learning from partially supervised data using mixture models and belief functions this paper addresses classification problems in which the class membership of training data are only partially known each learning sample is assumed to consist of a feature vector x email protected x and an imprecise and or uncertain soft label m i defined as a dempster shafer basic belief assignment over the set of classes this framework thus generalizes many kinds of learning problems including supervised unsupervised and semi supervised learning here it is assumed that the feature vectors are generated from a mixture model using the generalized bayesian theorem an extension of bayes theorem in the belief function framework we derive a criterion generalizing the likelihood function a variant of the expectation maximization em algorithm dedicated to the optimization of this criterion is proposed allowing us to compute estimates of model parameters experimental results demonstrate the ability of this approach to exploit partial information about class labels

maximum likelihood estimation from fuzzy data using the em algorithm a method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample this method is based on maximizing the observed data likelihood defined as the probability of the fuzzy data it is shown that the em algorithm may be used for that purpose which makes it possible to solve a wide range of statistical problems involving fuzzy data this approach called the fuzzy em fem method is illustrated using three classical problems normal mean and variance estimation from a fuzzy sample multiple linear regression with crisp inputs and fuzzy outputs and univariate finite normal mixture estimation from fuzzy data

clustering of fuzzy data and simultaneous feature selection a model selection approach abstract fuzzy data occurs frequently in the fields of decision making social sciences and control theory we consider the problem of clustering fuzzy data along with automatic component number detection and feature selection a model selection criterion called minimum message length is used to address the problem of component number selection the bayesian framework can be adopted here by applying an explicit prior distribution over the parameter values we discuss both uninformative and informative priors for the latter a gradient descent algorithm for automatic optimization of the prior hyper parameters is presented the problem of simultaneous feature selection involves ordering the discriminative features according to their relative importance and at the same time eliminating non discriminative features the feature selection problem is also formulated as a parameter estimation problem by extending the concept of feature saliency then the estimation can be computed simultaneously with the clustering steps by combining the clustering the cluster number detection and the feature selection into one estimation problem we modified the fuzzy expectation maximization em algorithm to perform all of the estimation evaluation criteria are proposed and empirical study results are reported to showcase the efficacy of our proposals

first order semi local invariants of stable maps of 3 manifolds into the plane in the late 1980s vassiliev introduced new graded numerical invariants of knots which are now called vassiliev invariants or finite type invariants since he made this definition many people have been trying to construct vassiliev type invariants for various mapping spaces in the early 1990s arnold and goryunov introduced the notion of first order local invariants of stable maps in this paper we define and study it first order semi local invariants of stable maps and those of stable fold maps of a closed orientable 3 dimensional manifold into the plane we show that there are essentially eight first order semi local invariants for a stable map one of them is a constant invariant six of them count the number of singular fibers of a given type which appear discretely there are exactly six types of such singular fibers and the last one is the euler characteristic of the stein factorization of this stable map besides these invariants for stable fold maps the bennequin invariant of the singular value set corresponding to definite fold points is also a first order semi local invariant our study of unstable fold maps with codimension 1 provides invariants for the connected components of the set of all fold maps

clustering and classification of fuzzy data using the fuzzy em algorithm in this article we address the problem of clustering imprecise data using a finite mixture of gaussians we propose to estimate the parameters of the model using the fuzzy em algorithm this extension of the em algorithm allows us to handle imprecise data represented by fuzzy numbers first we briefly recall the principle of the fuzzy em algorithm then we provide closed forms for the parameter estimates in the case of gaussian fuzzy data we also describe a monte carlo procedure for estimating the parameter updates in the general case experiments carried out on synthetic and real data demonstrate the interest of our approach for taking into account attribute and label uncertainty

