fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

error bounds for the large time step glimm scheme applied to scalar conservation laws in this paper we derive an l 1 error bound for the large time step i e large courant number version of the glimm scheme when used for the approximation of solutions to a genuinely nonlinear i e convex scalar conservation law for a generic class of piecewise constant data we show that the error is bounded by o delta x 1 2 vert log delta x vert for courant numbers up to 1 the order of the error is the same as that given by hoff and smoller 5 in 1985 for the glimm scheme under the restriction of courant numbers up to 1 2

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

on the accuracy of fractional step methods for conservation laws in two dimensions in this paper it is shown that on the assumption of the initial data belonging to bv bf r 2 cap l infty bf r 2 the convergence rates in l 1 bf r 2 for both godunov s spatial splitting method and strang s splitting method for scalar conservation laws in two dimensional spaces are of o delta t 1 2 where delta t is the splitting time step the author illustrates by numerical experiments that for some two dimensional riemann problems the rate is of the order o delta t it is also shown by counterexamples that if the initial data belongs to l infty bf r 2 no error estimate of the type o delta t beta for any beta 0 is valid

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

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

stability theorem and truncation error analysis for the glimm scheme and for a front tracking method for flows with strong discontinuities typical nonlinear wave interaction problems involve strong waves moving through a background of weak disturbance previous existence theorems and error analysis for computations are usually restricted to more idealized situations such as small data or single equations we consider here the problem of a single strong discontinuity interacting with a weak background for general hyperbolic systems of conservation laws we obtain the stability consistency theorems and upper bounds of the truncation errors for the glimm scheme and for a front tracking method the major error in the glimm scheme is the error generated by the strong discontinuity this error is reduced when a front tracking method is applied to follow the location of the strong discontinuity this demonstrates an advantage of front tracking methods in one space dimension

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

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

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

error bounds for a deterministic version of the glimm scheme consider the hyperbolic system of conservation laws u t f u x 0 let u be the unique viscosity solution with initial condition u 0 x bar u x and let ue be an approximate solution constructed by the glimm scheme corresponding to the mesh sizes delta x delta t o delta x with a suitable choice of the sampling sequence we prove the estimate big u ve t cdot u t cdot big bf l 1 strut l 1 o 1 cdot sqrt delta x big ln delta x big

geometric optics in a phase space based level set and eulerian framework we introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics this is important in a wide variety of applications in wave propagation our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts this is achieved through solving for the bicharacteristic strips whose projection to spatial space gives the wavefronts in a reduced phase space under a eulerian and partial differential equation based framework the bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the liouville equations results for the two dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field results are also introduced to show the ability to handle reflection and to extend the method to the three dimensional case

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

fast computation of weighted distance functions and geodesics on implicit hyper surfaces 730 an algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper surfaces is introduced in this paper the basic idea is to approximate the intrinsic weighted distance by the euclidean weighted distance computed in a band surrounding the implicit hyper surface in the embedding space thereby performing all the computations in a cartesian grid with classical and efficient numerics based on work on geodesics on riemannian manifolds with boundaries we bound the error between the two distance functions we show that this error is of the same order as the theoretical numerical error in computationally optimal hamilton jacobi based algorithms for computing distance functions in cartesian grids therefore we can use these algorithms modified to deal with spaces with boundaries and obtain also for the case of intrinsic distance functions on implicit hyper surfaces a computationally efficient technique the approach can be extended to solve a more general class of hamilton jacobi equations defined on the implicit surface following the same idea of approximating their solutions by the solutions in the embedding euclidean space the framework here introduced thereby allows for the computations to be performed on a cartesian grid with computationally optimal algorithms in spite of the fact that the distance and hamilton jacobi equations are intrinsic to the implicit hyper surface c 2001 academic press

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

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

numerical approximation of the euler maxwell model in the quasineutral limit we derive and analyze an asymptotic preserving scheme for the euler maxwell system in the quasi neutral limit we prove that the linear stability condition on the time step is independent of the scaled debye length l when l 0 numerical validation performed on riemann initial data and for a model plasma opening switch device show that the ap scheme is convergent to the euler maxwell solution when dx l 0 where dx is the spatial discretization but when l dx 0 the ap scheme is consistent with the quasi neutral euler maxwell system the scheme is also perfectly consistent with the gauss equation the possibility of using large time and space steps leads to several orders of magnitude reductions in computer time and storage

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

statistical behavior and consistency of classification methods based on convex risk minimization we study how closely the optimal bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function the measurement of closeness is characterized by the loss function used in the estimation we show that such a classification scheme can be generally regarded as a nonmaximum likelihood conditional in class probability estimate and we use this analysis to compare various convex loss functions that have appeared in the literature furthermore the theoretical insight allows us to design good loss functions with desirable properties another aspect of our analysis is to demonstrate the consistency of certain classification methods using convex risk minimization this study sheds light on the good performance of some recently proposed linear classification methods including boosting and support vector machines it also shows their limitations and suggests possible improvements

