a characterization of markov equivalence classes for acyclic digraphs undirected graphs and acyclic digraphs adgs as well as their mutual extension to chain graphs are widely used to describe dependencies among variables in multivariate distributions in particular the likelihood functions of adg models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building bayesian networks for expert systems whereas the undirected graph associated with a dependence model is uniquely determined there may however be many adgs that determine the same dependence markov model thus the family of all adgs with a given set of vertices is naturally partitioned into markov equivalence classes each class being associated with a unique statistical model statistical procedures such as model selection or model averaging that fail to take into account these equivalence classes may incur substantial computational or other inefficiencies here it is shown that each markov equivalence class is uniquely determined by a single chain graph the essential graph that is itself simultaneously markov equivalent to all adgs in the equivalence class essential graphs are characterized a polynomial time algorithm for their construction is given and their applications to model selection and other statistical questions are described

robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems we present a patch based equilibrated flux recovery procedure for the conforming finite element approximation to diffusion problems the recovered flux is computed as the solution to a local constraint free minimization problem on each patch the approach is valid for higher order conforming elements in both two and three dimensions the resulting estimator admits guaranteed reliability and the robust local efficiency is proved under the quasi monotonicity condition of the diffusion coefficient numerical experiments are given to confirm the theoretical results

propagation of probabilities means and variances in mixed graphical association models abstract a scheme is presented for modeling and local computation of exact probabilities means and variances for mixed qualitative and quantitative variables the models assume that the conditional distribution of the quantitative variables given the qualitative is multivariate gaussian the computational architecture is set up by forming a tree of belief universes and the calculations are then performed by local message passing between universes the asymmetry between the quantitative and qualitative variables sets some additional limitations for the specification and propagation structure approximate methods when these are not appropriately fulfilled are sketched it has earlier been shown how to exploit the local structure in the specification of a discrete probability model for fast and efficient computation thereby paving the way for exploiting probability based models as parts of realistic systems for planning and decision support the purpose of this article is to extend this computational s

adaptive finite element methods for elliptic equations with non smooth coefficients nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3 et sa discretisation par elements finis le but de cet article est de demontrer des estimations d erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients

distance between populations using mixed continuous and categorical variables summary a general form of the location model is considered for mixed continuous and categorical variables observed in a number of different populations and some special cases of practical interest are cited the distance between any two populations is derived for each of these models estimation of parameters in these distance measures is discussed and the methods are illustrated by application to previously published data

the tm algorithm for maximising a conditional likelihood function summary this paper describes an algorithm for maximising a conditional likelihood function when the corresponding unconditional likelihood function is more easily maximised the algorithm is similar to the em algorithm but different as the parameters rather than the data are augmented and the conditional rather than the marginal likelihood function is maximised in exponential families the algorithm takes a particular simple form which is computationally very close to the em algorithm the algorithm alternates between a t step which calculates a tilted version of the unconditional likelihood function and an m step which maximises it the algorithm applies to mixed graphical chain models lauritzen wermuth 1989 and their generalisations edwards 1990 and was developed with these in mind but it may have applications beyond these the algorithm has been implemented in the most recent version of the mim software edwards 2000 where it was named the me algorithm the name has been changed to avoid confusion with the algorithm described by marschner 2001

performance limitations of adaptive parameter estimation and system identification algorithms in the absence of excitation we address the problem of fundamental performance limitations in adaptive parameter estimation and system identification occurring in environments where perturbations are present but there is lack of sufficient excitation we construct a simple yet general bursting scenario to derive an analytical lower bound on the worst case peak steady state error for a wide class of parameter estimation and system identification algorithms our results show that in the absence of any input constraints arbitrarily small perturbations impose a serious performance limitation in the sense that the worst case performance deteriorates proportionally with the size of the parametric uncertainty set

chain graph models and their causal interpretations chain graphs are a natural generalization of directed acyclic graphs and undirected graphs however the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent there are many simple and apparently plausible but ultimately fallacious interpretations of chain graphs that are often invoked implicitly or explicitly these interpretations also lead to flawed methods for applying background knowledge to model selection we present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated from the equilibrium distributions of dynamic models with feed back these dynamic interpretations lead to a simple theory of intervention extending the theory developed for directed acyclic graphs finally we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feed back in econometrics

