approximation of fuzzy numbers by max product bernstein operators in this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear bernstein operators of max product kind by proving that their order of uniform approximation is the same as in the particular case of the unit interval then similarly to the particular case of the unit interval we will prove that these operators preserve the quasi concavity too since these properties will help us to generate in a simple way fuzzy numbers of the same support it turns out that these results are very suitable in the approximation of fuzzy numbers moreover these operators approximate the non degenerate segment core with a good order of approximation in addition in the case when the fuzzy numbers are given in parametric form the bernstein max product operator generates a sequence of fuzzy numbers whose widths expected intervals ambiguities and values approximate with a convergent rate the width the expected interval the ambiguity and the value of the approximated fuzzy number finally we obtain a quantitative approximation with respect to a well known integral type metric for some subclasses of functions leading to applications to the approximation of some subclasses of fuzzy numbers

a note on trapezoidal approximations of fuzzy numbers in this paper we propose some generalized and new properties of the trapezoidal approximations of fuzzy numbers first we claim that the trapezoidal approximation operator is linear with respect to the operations of fuzzy numbers second we show a new property concerning the centroid points of fuzzy numbers furthermore a comment for computing the trapezoidal approximations is provided finally some properties about the distance between a fuzzy number and its trapezoidal approximation are proved

m estimates of location for the robust central tendency of fuzzy data the aumann type mean has been shown to possess valuable properties as a measure of the location or central tendency of fuzzy data associated with a random experiment however concerning robustness its behavior is not appropriate the aumann type mean is highly affected by slight changes in the fuzzy data or when outliers arise in the sample robust estimators of location on the other hand avoid such adverse effects for this purpose this paper considers the m estimation approach and discusses conditions under which this alternative yields valid fuzzy valued m estimators the resulting m estimators are applied to a real life example finally some simulation studies show empirically the suitability of the introduced estimators

statistical estimation in varying coefficient models varying coefficient models are a useful extension of classical linear models they arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age the appeal of these models is that the coef cient functions can easily be estimated via a simple local regression this yields a simple one step estimation procedure we show that such a one step method cannot be optimal when different coefficient functions admit different degrees of smoothness this drawback can be repaired by using our proposed two step estimation procedure the asymptotic mean squared error for the two step procedure is obtained and is shown to achieve the optimal rate of convergence a few simulation studies show that the gain by the two step procedure can be quite substantial the methodology is illustrated by an application to an environmental data set

dealing with label switching in mixture models summary in a bayesian analysis of finite mixture models parameter estimation and clustering are sometimes less straightforward than might be expected in particular the common practice of estimating parameters by their posterior mean and summarizing joint posterior distributions by marginal distributions often leads to nonsensical answers this is due to the so called label switching problem which is caused by symmetry in the likelihood of the model parameters a frequent response to this problem is to remove the symmetry by using artificial identifiability constraints we demonstrate that this fails in general to solve the problem and we describe an alternative class of approaches relabelling algorithms which arise from attempting to minimize the posterior expected loss under a class of loss functions we describe in detail one particularly simple and general relabelling algorithm and illustrate its success in dealing with the label switching problem on two examples

trapezoidal approximations of fuzzy numbers the problem of the trapezoidal approximation of fuzzy numbers is discussed a set of criteria for approximation operators is formulated these constraints can be used for direct operator derivation a new nearest trapezoidal approximation operator preserving expected interval is suggested

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

smoothing spline estimation in varying coefficient models summary smoothing spline estimators are considered for inference in varying coefficient models with one effect modifying covariate bayesian confidence intervals are developed for the coefficient curves and efficient computational methods are derived for computing the curve estimators fitted values posterior variances and data adaptive methods for selecting the levels of smoothing the efficacy and utility of the methodology proposed are demonstrated through a small simulation study and the analysis of a real data set

interval random sets and entropy abstract the problem of proper choice from the class of interval random sets generated by some fuzzy set is considered this problem is determined for the maximum entropy criterion the class of interval random sets is studied

conditioned weighted l r approximations of fuzzy numbers we compute the extended weighted l r approximation of a given fuzzy number by a method based on general results in hilbert spaces the weighted average euclidean distance being considered the metric properties of the extended weighted l r approximation of a fuzzy number are proved we elaborate on a general method to study the existence uniqueness and to calculate the l r approximations of fuzzy numbers under the preservation of some parameters we apply the results to find the weighted l r approximations preserving ambiguity and value and respectively width in the general and unimodal case

smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables longitudinal samples i e datasets with repeated measurements over time are common in biomedical and epidemiological studies such as clinical trials and cohort observational studies an exploratory tool for the analyses of such data is the varying coefficient model y t xt t t t where y t and x t x 0 t x k t t are the response and covariates at time t t 0 t k t t are smooth coefficient curves of t and t is a mean zero stochastic process a special case that is of particular interest in many situations is data with time dependent response and time independent covariates we propose in this article a componentwise smoothing spline method for estimating 0 t k t nonparametrically based on the previous varying coefficient model and a longitudinal sample of t y t x with time independent covariates x x 0 x k t from n independent subjects a leave one subject out cross validation is suggested to choose the smoothing parameters asymptotic properties of our

fuzzy number approximation via shadowed sets interval and trapezoidal approximation methods for fuzzy numbers have been intensively developed and examined recently in this paper another approach to simplify fuzzy numbers called shadowed set approximation is suggested some basic properties of the proposed method is also discussed it seems that the proposed approximation might be useful for granular computing as a tool that removes excessive precision in describing imprecise phenomena

varying coefficient models we explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables general algorithms are presented for estimating the models flexibly and some examples are given this class of models ties together generalized additive models and dynamic generalized linear models into one common framework when applied to the proportional hazards model for survival data this approach provides a new way of modelling departures from the proportional hazards assumption

a new version of 2 tuple fuzzy linguistic representation model for computing with words in this paper we provide a new proportional 2 tuple fuzzy linguistic representation model for computing with words cw which is based on the concept of symbolic proportion this concept motivates us to represent the linguistic information by means of 2 tuples which are composed by two proportional linguistic terms for clarity and generality we first study proportional 2 tuples under ordinal contexts then under linguistic contexts and based on canonical characteristic values ccvs of linguistic labels we define many aggregation operators to handle proportional 2 tuple linguistic information in a computational stage for cw without any loss of information our approach for this proportional 2 tuple fuzzy linguistic representation model deals with linguistic labels which do not have to be symmetrically distributed around a medium label and without the traditional requirement of having equal distance between them moreover this new model not only provides a space to allow a continuous interpolation of a sequence of ordered linguistic labels but also provides an opportunity to describe the initial linguistic information by members of a continuous linguistic scale domain which does not necessarily require the ordered linguistic terms of a linguistic variable being equidistant meanwhile under the assumption of equally informative which is defined by a condition based on the concept of ccv we show that our model reduces to herrera and mart spl inodot spl acute nez s translational 2 tuple fuzzy linguistic representation model

mixtures of varying coefficient models for longitudinal data with discrete or continuous nonignorable dropout the analysis of longitudinal repeated measures data is frequently complicated by missing data due to informative dropout we describe a mixture model for joint distribution for longitudinal repeated measures where the dropout distribution may be continuous and the dependence between response and dropout is semiparametric specifically we assume that responses follow a varying coefficient random effects model conditional on dropout time where the regression coefficients depend on dropout time through unspecified nonparametric functions that are estimated using step functions when dropout time is discrete e g for panel data and using smoothing splines when dropout time is continuous inference under the proposed semiparametric model is hence more robust than the parametric conditional linear model the unconditional distribution of the repeated measures is a mixture over the dropout distribution we show that estimation in the semiparametric varying coefficient mixture model can proceed by fitting a parametric mixed effects model and can be carried out on standard software platforms such as sas the model is used to analyze data from a recent aids clinical trial and its performance is evaluated using simulations

trapezoidal approximations of fuzzy numbers revisited fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed the previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number

fuzzy multi item economic production quantity model under space constraint a geometric programming approach in this study we have formulated an economic production quantity epq model with flexibility and reliability consideration of production process and demand dependent unit production cost with fuzzy parameters the model is restricted with available limited storage space constraint the inventory related costs storage spaces and others parameters are taken here as triangular fuzzy numbers the problem is solved by modified geometric programming method finally the model is illustrated by numerical example

