learning equivalence classes of bayesian network structures two bayesian network structures are said to be equivalent if the set of distributions that can be represented with one of those structures is identical to the set of distributions that can be represented with the other many scoring criteria that are used to learn bayesian network structures from data are score equivalent that is these criteria do not distinguish among networks that are equivalent in this paper we consider using a score equivalent criterion in conjunction with a heuristic search algorithm to perform model selection or model averaging we argue that it is often appropriate to search among equivalence classes of network structures as opposed to the more common approach of searching among individual bayesian network structures we describe a convenient graphical representation for an equivalence class of structures and introduce a set of operators that can be applied to that representation by a search algorithm to move among equivalence classes we show that our equivalence class operators can be scored locally and thus share the computational efficiency of traditional operators defined for individual structures we show experimentally that a greedy model selection algorithm using our representation yields slightly higher scoring structures than the traditional approach without any additional time overhead and we argue that more sophisticated search algorithms are likely to benefit much more

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

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

scaling properties of a scattering system with an incomplete horseshoe we investigate a parameter dependent map describing a chaotic scattering system in parameter ranges leading to an incomplete horseshoe we construct an approximate symbolic dynamics which describes quite well the hyperbolic component of the invariant set its grammatical rules are correlated with the convergence properties of the thermodynamical formalism for the measures characterizing the invariant set

estimating high dimensional directed acyclic graphs with the pc algorithm we consider the pc algorithm spirtes et al 2000 for estimating the skeleton and equivalence class of a very high dimensional directed acyclic graph dag with corresponding gaussian distribution the pc algorithm is computationally feasible and often very fast for sparse problems with many nodes variables and it has the attractive property to automatically achieve high computational efficiency as a function of sparseness of the true underlying dag we prove uniform consistency of the algorithm for very high dimensional sparse dags where the number of nodes is allowed to quickly grow with sample size n as fast as o na for any 0 a the sparseness assumption is rather minimal requiring only that the neighborhoods in the dag are of lower order than sample size n we also demonstrate the pc algorithm for simulated data

the numerical computation of homoclinic orbits for maps transversal homoclinic orbits of maps are known to generate shift dynamics on a set with cantor like structure in this paper a numerical method is developed for computation of the corresponding homoclinic orbits they are approximated by finite orbit segments subject to asymptotic boundary conditions we provide a detailed error analysis including a shadowing type result by which one can infer the existence of a transversal homoclinic orbit from a finite segment this approach is applied to several examples in some of them parameters appear and closed loops of homoclinic orbits are found by a path following algorithm

counting and exploring sizes of markov equivalence classes of directed acyclic graphs when learning a directed acyclic graph dag model via observational data one generally cannot identify the underlying dag but can potentially obtain a markov equivalence class the size the number of dags of a markov equivalence class is crucial to infer causal effects or to learn the exact causal dag via further interventions given a set of markov equivalence classes the distribution of their sizes is a key consideration in developing learning methods however counting the size of an equivalence class with many vertices is usually computationally infeasible and the existing literature reports the size distributions only for equivalence classes with ten or fewer vertices in this paper we develop a method to compute the size of a markov equivalence class we first show that there are five types of markov equivalence classes whose sizes can be formulated as five functions of the number of vertices respectively then we introduce a new concept of a rooted sub class the graph representations of rooted subclasses of a markov equivalence class are used to partition this class recursively until the sizes of all rooted subclasses can be computed via the five functions the proposed size counting is efficient for markov equivalence classes of sparse dags with hundreds of vertices finally we explore the size and edge distributions of markov equivalence classes and find experimentally that in general 1 most markov equivalence classes are half completed and their average sizes are small and 2 the sizes of sparse classes grow approximately exponentially with the numbers of vertices

weighted least squares solutions of the equation axb c 0 abstract let h be a hilbert space l h the algebra of bounded linear operators on h and w l h a positive operator such that w 1 2 is in the p schatten class for some 1 p given a b l h with closed range and c l h we study the following weighted approximation problem analyze the existence of 0 1 m i n x l h a x b c p w where x p w w 1 2 x p we also study the related operator approximation problem analyze the existence of 0 2 m i n x l h a x b c w a x b c where the order is the one induced in l h by the cone of positive operators in this paper we prove that the existence of the minimum of 0 2 is equivalent to the existence of a solution of the normal equation a w a x b c 0 we also give sufficient conditions for the existence of the minimum of 0 1 and we characterize the operators where the minimum is attained

active learning of causal networks with intervention experiments and optimal designs the causal discovery from data is important for various scientific investigations because we cannot distinguish the different directed acyclic graphs dags in a markov equivalence class learned from observational data we have to collect further information on causal structures from experiments with external interventions in this paper we propose an active learning approach for discovering causal structures in which we first find a markov equivalence class from observational data and then we orient undirected edges in every chain component via intervention experiments separately in the experiments some variables are manipulated through external interventions we discuss two kinds of intervention experiments randomized experiment and quasi experiment furthermore we give two optimal designs of experiments a batch intervention design and a sequential intervention design to minimize the number of manipulated variables and the set of candidate structures based on the minimax and the maximum entropy criteria we show theoretically that structural learning can be done locally in subgraphs of chain components without need of checking illegal v structures and cycles in the whole network and that a markov equivalence subclass obtained after each intervention can still be depicted as a chain graph

