a new framework for the bayesian analysis of single stage decision problems with imprecise utilities a mathematical framework to model the bayesian analysis of single stage decision problems with imprecise utilities is proposed the main advantage of this model with respect to previous models for such a problem is that product measurability of the utility function is not necessary since this model involves iterated expectations instead of an integral over a product space conditions for the equivalence between the extensive and normal forms of the bayesian analysis within the proposed framework are obtained the model is illustrated with an example in which the utility function is not product measurable

a note on approximate equality versus the poincare paradox the purpose of this note is to comment on a paper by m de cock and e e kerre that was recently accepted for publication in fuzzy sets and systems the paper in contrast to the view of several other researchers states that fuzzy equivalence relations are not always able to resolve the poincare paradox this brief paper discusses this controversial contribution from the present author s subjective point of view

newton method for reactive solute transport with equilibrium sorption in porous media we present a mass conservative numerical scheme for reactive solute transport in porous media the transport is modeled by a convection diffusion reaction equation including equilibrium sorption the scheme is based on the mixed finite element method mfem more precisely the lowest order raviart thomas elements and one step euler implicit the underlying fluid flow is described by the richards equation a possibly degenerate parabolic equation which is also discretized by mfem this work is a continuation of radu et al 2008 and radu et al 2009 1 2 where the algorithmic aspects of the scheme and the analysis of the discretization method are presented respectively here we consider the newton method for solving the fully discrete nonlinear systems arising on each time step after discretization the convergence of the scheme is analyzed in the case when the solute undergoes equilibrium sorption of freundlich type the problem becomes degenerate and a regularization step is necessary we derive sufficient conditions for the quadratic convergence of the newton scheme

soft and hard classification by reproducing kernel hilbert space methods reproducing kernel hilbert space rkhs methods provide a unified context for solving a wide variety of statistical modelling and function estimation problems we consider two such problems we are given a training set yi ti i 1 n where yi is the response for the ith subject and ti is a vector of attributes for this subject the value of yi is a label that indicates which category it came from for the first problem we wish to build a model from the training set that assigns to each t in an attribute domain of interest an estimate of the probability pj t that a future subject with attribute vector t is in category j the second problem is in some sense less ambitious it is to build a model that assigns to each t a label which classifies a future subject with that t into one of the categories or possibly none of the above the approach to the first of these two problems discussed here is a special case of what is known as penalized likelihood estimation the approach to the second problem is known as the support vector machine we also note some alternate but closely related approaches to the second problem these approaches are all obtained as solutions to optimization problems in rkhs many other problems in particular the solution of ill posed inverse problems can be obtained as solutions to optimization problems in rkhs and are mentioned in passing we caution the reader that although a large literature exists in all of these topics in this inaugural article we are selectively highlighting work of the author former students and other collaborators

pac bayes risk bounds for stochastic averages and majority votes of sample compressed classifiers we propose a pac bayes theorem for the sample compression setting where each classifier is described by a compression subset of the training data and a message string of additional information this setting which is the appropriate one to describe many learning algorithms strictly generalizes the usual data independent setting where classifiers are represented only by data independent message strings or parameters taken from a continuous set the proposed pac bayes theorem for the sample compression setting reduces to the pac bayes theorem of seeger 2002 and langford 2005 when the compression subset of each classifier vanishes for posteriors having all their weights on a single sample compressed classifier the general risk bound reduces to a bound similar to the tight sample compression bound proposed in laviolette et al 2005 finally we extend our results to the case where each sample compressed classifier of a data dependent ensemble may abstain of predicting a class label

on representing and generating kernels by fuzzy equivalence relations kernels are two placed functions that can be interpreted as inner products in some hilbert space it is this property which makes kernels predestinated to carry linear models of learning optimization or classification strategies over to non linear variants following this idea various kernel based methods like support vector machines or kernel principal component analysis have been conceived which prove to be successful for machine learning data mining and computer vision applications when applying a kernel based method a central question is the choice and the design of the kernel function this paper provides a novel view on kernels based on fuzzy logical concepts which allows to incorporate prior knowledge in the design process it is demonstrated that kernels mapping to the unit interval with constant one in its diagonal can be represented by a commonly used fuzzy logical formula for representing fuzzy rule bases this means that a great class of kernels can be represented by fuzzy logical concepts apart from this result which only guarantees the existence of such a representation constructive examples are presented and the relation to unlabeled learning is pointed out

