positivity of solutions of elliptic equations with nonlocal terms in this paper we study a nonlocal problem for a second order partial differential equation which depends on a parameter n we prove the existence of critical values 0 and 0 such that for all and for all non negative right hand sides our problem has nonnegative solutions we obtain a formula for 0 the maximal possible value of and find the exact value of for spherical we also study the corresponding eigenvalue problem at the end of the paper we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor

distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution summary a fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry and to generate a whole set of non symmetric densities the approach is sufficiently general to encompass some recent proposals in the literature variously related to the skew normal distribution the special case of skew elliptical densities is examined in detail establishing connections with existing similar work the final part of the paper specializes further to a form of multivariate skew t density likelihood inference for this distribution is examined and it is illustrated with numerical examples

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

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

local classifier weighting by quadratic programming it has been widely accepted that the classification accuracy can be improved by combining outputs of multiple classifiers however how to combine multiple classifiers with various potentially conflicting decisions is still an open problem a rich collection of classifier combination procedures many of which are heuristic in nature have been developed for this goal in this brief we describe a dynamic approach to combine classifiers that have expertise in different regions of the input space to this end we use local classifier accuracy estimates to weight classifier outputs specifically we estimate local recognition accuracies of classifiers near a query sample by utilizing its nearest neighbors and then use these estimates to find the best weights of classifiers to label the query the problem is formulated as a convex quadratic optimization problem which returns optimal nonnegative classifier weights with respect to the chosen objective function and the weights ensure that locally most accurate classifiers are weighted more heavily for labeling the query sample experimental results on several data sets indicate that the proposed weighting scheme outperforms other popular classifier combination schemes particularly on problems with complex decision boundaries hence the results indicate that local classification accuracy based combination techniques are well suited for decision making when the classifiers are trained by focusing on different regions of the input space

estimation of positive definite m matrices and structure learning for attractive gaussian markov random fields abstract consider a random vector with finite second moments if its precision matrix is an m matrix then all partial correlations are non negative if that random vector is additionally gaussian the corresponding markov random field gmrf is called attractive we study estimation of m matrices taking the role of inverse second moment or precision matrices using sign constrained log determinant divergence minimization we also treat the high dimensional case with the number of variables exceeding the sample size the additional sign constraints turn out to greatly simplify the estimation problem we provide evidence that explicit regularization is no longer required to solve the resulting convex optimization problem we propose an algorithm based on block coordinate descent in which each sub problem can be recast as non negative least squares problem illustrations on both simulated and real world data are provided

drcw ovo one vs one strategy is a common and established technique in machine learning to deal with multi class classification problems it consists of dividing the original multi class problem into easier to solve binary subproblems considering each possible pair of classes since several classifiers are learned their combination becomes crucial in order to predict the class of new instances due to the division procedure a series of difficulties emerge at this stage such as the non competence problem each classifier is learned using only the instances of its corresponding pair of classes and hence it is not competent to classify instances belonging to the rest of the classes nevertheless at classification time all the outputs of the classifiers are taken into account because the competence cannot be known a priori the classification problem would be solved on this account we develop a distance based combination strategy which weights the competence of the outputs of the base classifiers depending on the closeness of the query instance to each one of the classes our aim is to reduce the effect of the non competent classifiers enhancing the results obtained by the state of the art combinations for one vs one strategy we carry out a thorough experimental study supported by the proper statistical analysis showing that the results obtained by the proposed method outperform both in terms of accuracy and kappa measures the previous combinations for one vs one strategy highlightsthe non competence is an important problem in one vs one strategy we develop a distance based combination strategy based on dynamic classifier weighting strategies weights are settled depending on the closeness of the test instance to each one of the classesthe effect of the non competent classifiers is reduced the new strategy enhances the results obtained w r t the state of the art aggregations

blowing up of solution for a nonlocal reaction diffusion problem in combustion theory abstract in a nonlocal reaction diffusion model in combustion theory the reaction function involves a physical parameter which is a measure of the strength of the reaction mechanism the purpose of this paper is to show the existence of a critical value such that for a unique global time dependent solution exists and converges to a steady state solution as t and for the solution blows up in finite time a characterization as well as upper and lower bounds of are given

