rational matrices as the first main result of this paper we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices we will also establish conditions for the equivalence of rational image representations finally we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation

driving variable realizations and the non negative realization problem for controllable behaviours non negative linear systems traditionally investigated within the state space framework have been recently analysed within the behavioural setting in a couple of recent papers j w nieuwenhuis when to call a linear system nonnegative linear algebra its appl 281 1998 pp 43 58 m e valcher nonnegative linear systems in the behavioural approach the autonomous case linear algebra and its appl 2000 319 pp 147 162 several definitions and results about non negative behaviours and in particular about non negative autonomous behaviours have been derived moreover the non negative realization problem for autonomous behaviours has been fully explored in m e valcher non negative realization of autonomous systems in the behavioral approach siam j on control and opt 2001 40 pp 540 556 thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non negative realization here we focus our attention on the non negative re

smooth impulsive linear systems axiomatic description abstract paper concerns with linear systems having arbitrary singularity at infinity these systems deserve to study since they are limits of familiar classical linear systems an axiomatic description of such systems is obtained

geometric minimization under external equivalence for implicit descriptions we consider minimality under external equivalence for linear time invariant implicit e a b c descriptions of the type ex t ax t bu t y t cx t necessary and sufficient conditions for external minimality have been previously given from an algebraic point of view by kuijper we propose here alternative geometric characterizations as well as a geometric reduction procedure that extracts from a given e a b c description a minimal model of the same type we also geometrically characterize in terms of the geometry of e a b c the corresponding minimal dimensions for the state space and for the state equation space

causal inference using potential outcomes design modeling decisions causal effects are defined as comparisons of potential outcomes under different treatments on a common set of units observed values of the potential outcomes are revealed by the assignment mechanism a probabilistic model for the treatment each unit receives as a function of covariates and potential outcomes fisher made tremendous contributions to causal inference through his work on the design of randomized experiments but the potential outcomes perspective applies to other complex experiments and nonrandomized studies as well as noted by kempthorne in his 1976 discussion of savage s fisher lecture fisher never bridged his work on experimental design and his work on parametric modeling a bridge that appears nearly automatic with an appropriate view of the potential outcomes framework where the potential outcomes and covariates are given a bayesian distribution to complete the model specification also this framework crisply separates scientific inference for causal effects and decisions based on s

structural matrix minimization algorithm for implicit descriptions in this paper we propose a matrix minimization algorithm for linear time invariant implicit e a b c descriptions of the type ex t ax t bu t y t cx t given the initial matrices e a b c we obtain an externally equivalent minimal realization em am bm cm after applying a matricial procedure based on maximal row and column compressions

a cell structure for the set of autoregressive systems the set of autoregressive systems generalizes the set of transfer functionsin a natural way we describe a topology for the set of all autoregressive systems of fixed size and bounded mcmillan degree we show that this topological space has the structure of a finite cw complex

a class of pattern mixture models for normal incomplete data summary likelihood based methods are developed for analyzing a random sample on two continuous variables when values of one of the variables are missing normal maximum likelihood estimates when values are missing completely at random were derived by anderson 1957 they are also maximum likelihood providing the missing data mechanism is ignorable in rubin s 1976 sense that the mechanism depends only on observed data a new class of pattern mixture models little 1993 is described for the situation where missingness is assumed to depend on an arbitrary unspecified function of a linear combination of the two variables maximum likelihood for models in this class is straightforward and yields the estimates of anderson 1957 when missingness depends solely on the completely observed variable and the estimates of brown 1990 when missingness depends solely on the incompletely observed variable another choice of linear combination yields estimates from complete case analysis large sample and bayesian methods are described for this model the data do not supply information about the ratio of the coefficients of the linear combination that controls missingness if this ratio is not welldetermined based on prior knowledge a prior distribution can be specified and bayesian inference is then readily accomplished alternatively sensitivity of inferences can be displayed for a variety of choices of the ratio

bayesian information criterion for censored survival models we investigate the bayesian information criterion bic for variable selection in models for censored survival data kass and wasserman 1995 journal of the american statistical association 90 928 934 showed that bic provides a close approximation to the bayes factor when a unit information prior on the parameter space is used we propose a revision of the penalty term in bic so that it is defined in terms of the number of uncensored events instead of the number of observations for a simple censored data model this revision results in a better approximation to the exact bayes factor based on a conjugate unit information prior in the cox proportional hazards regression model we propose defining bic in terms of the maximized partial likelihood using the number of deaths rather than the number of individuals in the bic penalty term corresponds to a more realistic prior on the parameter space and is shown to improve predictive performance for assessing stroke risk in the cardiovascular health study

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

when are linear differentiation invariant spaces differential it is shown that a linear differentiation invariant subspace of a c trajectory space is differential i e can be represented as the kernel of a linear constant coefficient differential operator if and only if its mcmillan degree is finite

impulsive smooth behavior in multimode systems part ii minimality and equivalence this is the second part of a two part study of linear multimode systems in the first part it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant and both first order and polynomial representations were introduced that satisfy this requirement here we determine the conditions under which first order representations are minimal we also show how two minimal representations of the same behavior are related this leads in particular to an appropriate state space isomorphism theorem the minimality conditions are given a dynamic interpretation

statistics and causal inference abstract problems involving causal inference have dogged at the heels of statistics since its earliest days correlation does not imply causation and yet causal conclusions drawn from a carefully designed experiment are often valid what can a statistical model say about causation this question is addressed by using a particular model for causal inference holland and rubin 1983 rubin 1974 to critique the discussions of other writers on causation and causal inference these include selected philosophers medical researchers statisticians econometricians and proponents of causal modeling

doubly robust estimation in missing data and causal inference models the goal of this article is to construct doubly robust dr estimators in ignorable missing data and causal inference models in a missing data model an estimator is dr if it remains consistent when either but not necessarily both a model for the missingness mechanism or a model for the distribution of the complete data is correctly specified because with observational data one can never be sure that either a missingness model or a complete data model is correct perhaps the best that can be hoped for is to find a dr estimator dr estimators in contrast to standard likelihood based or nonaugmented inverse probability weighted estimators give the analyst two chances instead of only one to make a valid inference in a causal inference model an estimator is dr if it remains consistent when either a model for the treatment assignment mechanism or a model for the distribution of the counterfactual data is correctly specified because with observational data one can never be sure that a model for the treatment assignment mechanism or a model for the counterfactual data is correct inference based on dr estimators should improve upon previous approaches indeed we present the results of simulation studies which demonstrate that the finite sample performance of dr estimators is as impressive as theory would predict the proposed method is applied to a cardiovascular clinical trial

behaviors and symbols of rational matrices abstract recently we have defined rational differential equations and derived conditions when two such equations have the same solution set in this article we study the same question but present a different approach based on the new notion of symbols also we try to describe the relationship with the work behaviors defined by rational functions of willems and yamamoto

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score