a robust direct adaptive controller this paper proposes a new direct adaptive control algorithm which is robust with respect to additive and multiplicative plant unmodeled dynamics the algorithm is designed based on the reduced order plant which is assumed to be minimum phase and of known order and relative degree but is analyzed with respect to the overall plant which due to the unmodeled dynamics may be nonminimum phase and of unknown order and relative degree it is shown that if the unmodeled dynamics are sufficiently small in the low frequency range then the algorithm guarantees boundedness of all signals in the adaptive loop and small residual tracking errors for any bounded initial conditions in the absence of unmodeled dynamics the residual tracking error is shown to be zero

a saddle point least squares approach for primal mixed formulations of second order pdes abstract we present a saddle point least squares spls method for discretizing second order elliptic problems written as primal mixed variational formulations a stability lbb condition and a data compatibility condition at the continuous level are automatically satisfied the proposed discretization method follows a general spls approach and has the advantage that a discrete inf sup condition is automatically satisfied for standard choices of the test and trial spaces for the proposed iterative processes a nodal basis for the trial space is not required efficient preconditioning techniques that involve inversion only on the test space can be considered stability and approximation properties for two choices of discrete spaces are investigated applications of the new approach include discretization of second order problems with highly oscillatory coefficient interface problems and higher order approximation of the flux for elliptic problems with smooth coefficients

a note on constant free a posteriori error estimates in this note we look at constant free a posteriori error estimates from a different perspective we show that they can be interpreted as an alternative way of expressing the residual of a finite element approximation and thus fit into the same framework as other a posteriori error estimates such as residual error indicators our approach also reveals that when applied to singularly perturbed reaction diffusion or convection diffusion problems constant free a posteriori error estimates will not be fully robust unless extra measures are taken

a new finite element gradient recovery method superconvergence property this is the first in a series of papers in which a new gradient recovery method is introduced and analyzed it is proved that the method is superconvergent for translation invariant finite element spaces of any order the method maintains the simplicity efficiency and superconvergence properties of the zienkiewicz zhu patch recovery method in addition for uniform triangular meshes the method is superconvergent for the linear element under the chevron pattern and ultraconvergent at element edge centers for the quadratic element under the regular pattern applications of this new gradient recovery technique will be discussed in forthcoming papers

flux recovery and a posteriori error estimators conforming elements for scalar elliptic equations in this paper we first study two flux recovery procedures for the conforming finite element approximation to general second order elliptic partial differential equations one is accurate in a weighted l 2 norm studied in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 for linear elements and the other is accurate in a weighted h mathrm div norm up to the accuracy of the current finite element approximation for the l 2 recovered flux we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual based estimator based on the h mathrm div recovered flux we introduce two a posteriori error estimators one estimator may be regarded as an extension of the recovery based estimator studied in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 to higher order conforming elements the global reliability and the local efficiency bounds for this estimator are established provided that the underlying problem is neither convection nor reaction dominant the other is proved to be exact locally and globally on any given mesh with no regularity assumptions with respect to a norm depending on the underlying problem numerical results on test problems for these estimators are also presented

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

superconvergent derivative recovery for lagrange triangular elements of degree p on unstructured grids in this paper we develop a postprocessing derivative recovery scheme for the finite element solution u h on general unstructured but shape regular triangulations in the case of continuous piecewise polynomials of degree p geq 1 by applying the global l 2 projection q h and a smoothing operator s h the recovered p th derivatives s h m q h partial p u h superconverge to the exact derivatives partial p u based on this technique we are able to derive a local error indicator depending only on the geometry of corresponding element and the p 1 st derivatives approximated by partial s h m q h partial p u h we provide several numerical examples illustrating the effectiveness of our schemes we also observe that higher order elements are likely to require more conservative refinement strategies to create meshes corresponding to optimal orders of convergence

