quantity of information comparison between information systems 1 non fuzzy states abstract in statistical theory experiments or probabilistic information systems are supposed to be informative since they reduce the amount of uncertainty associated with the states of nature for the case that the available information systems are vague fuzzy information systems h tanaka t okuda and k asai have proven using the measure of information as provided by entropy that the fuzzy information systems are informative too now we wish to state and to study a criterion in order to compare fuzzy information systems by the quantity of information of a fuzzy information system defined by tanaka et al in this first paper we consider the situation where we require information about the original state space non fuzzy state space the second paper will deal with the situation where we require only information on certain vague states fuzzy states

a note on interval estimation with fuzzy data abstract in this note we formulate the problem of interval estimation when the experimental outcomes cannot be exactly perceived but they may be assimilated with fuzzy information then we suggest a method to derive crude confidence intervals but always possible to apply

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

geometric mean flows and the cartan barycenter on the wasserstein space over positive definite matrices abstract we introduce a class of flows on the wasserstein space of probability measures with finite first moment on the cartan hadamard riemannian manifold of positive definite matrices and consider the problem of differentiability of the corresponding cartan barycentric trajectory as a consequence we have a version of lie trotter formula and a related unitarily invariant norm inequality furthermore a fixed point theorem related to the karcher equation and the cartan barycentric trajectory is also presented as an application

the fuzzy decision problem an approach to the problem of testing statistical hypotheses with fuzzy information abstract this paper is devoted to the problems of testing statistical hypotheses about an experiment when the available information from its sampling is vague when the information supplied by the experimental sampling is exact the problems of testing statistical hypotheses about the experiment can be regarded as a particular statistical decision problem in addition decision procedures may be used in problems of testing hypotheses in a similar manner the problem of testing statistical hypotheses about an experiment when the available sample information is vague is approached in this paper as a particular fuzzy decision problem as defined by tanaka okuda and asai this approach assumes that the previous information about the experiment can be expressed by means of certain conditional probabilistic information whereas the present information about it can be expressed by means of fuzzy information the preceding framework allows us to extend the notion of risk function and some nonfuzzy decision procedures to the fuzzy case and particularize them to the problem of testing finally several illustrative examples are presented

simulation of random fuzzy variables an empirical approach to statistical probabilistic studies with fuzzy experimental data in this paper we simulate different types of random fuzzy variables to get some conclusions concerning fuzzy valued random variables this simulation was carried out to illustrate certain limit results formalizing the convergence of the arithmetic mean of sampled fuzzy data to the population mean or an expected value of the random fuzzy variable like the well known strong law of large numbers and the law of iterated logarithm since the theoretical results have recently been proved this simulation analysis represents a complementary study in which we can determine sample sizes providing us with suitable approximations finally future directions regarding statistical inference with fuzzy data and other applications are commented

an effective matrix geometric mean satisfying the ando li mathias properties we propose a new matrix geometric mean satisfying the ten properties given by ando li and mathias linear alg appl 2004 this mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by ando li and mathias is the limit of a sequence having order of convergence 1 this makes this new mean very easily computable we provide a geometric interpretation and a generalization which includes as special cases our mean and the ando li mathias mean

barycentric maps for compactly supported measures abstract after outlining the way in which an intrinsic mean g g n on a complete metric space gives rise to a contractive barycentric map on some class of borel probability measures and some basic examples of this process we show how the resulting barycentric map gives rise to a general theory of integration of measurable functions into the space we apply this machinery to the cone of positive invertible elements of a c algebra equipped with the thompson metric to derive barycentric maps and their basic properties arising from the power means finally we derive basic results for the karcher barycenter including its approximation by the barycentric maps for power means and its satisfaction of the karcher equation

bootstrap method for some estimators based on fuzzy data abstract the bootstrap efron ann statist 7 1979 1 26 is a simple and straightforward method for calculating approximated biases standard deviations confidence intervals and so forth in almost any nonparametric estimation problem in this paper we consider another example namely fuzzy data and use the bootstrap to answer several questions concerning the minimum inaccuracy estimator corral and gil stochastica 8 1984 63 81 a what is the standard error of this estimator b what is a reasonable confidence interval for such an estimate the validity of bootstrap method is investigated using a real data and computer simulation