finite mixture varying coefficient models for analyzing longitudinal heterogenous data this paper aims to develop a mixture model to study heterogeneous longitudinal data on the treatment effect of heroin use from a california civil addict program each component of the mixture is characterized by a varying coefficient mixed effect model we use the bayesian p splines approach to approximate the varying coefficient functions we develop markov chain monte carlo algorithms to estimate the smooth functions unknown parameters and latent variables in the model we use modified deviance information criterion to determine the number of components in the mixture a simulation study demonstrates that the modified deviance information criterion selects the correct number of components and the estimation of unknown quantities is accurate we apply the proposed model to the heroin treatment study furthermore we identify heterogeneous longitudinal patterns copyright 2011 john wiley sons ltd

solution of a possibilistic multiobjective linear programming problem abstract the estimate of the parameters which define a conventional multiobjective decision making model is a difficult task normally they are either given by the decision maker who has imprecise information and or expresses his considerations subjectively or by statistical inference from the past data and their stability is doubtful therefore it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets and several fuzzy approaches to multiobjective programming have been developed 1 9 10 11 the fuzziness of the parameters gives rise to a problem whose solution will also be fuzzy see 2 3 and which is defined by its possibility distribution once the possibility distribution of the solution has been obtained if the decision maker wants more precise information with respect to the decision vector then we can pose and solve a new problem in this case we try to find a decision vector which approximates as much as possible the fuzzy objectives to the fuzzy solution previously obtained in order to solve this problem we shall develop two different models from the initial solution and based on goal programming an interval goal programming problem if we define the relation as accurate as possible based on the expected intervals of fuzzy numbers as we showed in 4 and an ordinary goal programming based on the expected values of the fuzzy numbers that defined the goals finally we construct algorithms that implement the above mentioned solution method our approach will be illustrated by means of a numerical example

trapezoidal and triangular approximations preserving the expected interval in this paper an improvement of the nearest trapezoidal approximation operator preserving the expected interval is studied which is proposed by grzegorzewski and mrowka a formula for computing the improved approximation is presented moreover the nearest triangular approximation operator preserving the expected interval is also investigated

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

time varying effect models for ordinal responses with applications in substance abuse research ordinal responses are very common in longitudinal data collected from substance abuse research or other behavioral research this study develops a new statistical model with free sas macro s that can be applied to characterize time varying effects on ordinal responses our simulation study shows that the ordinal scale time varying effects model has very low estimation bias and sometimes offers considerably better performance when fitting data with ordinal responses than a model that treats the response as continuous contrary to a common assumption that an ordinal scale with several levels can be treated as continuous our results indicate that it is not so much the number of levels on the ordinal scale but rather the skewness of the distribution that makes a difference on relative performance of linear versus ordinal models we use longitudinal data from a well known study on youth at high risk for substance abuse as a motivating example to demonstrate that the proposed model can characterize the time varying effect of negative peer influences on alcohol use in a way that is more consistent with the developmental theory and existing literature in comparison to the linear time varying effect model

penalized clustering of large scale functional data with multiple covariates in this article we propose a penalized clustering method for large scale data with multiple covariates through a functional data approach in our proposed method responses and covariates are linked together through nonparametric multivariate functions fixed effects which have great flexibility in modeling various function features such as jump points branching and periodicity functional anova is used to further decompose multivariate functions in a reproducing kernel hilbert space and provide associated notions of main effect and interaction parsimonious random effects are used to capture various correlation structures the mixed effects models are nested under a general mixture model in which the heterogeneity of functional data is characterized we propose a penalized henderson s likelihood approach for model fitting and design a rejection controlled em algorithm for the estimation our method selects smoothing parameters through generalized cross validation furthermore bayesian confidence in

bayesian identification selection and estimation of semiparametric functions in high dimensional additive models in this paper we propose an approach to both estimate and select unknown smooth functions in an additive model with potentially many functions each function is written as a linear combination of basis terms with coefficients regularized by a proper linearly constrained gaussian prior given any potentially rank deficient prior precision matrix we show how to derive linear constraints so that the corresponding effect is identified in the additive model this allows for the use of a wide range of bases and precision matrices in priors for regularization by introducing indicator variables each constrained gaussian prior is augmented with a point mass at zero thus allowing for function selection posterior inference is calculated using markov chain monte carlo and the smoothness in the functions is both the result of shrinkage through the constrained gaussian prior and model averaging we show how using non degenerate priors on the shrinkage parameters enables the application of substantially more computationally efficient sampling schemes than would otherwise be the case we show the favourable performance of our approach when compared to two contemporary alternative bayesian methods to highlight the potential of our approach in high dimensional settings we apply it to estimate two large seemingly unrelated regression models for intra day electricity load both models feature a variety of different univariate and bivariate functions which require different levels of smoothing and where component selection is meaningful priors for the error disturbance covariances are selected carefully and the empirical results provide a substantive contribution to the electricity load modelling literature in their own right