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part ii higher order fem averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing in the second paper of our analysis of their reliability we consider conforming h fem of higher i e not of lowest order in two or three space dimensions in this paper reliablility is shown for conforming higher order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of local averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

markov properties of nonrecursive causal models this paper aims to solve an often noted incompatibility between graphical chain models which elucidate the conditional independence structure of a set of random variables and simultaneous equations systems which focus on direct linear interactions and correlations between random variables various authors have argued that the incompatibility arises mainly from the fact that in a simultaneous equations system e g a lisrel model reciprocal causality is possible whereas this is not so in the case of graphical chain models in this article it is shown that this view is not correct in fact the definition of the markov property embodied in a graph can be generalized to a wider class of graphs which includes certain nonrecursive graphs the resulting class of reciprocal graph probability models strictly includes the class of chain graph probability models the class of lattice conditional independence probability models is also strictly included it is shown that the resulting methodology is directly applicable to quite general simultaneous equations systems that are subject to mild restrictions only provided some adjustments are made general simultaneous equations systems can be handled as well in all cases consistency with the lisrel methodology is maintained

response models for mixed binary and quantitative variables summary a number of special representations are considered for the joint distribution of qualitative mostly binary and quantitative variables in addition to the conditional gaussian models and to conditional gaussian regression chain models some emphasis is placed on models derived from an underlying multivariate normal distribution and on models in which discrete probabilities are specified linearly in terms of unknown parameters the possibilities for choosing between the models empirically are examined as well as the testing of independence and conditional independence and the estimation of parameters often the testing of independence is exactly or nearly the same for a number of different models

marginalization and collapsibility in graphical interaction models the behaviour of a graphical interaction model under marginalization is discussed a graphical interaction model is called collapsible onto a set of variables if the class of marginal distributions is the same as that implied by the related subgraph the necessary and sufficient condition for collapsibility is found and it is shown that collapsibility is equivalent to a range of other important statistical properties of the model

on tuning leakage for performance robust adaptive control the behavior of a model reference adaptive control mrac system is examined the effect of the inclusion of leakage in the adaptive algorithm is studied and the leakage mechanism of the sigma modification and the e sub 1 modification are discussed a useful approximation is given for a particular mrac problem for the small step size average parameter estimate motion away from an arbitrary stabilizing adaptive controller parameterization arrow diagrams verify the conclusion that designer selected leakage constants must be well chosen in relation to the unknown plant and environment to avoid bursting by an adaptive controller in the absence of adequate excitation by the external forcing function s

a simple approach to reliable and robust a posteriori error estimation for singularly perturbed problems abstract a simple flux reconstruction for finite element solutions of reaction diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction diffusion problem the flux reconstruction is based on simple independent post processing operations over patches of elements in conjunction with standard raviart thomas vector fields we prove that the corresponding local error indicators are locally efficient and robust up to a nonstandard oscillation term with respect to any mesh size and any size of the reaction coefficient including the singularly perturbed limit

ideal parent structure learning for continuous variable bayesian networks bayesian networks in general and continuous variable networks in particular have become increasingly popular in recent years largely due to advances in methods that facilitate automatic learning from data yet despite these advances the key task of learning the structure of such models remains a computationally intensive procedure which limits most applications to parameter learning this problem is even more acute when learning networks in the presence of missing values or hidden variables a scenario that is part of many real life problems in this work we present a general method for speeding structure search for continuous variable networks with common parametric distributions we efficiently evaluate the approximate merit of candidate structure modifications and apply time consuming exact computations only to the most promising ones thereby achieving significant improvement in the running time of the search algorithm our method also naturally and efficiently facilitates the addition of useful new hidden variables into the network structure a task that is typically considered both conceptually difficult and computationally prohibitive we demonstrate our method on synthetic and real life data sets both for learning structure on fully and partially observable data and for introducing new hidden variables during structure search

robust error bounds for finite element approximation of reaction diffusion problems with non constant reaction coefficient in arbitrary space dimension abstract we present a fully computable a posteriori error estimator for piecewise linear finite element approximations of reaction diffusion problems with mixed boundary conditions and piecewise constant reaction coefficient formulated in arbitrary dimension the estimator provides a guaranteed upper bound on the energy norm of the error and it is robust for all values of the reaction coefficient including the singularly perturbed case the approach is based on robustly equilibrated boundary flux functions of ainsworth and oden 2000 and on subsequent robust and explicit flux reconstruction this paper simplifies and extends the applicability of the previous result of ainsworth and vejchodsk 2011 in three aspects i arbitrary dimension ii mixed boundary conditions and iii non constant reaction coefficient it is the first robust upper bound on the error with these properties an auxiliary result that is of independent interest is the derivation of new explicit constants for two types of trace inequalities on simplices

