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

global identifiability of linear structural equation models structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables we consider the classical case of linear relationships and additive gaussian noise terms we give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms global identifiability is understood to mean injectivity of the parametrization of the model and is fundamental in particular for applicability of standard statistical methodology

effective connectivity of fmri data using ancestral graph theory dealing with missing regions most of the current methods to assess effective connectivity from functional magnetic resonance imaging fmri rely on the assumption that all relevant brain regions are entered into the analysis if this assumption is untenable which we believe is most often the case then spurious connections between brain regions can appear in this paper we propose to use an ancestral graph to model connectivity which provides a way to avoid spurious connections the ancestral graph is determined from trial by trial variation and not from the time series a random effects model is defined for ancestral graphs which allows for individual differences in terms of graph parameters e g connection strength procedures for model selection model fit and hypothesis testing of ancestral graphs are proposed the hypothesis test can be used to find differences in connection strength between for example conditions monte carlo simulations show that the ancestral graph is appropriate to model connectivity from fmri condition specific trial data to assess the accuracy further the proposed method is applied to real fmri data to determine how brain regions interact during speech monitoring

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

random networks graphical models and exchangeability we study conditional independence relationships for random networks and their interplay with exchangeability we show that for finitely exchangeable network models the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts we then characterize all possible markov structures for finitely exchangeable random graphs thereby identifying a new class of markov network models corresponding to bidirected kneser graphs in particular we demonstrate that the fundamental property of dissociatedness corresponds to a markov property for exchangeable networks described by bidirected line graphs finally we study those exchangeable models that are also summarized in the sense that the probability of a network depends only on the degree distribution and we identify a class of models that is dual to the markov graphs of frank and strauss particular emphasis is placed on studying consistency properties of network models under the process of forming subnetworks and we show that the only consistent systems of markov properties correspond to the empty graph the bidirected line graph of the complete graph and the complete graph

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

multi particle diffusion limited aggregation we consider a stochastic aggregation model on mathbb z d start with particles distributed according to the product bernoulli measure with parameter mu in addition start with an aggregate at the origin non aggregated particles move as continuous time simple random walks obeying the exclusion rule whereas aggregated particles do not move the aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it this evolution is called multi particle diffusion limited aggregation our main result states that if on d 1 the initial density of particles is large enough then with positive probability the aggregate has linearly growing arms that is there exists a constant c 0 so that at time t the aggregate contains a point of distance at least ct from the origin for all t the key conceptual element of our analysis is the introduction and study of a new growth process consider a first passage percolation process called type 1 starting from the origin whenever type 1 is about to occupy a new vertex with positive probability instead of doing it it gives rise to another first passage percolation process called type 2 which starts to spread from that vertex each vertex gets occupied only by the process that arrives to it first this process may have three phases extinction type 1 gets eventually surrounded by type 2 coexistence infinite clusters of both types emerge and strong survival type 1 produces an infinite cluster which entraps all type 2 clusters understanding the various phases of this process is of mathematical interest on its own right we establish the existence of a strong survival phase and use this to show our main result

on the uniqueness of solution for all wavenumbers in acoustic radiation the integral relations equivalent to helmholtz s equation with closed boundary conditions possess nonunique exterior solutions at eigenwavenumbers of the corresponding interior problems if the green s function for helmholtz s equation can be expanded into a series of orthogonal functions then an infinite set of integral equations at the boundary results these null field equations are shown to possess unique solutions for all wavenumbers furthermore these solutions can be used to represent the solutions in the exterior domain as algebraic series

structural intervention distance for evaluating causal graphs causal inference relies on the structure of a graph often a directed acyclic graph dag different graphs may result in different causal inference statements and different intervention distributions to quantify such differences we propose a pre metric between dags the structural intervention distance sid the sid is based on a graphical criterion only and quantifies the closeness between two dags in terms of their corresponding causal inference statements it is therefore well suited for evaluating graphs that are used for computing interventions instead of dags it is also possible to compare cpdags completed partially dags that represent markov equivalence classes the sid differs significantly from the widely used structural hamming distance and therefore constitutes a valuable additional measure we discuss properties of this distance and provide a reasonably efficient implementation with software code available on the first author s home page

binary models for marginal independence summary log linear models are a classical tool for the analysis of contingency tables in particular the subclass of graphical log linear models provides a general framework for modelling conditional independences however with the exception of special structures marginal independence hypotheses cannot be accommodated by these traditional models focusing on binary variables we present a model class that provides a framework for modelling marginal independences in contingency tables the approach that is taken is graphical and draws on analogies with multivariate gaussian models for marginal independence for the graphical model representation we use bidirected graphs which are in the tradition of path diagrams we show how the models can be parameterized in a simple fashion and how maximum likelihood estimation can be performed by using a version of the iterated conditional fitting algorithm finally we consider combining these models with symmetry restrictions

a recursive method for structural learning of directed acyclic graphs in this paper we propose a recursive method for structural learning of directed acyclic graphs dags in which a problem of structural learning for a large dag is first decomposed into two problems of structural learning for two small vertex subsets each of which is then decomposed recursively into two problems of smaller subsets until none subset can be decomposed further in our approach search for separators of a pair of variables in a large dag is localized to small subsets and thus the approach can improve the efficiency of searches and the power of statistical tests for structural learning we show how the recent advances in the learning of undirected graphical models can be employed to facilitate the decomposition simulations are given to demonstrate the performance of the proposed method