analysis of an upwind mixed finite element method for nonlinear contaminant transport equations in this paper the numerical approximation of a nonlinear diffusion equation arising in contaminant transport is studied the equation is characterized by advection diffusion and adsorption assuming the adsorption term is modeled by a freundlich isotherm it can be nonlinear in concentration and nondifferentiable as the concentration approaches zero we consider the approximation of this equation by a method which upwinds the advection and incorporates diffusion using a mixed finite element method error estimates for a semidiscrete formulation are derived and numerical results for a fully discrete formulation are given

learning intransitive reciprocal relations with kernel methods abstract in different fields like decision making psychology game theory and biology it has been observed that paired comparison data like preference relations defined by humans and animals can be intransitive intransitive relations cannot be modeled with existing machine learning methods like ranking models because these models exhibit strong transitivity properties more specifically in a stochastic context where often the reciprocity property characterizes probabilistic relations such as choice probabilities it has been formally shown that ranking models always satisfy the well known strong stochastic transitivity property given this limitation of ranking models we present a new kernel function that together with the regularized least squares algorithm is capable of inferring intransitive reciprocal relations in problems where transitivity violations cannot be considered as noise in this approach it is the kernel function that defines the transition from learning transitive to learning intransitive relations and the kronecker product is introduced for representing the latter type of relations in addition we empirically demonstrate on two benchmark problems one in game theory and one in theoretical biology that our algorithm outperforms methods not capable of learning intransitive reciprocal relations

finite element approximation of the transport of reactive solutes in porous media part 1 error estimates for nonequilibrium adsorption processes in this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration backward euler time discretization and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media find u x t v x t such that begin eqnarray partial t u partial t v delta u f quad mbox in omega times 0 t qquad u 0 quad mbox on partial omega times 0 t partial tv k varphi u v quad mbox in omega times 0 t u cdot 0 g 1 cdot quad v cdot 0 g 2 cdot quad mbox in omega subset real d quad 1 le d le 3 end eqnarray for given data k in real f g 1 g 2 and a monotonically increasing varphi in c 0 real cap c 1 infty 0 cup 0 infty satisfying varphi 0 0 which is only locally holder continuous with exponent p in 0 1 at the origin e g varphi s equiv s p this lack of lipschitz continuity at the origin limits the regularity of the unique solution u v and leads to difficulties in the finite element error analysis

classifier selection using the predicate depth typically one approaches a supervised machine learning problem by writing down an objective function and finding a hypothesis that minimizes it this is equivalent to finding the maximum a posteriori map hypothesis for a boltzmann distribution however map is not a robust statistic we present an alternative approach by defining a median of the distribution which we show is both more robust and has good generalization guarantees we present algorithms to approximate this median one contribution of this work is an efficient method for approximating the tukey median the tukey median which is often used for data visualization and outlier detection is a special case of the family of medians we define however computing it exactly is exponentially slow in the dimension our algorithm approximates such medians in polynomial time while making weaker assumptions than those required by previous work

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

an existence theorem for fuzzy utility functions a new elementary proof first our paper consists in extending the ponsard s result to a convex set of objects by giving conditions under which a fuzzy complete preorder can be represented by a numerical function here we propose a new proof based on a fuzzy definition for topological separability basically the utility function we obtain is formally identical using homeomorphism of 0 1 and r and to a standard one but its foundations and thus the interpretation are different because of behavior fuzziness on the one hand and perfect separability property of a fuzzy preordered set on the other second we prove that it is always possible to exhibit an implicit fuzzy model that is hidden behind the standard one by means of the way the boolean standard models rank the elements of the quotient set

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

characteristic galerkin methods for contaminant transport with nonequilibrium adsorption kinetics a procedure based on combining the method of characteristics with a galerkin finite element method is analyzed for approximating reactive transport in groundwater in particular equations modeling

pictorial representations of fuzzy connectives part i cases of t norms t conorms and averaging operators part i of this paper shows a number of examples of existing t norms and t conorms which are known as good models of fuzzy set theoretic intersection and union and then the pictorial representations of t norms and t conorms are made with the aid of a computer several new examples of t norms and t conorms are also given averaging operators between min and max are summarized and their pictorial representations are also made

