automatic mesh generation and adaptation by using contours a general and efficient remeshing algorithm is presented for the discretization of arbitrary planar domains into triangular elements in consistency with the given node spacing funstion the contour lines of the node spacing function at suitable calculated levels provide the natural lines of division of the problem domain into subregions where finite elements meshes of different element sizes zre generated using the available general purpose mesh generators

on line computation of minimal and maximal length paths abstract we consider the problem of maintaining minimum length paths in a directed graph g v e with n nodes while inserting new arcs a data structure which supports the following operations is presented an add operation which inserts an arc in the digraph and a minpath operation which returns a minimal length path between a pair of nodes the data structure supports each minpath operation in o k worst case time where k n is the length of the returned path moreover if we assume that the weights of the arcs are integer numbers in the range 1 w then the expected cost of any sequence of add operations is o min n 4 n 3 max w log n time the space complexity is o n 2 the same algorithm can be used for solving the problem of maintaining maximum length paths when the digraph is acyclic and add operations preserve acyclicity

learning minimal latent directed information polytrees we propose an approach for learning latent directed polytrees as long as there exists an appropriately defined discrepancy measure between the observed nodes specifically we use our approach for learning directed information polytrees where samples are available from only a subset of processes directed information trees are a new type of probabilistic graphical models that represent the causal dynamics among a set of random processes in a stochastic system we prove that the approach is consistent for learning minimal latent directed trees we analyze the sample complexity of the learning task when the empirical estimator of mutual information is used as the discrepancy measure

short run and long run causality in time series theory causality in the sense of granger is typically defined in terms of predictibility of a vector of variables one period ahead recently liitkepohl 1993 proposed to define noncausality between two variables in terms of nonpredictibility at any number of periods ahead when more than two vectors are considered i e when the information set contains auxiliary variables these two notions are not equivalent in this paper we first generalize the notion of causality by considering causality at a given arbitrary horizon h then we derive necessary and sufficient conditions for noncausality between vectors of variables inside a larger vector up to any given horizon h where h can be infinite in particular for general possibly nonstationary processes with finite second moments relatively simple exhaustivity and separation conditions which are sufficient for noncausality at all horizons are provided to deal with cases where such conditions do not apply we consider a more specific although still very wide class of vector autoregressive processes possibly of infinite order stationary or nonstationary which include multivariate arima processes and we derive general parametric characterizations of noncausality at various horizons for this class including a causality chain characterization we also observe that the coefficients of lagged variables in forecasts at various horizons h 1 can be interpreted as generalized impulse response coefficients which yield a complete picture of linear causality properties in contrast with usual response coefficients which can be quite misleading in this respect

fully dynamic planarity testing with applications this paper introduces compressed certificates for planarity biconnectivity and triconnectivity in planar graphs and proves many structural properties of certificates in planar graphs as an application of our compressed certificates we develop efficient dynamic planar algorithms in particular we consider the following three operations on a planar graph g i insert an edge if the resultant graph remains planar ii delete an edge and iii test whether an edge could be added to the graph without violating planarity we show how to support each of the above operations in o n 2 3 time where n is the number of vertices in the graph the bound for tests and deletions is worst case while the bound for insertions is amortized this is the first algorithm for this problem with sub linear running time and it affirmatively answers a question posed in epstein et al 1992 we use our compressed certificates for biconnectivity and triconnectivity to maintain the biconnected and triconnected components of a dynamic planar graph the time bounds are the same o n 2 3 worst case time per edge deletion o n 2 3 amortized time per edge insertion and o n 2 3 amortized time per edge insertion and o n 2 3 worst case time to check whether two vertices are either biconnected or triconnected

plotting of contours in a natural way a simple and straightforward scheme for plotting contours is introduced for any function the appropriate contours are traced on the paths of constant function values disclosing the information contained in the shape functions the exactness of the contours matches that of the finite element analysis extreme value considerations must be carried out to decide whether contours appear in an element at all three co ordinate systems are dealt with the local the global and the screen co ordinate system and the necessary mutual transformations a short section deals with the treatment of contour discontinuities across element borders which frequently occur with stresses and heat fluxes finally some examples are shown