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

some norm inequalities concerning generalized inverses 2 abstract we extend to the von neumann schatten classes c p and norms psome inequalities concerning generalized inverses we extend penrose s result on minimizing ax b 2 to the p norm we also minimize x p where x varies such that 1 b a ax a ab and 2 a ax a in both cases a b fixed local considerations are adduced in each result here global minima and critical points coincide

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

nonlinear modes in coupled rotator models abstract we compare the properties of generalized nonlinear klein gordon equations with a sinusoidal coupling sine lattice equations to those of the usual nonlinear klein gordon equation with harmonic coupling for three equations leading to the same sine gordon equation in the continuum limit using hirota s bilinear method we find a strong analogy between the properties of nonlinear oscillatory modes in the three equations numerical simulations confirm this analogy even for very discrete cases however we point out a fundamental difference between the harmonic and sinusoidal coupling by exhibiting a new class of localized nonlinear excitations which are specific to the sinusoidal coupling the rotating modes they can be created thermally an approximate analytical solution is obtained for these modes which are intrinsically discrete and appear to be extremely stable even in collisions with breathers possible implications for physics are discussed

breathers and multibreathers in a periodically driven damped discrete nonlinear schrodinger equation we study an integrable discretization of the nonlinear schrodinger equation nls under the effects of damping and periodic driving from the point of view of spatially localized solutions oscillating in time with the driver s frequency we locate the equilibrium states of the discretized dnls system in the plane of its dissipation gamma and forcing amplitude h parameters and use a shooting algorithm to construct the desired solutions psi n t phi n exp it as homoclinic orbits of a four dimensional symplectic map in the complex phi n phi n 1 space for infinity n infinity we derive in the gamma 0 case closed form expressions for two fundamental such solutions having a single hump in n psi n and psi n and determine analytically their threshold of existence in the gamma h plane using mel nikov s theory then we demonstrate numerically that above this threshold a remarkable variety of multihump structures appear whose complexity in terms of their spatial extrema grows with increasing h all these solutions are numerically found to be unstable in time except for psi n which is seen to be stable over a certain region in the gamma h plane in the continuum limit our results are in close agreement with recent studies on the nls equation from a more general perspective we view these dnls multihump solutions as homoclinic orbits of a higher dimensional map thereby providing a possible mechanism for explaining the occurrence of similar structures called discrete multi breathers found in a wide variety of one dimensional nonlinear lattices

dynamics for a two dimensional antisymmetric map a two dimensional representation of symbolic dynamics for a two dimensional map with antisymmetric property is constructed the behaviour of symmetry breaking periodic orbits symmetry breaking bifurcations symmetry breaking attractors and the boundaries of basins between co existing attractors are then discussed using this representation of symbolic dynamics

conditions on the existence of localized excitations in nonlinear discrete systems we use recent results that localized excitations in nonlinear hamiltonian lattices can be viewed and described as multiple frequency excitations their dynamics in phase space takes place on tori of corresponding dimension for a one dimensional hamiltonian lattice with nearest neighbour interaction we transform the problem of solving the coupled differential equations of motion into a certain mapping m l 1 f m l m l 1 where m l for every l lattice site is a function defined on an infinite discrete space of the same dimension as the torus we consider this mapping in the tails of the localized excitation i e for l rightarrow pm infty for a generic hamiltonian lattice the thus linearized mapping is analyzed we find conditions of existence of periodic one frequency localized excitations as well as of multiple frequency excitations the symmetries of the solutions are obtained as a result we find that the existence of localized excitations can be a generic property of nonlinear hamiltonian lattices in contrast to nonlinear hamiltonian fields

weighted procrustes problems abstract let h be a hilbert space l h the algebra of bounded linear operators on h and w l h a positive operator such that w 1 2 is in the p schatten class for some 1 p given a l h with closed range and b l h we study the following weighted approximation problem analyze the existence of m i n x l h a x b p w where x p w w 1 2 x p in this paper we prove that the existence of this minimum is equivalent to a compatibility condition between r b and r a involving the weight w and we characterize the operators which minimize this problem as w inverses of a in r b

a mel nikov approach to soliton like solutions of systems of discretized nonlinear schro dinger equations we investigate a class of n coupled discretized nonlinear schrodinger equations of interacting chains in a nonlinear lattice which in the limit of zero coupling become integrable ablowitz ladik differential difference equations we study the existence of stationary localized excitations in the form of soliton like time periodic states by reducing the system to a perturbed 2n dimensional symplectic map whose homoclinic orbits are obtained by a recently developed mel nikov analysis we find that depending on the perturbation homoclinic orbits can be accurately located from the simple zeros of a mel nikov vector and illustrate our results in the cases n 2 and 3