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

a fuzzy rough set approach for incremental feature selection on hybrid information systems in real applications there may exist many kinds of data e g boolean categorical real valued and set valued data and missing data in an information system which is called as a hybrid information system his a new hybrid distance hd in his is developed based on the value difference metric and a novel fuzzy rough set is constructed by combining the hd distance and the gaussian kernel considering the information systems often vary with time the updating mechanisms for attribute reduction feature selection are analyzed with the variation of the attribute set fuzzy rough set approaches for incremental feature selection on his are presented then two corresponding incremental algorithms are proposed respectively finally extensive experiments on eight datasets from uci and an artificial dataset show that the incremental approaches significantly outperform non incremental approaches with feature selection in the computational time

diffusion and dissolution in a reactive porous medium mathematical modelling and numerical simulations in this work a simple mathematical model for diffusion and dissolution in reactive porous medium is presented the case of lime and lead in solid phase enclosed in cement matrices is considered more specifically a numerical method based on finite difference and on a marching technique is proposed and some numerical results are provided in a simple case the results obtained are compared with numerical results available in literature

a class of single class minimax probability machines for novelty detection single class minimax probability machines mpms offer robust novelty detection with distribution free worst case bounds on the probability that a pattern will fall inside the normal region however in practice they are too cautious in labeling patterns as outlying and so have a high false negative rate fnr in this paper we propose a more aggressive version of the single class mpm that bounds the best case probability that a pattern will fall inside the normal region these two mpms can then be used together to delimit the solution space by using the hyperplane lying in the middle of this pair of mpms a better compromise between false positives fps and false negatives fns and between recall and precision can be obtained experiments on the real world data sets show encouraging results

positive definite indistinguishability operators abstract two geometric characterizations of positive definite reflexive and symmetric fuzzy relations are provided based on the embeddability of their naturally associated pseudodistances with respect to the continuous archimedean t norms with additive generators t x arccos x and t x 1 x due to the important role of the t norm t arccos x with additive generator t x arccos x in these characterizations different characterizations of it are given in particular a left continuous t norm t is t arccos x if and only if the rank of the matrices of all one dimensional indistinguishability operators with respect to t are less than or equal to 2

functions that preserve families of positive semidefinite matrices we study various notions of multivariate functions which map families of positive semidefinite matrices or of conditionally positive semidefinite matrices into matrices of the same type

generalization as search abstract the problem of concept learning or forming a general description of a class of objects given a set of examples and non examples is viewed here as a search problem existing programs that generalize from examples are characterized in terms of the classes of search strategies that they employ several classes of search strategies are then analyzed and compared in terms of their relative capabilities and computational complexities

fuzzy mathematical models in economics abstract this paper is a survey of three fuzzy mathematical models dealing with economic choice economic calculation and general economic equilibrium they are the elements of a fuzzy theory of value the aim is to test the efficiency of the application of fuzzy subsets theory to the analysis of the foundations of economics and thus to legitimate its use in the area

a transitivity analysis of bipartite rankings in pairwise multi class classification many multi class classification algorithms in statistics and machine learning typically combine several binary classifiers in order to construct an overall classifier in the popular pairwise ensemble one classifier is built for each pair of classes resulting in pairwise bipartite rankings in contrast ordinal regression algorithms consider a single ranking function for several ordered classes it is known in the literature that pairwise ensembles can be useful for ordinal regression however can single ranking models make a contribution to multi class classification the answer to this question should be affirmative as supported by theoretical results presented in this article we conduct a formal analysis of the consistency of pairwise bipartite rankings by uncovering the conditions under which they can be equivalently expressed in terms of a single ranking similar to the utility representability of pairwise preference relations it turns out that transitivity plays a crucial role in the characterization of the ranking representability of pairwise bipartite rankings to this end we introduce the new concepts of strict ranking representability a restrictive condition that can be verified easily and auc ranking representability a practically more useful condition that is more difficult to verify however the link between pairwise bipartite rankings and dice games allows us to formulate necessary transitivity conditions for auc ranking representability a sufficient condition on the other hand is obtained by introducing a new type of transitivity that can be verified by solving an integer quadratic program