the fuzzy estimators of fuzzy parameters based on fuzzy random variables abstract the concept for fuzzy estimators of fuzzy parameters in fuzzy random variables is proposed in this paper due to the resolution identity the construction of a fuzzy number from a family of closed intervals is developed the construction of the fuzzy estimators of fuzzy parameters from the known conventional estimators inherits evidently from the aforementioned technique in the meantime we prove that the fuzzy estimator is a fuzzy random variable in order to provide the notion of fuzzy expectation and variance of fuzzy estimator the computational procedure is also provided by solving the related nonlinear programming problems that are transformed from the original problem for the purpose of evaluation of the membership values of the fuzzy estimates

bayesian estimation in forest surveys when samples or prior information are fuzzy abstract fuzzy phenomena are frequently encountered in forestry due to the complexity of the systems under investigation in this paper existing inference methods are presented to accommodate situations in which random events take place in fuzzy circumstances fuzzy concepts are incorporated into these commonly accepted approaches to account for uncertainties used by factors other than randomness which traditional statistical approaches deal primarily with based on two extensions of likelihood functions for fuzzy samples bayesian estimates are generalized for use when sample information and prior distribution of parameters are fuzzy the estimators are developed for some typical membership functions for fuzzy samples and fuzzy priors applications are illustrated using examples for forestry

repetition invariant geometric means abstract we introduce a class of repetition invariant geometric means and obtain corresponding contractive barycentric maps of integrable borel probability measures on the cartan hadamard riemannian manifold of positive definite matrices they retain most of the properties of the cartan barycenter and lead to the conclusion that there are infinitely many distinct contractive barycentric maps inequalities from the derived geometric means including the yamazaki inequality and unitarily invariant norm inequalities are presented

ordered probability spaces abstract let c be an open cone in a banach space equipped with the thompson metric with closure a normal cone the main result gives sufficient conditions for borel probability measures on c with finite first moment for which in the stochastic order induced by the cone to be order approximated by sequences n n of uniform finitely supported measures in the sense that n n for each n and n n in the wasserstein metric this result is the crucial tool in developing a pathway for extending various inequalities on operator and matrix means which include the harmonic geometric and arithmetic operator means on the cone of positive elements of a c algebra to the space p 1 c of borel measures of finite first moment on c as an illustrative and important particular application we obtain the monotonicity of the karcher geometric mean on p 1 a for the positive cone a of a c algebra a

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

contractive barycentric maps and l1 ergodic theorems on the cone of positive definite matrices abstract we are concerned with contractive with respect to the wasserstein metric barycenters of probability measures with bounded support on the convex cone of positive definite matrices equipped with the thompson metric based on the important construction schemes of multivariate matrix means namely the proximal average and the cartan mean the least squares average for the cartan hadamard metric we construct a one parameter family of contractive barycentric maps interpolating continuously and monotonically the harmonic arithmetic and cartan barycenters we show that each contractive barycentric map is monotonic for the stochastic order induced by the cone and establish stochastic approximations and l 1 ergodic theorems for the parameterized contractive barycenters

monotonic properties of the least squares mean we settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual loewner order indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity moreover we introduce a weighted least squares mean and derive our results in this more general setting

the fuzzy decision problem an approach to the point estimation problem with fuzzy information abstract this paper is devoted to the extension of the bayesian method for the point estimation when the available information is vague in the nonfuzzy case the parametric estimation can be approached as a particularization in the statistical decision problem this motivates us to accomplish the mentioned extension by looking at the parametric estimation in the fuzzy case as a special situation in the fuzzy decision problem defined by tanaka okuda and asia in this way concepts in the fuzzy decision problem are first expressed in the estimation terminology then on the basis of these concepts we shall introduce some notions and state some interesting results finally several illustrative examples will be exposed

monotonicity of the matrix geometric mean an attractive candidate for the geometric mean of m positive definite matrices a1 am is their riemannian barycentre g one of its important operator theoretic properties monotonicity in the m arguments has been established recently by lawson and lim we give an elementary proof of this property using standard matrix analysis and some counting arguments we derive some new inequalities for g one of these says that for any unitarily invariant norm g is not bigger than the geometric mean of a1 am

pcg methods in transient fe analysis part i first order problems the performances of some pcg preconditioned conjugate gradient algorithms are evaluated in the solution of first order time dependent parabolic partial differential equations such as the heat conduction equation which have been spatially discretized using finite elements consistent mass discretizations are preferred by the authors to lumped mass ones and various preconditioners are then compared diagonal incomplete choleski and ebe element by element recommendations are made and implications for parallel computation outlined