causal inference by using invariant prediction identification and confidence intervals summary what is the difference between a prediction that is made with a causal model and that with a non causal model suppose that we intervene on the predictor variables or change the whole environment the predictions from a causal model will in general work as well under interventions as for observational data in contrast predictions from a non causal model can potentially be very wrong if we actively intervene on variables here we propose to exploit this invariance of a prediction under a causal model for causal inference given different experimental settings e g various interventions we collect all models that do show invariance in their predictive accuracy across settings and interventions the causal model will be a member of this set of models with high probability this approach yields valid confidence intervals for the causal relationships in quite general scenarios we examine the example of structural equation models in more detail and provide sufficient assumptions under which the set of causal predictors becomes identifiable we further investigate robustness properties of our approach under model misspecification and discuss possible extensions the empirical properties are studied for various data sets including large scale gene perturbation experiments

distinguishing cause from effect using observational data methods and benchmarks the discovery of causal relationships from purely observational data is a fundamental problem in science the most elementary form of such a causal discovery problem is to decide whether x causes y or alternatively y causes x given joint observations of two variables x y an example is to decide whether altitude causes temperature or vice versa given only joint measurements of both variables even under the simplifying assumptions of no confounding no feedback loops and no selection bias such bivariate causal discovery problems are challenging nevertheless several approaches for addressing those problems have been proposed in recent years we review two families of such methods methods based on additive noise models anms and information geometric causal inference igci we present the benchmark causeeffectpairs that consists of data for 100 different causee ffect pairs selected from 37 data sets from various domains e g meteorology biology medicine engineering economy etc and motivate our decisions regarding the ground truth causal directions of all pairs we evaluate the performance of several bivariate causal discovery methods on these real world benchmark data and in addition on artificially simulated data our empirical results on real world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data although more benchmark data would be needed to obtain statistically significant conclusions one of the best performing methods overall is the method based on additive noise models that has originally been proposed by hoyer et al 2009 which obtains an accuracy of 63 10 and an auc of 0 74 0 05 on the real world benchmark as the main theoretical contribution of this work we prove the consistency of that method

a new criterion for confounder selection we propose a new criterion for confounder selection when the underlying causal structure is unknown and only limited knowledge is available we assume all covariates being considered are pretreatment variables and that for each covariate it is known i whether the covariate is a cause of treatment and ii whether the covariate is a cause of the outcome the causal relationships the covariates have with one another is assumed unknown we propose that control be made for any covariate that is either a cause of treatment or of the outcome or both we show that irrespective of the actual underlying causal structure if any subset of the observed covariates suffices to control for confounding then the set of covariates chosen by our criterion will also suffice we show that other commonly used criteria for confounding control do not have this property we use formal theory concerning causal diagrams to prove our result but the application of the result does not rely on familiarity with causal diagrams an investigator simply need ask is the covariate a cause of the treatment and is the covariate a cause of the outcome if the answer to either question is yes then the covariate is included for confounder control we discuss some additional covariate selection results that preserve unconfoundedness and that may be of interest when used with our criterion

marginal log linear parameters for graphical markov models marginal log linear mll models provide a flexible approach to multivariate discrete data mll parametrizations under linear constraints induce a wide variety of models including models defined by conditional independences we introduce a subclass of mll models which correspond to acyclic directed mixed graphs admgs under the usual global markov property we characterize for precisely which graphs the resulting parametrization is variation independent the mll approach provides the first description of admg models in terms of a minimal list of constraints the parametrization is also easily adapted to sparse modelling techniques which we illustrate using several examples of real data

graphical methods for efficient likelihood inference in gaussian covariance models in graphical modelling a bi directed graph encodes marginal independences among random variables that are identified with the vertices of the graph we show how to transform a bi directed graph into a maximal ancestral graph that i represents the same independence structure as the original bi directed graph and ii minimizes the number of arrowheads among all ancestral graphs satisfying i here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi directed edges in gaussian models this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts

marginalization and conditioning for lwf chain graphs supported by grant fa9550 12 1 0392 from the u s air force office of scientific research afosr and the defense advanced research projects agency darpa

estimation of a covariance matrix with zeros summary we consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero we first present an algorithm which we call iterative conditional fitting for computing the maximum likelihood estimate of the constrained covariance matrix under the assumption of multivariate normality in contrast to previous approaches this algorithm has guaranteed convergence properties dropping the assumption of multivariate normality we show how to estimate the covariance matrix in an empirical likelihood approach these approaches are then compared via simulation and on an example of gene expression

half trek criterion for generic identifiability of linear structural equation models a linear structural equation model relates random variables of interest and corresponding gaussian noise terms via a linear equation system each such model can be represented by a mixed graph in which directed edges encode the linear equations and bidirected edges indicate possible correlations among noise terms we study parameter identifiability in these models that is we ask for conditions that ensure that the edge coefficients and correlations appearing in a linear structural equation model can be uniquely recovered from the covariance matrix of the associated distribution we treat the case of generic identifiability where unique recovery is possible for almost every choice of parameters we give a new graphical condition that is sufficient for generic identifiability and can be verified in time that is polynomial in the size of the graph it improves criteria from prior work and does not require the directed part of the graph to be acyclic we also develop a related necessary condition and examine the gap between sufficient and necessary conditions through simulations on graphs with 25 or 50 nodes as well as exhaustive algebraic computations for graphs with up to five nodes