solute transport in porous media with equilibrium and non equilibrium multiple site adsorption travelling waves we study travelling wave solutions of a model describing solute transport through porous media where the chemical species undergoes absorption i e a retention release reaction with the surface of the porous sceleton processes of this type are of fundamental importance in different fields we do not consider the process at the microscopic scale of single grains and pores but at the microscopic scale appropriate for the measurement of the phenomena described above

gaussian processes for classification mean field algorithms we derive a mean field algorithm for binary classification with gaussian processes that is based on the tap approach originally proposed in statistical physics of disordered systems the theory also yields an approximate leave one out estimator for the generalization error which is computed with no extra computational cost we show that from the tap approach it is possible to derive both a simpler naive mean field theory and support vector machines svms as limiting cases for both mean field algorithms and support vector machines simulation results for three small benchmark data sets are presented they show that one may get state of the art performance by using the leave one out estimator for model selection and the built in leave one out estimators are extremely precise when compared to the exact leave one out estimate the second result is taken as strong support for the internal consistency of the mean field approach

numerical methods for the simulation of flow in root soil systems the numerical properties of approximation schemes for a model that simulates water transport in root soil systems are considered the model is derived in detail it is based on a previously proposed model which is reformulated completely in terms of the water potential the system of equations consists of a parabolic partial differential equation that contains a nonlinear capacity term coupled to two linear ordinary differential equations a closed form solution is obtained for one of the latter equations finite element and finite difference schemes are defined to approximate the solution of the coupled system some new techniques which have wide applicability for analyzing the nonlinear capacity term are used and optimal order error estimates are derived a postprocessed water mass flux computation is also presented and shown to be superconvergent to the true flux computational results which verify the theoretical convergence rates are given

analysis of a model for precipitation and dissolution coupled with a darcy flux abstract in this paper we deal with the numerical analysis of an upscaled model of a reactive flow in a porous medium which describes the transport of solutes undergoing precipitation and dissolution leading to the formation degradation of crystals inside the porous matrix the model is defined at the darcy scale and it is coupled to a darcy flow characterized by a permeability field that changes in space and time according to the precipitated crystal concentration the model involves a non linear multi valued reaction term which is treated exactly by solving an inclusion problem for the solutes and the crystals dynamics we consider a weak formulation for the coupled system of equations expressed in a dual mixed form for the darcy field and in a primal form for the solutes and the precipitate and show its well posedness without resorting to regularization of the reaction term convergence to the weak solution is proved for its finite element approximation we perform numerical experiments to study the behavior of the system and to assess the effectiveness of the proposed discretization strategy in particular we show that a method that captures the discontinuity yields sharper dissolution fronts with respect to methods that regularize the discontinuous term

fuzzy logic based generalized decision theory with imperfect information the existing decision models have been successfully applied to solving many decision problems in management business economics and other fields but nowadays arises a need to develop more realistic decision models the main drawback of the existing utility theories starting from von neumann moregnstern expected utility to the advanced non expected models is that they are designed for laboratory examples with simple well defined gambles which do not adequately enough reflect real decision situations in real life decision making problems preferences are vague and decision relevant information is imperfect as described in natural language nl vagueness of preferences and imperfect decision relevant information require using suitable utility model which would be fundamentally different to the existing precise utility models precise utility models cannot reflect vagueness of preferences vagueness of objective conditions and outcomes imprecise beliefs etc the time has come for a new generation of decision theories in this study we propose a decision theory which is capable to deal with vague preferences and imperfect information the theory discussed here is based on a fuzzy valued non expected utility model representing linguistic preference relations and imprecise beliefs

convergence analysis of mixed numerical schemes for reactive flow in a porous medium this paper deals with the numerical analysis of an upscaled model describing the reactive flow in a porous medium the solutes are transported by advection and diffusion and undergo precipitation and dissolution the reaction term and in particular the dissolution term have a particular multivalued character which leads to stiff dissolution fronts we consider the euler implicit method for the temporal discretization and the mixed finite element for the discretization in space more precisely we use the lowest order raviart thomas elements as an intermediate step we consider also a semidiscrete mixed variational formulation continuous in space we analyze the numerical schemes and prove the convergence to the continuous formulation apart from the proof for the convergence this also yields an existence proof for the solution of the model in mixed variational formulation numerical experiments are performed to study the convergence behavior