bayesian sequential test for fuzzy parametric hypotheses from fuzzy information abstract this paper is devoted to the sequential problem of testing fuzzy hypotheses concerning an experiment the outcomes of which provide fuzzy information this problem will be formalized as a special fuzzy sequential decision problem to which it is possible to extend the bayes nonfuzzy decision procedures when the bayesian framework is assumed

a numerical study of various algorithms related to the preconditioned conjugate gradient method we present the results of a numerical study of the preconditioned conjugate gradient algorithm the minimal residual algorithm the biconjugate gradient algorithm and the bi minimal residual algorithm using both simple test matrices and more realistic test matrices derived from physical problems the application of the methods to unsymmetric matrices is considered we emphasize the importance of a good preconditioning look at various methods including icgg n and incomplete block factorization and make some practical recommendations some of the folk lore surrounding the semi iterative methods is dispelled

log majorization and lie trotter formula for the cartan barycenter on probability measure spaces abstract we extend ando hiai s log majorization for the weighted geometric mean of positive definite matrices into that for the cartan barycenter in the general setting of probability measures on the riemannian manifold of positive definite matrices equipped with trace metric the main key is the settlement of the monotonicity problem of the cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone we also derive a version of lie trotter formula and related unitarily invariant norm inequalities for the cartan barycenter as the main application of log majorization

non asymptotic fuzzy estimators based on confidence intervals abstract in this paper we study and generalize the method of fuzzy estimation which is based on confidence intervals we provide definitions and the corresponding formulas the method we propose allows new additional ways in constructing fuzzy estimators in particular instead of constructing fuzzy estimators as the exact set of confidence intervals of the parameters to be estimated we generalize them in such a way that we eliminate discontinuities and ensure compact support preserving the statistically derived triangular shape of the estimators our approach is particularly useful in critical situations where subtle fuzzy comparisons between almost equal statistical quantities have to be made an example of this application is provided

operator means of probability measures and generalized karcher equations abstract in this article we consider means of positive bounded linear operators on a hilbert space we present a complete theory that provides a framework which extends the theory of the karcher mean its approximating matrix power means and a large part of kubo ando theory to arbitrary many variables in fact to the case of probability measures with bounded support on the cone of positive definite operators this framework characterizes each operator mean extrinsically as unique solutions of generalized karcher equations which are obtained by exchanging the matrix logarithm function in the karcher equation to arbitrary operator monotone functions over the positive real half line if the underlying hilbert space is finite dimensional then these generalized karcher equations are riemannian gradients of convex combinations of strictly geodesically convex log determinant divergence functions hence these new means are their global minimizers in analogy to the case of the karcher mean as pointed out our framework is based on fundamental contraction results with respect to the thompson metric which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above

a bayesian approach to fuzzy hypotheses testing in statistical decisions we may come across imprecise fuzzy concepts one important case is a situation where we are interested in testing the hypotheses that are fuzzy rather than crisp in this paper we consider and analyze the testing of fuzzy hypotheses on the basis of a bayesian approach we illustrate our method by some examples and compare it with previous works on this topic

optimal transportation with infinitely many marginals abstract we formulate and study an optimal transportation problem with infinitely many marginals this is a natural extension of the multi marginal problem studied by gangbo and wi ch 1998 15 we prove results on the existence uniqueness and characterization of the optimizer which are natural extensions of the results in gangbo and wi ch 1998 15 the proof relies on a relationship between this problem and the problem of finding barycenters in the wasserstein space a connection first observed for finitely many marginals by agueh and carlier 2011 1

on the use of zadeh s probabilistic definition for testing statistical hypotheses from fuzzy information abstract a statistical hypothesis is an assertion about the distribution of an experiment we consider the study of the problem of testing a statistical hypothesis that is the problem of concluding whether or not the hypothesis is correct on the basis of data from the experiment when its outcomes do not provide exact but rather fuzzy information for establishing optimality criteria of testing we will use the definition of probability of a fuzzy event given by zadeh in order to extend both neyman pearson and bayes theories to the fuzzy framework then we will analyze several properties for the new criteria particularly the goodness of optimal procedures in both the fuzzy and the nonfuzzy situation will be compared for each criterion finally we will apply the extended criteria for testing simple hypotheses this application leads us to prefer bayesian procedures to neyman pearson procedures in the fuzzy context

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

a fuzzy decision problem based on the generalized neyman pearson criterion korea science and engineering foundation kosef under grant 893 0801 011 2 and korea advanced institute of science and technology kaist under an internal grant