constraint based causal discovery from multiple interventions over overlapping variable sets scientific practice typically involves repeatedly studying a system each time trying to unravel a different perspective in each study the scientist may take measurements under different experimental conditions interventions manipulations perturbations and measure different sets of quantities variables the result is a collection of heterogeneous data sets coming from different data distributions in this work we present algorithm combine which accepts a collection of data sets over overlapping variable sets under different experimental conditions combine then outputs a summary of all causal models indicating the invariant and variant structural characteristics of all models that simultaneously fit all of the input data sets combine converts estimated dependencies and independencies in the data into path constraints on the data generating causal model and encodes them as a sat instance the algorithm is sound and complete in the sample limit to account for conflicting constraints arising from statistical errors we introduce a general method for sorting constraints in order of confidence computed as a function of their corresponding p values in our empirical evaluation combine outperforms in terms of efficiency the only pre existing similar algorithm the latter additionally admits feedback cycles but does not admit conflicting constraints which hinders the applicability on real data as a proof of concept combine is employed to co analyze 4 real mass cytometry data sets measuring phosphorylated protein concentrations of overlapping protein sets under 3 different interventions

finite size effect of harmonic measure estimation in a dla model variable size of probe particles abstract a finite size effect in the probing of the harmonic measure in simulation of diffusion limited aggregation dla growth is investigated we introduce a variable size of probe particles to estimate harmonic measure and extract the fractal dimension of dla clusters taking two limits of vanishingly small probe particle size and of infinitely large size of a dla cluster we generate 1000 dla clusters consisting of 50 million particles each using an off lattice killing free algorithm developed in the early work the introduced method leads to unprecedented accuracy in the estimation of the fractal dimension we discuss the variation of the probability distribution function with the size of probing particles

markov equivalence for ancestral graphs ancestral graphs can encode conditional independence relations that arise in directed acyclic graph dag models with latent and selection variables however for any ancestral graph there may be several other graphs to which it is markov equivalent we state and prove conditions under which two maximal ancestral graphs are markov equivalent to each other thereby extending analogous results for dags given by other authors these conditions lead to an algorithm for determining markov equivalence that runs in time that is polynomial in the number of vertices in the graph

the wald test and crame spl acute r rao bound for misspecified models in electromagnetic source analysis by using signal processing techniques an estimate of activity in the brain from the electro or magneto encephalogram eeg or meg can be obtained for a proper analysis a test is required to indicate whether the model for brain activity fits a problem in using such tests is that often not all assumptions are satisfied like the assumption of the number of shells in an eeg in such a case a test on the number of sources model order might still be of interest a detailed analysis is presented of the wald test for these cases one of the advantages of the wald test is that it can be used when not all assumptions are satisfied two different previously suggested wald tests in electromagnetic source analysis emsa are examined a test on source amplitudes and a test on the closeness of source pairs the wald test is analytically studied in terms of alternative hypotheses that are close to the null hypothesis local alternatives it is shown that the wald test is asymptotically unbiased that it has the correct level and power which makes it appropriate to use in emsa an accurate estimate of the crame spl acute r rao bound crb is required for the use of the wald test when not all assumptions are satisfied the sandwich crb is used for this purpose it is defined for nonseparable least squares with constraints required for the wald test on amplitudes simulations with eeg show that when the sensor positions are incorrect or the number of shells is incorrect or the conductivity parameter is incorrect then the crb and wald test are still good with a moderate number of trials additionally the crb and wald test appear robust against an incorrect assumption on the noise covariance a combination of incorrect sensor positions and noise covariance affects the possibility of detecting a source with small amplitude

on the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias causal discovery becomes especially challenging when the possibility of latent confounding and or selection bias is not assumed away for this task ancestral graph models are particularly useful in that they can represent the presence of latent confounding and selection effect without explicitly invoking unobserved variables based on the machinery of ancestral graphs there is a provably sound causal discovery algorithm known as the fci algorithm that allows the possibility of latent confounders and selection bias however the orientation rules used in the algorithm are not complete in this paper we provide additional orientation rules augmented by which the fci algorithm is shown to be complete in the sense that it can under standard assumptions discover all aspects of the causal structure that are uniquely determined by facts of probabilistic dependence and independence the result is useful for developing any causal discovery and reasoning system based on ancestral graph models

general multilevel linear modeling for group analysis in fmri this article discusses general modeling of multisubject and or multisession fmri data in particular we show that a two level mixed effects model where parameters of interest at the group level are estimated from parameter and variance estimates from the single session level can be made equivalent to a single complete mixed effects model where parameters of interest at the group level are estimated directly from all of the original single sessions time series data if the co variance at the second level is set equal to the sum of the co variances in the single level form using the blue with known covariances this result has significant implications for group studies in fmri since it shows that the group analysis requires only values of the parameter estimates and their co variance from the first level generalizing the well established summary statistics approach in fmri the simple and generalized framework allows different prewhitening and different first level regressors to be used for each subject the framework incorporates multiple levels and cases such as repeated measures paired or unpaired t tests and f tests at the group level explicit examples of such models are given in the article using numerical simulations based on typical first level covariance structures from real fmri data we demonstrate that by taking into account lower level covariances and heterogeneity a substantial increase in higher level z score is possible