a numerical scheme for the pore scale simulation of crystal dissolution and precipitation in porous media in this paper we analyze a numerical scheme for a dissolution and precipitation model in porous media we focus here on the chemistry which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion defined on the boundary we use a regularization approach for constructing a semi implicit scheme that is stable and convergent for dealing with the emerging time discrete nonlinear problems we propose a simple fixed point iterative procedure the paper is concluded by numerical results

necessary and sufficient conditions for the existence of solution of generalized fuzzy relation equations a x b abstract in 2013 li and jin studied a particular type of fuzzy relational equations on finite sets where the introduced min bi implication composition is based on ukasiewicz equivalence in this paper such fuzzy relation equations are studied on a more general level namely complete residuated lattice valued fuzzy relation equations of type y y a x y x y b x are analyzed and the existence of solutions s is studied first a necessary condition for the existence of solution is established then conditions for lower and upper limits of solutions are given and finally sufficient conditions for the existence of the smallest and largest solutions respectively are characterized if such general or global solutions do not exist there might still be partial or point wise solutions this is a novel way to study fuzzy relation equations such point wise solutions are studied on ukasiewicz product and godel t norm based residuated lattices on the real unit interval

on the ability of the optimal perceptron to generalise a linearly separable boolean function is derived from a set of examples by a perceptron with optimal stability the probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method

error estimates for a mixed finite element discretization of some degenerate parabolic equations we consider a numerical scheme for a class of degenerate parabolic equations including both slow and fast diffusion cases a particular example in this sense is the richards equation modeling the flow in porous media the numerical scheme is based on the mixed finite element method mfem in space and is of one step implicit in time the lowest order raviart thomas elements are used here we extend the results in radu et al siam j numer anal 42 1452 1478 2004 schneid et al numer math 98 353 370 2004 to a more general framework by allowing for both types of degeneracies we derive error estimates in terms of the discretization parameters and show the convergence of the scheme the features of the mfem especially of the lowest order raviart thomas elements are now fully exploited in the proof of the convergence the paper is concluded by numerical examples

maximum relative margin and data dependent regularization leading classification methods such as support vector machines svms and their counterparts achieve strong generalization performance by maximizing the margin of separation between data classes while the maximum margin approach has achieved promising performance this article identifies its sensitivity to affine transformations of the data and to directions with large data spread maximum margin solutions may be misled by the spread of data and preferentially separate classes along large spread directions this article corrects these weaknesses by measuring margin not in the absolute sense but rather only relative to the spread of data in any projection direction maximum relative margin corresponds to a data dependent regularization on the classification function while maximum absolute margin corresponds to an l2 norm constraint on the classification function interestingly the proposed improvements only require simple extensions to existing maximum margin formulations and preserve the computational efficiency of svms through the maximization of relative margin surprising performance gains are achieved on real world problems such as digit text classification and on several other benchmark data sets in addition risk bounds are derived for the new formulation based on rademacher averages

a characteristics mixed finite element method for advection dominated transport problems we define a new finite element method called the characteristics mixed method for approximating the solution to an advection dominated transport problem the method is based on a space time variational form of the advection diffusion equation our test functions are piecewise constant in space and in time they approximately follow the characteristics of the advective i e hyperbolic part of the equation thus the scheme uses a characteristic approximation to handle advection in time this is combined with a low order mixed finite element spatial approximation of the equation boundary conditions are incorporated in a natural and mass conservative fashion the scheme is completely locally conservative in fact on the discrete level fluid is transported along the approximate characteristics a postprocessing step is included in the scheme in which the approximation to the scalar unknown is improved by utilizing the approximate vector flux this has the effect of improving the rate of convergence of

possibilistic risk aversion risk theory is usually developed within probability theory the aim of this paper is to propose an approach of the risk aversion by possibility theory initiated by zadeh in 1978 the main notion studied in this paper is the possibilistic risk premium associated with a fuzzy number a and a utility function u under the hypothesis that the utility function u verifies certain hypotheses one proves a formula to evaluate the possibilistic risk premium in terms of u and of some possibilistic indicators