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

liftable vector fields over corank one multigerms in this paper a systematic method is given to construct all liftable vector fields over an analytic multigerm f mathbb k n s rightarrow mathbb k p 0 of corank at most one admitting a one parameter stable unfolding

new criteria for exponential stability of nonlinear difference systems with time varying delay general nonlinear time varying difference systems with time varying delay are considered some new explicit criteria for global exponential stability are given two examples are given to illustrate the obtained results

classifying codimension two multigerms we generalise the operations of augmentation and concatenations defined in cooper et al compos math 131 2 121 160 2002 in order to obtain multigerms of analytic or smooth maps mathbb k n s rightarrow mathbb k p 0 with mathbb k mathbb c or mathbb r from monogerms and some special multigerms we then prove that any corank 1 codimension 2 multigerm in mather s nice dimensions n p with n ge p 1 can be constructed using augmentations and these operations

assessing a mixture model for clustering with the integrated completed likelihood we propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood for this purpose the observed data are assigned to unknown clusters using a maximum a posteriori operator then the integrated completed likelihood icl is approximated using the bayesian information criterion bic numerical experiments on simulated and real data of the resulting icl criterion show that it performs well both for choosing a mixture model and a relevant number of clusters in particular icl appears to be more robust than bic to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data

fuzziness in data analysis the first aim is to emphasize the use of fuzziness in data analysis to capture information that has been traditionally disregarded with a cost in the precision of the conclusions fuzziness can be considered in the data analysis process at various stages but the main target in this paper will be fuzziness in the data depending on the nature of the fuzzy data or the aim to which they are handled different approaches should be applied we attempt to contribute to the clarification of such a difference while focusing on the so called ontic approach in contrast to the epistemic approach the second aim is to underline the need of considering robust methods to reduce the misleading impact of outliers in fuzzy data analysis we propose trimming as a general and intuitive method to discard outliers we exemplify this approach with the case of the ontic fuzzy trimmed mean variance and highlight the differences with the epistemic case all the discussions and developments are illustrated by means of a case study concerning the perception of lengths of men and women

overview on the development of fuzzy random variables this paper presents a backward analysis on the interpretation modelling and impact of the concept of fuzzy random variable after some preliminaries the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained we also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications

making use of partial knowledge about hidden states in hmms an approach based on belief functions this paper addresses the problem of parameter estimation and state prediction in hidden markov models hmms based on observed outputs and partial knowledge of hidden states expressed in the belief function framework the usual hmm model is recovered when the belief functions are vacuous parameters are learned using the evidential expectation maximization algorithm a recently introduced variant of the expectation maximization algorithm for maximum likelihood estimation based on uncertain data the inference problem i e finding the most probable sequence of states based on observed outputs and partial knowledge of states is also addressed experimental results demonstrate that partial information about hidden states when available may substantially improve the estimation and prediction performances

principal component analysis of fuzzy data using autoassociative neural networks this paper describes an extension of principal component analysis pca allowing the extraction of a limited number of relevant features from high dimensional fuzzy data our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as pca without explicit matrix diagonalization fuzzy input values are propagated through the network using fuzzy arithmetics and the weights are adjusted to minimize a suitable error criterion the inputs being taken as target outputs the concept of correlation coefficient is extended to fuzzy numbers allowing the interpretation of the new features in terms of the original variables experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data

exponential ultimate boundedness of nonlinear stochastic difference systems with time varying delays in this paper we discuss the boundedness of nonlinear stochastic difference systems with time varying delays several criteria on exponential ultimate boundedness in mean square are derived examples are also discussed to illustrate the effectiveness of the obtained results

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

statistical inference about the means of fuzzy random variables applications to the analysis of fuzzy and real valued data the expected value of a fuzzy random variable plays an important role as central summary measure and for this reason in the last years valuable statistical inferences about the means of the fuzzy random variables have been developed some of the main contributions in this topic are gathered and discussed concerning the hypothesis testing the bootstrap techniques have empirically shown to be efficient and powerful algorithms to apply these techniques in practice and some illustrative real life examples are included on the other hand it has been recently shown that the distribution of any real valued random variable can be represented by means of a fuzzy set the characterizing fuzzy sets correspond to the expected value of a certain fuzzy random variable based on a family of fuzzy valued transformations of the original real valued ones they can be used for descriptive exploratory or inferential purposes this fact adds an extra value to the fuzzy expected value and the preceding statistical procedures that can be used in statistics about real distributions

two and three way component models for lr fuzzy data in a possibilistic framework abstract in this work we address the data reduction problem for fuzzy data in particular following a possibilistic approach several component models for handling two and three way fuzzy data sets are introduced the two way models are based on classical principal component analysis pca whereas the three way ones on three way generalizations of pca as tucker3 and candecomp parafac the here proposed models exploit the potentiality of the possibilistic regression in fact the component models for fuzzy data can be seen as particular regression analyses between a set of observed fuzzy variables response variables and a set of unobservable crisp variables explanatory variables in order to show how the models work the results of an application to a three way fuzzy data set are illustrated