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

feature extraction for classification problems and its application to face recognition this study investigates a new method of feature extraction for classification problems the method is based on the independent component analysis ica however unlike the original ica one of the unsupervised learning methods it is developed for classification problems by utilizing class information the proposed method is an extension of our previous work on binary class problems to multi class problems it treats the class labels as input features in order to produce two sets of new features one that carries much information on the class labels and the other that is irrelevant to the class the learning rule for this method is obtained using the stochastic gradient method to maximize the likelihood of the observed data among the new features using only class relevant ones the dimension of the feature space can be greatly reduced in line with the principle of parsimony resulting better generalization this method was applied to recognize face identities and facial expressions using various databases such as the yale at t former orl color feret face databases and so on the performance of the proposed method was compared with those of conventional methods such as the principal component analysis pca fisher s linear discriminant fld etc the experimental results show that the proposed method performs well for face recognition problems

on the design of an ecoc compliant genetic algorithm genetic algorithms ga have been previously applied to error correcting output codes ecoc in state of the art works in order to find a suitable coding matrix nevertheless none of the presented techniques directly take into account the properties of the ecoc matrix as a result the considered search space is unnecessarily large in this paper a novel genetic strategy to optimize the ecoc coding step is presented this novel strategy redefines the usual crossover and mutation operators in order to take into account the theoretical properties of the ecoc framework thus it reduces the search space and lets the algorithm to converge faster in addition a novel operator that is able to enlarge the code in a smart way is introduced the novel methodology is tested on several uci datasets and four challenging computer vision problems furthermore the analysis of the results done in terms of performance code length and number of support vectors shows that the optimization process is able to find very efficient codes in terms of the trade off between classification performance and the number of classifiers finally classification performance per dichotomizer results shows that the novel proposal is able to obtain similar or even better results while defining a more compact number of dichotomies and svs compared to state of the art approaches highlightsa novel genetic algorithm to optimize the ecoc coding step is presented the crossover and mutation operators are redefined taking into account the ecoc properties a new operator that is able to extend the ecoc code is developed we introduce a novel regularization parameter that is able to control the number of dichotomies

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

combining predictions in pairwise classification an optimal adaptive voting strategy and its relation to weighted voting weighted voting is the commonly used strategy for combining predictions in pairwise classification even though it shows good classification performance in practice it is often criticized for lacking a sound theoretical justification in this paper we study the problem of combining predictions within a formal framework of label ranking and under some model assumptions derive a generalized voting strategy in which predictions are properly adapted according to the strengths of the corresponding base classifiers we call this strategy adaptive voting and show that it is optimal in the sense of yielding a map prediction of the class label of a test instance moreover we offer a theoretical justification for weighted voting by showing that it yields a good approximation of the optimal adaptive voting prediction this result is further corroborated by empirical evidence from experiments with real and synthetic data sets showing that even though adaptive voting is sometimes able to achieve consistent improvements weighted voting is in general quite competitive all the more in cases where the aforementioned model assumptions underlying adaptive voting are not met in this sense weighted voting appears to be a more robust aggregation strategy

maximum likelihood estimation for linear gaussian covariance models summary we study parameter estimation in linear gaussian covariance models which are p dimensional gaussian models with linear constraints on the covariance matrix maximum likelihood estimation for this class of models leads to a non convex optimization problem which typically has many local maxima using recent results on the asymptotic distribution of extreme eigenvalues of the wishart distribution we provide sufficient conditions for any hill climbing method to converge to the global maximum although we are primarily interested in the case in which n p the proofs of our results utilize large sample asymptotic theory under the scheme n p 1 remarkably our numerical simulations indicate that our results remain valid for p as small as 2 an important consequence of this analysis is that for sample sizes n 14p maximum likelihood estimation for linear gaussian covariance models behaves as if it were a convex optimization problem

sparse estimation of a covariance matrix we suggest a method for estimating a covariance matrix on the basis of a sample of vectors drawn from a multivariate normal distribution in particular we penalize the likelihood with a lasso penalty on the entries of the covariance matrix this penalty plays two important roles it reduces the effective number of parameters which is important even when the dimension of the vectors is smaller than the sample size since the number of parameters grows quadratically in the number of variables and it produces an estimate which is sparse in contrast to sparse inverse covariance estimation our method s close relative the sparsity attained here is in the covariance matrix itself rather than in the inverse matrix zeros in the covariance matrix correspond to marginal independencies thus our method performs model selection while providing a positive definite estimate of the covariance the proposed penalized maximum likelihood problem is not convex so we use a majorize minimize approach in which we iteratively solve convex approximations to the original nonconvex problem we discuss tuning parameter selection and demonstrate on a flow cytometry dataset how our method produces an interpretable graphical display of the relationship between variables we perform simulations that suggest that simple elementwise thresholding of the empirical covariance matrix is competitive with our method for identifying the sparsity structure additionally we show how our method can be used to solve a previously studied special case in which a desired sparsity pattern is prespecified