spatial modeling with spatially varying coefficient processes in many applications the objective is to build regression models to explain a response variable over a region of interest under the assumption that the responses are spatially correlated in nearly all of this work the regression coefficients are assumed to be constant over the region however in some applications coefficients are expected to vary at the local or subregional level here we focus on the local case although parametric modeling of the spatial surface for the coefficient is possible here we argue that it is more natural and flexible to view the surface as a realization from a spatial process we show how such modeling can be formalized in the context of gaussian responses providing attractive interpretation in terms of both random effects and explaining residuals we also offer extensions to generalized linear models and to spatio temporal setting we illustrate both static and dynamic modeling with a dataset that attempts to explain log selling price of single family houses

nearest interval approximation of a fuzzy number the problem of the interval approximation of fuzzy numbers is discussed a new interval approximation operator which is the best one with respect to a certain measure of distance between fuzzy numbers is suggested

technical communique new design of linear least squares fixed interval smoother using covariance information this paper derives recursive linear least squares fixed interval smoothing algorithm using covariance information by applying an invariant imbedding method to a wiener hopf integral equation the algorithm is obtained for the white plus coloured observation noise the signal process is nonstationary stochastic autocovariance functions of the signal and the coloured noise are expressed using a degenerate kernel the degenerate kernel can represent general covariance functions of nonstationary stochastic processes by a finite sum of nonrandom functions

trapezoidal approximations of fuzzy numbers preserving the expected interval algorithms and properties fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is discussed algorithms for calculating the proper approximations are proposed and some properties of the approximation operators are discussed it is shown that an adequate approximation operator might be chosen through the comparisons of some characteristics of the fuzzy number like its ambiguity width its value and weighted expected value

nonparametric smoothing estimates of time varying coefficient models with longitudinal data this paper considers nonparametric estimation in a varying coefficient model with repeated measurements y ij x ij t ij for i 1 n and j 1 n i where x ij x ij x ijk t and y ij x ij t ij denote the jth outcome covariate and time design points respectively of the ith subject the model considered here is y ij x ij t t ij e i t ij where t 0 t k t t for k 0 are smooth nonparametric functions of interest and e i t is a zero mean stochastic process the measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject two nonparametric estimators of t namely a smoothing spline and a locally weighted polynomial are derived for such repeatedly measured data a crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters asymptotic properties such as consistency rates of convergence and asymptotic mean squared errors are established for kernel estimators a special case of the local polynomials these asymptotic results give useful insights into the reliability of our general estimation methods an example of predicting the growth of children born to hiv infected mothers based on gender hiv status and maternal vitamin a levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies

new axiomatisations of discrete quantitative and qualitative possibilistic integrals necessity resp possibility measures are very simple min decomposable resp max decomposable representations of epistemic uncertainty due to incomplete knowledge they can be used in both quantitative and qualitative settings in the present work we revisit choquet and sugeno integrals as criteria for decision under uncertainty and propose new axioms when uncertainty is representable in possibility theory first a characterization of choquet integral with respect to a possibility or a necessity measure is proposed we respectively add an optimism or a pessimism axiom to the axioms of the choquet integral with respect to a general capacity this new axiom enforces the maxitivity or the minitivity of the capacity without requiring the same property for the functional it essentially assumes that the decision maker preferences only reflect the plausibility ordering between states of nature the obtained pessimistic resp optimistic criterion is an average maximin resp maximax criterion of wald across cuts of a possibility distribution on the state space the additional axiom can be also used in the axiomatic approach to sugeno integral and generalized forms thereof to justify possibility and necessity measures the axiomatization of these criteria for decision under uncertainty in the setting of preference relations among acts is also discussed we show that the new axiom justifying possibilistic choquet integrals can be expressed in this setting in the case of sugeno integral we correct a characterization proof for an existing set of axioms on acts and study an alternative set of axioms based on the idea of non compensation