granger causality and path diagrams for multivariate time series in this paper we introduce path diagrams for multivariate time series which visualize the dynamic relationships among the variables in these path diagrams the vertices represent the components of the time series and are connected by arrows or lines according to the nonvanishing parameters in the autoregressive representation of the time series we show that these path diagrams provide a framework for the analysis of the dependence structure of the time series in particular we give sufficient graphical conditions for granger noncausality and granger noncausality up to a certain horizon

evolutionary coherence of nonstationary signals coherence is a widely used measure for characterizing linear dependence between a pair of signals for nonstationary signals the autospectrum cross spectrum and coherence between signals may evolve over time a standard approach is to divide the signals into overlapping blocks of fixed width and then smooth over frequency the periodogram matrix at each time block in this paper a consistent estimation procedure is developed using time localized linear filtering the proposed method automatically selects via repeated tests of homogeneity the optimal window width for estimating local coherence it is pointwise adaptive in the sense that the width of the optimal interval is allowed to change across time under the locally stationary process framework we develop a central limit theorem on the fisher z transform of our time localized band coherence we apply our method to a pair of highly dynamic brain waves signals whose coherence is shown to evolve during an epileptic seizure

causal diagrams for empirical research summary the primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject matter information in particular the paper develops a principled nonparametric framework for causal inference in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data if so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions otherwise the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained

algorithms and data structures for an expanded family of matroid intersection problems consider a matroid of rank n in which each element has a real valued cost and one of d 1 colors a class of matroid intersection problems is studied in which one of the matroids is a partition matroid that specifies that a base has q j elements of color j for j 1 2 cdots d relationships are characterized among the solutions to the family of problems generated when the vector q 1 q 2 cdots q d is allowed to range over all values that sum to n a fast algorithm is given for solving such matroid intersection problems when d is small a characterization is presented for how the solution changes when one element changes in cost data structures are given for updating the solution on line each time the cost of an arbitrary matroid element is modified efficient update algorithms are given for maintaining a color constrained minimum spanning tree in either a general or a planar graph an application of the techniques to the problem of finding a minimum spanning tree with several degree

framework to study dynamic dependencies in networks of interacting processes the analysis of dynamic dependencies in complex systems such as the brain helps to understand how emerging properties arise from interactions here we propose an information theoretic framework to analyze the dynamic dependencies in multivariate time evolving systems this framework constitutes a fully multivariate extension and unification of previous approaches based on bivariate or conditional mutual information and granger causality or transfer entropy we define multi information measures that allow us to study the global statistical structure of the system as a whole the total dependence between subsystems and the temporal statistical structure of each subsystem we develop a stationary and a nonstationary formulation of the framework we then examine different decompositions of these multi information measures the transfer entropy naturally appears as a term in some of these decompositions this allows us to examine its properties not as an isolated measure of interdependence but in the context of the complete framework more generally we use causal graphs to study the specificity and sensitivity of all the measures appearing in these decompositions to different sources of statistical dependence arising from the causal connections between the subsystems we illustrate that there is no straightforward relation between the strength of specific connections and specific terms in the decompositions furthermore causal and noncausal statistical dependencies are not separable in particular the transfer entropy can be nonmonotonic in dependence on the connectivity strength between subsystems and is also sensitive to internal changes of the subsystems so it should not be interpreted as a measure of connectivity strength altogether in comparison to an analysis based on single isolated measures of interdependence this framework is more powerful to analyze emergent properties in multivariate systems and to characterize functionally relevant changes in the dynamics

fully dynamic algorithms for bin packing being mostly myopic helps the problem of maintaining an approximate solution for one dimensional bin packing when items may arrive and depart dynamically is studied in accordance with various work on fully dynamic algorithms and in contrast to prior work on bin packing it is assumed that the packing may be arbitrarily rearranged to accommodate arriving and departing items in this context our main result is a fully dynamic approximation algorithm for bin packing mmp that is frac 5 4 competitive and requires theta log n time per operation i e for an insert or a delete of an item this competitive ratio of frac 5 4 is nearly as good as that of the best practical off line algorithms our algorithm utilizes the technique introduced here whereby the packing of an item is done with a total disregard for already packed items of a smaller size this myopic packing of an item may then cause several smaller items to be repacked in a similar fashion with a bit of additional sophistication to avoid certain bad cases the number of items either individual items or bundles of very small items treated as a whole that needs to be repacked is bounded by a constant