global existence and divergence of critical solutions of a non local parabolic problem in ohmic heating process abstract we investigate the behaviour of some critical solutions of a non local initial boundary value problem for the equation u t u f u f u d x 2 r n n 1 2 under specific conditions on f there exists a such that for each 0 there corresponds a unique steady state solution and u u x t is a global in time bounded solution which tends to the unique steady state solution as t uniformly in x whereas for there is no steady state and if then u blows up globally here we show that when a n 1 1 1 and f s 0 f s s 0 or b n 2 b 0 1 and f s e s the solution u u x t is global in time and diverges in the sense u t as t moreover it is proved that this divergence is global i e u x t as t for all x the asymptotic form of divergence is also discussed for some special cases

classification by pairwise coupling we discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes and then coupling the estimates together the coupling model is similar to the bradley terry method for paired comparisons we study the nature of the class probability estimates that arise and examine the performance of the procedure in real and simulated data sets classifiers used include linear discriminants nearest neighbors adaptive nonlinear methods and the support vector machine

a linear recognition algorithm for cographs cographs are the graphs formed from a single vertex under the closure of the operations of union and complement another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms furthermore it is known that cographs have a unique tree representation called a cotree using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general such problems include chromatic number clique determination clustering minimum weight domination isomorphism minimum fill in and hamiltonicity in this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation

on the toric algebra of graphical models we formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model or a log linear model or other more general exponential models for decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the hammersley clifford theorem however we show that for nondecomposable graphical models they are not we also show that nondecomposable models can have nonrational maximum likelihood estimates these results are used to give several novel characterizations of decomposable graphical models

covariance hypotheses which are linear in both the covariance and the inverse covariance both the covariance and the inverse covariance are products of models each of which consists of either i independent identically distributed random vectors which have a covariance with a real complex or quaternion structure or ii independent identically distributed random vectors with a parametrization of the covariance which is given by means of the clifford algebra the models i are well known for models ii we have found under the assumption that the distribution is nornal the exact distributions of the maximum likelihood estimates and the likelihood ratio test statistics 1 introduction the purpose of the present paper is to describe for the family of normal distributions the structure of those statistical hypotheses models which are linear in both the covariance and the inverse covariance in section 2 we show that under such a hypothesis the problem of maximum likelihood estimation of the covariance has an explicit solution and that these hypotheses are parametrized by jordan algebras quadratic subspaces based on the theory of jordan algebras it is then possible to give a canonical form for such hypotheses in section 3 we show that the jordan algebras are products of so called simple jordan algebras and in section 4 we prove that the hypotheses can be decomposed into products of hypotheses parametrized by simple jordan algebras the simple jordan algebras are classified according to their so called degree the only simple jordan algebras of degree 1 is the set of real numbers and corresponding to that we have the hypothesis under which the covariance is proportional to a known covariance in section 5 we give a canonical form for the hypotheses which are parametrized by simple jordan algebras of degree greater than 3 it tums out that these hypotheses are equivalent to those consisting of independent identically distributed random vectors which have a covariance with a real complex or quatemion structure and the distributions of the maximum likelihood estimates are given by wishart 1928 goodman 1963 and andersson 1975 respectively the statistical hypotheses which are parametrized by simple jordan algebras of degree 2 are new and in section 6 we give a complete solution to the problems of maximum likelihood inference it turns out that the distribution problems are not more complicated than those for the two dimensional normal distribution the mathematical structure of the hypotheses is closely related to the theory of spinors in quantum mechanics based on that theory we give a canonical form for such hypotheses in section 7

