recursive maximum likelihood identification of jump markov nonlinear systems we present an online method for joint state and parameter estimation in jump markov non linear systems jmnls state inference is enabled via the use of particle filters which makes the method applicable to a wide range of non linear models to exploit the inherent structure of jmnls we design a rao blackwellized particle filter rbpf where the discrete mode is marginalized out analytically this results in an efficient implementation of the algorithm and reduces the estimation error variance the proposed rbpf is then used to compute recursively in time smoothed estimates of complete data sufficient statistics together with the online expectation maximization algorithm this enables recursive identification of unknown model parameters including the transition probability matrix the method is also applicable to online identification of jump markov linear systems jmls the performance of the method is illustrated in simulations and on a localization problem in wireless networks using real data

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

open fuzzy cubature rule with application to nonlinear fuzzy volterra integral equations in two dimensions abstract in this paper we propose an open fuzzy cubature formula and as application we construct an iterative numerical method for solving nonlinear fuzzy volterra integral equations in two dimensions the convergence of the method is proved by providing the error estimate and it is tested on two numerical experiments

weak laws of large numbers for fuzzy random variables abstract in this paper we establish weak laws of large numbers for sums of convex compactly uniformly integrable fuzzy random variables taking values in the space of fuzzy numbers in rp as a result weak laws of large numbers for sums of convexly tight fuzzy random variables are given

monte carlo methods for estimating smoothing and filtering one and two factor stochastic volatility models one and two factor stochastic volatility models are assessed over three sets of stock returns data sp the problem is to get the shape of the conditional returns distribution right none of the models come close to matching the tails of this distribution including a second factor provides only a relatively small improvement over the single factor models fitting this aspect of the data is important for option pricing and risk management

fast filtering in switching approximations of non linear markov systems with applications to stochastic volatility we consider the problem of optimal statistical filtering in general non linear non gaussian markov dynamic systems the novelty of the proposed approach consists in approximating the non linear system by a recent markov switching process in which one can perform exact and optimal filtering with a linear time complexity all we need to assume is that the system is stationary or asymptotically stationary and that one can sample its realizations we evaluate our method using two stochastic volatility models and results show its efficiency

a class of fast exact bayesian filters in dynamical models with jumps we address the statistical filtering problem in dynamical models with jumps when a particular application is adequately modeled by linear and gaussian probability density functions with jumps a usual method consists in approximating the optimal bayesian estimate in the sense of the minimum mean square error mmse in a linear and gaussian jump markov state space system jmss practical solutions include algorithms based on numerical approximations or on sequential monte carlo smc methods in this paper we propose a class of alternative methods which consists in building statistical models which locally similarly model the problem of interest but in which the computation of the mmse estimate can be be computed exactly without numerical nor smc approximations and at a computational cost which is linear in the number of observations

efficient particle filtering for jump markov systems application to time varying autoregressions we present an efficient particle filtering method to perform optimal estimation in jump markov nonlinear systems jmss such processes consist of a mixture of heterogeneous models and possess a natural hierarchical structure we take advantage of these specificities in order to develop a generic filtering methodology for these models the method relies on an original and nontrivial combination of techniques that have been presented recently in the filtering literature namely the auxiliary particle filter and the unscented transform this algorithm is applied to the complex problem of time varying autoregressive estimation with an unknown time varying model order more precisely we develop an attractive and original probabilistic model that relies on a flexible pole representation that easily lends itself to interpretations we show that this problem can be formulated as a jms and that the associated filtering problem can be efficiently addressed using the generic methodology developed in this paper simulations demonstrate the performance of our method compared to standard particle filtering techniques

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

topological properties on the space of fuzzy sets abstract in this paper we introduce a new metric on the space f rp of fuzzy sets and prove that f rp is separable and complete also a characterization of compact subsets of f rp is presented and some examples are given