an algorithm for finding k minimum spanning trees this paper presents an algorithm for finding k minimum spanning trees in an undirected graph the required time is o km min n 2 m log log n and the space is o k m where n is the number of vertices and m is the number of edges the algorithm is based on three subroutines the first two subroutines are used to obtain the second minimum spanning tree in o min n 2 m alpha m n steps where alpha m n is tarjan s inverse of ackermann s function 12 which is very slowly growing the third one obtains the kth minimum spanning tree in o m steps when the jth minimum spanning trees for j 1 2 cdots k 1 are given

on line updating of solutions to a class of matroid intersection problems abstract the class of matroid intersection problems is considered in which one of the matroids is a partition matroid specifying that exactly q elements in the solution must be red and the rest green a characterization is presented for how the solution changes when one element changes in cost data structures are given for updating the solution on line each time the cost of an arbitrary matroid element is modified efficient update algorithms are given for maintaining a red green minimum spanning tree in both a general and a planar graph and a red green job schedule for unit time jobs with deadlines

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

an on line edge deletion problem there is given an undirected graph g v e from which edges are deleted one at a time and about which questions of the type are the vertices u and v in the same connected component have to be answered on line there is presented an algorithm which maintains a data structure in which each question is answered in constant time and for which the total time involved in answering q questions and maintaining the data structure is o q i vi led

maintaining minimum spanning forests in dynamic graphs we present the first fully dynamic algorithm for maintaining a minimum spanning forest in time o sqrt n per operation to be precise the algorithm uses o n1 3 log n amortized time per update operation the algorithm is fairly simple and deterministic an immediate consequence is the first fully dynamic deterministic algorithm for maintaining connectivity and bipartiteness in amortized time o n1 3 log n per update with o 1 worst case time per query

improved data structures for fully dynamic biconnectivity we present fully dynamic algorithms for maintaining the biconnected components in general and plane graphs a fully dynamic algorithm maintains a graph during a sequence of insertions and deletions of edges or isolated vertices let m be the number of edges and n be the number of vertices in a graph the time per operation of the best deterministic algorithms is o sqrt n in general graphs and o log n in plane graphs for fully dynamic connectivity and o min m2 3 n in general graphs and o sqrt n in plane graphs for fully dynamic biconnectivity we improve the later running times to o sqrt m log n in general graphs and o log 2 n in plane graphs our algorithm for general graphs can also find the biconnected components of all vertices in time o n

separator based sparsification ii edge and vertex connectivity we consider the problem of maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding we describe algorithms and data structures for maintaining information about 2 and 3 vertex connectivity and 3 and 4 edge connectivity in a planar graph in o n1 2 amortized time per insertion deletion or connectivity query all of the data structures handle insertions that keep the graph planar without regard to any particular embedding of the graph our algorithms are based on a new type of sparsification combined with several properties of separators in planar graphs

graphical modelling of multivariate time series we introduce graphical time series models for the analysis of dynamic relationships among variables in multivariate time series the modelling approach is based on the notion of strong granger causality and can be applied to time series with non linear dependences the models are derived from ordinary time series models by imposing constraints that are encoded by mixed graphs in these graphs each component series is represented by a single vertex and directed edges indicate possible granger causal relationships between variables while undirected edges are used to map the contemporaneous dependence structure we introduce various notions of granger causal markov properties and discuss the relationships among them and to other markov properties that can be applied in this context examples for graphical time series models include nonlinear autoregressive models and multivariate arch models

conditional independence graph for nonlinear time series and its application to international financial markets abstract conditional independence graphs are proposed for describing the dependence structure of multivariate nonlinear time series which extend the graphical modeling approach based on partial correlation the vertexes represent the components of a multivariate time series and edges denote direct dependence between corresponding series the conditional independence relations between component series are tested efficiently and consistently using conditional mutual information statistics and a bootstrap procedure furthermore a method combining information theory with surrogate data is applied to test the linearity of the conditional dependence the efficiency of the methods is approved through simulation time series with different linear and nonlinear dependence relations finally we show how the method can be applied to international financial markets to investigate the nonlinear independence structure