on weighted possibilistic mean and variance of fuzzy numbers dubois and prade defined an interval valued expectation of fuzzy numbers viewing them as consonant random sets carlsson and fuller defined an interval valued mean value of fuzzy numbers viewing them as possibility distributions in this paper we shall introduce the notation of weighted interval valued possibilistic mean value of fuzzy numbers and investigate its relationship to the interval valued probabilistic mean we shall also introduce the notations of crisp weighted possibilistic mean value variance and covariance of fuzzy numbers which are consistent with the extension principle furthermore we show that the weighted variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

why triangular membership functions abstract in this note we look at a certain theoretically sound motivation behind the common use of triangular and trapezoidal membership functions the studies are completed within a conceptual framework of fuzzy modelling whose structure comprises of input and output interfaces linked with a single transformation module aimed at processing linguistic information it is shown that under some additional mild assumptions these triangular fuzzy sets comply with a request for a uniformly excited codebook in the case of the input interfaces and a satisfaction of a zero error reconstruction criterion being formulated for the output interface

general approximation of fuzzy numbers by f transform in this paper we will prove that in most of the cases the extended inverse fuzzy transform preserves the quasi concavity of a fuzzy number and hence it can be used to generate fuzzy numbers by approximating the restriction of the membership function to its support in the case of continuous fuzzy numbers with cores containing more than one element the rate of uniform convergence is of linear type and the same holds when we approximate the important characteristics of a fuzzy number such as the value or the ambiguity moreover we have the preservation of the support and the convergence of the core which in addition can be determined precisely in the case of continuous fuzzy numbers with one element core it is in general necessary to normalize the approximation but the support is preserved again and the core can be determined exactly in this case too moreover the approximations have again linear rate of uniform convergence

metrics and orders in space of fuzzy numbers abstract two classes of metrics p q and p where 1 p 0 q 1 in space of fuzzy numbers are introduced then a method of ranking fuzzy numbers based on these metrics is proposed and investigated not only fuzzy numbers but also the left sided and right sided fuzzy numbers are considered applications in making decisions and statistics are suggested

fuzzy numbers from raw discrete data using linear regression this paper focuses on modelling fuzzy numbers with meaningful membership functions more precisely it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data in many applications input information is numerical and therefore particular fuzzy sets such as fuzzy numbers hold great interest and relevance in managing data imprecision and vagueness the proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression the technique is simple fast and effective preliminary tests are performed using different types of input data a gaussian function a sigmoidal function three datasets of synthetic discrete data and an histogram obtained from a colour satellite image

brief papers relation between filter using covariance information and kalman filter this paper presents sequential algorithms for the optimal impulse function kalman gain and the error variance in linear least squares filtering problems when the autocovariance function of the signal is given in the form of a semi degenerate kernel and the additive observation noise in white gaussian a digital simulation result indicates that the algorithms presented in this paper are feasible and that the values of kalman gain and the error variance calculated by these algorithms approach to those obtained by the kalman filter theory for time sufficiently large

markov chain monte carlo estimation of classical and dynamic switching and mixture models bayesian estimation of a very general model class where the distribution of the observations depends on a latent process taking values in a discrete state space is discussed in this article this model class covers finite mixture modeling markov switching autoregressive modeling and dynamic linear models with switching the consequences the unidentifiability of this type of model has on markov chain monte carlo mcmc estimation are explicitly dealt with joint bayesian estimation of all latent variables model parameters and parameters that determine the probability law of the latent process is carried out by a new mcmc method called permutation sampling the permutation sampler first samples from the unconstrained posterior which often can be done in a convenient multimove manner and then applies a permutation of the current labeling of the states of the latent process in a first run the random permutation sampler used selected the permutation randomly the mcmc output of the random permutation s

on bayesian analysis of mixtures with an unknown number of components discussion author s reply new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

the fuzzy kalman filter state estimation using possibilistic techniques a new method to implement fuzzy kalman filters is introduced the combination of possibilistic techniques and the extended kalman filter has special application in fields where inaccurate information is involved the novelty of this article comes from the fact that by using possibility distributions instead of gaussian distributions a fuzzy description of the expected state and observation is sufficient to obtain a good estimation some characteristics of this approach are that uncertainty does not need to be symmetric and that a wide region of possible values for the expectations is allowed to implement the algorithm this approach also contributes a method to propagate uncertainty through the process model and the observation model based on trapezoidal possibility distributions finally several examples of a real mobile robot moving through a localization process while using qualitative landmarks are shown