wald statistic for model order selection in superposition models a consistent model selection algorithm is presented for superimposed signal models the proposed method is motivated by the wald statistic and reduces the computational complexity of procedures based on the minimum description length mdl principle the procedure is suggested when a noncyclostationary signal model or short data length prevents use of covariance rank test for maximum model order k the procedure provides o k computational savings over an mdl test additionally a proof establishes the consistency of a least squares estimator using overparametrized models finite sample performance of the proposed model selection method is studied via monte carlo simulations for estimating the multipath delays and amplitudes of a chirp signal

causal reasoning with ancestral graphs causal reasoning is primarily concerned with what would happen to a system under external interventions in particular we are often interested in predicting the probability distribution of some random variables that would result if some other variables were forced to take certain values one prominent approach to tackling this problem is based on causal bayesian networks using directed acyclic graphs as causal diagrams to relate post intervention probabilities to pre intervention probabilities that are estimable from observational data however such causal diagrams are seldom fully testable given observational data in consequence many causal discovery algorithms based on data mining can only output an equivalence class of causal diagrams rather than a single one this paper is concerned with causal reasoning given an equivalence class of causal diagrams represented by a partial ancestral graph we present two main results the first result extends pearl 1995 s celebrated do calculus to the context of ancestral graphs in the second result we focus on a key component of pearl s calculus the property of invariance under interventions and give stronger graphical conditions for this property than those implied by the first result the second result also improves the earlier similar results due to spirtes et al 1993

instrumental variable approach to array processing in spatially correlated noise fields high performance signal parameter estimation from sensor array data is a problem which has received much attention a number of so called eigenvector ev techniques such as music esprit wsf and mode have been proposed in the literature the ev techniques for array processing require knowledge of the spatial noise correlation matrix that constitutes a significant drawback a novel instrumental variable iv approach to the sensor array problem is proposed the iv technique relies on the same basic geometric properties as the ev methods to obtain parameter estimates however by exploiting the temporal correlation of the source signals no knowledge of the spatial noise covariance is required the asymptotic properties of the iv estimator are examined and an optimal iv method is derived computer simulations are presented to study the properties of the iv estimators in samples of practical length the proposed algorithm is also shown to perform better than music on a full scale passive sonar experiment

discussion latent variable graphical model selection via convex optimization we want to congratulate the authors for a thought provoking and very interesting paper sparse modeling of the concentration matrix has enjoyed popularity in recent years it has been framed as a computationally convenient convex 1constrained estimation problem in yuan and lin 2007 and can be applied readily to higher dimensional problems the authors argue we think correctly that the sparsity of the concentration matrix is for many applications more plausible after the effects of a few latent variables have been removed the most attractive point about their method is surely that it is formulated as a convex optimization problem latent variable fitting and sparse graphical modeling of the conditional distribution of the observed variables can then be obtained through a single fitting procedure practical aspects the method deserves wide adoption but this will only be realistic if software is made available for example as an r package not many users will go to the trouble of implementing the method on their own so we will strongly urge the authors to do so an imputation method in the absence of readily available software it is worth thinking whether the proposed fitting procedure can be approximated by methods involving known and well tested computational techniques the concentration matrix of observed and hidden variables is k ko koh k ho kh where we have deviated from the notation in the paper by omitting the asterisk the proposed estimator sn ko of ko was defined as

test retest reliability of resting state connectivity network characteristics using fmri and graph theoretical measures characterizing the brain connectome using neuroimaging data and measures derived from graph theory emerged as a new approach that has been applied to brain maturation cognitive function and neuropsychiatric disorders for a broad application of this method especially for clinical populations and longitudinal studies the reliability of this approach and its robustness to confounding factors need to be explored here we investigated test retest reliability of graph metrics of functional networks derived from functional magnetic resonance imaging fmri recorded in 33 healthy subjects during rest we constructed undirected networks based on the anatomic automatic labeling aal atlas template and calculated several commonly used measures from the field of graph theory focusing on the influence of different strategies for confound correction for each subject method and session we computed the following graph metrics clustering coefficient characteristic path length local and global efficiency assortativity modularity hierarchy and the small worldness scalar reliability of each graph metric was assessed using the intraclass correlation coefficient icc overall iccs ranged from low to high 0 to 0 763 depending on the method and metric methodologically the use of a broader frequency band 0 008 0 15 hz yielded highest reliability indices mean icc 0 484 followed by the use of global regression mean icc 0 399 in general the second order metrics small worldness hierarchy assortativity studied here tended to be more robust than first order metrics in conclusion our study provides methodological recommendations which allow the computation of sufficiently robust markers of network organization using graph metrics derived from fmri data at rest