randomized fully dynamic graph algorithms with polylogarithmic time per operation this paper solves a longstanding open problem in fully dynamic algorithms we present the first fully dynamic algorithms that maintain connectivity bipartiteness and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion the algorithms are designed using a new dynamic technique that combines a novel graph decomposition with randomization they are las vegas type randomized algorithms which use simple data structures and have a small constant factor let n denote the number of nodes in the graph for a sequence of o m 0 operations where m 0 is the number of edges in the initial graph the expected time for p updates is o p log 3 n througout the paper the logarithms are based 2 for connectivity and bipartiteness the worst case time for one query is o log n log log n for the k edge witness problem does the removal of k given edges disconnect the graph the expected time for p updates is o p log 3 n and the expected time for q queries is o qk log 3 n given a graph with k different weights the minimum spanning tree can be maintained during a sequence of p updates in expected time o pk log 3 n this implies an algorithm to maintain a 1 e approximation of the minimum spanning tree in expected time o p log 3 n log u e for p updates where the weights of the edges are between 1 and u

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

maintenance of 2 and 3 edge connected components of graphs ii data structures and algorithms are presented to efficiently maintain the 2 and 3 edge connected components of a general graph under insertions of edges and nodes in the graph at any moment the data structure can answer whether two nodes are 2 or 3 edge connected the algorithms run in o n alpha m n time where m is the total number of queries and edge insertions furthermore a linear time algorithm is presented for maintaining the 2 edge connected components in case the initial graph is connected finally a new solution is presented for the 2 vertex connected components of a graph

preemptive ensemble motion planning on a tree consider the problem of finding a minimum cost tour to transport a set of objects between the vertices of a tree by a vehicle that travels along the edges of the tree the vehicle can carry only one object at a time and it starts and finishes at the same vertex of the tree it is shown that if objects can be dropped at intermediate vertices along its route and picked up later then the problem can be solved in polynomial time two efficient algorithms are presented for this problem the first algorithm runs in o k qn time where n is the number of vertices in the tree k is the number of objects to be moved and q leq min k n is the number of nontrivial connected components in a related directed graph the second algorithm runs in o k n log n time

causal conditioning and instantaneous coupling in causality graphs this study aims at providing the definitive links between massey and kramer s directed information theory and granger causality graphs recently formalized by eichler this naturally leads to consider two concepts of causality that can occur in physical systems dynamical causality and instantaneous coupling although it is well accepted that the former is assessed by conditional transfer entropy the latter is often overlooked even if it was clearly introduced and understood in seminal studies in the bivariate case we show that directed information is decomposed into the sum of transfer entropy and instantaneous information exchange in the multivariate case encountered for conditional graph modeling such a decomposition does not hold anymore we provide a new insight into the problem of instantaneous coupling and show how it affects the estimated structure of a graphical model that should provide a sparse representation of a complex system this result is discussed and analyzed for the inference of causality graphs two synthetic examples are developed to illustrate the theoretical concepts practical issues are also briefly discussed on these examples and an extension of leonenko s k nearest neighbors based entropy estimators is used to derive a nonparametric estimator of directed information

impulse response functions based on a causal approach to residual orthogonalization in vector autoregressions abstract a data determined method for testing structural models of the errors in vector autoregressions is discussed the method can easily be combined with prior economic knowledge and a subjective analysis of data characteristics to yield valuable information concerning model selection and specification in one dimension it turns out that standard t statistics can be used to test the various overidentifying restrictions that are implied by a model in another dimension the method compares a priori knowledge of a structural model for the errors with the properties exhibited by the data thus this method may help to ensure that orderings of the errors for impulse response and forecast error variance decomposition analyses are sensible given the data two economic examples are used to illustrate the method

dynamic trees as search trees via euler tours applied to the network simplex algorithm thedynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges linking and splitting by deleting edges cutting while at the same time allowing reporting of certain combinations of vertex or edge values for many applications of dynamic trees values must be combined along paths for other applications values must be combined over entire trees for the latter situation an idea used originally in parallel graph algorithms to represent trees by euler tours leads to a simple implementation with a time of o logn per tree operation wheren is the number of tree vertices we apply this representation to the implementation of two versions of the network simplex algorithm resulting in a time of o logn per pivot wheren is the number of vertices in the problem network