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

measurablility concepts for fuzzy mappings abstract in a series of recent works alternative measurability concepts for fuzzy mappings were introduced and studied in this paper we show that some of these concepts are equivalent or strongly inter related

some properties of strong solutions to stochastic fuzzy differential equations we consider stochastic fuzzy differential equations driven by m dimensional brownian motion such equations can be useful in modeling of hybrid dynamic systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously under a boundedness condition which is weaker than linear growth condition and the lipschitz condition we obtain existence and uniqueness of solution to stochastic fuzzy differential equations solutions which are fuzzy stochastic processes and their uniqueness are considered to be in a strong sense an estimation of error of the picard approximate solution is established we give a boundedness type result for the solution defined on finite time interval also the stabilities of solution on initial condition and coefficients of the equation are shown the existence and uniqueness of a solution defined on infinite time interval is proven finally some applications of fuzzy stochastic differential equations are considered all the results presented in this paper apply to set valued stochastic differential equations

fuzzy and set valued stochastic differential equations with local lipschitz condition we are concerned with the fuzzy stochastic differential equations driven by multidimensional brownian motion viewed as a tool used to describe the behavior of dynamic systems operating in fuzzy environments with stochastic noises under the uniform lipschitz condition we prove the local uniqueness theorem for the solutions of fuzzy stochastic differential equations next we show assuming the lipschitz condition is satisfied only locally that these equations have a unique solution the fact that the solution is bounded is also proved we conclude the paper with a number of corresponding results holding for the deterministic fuzzy differential equations and set valued stochastic differential equations with local lipschitz condition

almost periodic solutions of fuzzy systems we prove the existence of almost periodic solutions and asymptotically almost periodic solutions for the fuzzy functional differential equations moreover we consider uniform stable and uniformly asymptotically stable of almost periodic solutions for the fuzzy system

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

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

strong convergence for weighted sums of fuzzy random sets in this paper we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of upper semicontinuous fuzzy sets in r p

on borel measurability and large deviations for fuzzy random variables fuzzy random variables are a special case of measurable mappings taking on values in a non separable metric space due to that non separability the two main definitions in the literature turn out not to be equivalent we give an easy necessary and sufficient condition for their equivalence an intermediate result of independent value is a characterization of separable subsets of that metric space several probabilistic applications are proposed notably a generalization to this setting of the large deviations principle of bolthausen

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

a new bernoulli wavelet method for accurate solutions of nonlinear fuzzy hammerstein volterra delay integral equations in this article bernoulli wavelet method has been developed to solve nonlinear fuzzy hammerstein volterra integral equations with constant delay this type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology bernoulli wavelets have been generated by dilation and translation of bernoulli polynomials the properties of bernoulli wavelets and bernoulli polynomials are first presented the present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by newton s method convergence analysis of the present method has been discussed in this article also the results obtained by present wavelet method have been compared with that of by b spline wavelet method some illustrative examples have been demonstrated to show the applicability and accuracy of the present method

markov chain monte carlo methods for stochastic volatility models this paper is concerned with simulation based inference in generalized models of stochastic volatility defined by heavy tailed student t distributions with unknown degrees of freedom and exogenous variables in the observation and volatility equations and a jump component in the observation equation by building on the work of kim shephard and chib rev econom stud 65 1998 361 we develop efficient markov chain monte carlo algorithms for estimating these models the paper also discusses how the likelihood function of these models can be computed by appropriate particle filter methods computation of the marginal likelihood by the method of chib j amer statist assoc 90 1995 1313 is also considered the methodology is extensively tested and validated on simulated data and then applied in detail to daily returns data on the s p 500 index where several stochastic volatility models are formally compared under different priors on the parameters

the skorokhod topology on space of fuzzy numbers in this paper we introduce the skorokhod metric on the space f r of fuzzy numbers and prove that f r is separable and complete

continuity of functions defined on the space of fuzzy sets in this paper we investigate the continuity of many useful functions defined on the space f rp of normal and upper semicontinuous fuzzy sets in rp with compact support

planning and monitoring of equivalence studies demonstrating therapeutic equivalence of two treatments is the goal of many clinical trials for instance when the toxicity of an effective standard treatment is of concern much effort is devoted to developing new therapies that would be both as effective and less toxic in this paper we review the special characteristics of these trials and describe sequential monitoring of equivalence studies using repeated confidence intervals we show how sequential monitoring may be of particular value in this setting and critically discuss the choice of some important design parameters we also provide tables for use when planning a sequential equivalence trial

an improvement of commonly used tests in bioequivalence assessment if equivalence of two treatments has to be proved by an experiment with normally distributed observations and fixed sample size there are two usual methods on the one hand there is the double t test and on the other hand a test procedure suggested by anderson hauck which is uniformly more powerful than the first one but only capable of keeping asymptotically to the nominal level based on these two methods a new test is presented which is uniformly more powerful than the double t test and also keeps the preassigned level