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

a hybrid a posteriori error estimator for conforming finite element approximations abstract this paper introduces a hybrid a posteriori error estimator for the conforming finite element method which may be regarded as a combination of the explicit residual and the improved zz error estimators with comparable cost the hybrid estimator is more accurate than the residual estimator it is shown that the hybrid estimator is reliable on all meshes unlike estimators of the zz type moreover the reliability constant is independent of the jump of the diffusion coefficients for elliptic interface problems under the monotonicity assumption of the coefficients finally numerical examples confirm the robustness of the estimator with respect to coefficient jumps and also better effectivity index compared to the residual estimator

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

robust equilibrated residual error estimator for diffusion problems conforming elements this paper analyzes an equilibrated residual a posteriori error estimator for the diffusion problem the estimator which is a modification of those in d braess and j schoberl math comput 77 2008 pp 651 672 r verfurth siam j numer anal 47 2009 pp 3180 3194 is based on the prager synge identity and on a local recovery of an equilibrated flux numerical results for an interface test problem show that the modification is necessary for the robustness of the estimator when the distribution of diffusion coefficients is local quasi monotone it is shown theoretically that the estimator is robust with respect to the size of jumps

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

recovery based error estimator for stabilized finite element methods for the stokes equation abstract a recovery based error estimator is proposed and analyzed for stabilized p 1 p 0 continuous linear velocity constant pressure finite element approximations to the stokes equation reliability and efficiency of the estimator are established for various stabilized methods for several test problems numerical results show that our estimator is more accurate than the classical residual error estimator

hierarchical interaction models lauritzen and wermuth have proposed a class of models for mixed qualitative and continuous data defined by two properties that the continuous variables are normally distributed given the qualitative variables and that a set of conditional independence relations hold between specified pairs of variables the present paper examines an extension to this class called hierarchical interaction models a compact form for model representation is described and an estimation algorithm is given some properties of the models a compact form for model representation is described and an estimation algorithm is given some properties of the models concerning marginalization and conditioning are examined the class includes and generalizes hierarchical log linear models standard fixed effect anova multivariate anova and multivariate regression models

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

gibbs sampling in bayesian networks research note abstract posterior probabilities in bayesian networks can be evaluated by stochastic simulation it is shown that the stochastic simulation can be viewed as a sampling from the gibbs distribution this view is useful in 1 making statements about convergence of the simulation and 2 finding the most likely instantiation of the bayesian network

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

asymptotically exact a posteriori error estimators part i grids with superconvergence in part i of this work we develop superconvergence estimates for piecewise linear finite element approximations on quasi uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms in particular we first show a superconvergence of the gradient of the finite element solution uh and to the gradient of the interpolant ui we then analyze a postprocessing gradient recovery scheme showing that q h nabla u h is a superconvergent approximation to nabla u here qh is the global l2 projection in part ii we analyze a superconvergent gradient recovery scheme for general unstructured shape regular triangulations this is the foundation for an a posteriori error estimate and local error indicators

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

graphical regression models for polytomous variables when modeling the relationship between two nominal categorical variables it is often desirable to include covariates to understand how individuals differ in their response behavior typically however not all the relevant covariates are available with the result that the measured variables cannot fully account for the associations between the nominal variables under the assumption that the observed and unobserved variables follow a homogeneous conditional gaussian distribution this paper proposesrc m regression models to decompose the residual associations between the polytomous variables based on goodman s 1979 1985 rc m association model a distinctive feature ofrc m regression models is that they facilitate the joint estimation of effects due to manifest and omitted continuous variables without requiring numerical integration therc m regression models are illustrated using data from the high school and beyond study tatsuoka lohnes 1988

log multiplicative association models as item response models log multiplicative association lma models which are special cases of log linear models have interpretations in terms of latent continuous variables two theoretical derivations of lma models based on item response theory irt arguments are presented first we show that anderson and colleagues anderson vermunt 2000 anderson bockenholt 2000 anderson 2002 who derived lma models from statistical graphical models made the equivalent assumptions as holland 1990 when deriving models for the manifest probabilities of response patterns based on an irt approach we also present a second derivation of lma models where item response functions are specified as functions of rest scores these various connections provide insights into the behavior of lma models as item response models and point out philosophical issues with the use of lma models as item response models we show that even for short tests lma and standard irt models yield very similar to nearly identical results when data arise from standard irt models log multiplicative association models can be used as item response models and do not require numerical integration for estimation