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

partition mcmc for inference on acyclic digraphs abstractacyclic digraphs are the underlying representation of bayesian networks a widely used class of probabilistic graphical models learning the underlying graph from data is a way of gaining insights about the structural properties of a domain structure learning forms one of the inference challenges of statistical graphical models markov chain monte carlo mcmc methods notably structure mcmc to sample graphs from the posterior distribution given the data are probably the only viable option for bayesian model averaging score modularity and restrictions on the number of parents of each node allow the graphs to be grouped into larger collections which can be scored as a whole to improve the chain s convergence current examples of algorithms taking advantage of grouping are the biased order mcmc which acts on the alternative space of permuted triangular matrices and nonergodic edge reversal moves here we propose a novel algorithm which employs the underlying combinatorial structure of dags t

obtaining breathers in nonlinear hamiltonian lattices we present a numerical method for obtaining high accuracy numerical solutions of spatially localized time periodic excitations on a nonlinear hamiltonian lattice we compare these results with analytical considerations of the spatial decay we show that nonlinear contributions have to be considered and we obtain very good agreement between the latter and numerical results we discuss further applications of the method and results

reversible mcmc on markov equivalence classes of sparse directed acyclic graphs author s he y jia j yu b abstract graphical models are popular statistical tools which are used to represent dependent or causal complex systems statistically equivalent causal or directed graphical models are said to belong to a markov equivalent class it is of great interest to describe and understand the space of such classes however with currently known algorithms sampling over such classes is only feasible for graphs with fewer than approximately 20 vertices in this paper we design reversible irreducible markov chains on the space of markov equivalent classes by proposing a perfect set of operators that determine the transitions of the markov chain the stationary distribution of a proposed markov chain has a closed form and can be computed easily specifically we construct a concrete perfect set of operators on sparse markov equivalence classes by introducing appropriate conditions on each possible operator algorithms and their accelerated versions are provided to efficiently generate markov chains and to explore properties of markov equivalence classes of sparse directed acyclic graphs dags with thousands of vertices we find experimentally that in most markov equivalence classes of sparse dags 1 most edges are directed 2 most undirected subgraphs are small and 3 the number of these undirected subgraphs grows approximately linearly with the number of vertices institute of mathematical statistics 2013

a method for locating symmetric homoclinic orbits using symbolic dynamics in this paper we present a method which can identify and locate symmetric homoclinic orbits in a homoclinic tangle formed by the intersecting stable and unstable manifolds of a symmetric 2d map the method consists of a systematic search in parameter space and determination of the order in which these orbits arise using symbolic dynamics each orbit corresponds to a unique sequence and it is computed by iterating the map along the unstable manifold to match a specific symmetry at the middle of the orbit an application of the method to the determination of multibreather solutions of 1d lattices is discussed

the size distribution for markov equivalence classes of acyclic digraph models bayesian networks equivalently graphical markov models determined by acyclic digraphs or adgs also called directed acyclic graphs or dags have proved to be both effective and efficient for representing complex multivariate dependence structures in terms of local relations however model search and selection is potentially complicated by the many to one correspondence between adgs and the statistical models that they represent if the adgs models ratio is large search procedures based on unique graphical representations of equivalence classes of adgs could provide substantial computational efficiency hitherto the value of the adgs models ratio has been calculated only for graphs with n 5 or fewer vertices in the present study a computer program was written to enumerate the equivalence classes of adg models and study the distributions of class sizes and number of edges for graphs up to n 10 vertices the ratio of adgs to numbers of classes appears to approach an asymptote of about 3 7 distributions of the classes according to number of edges and class size were produced which also appear to be approaching asymptotic limits imposing a bound on the maximum number of parents to any vertex causes little change if the bound is sufficiently large with four being a possible minimum the program also includes a new variation of orderly algorithm for generating undirected graphs

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived

norm inequalities for certain operator sums abstract we extend to larger classes of norms some inequalities concerning sums of positive operators and sums of operators having orthogonal ranges these inequalities are useful in the theories of best approximation in c algebras and generalized inverses

anti integrability in dynamic and variational problems abstract we show using a series of examples that the existence of a singular limit called anti integrable can be useful for proving the existence of chaotic solutions and elucidating some of their properties leading to some new and unexpected results in particular the existence proof of chaotic trajectories in large classes of dynamical systems with discrete time becomes straightforward the method applies even for maps in infinite dimension symplectic or not similar ideas can be used for proving the existence of breathers or self localised modes and of chaotic distributions of breathers in any array of weakly coupled non linear oscillators with continuous time the existence of insulating bipolaronic structures in electron phonon systems electronic self localisation at large electron phonon coupling can also be proven on the same basis