a novel svm nda model for classification with an application to face recognition support vector machine svm is a powerful classification methodology where the support vectors fully describe the decision surface by incorporating local information on the other hand nonparametric discriminant analysis nda is an improvement over lda where the normality assumption is relaxed nda also detects the dominant normal directions to the decision plane this paper introduces a novel svm nda model which can be viewed as an extension to the svm by incorporating some partially global information especially discriminatory information in the normal direction to the decision boundary this can also be considered as an extension to the nda where the support vectors improve the choice of k nearest neighbors on the decision boundary by incorporating local information being an extension to both svm and nda it can deal with heteroscedastic and non normal data it also avoids the small sample size problem moreover it can be reduced to the classical svm model so that existing softwares can be used a kernel extension of the model called ksvm knda is also proposed to deal with nonlinear problems we have carried an extensive comparison of the svm nda to the lda svm heteroscedastic lda hlda nda and the combined svm and lda on artificial real and face recognition data sets results for ksvm knda have also been presented these comparisons demonstrate the advantages and superiority of our proposed model

single point blowup for nonlocal parabolic problems abstract the formation of shear bands in materials has important implications to a variety of physical processes these bands are observed in very thin zones and are generally regarded as a precursor to material failure shear band formation is caused by the heat generated in regions with highest strain rate with insufficient time for diffusion of this heat a localized thermal softening of the material occurs which enhances plastic flow in a thin zone this adiabatic strain localization can be modelled as nonlinear thermally activated reaction diffusion equations this leads to a class of nonlocal parabolic problems and their associated time independent steady state counterparts a discussion of the models and an analysis of the problems will be presented

statistical applications of the multivariate skew normal distribution azzalini and dalla valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter the first part of the present paper examines further probabilistic properties of the distribution with special emphasis on aspects of statistical relevance inferential and other statistical issues are discussed in the following part with applications to some multivariate statistics problems illustrated by numerical examples finally a further extension is described which introduces a skewing factor of an elliptical density

a new approach to cholesky based covariance regularization in high dimensions summary in this paper we propose a new regression interpretation of the cholesky factor of the covariance matrix as opposed to the well known regression interpretation of the cholesky factor of the inverse covariance which leads to a new class of regularized covariance estimators suitable for high dimensional problems regularizing the cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator in particular one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of bickel levina 2008b which is not guaranteed to be positive definite we also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint and compare the numerical performance of several methods in simulations and on a sonar data example

a nonlocal sturm liouville eigenvalue problem a nonlocal eigenvalue problem of the form u a x u bu u with homogeneous dirichlet boundary conditions is considered where b is a rank one bounded linear operator and x belongs to some bounded interval on the real line the behaviour of the eigenvalues is studied using methods of linear perturbation theory in particular some results are given which ensure that the spectrum remains real a sturm type comparison result is obtained finally these results are applied to the study of some nonlocal reaction diffusion equations

a non parametric approach to extending generic binary classifiers for multi classification ensemble methods which combine generic binary classifier scores to generate a multi classification output are commonly used in state of the art computer vision and pattern recognition systems that rely on multi classification in particular we consider the one vs one decomposition of the multi class problem where binary classifier models are trained to discriminate every class pair we describe a robust multi classification pipeline which at a high level involves projecting binary classifier scores into compact orthogonal subspaces followed by a non linear probabilistic multi classification step using kernel density estimation kde we compare our approach against state of the art ensemble methods dcs drcw on 16 multi class datasets we also compare against the most commonly used ensemble methods vote nest on 6 real world computer vision datasets finally we measure the statistical significance of our approach using non parametric tests experimental results show that our approach gives a statistically significant improvement in multi classification performance over state of the art highlightsensemble methods combine binary classifiers to yield a multi classification output one vs one ensemble binary classifiers trained to discriminate each class pair we propose a robust non parametric probabilistic one vs one ensemble method kdemrp kdemrp improves classification performance over state of the art dcs drcw kdemrp improvements are statistically significant

generalized linear models incorporating population level information an empirical likelihood based approach summary in many situations information from a sample of individuals can be supplemented by population level information on the relationship between a dependent variable and explan atory variables inclusion of the population level information can reduce bias and increase the efficiency of the parameter estimates population level information can be incorporated via con straints on functions of the model parameters in general the constraints are non linear making the task of maximum likelihood estimation more difficult we develop an alternative approach exploiting the notion of an empirical likelihood it is shown that within the framework of gener alized linear models the population level information corresponds to linear constraints which are comparatively easy to handle we provide a two step algorithm that produces parameter estimates by using only unconstrained estimation we also provide computable expressions for the standard errors we give an application to demographic hazard modelling by combining panel survey data with birth registration data to estimate annual birth probabilities by parity