sparsification a technique for speeding up dynamic graph algorithms we provide data strutures that maintain a graph as edges are inserted and deleted and keep track of the following properties with the following times minimum spanning forests graph connectivity graph 2 edge connectivity and bipartiteness in time o n 1 2 per change 3 edge connectivity in time o n 2 3 per change 4 edge connectivity in time o n n per change k edge connectivity for constant k in time o n log n per change 2 vertex connectivity and 3 vertex connectivity in the o n per change and 4 vertex connectivity in time o n n per change further results speed up the insertion times to match the bounds of known partially dynamic algorithms all our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph which we call sparsification

data structures for on line updating of minimum spanning trees with applications data structures are presented for the problem of maintaining a minimum spanning tree on line under the operation of updating the cost of some edge in the graph for the case of a general graph mai

dynamic 2 connectivity with backtracking we give algorithms and data structures that maintain the 2 edge and 2 vertex connected components of a graph under insertions and deletions of edges and vertices where deletions occur in a backtracking fashion i e deletions undo the insertions in the reverse order our algorithms run in theta log n worst case time per operation and use theta n space where n is the number of vertices using our data structure we can answer queries which ask whether vertices u and v belong to the same 2 connected component in theta log n worst case time

continuous stress fields by the finite element difference method standard finite elements yield a stress field that is discontinuous across interelement boundaries in this paper results given by standard finite elements are postprocessed by an efficient iterative scheme that makes use of nodal strains finite difference operations and interpolations that augment and may differ from the shape functions used to generate the element stiffness matrix various types of interpolation are considered each leads to a continuous stress field that is more accurate than the unprocessed stress field

on line planarity testing the on line planarity testing problem consists of performing the following operations on a planar graph g i testing if a new edge can be added to g so that the resulting graph is itself planar ii adding vertices and edges such that planarity is preserved an efficient technique for on line planarity testing of a graph is presented that uses o n space and supports tests and insertions of vertices and edges in o log n time where n is the current number of vertices of g the bounds for tests and vertex insertions are worst case and the bound for edge insertions is amortized we also present other applications of this technique to dynamic algorithms for planar graphs

fast algorithms for shortest paths in planar graphs with applications graph decomposition and data structures techniques are presented that exploit the structure of planar graphs to yield faster algorithms for a number of shortest path and related problems improved

geometric and long run aspects of granger causality abstract this paper extends multivariate granger causality to take into account the subspaces along which granger causality occurs as well as long run granger causality the properties of these new notions of granger causality along with the requisite restrictions are derived and extensively studied for a wide variety of time series processes including linear invertible processes and varma using the proposed extensions the paper demonstrates that i mean reversion in l 2 is an instance of long run granger non causality ii cointegration is a special case of long run granger non causality along a subspace iii controllability is a special case of granger causality and finally iv linear rational expectations entail possibly testable granger causality restriction along subspaces

dynamic and static algorithms for optimal placement of resources in a tree abstract we consider the problem of optimally placing identical resources at the nodes of a weighted tree shaped network of size n the resources satisfy requests issued from the nodes of the network the cost of a placement of the resources is the sum over all nodes of the network of the product of the node weight times the distance from the closest resource the static problem consists in determining the minimal cost placement of the resources the dynamic version consists in recomputing such a cost when a node weight has changed we present a linear time algorithm that computes the optimal placement of two resources in a tree for the dynamic version we give an o log n time algorithm for placing one resource and for the case of complete binary trees an o log n 3 time algorithm for two resources the static algorithm is faster than the algorithms found in literature while the dynamic algorithms are the first for this problem

a linear theory for noncausality abstract different definitions of noncausality according to granger sims haugh and pierce are analyzed in terms of orthogonality in the hilbert space of square integrable variables conditions when necessary are given for their respective equivalence some problems of testability are mentioned finally noncausality is also analyzed in terms of rational expectations extending previous results of sims author