on the era ranking representability of pairwise bipartite ranking functions in domains like decision theory and social choice theory it is known for a long time that stochastic transitivity properties yield necessary and sufficient conditions for the ranking or utility representability of reciprocal preference relations in this article we extend these results for reciprocal preference relations originating from the pairwise comparison of random vectors in a machine learning context more specifically the expected ranking accuracy era is such a reciprocal relation that occurs in multi class classification problems when ranking or utility functions are fitted to the data in a pairwise manner we establish necessary and sufficient conditions for which these pairwise bipartite ranking functions can be simplified to a single ranking function such that the pairwise expected ranking accuracies of both models coincide similarly as for more common reciprocal preference relations cycle transitivity plays a crucial role in this new setting we first consider the finite sample case for which expected ranking accuracy can be estimated by means of the area under the roc curve auc and subsequently we further generalize these results to the underlying distributions it turns out that the ranking representability of pairwisely compared random vectors can be expressed elegantly in a distribution independent way by means of a specific type of cycle transitivity defined by a conjunctor that is closely related to the algebraic product

frsvms fuzzy rough set based support vector machines this paper aims to improve hard margin support vector machines svms by considering the membership of every training sample in constraints the membership is computed by employing the technique of fuzzy rough sets so that hard margin svms can be combined with fuzzy rough sets and the inconsistence between conditional features and decision labels can be taken into account at the same time in this paper we first propose fuzzy transitive kernel based fuzzy rough sets for binary classification we use a lower approximation operator in fuzzy transitive kernel based fuzzy rough sets to compute the membership for every training input and then we reformulate hard margin support vector machines into fuzzy rough set based svms frsvms with new constraints in which the membership is taken into account finally comparisons with soft margin svms and fuzzy svms are made the experimental results show that the proposed approach is feasible and valid it significantly improved the performance of the hard margin svms

generalization performance of bayes optimal classification algorithm for learning a perceptron the generalization error of the bayes optimal classification algorithm when learning a perceptron from noise free random training examples is calculated exactly using methods of statistical mechanics it is shown that if an assumption of replica symmetry is made then in the thermodynamic limit the error of the bayes optimal algorithm is less than the error of a canonical stochastic learning algorithm by a factor approaching ensuremath surd 2 as the ratio of the number of training examples to perceptron weights grows in addition it is shown that approximations to the generalization error of the bayes optimal algorithm can be achieved by learning algorithms that use a two layer neutral net to learn a perceptron

learning iteratively a classifier with the bayesian model averaging principle we present a learning algorithm for nominal vector data it builds a complex classifier by adding iteratively a simple function that modifies the current classifier in order to limit overtraining problem we focus on a class of such functions for which optimal bayesian learning is tractable we investigate a few classes of functions that yield to models that are similar to nai ve bayes and logistic classification we report experimental results for a collection of standard data sets that show that our learning algorithm outperforms standard learning of such these standard models

a kernel based framework for learning graded relations from data driven by a large number of potential applications in areas such as bioinformatics information retrieval and social network analysis the problem setting of inferring relations between pairs of data objects has recently been investigated intensively in the machine learning community to this end current approaches typically consider datasets containing crisp relations so that standard classification methods can be adopted however relations between objects like similarities and preferences are often expressed in a graded manner in real world applications a general kernel based framework for learning relations from data is introduced here it extends existing approaches because both crisp and graded relations are considered and it unifies existing approaches because different types of graded relations can be modeled including symmetric and reciprocal relations this framework establishes important links between recent developments in fuzzy set theory and machine learning its usefulness is demonstrated through various experiments on synthetic and real world data the results indicate that incorporating domain knowledge about relations improves the predictive performance