support vector learning for fuzzy rule based classification systems to design a fuzzy rule based classification system fuzzy classifier with good generalization ability in a high dimensional feature space has been an active research topic for a long time as a powerful machine learning approach for pattern recognition problems the support vector machine svm is known to have good generalization ability more importantly an svm can work very well on a high or even infinite dimensional feature space this paper investigates the connection between fuzzy classifiers and kernel machines establishes a link between fuzzy rules and kernels and proposes a learning algorithm for fuzzy classifiers we first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function fuzzy inference on the if part of a fuzzy rule can be viewed as evaluating the kernel function the kernel function is then proven to be a mercer kernel if the reference functions meet a certain spectral requirement the corresponding fuzzy classifier is named positive definite fuzzy classifier pdfc a pdfc can be built from the given training samples based on a support vector learning approach with the if part fuzzy rules given by the support vectors since the learning process minimizes an upper bound on the expected risk expected prediction error instead of the empirical risk training error the resulting pdfc usually has good generalization moreover because of the sparsity properties of the svms the number of fuzzy rules is irrelevant to the dimension of input space in this sense we avoid the curse of dimensionality finally pdfcs with different reference functions are constructed using the support vector learning approach the performance of the pdfcs is illustrated by extensive experimental results comparisons with other methods are also provided

reducing multiclass to binary a unifying approach for margin classifiers we present a unifying framework for studying the solution of multiclass categorization problems by reducing them to multiple binary problems that are then solved using a margin based binary learning algorithm the proposed framework unifies some of the most popular approaches in which each class is compared against all others or in which all pairs of classes are compared to each other or in which output codes with error correcting properties are used we propose a general method for combining the classifiers generated on the binary problems and we prove a general empirical multiclass loss bound given the empirical loss of the individual binary learning algorithms the scheme and the corresponding bounds apply to many popular classification learning algorithms including support vector machines adaboost regression logistic regression and decision tree algorithms we also give a multiclass generalization error analysis for general output codes with adaboost as the binary learner experimental results with svm and adaboost show that our scheme provides a viable alternative to the most commonly used multiclass algorithms

in defense of one vs all classification we consider the problem of multiclass classification our main thesis is that a simple one vs all scheme is as accurate as any other approach assuming that the underlying binary classifiers are well tuned regularized classifiers such as support vector machines this thesis is interesting in that it disagrees with a large body of recent published work on multiclass classification we support our position by means of a critical review of the existing literature a substantial collection of carefully controlled experimental work and theoretical arguments

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

on identification of multi factor models with correlated residuals summary we specify some conditions for the identification of a multi factor model with correlated residuals uncorrelated factors and zero restrictions in the factor loadings these conditions are derived from the results of stanghellini 1997 and vicard 2000 which deal with single factor models with zero restrictions in the concentration matrix like these authors we make use of the complementary graph of residuals and the conditions build on the role of odd cycles in this graph however in contrast to these authors we consider the case where the conditional dependencies of the residuals are expressed in terms of a covariance matrix rather than its inverse the concentration matrix we first derive the corresponding condition for identification of single factor models with structural zeros in the covariance matrix of the residuals this is extended to the case where some factor loadings are constrained to be zero we use these conditions to obtain a sufficient and a necessary condition for identification of multi factor models

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

groups acting on gaussian graphical models gaussian graphical models have become a well recognized tool for the analysis of conditional independencies within a set of continuous random variables from an inferential point of view it is important to realize that they are composite exponential transformation families we reveal this structure by explicitly describing for any undirected graph the maximal matrix group acting on the space of concentration matrices in the model the continuous part of this group is captured by a poset naturally associated to the graph while automorphisms of the graph account for the discrete part of the group we compute the dimension of the space of orbits of this group on concentration matrices in terms of the combinatorics of the graph and for dimension zero we recover the characterization by letac and massam of models that are transformation families furthermore we describe the maximal invariant of this group on the sample space and we give a sharp lower bound on the sample size needed for the existence of equivariant estimators of the concentration matrix finally we address the issue of robustness of these estimators by computing upper bounds on finite sample breakdown points keywords gaussian graphical models covariance matrix concentration matrix robust estimator breakdown point equivariant estimator transformation families