single value simulation of fuzzy variables abstract the basic ideas of the present paper arise from relations between random sets and fuzzy ones the suggested approach which may be called fuzzy numerical simulation allows for ascribing a precise numerical value to a fuzzy variable by generating a value of a random variable related in some way to the fuzzy variable the so called generative characteristics of a fuzzy variable being the expected value and variance of the generating random variable are introduced the problem of simultaneous generating values of two fuzzy variables is also discussed finally methods of generating when a fuzzy variable is described with a complex body of evidence are presented

computational and inferential difficulties with mixture posterior distributions abstract this article deals with both exploration and interpretation problems related to posterior distributions for mixture models the specification of mixture posterior distributions means that the presence of k modes is known immediately standard markov chain monte carlo mcmc techniques usually have difficulties with well separated modes such as occur here the mcmc sampler stays within a neighborhood of a local mode and fails to visit other equally important modes we show that exploration of these modes can be imposed using tempered transitions however if the prior distribution does not distinguish between the different components then the posterior mixture distribution is symmetric and standard estimators such as posterior means cannot be used we propose alternatives for bayesian inference for permutation invariant posteriors including a clustering device and alternative appropriate loss functions

dynamic conditionally linear mixed models for longitudinal data we develop a new class of models dynamic conditionally linear mixed models for longitudinal data by decomposing the within subject covariance matrix using a special cholesky decomposition here dynamic means using past responses as covariates and conditional linearity means that parameters entering the model linearly may be random but nonlinear parameters are nonrandom this setup offers several advantages and is surprisingly similar to models obtained from the first order linearization method applied to nonlinear mixed models first it allows for flexible and computationally tractable models that include a wide array of covariance structures these structures may depend on covariates and hence may differ across subjects this class of models includes e g all standard linear mixed models antedependence models and vonesh carter models second it guarantees the fitted marginal covariance matrix of the data is positive definite we develop methods for bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited

fuzzy entropy a more comprehensible perspective for interval shadowed sets of fuzzy sets shadowed sets proposed by pedrycz provide a three way approximation scheme for transforming the universe of a fuzzy set into three disjoint areas i e elevated area reduced area and shadow area to calculate a pair of decision making thresholds an analytic method was proposed by solving a minimization problem of the uncertainty arising from the three areas however some uncertainty will be lost in the process of constructing the shadowed set model using pedrycz s method moreover few references on how to measure the uncertainty of shadow sets exist in this article a comprehensible method for measuring the fuzzy entropy of a shadowed set i e interval fuzzy entropy is defined based on the interval fuzzy entropy a new shadowed set model namely interval shadowed sets is proposed compared with pedrycz s model the main difference is that the range of the shadow area in this model is beta alpha 0 leq beta while not 0 1 by solving a fuzzy entropy loss minimization problem a pair of optimal thresholds alpha and beta can be obtained finally the results of the instance analysis of different types of representative membership functions and many experiments show that the fuzzy entropy loss of the interval shadowed set is lower than that of the traditional shadowed set of a fuzzy set these results enrich shadowed set theory from a new perspective

mean semivariance models for fuzzy portfolio selection this paper discusses portfolio selection problem in fuzzy environment in the paper semivariance is originally presented for fuzzy variable and three properties of the semivariance are proven based on the concept of semivariance of fuzzy variable two fuzzy mean semivariance models are proposed to solve the new models in general cases a fuzzy simulation based genetic algorithm is presented in the paper in addition two numerical examples are also presented to illustrate the modelling idea and the effectiveness of the designed algorithm

decision theoretic three way approximations of fuzzy sets abstract a three way three valued or three region approximation of a fuzzy set is constructed from a pair of thresholds on the fuzzy membership function an element whose membership grade equals to or is greater than is put into the positive region an element whose membership grade equals to or is less than is put into the negative region and an element whose membership grade is between and is put into the boundary region a fundamental issue is the determination and interpretation of the required pair of thresholds in the framework of shadowed sets i e an example of three way approximations of fuzzy sets pedrycz provides an analytic solution to computing the thresholds by searching for a balance of uncertainty introduced by the three regions to gain further insights into three way approximations of fuzzy sets we introduce an alternative decision theoretic formulation in which the required thresholds are computed by minimizing decision cost