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

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

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

optimal visibility based path and motion planning of mobile observers for 3 d objects abstract various optimization problems derived from visibility based path and motion planning of mobile observers for three dimensional objects such as asteroids are considered first the mathematical formulations of these problems are presented these problems are generally non smooth then sufficient conditions for the existence of solutions are established the nature of the optimal solutions to some of the posed problems is illustrated by simple examples next the computational complexities in the solution of these problems are discussed briefly finally explicit necessary optimality conditions for asteroid observer optimal motion planning problems are presented

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

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

a variational approach to path planning in three dimensions using level set methods in this paper we extend the two dimensional methods set forth in t cecil d marthaler a variational approach to search and path planning using level set methods ucla cam report 04 61 2004 proposing a variational approach to a path planning problem in three dimensions using a level set framework after defining an energy integral over the path we use gradient flow on the defined energy and evolve the entire path until a locally optimal steady state is reached we follow the framework for motion of curves in three dimensions set forth in p burchard l t cheng b merriman s osher motion of curves in three spatial dimensions using a level set approach j comput phys 170 2 2001 720 741 modified appropriately to take into account that we allow for paths with positive varying widths applications of this method extend to robotic motion and visibility problems for example numerical methods and algorithms are given and examples are presented

simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension we introduce simplex free adaptive tree numerical methods for solving static and time dependent hamilton jacobi equations arising in level set problems in arbitrary dimension the data structure upon which our method is built in a generalized n dimensional binary tree but it does not require the complicated splitting of cubes into simplices aka generalized n dimensional triangles or hypertetrahedrons that current tree based methods require it has enough simplicity that minor variants of standard numerical hamiltonians developed for uniform grids can be applied yielding consistent monotone convergent schemes combined with the fast sweeping strategy the resulting tree based methods are highly efficient and accurate thus without changing more than a few lines of code when changing dimension we have obtained results for calculations in up to n 7 dimensions

error bounds for glimm difference approximations for scalar conservation laws on deduit des bornes d erreur pour l approximation aux differences de glimm pour la solution d une loi de conservation scalaire non lineaire

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

visibility and its dynamics in a pde based implicit framework we investigate the problem of determining visible regions given a set of moving obstacles and a moving vantage point our approach to this problem is through an implicit framework where the obstacles are represented by a level set function the visibility problem is formally formulated as a boundary value problem bvp of a first order partial differential equation it is based on the continuation of values along the given ray field we propose a one pass multi level algorithm for the construction of the solution on a grid furthermore we study the dynamics of shadow boundaries on the surfaces of the obstacles when the vantage point moves along a given trajectory in all of these situations topological changes such as merging and breaking occur in the regions of interest these are automatically handled by the level set framework proposed here finally we obtain additional useful information through simple operations in the level set framework

mm algorithms for generalized bradley terry models the bradley terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs among the many studies of the model in the past 75 years numerous authors have generalized it in several directions sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations building on a theory of algorithms known by the initials mm for minorization maximization this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms for a wide class of generalizations of the bradley terry model while algorithms for problems of this type have tended to be custom built in the literature the techniques in this paper enable their mass production simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator several of the algorithms and convergence results herein are new

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

motion of curves constrained on surfaces using a level set approach the level set method has been successfully applied to a variety of problems that deal with curves in r2 or surfaces in r3 we present here a combination of these two cases creating a level set representation for curves constrained to lie on surfaces we study primarily geometrically based motions of these curves on stationary surfaces while allowing topological changes in the curves i e merging and breaking to occur applications include finding geodesic curves and shortest paths and curve shortening on surfaces further applications can be arrived at by extending those for curves moving in r2 to surfaces the problem of moving curves on surfaces can also be viewed as a simple constraint problem and may be useful in studying more difficult versions results show that our representation can accurately handle many geometrically based motions of curves on a wide variety of surfaces while automatically enforcing topological changes in the curves when they occur and automatically fixing the curves to lie on the surfaces the method can also be easily extended to higher dimensions

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

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

motion of curves in three spatial dimensions using a level set approach the level set method was originally designed for problems dealing with codimension one objects where it has been extremely succesful especially when topological changes in the interface i e merging and breaking occur attempts have been made to modify it to handle objects of higher codimension such as vortex filaments while preserving the merging and breaking property we present numerical simulations of a level set based method for moving curves in r3 the model problem for higher codimension that allows for topological changes a vector valued level set function is used with the zero level set representing the curve our results show that this method can handle many types of curves moving under all types of geometrically based flows while automatically enforcing merging and breaking