solving multi class problems with linguistic fuzzy rule based classification systems based on pairwise learning and preference relations this paper deals with multi class classification for linguistic fuzzy rule based classification systems the idea is to decompose the original data set into binary classification problems using the pairwise learning approach confronting all pair of classes and to obtain an independent fuzzy system for each one of them along the inference process each fuzzy rule based classification system generates an association degree for both of its corresponding classes and these values are encoded into a fuzzy preference relation our analysis is focused on the final step that returns the predicted class label specifically we propose to manage the fuzzy preference relation using a non dominance criterion on the different alternatives contrasting the behaviour of this model with both the classical weighted voting scheme and a decision rule that combines the fuzzy relations of preference conflict and ignorance by means of a voting strategy our experimental study is carried out using two different linguistic fuzzy rule learning methods for which we show that the non dominance criterion is a good alternative in comparison with the previously mentioned aggregation mechanisms this empirical analysis is supported through the corresponding statistical analysis using non parametrical tests

stochastic independence algebraic independence and abstract connectedness abstract mutual stochastic independences among algebras and mutual algebraic independences among elements of semimodular lattices are observed to have a very similar behaviour we suggest abstract independence structures called i relations describing it presented examination of i relations resembles a theory of abstract connectedness a dual characterization of i relations by families of connected sets is found by means of a special galois connection representations of i relations in the matroid theory sense by algebras and by elements of lattices are discussed

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

wishart distributions for decomposable graphs when considering a graphical gaussian model n g markov with respect to a decomposable graph g the parameter space of interest for the precision parameter is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g the parameter space for the scale parameter of n g is the cone q g dual to p g of incomplete matrices with submatrices corresponding to the cliques of g being positive definite in this paper we construct on the cones q g and p g two families of wishart distributions namely the type i and type ii wisharts they can be viewed as generalizations of the hyper wishart and the inverse of the hyper inverse wishart as defined by dawid and lauritzen ann statist 21 1993 1272 1317 we show that the type i and ii wisharts have properties similar to those of the hyper and hyper inverse wishart indeed the inverse of the type ii wishart forms a conjugate family of priors for the covariance parameter of the graphical gaussian model and is strong directed hyper markov for every direction given to the graph by a perfect order of its cliques while the type i wishart is weak hyper markov moreover the inverse type ii wishart as a conjugate family presents the advantage of having a multidimensional shape parameter thus offering flexibility for the choice of a prior both type i and ii wishart distributions depend on multivariate shape parameters a shape parameter is acceptable if and only if it satisfies a certain eigenvalue property we show that the sets of acceptable shape parameters for a noncomplete g have dimension equal to at least one plus the number of cliques in g these families as conjugate families are richer than the traditional diaconis ylvisaker conjugate families which all have a shape parameter set of dimension one a decomposable graph which does not contain a three link chain as an induced subgraph is said to be homogeneous in this case our wisharts are particular cases of the wisharts on homogeneous cones as defined by andersson and wojnar j theoret probab 17 2004 781 818 and the dimension of the shape parameter set is even larger than in the nonhomogeneous case it is indeed equal to the number of cliques plus the number of distinct minimal separators using the model where g is a three link chain we show by computing a 7 tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one

maximum likelihood estimation of the parameters of a multivariate normal distribution abstract this paper provides an exposition of alternative approaches for obtaining maximum likelihood estimators mle for the parameters of a multivariate normal distribution under different assumptions about the parameters a central focus is on two general techniques namely matrix differentiation and matrix transformations these are systematically applied to derive the mle of the means under a rank constraint and of the covariances when there are missing observations derivations using induction and inequalities are also included to illustrate alternative methods other examples such as a connection with an econometric model are included although the paper is primarily expository some of the proofs are new

on logistic diffusion equations with nonlocal interaction terms abstract this paper is concerned with logistic diffusion equations with nonlocal interaction terms appearing in population biology we intend to study effects of nonlocal terms and discuss the similarity and difference between local problems and nonlocal problems mainly the stationary problem is investigated for a certain class of nonlocal terms a constructive approach is proposed to look for positive stationary solutions and the unique existence of such a positive solution is established the analysis of stationary solutions depends on the spectrum for the linearized operator around the stationary solution however the linearized operator contains a nonlocal term which makes the spectral analysis delicate and difficult putting some additional assumptions we will derive the asymptotic stability of the unique positive solution and furthermore its global attractivity finally it will be seen that some arguments are valid to show the unique existence of a positive stationary solution for a considerably general class of nonlocal terms