varying coefficient models and basis function approximations for the analysis of repeated measurements summary a global smoothing procedure is developed using basis function approximations for estimating the parameters of a varying coefficient model with repeated measurements inference procedures based on a resampling subject bootstrap are proposed to construct confidence regions and to perform hypothesis testing conditional biases and variances of our estimators and their asymptotic consistency are developed explicitly finite sample properties of our procedures are investigated through a simulation study application of the proposed approach is demonstrated through an example in epidemiology in contrast to the existing methods this approach applies whether or not the covariates are timeinvariant and does not require binning of the data when observations are sparse at distinct observation times

on the nearest parametric approximation of a fuzzy number many nearest parametric approximation methods of fuzzy sets are proposed in the literature it is clear that the specific approximations may lead to the loss of information about fuzziness to overcome this problem most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation but these approximations mostly are not flexible to the decision maker s choice hence in this paper we offer a parametric fuzzy approximation method based on the decision maker s strategy as an extension of trapezoidal approximation of a fuzzy number this method comprises the selection of the form of the parametric membership function and its evaluation

the nearest approximation of a fuzzy quantity in parametric form applications of fuzzy logic and fuzzy mathematics are increasing widely all around the world thus working with fuzzy numbers are very important in many applications of fuzzy mathematics we need or it is better to work with the same fuzzy numbers in this work we approximate parametric fuzzy numbers with polynomial parametric fuzzy numbers

approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval the problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well known metric which preserves the expected interval of the fuzzy number is completely solved the previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number properties of this new trapezoidal approximation operator are studied

principles for constructing three way approximations of fuzzy sets a comparative evaluation based on unsupervised learning abstract three way approximations of fuzzy sets are an important scheme of granular computing by abstracting a fuzzy set to its discrete three option alternatives which adhere to human cognitive behaviors and reduce the computational burden the key point of such three way approximations of fuzzy sets is how to choose a suitable design leading to their realization undesired three way approximations might be produced if the selected mechanism is unsuitable to data distribution in this study the principles for constructing three way approximations of fuzzy sets are summarized the following taxonomy of these principles is provided namely i uncertainty balance based principle ii prototype based principle and iii model based invoking the tradeoff between classification error and the number of data that have to be classified a number of detailed optimization models are discussed in detail to evaluate the performance of different construction principles a general unsupervised learning framework based on three way approximations of fuzzy sets is exhibited some synthetic data sets along with sixteen data sets from uci repository are involved for experiments friedman testing followed by holm bonferroni testing are exploited to test the performance significance of the proposed criteria which can provide insights and deliver guidance when choosing a principle for constructing three way approximations of fuzzy sets in the real world scenarios the research methods in this paper can also be extended to supervised and semi supervised learning areas

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

bayesian smoothing and regression splines for measurement error problems in the presence of covariate measurement error estimating a regression function nonparametrically is extremely difficult the problem being related to deconvolution various frequentist approaches exist for this problem but to date there has been no bayesian treatment in this article we describe bayesian approaches to modeling a flexible regression function when the predictor variable is measured with error the regression function is modeled with smoothing splines and regression p splines two methods are described for exploration of the posterior the first called the iterative conditional modes icm is only partially bayesian icm uses a componentwise maximization routine to find the mode of the posterior it also serves to create starting values for the second method which is fully bayesian and uses markov chain monte carlo mcmc techniques to generate observations from the joint posterior distribution use of the mcmc approach has the advantage that interval estimates that directly model and

on a canonical representation of fuzzy numbers abstract fuzzy numbers and more generally linguistic values are approximate assessments given by experts and accepted by decision makers when obtaining more accurate values is impossible or unnecessary to simplify the task of representing and handling fuzzy numbers several authors have introduced real indices in order to capture the information contained in a fuzzy number in this paper we propose two parameters value and ambiguity for this purpose we use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision making problems several examples illustrate these ideas