fractal dimension and unscreened angles measured for radial viscous fingering we have examined fractal patterns formed by the injection of air into oil in a thin 0 127 phantom rule 0 3em 0ex mathrm mm layer contained between two cylindrical glass plates of 288 phantom rule 0 3em 0ex mathrm mm diameter a hele shaw cell for pressure differences in the range 0 25 ensuremath leqslant ensuremath delta p ensuremath leqslant 1 75 phantom rule 0 3em 0ex mathrm atm we find that an asymptotic structure is reached at large values of the ratio r b where r is the pattern radius and b the gap between the plates both the driving force and the size of the pattern which reaches r b 900 are far larger than in past experiments the fractal dimension d 0 of the pattern for large r b is 1 70 ifmmode pm else textpm fi 0 02 further the generalized dimensions d q of the pattern are independent of q d q ensuremath simeq 1 70 for the range examined ensuremath 11lql17 thus the pattern is self similar within the experimental uncertainty the results for d q agree well with recent calculations for diffusion limited aggregation dla clusters we have also measured the probability distribution of unscreened angles at late times the distribution approaches a universal i e forcing and size independent asymptotic form that has mean 145 ifmmode circ else textdegree fi and standard deviation 36 ifmmode circ else textdegree fi these results indicate that the distribution function for the unscreened angle is an invariant property of the growth process

stratified exponential families graphical models and model selection we describe a hierarchy of exponential families which is useful for distinguishing types of graphical models undirected graphical models with no hidden variables are linear exponential families lefs directed acyclic graphical dag models and chain graphs with no hidden variables including dag models with several families of local distributions are curved exponential families cefs graphical models with hidden variables are what we term stratified exponential families sefs a sef is a finite union of cefs of various dimensions satisfying some regularity conditions we also show that this hierarchy of exponential families is noncollapsing with respect to graphical models by providing a graphical model which is a cef but not a lef and a graphical model that is a sef but not a cef finally we show how to compute the dimension of a stratified exponential family these results are discussed in the context of model selection of graphical models

sound radiation from a finite fluid filled submerged cylindrical shell with porous material sandwich a theoretical model is developed to predict the far field sound radiation from a finite fluid filled submerged cylindrical thin shell with porous material sandwich a combination of the wave number domain approach and the transfer matrix method is presented which is convenient to analyze the vibratory responses in terms of wave number expressions for the spectral radial velocity of the outer surface of the shell are also formulated a prediction model for sound radiation at far field is given by using the boundary integral equation extensive numerical results are also presented to illustrate the general characteristics of the far field sound pressure as a function of frequency an experimental verification is performed and a good correlation between the theoretical results and the experimental results shows that the theoretical study work in this paper is correct

towards integrative causal analysis of heterogeneous data sets and studies we present methods able to predict the presence and strength of conditional and unconditional dependencies correlations between two variables y and z never jointly measured on the same samples based on multiple data sets measuring a set of common variables the algorithms are specializations of prior work on learning causal structures from overlapping variable sets this problem has also been addressed in the field of statistical matching the proposed methods are applied to a wide range of domains and are shown to accurately predict the presence of thousands of dependencies compared against prototypical statistical matching algorithms and within the scope of our experiments the proposed algorithms make predictions that are better correlated with the sample estimates of the unknown parameters on test data this is particularly the case when the number of commonly measured variables is low the enabling idea behind the methods is to induce one or all causal models that are simultaneously consistent with fit all available data sets and prior knowledge and reason with them this allows constraints stemming from causal assumptions e g causal markov condition faithfulness to propagate several methods have been developed based on this idea for which we propose the unifying name integrative causal analysis inca a contrived example is presented demonstrating the theoretical potential to develop more general methods for co analyzing heterogeneous data sets the computational experiments with the novel methods provide evidence that causally inspired assumptions such as faithfulness often hold to a good degree of approximation in many real systems and could be exploited for statistical inference code scripts and data are available at www mensxmachina org

mean and covariance structure analysis theoretical and practical improvements abstract the most widely used multivariate statistical models in the social and behavioral sciences involve linear structural relations among observed and latent variables in practice these variables are generally nonnormally distributed hence classical multivariate analysis based on multinomial error free variables having no simultaneous interrelations is not adequate to deal with such data a promising alternative based on asymptotically distribution free adf covariance structure analysis has been found to be virtually useless in practical model evaluation at finite sample sizes with nonnormal data we take a new look at the basic statistical theory of structural models under arbitrary distributions using the methodology of nonlinear regression and generalized least squares estimation for example we adopt the use of residual weight matrices from regression theory we develop a series of estimators and tests based on arbitrary distribution theory we obtain a type of probabilistic bartlett co

statistical physics of fracture surfaces morphology experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics one finds scaling exponents associated with correlation and structure functions indicating a rich phenomenology of anomalous scaling we argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for we present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces successfully reproducing the multiscaling phenomenology at least in 1 1 dimensions for surfaces in 2 1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the so 2 symmetry group it appears that this approach organizes effectively the rich scaling properties we end up proposing new experiments in which the rotational symmetry is not broken such that the scaling properties should be particularly simple

graphical gaussian modelling of multivariate time series with latent variable in time series analysis inference about causeeffect relationships among multiple times series is commonly based on the concept of granger causality which exploits temporal structure to achieve causal ordering of dependent variables one major problem in the application of granger causality for the identification of causal relationships is the possible presence of latent variables that affect the measured components and thus lead to so called spurious causalities in this paper we describe a new graphical approach for modelling the dependence structure of multivariate stationary time series that are affected by latent variables to this end we introduce dynamic maximal ancestral graphs dmags in which each time series is represented by a single vertex for gaussian processes this approach leads to vector autoregressive models with errors that are not independent but correlated according to the dashed edges in the graph we discuss identifiability of the parameters and show that these models can be viewed as graphical arma models that satisfy the granger causality restrictions encoded by the associated dynamic maximal ancestral graph