a new adaptive law for robust adaptation without persistent excitation a new adaptive law motivated by the work of loannou and kokotovic 1983 is proposed for the robust adaptive control of plants with unknown parameters in this adaptive law the output error plays a dual role in the adjustment of the control parameter vector the advantages of using the adaptive law over others proposed in the literature are discussed in the ideal case the adaptive system has bounded solutions in addition the origin of the error equations is exponentially stable when the reference input is persistently exciting and has a sufficiently large amplitude the adaptive system is also shown to be robust under bounded external disturbances finally it is shown that by suitably modifying the adaptive law the overall system can be made robust in the presence of a class of unmodeled dynamics of the plant simulation results are presented throughout the paper to complement the theoretical developments

regression models for a bivariate discrete and continuous outcome with clustering abstract developmental toxicity studies of laboratory animals play a crucial role in the testing and regulation of chemicals and pharmaceutical compounds exposure to developmental toxicants typically causes a variety of adverse effects such as fetal malformations and reduced fetal weight at term in this article we discuss regression methods for jointly analyzing bivariate discrete and continuous outcomes that are motivated by the statistical problems that arise in analyzing data from developmental toxicity studies we focus on marginal regression models that is models in which the marginal expectation of the bivariate response vector is related to a set of covariates by some known link functions in these models the regression parameters for the marginal expectation are of primary scientific interest whereas the association between the binary and continuous response is considered to be a nuisance characteristic of the data we describe a likelihood based approach based on the general location mode

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

joint response graphs and separation induced by triangular systems summary we consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions the independences are captured by missing edges in a directed graph a matrix form of such a graph called the generating edge matrix is triangular so the distributions that are generated over such graphs are called triangular systems we study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models i e for alternative recursive factorizations of the given density using joint conditional distributions for this we introduce families of linear triangular equations which do not require assumptions of distributional form the strength of the associations that are implied by such linear families for chain graph models is derived the edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix it is shown how induced independences and dependences can be studied by graphs by edge matrix calculations and via the properties of densities some ways of using the results are illustrated

graphical and recursive models for contingency tables summary we discuss two classes of models for contingency tables graphical and recursive models both of which arise from restrictions that are expressible as conditional independencies of variable pairs the first of these is a subclass of hierarchical log linear models each of its models can be represented by an undirected graph in the second class each model corresponds to a particular kind of a directed graph instead and can be characterized by a nontrivial factorization of the joint distribution in terms of response variables we derive decomposable or multiplicative models as the intersecting class this result has useful consequences for exploratory types of analysis as well as for the model interpretation we can give an aid for detecting well fitting decomposable models in a transformation of the observed contingency table and each decomposable model may be interpreted with the help of an undirected or directed graph

moderating effects of subgroups in linear models summary possibilities for moderating effects of a subgrouping variable on strength or direction of an association have been much discussed by social scientists but have not been given satisfactory statistical formulations the results concern directed measures of associations in linear models containing just three variables some key words analysis of covariance analysis of variance cg distribution conditional independence graphical chain model parallel regressions yule simpson paradox 1 introduction linear models are commonly used as a framework to estimate and test how a continuous response variable depends on potential influencing variables this paper is concerned with the situation in which two influences are random variables as well which may be discrete or continuous the continuous random variables are to stand for quantitative properties of observational units while the discrete variables either capture qualitative properties or they represent subgroups of the population to which the observational units belong to think of all variables as random variables is usually appropriate for data obtained from observational studies where one cannot control which levels or values of the influencing variables are to be observed the linear model permits an analysis conditional on fixed levels and values of the influences but frequently the association structure among the influences is in social science applications of interest itself and as it will turn out it is important for a correct interpretation of how the response depends on the influences awareness of such problems in interpretation is widespread among social scientists as is documented by extensive discussions of moderating effects see for example saunders 1956 zedeck 1971 and baron kenny 1986 there a variable is called a moderator if its presence changes the strength or direction of an association however methods recommended to social scientists for identifying moderating effects given in the above literature or by cohen cohen 1983 pp 310 4 have been shown to be seriously deficient wermuth 1988 thus there is need for clarification we treat the case of a potential discrete moderator variable and call its categories or levels the subgroups in the linear model a moderating effect is closely tied to the notion of consistent results on an association wermuth 1987 in the present context results are said to be weakly consistent if the associations within subgroups coincide in direction and strongly consistent if the associ ations within subgroups coincide in direction and strength this notion depends typically on the chosen measure of association in linear models we look at directed measures of