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

a set of practical experiments to validate the fuzzy kalman filter abstract the fuzzy kalman filter fkf was recently reformulated with a number of basic examples included in order to demonstrate its applicability we now look ahead to its practical implementation introducing deep analysis which includes very detailed simulations and complete real applications this will allow us to not only validate the fkf but also to draw important conclusions which will help users in the adjustment of the fuzzy filter parameters and the choosing of possibilistic distributions

technical communique new design of linear discrete time predictor using covariance information a discrete one stage predictor algorithm using covariance information in linear systems is derived the algorithm is obtained for white gaussian observation noise the signal is a nonstationary or stationary stochastic process the auto covariance function of the signal is expressed using a semi degenerate kernel of discrete time systems the semi degenerate kernel can represent general covariance functions of random processes by a finite sum of nonrandom functions

linear programming with fuzzy parameters an interactive method resolution this paper proposes a method for solving linear programming problems where all the coefficients are in general fuzzy numbers we use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints it allows us to work with the concept of feasibility degree the bigger the feasibility degree is the worst the objective value will be we offer the decision maker dm the optimal solution for several different degrees of feasibility with this information the dm is able to establish a fuzzy goal we build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal a reasonable solution is the one that has the biggest membership degree to this fuzzy subset finally to illustrate our method we solve a numerical example

the expected value of a fuzzy number abstract the notions of expected interval and expected value of fuzzy number are introduced the expected interval is defined as the expected value of an interval random set generated by the fuzzy number and the expected value of this number is defined as the centre of the expected interval the linearity of the expected interval is studied

a general model of decision theoretic three way approximations of fuzzy sets based on a heuristic algorithm abstract a model of decision theoretic three way approximations of fuzzy sets that can calculate a pair of interpretable thresholds by combining shadowed sets with three way decisions was established by deng in that model the significant contribution is that the value 0 5 which denotes a membership grade with the highest uncertainty is used to replace the uncertain region with the interval 0 1 in shadowed sets from the principle of the minimum decision cost although the lowest overall cost can be achieved precisely in some data distributions based on the value 0 5 there is a difference between the obtained overall cost and the least cost in some cases therefore in this paper based on deng s model the concept of a general three way approximation of a fuzzy set is proposed to replace 0 5 with a variable value c 0

on the interval approximation of a fuzzy number the notion of an approximation interval of a fuzzy number is introduced it is the interval which fulfills two conditions in the first its width is equal to the width of a fuzzy number being approximated in the second the hamming distance between this interval and the approximated number is minimal the introduced notion is compared with the notion of expected interval of a fuzzy number known in literature

a bayesian modeling approach for generalized semiparametric structural equation models in behavioral biomedical and psychological studies structural equation models sems have been widely used for assessing relationships between latent variables regression type structural models based on parametric functions are often used for such purposes in many applications however parametric sems are not adequate to capture subtle patterns in the functions over the entire range of the predictor variable a different but equally important limitation of traditional parametric sems is that they are not designed to handle mixed data types continuous count ordered and unordered categorical this paper develops a generalized semiparametric sem that is able to handle mixed data types and to simultaneously model different functional relationships among latent variables a structural equation of the proposed sem is formulated using a series of unspecified smooth functions the bayesian p splines approach and markov chain monte carlo methods are developed to estimate the smooth functions and the unknown parameters moreover we examine the relative benefits of semiparametric modeling over parametric modeling using a bayesian model comparison statistic called the complete deviance information criterion dic the performance of the developed methodology is evaluated using a simulation study to illustrate the method we used a data set derived from the national longitudinal survey of youth

fuzzy linguistic pert a model for program evaluation and review technique pert under fuzzy linguistic contexts is introduced in this fuzzy linguistic pert network model each activity duration is represented by a fuzzy linguistic description aggregation and comparison of the estimated linguistic expectations of activity durations are manipulated by the techniques of computing with words cw to provide suitable contexts for this purpose we first introduce several variations of basic linguistic labels of a linguistic variable such as weighted linguistic labels generalized linguistic labels and weighted generalized linguistic labels etc and then based on the notion of canonical characteristic value ccv function of a linguistic variable we develop some related cw techniques for aggregation and comparison of these linguistic labels afterward using a computing technique of linguistic probability introduced by zadeh and based on the new developed cw techniques for weighted generalized linguistic labels we investigate the associated linguistic expectation pert network of a fuzzy linguistic pert network also throughout the paper several examples are used to illustrate related notions and applications