markov properties for linear causal models with correlated errors a linear causal model with correlated errors represented by a dag with bi directed edges can be tested by the set of conditional independence relations implied by the model a global markov property specifies by the d separation criterion the set of all conditional independence relations holding in any model associated with a graph a local markov property specifies a much smaller set of conditional independence relations which will imply all other conditional independence relations which hold under the global markov property for dags with bi directed edges associated with arbitrary probability distributions a local markov property is given in richardson 2003 which may invoke an exponential number of conditional independencies in this paper we show that for a class of linear structural equation models with correlated errors there is a local markov property which will invoke only a linear number of conditional independence relations for general linear models we provide a local markov property that often invokes far fewer conditional independencies than that in richardson 2003 the results have applications in testing linear structural equation models with correlated errors

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

trek separation for gaussian graphical models gaussian graphical models are semi algebraic subsets of the cone of positive definite covariance matrices submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables we give a precise graph theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases our new trek separation criterion generalizes the familiar d separation criterion proofs are based on the trek rule the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials

a combined order selection and parameter estimation algorithm for undamped exponentials we propose an approximate maximum likelihood parameter estimation algorithm combined with a model order estimator for superimposed undamped exponentials in noise the algorithm combines the robustness of fourier based estimators and the high resolution capabilities of parametric methods we use a combination of a wald 1945 statistic and a map test for order selection and initialize an iterative maximum likelihood descent algorithm recursively based on estimates at higher candidate model orders experiments using simulated data and synthetic radar data demonstrate improved performance over mdl map and aic in places of practical interest

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

on the choice of a model to fit data from an exponential family let x 1 cdots x n be iid observations coming from an exponential family the problem of interest is this given a finite number of models m j smoothly curved manifolds in mathbb r k choose the best model to fit the observations with some penalty for choosing models with dimensions which are too large a result of schwarz is made more specific and is extended to the case where the models are curved manifolds if s y n j is up to a constant c n independent of the model the log of the posterior probability of the j th model where the sample mean y n 1 n sum n i 1 x i has been replaced by y schwarz suggested an asymptotic expansion of s y n j whose leading terms are gamma y n j n sup psi in m j cap theta y psi b psi frac 1 2 k j log n in the case where the models are affine subspaces of mathbb r k we establish a similar asymptotic expansion including the next term with uniform bounds for y in a compact neighborhood of nabla b theta where theta is the true value of the parameter we suggest a criterion for the choice of the best model that consists of maximizing the three leading terms in the expansion s y n j we show that the criterion gives the correct model with probabilities p n theta rightarrow 1 as n rightarrow infty

a note on the efficiency of sandwich covariance matrix estimation the sandwich estimator also known as robust covariance matrix estimator heteroscedasticity consistent covariance matrix estimate or empirical covariance matrix estimator has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified surprisingly though there has been little discussion of properties of the sandwich method other than consistency we investigate the sandwich estimator in quasi likelihood models asymptotically and in the linear case analytically we show that under certain circumstances when the quasi likelihood model is correct the sandwich estimate is often far more variable than the usual parametric variance estimate the increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the p

source localization using recursively applied and projected rap music a new method for source localization is described that is based on a modification of the well known music algorithm in classical music the array manifold vector is projected onto an estimate of the signal subspace errors in the estimate of the signal subspace can make localization of multiple sources difficult recursively applied and projected rap music uses each successively located source to form an intermediate array gain matrix and projects both the array manifold and the signal subspace estimate into its orthogonal complement the music projection to find the next source is then performed in this reduced subspace special assumptions about the array manifold structure such as vandermonde or shift invariance are not required using the metric of principal angles we describe a general form of the rap music algorithm for the case of diversely polarized sources through a uniform linear array simulation with two highly correlated sources we demonstrate the improved monte carlo error performance of rap music relative to music and two other sequential subspace methods s and ies music we then demonstrate the more general utility of this algorithm for multidimensional array manifolds in a magnetoencephalography meg source localization simulation

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

computing maximum likelihood estimates in recursive linear models with correlated errors in recursive linear models the multivariate normal joint distribution of all variables exhibits a dependence structure induced by a recursive or acyclic system of linear structural equations these linear models have a long tradition and appear in seemingly unrelated regressions structural equation modelling and approaches to causal inference they are also related to gaussian graphical models via a classical representation known as a path diagram despite the models long history a number of problems remain open in this paper we address the problem of computing maximum likelihood estimates in the subclass of bow free recursive linear models the term bow free refers to the condition that the errors for variables i and j be uncorrelated if variable i occurs in the structural equation for variable j we introduce a new algorithm termed residual iterative conditional fitting ricf that can be implemented using only least squares computations in contrast to existing algorithms ricf has clear convergence properties and yields exact maximum likelihood estimates after the first iteration whenever the mle is available in closed form