nonparametric rank based statistics and significance tests for fuzzy data nonparametric rank based statistics depending only on linear orderings of the observations are extended to fuzzy data the approach relies on the definition of a fuzzy partial order based on the necessity index of strict dominance between fuzzy numbers which is shown to contain in a well defined sense all the ordinal information present in the original data a concept of fuzzy set of linear extensions of a fuzzy partial order is introduced allowing the approximate computation of fuzzy statistics alpha cutwise using a markov chain monte carlo simulation approach the usual notions underlying significance tests are also extended leading to the concepts of fuzzy p value and graded rejection of the null hypothesis quantified by a degree of possibility and a degree of necessity at a given significance level this general approach is demonstrated in two special cases kendall s rank correlation coefficient and wilcoxon s two sample rank sum statistic

a hypothesis testing based discussion on the sensitivity of means of fuzzy data with respect to data shape abstract in assessing fuzzy numbers to model imprecise data associated with random experiments trapezoidal fuzzy numbers are often considered such an assessment is mainly due to easing both interpretation and computation this becomes especially noticeable when those assessing fuzzy numbers to data have a weak knowledge low background and little or no expertise in using fuzzy sets as it happens when questionnaires whose responses involve a free fuzzy rating are conducted since the required training to explain the meaning and use of trapezoidal fuzzy numbers is definitely lower than that associated with other shapes nevertheless a question that constantly arises in connection with this trapezoidal assessment is whether it can importantly affect the conclusions of the study involving such data this paper aims to answer the last question from a statistical perspective more concretely the analysis of the influence of the choice of trapezoidal fuzzy numbers to model data is to be based on the conclusions from statistical hypothesis testing about the mean values of the involved fuzzy datasets for this purpose the p values of tests have been compared for trapezoidal assessment vs other frequently used ones like some lu assessments the analysis is first performed by developing simulation based pairwise comparisons and it is later illustrated and corroborated to some extent with a real life example the analysis indicates that the shape of the fuzzy assessment scarcely affects statistical conclusions

stability analysis of a general family of nonlinear positive discrete time delay systems in this paper we propose a new approach to analyse the stability of a general family of nonlinear positive discrete time delay systems first we introduce a new class of nonlinear positive discrete time delay systems which generalises some existing discrete time delay systems second through a new technique that relies on the comparison and mathematical induction method we establish explicit criteria for stability and instability of the systems three numerical examples are given to illustrate the feasibility of the obtained results

machine learning models epistemic set valued data and generalized loss functions we study those problems where the goal is to find optimal models with respect to some specific criterion in regression and supervised classification problems alternatives to the usual expected loss minimization criterion are proposed and a general framework where this criterion can be seen as a particular instance of a general family of criteria is provided in the new setting each model is formally identified with a random variable that associates a loss value to each individual in the population based on this identification different stochastic orderings between random variables lead to different criteria to compare pairs of models our general setting encompasses the classical criterion based on the minimization of the expected loss but also other criteria where a numerical loss function is not available and therefore the computation of its expectation does not make sense the presentation of the new framework is divided into two stages first we consider the new framework under standard situations about the sample information where both the collection of attributes and the response variables are observed with precision then we assume that just incomplete information about them expressed in terms of set valued data sets is provided we cast some comparison criteria from the recent literature on learning methods from low quality data as particular instances of our general approach

bootstrap techniques and fuzzy random variables synergy in hypothesis testing with fuzzy data in previous studies we have stated that the well known bootstrap techniques are a valuable tool in testing statistical hypotheses about the means of fuzzy random variables when these variables are supposed to take on a finite number of different values and these values being fuzzy subsets of the one dimensional euclidean space in this paper we show that the one sample method of testing about the mean of a fuzzy random variable can be extended to general ones more precisely to those whose range is not necessarily finite and whose values are fuzzy subsets of finite dimensional euclidean space this extension is immediately developed by combining some tools in the literature namely bootstrap techniques on banach spaces a metric between fuzzy sets based on the support function and an embedding of the space of fuzzy random variables into a banach space which is based on the support function