minimal sufficient causation and directed acyclic graphs notions of minimal sufficient causation are incorporated within the directed acyclic graph causal framework doing so allows for the graphical representation of sufficient causes and minimal sufficient causes on causal directed acyclic graphs while maintaining all of the properties of causal directed acyclic graphs this in turn provides a clear theoretical link between two major conceptualizations of causality one counterfactual based and the other based on a more mechanistic understanding of causation the theory developed can be used to draw conclusions about the sign of the conditional covariances among variables

symmetry and lattice conditional independence in a multivariate normal distribution a class of multivariate normal models with symmetry restrictions given by a finite group and conditional independence restrictions given by a finite distributive lattice is defined and studied the statistical properties of these models including maximum likelihood inference invariance and hypothesis testing are discussed

on chain graph models for description of conditional independence structures a chain graph cg is a graph admitting both directed and undirected edges with partially directed cycles forbidden it generalizes both the concept of undirected graph ug and the concept of directed acyclic graph dag a chain graph can be used to describe efficiently the conditional independence structure of a multidimensional discrete probability distribution in the form of a graphoid that is in the form of a list of statements x is independent of y given z obeying a set of five properties axioms an input list of independency statements for every cg is defined and it is shown that the classic moralization criterion for cgs embraces exactly the graphoid closure of the input list a new direct separation criterion for reading independency statements from a cg is introduced and shown to be equivalent to the moralization criterion using this new criterion it is proved that for every cg there exists a strictly positive discrete probability distribution that embodies exactly the independency statements displayed by the graph thus both criteria are shown to be complete and the use of cgs as tools for description of conditional independence structures is justified

recovery based error estimator for interface problems conforming linear elements this paper studies a new recovery based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems instead of recovering the gradient in the continuous finite element space the flux is recovered through a weighted l 2 projection onto h mathrm div conforming finite element spaces the resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results this paper also proposes an adaptive finite element method based on either the recovery based estimators or the edge estimator through local mesh refinement and establishes its convergence in particular it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps

collapsibility and response variables in contingency tables summary various definitions of the collapsibility of a hierarchical log linear model for a multidimensional contingency table are considered and shown to be equivalent necessary and sufficient conditions for collapsibility are found in terms of the generating class it is shown that log linear models are appropriate for tables with response and explanatory variables if and only if they are collapsible onto the explanatory variables some key word8 collapsibility contingency table graphical model interaction graph log linear model response variable s sufficiency shown to be closely related some models have the property that relations between a set of the classifying factors may be studied by examination of the table of marginal totals formed by summing over the remaining factors such models are said to be collapsible onto the given set of factors collapsibility has important consequences for hypothesis testing and model selection and can be useful in data reduction we consider various definitions of collapsibility and show their equivalence furthermore necessary and sufficient conditions for collapsi bility are found in terms of the generating class many tables analysed in practice involve response variables simple examples one of which is given in 3 suffice to show the importance of distinguishing between response and explanatory variables first that inappropriate models may be avoided and second that natural and relevant models that are not log linear may be considered this paper characterizes appropriate and inappropriate log linear models for tables with response variables and some alternative approaches for the analysis of such tables are briefly considered we consider a multidimensional contingency table n based on a set of classifying factors f for a given subset a of f we are interested in the table of marginal totals na that is to say the table of cell counts summed over the remaining factors ac that is the complement of a in f we identify a hierarchical log linear model l that is the set of probabilities p e l with its generating class whose elements generators are given in square brackets thus for example the model ab bcd for a 4 way table corresponds

characterizing persistent excitation for the sign sign equation error identifier abstract the sign sign ss algorithm is a computationally efficient adaptive identifier often used in signal processing tasks it is obtained by introducing signum functions on both the regressor and the prediction error multiplicands in the update kernel of the well known lms algorithm this paper gives a deterministic persistent excitation condition on the regressor sequence which guarantees ss convergence it also gives conditions under which ss may diverge and discusses how these persistent excitation conditions may be verified through a finite amount of computations