array processing in correlated noise fields based on instrumental variables and subspace fitting accurate signal parameter estimation from sensor array data is a problem which has received much attention in the last decade a number of parametric estimation techniques have been proposed in the literature in general these methods require knowledge of the sensor to sensor correlation of the noise which constitutes a significant drawback this difficulty can be overcome only by introducing alternative assumptions that enable separating the signals from the noise in some applications the raw sensor outputs can be preprocessed so that the emitter signals are temporally correlated with correlation length longer than that of the noise an instrumental variable iv approach can then be used for estimating the signal parameters without knowledge of the spatial color of the noise a computationally simple iv approach has recently been proposed by the authors herein a refined technique that can give significantly better performance is derived a statistical analysis of the parameter estimates is performed enabling optimal selection of certain user specified quantities a lower bound on the attainable error variance is also presented the proposed optimal iv method is shown to attain the bound if the signals have a quasideterministic character

causal discovery with continuous additive noise models we consider the problem of learning causal directed acyclic graphs from an observational joint distribution one can use these graphs to predict the outcome of interventional experiments from which data are often not available we show that if the observational distribution follows a structural equation model with an additive noise structure the directed acyclic graph becomes identifiable from the distribution under mild conditions this constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the markov equivalence class of the graph thus leaving some edges undirected we provide practical algorithms for finitely many samples resit regression with subsequent independence test and two methods based on an independence score we prove that resit is correct in the population setting and provide an empirical evaluation

markovian acyclic directed mixed graphs for discrete data acyclic directed mixed graphs admgs are graphs that contain directed rightarrow and bidirected leftrightarrow edges subject to the constraint that there are no cycles of directed edges such graphs may be used to represent the conditional independence structure induced by a dag model containing hidden variables on its observed margin the markovian model associated with an admg is simply the set of distributions obeying the global markov property given via a simple path criterion m separation we first present a factorization criterion characterizing the markovian model that generalizes the well known recursive factorization for dags for the case of finite discrete random variables we also provide a parameterization of the model in terms of simple conditional probabilities and characterize its variation dependence we show that the induced models are smooth consequently markovian admg models for discrete variables are curved exponential families of distributions

learning high dimensional directed acyclic graphs with latent and selection variables we consider the problem of learning causal information between random variables in directed acyclic graphs dags when allowing arbitrarily many latent and selection variables the fci fast causal inference algorithm has been explicitly designed to infer conditional independence and causal information in such settings however fci is computationally infeasible for large graphs we therefore propose the new rfci algorithm which is much faster than fci in some situations the output of rfci is slightly less informative in particular with respect to conditional independence information however we prove that any causal information in the output of rfci is correct in the asymptotic limit we also define a class of graphs on which the outputs of fci and rfci are identical we prove consistency of fci and rfci in sparse high dimensional settings and demonstrate in simulations that the estimation performances of the algorithms are very similar all software is implemented in the r package pcalg

ancestral graph markov models this paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all dag independence models this class of graphs called maximal ancestral graphs has two attractive features there is at most one edge between each pair of vertices every missing edge corresponds to an independence relation these features lead to a simple parameterization of the corresponding set of distributions in the gaussian case

resonant vibrations of a submerged beam abstract forced vibration of a simply supported submerged beam of circular cross section is investigated by the use of two mathematical methods in the first approach the problem formulation is reduced to a singular integro differential equation for the transverse deflection in the second approach the method of matched asymptotic expansions is employed the integro differential equation is solved numerically to yield an exact solution for the frequency response subsequent use of a representation integral yields the radiated far field acoustic pressure the exact results for the beam deflection are compared with approximate results that are available in the literature next a matched asymptotic expansion is worked out by constructing inner and outer expansions for frequencies near and not near resonance frequencies respectively the two expansions are matched in an appropriate manner to yield a uniformly valid solution the leading term of the matched asymptotic solution is compared with exact numerical results

multimodality of the likelihood in the bivariate seemingly unrelated regressions model summary we analyse the simplest two equation seemingly unrelated regressions model and demonstrate that its likelihood may have up to five stationary points and thus there may be up to three local modes consequently the estimates obtained via iterative estimation methods may depend on starting values we further show that the probability of multi modality vanishes asymptotically monte carlo simulations suggest that multimodality rarely occurs if the seemingly unrelated regressions model is true but can become more frequent if the model is misspecified the existence of multimodality in the likelihood for seemingly unrelated regressions models contradicts several claims in the literature

cumulative distribution networks and the derivative sum product algorithm models and inference for cumulative distribution functions on graphs we present a class of graphical models for directly representing the joint cumulative distribution function cdf of many random variables called cumulative distribution networks cdns unlike graphs for probability density and mass functions for cdfs the marginal probabilities for any subset of variables are obtained by computing limits of functions in the model and conditional probabilities correspond to computing mixed derivatives we will show that the conditional independence properties in a cdn are distinct from the conditional independence properties of directed undirected and factor graphs but include the conditional independence properties of bi directed graphs in order to perform inference in such models we describe the derivative sum product dsp message passing algorithm in which messages correspond to derivatives of the joint cdf we will then apply cdns to the problem of learning to rank players in multiplayer team based games and suggest several future directions for research