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

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

probability estimates for multi class classification by pairwise coupling pairwise coupling is a popular multi class classification method that combines all comparisons for each pair of classes this paper presents two approaches for obtaining class probabilities both methods can be reduced to linear systems and are easy to implement we show conceptually and experimentally that the proposed approaches are more stable than the two existing popular methods voting and the method by hastie and tibshirani 1998

an extension on statistical comparisons of classifiers over multiple data sets for all pairwise comparisons in a recently published paper in jmlr dem sar 2006 recommends a set of non parametric statistical tests and procedures which can be safely used for comparing the performance of classifiers over multiple data sets after studying the paper we realize that the paper correctly introduces the basic procedures and some of the most advanced ones when comparing a control method however it does not deal with some advanced topics in depth regarding these topics we focus on more powerful proposals of statistical procedures for comparing n n classifiers moreover we illustrate an easy way of obtaining adjusted and comparable p values in multiple comparison procedures

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

enhancing directed binary trees for multi class classification one approach to multi class classification consists in decomposing the original problem into a collection of binary classification tasks the outputs of these binary classifiers are combined to produce a single prediction winner takes all max wins and tree voting schemes are the most popular methods for this purpose however tree schemes can deliver faster predictions because they need to evaluate less binary models despite previous conclusions reported in the literature this paper shows that their performance depends on the organization of the tree scheme i e the positions where each pairwise classifier is placed on the graph different metrics are studied for this purpose proposing a new one that considers the precision and the complexity of each pairwise model what makes the method to be classifier dependent the study is performed using support vector machines svms as base classifiers but it could be extended to other kind of binary classifiers the proposed method tested on benchmark data sets and on one real world application is able to improve the accuracy of other decomposition multi class classifiers producing even faster predictions

an extension of the results of asmussen and edwards on collapsibility in contingency tables summary asmussen edwards 1983 defined necessary and sufficient conditions for collapsibility of a hierarchical log linear model for a multidimensional contingency table we have shown that for decomposable log linear models these conditions can be combined with various graph theoretic algorithms to provide useful classes of sub tables which are collapsible onto in particular the sahr algorithm finds the minimal set onto which the model can be collapsed and which contains a sub table of interest in the context of expert systems by reducing a probabilistic influence network onto only the relevant nodes the algorithms reduce the required computation and simplify interpretation

further spectral properties of uniformly elliptic operators that include a non local term in this paper we consider the eigenvalues of a class of non local differential operators the real eigenvalues of certain examples of these operators have previously been plotted as functions of a fundamental parameter in the operator using appropriate computer software here we show how the complex eigenvalues of such operators may also be plotted we use the plots of both the real and the complex eigenvalues to motivate a number of new theoretical results regarding the structure of the spectrum which we subsequently prove under suitable hypotheses

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

sparse matrix decompositions and graph characterizations abstract zeros in positive definite correlation matrices arise frequently in probability and statistics and are intimately related to the notion of stochastic independence the question of when zeros i e sparsity in a positive definite matrix a are preserved in its cholesky decomposition and vice versa was addressed by paulsen et al v i paulsen s c power r r smith schur products and matrix completions j funct anal 85 1989 151 178 in particular they prove that for the pattern of zeros in a to be retained in the cholesky decomposition of a the pattern of zeros in a has to necessarily correspond to a chordal or decomposable graph associated with a specific type of vertex ordering this result therefore also yields a characterization of chordal graphs in terms of sparse positive definite matrices and has proved to be extremely useful in probabilistic and statistical analysis of markov random fields now consider a positive definite matrix a and its cholesky decomposition given by a ldl t where l is lower triangular with unit diagonal entries and d a diagonal matrix with positive entries in this paper we prove that a necessary and sufficient condition for zeros i e sparsity in a positive definite matrix a to be preserved in its associated cholesky matrix l and in addition also preserved in the inverse of the cholesky matrix l 1 is that the pattern of zeros corresponds to a co chordal or homogeneous graph associated with a specific type of vertex ordering we proceed to provide a second characterization of this class of graphs in terms of determinants of submatrices that correspond to cliques in the graph these results add to the growing body of literature in the field of sparse matrix decompositions and also prove to be critical ingredients in the probabilistic analysis of an important class of markov random fields