parameter updating in a bayes network abstract a bayes network is a directed acyclic graph in which the links are quantified by fixed conditional probabilities and the nodes represent random variables the primary use of the network is to provide an efficient method for updating conditional probabilities in the graph we consider the consequences of using the network as the computational device for updating parameter estimates in the dynamic linear model a discrete time bayesian model we show that using the network characterizes the dynamic linear model and its computations in a unified way the generality of the network permits nonsequential data collection and thereby provides a straightforward method of incorporating delayed data an on line diagnostic is offered to complement the conventional forecast error and an approximation to the posterior distribution is proposed algorithms for data propagation in multivariate gaussian causal trees are presented

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

improved zz a posteriori error estimators for diffusion problems conforming linear elements abstract in cai and zhang 2009 we introduced and analyzed an improved zienkiewicz zhu zz estimator for the conforming linear finite element approximation to elliptic interface problems the estimator is based on the piecewise constant flux recovery in the h div conforming finite element space this paper extends the results of cai and zhang 2009 to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear h div space

mixed multivariate generalized linear models for assessing lower limb arterial stenoses experiments and observational studies often involve gathering information on several response variables enabling us to model their dependence on observable predictor variables despite associations between the response variables they are often analysed separately using general and generalized linear models this paper investigates applications of multivariate regression analysis to improve the accuracy of predictions and decisions in the specific context of diagnosing arterial stenoses in human legs two basic models are developed for this application using i four binary responses and ii a mixture of two binary and two normal responses the results clearly demonstrate the potential advantages offered by this approach

globally stable direct fuzzy model reference adaptive control abstract in the paper a fuzzy adaptive control algorithm is presented it belongs to the class of direct model reference adaptive techniques based on a fuzzy takagi sugeno model of the plant the plant to be controlled is assumed to be nonlinear and predominantly of the first order consequently the resulting adaptive and control laws are very simple and thus interesting for use in practical applications the system remains stable in the presence of unmodelled dynamics disturbances parasitic high order dynamics and reconstruction errors are treated explicitly the global stability of the overall system is proven in the paper i e it is shown that all signals remain bounded while the tracking error and estimated parameters converge to some residual set that depends on the size of disturbance and high order parasitic dynamics the proposed algorithm is tested on a simulated three tank system its performance is compared to the performance of a classical mrac

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

gaussian markov distributions over finite graphs on s interesse aux modeles de selection de covariance introduites par dempster 1972 et etudies par wermuth 1976

on substantive research hypotheses conditional independence graphs and graphical chain models summary graphs consisting of points and lines or arrows as connections between selected pairs of points are used to formulate hypotheses about relations between variables points stand for variables connections represent associations when a missing connection is interpreted as a conditional independence the graph characterizes a conditional independence structure as well statistical models called graphical chain models correspond to special types of graphs which are interpreted in this fashion examples are used to illustrate how conditional independences are reflected in summary statistics derived from the models and how the graphs help to identify analogies and equivalences between different models graphical chain models are shown to provide a unifying concept for many statistical techniques that in the past have proven to be useful in analyses of data they also provide tools for new types of analysis

log linear modelling of pairwise interobserver agreement on a categorical scale this article uses log linear models to describe pairwise agreement among several raters who classify a sample on a subjective categorical scale the models describe agreement structure simultaneously for second order marginal tables of a multidimensional cross classification of ratings practical difficulties arise in fitting the models because models refer to pairwise marginal tables of a very large and sparse table a standard analysis that treats the marginal tables as independent yields consistent estimates of model parameters but not of the covariance matrix of the estimates we estimate the covariance matrix using the jackknife we apply the models to describe agreement between evaluations made by seven pathologists of carcinoma in situ of the uterine cervix using a five level ordinal scale previous analyses showed differences among the pathologists in their pairwise levels of agreement but we observe near homogeneity in the dependence structure of their ratings