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

convergence in distribution for level continuous fuzzy random sets in this paper we give the relationships between tightness and convergence in distribution for level continuous fuzzy random sets to this end we first obtain some characterizations of compact subsets of the space of level continuous fuzzy sets in r p and establish the tightness for level continuous fuzzy random sets

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

dynamic linear models with switching abstract the problem of modeling change in a vector time series is studied using a dynamic linear model with measurement matrices that switch according to a time varying independent random process we derive filtered estimators for the usual state vectors and also for the state occupancy probabilities of the underlying nonstationary measurement process a maximum likelihood estimation procedure is given that uses a pseudo expectation maximization algorithm in the initial stages and nonlinear optimization we relate the models to those considered previously in the literature and give an application involving the tracking of multiple targets

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

an unbiased test for the bioequivalence problem it is shown that the standard two one sided tests procedure for bioequivalence is a biased test better tests exist in this paper an unbiased level test and other tests which are uniformly more powerful than the two one sided tests procedure are constructed its power can be noticeably larger than that of the level two one sided tests procedure

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

on testing for bioequivalence a number of methods have been proposed for statistical analysis of bioequivalence trials a new method is proposed and compared to three previously suggested techniques this comparison clarifies the relationship among the four methods and yields practical suggestions regarding statistical tests

some properties of a new metric on the space of fuzzy numbers abstract this paper deals with some properties of a new metric here called the hausdorff skorokhod metric previously introduced by joo and kim on the space of fuzzy numbers in the euclidean space r k we first prove that convergence in the hausdorff skorokhod metric implies convergence in the sendograph metric also we show that the hausdorff skorokhod metric is not absolutely homogeneous and not translation invariant and investigate the continuity of operations and convexity on the space of fuzzy numbers equipped with the hausdorff skorokhod metric

measurability for fuzzy valued functions in this paper we characterize the borel field of the space of fuzzy sets endowed with the skorokhod metric as a result the measurability for a fuzzy valued function is discussed

some bayesian and non bayesian procedures for the analysis of comparative experiments and for small area estimation computational aspects frequentist properties and relationships abstract the estimation of a treatment contrast from experimental data and the estimation of a small area mean are special cases of the prediction of the realization of a linear combination of fixed and random effects in a possibly unbalanced two part mixed linear model in this article a bayesian approach to point and interval prediction is presented and its computational requirements are examined differences between the bayesian approach and the traditional classical approach are discussed in general terms and in addition in terms of two examples taken from the literature 1 the comparison of drug formulations in a bioavailability trial westlake and 2 the estimation of corn crop areas using satellite data battese harter and fuller some deficiencies in the classical approach are pointed out and the bayesian approach is considered from a frequentist perspective it is shown via a monte carlo study that for certain noninformative choices of the prior distribution the frequentist prop

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

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

hukuhara derivative of the fuzzy expected value in this paper we establish sufficient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its hukuhara derivative with respect to the parameter main results are obtained for two different sets of general conditions one of them requiring measurability conditions with respect to the product space the fuzzy random variable is defined from and the second one just requiring conditions over the projections of the fuzzy random variables

optimal filter approximations in conditionally gaussian pairwise markov switching models we consider a general triplet markov gaussian linear system bf x bf r bf y where bf x is an hidden continuous random sequence bf r is an hidden discrete markov chain bf y is an observed continuous random sequence when the triplet bf x bf r bf y is a classical conditionally gaussian linear state space model cglssm the mean square error optimal filter is not workable with a reasonable complexity and different approximate methods e g based on particle filters are used we propose two contributions the first one is to extend the cglssm to a new more general model called the conditionally gaussian pairwise markov switching model cgpmsm in which bf x is not necessarily markov given bf r the second contribution is to consider a particular case of cgpmsm in which bf r bf y is markov and in which an exact filter optimal in the sense of mean square error can be performed with linear time complexity some experiments show that the proposed method and the suited particle filter have comparable efficiency while the second one is much faster

the calculation of probabilities in rejecting bioequivalence methods are given for deriving the lower and upper bounds of the probabilities for rejecting bioequivalence except when two formulations are nearly equal the lower and upper bounds become very close and thus give an exact probability the methods can be used in two or higher way crossover or parallel groups designs one can also determine the sample size from the equations the results are compared with the probabilities obtained by simulation