hyper markov laws in the statistical analysis of decomposable graphical models this paper introduces and investigates the notion of a hyper markov law which is a probability distribution over the set of probability measures on a multivariate space that i is concentrated on the set of markov probabilities over some decomposable graph and ii satisfies certain conditional independence restrictions related to that graph a stronger version of this hyper markov property is also studied our analysis starts by reconsidering the properties of markov probabilities using an abstract approach which thereafter proves equally applicable to the hyper markov case next it is shown constructively that hyper markov laws exist that they appear as sampling distributions of maximum likelihood estimators in decomposable graphical models and also that they form natural conjugate prior distributions for a bayesian analysis of these models as examples we construct a range of specific hyper markov laws including the hyper multinomial hyper dirichlet and the hyper wishart and inverse wishart laws these laws occur naturally in connection with the analysis of decomposable log linear and covariance selection models

estimating high dimensional intervention effects from observational data we assume that we have observational data generated from an unknown underlying directed acyclic graph dag model a dag is typically not identifiable from observational data but it is possible to consistently estimate the equivalence class of a dag moreover for any given dag causal effects can be estimated using intervention calculus in this paper we combine these two parts for each dag in the estimated equivalence class we use intervention calculus to estimate the causal effects of the covariates on the response this yields a collection of estimated causal effects for each covariate we show that the distinct values in this set can be consistently estimated by an algorithm that uses only local information of the graph this local approach is computationally fast and feasible in high dimensional problems we propose to use summary measures of the set of possible causal effects to determine variable importance in particular we use the minimum absolute value of this set since that is a lower bound on the size of the causal effect we demonstrate the merits of our methods in a simulation study and on a data set about riboflavin production

large sample properties of separable nonlinear least squares estimators in this paper the large sample properties of the separable nonlinear least squares algorithm are investigated unlike the previous results in the literature the data are assumed to be complex valued and the whiteness assumption on the measurement noise sequence has been relaxed convergence properties of the parameter estimates are established asymptotic accuracy analysis has been carried out in which the assumptions used are relatively weaker than the assumptions in the previous related works it is shown under quite general conditions that the parameter estimates are asymptotically circular conditions for asymptotic complex normality are also established next a bound on the deviation of the asymptotic covariance matrix from the crame spl acute r rao bound crb is derived finally a sufficient condition for the nonlinear least squares estimate to achieve the crame spl acute r rao lower bound is established the results presented in this paper are general and can be applied to any specific application where separable nonlinear least squares is employed

covariate selection for the nonparametric estimation of an average treatment effect observational studies in which the effect of a nonrandomized treatment on an outcome of interest is estimated are common in domains such as labour economics and epidemiology such studies often rel

laplacian growth and diffusion limited aggregation different universality classes it had been conjectured that diffusion limited aggregates and laplacian growth patterns with small surface tension are in the same universality class using iterated conformal maps we construct a one parameter family of fractal growth patterns with a continuously varying fractal dimension this family can be used to bound the dimension of laplacian growth patterns from below the bound value is higher than the dimension of diffusion limited aggregates showing that the two problems belong to two different universality classes

on the causal interpretation of acyclic mixed graphs under multivariate normality abstract in multivariate statistics acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden i e latent or unobserved variables indeed under multivariate normality every mixed graph corresponds to a set of covariance matrices that contains as a full dimensional subset the covariance matrices associated with a causally interpretable acyclic digraph this digraph generally has some of its nodes corresponding to hidden variables we seek to clarify for which mixed graphs there exists an acyclic digraph whose hidden variable model coincides with the mixed graph model restricting to the tractable setting of chain graphs and multivariate normality we show that decomposability of the bidirected part of the chain graph is necessary and sufficient for equality between the mixed graph model and some hidden variable model given by an acyclic digraph

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

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

model selection for gaussian concentration graphs gga multivariate gaussian graphical markov model for an undirected graph g also called a covariance selection model or concentration graph model is defined in terms of the markov properties i e conditional independences associated with g which in turn are equivalent to specified zeros among the set of pairwise partial correlation coefficients by means of fisher s z transformation and sidak s correlation inequality conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion the simultaneous p values corresponding to the partial correlations are partitioned into three disjoint sets a significant set s an indeterminate set i and a nonsignificant set n our model selection method selects two graphs a graph g si whose edges correspond to the set s i and a more conservative graph g s whose edges correspond to s only similar considerations apply to covariance graph models which are defined in terms of marginal independence rather than conditional independence the method is applied to some well known examples and to simulated data

electrical neuroimaging based on biophysical constraints this paper proposes and implements biophysical constraints to select a unique solution to the bioelectromagnetic inverse problem it first shows that the brain s electric fields and potentials are predominantly due to ohmic currents this serves to reformulate the inverse problem in terms of a restricted source model permitting noninvasive estimations of local field potentials lfps in depth from scalp recorded data uniqueness in the solution is achieved by a physically derived regularization strategy that imposes a spatial structure on the solution based upon the physical laws that describe electromagnetic fields in biological media the regularization strategy and the source model emulate the properties of brain activity s actual generators this added information is independent of both the recorded data and head model and suffices for obtaining a unique solution compatible with and aimed at analyzing experimental data the inverse solution s features are evaluated with event related potentials erps from a healthy subject performing a visuo motor task two aspects are addressed the concordance between available neurophysiological evidence and inverse solution results and the functional localization provided by fmri data from the same subject under identical experimental conditions the localization results are spatially and temporally concordant with experimental evidence and the areas detected as functionally activated in both imaging modalities are similar providing indices of localization accuracy we conclude that biophysically driven inverse solutions offer a novel and reliable possibility for studying brain function with the temporal resolution required to advance our understanding of the brain s functional networks