mixed finite element discontinuous finite volume element discretization of a general class of multicontinuum models we present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua in the framework of mixture theory we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases the model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture barycentric velocities phase pressure and saturation we construct an accurate robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume element discretization of the mass or transport equations the scheme is devised for unstructured meshes and relies on mixed brezzi douglas marini approximations of phase and total velocities on piecewise constant elements for the approximation of phase or total pressures as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest such as volume fraction total density saturation etc as the discretization scheme is derived for a general formulation of multicontinuum physical systems it can be readily applied to a large class of simplified multiphase models on the other the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate an extensive set of numerical test cases involving two and three dimensional porous media are presented to demonstrate the accuracy of the method displaying an optimal convergence rate the physics preserving properties of the mixed primal scheme as well as the robustness of the method which is successfully used to simulate diverse physical phenomena such as density fingering terzaghi s consolidation deformation of a cantilever bracket and boycott effects the applicability of the method is not limited to flow in porous media but can also be employed to describe many other physical systems governed by a similar set of equations including e g multi component materials

crystal dissolution and precipitation in porous media pore scale analysis in this paper we discuss a pore scale model for crystal dissolution and precipitation in porous media we consider first general domains for which existence of weak solutions is proven for the particular case of strips we show that free boundaries occur in the form of dissolution precipitation fronts as the ratio between the thickness and the length of the strip vanishes we obtain the upscaled reactive solute transport model proposed in 12

playing billiards in version space a ray tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a given set of linear separable examples for randomly distributed examples the billiard estimate of the single perceptron with best average generalization probability agrees with known analytic results while for real life classification problems the generalization probability is consistently enhanced when compared to the maximal stability perceptron

bayes point machines kernel classifiers comprise a powerful class of non linear decision functions for binary classification the support vector machine is an example of a learning algorithm for kernel classifiers that singles out the consistent classifier with the largest margin i e minimal real valued output on the training sample within the set of consistent hypotheses the so called version space we suggest the bayes point machine as a well founded improvement which approximates the bayes optimal decision by the centre of mass of version space we present two algorithms to stochastically approximate the centre of mass of version space a billiard sampling algorithm and a sampling algorithm based on the well known perceptron algorithm it is shown how both algorithms can be extended to allow for soft boundaries in order to admit training errors experimentally we find that for the zero training error case bayes point machines consistently outperform support vector machines on both surrogate data and real world benchmark data sets in the soft boundary soft margin case the improvement over support vector machines is shown to be reduced finally we demonstrate that the real valued output of single bayes points on novel test points is a valid confidence measure and leads to a steady decrease in generalisation error when used as a rejection criterion

bayesian classification with gaussian processes we consider the problem of assigning an input vector to one of m classes by predicting p c x for c 1 m for a two class problem the probability of class one given x is estimated by spl sigma y x where spl sigma y 1 1 e sup y a gaussian process prior is placed on y x and is combined with the training data to obtain predictions for new x points we provide a bayesian treatment integrating over uncertainty in y and in the parameters that control the gaussian process prior the necessary integration over y is carried out using laplace s approximation the method is generalized to multiclass problems m 2 using the softmax function we demonstrate the effectiveness of the method on a number of datasets

an exponential integrator for finite volume discretization of a reaction advection diffusion equation we consider the numerical approximation of a general second order semi linear parabolic partial differential equation equations of this type arise in many contexts such as transport in porous media using the finite volume with two point flux approximation on regular mesh combined with exponential time differencing of order one etd1 for temporal discretization we derive the l 2 estimate under the assumption that the non linear term is locally lipschitz numerical simulations to sustain the theoretical results are provided

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data

on the t transitivity of kernels abstract in this paper we present a view of kernels from a fuzzy set theoretical perspective indeed it turns out that kernels which are positive definite functions have to fulfill a consistency property given by the so called t transitivity of a fuzzy t equivalence relation with respect to the triangular norm t as a result we introduce a triangular norm t cos which is characterized as being the greatest one for which all kernels mapping to the unit interval and having constant 1 in its diagonal are t cos equivalences finally some applied relevant aspects are discussed as a starting point for further research

convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media in this paper we discuss the numerical analysis of an upscaled core scale model describing the transport precipitation and dissolution of solutes in a porous medium the particularity lies in the modeling of the reaction term especially the dissolution term which has a multivalued character we consider the weak formulation for the upscaled equation and provide rigorous stability and convergence results for both the semi discrete time discretization and the fully discrete schemes in doing so compactness arguments are employed