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

recovery based error estimators for interface problems mixed and nonconforming finite elements in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 we introduced and analyzed a recovery based a posteriori error estimator for conforming linear finite element approximation to interface problems it was shown theoretically that the estimator is robust with respect to the size of jumps provided that the distribution of coefficients is locally monotone numerical examples showed that this condition is unnecessary this paper extends the idea in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 to mixed and nonconforming finite element methods for developing and analyzing robust estimators numerical results on test problems are also presented moreover an a priori error estimate is obtained when the underlying problem has low regularity

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

an approximation to maximum likelihood estimates in reduced models summary an approximation to the maximum likelihood estimates of the parameters in a model can be obtained from the corresponding estimates and information matrices in an extended model i e a model with additional parameters the approximation is close provided that the data are consistent with the first model applications are described to log linear models for discrete data to models for multivariate normal distributions with special covariance matrices and to mixed discrete continuous models

improved likelihood ratio statistics for covariance selection models summary the expected value of the likelihood ratio statistic for covariance selection models when this statistic has a closed form is derived correct to terms of order n 2 n being the sample size then a bartlett type correction factor is available for such statistics by simulation the practical use of such a correction factor is illustrated and is compared with an alternative correction factor in a special case the null density of the corrected statistic is also considered

marginal log linear parameterization of conditional independence models models defined by a set of conditional independence restrictions play an important role in statistical theory and applications especially but not only in graphical modelling in this paper we identify a subclass of these consisting of hierarchical marginal log linear models as defined by bergsma rudas 2002a such models are smooth which implies the applicability of standard asymptotic theory and simplifies interpretation furthermore we give a marginal log linear parameterization and a minimal specification of the models in the subclass which implies the applicability of standard methods to compute maximum likelihood estimates and simplifies the calculation of the degrees of freedom of chi squared statistics to test goodness of fit the utility of the results is illustrated by applying them to block recursive markov models associated with chain graphs

maximum likelihood estimation in graphical models with missing values in this paper we discuss maximum likelihood estimation when some observations are missing in mixed graphical interaction models assuming a conditional gaussian distribution as introduced by lauritzen wermuth 1989 for the saturated case ml estimation with missing values via the em algorithm has been proposed by little schluchter 1985 we expand their results to the special restrictions in graphical models and indicate a more efficient way to compute the e step the main purpose of the paper is to show that for certain missing patterns the computational effort can considerably be reduced

application of fuzzy naive bayes and a real valued genetic algorithm in identification of fuzzy model we present a method to identify a fuzzy model from data by using the fuzzy naive bayes and a real valued genetic algorithm the identification of a fuzzy model is comprised of the extraction of if then rules that is followed by the estimation of their parameters the involved parameters include those which determine the membership function of fuzzy sets and the certainty factors of fuzzy if then rules in our method as long as the fuzzy partition in the input output space is given the certainty factor of each rule is computed with the fuzzy conditional probability of the consequent conditioned on the antecedent by using the fuzzy naive bayes which is a generalization of naive bayes the fuzzy model involves the rules characterized by the highest values of certainty factors the certainty factor of each rule is the fuzzy conditional probability and it reflects the inner relationship between the antecedent and the consequent in order to improve the accuracy of the fuzzy model the real valued genetic algorithm is incorporated into our identification process this process concerns the optimization of the membership functions occurring in the rules we just involve the parameters of membership function of the fuzzy sets into the real valued genetic algorithm since the certainty factor of each rule can be computed automatically the performance of the model is shown for the backing truck problem and the prediction of mackey glass time series

propagating imprecise probabilities in bayesian networks abstract often experts are incapable of providing exact probabilities likewise samples on which the probabilities in networks are based must often be small and preliminary in such cases the probabilities in the networks are imprecise the imprecision can be handled by second order probability distributions it is convenient to use beta or dirichlet distributions to express the uncertainty about probabilities the problem of how to propagate point probabilities in a bayesian network now is transformed into the problem of how to propagate dirichlet distributions in bayesian networks it is shown that the propagation of dirichlet distributions in bayesian networks with incomplete data results in a system of probability mixtures of beta binomial and dirichlet distributions approximate first order probabilities and their second order probability density functions are obtained by stochastic simulation a number of properties of the propagation of imprecise probabilities are discussed by the use of examples an important property is that the imprecision of inferences increases rapidly as new premises are added to an argument the imprecision can be used as a pruning criterion in a network to keep the number of variables involved in an inferential argument small thus imprecision may be used as an ockam s razor in bayesian networks